Canonical Quantization of Two-Dimensional Gravity S

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 1Ð16. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 5Ð21. Original Russian Text Copyright © 2000 by Vergeles. GRAVITATION, ASTROPHYSICS

Canonical Quantization of Two-Dimensional Gravity S. N. Vergeles Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia; e-mail: [email protected] Received March 23, 1999

Abstract—A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana ﬁelds is presented. The physical state space of the theory is completely described and calculations are also made of the average values of the metric tensor relative to states close to the ground state. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION quantization) for observables in the highest orders, will The quantum theory of gravity in four-dimensional serve as a criterion for the correctness of the selected spaceÐtime encounters fundamental difficulties which quantization route. have not yet been surmounted. These difficulties can be The progress achieved in the construction of a two- arbitrarily divided into conceptual and computational. dimensional quantum theory of gravity is associated The main conceptual problem is that the Hamiltonian is with two ideas. These ideas will be formulated below a linear combination of first-class constraints. This fact after the necessary notation has been introduced. makes the role of time in gravity unclear. The main computational problem is the nonrenormalizability of We shall postulate that spaceÐtime is topologically gravity theory. These difficulties are closely inter- equivalent to a two-dimensional cylinder. The time twined. For instance, depending on the computational coordinate t varies between minus infinity and plus procedure, the constraint algebra may or may not con- infinity while the spatial coordinate σ varies between 0 tain an anomalous contribution (central charge). The and 2π. All these functions are periodic with respect to the coordinate σ. The set of coordinates (t, σ) is presence or absence of an anomaly in the first-class µ constraint algebra has a decisive influence on the quan- denoted as {x }. The metric tensor in spaceÐtime is tization procedure and the ensuing physical picture. denoted by gµν so that the square of the interval is writ- These fundamental problems can be successfully ten as resolved using relatively simple models of generally 2 2 σ2 σ covariant theories in two-dimensional spaceÐtime. ds =g00dt ++g11d 2g01dtd . (1.1) These models particularly include two-dimensional gravity models, both pure and interacting with matter, Most of the formulas and notation in the Introduction and also two-dimensional string models (see, for exam- are taken from [2]. The metric tensor is then parame- ple [1Ð6] and the literature cited therein). trized as follows: In the present paper we present a canonical quantization of two-dimensional gravity minimally coupled 22 2ρuÐvv to real scalar and spinor Majorana fields. All the con- gµν =e, structions and calculations are given before the final vÐ1 (1.2) result is obtained. The physical states of the theory are 24ρ fully described. The complete state space has similar g≡detgµν =Ðue. properties to the multidimensional Fock space in which η boson and fermion operators are acting. The calcula- Let i, j = 0, 1 and ij = diag(1, Ð1). We introduce the µ tions begin with the average values of the metric tensor orthonormalized basis {}e so that relative to states close to the ground state. i However, the present study should merely be per- µ ν η ceived as one of the first steps along the way to the gµνei ej = ij. (1.3) anomaly-free quantization (if this is at all possible) of some of the theories which, in more traditional quanti- To be specific we take zations, are anomalous. Only further studies, including a systematic study of the discrepancies in the predic- µ 1 Ð ρ 1 µ 0 e0 ==--- e , e1 Ðρ. (1.4) tions (which are encountered in different approaches to u v e

2 VERGELES

i The dyad {eµ} is uniquely determined by the equa- Cartan structure equation, we can easily establish the i ν ν i µ i relationship δ δ tions eµei = µ eµe j = j . Taking into account (1.4), we have ÐgRdt ∧ dσ = 2dω01. (1.11) ρ 0 ue  1 ρÐv Since the fields φ and χ in (1.8) are real and belong eµ = , eµ = e . (1.5)  0   1  to Grassmann algebra, we have We analyze the action φ(x)φ(x) = 0, χ(x)χ(x) = 0. (1.12) 2π Using (1.12), we can make the substitution  1 Ᏸ ψ ∂ ∂ µ ψ S = ∫dt ∫ dσ Ðg------(ηR Ð 2λ) µ ( / x ) in (1.6). Thus, the fermion compo- 4πG nent of the action is proportional to the expression 0 (1.6) ρ  e {φφú + (u + v )φφ' + χχú Ð (u Ð v )χχ'}. 1 µν∂ ∂ i µψγ jᏰ ψ + --g µ f ν f + --e j µ . ρ 2 2  In this last expression the factor e may be eliminated by substituting λ Here G is the gravitational constant, is the cosmo- φ eÐρ/2φ, χ eÐρ/2χ. logical constant, R is the scalar curvature of spaceÐ time, η and f are the real scalar fields, ψ is the two-com- On account of (1.12), no additional derivatives of the ponent spinor Majorana field, and {γi} are two-dimen- field ρ appear in the action when this substitution is sional Dirac matrices. We then assume that made. Consequently, the Lagrangian of this system has the form     γ 0 =  0 1  , γ 1 =  0 Ð1  ,  1 Ð1  1 0   1 0  ᏸ = ∫dσ------u ηú (ρú + v ρ' + v') (1.7) 2πG   γ 5 ≡ γ 0γ 1 =  1 0  . v  2ρ  0 Ð1  + η' ---(ρú + v ρ' + v') Ð u' Ð uρ' Ð λue  u  (1.13) The Majorana nature of the spinor field implies that 1 2 2 2 2 t + ----- [ fú + 2v fúf ' Ð (u Ð v ) f ' ] ψ = γ0 ψ (the superscript t indicates transposition). In 2u our case, we have φ i  ψ = , φ = φ†, χ = χ†. (1.8) + --[φφú + (u + v )φφ' + χχú Ð (u Ð v )χχ'] . χ 2  The covariant differentiation operation of the spinor π π π field is determined in accordance with the formula We denote by η, ρ, and the fields canonically conjugate to the fields π, ρ, and f, respectively. The ∂ Ᏸ ψ  1ω σij ψ fields u and v in (1.13) are Lagrangian factors. We µ =  µ + -- ijµ  , ∂x 2 obtain the Hamiltonian of the system (1.13) by a stan- (1.9) dard procedure: ij 1 i j σ = --[γ , γ ]. 4 Ᏼ = ∫dσ(uᏱ + v ᏼ), ω The form of connectedness ijµ is obtained uniquely 1 2ρ from Ᏹ = 2πGπηπρ + ------[Ð (η'' Ð η'ρ') + λe ] 2πG ωi ωi ∧ ω j (1.14) d + j = 0, 1 2 2 i + --(π + f ' ) + --(Ð φφ' + χχ'), µ i i 2 2 where ω ≡ eµdx . Hence, we find  i i  v ᏼ = Ð πηη' + πρρ' + π f ' + --φφ' + --χχ' . ω = u' + uρ' Ð ---(ρú + ρ'v + v') dt  2 2  01 u (1.10) 1 We make the following canonical transposition of vari- + --(ρú + ρ'v + v')dσ. ables: u 0 1 λ Ðρ Here and subsequently the dot and prime denote the λr = Ð------e (2η'coshΣ Ð 4πGπρ sinhΣ), partial derivatives ∂/∂t and ∂/∂σ, respectively. Using the 2 4πG

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GRAVITY 3

charge is present in the quantum algebra of the quanti- 1 1 λ Ðρ λr = Ð------e (4πGπρ coshΣ Ð 2η'sinhΣ), ties (1.16). This implies that all the operators (1.16) 2 4πG (1.15) may be treated as ﬁrst-class constraints in the Dirac 1 λ ρ sense. This new approach is used in the present study. π Ð r ' = ------e sinhΣ, 0 πG The idea of a new approach to quantization arose when studying a model describing pure gravity. This 0 λ ρ model is obtained from the model (1.16) by deleting the π + r ' = Ð ------e coshΣ. 1 πG second terms on the right-hand sides of the system (1.16). It was shown in [1Ð4, 6] that in pure gravity the- Here we have ory the central charge is zero if the scalar product is σ positive deﬁnite in the entire scalar state space. The rea- Σ σ π σ˜ π σ˜ son for this phenomenon is that in the new approach the ( ) = 2 G∫d η( ). operator ordering procedure in the quantities Ᏹ and ᏼ 0 differs radically from the ordering in traditional quanti- The variables describing the matter remain unchanged. zation. In the new variables the Hamiltonian (1.14) has the We now formulate the assumptions used as the basis form to develop the new quantization method.

1) The entire state space HC in which the fundamen- 1  2 0 2  2 1 2 a π π ψ Ᏹ = -- Ð π + (r ') + π + (r ') tal operator fields r , a, f, , and act has a positive 2  0   1  definite scalar product. No indefinite metric is present in the HC space. 1 2 2 + --[π + f ' + i(Ð φφ' + χχ')], (1.16) In order to formulate the next assumption, we 2 denote the set of operators (1.16) by L and the set of all the material fields f, ψ by ψ. From the operators L we ᏼ (π 0' π 1') π i φφ i χχ  = Ð 0r + 1r Ð  f ' + -- ' + -- ' . eliminate the degrees of freedom describing the mate- 2 2 rial fields and we denote the set of operators thus So far the analysis has been classical. In order to quan- obtained by L(0). Hence the operators L(0) only deter- tize the system, we must first begin by defining the mine the dynamics of the gravitational degrees of free- commutation relations for the canonically conjugate dom and variables. In our case, we have ( ) [L 0 , ψ] = 0. (1.19) [r0(σ), π (σ')] = [r1(σ), π (σ')] 0 1 (1.17) 2) In the theory (1.16) there exists a unitary transfor- = [ f (σ), π(σ')] = iδ(σ Ð σ'). mation U such that ( ) For the fermion degrees of freedom we have the anti- UL 0 U† = L. (1.20) commutation relations As a result of (1.19) and (1.20) the fields {φ(σ), φ(σ')} = {χ(σ), χ(σ')} = δ(σ Ð σ'). (1.18) Ψ = UψU† (1.21) All the other commutators or anticommutators of the a π π ψ commute with all the operators L. fundamental fields r , a, f, , and are zero. It is easy ᏻú ᏻ Ᏼ We shall clarify the important role of this last to check that the Heisenberg equations i = [ , ] assumption in the quantization of this system. We pos- obtained using the commutation relations (1.17) and tulate that in the theory (1.16) there is a state |0〉 which (1.18) are the same as the Lagrange equations, where ᏻ annuls all the operators L(0) and all the annihilation is any operator. operators of the fields ψ. According to the reasoning put Since the fields u and v in (1.14) are Lagrangian fac- forward above this is possible. Then the state U|0〉 is tors, the quantities (1.16) are constraints In terms of annulled by all the operators L and all the annihilation classical analysis, they are first-class constraints. How- operators of the fields Ψ. The physical space of the ever, it is well-known that as a result of quantization, states annulling all the operators L is constructed from anomalies or a central charge may appear in this sys- the ground state U|0〉 using the creation operators of the tem: the algebra of simultaneous commutators of Ᏹ and fields Ψ. Consequently the problem of quantizing the ᏼ contains a central charge. The existence of this cen- system (1.16)Ð(1.18) is solved completely. tral charge in the constraint algebra radically compli- In the second assumption, the properties of the uni- cates the quantization problem. In particular, the sys- tary transformation U of interest to us are only tem (1.16)Ð(1.18) may be nonself-consistent. described in broad outline. Subsequently this unitary Recently an anomaly-free approach to the quantiza- transformation is constructed explicitly for the model tion of the system (1.16)Ð(1.18) has been proposed in of two-dimensional gravity studied in the present arti- various studies [1Ð4,6]. In this approach no central cle. The equivalent of formula (1.20) then has a more

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 4 VERGELES complex form. Nevertheless, fairly good progress can Using (2.2) we find be made in the calculations by using the constructed unitary transformation. 1 a L = -- : ∑ α α : , n 2 n Ð m am m (2.6) 2. QUANTIZATION OF PURE GRAVITY 1 a Ln = -- : ∑ α α : . 2 n Ð m am The problem of quantizing two-dimensional pure m gravity was studied in [1Ð4]. In [4, 6] the author The ordering of the operators in (2.6) is determined in described an anomaly-free quantization of a two- accordance with the general quantization conditions dimensional string whose constraint system is the same and plays a decisive role. The aim of quantization is to as the constraint system of two-dimensional pure grav- search for that space of physical states on which all the ity in the representation (1.16). This gives us the possi- operators (2.6) go to zero and in which there is a math- bility of using the methods developed in [4, 6]. ematically correct and positive definite scalar product. η In the present article we adopt two approaches to the Let us assume that a = 0, 1 and ab = diag(Ð1, 1). In quantization of this system. the gauge u = 1, v = 0 the Heisenberg equations for the The first approach is well-known. It was formulated a πa ηabπ fields r , = b have the form by Dirac and is described by the following scheme. Let χ us assume that { n} is a complete set of first-class con- 2 2 2 2  ∂ ∂  a  ∂ ∂  a straints. The physical states then satisfy the conditions Ð r = 0, Ð π = 0. (2.1) ∂ 2 ∂σ2 ∂ 2 ∂σ2 χ | 〉 t t n PD = 0. (2.7) Consequently, the fields ra and πa contain both pos- The conditions for consistency of the theory then fol- itive- and negative-frequency modes: low from (2.7) [χ , χ ] l χ m n = cmn l. (2.8) a a x i 1 a inσ a Ðinσ l r (σ) = ------+ ------∑ --(α e + α e ), In (2.8) the coefficients cmn may be operator quantities π π n n n χ 4 4 ≠ and should be positioned to the left of the constraints l. n 0 (2.2) a The following difficulty is encountered when quan- πa σ p i (αa inσ αa Ðinσ) tizing (2.2)Ð(2.8) (for further details see [6]). It follows ( ) = ------+ ------∑ ne + ne . π 4π from the conditions (2.7) that all the physical states do n ≠ 0 not depend on certain initial dynamic variables. For this reason the following problems arise: αa ≡ αa ≡ a We assume that 0 0 p . From the conditions a) Determining the scalar product on the physical that the fields (2.2) are real it follows that state space; b) Calculating the matrix elements relative to the a† a αa† αa αa† αa x = x , n = Ðn, n = Ðn. (2.3) physical states. This is because not all the initial dynamic variables In order to satisfy the commutation relations (1.17), we are operators in the physical state space. Hence the impose the constraint that the nonzero commutators of matrix elements of these variables are not determined the new variables should have the form in physical space. Although the observable quantities do not depend on these dynamic variables, neverthe- a b a b ab less, serious difficulties may arise when the matrix ele- [α , α ] = [α , α ] = mη δ , m n m n m + n (2.4) ments of the observable quantities are calculated in [xa, pb] = iηab. physical space. We shall subsequently call the quantization (2.7)Ð The set of operators (1.16) is equivalent to the two (2.8) the first method of quantization. series of operators In [6] a different method of quantization was applied to the system (2.4), (2.6). The idea of this 2π method involves somewhat weakening the Dirac condi- 1 σ Ðinσ Ᏹ ᏼ tions (2.7) by replacing them with the conditions Ln = -- ∫ d e ( + ), 2 〈 χ 〉 0 P m P = 0. (2.9) π (2.5) 2 Here P numbers the physical states. The quantization 1 σ inσ Ᏹ ᏼ , ± , … Ln = -- ∫ d e ( Ð ), n = 0 1 conditions (2.9) are similar to the GuptaÐBleuler con- 2 µ 0 ditions in electrodynamics when the equality ∂µA = 0

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GRAVITY 5 is only satisfied in the sense of the mean value, and also We perform the canonical transformation to the quantization conditions in ordinary strong theory α α † α α † † when the Virasoro algebra generators also only satisfy UM UM, UM UM, x UM xUM, the conditions L = 0 in the sense of the mean value. In n 2 this case, averaging is performed relative to the physi- iM x+ UM = exp------ . cal states. 2 p+ The fundamental difference between the quantiza- α(Ð) tion method proposed here and the GuptaÐBleuler In this canonical transformation only the variables 0 quantization and the generally accepted string quanti- and x change, as given by the formulas zation is that in our approach the entire state space has Ð 2 a positive definite scalar product. We subsequently (Ð) (Ð) † (Ð) M α˜ = U α U = α Ð ------, show that this fact can be used to make an anomaly-free 0 M 0 M 0 p quantization of a two-dimensional string. + (2.14) The conditions for consistency of the theory, which † 2 x+ x˜ Ð = UM xÐUM = xÐ + M -----. replace the Dirac conditions (2.8), now have the form p+ 〈 [χ , χ ] 〉 P m n P = 0. (2.10) For uniformity, in this section we introduce the notation α(Ð) α(Ð) ≠ The physical meaning of the conditions (2.10) is as fol- ˜ m = m for m 0. lows. Let us assume that the Hamiltonian of the system Subsequently, instead of the operators (2.13) we use has the form used in the generally covariant theories: the operators Ᏼ χ T = ∑v m m. 1 (+) (Ð) L = Ð--∑α α˜ . (2.15) We assume that at time t the conditions (2.9) are satis- n 2 n Ð m m δ χ m fied. At an infinitely close time t + t the constraint n is given by The reason for this substitution will become clear in the following sections. χ (t + δt) = χ (t) + iδt∑v [χ , χ ](t). We define the vector state space in which the n n m m n dynamic variables of the system act, as linear operators. m We represent the entire state space HCD as the tensor Thus the self-consistency conditions (2.10) yield the product of the gauge state space HG and the physical equality (2.9) at any time. The method of quantization state space HPD: (2.9), (2.10) is subsequently called the second method ⊗ of quantization. We initially apply the first method of HCD = HG HPD. (2.16) quantization to the model (2.6). We introduce the nota- The space H is generated by its vacuum vector |0; G〉 tion G which is determined by the following properties: 0 ± 1 α( ±) α0 ± α 1 x± = x x , m = m m , α0 | 〉 α1 | 〉 > (2.11) Ðm 0; G = 0, m 0; G = 0, m 0, α( ±) α0 ± α 1 (2.17) m = m m . 〈0; G 0; G 〉 = 1. The nonzero commutation relations of the new vari- The basis of the space H consists of vectors of the type ables are obtained using (2.4): G α0 …α0…α1 …α1 | 〉 , , , > [α(+), α(Ð)] δ [α(+), α(Ð)] δ m n Ðl Ðr 0; G , m n l r 0. (2.18) m n = Ð2m m + n, m n = Ð2m m + n, Therefore H is a Fock space with a positive-definite [ , α(Ð)] [ , α(+)] δ (2.12) G x+ n = xÐ n = Ð2i n. scalar product.

We write the operators (2.6) in the new variables: The basis in the physical Dirac state space HPD con- | 〉 0 | sists of two series of states k D, k = (k , k') possessing the following properties: 1 α(+) α(Ð) 1 α(Ð) α(+) Ln = Ð--∑ n Ð m m = Ð--∑ n Ð m m . (2.13) 2 2 α(Ð)| 〉 α˜ (Ð)| 〉 , ± , … m m ˜ m k D = m k D = 0, m = 0 1 (2.19) Where possible subsequently, for the operators with The relationships (2.12) with m = 0 are rewritten in the a bar we do not write those relations which are exactly form the same as those for the operators without a bar. By ( 2 2)| 〉 deﬁnition, in (2.13) the ordering operation implies that pa + M k D = 0. (2.20) either the elements α(+) are positioned to the left of all (Ð) | 〉 the operators α or the converse is true. Both these From this it can be seen that vectors of the basis set k D | ±〉 orders are equivalent (see [4, 6]). are split into two series of vectors k D, each parame-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 6 VERGELES trized by a single continuous real parameter. For In our opinion, in this particular model the most instance, convenient physical states satisfying the conditions (2.9) are states which are coherent in terms of gauge 1| ±〉 ± | ±〉 p k D = k k D, degrees of freedom. In HG space we analyze the coherent state 0| ±〉 ± 2 2| ±〉 (2.21) p k D = k + M k D,  1 0 2 1 2 0 2 2 2 Ð∞ < k < +∞. | , 〉 ( ) z z; G = ∏ exp  Ð ------z Ð m + zm + zÐm + zm  2m Since the operators pa are Hermitian, the scalar m > 0 products (2.26)  1 ( 0 α0 1 α1 0 α0 1 α1 ) | 〉 〈k± k'±〉 = δ(k Ð k'), 〈kÐ k'+〉 = 0 (2.22) + --- zÐm m + zm Ðm + zÐm m + zm Ðm  0; G . D D m  are self-consistent. Quantization conditions similar to (2.19) were used Here za and za are complex numbers. We subse- in [1, 2, 4, 6] and much earlier by Dirac in electrody- m m a a a* a namics (see [8]). quently assume that z0 = z0 and zm = zÐm . The aster- αa αa ≠ isk indicates complex conjugation. By definition we Note. We stress that the variables n , n , n 0, have being linear operators in the space HG, are not generally ( ±) 0 ± 1 ( ±) (0) ± 1 operators in the space HPD. In fact, as a result of the action zm = zm zm, zm = zm zm. α(+) α(+) | −〉 of the operators n and n on the vectors k D we We also introduce the notation obtain vectors which do not belong to the physical ( ) ( ) (Ð) ( ) z˜ Ð = z Ð , z˜ = z Ð , m ≠ 0, space HPD. m m m m As a result of (2.13) and (2.19) the following equal- 2 (2.27) (Ð) (Ð) M ities hold z˜0 = z0 Ð ---(----). z + | 〉 | 〉 | 〉 ∈ 0 Ln p D = Ln p D = 0, p D HPD. (2.23) 0 Hence the model (2.6) is quantized using the first Throughout this article it is assumed that z0 > 0. | ±〉 method. We denote the basis vectors in HP space as z0 P , We shall now quantize this model using the second a ≡ a z0 k . We have method. We postulate that the entire state space HC in which the initial variables act, is expressed as the tensor ( ) a| ±〉 ± a| ±〉 ˜ Ð product p z0 P = z0 z0 P, z0 = 0. (2.28) H = H ⊗ H . (2.24) The last equality in (2.28) is a consequence of the rela- C G P tionship (2.19) with m = 0. Here the space HG is defined in accordance with (2.17), Let us assume that the sets of complex numbers (2.18). The space HP has a basis with the properties { , } | 〉 ∈ z z satisfy the equations (2.28) with the upper sign (2.20)Ð(2.22). If the vector is p HP, it satisfies the and conditions (2.19) with m = 0. (+) α 1 (+) (Ð) We draw attention to the fact that the operators n ≡ Ln(z) Ð--∑zn Ð mz˜m = 0, ( ) α + ≠ 2 (2.29) and n with n 0 (or combinations of these) act in the m , ± , … space HG but their action is not determined on any one Ln(z) = 0, n = 0 1 vector from the space H . This distinguishes H from P P Using the notation HPD (see (2.19)). Thus, the entire state space (2.24) is the tensor product of the spaces in which the corre- (+) 1 inσ (+) sponding operators act. Quite clearly, the space (2.24) z (σ) ≡ ------∑e z , π n has a positive definite scalar product. 2 n For the following calculations we need to determine the ordering of the operators. The ordering (2.15) is (+) 1 Ðinσ (+) z (σ) ≡ ------∑e z equivalent to the ordering π n 2 n 1 2 2 0 0 1 1 L = --( p + M ) Ð ∑ (α α Ð α α ), (2.25) these equations can be rewritten in the more convenient 0 2 a m Ðm Ðm m form: m > 0 (+) (Ð) (+) (Ð) which we shall use subsequently. z (σ)z˜ (σ) = 0, z (σ)z˜ (σ) = 0. (2.29')

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GRAVITY 7 ( ) ( ) (Ð) The functions z(+)(σ), z˜ Ð (σ) , z + (σ) and z˜ (σ) are The ordering on the right-hand side of equality real and periodic and their zero harmonics satisfy (2.36) is defined in accordance with (2.25). It can be (2.27) and (2.28). seen from (2.36) that the equations (2.33) are satisfied, The states i.e., the self-consistency conditions (2.10) are satisfied. Note that generally |z, z±〉 ≡ |±z, ±z; G 〉 ⊗ |z ±〉 (2.30) P 0 P 〈 , ± , ±〉 ≠ z z Ln z z P 0. (2.37) are called basis physical states if conditions (2.28) and (2.29') are satisfied. We can see that the second method also yields a self- According to these definitions, the basis physical consistent quantum theory of the model (2.6). states have the following properties: We shall briefly discussion the superposition principle in the second method of quantization. α0 | , ±〉 ± 0 | , ±〉 Ðm z z P = zÐm z z P, | , ±〉 | , ±〉 (2.31) We assume that the states z z P and z' z' P are α1 | , ±〉 ± 1 | , ±〉 ≥ physical. Is the state m z z P = zm z z P, m 0. | , ±〉 | , ±〉 It rapidly follows from formulas (2.25), (2.29), and z z P + z' z' P (2.38) (2.31) that in our case the conditions (2.9) are satisfied: physical? 〈 , ± , ±〉 z z Ln z z P = 0, In our view, the superposition principle need not (2.32) necessarily be extended to nonphysical gauge degrees 〈z, z± Ln z, z±〉 P = 0. of freedom. Thus, if in more complex theories using the second method of quantization, the superposition prin- We check whether the self-consistency conditions ciple in HG space is bounded, in our opinion this does (2.10) are satisfied. For this it is sufficient to confirm not invalidate the method. In physical state space the that superposition principle is fully obeyed. 〈 , ± ( ) , ±〉 z z Ln LÐn Ð LÐn Ln z z P = 0. (2.33) Simple calculations show that 3. INCLUSION OF MATTER

n n It can be seen from the commutation relations (1.17) 1 1 2 1 1 L L = -- ∑ m(n Ð m) + n(α˜ ) + 2n ∑ α˜ α˜ and the anti-commutation relations (1.18) that boson n Ðn 2 0 Ðm m and fermion ﬁelds have the following expansions in m = 1 m = 1 terms of modes [cf. (2.2)Ð(2.4)]: ∞ ( )α˜ 1 α˜ 1 + ∑ n + m Ðm m (2.34) σ x i 1(α inσ α Ðinσ) f ( ) = ------+ ------∑ -- ne + ne , m = n + 1 π π n 4 4 ≠ ∞ n 0 (3.1) 0 0 ( )α˜ α˜ + ∑ m Ð n m Ðm + : L n LÐn : . π σ p 1 (α inσ α Ðinσ) ( ) = ------+ ------∑ ne + ne , m = n + 1 π π 4 n ≠ 0 Similarly n n φ σ 1 β inσ 1 0 2 0 0 ( ) = ------∑ ne , L L = -- ∑ m(n Ð m) + n(α˜ ) + 2n ∑ α˜ α˜ 2π Ðn n 2 0 m Ðm n m = 1 m = 1 (3.2) ∞ 1 Ðinσ χ(σ) = ------∑βne . + (n + m)α˜ 0 α˜ 0 (2.35) π ∑ m Ðm 2 n m = n + 1 α ≡ α ≡ ∞ We further assume that 0 0 p. Since the ﬁelds ( )α1 α1 (3.1) and (3.2) are real, we have + ∑ m Ð n ˜ Ðm ˜ m + : L Ð n Ln : . m = n + 1 † α† α α† α x = x, n = Ðn, n = Ðn, a (3.3) Here the operators α˜ are expressed in terms of the β† β β† β n = Ðn, n = Ðn. ( ) α(+) α˜ Ð operators , in the same way that the operators The commutation relationships (1.17) and (1.18) are αa are expressed in terms of the operators α(+), α(Ð). equivalent to ≡ Since : LnLÐn : : LÐnLn :, from the last two equalities [α , α ] [α , α ] δ [ , ] we have m n = m n = m m + n, x p = i, (3.4) {β , β } {β , β } δ LnlÐn Ð LÐn Ln = 2nL0. (2.36) m n = m n = m + n. (3.5)

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We only write the nonzero commutation relationships. The equalities (3.9a) are thereby established. As a We shall subsequently denote the operators (2.6) by result of (3.5), (3.8), and (A.14') we have (0) (0) Ln and Ln , respectively. Taking into account the contribution of the material degrees of freedom, the { , } Fᏹ Fᏹ Fᏹ FᏹÐ1 Bm Bn = ∑ m, l n, Ðl = ∑ m, l l, Ðn. Fourier components (2.5) have the form l l (3.11)

(0) Ln = Ln From this it follows that the commutation relations (3.9b) are valid. (3.6) 1 α α  n β β The operators (3.7) and (3.8) introduced here differ + --∑ n Ð m m + m Ð -- n Ð m m . 2 2 only negligibly from the DDF operators used in the m string theory (see [9, 10]). It is convenient to begin constructing the unitary It is easy to see from these definitions that: transformation indicated at the end of the Introduction, which solves the quantization problem, by defining the [α(+), ] [α(+), ] m An = m An creation and annihilation operators of the field (1.21). (3.12) (+) (+) In other words, our first task is to construct boson and = [α , B ] = [α , Bn] = 0. fermion creation and annihilation operators of matter m n m which commute with all the operators (3.6). We can see α(+) that the problem with slightly weaker conditions has a The relations (3.12) remain valid if instead of m we solution. This is sufficient for our purposes. α(+) substitute m or x+. We shall analyze the “gravitationally dressed” oper- α(Ð) ators of the material fields: Now, instead of m we must introduce the vari- α(Ð) ables ˜ m into the theory, which conserve the previous form of the commutation relations with the variables A = ∑ᏹ , α , Am = ∑ᏹm, nα , (3.7) m m n n n α(+) n n m and have zero commutators with the new variables (3.7), (3.8). Fᏹ β Fᏹ β From the definition (3.7) we find Bm = ∑ m, n n, Bm = ∑ m, n n. (3.8) n n [α(Ð), ] [α(Ð), ᏹ ]α m An = ∑ m n, l l. ᏹ ᏹ Fᏹ The infinite-dimensional matrices m, n, m, n , m, n , l Fᏹ and m, n in (3.7) and (3.8) are determined in the We use (A.16) and also reverse the inequalities (3.7). Appendix. The elements of these matrices depend on As a result, we obtain α(+) α(+) the operators (x+/p+, m /p+, m /p+) whose relative   commutators are zero. Thus, all the matrix elements in [α(Ð), ] 2n ᏹ ᏹÐ1 ≠ m An = Ð----- ∑∑ n, l Ð m l, p A p, m 0. (3.7) and (3.8) mutually commute p+   p l It is easy to check by means of direct calculations that the nonzero commutators of the operators (3.7) and Here the sum in brackets is calculated using formulas (3.8) have the following form: (A.3), (A.10), and (A.12). Thus, we find

[ , ] [ , ] δ (Ð) 2n Ð1 Am An = Am An = m m + n, (3.9a) [α , ] ᏹ ≠ m An = Ð----- ∑ m, p Ð n A p, m 0, p+ { , } { , } δ p (3.13) Bm Bn = Bm Bn = m + n. (3.9b) [α(Ð), ] n 0 An = Ð-----An. Using (3.4), (3.7), and (A.14) we ﬁnd p+ Here the equality (3.11) was taken into account. The [ , ] ᏹ ᏹ following relations also hold Am An = ∑l m, l n, Ðl l (3.10) [α(Ð), ] n 0 An = Ð-----An, Ð1 p+ (3.14) = Ðn∑ᏹ , ᏹ , = mδ . m l l Ðn m + n [α(Ð), ] ≠ l m An = 0, m 0.

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Similarly, using formulas (3.8), (A.4), (A.11), mally ordered, since the following equalities are satis- (A.12), and (A.14) we obtain ﬁed

(Ð) 1 Ð1 ᏹÐ1  p Ð q  [α , ] ----- ( )ᏹ , ∑ m, p + q A p Aq Ð ------Bp Bq m Bn = Ð ∑ n + p m p Ð n B p,  2  p+ , p p q (3.18) (Ð) Ð1  p Ð q 

[α , Bn] = 0, m ≠ 0, =: ∑ᏹ , A A Ð ------B B : . m m p + q p q 2 p q (3.15) p, q [α(Ð), ] n 0 Bn = Ð-----Bn, p+ In order to prove the equalities (3.18) it is sufficient to establish that [α(Ð), ] n 0 Bn = Ð-----Bn. p+ ( ) ( ) ∑ A p AÐp + pBÐp B p = ∑: A p A Ðp + pBÐp B p It follows directly from formulas (3.13)Ð(3.15) and p p (3.19) also (3.9) and (3.12) that the variables ( ) :=: ∑ A p AÐp + pBÐp B p : . p (Ð) (Ð) 1 Ð1  p Ð q  α˜ = α Ð -----∑ᏹ , A A Ð ------B B , m m p m p + q p q 2 p q The first equality in (3.19) follows directly from the + , p q definition of ordering and the commutation relations m ≠ 0, (3.9). In order to prove the second equality in (3.19), it should be borne in mind that formally (Ð) (Ð) (Ð) 1 (3.16) α˜ ≡ α˜ = α Ð ------∞ 0 0 0 2 p + ∑ n = ζ(Ð1), n = 1 × { 2} ∑ A p AÐp + A p AÐp + p(BÐp B p + BÐp B p) + 2M where ζ(s) is a Riemann zeta function: p ∞ ζ Ðs commute with all the operators An, An , Bn, and Bn : (s) = ∑ n . (3.20) n = 1 [α˜ (Ð), ] [α˜ (Ð), ] m An = m An The zeta function has a unique analytic continuation to (3.17) ζ [α(Ð), ] [α(Ð), ] the point s = Ð1 where (Ð1) = Ð1/12. This regulariza- = ˜ m Bn = ˜ m Bn = 0. ∞ tion of the divergent sum ∑n = 1 n is now generally The number M2 in braces in (3.16) can also be taken as accepted. Thus, we can assume that the result of normal ordering [cf. (2.14)].  ∞   ∞  If all quantities without a bar in formulas (3.13)Ð  ∑ n Ð  ∑ n = ζ(Ð1) Ð ζ(Ð1) = 0. (3.17) are replaced by the same quantities with a bar     and at the same time all quantities with a bar are n = 1 n = 1 replaced by the same quantities without a bar, these for- Here the first divergent sum appears as a result of the mulas remain valid. ordering of the boson operators and the second appears We shall now determine the normal ordering of the as a result of the ordering of the fermion operators. creation and annihilation operators (3.7) and (3.8). From this follows the second equality in (3.19). These operators are by definition assumed to be nor- Thus, the right-hand sides of the equalities (3.16) mally ordered if all the creation operators AÐ|n|, AÐ n , can equally well be taken to be normally ordered relative to the operators (3.7) and (3.8) or disordered and B−|n|, BÐ n are positioned to the left of all the annihila- written in the form of sums contained in (3.16). For tion operators A|n|, A n , B|n|, and B n . some calculations the disordered variant of the right- hand sides of (3.16) is more convenient. The right-hand sides of the equalities (3.16) contain quadratic forms of the operators (3.7) and (3.8). These We shall now prove the following commutation quadratic forms are represented as sums which are not relation; normally ordered. However the right-hand sides of the ( ) ( ) ( ) [α(Ð), α(Ð)] [α˜ Ð , α˜ Ð ] [α(Ð), α˜ Ð ] equalities (3.16) can in fact be considered to be nor- ˜ m ˜ n = m n = ˜ m n = 0. (3.21)

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( ) Let m ≠ 0 and n ≠ 0. We take the variables α˜ Ð in the follow trivially from the fundamental commutation m relations (3.4), (3.5) and the definitions (3.16). The form (3.16). We then have validity of the commutation relations (3.21) is thus ( ) ( ) ( ) ( ) [α˜ Ð , α˜ Ð ] = [α˜ Ð , α Ð ] completely proven. In addition, from the definitions m n m n (3.16) and the commutation relations (3.12) we have (3.22) ( ) 1 [α˜ (Ð), ᏹÐ1 ] p Ð q  [α(+), α˜ (+)] [α(+), α˜ Ð ] δ Ð -----∑ m n, p + q A p Aq Ð ------Bp Bq . m n = m n = Ð2m m + n, p+ 2 (3.24) p, q [ , α(Ð)] [ , α(+)] δ x ˜ n = x˜ ˜ n = Ð2i . ( ) + Ð n Here we use the definition (3.16) for α˜ Ð and the com- n The explicit form of the new variable x˜ is not given mutation relations (3.17). On the right-hand side of Ð ( ) here since this variable is not used below. (3.22) we replace α˜ Ð by its value as given by (3.16) m α( ±) α( ±) α α β The initial variables {x±, m , m , m, m , m, [α(Ð), α(Ð)] ˜ m ˜ n βm }, (or more accurately their linear combinations) are canonical. It follows from the commutation relations 1 (Ð) Ð1 (Ð) Ð1 = -----∑{[α˜ , ᏹ , ] Ð [α˜ , ᏹ , ]} (3.9), (3.12), (3.17), and (3.21) that the set of variables p n m p + q m n p + q + , p q { , , α(+), α(Ð), α(+), α˜ (Ð), , , , } x+ x˜ Ð m ˜ m m m Am Am Bm Bm (3.25)  p Ð q  × A A Ð ------B B  p q 2 p q is also canonical. We shall now determine the unitary transformation   appearing in (1.21). We define the unitary operator U 1 ᏹÐ1 [α , ] p Ð q[α(Ð), ] + -----∑ m, p + q n A p Aq Ð ------n B p Bq . using the following equalities: p+  2  p, q Ux+ = x+U, UxÐ = x˜ ÐU, Using formulas (A.17), (3.13), and (3.15), after redefin- α(+) α(+) α(Ð) α(Ð) U m = m U, U m = ˜ m U, ing the notation of the indices in some sums we trans- (3.26) form this last expression to give α β U m = AmU, U m = BmU,  α(+) α(+) , … 1 ᏹ  p Ð q  U m = m U -----∑ 2m Ð( p + q), Ð(m + n)A p Aq Ð ------Bp Bq 2  2 p+ p, q and so on for the other operators with a bar. We know that equalities of the type (3.26) uniquely determine the linear operator U and this operator is unitary [11]. ᏹÐ1 ᏹÐ1 Ð 2∑r m, q + r n, p Ð r A p Aq We must express the operators (3.6) in terms of the r new variables. To do this we represent the operators (3.23) ( ) L 0 from (3.6) in the form (2.13) and express the oper- 1 Ð1 Ð1  n -- ( ) ᏹ , ᏹ ,  ( ) ( ) + ∑ r + p (r Ð q) m q + r n p Ð r B p Bq α Ð α β α + 2  ators m , m, and m in terms of the operators x+, m , r α(Ð) ˜ m , Am, and Bm using formulas (3.7), (3.8), and (3.16). 1 { } Ð ----2- m n . As a result of simple calculations using the sum rules p+ (A.20) and (A.21) we arrive at the following formulas: Here the components of the sums over r which are antisymmetric in terms of the indices m and n are obtained 1 α(+) α(Ð) 1 α(+)ᏺ Ln = Ð --∑ n Ð m ˜ m + ------n , using the relations (A.18). As a result, all the terms in 2 4 p+ (3.23) are mutually reduced. Thus, we have proven that m the commutator (3.22) is zero. The relations 1 α(+) α˜ (Ð) 1 α(+)ᏺ (Ð) (Ð) Ln = Ð --∑ n Ð m m Ð ------n , (3.27) [α˜ , α˜ ] ≠ ≠ 2 4 p+ m n = 0, m 0, n 0, m are proven by exactly repeating these procedures. Similarly, it is established that ᏺ { } = ∑ AÐl Al Ð AÐl Al + l(BÐl Bl Ð BÐl Bl) . ( ) [α(Ð), α(Ð)] [α(Ð), α˜ Ð ] l 0 ˜ m = ˜ 0 m = 0. Repeating the reasoning put forward to prove (3.19), The equalities we arrive at the equality [α(Ð), α˜ (Ð)] ≠ ≠ ˜ m n = 0, m 0, n 0 ᏺ =: ᏺ :. (3.28)

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The validity of the equalities (3.27) can be con- 4. PHYSICAL STATE SPACE firmed by a simpler method. This involves calculating the commutators of the operators (3.7), (3.8), (3.16) The formulas obtained in Sections 2 and 3 can be used to quantize the model being studied. with the operators Ln in the representations (3.26) and (3.27). The results of these calculations agree. As a result of the existence of a unitary transformation having the properties (3.26) and (3.29), we can Using (3.26) and (3.27), we find that confirm that the state space of the system (3.6) is iso-   morphic with the state space of the two interacting sys- (0) 1 α(+)ᏺ † tems: pure gravity and free fields (3.1), (3.2). Bearing Ln = ULn + ------n 0 U ,  2 p+  in mind this unitary transformation, we can construct the physical state space directly in the theory with interaction.  (0) 1 (+)  L = UL Ð ------α ᏺ U†, (3.29) n n 2 p n 0 We first apply the first quantization method. For this  +  we define two families of states using the formulas [cf. (2.19)] ᏺ [α α α α β β β β ] ( ) 0 = ∑ Ðl l Ð Ðl l + l( Ðl l Ð Ðl l) . α(Ð)| ±〉 α˜ Ð | ±〉 ˜ m k D = m k D = 0, l > 0 (4.1) m = 0, ±1, …, The relations (3.29) are an exact variant of (1.20). Although the formulas (3.29) are slightly more com- A |k±〉 = A |k±〉 plex than (1.20), the new method of quantization based n D n D | ±〉 | ±〉 (4.2) on the assumptions put forward at the end of the Intro- = Bn k D = Bn k D = 0, duction can be developed further. n > 0. We note that the operators (3.7), (3.8) used here differ substantially from the gravitationally dressed oper- We also impose the constraint ators in [2]. For the operators (3.7), (3.8) we have | ±〉 | ±〉 A0 k D = A0 k D = 0, m (+) [A , L ] = ------α A (3.30) m n 2 p n m which is not necessary and is merely used to simplify + the formulas. In addition, we assume that relations and so on. We stress that it is impossible to construct a (2.21) and (2.22) are satisfied. set of operators which can be expressed linearly in The reason why the quantity M2 > 0 was introduced terms of the operators of the material fields and also in formulas (2.14) and (3.16) now becomes clear: as a commute with all the operators Ln and Ln . (A similar result of this constraint and condition (4.1) with m = 0 situation is encountered in closed string theory when the operator p+ has no zero eigenvalues in the physical the transverse degrees of freedom are described by space HPD. Thus, the unitary transformation (3.26) is DDF operators.) A different approach to the model defined correctly and the operators An can act on the | ±〉 being studied was applied in [2]: the authors introduced states k D . new operators Ln' and Ln' which differ from the opera- All the physical states are linear combinations of basis states having the form tors Ln and Ln by values proportional to the Planck | ± , , , 〉 k ; n i n i m i m i constant. The operators Ln' and Ln' have the same alge- D ( ±) bra as L and Ln . At the same time there is a set of grav- = (A …A …B …B …)|k±〉 ∈ H , (4.3) n Ðn1 Ðn1 Ðm1 Ðm1 D PD itationally dressed operators {C , Cn } which describe , , , > n ni ni mi mi 0, the degrees of freedom of the material fields and are linearly coupled to them, and also commute with all oper-   ators L' and Ln' . This factor simplifies the formal ∑n + ∑m Ð ∑n Ð ∑m  = 0. (4.4) n  i i i i quantization procedure. However, significant difficul- i i i i ties are encountered when we attempt to express the initial dynamic variables in terms of those operators The entire physical state space is expressed as the direct used for quantization. sum ( ) ( ) In the present paper, unlike [2], we give explicit for- H = H + ⊗ H Ð . (4.5) mulas which can be used to express the initial variables PD PD PD in terms of the new variables (see (3.16)). This makes it As a result of relations (4.1), (4.4), and (3.9), we have possible to calculate the matrix elements of the metric ᏺ| 〉 | 〉 ∈ tensor (1.2) (see Section 5). P = 0, P H P D . (4.6)

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Now using (3.27), (4.1), (4.2), and (4.6) we obtain we need to calculate the average of the expression exp(2ρ) since the parameters u and v are Lagrange | 〉 | 〉 Ln P = Ln P = 0. (4.7) numerical factors. It follows from the commutation relations (3.9) and In order to begin our calculations we postulate that equalities (4.2), (2.22) that the scalar product in the the formulas (1.15) valid in classical theory also hold in (+) (Ð) quantum theory. This assumption allows us to express HPD and HPD states is positive definite and these states the unknown quantity in terms of the quantum fields πa, are mutually orthogonal. ra (2.2) as follows: This implies that an anomaly-free quantization of λ 2ρ π πa a π0 1 π1 0 the system (3.6) has been performed using the first ------e = a Ð ra' r ' + 2( r ' Ð r '). method. πG We now apply the second method of quantization. Taking into account (2.2) this inequality is rewritten in By definition, the state space is generated by two series the convenient form for us: of states |z, z±〉 having the following properties. The P     | , ±〉 2ρ(σ) G (+) imσ (Ð) Ðimσ states z z P satisfy equations (2.28), (4.2) and also e = Ð--- ∑α e  ∑α e  . (5.1) λ  m   n  [cf. (2.31)] the equalities m n

1 (+) (Ð) 0 ( ) --(α + α˜ )|z, z±〉 = ±z |z, z±〉 , In (5.1) the variables α Ð should be expressed in terms 2 Ðm Ðm P Ðm P n α(+) α˜ (Ð) 1 (+) (Ð) 1 (4.8) of the new variables n , n and Am , Bm , Am, and Bm --(α Ð α˜ )|z, z±〉 = ±z |z, z±〉 , 2 m m P m P in accordance with (3.16). Then we can calculate the average values of the expression (5.1) using the results > m 0, of the previous section. and similarly for the quantities with a bar. On the right-hand side of (5.1) all the variables in It follows from (2.28) that the first set of brackets commute with all the variables in the second set of brackets. Bearing this in mind and α(Ð)| , ±〉 ˜ 0 z z P = 0. (4.9) also the results of Section 4, we can confirm that the following formula is valid to calculate the averages rel- The basis states in the physical space are denoted by ative to the basis vectors | , ± , , , 〉 z z ; n i n i mi mi P . The y are constructed using the ρ ( ) σ ( ) σ operators AÐm, …, m > 0 in accordance with (4.3) and 〈 2 〉 G α + im α Ð Ðim e = Ð-λ-- ∑ m e ∑ n e . (5.2) (4.4). The physical state space HP is expanded as a m n (+) (Ð) direct sum of the orthogonal subspaces HP and HP . We use the second method of quantization to calcu- The scalar product in the HP space is positive definite. late the averages in expression (5.2). We first calculate | , ±〉 As a result of the commutation relations (3.12) and the averages relative to the ground state z z P . We (3.17), equations (4.8) and (4.9) also hold for physical α(+) states if these physical states are “pure” in relation to express the variables m in the form the gauge degrees of freedom. Here we take “purity” to (+) 0 1 0 1 (+) (Ð) mean that all the physical states have the same set of α = a + a , a = --(α + α˜ ), m m m m 2 m m { a , a } parameters zm zm . (5.3) 1 1(α(+) α(Ð)) We shall not demonstrate in detail that in this case am = -- m + ˜ m 2 the quantization conditions (2.9) and (2.10) are satisfied. This follows directly from all the reasoning put and use formulas (4.8) and (2.28). Thus, we obtain forward above. Here we merely stress the following important fact: averaging in (2.9) and (2.10) can only , α(+) imσ , (+) imσ be performed in the space of the gauge degrees of free- z z+ ∑ m e z z+ = ∑zm e . (5.4) P dom HG [see (2.17), (2.18)]; averaging in (2.9) and m m (2.10) need not be performed over variables described Similarly we find by the operators Am, Am , Bm, Bm . ˜ (Ð) Ðinσ ˜(Ð) Ðinσ z, z+ ∑αn e z, z+ = ∑zn e . (5.5) 5. CALCULATION OF AVERAGES n P n In the model studied the most interesting quantity is It follows from (3.16), (3.18), and (4.2) that the average value of the metric tensor (1.2) relative to ( ) , (α(Ð) α˜ Ð ) , the physical states. For this, in accordance with (1.2), z z+ n Ð n z z+ P = 0. (5.6)

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Combining formulas (5.2), (5.4), (5.5), (5.6), and Bearing this observation in mind and also formula (2.29'), we obtain (A.12) we note that the matrix elements should be calculated using the following relations 2ρ(σ) 2πG (+) (Ð) z, z+ e z, z+ = Ð------z (σ)z˜ (σ). (5.7) P λ α(+) α(Ð) α˜ (Ð) m ( ) m = m + ------: ∑ A Ðp A p + pBÐp B p : , In order to ﬁnd more complex matrix elements, we p2 (5.13) need to make additional calculations. + p ≠ As a result of (A.2), we have m 0, x and the second formula (3.16) (for m = 0). Using (5.2), l l  +  (ξ ξ ) (5.4), and (5.13), we can calculate the diagonal matrix u = z expil------ exp + + Ð , 2 p+ element of the metric tensor relative to the basis state | 〉 | 〉 p = z, z+ ; ni, ni , mi, mi in the form (4.3), (4.4): ξ l 1 ( Ðm 0 m 1 ) + = ----- ∑ --- Ðz am + z aÐm , (5.8) 2ρ(σ) 2πG (+) p+ m σ m > 0 p e p = Ð-----λ-----z ( )

l 1 m 0 Ðm 1 (+) (5.14) ξ = ----- ∑ --- (z a Ð z a ).  ( ) σ    Ð p m Ðm m × ˜ Ð σ z ( )  + > z ( ) + ------∑ni + ∑mi . m 0  (+) 2    (z ) i i 0 1 0 Here am and am are defined in accordance with (5.3). (+) (Ð) As a result of the relations (4.8), when calculating the The functions z(+)(σ), z (σ) , and z˜ (σ) satisfy matrix elements we need to order the operators in (5.8) (2.29'). To derive formula (5.14) we used equality (4.4) ξ so that the functions of + are to the left of the functions and assumed that the basis vector is normalized. ξ of Ð. This is easily achieved since the commutator Further calculations of averages together with their 2 study and interpretation are outside the scope of the 1[ξ , ξ ]  l  1 present study. -- + Ð = ----- ∑ --- (5.9) 2 p+ m Calculations of the matrix elements from the metric m > 0 tensor using the first method encounter serious difficul- ξ ξ ( ) is a c-number which commutes with + and Ð. Then ties. This is because the variables α + are not operators using the well-known BakerÐHausdorff formula we m obtain in the physical state space HPD. Thus averages having the form 2 ξ ξ l  l  1 l  x+  + Ð u = exp Ð----- ∑ --- z expil------ e e . (5.10) α(+) imσ α(+) Ðimσ p+ m 2 p+ ∑ m e , ∑ m e (5.15) m > 0 m PD m PD Now the averages of expression (5.10) are calculated are not determined. As a result of the quantization con- using relations (4.8): ditions (4.1) and (4.2) we could assume that the average 2 (5.2) is zero if the matrix element is calculated relative l  l  1 | ±〉 , , ∼ to the generating states k D. However, this does not z z+ u z z+ P exp Ð----- ∑ --- p+ m rescue the general situation since averages of the type m > 0 (5.11) (5.15) (see (5.12) and (5.14)) must be calculated to cal-  (+)  culate the matrix elements relative to the excited states. l zn Ðn × expÐ----- ∑ ------z  .  p+ n  n ≠ 0 6. CONCLUSIONS Since the argument of the first exponential function on In the present paper we have applied two methods of the right-hand side of (5.11) is an infinitely large nega- quantization to the theory of two-dimensional gravity. tive number, the average of (5.11) is zero. This implies The first method using the Dirac approach allows us to that in calculations of the matrix elements below the achieve complete quantization. This implies ᏹÐ1 average sign the elements of the matrix n, l are in fact (a) constructing a physical state space with a posi- only nonzero for l = 0. This fact which applies equally tive definite scalar product; ᏹÐ1 (b) explicitly expressing physically meaningful to the elements of the matrix n, l substantially simpli- quantities in terms of those operators used to construct fies the calculations. Using (A.9) and (A.10) we obtain the physical state space. α(+) However the averages of the metric tensor cannot be ᏹÐ1 n n, l = ------. (5.12) calculated using the first method for fundamental rea- p+ sons. This statement holds to a lesser extent when the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 14 VERGELES physical state space is constructed using the operators 1 dz Ðn m ᏹ , = ------z u , (A.3) An, Bn, …, determined in Section 3. m n 2πi°∫ z In addition to the problems (a) and (b) the second C method of quantization can also solve the following F 1 dz Ðn m ᏹ , = ------z u qú. (A.4) problem: m n 2πi°∫ z (c) calculating the averages of the metric tensor rel- C ative to the physical states. F The definitions of the matrixes ᏹm, n and ᏹm, n are On the basis of these results we can conclude that obtained from (A.3) and (A.4) by substituting q q the second method of quantization should be used subsequently to study other models. and u u . Here and subsequently we have To conclude we make the following observation. d d 1 α(+) Ðn The model (3.6) can easily be quantized in a “light qú = τq = iz q = 1 + ----- ∑ n z . (A.5) d dz p+ cone” gauge. In the terms used in the present study n ≠ 0 using this gauge implies We draw attention to the fact that all the quantities imposing second-class constraints: (A.1)Ð(A.5) should be considered to be formal series in (+) (Ð) (+) the elements of free associative commutative involutive α = 0, α˜ = 0, α = 0, ( ) m m m Ꮽ(+) α + algebra with the generatrices {x+/p+, m /p+, ( ) (6.1) ˜ Ð ( ) αm = 0, m ≠ 0. α + m /p+}. This assumption holds until any averages are Then, in accordance with (5.2) and (5.6), the average of calculated relative to physical states. The coefficients at the metric tensor relative to the ground state is zero. On the monomials relative to the generatrices of the alge- the other hand, in the second method of quantization, in bra Ꮽ(+) in the expansions of (A.1)–(A.5) are finite accordance with (5.7) and (2.29'), the average of the polynomials in z and zÐ1. Hence, the integrals in (A.3) metric tensor relative to the ground state is generally and (A.4) are determined correctly. The matrix ele- nonzero. From this it can be seen that imposing special ments of the matrices (A.3) and (A.4) thus belong to the calibration conditions which simplify the solution of algebra Ꮽ(+). the quantization problem should be avoided if possible. From (A.2) we have In our case, imposing the constraints (6.1) significantly degrades the results.  α(+)  1 Ðn This work was supported by the VNSh program limvar lnu(z) = limvar lnz Ð limvar ∑ ------n----z  . C C C  n p+  (grant No. 96-1596821). n ≠ 0

Here varCF(z) implies the change in the function F(z) APPENDIX (generally nonunique around the contour C) for a single Let us assume that τ is some “time-like” parameter counterclockwise circuit around the contour C. For this and z = eiτ. We introduce the following operator func- deﬁnition of the contour the second term on the right- tions: hand side of the last equality makes no contribution and we have

x 1 i 1 (+) Ðn π q(τ) ≡ q(z) = ------+-- + -- lnz + ----- ∑ --α z , limvar lnu(z) = vlimar lnz = 2 i. (A.6) 2 p i p n n C C + + ≠ n 0 (A.1) From (A.2) and (A.5) we find x ( ) ∞ τ ≡ + 1 i 1α + Ðn + α(+) q( ) q(z) = ------+ -- lnz + ----- ∑ -- n z . du(z) u(z) n Ðn 2 p+ i p+ n ------= ------z ≠ 0. (A.7) n ≠ 0 ∑ dz z p+ n = Ð∞ By definition we have This last inequality is a consequence of the fact that in  α(+)  Ꮽ(+) algebra there are no relations apart from those ≡ iq(z)  x+  n Ðn u(z) e = zexpi------ expÐ∑ ------z  , derived from its commutativeness. 2 p+  n p+  (A.2) n ≠ 0 The transform of the contour C in Ꮽ(+) algebra for the mapping (A.2) is denoted by C*. u(z) ≡ eiq(z). The mapping (A.2) can be inverted. This is a conse- Let the contour C in the plane of the complex vari- quence of inequality (A.7). This inversion implies that able z go counterclockwise once around the point z = 0. there is an analytic function z(u) of the variable u which We define four infinite-dimensional matrices according inverts the equation (A.2) into an identity. The function α(+) to the formulas z(u) is a formal series of the variables { /p+}. Equa-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GRAVITY 15 tion (A.2) may be inverted by an iterative method in s α(+)  d  powers of the variables /p+. Let us assume that u----- f (u, z(u))  du (0) (1) z(u) = z (u) + z (u) + …, (A.12) +∞ n where 1 du1 , s u  = ----π---- °∫ ------f (u1 z(u1)) ∑ n ---- , ∞ 2 i u1 u1 + C* n = Ð∞ (i) (i) Ðn z (u) = ∑ zn u , s = 0, 1, … ∞ n = Ð In (A.12) the function f(z, u) is expanded as a Laurent (i) series in terms of its arguments and integration is per- and zn are homogeneous functions of the variables α(+) formed in the counterclockwise direction. The identi- /p+ to the power i. Using equation (A.2) we obtain ties (A.12) are based on the obvious equalities ( ) δ x z 0 (u) = ueÐi , δ = ------+--, dz n du n 2 p+ ---- z = -----u = 2πiδ . °∫ z °∫ u n α(+) C C* (1) Ðiδ n Ðiδ Ðn z (u) = ue ∑ ------(e u) , As a result of (A.6) and (A.12) for s = 0 we have n p+ n ≠ 0 (A.8) +∞ α(+) 2 ᏹ ᏹÐ1 1 du1 Ðn (2) Ðiδ1 Ðiδ Ðn ∑ m, l l, n = ------u z (u) = ue ------n----(e u) 2πi °∫ u 1 ∑ ∞ 1 2 n p+ l = Ð C∗ n ≠ 0 (A.13) +∞ l 1 dz m z  1 du m Ð n α(+) α(+)  × ------u ∑ ----1 = ------1u = δ . n Ðiδ Ðn m Ðiδ Ðm 2πi°∫ z  z  2πi °∫ u 1 m Ð n Ð ∑ ------(e u) ∑ ------(e u)  ∞ 1 C l = Ð C∗ p+ m p+  n ≠ 0 m ≠ 0 In (A.13) it is implied that u = u(z) and z1 = z(u1). Thus, and so on. Equation (A.2) uniquely determines each the equality ᏹᏹÐ1 = 1 is established. The relations (i) successive term z (u) in terms of the previous ones. ᏹ−1ᏹ = 1, FᏹFᏹÐ1 = 1, and FᏹÐ1Fᏹ = 1 are estab- It also follows from (A.6) that for a single counter- lished similarly. clockwise circuit of the variable u around the circuit Formulas (A.3) and (A.10) directly yield the rela- C*, the variable z goes once counterclockwise around tion the circuit C. From this it follows that a closed integral over the variable z along the circuit C may be converted Ð1 lᏹ , = nᏹ , , (A.14) into a closed integral over the variable u along the cir- n l Ðl Ðn cuit C* and conversely. In accordance with (A.7) and which is valid for all l and n. From (A.4), (A.9), and (A.5) we have (A.11) we obtain  α(+)  FᏹÐ1 Fᏹ du dz n Ðn dz n, l = Ðl, Ðn. (A.14') ----- = -----1 + ∑ ------z  = -----qú. (A.9) u z  p+  z (Ð) (Ð) n ≠ 0 α α The nonzero commutators m and m with u(z) and The matrices which are the inverse of (A.3) and (A.4) u(z) are obtained using (A.2) and (2.12): are represented most simply in the form (Ð) 2 m [α , u(z)] = Ð-----z u(z), m ≠ 0, m p ᏹÐ1 1 du Ðl n + n, l = ----π---- °∫ ----- u z , (A.10) (A.15) 2 i u [α(Ð), ] 1 m C∗ 0 u(z) = Ð-----z u(z) p+ FᏹÐ1 1 du Ðl n( )Ð1/2 and similarly for barred quantities. n, l = ------∫ ----- u z qú . (A.11) 2πi ° u From the deﬁnitions (A.3) and (A.4) allowing for C∗ (A.15) it follows that In order to prove the equalities ᏹᏹÐ1 = ᏹÐ1ᏹ = 1, FᏹFᏹÐ1 FᏹÐ1Fᏹ [α(Ð), ᏹ ] 2nᏹ = = 1 we use the identities m n, l = Ð----- n, l Ð m, p+ s +∞ n  d  1 dz1 s z  z f (z, u(z)) = ------f (z , u(z )) ∑ n ---- , (Ð) F 1   π °∫ 1 1   [α , ᏹ , ] = Ð------dz 2 i z1 z1 m n l π C n = Ð∞ 2 i p+

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dz Ð l + m n 1/2 Ð1/2 that (dz/du)du = dz we obtain (A.18b). The other for- × ∫---- z u (2n(qú) + m(qú) ), ° z (A.16) mulas [(A.18a) and (A.18c)] are proven similarly. C The following formulas also hold: m ≠ 0, +∞ +∞ Ð1 Ð1 1 (+) Ð1 ( ) ᏹ ᏹ α ᏹ [α Ð , ᏹ ] n ᏹ ∑ m, s n Ð m, r = ----- ∑ n Ð m m, r + s, (A.20) 0 n, l = Ð----- n, l, p+ p+ m = Ð∞ m = Ð∞

(Ð) F n F [α , ᏹ , ] = Ð----- ᏹ , . F Ð1 F Ð1 F Ð1 F Ð1 0 n l n l ∑l( ᏹ , ᏹ , Ð ᏹ , ᏹ , ) p+ n Ð l p l q n Ð l q l p l The commutation relations (A.16) conserve their form (A.21) ( ) if barred quantities are substituted everywhere in q Ð p α(+)ᏹÐ1 (A.16). The relations (A.16) exhaust all the nonzero = ------∑ l (n Ð l), ( p + q), p+ ( ) F l commutators between α(Ð), α Ð , ᏹ, ᏹ , Fᏹ, and ᏹ . which we derived using formulas (A.9), (A.10), and From (A.14) and (A.16) we then obtain (A.12). [α(Ð), ᏹÐ1 ] 2nᏹ ≠ m n, l = ----- Ðl, Ð(m + n), m 0, p+ (A.17) REFERENCES [α(Ð), ᏹÐ1 ] n ᏹ l ᏹÐ1 1. R. Jackiw, E-print Archive, gr-qc/9612052. 0 n, l = ----- Ðl, Ðn = ----- n, l. p+ p+ 2. E. Benedict, R. Jackiw, and H.-J. Lee, Phys. Rev. D 54, 6213 (1996). In Section 3 we use the following formulas: 3. D. Cangemi, R. Jackiw, and B. Zwiebach, Ann. Phys. 245, 408 (1996); D. Cangemi and R. Jackiw, Phys. Lett. ᏹÐ1 ᏹÐ1 ᏹÐ1 ∑ m, l + q n, p Ð q = (m + n), (l + p), (A.18a) B 337, 271 (1994); D. Cangemi and R. Jackiw, Phys. Rev. D 50, 3913 (1994); D. Amati, S. Elitzur, and q E. Rabinovici, Nucl Phys. B 418, 45 (1994); D. Louis- Martinez, J. Gegenberg, and G. Kunstatter, Phys. Lett. B (ᏹÐ1 ᏹÐ1 ᏹÐ1 ᏹÐ1 ) 321, 193 (1994); E. Benedict, Phys. Lett. B 340, 43 ∑q m, l + q n, p Ð q Ð n, l + q m, p Ð q (A.18b) (1994); T. Strobl, Phys. Rev. D 50, 7346 (1994). q 4. S. N. Vergeles, Zh. Éksp. Teor. Fiz. 113, 1566 (1998). = (m Ð n)ᏹ ( ), ( ), Ð l + p Ð m + n 5. O. Andreev, Phys. Rev. D 57, 3725 (1998). 6. S. N. Vergeles, E-Print Archive, hep-th/9906024. 2(ᏹÐ1 ᏹÐ1 ᏹÐ1 ᏹÐ1 ) ∑q m, l + q n, p Ð q Ð n, l + q m, p Ð q 7. P. A. M. Dirac, Lectures on Quantum Mechanics (A.18c) q (Yeshiva Univ., New York, 1964). 8. P. A. M. Dirac, Lectures on Quantum Field Theory = (m Ð n)(p Ð l)ᏹ ( ), ( ). Ð l + p Ð m + n (Yeshiva Univ., New York, 1967; Mir, Moscow, 1971). Using formulas (A.10) and (A.12) for s = 1 we can 9. E. Del Guidice, P. Di Vecchia, and S. Fubini, Ann. Phys. obtain the following relation: 70, 378 (1972).

+∞ 10. M. B. Green, J. H. Schwarz, and E. Witten, Superstring Ð1 Ð1 1 Ðp n d Ðl m Theory (Cambridge Univ. Press, Cambridge, 1987; Mir, ∑ qᏹ , ᏹ , = ------duu z (u z ). m l + q n p Ð q 2πi °∫ du Moscow, 1990). ∞ (A.19) q = Ð C∗ 11. P. A. M. Dirac, The Principles of Quantum Mechanics Since we are only interested in the antisymmetric com- (Clarendon Press, Oxford, 1958; Nauka, Moscow, ponent of the relation (A.19) in terms of the indices m 1970). and n, on the right-hand side we can remove uÐl from beneath the differentiation sign. Also bearing in mind Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 102Ð108. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 115Ð121. Original Russian Text Copyright © 2000 by Eritsyan. FLUIDS

Diffraction Reflection of Light in a Cholesteric Liquid Crystal in the Presence of Wave Irreversibility and Bragg Formula for Media with Nonidentical Forward and Return Wavelengths O. S. Eritsyan Yerevan State University, Yerevan, 375049 Armenia; e-mail: [email protected] Received October 22, 1998

Abstract—An analysis is made of the diffraction reﬂection of light in a cholesteric liquid crystal in the presence of magnetooptic activity which leads to wave irreversibility and in particular to nonidentical forward and return wavelengths. It is shown that in this particular case, the Bragg formula containing a single wavelength which is the same for the forward and return waves should be written to include two wavelengths. Relations are put forward which generalize the Bragg formula for media with nonidentical forward and return wavelengths and examples of using these relations are considered. A boundary-value problem is solved for a layer of cholesteric liquid crystal. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Direct calculations show that the wavelengths of the The propagation of light in cholesteric liquid crys- forward and return waves differ and the form of the tals (CLCs) is known to have diffraction properties [1, 2]. relationship which expresses the condition for amplifi- In media whose spatial structure exhibits rightÐleft cation of waves reflected at periodic inhomogeneities in asymmetry such as gyrotropic media and CLCs, which the medium differs from the Bragg formula. Accord- by definition should also be classed as gyrotropic [3], ingly, in the Laue equation [5] the moduli of the wave wave irreversibility occurs in the presence of magne- vectors of the forward and return waves are different tooptic activity [4]: the wave vector surface has no cen- and the diagram which expresses this equation geomet- ter of symmetry with the result that the quantities char- rically [5] is asymmetric relative to the plane perpen- acterizing the properties of the medium (phase velocity, dicular to the vector of the reciprocal grating and moduli of the angles of rotation of the polarization bisecting it. plane and circular dichroism, and so on) differ for mutually opposite directions of propagation. It is found that wave irreversibility in the sense of no center of symmetry at the wave vector surface is a strin- If diffraction reflection is analyzed in a CLC as light gent constraint for changing the form of the Bragg for- propagates along the axis of the medium in the pres- mula. Specifically, this change also occurs in a natu- ence of a magnetic field, the field induces magnetooptic activity in the medium which leads to the wave irrevers- rally gyrotropic medium in the presence of periodic ibility noted above. In order to avoid effects involving inhomogeneity. The different wavelengths of the for- distortion of the structure which occur under the influ- ward and return waves in this medium is attributed to ence of the field, it is advisable to select CLCs with the different polarization of these waves. Periodically inhomogeneous naturally gyrotropic media are investi- Franck moduli K22, K33 [2, p. 244], for which no distortion occurs until the field reaches a certain critical gated in Section 3. value, and then the Bragg formula In Section 4 we analyze diffraction reflection for ≠ 2d sinϕ = nλ ki ks (ki and ks are the wave vectors of the forward (incident) and return (scattered) waves) for propagation becomes meaningless since there is not one wavelength inclined toward the planes of the layers neglecting but two different ones (as a result of the wave irrevers- interaction between the forward and return waves in the ibility) for the forward and return directions of propa- dispersion equation. We also briefly consider the case gation. where an external magnetic field is present in a natu- In Section 2 we shall analyze diffraction reflection rally gyrotropic medium. in a CLC under conditions of wave irreversibility and we shall calculate the wavelengths of the forward and In Section 5 we present results for the propagation return waves at frequencies coinciding with the bound- of light across a CLC layer possessing wave irrevers- aries of the frequency range of diffraction reflection. ibility.

DIFFRACTION REFLECTION OF LIGHT IN A CHOLESTERIC LIQUID CRYSTAL 103

2. DIFFRACTION REFLECTION IN A determined from the constraint that equation (1) has MAGNETOACTIVE CHOLESTERIC LIQUID zero roots [1, 2] and these are automatically multiples.) CRYSTAL AND THE BRAGG FORMULA We denote the multiple roots of equation (1) for g = 0 ᏷ ᏷ ω 2.1. Dispersion Equation by m. For g = 0 we have m = 0 at the boundaries 1 ω We consider the propagation of light at frequency ω and 2 of the diffraction reflection region. In view of ᏷ along the axis of a CLC (z axis) in the presence of an the smallness of g, the values of mg will differ little external magnetic field directed along this axis. Using from their values for g = 0, i.e., from zero. (The magne- the method of circular components [1] or using a tooptic rotation of the plane of polarization in fields of method of converting in the wave equation to the field ~104 G in nonmagnetic dielectrics in the optical fre- components relative to the x' and y' axes which rotate quency range is ~10 deg/cm. At the wavelength (in vac- together with the structure [6] (the x' axis is everywhere uum) λ Ӎ 5 × 10Ð5 cm and the permittivity ε ~ 5, we oriented along the director, the y' axis is perpendicular Ð5 ᏷ obtain g ~ 10 .) Neglecting the fourth power of g in to the x' axis, and the three axes x', y', and z form a right- (1), we arrive at the following expression for ᏷ : handed system), in either case we arrive at the follow- g ing equation: Ð1 ω2 ω2 ᏷  ± η (ε ε ) 2 g = Ð 2q-----g  ----- 1 + 2 + 2q , (3) ω2 ω2 2 2 ᏷ 4 (ε ε ) 2 ᏷ 2 ᏷ c c g Ð ----- 1 + 2 + 2q g Ð 4q-----g g c2 c2 (1) where 2 2 4 ω 2 ω 2 ω 2 ε ε 2 2 2 + ----- 1 Ð q ----- 2 Ð q Ð -----g = 0.  ω ω ω  2   2  4 η (ε ε ) 2  ε 2  ε 2 c c c =  ----- 1 + 2 + 2q ----- 1 Ð q  ----- 2 Ð q   c2 c2 c2 π ᏷ Here 2 / g is the spatial period of the field in the local 1/2 (4) ε ε 4 2 system of the medium, 1 and 2 are the principal values ω ω  2 2 (ε ε )  of the permittivity tensor of the CLC along the x' and y' + ----4-g 2q Ð ----2- 1 + 2 . axes, respectively, g is the z-component of the gyration c c  vector directed along the z axis, which is responsible for the magnetooptic activity, q = 2π/σ, and σ is the The roots will be multiples if η = 0. From the equation η ω ω pitch of the helix. The projections kmzof the wave vec- = 0 we determine the boundaries 1g and 2g of the tors of the circular components of the field are related frequency region of diffraction reflection. To within to the roots of equation (1) by terms containing g to the second power, we have ± ᏷ ± 2 kmz = mg q, (2) qc g ω = ------1 + ------, (5) 1g ε 2ε (3ε + ε ) as in the case of no magnetooptic activity analyzed in 1 1 1 2 [1, 2]: the difference is that the roots of equation (1) for ≠ 2 g 0 differ from the roots of this equation for g = 0. qc g ω = ------1 + ------. (6) 2g ε 2ε (3ε + ε ) 2 2 2 1 2.2. Determination of the Boundaries of the Diffraction Reflection Region and the Values Substituting into (3) η = 0 and the values of the fre- ᏷ ᏷ ω of mg at these Boundaries quencies (5) and (6), we obtain the values 1, 2g( 1g) ω ω and ᏷ (ω ) of the multiple roots, confining our- We shall now determine the boundaries 1g and 2g 1, 2g 2g of the diffraction reflection region (the subscript g indi- selves to quantities linear in g: cates the presence of magnetooptic activity). We shall assume that ε , ε , and g are real. Outside the region of ᏷ (ω ) 2qg 1 2 1, 2g 1g = Ð----ε------ε--- (7) ᏷ 3 1 + 2 diffraction reflection the values of mg are real, while inside this region they have an imaginary part, i.e., they ω ω ± at the frequency = 1g and are complex. The same applies to kmz since the value of q in (2) is real. Since the coefficients of equation (1) ᏷ (ω ) 2qg 1, 2g 2g = Ð----ε------ε--- (8) are real, its complex roots should be complex-conju- 3 2 + 1 gate. Thus, at the boundaries of the diffraction reflec- ω ω tion region at which the imaginary parts of the complex at = 2g. The relative error in the determination of the roots of equation (1) vanish on leaving this region, ᏷ 4 these roots should be multiples. (For g = 0 the bound- multiple roots associated with the neglect of g in (1) 2 ε2 ε ε ε aries of the region of diffraction reflection are usually is of the order of g / ( ~ 1 ~ 2).

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A characteristic feature of this diagram is that the moduli of the vectors ki and ks differ so that there is no ks symmetry relative to the plane perpendicular to the vector t which bisects the latter (this plane is indicated by the dashed line in the ﬁgure). τ Projecting the Laue equation onto the direction t, we obtain 2π 2π 2π 0 -λ----- + --λ---- = ------, (15) s i d which merely yields the Bragg formula in its usual ki form 2d sinϕ = nλ (16) Fig. 1. Diagram describing the Laue equation for a magne- ϕ π λ λ toactive CLC. As a result of the wave irreversibility, the (in this case n = 1, = /2) when i = s. moduli of the wave vectors of the forward and return waves In the following section we consider an example of (ki and ks, respectively) differ. another medium for which the Bragg formula in its usual form also cannot be applied because of a differ- λ λ ence between i and s. 2.3. Laue Diagram and Bragg Formula Using (2), (7), and (8) we obtain for the z-compo- 3. PERIODICALLY INHOMOGENEOUS nents of the wave vectors of waves with diffraction ISOTROPIC NATURALLY GYROTROPIC polarization MEDIUM

± ± 3.1. Material Equations k = k = Ð 2qg(3ε + ε )Ð1 ± q (9) 1z 2z 1 2 We shall analyze the propagation of an electromag- ω netic wave of frequency ω in a medium described by at frequency 1g and the material equations ± ± k = k = Ð 2qg(3ε + ε )Ð1 ± q (10) 1z 2z 2 1 1 ÐiΩt iτz Ðiτz D = ε E + --(∆ε)e [e + e ]E + γ rotE, ω 0 2 (17) at frequency 2g. To be specific, we shall analyze one of these fre- B = H, ω ω quencies, say = 1g. From (9) we have for the z-com- where Ω is the frequency of the wave modulating the ponent of the wave vector of the forward wave (propa- permittivity of the medium, ∆ε is the percent modula- gating in the direction of the z axis) tion, τ = 2π/d, and d is the period of the inhomogeneity ( ε ε ) of the medium. The equations (17) describe a naturally kzi = Ð 2qg 3 1 + 2 + q. (11) gyrotropic isotropic medium [3, 5, 7, 8] with spatially For the return wave (k < 0) we have modulated permittivity. This modulation may be cre- zs ated, for example, by a plane ultrasonic wave. We shall k = Ð 2qg(3ε + ε ) Ð q. (12) assume that in the wave equation for the electromag- zs 1 2 netic wave we can neglect the time dependence of the | | ≠ | | In accordance with (11) and (12) we have kzi kzs . parameters of the medium but allow for their time Thus, the wavelengths of the forward and return waves, dependence in the final results. Whereas in the absence λ and λ also differ: of spatial dispersion (γ = 0) this procedure can be i s adopted for Ω/ω Ӷ 1 [9], for γ ≠ 0 we also need to π π σ impose the constraint λ 2 2 i, s = ------= ------= ------Ð--1 . (13) k , k , ± ( ε ε ) 2 i s zi s 1 2g 3 1 + 2 Ω ω ----∆ε Ӷ -----γ , (18) ω 2 Figure 1 gives a diagram illustrating the Laue equa- c tion in order to correctly conserve γ in the wave equation, ks Ð ki = t, (14) neglecting the time derivatives given above. Assuming that the length of the ultrasonic wave is of the order of where τ = 2πn/d, d = σ/2 is the period of inhomogene- the wavelength of light (which is required for diffrac- ity of the medium. tion reflection, i.e., Ω/v ~ ω/c, where v is the velocity

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2 of the mechanical wave), we write condition (18) in the 2 2 4 2 2 4 ω τ  ω (∆ε) τ ω 2 form ε ------γ + ----2- 0 Ð ---- Ð ----4- Ð ----4- = 0. 2 c 4 c 4 4 c v ω -----∆ε Ӷ -----γ . (19) λ 2 c c In an isotropic homogeneous medium the spatial dispersion leads to a change in the moduli of the wave ϑ 2 2 The usual values of the angle 0 of rotation of the plane vectors by a value of the order of ω γ/c [5]. Assuming of polarization per unit path length of the ray are a few that x in (25) is a value of this order and neglecting x to ϑ Ӎ degrees per centimeter. Assuming 0 5 deg/cm, we the fourth and third powers, we obtain from (25) obtain |ω2γ/c2| ~ 10Ð1. Consequently relation (19) is ω2 τ2 ω2 reduced to the inequality  ε  γ ± x = ----2- 0 + ---- ----2- u ω c 4 c ∆ε Ӷ --- 0.1, Ð1 (26) c 2 2 4 ω τ  ω 2 × 2 -----ε + ---- + -----γ , which is easily satisfied.  2 0  4 c 4 c ω2 τ2 ω4 3.2. Diffraction Reflection  ε  γ 2 u = ----- 0 + ---- ----- We shall express the field of a monochromatic wave c2 4 c4 propagating in a medium along the z axis in the form ω2 τ2 ω4 E(z, t) ×  ε  γ 2 2----- 0 + ---- + ----- (27) c2 4 c4 (20) = E exp(ik z) + ∑E exp(ik z) exp(Ðiωt). 2 0 0z m mz 2 2 4 2 2 4 ω τ  ω (∆ε) τ ω 2 m × -----ε Ð ---- Ð ------Ð ------γ .  2 0  4 4 4 4 Using equations (17) we obtain from the wave equation c 4 c c τ ± , ± , … kmz = k0z + m , m = 1 2 . (21) Assuming a relative error of the order Ð1 In a two-wave approximation which takes into account 4 2 2 ω ω 2ω τ  the spatial components E0exp[i(k0zz Ð t)] and -----γ -----ε + ---- , ω 4  2 0 4  E−1exp[i(kÐ1zz Ð t)] we obtain the following equation c c for k : 0z we obtain from (27) the following expression for the ω2 ω2 multiple roots: ε 2 γ ----- 0 Ð k0z + ----- k0z c2 c2 ω2 1 γ (22) x1, 2 = ------. (28) ω2 ω2 ω4 (∆ε)2 2 2 × ε ( τ)2 γ( τ) c ----- 0 Ð k0z Ð + ----- k0z Ð Ð ------= 0. c2 c2 c4 4 Thus, in accordance with (23), (24), and (28), we have

In order to determine k0z and kÐ1z at the boundaries of τ 1ω2 τ 1ω2 the diffraction reﬂection region, we express k in the k = -- + ------γ , k = Ð -- + ------γ . (29) 0z 0z 2 2 Ð1z 2 2 form 2 c 2 c τ Formulas (29) are satisﬁed when the polarization of the k = -- + x. (23) 0z 2 waves is given by

For kÐ1z we then have E0x Ð iE0y = 0, EÐ1x Ð iEÐ1y = 0 (30) τ (the forward wave is right circularly polarized and the k = Ð -- + x. (24) Ð1z 2 return wave is left circularly polarized). When the polarizations of both waves are the reverse, γ in (29) Substituting (23) into (22) we obtain the equation for x: should be replaced by Ðγ. ω2 ω2 τ2 ω4 According to (29), the wavelengths of the forward 4 γ 3  ε  γ 2 2 x Ð 2----- x Ð 2 ----- 0 + ---- Ð ----- x and return waves differ so that the Bragg formula 2  2 4  4 c c c should contain two wavelengths, as in the case of a CLC possessing wave irreversibility. ω2 τ2 ω2  ε  γ + 2----2- 0 + ---- ----2- x (25) The frequency boundaries of the diffraction reﬂec- c 4 c tion region are determined from the equation u = 0 in

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periodic inhomogeneities. Substituting into (29) the + value of τ/2 from (31) we obtain – τ ϕ 2 ϕ ks s ω 1ω ω ∆ε i k = --- ε + ------γ ± ------(33) 0z c 0 2 2 c ε c 4 0 0 k kx i ψ (the two signs correspond to the two frequency bound- ψ s i aries of the diffraction reflection region). Hence, at the boundaries of the diffraction reflection region wave interaction produces a correction to the wave vector ω ∆ε ε kz moduli given by ( /c) /4 0 and the spatial dispersion changes these moduli by ω2γ/2c2. Thus, it is justi- Fig. 2. Diagram describing the Laue equation for an isotropic naturally gyrotropic periodic inhomogeneous medium. fiable to neglect wave interaction when determining the The circles give the lines of intersection of the wave vector wavelengths if surface with the plane k and k . x z ω ∆ε ---γ ӷ ------. (34) c 2 ε (27). Assuming a relative error of the order of τ2γ2/∆ε, 0 for these boundaries we have For light having the wavelength in vacuum λ Ӎ 6 × 10−5 cm relations (32) and (34) for ω2γ/c2 ~ 0.1 give ω Ð1/2 1, 2 τ ∆ε -- ε ± ------∆ε Ð6 ------=  0  . (31) Ӷ 10 Ӷ 2 ∆ε . (35) c 2 2 -ε----- 0 The following relation was assumed to derive (31) These relations are satisfied, for example, for ω ---γ Ӷ 2 ∆ε . (32) ε ∼ 5, ∆ε Ӎ 5 × 10Ð7. c 0 Figure 2 shows a cross section over the plane (passing through the z axis) of the wave vector surface for 4. OBLIQUE PROPAGATION OF LIGHT. medium (17) neglecting periodic inhomogeneity. The GENERALIZATION OF THE BRAGG FORMULA two spheres correspond to right and left circularly In Sections 2 and 3 we considered the propagation polarized waves: the moduli of the wave vectors for of light perpendicular to the layers and we allowed for these waves (i.e., the radii of the spheres) are given by: the interaction of the forward and return waves. For a ω ω2 ω ω2 Ð ε 1 γ + ε 1 γ CLC this can be seen from the fact that we used an k = --- 0 + ------, k = --- 0 Ð ------(36) exact dispersion equation which allows for all the c 2 c2 c 2 c2 waves (we selected those solutions corresponding to diffraction polarization). For a medium described by (the superscripts “–” and “+” correspond to left and equations (17), we obtained a dispersion equation to right circular polarizations). determine k0z and kÐ1z allowing for both waves with the z-components of the wave vectors k0z and kÐ1z. We shall 4.2. Laue Diagram now consider oblique propagation with respect to the layers neglecting interaction of the waves. We shall now assume that a periodic inhomogeneity is created in the medium and relation (34) is satisfied. Then, assuming a relative error of the order ∆ε ε ωγ Ð1 + 4.1. Condition for which Wave Interaction ( / 0)( /c) we can use the expressions (36) for k at Periodic Inhomogeneities and kÐ and construct a diagram which geometrically in a Medium Can Justifiably Be Neglected describes the Laue equation, assuming that the moduli of the wave vectors are equal to the radii of the spheres. According to (29), the difference between the wavelengths of the forward and return waves is caused by the Figure 2 shows a situation which satisfies the Laue presence of spatial dispersion which makes the contri- equation. Since no inhomogeneities occur in the direc- butions ~ω2γ/2c2 to the moduli of the wave vectors of tion of the x axis, the tangential components of the these waves. In order to conserve the effect of different wave vectors are the same: wavelengths we must retain the quantities ω2γ/2c2 in ϕ ϕ ks cos s = ki cos i. (37) k0z and kÐ1z. We now determine the conditions under which the contribution of the spatial dispersion can be This relation is also obtained by projecting the Laue ≠ correctly retained while neglecting wave interaction at equation onto the plane perpendicular to t. Since ks ki,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 DIFFRACTION REFLECTION OF LIGHT IN A CHOLESTERIC LIQUID CRYSTAL 107 we have ϕ ≠ ϕ s i. (38) ϕ ϕ k s i s Projecting the Laue equation onto the t direction, we 02 τ obtain 0 kx 01 ki 2π ψ k sinϕ + k sinϕ = ------n. (39) ψ s s s i i d i This relation is the phase condition for amplification of waves scattered at periodic inhomogeneities in a ϕ ≠ ϕ kz medium. Since i s, formula (39) does not reduce to the usual form of the Bragg formula. Fig. 3. As Fig. 2 for the case of an external magnetic field which produces wave irreversibility. 4.3. Case Where a Magnetic Field is Present for light incident normal to the layers have the same In the presence of a magnetic field directed along form as the diagram in Fig. 1 for a CLC. the z axis, the term i[gE] responsible for the magnetooptic activity formed as a result of the presence of this field is added to the right-hand side of the first of the 4.4. Generalization of the Bragg Formula material equations (17). Assuming a relative error of The Bragg formula for a periodically inhomoge- the order g2 in the moduli of the wave vectors, for the neous medium in which the wavelengths of the forward moduli k+ and kÐ we have [4, 10] and return waves differ has the form π π π ± ω −  g (ω ⁄ c)γ 2 ϕ 2 ϕ 2 k = --- ε 1 + ---- cosψ + ------, (40) -λ---(---ϕ-----) sin i + λ----(---ϕ-----)-sin s = ------n, (43) c 0 ε ε  i s d 0 0 to which we need to add the following formula because ψ ϕ ϕ where is the angle between the direction of propaga- of the presence of the two angles i and s which gen- tion of the wave and the z axis along which the external erally differ field is applied. 2π 2π ϕ ψ ------cosϕ = ------cosϕ (44) The Bragg formula has the form (sin i = cos i, λ(ϕ ) i λ(ϕ ) s cosϕ = sinψ , sinϕ = Ðcosψ , cosϕ = sinψ , see i s i i s s s s ϕ ϕ Fig. 3) (instead of i = s). The dependence of λ on ϕ is given by the dispersion ω 1 g ω 1  --- ε 1 +− ------sinϕ + ---γ ------sinϕ equation. c 0 2ε i c ε  i 0 0 (41) 5. PROPAGATION OF LIGHT ACROSS ω 1 g ω 1  2π + --- ε 1 +− -- Ð---- sinϕ + ---γ ------sinϕ = ------n. A PLANE-PARALLEL LAYER c 0 2 ε s c ε  s d 0 0 OF CHOLESTERIC LIQUID CRYSTAL WITH WAVE IRREVERSIBILITY To this formula we need to add the following (which ϕ ϕ λ λ We analyze the normal propagation of plane-polar- replaces the relationship s = i satisfied for s = i): ized light across a plane-parallel CLC layer possessing wave irreversibility. We shall study two variants: prop- 1 g ω 1  1 +− ------sinϕ + ---γ ------cosϕ agation in two mutually opposite directions. Figure 4b 2ε i c ε  i 0 0 gives the results of calculating the difference between (42) the transmission coefficients for different wavelengths 1 g ω 1  = 1 +− -- Ð---- sinϕ + ---γ ------cosϕ . outside the region of diffraction reflection and Fig. 4a 2 ε s c ε  s ∆ 0 0 gives the results for this region. The difference T = T1 Ð T2 is plotted on the ordinate where T1 is the trans- The four variants of the signs on the left- and right- mission coefficient when light is incident on the layer hand sides of expressions (41) and (42) correspond to in the direction of the z axis which lies in the direction the four situations: right and left polarizations of the of the gyration vector g perpendicular to the layer scattered wave for right and left polarizations of the boundaries; T2 is the transmission coefficient for the incident wave. opposite direction of propagation of the incident light. ε A diagram expressing the Laue equation is shown in The components of the permittivity tensor are 1 = 2.290, ε Ð4 µ Fig. 3. Note that the diagrams plotted in Figs. 2 and 3 2 = 2.143; g = 10 , the helix pitch is 0.42 m; the layer

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–4 ∆T, 10 in different coefﬁcients T1 and T2 for the two mutually opposite directions of propagation of plane-polarized (a) light across the layer. 0.08

0 ACKNOWLEDGMENTS The author would like to thank A. A. Gevorgyan for –0.08 the computer calculations and discussions of various results. –0.20

0.62 0.63 λ, µm REFERENCES ∆T 1. E. I. Kats, Zh. Éksp. Teor. Fiz. 59, 1854 (1970). 2. V. A. Belyakov and A. S. Sonin, Optics of Cholesteric 0.01 (b) Liquid Crystals (Nauka, Moscow, 1982). 3. F. I. Feodorov, Gyrotropy Theory (Nauka i Tekhnika, Minsk, 1976). 4. O. S. Eritsyan, Usp. Fiz. Nauk 138, 645 (1982). 0 5. L. D. Landau and E. M. Lifshits, Électrodynamics of Continuous Media (Pergamon, Oxford, 1984; Nauka, Moscow, 1982). 6. C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933). –0.01 7. V. M. Agranovich and V. L. Ginzburg, Spatial Disper- 0.60 0.61 0.64 0.65 sion in Crystal Optics and Theory of Excitons (Nauka, λ, µm Moscow, 1979). Fig. 4. Difference between the transmission coefficients for 8. B. V. Bokut’, A. N. Serdyukov, F. I. Feodorov, et al., different wavelengths inside (a) and outside (b) the region of Kristallografiya 18, 227 (1973). diffraction reflection. 9. L. A. Ostrovskiœ and B. N. Stepanov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 484 (1981). thickness is 200 µm, and the boundaries of the diffrac- 10. O. S. Eritsyan, Optics of Gyrotropic Media and Choles- tion reflection region 615 and 630 µm. It can be seen teric Liquid Crystals (Aœastan, Yerevan, 1988), p. 68. from the figures that the wave irreversibility (different wavelengths of the forward and return waves) resulted Translation was provided by AIP

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 109Ð120. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 122Ð135. Original Russian Text Copyright © 2000 by Bogdanov, Men’shikov. SOLIDS

Multiple Small-Angle Neutron Scattering for an Arbitrary Value of the Born Parameter S. G. Bogdanov and A. Z. Men’shikov Institute of Metal Physics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620219 Russia Received May 11, 1999

Abstract—Computer calculations are made of the intensity of multiple small-angle neutron scattering using the general Moliére formula [8] over a wide range of variation of the Born parameter, embracing the diffraction and refraction regimes, and a transition region between diffraction and reﬂection. A comparison is made with approximate formulas obtained earlier by Maleev et al. [9, 10] in the limiting cases of the Born parameter α Ӷ 1 and α ӷ 1 for the diffraction and refraction regimes, respectively. It is shown that over a wide range of values of α the results of the calculations using the approximate and general formulas are the same. The theoretical conclusions were checked experimentally using data from measurements of small-angle neutron scattering for the domain structure of ferromagnets. Measurements were made of the neutron beam broadening for samples of different thickness and these were used to determine the effective domain sizes in pure iron and nickel exposed to thermal treatment and plastic deformation, and also in the Invar alloys Fe65Ni35 and Fe3Pt. An analysis is made of the angular dependence of magnetic small-angle neutron scattering at the asymptote. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION processes written in an integral form, which is valid for Multiple small-angle scattering is one of the main an arbitrary value of the Born parameter. However, this physical processes accompanying the propagation of theory has not been widely used to analyze the experi- corpuscular or wave radiation through inhomogeneous mental results because of the complexity of analyzing a media. It is observed in the broadening of the primary general expression. In order to simplify this, the authors beam as a result of multiple refraction or diffraction of [9, 10] used various approximate solutions of this from various types of inhomogeneities whose effective problem which can give expressions for the multiple radius R is considerably greater than the wavelength scattering intensity in an easier form for analysis. In λ eff ӷ λ of the incident radiation (Reff ). This phenomenon particular, the case α Ӷ 1 was considered in [9] (the was first analyzed in 1926 by Nardroff [1] for the mul- Born approximation or diffraction regime) and the case tiple refraction of X-rays. Nardroff derived a formula α ӷ 1 (non-Born approximation or refraction regime) linking the broadening of the primary beam with the in [10]. In both approximations multiple scattering was effective size of the particles at which refraction takes considered as random processes described by a Gauss- place. Subsequently, an effect involving changes in the ian function. Both solutions yielded the conclusion that width of the X-ray scattering curve was analyzed for the broadening of a primary neutron beam is propor- diffraction processes of various orders [2]. tional to the square root of the scattering order: Multiple neutron scattered was first observed by ∝(L/l)1/2, where L is the sample thickness and l) is the Hughes et al. [3] when neutrons passed through an neutron mean free path. This showed fairly good agree- unmagnetized pure iron plate. In this case, the broaden- ment with the experimental results [4Ð6]. However, in ing of the primary neutron beam occurred as a result of the refraction regime where the sample thickness multiple refraction and diffraction at the domain struc- ӷ ture. This effect was then studied in greater detail in [4, 5] exceeded some critical value (L L0) another law was where the effective domain size was determined from identified, i.e., a linear dependence of the broadening the results of the broadening of a primary beam of on thickness. However, the experimental confirmation monochromatic neutrons. Recently this method has reported in [11] was cast into doubt by these same also been widely used to determine the size of the scat- authors because of the large experimental error associ- tering particles in nonmagnetic powder and porous ated with the nonidentical domain structure in samples materials [6, 7]. A method based on an analysis of the of different thickness. In addition, the dependence of line curvature near the zero scattering angle developed the broadening on the thickness in the transition region in [7] is highly original in this respect. from refraction to diffraction where the parameter α is The problem of multiple particle collisions was con- close to one was left undefined. The angular depen- sidered theoretically by Moliére (see, for example, [8]) dence of the multiple scattering intensity for large scat- who obtained an expression for the intensity of multiple tering vectors was also not determined in this region.

110 BOGDANOV, MEN’SHIKOV

The aim of the present paper is to study multiple In a different formulation this parameter α has the fol- small-angle neutron scattering over a wide range of lowing form [10]: α particle sizes or the Born parameter , covering the dif- α λ δ( ) fraction and refraction regimes, and also the transition = 2 R Nb , (6) region from diffraction to refraction, by means of δ δ | | where (Nb) is the contrast, (Nb) = N1b1 Ð N0b0 . Here numerical calculations using the Moliére formula. On N is the nuclear density and b is the amplitude of coher- the basis of these calculations and an analysis of sim- ent neutron scattering. The subscripts “0” and “1” refer pler analytic expressions obtained in [9, 10] we discuss the matrix and the inhomogeneity, respectively. the results of experimental investigations of multiple magnetic scattering of neutrons at the domain structure Formulas from [9, 10] written in the following form in various ferromagnetic materials such as pure iron were used as approximate solutions of the multiple neutron scattering problem: and nickel, and also in the Invar alloys Fe65Ni35 and Fe3Pt. ( , ) ( , ) ( , ) ( , ) I q L = I0 q L + I1 q L + I2 q L , (7) α Ӷ 2. NUMERICAL CALCULATIONS where for 1 we have ( , ) ( , ) ( 2 ⁄ 2) In general, the intensity of multiple small-angle I0 q L = I0 0 L exp Ðq q1 , (8) scattering for an arbitrary value of the Born parameter α has been calculated using the Moliére formula [8] 1 BL written in the following form: q1 = ------. (9) R ld ∞ σ ( , ) S Λ Λq  Λ L Λ In this case, the parameter B was defined as B Ӎ I q L = ----π--∫ J0 -- exp Ð1 Ð -σ----- --- d , (1) σ 2 k 0 l ln(4L/ld) + lnln(4L/ld), where ld = 1/(n d) is the mean 0 free path, n is the volume density of the inhomogene- σ πα2 2 where ities, and d = R /2 is the integral cross section for ∞ single scattering. 2π  q σ Λ σ( ) σ σ We used the following expressions for the terms I1 Λ = ----2--∫ J0 -- q dq, 0 = Λ Λ = 0, (2) k k and I2 [5]: 0 π θ λ 1 dσ S is the area of the beam cross section, q = 4 sin / is I (q, L) = --SLn------a--s, (10a) the transferred neutron momentum, θ is the scattering 1 2 dΩ π λ angle, k = 2 / is the wave number, J0(x) is a Bessel function, and σ(q) ≡ dσ/dΩ is the differential cross sec- 4ν2 dσ 1 I (qL) Ӎ ------SLn------a--s ------, (10b) tion for single scattering which is calculated as the 2 π dΩ qR square of the scattering amplitude. Following [12], in σ Ω α2 2 4 general we can write this in the form where d as/d = k /2q is the differential cross sec- 2 4 tion for single scattering at the asymptotic limit for σ(q) = k R spherical particles of radius R having smooth surfaces. 1 ν ӷ  2 For the case 1 (refraction and diffraction in the × (α 2) ) ( ) non-Born approximation) the differential and total  ∫(cos 1 Ð x Ð 1 J0 qRx xdx (3)  cross section for single scattering was calculated in 0 accordance with [13] using the formulas 1 2  2 σ 2 2( ) (α ) ( )  d r 2 4 α J1 qR + ∫ sin 1 Ð x J0 qRx xdx . ------= k R ------+ ------ Ω 2 2 2 2 0 d (α + (qR) ) (qR) In accordance with [13], the total cross section for sin- (11) ( ) gle scattering per particle σ is given by 2α J qR 2 2 0 ----Ð ------1------sin α + (qR) , α2 ( )2 qR 21 sinα cosα Ð 1 + qR σ = 4πR -- Ð ------Ð ------, (4) 0 2 α α2  σ π 2 r = 2 R . (12) where α is the Born parameter, defined as In expression (11), the first term describes refraction, U α ≡ 2ν = ----kR, (5) the second describes Fraunhofer diffraction, and the E third has an interference nature. U is the energy associated with the inhomogeneity, E is Neglecting the third term in (11), the authors of [10] the neutron energy, and R is the inhomogeneity radius. obtained different expressions for the multiple scatter-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 MULTIPLE SMALL-ANGLE NEUTRON SCATTERING 111 Ӷ ӷ ing intensity for L L0 and L L0, where L0 is a cer- R, cm tain critical thickness 10–6 10–5 10–4 10–3 10–2 10–1 α2 α L0 = lr ln . (13) l, cm 1 σ 0 Here lr = 1/(n r) is the neutron mean free path in the 10 Ӷ refraction regime. For L L0 the expressions for the intensity of the multiple processes, and therefore the broadening of the primary neutron beam, were similar to these expressions for the diffraction regime: 10–1 2 2  2  r S  k  q I (q, L) = ------expÐ------ , (14) 0 π  2 2 q2   2q2

α L L –2 q2 = ------ln------, (15) 10 2R lr 2lr r and the expression for I1 (q, L) is the same as formula (10a).1 ӷ –3 For samples of thickness L L0 the multiple scat- 10 tering intensity was described by formulas of a different type: 10–2 10–1 100 101 102 α 2 d S k q I (q, L) = ------3------, (16) 2π( 2 2)3/2 Fig. 1. Dependences of neutron mean free path l on Born q + q3 parameter α, calculated using various formulas: (18), (4)— solid curves, (19)—dashed curves, (20)—dot-dash curves.  1 1 L q = 1 + ------. (17) The scale for the inhomogeneity radius is shown above. The 3  π 2Rl calculations were made for volume concentrations of inho- r mogeneities ρ = 0.02 (1) and 0.3 (2). Here the multiple scattering intensity at the asymptotic limit is proportional to qÐ3 and the broadening has a linear dependence on the sample thickness. experimental conditions λ = 0.5 and 1 nm. The particle or pore size was varied between 10 nm and 1 mm. The Thus, in addition to comparing the results of the cal- ρ culations using general and approximate formulas, it volume fraction of inhomogeneities was taken to be would be interesting to check the validity of the conclu- 0.02 and 0.3 which is valid for systems having low and sions that the different types of dependences of the high inhomogeneity concentrations, respectively. broadening on the thickness and of the intensities on The neutron mean free path was calculated using the the wave vector exist for the refraction regime when formula L < L0 and L > L0. 1 l = ----σ----, (18) n 0 2.1. Dependence of Neutron Beam Broadening π 3 ρ σ on Sample Thickness where 1/n = 4 R /(3 ) and 0 is described by formula (4). The mean free path was calculated as a function of Numerical calculations of neutron beam broadening α or R and is given by the solid curve in Fig. 1 on a log- as a result of multiple processes were made for a hypo- arithmic scale for two values of ρ: 0.02 (1) and 0.3 (2). thetical sample comprising a system of spherical parti- The dashed and dot-dash curves also give the mean free cles of the same size, distributed uniformly in vacuum. paths ld and lr calculated using the approximate formu- The density of nuclei inside the particle was assumed to las valid in the diffraction and refraction regimes, 22 Ð3 be N = 8 × 10 cm and the amplitude of coherent neu- respectively: tron scattering was assumed to be b = 1 × 10Ð12 cm. According to the Babinet principle, this system may be 2 1 ld = ------, (19) represented as a matrix having the appropriate density 3ρλ2 N2b2 R in which pores of radius R are uniformly distributed. In δ 2R both cases the nuclear contrast is (NB) = NB. The sam- l = -----. (20) ple thickness was taken to be 2 and 10 cm. The calcula- r 3 ρ tions were made for neutron wavelengths close to the α It can be seen that the lines ld and lr intersect at = 2 ν 1 r ( = 1) whereas the true curve l(R) has a minimum at a An expression for I1 (q, L) was obtained by Yu. N. Skryabin. slightly higher value of α and exhibits oscillating

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R, cm L –6 –5 –4 –3 –2 –1 w = 0.241ρλ-----, α ӷ 1, L ӷ L , (21c) 10 10 10 10 10 10 3 2 0 101 R 102 where wi is the line width at half height. The results of the calculations for the volume concentration of inho- 101 0 10 mogeneities ρ = 0.3 are plotted in Fig. 2 (solid curves ρ

, cm 1, 2, and 3). Similar dependences were obtained for = , mrad 0 10 0 w L 0.02. The dashed curve gives the dependence of the 10–1 critical thickness L on the inhomogeneity radius R or –1 0 10 the Born parameter α calculated using formula (13). It –2 was found that the range of sizes where the condition 10 10–2 L ӷ L can be satisfied in practice and where the depen- 10–2 10–1 100 101 102 α 0 dence w3(R) should be used, is very narrow and corre- Fig. 2. Dependences of the primary neutron beam broaden- sponds to α ~ 2Ð10. ing on the inhomogeneity size or Born parameter α for a hypothetical sample of thickness L = 2 cm in the diffraction Naturally the reliability of these dependences can be (1) and refraction (2 and 3) regimes for a volume inhomoge- ρ checked by means of numerical calculations, using the neity concentration = 0.3. The dashed curve gives the crit- general formulas (1)Ð(3) which are valid for an arbi- ical thickness L0 introduced in [10] to determine the range of validity of the dependence w (L). trary value of the Born parameter. For this purpose we 3 first calculated the differential cross section for single scattering dσ/dΩ for each particle size R and then cal- R, µm culated the Fourier transform of the scattering cross 0 5 10 15 20 25 30 section σΛ. We then determined the angular dependence of the multiple scattering cross section I(q) and 1.5 the full width at half-height of this curve w(R). The calculations were made for the sample thickness L = 2 cm, ӷ Ӎ × Ð4 2 which satisfied the condition L L0 for R 5 10 cm. R 1.0 The results of these calculations are given by the circles π

/2 in Fig. 2. It can be seen that the values of w(R) obtained 0

σ 0.5 using the Moliére formula are accurately described by the sections of the curves w1(R) and w2(R) calculated 0 using the approximate formulas (21a) and (21b). More- over, the broadening only depends monotonically on 0 5 10 15 20 α the size and no anomalies are observed in the transition region. Similar calculations made for a sample of thick- Fig. 3. Integral cross section for single scattering calculated ness L = 10 cm and for λ = 1 nm revealed the same using the general formula (4) (solid curve) and in the refraction dependence. This suggests that the dependence w (L) limit using formula (2) allowing for (circles) and neglecting ӷ 3 (dashed curve) the interference term in formula (11) for the sin- for L L0 does not in fact exist, although the mathe- gle differential cross section for neutron scattering. matical solution of the problem in [10], made assuming that the third term in formula (11) makes a small contribution, was performed correctly. α Ӎ behavior in the range 2Ð20. It can be seen that the In order to show the influence of the third term in character of the curves depends weakly on the volume ρ expression (11) on the final results of the calculations concentration of inhomogeneities . of the total neutron scattering cross section, we made In order to explain the dependences of the primary numerical calculations of the single scattering cross neutron beam broadening on the size of the inhomoge- section in the crossover region. Figure 3 gives various neities, we first calculated the angular broadening of dependences of this cross section in arbitrary units. The the neutron beam using formulas (9), (15), and (17) first, shown by the solid curve, was calculated using the obtained in the diffraction and refraction regimes general formula (4). The second dependence, shown by the circles, was obtained from formula (2) using ρ λ2 L 4L 4L expression (11) where all terms including the interfer- w1 = 0.325 Nb ---ln------+ ln ln------ , R ld ld (21a) ence term were taken into account. It can be seen that for α > 2 these two dependences are very similar. How- α Ӷ 1, ever, if we neglect the third term in formula (11), we σ π 2 obtain the expression 0 = 2 R , shown by the dashed ρ λ2 L L L  curve in Fig. 3, which differs substantially from the true w1 = 0.459 Nb ---ln------+ ln ln------ , R 2lr 2lr (21b) scattering cross section given by the solid curve in the range of α values between 2 and 30. Hence, neglecting α ӷ Ӷ 1, L L0, the interference term in formula (11) leads to apprecia-

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Table 1. Beam broadening calculated using formulas (1) and (2) using the single scattering cross section in the refraction limit (11) allowing for (column 3) and neglecting (column 5) the interference term in the cross section (11) and results obtained using formulas (21) α µ R, m w(3), mrad w2, mrad w(2), mrad w3, mrad 1 2 3 4 5 6 2.4 3 15 13 83 81 4.8 6 10 8.5 24 20 8 10 6.4 6.3 11 7.2 16 20 5.0 4.2 5.0 1.8 24 30 3.5 3.3 3.5 0.8 80 100 1.4 1.6 1.4 0.07 ble inaccuracies in the calculations of the total scatter- ferred momentum. It can be seen from expression (7) ing cross section in this range of α values. that the multiple scattering intensity can be represented Similar inaccuracies are observed in the calcula- as the sum of three components I0, I1 and I2. The first of tions of the primary beam broadening. Table 1 gives the these accurately describes the broadening of the pri- primary beam broadening calculated from formula (1) mary neutron beam as a result of multiple processes using the differential single scattering cross section since the profile of the multiple scattering curve (1) is (11) allowing for (column 3) and neglecting (column 5) close to Gaussian. Differences between the curves I(q) the third term; results of calculations using the formu- calculated using the general formula (1) and formula las (21) are also given for comparison. It can be seen (8) only occur for large and very small q. We shall first that if the third term is neglected [w(2), column 5], then discuss the difference between these curves for large q α ӷ which is directly relevant to the experimentally mea- for < 10 (the condition L L0 is satisfied), the values of the neutron beam broadening are in fact close to those sured angular dependences of small-angle neutron scattering. The range of very small transferred momenta is obtained from the formula for w3(L) (column 6). If this term is taken into account in the calculations, the values less easily measured. of w [w(3), column 3] are closer to the broadening val- In the approximate theories [9, 10], the deviation of ues corresponding to the curve w2(R) (column 4). At the the dependences I (q, L) from the Moliére line profile is same time, as α increases, the contribution of the inter- 0 taken into account by the terms I1 and I2, for which ana- ference term to the cross section (11) decreases and the lytic expressions are given by formulas (10a) and (10b) values given in columns 3 and 5 become comparable Ð4 Ð5 where I1 and I2 are proportional to q and q , respec- and close to the values of w2(R) calculated using formula (21b). tively. Calculations of the dependence of the multiple scattering cross section on the transferred momentum Consequently, the conclusion reached in [10] that a for particles of different sizes using the Moliére for- critical thickness L0 exists and that the dependences ∝ ∝ Ð3 ӷ mula showed that at the asymptote the multiple scatter- w3 L and I(q) q are obtained for L L0 should be ing intensity is described by the law I(q) ∝ qÐ4. In this attributed to neglecting the interference term in the sin- case, the contribution from I is not significant. This can gle scattering cross section in the refraction regime. 2 However, this neglect of the interference term does not be seen clearly from Fig. 4 which gives various depen- distort the main results derived from the numerical cal- dences I(q) for 100 nm particles. culations of broadening using the general Moliére for- It also follows from the results of the numerical cal- mula. This is that the approximate formulas obtained in culations that the macroscopic cross sections for single [9, 10] for broadening in the diffraction regime for any and multiple neutron scattering at the asymptote have L and in the refraction regime for L < L0 can be jointly the same value. From the data plotted in Fig. 4 we can used over the entire range of variation of α, including form an opinion on the quantitative determination of the transition regime from diffraction to refraction. the “asymptotic limit.” If the value of the scattering The diffraction approximation can be applied as far as vector qas at which the multiple scattering cross section α ~ 10Ð15 while the formulas obtained in the refraction agrees with its asymptotic value to within 10% is taken α Ӎ approximation hold for larger . as the asymptotic limit, we obtain qas 10w. However, on the experimental curves for multiple neutron scattering typical dependences for the asymptotic limit are 2.2. Angular Dependence of Multiple Scattering observed at angles an order of magnitude smaller. This We shall now check the conclusions of the approxi- characteristic of the experimental dependences can nat- mate theories [9, 10] in relation to the predicted depen- urally be explained by the experimental line profile for dences of the multiple scattering intensity on the trans- multiple scattering at the domain structure of ferromag-

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I, cm2 In order to use the conclusions of the theory described above to analyze magnetic multiple neutron scattering at the domain structure in a ferromagnet, we R = 100 nm 2 need to make some observations. The first relates to the 10 fact that in general the domains are not spherical. How- ever, all the formulas given above imply that they are. 2 The possibility of applying the theory to aspherical 101 inhomogeneities was discussed in [9, 13] for chaoti- cally oriented particles of arbitrary shape. It was shown in analyses of multiple scattering using this theory that 100 4 the shape of the inhomogeneities does not play a significant role. Particles of any shape can be considered as spherical with the characteristic dimension Reff, which 10–1 3 5 is of the order of magnitude of the cube root of the inhomogeneity volume: R ~ V1/3. 1 eff The second observation relates to the concept of 10–2 contrast in multiple scattering at a domain structure. Since in this case the magnetic domains themselves are the inhomogeneities, the contrast is determined by the 10–3 magnetic induction of the domain which is considered to be a vector. Then for the two neighboring domains in a ferromagnet with 180-degree domain walls we have –4 10 B1 = ÐB2 when the inhomogeneity dimension R is the 10–1 100 101 same as the domain thickness. For a ferromagnet with q, nm–1 a chaotic domain structure the domain thickness is replaced by some effective radius Reff which is equal to Fig. 4. Differential cross sections for multiple scattering half the wavelength of an averaged alternating-sign I0(q) (curve 1), I1(q) (2), I2(q) (3), and scattering cross sec- step function of amplitude is B, and the corresponding tion calculated using the general Moliére formula (curve 4) contrast is |δB| = 2B. for 100 nm particles. Curve 5 gives the squared Lorentz function most frequently used to describe the experimental The third observation concerns the initial conditions curves for multiple scattering at the domain structure of fer- of the theory which assumes that multiple neutron scat- romagnets. tering takes place at dilute systems, for which the ratio of the volume occupied by the inhomogeneities ∆V to the sample volume V is much less than one (∆V/V Ӷ 1). nets. This is usually described by a squared Lorentz A multidomain ferromagnet is undoubtedly a concen- Ð4 function which reaches its asymptotic limit I ∝ q trated system since, as follows from the above reason- more rapidly that the Moliére multiple scattering curve. ing, the analysis of the domain structure as an inhomogeneous system is made using the Ising model where ρ = 0.5. This may give rise to some characteristic fea- 3. MULTIPLE MAGNETIC tures mainly associated with the experimental profile of SMALL-ANGLE NEUTRON SCATTERING the multiple scattering curve. This may be why the pro- Multiple small-angle neutron scattering effects at file of the experimental curve is closer to the squared magnetic inhomogeneities can be analyzed theoreti- Lorentz function than the Moliére scattering curve. cally as for nuclear ones. The only difference is that in this case, the energy associated with the inhomogeneity is given by U = Ðm á δB and the Born parameter α in the 4. EXPERIMENT formulas for the single scattering cross sections (3) and In order to check the conclusions of this multiple (4) is written in the form scattering theory experimentally, we used the results of 2πm ⋅ δB an investigation of small-angle magnetic neutron scat- α = ------R. (22) tering in ferromagnetic materials such as pure nickel λ E and iron, and also in the alloys Fe65Ni35 and Fe3Pt. Here E and µ are the neutron energy and magnetic The experiment was carried out using the D6 small- δ moment, respectively, B = B1 Ð B0 is the magnetic angle neutron scattering device mounted in the hori- contrast, and B0 and B1 are the magnetic inductions of zontal channel of the IVV-2M reactor (Zarechnyœ) at the matrix and the inhomogeneity, respectively. For a wavelength λ = 0.478 nm and angular instrumental line Ӎ ferromagnet with a chaotic domain structure the mag- width w0 8 min. We used a neutron beam formed in netic domains at whose boundaries multiple refraction a slit geometry. The beam width was 1 mm and the takes place act as inhomogeneities. height varied between 8 and 30 mm. The samples were

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w, mrad R, µm w, mrad 100 Fe Fe3Pt 200 100

150 60

100 0.57 β = 0.54 40 0.51

50 0.61 20

10 0 0 0.1 1.0 3.0 L, cm 0 2 4 6 H, kOe Fig. 6. Experimental dependences of neutron beam broadening on the thickness of pure iron samples after various Fig. 5. Field dependences of neutron beam broadening and treatment: (᭹) annealing at 1000°C for 2 h; (᭺) 10% defor- domain size for the Invar alloy Fe3Pt. The domain size was mation; (ᮀ) 15% deformation; (᭝) 25% deformation. The calculated assuming that the magnetic contrast in the sample straight lines show the fitting of the experimental points to remains constant as the external magnetic field increases, power dependences. parallelepipeds or disks of various thickness. In order that for all the iron samples exposed to various treat- to vary the domain size, the metal samples were sub- ment to change the domain structure, the dependence of jected to plastic deformation and heat treatment, and a the broadening on the thickness is described by the nickel sample with a fine-grained structure was power function w ∝ Lβ with the exponent β close to obtained by thermoplastic deformation. All the mea- 0.5Ð0.6. This agrees with the conclusions of the surements were made at room temperature. approximate theories put forward above, where the rad- The experimentally measured neutron beam broad- icand expression in formulas (21a) and (21b) has a log- ening effects are only of a magnetic nature. This can be arithmic cofactor in addition to the thickness L. In order seen from Fig. 5 which gives the field dependence of to determine how significantly this influences the pro- the neutron beam broadening for Invar alloy Fe3Pt as an file of the dependence w(L), we calculated the broaden- example. Here and subsequently the neutron beam broad- ing directly using the Moliére formula (1) for this hypo- 2 2 thetical sample and expressed the results in two vari- ening is taken as the angular value w = (w Ð w )1/2, 1 0 ants: w = f(Lβ) and w = f(Lβ[ln(4L/l) + lnln(4L/l)]β). It where w1 and w0 are the total half-height widths of the was found that in the first case the exponent β is curve giving the intensity as a function of the scatter- 10−15% higher than 0.5. Thus, we can assume that the ing angle with and without the sample, respectively. experimentally determined values of β fall within the An external magnetic field of several kilo-oersted limits of the theoretical thickness dependences and the causes the neutron beam broadening effect to disappear experimental accuracy. The spread of points in Fig. 6 and appreciably reduces the small-angle scattering can be attributed to the fact that when the thickness intensity. dependences of the neutron beam broadening are measured using a set of samples, the domain structure is not identical. For this reason it is best to study the thickness 4.1. Dependence of the Neutron Beam Broadening dependence of the broadening by rotating a plane-par- on the Sample Thickness allel sample about the vertical axis in the beam. The Typical dependences of the neutron beam broaden- results of such an experiment obtained using a plate of ing as a function of the sample thickness for pure iron ordered Fe3Pt alloy are plotted in Fig. 7. The experi- are shown in Fig. 6 using a logÐlog plot. It can be seen mental dependence is accurately described by the

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w, mrad ear dependence which, when extrapolated to L = 0, passes through the origin. (Note that the value of w(0) Fe Pt is not determined since the multiple scattering theory is 120 3 valid for L ӷ l). This dependence is observed for any value of the parameter α. On the basis of the agreement between the experimental and calculated values of the thickness dependences, we subsequently made a complex analysis of the experimental data in order to determine the effec- 80 tive sizes of the magnetic domains and the neutron mean free path using formulas similar to (21a) and (21b). In this case, we expressed these in the form: δ B L 4L 4L α Ӷ wd = kd ------ln------+ ln ln------ , 1, (23a) E R ld ld 40 δ B L L L  α ӷ wr = kr------ln------+ ln ln------ , 1, (23b) E R 2lr 2lr × Ð24 × where kd = 6.97 10 erg/G and kr = 9.85 10−24 erg/G. The procedure involved first defining the magnetic 0 0.5 1.0 L1/2, cm1/2 contrast in accordance with the known values of the induction for these materials, and constructing graphs Fig. 7. Thickness dependence of the broadening obtained by of the broadening as a function of the inhomogeneity × × rotating a planar sample of Fe3Pt alloy measuring 3 15 size for a single sample thickness over a wide range of 15 mm3 about the vertical axis. sizes similar to the graphs plotted in Fig. 2. Then, the experimental values of the broadening were used to determine the typical domain size and scattering regime. The ∝ 0.61 power law w L which is similar to that observed formulas (23) were then used to fit the theoretical thick- for pure iron. In addition, it is easily seen that extrapo- ness dependences to the experimental ones to determine lating the experimental dependence to zero thicknesses the final domain size and the neutron mean free path. An gives a nonzero cutoff on the abscissa. The existence of example of this fitting is shown in Fig. 8. a cutoff was discussed in [9] and is a consequence of A similar treatment of the experimental data was the fact that the argument of the root function in the made for all the samples. The results presented in Table 2 expression for w(L) has a logarithmic cofactor in addi- indicate that iron samples annealed at 1000°C and tion to the thickness. After calculating the broadening deformed by 10% and 15% scatter in the refraction using the general formula (1) and expressing this as a regime and the deformed nickel samples scatter in the 1/2 function of the argument (Lln(…)) we obtained a lin- diffraction regime. However, the alloys Fe65Ni35 and

Table 2. Results of an analysis of experimental data on the thickness dependence of broadening. The second column gives the experimental broadening for samples of thickness L Ӎ 1 cm, the third column gives the exponent in the thickness dependence of the broadening, the fourth column gives the Born parameter, the ﬁfth column gives the effective domain thickness, and the sixth gives the neutron mean free path. The error in the determination of the exponent is ∆β = ±0.05 Sample w, mrad β α R, µm l, µm Scattering regime 1 2 3 4 5 6 7 Fe annealed 20 0.51 60 62 82 refraction Fe 10% deformed 34 0.57 38 39 52 refraction Fe 15% deformed 36 0.61 22 22 29 refraction Fe 25% deformed 47 0.54 17 18 23 transition Ni 40% deformed 38 0.57 1 4 19 diffraction Ni ﬁne-grained 22 0.45 0.3 1 76 diffraction

Fe65Ni35 31 0.52 6 11 15 transition Fe3Pt 65 0.52 8 8 12 transition

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Fe3Pt and 25% deformed iron are located in the transi- w, mrad tion region. It can also be seen from Table 2 that sub- 120 stantial neutron beam broadening is observed for Fe3Pt. This can be attributed to the fairly small effective size of the magnetic domains R Ӎ 8 µm and the high magnetic induction B = 21600 G. The smallest average domain size R Ӎ 1 µm with a mean free path l Ӎ 76 µm was obtained for fine-grained nickel. 1 2 By analyzing the neutron beam broadening in an external magnetic field, we can follow the change in the 80 domain size during magnetization of the sample. This 3 is shown in Fig. 5 for Fe3Pt alloy. It can be seen that initially when grain boundary displacement processes predominate, the size of the domains varies only slightly whereas when a field of around 2 kOe is reached, the domain size increases sharply. This result was obtained assuming that the magnetic contrast 40 remains constant with increasing magnetic field. In fact in the region of the rotation processes the contrast decreases so that the increase in the domain size at the concluding stage was not as steep as that shown in the figure. 0 1 2 3 4 L, cm

4.2. Dependence of Scattering Intensity Fig. 8. Dependences of neutron beam broadening on thick- on Transferred Neutron Momentum ness. Circles—measured values, solid curves—fitting using formulas (23): (1) Fe Pt; (2) 25%-deformed iron; (3) Fe Ni . It follows from the previous analysis that the broad- 3 65 35 ening of a primary neutron beam after its propagation through an unmagnetized ferromagnetic plate fairly accurately confirms the conclusions of the theory of experiment. Taking into account the specific geometry, multiple neutron scattering at domains as at inhomoge- we made a theoretical estimate of the influence of ver- neities of effective radius Reff. The experimentally tical divergence on the angular intensity distribution observed angular dependence of the magnetic scatter- using well-known techniques [16]. This showed that ing intensity is not so obviously related to the multiple for the geometric parameters of our system the influ- processes. This is primarily because the multiple scat- ence of vertical divergence on the function I(q) is neg- tering cross section at the asymptote goes over to the ligible. This conclusion also follows from the results of single scattering cross section, which also varies as qÐ4 checking these dependences experimentally for various (Porod law). In addition, for high values of q single slit heights before the sample and the counter. These scattering effects may be observed at small-scale inho- experiments showed that in our geometry the resolution mogeneities located within a single domain. Finally, for of the diffractometer in the vertical plane has no signif- small transferred momenta below the critical value qcr, icant influence on the measured angular dependence of inelastic scattering effects take place at spin waves the intensity, which distinguished our system from a which may distort the angular dependences of the mul- double crystal spectrometer where these effects are tiple scattering intensity. substantial [17]. In the present study we attempted to isolate these The results of an experimental investigation of the contributions and estimate their absolute values for dependences of the small-angle neutron scattering inten- pure metals (iron and nickel) where no small-scale fluc- sity on q can be seen for samples of pure iron of different tuations of the magnetization occur inside the domain, thickness in the annealed and deformed states (Fig. 9). In and also for Invar alloys Fe65Ni35 and Fe3Pt where such the range of transferred momenta q < 0.2 nmÐ1 these fluctuations may occur as a result of the presence of an dependences are described by the power law I(q) ∝ qÐ4, inhomogeneous magnetic structure [14] attributed, for which agrees with the conclusions of the approximate example, to antiferromagnetic interaction between − theories [9, 10] and with the numerical calculations Fe Fe atoms against the background of ferromagnetic using the Moliére formula (1). It can be seen from interaction between NiÐNi and NiÐFe atoms [15]. Fig. 10 that the dependences on q are very sensitive to Before analyzing the experimentally measured the external magnetic field as a result of changes in the angular dependences, we make some observations on domains under magnetization. The absolute value of the possible influence of vertical divergence on the the intensity in small angles is reduced by several fac- dependence I(q) since a slit geometry was used in our tors of 10 when the sample is transferred from the

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dΣ/dΩ, cm–1 dΣ/dΩ, cm–1

Fe 102 Fe 103 q–3.8 q–4.1 q–3.9 q–3.4 102 101

101 q–2.0 q–3.8 100 q–4.1 100

q–1.6 10–1 10–1 q–2.3

10–2 –1 0 10–2 10 10 –1 q, nm 10–1 100 q, nm–1 Fig. 9. Experimental dependences of the differential neutron scattering cross section on the transferred momentum at the Fig. 10. Angular dependences of the multiple neutron scat- asymptote for annealed (᭺, ᭹) and 25% deformed (ᮀ, ᭿) iron tering cross section for an annealed iron sample in a zero mag- samples of thickness L = 1.2 cm (᭺, ᮀ) and 2.5 cm (᭹, ᭿). netic field ( ) and in fields of 0.5 ( ) and 2 kOe ( ). demagnetized to the magnetized state and the depen- tion from inelastic scattering of neutrons at spin waves. dence I(q) ∝ qÐn with n Ӎ 4 becomes distorted in the This scattering exists in the range of angles below the θ ប2 direction of decreasing n when a many-domain sample critical value c = /(2mD), where D is the coefficient is converted to a single-domain one. of exchange hardness in the quadratic dispersion law 2 Similar dependences are observed for Fe65Ni35 for the spin waves E = Dq , m is the neutron mass, and alloy, as can be seen from Fig. 11 which gives the pro- ប is Planck’s constant. The procedure for isolating the file of the scattering curves as a function of the direc- small-angle magnetic inelastic scattering described in tion of the magnetic field relative to the neutron scatter- [19] involves investigating the angular dependences for ing vector. However, in the range of small transferred two directions of the external magnetic field on the momenta where multiple processes are observed at the sample, parallel and perpendicular to the scattering domain structure, an increase in the external magnetic vector. Figure 11 gives these angular dependences for field changes the behavior of I(q) whereas for large q Fe65Ni35 alloy in the magnetic field H = 4 kOe. It can be this influence is small. Here the dependence I(q) is seen that for scattering vectors q < 0.1 nmÐ1 in the absence described by a power law with n ~ 1.5Ð2 and is attrib- of a magnetic field I (q) reaches 105Ð106 arb. units uted to single neutron scattering at magnetic fluctua- H = 0 whereas the values of IH || q(q) and IH ⊥ q(q) do not tions of the paramagnetic inclusion type in a ferromag- 4 netic matrix. It was shown in [18] that in Invar alloys exceed 10 arb. units. Hence, the inelastic neutron scat- scattering at these inhomogeneities is described by a tering intensity is approximately two orders of magni- Lorentz function and the characteristic size of the inho- tude lower than the multiple scattering intensity at the mogeneities is of the order of a few nanometers. This is domain structure. significantly smaller than the effective radius of the fer- By investigating the angular dependences of the romagnetic domain which, according to Table 2, is tens intensity for two directions of the external magnetic of micron for these alloys. θ field, we can determine the critical angle cof the scat- Experiments to study the angular dependences of tering processes at spin waves. It can be seen from the scattering intensity in a magnetic field in various Fig. 12 that the difference scattering intensity near zero directions relative to the scattering vector can also be angle is negative, then changes sign as the angle used to estimate the contribution to small-angle diffrac- increases, passes through a maximum and then, after

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Intensity, arb. units Intensity, arb. units

Fe65Ni35 105 0.5 q–3.1

104 q–1.6 0

q–1.0

103

–0.5

102 0 50 100 150 2θ, min 10–1 100 q, nm–1 Fig. 12. Angular dependences of the difference intensity of neutron scattering IH || q(q) and IH ⊥ q(q) measured in a ᭺ Fig. 11. Experimental dependences of the neutron scattering magnetic field of 4 kOe for Invar alloys Fe65Ni35 ( ) and ᭹ θ intensity on the transferred momentum at the asymptote for Fe3Pt ( ). The arrows indicate the critical angles c for the × Fe65Ni35 alloy in a zero magnetic field ( ) and in a field of existence of long-wavelength spin waves. The dashed curve 4 kOe parallel (᭹) and perpendicular (᭺) to the scattering gives the difference intensity for Fe50Ni50 alloy calculated vector. Sample thickness L = 0.7 cm. theoretically in [19].

reaching a critical value, becomes zero. The dashed and Fe3Pt alloys for which the angular dependence at curve in this figure gives the similar difference intensity the asymptote obeys the law I(q) ∝ qÐ3 rather than for spin-wave neutron scattering from [19] calculated I(q) ∝ qÐ4. In addition the angular dependence of the for Fe50Ni50 alloy allowing for the influence of the small-angle scattering intensity reveals characteristics external magnetic field and dipoleÐdipole interaction. which are observed as steps in a logÐlog plot. However, Also plotted are the results of studying magnetic small- these aspects require further study. angle neutron scattering in a sample of ordered Fe3Pt alloy. The arrows indicate the angular positions corresponding to the experimental values of the critical 5. CONCLUSION angle. These are 80 and 120 min for Fe65Ni35 and Fe3Pt, These numerical calculations of the general Moliére respectively. The exchange hardness coefficients for the formula for the multiple neutron scattering intensity spin waves D0 calculated from these data are 120 and show that its analytic analogs obtained in the approxi- 90 meV Å2, respectively, which satisfactorily agree mations α Ӷ 1 [9] and α ӷ 1 [10] can be used jointly with the published data [20]. to analyze the experimental data on small-angle magnetic neutron scattering. Multiple scattering is observed Thus, in measurements of the angular dependence as a broadening of the primary beam whose thickness of magnetic small-angle neutron scattering for unmag- dependence is expressed by the power function w ∝ Lβ, netized samples of ferromagnetic materials inelastic where β = 0.5Ð0.6. Using the corresponding experi- spin-wave scattering effects are observed but their mental dependences of the broadening on the sample intensity is fractions of percent of the total intensity of thickness, we can make approximate estimates of the multiple neutron scattering at the domain structure. effective size of the magnetic domains. Nevertheless, we can assume that spin wave scattering In measurements of the angular dependence of the distorts the behavior of the dependence I(q) ∝ qÐ4 when small-angle magnetic neutron scattering intensity for θ the angle c becomes appreciable as a result of the unmagnetized samples, we must bear in mind that there smallness of the exchange hardness coefficient D. For is always a range of angles where multiple neutron example, this effect is observed in Fe65Ni35 (Fig. 11) scattering is observed at the domain structure. This

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 120 BOGDANOV, MEN’SHIKOV generally corresponds to the range of transferred 5. S. G. Bogdanov, E. Z. Valiev, and A. Z. Menshikov, Solid momenta q < 1 nmÐ1 where the small-angle scattering State Commun. 76, 809 (1990). intensity is extremely sensitive to the external magnetic 6. Yu. I. Smirnov, P. Yu. Peksheev, A. O. Éœdlin, et al., Fiz. field and is approximately two orders of magnitude Tverd. Tela. 33, 2273 (1991); Yu. G. Abov, Yu. I. Smirnov, higher than the intensity of inelastic scattering at spin D. S. Denisov, et al., Fiz. Tverd. Tela 34, 1408 (1992). waves which takes place in this range of angles. 7. N. F. Berk and K. A. Hardman-Rhyne, J. Appl. Crystal- logr. 18, 467 (1985); J. Appl. Crystallogr. 18, 473 The question naturally arises as to whether it is pos- (1985); Physica B 136, 223 (1988); J. Appl. Crystallogr. sible to carry out small-angle neutron scattering exper- 21, 645 (1988); A. Z. Allen and N. F. Berk, Neutron iments to identify spin wave fluctuations without News 9, 13 (1988). applying an external magnetic field to the sample. This 8. N. F. Mott and G. Massey, The Theory of Atomic Colli- need frequently arises because of the design character- sions (Clarendon, Oxford, 1965; Mir, Moscow, 1965). istics of a device where it is difficult to make tempera- 9. S. V. Maleev and B. P. Toperverg, Zh. Éksp. Teor. Fiz. 78, ture measurements in an external magnetic field. The 315 (1980). answer to this question may be positive if we only take 10. S. V. Maleev, R. V. Pomortsev, and Yu. N. Skryabin, into account the range of small transferred momenta in Phys. Rev. B 50, 7133 (1994). which multiple neutron scattering effects are small. 11. A. Z. Menshikov, S. G. Bogdanov, and Yu. N. Skryabin, Information on spin fluctuations is usually only con- Physica B 234Ð236, 584 (1997). centrated in that part of the angular dependence where 12. Yu. N. Skryabin, Solid State Commun. 88, 747 (1993). the exponent in the law I ∝ qÐn is close to two. 13. R. J. Weiss, Phys. Rev. 83, 379 (1951). The authors are grateful to É. Z. Valiev and 14. A. Z. Men’shikov, S. K. Sidorov, and V. E. Arkhipov, Yu. N. Skryabin for numerous useful discussions of the Zh. Éksp. Teor. Fiz. 61, 311 (1971). aspects considered here. 15. M. Hatherly, K. Hirakawa, R. D. Lowde, et al., Proc. This work was supported by the State Scientific-Tech- Phys. Soc. 84, 55 (1964). nical Program “Topical Directions in the Physics of Con- 16. D. I. Svergun and L. A. Feœgin, X-ray and Neutron Small- densed Media”, “Neutron Research” (project no. 4). Angle Scattering (Nauka, Moscow, 1986). 17. Yu. G. Abov, D. S. Denisov, F. S. Dzheparov, et al., Zh. Éksp. Teor. Fiz. 114, 2194 (1998). REFERENCES 18. A. Z. Men’shikov and V. A. Shestakov, Fiz. Metall. Met- 1. R. Von Nardroff, Phys. Rev. 28, 240 (1927). alloved. 43, 722 (1977). 2. D. L. Dexter and W. W. Beeman, Phys. Rev. 76, 1782 19. M. W. Stringfellow, Report of Atomic Energy Research (1949). Establishment (Harwell, UK, 1966), No. AERE-R 4535. 3. D. J. Hughes, M. T. Burgy, R. B. Heller, et al., Phys. Rev. 20. I. Ishikawa, S. Onodara, and R. Tajima, JMMM 10, 183 75, 565 (1949). (1979). 4. S. Sh. Shil’shtein, V. A. Somenkov, M. Kalanov, et al., Fiz. Tverd. Tela 18, 3231 (1976). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 121Ð128. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 136Ð144. Original Russian Text Copyright © 2000 by Bykovskiœ, Karavanskiœ, Kotkovskiœ, Kuznetsov, Chistyakov, Lomov, Gavrilov. SOLIDS

Photophysical Processes Stimulated in Nanoporous Silicon by High-Power Laser Radiation Yu. A. Bykovskiœ1, V. A. Karavanskiœ*2, G. E. Kotkovskiœ1, M. B. Kuznetsov1, A. A. Chistyakov1, A. A. Lomov3, and S. A. Gavrilov4 1 Moscow State Engineering-Physics Institute (Technical University), Moscow, 115409 Russia 2 Institute of General Physics, Russian Academy of Sciences, Moscow, 117924 Russia 3 Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, 177333 Russia 4 Moscow Institute of Electronics, Moscow, 111250 Russia *e-mail: [email protected] Received July 15, 1999

Abstract—Photoprocesses initiated on the surface of porous silicon irradiated with laser radiation with various wavelengths (λ = 266, 337, and 532 nm) in a wide range of intensities (up to 2 × 107 W/cm2) were investigated. Laser-induced luminescence and laser mass-spectrometry were used as experimental procedures. X-ray reﬂec- tion was used to determine the parameters of the porous silicon ﬁlms. The photoluminescence spectra obtained at different wavelengths and low intensities were analyzed. This analysis showed that for an optically thin layer of porous silicon the luminescence spectrum does not depend on the wavelength of the exciting radiation. This indicates the existence of a separate system of levels in porous silicon that are responsible for the luminescence. The behavior of the photoluminescence spectra as a function of the intensity q of the exciting radiation was investigated. It was shown that the luminescence intensity is a nonlinear function of q. At high intensities of the exciting radiation, the luminescence intensity saturates and a short-wavelength shift of the spectra is observed; this is due to the high concentrations of photoexcited carriers. This increases the probability of the experimentally observed nonequilibrium photodesorption of H2 and Si from the surface of porous silicon. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION presses luminescence. Here, the smaller the characteristic size of the nanocrystals, the shorter the wave- The steady interest in the photophysical properties length of the observed photoluminescence is. In the of nanoporous silicon is due primarily to the fact that second model, silicon compounds formed as a result this material is viewed as a promising material for of anodic etching or localized states on the surface of developing a new generation of optoelectronic devices nanocrystals (surface complexes SiÐHx and oxygen [1]. A unique photophysical property of nanoporous vacancies) play the main role in the photolumines- silicon is its photoluminescence, discovered by Can- cence [10, 11]. In this case, the size-quantization lev- ham in 1990 [2] (bulk silicon is an indirect gap semiconductor, and radiative transitions are strongly sup- els determine the absorption, but they bear no relation pressed in it). However, even though there are many to luminescence. works devoted to this problem, the reason for and the There recently have appeared works endeavoring to physical mechanisms of photoluminescence of porous combine certain characteristic features of these two silicon are still not completely understood [3]. models in order to explain the photoluminescence of As is well known [3Ð6], porous silicon consists of porous silicon [12, 13]. In this approach it is assumed an aggregate of nanocrystals separated by a character- that the absorption spectrum of porous silicon can istic distance which is also of the order of several indeed be explained by size-quantization effects, and nanometers. Such a structure possesses a very high spe- surface complexes of the type SiÐHx passivate the dan- cific surface area, reaching of the order of 300 m2/cm3. gling bonds and close the nonradiative relaxation chan- Existing models explaining the intense visible-range nels, while a radiative transition occurs with participa- photoluminescence from porous silicon can be tion of localized levels associated with the surface and reduced to two alternatives. The first model asserts with the position of the size-quantization levels in the that transitions between size-quantization levels in sil- nanocrystals themselves. This makes it possible to icon nanocrystals are radiative [7Ð9], and the surface unite the size-quantization shift of the photolumines- is treated as a source of a nanoradiative recombination cence band and the high efficiency of the photolu- channel, where the presence of dangling bonds sup- minscence.

122 BYKOVSKIŒ et al. Most works on the photoluminescence of porous sil- regime in which extended pores are formed, should icon have been performed for low (up to 3 W/cm2) exist on the outer boundary. The thickness of this layer power densities of continuous-wave exciting laser radi- presumably could be several diameters of an average ation But, even in these works [7, 10] have shown that pore. Several samples were prepared using different as the radiation power increases, the photolumines- current density j and anodization duration t with a sur- cence spectra undergoes a shift and distortion, which face layer in the form of a thin film of porous silicon and seem to be due to a characteristic feature of the energy one sample was prepared in the form of a thick film. The structure of the nanocrystals and its behavior under the constant value of the charge Q = jt = 0.15 C/cm2 with dif- conditions of strong excitation at high intensities ferent current densities was chosen as the parameter for (103−106 W/cm2). Investigation of the processes giving thin films: j = 1, 10, and 50 mA/cm2. In what follows rise to such evolution of the photoluminescence spectra these samples are designated as nos. 1, 2, and 3. Wafers can give additional information about its mechanism in of standard (100) p-type KDB10 silicon (resistivity ρ ~ porous silicon. This is now of great interest. Moreover, 10 Ω cm) were used as substrates. To prevent variance these investigations make it possible to understand the of the parameters, due to differences between wafers, characteristic features of the electronic excitation relax- from having an effect, the entire series of samples was ation and transfer processes and to answer questions prepared from the same wafer. The sample no. 4 with a concerning the possibility of photoprocesses, such as thick film was obtained with Q = 0.85 C/cm2 (j = photodesorption, photodissociation, and ablation, 12 mA/cm2, t = 425 s, KDB12 substrate). occurring on the surfaces of nanocrystals of porous silicon [12, 14]. For thin films, the thickness and effective porosity were determined by measuring the angular dependence In summary, our objective in this work was to inves- of the intensity of the mirror X-ray reflection near the tigate the photoprocesses occurring on the surface of angles of total external reflection [16]. The thickness of porous silicon under irradiation with high-power laser the thick film was estimated from the image of the radiation. cleavage surface in an optical microscope and was ~10 µm. To study the photoluminescence of porous silicon in 2. EXPERIMENTAL PROCEDURE a wide range of intensities of the exciting radiation and AND SAMPLES for various wavelengths, a laser fluorimeter, equipped Layers of porous silicon were obtained by anodiza- with a repetitive-pulse YAG-Nd3+ laser with the har- λ λ tion in an electrolyte (HF(49%) : C2H5OH in a 1 : 2 monics = 532 and 266 nm and a nitrogen laser ( = volume ratio) in a two-chamber electrochemical cell. 337 nm). The pulse duration was 10 ns. A collection of Platinum was used as the cathode. After anodization, interchangeable filters and a system of focusing lenses the samples were rinsed in deionized water and dried in made it possible to change the power density of the a stream of dry air. laser radiation at the sample from 7 × 104 to 107 W/cm2 for λ = 532 nm and from 103 to 4 × 105 W/cm2 for λ = One of the main steps in the sample preparation pro- 266 nm. The power density of the nitrogen laser was cess is choosing the film formation regimes. On the one 4 2 hand, it is known that porous silicon films can possess 10 W/cm . a layered structure: different layers have different pho- The possible irreversible photoprocesses that could toluminescence properties. Indeed, the properties of the be initiated on the surface of porous silicon by high- porous-silicon layer formed can vary during the anod- power laser radiation (photodesorption, photodissocia- ization process [15], and under certain conditions even tion, and ablation) were traced by laser-mass spectrom- the etching mechanism changes. On the other hand, it etry [12]. has been shown that the photoluminescence spectra can In summary, the combination of methods used made depend strongly on the wavelength of the exciting radi- it possible to monitor the development and structure of ation. Effects associated with the nonuniformity and the porous silicon films and to investigate the photopro- different excitation apparently can be distinguished if cesses occurring in porous silicon irradiated with high- the film of porous silicon is sufficiently thin so that it power laser radiation, starting from photoluminescence can be treated as being uniform and so that the excita- and ending with photodesorption and ablation. tion conditions at different wavelengths do not differ much. At the same time, preliminary investigations have shown that immediately after the onset of anod- 3. EXPERIMENTAL RESULTS ization there exists a certain period during which a stationary state of etching is established (this period The results of X-ray reflection measurements of the decreases with increasing anodization current density). structural parameters of the films are presented in the This suggests that a layer with variable characteristics, table. which is determined by a transition from the etching Experiments with pulsed pumping were performed regime of the initial smooth surface to a stationary using the following wavelengths of the exciting radia-

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PHOTOPHYSICAL PROCESSES STIMULATED IN NANOPOROUS SILICON 123 tion: λ = 532, 337, and 266 nm. Since specially pre- I, arb. units pared samples with very thin porous layers were used 1.0 (d = 105, 92, and 85 nm), the radiation with each wave- 1' (×1.3) 1 length could be absorbed over the entire depth of the porous layer. This made it possible to decrease (a) the 0.8 effect of the structural nonuniformity of the porous silicon over depth on the experimental results [17] and 0.6 (b) the effect of a difference in the absorption coeffi- 2' 2 cients for the first two wavelengths, which in the case of thick films always results in a mismatch between the 0.4 depths of the luminescing regions. We also note that by 3 choosing samples with a thin porous layer it is possible, 3'' (×0.5) according to our estimates, to eliminate any effect due 0.2 to the reabsorption of the characteristic luminescence 3' in the porous-silicon layer. Reabsorption of photolumi- 0 nescence must be taken into account only for thick- 500 550 600 650 700 750 800 nesses of such samples greater than 10 µm. λ, nm The luminescence spectra for thin films are presented in Fig. 1. It is evident that the positions of the Fig. 1. Photoluminescence spectra of samples of porous silicon. (1) sample no. 1, λ = 532 nm, (1') sample no. 1, λ = maxima in the photoluminescence spectra of porous 266 nm; (2) sample no. 2, λ = 532 nm, (2') sample no. 2, λ = silicon films obtained with different current densities 266 nm; (3) sample no. 3, λ = 532 nm, (3') sample no. 3, λ = are different. A large shift into the short-wavelength 266 nm, (3'') sample no. 3, λ = 337 nm. Laser power flux × 3 2 × 5 2 region corresponds to a high current density. This density q266 = 2.5 10 W/cm and q532 = 2 10 W/cm . agrees with the well-known result that films formed with a high current density have smaller silicon nanocrystals. It is also evident that on switching from λ = 532 nm silicon, whose luminescence spectrum with a maxi- to λ = 266 nm the maximum of the spectra for samples mum in the region 580Ð610 nm indicates that the char- nos. 1 and 2 shifts into the short-wavelength region, acteristic size of the crystallites is very small. However, while the position of the spectrum of sample no. 3 even in this case, good agreement was observed remains virtually unchanged. For comparison, the between the luminescence spectra for excitation with spectra for sample no. 4 with a thick film of porous sil- radiation with λ = 532 and 266 nm. icon, for which this shift is greatest, 34 nm, is pre- Therefore it can be concluded that the shift of the sented in Fig. 2. The results obtained for the shift photoluminescence spectra in the thicker samples agree qualitatively with the results of other investiga- (nos. 1 and 2 and also no. 4) is due precisely to the dif- tions for comparatively thick films of porous silicon µ ferent absorption coefficient, more accurately, the dif- (d > 1 m) [9]. Apparently, the observed dependence ferent “optical thickness” for different wavelengths. of the shift on the film thickness and on the conditions For sample no. 3 the optical thicknesses for all excita- of formation (porosity and current density) is more tion wavelengths are the same. important. As one can see from the plots in Figs. 1 and 2, the shift of the maximum of the photoluminescence In thick samples the short-wavelength radiation λ spectrum decreases monotonically with thickness. It (with = 266 and 337 nm and two to three orders of λ is 15 and 10 nm for samples nos. 1 and 2, respectively. magnitude larger absorption coefficient than for = For the thinnest sample (no. 3, thickness d = 85 nm) 532 nm) excites the top layer, where the porosity is this shift is virtually absent, and the spectra obtained greater than the average porosity of the sample and at three wavelengths (266, 337, and 532 nm) are iden- which is more strongly oxidized. For this reason, the tical to within the limits of the experimental error. spectrum of such samples will be shifted into the short- The physical reason for the observed shift could be the difference not only in the optical thickness of the Structural parameters P and d of films of porous silicon sample but also in their porosity. Indeed, samples with according to measurements performed by the method of total a higher porosity typically have smaller nanocrystals external reflection of X-rays [21] and, as is well known, long-wavelength radiation (λ = 532 nm in our case) will be absorbed only by relatively Sample no. P, % d, nm “large” crystallites, whose luminescence spectrum is shifted into the “red” region of the spectrum. But, then, 1 58 ± 1 105 ± 1 the largest shift of the spectra should be expected for 2 38 ± 1 92 ± 1 the third sample, for which this shift is absent. To con- 3 55 ± 1 85.0 ± 0.5 firm the arguments presented above, a control experiment was performed using thin samples of nanoporous Note: P is the porosity; d is the average film thickness.

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I, arb. units Thus, we arrive at the following important conclu- 1.0 sion. In samples with the same optical thickness, for the λ = 266 nm (×12) λ = 532 nm entire excitation spectrum lying above the absorption q = 2 × 103 W/Òm2 q = 104 W/Òm2 edge, the photoluminescence spectrum of porous sili- 0.8 con does not depend on the excitation wavelength (in our case for radiation with λ ≤ 532 nm). Under these conditions, irrespective of the photon energy, after 0.6 excitation the charge carriers relax nonradiatively on a separate system of levels in nanocrystals of porous silicon, and luminescence occurs from this system. 0.4 We shall now examine the behavior of the photoluminescence for various intensities of the exciting radi- 0.2 ation. Figures 3 and 4 show the dependence of the photoluminescence intensity of the same samples on the 0 radiation power q for wavelengths λ = 266 and 532 nm, 500 550 600 650 700 750 800 respectively. The laser ﬂuorimeter was tuned to the λ, nm wavelengths corresponding to the maxima of the photoluminescence spectra for small q. The nonlinear Fig. 2. Photoluminescence spectra of a sample of porous sil- dependence of the photoluminescence intensity for q > icon under excitation with radiation with λ = 532 and 7 × 103 W/cm2 at λ = 266 nm and q > 2 × 105 W/cm2 at 266 nm. λ = 532 nm with the intensity I subsequently reaching values for which the photoluminescence saturates is especially interesting. wavelength region compared with the spectrum obtained by excitation with λ = 532 nm radiation, for One explanation for the experimentally observed which nanocrystals are excited on the surface and in the character of the dependence of I on q is saturation of the interior volume of the porous layer. excited states. Indeed, since the lifetime of the photoex-

I, arb. units 1.00 d = 105 nm d = 92 nm d = 85 nm

0.75

0.50 0.2

0.1 0.25

0 5 10 15 20 0 100 200 300 400 q, 103 W/cm2

Fig. 3. Photoluminescence intensity versus the power density q of the exciting radiation for samples nos. 1, 2, and 3. The wavelength is λ = 266 nm.

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I, arb. units 1.00 d = 105 nm d = 92 nm d = 85 nm

0.75

0.4 0.50 0.3

0.2 0.25 0.1

0 5 10 0 20 40 60 80 100 q, 105 W/cm2 Fig. 4. Photoluminescence intensity versus the power density q of the exciting radiation for samples nos. 1, 2, and 3. The wavelength is λ = 532 nm. cited carriers in porous silicon is of the order of τ ~ 10Ð intensity of the laser radiation increases, the spectra 100 µs [3, 4], it can be expected that carriers are effec- shift into the short-wavelength region. The shift of the tively transferred into excited states under our experi- maxima of the photoluminescence spectra with excita- mental conditions. Estimates made on the basis of [18] tion by light with wavelength λ = 266 nm is ∆λ = 35, show that an appreciable deviation from a linear dependence I(q) should be expected for I, arb. units បω ≥ 1.0 q0 -τ---σ---, 1' where σ is the absorption cross section of the radiation 0.8 exciting the luminescence. 2' λ > 6 2 λ 3' For = 532 nm q0 10 W/cm , and for = 266 nm 0.6 1 > × 4 2 q0 2 10 W/cm . It is evident that the experimental dependences confirm these estimates. 0.4 Thus, high concentrations of photoexcited carriers, for which, for example, nonradiative Auger recombina- 0.2 tion of carriers can be significant [19], will correspond 3 to the flux densities of the exciting radiation (q = 2 7 2 λ 10 W/cm for = 532 nm) which were realized under 0 our experimental conditions. 500 550 600 650 700 750 800 λ To obtain a more complete picture of the behavior of , nm the photoluminescence for various intensities we must Fig. 5. Photoluminescence spectra of samples of porous sil- examine the photoluminescence spectra. icon under excitation with λ = 266 nm radiation for two val- × 3 2 The photoluminescence spectra obtained for the ues of the power density: q1 = 2.5 10 W/cm and q2 = 4 2 same samples with various limits of q and pump wave- 10 W/cm : (1) sample no. 1 with q1, (1') sample no. 1 with lengths λ = 266 and 532 nm, respectively, are displayed q2; (2) sample no. 2 with q1, (2') sample no. 2 with q2; in Figs. 5 and 6. It is evident in the figures that as the (3) sample no. 3 with q1, (3') sample no. 3 with q2.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 126 BYKOVSKIŒ et al.

I, arb. units 30, and 25 nm, respectively, for samples nos. 1, 2, and 1.0 3 (Fig. 5); for excitation with light with wavelength λ = 532 nm the shift is ∆λ = 45, 40, and 5 nm, respectively, for samples nos. 1, 2, and 3, respectively (Fig. 6). Time-resolved spectroscopy [9, 10, 20] records the shift of the maximum of the photoluminescence with 0.8 1' time into the long-wavelength region, i.e., “fast” relaxation, with a time of the order of 10 ns; radiative relaxation occurs in the short-wavelength range; “slow” relaxation (τ ~ 10Ð100 µs) occurs in the long-wavelength range [3, 9]. Some investigators [9] attribute this 0.6 to the fact that in smaller nanocrystals (and, corre- 2' (×2) spondingly, shorter-wavelength photoluminescence 1 spectrum), the radiative relaxation time is shorter than in nanocrystals with a larger characteristic size. 0.4 Thus, the short-wavelength shift of the photolumi- 3' nescence spectrum of porous silicon with increasing q × can be explained by the fact that in this case the smaller 2( 2) nanocrystals, for which the dependence of I on q is still linear, start to make a large contribution to the photolu- 0.2 minescence, while in large nanocrystals photoluminescence starts to saturate. 3(×2) 2 It should also be noted that the magnitude of the shift of the maximum of the photoluminescence with 0 increasing q is directly proportional to the thickness of 600 700 800 the sample. This can be explained by the fact that the λ, nm nanocrystals in a thinner sample are more uniform with respect to their characteristic size and therefore Fig. 6. Photoluminescence spectra of samples of porous sil- undergo saturation for close values of q. icon under excitation with λ = 532 nm radiation for two values of the power density: q = 2 × 105 W/cm2 and q = Mass-spectrometric investigations showed that for q 1 2 6 6 2 × 6 2 ranging from 3 × 10 to 8 × 10 W/cm photodesorption 2.5 10 W/cm : (1) sample no. 1 with q1, (1') sample no. 1 with q ; (2) sample no. 2 with q , (2') sample no. 2 with q ; of H2 and Si is observed simultaneously at values for 2 1 2 which photoluminescence saturates. The characteristic (3) sample no. 3 with q1, (3') sample no. 3 with q2. The distortion of the spectra in the range 550Ð570 nm is caused by mass spectrum is shown in Fig. 7, and the interpreted the OS-23-1 ﬁlter, which cuts off λ = 532 nm radiation. mass spectrum is displayed in Fig. 8. Generally speaking, the high carrier densities realized under our experimental conditions on nanocrystals should lead to a A, arb. units sharp increase in the probability of surface photoprocesses, such as photodesorption, observed in our exper- Si and N H2 2 iments, and photodissociation. Efﬁcient laser photoox- 30 idation is possible. On the other hand, the observed photodesorption, in principle, increases the number of dangling bonds and therefore also the number of non- 20 radiative relaxation channels. This mechanism can lead to an irreversible degradation of luminescence. 10 The dependence of the luminescence intensity at the maximum of the spectrum on the time or on the number of laser pulses is displayed in Fig. 9 for λ = 532 nm and 0 q = 8 × 106 W/cm2. It is evident that for 1000 pulses the intensity decreases by 40%. 0 2 18 28 44 m, au As the intensity of the exciting radiation increases further (above 8 × 106 W/cm2 for λ = 532 nm), a sharp Fig. 7. Characteristic mass spectrum under the action of and irreversible decrease in the photoluminescence radiation with q = 5 × 106 W/cm2 on the surface of the intensity occurs. Mass-spectrometric investigations porous-silicon sample. show that in this case the following products of laser

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 PHOTOPHYSICAL PROCESSES STIMULATED IN NANOPOROUS SILICON 127

A, arb. units I, arb. units 12 Si 1.0 10 0.8 8 H2 0.6 6 4 0.4 0 500 1000 1500 2000 2500 3000 2 Number of laser pulses

0 5 10 15 20 25 30 35 40 45 50 m, au Fig. 8. Interpreted mass spectrum under the action of radia- Fig. 9. Peak intensity of the luminescence of sample no. 3 tion with q = 5 × 106 W/cm2 on the surface of a porous-sil- versus the number of pulses of laser radiation with λ = icon sample. 532 nm for q = 8 × 106 W/cm2.

A, arb. units I, arb. units 120 120 Si and N2 100 100 Si 80 80 H2 60 60 H2 C2H2 SiO and CO2 40 40 C2H2 SiO 20 20 0 0 10 20 30 40 50 2 26 28 44 m, au m, au

Fig. 10. Characteristic mass spectrum under action of radi- Fig. 11. Interpreted mass spectrum under action of radiation ation with q = 2 × 107 W/cm2 on the surface of a sample of with q = 2 × 107 W/cm2 on the surface of a sample of porous porous silicon. silicon.

action are formed: H2, Si, C2H2, and SiO (Figs. 10, 11), separate system of levels in nanocrystals of porous sil- which are characteristic for ablation and destruction of icon, and luminescence occurs from this system. the porous layer [21]. The nonlinear character of the dependence of the photoluminescence intensity on q, due to the high level of excitation of the semiconductor, was observed 4. CONCLUSIONS experimentally. Photoprocesses occurring under the action of laser The shift of the luminescence spectrum as a func- radiation on the surface of porous silicon were investi- tion of the intensity of the exciting radiation into the gated in a wide range of intensities, up to ~107 W/cm2, short-wavelength region of the spectrum was observed for wavelengths λ = 532, 337, and 266 nm. and investigated.

It was shown that samples with the same optical Photodesorption with formation of H2 and Si was thickness for the entire excitation spectrum lying above observed for q ~ 8 × 106 W/cm2. the absorption edge, the photoluminescence spectrum of porous silicon does not depend on the excitation It was shown that nonequilibrium photodesorption wavelength (in our case for radiation with λ ≤ 532 nm) gives rise to new nonradiative relaxation channels on in the linear region of the dependence I(q). Under these account of an increase in the number of dangling conditions, irrespective of the photon energy, after bonds. For q > 9 × 106 W/cm2 and λ = 532 nm, a sharp excitation the charge carriers relax nonradiatively on a decrease in the photoluminescence intensity on account

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 128 BYKOVSKIŒ et al. of an increase in the nonradiative relaxation rate as a 10. K. L. Narasimhan, S. Banerjee, A. K. Srivastava, et al., result of heating and ablation was observed. Appl. Phys. Lett. 62, 331 (1993). 11. M. Ohmukai and Y. Tsutsumi, J. Appl. Phys. 84, 4459 (1998). ACKNOWLEDGMENTS 12. Yu. A. Bykovskiœ, G. E. Kotkovskiœ, M. B. Kuznetsov, This research was supported in part by the Ministry et al., in XVI International Conference on Coherent and of Science and Technologies of the Russian Federation Nonlinear Optics Technical Digest (Moscow, 1998), (projects no. 08-02-48 and no. 97-10-73) and the pro- p. 243. gram for Integration of Higher Education and Funda- 13. M. Kondo, J. Non-Crystalline Solids 164Ð166, 941 mental Science (project no. A0103). (1993). 14. A. Hashimoto, K. Iwata, M. Ohkubo, et al., J. Appl. Phys. 75, 5447 (1994). REFERENCES 15. S. A. Gavrilov, T. N. Zavaritsakaya, V. A. Karavanskiœ, 1. N. Koshida and H. Koyama, Appl. Phys. Lett. 60, 347 et al., Élektrokhimiya 33, 1064 (1997). (1992). 16. V. A. Karavanskiœ, A. A. Lomov, E. V. Rakova, et al., 2. L. T. Canham, Appl. Phys. Lett. 57, 1046 (1990). Poverkhnost, no. 12, 1999 (in press). 3. A. G. Cullis, L. T. Canham, and P. D. J. Calcott, J. Appl. 17. A. N. Obraztsov, V. A. Karavanskiœ, H. Okushi, et al., Phys. 82 (3), 909 (1997). Fiz. Tekh. Poluprovodn. 32, 1001 (1998) [Semiconduc- 4. B. M. Kostishko, A. M. Orlov, S. N. Mikov, et al., Non- tors 32, 896 (1998)]. organic Materials 31, 444 (1995). 18. V. P. Gribkovskiœ, The Theory of Absorption and Emis- 5. S. M. Prokes, O. J. Glembocki, V. M. Bermúdez, et al., sion of Light in Semiconductors (Nauka i Tekhnika, Phys. Rev. B 45, 13 788 (1992). Minsk, 1975). 6. C. Tsai, K. H. Li, J. Sarathy, et al., Appl. Phys. Lett. 59, 19. V. P. Gribkovskiœ and G. N. Yaskevich, Zh. Prikl. Spek- 2814 (1991). trosk. 12, 231 (1970). 7. M. Koós, I. Pócsik, and É. B. Vázsonyi, Appl. Phys. Lett. 20. R. Laiho, A. Pavlov, O. Hovi, et al., Appl. Phys. Lett. 63, 62, 1797 (1993). 275 (1993). 8. V. Lehmann and U. Gösele, Appl. Phys. Lett. 58 (8), 856 21. Yu. A. Bykovskiœ, V. A. Karavanskiœ, G. E. Kotkovskiœ, (1991). et al., Poverkhnost, No. 9, 23 (1999) 9. M. S. Bresler and I. N. Yassievich, Fiz. Tekh. Polupro- vodn. 27, 871 (1993) [Semiconductors 27, 475 (1993)]. Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 129Ð132. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 145Ð148. Original English Text Copyright © 2000 by Kageyama, Onizuka, Ueda, Nohara, Suzuki, Takagi. SOLIDS

Low-Temperature Specific Heat Study of SrCu2(BO3)2 with an Exactly Solvable Ground State¦ H. Kageyama*1, K. Onizuka1, Y. Ueda1, M. Nohara2, H. Suzuki2, and H. Takagi2 1 Institute for Solid State Physics, University of Tokyo, Tokyo 106Ð8666, Japan 2 Graduate School of Frontier Science, University of Tokyo, Tokyo 113Ð8656, Japan *e-mail: [email protected] Received July 22, 1999

Abstract—The specific heat of a two-dimensional spin gap system SrCu2(BO3)2 realizing the ShastryÐSuther- land model was measured between 1.3 and 25 K under various magnetic fields up to 12 T. The analysis based on an isolated dimer model in a low temperature region revealed that the value of the spin gap at zero field is ∆ = 34.4 K. It turned out that ∆ decreases in proportion to H due to the Zeeman splitting of the excited triplet levels. This simplest model, however, fails to reproduce the result in a high-temperature region, suggesting rather strong spinÐspin correlation of the system. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ments of the temperature variation of the magnetic susceptibility (34 K) [6] and electron spin resonance (ESR; Exactly solvable models have been extensively 34.7 K) [7]. It was also found that the spin system for studied in the area of strongly correlated electron sys- SrCu2(BO3)2 is fairly frustrated, located very close to the tems for the purpose of elucidating various exotic phys- critical point (J'/J)c = 0.70 between the exact dimer state ical phenomena because some rigorous results can be and the Néel-ordered state [3, 5]: the ratio of intradimer derived from them, sometimes providing us a crucial and interdimer interactions, respectively, J = 100 K and key to solve underlying problems of the phenomena. J' = 68 K, is 0.68. Furthermore, several quantized plateaux Such models, even if being far from realistic, can were observed in the magnetization [3, 6, 8], which origi- remain tantalizing theoretical subjects owing to the nates from the extremely localized triplet excitations [5]. beauty of the solutions. For example, Majumdar and Ghosh first proved that an exact dimer ground state for In the present paper, we performed the specific heat z one-dimensional spin chain imposed a stringent con- measurement of SrCu2(BO3)2 under magnetic fields H dition on the first and second nearest neighbor interac- in order to obtain more information on the exchange inter- tions [1]. Stimulated by this discovery, a number of sys- actions as well as the effect of the spin-gapped behavior tems with the identical exact wave function have been upon H. The data were analyzed in terms of an isolated explored from the theoretical point of view for one-, dimer model, and the spin gap in the absence of the field two,- and three-dimensions (see, for example [2] and was evaluated to be 34.4 K. Furthermore, it was found that references therein). However, in spite of extensive application of magnetic fields causes the Zeeman splitting efforts by chemists to tailor experimental examples, no of the excited triplet states, leading to a H-linear decrease material had been discovered for a long time. in the value of the spin gap. Recently, we reported the magnetic properties of an 2. EXPERIMENT inorganic compound SrCu2(BO3)2, which consists of a two-dimensional orthogonal dimer lattice, concluding The specific heat measurement was performed by a that this material verifies the Shastry–Sutherland heat-relaxation method [9] in a temperature range model, which has the exact dimer ground state [3Ð5]. between 1.3 and 25 K under magnetic fields between 0 Although an imaginary lattice Shastry and Sutherland and 12 T. A bulk single crystal of SrCu2(BO3)2 was considered—i.e., a two-dimensional square lattice with used, which was grown by the traveling solvent floating some additional diagonal bonds—differs from the real zone (TSFZ) method with an image furnace using a sol- one of SrCu2(BO3)2, these two are equivalent from a vent, LiBO2 under flowing O2 gas (P = 1 atm, topological point view. The value of the spin gap was O2 estimated from various measurements like measure- 99.99%). For a detailed procedure of the crystal growth, see [10]. A piece of the crystal with the ¦This article was submitted by the authors in English. dimensions of 2 × 2 × 1 mm was attached to a sapphire

130 KAGEYAMA et al.

C/T, mJ/(K2 mol) substrate by a small amount of Apiezon N grease. The 1000 magnetic fields were applied perpendicular to the ab plane, i.e., the ShastryÐSutherland lattice. The substrate SrCu2(BO3)2 0 T was weakly coupled by tungsten wires to a copper heat 6 T sink. A bare chip of Cernox resistance sensor (Lake 800 9 T 12 T Shore) was used as a thermometer to minimize the addenda heat capacity. The magnetic field dependence of the thermometer was calibrated using a capacitance 600 thermometer. The heat capacity of the sample was obtained by subtracting the addenda heat capacity, which was determined in a separate run without the sample. No appreciable magnetic field dependence was 400 observed for the addenda heat capacity. The resolution of the measurement was about 0.5%, and the absolute accuracy determined from the measurement of a Cu 200 standard was better than 5%. The measurements were Photon performed with increasing temperature.

0 5 10 15 20 25 3. RESULTS AND DISCUSSION T, K A total specific heat divided by T, C/T, measured in the absence of a magnetic field is plotted as a function Fig. 1. C/T versus T measured at H = 0 (᭹), 6 (᭺), 9 (᭡), and of T by closed circles in Fig. 1. With decreasing T from 12 T (᭝). Dotted curves are the calculations based on the isolated dimer model for ∆(0) = 34.4 K. Dot-dashed curve 15 K, C/T rises, reaches a round maximum at 7.5 K, and represents the phonon term, βT3 (β = 0.460 mJ/K4 mol). then falls rapidly, approaching naught. These behav- iors, that is to say, the so-called Schottky anomalies are typical of spin–singlet system with a finite spin gap to a lowest exited state. A gradual increase in C/T with T C/T2, mJ × K/mol ∆, K above 15 K comes from the phonon term, with is in ∝ β 3 108 40 general known to vary as C T . As also shown in Fig. 1, qualitatively similar features described above 30 appear even when magnetic fields are applied, indicat- 107 ing that the system still has a spinÐgapped ground state 20 at least for H < 12 T. A prominent difference is that a 6 peak of C/T shifts to lower temperature with rising H: 10 10 the temperature at which C/TÐT curve reaches a maximum (=Tmax) for H = 6.9 and 12 T is, respectively, 7.3, 105 6.9 and 6.8 K, implying a reduction in the actual size of 0 5 10 15 20 the spin gap ∆(H) with H. This is quantitatively dis- , T H cussed below. 104 12 T Because of a lack of an appropriate theory for the specific heat from the standpoint of the ShastryÐSuther- 103 9 T land model, we will analyze the experimental data uti- 6 T lizing the isolated dimer model, where J' is neglected 0 T 2 and only J is taken into consideration. Let us define the 10 magnetic specific heat under a certain magnetic field H as C(H). Take the example of H = 0, C(0) is given by SrCu (BO ) 101 2 3 2 the following formula,

3R(∆(0)/T)2 exp(∆(0)/T) 100 C(0) = ------, (1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 [1 + 3exp(∆(0)/T)]2 1/T, K–1 where R is 8.30 J/(K mol) (see, for example, [11]). Fig. 2. Logarithmic plot of CT2 as a function of 1/T. Solid Likewise, C(H) for a finite magnetic fields is easily cal- lines denote the fit to equation (2). Inset shows the magnetic culated. In the low temperature limit, the magnetic spe- field variation of ∆(H). cific heat the isolated dimer model can be reduced to

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 LOW-TEMPERATURE SPECIFIC HEAT STUDY OF SrCu2(BO3)2 131 the following expression as long as the system is in a S, J/(K mol) gapful state: 12 C(H) ∝ T Ð2 exp(Ð∆(H)/T). (2) 2Rln2 10 Thus CT2 is plotted against 1/T in a logarithmic scale as shown in Fig. 2. One can see all data roughly follows a linear reversal-temperature dependency. Using the 8 reduced expression of equation (2), we obtained ∆(0) = 35.9 K, ∆(6 T) = 27.5 K, ∆(9 T) = 22.5 K, and ∆(12 T) = 6 16.8 K. The deviation from the calculations (the solid lines in Fig. 2) in lower temperature region, more prominent in case of lower field, is for a most part due to the 4 phonon contribution which is neglected here and will SrCu2(BO3)2 ∆ be included later. The obtained values of (H) are plot- 2 ted in the inset of Fig. 2 as a function of H. It is clear that ∆(H) decreases nearly in proportion to H. The origin of the decrease should be the Zeeman splitting of 0 the excited states. Namely, a three-fold degeneracy of 0 10 20 30 the lowest excited triplet states (S = 1) in the absence of T, K the magnetic field is lifted up by applied magnetic field. ∆(0) was estimated to be 35.0 K using the following Fig. 3. Magnetic entropy of SrCu2(BO3)2 at H = 0 (circles). relation: ∆(H) = ∆(0) Ð gµ H, where g is the g-factor of Solid curve represents the magnetic entrope for the isolated B dimer model for ∆(0) = 34.4 K. 2+ µ the Cu electron spin and B is the Bohr magneton. An isotropic g-value, i.e., g = 2.0 was assumed. The obtained value of ∆(0) is consistent with that obtained in other measurements using a single crystalline SrCu2(BO3)2 such as retical one at around 10 K. For example, the magnetic the magnetic susceptibility (34 K) [6], ESR (34.7 K) [7] entrope at 25 K is still about and 74% of that for the iso- and Boron nuclear magnetic resonance (B-NMR; 36 K) lated dimer model and 62% of the full entropy. This [12], Cu-NMR (35 K) [12], and neutron scattering indicated that the spin system of SrCu2(BO3)2 is effec- (34 K) [13]. tively correlated over much higher temperatures, and Next, let us take a phonon term into consideration. thus consistent with the estimation of exchange con- Then, the total specific heat is given by the sum of the stants by Miyahara and Ueda; J = 100 K and J' = 68 K magnetic and phonon terms, C = C(H) + βT3. Dotted [5]. It is noteworthy that the value of J is identical with curves in Fig. 1 denote the results of the global least- that of ∆(0) for the isolated dimer model, and J = 35 K square fit in the T range well below the spin-gap size, (=∆(0)) derived from the isolated dimer model is too namely, 2.6 K < T < 4.8 K for 0 T, 2.4 K < T < 4.1 K for much smaller. 6 T, 2.1 K < T < 3.5 K for 9 T, and 1.5 K < T < 2.8 K for 12 T, from which we obtained once again a reasonable To summarize, we have measured the specific heat of SrCu (BO ) under various magnetic fields. From value of ∆(0) = 34.4 K together with β = 0.460 mJ/(K4 mol) 2 3 2 and g =2.03. The phonon contribution is independently the fitting based on the isolated dimer model, the gap shown by the dot-dashed curve in Fig. 1, which also was estimated to be 34.4 K, which is in good agreement seems to reproduce the temperature dependence of the with the values determined from other physical mea- experiment above 15 K. surements. With increasing H, the gap decreases in proportion with H. The simple dimer model, however, can As demonstrated above, it seems that the isolated not explain the data at all in higher-temperature, region, dimer model nicely reproduces the experimental data, ∆ suggesting rather stronger correlation of the spin sys- providing a consistent value of (0). In a higher temper- tem. We are looking forward to a theory based on the ature region, however, the deviation between the exper- ShastryÐSutherland model with J and J' to reproduce iment and the theory is appreciable. One can notice our specific heat data over the whole T range. from Fig. 1 that experimental Tmax is lower as compared with the theoretical one in any magnetic field, and above T the value of experimental C/T is much sup- max REFERENCES pressed. In Fig. 3, we show the T variation of the magnetic entrope of the system for H = 0, which should 1. C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, reach 2Rln2 ideally in the high-T limit. For compari- 1399 (1969). son, a theoretical curve for the isolated dimer model for ∆(0) = 34.4 K is shown by the solid line. The experi- 2. M. P. Gelfand, Phys. Rev. B 43, 8644 (1991); T. Oguchi mental entropy starts to deviate largely from the theo- and H. Kitatani, J. Phys. Soc. Jpn. 64, 1403 (1994).

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3. H. Kageyama, K. Yashimura, R. Stern, et al., Phys. Rev. 8. H. Kageyama, K. Onizuka, Y. Ueda, et al., J. Phys. Soc. Lett. 82, 3168 (1999). Jpn. 67, 4304 (1998). 4. B. S. Shastry and B. Sutherland, Physica B 108, 1069 9. R. Bachmann, E. J. DiSalvo, Jr., T. H. Geballe, et al., (1981). Rev. Sci. Instrum. 43, 205 (1972). 5. S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 10. H. Kageyama, K. Onizuka, T. Yamauchi, et al. (submit- (1999). ted to J. Crystal Growth). 6. H. Kageyama, K. Onizuka, T. Yamauchi, et al., J. Phys. 11. R. L. Carlin, Magnetochemistry (Springer-Verlag, Ber- Soc. Jpn. 68, 1821 (1999). lin, 1986). 7. H. Nojiri, H. Kageyama, K. Onizuka, et al. (submitted to 12. M. Takigawa, private communication. J. Phys. Soc. Jpn.); cond-mat/9906072. 13. H. Kageyama et al. (in preparation).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 133Ð143. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 149Ð161. Original Russian Text Copyright © 2000 by Babushkin, Logvin, Loœko. SOLIDS

Interrelation of Spatial and Temporal Instabilities in a System of Two Nonlinear Thin Films I. V. Babushkin*, Yu. A. Logvin**, and N. A. Loœko*** Institute of Physics, Belarussian National Academy of Sciences, Minsk, 220072 Belarus *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received May 26, 1999

Abstract—The spatiotemporal dynamics of a system of two thin films possessing a resonance nonlinearity and irradiated on both sides with spatially uniform monochromatic light with the same intensity is investigated. The conditions under which bistability and symmetry breaking occur in the system are obtained. It is shown that self-pulsations can arise in the system as a result of the retardation of the light between the films, if the aperture of the incident beam is sufficiently small, and the dynamical regimes arising in the process are investigated numerically. As the beam aperture increases, the pulsations break down and a stationary spatially nonuniform field distribution is established. The transverse structures arising in this case are studied, and the relation between the symmetry breaking, bistability, self-pulsations, and spatial structures in the system investigated is established. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION As is well known, the appearance of spatial structures is closely related with the existence of bistability In recent years, a great deal of attention has been in the system [7, 8]. This bistability is even observed in devoted to the development of optical methods for the presence of a single thin film of two-level atoms transferring and processing information. One of the [9−12]. When external feedback, accomplished using a main advantages of optical over traditional methods is mirror, is introduced into the system, more complicated the possibility of using spatially distributed signals, dynamical regimes, such as the appearance of self-pul- which is impossible in ordinary electronics because of sations [13Ð15] and formation of transverse static and the extremely long wavelength of the carrier radiation. traveling spatial structures [16Ð19], can appear. A sys- In this connection, it is now urgent to study the mecha- tem consisting of two thin bistable films is an elemen- nisms leading to the formation of spatial light struc- tary object for studying the effect of feedback in multi- tures (patterns) in laser and nonlinear-optical systems film systems. [1, 2]. The investigations of the spontaneous formation of patterns in optics is also important from the standpoint In the present paper symmetry breaking in a system of the general theory of nonequilibrium systems, since it of two films is investigated and the relation of symme- is possible to determine the mechanisms leading to self- try breaking with bistability is determined in a more organization, which are common to optical and hydrody- general case than in [4], specifically, without imposing namic, chemical, and biological systems [3]. any conditions on the phase relations in the films. This Our objective in the present work is to investigate paper is organized as follows. The model describing the the spatiotemporal dynamics of the transmission of system under study is presented in Section 2. In Sec- light through a system of two thin films of a nonlinear tions 3 and 4 the stationary states are determined ana- medium in a more general form than in preceding lytically and their stability is analyzed on the symmet- works. It has been shown previously that, together with ric and asymmetric branches of the solutions, and the bistability, in such a system symmetry breaking, where regions of spatial and temporal instabilities are deter- for identical fields incident on a system from both sides mined. In Section 5 the temporal dynamics generated the reflected fields have different amplitudes [4], and by instabilities on the symmetric and asymmetric the appearance of asymmetric structures [5], instability, branches are compared. It is found that stable periodic and chaos [6] can be observed. We shall show, taking pulsations can appear only on the asymmetric branch of into consideration the spatial degrees of freedom and the solution, while on the symmetric branch we retardation effects, that the spontaneous formation of observed only quasistable pulsations, which are due to spatial structures and temporal pulsations are occur by switching from one bistable branch to another and the same feedback mechanism in which the phase rela- whose decay time increases exponentially as the dis- tions in the feedback circuit play a key role. tance between the films increases. Quasipulsations of

134 BABUSHKIN et al.

n – in' the longitudinal and transverse relaxation times; ω is ω the frequency of the incident radiation; and, 0 is the (+) (+) (+) (+) E 0 E 1 E 2 E resonance transition frequency of the atoms in the film. The polarization in the film is P = µNR, where N is the (–) (–) (–) (–) density of atoms in the film. Taking account the nor- E E 2 E 1 E 0 malization relations ω ω d γ T2 ∆ Ð 0 R j = -----, = ------, w j = W j, r j = ------, T 1 γ γ (3) Fig. 1. A system consisting of two films separated by a dis- µ (±) T 1T 2 tance d; E are the amplitudes of the fields incident on the ------, 0 e j = E j ប , j = 1 2, (±) first (+) and second (–) films, E1 are the amplitudes of the (±) equations (1) and (2) transform into the following equa- fields transmitted through the films, and E2 are the ampli- tions for the normalized polarization and population tudes of the fields which reach the opposite film. difference of atoms in the films: rú = γ (Ð1 + i∆)r + iγ e w , (4) this kind have been investigated in the general theory of j j j j differential equations with retardation [20]. i wú = Ð(w + 1) + --(e*r Ð r*e ). (5) In Section 6 it is shown that the temporal dynamics j j 2 j j j j of the system becomes qualitatively different when the transverse spatial coordinates are taken into account. In It was shown [6, 11] that the effective field in a film this case the oscillatory solutions break down and the can be represented in the form system passes into a stationary spatially nonuniform ( ) ( ) state, and even an attempt to stabilize the most unstable e = e + + e Ð Ð iαr , (6) static spatial harmonics does not lead to the appearance 1 0 2 1 of oscillatory solutions, but rather only new classes of (Ð) (+) α stationary spatially nonuniform solutions arise. The e2 = e0 + e2 Ð i r2, (7) general conclusions are given in Section 7. where α is the nonlinearity parameter in the film and is given by 2. MODEL 2πnNLωµ2T Let us consider a system of two thin nonlinear films α = ------2-, (8) located at a distance d from one another and separated by បc a linear medium with a complex refractive index n + in' (Fig. 1). The system is illuminated from both sides by where L is the film thickness. The relations (6) and (7) monochromatic spatially uniform light fields with were obtained in the approximation where the film ( ) ( ) amplitudes E + and E Ð , respectively. The amplitudes thickness is small compared with the wavelength of the 0 0 incident radiation. of the fields transmitted through the first and second (+) (Ð) (Ð) (+) The propagation of the field in the linear medium films are E1 and E1 , respectively, and E2 and E2 between the films satisfies the diffraction equation, are the amplitudes of the fields reaching the opposite which can be written in the following operator form as films (see Fig. 1). We shall use the Bloch equations for the polariza- (±)  d  (±) e (r⊥, τ) = ρexp(is)exp Ði--∆⊥ e (r⊥, τ), (9) tion and the population difference between the levels of 2  k  1 two-level atoms to describe the interaction of light with the films: where, respectively, ρ = exp(kn'd) are the losses, s = µ knd is the phase shift, τ is the propagation time of the dR j  (ω ω ) 1  W jE j ------= i Ð 0 Ð ----- Ð i------ប------, (1) light between the fields, r⊥ = (x, y) is the transverse dt T 2 component of the radius vector of the point (x, y, z), k⊥ = (kx, ky) is the transverse component of the total dW j W j + 1 µ ( * * ) wave vector k of the light field, and ∆⊥ is the transverse ------= Ð------+ i----ប-- E j R j + R j E j , (2) dt T1 2 part of the Laplacian. The amplitudes of the fields where R , j = 1, 2, is the slowly varying part of the off- transmitted through the first and second films, accord- j (+) (+) α (Ð) (Ð) α diagonal element of the density matrix of a two-level ing to [11], are e1 = e0 Ð i r1 and e1 = e0 Ð i r2, µ atom; is the transition dipole moment; T1 and T2 are respectively.

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w1 –0.1 (a) (b) (c)

–0.2

–0.3

–0.4

–0.5

8.8 9.0 9.2 6.0 6.4 6.8 4 5 6 ein

Fig. 2. Stationary states for the parameters α = 10, ∆ = 2, ρ = 0.5, and γ = 0.1. The solid lines mark stable states and dashed lines mark states which are unstable with respect to perturbations with θ = 0. (a) s = 0: bistability on the symmetric branch of the solution; (b) s = 0.5π: no symmetry breaking or bistability; (c) s = π: symmetry breaking.

3. STATIONARY UNIFORM STATES When the incident fields are the same (e01 = e02), the OF THE SYSTEM stationary symmetric state e = e1 = e2 can be found from It has been demonstrated in [6, 4] in the particular the equation case of no offset of the frequency from resonance and s αη[ ρ ( )] π i 1 + exp is e a multiple of that bistability and symmetry breaking e01 = e02 = e + ------2------, (16) are possible in a system of two films. In what follows, 1 + β e we shall study the relation between these two phenom- which is well known from the theory of bistability [7, 8]. ena in a general form. To analyze symmetry breaking we substitute the We obtain the following relations for the stationary expressions (14) and (15) into equations (10) and (11), values of the fields in the films: and after simple algebraic transformations we obtain e = e + iαr + iαρexp(is)r , (10) iαη[1 Ð ρexp(is)]e 01 1 1 2 e + ------1- 1 β 2 e = e + iαr + iαρexp(is)r , (11) 1 + e1 02 2 2 1 (17) iαη[1 Ð ρexp(is)]e (+) ρ ( ) (Ð) = e + ------2-. e01 = e0 + exp is e0 , (12) 2 β 2 1 + e2 (Ð) ρ ( ) (+) e02 = e0 + exp is e0 . (13) It is evident from the expression (17) that both parts contain the same bistable function as the first part of equa- Here the stationary values of the polarization and pop- tion (16) with one difference: s is replaced by s + π. It is ulation difference have the form obvious from the S shape of this function that the same ηe value of the right(left)-hand hand part of the expres- r = ------j------, (14) sion (17) can be obtained for different values of e (e ). j β 2 1 2 1 + e j This situation corresponds to the equality (17) with different values of e on the left-hand side and e on the 1 1 2 w = Ð------, (15) right-hand side, i.e., symmetry breaking with the same j β 2 1 + e j incident fields. It is found that bistability of the symmetric solution (16) is related with symmetry breaking, where described by the expression (17) by replacing s by s + π. This can be seen in Fig. 2, which shows the dependence 1 i Ð ∆ β = ------, η = ------j = 1, 2. of the stationary population difference in the films on ∆2 ∆2 1 + 1 + the magnitude of the incident field ein.

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w in nonlinear systems: bistability, symmetry breaking, –0.2 and 2τ pulsations.

4. STABILITY OF STATIONARY STATES To investigate the stability of a state of equilibrium δ δ with respect to spatiotemporal perturbations ( rj, wj), we shall linearize the system (4)Ð(5). Linearizing relative to disturbances proportional to exp(λt + r⊥k⊥) gives a system of equations for λ: –0.3  D U   F   F   1 1   1  = λ 1  , (19)       U2 D2 F2 F2

 ∆ α   Ð1 + i + w j 0 ie j    ∆ α * D =  0 Ð1 Ð i + w j Ðie j  , (20) –0.4 j   0 0.5 1.0 1.5 2.0  γ γ  θ π --(ie* Ð αr*) --(Ðie Ð αr ) γ /  2 j j 2 j j 

Fig. 3. Regions of static instability for the parameters in τ  α ( Θ)  Fig. 2a, = 50. The dots show the boundaries of the regions w j exp i 0 0 where instability with respect to AH perturbations occurs;    α ( Θ)  triangles and crosses show the boundaries of regions with U = 0 w j exp Ði 0 , (21) instability with respect to symmetric (δw = δw ) and asym- j   δ δ 1 2  γα γ  metric ( w1 = Ð w2) static disturbances, respectively. Ð------e*exp(iΘ) Ð--e exp(ÐiΘ) 0  2 j 2 j 

It is evident that as s changes from s = 0 (Fig. 2a) to δ δ * s = π (Fig. 2c), bistability is replaced by symmetry where Fj is the perturbation vector: Fj = ( rj, r j , δ T Θ θ λτ θ 2 breaking for the same values of the population differ- wj) , = + s + , and = k⊥ d/k characterizes the ence (or light intensity) in the films. For an intermediate transverse perturbation with wave number k⊥. Since θ value of the phase s (see Fig. 2b) neither bistability nor and λ appear in the expression for Θ in the same way, symmetry breaking occur in the system. The condition the spatial and temporal instabilities manifest them- for the existence of bistability (symmetry breaking) is selves in the same way. determined by the inequality The equation (19) is a characteristic quasipolyno- λ [2cos(argΛ) Ð Λ ]3 + 27 Λ < 0, (18) mial, in which the unknown quantity appears in the form of powers λn and in the form of terms of the form where Λ = iαη[1 Ð ρexp(is)]. It transforms into a simi- exp(Ðλτ). The boundaries of stability are obtained from lar condition obtained in [4] under the corresponding equation (19) by making the substitution λ = iΩ, where simplifying assumptions. Ω is a real quantity. If Ω = 0, we have a region of instability relative to “static” disturbances (with zero tempo- Figure 2 demonstrates the important role of phase ral frequency), and in the opposite case we have regions effects in the transmission of light between the films. of instability relative to disturbances of the AndronovÐ These effects determine the development of spatial and Hopf (AH) type. Curves of neutral stability of the sys- temporal instabilities, as will be shown below. We note tem with respect to static (marked by triangles and that an equality similar to equation (17) has been inves- crosses) and AH (marked by dots) instabilities on the tigated in [13] in a study of 2τ pulsations in a system symmetric branch for parameters corresponding to consisting of a film deposited on a dielectric substrate. Fig. 2a are displayed in Fig. 3. It is evident that the AH In this work e1 and e2 represent the values of the field in zones are embedded in the static zones, so that their successive half-periods of the pulsations, and the tran- presence results only in an increase in the order of the sition from a regime with pulsations to a regime where instability of the system. A numerical experiment only bistability is present, just as in our system a tran- shows that this has no qualitative effect on the dynam- sition from symmetry breaking to bistability, can be ics of the system. As the retardation time τ increases, accomplished only by changing the phase relations in the number of such zones increases, as all new roots the feedback circuit. Thus, the same algebraic relations intersect the imaginary axis, while existing zones describe the mutually complementary effects observed approach the boundary of static instability.

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w1 0 (a)

–0.4

–0.8

4 6 8 10 12 ein w1 0

w1 0 (b) –0.2 –0.1 (c)

–0.2 –0.4

–0.3 0 1 2 0 1 2 θ/π θ/π

Fig. 4. (a) Stationary curve for the parameters α = 20, ∆ = 2, ρ = 0.5, s = π, γ = 0.1, and τ = 10. The solid lines mark stable states, the dashed lines mark states whose instability is determined primarily by static perturbations with θ = 0, and the thick lines show states which are unstable with respect to static perturbations with θ ≠ 0 and with respect to AH perturbations. (b) Boundaries of the neutral stability of states corresponding to the part of the asymmetric branch in the left-hand rectangle in Fig. 4a. (c) Boundaries of neutral stability of the states lying on the part of the asymmetric branch in the right-hand rectangle in Fig. 4a. The triangles show the boundaries of stability with respect to static perturbation; the dots show the boundaries of stability with respect to AH perturbations.

As one can see from the expression for Θ, a dis- spatiotemporal instability on the asymmetric branch is placement along s in it is compensated by an opposite possible only if a fork-type bifurcation, corresponding displacement along θ. If for such a change in s the to the appearance of asymmetric solutions, becomes quantities ej(wj, rj), satisfying equation (19), once again subcritical, which is achieved by increasing the nonlin- correspond to stationary states of the system, then the earity parameter α. An example of this is presented in instability zones merely shift along the θ axis without a Fig. 4. Figure 4a displays the stationary dependences of change in form. Such a situation occurs in the case of w1 on the incident ﬁeld ein. Regions of instability, cor- the symmetric branch considered above, since the val- responding to the stationary states on the asymmetric ues of ej, determined by equation (19), remain station- branch located in the left- and right-hand rectangles in ary values of the system as a function of s because the Fig. 4a are presented in Fig. 4b. To save space, we show incident ﬁeld is correspondingly adjusted according to only the zones corresponding to the upper part of the the expression (16). asymmetric branch, since the zone on the bottom part For parameters corresponding to the case in Fig. 2a, of the branch has a similar form. It follows from Fig. 4b that the states marked by the thick line in Fig. 4a are the regions of instability marked by triangles in Fig. 3 θ ≠ are responsible for the development of symmetric per- unstable with respect to static perturbations with 0, δ δ i.e., here spatial structures can arise. Moreover, these turbations ( w1 = w2), while the region marked by the crosses corresponds to the development of asymmetric states are unstable with respect to AH perturbations, perturbations (δw = Ðδw ). and zones of this instability do not overlap with regions 1 2 of static instability for 0 ≤ θ ≤ 0.25, 0.75 ≤ θ ≤ 1.25, and We shall now investigate the instability on the asym- so on, which indicates the possibility of the excitation metric branch of the solution. Analysis shows that a of pulsations. In contrast to the situation on the sym-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 138 BABUSHKIN et al. metric branch, the regions of instability on the asym- polarization and population of the nonlinear medium in θ metric branch do not shift along the axis as s varies, films, which is determined by the relaxation times T1 but rather they are compressed or stretched in the verti- and T2. As the retardation increases compared with cal direction, since the relations between the stationary these quantities, the effect of the inertia decreases and values of ej(wj, rj) in the two films change. One of the the oscillatory solutions, because of the continuous minima of the curve of neutral stability with respect to dependence of solutions of equations with a retarded the static perturbations always lies on the θ = 0 axis, argument on the retardation [21, 22], can asymptoti- just as in Fig. 4b. cally approach a discontinuous solution obtained in the The stationary states on the part of the asymmetric case of discrete mappings. branch located in the right-hand rectangle in Fig. 4a are For sufficiently large τ (1/τ Ӷ 1) the system (4)Ð(5) unstable for w1 ranging from Ð0.41 to Ð0.03 with is singularly perturbed with respect to the correspond- respect to static disturbances with any value of θ, ing discrete mapping. It has been shown for very simple including θ = 0. As a result of this, spatially uniform one-dimensional systems of this kind [20] that in the disturbances, transferring the system into stable sta- case of a mapping possessing two attractors—a station- tionary states existing for given parameters, will ary point with a quite large region of attraction and a develop. In turn, the regions of static instability also cycle with period 2τ, the following behavior is typical contain regions of AH instability and a secondary static for the corresponding differential-difference equation: instability, which only increase the order of the instabil- If the system is initially in the region of attraction of a ity of the solutions, and just as in the case of embedded 2τ periodic attractor of the corresponding limiting map- AH zones on the symmetric branch, they have no qual- ping (τ = ∞), then its oscillations break down after a itative effect on the dynamics of the system. time exp(aτ) has elapsed (where a is a constant), and it passes into a stationary state. This means that if the constant a is sufficiently large, then even for small τ we 5. TEMPORAL DYNAMICS obtain a trajectory which can be assumed to be periodic We shall consider first the case where the aperture of for a quite long period of time. A numerical experiment the incident light beam is comparatively small, so that showed that such behavior is also characteristic of the only the zero spatial harmonic can be excited. In this system (4)Ð(5). Figure 5 shows quasipulsations of this case, only the instability on the θ = 0 axis in Figs. 3 and kind for the system parameters presented in [6] and the 4 need be considered. values of s for which symmetry breaking (Fig. 5a) or bistability (Fig. 5b) occurs. In the first case the station- As already mentioned, the regions of AH instability ary characteristic is similar to that shown in Fig. 2c, and on the symmetric branch always lie inside the static in the second case it is similar to that presented in zones, in contrast to the related system of a thin film Fig. 2a. For sufficiently large τ these quasipulsations with a mirror [16], which makes it difficult to obtain acquire a squared form. Figures 4a and 4b show only AH pulsations, similarly to [16], since after the tran- several periods before the pulsations break down, sient process a stationary stable state is established in which occurs after a comparatively long time has the system. However, in [6, 13], where differential elapsed. The process leading to the breakdown of qua- equations with a retarded argument reduced in the limit τ τ ∞ sipulsations for small is shown in Fig. 5c. Figure 5d of very large retardation to discrete mappings, shows the dependence of their lifetime on the retarda- it was shown that discontinuous pulsations with period tion time τ. 2τ can exist. In [6] more complicated regimes, right up to chaos, were also investigated. In these works the We note that to obtain quasipulsations it is sufficient question of whether or not the dynamics of the system to prescribe special boundary conditions. If the station- reduces for very large but finite τ to the dynamics ary curve exhibits bistability, the state for the first film obtained in the limit τ ∞, in other words, whether must correspond to the lower branch of the bistability or not a finite τ for which this limit cycle or a set close curve and the state for the second film must correspond to it is a stable attractor exists, remained open. It is to the upper branch. After a time τ has elapsed, switch- obvious that for arbitrary finite τ this attractor of a dis- ing occurs (the films change places), after a time 2τ the crete mapping, which is a piecewise-continuous func- system returns into the initial state, and so on. This cor- tion, will not be an attractor of the system (4)Ð(5), sta- responds to a stable 2τ cycle, obtained in the limit map- ble or unstable, since, as follows from the general the- ping (τ ∞). Such a sequence of switchings results ory of differential equations with retardation [21], the in the asymmetric quasipulsations shown in Fig. 5b. solutions of the system (4)Ð(5) smooth out with time (if Conversely, for a stationary dependence with broken at time t = 0 the solution is discontinuous, then at time symmetry, both films must be placed on the same asym- t = nτ it will be differentiable n Ð 1 times) and, thus, metric branch. After a time τ has elapsed, they switch only an infinitely differentiable function can be a peri- onto the second branch together; this corresponds to odic attractor of the system (4)Ð(5). The process of symmetric pulsations (Fig. 5a). Nonetheless, the region smoothing of the solutions reflects the presence of tem- of attraction of such quasipulsations is quite large. poral dispersion in the system due to the inertia of the Even if a state in which the films are located in the

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wi wi 0 (a) 0 (b)

–0.5 –0.4

–1.0 –0.8 0 500 1000 t 0 500 1000 t wi t × 10–6 0 (c) (d) 2

–0.4 1

–0.8 0 0 200 400 t 40 80 120 τ

Fig. 5. Quasipulsations generated by switchings from one part of the branch of a stationary curve to another. The values of the parameters are the same as in [6] (α = 8, ∆ = 0, ρ = 0.5, and γ = 0.1). The solid line shows the time dependence for w and the dashed line π τ 1 shows the saem for w2. (a) s = and = 100: switchings between asymmetric branches (stationary characteristic, just as in Fig. 2c); (b) s = 0 and τ = 100: switchings between the upper and lower branches of the bistable curve (stationary characteristic, just as in Fig. 2a); (c) s = π and τ = 20: damping of quasipulsations for small values of τ. (d) Lifetime of quasipulsations (dashed curve with triangles) and the best exponential fit to this curve, Cexp(6.7τ) (the solid curve with the asterisks). absence of an external field is taken as the initial condi- branch that is unstable with respect to temporal pertur- tions and the field is switched on instantaneously at bations. Figure 6 shows examples of oscillations arising time t = 0 (as in Fig. 5a), the system nonetheless dem- from such stationary states for increasing values of τ. onstrates 2τ quasipulsations. At the initial stage (Fig. 6a, t < 400), one can see the development of AH pulsations, but then the system It is evident from Fig. 5d that the lifetime of the qua- passes into a regime with large-amplitude oscillations sipulsations actually increases exponentially. In this that correspond to switching between branches of the figure the asterisks connected by a solid line show the stationary characteristic in Fig. 4a. As one can see from data obtained from the numerical experiment; the trian- Fig. 6b, as τ increases, the pulsations arising on the gles connected by a dashed line represent the curve t = asymmetric branch can be symmetric. As τ increases C*exp(6.7τ), where C* is a constant. Even for not very τ further, the pulsations approache a square form with large the lifetime is several thousandths of periods, τ and as τ increases, the ratio of the lifetime to the period period 4 (Fig. 5d), but, in contrast to pulsations on the symmetric branch, they are stable. Figure 6c shows increases rapidly. Thus, for appropriate retardation τ times, pulsations exist for such a long time that they are oscillations for an intermediate value of , which effectively full-fledged periodic oscillations (which is explain the transition from Fig. 6b to Fig. 6d. As one an argument for applicability of switching to limit map- can see by comparing Fig. 6d and Fig. 4a, where the pings in [6, 13]). straight vertical line indicates the value of the incident field (the initial conditions of the system are found as Returning to Fig. 4b, we shall investigate the tempo- the point of intersection of the upper part of the asym- ral dynamics arsing in the system if the initial condi- metric branch with this line for the first film and the tions are chosen on the section of the asymmetric lower part of the asymmetric branch for the second

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wi wi 0 (a) 0 (b)

–0.4 –0.4

–0.8 –0.8

400 600 800 t 150 300 450 t w i wi 0 (c) 0 (d)

–0.4 –0.4

–0.8 –0.8

20 40 60 80 100 20 30 40 50 t × 10–2 t × 10–3

Fig. 6. Stable pulsations for values of the incident ﬁeld marked by the straight line in Fig. 4a: α = 20, ∆ = 2, ρ = 0.5, γ = 0.1, s = π and τ = 4 (a), 100 (b), 200 (c), and 1000 (d).

τ film), after each time interval a switching occurs symmetric disturbances, and k1 and k2 belong to the between the stationary states of the system, and four regions responsible for the asymmetric disturbances. switchings correspond to the period 4τ. As one can see from Figs. 7a and 7b, the maxima in the first film occur at the minima in the second film and vice versa. 6. TAKING ACCOUNT It follows from Fig. 4b that transverse spatial struc- OF THE TRANSVERSE DEGREE OF FREEDOM tures can appear on the asymmetric branch as a result θ We shall now consider the effect of the transverse of the instability of static modes with nonzero . It has degrees of freedom on the dynamics of the system. This been shown in [5] that in the absence of retardation corresponds to increasing the aperture of the incident asymmetric hexagonal structures arise, which in con- light beam. In this case, the harmonics with nonzero k⊥ tradistinction to structures on the symmetric branch can be most unstable. have a different contrast in the first and second films. There arises the question of what happens when retar- Let us consider first the situation that arises for dation is taken into account and the temporal effects parameters corresponding to the symmetric branch of start to influence the spatial dynamics of the system. To the solution in Fig. 2b. A numerical experiment, whose answer this question, a numerical experiment was per- result is presented in Fig. 7, shows that spatial struc- formed with parameters and initial conditions corre- tures can arise in this case (Figs. 7a, 7b). As they form, sponding to the case in Fig. 6. It turns out that in con- the unstable harmonics belonging to the regions trast to the case of a small-aperture beam, when the pul- responsible for the symmetric perturbations and sations arising remain stable (Fig. 6), in the present regions responsible for the asymmetric perturbations case the development of pulsations drives the system can interact with one another, as a result of which into a spatially uniform but stationary state (Fig. 8). It rhombic structures arise. Figure 7c shows how the is evident from the figure that as a result of the transient interaction of the spatial modes occurs. Here k3 process (Fig. 8d), symmetric hexagonal structures belongs to the region of instability responsible for the (Figs. 8a, 8b), which are identical in both films, are

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ky (c)

k3 k1

(a) (b)

kx Fig. 7. Asymmetric rhombic structures, generated from a symmetric state for parameters corresponding to the case in Fig. 2c. The population differences in the ﬁrst (a) and second (b) ﬁlms and the spatial spectrum of the structure (c): k3 lies in the region responsible for the development of symmetric disturbances and shown in Fig. 3; k1 and k2 belong to the region responsible for the development of asymmetric disturbances.

y ky

(b)

kx (a) x wi (c) (d) 0

–0.4

–0.8

0 0.4 0.8 y/d 100 300 t

Fig. 8. Structures arising from a uniform state on the asymmetric branch (the value of the incident field is marked by the vertical line in Fig. 4a), for parameters corresponding to Fig. 4 and τ = 4. (a) Spatial distribution of the population difference in the first film. (b) Spatial spectrum of this structure. (c) Section of the spatial distribution in Fig. 8a for y = 32. The distribution in the first film is completely identical to the distribution in the second film. (d) Transient process driving the system to the indicated spatial structure. The solid line shows the dynamics at the point (0, 0) of the first film and the dashed line shows the dynamics at the same point of the second film.

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y y

(a) (b) x x

wi (c) (d)

–0.2

–0.4

–0.6

–0.8 0 0.4 0.8 y/d 0 200 400 t

Fig. 9. Spatial structures arising for the same values of the initial parameters as in Fig. 8 but with oscillations on the mode θ = 0 suppressed and the spatial modes which are unstable with respect to static disturbances cut out (see Fig. 4b): (a, b) spatial distributions of the population differences in the first and second films, respectively; (c) section of Fig. 9a (dashed line) and Fig. 9b (solid line) for y = 32: (d) temporal dynamics of the transient process for one point (1, 1) in the first (dashed line) and second (solid line) films. obtained. The maxima and minima of these structures Fig. 6. The restrictions imposed on the system have the (Fig. 8c) correspond to the upper and lower asymmetric effect that the pulsations start to develop only on the branches of the stationary curve (Fig. 4a). In this case spatial harmonics with Ω ≠ 0. However, the pulsations symmetry breaking seems to be distributed in space. arising in the spatial structures decay, and with time the This picture is also qualitatively identical to Fig. 6b, the system passes into a spatially nonuniform stationary difference being that in Fig. 6b the time dependence is state. Figure 9 shows the transient process (d) and the shown, while in Fig. 8c the spatial dependence is steady state (aÐc) established in the system. Just as the shown. Thus, symmetry breaking, pulsations, and spa- preceding case, a large part of the transverse distribu- tial structures in this system are seen to be interrelated. tion of the system is close to one of the stationary states, This complements the fact already mentioned above but here sharp, soliton-like peaks, corresponding to that the factors responsible for the spatial and temporal switching between the upper and lower solutions in the instability enter in the characteristic quasipolynomial region of symmetry breaking in Fig. 4a, appear in one of the system in the same manner. of the films (Fig. 9c). Hopf instability zones, for which θ ≠ 0, are seen in Fig. 4b. There arises the question of how these instabil- 7. CONCLUSIONS ity zones influence the dynamics when other competing instabilities are suppressed. To clarify this question we In summary, the results presented above attest to the suppressed in the numerical experiment the instability fact that a system of two thin bistable films irradiated of the main AH mode with θ = 0; this corresponds to with monochromatic light demonstrates a rich dynam- introducing into the system an additional apparatus, a ics. Specifically, bistability can arise in the system as a stabilizer. Once again, initially, the system was in a uni- result of the characteristic nonlinearity of each film. form asymmetric state, the same as in the case shown in The presence of feedback, which is a result of the inter-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INTERRELATION OF SPATIAL AND TEMPORAL INSTABILITIES 143 action of the films, also leads to the appearance of sym- 5. Yu. A. Logvin, Phys. Rev. A 57, 1219 (1998). metry breaking. An increase in the distance between the 6. Yu. A. Logvin and A. M. Samson, Zh. Éksp. Teor. Fiz. films and therefore the retardation time in the feedback 102, 472 (1992) [Sov. Phys. JETP 75, 250 (1992)]. circuit results in the appearance of pulsations in the sys- 7. L. A. Lugiato, Prog. Opt. 21 (1984). tem, when the beam aperture is sufficiently small so 8. H. M. Gibbs, Optical Bistability: Controlling Light with that the transverse effects do not influence the dynam- Light (Mir, Moscow, 1988; Academic Press, Florida, ics of the system. The period of the pulsations is related Orlanddo, 1985). with the distance between the films. This makes it pos- 9. S. M. Zakharov, Zh. Éksp. Teor. Fiz. 108, 829 (1995) sible to use this system to obtain a periodic sequence of [JETP 81, 452 (1995)]. Π-shaped pulses with a prescribed period. An increase 10. S. M. Zakharov and É. A. Manykin, Poverkhnost, No. 2, of the beam aperture, which can be described by taking 137 (1988). account of the spatial modes with nonzero k⊥, substan- 11. M. Benedict, V. A. Malyshev, E. D. Trifonov, et al., Phys. tially changes the behavior of the system. Specifically, Rev. A 43, 3845 (1991). for a short delay, formation of asymmetric rhombic 12. L. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Com- structures on the symmetric branch is possible in the mun. 111, 163 (1994). system as a result of the interaction of the spatial modes 13. A. M. Basharov, Zh. Éksp. Teor. Fiz. 94, 12 (1988) [Sov. from the instability regions responsible for the develop- Phys. JETP 67, 1741 (1988)]. ment of asymmetric perturbations and the regions 14. N. A. Loœko, Yu. A. Logvin, and A. M. Samson, Kvan- responsible for the development of symmetric pertur- tovaya Élektron. 22, 389 (1995). bations. As the delay time increases, other stationary 15. A. N. Loœko, Yu. A. Logvin, and A. M. Samson, Opt. states are established, for example, symmetric hexago- Commun. 124, 383 (1996). nal structures on the asymmetric branch. Temporal 16. Yu. A. Logvin and N. A. Loœko, Phys. Rev. E 56, 3803 Hopf oscillations arising on the asymmetric branch of (1997). the stationary solution of the system with a small aper- 17. Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, et al., ture of the incident beam collapse. As a rule, this spatial Phys. Rev. E 54, R4548 (1996). distribution of the field is irregular and has defects. For 18. N. A. Loœko and Yu. A. Logvin, Laser Phys. 8, 322 example, soliton-like spatial structures appear. (1998). 19. A. A. Afanas’ev, Yu. A. Logvin, A. M. Samson, et al., REFERENCES Opt. Commun. 115, 559 (1995). 20. A. M. Sharkovskiœ, Yu. L. Maœstrenko, and E. Yu. Ro- 1. N. N. Rozanov, Optical Bistability and Hysteresis in manenko, Difference Equations and Their Applications Distributed Nonlinear Systems (Nauka, Moscow, 1997). (Naukova Dumka, Kiev, 1986), p. 203. 2. Self-Organization in Optical Systems and Applications 21. L. É. Éksgol’ts and S. B. Norkin, Introduction to the in Information Technology, Ed. by M. A. Vorontsov and Theory of Differential Equations with Deviating Argu- W. B. Miller (Springer, Berlin, 1995). ments (Nauka, Moscow, 1971, p. 19; Academic Press, 3. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 New York, 1973). (1993). 22. A. D. Myshkis, Usp. Mat. Nauk 4, 33 (1949). 4. I. V. Babushkin, Yu. A. Logvin, and N. A. Loœko, Kvan- tovaya Élektron. 25, 110 (1998). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 144Ð152. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 162Ð174. Original Russian Text Copyright © 2000 by Ovchinnikov, Dyugaev. SOLIDS

Breakdown of Renormalizability in the Kondo Problem in the High-Temperature Expansion of the Free Energy Yu. N. Ovchinnikov and A. M. Dyugaev Max-Planck-Institute for the Physics of Complex Systems, Dresden, D-01187 Germany Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia Received July 15, 1999

Abstract—It is shown that there are two energy scales in the Kondo problem: Tk and T0, one of which (Tk) is exponentially small in the coupling constant g. The second scale T is proportional to the squared coupling con- ӷ 0 stant. Perturbation theory is valid only in the region T T0. The point T0 is apparently the crossover from weak to strong coupling. The ﬁrst indications of the breakdown of the hypothesis of only one energy scale in the Kondo problem appear in fourth order of perturbation theory. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION model is nonrenormalizable and the expression (1) for In investigations of the thermodynamics of the the free energy is incorrect. In the Kondo model there 2 ӷ Kondo model, it is ordinarily assumed that there is only exists a second energy scale T0 ~ g D Tk. This energy one energy scale and that the theory is renormalizable scale was obtained in [6] in the self-consistent field [1–3]. Specifically, it is assumed that the dependence of approximation. Perturbation theory is valid only in the ӷ Ӷ the energy Ᏺ on the cutoff parameter D has the form temperature range T T0. In the range T T0 strong coupling is realized. µ Ᏺ n H, T =Dg∑Cn+Tkf------, (1) TkT k n 2. MODEL where g is a dimensionless coupling constant, which We shall assume that the interaction of an electron localized on an impurity atom with the conduction elec- will be determined below; Tk ~ Dexp(Ð1/2g) is the µ trons can be described by the exchange Hamiltonian Kondo temperature; Cn are numerical coefficients; is the Bohr magneton; and, H is the external magnetic 3 3 + + Hˆ = Hˆ +drr d Vr()Ðr χα()rϕβ()χr β()rϕα()r field. 0 ∫ 1 2 1 2 1 2 2 1 (2) The first term in equation (1) is assumed to be tem- µH + + 3 Ð------()ϕ ↑ ()r ϕ↑(r ) Ð ϕ↓()r ϕ↓()r dr. perature-independent. Physically, it is the shift of the 2∫1 1 1 1 1 ground state of the system as a function of the coupling ++ constant g. Investigation of the dependence of the aver- In equation (2) the operators ϕα and χα create an age electron spin of an impurity atom on the magnetic electron in a state localized on an impurity and in the field strength T = 0 [4] has shown that perturbation the- continuous spectrum, respectively. ory is inapplicable even for strong magnetic fields, µ ӷ At finite temperature the partition function Z can be H Tk. As a result, in strong magnetic fields, instead represented in the form [7] of very slowly decreasing logarithmic corrections to the 1/T saturated value of the impurity spin, power-law correc-  ˆ⁄ ˆ ⁄ ÐH0T τˆτ tions arise [4], which agrees much better with the Z==Tr exp()TrÐH T e1Ð ∫ d1V()1 experimental data [5].  0 (3) In this connection, it makes sense to check the 1/T τ hypothesis that only one energy scale exists and the 1  ττˆτˆτ … renormalizability of the exchange interaction on the +∫ d1∫d2V()1V()2Ð ,  basis of finite-temperature perturbation theory. The 0 0 problem is complicated by the fact that in the Kondo problem the free energy is not a sum of irreducible dia- where the operator Vˆ ()τ is determined by the expres- grams, and the perturbation series for the partition sion function Z must be investigated in order to calculate the Hˆ τ ÐHˆ τ free energy. As a result, we shall show that the Kondo Vˆ()τ =e 0 Vr()e 0 . (4)

In what follows, we shall use the symmetric model where for cutting off the matrix elements with cutoff energy D. τ 1/T 1 In the calculation of the perturbation-theory series J = dτ dτ Φ2(τ Ð τ )ᏺ(τ Ð τ ), (11) in the expression (3) for the partition function there H ∫ 1 ∫ 2 1 2 1 2 arises the quantity 0 0 and J = J , Zi = 2cosh(µH ⁄ 2T), and Ze is the par- τε 0 H = 0 0 0 Φ τ ( ) Ð k ( ) = ∑ 1 Ð nk e tition function of a free electron gas. The function ᏺ(τ) k and the dimensionless coupling constant g are defined D (5) as follows: Ðτξ 1  = dξe 1 Ð ------, ᏺ τ ( ÐµHτ ÐµH ⁄ T + µHτ) ⁄ ( ÐµH ⁄ T ) ∫  1 + exp(ξ ⁄ T) ( ) = e + e 1 + e , (12) ÐD D where nk is the family distribution function, and the 〈V〉(…) g dξ(…) . (13) ε ∑ ∫ electron energy k is measured from the Fermi surface. k ÐD Simple calculations show that the following two relations hold: The expression (10) is identical to the results obtained in [3]. We shall not require the explicit form of D Ð1 the function JH, since in the free energy the higher- τε τξ ξ k ξ   Φ τ order and subsequent terms in powers of (D/T) cancel ∑nke = ∫ d e 1 + exp--- = ( ), (6) T for any function Φ(τ) of the form (7) and (9). k ÐD In third order of perturbation theory the correction Φ τ Φ ⁄ τ ( ) = (1 T Ð ). (7) Z3 to the partition function is τ τ The relations (6) and (7) greatly simplify the calcula- 1/T 1 2 tion of the perturbation-theory series in the expression (3) e i 3 τ τ τ Φ τ τ Φ τ τ Z3 = ÐZ0Z0g ∫ d 1 ∫ d 2 ∫ d 3 ( 1 Ð 2) ( 2 Ð 3) for the partition function. (14) The main results—the nonrenormalizability of the 0 0 0 × Φ τ τ {ᏺ τ τ ᏺ τ τ ᏺ τ τ } Kondo model and the existence of a second energy ( 1 Ð 3) ( 1 Ð 2) + ( 2 Ð 3) + ( 1 Ð 3) . scale T0—do not depend on the specific form of the cut- We note that the function ᏺ(τ) can appear in the off of the matrix elements. expressions for the corrections to the partition function From equations (5) and (6) we find an explicit in order no higher than linear. This is because in a local- expression for the function Φ(τ): ized state there is always only one electron. In the fourth-order correction it is convenient to sin- 1 ÐDτ Φ(τ) = --(1 Ð e ) 2 τ gle out the large term 0.5(J0 + JH) , which is propor- 2 tional to (D/T) . Then, the quantity Z4 can be repre- 1 Ð exp(ÐD(1 ⁄ T Ð τ)) 1  + ------Ð ------sented in the form  1 ⁄ T Ð τ 1 ⁄ T + τ (8)  ∞ e i 4 1 2 n Z = ÐZ Z g --(J + J ) (Ð1) 4 0 0 0 H Ð 2τ ∑ ------. 2 ( ⁄ )2 τ2 n = 2 n T Ð τ τ τ 1/T 1 2 3 τ Ӷ Φ τ 2 2 Near the singularity ( T 1) the function ( ) can + dτ dτ dτ dτ [Φ (τ Ð τ )Φ (τ Ð τ ) be represented in the form ∫ 1 ∫ 2 ∫ 3 ∫ 4 1 2 3 4 0 0 0 0 2 1 ÐDτ π 2 3 × (ᏺ (τ τ ) (τ τ ) ᏺ τ τ ᏺ τ τ ) Φ(τ) = --(1 Ð e ) + -----T τ + O(T(Tτ) ). (9) ( 1 Ð 2 + 3 Ð 4 ) Ð ( 1 Ð 2) ( 3 Ð 4) τ 6 + Φ2(τ Ð τ )Φ2(τ Ð τ ) An important property of the function Φ(τ) is that there 1 4 2 3 is no term of the form const á T on the right-hand side × (ᏺ (τ τ ) (τ τ ) ᏺ τ τ ᏺ τ τ ) ( 1 Ð 2 + 3 Ð 4 ) Ð ( 1 Ð 4) ( 2 Ð 3) of equation (9). Φ2 τ τ Φ2 τ τ Ð ( 1 Ð 3) ( 2 Ð 4) × (ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ ) ] 3. PERTURBATION THEORY ( 1 Ð 3) + ( 2 Ð 4) + ( 1 Ð 3) ( 2 Ð 4)

In the second order of perturbation theory, we ﬁnd 1/T τ τ τ from equations (2) and (3) 1 2 3 τ τ τ τ [Φ τ τ Φ τ τ + ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 ( 1 Ð 4) ( 3 Ð 4) (15) e i 2( ) Z2 = Z0Z0g J0 + J H , (10) 0 0 0 0

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× Φ τ τ Φ τ τ ( ᏺ τ τ τ τ ) ˜ ( 2 Ð 3) ( 1 Ð 2) 2 + 2 ( 1 Ð 2) + N( 3 Ð 4) The shift δE of the ground state are T = 0 was found in [4] (see Appendix 2). Using the results of [4], we ᏺ τ τ ᏺ τ τ ᏺ τ τ + ( 1 Ð 2) + ( 1 Ð 4) + ( 2 Ð 4) write the expression for δE˜ in the form + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) )  1 ξ ξ ξ 3 4 1 3 2 3 δ ˜ 2 3 d 1d 2d 3 Ð E = Dg 4ln2 Ð 3g ∫(---ξ------ξ------)----ξ------ξ------Φ τ τ Φ τ τ Φ τ τ Φ τ τ  1 + 2 ( 2 + 3) Ð ( 2 Ð 4) ( 3 Ð 4) ( 1 Ð 3) ( 1 Ð 2) 0 1 × ( ᏺ τ τ ᏺ τ τ ) 2 + ( 1 Ð 2) + ( 3 Ð 4) 4 ξ ξ ξ ξ 10 + g ∫d 1d 2d 3d 4 (---ξ------ξ------)----ξ------ξ------ξ------ξ----- 1 + 2 ( 1 + 3)( 1 + 4) Φ τ τ Φ τ τ Φ τ τ Φ τ τ 0 Ð ( 1 Ð 4) ( 2 Ð 4) ( 2 Ð 3) ( 1 Ð 3) 4  Ð (---ξ------ξ------)----ξ------ξ------ξ------ξ----- (17) × ( ᏺ τ τ ᏺ τ τ ) ] 1 + 2 ( 1 + 3)( 3 + 4) 2 + ( 1 Ð 4) + ( 2 Ð 3)  .  4 Ð (---ξ------ξ------)----ξ------ξ------ξ------ξ------ξ------ξ------To establish the existence of a second energy scale 1 + 2 ( 1 + 3)( 1 + 2 + 3 + 4) in the Kondo problem, it is sufficient to known the par-  tition function Z up to terms of fourth order. For this 1 + (---ξ------ξ------)----ξ------ξ------ξ------ξ------ξ------ξ------. reason, we shall present in Appendix 1 an expression 1 + 2 ( 3 + 4)( 1 + 2 + 3 + 4)  for the fifth-order correction Z5 to the partition func- Calculating the integrals in the third-order terms in tion. equations (16) and (17), we obtain The expressions (10), (14), and (15) make it possi- 1 ξ ξ ξ 2 ble to find the correction arising in the free energy d 1d 2d 3 2 π because of the interaction up to terms of fourth order ∫(---ξ------ξ------)----ξ------ξ------= 2ln 2 + -----, 1 + 2 ( 1 + 3) 6 in g, inclusively. The correction to the free energy con- 0 ∞ tains a large term that is proportional to the cutoff x ( ) dx dy(1 Ð eÐx)(1 Ð eÐy)(1 Ð eÐ x Ð y ) energy D and depends on the interaction constant g. ------(18) This contribution is ordinarily treated as a shift δE of ∫ x ∫ y x Ð y the ground-state energy and is assumed to be tempera- 0 0 ture-independent to all orders in the coupling constant. ∞ π2 π2 1  Then, the quantity δE so determined should be identical = ----- + ----- Ð 4 ∑ ------ . to all orders in g (for the same cutoff of the matrix ele- 6  3 2nn2 ments) to the quantity δE, which is the shift of the n = 1 ground state and is found at T = 0. Since ∞ We shall show that this agreement occurs only in 2 π 1 2 second and third orders in g. In fourth order in the cou------Ð 4 ∑ ------= 2ln 2, (19) pling constant g, there arises a difference that leads to 3 2nn2 the second energy scale in the Kondo problem. We shall n = 1 prove this assertion. From equations (10), (14), and the expressions for δE and δE˜ are identical in the sec- (15) we find the following expression for δE: ond and third orders of perturbation theory. In fourth order we find from equations (16) and (17) that  ÐδE = Dg24ln2 Ð(δE˜ Ð δE) = 4g4 D  1  1 ∞ x × dξ dξ dξ dξ ------Ðx Ðy Ð(x Ð y) ∫ 1 2 3 4 (ξ + ξ )(ξ + ξ )(ξ + ξ ) 3 dx dy(1 Ð e )(1 Ð e )(1 Ð e )  1 2 1 3 1 4 Ð 3g ------0 ∫ x ∫ y(x Ð y) 0 0 (16) 1 ------∞ Ð (ξ ξ ) ξ ξ ξ ξ (20) Ðx Ðy Ðz Ð(x + y + z) 1 + 2 ( 1 + 3)( 3 + 4) 4 (1 Ð e )(1 Ð e )(1 Ð e )(1 Ð e ) +g 6 dxdydz------∫ xyz(x + y + z) 1 0 Ð (---ξ------ξ------)----ξ------ξ------ξ------ξ------ξ------ξ------1 + 2 ( 1 + 3)( 1 + 2 + 3 + 4) ∞ Ð(x + y) 2 Ð(y + z) 2  (1 Ð e ) (1 Ð e ) 1  Ð 3∫dxdydz------. + ------. (x + y)2(y + z)2  (ξ + ξ )(ξ + ξ )(ξ + ξ + ξ + ξ ) 0 1 2 3 4 1 2 3 4 

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The second and third integrals in equation (20) are investigate the higher-order terms in the perturbation related by the simple relation theory. We shall show below that in the presence or absence of a magnetic field the anomalous terms appear 1 ξ ξ ξ ξ d 1d 2d 3d 4 only in eighth order of perturbation theory [8]. ∫(---ξ------ξ------)----ξ------ξ------ξ------ξ------ξ------ξ------1 + 2 ( 1 + 3)( 1 + 2 + 3 + 4) 0 (21) 4. ANOMALOUS TERMS 1 ξ ξ ξ ξ IN THE FREE ENERGY 1 d 1d 2d 3d 4 = --∫(---ξ------ξ------)----ξ------ξ------ξ------ξ-----. 2 1 + 3 ( 1 + 2)( 3 + 4) Anomalous terms arise only in eighth order of per- 0 turbation theory, irrespective of whether or not an Using the relation (21) and simple transformations we external magnetic field is present. On the other hand, it write equation (20) in the form is much easier to investigate the perturbation-theory series for H = 0. For this reason, since we intend to  1 investigate the eight-order terms of the perturbation (δ ˜ δ ) 4 31 + x Ð E Ð E = 4g D∫dxln ------ theory, we shall confine our attention in this section to  x the case H = 0. 0 (22) 1 For H = 0 the expression for the partition function Z, 3 ln((1 + x) ⁄ x)ln((1 + y) ⁄ y)  up to terms of ninth order in g, can be written in the Ð --∫dxdy------+ I . 2 x + y  form 0 e i { 2 3 4 5 6 The integral I in equation (22) is determined by the Z = Z0Z0 2g J0 Ð 3g f 3 + g f 4 + g f 5 + g f 6 expression 7 8 9 } e i { 2 + g f 7 + g f 8 + g f 9 = Z0Z0 exp 2g J0 (26) 1 ξ ξ ξ ξ 3 4 5 6 7 8 9 d 1d 2d 3d 4 Ð 3g f + g F + g F + g F + g F + g F + g F }. I = ∫(---ξ------ξ------)----ξ------ξ------ξ------ξ------ξ------ξ------3 4 5 6 7 8 9 1 + 2 ( 3 + 4)( 1 + 2 + 3 + 4) 0 (23) From equation (26) we find a relation between the ∞ functions F and f : 1 i i = 40ln2 Ð 24ln3 + 2 ∑ ------. 2 n 2 n = 1 n 2 F4 = f 4 Ð 2J0, It is convenient to calculate the first two integrals in F = f + 6 f f , equation (22) together: 5 5 2 3

1 1 9 2 8 3 1 + x 3 dxdy 1 + x 1 + y F6 = f 6 Ð 2J0 f 4 Ð -- f 3 + -- f 2, dxln 3 ------Ð ------ln ------ln ------2 3 ∫  x  2∫ x + y  x   y  0 0 F = f Ð 2J f + 3 f f Ð 12J2 f , ∞ 7 7 0 5 3 4 0 3 (27) 3 dxlnxln(x ⁄ 2) = ln 2 Ð 6 ------(24) F8 = f 8 Ð 2J0 f 6 ∫ x(x Ð 1)(x Ð 2) 2 1 2 2 2 4  ∞  Ð -- f 4 + 3 f 3 f 5 + 4 f 2 f 4 + 18J0 f 3 Ð 4J0, 3 1 2 = Ð2ln 2 Ð 6ζ(3) Ð 2 ∑ ------ ,  2nn3 n = 1 F9 = f 9 Ð 2J0 f 7 + 3 f 3 f 6 where ζ(x) is the Riemann zeta function. From + 4J2 f Ð 12J f f Ð 9 f 2 + 24J3 f Ð f f . equations (22)–(24) we ﬁnd 0 5 0 3 4 3 0 3 4 5 ∞ In equations (27) for the functions F6 and F8 we express 4  1 the functions f in terms of F . The result is Ð(δE˜ Ð δE) = 4g D40ln2 Ð 24ln3 + 2 ∑ ------4, 5 4, 5  n222n n = 1 9 2 4 3 (25) F6 = f 6 Ð 2J0F4 Ð -- f 3 Ð -- f 2,  ∞  2 5 3 1  4 Ð 2ln 2 Ð 6ζ(3) Ð 2 ------  = g D × 3.8506. 2 ∑ n 3 (28)    F8 = f 8 Ð 2J0F6 Ð 9J0 f 3 n = 1 2 n 2 4 1 2 2 Ð --J + 3 f F Ð --F Ð 2J F . The fact that in equation (25) the right-hand side is dif- 3 0 3 5 2 4 0 4 ferent from zero means that in the Kondo problem there exists a second characteristic energy scale T0 such that To simplify the formulas and make them easier to Ӷ Ӷ ε Tk T0 F. To determine this scale it is necessary to interpret, we switch to a diagrammatic representation

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 148 OVCHINNIKOV, DYUGAEV of the equations for Fi. From equations (14) and (26) Using equations (28)–(34) we ﬁnd the following we ﬁnd representation for the function F6:

6 6 f 3 = τ τ F6 = + ∑ + ∑ Ð 2 Ð 9 ∑ 1/T 1 2 (29) 3 = dτ dτ dτ Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ ). ∫ 1 ∫ 2 ∫ 3 1 2 2 3 1 3 + 4 + 10 0 0 0 (35)   Each line in a diagram represents Φ(τ), each vertex cor- + 6 +  τ τ responds to the time i, and all times i are ordered in accordance with the diagram. Two lines always meet at   each point. From equations (15) and (27) we find Ð 2 +  . F = ᮀ Ð 3 . (30) 4 All diagrams in equation (35) are irreducible. In equation (30) a square ᮀ represents a sum of all con- Therefore, no anomalous terms appear in the free nected diagrams in equation (15) which are constructed energy in sixth order in the coupling constant. The from single lines: equations (28) and (35) make it possible to find the anomalous contribution to the function F8. The irreduc- ᮀ ible diagrams do not contain an anomalous contribu- = 10 Ð 4 Ð 4 . (31) tion. For this reason, the right-hand side of equation (28) for F should be reprojected on all possible reduc- The second term in equation (30) consists of double 8 ible diagrams contained in f8 or in any other term on the lines. As an example, we give its analytic expression: right-hand side of equation (28). τ τ τ 1/T 1 2 3 To investigate the function F8 it is necessary to find a 2 2 τ τ τ τ Φ τ τ Φ τ τ representation for the function F5 that is similar to equa- = ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 ( 1 Ð 3) ( 2 Ð 4). tion (35) for the function F6. Such a representation is 0 0 0 0 (32) obtained in Appendix 1 (equation (A.4)).

We shall also require the diagrammatic representa- Analysis shows that all reducible diagrams in F8, 2 3 4 with the exception of diagrams of the form tion of the quantities J0 , J0 , and J0 . As an example, we have + , (36) 2 ≡ [ ] J0 2 + + . (33) cancel. As a result, we ﬁnd the following expression for δᏲ the anomalous contribution to the free energy an: We shall also introduce the following notations: τ τ is the set of all irreducible diagrams in f that are con- 1/T 1 2 6 δᏲ 8 τ τ τ 6 an = 24J0Tg ∫ d 1 ∫ d 2 ∫ d 3 structed only with single lines; ∑3 is the set of all irre- 0 0 0 ducible diagrams in f6 that are constructed only with τ τ τ (37) 6 3 4 5 double lines; ∑ is the set of all irreducible dia- × τ τ τ Φ2 τ τ Φ2 τ τ Φ2 τ τ ∫ d 4 ∫ d 5 ∫ d 6 ( 1 Ð 4) ( 2 Ð 5) ( 3 Ð 6). grams in f that are constructed with one double and 6 0 0 0 four single lines; and, is the set of all diagrams in Substituting the explicit expression (8) for the func- equation (31) in which one or several vertices are Φ τ encompassed by a double line (without all vertices tion ( ) into equation (37), we obtain [8] being encompassed). δᏲ 8 an = 24J0 Dg ∞ We also introduce the quantity ∑ using the Ð(x + y + z) 2 (38) I1(x + y)I1(y + z)(1 Ð e ) relation × ∫dxdydz------, (x + y + z)2 0 1 2 -- f = where 2 3 (34) ∞ dx( Ðx)2 + + + ∑ . I1(a) = ∫----- 1 Ð e . (39) x2 a

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In order of magnitude, the anomalous contribution × [ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) to the free energy is 1 2 2 3 3 4 4 5 + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) δᏲ ∼ 8 2 ⁄ 1 5 1 3 1 4 2 4 an g D T. (40) + ᏺ(τ Ð τ ) + ᏺ((τ Ð τ ) + (τ Ð τ )) + ᏺ(τ Ð τ ) Comparing the corrections arising in the specific heat 2 5 1 2 3 4 3 5 from the fourth-order terms (in equation (15)) and + ᏺ((τ Ð τ ) + (τ Ð τ )) + ᏺ((τ Ð τ ) + (τ Ð τ )) eighth-order terms (equation (40)) shows that the sec- 1 2 3 5 1 3 4 5 ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) ] ond characteristic energy scale T0 in the Kondo prob- + ( 1 Ð 2 + 4 Ð 5 ) + ( 2 Ð 3 + 4 Ð 5 ) lem is, in order of magnitude, Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ + ( 1 Ð 5) ( 1 Ð 3) ( 2 Ð 3) ( 2 Ð 4) ( 4 Ð 5) T ∼ g2 D. (41) 0 × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ ( 1 Ð 5) + ( 1 Ð 4) + ( 4 Ð 5) Ð ( 2 Ð 3) This value of T0 is identical to the expression obtained (ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) ) ] in [6] in the self-consistent field approximation. The Ð ( 1 Ð 3 + 4 Ð 5 ) Ð ( 1 Ð 2 + 4 Ð 5 ) point T seems to be the crossover from perturbation 0 Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ theory to strong coupling. Perturbation theory is valid + ( 1 Ð 2) ( 1 Ð 3) ( 3 Ð 5) ( 2 Ð 4) ( 4 Ð 5) ӷ only in the region T T0. Strong coupling is realized in × [ᏺ τ τ ᏺ τ τ ᏺ (τ τ ) (τ τ ) ] Ӷ ( 1 Ð 2) + ( 4 Ð 5) + ( 1 Ð 2 + 4 Ð 5 ) the region T T0. Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ In Appendix 2 we show that a magnetic field does + ( 1 Ð 2) ( 1 Ð 3) ( 3 Ð 4) ( 4 Ð 5) ( 2 Ð 5) not lead to the appearance of anomalous terms in sixth × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ order of perturbation theory. ( 3 Ð 4) + ( 4 Ð 5) + ( 3 Ð 5) Ð ( 1 Ð 2) ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) ] Ð ( 1 Ð 3 + 4 Ð 5 ) Ð ( 2 Ð 3 + 4 Ð 5 ) ACKNOWLEDGMENTS + Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ ) We are grateful to P. Fulde for helpful remarks and 1 2 2 3 3 5 4 5 1 4 × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ for encouragement. The bulk of this work was per- ( 1 Ð 2) + ( 2 Ð 3) + ( 1 Ð 3) Ð ( 4 Ð 5) formed at the Max-Planck-Institute for the Physics of ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) ] Complex Systems (Dresden). Ð ( 1 Ð 2 + 3 Ð 4 ) Ð ( 1 Ð 2 + 3 Ð 5 ) Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ + ( 1 Ð 2) ( 2 Ð 5) ( 1 Ð 4) ( 3 Ð 4) ( 3 Ð 5) APPENDIX 1 × [ᏺ τ τ ᏺ τ τ ᏺ (τ τ ) (τ τ ) ] We present an expression for the fifth-order correc- ( 1 Ð 2) + ( 3 Ð 4) + ( 1 Ð 2 + 3 Ð 4 ) tion to the partition function (equation (26)) in an exter- + Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ ) nal magnetic field: 1 3 2 3 2 5 1 4 4 5 τ τ τ τ × [ᏺ τ τ ᏺ τ τ 1/T 1 2 3 4 ( 2 Ð 3) + ( 4 Ð 5) (A.1) f = dτ dτ dτ dτ dτ ᏺ (τ τ ) (τ τ ) ] } 5 ∫ 1 ∫ 2 ∫ 3 ∫ 4 ∫ 5 + ( 2 Ð 3 + 4 Ð 5 ) 0 0 0 0 0 τ τ τ τ 1/T 1 2 3 4 × {Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ ( 1 Ð 5) ( 1 Ð 4) ( 2 Ð 3) ( 3 Ð 4) ( 2 Ð 5) τ τ τ τ τ Ð ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 ∫ d 5 × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ 0 0 0 0 0 ( 2 Ð 3) + ( 2 Ð 4) + ( 3 Ð 4) Ð ( 1 Ð 5) 2 ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) ] × {Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ )Φ (τ Ð τ ) Ð ( 1 Ð 2 + 3 Ð 4 ) Ð ( 2 Ð 3 + 4 Ð 5 ) 1 2 2 5 1 5 3 4 Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ × [ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ᏺ(τ Ð τ ) + ( 1 Ð 5) ( 2 Ð 5) ( 2 Ð 4) ( 1 Ð 3) ( 3 Ð 4) 1 2 2 5 1 5 ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) × [ᏺ τ τ ᏺ τ τ ᏺ (τ τ ) (τ τ ) ] + ( 1 Ð 3 + 4 Ð 5 ) + ( 1 Ð 2 + 3 Ð 4 ) ( 1 Ð 5) + ( 3 Ð 4) + ( 1 Ð 3 + 4 Ð 5 ) Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ + ᏺ((τ Ð τ ) + (τ Ð τ )) ] + ( 1 Ð 5) ( 1 Ð 2) ( 2 Ð 4) ( 3 Ð 4) ( 3 Ð 5) 2 3 4 5 × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ ( 1 Ð 5) + ( 1 Ð 2) + ( 2 Ð 5) Ð ( 3 Ð 4) + ( 1 Ð 3) ( 1 Ð 5) ( 3 Ð 5) ( 2 Ð 4) ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) ] × [ᏺ(τ Ð τ ) Ð ᏺ((τ Ð τ ) + (τ Ð τ ))] Ð ( 1 Ð 2 + 4 Ð 5 ) Ð ( 1 Ð 2 + 3 Ð 5 ) 1 5 1 2 4 5 Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ + ( 1 Ð 5) ( 1 Ð 4) ( 2 Ð 4) ( 2 Ð 3) ( 3 Ð 5) + ( 1 Ð 2) ( 2 Ð 3) ( 1 Ð 3) ( 4 Ð 5) × [ᏺ τ τ ᏺ τ τ ᏺ (τ τ ) (τ τ ) ] × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ( 1 Ð 5) + ( 2 Ð 3) + ( 1 Ð 2 + 3 Ð 5 ) ( 1 Ð 2) + ( 2 Ð 3) + ( 1 Ð 3) Φ τ τ Φ τ τ Φ τ τ Φ τ τ Φ τ τ ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) Ð ( 1 Ð 5) ( 1 Ð 2) ( 2 Ð 3) ( 3 Ð 4) ( 4 Ð 5) + ( 1 Ð 2 + 4 Ð 5 ) + ( 1 Ð 3 + 4 Ð 5 )

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ᏺ (τ τ ) (τ τ ) ] From equations (27) and (A.2) we also find an + ( 1 Ð 3 + 4 Ð 5 ) expression for the function F5 in a zero magnetic field: Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ + ( 1 Ð 2) ( 1 Ð 4) ( 2 Ð 4) ( 3 Ð 5) × [ᏺ τ τ ᏺ (τ τ ) (τ τ ) ] ᮀ ( 1 Ð 2) Ð ( 1 Ð 2 + 3 Ð 4 ) F5 = + 6 + (A.4) Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ + ( 3 Ð 4) ( 4 Ð 5) ( 3 Ð 5) ( 1 Ð 2) + + + . × [ᏺ τ τ ᏺ τ τ ᏺ τ τ ( 3 Ð 4) + ( 4 Ð 5) + ( 3 Ð 5) + ᏺ((τ Ð τ ) + (τ Ð τ )) + ᏺ((τ Ð τ ) + (τ Ð τ )) 1 2 4 5 1 2 3 4 APPENDIX 2 + ᏺ((τ Ð τ ) + (τ Ð τ )) ] 1 2 3 5 In Appendix 2 we show that a magnetic field does + Φ(τ Ð τ )Φ(τ Ð τ )Φ(τ Ð τ )Φ2(τ Ð τ ) not lead to the appearance of anomalous terms in the 2 4 4 5 2 5 1 3 free energy in sixth order in the coupling constant g. To × [ᏺ(τ Ð τ ) Ð ᏺ((τ Ð τ ) + (τ Ð τ ))] check this assertion, we write the sixth-order correction 4 5 1 3 4 5 δᏲ 6 to the free energy in the form Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ + ( 1 Ð 4) ( 4 Ð 5) ( 1 Ð 5) ( 2 Ð 3) δᏲ = ÐTg6F (H), (A.5) × [ᏺ τ τ ᏺ τ τ ᏺ τ τ 6 6 ( 1 Ð 4) + ( 4 Ð 5) + ( 1 Ð 5) ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) where the function F6(H) is determined by an equation + ( 1 Ð 2 + 3 Ð 4 ) + ( 1 Ð 2 + 3 Ð 5 ) of the form (27) in a magnetic field ᏺ (τ τ ) (τ τ ) ] + ( 2 Ð 3 + 4 Ð 5 ) 1( δ )3 1 2 ( δ )ᮀ F6 = Ð-- 2J0 + J H Ð --J3(H) Ð 2J0 + J(H) Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ 6 2 + ( 2 Ð 3) ( 3 Ð 4) ( 2 Ð 4) ( 1 Ð 5) τ τ τ × [ᏺ τ τ ᏺ τ τ ᏺ τ τ 1/T 1 2 3 ( 2 Ð 3) + ( 3 Ð 4) + ( 2 Ð 4) Ð (2J + δJ ) dτ dτ dτ dτ ᏺ (τ τ ) (τ τ ) ᏺ (τ τ ) (τ τ ) 0 H ∫ 1 ∫ 2 ∫ 3 ∫ 4 + ( 1 Ð 2 + 4 Ð 5 ) + ( 1 Ð 3 + 4 Ð 5 ) 0 0 0 0 ᏺ (τ τ ) (τ τ ) ] + ( 1 Ð 2 + 3 Ð 5 ) × {[ᏺ (τ τ ) (τ τ ) ᏺ τ τ ᏺ τ τ ] ( 1 Ð 2 + 3 Ð 4 ) Ð ( 1 Ð 2) ( 3 Ð 4) Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ + ( 1 Ð 3) ( 1 Ð 4) ( 3 Ð 4) ( 2 Ð 5) × Φ2 τ τ Φ2 τ τ ( 1 Ð 2) ( 3 Ð 4) × [ᏺ(τ Ð τ ) Ð ᏺ((τ Ð τ ) + (τ Ð τ ))] (A.6) 3 4 2 3 4 5 [ᏺ (τ τ ) (τ τ ) ᏺ τ τ ᏺ τ τ ] + ( 1 Ð 2 + 3 Ð 4 ) Ð ( 1 Ð 4) ( 2 Ð 3) Φ τ τ Φ τ τ Φ τ τ Φ2 τ τ + ( 2 Ð 3) ( 3 Ð 5) ( 2 Ð 5) ( 1 Ð 4) × Φ2 τ τ Φ2 τ τ × [ᏺ τ τ ᏺ (τ τ ) (τ τ ) ] } ( 1 Ð 4) ( 2 Ð 3) ( 2 Ð 3) Ð ( 1 Ð 2 + 3 Ð 4 ) . [ᏺ τ τ ᏺ τ τ ᏺ τ τ ᏺ τ τ ] In a zero magnetic field the expression for the func- + ( 1 Ð 3) + ( 2 Ð 4) + ( 1 Ð 3) ( 2 Ð 4) tion f5 simplifies substantially and its diagrammatic representation is presented below: × Φ2 τ τ Φ2 τ τ } ( 1 Ð 3) ( 2 Ð 4) + f 6(H). δ δ f = ᮀ Ð 6 + In equation (A.6) the function JH is J(H) = JH Ð J0, 5 and the functions J (H) and f (H) are related with the (A.2) 3 6 third- and sixth-order corrections to the partition func- + + + . tion by the relations

e i 3 e i 6 In equation (A.2) the symbol ᮀ represents the set of Z3 = ÐZ0Z0g J3(H), Z6 = Z0Z0g f 6(H). (A.7) irreducible diagrams appearing in f5 (equation (A.1)) and consisting only of single lines. From equation (A.1) The square in equation (A.6) must be removed in a we find its diagrammatic representation in a zero mag- magnetic field. Its expression is determined by the last netic field: term in equation (15). We have shown above (see equation (35)) that in a ᮀ = 3 + 3 Ð 15 (A.3) zero magnetic field the function F6(0) does not contain anomalous terms. We shall show that there are also no + 3 + 3 + 3 . anomalous terms in F6(H). Retaining only the anoma-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 BREAKDOWN OF RENORMALIZABILITY IN THE KONDO PROBLEM 151 lous terms on the right-hand side of equation (A.6), we + * + * + * + * put the expression for F6(H) in the form τ τ τ * * 1/T 1 2 3 + + * + ∼ 2δ τ τ τ τ F6(H) Ð2J0 J H Ð 2J0 ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 * * 0 0 0 0 + + × {Φ2(τ Ð τ )Φ2(τ Ð τ ) 1 2 3 4 * + + + × [δᏺ((τ Ð τ ) + (τ Ð τ )) Ð δᏺ(τ Ð τ ) 1 2 3 4 1 2 * * 2 2 Ð δᏺ(τ Ð τ ) ] + Φ (τ Ð τ )Φ (τ Ð τ ) * 3 4 1 4 2 3 + + * + + × [δᏺ (τ τ ) (τ τ ) δᏺ τ τ ( 1 Ð 4 Ð 2 Ð 3 ) Ð ( 1 Ð 4) * * τ τ τ 1/T 1 2 3 (A.8) + + * + + δᏺ τ τ ] } τ τ τ τ Ð ( 2 Ð 3) + 4J0 ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 * * * 0 0 0 0 + * + + * . × Φ2(τ Ð τ )Φ2(τ Ð τ )[δᏺ(τ Ð τ ) + δᏺ(τ Ð τ )] 1 3 2 4 1 3 2 4 * τ τ τ 1/T 1 2 3 * δ τ τ τ τ Φ2 τ τ Φ2 τ τ In equation (A.10) the symbol represents a + 3 J H ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 ( 1 Ð 2) ( 2 Ð 4) double line with the factor δᏺ(τ) 0 0 0 0 * = Φ2(τ)δᏺ(τ). (A.11) 1 2 2 + ( f (H) Ð f (0)) Ð --(J (H) Ð J (0)) 6 6 2 3 3 We also present an expression for the anomalous part of the function f (H) consisting of blocks only with δ[( δ )ᮀ] 6 Ð 2J0 + J(H) . double lines: In equation (A.8) the function δᏺ(τ) is given by  ( ) * δᏺ(τ) = ᏺ(τ) Ð 1. (A.9) f 6(H) Ð f 6(0) an = 4  The symbol δ in front of the last term in equation (A.9) means that its value for H = 0 should be subtracted from * *  the expression in parentheses. + +  In the subsequent analysis each term in equation (A.8)  is represented as a sum of time-ordered diagrams. For example, we have τΦ2 τ τ Φ2 τ τ Φ2 τ τ + ∫d ( 1 Ð 2) ( 3 Ð 4) ( 5 Ð 6)

1 2 --J δJ = ∗ + ∗ × {[δᏺ((τ Ð τ ) + (τ Ð τ ) + (τ Ð τ )) 2 0 H 1 2 3 4 5 6 δᏺ τ τ δᏺ τ τ δᏺ τ τ ] Ð ( 1 Ð 2) Ð ( 3 Ð 4) Ð ( 5 Ð 6) + * + * + * [δᏺ (τ τ ) (τ τ ) δᏺ τ τ + ( 1 Ð 2 + 3 Ð 4 ) Ð ( 1 Ð 2) * + * + * + δᏺ τ τ ] [δᏺ (τ τ ) (τ τ ) Ð ( 3 Ð 4) + ( 1 Ð 2 + 5 Ð 6 ) * * * δᏺ τ τ δᏺ τ τ ] + + + Ð ( 1 Ð 2) Ð ( 5 Ð 6) * [δᏺ (τ τ ) (τ τ ) δᏺ τ τ + * + * + + ( 3 Ð 4 + 5 Ð 6 ) Ð ( 3 Ð 4) * *  δᏺ τ τ ] } * + + Ð ( 5 Ð 6) + 4  * * * + + +  + * + *  * + + * + * (A.10) 

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τΦ2 τ τ Φ2 τ τ Φ2 τ τ τΦ2 τ τ Φ2 τ τ Φ2 τ τ + ∫d ( 2 Ð 3) ( 1 Ð 4) ( 5 Ð 6) + ∫d ( 1 Ð 6) ( 2 Ð 5) ( 3 Ð 4) × {[δᏺ (τ τ ) (τ τ ) (τ τ ) × {[δᏺ (τ τ ) (τ τ ) (τ τ ) ( 1 Ð 4 Ð 2 Ð 3 + 5 Ð 6 ) ( 1 Ð 6 Ð 2 Ð 5 + 3 Ð 4 ) δᏺ τ τ δᏺ τ τ δᏺ τ τ ] δᏺ τ τ δᏺ τ τ δᏺ τ τ ] Ð ( 1 Ð 4) Ð ( 2 Ð 3) Ð ( 5 Ð 6) Ð ( 1 Ð 6) Ð ( 2 Ð 5) Ð ( 3 Ð 4) [δᏺ (τ τ ) (τ τ ) δᏺ τ τ [δᏺ (τ τ ) (τ τ ) δᏺ τ τ δᏺ τ τ ] + ( 1 Ð 4 Ð 2 Ð 3 ) Ð ( 1 Ð 4) + ( 2 Ð 5 Ð 3 Ð 4 ) Ð ( 2 Ð 5)Ð ( 3 Ð 4) δᏺ τ τ ] [δᏺ (τ τ ) (τ τ ) [δᏺ (τ τ ) (τ τ ) δᏺ τ τ Ð ( 2 Ð 3) + ( 1 Ð 4 + 5 Ð 6 ) + ( 1 Ð 6 Ð 2 Ð 5 ) Ð ( 1 Ð 6) δᏺ τ τ δᏺ τ τ ] δᏺ τ τ ] [δᏺ (τ τ ) (τ τ ) Ð ( 1 Ð 4) Ð ( 5 Ð 6) Ð ( 2 Ð 5) + ( 1 Ð 6 Ð 3 Ð 4 ) [δᏺ (τ τ ) (τ τ ) δᏺ τ τ δᏺ τ τ δᏺ τ τ ] } + ( 2 Ð 3 + 5 Ð 6 ) Ð ( 2 Ð 3) Ð ( 1 Ð 6) Ð ( 3 Ð 4)

 *  δᏺ τ τ ] } * Ð ( 5 Ð 6) + 4 +  + * + *  

*  * *  + + *  + * + + .  

τΦ2 τ τ Φ2 τ τ Φ2 τ τ τ + ∫d ( 1 Ð 2) ( 3 Ð 6) ( 4 Ð 5) In equation (A.12) the symbol ∫d represents a time-ordered integral × {[δᏺ (τ τ ) (τ τ ) (τ τ ) ( 1 Ð 2 + 3 Ð 6 Ð 4 Ð 5 ) τ τ τ τ τ (A.12) 1/T 1 2 3 4 5 δᏺ τ τ δᏺ τ τ δᏺ τ τ ] Ð ( 1 Ð 2) Ð ( 3 Ð 6) Ð ( 4 Ð 5) τ τ τ τ τ τ τ ∫d = ∫ d 1 ∫ d 2 ∫ d 3 ∫ d 4 ∫ d 5 ∫ d 6. [δᏺ (τ τ ) (τ τ ) δᏺ τ τ 0 0 0 0 0 0 + ( 1 Ð 2 + 3 Ð 6 ) Ð ( 1 Ð 2) The remaining anomalous blocks consisting of double δᏺ τ τ ] [δᏺ (τ τ ) (τ τ ) Ð ( 3 Ð 6) + ( 1 Ð 2 + 4 Ð 5 ) lines in equation (A.8) are quite simple, and we shall δᏺ τ τ δᏺ τ τ ] not write them out explicitly. It follows from equations Ð ( 1 Ð 2) Ð ( 4 Ð 5) (A.8), (A.10), and (A.12) that all anomalous terms in δᏲ + [δᏺ((τ Ð τ ) Ð (τ Ð τ )) Ð δᏺ(τ Ð τ ) the function 6(H) that contain only double lines can- 3 6 4 5 3 6 cel. The anomalous terms containing a square ᮀ or  can be analyzed quite simply. These terms in δᏺ τ τ ] } Ð ( 4 Ð 5) + 4 * equation (A.8) also all cancel. We have thereby shown  that there are no anomalous terms in the function F6(H).

 + * + ++ * REFERENCES  1. A. A. Abrikosov and A. A. Migdal, J. Low Temp. Phys. 3, 519 (1970). + dτΦ2(τ Ð τ )Φ2(τ Ð τ )Φ2(τ Ð τ ) 2. A. M. Tsvelick and P. B. Wigmann, Adv. Phys. 32, 453 ∫ 1 6 2 3 4 5 (1983). × {[δᏺ((τ Ð τ ) Ð (τ Ð τ ) Ð (τ Ð τ )) 3. N. Andrei and K. Furuya, Physics 55, 331 (1983). 1 6 2 3 4 5 4. Yu. N. Ovchinnikov and A. M. Dyugaev, Zh. Éksp. Teor. Ð δᏺ(τ Ð τ ) Ð δᏺ(τ Ð τ ) Ð δᏺ(τ Ð τ ) ] Fiz. 115, 1263 (1999) [JETP 88, 696 (1999)]. 1 6 2 3 4 5 5. W. Felsch, Zeitschrift fur Physik B 29, 212 (1978). + [δᏺ((τ Ð τ ) + (τ Ð τ )) Ð δᏺ(τ Ð τ ) 6. Yu. N. Ovchinnikov, A. M. Dyugaev, P. Fulde, et al., 2 3 4 5 2 3 Pis’ma Zh. Éksp. Teor. Fiz. 66, 184 (1997) [JETP Lett. Ð δᏺ(τ Ð τ ) ] + [δᏺ((τ Ð τ ) Ð (τ Ð τ )) 66, 195 (1997)]. 4 5 1 6 2 3 7. A. A. Abrikosov, A. P. Gor’kov, and I. E. Dzyaloshinskiœ, Ðδᏺ(τ Ð τ ) Ð δᏺ(τ Ð τ ) ] + [δᏺ((τ Ð τ ) Ð (τ Ð τ )) Methods of Quantum Field Theory in Statistical Physics 1 6 2 3 1 6 4 5 (Fizmatgiz, Moscow, 1962; Prentice-Hall, Englewood Ð δᏺ(τ Ð τ ) Ð δᏺ(τ Ð τ ) ] } Cliffs, NJ, 1963). 1 6 4 5 8. Yu. N. Ovchinnikov and A. M. Dyugaev, Pis’ma Zh. Éksp. Teor. Fiz. 70, 106 (1999) [JETP Lett. 70, 111  * + 4 * + * +  (1999)].   Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 153Ð159. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 175Ð181. Original Russian Text Copyright © 2000 by Mamaluœ, Chupis. SOLIDS

Surface Polaritons in a Dielectric at a Boundary with a Metal in Crossed Electric and Magnetic Fields D. A. Mamaluœ* and I. E. Chupis** Verkin Physicotechnical Institute for Low Temperatures, Ukrainian National Academy of Sciences, Khar’kov, 310164 Ukraine *e-mail: [email protected] **e-mail: [email protected] Received May 7, 1999

Abstract—The spectrum of surface polaritons in a dielectric at a boundary with an ideal metal or a superconductor in crossed constant electric and magnetic fields is studied. It is shown that the polariton spectrum possesses strong nonreciprocity (polaritons with fixed frequency propagate only in one direction; this is the rectification effect) and depends strongly on the directions of the external fields and their ratios H0 /E0. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION charge, and V0 is the volume of the unit cell. In our case, Surface polaritons, i.e., electromagnetic waves on the electric polarization P consists of electronic and surfaces and interfaces between different media, are ionic parts. The indicated dynamical energy is a scalar, being actively studied theoretically and experimentally i.e., it is present in crystals with any symmetry. (see, for example, [1]). Coupled electromagnetic and Taking account of this dynamical magnetoelectric electric polarization waves in a dielectric are called energy in the analysis of polaritons in a dielectric at a phonon polaritons. In metals and semiconductors, plas- boundary with vacuum in the presence of a constant mon polaritons result from the action of an electromag- electric (magnetic) field has shown the existence of netic wave on free charge carriers. In magnets, the cou- interesting effects. Thus, in the presence of a constant pled states of electromagnetic and spin excitations are electric field oriented in a direction normal to the sur- called magnetic polaritons (see the review in [2]). The face of the dielectric, the spectrum of phonon polaritons is different for opposite directions of this field (or linear magnetoelectric effect in magnetic crystals with ° a definite symmetry introduced its own characteristic in 180 domains of a ferroelectric), and birefringence is features into the spectrum of polariton excitations. Spe- possible in certain frequency ranges [7]. A magnetic cifically, it leads to nonreciprocity of the spectrum, so field makes possible the conversion of virtual phonon that ω(Ðk) ≠ ω(k), where k is the wave vector [3Ð5]. polaritons into real polaritons [8]. However, these new effects are weak because the relativistic magnetostatic It is known that nonreciprocity in the spectrum energy W is small. exists not only in magnets but also in nonmagnetic int crystals in the presence of an external magnetic field. Nonetheless, the effects caused by the influence of The effect of an external magnetic field on surface the dynamical magnetoelectric interaction are no polaritons has been studied with respect to its action on longer weak at the contact surface of a dielectric with an electron plasma (magnetoplasmon polaritons), i.e., an ideal metal or superconductor. in metals and semiconductors (see references in [1]). It is known that surface polaritons do not exist in a In nonmagnetic dielectrics, the interaction of the dielectric at a boundary with a metal because of “metal- electric polarization P with a magnetic field H can be lic quenching” (the requirement that the tangential described [6] by a universal scalar magnetoelectric components of the electric field be zero) [1]. As shown energy of dynamical origin: in [9], in the presence of a constant electric field the indicated dynamical magnetoelectric energy Wint lifts V this forbiddenness. It turns out that at a contact with an W =------0- P ⋅[]P × H . int mc ideal metal the penetration depth of phonon polaritons in the dielectric is inversely proportional to Wint. Since This is the interaction energy of the electric polariza- the energy W is proportional to the constant electric × int tion P with the effective electric field Eeff = Ð(1/c)v H field, the penetration depth is all the smaller, the stron- arising when a charge moves with velocity v in a mag- ger this field is. While Wint substantially determines the netic field (c is the speed of light). The vector P = penetration depth of polaritons in a dielectric, the dis- (m/V0)v is the momentum density, m is the mass of the persion relation for polaritons is essentially indepen-

154 MAMALUŒ, CHUPIS dent of the energy Wint, because this energy is so small. along the y and z axes, respectively. The term in the The dispersion law for surface polaritons has the same expression for W with the coefficient ξ is the scalar analytic form as for volume polaritons. The frequency magnetoelectric energy, mentioned above, of the ξ ρ ranges of phonon polaritons in an electric field oriented dynamical origin, = V0/mc and = m/V0. Generally in the direction of the outer and inner normals to the speaking, the polarization P contains both ionic and interface are not small and differ substantially, so that electronic contributions. In the infrared region of the the switching of the direction of the electric field indi- spectrum, especially near the characteristic ionic fre- cates that surface polaritons with this frequency are quencies, the ionic polarizability prevails, the ions “switched on” or “switched off” [9]. This large effect make the largest contribution to the dynamical magne- arises because of the contact of the dielectric with an toelectric energy, and therefore m and P are the ion ideal metal even though the magnetoelectric interaction mass and momentum, respectively. In the optical region is weak. However, if a constant magnetic field H is of the spectrum, where the electronic polarizability is applied in the plane of contact of the dielectric with an much larger than the ionic polarizability, m is the elec- ideal metal or superconductor, then the existence of tron mass and P is the electron momentum. surface polaritons is also found to be possible, and their frequency ranges (with wave vector k ⊥ H) are substan- If the dielectric is not a ferroelectric, then c1 > 0 and tially different not only for opposite directions of the the equilibrium value of the electric polarization P0 in a || magnetic field but also for opposite directions of wave constant electric field E0 z is, as follows from equa- propagation. The spectrum exhibits strong nonreciproc- tion (1), ity (in contrast to the ordinarily weak nonreciprocity E [1])—phonon polaritons with a fixed frequency propa- P = P = ------0------. gate only in one direction, i.e., rectification occurs. 0 0z ξ2ρ 2 c1 Ð H0 In the present paper, phonon polaritons of a dielectric at its boundary with an ideal metal or superconduc- For a ferroelectric, the constant c1 < 0, the anharmonic tor in the presence of a constant magnetic field applied term δP4 /4 must be taken into account in the expres- || z in the contact plane (H0 y) together with a constant sion (1), and the spontaneous polarization vector is electric field oriented in a direction normal to the con- || tact boundary (E0 z), i.e., in crossed electric and mag- ξ2ρ 2 ± c1 Ð H0 netic fields, are studied. The spectrum obtained depends P0 = P0z = Ð------. strongly not only on the orientations of the electric and δ magnetic fields but also on their ratio H0/E0. The spec- If the dielectric is a ferroelectric, we assume that there trum is characterized by strong nonreciprocity and con- is no constant electric field. A characteristic feature of sists of three well-separated frequency ranges. Breaks the crossed-field case is the existence of an equilibrium appear in the dispersion curves, and a gap appears in the value of the momentum density lower nonactivational branch. The widths of the gap, the breaks, and the frequency ranges characterized by Π Π ρξ 1 0 = 0x = Ð H0yP0z = Ð--H0P0, strong nonreciprocity depend on the ratio H0/E0 and c can be regulated over wide limits. which means that toroidal ordering with moment den- ∝ sity T P0 is induced in the dielectric by the external 2. LINEAR RESPONSE OF A DIELECTRIC fields. IN CROSSED ELECTRIC The linear response of a nonmagnetic dielectric with AND MAGNETIC FIELDS magnetic permeability µ = 1 in the field of an electro- Let us consider a dielectric occupying the half-space magnetic wave in the absence of damping and neglect- z > 0 and bounding a metal (z < 0). For definiteness, we ing spatial dispersion in the case of crossed fields can assume the dielectric to be uniaxial (z is the “easy be obtained similarly to [9]. In so doing, the existence Π axis”), though the results obtained below are general. of an equilibrium value 0 must be taken into account. χ e We write the energy density W of the dielectric in Then, the nonzero components of the electric ik and the form χ em ∂ ∂ χ me magnetoelectric ik = Pi/ hk = ( ki )* susceptibili- c c Π2 ties are as follows: W = ----1 P 2 + ----2(P 2 + P 2) + ------z x y ρ (1) 2 2 2 πω 2(ω 2 ω2) ε πχ e 4 0 e Ð Ð E ⋅ P + ξP ⋅ [P × H], 1 = 1 + 4 xx = 1 + ------, (ω 2 ω2)(ω 2 ω2) 1 Ð 2 Ð where P is the electric polarization, P is the momentum density, H = H + h, E = E + e, and e and h are the ac 2 0 0 e 4πω electric and magnetic fields. The constant magnetic ε = 1 + 4πχ = 1 + ------0------, 2 yy (ω 2 ω2) field H0 and the constant electric field E0 are applied 0 Ð

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(Ω 2 Ð ω2)(Ω 2 Ð ω2) where k Ð1 is the penetration depth of surface phonon ε = 1 + 4πχ e = ------1------2------, 0 3 zz (ω 2 ω2)(ω 2 ω2) polaritons in the dielectric. 1 Ð 2 Ð The following conditions must be satisfied at the 8πω 2ωgH interface with an ideally conducting metal: ε' = 4πiχe = Ð4πiχ e = ------0------0------, xz zx (ω 2 ω2)(ω 2 ω2) ˜ ˜ ˜ 1 Ð 2 Ð bz = bz, ht = ht, dz = dz, et = 0, (5) (2) 4πωgP (ω 2 + 3g2 H 2 Ð ω2) where d is the electric induction, b is the magnetic γ = 4πiχ em = ------0------e------0------, induction, and e and h are, respectively, the electric and 1 xy (ω 2 ω2)(ω 2 ω2) 1 Ð 2 Ð magnetic field intensities. In equation (5) the letters with an overtilde designate quantities referring to the em 4πωgP metal and the quantities labeled with an index t are the γ = 4πiχ = ------0--, 2 yx (ω2 ω 2) tangential components of the fields. For a superconduc- Ð 0 tor, in equation (5) bz = 0, but in the case at hand this 2 2 2 2 equality does not introduce additional conditions in 8πg P H (ω Ð g H ) γ = 4πχ em = ------0------0------0------0----, Maxwell’s equations, so that all results obtained also 3 zy (ω 2 ω2)(ω 2 ω2) 1 Ð 2 Ð hold for a dielectric in contact with a superconductor. Taking account of the equalities (2), (4), and (5), we π 2 can write Maxwell’s equations in the form γ πχ em 4 g P0 H0 4 = 4 yz = ------, (ω2 Ð ω 2) ω  ω  0 e ---ε' + h k + ---γ = 0, z c y 0 c 1 where ω ω 2 1 2 2 2 2  γ  ω , = --[ω + ω + 2g H e k + --- + h --- = 0, (6) 1 2 2 0 e 0 z x c 3 y c

− 2 2 2 2 2 2 2 ω  ω  + (ω Ð ω ) + 8g H (ω + ω ) ], e ---ε + h k + ---γ = 0. 0 e 0 0 e z c 3 y x c 3 2 1 2 2 2 2 Ω , = --[ω + Ω + 2g H As one can see from equations (6), only the compo- 1 2 2 0 e 0 (3) nents ez and hy of the electromagnetic field are different from zero. The dispersion law and the penetration −+ (ω 2 Ω 2)2 2 2(ω 2 Ω 2) ] 0 Ð e + 8g H0 0 + e , depth of the electromagnetic field are as follows: 2 ω ω ε'  e 2 e 2 2 (± ε γ ) ± γ ω ω ω kx = --- 3 Ð 3 , k0 = --- ------Ð 1 . (7) g = ------, 0 = ------, 0 = 0 c2, c c ε mc mV 0 3 ω 2 ω 2 Ω 2 ω 2 πω 2 Analysis of equations (7), the condition for k0 to be pos- e = 0 c1, e = e + 4 0 . itive, and taking account of the frequency dependences ω (2) makes it necessary to distinguish the four possible Here e is the excitation frequency of Pz (for a ferro- cases of orientation of the external fields H0 = Hy and electric ω 2 = Ð2ω 2 c ), and ω is the excitation fre- e 0 1 0 E0 = Ez. In what follows, it is convenient for us to use quency for the transverse components Px and Py of the instead of E0 the value P0 proportional to it and the fol- polarization in the absence of a magnetic field. For a lowing notations for possible orientations of the exter- uniaxial dielectric, which is the case considered here, nal fields: ω > ω , and a natural condition is Ω > ω , i.e., χ Ð1 (0) + > > 0 e e 0 zz I gP0 0, gH0 0; π χ Ð1 χ Ð1 χ Ð1 4 > xx (0), where zz(0) = c1 and xx(0) = c2 are the > < II gP0 0, gH0 0; static dielectric susceptibilities. < > III gP0 0, gH0 0; < < 3. SPECTRUM OF SURFACE POLARITONS IV gP0 0, gH0 0. 3.1. Dispersion Law. Penetration Depth In addition, it should be kept in mind that the gyromag-

We seek the solution of Maxwell’s equations for netic ratio g = e/mc is positive for ionic excitations and waves propagating along the boundary with a metal in negative for electronic excitations. the direction of the x (k = kx) axis in the form The dispersion laws for surface phonon polaritons for various orientations of the external ﬁelds are shown , ∝ [ ( ω ) ] e h exp i kx x Ð t Ð k0z , (4) in Fig. 1. Figure 1a corresponds to the case I, and its

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 156 MAMALUŒ, CHUPIS ω ω

(a) (b) ω~ ω~ 2 2 Ω ck Ω ck 2 = 2 = ω ω ω ω ------ω2 ------ω2 q q ------Ω ------Ω 1 1 ------ω ------ω 1 1 ω~ ω~ 1 1 ~ ~ ~ ~ kq k2 k20 k1 k1 kx k1k1 0 k2 k2 kq kx

Fig. 1. Spectra of surface polaritons in the cases I (gP0 > 0, gH0 > 0) (a) and III (gP0 < 0, gH0 > 0) (b). In the cases II (gP0 > 0, gH0 < 0) and IV (gP0 < 0, gH0 < 0) the spectra are mirror images with respect to the frequency axis of the spectra in the cases I and III, respectively. mirror image with respect to the frequency axis corre- 3.2. Lower Branch of the Spectrum sponds to the case II. Figure 1b refers to the case III, The lower branch of the spectrum lies in the fre- and its mirror image with respect to the frequency axis ω ω ω quency range < 1, where, because the ratio gH0/ 0 corresponds to the case IV. The spectrum of surface ω polaritons consists of three well-separated frequency is small, the approximate value of 1, as follows from ω ω ≈ ω Ω ω ω Ω ω ω equation (3), is ranges: < 1 e, 1 < < 2 ( 1 and 2 near 0) ω Ω ≈ Ω and > 2 e. In the IR range, for ionic excitations ω2 ω 2 ω 13 1 0 + 3 0 2 2 (g > 0) e ~ 10 rad/s. In the optical range, where elec- ω ≈ ω Ð ------g H . 1 e 2ω ω 2 ω 2 0 trons make the main contribution to the electric polar- e 0 Ð e ization (g < 0), these frequencies are at least an order of ω magnitude higher. For certain values of the frequencies As follows from equation (8), the frequency ˜ 1 is ω ω ω ˜ q and ˜ 1, 2 and the wave vectors kq = k( q), k1, 2 = real, if ω˜ Ω k( 1, 2 ), and k1, 2 = k( 1, 2) we employed the smallness 4πχ + 1 H0 < ν = ------, χ = χ (0). (9) of the ratio of the cyclotron frequency gH0 to the optical ------2χ zz ω ω Ӷ P0 frequency 1, 2: gH0/ 1, 2 1. We have approximately

2 2 2 2 2 2 Far from the ferroelectric transition temperature, the 2 ω q + Ω ω (1 + 2q ) Ω Ð ω χ ω 2 1 -----q e 0 static dielectric susceptibility ordinarily is essentially 1, q = ------2------, kq = Ð ------, ν | | ν c q ω 2 ω 2 so that ~ 1. If H0/P0 > , then the lower polariton 1 + q 0 Ð e ω branch exists in the range [0, 1], characterized by ω2 strong nonreciprocity: In the cases I and III this mode 2H0 0 q = ------, exists only for kx > 0, and in the cases II and IV it exists P0 (ω 2 ω 2)(Ω 2 ω 2) 0 Ð e e Ð 0 only for kx < 0 (see Fig. 1), i.e., rectification of the (8) polaritons occurs: They propagate only in one direc-  2 tion. The lower polariton branch also has a similar form 2 2 2  H  ω˜ ω πω  +− 0  in the absence of an electric field (P = 0). The variation 1, 2 = e + 2 0 1 1 + π------ , 0  P0  of the lower polariton mode in an electric field is a

2 2 2 2 threshold process, since ω˜ ∝ E Ð Eν arises only in ω P (Ω Ð ω ) 2Ω g P H ω 1 0 -----0 -----0------e------0------e ------0------0------0--- | | | |χÐ1νÐ1 k1 = 2 , k2 = 2 2 2 . a field E0 > Eν = H0 . The character of the 2c H0 ω c (Ω Ð ω )ω 0 e 0 0 changes in the spectrum for E0 > Eν depends on the ˜ direction of the electric field. If gP0 > 0 (the cases I and II), The expressions for k , = k(ω˜ , ) are too complicated 1 2 1 2 then a gap ω˜ , whose width increases with the electric to present here. Then can be calculated from equation (7) 1 using equations (2), (3), and (8). The sequence of the field, appears in the lower branch, while the allowed ω ω frequency range [ ˜ 1 , 1] decreases. However, if gP0 < 0 arrangement of k , k , and k˜ , is not necessarily as 1, 2 q 1 2 (the cases III and IV), then the electric field “symme- | ˜ | | | shown in the figures (though k2 > k2 ); it depends on trizes” the lower branch, adjusting the frequency inter- the ratio (H0/P0) of the magnetic and electric fields. val for kx < 0 (III) and kx > 0 (IV). In the absence of a

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 SURFACE POLARITONS IN A DIELECTRIC AT A BOUNDARY WITH A METAL 157 magnetic field (H0 = 0), the lower branch is symmetric, The plus sign in equation (12) corresponds to the radi- the spectrum is reciprocal (III and IV) or there are no ation point lying on the right-hand side (kx > 0) and a surface polaritons (I and II) [9]. A magnetic field in the minus sign corresponds to the left-hand side (kx < 0). The cases I and II creates a nonreciprocal gap mode at loca- radiation point can be moved into or out of the range Ω ω tions where there was no mode for H0 = 0, and in the [ 1, 2] by varying the ratio of the electric and mag- cases III and IV it “cuts off” the symmetric spectrum netic fields. earlier, creating frequency ranges [ω˜ , ω ] where 1 1 As one can see from Fig. 1 , reversing the direction waves propagate in only one direction. The magnitude of the magnetic field is equivalent to reversing the ω ω of the frequency interval [ ˜ 1 , 1] is direction of wave propagation. In the absence of an ω ω electric field (P0 = 0, q = 2, and k1 = 0), the polariton 2  H  mode under study exists in the cases I and III only on ω 2 ω˜ 2 πω 2  0   1 Ð 1 = 2 0 1 + π------ Ð 1 . (10) the left-hand side (kx < 0), and in the cases II and IV it  P0  exists only on the right-hand side (kx > 0). The spectrum As is evident from equation (10), it depends on the ratio is characterized by strong nonreciprocity: polaritons of the electric and magnetic fields and therefore can propagate only in one direction (rectification effect). In the cases I and II (Fig. 1a), an electric field deforms the vary over wide limits: from zero (for H0/P0 = 0) up to ω (for |H /P | ≥ ν). polariton branch, leaving it completely nonreciprocal. 1 0 0 In the cases III and IV (Fig. 1b) it symmetrizes the ω ω spectrum, additionally adjusting the ranges [ q, 2] Ω ω 3.3. Radiation Branch and, therefore, decreasing the frequency range [ 1, q] Ω ω where rectification is possible. The position of the fre- The frequency range [ 1, 2] is shown in Fig. 1 as quency ω in the range [Ω , ω ] depends on the value of being disproportionately large. Actually, as follows q 1 2 ω q (8), i.e., on the ratio H0/P0 of the fields. from equation (3), because the ratio gH0/ 1, 2 is small Ω ω ω Ω the frequencies 1 and 2 lie near 0, and the range Near the frequency 1, i.e., a zero of the permittivity Ω ω ε [ 1, 2] is small: 3, as follows from equations (6) and (7), the electric 2 2 2 2 field h = ± ε e predominates in the wave. Con- ω Ð Ω 8πg H ω gH  y 3 z -----2------1 ≈ ------0------0------∼ ------0 . (11) versely, near the characteristic frequencies ω the ω 2 2 2 2  ω  1, 2 0 (Ω ω )(ω ω ) 0 ω ω e Ð 0 0 Ð e wave is predominantly magnetic, and for = q the ratio of the fields in the wave depends on the ratio This frequency range arises only in the presence of a Ð1 hy/ez = (2H0/P0)(c2 Ð c1) of the constant fields. magnetic field, since for H0 = 0 the necessary condition ε 3 > 0 is not satisfied (see equation (7)). It lies in a previously forbidden (for H0 = 0) zone, where strong damping occurs. 3.4. Upper Branch of the Spectrum However, this frequency branch is interesting because, The upper branch of the polariton spectrum, ω Ω ∞ in contrast to the lower [0, 1] and upper [ 2, ] branches, it is a radiation branch: It can intersect with 2 2 1 ω + 3Ω 2 2 the electromagnetic mode ω = ck. This means that, in ω > Ω ≈ Ω + ------0------e-g H , 2 e Ω Ω 2 ω 2 0 principle, resonance excitation of such a polariton e e Ð 0 mode of an electromagnetic wave is possible. In the absence of an electric field (P0 = 0), the radiation point in a magnetic field and in the absence of an electric field always exists, and its frequency is ω 2 = ω 2 Ð g2 H 2 . In is characterized by strong nonreciprocity and exists r 0 0 only for k > 0 in the cases I and III and for k < 0 in the crossed fields, the point of intersection of the electro- x x ω˜ ∞ magnetic and polariton modes (the radiation point) can cases II and IV, and k2 = 0, 2 . The electric field occur for both positive and negative values of kx limits the spectrum (cases III, IV, Fig. 1b) or symme- depending on the ratio P0/H0 of the fields. Thus, in the trizes it (cases I, II, Fig. 1a). When only an electric field ω ω Ω Ω case I the position of the radiation point r depends on (H0 = 0, ˜ 2 = 2 = e) is present, the polariton spec- ε γ | | the relation 3 + 3 = Ðkx/ kx , which, for example, for trum is symmetric and exists only in the cases I and II. ω ω ω The addition of a magnetic field covers the spectrum: in the frequency 0 (i.e., when the radiation point r = 0) means the following condition for P /H : the cases I and II (Fig. 1a) it cuts out sections of the 0 0 Ω ω curve in the range [ 2, ˜ 2 ] (the width of this range is 2 2 P 3ω + ω 2 2 2 2 determined by the same expression as in equation (10)) -----0- ≈ ------0------e-( 3ω + Ω ± 3ω + ω ). (12) and adds these same sections in the cases III and IV H πω 2 0 e 0 e 0 8 0 (Fig. 1b).

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Table Although the dynamic magnetoelectric interaction Wint is weak, it produces effects which are not small for I II III IV phonon polaritons of a dielectric at the surface of con- [0, ω˜ ] 0 0 1 1 tact with an ideal metal or a superconductor in external 1 electric and magnetic fields. If only an electric field is ω ω [ ˜ 1 , 1] 1 0 1 0 present, the spectrum of surface phonon polaritons has only one (lower or upper) branch, depending on the [Ω , ω˜ ] 1 0 1 0 2 2 direction of the external electric field E0, so that when [ω˜ , ∞] 1 1 0 0 the direction of E0 is switched, the polaritons with a given 2 frequency are switched on (off) [9]. If only a magnetic field is applied to the system, then the polariton spectrum possesses three branches characterized by strong nonreci- 4. DISCUSSION procity: a polariton on any of these branches propagates We note first the following properties of the spectra only in one direction. presented in Fig. 1. When electric and magnetic fields are present together, the spectrum consists of three frequency ranges and 1. The substitution H0 ÐH0 is equivalent to k Ð k , so that the plots for the cases I and II, III and depends not only on the direction of the fields but also on x x their ratio H /P . IV are mirror images of one another. 0 0 When electric and magnetic fields are present at the 2. In the cases differing by the signs of the electric and same time, breaks appear in the frequency branches and magnetic fields simultaneously (I and IV, II and III), the ω˜ spectra mutually complement one another up to a com- a gap 1 appears in the previously nonactivational plete symmetric spectrum. For example, the spectrum lower branch. The spectrum is characterized by strong ω ω of the upper branch in the case IV is a “cutout” piece of nonreciprocity in the wide frequency ranges [ ˜ 1 , 1] the spectrum of this branch in the case I and so on. For and [Ω , ω˜ ]. The width of these ranges depends on the a fixed orientation of the external fields the transitions 2 2 ratio H /P . For example, for the tetragonal crystal I IV and II III correspond to a transition from 0 0 ε ≈ ≈ π the IR region of the spectrum, where the ionic contribu- MgF2 we have zz(0) 5 [11] (i.e., c1 ), while for tion to the polarizability predominates and therefore fields H0 = 1 T and E0 = 1 mV/m the relative width of ω Ω Ω ≈ g > 0, into the optical region where the electronic polar- the gap in the upper branch is ( ˜ 2 Ð 2)/ 2 2.6. Since izability predominates (g < 0). Here, one should bear in the widths of the gap and the breaks in the spectrum and mind the change in the order of the characteristic fre- the nonreciprocity ranges depend on the ratio H0/P0, quencies. they can be regulated by varying the intensities of the The penetration depth of the electromagnetic field in fields. This property could be helpful for producing fil- Ð1 ters, whose frequency can be tuned by means of exter- the dielectric, k (see equations (7) and (2)), can be 0 nal fields, and rectifiers for surface electromagnetic finite only in the presence of at least one field (electric waves. or magnetic). If there is no field, then k = 0, i.e., the 0 The middle branch of the polariton spectrum in the waves are volume waves. In our case of crossed fields, range [Ω , ω ] is interesting in that it can interact reso- the electric and magnetic fields make additive contribu- 1 2 tions to k , since ε' ∝ H and γ ∝ P . If these contribu- nantly with an electromagnetic wave, i.e., it is a radia- 0 0 1 0 tion branch, a fact that is very important for studying a tions have the same sign, then both fields together dis- polariton spectrum in practice. The position of the place the electromagnetic wave out of the dielectric, “radiation point” (resonance point) in crossed fields can converting it into a surface wave. The penetration depth be controlled by these external fields, since it depends decreases with increasing P0 and H0. However, if the on their ratio P0/H0 (see (12)). Unfortunately, however, signs of the terms in k0 (7) are different, then the elec- this radiation mode lies in the region of strong absorp- tric and magnetic fields act oppositely and k0 can van- tion. ish, i.e., the wave becomes a volume wave. Such a sit- We note one other possibility for using the strong Ð1 ∞ ω ω uation (k0 ) occurs for frequencies q, ˜ 1, and dependence of the polariton spectrum on the external ω˜ in Fig. 1. Even though the penetration depth of the fields that arises in crossed electric and magnetic fields. 2 In the table the number 1 (0) denotes the existence field is inversely proportional to the small magnetoelec- (absence) of surface polaritons for various orientations tric interaction energy Wint, the penetration depth can ε ε of the fields IÐIV in wide frequency ranges. The depen- be small because of the frequency dispersion of 3, ', dence of the spectrum on the direction of the external γ and 1 (7). Thus, the penetration depth of an electro- fields means that the polariton modes in 180° domains magnetic wave in the dielectric approaches zero near of ferroelectrics and magnets are different. Spectral ω Ω ω the poles of k0 (7), i.e., near the frequencies 1, 1, 2, investigations of surface polaritons in the system under Ω and 2. study make it possible to determine the relative orienta-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 SURFACE POLARITONS IN A DIELECTRIC AT A BOUNDARY WITH A METAL 159 tion of the electric polarization and the magnetization. 3. V. N. Lyubimov, Dokl. Akad. Nauk SSSR 181, 858 It is evident from the table that if the two ﬁxed frequen- (1968) [Sov. Phys. Dokl. 13, 739 (1969)]. ω cies are chosen from the pair of ranges [0, ˜ 1 ] and 4. V. A. Markelov, M. A. Novikov, and A. A. Turkhin, ω˜ ω ω˜ ω ω˜ ∞ ω˜ Ω Pis’ma Zh. Éksp. Teor. Fiz. 25, 404 (1977) [JETP Lett. [ 1 , 1], or [ 1 , 1] and [ 2 , ], or [0, 1 ] and [ 2, 25, 378 (1977)]. ω Ω ω ω ∞ ˜ 2 ], or [ 2, ˜ 2 ] and [ ˜ 2 , ], then one of the four pos- 5. V. D. Buchel’nikov and V. G. Shavrov, Zh. Éksp. Teor. sible methods of orientation of the polarization and mag- Fiz. 109, 706 (1996) [JETP 82, 380 (1996)]. netization can be determined on the basis of the presence 6. I. E. Chupis, Ferroelectrics 204, 173 (1997). or absence of a surface wave at these two frequencies. 7. I. E. Chupis and D. A. Mamaluœ, Fiz. Nizk. Temp. 24, Indeed, in this case a binary code, uniquely determining 1010 (1998) [Low Temp. Phys. 24, 762 (1998)]. this orientation, corresponds to each of the orientations IÐIV (see table). This property could be helpful in com- 8. I. E. Chupis and N. Ya. Alexandrova, J. Korean Phys. puter technology, for example, when ferromagnets are Soc. 32, 51 134 (1998). used as elements with binary (electric and magnetic) 9. I. E. Chupis and D. A. Mamaluœ, Pis’ma Zh. Éksp. Teor. memory. Fiz. 68, 876 (1998) [JETP Lett. 68, 922 (1998)]. 10. I. E. Chupis and D. A. Mamaluœ, Fiz. Nizk. Temp. 25, 1112 (1999) [Low Temp. Phys. 25, 833 (1999)]. REFERENCES 11. V. V. Bryskin, D. N. Mirlin, and I. I. Reshina, Fiz. Tver. 1. Surface Polaritons, Ed. by V. M. Agranovich and Tela 15, 1118 (1973) [Sov. Phys. Solid State 15, 760 D. L. Mills (Nauka, Moscow, 1985; North-Holland, (1973)]. Amsterdam, 1982). 2. M. I. Kaganov, N. B. Pustyl’nik, and T. I. Shalaeva, Usp. Fiz. Nauk 167, 191 (1997). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 160Ð172. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 182Ð195. Original Russian Text Copyright © 2000 by Agafonov, Manykin. SOLIDS

InsulatorÐSuperconductorÐMetal Transitions Induced by Spin Fluctuations in the Paramagnetic Phase of Doped Insulators A. I. Agafonov* and É. A. Manykin** Institute of Superconductivity and Solid-State Physics, Kurchatov Institute of Atomic Energy (All-Russia Scientiﬁc Center), Moscow, 123182 Russia *e-mail: [email protected] **e-mail: [email protected] Received May 31, 1999

Abstract—The band structure of cuprates as a doped 2D insulator is modeled assuming that the excess charge carriers are associated with the corresponding substitution atoms, and the phase diagram of the paramagnetic states as a function of the degree x of doping at zero temperature is studied. The Hamiltonian contains electronic correlations on impurity orbitals and hybridization between them and the initial band states of the insulator. It is shown that the change in the electronic structure of a doped compound includes the formation of impurity bands of distributed and localized electronic states in the initial insulator gap. It is established that in the case of one excess electron per substitution atom the spin ﬂuctuations (1) give rise to an insulator state of the doped compound for x < xthr, 1, (2) lead to a superconducting state for xthr, 1 < x < xthr, 2, and (3) decay as x > xthr, 2 increases further, and the doped compound transforms into a paramagnetic state of a “poor” metal with a high density of localized electronic states at the Fermi level. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION After it was found that the mobility of charge carriers in cuprates is comparable to the MottÐIoffeÐRegel’ Phase transitions in doped cuprates are now being criterion, it became obvious that localization effects intensively studied. As a rule, the initial compounds play an important role in these systems [15, 16]. As a possess an antiferromagnetic insulator state. It is result of this, a different direction arose in the investi- known that the magnetic state vanishes for low concen- gation of HTSC, in which HTSC materials are regarded trations of the substitution impurity [1, 2]. As the impu- as doped semiconductors [15, 17Ð20]. This approach is rity concentration increases further, insulatorÐsuper- based on the assumption that the excess charge carriers conductorÐmetal phase transitions occur in the para- are initially associated with the corresponding impurity magnetic phase. However, this paramagnetic phase atoms. An important circumstance for such an analysis exhibits unusual spin-dynamic properties, which can is that in the paramagnetic insulator phase, which is fol- contain substantial information about the orgin of the lowed immediately by the superconducting phase, the normal state and about the mechanism of superconduc- electronic properties (specifically, the conductivity) of tivity [2Ð9]. doped cuprates are determined by activated hops of charge carriers between localized states [17, 21Ð24], The construction of the theory of normal and super- which is typical for doped semiconductors. conducting states is largely based on an assumption The ARPES data for Bi2Sr2CaCu2O8 + y showed that about the nature of the excess charge carriers intro- in the normal and in the superconducting states the duced into the electronic system by the impurity electronic structure near the Fermi level is formed by ensemble. At present, most works assume that these two types of bands (see [25] and the references cited excess charge carriers (1) are free, (2) leave the inter- there). The first type corresponds to narrow bands lying mediate layers of the cuprates, which are regarded as in the range ±40 meV at the Fermi level and possessing reservoirs for charge carriers, and (3) migrate in the a well-determined electronic dispersion, i.e., these CuO2 plane [1, 2, 6, 7, 10]. The most common direction bands correspond to distributed states. Bands of the of investigations of HTSCs is based on this and consists second type have no dispersion in a wide range of val- in the fact that HTSC is due to processes occurring in ues of the wave vectors. Smoothing of the dispersion the regular 2D lattice of the CuO2 planes with variable dependence in certain regions of the spectrum can be filling [1–8, 11–14]. To describe these processes, mod- obtained on the basis of Emery’s model [2, 13, 25]. els of strongly correlated electrons, based on the Hub- However, considering the properties of cuprates in the bard model in the site representation, are used. paramagnetic insulator phase, it can be inferred [26]

INSULATORÐSUPERCONDUCTORÐMETAL TRANSITIONS 161 that these bands belong to localized states, which coex- Nd2 − xCexCuO4. For La2CuO4 + y, the excess oxygen ist with the distributed states near the Fermi level and atoms can occupy interstitial positions near the CuO2 contribute to the formation of the superconducting planes. Then all valence electrons of the O atoms can state. participate in the formation of the impurity bands. At present there is no single theory of HTSC. This In equation (1) the impurity orbital is characterized is due primarily to the fact that the theory must describe ε adequately all unusual properties of cuprates in the by two parameters, 0 and U. The latter parameter is an insulator, metallic, and superconducting phases. For important parameter of the model, and it can be esti- this reason, the approach which we have developed mated as provides one possibility for describing the insulator, 2 superconducting, and metallic states. Ӎ e U ε------, (2) optaeff 2. MODEL where aeff is the effective Bohr radius of the impurity ε The initial cuprates belong to systems of strongly orbital and opt is the high-frequency permittivity of the correlated electrons and are insulators, in which prima- compound. The estimated value of the effective radius rily the O 2p and Cu 3d orbitals form an important part for doped cuprates in an insulator state is well known: of the electronic structure near the insulator gap, due to aeff = 4–8 Å [17, 21–24]. Substituting into equation (2) charge transfer [2]. The unit cell of the compound ε Ӎ the typical value opt = 3 [28], we obtain U 0.6 eV. At La2CuO4 contains a single CuO2 plane, lying between the same time, according to the ARPES data, the typical two LaO planes. When a substitution impurity is intro- width of the main peak of the valence band, determined 3+ duced into La2 Ð xSrxCuO4, the La atoms randomly by the spectrum of single-electron states of the cuprate 2+ Ӎ replace the Sr atoms. Both valence electrons of Sr go plane, is 2Db 3 eV [25]. to the formation of valence bonds, and therefore it can be expected that a singly filled impurity orbital of acceptor It was noted above that the change in the electronic type will arise near the cuprate plane [17, 19, 27]. structure of doped cuprates should include the formation of impurity bands of both distributed and localized Since the basis of initial band states is determined states near the insulator gap. Formally, Hamiltonian (1) by the CuO2 layer, which does not change on doping, is the Anderson Hamiltonian [29] extended to an the interaction of the impurity orbitals with the band ensemble of substitution atoms randomly distributed in states of the CuO2 plane can be described by the hybrid- the initial lattice. As is well known, the Anderson ization between them. Here, for simplicity, we confine Hamiltonian for a single impurity atom possesses a our attention to single-band approximation (valence solution only for localized states in the gap. When the band) for the density of states in the CuO2 plane. Taking model is extended to an ensemble of impurities, Hamil- account of the strong anisotropy of the physical charac- tonian (1) contains solutions corresponding to the for- teristics of cuprates, the Hamiltonian for one structural mation of impurity bands of both localized and distrib- cell, not coupled with other cells by charge transfer uted electronic states in the gap. This was demonstrated along the c(z) axis, can be represented in the form [27] by solving equation (1) in the HartreeÐFock approximation [30]. The formation of narrow impurity bands of + + distributed states with a high density is due to hybridiza- H = ∑ε a σ a σ + ∑ε a σ a σ k k k 0 j j tion, which results in virtual single-electron transitions: σ σ k j (1) initial impurity site k-band state difference site k -band state, and so on. 1 + 1 + --∑Un σn , σ + ∑ {V a σ a σ + h.c.}, 2 j j Ð kj k j σ , σ As is well known, spin fluctuations are neglected in j j k the HartreeÐFock approximation. It was found [30] that where akσ and ajσ are the standard annihilation opera- in the case of a single excess electron on an impurity the tors for the initial band state of the insulator and for the paramagnetic solution (1) in this approximation always impurity orbital with number j, respectively; σ = ±1/2 corresponds to a metallic state, in which the Fermi level is the spin index; k is the 2D wave vector of an electron lies in two overlapping bands of localized and distrib- ε ε in a band state with energy k; 0 is the energy of the uted states, i.e., the insulatorÐmetal paramagnetic tran- impurity orbital; Vkj is the hybridization matrix ele- sition accompanying a change in the degree of doping ment; and, U is a seed parameter for the intracenter cannot be described in the HartreeÐFock approxima- interaction of electrons in an impurity orbital (U > 0). tion. The structure of Nd2CuO4 is similar to La2CuO4 However, the situation changes radically when the with the exception of the positions of the O atoms out spin fluctuations are taken into account. In this case it of the CuO2 planes. For “electronic doping,” the is possible to obtain insulator, superconducting, and replacement of Nd3+ atoms by Ce4+ atoms should result metallic states in the paramagnetic phase of the doped in the appearance of a singly filled donor level in insulator. This will be presented below.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 162 AGAFONOV, MANYKIN

3. PARAMAGNETIC SOLUTION σ, Ðσ ω F jj ( ) is important. These functions describe local- We introduce the normal Green’s functions ized bosons with zero spin. We introduce the notation

σσ 1 + σ, Ðσ + dω σ, Ðσ σ, Ðσ Gηη (t, t ) = Ði Nσ σ Ta˜ ησ(t)a˜ η σ (t ) N , ω ( ω ) β 1 1 1 Ð 1 1 1 0 F jj (0 ) = lim ∫------F jj ( )exp Ði t = i j . (5) + π t → 0 2 and the anomalous Green’s functions The integral equations (3)Ð(5) must be supple- ( )σσ + 1 mented by an equation for the Fermi energy µ: Fηη (t, t ) 1 1 µ ( µ ) ( ) + ( ) + ( ) 1 = exp Ð2i t N + 2 σ + σ Ta˜ ησ t a˜ η σ t1 N0 , ( ) -- ω (ω) 1 1 1 1 + x Nt = Ðπ ∫ d ImTrG , (6) σσ Ð∞ 1 Fηη (t, t ) 1 1 where Nt is the total number of electrons in the initial band of the insulator (for definiteness, the valence band = exp(2iµt) N Ta˜ η, σ(t)a˜ η σ (t ) (N + 2)σ σ , 0 1 1 1 + 1 of the CuO2 plane, hole doping) and Nim = xNt is the where the operators in the Heisenberg representation density of substitution atoms. Here it is assumed that with Hamiltonian (1) are used; η assumes the values k the impurity level is singly filled. The quantity x will be or j; µ is the chemical potential of the system. Averag- interpreted as the degree of doping. ing is performed over the spin states of the system with We assume that the impurity atoms occupy ran- the total number of particles N and N + 2 [31]. The bot- domly the equivalent sites of the crystal lattice of the tom index in the ground states |Nσ σ 〉 and 1 Ð material. In this case the order parameters (3)Ð(5) do not depend on j. For the paramagnetic phase, the spin |(N + 2)σ σ 〉 indicates the presence of excess spins + 1 indices for the parameters A, λ, and β can be dropped with a fixed total number of particles compared with [32]. The quantity A is the filling factor of a localized | 〉 | 〉 the spin state in N0 and (N + 2)0 . orbital and it is therefore real and positive. Since the Using equation (1) it is easy to obtain a closed sys- Hamiltonian (1) does not change as a result of the fol- tem of equations for the Green’s functions. The para- lowing transformation in the two spin subspaces magnetic solution of this system for the Green’s func- ( φ) ( φ) tions, averaged over a random distribution of substitu- aη, ↑ aη, ↑ exp i , aη, ↓ aη, ↓ exp Ði tion atoms, was found in [32]. This solution will be φ λ β presented below, but we shall first discuss the order with constant phase , one of the quantities (4) or (5) can be chosen to be real. For definiteness, we parameters in the model under study. Here the fixed λ particle number representation is used. choose to be real. The characteristics of the superconducting state are determined only by the modulus of the The first parameter is determined by the average fill- complex quantity β. ing number of the localized orbitals: For the impurity ensemble under consideration, the µ hybridization matrix element has the form σσ( +) 1 ω σσ(ω) G jj Ð0 = Ði-- ∫ d ImG jj = iA j, σ. (3) ( ) π V kj = V kl exp ikR j , Ð∞ In addition, the system contains two different types where Rj is a two-dimensional radius vector of the of order parameters, associated with the spin fluctua- jth substitution atom. tions [32]. One of them is determined by the spin-off- Taking account of equation (1), the solution of the diagonal normal Green’s function for localized states: closed system of equations for the Green’s functions can be represented in the form of series whose terms +∞ possess a similar structure: σ, Ðσ( +) 1 ω σ, Ðσ(ω) λ σ, Ðσ G jj 0 = Ðiπ-- ∫ d ImG jj = i j . (4) µ ∑ V V V V …V V ∏F(k ), k j1 j1k1 k1 j2 j2k2 km jm + 1 jm + 1k m { }, { } m It is easy to see that for the normal state of the system jm + 1 km the effect of the spin fluctuations is determined by λ. where F(km) are known functions that depend on the The possibility of a superconducting state in the sys- wave vectors. tem is due to the anomalous Green’s functions. Here, Taking account of V , in performing the summation the order parameter determined by the spin-nondiago- kj over {j} the result obtained must be averaged over the (+)σ, Ðσ ω nal variable site Green’s functions F jj ( ) and random distribution of the impurity atoms {Rj}. In so

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INSULATORÐSUPERCONDUCTORÐMETAL TRANSITIONS 163 doing, it is necessary to calculate the sth moments of the following type [33]: ∑ exp(i(k Ð k )R ) 1 j1 ≠ , ≠ v ( , , …, ) k1 k j1 j a Ms k1 k2 ks   = N ∑ δ(k Ð k) = 0. = … expÐi k R  , im 1 ∑∑ ∑ ∑ m nm ≠   k1 k { } { } { } n1 n2 ns m av In second order, corrections ∝x2 to the Green’s func- 〈 〉 where … av denotes averaging over all possible con- tions which are nondiagonal in the bottom indices arise. figurations of the impurity ensemble. Then, the calcula- Since, in reality x Ӷ 1, such double sums were tion of the configuration-averaged Green’s functions neglected in [32], and diagrams ∝xn with n ≥ 2 were reduces to summation of the series arising, whose sth summed only partially. The computational details can term is a polynomial of degree s in the impurity concen- be found in [32]. tration [30, 33]. We introduce the notation ( ) It is convenient to represent this solution in matrix G 0 (ω) = (ω Ð ε )Ð1, form, in which the Green’s functions which are not kk k ( ) diagonal in the bottom indices are expressed in terms of F 0 (ω) = (ω + ε Ð 2µ)Ð1, the diagonal Green’s functions [32]. There arise resid- kk k ual terms proportional to double sums of the following G˜ (ω, A, λ) = (ω Ð ε Ð U(A Ð λ))Ð1, type: ll 0 ˜ (ω, , λ, µ) (ω ε µ ( λ))Ð1 Fll A = + 0 Ð 2 + U A + . σσ ∑ V V G 1(ω). Now, we present the expressions for the site Green’s k j1 j1k1 k1 j ≠ , ≠ functions for the paramagnetic phase [32]: k1 k j1 j σσ 1 1( (ω µ, , λ, β) The first correction due to such terms to the Green’s G jj = -- Gloc ; A functions which are nondiagonal with respect to the 2 (7) σ Ð σ bottom indices is zero, since the summation over j1 and + (Ð1) 1G (ω; µ, A, Ðλ, β) ), averaging over the random distribution of impurity loc atoms give where

ω + ε Ð 2µ Ð S (λ) G = ------0------2------(8) loc (ω ε µ (λ))(ω ε (λ)) β 2 2 2(λ) + 0 Ð 2 Ð S2 Ð 0 Ð S1 Ð U Z and

( )σσ σ σ + 1 1 Ð 1 F = --((Ð1) F (ω; µ, A, λ, β) jj 2 loc (9) (ω µ, , λ, β) ) Ð Floc ; A Ð , where β∗UZ(ω; A, µ, λ; β) F = ------. (10) loc (ω ε µ (λ))(ω ε (λ)) β 2 2 2(λ) + 0 Ð 2 Ð S2 Ð 0 Ð S1 Ð U Z

In the expressions (8) and (10) S1 and S2 are the self-energy parts, which have the form

Ð1 2 [ (0) Ð1(ω) ˜ Ð1 2 ] (ω , λ, µ, β) ( λ) V kl FÐk Ð k Ð xNtT ll Gll V kl S1 ; A = U A Ð + ∑------Ð--1------Ð--1------Ð--1------Ð--1------(11) (0) (0) 2 Ð1(ω)[ ˜ Ð1 (0) ˜ Ð1 (0) 2 ] k Gkk F Ðk Ð k Ð xNtV kl T ll Gll Gkk + Fll F Ðk Ð k Ð xNtV kl and

Ð1 2 [ (0) Ð1(ω) ˜ Ð1 2 ] (ω , λ, µ, β) ( λ) V kl Gkk Ð xNtT ll Fll V kl S2 ; A = ÐU A + + ∑------Ð--1------Ð--1------Ð--1------Ð--1------, (12) (0) (0) 2 Ð1(ω)[ ˜ Ð1 (0) ˜ Ð1 (0) 2 ] k Gkk F Ðk Ð k Ð xNtV kl T ll Gll Gkk + Fll F Ðk Ð k Ð xNtV kl

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 164 AGAFONOV, MANYKIN where (ω) ˜ Ð1 ˜ Ð1 β 2 2 Tll = Gll Fll Ð U . (13) The function Z(ω; A, λ, µ, β) has the following form: xN V 4 Z(ω; A, λ, µ, β) = 1 + ------t------k---l------. (14) (ω)∑ ( )Ð1 ( )Ð1 Ð1 ( )Ð1 Ð1 ( )Ð1 T ll 0 0 2 Ð1(ω)[ ˜ 0 ˜ 0 2 ] k Gkk F Ðk Ð k Ð xNtV kl T ll Gll Gkk + Fll F Ðk Ð k Ð xNtV kl The one- and two-particle distributed states are determined by the following Green’s functions [32]: σ, σ 1 1( ( , ω µ, , λ, β) Gkk = -- Gext k ; A 2 (15) σ σ ( ) Ð 1 ( , ω µ, , λ, β) ) + Ð1 Gext k ; A Ð , where

Ð1 (0) (λ) (λ) F Ðk Ð k Ð b1 W Gext = ------Ð-1------Ð-1------, (16) ( (0) (λ) (λ))( (0) (λ) (λ)) β 2 2 2(λ) Gkk Ð b2 W F Ðk Ð k Ð b1 W Ð U W and ( )σ, σ σ σ + 1 1 + 1 F = --((Ð1) F (k, ω; µ, A, λ, β) Ðk k 2 ext (17) ( , ω µ, , λ, β) ) + Fext k ; A Ð , where β∗UW(λ) Fext = ------Ð-1------Ð-1------. (18) ( (0) (λ) (λ))( (0) (λ) (λ)) β 2 2 2(λ) Gkk Ð b2 W F Ðk Ð k Ð b1 W Ð U W

The following notation is used in equations (15)Ð(18): above that the formation of the impurity bands of distributed single-particle states is due to hybridization, ˜ Ð1 2 (0) which results in single-electron transitions: initial b (λ) = Gll Ð ∑V G , (19) 1 k1l k1k1 impurity site band state other sites band k1 state, and so on. The appearance of a superconducting state is deter- (λ) ˜ Ð1 2 (0) mined only by the existence of a nontrivial solution β ≠ 0 b2 = Fll Ð ∑V k l Fk k (20) 1 1 1 of equation (5) for the anomalous singlet Green’s func- k 1 (+)σ, Ðσ and tion F jj . Such a solution indicates the formation of localized bosons with spin 0. In addition, localized N V 2 spin-1 bosons should also be present, as is evident from (λ) im k1l W = ------2------2-. (21) equations (9) and (10). This spin-ﬂuctuation parameter b (λ)b (λ) Ð β U β 1 2 gives the function Fext (18), in terms of which the We shall now analyze the solution of equations (3)Ð superconducting condensate is determined in the singlet (21). This solution is determined by four self-consistent and in the triplet quasiparticle pairing channels (17). parameters, µ, A, λ, and β, which are found from the Thus, in this theory of spin-ﬂuctuation supercon- four integral equations (3)Ð(6). If equation (5) pos- ductivity in doped insulators equations of the BCS type sesses only the trivial solution β = 0, then, as one can for the energy gap do not arise. Instead, the key equa- easily see, the state of the doped material is normal. The tion for the superconducting state is equation (5), electronic structure is determined only by the single- describing localized spin-zero bosons. The mechanism particle normal Green’s functions for localized (7) and leading to the formation of a superconducting conden- distributed (15) electronic states. These functions have sate is due to two-quasi-particle transitions along an poles in the region of the initial band and in the region impurity ensemble: localized boson on an impurity site of the initial insulator gap, where impurity bands of pair of coupled quasiparticles (Ðkk) in the impu- localized and distributed states are formed. It was noted rity band of the distributed states localized boson

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INSULATORÐSUPERCONDUCTORÐMETAL TRANSITIONS 165 on another impurity site, and so on. The singlet and try in a doped insulator and is a consequence of averag- triplet quasiparticle pairing channels are intercoupled ing over an ensemble of substitution atoms. (see equation (17)). We note that in this model superconducting bosons arise only if the center of mass is at Using equations (9) and (10), we rewrite equation (5) rest. This is due to the absence of translational symme- in the form

∞ dω (U ⁄ 2)Z(ω; A, µ, (Ð1)nλ; β) β = β ∑ ------Im------. (22) ∫ π (ω ε µ (( )nλ))(ω ε (( )nλ)) β 2 2 2 + n = 0, 1 µ + 0 Ð 2 Ð S2 Ð1 Ð 0 Ð S1 Ð1 Ð U Z + i0

The existence of a solution β ≠ 0 of equation (22) is To show clearly the inﬂuence of doping on the den- determined by the values of the complex function Z sity of distributed states in the valence band, we intro- (14) at the poles or their regions of the anomalous sin- duce the change in the density of states per substitution glet Green’s function (9), which lie near the Fermi level atom on the Reω ≥ µ semiaxis in the lower part of the com- ω plex plane (Im < 0). The function UZ can be regarded ∆ρ (ω) 1 ( σσ (0)) as an intrasite interaction between localized electrons, ext = Ðπ------∑Im Gkk Ð Gkk . (24) xNt renormalized by spin ﬂuctuations and hybridization in k the quasiparticle pairing channel in the distributed states. In the region of the initial gap it is convenient to nor- If the second term on the right-hand side of equation (14) malize the density of the localized and distributed states is less than 1, which arises here from the seed correla- to the density of substitution atoms: tion energy, then, as one can easily see, equation (22) will have only the trivial solution β = 0 and, corre- ρ (ω) 1 σσ spondingly, the system will be in the normal state (or ext = Ðπ------∑ImGkk (25) β ≠ xNt metallic, or insulator). The solution 0 can exist if k there exists at least a region of poles of the Green’s function (9) near the Fermi level, where ReZ < 0. This and would mean that in this region of the spectrum the cor- ρ (ω) 1 σσ relation energy UZ changed its initial positive sign and loc = Ðπ--ImG jj . (26) started to correspond to attraction. The superconducting condensate was calculated as follows. The k distribution of the singlet and triplet 4. COMPUTATIONAL DETAILS bosons has the form For the calculations, we chose a model of a symmet- µ 2 ric valence band with density of k states per spin σ, σ ω ( )σ, σ ρ 1( ) d + 1(ω) k = ∫ --π----ImFÐkk . (27)  N 2 2 1/2 ------t--[D Ð ε ] , ε ≤ D Ð∞ ρ(0)(ε) π 2 b b =  Db (23) We denote the branches of the poles of the anoma-  ( )σ, σ 0, ε > D . + 1 ω ε b lous Green’s function FÐkk as k = fn( k), where n is the branch number. Then, introducing the inverse func- Here 2D is the width of the band and N is the total b t ε Ð1 ω number of states in the band. Doping will be described tion k = f n ( k), we obtain from equation (27) the in terms of the relative density of substitution atoms: energy distribution of the superconducting condensate, x = Nim/Nt. normalized to the density of substitution atoms:

The poles of the Green’s function for localized and σ, σ σ, σ ρ 1(ω ) 1 ρ 1( Ð1(ω )) distributed electronic states lie in the region of the ini- n k = ------f n k tial band and in the gap. According to equations (7)Ð(10) xNt and (15)Ð(18), each of these functions is represented as Ð1 (28) ( ) d f (ω ) a sum of two terms. The difference between these terms × ρ 0 ( Ð1(ω )) n k f n k ------ω------, is that the spin-ﬂuctuation parameter λ occurs in them d k ± with different signs . For this reason, the electronic ρ(0) structure divides into pairs of bands for which the sub- where the density of states is given by equation (23). stitution λ Ðλ transforms one band into another. The substitution For such pairs we shall use the same notations, which × Ð1/2 differ only by the upper index ±. V kl = V k Nt ,

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µ 5. RESULTS AND DISCUSSION – – – + + 1.8 D0 Lg 6 Dm D0 Dm (×15) The model presented makes it possible to describe – + at the same time insulatorÐsuperconductorÐmetal tran- 1

– L + L 0.9 b b sitions in a doped compound. We begin with a discus- (×15) 4 sion of the normal paramagnetic phase. In this state the 0 spin ﬂuctuations are represented by the parameter λ, which leads to the above-mentioned doubling of the 2 spectral features of the electronic structure. We shall –0.9 – + – + + Eb + Eb Eg Lg Eg show that the spin ﬂuctuations (1) lead to an insulator (×50) state of the material for low values of x and (2) give rise

–1.8 0 ~ ~ Density of states, eV to an insulatorÐmetal phase transition at some threshold –0.9 0 0.9 2.0 2.1 2.6 ω, eV value of the degree of doping.

Fig. 1. Electronic structure of the insulator state for x = 0.02. 5.1. InsulatorÐMetal Transition The Fermi energy µ = 2.1 eV and coincides with the position + Ð + The electronic structure of the paramagnetic insula- of the peak D0 . The curve Eb + Eb represents the change Ð tor state with x = 0.02 is shown in Fig. 1. The curve Eb + in the density of states in the initial band. Parameters: Db = 1.5 eV, ε = D Ð 0.4 eV, V = 1.5 eV, and U = 1 eV. + 0 b k Eb represents the change in the density of distributed states in the valence band. A common feature for various values of the parameters of the model is that the doping-induced change in the total number of band where Vk has the dimension of energy, is used in the states is negative. For the result presented, this change expressions obtained for the Green’s functions. In the is −0.2531 per spin per substitution atom. This deficit calculations, V is assumed to be independent of k. of states is exactly compensated by the distributed k states, which form in the region of the insulator gap. Thus, the four constants with the dimension of Spin fluctuations lead to splitting of these states into energy D , U, ε , and V completely determine the Ð + b 0 k two narrow bands, Eg and Eg , shown in Fig. 1. model (1). In the calculations the values of the con- An important circumstance is that the maximum stants Db and U which agree with the existing experi- ± mental data (see Section 2) were used. The values of the density of distributed states in each band Eg , ε two remaining constants 0 and Vk were chosen so that ρ max Nt (1) the Fermi level in the paramagnetic insulator phase xNt g, ext = ------, (29) lies above the valence-band top by Ӎ0.4Ð0.6 eV, which 4.712 eV corresponds to the position of the known spectral fea- is equal to half the maximum density of states in the ture (MIR band) in the doped cuprates (see [2] and ref- valence band, while the total number of states in these erences cited there) and (2) the insulatorÐsuperconduc- bands is small compared with the density xNt of substi- tor phase transition occurs at a reasonable impurity tution atoms. In the insulator state, the completely filled Ӎ concentration, x 0.05 [2]. Ð µ Eg band lies below the Fermi energy = 2.1 eV, and The solution of (1) depends on the four self-consis- + tent parameters A, µ, λ, and β. These parameters were the Eg band is unfilled. found from the four integral equations (3)Ð(6). An iter- In the notations used above, the density of localized ation procedure was used to solve these equations. Tak- Ð + states is normalized to 1. The curve Lb + Lb in Fig. 1 ing into account the fact that the exact single-particle corresponds to the density of localized states near the Green’s function must satisfy Levinson’s theorem, initial band. The total number of these states is which means that the total number of electronic states 0.2414 per spin per substitution atom. This band is due is conserved, the accuracy of our calculations using the to hybridization, and it arises even when the Anderson Green’s functions obtained can be estimated as follows: Hamiltonian is solved with a single impurity atom [29]. In addition, the energy distribution of localized 1 (0) δ γ = ------Im dωTr(G(ω) Ð G (ω)), states contains four functions, which correspond to π ∫ σσ Nim ω the simple poles of G jj ( ) (7). Two of these poles, ω ± ε ± λ δ where G(0) is the unperturned Green’s function, which 0 = 0 + U(A ), and the corresponding peaks are is determined by the first and second terms on the right- ± labelled in Fig. 1 as D0 . In the insulator state the posi- hand side of the Hamiltonian (1). The computational + accuracy decreases as the degree of doping x increases. tion of the partially filled peak D0 in the gap always For the results presented below γ ≤ 0.7 × 10Ð3. coincides with the Fermi energy. The number of states

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INSULATORÐSUPERCONDUCTORÐMETAL TRANSITIONS 167 in each of these peaks is x/(1 + 2x) per spin per substi- µ 2.5 tution atom. Taking account of the normalization of the – + + density to 1, these peaks are low-concentration peaks Dm D0 Dm for x Ӷ 1. 2.0 1 – The two other simple poles correspond to high-con- 1.5 δ ± centration peaks, which are denoted as Dm . The number of localized states per spin per substitution atom is 1.0 Ð + E– L+ E+ 0.3052 for the Dm peak and 0.3868 for the Dm peak. 0.5 g g g A general feature of the model for various values of the (×20) ± ~ ~ parameters is that the D peaks lie inside the band of Density of states, eV 0 m 2.03 2.10 2.17 2.24 2.62 2.64 2.66 ± ω distributed states Eg , as shown in Fig. 1. The electronic , eV ± structure also includes two bands Lg of localized Fig. 2. Density of states in the initial insulator gap for x = states. The number of states in these bands is usually 0.0524. This doping level lies near the insulatorÐmetal relatively small (Ӎ10Ð2 in the units adopted). boundary xthr, 1 in the phase diagram of normal states. The parameters are the same as in Fig. 1. We note that in the region of the gap all bands of localized states with upper index “+” lie above all bands of localized states with upper index “–” (see Fig. 1). Then, µ µ 1.5 0.6 using the expressions for the Green’s function (7) and D – D+ D – D + (8) and the definitions (3) and (4), it can be shown that m 0 m m in the insulator state the following relation holds (a) (b) 1 between the filling factor A and the spin-fluctuation – 1.0 0.4 parameter λ: A + λ = 1. (30) 0.5 – + + 0.2 – + For the electronic structure presented in Fig. 1, A = Eg Lg Eg Eg Eg 0.5904 and λ = 0.4096. Therefore the spin fluctuations (×12) in the model (1) cannot be neglected. ~ ~

Density of states, eV 0 0 The change in the electronic structure in the region 2.16 2.24 2.32 2.65 2.70 2.34 2.52 of the valence band with increasing doping x is qualita- ω, eV tively similar to the result presented in Fig. 1, and it will not be shown below. Fig. 3. Electronic structure of the metallic state in the region of the initial insulator gap for various degrees of doping x: For the normal phase, the behavior of the electronic (a) x = 0.085; (b) x = 0.22. The parameters are the same as structure in the region of the initial gap with increasing in Fig. 1. x gives rise to an insulatorÐmetal transition in the doped compound. In the insulator state the Fermi level coincides with the position of the D + peak at ω + = ε + 0 0 0 For x = 0.085 the D + peak lies below the E Ð band, λ ω + 0 g U(A + ). Because of the relation (30), the value of 0 and the Fermi level now coincides with the position of does not change with increasing x. However, the posi- the high-concentration peak D Ð of localized states tion of the E Ð band of distributed states depends on x, m g (Fig. 3a). This partially filled peak lies in two overlap- and this band shifts with increasing x in the direction of Ð + ping bands, one of which is the band Eg of distributed the partially filled peak of localized D0 states. Figure 2 shows the computed electronic structure near the Fermi + states and the other is the band Lg of localized states. Ð level for x = 0.0524. It is evident that the top of the Eg Thus, the electronic structure presented here in the + region of the initial insulator gap corresponds to the band reached the D0 peak. Here the relation (30) is still satisfied: λ = 0.4135 and A = 0.5865. The number of state of a poor metal, in which the distributed and local- localized states per spin per substitution atom is 0.2755 ized states coexist near the Fermi level. Moreover, the Ð + total number of localized D Ð states is 2.12 times for the Dm peak and 0.3572 for Dm . The number of m distributed states per spin per substitution atom is Ð greater than the number of states in the Eg band, which Ð + 0.1484 for the Eg band and 0.0865 for Eg . is 0.1265 per spin per substitution atom. The number of

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5 5 5 Ð (a) (b) (c) tution atom is 0.2103 for the Dm peak and 0.2229 for 4 4 4 the D + peak. Here the ﬁlling A Ð = 0.2785, while for 3 ReZ– µ 3 ReZ– µ 3 ReZ– µ m m Ð 2 2 2 x = 0.085 we have Am = 0.9743. 1 1 1 It should be noted that the Fermi energy is essen- 0 0 0 tially independent of x. According to Figs. 1Ð3 µvaries in a Ӎ10% range, while the doping changes by a factor –1 –1 –1 – – – of 11. It also follows from Figs. 1Ð3 that, although the –2 ImZ –2 ImZ –2 ImZ ± width of the E bands increases with increasing x, the –3 –3 –3 g 1.2 1.8 2.4 1.2 1.8 2.4 1.2 1.8 2.4 maximum density of states in these bands does not ω, eV depend on doping and is given by equation (29).

Fig. 4. The function Z, representing the renormalization of the intrasite interelectronic interaction in the quasiparticle 5.2. Renormalization of the Intrasite Interelectron pairing channel in the distributed states, versus the energy ω Interaction in the Quasiparticle Pairing Channel for various degrees of doping x: (a) x = 0.0524; (b) x = 0.085; in Distributed States (c) x = 0.107. The parameters are the same as for Fig. 1. It was shown above that the possibility of a nontrivial solution of equation (22) for the spin-fluctuation parameter β ≠ 0 is determined by the function Z(ω) (14) + on the semiaxis ω ≥ µ. If such a solution exists, then the states in the Lg band is relatively small, 0.0148 in the units adopted. It was found that A = 0.6494 and λ = doped material is a superconductor. The quantity UZ can be interpreted as the effective intrasite interaction. 0.3368. Thus, equation (30) is not satisfied in the metal- λ lic state. Since the other spin fluctuation parameter leads to doubling of the spectral features of the electronic struc- As x increases further, the L + band also shifts below ture, the solution (22) depends on the two functions g Z(ω; A, µ, ±λ, β). However, it is easy to show from the Fermi level. Then we obtain from equations (4) and equation (14) that Z+(µ + ω; A, µ, λ, β) = ZÐ(µ Ð ω; (7), (8) the following expression for the spin-fluctua- µ λ β tion parameter λ: A, , Ð , ). For this reason, in what follows, we shall present the computational results only for one of them. λ = N + Ð N Ð (1 Ð A Ð ), (31) We have shown that in the normal phase, near the m m m Fermi level, there is always a density of localized sin- + Ð gle-particle states that is determined by the normal where Nm and Nm are the number of states per spin per Green’s function (7), and therefore there exists a region substitution atom in the D + and D Ð bands, respec- of poles of the anomalous Green’s function (9) for m m β ≠ Ð Ð localized bosons. For this reason, the solution 0 can tively, and Am is the filling factor of the Dm peak. All exist, if there exists a finite energy range near the Fermi ± + level where ReZ < 0. We shall present below the func- bands of localized states, with the exception of D and m tion ZÐ(ω) for the normal states (β = 0) for various val- D Ð , lie below the Fermi level, and there total number ues of x. Actually, in the superconducting state the β is m λ of states increases with x. Then, the contribution from much less than A and , and it does not substantially ω these bands to the filling factor A also increases and, affect Z( ). Ð ω Ð ω correspondingly, Am decreases as x increases. Since the Figure 4a shows the functions ReZ ( ) and ImZ ( ) + Ð for x = 0.0524 close to the threshold doping level x values of N and N are close, λ 0 in the metallic thr, 1 m m for the insulatorÐmetal transition (see Fig. 2). Here µ = phase as x increases. In consequence, there is a ten- 2.1 eV and ReZÐ(ω = µ) = 0.0771, ImZÐ(ω = µ) = 0. It dency for any pair of bands whose designations differ Ð only by the upper index ± to merge, and the electronic is evident that there exists a region where ReZ < 0. The width of this region is 49 meV, but it lies below the structure has its own asymptotic solution (1) in the Har- Ð µ treeÐFock approximation (λ = 0). This tendency for the Fermi level. A region where ImZ < 0 also lies below . For smaller values of x these regions become narrower electronic structure to become modified is represented µ in Fig. 3b. For x = 0.22 we obtain λ = 0.07115 and A = and shift to the left from = 2.1 eV. We have estab- ± lished that for x < 0.0524 there exists only the trivial 0.6254. Both bands of distributed states Eg overlap solution (22), and the normal state of the doped com- with one another and are partially filled. The peak of pound is an insulator (see Figs. 1, 2). + As x increases, the region ReZÐ < 0 shifts to the right localized states Dm is still not filled, but it lies closer to µ Ð ω Ð with respect to . For x = 0.085 the function Z ( ) also Dm . The number of localized states per spin per substi- is presented in Fig. 4b. Here µ = 2.1513 eV lies in the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INSULATORÐSUPERCONDUCTORÐMETAL TRANSITIONS 169 region of negative values of ReZÐ and ImZÐ, and 1.5 E D– D+ D – D– D + D+ ReZ−(ω = µ) = Ð1.2675 and ImZÐ(ω = µ) = Ð1.8937. b 0 0 m s m s Ð 1.2 – × The width of the region where ReZ < 0 is 84 meV. 1 Lg ( 10) Here, one should note also another region, ReZÐ < 0, – lying below 1.5 eV. This region does not contribute to 0.9 the integral of the expression containing ZÐ on the right- hand side of equation (22). However, because of the 0.6 ± + – – + + relation between Z , the function ReZ , together with Lb Es Eg Lg Eg + µ 0.3 (×10) (×104) (×10) the region of negative values of ReZ near , acquires + a second region where ReZ+ < 0, located near the peak Es Density of states, eV 0 of localized states D + . For this reason, for fixed x it can 1.2 1.4 1.6 1.8 2.10 2.24 2.6 2.7 m ω be expected that there exist at least three positive con- , eV tributions to the right-hand side of equation (22), one of Ð Fig. 5. Density of single-particle electronic states in the which is related with Z and the other two are related superconducting state of a doped compound with x = 0.085. with Z+. If β = 0, the doped material would be in a The parameters are the same as for Fig. 1. metallic state, whose electronic structure is presented in Fig. 3. However, for this degree of doping, there exists a solution β ≠ 0 of equation (22) and, corre- The following changes in the parameters were spondingly, the doped compound is a superconductor. obtained in the superconducting state: ∆A = Ð2.7 × 10Ð4, As x increases further, this region passes through the ∆λ = 3.2 × 10Ð4, and ∆µ = Ð0.28 meV with respect to Fermi surface and for x = 0.107 it lies to the right of µ = their values in the normal state. The electronic structure 2.2173 eV (Fig. 4c). Here ReZÐ(ω = µ) = 1.0671 and of the single-particle states is shown in Fig. 5. Here the ImZÐ(ω = µ) = Ð2.1392. For x > 0.107 we obtained only ± ± curves Eb and Lb are fragments of the distribution of the trivial solution (22). Correspondingly, the normal distributed and localized states in the region of the ini- state of the doped compound is metallic (see Fig. 3). tial band near its boundary with the gap at 1.5 eV. Com- Thus, superconductivity can exist in a bounded paring this result with the results shown in Fig. 3, it is range of doping. Comparing the results presented in evident that in the superconducting state, two new Figs. 2 and 4, it can be concluded that the onset of ± ± superconductivity is associated with the first threshold bands Es of distributed states and two new peaks Ds doping level xthr, 1. For the model parameters used, of localized states near the gap have appeared. The Ӎ number of states per spin per substitution atom is xthr, 1 0.0524. As the normal phase is approached, × Ð5 Ð × Ð5 near xthr, 1, an insulatorÐmetal transition occurs and for 0.7377 10 for the Es band and 0.4497 10 for the Ð the same doping the region with ReZ < 0 approaches µ. + ± We were not able to determine whether a transition Es band. The number of states in the Ds peaks is the from the insulator into the superconducting state occurs same and is equal to 1.755 × 10Ð6. We note that small immediately or there exists an intermediate metallic but finite widths have appeared for the high-concentra- ± phase. The solution for the superconducting state for tion peaks of the localized states D . Near the Fermi x = 0.085, i.e., quite far from the threshold value, will be m presented below. We have shown that the model is also level there are six overlapping bands. Two of them are Ð + characterized by a second threshold doping level xthr, 2, bands of the distributed states Eg and Es , and the oth- for which a superconductorÐmetal transition occurs. ers are bands of localized states. We note that x increases with D . For a width of thr, 1 b Now, we shall discuss the solution of equation (22). the initial band 2Db = 6 eV and the same values of the other model parameters, we obtain x = 0.134. For This equation contains the anomalous Green’s function thr, 1 (+)σ, Ðσ this reason, the superconducting state induced by the F jj . Its poles coincide with the poles of the nor- spin fluctuations actually can arise in doped insulators (+)σσ with initially narrow allowed bands. mal Green’s function G jj , as one can see from equations (7), (8) and (9), (10). Therefore, for this solution with β ≠ 0 we have 5.3. Superconducting State Assuming β = 0, the normal state of the doped com- ( , µ, λ β) pound is metallic with x = 0.085 (see Fig. 3). However, 1 = ∑Qm A ; , (32) equation (22) possesses the solution |β| = 0.00395. We m note that here no cutoff of the integral on the right-hand side of equation (22) was used, and the integration was where Qm is the contribution of the mth band of single- performed along the entire semiaxis ω ≥ µ. particle localized states, which lies no lower than the

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– + + Q1(Dm) (a) Q2(Lg ) (b) Q3(Dm) (c) 2000 0.5 2000 Q1 = 0.7925 Q2 = 0.0545 Q3 = 0.3398 1500 0.4 1500 1000 1000 0.3 500 500 0.2 0 0 –500 0.1 –500 –1000 0 –1000 2.150 2.155 2.139 2.232 2.63 2.65 ω, eV

|β| Ð + Fig. 6. The solution = 0.00395 of equation (22) with x = 0.085: (a) contribution from the Dm band; (b) contribution from Lg ; + ± − (c) contribution from Dm . The contributions from Ds are the same and are equal to 0.0904. The corresponding bands are shown in Fig. 5.

Fermi level µ = 2.1511 eV, to the integral on the right- regions of these bands. Considering the relation hand side of equation (22). between Z+ and ZÐ (Fig. 4), it is easy to see that Z+ pos- It follows from Fig. 5 that there are five such bands sesses two regions where ReZ+ < 0. The first region lies (or regions of integration in equation (22)). Three of + near the Fermi energy, where the band Lg lies. The Ð + + them are the bands Dm , Dm , and Lg , which arise in the integrand for this band is shown in Fig. 6b and its con- solution for the normal phase and which are also shown in + β tribution is relatively small Q2(Lg ) = 0.0545 for = Ð Fig. 3a. The contribution of the peak Dm , which coincides 0.00395. This is due to the fact that the amplitude of the with the Fermi level µ and now has a finite width, in equa- poles for the single-particle states in this region is also tion (32) is determined by the function ZÐ. As shown in small, as shown in Figs. 3a and 5. The second region Ð Ð + + Fig. 4, ReZ < 0 near the Fermi level. Here also ImZ < 0, with ReZ < 0 lies near the peak Dm and corresponds which is due to the damping of the quasiparticles. The to the region ReZÐ < 0, lying below 1.5 eV (Fig. 4). For integrand for this band is shown in Fig. 6a and its con- this band, the integrand is shown in Fig. 6c and its con- Ð |β| + tribution is Q1(Dm ) = 0.7925 for = 0.00395. β tribution is Q3(Dm ) = 0.3338 for = 0.00395. + + The two remaining regions of integration in equa- The contribution of Dm and Lg in equation (32) is + tion (22) are related with the two new peaks of local- determined by the values of the function Z in the ± ized states Ds , which have appeared, in the region of the gap. Their contributions are the same and negative: ρ σσ –7 –1 ρ σσ –6 –1 ρ σσ –6 –1 ± 1 1, 10 eV 2 1, 10 eV 3 1, 10 eV Q4, 5(Ds ) = Ð0.0904. 1.5 4 1.5 (a) (b) (c)µ Thus, we have obtained the solution of equation (32), and therefore the doped compound is a superconductor. 3 1.0 1.0 The energy distribution of the distributed bosons in the singlet and triplet pairing channels were calculated 1 2 1, 2 1, 2 according to equation (28). The results are presented in 0.5 0.5 Fig. 7. An unexpected result is that there is a distribu- 1 tion of bosons in the region of the initial valence band 2 far from the Fermi surface (Fig. 7a). Here the number 0 0 0 of pairs per spin per substitution atom is relatively 0.6 0.9 1.2 1.5 1.60 1.65 1.70 2.08 2.16 small and is equal to 3.46 × 10Ð8 in the singlet channel ω, eV and 5.57 × 10Ð9 in the triplet channel. In the region of the initial gap there are two regions Fig. 7. Energy distribution of the distributed bosons in the of distribution of the condensate. The first region also region of the initial band of the insulator (a), in the region of µ Ð + lies far from , and the corresponding distribution of the band Es (b), and in the region of the band Es (c). The the superconducting bosons coincides with the band of corresponding bands are shown in Fig. 5: curve 1, spin zero; Ð curve 2, spin 1. single-particle distributed states Es (Fig. 7b). Here the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INSULATORÐSUPERCONDUCTORÐMETAL TRANSITIONS 171 number of pairs per spin per substitution atom in the near the energy µ in the normal state, if the interaction singlet and triplet channels is the same and is equal to with phonons in the form 1.821 × 10Ð7. The second region in the gap lies near µ, and its position coincides with the band of single-parti- κ φ + HelÐph = ∑ qjnˆ jσ q, cle distributed states Es (Fig. 7c). The energy distribution of the pairs in this region is cut off by the Fermi qjσ φ energy. The number of pairs per spin per substitution where q is the field operator for phonons, is introduced atom in the singlet and triplet channels is the same and into equation (1). This is due to the fact that in the is equal to 1.053 × 10Ð7. metallic state there is a high density of localized elec- Ð This model of superconductivity is also character- tronic states Dm at the Fermi surface. This interaction ized by the number of localized bosons with spin 0 and 1. results in dephasing of the site electrons. Then the Using equations (9) and (10), this number per spin per Ð substitution atom is localized states Dm will be characterized by damping, and the peak will be broadened. As a result of hybrid- µ 2 ization, a corresponding imaginary part will also ( )σσ dω + 1 N = ∑ ------ImF (ω) . appear in the energies of electrons in the band of dis- LB ∫ π jj Ð σ, σ tributed states E . This will lead to a dip in the density 1 Ð∞ g Ð µ β × Ð5 of states in the Eg band near , the depth of the depth For = 0.00395 we obtain NLB = 4.50 10 . being determined by the strength κ of the interaction. We now obtain the following relation between the qj quasiparticle concentrations in the superconducting state. Summing over the three bands shown in Fig. 7, REFERENCES we obtain that the total number of pairs per substitution 1. G. M. Éliashberg, in Physical Properties of High-Tem- × Ð6 atom is 0.64 10 in the singlet pairing channel and perature Superconductors, Ed. by D. M. Ginzberg (Mir, 0.58 × 10Ð6 in the triplet channel. Therefore the concen- Moscow, 1990), p. 505. tration of distributed bosons is much lower than the 2. E. Dagotto, Rev. Mod. Phys. 66, 673 (1994). concentration of localized bosons. The latter concentra- 3. N. Bulut and D. J. Scalapino, Phys. Rev. B 50, 16078 tion, in turn, is much less than the concentration of (1994). filled single-particle states near the Fermi energy. The 4. Q. Chen, I. Kosztin, B. Jankó, et al., Phys. Rev. Lett. 81, total number of single-particle distributed states near 4708 (1998). Ð the Fermi energy is largely determined by the Eg band 5. Yu. A. Uzyumov, Usp. Fiz. Nauk 161, 2 (1991) [Sov. and is 0.253 per substitution atom. This value is Phys. Usp. 34, 361 (1991)]; Usp. Fiz. Nauk 165, 403 2.12 times less than the total number of localized states, (1995) [Sov. Phys. 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Phys. 64, 503 density in the region of the initial insulator gap, and (1995). therefore difficulties appear in the explanation of the 12. D. J. Scalapino and S. R. White, Phys. Rev. B 58, 8222 results obtained on the basis of the Hubbard model (1998). [34, 35]. Our approach makes it possible to describe 13. V. J. Emery and S. A. Kivelson, Phys. Rev. Lett. 74, 3253 the insulatorÐmetal phase transition in the case where (1995). equation (5) possesses only the trivial solution β = 0. 14. Q. Si, J. Phys. Cond. Matt. 8, 9953 (1996). The mechanism leading to the formation of a super- 15. A. S. Alexandrov, Physica C 274, 237 (1997). conducting state is electronic and is related with spin 16. M. V. Sadovskiœ, Sverkhprovodimost’: Fiz., Khim., fluctuations in the doped compound. However, in this Tekh. 3, 337 (1995). model the electronÐphonon interaction should have a 17. P. P. Edwards, N. F. Mott, and A. S. Alexandrov, J. 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19. A. I. Agafonov and É. A. Manykin, Pis’ma Zh. Éksp. 29. F. D. M. Haldane and P. W Anderson, Phys. Rev. B 13, Teor. Fiz. 65, 419 (1997) [JETP Lett. 65, 439 (1997)]. 2553 (1976). 20. M. I. Ivanov, V. M. Loktev, and Yu. G. Pogorelov, Fiz. 30. A. I. Agafonov and E. A. Manykin, Phys. Rev. B 52, Nizk. Temp. 24, 615 (1998) [Low Temp. Phys. 24, 463 14571 (1995); A. I. Agafonov and É. A. Manykin, (1998)]. Zh. Éksp. Teor. Fiz. 109, 1405 (1996) [JETP 82, 758 (1996)]. 21. N. F. Mott, Metal-Insulator Transition (Taylor and Fran- cis, London, 1990). 31. A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskiœ, Methods of Quantum Field Theory in Statistical Physics 22. P. P. Edwards, T. V. Ramakrishnan, and C. N. R. Rao, (Fizmatgiz, Moscow, 1962; Prentice-Hall, Englewood J. Phys. Chem. 99, 5228 (1995). Cliffs, New Jersey, 1963). 23. C. Y. Chen, R. J. Birgeneau, M. A. Kastner, et al., Phys. 32. A. I. Agafonov and É. A. Manykin, Zh. Éksp. Teor. Fiz. Rev. B 43, 392 (1991). 114, 1765 (1998) [JETP 87, 956 (1998)]. 24. G. A. Thomas, in High Temperature Superconductivity, 33. F. Yonezawa and T. Matsubara, Prog. Theor. Phys. 35, Ed. by D. P. Tunstall and W. Barford (Adam Hilger, Bris- 357 (1966). tol, 1991), p. 169. 34. S. W. Robey, L. T. Hudson, C. Eylem, et al., Phys. Rev. 25. Z.-X. Shen and D. S. Dessau, Phys. Rep. 253, 1 (1995). B 48, 562 (1993). 26. C. Quitmann, J. Ma, R. J. Kelly, et al., Physica C 235- 35. T. Katsufuji, Y. Okimoto, and Y. Tokura, Phys. Rev. Lett. 240, 1019 (1994). 75, 3497 (1995); M. Kasuya, Y. Tokura, T. Arima, et al., Phys. Rev. B 47, 6197 (1993); Y. Taguchi, Y. Tokura, 27. A. I. Agafonov and E. A. Manykin, Physica B 259Ð261, T. Arima, et al., Phys. Rev. B 48, 511 (1993). 458 (1999). 28. T. Timusk, S. L. Herr, K. Kamaras, et al., Phys. Rev. B 38, 6683 (1988). Translation was provided by AIP

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 17Ð23. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 22Ð28. Original Russian Text Copyright © 2000 by Achasov, Beloborodov, Berdyugin, Bozhenok, A. Bukin, D. Bukin, Burdin, Vasil’ev, Ganyushin, Gaponenko, Golubev, Dimova, Dolinskiœ, Druzhinin, Dubrovin, Zakharov, Ivanchenko, Ivanov, Korol’, Koshuba, Kukhartsev, Mamutkin, Otboev, Pakhtusova, Sal’nikov, Serednyakov, Sidorov, Silagadze, Sharyœ, Shatunov. NUCLEI, PARTICLES, AND THEIR INTERACTION Measurement of the Relative Probability of f hg Decay in the h p+pÐp0 Channel M. N. Achasov, K. I. Beloborodov, A. V. Berdyugin, A. V. Bozhenok, A. D. Bukin*, D. A. Bukin, S. V. Burdin, A. V. Vasil’ev, D. I. Ganyushin, I. A. Gaponenko, V. B. Golubev, T. V. Dimova, S. I. Dolinskiœ, V. P. Druzhinin, M. S. Dubrovin, A. S. Zakharov, V. N. Ivanchenko, P. M. Ivanov, A. A. Korol’, S. V. Koshuba, G. A. Kukhartsev, A. A. Mamutkin, A. V. Otboev, E. V. Pakhtusova, A. A. Sal’nikov, S. I. Serednyakov, V. A. Sidorov, Z. K. Silagadze, V. V. Sharyœ, and Yu. M. Shatunov Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk State University, Novosibirsk, 630090 Russia *e-mail: [email protected] Received July 28, 1999

Abstract—The relative decay probability B(φ ηγ) = (1.259 ± 0.030 ± 0.059)% in the decay channel η π+πÐπ0 has been measured in an experiment using a spherical neutral detector in the VEPP-2M electron- positron storage ring. The result agrees with the tabulated value and with measurements of this probability using the spherical neutral detector in other η-meson decay channels. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION lution in the detector calorimeter is given by φ ηγ σ 4.2% The magnetic-dipole transition has been -----E- = ------, studied in several experiments, mainly using counter- E 4 E []GeV propagating electron-positron beams [1]. In these experiments the systematic error is comparable to or and the angular resolution as a function of the photon greater than the statistical error. Since it is more difﬁ- energy is cult to estimate systematic errors, in order to improve 2 ()0.82° 2 the experimental accuracy of the averaged result it is σϕ =------+.() 0.63° important to make independent measurements of this E[]GeV value for different detectors and in different η-meson η The coordinate resolution of the drift chamber for the decay modes. So far the neutral -meson decay modes: charged particles determines the angular resolution η γγ η π0 [2, 3] and 3 have mainly been used with respect to the azimuthal angle σϕ = 0.5° and the although the η π+πÐπ0 decay channel was used in polar angle σθ = 1.8° (the polar axis is directed along a recent experiment using the KMD-2 detector [5]. For the beam axis in the collider), the spherical neutral detector (SND) measurements of φ the decay probability B(φ ηγ) in the η π+πÐπ0 Seven sweeps of the -meson energy range were made in 1996 and two sweeps were made in 1998 in the channel are of additional value for studying systematic range 2E = 980Ð1060 MeV. The total integrated lumi- errors when recording events not only with neutral par- nosity is 13.2 pbÐ1 and the total number of created ticles but also with charged ones. In addition, φ ηγ φ-mesons was 2.1 × 107. For conciseness the sweeps decay has now been measured using the SND in all sig- η will be arbitrarily designated PHI9601, PHI9602, …, niﬁcant -meson decay channels [3,4] so that errors in PHI9802. The luminosity was measured for large-angle measurements of η-meson decay probabilities in differ- electron and positron elastic scattering processes and ent channels can be almost completely eliminated. also for two-quantum annihilation. The difference between the results of the measurements by these two methods does not exceed 1%. In the present study we 2. EXPERIMENT used the luminosity measured using elastic scattering, as this process is closer to that being studied. The accu- Experiments were carried out in 1996 [6] and 1998 racy of the theoretical formulas used to model the elas- [7] using the VEPP-2M electron-positron storage ring tic scattering allow us to estimate the accuracy of the [8] with the SND detector [9]. The photon energy reso- luminosity measurements at around 2%.

18 ACHASOV et al.

Number of events reconstruction we obtained three parameters which 1000 were used for the ensuing selection: the two plausibility functions Lηγ and L3π, and the invariant mass mη of the π+ system. In order to simplify the profile of the plausi- 800 bility function in multidimensional space and to improve the results of the numerical minimization, the minimum with respect to three angular variables corre- 600 sponding to the rotation of the event as a whole, was determined analytically [10]. We give a complete list of the selection criteria: 400 < < α < ° < R1 0.5 cm, R2 0.5 cm, 12 130 , Lηγ 7, < < 200 Lηγ Ð L3π Ð1, mη Ð 550 50 MeV, (1) ° < θ < ° 27 i 153 . 0 2 4 6 8 10 Lηγ Here R1, 2 is the minimum distance between the charged particle trajectory and the beam axis (the half-height width of the distribution is approximately 0.1 cm and Fig. 1. Distribution of modeled events (histogram) and α experiment (circles with error bars) over the parameter Lηγ. 50% of the events have R1, 2 < 0.06 cm) and 12 is the spatial angle between the charged particles (this condition can suppress the background from collinear events, + Ð 0 0 3. EVENT SELECTION mainly from K K and also from KSK L events). The The events of the process being studied φ ηγ, kinematic reconstruction of events in the fundamental η π+πÐπ0 π0 γγ hypothesis φ Xγ was carried out using only parti- , and have two charged particles ° < θ < ° (pions) and three photons in the final state. For the anal- cles having the polar angle 27 i 153 and the other ysis we selected events with two charged noncollinear particles were neglected. This selection was carried out particles and three or more γ-quanta. The main back- to isolate small angles for background particles and ground process is π+πÐπ0 creation when an “excess” also because of the lower reliability in modeling the photon is formed in the event by means of one of sev- detector edges. eral possible mechanisms: “splitting” of photons in the These sampling conditions are weak for the process calorimeter, “noise” triggering of crystals, large-angle being studied. The recording probability was obtained photon emission by initial particles, and the “superpo- as 16–21% (the fraction of the solid angle to the fifth sition” of particles from other events which are simu- power is 0.8915 = 56%). lated with poor accuracy. However, because of errone- Although the distributions of the events over the ous reconstruction there is a small spurious contribu- parameters specified in the list of selection criteria φ 0 0 φ + Ð show fairly good agreement with the experimental dis- tion of events from KSK L , K K , and ωπ0 tributions, for each parameter we can identify a selec- ee . tion limit of the modeling events which is more consis- For the selected events we made a kinematic recon- tent with the experimental selection limit. For these struction which uses the measured angles of all parti- refined selection boundaries we obtain a certain event cles and the photon energies taking two hypotheses: recording efficiency. A change in the efficiency when (1) The event is a decay process the standard selection conditions were replaced by the refined ones served as the basis for introducing a cor- φ Xγ π+πÐπ0γ π+πÐγγγ, rection to the efficiency and the measure of systematic (2) The event is a background event error, which is determined by the difference between the distributions for the modeling and the experiment. φ π+πÐπ0. These corrections to the selection limits can be selected In both cases, from all the recorded photons in the most easily for the distribution over the mass mη where, event we selected those which best satisfy the hypothe- as a result of fitting the distribution, we can obtain its sis and neglected the remaining photons (in the selected full width at half-maximum and the coordinate of the events of the effect in the experiment 27% of the events distribution maximum in the form of peak parameters. have an excess photon whereas in the φ ηγ model- The greatest efficacy in suppressing the background ing this figure is only 13%). In both hypotheses the π0 is obtained by sampling over the parameter Lηγ (Fig. 1), mass was fixed whereas in the first hypothesis the which requires the energy and momentum conservation η-meson mass (the invariant mass of the π+πÐπ0 sys- laws to be satisfied and the invariant mass of two pho- tem) was a free parameter. As a result of kinematic tons to be equal to the pion mass. The background from

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 MEASUREMENT OF THE RELATIVE PROBABILITY 19 the process e+eÐ π+πÐπ0(γ) is mainly suppressed by Number of events sampling over Lηγ and Lηγ Ð L3π and for the remainder 1400 (around 10%) we made a correction based on the mη distribution (Fig. 2). This figure gives a histogram 1200 showing the ηγ modeling and the circles with the error bars give the experimental results (sweep PHI9801). 1000 The full width at half-maximum of the mη distribution 800 is hη = 29.5 ± 0.6 MeV for the modeling and for the experimental distribution of the events for all the 1996 600 sweeps we have hη = 34.7 ± 0.9 MeV, for sweep PHI9801 we have hη = 29.3 ± 1.0 MeV, and for 400 PHI9802 we have hη = 29.8 ± 1.4 MeV. 200 By approximating the experimental mη distribution and assuming that the distribution of background events is close to a straight line near mη ~ 550 MeV, we 400 450 500 550 600 650 700 750 can obtain an estimate of the total number of back- mηγ, MeV ground events in the range 500 MeV < mη < 600 MeV. By using the results of fitting the distributions of these Fig. 2. Distribution of φ ηγ modeling events (histogram) and experiment (circles with error bars) over the experimental events over the energy 2E (Fig. 3), we can parameter m . obtain the number of nonresonant background events. η Now the number of resonant background events can be determined as the difference between the total number Observed cross section, nb of background events and the number of nonresonant 2.0 events. The final error in calculations of this correction in our case is largely determined by the statistical error of the nonresonant cross section. The corrections to the 1.5 resonant residual background were determined separately for each sweep. Statistically these all agree with their average value (9.4 ± 0.7)%. Finally for all the sweeps we used this average correction to the residual 1.0 resonant background.

The coefficient of suppression of the main back- 0.5 ground processes associated with the φ-meson resonance is extremely large (the recording probability is ε × Ð5 ε × Ð5 ε × Ð4 + Ð ~ 4 10 , 0 0 ~ 3 10 , 3π ~ 2.4 10 ) and K K KSK L 0 provides no basis for subtracting the remaining back- 980 1000 1020 1040 1060 ground according to the modeling data. The nonreso- 2E, MeV nant background was subtracted automatically when φ Fig. 3. Result of fitting φ-meson resonance excitation curve fitting the -meson resonance curve with the free back- for sweep PHI9801. ground. [11] taking into account the contribution of ω- and ρ-mesons: 4. RESONANCE CURVE FITTING π 2 σ( ) 12 mere The detector state (the presence of nonoperating cal- s = ----------α------ orimeter channels, the detector triggering conditions, s ϕ (2) and so on) changed significantly from one sweep to i V 2 F(s) Γ → Γ → ηγ (s)m e another. Hence, separate results were obtained for each × ------b + ∑ ------V------e--e------V------V------. sweep so that it was ultimately possible to obtain not s 2 Γ ( ) ρ, ω, φ s Ð mV + imV V s only an average result but also to estimate any system- V = atic error of unknown origin which could be manifest in Here the possible nonresonant contribution, in the instability of the result from one sweep to another. accordance with [13], is written with an arbitrary com- The cross section was parametrized in accordance with plex constant b. The phase volume F(s) is given by standard formulas for the phase volumes and the depen- ( ) [( 2 ) ]3 dence of the resonance width on the energy Γ(s) from F s = s Ð mη /2 s .

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Table summarizing results of analysis of each sweep of the φ-meson energy range (statistical errors) Ð1 χ2 ε φ ηγ σ φ ηγ Sweep L, pb /nD mφ, MeV , % B( ), % ( ), nb PHI9601 0.459 6.4/7 1018.96 ± 0.23 3.877 1.380 ± 0.098 58.4 ± 4.1 PHI9602 0.494 6.4/7 1019.420 ± 0.030 4.044 1.247 ± 0.074 52.6 ± 3.1 PHI9603 0.807 20.8/9 1019.366 ± 0.023 3.652 1.379 ± 0.050 58.2 ± 2.1 PHI9604 0.707 7.0/10 1019.130 ± 0.070 3.630 1.180 ± 0.068 49.8 ± 2.9 PHI9605 1.037 5.0/10 1019.053 ± 0.050 3.666 1.333 ± 0.060 56.4 ± 2.5 PHI9606 0.666 8.0/8 1019.08 ± 0.26 3.641 1.165 ± 0.083 49.3 ± 3.5 PHI9608 0.222 1019.413 3.770 1.294 ± 0.074 54.7 ± 3.1 PHI9801 4.226 16.0/13 1019.830 ± 0.108 4.750 1.179 ± 0.032 49.8 ± 1.4 PHI9802 4.541 17.3/13 1019.848 ± 0.091 4.709 1.251 ± 0.032 52.8 ± 1.4

The observed reaction cross section was ﬁtted tak- The free parameters used to minimize the plausibil- ing into account the radiation corrections [12] and the ity function were the φ-meson mass, the decay proba- particle energy spread in the beam. The energy depen- bility B(φ ηγ), and the nonresonant background dence of the recording efﬁciency was neglected: near a level. If the width of the φ-resonance is left as a free φ-meson ±20 MeV estimates using modeling events parameter, its statistical error for sweeps PHI9801 and give PHI9802 is found to be of the order of 0.3 MeV (even

Ð5 Ð1 larger for the 1996 sweeps) whereas the tabulated value dε/d s = (6.5 ± 8.5) × 10 MeV , of the width is considerably more accurate. The distri- which is of the order of zero. In addition, if the effi- bution of the optimum values of the width in different ciency has a weak, first-order dependence on energy, sweeps around the tabulated value does not contradict the contribution to the cross section is zero. the assumption that these fluctuations are of a statistical nature and thus the width, together with the two other Fitting the resonance curve, even in sweeps fitting parameters, was determined taking into account PHI9801 and PHI9802 which have the largest number the experimental accuracy of the measurements [1]: of energy points and a large integrated luminosity, yielded values of the constant b of the order of zero: Γφ = 4.43 ± 0.05 MeV, PHI9801: Re(b) = (Ð0.9 ± 1.9) × 10Ð7 MeVÐ1, B(ρ ηγ) = (2.4 ± 0.9) × 10Ð4, ω ηγ ± × Ð4 Im(b) = (0.4 ± 5.5) × 10Ð7 MeVÐ1, B( ) = (6.5 1.0) 10 . (3) PHI9802: Re(b) = (Ð0.1 ± 2.4) × 10Ð7 MeVÐ1, Allowance for the tabulated accuracy of these parameters was standard: the parameter was assumed to be free Im(b) = (0.3 ± 14.8) × 10Ð7 MeVÐ1. but half the square of the deviation of the parameter from the tabulated value in units of the measurement Thus, the value of this constant was subsequently fixed error was added to the logarithmic plausibility func- at zero and the cross section was parametrized using the tion. simple model of vector dominance. Quite clearly more The optimum values of the mass in different sweeps accurate experiments and possibly a wider energy have a spread considerably greater than the statistical range are needed to study this anomalous contribution. error. This can be attributed to the inaccuracy in reproduc- In addition to these fitting parameters, we can also ing the energy after tuning the magnetic fields in the col- construct another parameter lider before the beginning of sweeping. The observed spread of mass values is consistent with this additional m r  2 σ(φ ηγ ) = 12π -----e-----e error of the order of 0.3 MeV (allowance for this error  α χ2 mφ (4) gives /nD = 1). If this error is added when averaging the mass values over different sweeps, we obtain the result × B(φ ee)B(φ ηγ ), mφ = 1019.36 ± 0.12 MeV, which is the reaction cross section at the energy s = mφ assuming that Γ(ρ ηγ) and Γ(ω ηγ) are which agrees with the tabular value. zero. The phases of the resonances in (2) were selected as The table gives the results of fitting for each sweep. follows: ϕω = ϕρ = 0, ϕφ = π. The graph of the plausibil- A graphical example of the fitting result is plotted in ity function as a function of the phase ϕφ plotted in Fig. 3. Fig. 4 shows that the interference with ρ and ω must

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 MEASUREMENT OF THE RELATIVE PROBABILITY 21 deﬁnitely be taken into account, but the statistics at L this level are insufﬁcient to make the phase a free parameter.

The recording efficiency is given in Table 1 allowing 90 for B(η π+πÐπ0) = 0.231 ± 0.005. Different values of the efficiency for different sweeps are obtained because of changes in the trigger and differences in the list of inoperative calorimeter crystals. In sweep PHI9608 measurements were only made at the resonance maximum so that the mass and back- 85 ground level were fixed. In view of the difficulties involved in making a correct choice of values of these parameters and the small fraction of statistics in it, sweep PHI9608 was not included in the final result. Averaging σ(φ ηγ) and B(φ ηγ) over all 80 0 100 200 300 sweeps (except PHI9608), we obtain ϕφ, deg σ(φ ηγ) = 52.77 ± 0.75 nb, Fig. 4. Graph of plausibility function as a function of the B(φ ηγ) = (1.248 ± 0.017)% phase ϕφ for sweep PHI9801.

χ2 with /nD = 17/7. The level of statistical agreement φ χ2 χ2 Finally, taking from the tables [1] B( ee) = according to the criterion (P( ) = 1.7%) is satisfac- (2.99 ± 0.08) × 10Ð4, we obtain tory, although the observed appreciable spread of the results from one sweep to another can be taken into B(φ ηγ) = (1.259 ± 0.030 ± 0.059)%. account to some extent in the statistical error. Let us assume that there is an unknown random shift of the order of 5%. If the unknown statistical error of this 5. DISCUSSION OF RESULTS quantity is added quadratically to the intrinsic statistical error of each sweep, the ratio χ2/n = 7/7 is obtained We shall analyze several of the most accurate mea- D surements of the decay probability B(φ ηγ): and, as a result of averaging, we obtain σ(φ ηγ) = 53.2 ± 1.2 nb, B(φ ηγ ) ( ± ± ) B(φ ηγ) = (1.259 ± 0.030)%. 1.246 0.025 0.057 %, (η π0) [ ] In order to obtain the ﬁnal result, we need to take SND 3 4 , into account various systematic errors. The statistical (1.338 ± 0.012 ± 0.052)%, modeling error (1%), the systematic error in modeling (η γ ) [ ] the distributions used to sample events (1%), the sys- SND 2 3 , ( ± ± ) tematic error in the luminosity measurements (2%), the = 1.259 0.030 0.059 %, reconstruction errors and the inefﬁciency of the cham- + Ð 0 bers (1%), the trigger (1%), the indeterminacy of the SND(η π π π ), phase ϕφ (1%), and the corrections for the residual res- (1.18 ± 0.03 ± 0.06)%, onance background (0.5%) together give 3.0%. Taking + Ð 0 this into account we have CMD-2(η π π π ) [5], ( ± ) [ ] σ(φ ηγ) = 53.2 ± 1.2 ± 1.6 nb, 1.26 0.06 %, World aver. 1 .

φ φ ηγ η π+πÐπ0 B( ee)B( )B( ) When results obtained in the same π+πÐπ0 channel ± ± × Ð6 using the SND and CMD-2 detectors are compared, it = (0.870 0.021 0.027) 10 . must be borne in mind that most of the systematic error φ η π+πÐπ0 Using the tabulated value B(η π+πÐπ0) = (2.7% B( ee) and 2.2% B( ) makes 0.231 ± 0.005, we obtain no contribution to the ratio of these two results:

B(φ ee)B(φ ηγ) B(φ ηγ )[SND] ------= 1.067 ± 0.037 ± 0.052. (5) B(φ ηγ )[CMD-2] = (3.765 ± 0.092 ± 0.143) × 10Ð6.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 22 ACHASOV et al. Quite clearly the agreement between these two SND detector [9], carried out over the period 1996Ð results is good. 1998: Similarly, the ratio of the results of measurements σ(φ ηγ) = 53.2 ± 1.2 ± 1.6 nb, for different η-meson decay channels using the same B(φ ee)B(φ ηγ)B(η π+πÐπ0) SND detector contains no systematic error from Ð6 B(φ ee) and provides information on the ratio of = (0.870 ± 0.021 ± 0.027) × 10 , the decay probabilities η in different channels: B(φ ee)B(φ ηγ) Ð6 + Ð 0 = (3.765 ± 0.092 ± 0.143) × 10 , B(η π π π ) ------B(φ ηγ) = (1.259 ± 0.030 ± 0.059)%. B(η 2γ ) + B(η 3π0) (6) Measurements of the probability B(φ ηγ) using ± ± = 0.3141 0.0081 0.0058 the same detector simultaneously in the three most significant η-meson decay channels can be used to deter- (the contributions to the systematic error are: 1% statis- mine various quantities more accurately. The value of tical modeling error, 1% reconstruction and ineffi- the ratio ciency of the chambers, 0.5% residual resonance background, and 1% trigger), which should be compared B(η π+πÐπ0) with the tabulated data [1]: ------B(η 2γ) + B(η 3π0) Γ(η 2γ ) Γ(η 3π0) Ð1 = 0.3141 ± 0.0081 ± 0.0058 ------+ ------Γ(η π+πÐπ0) Γ(η π+πÐπ0) (7) agrees with the world average of 0.324 ± 0.007 and is of comparable accuracy. [ ± ± ]Ð1 ± = 1.70 0.05 + 1.39 0.05 = 0.324 0.007. From the sum of three measurements using the SND detector in different η-meson decay channels we can The accuracy of the result was slightly worse than also obtain the world average, but the statistical agreement is good. B(φ ee)B(φ ηγ) The fact that the probability B(φ ηγ) in all sig- ± ± × Ð6 nificant η-meson decay modes is measured using the = (3.848 0.036 0.070) 10 , same detector can also be utilized to improve the accu- B(φ ηγ) = (1.287 ± 0.012 ± 0.042)%, racy: which is the most accurate measurement made using a B(φ ee)B(φ ηγ ) single detector. (8) This work was supported by the STP Integration = (3.848 ± 0.036 ± 0.070) × 10Ð6 Fund (Grant no. 274) and by the Russian Foundation for Basic Research (Grants nos. 96-15-96327, 97-02- (here the systematic error includes: 1% modeling, 1% 18563, and 99-02-17155). trigger, 1% luminosity measurements and 0.5% indeterminacy of B(η 2γ) + B(η 3π0) + B(η π+πÐπ0)). The error in the luminosity was reduced REFERENCES because the luminosity in the neutral channels was 1. Review of Particle Physics, Eur. Phys. J. C 3 (1998). measured using the two-quantum annihilation process 2. V. P. Druzhinin, V. N. Golubev, V. N. Ivanchenko, et al., whereas that in the charged channel was measured Phys. Lett. B 144, 136 (1984). using large-angle elastic scattering. 3. M. N. Achasov, A. V. Berdyugin, A. V. Boshenok, et al., Preprint No. 99-39 (Novosibirsk, Budker Institute of Now, from the sum of the three measurements of Nuclear Physics, 1999). φ ηγ B( ) using the SND detector we can obtain 4. M. N. Achasov, S. E. Baru, A. V. Berdyugin, et al., Pis’ma Zh. Éksp. Teor. Fiz. 68, 573 (1998). B(φ ηγ) = (1.287 ± 0.012 ± 0.042)%, (9) 5. R. R. Akhmetshin, G. A. Aksenov, E. V. Anashkin, et al., which is the most accurate measurement made using a Preprint No. 99-11 (Novosibirsk, Budker Institute of Nuclear Physics, 1999); T. A. Purlatz, in Proc. Intern. single detector. The remaining indeterminacy is mainly + Ð φ Ψ φ Workshop on e e Collisions from to J/ (Novosibirsk, determined by the error in B( ee). 1999); R. R. Akhmetshin, E. V. Anashkin, V. S. Ban- zarov, et al., E-printes Archive no. hep-ex/9907003, submitted for publication in Phys. Lett. B. 6. CONCLUSIONS 6. M. N. Achasov, M. G. Beck, K. I. Beloborodov, et al., Preprint No. 97-78 (Novosibirsk, Budker Institute of + Ð 0 The probability B(φ ηγ) in the η π π π Nuclear Physics,1997); M. N. Achasov, K. I. Beloboro- channel has been measured in experiments on the dov, A. V. Berdyugin, et al., in Proc. Conf. HADRON-97 VEPP-2M electron-positron storage ring [8] using an (Brookhaven, 1997), p. 26.

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7. M. N. Achasov, V. M. Aulchenko, S. E. Baru, et al., Pre- 10. A. D. Bukin, Preprint No. 97-50 (Novosibirsk, Budker print No. 98-65 (Budker Institute of Nuclear Physics, Institute of Nuclear Physics, 1997); A. D. Bukin, in Proc. Novosibirsk, 1998). Computing in High-Energy Physics, CHEP-97 (Berlin, 1997). 8. G. M. Tumaœkin, in Proc. 10th Int. Conf. on High-Energy Particle Accelerators (Serpukhov, Russia, 1977), vol. 1, 11. N. N. Achasov, A. A. Kozhevnikov, M. S. Dubrovin, p. 443. et al., Int. J. Mod. Phys. A 7, 3187 (1992). 12. E. A. Kuraev and V. S. Fadin, Yad. Fiz. 41, 733 (1985). 9. V. M. Aul’chenko, B. O. Baœbusinov, T. V. Baœer, et al., 13. M. Benayoun, S. I. Eidelman, and V. N. Ivanchenko, Proc. Working Meeting on Physics and Detectors for Z. Phys. C 72, 221 (1996). DAPHNE (Frascati, Italy, 1991); M. N. Achasov, V. M. Aul’chenko, S. E. Baru, in Proc. Intern. Workshop on e+eÐ Collisions from φ to J/Ψ (Novosibirsk, 1999). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 173Ð182. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 196Ð206. Original Russian Text Copyright © 2000 by Kashurnikov, Rudnev, Gracheva, Nikitenko. SOLIDS

Phase Transitions in a Two-Dimensional Vortex System with Defects: Monte Carlo Simulation V. A. Kashurnikov, I. A. Rudnev*, M. E. Gracheva, and O. A. Nikitenko Moscow State Engineering-Physics Institute, Moscow, 115409 Russia *e-mail: [email protected] Received June 4, 1999

Abstract—The phase states and phase transitions in a system consisting of a two-dimensional vortex lattice with defects are studied by the Monte Carlo method. It is shown that a “rotating lattice” phase, which is an intermediate phase between the vortex crystal and vortex liquid phases, is present. The dependence of the temperature of the transition from the rotating lattice phase into a vortex liquid on the strength of the defect potential is determined. The current-voltage characteristics of the system are calculated at various temperatures for point, square, and linear defects. It is shown that the phase state of the system strongly affects its transport properties. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION presence of defects. Thus, the question of how melting of a vortex structure occurs in the presence of pinning centers In recent years, a great deal of attention has been has been discussed in our previous works [13Ð15]. It has devoted to investigations of the dynamics and phase been shown that there exist three stages of melting. transformations in the vortex lattice of high-tempera- First, at low temperature, a triangular lattice starts to ture superconductors (HTSCs) [1]. Experimental and break up far from the pinning centers. When strict long- theoretical investigations have been performed. The range order is completely absent, “islands” of the lat- complexity of the problems arising often makes it nec- tice form around the defects. The islands start to essary to use numerical simulation (see review [2]). It “rotate” around the secured pinning centers, forming a can now be positively stated that the Abrikosov lattice “rotating lattice” phase. At the third stage, complete [3] is not the only state of a vortex system in HTSCs. melting of the lattice occurs (vortex liquid phase) [11]. Actually, as the temperature increases, the vortex lattice melts and a vortex latticeÐvortex liquid phase transition The questions of the dynamic interaction of a vortex is observed. A convincing proof of the existence of a lattice with pinning centers in the presence of a trans- phase transition has been given, for example, in [4], port current and the current-voltage characteristics where the short-range correlations in the vortex lattice (IVCs) obtained are important for practical applica- were analyzed by numerical simulation and the melting tions of superconducting materials. The first Monte temperature of the vortex lattice was calculated. The Carlo calculations of the IVCs appeared only very melting temperature can be found from Lindemann’s recently [16Ð18]. The IVCs were calculated in the pres- criterion, i.e., by comparing the standard deviation of ence of a large number of defects with various potential the vortices with the period of the vortex lattice [5, 6]. energies (relative number of vortices) [16], but defects A convenient criterion of a phase transition has also with different activation energies and temperature been investigated in [5], where the melting temperature dependences of the currentÐvoltage characteristics was determined according to the features in the temper- were not investigated. ature dependence of the specific heat C(T) of the vortex system. The data, obtained by analyzing the IVCs of HTSCs, on various phase regimes of current flow [9] should be Investigations of phase transformations in a defect especially noted (see also [19Ð22]). Thus, pinned vor- field are of special interest. Thus, phase transitions tex glass regimes, a plastic flow regime of a vortex liq- from a triangular lattice of vortices into a vortex glass uid, and a moving vortex glass regime have been in the presence of a periodic system of pinning centers observed. These phase states of an Abrikosov lattice have been investigated [7, 8], the temperature depen- and the transitions between them are close to the phase dence of the pinning force has been taken into account transitions, which we have studied, between the states [9], and the temperature of the transition into a vortex of a rotating lattice and a vortex liquid only in the cur- glass has been estimated using various criteria [10Ð12]. rent state. Computational results for the IVCs and the One of the new and important results obtained is that critical current of two-dimensional HTSCs in various the picture of phase transformations changes in the phase states of the vortex structure are presented in [23].

174 KASHURNIKOV et al.

In the present paper, we present a systematic pic- where ture of the phase transitions and the dynamics of a 2 vortex structure in a two-dimensional system, simu- Φ d  r Ð r  H(r , r ) = ------0------K ----i------j-- lating a layer of the high-temperature superconductor i j πλ2( )µ 0 λ(T)  2 T 0 Bi2Sr2CaCu2Ox. A situation with a low density of point (2) defects, which act as pinning centers, is studied. Equi-  r Ð r  = U (T)K ----i------j-- , librium configurations of the vortex system that corre- 0 0 λ(T)  spond to various phase states will be demonstrated; the phase transition temperatures and their dependence on Up(ri) is the vortexÐdefect interaction energy at the site the defect density and the magnitude of the defect i, ni is the occupation number of the vortices (0 or 1) at Φ potential are calculated. Computational results are also the site i of the spatial lattice, 0 = hc/2e is the magnetic presented for the IVCs of model layered superconduc- flux quantum, K0 is a modified Bessel function of order tors, and they are compared with the experimental zero, d is the thickness of the superconducting layer, IVCs. It will be shown how the IVC is modified as a µ π × Ð7 λ λ ( )3.3 result of a change in temperature or an increase in the 0 = 4 10 H m, and (T) = (0)/ 1 Ð T/T c is number of defects, how the IVC depends on the depth the penetration depth of the magnetic field. of the defects, and how the form of the defects influences the IVC. 3. COMPUTATIONAL PROCEDURE All results will be obtained by Monte Carlo simula- For a specific calculation, parameters close to the tion of the vortex system. A detailed description of this characteristics of the layered Bi2Sr2CaCu2Ox HTSC method in application to calculations of vortex struc- were chosen: d = 2.7 Å, λ(T = 0) = 1800 Å and T = 84 K. tures is given in Secs. 2 and 3. c The external field was chosen in the range B ≈ 0.1 T, which corresponds to the real scale of the magnetic induction for which melting of a vortex lattice is 2. DESCRIPTION OF THE MODEL observed in bismuth HTSCs. The two-dimensional model is applicable in this case because the volume In most cases, the Monte Carlo (MC) method is vortices in Bi Sr CaCu O decompose in a wide range used to investigate the behavior of the lattice of Abriko- 2 2 2 x of fields (including at B = 0.1 T) and temperatures into sov vortices in HTSCs. flat two-dimensional “vortex pancakes” with weak cou- By choosing an ensemble, such as, for example, the pling between the planes (3DÐ2D transition). For this canonical ensemble, the MC method makes it possible reason, in the present case we can talk about the quasi- to calculate the observed physical quantities with a two- dimensionality of the vortex structure and con- fixed number of particles, volume, and temperature. sider an approximate two-dimensional model. The main advantage of the MC method is that it does The calculations were performed on a flat 200 × not require any approximations, and the interaction is 200 spatial grid with periodic boundary conditions. taken into account exactly. The system is described by The basic data were obtained for Nv = 150 vortices and a model Hamiltonian that is most convenient for the Nd = 5Ð200 defects (the two-dimensional defect density problem [2]. In the present work, the standard MC was ~1010Ð4 × 1011 cmÐ2). The defects were arranged method together with Metropolis’ algorithm for a randomly on the grid. The real vortex density corre- canonical ensemble was used [24]. sponding to a given field B was reproduced by varying the value of a division of the spatial cell so that the A two-dimensional system of Abrikosov vortices in period av of a triangular lattice of vortices would satisfy the form of model classical particles with a long-range Φ potential, which are arranged on a periodic square grid, the relation av = 2 0/B 3 , i.e., as the external mag- is studied. The discreteness of the spatial grid is chosen netic field varied, the vortex density varied. This is so that its period is much smaller than the period of a reflected in the model: As the magnetic field varies, the triangular lattice. In the present model, the contribution area of the region under study with a constant simula- of the interaction of vortices with one another and with tion parameter (the number of vortices) varies. There- defects to the energy is taken into account. In the calcu- fore, there was no need to include a direct interaction lations, the shortest distance between the vortices, tak- with an external field in the model (1), since the number ing account of the periodicity, is always chosen. On this of vortices is fixed. basis, the model Hamiltonian has the form We note that we are justified in treating the two- dimensional vortices as point quasiparticles because N N 1 the characteristic distance between the vortices (the lat- H = --∑H(r , r )n n + ∑ U (r )n , (1) tice period av) in a field B = 0.1 T is approximately 2 i j i j p i i i ≠ j i = 1 av ≈ 2000 Å, which is two orders of magnitude greater

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 PHASE TRANSITIONS IN A TWO-DIMENSIONAL VORTEX SYSTEM WITH DEFECTS 175 than the characteristic size of a vortex core ζ ≈ 20 Å. C, arb. units The linear size of the system is approximately 20000 Å. C, arb. units The defects were simulated in the form of point-like 0.2 wells with the same potentials, and the interaction energy of a vortex with a pinning center was taken to be 0.3 of the form 3 ( ) ( , ) U0 T δ 0.1 U p r T = Ð------r, ri. (3) 8 0.2 2 It should be underscored that the temperature dependence of the depth of the potential well (3) was chosen 1 to be the same as for the intervortex interaction potential. This is not always valid for a real situation, but the 0 5 10 15 20 T, ä dependence (3), as a rule, is used in computational 0.1 T T works (see, for example, [9, 18]). m m The real picture is such that the density of vortex filaments for the magnetic fields investigated is much greater than the defect density. In addition, in the calculations we employ the defect densities which are close 0 20 40 60 80 T, ä to realistic values [1]. The specific heat as a function of temperature was Tm Tm reproduced from the numerical results. The expression for the specific heat in terms of the fluctuation of the Fig. 1. Temperature dependence of the specific heat in a ᭺ ᮀ defect field: ( ) Up = 1 meV; ( ) Up = 100 meV. Inset: internal energy has the form (kB = 1) Effect of the vortex density Nv on the melting tempera- 2 2 〈 E 〉 Ð 〈 E〉 ture: (1) Nv = 50; (2) Nv = 100; (3) Nv = 150, Up = 70 meV. C(T) = ------. (4) T 2 Figure 1 shows examples of the calculation of C(T) according to the so-called Metropolis algorithm, which and the determination of the melting temperature T realizes a Gibbs distribution. The typical total number m of MC steps for the calculation was chosen to be 60 000. for two values of the defect potential: Up = 1 meV and The number of steps required for thermalization of the Up = 100 meV. The calculations were performed for system (when the system reaches stable configurations defect potentials ranging from 1 to 100 meV with Nv = with minimum energy) is 30 000, so that there was 150 vortices. enough time for the system to reach equilibrium in the To show that the number of vortices is macroscopi- chosen number of MC steps. The minimum change in cally large for this problem, a calculation of the specific an instantaneous configuration of the system is an ele- heat was performed for the same defect potential but mentary displacement of one vortex. By MC step we with a different number of vortices (for example, Nv = mean, as usual, a single displacement of all vortices in 50, 100, and 150). The results are presented in the inset the system, i.e., in a MC step each vortex was shifted on in Fig. 1. It is evident that even for Nv = 100 and Nv = the average once. Estimates showed that this number of MC steps is sufficient to calculate the specific heat with 150 the temperatures Tm are the same to within the limits of error, and as the number of vortices increases the minimum statistical errors. from 50 to 100, the peak in the specific heat shifts in the To investigate dynamical processes, a transport cur- direction of high temperatures and increases in height. rent was introduced into the system. In this case, a term For Nv = 150 vortices the temperature of the peak no due to the action of the Lorentz force on each vortex longer shifts. From this it can be concluded that the was added to the Hamiltonian describing the behavior number of vortex points Nv = 150 is macroscopically of the entire system. For an elementary motion of the δ Φ ∆ large for our problem. Test calculations of the specific vortex along the x axis, a term of the form U = 0J x heat for a different number of MC steps were per- was subtracted from the total energy, if the direction of formed in the same manner. motion of the vortex coincided with the Lorentz force The standard MC procedure consisted of the follow- acting on it, or added if the vortex moved in a direction ing. First, the vortices were distributed arbitrarily on a opposite to the Lorentz force. The transport current was grid in phase space, simulating the plane of the HTSC. directed along the y axis. For a nonzero current the vor- It is obvious that the initial arbitrary arrangement of tex filaments were allowed to move randomly over any vortices does not correspond to a minimum of the distance within the system. The voltage arising on the energy, so that the system is forced to evolve in order to boundaries of the system was calculated from the rela- sort the energetically most favorable configurations tion V = Bvdr, where B is the external field and

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(‡) (b)

(c)

Fig. 2. Form of the vortex system for weak pinning at T = (a) 1, (b) 5, and (c) 35 K. The total vertical and horizontal linear dimensions of the system are 2 µm.

τ vdr = Xs/ is the vortex drift velocity, i.e., the displace- of the current and does not reach the boundaries of the ment Xs of the center of mass of a vortex filament in system. unit time τ. Nominally, one elementary MC step is chosen as a unit of time, i.e., there is an uncertainty in the 4. COMPUTATIONAL RESULTS time scale that is reflected in the arbitrariness in the AND DISCUSSION choice of the dimension of the voltage scale. This uncertainty can later be removed by normalizing to a 4.1. Phase Transitions in the Vortex System real IVC. We shall consider the equilibrium configurations of the vortex system at various temperatures. In order to take account of the boundary conditions correctly, vortex filaments were not allowed to reach A virtually ideal triangular lattice is reproduced in the edge of the system by a single random displacement Fig. 2a (T = 1 K). We note that when the field was under the action of the current. This condition actually increased to 1 T, we observed a transition from a trian- determined the maximum possible current for the cal- gular into a square lattice at T = 2 K [15]. As one can see, not all defects are occupied by vortices; this attests culation and was always checked. Thus, in [22] a typi- to the stiffness of the lattice [1] at such temperatures. cal distribution of displacements of a vortex filament The defects would be occupied by vortices if the corre- relative to the center of mass (actually, the distribution spondence between the arrangement of the defects and of drift velocities) was presented for T = 70 K and the centers of the triangular lattice was ideal. At a tem- × 10 2 J/Jp = 5 (Jp = 5 10 A/m ). The distribution is ellip- perature of 3 K, all defects are occupied by vortices, soidal, and it is extended perpendicular to the direction which rigidly hold the lattice around themselves. As a

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 PHASE TRANSITIONS IN A TWO-DIMENSIONAL VORTEX SYSTEM WITH DEFECTS 177 result of the irregularity of their arrangement, the As already mentioned, an alternative method is to defects seemingly drag around pieces of the lattice, calculate the specific heat C(T) of the system taking breaking up the lattice far away from themselves. The account of the fact that, according to the fluctuation- boundaries between the coherent regions arising are dissipation theorem, this quantity is related with the melted, and the lattice loses its stiffness. We note that a fluctuation 〈E〉 of the total energy by the relation (4). similar “orientational melting” was observed in [24] in We reproduce the function C(T) for the case of weak pin- a model planar problem. As the temperature increases ning also in Fig. 1. A feature can be seen at Tm1 = 3 K, cor- further (up to 5 K, Fig. 2b), it is evident that the responding to the point of the transition into a rotating “islands” of the triangular lattice, which are confined lattice state. In addition, a quite sharp “front” is around the defects and possess mobility relative to a observed, proving that this transition is of a thermody- defect as a rotation axis, seemingly rotate, smearing the namic character. The function C(T) is divided sharply vortex density into concentric circles with maxima on into three stages: at temperature T = 3 K growth of the the coodination spheres. A region depleted of vortices specific heat corresponding to Tm1 starts (Fig. 2a); next, is formed near the defects themselves at a distance of after this weakly expressed maximum, a sharp jump is one coordination sphere, corresponding to the period of observed at T = 5 K; and, finally, the final transition into the ideal triangular lattice, since an immobile pinned the vortex liquid state corresponding to a sharp peak at vortex does not allow other vortices to approach closer. T = Tm2 = 7.5 K occurs. A calculation of the specific In this new phase state (the transition temperature is heat for strong pinning gives two clear peaks in the estimated to be T = 3 K), which we shall call a rotat- function C(T) that correspond to the temperatures Tm1 = m1 ing lattice by analogy to a floating lattice [7], there is 2.5 K and Tm2 = 73 K, which corresponds completely to still a long-range order in the coherent regions much the previously indicated differences of the weak and larger than the average distance between the vortices strong pinning regimes. and rigidly pinned on the pinning centers. As the temperature increases further (right up to T = T = 7.5 K), m2 4.2. Dependence of the Melting Temperature the vortices begin to detach from the defects, the coher- on the Strength of the Defect Potential ent regions break up, and, for example, at T = 35 K (already far from the transition point), a completely We have already examined two limiting cases: melted vortex liquid is observed (Fig. 2c). strong and weak pinning. To analyze the dependence of Thus, the melting of the lattice in the presence of the phase-transition temperature on the magnitude of pinning centers is a three-phase process: triangular lat- the defects, we calculated the temperature dependence ticeÐrotating latticeÐvortex liquid. of the specific heat of the system for various values of We also investigated melting in the presence of the defect potential. The transition temperature at defects with a 30 times higher intervortex attraction which the vortex lattice, divided into islands pinned on energy (i.e., U (T = 2 K) = 0.115 eV). The basic differ- defects, transforms into a vortex liquid phase can be p determined accurately according to the characteristic ences of this situation from the melting examined above in weaker pinning are as follows. features in the function C(T). This is the temperature Tm (or Tm2 according to the terminology already used (a) The transition point from a triangular lattice to a above). rotating lattice shifted in the direction of lower temperatures ( T = 2 K). Thus, displacement of the lattice Having determined Tm, we obtained the dependence m1 already starts at T = 2 K, while for weak pinning and the of the transition temperature on the defect potential same temperature we see an immobile triangular lat- (Fig. 3). It is evident that as the defect potential tice. Pinning centers with strong attraction and mini- increases, the melting temperature increases monoton- mum thermal activity of the vortices immediately break ically. The results obtained can be interpreted using a up the regular lattice into parts which then drift behind simple physical picture of the processes occurring with them. a vortex lattice in a HTSC in the presence of defects. Consider a vortex lattice with a certain defect potential. (b) Strong pinning results in an expansion of the Three interactions compete in it: the interaction of vor- temperature range of the rotating lattice phase. Almost tices with one another, which determines the stiffness ideal twisted regions at intermediate temperatures are of the lattice; thermal fluctuations; and, the interaction observed right up to T = 70Ð80 K. of vortices with defects. At low temperature the struc- For a quantitative investigation of the points of ture of the lattice is preserved. A vortex pinned in the phase transitions with melting of the vortex lattice, potential well of the defects is localized on the well ordinarily either the structure factor S(q) is calculated with a high probability. As the temperature increases, to estimate the degree of long-range order or the hexag- the lattice starts to lose its stiffness, at first far from the onal parameter S6 [5] is calculated in order to analyze defects. A vortex pinned on defects holds near it an the short-range correlations (equivalent to the Linde- island of a coherent region of the lattice of vortices mann criterion for the standard deviations). (rotating lattice phase [13]). Far from a defect the lat-

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Tm, ä Mirr is the width of the hysteresis loop and v is the scan 80 rate of the magnetic field. Three regions with different magnetic flux dynamics clearly stood out in the curve of S(T) versus H/Hmax. The first region, in the opinion of the authors of [25], was qualitatively consistent with 60 the elastic theory of a collective creep for small vortex bundles; the second was attributed by the authors to the appearance of plastic deformations in the vortex lattice; 40 and, the third corresponded to the appearance of a vortex liquid. The experimental proof of the presence of an intermediate phase in the phase diagram of bismuth HTSCs 20 was given in [26], where it was stated that the solid phase below the line of melting “is obviously divided into two phases.” In the first phase (I) the vortices are immobile, in the second phase (II) they begin to move, 0 0.04 0.08 Up, eV and as temperature increases further, a transition into the vortex liquid occurs. The phase I was determined to Fig. 3. Melting temperature versus the strength of the defect be a “weakly disordered Bragg glass,” and the phase II potential. was determined to be a “strongly disordered Josephson glass.” The effect of the point pinning on the system of vortices and the formation of a reverse line in the phase tice starts to break up. As temperature increases further, diagram were qualitatively investigated in [27]. It was thermal fluctuations grow and the size of the “confined” shown there that in the two-dimensional case the point region decreases. For this reason, the probability that a pinning influences the vortex liquid in a definite region vortex (together with the coherent region that is con- below the line of melting. In our case, this region cor- fined) leaves the potential well increases. Finally, at responds to the rotating lattice phase (orientational even higher temperatures, the islands break up, the vor- melting)—the phase II. tices detach from the defects, and complete melting of the vortex system occurs. It is obvious that for high val- In works on the experimental observation of the spe- ues of the defect potentials a defect will confine vorti- cific heat of a vortex system, peaks corresponding to a ces near itself at higher temperatures; this is what transition into a vortex liquid are also clearly seen in the temperature dependences of the specific heat of explains the increase in Tm with increasing depth of the potential well of defects in a complicated pattern of the YBa2Cu3O7 Ð δ single crystals [28, 29], corresponding to rotating lattice phase. A separate question is why a transition into a vortex liquid. The question of the nature of the transition remains open. However, know- slower growth of Tm is observed near the critical region (Fig. 3). We believe that this is explained simply by the ing that experimentally, together with a peak in the specific heat, a jump is observed in the magnetization of temperature dependence of the depth of the potential the system, it can only be inferred that this is a first- well of the defects that appears in the expression (3). order thermodynamic transition [28, 29]. Moreover, the calculations show that renormalization of the depth of the potential well, taking account of the temperature, “rectifies” the dependence Tm(Up). 4.3. Current-voltage Characteristics It should be noted especially that experiments have of a Vortex System repeatedly shown the existence of an intermediate The currentÐvoltage characteristics for systems con- phase, investigated in our model, with dynamics differ- taining from 1 to 100 defects were calculated at temper- ent from that of a triangular lattice or a vortex liquid. atures of 10, 20, 30 K, and so on (up to 83 K). Near the For example, in the experimental work [25] performed critical temperature the characteristics were calculated on TmBa2Cu3O6.9 crystals, a scaling behavior of the with an interval of 1 K right up to the critical region. normalized relaxation rate of the screening currents The typical IVCs are presented in Fig. 4 for T = 20 K was obtained: and different defect density. These data were compared with the experimental S H IVCs (for low currents) for Bi2Sr2CaCu2Ox films irradi- --- = f s------, T Hmax ated with high-energy ions [30]. Comparing the experimental and theoretical IVCs permits determining the where real scale of the electric field intensity (i.e., the quantity × Ð2 Ep). Thus, in our case we have Ep = 5 10 V/m, which dlgM dH S = ------i--r--r , v = ------, indeed corresponds to the actually observed values. dlgv dt Moreover, if the magnetic field is known, the relaxation

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 PHASE TRANSITIONS IN A TWO-DIMENSIONAL VORTEX SYSTEM WITH DEFECTS 179 τ time of the vortices can be estimated. In our calcula- J/Jp tions it is τ Ӎ 10Ð12 s, which also agrees well with phys- 5 ical estimates [1]. There is another area of agreement. According to 5 4 3 2 1 our estimates, the characteristic critical currents 4 obtained as a result of the calculation differ from the real currents in HTSCs by approximately a factor of 5. But, this numerical factor reflects the approximate fraction of superconducting layers (of thickness 2.7 Å) in 3 10 the total volume of the unit cell of the compound Nd = 300 J/Jp Bi2Sr2CaCu2Ox! Let us analyze the computational data. It is evident 2 that as the defect density increases, the IVC rectifies, 1 Nd = 100 increasingly demonstrating simple ohmic behavior of the system. This also occurs as the temperature 1 approaches the critical value (see Fig. 5 below). Analy- sis of the initial part of the IVC (at low currents) leads 0.1 to the following result: The IVC in double-logarithmic 0.001 0.01 0.1 1 E/Ep coordinates with the current and voltage (see inset in 0 Fig. 4) is strictly linear, which confirms the presence of 0.5 1.0 E/Ep magnetic flux creep in this regime. Figure 5 shows the IVCs of a system with a fixed Fig. 4. Typical IVCs for T = 20 K for various numbers of defect density but at different temperatures. defects Nd: (1) 1, (2) 10, (3) 40, (4) 60, (5) 100. Inset: IVCs for Nd = 100 and Nd = 300 in double-logarithmic coordi- Our computed data on the motion of a flux system nates. in a defect field in the presence of a current demonstrate a different behavior of the IVC depending on the phase state of the system. Thus, in Fig. 5 we see two groups J/Jp of curves, nominally separated by the temperature 5 Ӎ boundary Tm2 70 K. Visual analysis of the distribution density of the vortices (as in Fig. 2) shows that a < rotating lattice occurs for T Tm2, while a vortex glass 4 > occurs for T Tm2. In the rotating lattice phase, the IVCs change little with increasing temperature. This is explained by the still strong interaction with the pin- > 3 ning centers. Conversely, at temperatures T Tm2 we see a strong effect of the temperature (virtually equidis- tant increase of the voltage with the temperature increasing by only 2 degrees) right up to the critical 2 region. Thus, the observed differences in the temperature behavior of the IVCs of a real HTSC can be explained by the different phase states of the vortex 1 system which are described in Sec. 4.1. We calculated the IVCs of compounds not only with point-like defects with the same potentials but also with defects with different potentials. The potential was cho- 0 0.2 0.4 0.6 0.8 1.0 sen in the range from 20 to 50 meV. For such defects the E/Ep qualitative form of the IVC did not change. In this case two more important dependences were obtained: the Fig. 5. IVCs for Nd = 100 and various temperatures (in dependence of the critical current on the temperature degrees Kelvin): T = (᭿) 10, (᭺) 20, (᭡) 40, (+) 50, (᭜) 60, and the dependence of the differential resistance ρ = (᭢) 70, (ᮀ) 72, (᭹) 74, (᭝) 76, (᭞) 78, (᭪) 80, (Ճ) 80, ( ) 83. ∂ ∂ E/ J on the temperature. The critical current Jc was chosen just as it is chosen in an experiment Ð according ρ(T), shown in Fig. 7, on the basis of the results of the to a prescribed voltage threshold (in our case E/Ep = 0.1). well-known works [31, 32], where the pinning of a vor- The dependence Jc(T) is presented in Fig. 6. A kink can be seen in the dependence. This kink corresponds to a tex liquid in an HTSC was studied. It was shown ana- transition from the rotating lattice phase into a vortex lytically that in a vortex liquid in a HTSC there exist liquid phase. It is helpful to analyze the dependence two temperature ranges with different temperature

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J/J dependences of the resistance. At high temperatures the p ρ resistivity is equal to some value flow which is characteristic for the flux flow regime. As the temperature decreases, a transition occurs into a partially pinned, 2 thermally activated, flux flow regime with a linear resis- ρ tivity much less than flow and exponentially decreasing with temperature. It was found that the resistivity 1 which we calculated from the IVC follows an exponential law at temperature above the melting temperature of the vortex lattice; this is demonstrated in Fig. 7. In addition, the slope of the straight line in this figure is proportional to the activation energy of the vortex system. Therefore it is evident that the dependences constructed agree qualitatively with the theory of thermally 0.3 activated magnetic flux flow [31, 32]. The effect of the degree of disorder γ on ρ was also 0 20 40 60 80 examined in [31, 32], and it was stated that in the region T, ä of thermally activated flux flow the resistivity decreases Fig. 6. Critical current versus the temperature for point with increasing γ according to the law ρ ∝ γÐ1. We defects, Nd = 100. attempted to construct this dependence, taking as the degree of disorder the defect density Nd. Figure 8 shows lnρ [arb. units] Ð1 the dependence ρ(N ). It is evident that, qualitatively, 3.6 d the resistivity decreases with increasing defect density, but nonlinearly. The latter behavior seems to reflect the 3.4 fact that the degree of disorder is not linearly related with the defect density. In concluding this section, we shall examine the 3.2 effect of the form and size of the defects on the IVCs and the critical current of our vortex system, simulating various technological and natural defects occurring in 3.0 real conductors. Any defects, differing from point defects, can be represented as a set of a definite number of point defects. There arises the question of how point 2.8 defects arbitrarily distributed throughout the system and collected in clusters in the form of, for example, 2.6 squares or lines affect the IVCs. For this, we introduce 0 0.1 0.2 0.3 0.4 0.5 into the system 90 point defects by three different 1/T, ä–1 methods: Fig. 7. Resistivity versus the temperature. (1) ninety randomly distributed point defects with the same potentials (which can simulate vacancies and interstices); ρ, arb. units (2) ten square defects consisting of nine point 0.31 defects (which corresponds to pores or clusters); (3) ten linear defects consisting of nine point defects 0.30 (dislocations or twinning boundaries). Thus, we have the same density of point defects (90), but with 10 macrodefects in the second and third 0.29 cases. Likewise, to compare the same number of macrodefects and point defects, we add a fourth case: 0.28 10 point defects. For each situation, the IVCs at different temperatures were constructed. From these curves, the four 0.27 temperature dependences of the critical current shown in Fig. 9 were obtained. 0 0.01 0.02 0.03 0.04 1/ Let us see how the effectiveness of defects varies Nd with temperature. At low temperature there is virtually Fig. 8. Resistivity versus the number of defects. no difference in the square and linear defects, and point

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J/Jp ence of pinning the system passes through a rotating- lattice intermediate phase, whose existence can be 2 explained by the competition between two processes 1 against the background of an increasing thermal mobility of the vortices: the high stiffness of a triangular lat- 1 2 tice competes with the quite strong interaction with pinning centers, striving to fix a vortex together with its 3 surroundings, which results in the lattice breaking up into islands around defects. In layered HTSCs, which possess a lower lattice stiffness because of the weak bond between the superconducting layers, the pinning force competes with the elastic properties of the vortex structure, expanding the region of the rotating-lattice Ӎ phase up to the range T Tc. In conventional supercon- 0.1 ductors, the high stiffness of the volume Abrikosov lat- 0 20 40 60 80 tice weakens only near T [33], so that the temperature , ä c T range of the phase state mentioned is much narrower. Fig. 9. Critical current versus the temperature for various The fact that the temperature of the transition into the defects: (1) 90 point; (2) 10 linear; (3) 10 square. rotating phase is low (2Ð3 K) means that real and, therefore, defective HTSCs in a mixed state virtually immediately are in this state. This results in substantial defects give an approximately 20% higher critical cur- broadening of the IVCs, even in weak magnetic fields. rent. In the temperature range 20Ð60 K the effective- This could explain sometimes the absence of an Abri- ness of the point defects increases rapidly. This is kosov lattice, tested according to the broadening of the expressed as an increase in the critical current of a sys- IVCs, sometimes observed in thin films of conventional tem with point defects by almost a factor of 2 for square superconductors [33]. and linear defects. At the same time, a difference The defect potential strongly influences the temper- appears in the effectiveness of linear and square ature of the transition between the rotating-lattice and defects. Finally, near the critical temperature the effec- vortex-liquid phases. As the depth of the potential well tiveness of all defects decreases, and the difference in of a defect increases, the temperature of this transition the type of defects completely disappears. increases and the temperature range where an interme- The result obtained can be understood on the basis diate rotating-lattice phase occurs increases. of the previously considered phase states of a vortex system. Indeed, at low temperatures the vortex system The calculations of the IVCs and the temperature is close to a state of a vortex crystal, collective pinning dependences of the critical current for various types of is effective, and therefore the number and type of defects showed that the phase state of a vortex system defects play no substantial role. Conversely, at high has a determining effect on the transport properties of temperatures the vortex system is melted, pinning is of the system. It is in the rotating-lattice phase that differ- an individual character, and the type of defects once ences are observed in the effectiveness of point, square, again is not a determining factor. Differences in the and linear defects, which disappear when the vortex lat- types of defects are most noticeable only in the inter- tice melts. mediate region corresponding to the rotating lattice It was also shown, on the basis of a calculation of phase. Point defects arbitrarily distributed through the the resistivity versus the temperature, a thermally acti- system form centers of islands of a rotating lattice, on vated flux flow regime occurs. the whole confining the maximum number of vortices. This stabilizes the entire lattice. The total number of confined vortices is smaller for square and linear ACKNOWLEDGMENTS defects, since the possible centers of rotation are localized in macrodefects. We note that this is picture that This work was performed with the financial support we observe in the calculation of the visual distribution of the State Scientific and Technical Program “Topical of vortices at different temperatures for different Directions in Condensed-State Physics” (subprogram defects. “Superconductivity,” projects no. 96026 and no. 99011), and the Federal Target Program “Integration” (projects no. A-0133 and no. A-0099). 5. CONCLUSIONS We shall summarize the main results of this work. We investigated the phase transition triangular latticeÐ REFERENCES vortex liquid in a two-dimensional vortex system simu- 1. G. Blatter, M. V. Feigelman, V. B. Geshkenbein, et al., lating superconducting layers in an HTSC. In the pres- Rev. Mod. Phys. 66, 1125 (1994).

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2. M. E. Gracheva, M. V. Katargin, V. A. Kashurnikov, et al., 19. A. N. Lykov, C. Attanasio, L. Maritato, et al., Supercond. Fiz. Nizk. Temp. 23, 1151 (1997) [Low Temp. Phys. 23, Sci. Technol. 10, 119 (1997). 863 (1997)]. 20. B. Khaykovich, M. Konczykowski, E. Zeldov, et al., 3. A. A. Abrikosov, Zh. Éksp. Teor. Fiz. 32, 1442 (1957) Phys. Rev. B 56, R517 (1997). [Sov. Phys. JETP 5, 1174 (1957)]. 21. C. Goupil, A. Ruyter, V. Hardy, et al., Physica C 278, 23 4. J. W. Schneider, S. Schafroth, and P. F. Meier, Phys. Rev. (1997). B 52, 3790 (1995). 22. M. E. Gracheva, V. A. Kashurnikov, and I. A. Rudnev, 5. S. Ryu, S. Doniach, G. Deutscher, et al., Phys. Rev. Lett. Fiz. Nizk. Temp. 25, 148 (1999) [Low Temp. Phys. 25, 68, 710 (1992). 105 (1999)]. 6. K. Yates, D. J. Newman, and P. A. J. de Groot, Physica C 23. K. Binder and D. W. Heerman, Monte Carlo Simulation 241, 111 (1995). in Statistical Physics: An Introduction (Nauka, Moscow, 7. C. Reichardt, J. Groth, C. J. Olson, et al., Phys. Rev. B 1995; Springer, Berlin, 1997). 54, 16108 (1996). 24. Yu. E. Lozovik and E. A. Rakoch, Pis’ma Zh. Éksp. Teor. Fiz. 65, 268 (1997) [JETP Lett. 65, 282 (1997)]. 8. D. Ertas and D. R. Nelson, Physica C 272, 79 (1996). 25. G. K. Perkins, L. F. Cohen, A. A. Zhukov, et al., Phys. 9. S. Ryu, M. Hellerqvist, S. Doniach, et al., Phys. Rev. Rev. B 55, 8110 (1997). Lett. 77, 5114 (1996). 26. D. T. Fuchs, E. Zeldov, T. Tamegai, et al., Phys. Rev. 10. R. E. Hetzel, A. Sudbo, and D. A. Huse, Phys. Rev. Lett. Lett. 80, 4971 (1998). 69, 518 (1992). 27. D. R. Nelson, Physica C 263, 12 (1996). 11. K. Yates, D. J. Newman, and P. A. J. de Groot, Phys. Rev. B 52, 13149 (1995). 28. A. Schilling, R. A. Fisher, N. E. Phillips, et al., Phys. Rev. Lett. 78, 4833 (1997). 12. R. Sasik and D. Stroud, Phys. Rev. B 52, 3696 (1995). 29. M. Roulin, A. Junod, A. Erb, et al., Phys. Rev. Lett. 80, 13. M. E. Gracheva, V. A. Kashurnikov, and I. A. Rudnev, 1722 (1998). Pis’ma Zh. Éksp. Teor. Fiz. 66, 269 (1997) [JETP Lett. 30. V. F. Elesin, I. A. Esin, I. A. Rudnev, et al., Sverkhpro- 66, 291 (1997)]. vodimost: Fiz., Khim., Tekh. 6, 807 (1993). 14. M. E. Gracheva, V. A. Kashurnikov, and I. A. Rudnev, 31. V. M. Vinokur, M. V. Feigelman, V. B. Geshkenbein, et Phys. Low-Dim. Struct. 8/9, 125 (1997). al., Phys. Rev. Lett. 65, 259 (1990). 15. M. E. Gracheva, V. A. Kashurnikov, and I. A. Rudnev, 32. V. M. Vinokur, V. B. Geshkenbein, A. I. Larkin, et al., Phys. Low-Dim. Struct. 9/10, 193 (1998). Zh. Éksp. Teor. Fiz. 100, 1104 (1991) [Sov. Phys. JETP 16. E. Bonabeau and P. Lederer, Phys. Rev. Lett. 77, 5122 73, 610 (1991)]. (1996). 33. A. V. Nikulov, D. Yu. Remisov, and V. A. Oboznov, Phys. 17. R. Sugano, T. Onogi, and Y. Murayama, Physica C 263, Rev. Lett. 75, 2586 (1995). 17 (1996). 18. K. Moon, R. T. Scalettar, and G. T. Zimanyi, Phys. Rev. Lett. 77, 2778 (1996). Translation was provided by AIP

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 183Ð193. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 207Ð217. Original English Text Copyright © 2000 by Nagaev. SOLIDS

Bound Ferromagnetic and Paramagnetic Polarons as an Origin of the Resistivity Peak in Ferromagnetic Semiconductors and Manganites¦ E. L. Nagaev Institute for High Pressure Physics, Troitsk, Moscow oblast, 142092 Russia; e-mail: [email protected] Received 16 June 1999

Abstract—A theory of resistivity is developed for ferromagnetic semiconductors, possibly, including manganites. The theory is based on analysis of the interaction of the free and bound charge carriers with the magnetization of the crystal. The temperature dependence of free energy for nonionized donors and free electrons is calculated for the spin-wave and paramagnetic regions. In addition to the trapping by the ferromagnetic ﬂuctu- ations (the ferromagnetic polarons), the electron trapping by the random magnetization ﬂuctuations as T ∞ is taken into account (the paramagnetic polarons). For the nondegenerate semiconductors, the theory makes it possible to explain a nonmonotonic temperature dependence of the activation energy, with the value for T = 0 being lower than that for T ∞. For degenerate semiconductors, the theory explains a metalÐinsulator transition that occurs with increasing temperature in samples with relatively low charge carrier density. If the density is larger, a reentrant metalÐinsulator transition should take place, so that the crystal is highly conductive as T ∞. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION via free charge carries is exponentially small in nondegenerate semiconductors, the average ferromagnetic The present paper deals with ferromagnetic semi- ordering is supported only by the superexchange. In conductors, both degenerate and nondegenerate. All contrast, the magnetization near the donor is supported these semiconductors display a resistivity peak in the also by the indirect exchange via the donor electron. vicinity of the Curie point TC. The heavily doped semi- Hence, at finite temperatures, the donor magnetization conductors displaying the metallic conductivity at T = 0 is destroyed to a lesser degree than the average magne- can remain in the insulating state up to very high tem- tization. peratures after passing this peak; i.e., the metalÐinsula- tor transition takes place with increasing temperature. The donor overmagnetization means that, with Still more heavily doped semiconductors return to a increasing temperature, the donor level depth first highly conductive state after passing the peak, i.e., the increases, since the conduction band bottom rises much metalÐinsulator transition is reentrant in them. The more rapidly than the donor level. But, with further nondegenerate semiconductors have a temperature- increase in the temperature, the local ordering begins to dependent activation energy for the conductivity. This disappear. The level depth will then decrease with energy passes a maximum in the vicinity of the Curie increasing temperature. As a result, the charge carrier point. The high-temperature activation energy exceeds density is minimal and the resistivity is maximal at a its low-temperature value. All these materials display a certain temperature (to avoid a misunderstanding, the colossal negative magnetoresistance [1]. magnetization dependence of the donor level depth and of the free charge carrier density was not investigated in In what follows, we talk about the donors and con- [4, 5]). duction electrons, although all the results obtained below remain in force for the holes and acceptors. Unfortunately, the calculation [1Ð3] was carried out A semiqualitative explanation of the properties of non- under the assumption that the electron dwells only at degenerate ferromagnetic semiconductors was given in the magnetic atoms nearest to the donor atom, which [1Ð3]. The point is that the electron levels decrease with was also assumed in [4, 5]. On the other hand, the increasing magnetization. But the local magnetization orbital radius must depend on the magnetization for the in the vicinity of a nonionized donor is higher than the same reason as in the antiferromagnetic semiconduc- average magnetization in the crystal, which was first tors [6]: at finite temperatures, the overmagnetized pointed out in [4, 5]. In fact, since the indirect exchange region close to the donor is a potential well for the donor electron. Hence, the electron is attracted to the ¦This article was submitted by the author in English. donor not only by the Coulomb potential, but also by

184 NAGAEV the magnetic potential well. As a result, the orbital only for those systems in which the donor density radius must be magnetization-dependent and should be markedly exceeds the density of the metalÐinsulator found by a self-consistent calculation similar to that transition at T = 0. If this is not the case, the reverse carried out for the antiferromagnetic semiconductors in transition is prohibited due to the paramagnetic ferrons. [6]. In complete analogy with the case of donors in anti- This result agrees with the experimental data on degen- ferromagnetic semiconductors, where magnetized erate ferromagnetic semiconductors presented in [1]. regions arise close to the nonionized impurities, one It should be pointed out that earlier explanation of can use the term “the bound magnetic polaron”, or “the the temperature-induced metalÐinsulator transition in bound ferron” for the overmagnetized donors. degenerate ferromagnetic semiconductors was given in Calculations of the bound ferrons in the spin-wave terms of a modified Mott theory, in which only the sta- region will be carried out below. In this case the over- bility of the metallic state was investigated [1, 7]. But magnetized region is determined by enhanced ferro- this approach seems to be less fruitful than that used magnetic correlations in the vicinity of the nonionized here. In particular, it does not lead to the insulating state donor. The bound ferron radius and free energy are as T ∞, i.e., it does not allow one to explain some determined. This allows us to find the free charge car- essential features of the degenerate ferromagnetic semi- rier density and its activation energy as functions of conductors. In addition, it can be used only if the donor temperature. In essence, this part of the paper develops density is very close to the density at which the Mott ideas set forth for antiferromagnetic semiconductors in transition takes place at T = 0. The approach used here is [1, 2], although it requires a quite different calculation more general, allowing one to overcome these limita- procedure. tions and to prove similarity of the physical mecha- In addition to the already known low-temperature nisms leading to the resistivity peak in the nondegener- bound ferron, a new type of bound ferron will be con- ate ferromagnetic semiconductors and to the metalÐ sidered. It exists in the limit T ∞ and can be called insulator transitions in degenerate ferromagnetic semi- the paramagnetic bound polaron (ferron). While the conductors. But some problems treated in [1, 7] remain ferrons investigated so far are related to a self-consis- beyond the scope of our paper (e.g., the charge carrier tent enhancement of the ferromagnetic correlations in mobility). Thus, the present approach and that adopted in the region of the electron localization, the correlations [1, 7] are complementary. are absent here, and only the fluctuating magnetization of the region increases with decreasing size; it is of the 2. INDIRECT EXCHANGE HAMILTONIAN 1/2 order of 1/N I , where NI is the number of magnetic FOR A NONIONIZED DONOR atoms over which the donor electron is spread. To analyze the magnetic properties of the nonion- Although the mean local magnetization remains ized donors, it is advisable to begin with construction of zero, the electron spin adjusts to the fluctuating magne- the magnetic Hamiltonian describing the indirect tization, fluctuating jointly with it and thus ensuring the exchange via the donor electron. It must differ strongly gain in the exchange energy between the electron and from the RKKY indirect Hamiltonian since the latter the magnetic atoms. This means that the tendency assumes that the conduction electrons are completely arises for the electron to be concentrated inside a region spin-depolarized to the zero approximation. Certainly, as small as possible in size. This tendency competes the situation with a single donor electron is quite oppo- with the Coulomb interaction and kinetic energy in site. determining the equilibrium orbital radius. The shrink- As usual, the sÐd model is used. The Hamiltonian of ing of the electron orbit caused by the magnetization the system in the coordinate representation is given by fluctuations can lead to a considerable lowering of the ( ) ( ) donor level as compared with its depth at T = 0. Hence, H = Hs r + Hsd r + Hdd, the low-temperature activation energy for the resistivity is less than the high-temperature activation energy. The ∆ e2 H = Ð ------Ð ---- , uncorrelated fluctuations possibly also can trap the free s 2m εr charge carriers, but the binding energy of the free fluc- (1) tuation polarons, if it is nonzero, should be very small. H = ÐA (S s)D(r Ð g), Calculations carried out for nondegenerate ferro- sd ∑ g magnetic semiconductors are generalized for the g degenerate semiconductors in the following way. The free energy of the impurity metal consisting of the ion- I H = Ð--∑(S S ∆), ized donors and electrons is calculated and compared dd 2 g g + with the free energy of separate donors. If the former g, ∆ energy at T = 0 is lower than the latter value, then an where Sg is the d-spin of the atom g, s is the conduction increase in T can cause a crossover, which means that electron spin, D(r Ð g) is equal to unity inside the unit there is a transition to the insulating state. As for the cell g and zero outside it, m is the electron effective reverse transition at elevated temperatures, it can occur mass, ε is the dielectric constant, ∆ is the vector con-

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BOUND FERROMAGNETIC AND PARAMAGNETIC POLARONS 185 ប necting the nearest neighbors, = 1. The sÐd exchange In the particular case of w(g) = 1/NI, the system of integral A is assumed to be positive. equations (5) can be solved exactly. Accordingly, we Ӷ use the following relations, which are valid for any In what follows, the inequality AS W is assumed z z to be met, where S is the d-spin magnitude and W is the function f(S ) of S : conduction band width. This inequality is certainly met Ð z z Ð Ð + 2 z z in rare-earth semiconductors (EuO, EuS, and others) S f (S ) = f (S + 1)S , L L = L Ð L (L + 1). [1] and can also be satisfied in transition metal com- Ð pounds. In particular, it can possibly be met in colossal They follow from the definition of the S operator and from the commutation rules for the spin operators. The magnetoresistance manganites, although the experi- ≡ mental situation about them is not yet clear. Many inves- exact result for w(g) = const 1/NI is tigators believe that the holes in these systems move 2 2 not over the Mn ions but over the oxygen ions [8Ð10].  A  φ  A   2 1 φ − E + ------ = ------ L + -- , In this case, the s d exchange is relatively weak and the 2N I 2N I 4 band width is relatively large, in contrast to the double- exchange case where holes move over the Mn ions. which corresponds to the effective magnetic Hamilto- As, in the usual theory of indirect exchange, the adi- nian abatic approximation is used when, in dealing with the s-electron, the d-spins are considered as the classical A 1 ± 2 1 HmI = -------- L + -- . (6) vectors. In the first approximation in AS/W, the wave 2N I 2 4 function of the system cam be separated into the orbital part and the spin part: The double sign in equation (6) corresponds to two possible spin projections of the conduction electron z z Ψ(r, {S }, σ) = ψ(r)η({S }, σ), (2) onto the total moment L of the d-spins. As should be the η case, the exact eigenvalues of the Hamiltonian (6) are where the normalized magnetic wave function of the (ÐAL/2) and A(L + 1)/2. set of the d-spin variables {Sz} and s-electron spin variable σ will be specified below as a functional of the For an arbitrary w(g), the system of equations (5) donor ground-state orbital wave function ψ. After con- can be solved with accuracy of 1/2SNI, where NI is the structing the wave equation with Hamiltonian (1) and number of magnetic atoms over which the localized wave function (2), multiplying it by ψ(r) on the left electron is spread. The terms of this order (omitted side, and integrating over r, we obtain the wave equa- below) arise as a result of commuting LÐ and (E Ð tion for the magnetic subsystem ALz/2)Ð1 after exclusion of χ from the second equation in (5). In this case, one obtains the following relation η ( )η 3 ψ ψ with the accuracy indicated above: Hav = E Ð EI , EI = ∫d r Hs , (3) ( )( ) ( ) ψ2( ) 3 Hav = ÐA∑w g Sgs , w g = g a , A H = ±--- ∑w(g)w(f)(S S ). (7) mI 2 g f where EI is the energy of the s-electron bound to the g, f impurity, and a is the lattice constant. In contrast to the RKKY Hamiltonian, Hamilto- The eigenfunction of Hav is represented in the form nians (6) and (7) are linear and not quadratic in A. More importantly, they not only describe the bilinear η({Sz}, σ) = φ({Sz})δ(σ, 1 ⁄ 2) (4) exchange, but the multispin exchange as well in which + χ({Sz})δ(σ, Ð1 ⁄ 2), up to NI(NI Ð 1) spins take part simultaneously. Hamil- tonian (7) contains also the biquadratic terms and terms where δ(σ, ±1/2) is the s-electron spin wave function of higher orders in the scalar product of the spins, as with δ(x, y) = 1 for x = y and δ(x, y) = 0 for x ≠ y, (φ, χ) well as terms of still more complicated structure. This is the two-component wave function of the d-system. is seen from equation (7), if one separates the diagonal Using equations (3) and (4), we can represent the terms with g = f and then expands equation (7) in terms of the nondiagonal terms with g ≠ f. wave equation in the form (EI is omitted) The strength of indirect exchange between the spins AL+  ALz does not depend on the distance between them, but ------φ + E Ð ------χ = 0, 2  2  depends on the distance of each d-spin from the donor atom. Obviously, Hamiltonians (6) and (7) are isotro- Ð z pic, and there is no gap in the spectrum for the uniform AL χ  AL  φ ------+ E + ------ = 0, (5) spin rotation, as should be the case. At T = 0, Hamilto- 2 2 nians (6) and (7) correctly reproduce the s−d exchange ± energy for the complete ferromagnetic ordering (the L = ∑w(g)S , L = Lx ± iLy. g latter with accuracy of 1/2SNI).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 186 NAGAEV 3. DONOR STATES AND THE RESISTIVITY ized exactly. The perturbation theory also cannot be PEAK IN NONDEGENERATE FERROMAGNETIC used here. To carry out an approximate calculation, we SEMICONDUCTORS (SPIN-WAVE REGION) must replace the magnon potential hump H(g) of a complicated shape in equation (11) by a rectangular In this section, our ﬁrst task is to calculate the free ρ energy for a ferromagnetic system containing nonion- potential hump with the height h and radius equal to ized donors. This calculations is inapplicable to the the mean height of the hump (11) and the mean radius manganites, since the undoped manganites are antifer- of the electron wave function, respectively: romagnetic and only the heavily doped manganites are 3 2 Ax ferromagnetic. The problem can be solved by using a h = ∑H(g)ψ (g) = ------, variational procedure for the free energy under condi- π 3 16 b (12) tion that the donor electron is in the ground state with 3a a the wave function ρ = ∑gψ2(g) = ------B-, b = ----B-. 2x a x3  xr ε ψ(r) = ------exp Ð----- , a = ------, (8) This means that the magnon frequency in the region π 3  a  B 2 aB B me close to a nonionized donor is given by ω I ( ) Ω Ω ( γ ) where x is the variational parameter. q x = q + h, q = J 1 Ð q , In addition to the electron energy, the total free energy includes a contribution from the magnons, 1 (13) J = zIS, γ = --∑ exp(iq∆), whose frequencies are renormalized as a result of their q z interaction with the donor electron, realizing the indi- ∆ rect exchange between the d-spins in the vicinity of the where z is the coordination number. donor atom. The state of the magnon subsystem is To calculate the density of the conduction electrons determined from the spin-wave Hamiltonian, including in a nondegenerate semiconductor, it is necessary to the direct dÐd exchange from equation (1) and the indi- write the spin-wave Hamiltonian with an allowance for rect exchange (7). The Hamiltonian is obtained from the conduction electrons. The relative number of the these equations after the HolsteinÐPrimakoff transfor- donors ζ is assumed to be small. mation This makes it possible to disregard the interaction z + Ð between s-electrons. We can divide all regular magnetic S = S Ð b*b , S = 2Sb , S = 2Sb*, (9) g g Ðg g g g g atoms into those which enter spheres of radius ρ sur- where the electron distribution w(g) corresponds to rounding donors and those which are outside these equation (8): spheres (the number of the latter spheres greatly exceeds the total number of the former ones). Using the expression for the conduction-electron-magnon Hamil- H = IS∑(b*b Ð b*b ∆) mg g g g g + tonian Hmg (11), (13), we can represent the total elec- , ∆ g (10) tron-magnon Hamiltonian in the form A ( ω I ) + ---∑w(g)w(f)(b*b Ð b*b ). H = ∑nI, i EI + ∑ q mq, i 2 g g g f g, f ( )Ω + ∑ 1 Ð nI, i qmq, i + ∑Eknk (14) The last term ~bg* bf in equation (10) is basically Ω µ ( ) important to ensure the absence of the gap in the mag- + ∑Bqnkmq + ∑ qmq Ð ∑ nI, i + nk , non spectrum. But it does not inﬂuence the bulk of the where m and m are the magnon operators for the ith magnon frequencies. For example, if w(g) = 1/NI, only q, i q the q = 0 magnon has the zero frequency. In the absence sphere and outside the spheres that surrounds donors, of the d−d exchange, N Ð 1 magnons with other wave respectively. Since the magnon number operators for I different donor regions and outside them are con- vectors have the same frequency A/2NI. Hence, in calculating the free energy, we can use the following structed of the magnon operators bg* and bg with differ- Hamiltonian for the magnon frequencies: ent g, all the operators mq, i and mq are independent.

Further, nI, i and nk are the operators for an electron H = IS (b*b Ð b*b ∆) + H(g)b*b , mg ∑ g g g g + g g in the localized state at the donor i and for the delocal- g, ∆ (11) ized electrons with the quasimomentum k, respectively. Aψ2(g)a3 The spin index is absent from the electron operators, H(g) = ------. since the electrons are completely spin-polarized in the 2 spin-wave region. For the same reason, the sÐd However, Hamiltonian (11), written with an allowance exchange energy (ÐAS/2) is the same for all the electron for equation (8), is still too complicated to be diagonal- states considered and therefore can be omitted as an

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 BOUND FERROMAGNETIC AND PARAMAGNETIC POLARONS 187 additive constant. The quantity µ is the chemical poten- A similar calculation is carried out for the mean 〈 〉 tial. number nk of electrons with the quasimomentum k: The energy EI of an electron at the donor is given by equations (3) and (8). At low temperatures, we can set  E Ð µ 2 ε 〈 〉  k  x = 1 in equation (8), so that EI = ÐEB = Ðe /2 aB. The nk = 1 + exp------  T quantity Bq, which describes the sÐd interaction of the delocalized electrons with magnons when the electron quasimomenta are small compared to the magnon Ð1 quasimomenta, has the form [1, 11] Π [ ( Ω ⁄ ⁄ )]  × q 1 Ð exp Ð q T Ð Bq T ------Π------[------(------Ω------⁄------)--]------ 2 q 1 Ð exp Ð q T  Aq 2 (19) Bq = ------2------2---, p = 2mAS, (15) 2N( p + q ) Ð1 E + δF Ð µ = 1 + exp ----k------m---C------, where m is the s-electron effective mass, and N is the  T  total number of atoms. With an allowance for the mutual independence of δ 0 ( ) FmC = FmC Ð FmC = N f C Ð f 0 , mq, i and mq, the mean number of electrons at the donor is given by the expression (the index of the donor is 3 Ω omitted) a 3  q Bq f C = T------∫d qln 1 Ð expÐ ------Ð ----- . (20) (2π)3 T T 〈 〉 ( µ) ⁄ ω I ⁄ nI = ∑ exp Ð EI Ð T Ð ∑ q mq T m q Keeping in mind the fact that Bq ~ 1/N, we can write

 3 2  I Aa 3 q 1 ×  exp Ð (E Ð µ) ⁄ T Ð ω m ⁄ T (16) δF = ------d q------. (21) ∑ I ∑ q q mC 3∫ 2 2 [ (Ω ⁄ ) ]   ( π) exp q T Ð 1 m q 2 2 p + q

Ð1  Equating the number of ionized donors to the total Ω ⁄ number of the conduction electrons, we find an expres- + exp Ð∑ qmq T  .  sion for the charge carrier density n for E = k2/2m: q cc k In equation (16), the summation over m denotes ( δ ⁄ ) ncc = nneff exp Ð EB + Fm 2T , summation over mq. Carrying out the summation, we find ( )3/2 (22) mT δ δ δ Ð1 neff = ------Fm = FmI Ð FmC, I 2 2π  E Ð µ Π [1 Ð exp(Ðω ⁄ T)]  〈n 〉 = 1 + exp -----I------q------q------ I  T  Π [1 Ð exp(ÐΩ ⁄ T)]  q q  where neff is the effective density of states in the conduction band, and n = ζ/a3 is the donor density. δ µ Ð1 EI + FmI Ð  = 1 + exp------ , (17) It can be ascertained that the activation energy in T equation (22) increases with temperature in the spin- ӷ 0 wave region. It is sufficient to consider the case of J h, δ ( ) 2 FmI = FmI Ð FmI = N I f I Ð f 0 , p Ӷ 1. Using equation (17)–(22), we find 0 where FmI and FmI are the magnon free energies for a 3 7Aa 3 1 region of radius ρ containing the nonionized and ion- δF = Ð------d q------. (23) m 3∫ [ (Ω ⁄ ) ] ized donor, respectively, 32(2π) exp q T Ð 1 3 ω I a 3  q δ In other words, Fm is negative, and its absolute value f I = T------∫d qln 1 Ð expÐ------ , (2π)3 T increases with temperature. This conclusion is confirmed by numerical calculations. For example, for A = 2, 3 δ a 3  Ω  I = 0.02 (in the EB units), Fm = Ð0.022 at T = 0.01, but f = T------d qln 1 Ð exp Ð-----q- , (18) 0 3∫   δF = Ð0.214 at T = 0.03. The fact that the activation (2π) T m energy increases with temperature in the spin-wave 4πρ3 region suggests that the resistivity peak at elevated tem- N I = ------3---. peratures is caused by a minimum in the charge carrier 3a density.

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4. TEMPERATURE-INDUCED polarization, Eex(n) can be easily obtained by generali- METALÐINSULATOR TRANSITION zation of the corresponding Bloch expression for the (SPIN-WAVE REGION) completely spin-depolarized electron gas, e. g., in [12]: In this section we investigate the transition of a 1/3 36n e2 degenerate ferromagnetic semiconductor into the insu- E (n) = Ð------. (28) lating state, which occurs with increasing temperature. ex 4 π  ε We will compare the free energy of the highly conducting state with that of the insulating state. First, using To calculate the energy E(k = 0), we will use the equations (1), (8), (12), and (13), we write the total free WignerÐSeitz procedure (see, e. g., [13]). Each ionized energy of a separated nonionized donor in the E units: donor is surrounded by a sphere of radius L = B (3/4πn)1/3. Inside each WignerÐSeitz shell, the electron F (x) = (x2 Ð 2x) + δF (x). (24) wave function Φ corresponding to k = 0 satisfies the I mI wave equation If one considers the term δF (x) in equation (24) as a mI 2 small perturbation, the optimum value of x is  ∆ e  Ð ------Ð ---- Ð E(k = 0) Φ(r) = 0 (29)  2m εr  1dδF x = 1 Ð ------m---I (1) (25) 2 dx with the boundary condition dΦ and, to a first approximation, the optimum free energy ------(L) = 0. (30) is dr Fopt = Ð 1 + δF (1). (26) As is well known from the theory of metal adhesion, the I mI wave function Φ should be almost constant with the δ Since FmI(x) [equations (17) and (18)] decreases with boundary condition (30). A special analysis shows that, decreasing x and, hence, the last term in equation (25) for relative densities ζ between 0.001 and 0.1, the Φ = is negative, the parameter x increases, and, accordingly, const approximation ensures an accuracy in energy the electron radius decreases with the temperature. This higher than 1%. With sufficient accuracy, we can there- is a manifestation of the ferron effect: the electron is fore set dragged in by the region of the enhanced magnetization 1/3 and simultaneously supports it, realizing the indirect 4πn E(k = 0) = Ð3 ------E . (31) exchange inside this region. The temperature-induced  3  B decrease in the electron radius points to the tendency toward the temperature-induced transition from metal- With an allowance for equations (28) and (31), the lic to insulating state if the system at T = 0 is metallic. energy (27) in the EB units takes the form In fact, while the orbit overlapping of neighboring 1/3 atoms at T = 0 is sufficient for metallization, at finite F 3 2 2/3 2 1/3 36ζ E = --(6π ζ) b Ð (36πζ) Ð ------b, (32) temperatures this overlapping is insufficient and the M 5 2 π  transition to the insulating state should take place. To prove the possibility of such a transition, one should where ζ = na3, and b = a /a. compare the free energy of separated nonionized B donors to that of the impurity metal which consists of At finite temperatures, the free energy of a donor ionized donors and delocalized electrons. metal with the volume V is given by the expression Under typical conditions for degenerate ferromag- GF(n) = nVE F (n) + N f , netic semiconductors, due to a relatively small electron M M µ µ 3 M (33) density in them, the condition < AS is met (here is a 3  ω  f = T------d qln 1 Ð exp Ð-----q-- , the Fermi energy [1]). In other words, the electron gas M 3∫  T  is completely spin-polarized in the spin-wave region. (2π) Using expressions for the energy of the electron gas where, with an allowance for the non-RKKY indirect from [12, 13], we find the following expression for the exchange in our case (since µ < AS), the magnon fre- donor metal energy per donor atom (unlike the “mag- quencies are given by the following expression [1, 11] netic” index m, index M denotes metal): (see also equation (15)): 2/3 ( π2 ) 2ζ F ( ) 3 6 n ( ) ω M Ω Aq 2 EM = E k = 0 + ------+ Eex n , (27) q = q + ------, p = 2mAS. (34) 10m p2 + q2 where E(k = 0) is the electron energy at the conduction Equating the energy E F (n) (32) to the donor energy E band bottom, Eex is the exchange energy between con- M I duction electrons, and n is the electron (or donor) num- = ÐEB, we find that the density n0, at which the electron ber density. Under conditions of the complete spin delocalization takes place at T = 0, obeys the relation

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1/3 n0 aB = C, where C = 0.208, which is slightly lower but fluctuates freely, so that its mean value should van- than the value of 0.25 found by Mott. ish. But the spin of the s-electron adjusts to the direc- To find the temperature of the transition from highly tion of the fluctuating moment and fluctuates jointly conducting to insulating state for a material with n with it, providing the maximum gain in the sÐd exchange energy for the energetically favored direction exceeding n0, one should equate the metal free energy GF (33) to the free energy of the localized state found of the s-electron spin relative to the total spin of its with the use of equations (26) and (17): localization region. This gain should be on the order of the total moment per atom, i.e., ~AS/N1/2, as is the case FI = ζN(E + N f ) + N f (1 Ð ζN ). of equation (36). The term of the order of AS/W is I I I 0 I essential only for sufficiently small orbital radii. For For n sufficiently close to n0, we obtain the following larger radii, the terms of the next order in AS/W should implicit expression for the transition temperature: be taken into account. F ӷ ( ) Let us now consider the bound ferron at T TC, tak- (ζ ζ )d nEM ζ ( ) ( ) Ð 0 ------= NI f I Ð f 0 + f 0 Ð f M ing into account the fluctuation lowering of the energy dn (35) discussed above. When the correlations between the ζ = n a3. d-spins are weak, the donor magnetic Hamiltonian (7), 0 0 jointly with the direct dÐd exchange Hamiltonian (1), Numerical calculations based on equation (32) show can be represented in the Heisenberg form F ζ that the quantity d(nEM )/dn is negative for < 0.2. A H = Ð --- P + H , This does not mean that the system is unstable, since mP 2 H this derivative is not the electron Fermi energy. It does mean that the energy of the donor metal varies when the 1 number of the donor atoms changes by unity. The H = Ð--∑ I (g, f)(S S ), expression on the right-hand side of equation (35) is H 2 tI g f also negative for x close to unity, which is seen from g ≠ f (37) A numerical calculations. The proof of this statement is I (g, f) = ------w(g)w(f) + I(g Ð f), especially simple in the case S ӷ 1, if one considers the tI Ӷ Ӷ 2 P region TC/S T TC and uses equations (12), (13), (15), (18), and (33) (TC is the Curie point). This means ( ) 2( ) ζ ζ P = S S + 1 ∑w g . that the equality (35) can be met for that exceeds 0 only moderately, and the transition from metallic to The free energy of the system is obtained by the high- insulating state should take place with increasing tem- temperature expansion to the first-order terms in 1/T: perature. But for large densities, when ζ > 0.2, this transition is prohibited, at least in the spin-wave region, PI A which agrees with experimental data of the works cited F = EI Ð --- P + FmP, 2 in the Introduction.  H  F = ÐT lnTrexp Ð-----H-- (38) 5. FLUCTUATION TRAPPING mP  T  IN THE PARAMAGNETIC REGION AND RESISTIVITY OF NONDEGENERATE 2 2 S (S + 1) 2 SEMICONDUCTORS = ÐNT ln(2S + 1) Ð ------∑ I (g, f). 12T tI Calculations carried out in this section and in the next one are possibly also applicable to the manganites. Calculating the electron energy E using the Hamilto- First, expression (7) will be analyzed in the limit I ∞ nian Hs (1) and the trial wave function (8), and keeping T . Although correlations between the d-spins in mind that the direct exchange integral I(g Ð f) is non- are absent, the sÐd exchange energy remains nonzero in zero only in the nearest-neighbor approximation, we the first order in AS/W. We see from equation (6) that in can write the x-dependent portion of the free energy this case (38) in the form (for z = 6) ( ) ±A S S + 1 ( ) EmI = ------. (36) PI( ) ( 2 ) 3/2 Q x 2 N F x = x Ð 2x EB Ð Lx Ð ------, (39) I T The physical meaning of relation (36) is clear if we recall that, according to the mathematical statistics, a where system of N noninteracting spins should possess a total Ð1/2 A S(S + 1) Ð3/2 aB moment on the order of N of their maximum L = ------b , b = -----, moment NS. The direction of this moment is not fixed 2 8π a

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2 3 3 3 (ÐF∞/2) exceeds the low-temperature activation L x 3/2 3/2 x b Q(x) = ------+ AIS (S + 1) ------energy. 12 32π 2 2  4x b  Ð2xb × 1 + 2xb + ------e . 6. TEMPERATURE-INDUCED  3  METALÐINSULATOR TRANSITION IN THE PARAMAGNETIC REGION In writing equation (39), we calculate the integral At ﬁnite temperatures, from equations (38) and (39) ∑w(g)w(g + ∆) we obtain the total free energy of a system of N magnetic atoms and nV donors, in Q(x) in elliptic coordinates. Here and below, the − 2 2 2 entropy term NTln(2S + 1) is omitted from the free nVQ(x∞) + NS (S + 1) I ⁄ 2 energy. F(T) = nF∞ Ð ------T (44) Minimizing the free energy (39) with respect to x, we obtain its optimum value and the inverse orbital Ð NT ln(2S + 1), ∞ radius in the limit T (in the EB and 1/aB units, and for the donor orbital radius we have respectively): 1 dQ 8 3 2 8 2 2 x(T) = x∞ + ------(x∞), (45) F∞ = Ð --l [l + 1 + l ] Ð --l 1 + l Ð 4l Ð 1, (40) ( Ð1/2) dx 3 3 2T 1 Ð lx∞

2 where x∞ is given by equation (41). We can prove that [ 2] 3L ∼ AS x∞ = l + 1 + l ; l = ------. (41) the second term on the right-hand side of equation (45) 8E 2 1/2 B (We ⁄ εa) is positive if the parameter

If one sets a = a, then for AS/E varying from 1 to 5, 3 I B B c = 32πS(S + 1)b --- the energy F∞ varies from Ð1.104 to Ð1.659, and the A radius x∞, from 1.077 to 1.1445. Hence, the electron − interaction with random (uncorrelated) magnetization is in the range between 1 and 40. With I > 0, for any fluctuations leads to a marked decrease in the donor real parameter values, the inequality c < 40 is guaran- ionization energy and in the orbital radius; this applies teed. On the other hand, it can be satisfied even if I < 0, to any type of magnetic ordering at T = 0. The corre- but the indirect exchange dominates ensuring the total sponding electron state can be called the bound para- ferromagnetic ordering at T = 0. In fact, the intensity of 2 2ζ1/3 magnetic fluctuation polaron (ferron). the indirect exchange is proportional to A S /W, which can exceed the intensity of the dÐd exchange, Formally, random fluctuations could cause the trap- zIS2, provided that the latter quantity is small compared ping of a charge carrier in the absence of the impurity 2 2 potential (the free paramagnetic ferron). In contrast to with A S /W. The fact that the second term in equa- the ferron self-trapping, which occurs in the region of tion (45) is positive means that the radius of the donor the enhanced magnetization, no ferromagnetic correla- orbital state decreases with decreasing temperature. tions between d-spins appear in the region of the elec- This points to the tendency toward the electron localization at lower temperatures if the electrons are delo- tron localization. Mathematically, jointly with the solu- ∞ tion x = 0 corresponding to a free electron, a solution of calized in the limit T . equation (39) with x = 4l2 exists. The corresponding Let us now investigate in more detail the tempera- free energy of the trapped electron is ture-induced transition from metallic to insulating 3 state, which can occur with decreasing temperature. 16l4  AS In the high-temperature limit, the total free energy of F = Ð------∼ AS ------. (42) t 3  W  the donor metal is given by PM PM δ PM According to equation (42), the depth of the levels cor- G = nVEM + G . (46) responding to these trapped states is very small: it is beyond the accuracy in AS/W adopted here. For this rea- The energy of a nonmagnetized crystal per donor atom, son, the free fluctuation ferrons will not be considered instead of equation (32), is given by the following expres- in what follows. In the limit T ∞, we obtain the fol- sion, which includes the correlation contribution [12]: lowing equation for the charge carrier density similar to PM 3 2 2/3 2 1/3 E = --(3π ζ) b Ð (36πζ) b equation (22): M 5 F∞ 1/3 1/3 (47) n = nn exp ------. (43) 33ζ 0.113ζ b cc eff   Ð ------b Ð ------, 2T  π  1/3 2 0.1216 + ζ b A comparison of equations (43) and (22) shows that the ζ 3 high-temperature activation energy of the conductivity where = na and b = aB/a.

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In complete analogy with equation (38), the mag- In writing equation (50), we took into account that netic free energy is given by PM d(nEM )/dn is negative. This fact was established by ζ 2( )2 2 ( ) numerical calculations showing that, at least to = 0.2, S S + 1 ∑ItM q − δGPM = Ð NT ln(2S + 1) Ð ------, this derivative is about 2 in the EB units. 12T (48) Numerical calculations show also that, at I = 0 and ( ) ( ) ( ) ( ) γ ζ = ζ∞, the denominator in equation (50) for 1/T is pos- ItM q = I q + Iin q , I q = I q. itive, which accounts for the positive transition temper-

The structure of the indirect exchange integral Iin(q) ature Ttr. This temperature decreases with increasing corresponds to the RKKY theory which can be used density ζ and depends on the direct exchange integral I. ζ because electron gas is fully spin-depolarized in the For example, at AS/EB = 5, the difference Q Ð R is paramagnetic region: equal to 0.008 for I = 0.02, to 0.005 for I = 0, and to 0.001 for I = Ð0.02 (a negative I value corresponds to 2 3 2 2 initially antiferromagnetic systems such as manganites, ( ) 3nA a  4kF Ð q 2kF + q  which means that the transition form metallic to insu- Iin q = ------µ------1 + ------ln------ , 8 4kFq 2kF Ð q lating state in them is also possible). (49) 2/3 (3π2n) µ = ------, k = 2mµ. 7. DISCUSSION OF THE RESULTS 2m F The main results of the present treatment can be for- First, it will be proved that a sample, which was in mulated as follows. For the nondegenerate semicon- the highly conducting state at T = 0, can become insu- ductors, it is established that the activation energy of lating at an elevated temperature and remain nonmetal- the conductivity in the spin-wave region is determined lic up to arbitrarily high temperatures. This stems from not only by the depth of the donor level, but also by the the fact that fluctuations lower the donor level strongly, difference in the magnon free energies for delocalized while no such lowering takes place for delocalized and localized electrons. As this difference increases electrons. As a result, according to equation (32), the with temperature, the activation energy EA increases as delocalization of the donor electrons at T = 0 occurs at well. In the paramagnetic region, the activation energy the density n0 which corresponds to the Mott-like decreases with temperature. Qualitatively, the activa- equality n 1/3 a = 0.208. But, equating the energy (47) to tion energy behaves like the difference between the 0 B local magnetization in the vicinity of a nonionized the energy F∞ (40), we find that the delocalization den- donor and the mean magnetization over the crystal: sity n∞ at T ∞ exceeds the value n at T = 0 if the 0 with increasing temperature, it first increases and then ratio AS/EB exceeds 1.27. Normally, this ratio is essen- decreases, passing through a maximum at a tempera- tially greater, and for AS/EB = 5 the Mott-like relation ture comparable with the Curie temperature. The resis- 1/3 tivity peak for the nondegenerate semiconductors is takes the form n∞ aB = 0.378. Hence, normally, n∞ located at the temperature at which dEA(T)/dT = 0. exceeds n0 considerably. This fact results in a nontrivial temperature depen- A very important result is the fact that the high-tem- dence of the electrical properties of a degenerate ferro- perature activation energy exceeds the low-temperature magnetic semiconductor. For the donor density n in the value. This is a consequence of the fluctuation lowering of the donor level, which is caused by the fact that the range between n and n∞, the system behaves like a metal 0 moment of a region in which the localized electron at low temperatures, but remains insulating up to arbi- ∞ trarily high temperatures upon the transition from metal- dwells remains finite even when T . The direction of this moment fluctuates in space so that its mean lic to insulating state. If the density n exceeds n∞, then the reentrant metalÐinsulator transition takes place with value vanishes. But the s-electron spin adjusts to the increasing temperature. This suggests a high resistivity direction of the moment of the region and fluctuates peak at elevated temperatures of the order of the Curie jointly with the moment. The gain in the sÐd exchange temperature. Using equations (44), (46), and (48), we energy therefore remains finite for the localized elec- find the following expression for the temperature at tron, although it diminishes with increasing size of the which the temperature-induced metalÐinsulator transi- region. For a delocalized electron, such a fluctuation tion occurs when the donor density n exceeds n∞: lowering is absent. Obviously, the trapping by random fluctuations is possible not only in ferromagnetic (ζ ζ ) ( PM) semiconductors but also in all magnetic semiconduc- 1 Ð ∞ d nEM --- = Ð------, tors independently of their ground-state magnetic T ζQ Ð R dn ordering. (50) 2( )2 Let us now discuss in greater detail the case of more S S + 1 3 2 ( ) R = ------3---∫d qIin q . heavily doped ferromagnetic semiconductors, which 12(2π) are in the metallic state at T = 0. The same reason as for

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 192 NAGAEV nondegenerate semiconductors—increase in the stabil- clude that the resistivity peak should exist at intermadi- ity of the localized states as compared to the delocal- ate temperatures in these materials. Moreover, one can ized states—leads to their transition from metallic to state that, in the materials with the initial antiferromag- insulating state with increasing temperature. The high- netic ordering, the resistivity peak should be very close temperature fluctuation lowering of the donor levels to TC, in contrast to the materials with the initial ferro- again plays an important part. Because of this circum- magnetic ordering, where they can be well separated. In stance, the low-temperature electron delocalization fact, in the former case the localization of charge carri- density n0 turns out to be smaller than the high-temper- ers leads to disappearance of the indirect exchange pro- ature delocalization density n∞. ducing the ferromagnetic long-range order. Thus, after There are two possible scenarios of the temperature- the charge carrier localization in the initially ferromag- induced metalÐinsulator transition. The first corre- netic materials, the ferromagnetic order is supported by sponds to the case where the donor density exceeds n the dÐd exchange, and it is destroyed only due to ther- 0 mal fluctuations of magnetization. In the initially anti- but is less than n∞. Then, with increase in temperature, ferromagnetic materials, the ferromagnetic exchange the system undergoes a transition from metallic to insu- disappears simultaneously. lating state and remains in the latter state as the temperature is raised arbitrarily high. The second scenario cor- These theoretical results, which disregard the responds to the case where the donor density exceeds polaronic effects and are based only on the s−d model, both n0 and n∞. Then, with increase in temperature, make it possible to explain electric properties of degen- first, the transition from the highly conducting state to erate ferromagnetic semiconductors presented in [1], the insulating state takes place and then the reverse including doped manganites. Many investigators transition occurs. Obviously, the temperature range of believe that one should take into account the JahnÐ the insulating state should decrease with increasing Teller (JT) and lattice polarization effects to describe density. Then the reentrant metalÐinsulator transition properties of the manganites adequately. As for the should manifest itself as a resistivity peak, whose former, it should be pointed out that the resistivity peak height decreases with increasing density [14]. in the vicinity of TC and the colossal magnetoresistance The following remark is likely to be appropriate are observed in several tens of the non-JT systems [1], here. Many investigators use the terms “insulating” or so that the JT effect cannot be the origin of the specific “semiconducting” to denote the high-temperature state features of ferromagnetic semiconductors. The same of heavily doped feromagnetic semiconductors, since mechanisms as in other ferromagnetic semiconductors the resistivity ρ decreases with increasing temperature. are operative in the manganites, leading to the same In doing so, they ignore the fact that the resistivity specific features. exceeds the values typical of nondegenerate semicon- Search for additional mechanisms in the manganites ductors by many orders of magnitude; it is on the order would be justified, if their resistivity peaks in the vicin- of the resistivity typical of degenerate semiconductors. ity of TC were considerably higher than in other ferro- If one accepts this terminology, the state of a nondegen- magnetic semiconductors. The resistivity peak height erate semiconductor in the portion of the resistivity ρ in the manganites, however, is many orders of magni- peak where d /dT > 0 should be called metallic, tude lower than that in ferromagnetic semiconductors despite its giant value. Hence, this terminology is mis- such as EuO, EuS, and others. Therefore, one should for- leading. mally conclude that the JT and polaronic effects rather Strictly speaking, these results correspond to the hinder the manifestation of the particular properties of materials which are ferromagnetic when undoped. these materials. I do not insist on this conclusion but am A situation is more complicated in the case where the certain that the specific features of the ferromagnetic undoped crystal is antiferromagnetic and becomes fer- semiconductors are not related to the lattice effects. The romagnetic as a result of doping (e.g., the manganites). role of the polaron effects in manganites is discussed in The difficulty in finding n0 is attributed to the formation more detail in [15]. of bound magnetic polarons (ferrons) in the vicinity of the donors when electrons are localized, and to the fer- romagneticÐantiferromagnetic phase separation in the ACKNOWLEDGMENTS delocalized state of electrons which occurs at T = 0. It is still more difficult to investigate the temperature This investigation was supported in part by the dependence of the critical density at low temperatures. Grant no. 98-02-16148 of the Russian Foundation for Basic Research. A direct analysis of such materials is therefore carried out here only in the high-temperature limit. It is established on the basis of equation (50) that, as the REFERENCES temperature is lowered, such materials can undergo the transition from metallic to insulating state. The fact that 1. E. L. Nagaev, Physics of Magnetic Semiconductors (Mir, the electron is delocalized at T = 0 is sufficient to con- Moscow, 1983).

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2. E. L. Nagaev, Fiz. Tverd. Tela 11, 3428 (1969). 9. H. Ju, H.-C. Sohn, and K. Krishnan, Phys. Rev. Lett. 79, 3. A. P. Grigin and E. L. Nagaev, Phys. Status Solidi B 61, 3230 (1997). 65 (1974). 10. T. Saitoh, A. Sekiyama, K. Kobayashi, et al., Phys. Rev. B 4. A. Yanase and T. Kasuya, J. Phys. Soc. Jpn. 25, 1025 56, 8836 (1997). (1968). 11. E. L. Nagaev, Phys. Rev. B 58, 827 (1998). 5. T. Kasuya and A. Yanase, Rev. Mod. Phys. 40, 684 (1968). 12. P. Gombas, Theorie und Loesungmethoden des 6. E. L. Nagaev, Zh. Éksp. Teor. Fiz. 54, 228 (1968) [Sov. Mehrteilchenproblems der Wellenmechanik (Basel, 1950). Phys. JETP 27, 122 (1968)]. 13. C. Kittel, Quantum Theory of Solids (Wiley, New YorkÐ 7. E. L. Nagaev, Zh. Éksp. Teor. Fiz. 92, 569 (1987) [Sov. London, 1963). Phys. JETP 65, 322 (1987)]; Phys. Rev. B 54, 16608 (1996). 14. E. L. Nagaev, Usp. Fiz. Nauk 166, 833 (1996) [Phys. Usp. 39, 781 (1996)]. 8. T. Saitoh, A. Bocquet, T. Mizokawa, et al., Phys. Rev. B 51, 13942 (1995). 15. E. L. Nagaev, Phys. Lett. A 258, 65 (1999).

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 194Ð201. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 218Ð226. Original Russian Text Copyright © 2000 by Aplesnin. SOLIDS

Quantum Monte Carlo Investigation of the Magnetic Properties of Weakly Interacting Antiferromagnetic Chains with an Alternating Exchange Interaction with Spin S = 1/2 S. S. Aplesnin Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Krasnoyarsk, 660036 Russia; e-mail: [email protected] Received June 21, 1999

Abstract—An approximation dependence of the spontaneous magnetic moment at a site, σ/σ(0) Ð 1 = δ2.5(2) δ 0.71(6) , and the antiferromagnet-singlet state phase boundary, J2/J1 = 0.52(3) , are determined by the quantum Monte Carlo method in the self-consistent sublattice molecular field approximation for weakly inter- ± δ acting (J2) antiferromagnetic chains with spin S = 1/2 and alternating exchange interaction (J1 ). The Néél temperature and a number of critical temperatures which could be related with the filling energy of two singlets (∆Sz = 0) and one triplet (∆Sz = 1) spin bands, each of which is split by the sublattice field (hx, y ≠ hz) into two subbands, are determined on the basis of the computed correlation radii of the two- and four-spin correlation function, the squared total spin 〈(Sz)2〉 with respect to the longitudinal components, the dimerization parameter, and the correlation functions between the nearest neighbors with respect to longitudinal and transverse spin components. On the basis of the Monte Carlo calculations, the critical temperatures and possible energy gaps at the band center are determined for the antiferromagnets CuWO4 and Bi2CuO4 and for the singlet compounds (VO)2P2O7 and CuGeO3, agreeing satisfactorily with existing results, and new effects are also predicted. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ence of an energy gap at band center for ω = 1.4 meV in CuWO [2] and at ω = 0.7, 1.7, 2.3, 3.4, and 4 meV There exists a wide class of magnetic compounds 4 i with spatially anisotropic distribution of exchange in Bi2CuO4 [4, 7, 8] remain incomprehensible. Non- interactions and with a strong interaction between the monotonic temperature behavior of the susceptibility magnetic and elastic subsystems that in certain cases [9], the antiferromagnetic resonance field, and the line- results in a spin-Peierls transition. Ordinarily, a transi- width in Bi2CuO4, whose temperature derivatives have tion from a singlet state into the paraphrase is studied in several maxima [10], and an additional maximum of Ӎ the Hubbard or Heisenberg models with alternating the specific heat at T 17 K (TN = 45 K) in Bi2CuO4 exchange parameter J and hopping integral t using the [11], are observed in these antiferromagnets. mean-field theory or Green’s functions together with Several energy gaps in the spin excitation spectrum, perturbation theory. As a rule, spinon excitations are which do not fit either into the conventional theory of neglected in the analysis of these systems, resulting in the spin-Peierls transition with one triplet gap [12] or overestimation of the temperature of the spin-Peierls into the theory of the two-magnon excitation spectrum transition when interchain exchange is taken into [13], have also been found in the singlet magnets account, specifically, logarithmic behavior [1]. CuGeO3, Na2V2O5, and (VO)2P2O7. Of these com- Alternating exchange can also be achieved by pounds, CuGeO3 has been studied in greatest detail. In means of the geometry of the crystal lattice, such as in this compound three temperature ranges have been Ӎ CuWO4 [2], Bi2CuO4 [3, 4], (VO)2P2O7 [5], and found, Tc1 ~ (4Ð7) K, Tc2 14 K, and Tc3 ~ (20Ð25) K, (CH3)CHNH3CuBr3 [6]. These compounds are all where the EPR linewidth and intensity exhibit anoma- three-dimensional magnetic systems with alternating lous behavior [14, 15], and the magnetic thermal con- exchange. For most of them, the exchange interactions ductivity [16] and magnetostriction [17] possess max- in three directions of the corresponding crystal axes ima below and above the spin-Peierls transition temper- have been determined. The magnetic properties of the ature TN = 14 K. antiferromagnets CuWO4 and Bi2CuO4 are interpreted The present paper is devoted to an investigation of in a two-sublattice Heisenberg model, and the existence the region of stability of long-range antiferromagnetic of several branches of spin excitations, whose intensity order in an isotropic 3D antiferromagnet with a quite becomes zero at different temperatures, and the pres- strong anisotropic distribution of exchange interactions

QUANTUM MONTE CARLO INVESTIGATION OF THE MAGNETIC PROPERTIES 195 in the lattice relative to the magnitude of the alternating a quasi-one-dimensional antiferromagnet, while in the exchange, and the determination of the site magnetic paramagnetic and singlet states the instantaneous val- moment, the Néél temperature, and the critical temper- ues of the sublattice field are proportional to the magni- atures at which the correlation radii assume their max- tude of the short-range order, i.e., the spin-spin correla- imum values. According to the dynamic scaling tion function of the nearest neighbors. The average τ hypothesis, the relaxation time is proportional to the value is 〈h 〉 ≈ 0, and 〈h2〉 ≠ 0 in the singlet state. Tak- correlation radius, τ ∝ ξz, and the temperatures indi- i i cated above can be found from the temperature depen- ing account of the fluctuations of the sublattice field in dence of the EPR linewidth, antiferromagnetic reso- the singlet state leads to new effects, which will be nance, and diffuse neutron scattering. Additional spin described below. We shall study in the singlet and paramagnetic states two types of the sublattice field with excitations, spinons [18], will be proposed on the basis x of the four-spin correlation function. These excitations respect to transverse spin components: isotropic h = y z x have several excitation bands, which make it possible to h = h and anisotropic, characteristic for CuGeO3, h = explain previously reported experimental results and, hy = 1.4hz. In an antiferromagnet the sublattice fields using the computed values of the critical temperatures, have the form predict the existence of additional spin modes and a L/2 number of new effects. 2 z z m = --- ∑ abs(S S ), 0 L 0 i i = L 2. MODEL AND METHODS (3) z ( 〈 z z〉) ( z z) Let us consider a Heisenberg model with negative hi = 4J2sign S0Si abs S0Si , interactions between nearest neighbors with spin S = +, Ð ( )i ( + Ð) 1/2 in an external magnetic field oriented in the Z direc- b Ð hi = 4J2 Ð1 abs S0 S1 /i. tion. The alternating interaction is taken in the strong- δ δ coupling direction I = J1 + and K = J1 Ð . The Hamil- In this work, the quantum Monte Carlo method, tonian has the form which employs a trajectory algorithm of world lines, based on a transformation of a D-dimensional quantum system into a (D + 1)-dimensional classical system by H = ÐJ1∑Si, jSi + 1, j discretization of the path integral in the space (imagi- i, j nary time 0 < τ < 1/T, coordinate) [21, 22], is used. In (1) the Monte Carlo calculations, Trotter’s formula with α α α z the parameter m = 32, 64, 124, and 200 and periodic Ð ∑ J (γ )S , S , γ Ð ∑H S , 2 i j i j + i i boundary conditions on a chain of length L = 100, 200, i, j, γ = 1 i α = x, y, z and 400 is used. One Monte Carlo step was determined by rotating all spins on a L × 2m lattice. From 4000 to where J1 < 0 and J2 < 0 are the intra- and interchain 7000 Monte Carlo steps per spin were used to reach interactions, H is the external magnetic field, and γ sig- equilibrium, and 2000Ð5000 Monte Carlo steps per nifies summation over the nearest neighbors between spin were used for averaging. The autocorrelation time chains (z = 4). We transform the Hamiltonian of the 3D τ required to establish thermodynamic equilibrium was system to a one-dimensional chain of spins, which estimated from the relation ln(τ) = amT/J (T is the tem- interact with the effective field, by means of the self- perature) [23]. The systematic error due to quantum consistent molecular field approximation [19, 20]: fluctuations is proportional to ~1/(mT/J)2 and is of the L/2 L/2 order of 4% for the minimum temperature T/J = 0.025, used in the calculations. The rms errors of the com- H = Ð∑ I , S S Ð ∑ K , S S 2i 2i Ð 1 2i 2i Ð 1 2i 2i + 1 2i 2i + 1 puted quantities lie in the range (0.1Ð0.6)% for energy, i = 1 i = 1 (2) (6Ð11)% for susceptibility, and ~10% for the correla- L L α α z 2 tion radius. The errors due to the finite dimensions of Ð ∑ hi Si Ð ∑ HiSi Ð 2NJ2m0, the lattice can be neglected, since ξ < L/2. i = 1 i = 1 Let us consider the possible spin excitations in this where m0 and h are the sublattice magnetization and model. If the wave function of the ground state is sche- z + Ð field hi(h , h , h ), determined in [19, 20] as m0 = matically represented as a sum of the Néél configuration and a set of singlet states of spins with different L i 〈 z〉 (1/L)∑i = 1 (Ð1) Si and h = Ð4J2m0. To take account weight ratios, then, besides ordinary excitations of the of the quantum and temperature fluctuations, we shall spin-wave type, there can exist excitations in singlet determine these quantities from the spinÐspin correla- regions that can be divided into two groups: the longi- tion function, which is a power-law function of on dis- tudinal component of the spin vector does not change, tance in a 1D antiferromagnet at T = 0. We shall assume i.e., ∆Sz = 0 (we call such excitations singlet excita- that this dependence also holds also for the transverse tions), and the longitudinal component of the spin vec- spin components in the magnetically ordered region of tor changes by one unit, i.e., ∆Sz = 1, which correspond

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σ/σ(0) A classical excitation of the kink type, which in what 1.0 follows we shall call a spinon excitation, corresponds to σ the excited state in the simple RVB model with one 0.8 dangling bond. We shall determine the correlation 0.6 ξ η 0.8 radius 4 of the spinons and the parameter 4 from the 0.4 four-spin correlation function 0.2 0.6 〈 z z z z〉 〈 z z z z 〉 0 S0S1Sr Ð 1Sr Ð S0S1SL/2 Ð 1SL/2 δ δ 0 0.4 0.8 / c (5) A ( ξ ) 0.4 = ---η-- exp Ðr/ 4 , r 4

0.2 where r = 2i + 1 and i = 1, 2, 3 …. The following quantities will be calculated below: the energy, the specific heat C = dE/dT, the magnetization, 0 0.2 0.4 0.6 δ the susceptibility χ = M/H in an external field, the spin- spin correlation function between longitudinal 〈Sz(0)Sz(r)〉 2 〈(Sz) 〉 and transverse 〈S+(0)SÐ(1)〉 spin components, the Fourier L/2 z z spectrum S(q) = (2/L)∑r = 1 exp(Ðiqr)(S0Sr ), and the 0.6 magnetic structure factor. We shall determine the corre- ξ η lation radius 2 and the parameter 2 from the spin-spin correlation function 0.4 〈 z( ) z( )〉 〈 z( ) z( )〉 B ( ξ ) S 0 S r Ð S 0 S L/2 = ---η-- exp Ðr/ 2 . (6) r 2 0.2 The squared total spin will be calculated from the longitudinal component 〈(Sz)2〉. This parameter makes it 0 δ possible to distinguish a singlet state from a paramag- 0 0.2 0.4 0.6 0.8 netic state and is sensitive to a change in the spin exci- Fig. 1. Site magnetic moment (σ) and squared longitudinal tation spectrum. component of the total spin (〈(Sz)2〉) of an antiferromagnet with λ = 0.1 (᭿), 0.2 (ٗ), and 0.3 (᭹) versus the exchange alternation. Inset: Normalized values of the magnetization 3. DISCUSSION for the same parameters. We shall use a number of criteria to determine the antiferromagnetÐsinglet state phase boundary in the to triplet excitations. According to Anderson’s theory interchain interactionsÐalternating exchange plane: the [24], the singlet state is described well by a generalized sublattice magnetization is zero, σ 0, and the cor- resonance valence bond (RVB) model whose wave ξ ξ δ relation radii 2 and 4 for c have their maximum function is represented in the form of singlet pairs of value. The singlet state is distinguished from the para- spins over all possible configurations. In the presence magnetic state or the spin-glass state according to the of exchange alternation the generalized model reduces following indicators. In the singlet state, in a model to a simple RVB model in which pairing of nearest with alternating exchange, the total spin is zero, S = 0, spins is taken into account. Since two types of 2 exchange interactions, differing in magnitude, exist and the eigenvalue of the operator Sˆ z is also zero on the here, the energies of the singlet pairs and the corre- basis of the equality 〈(Sz)2〉 = S(S + 1)/3. The dimmer sponding excitations on K bonds differ from the energy ordering parameter is different from zero, q ≠ 0, and the on I interactions. For this reason, the characteristic fea- 〈 + Ð〉 Ӎ 〈 z z〉 tures of the temperature behavior of the magnetic char- relation S0 S1 2 S0S1 holds between the longitu- acteristics of an antiferromagnet with alternating dinal and transverse components of the spins. We shall exchange can be calculated and understood on the basis calculate the characteristics indicated above, some of of the four-spin correlation function of spin pairs which are displayed in Fig. 1, at low temperatures, 〈 〉 S0S1SrSr + 1 and the dimmer ordering parameter q: (0.1Ð0.2)TN, for a number of values of the interchain λ L/4 exchange parameters = J2/J1 = 0.05, 0.075, 0.1, 0.125, α 4 α α α α α α α α q = ---∑ ( 〈S S S S 〉 Ð 〈S S S S 〉), 0.15, 0.2, 0.25, and 0.3 as a function of the magnitude of L 0 1 2i Ð 2 2i Ð 1 1 2 2i Ð 1 2i (4) alternation. The normalized values of the sublattice mag- i = 2 netization and the energy can be approximated well by α = x, y, z. power-law dependences σ/σ(0) Ð 1 = 0.71(6)δ2.5(2) and

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ξ E/E(0) Ð 1 = 0.02δ3.6(3), where σ(0) = 1.9(1) λ , and are /a shown in the inset in Fig. 1. In the singlet state, the 20 (a) absolute value of the internal energy increases with the 1.2(1) exchange alternation, (E Ð 0.85) ≈ 0.63δ , which 15 agrees well with the results for a one-dimensional δ4/3 chain, ~ [25]. The difference in the exponent could 10 be due to the correlation effects of the interaction of the chains, which are taken into account in the form of the self-consistent sublattice fields h (3). The mag- 5 δ δ netic state for > c is a singlet state, and the finite quantity 〈(Sz)2〉 is due to the singlet excitations with 0 0.2 0.4 0.6 0.8 ∆Sz = 0, since Monte Carlo calculations are performed at finite temperatures. The phase boundary of the tran- ξ/a sition can be approximated well by the linear function λ = 0.52(3)δ. 15 (b) For an antiferromagnet with alternating exchange in the form of two sublattices with strong I and weak K 8 exchange interactions, three types of spinon (pair) excitations can be distinguished: IÐI, KÐK, and IÐK. In the 4 sublattice field (h+, Ð ≠ hz), each of these spinon bands can split into subbands with transverse and longitudinal spin excitations. The wave functions of these excita- 0 tions can be represented as 0.2 0.4 0.6 0.8 T/J ψs1 ∝ (|…↑↑↓↓…〉 |…↓↓↑↑…〉) c1 Ð Fig. 2. Temperature dependence of the correlation radii of (|…↓↑↓↑…〉 |…↑↓↑↓…〉) two-spin (solid line in Fig. a and ᭿ in Fig. b) and four-spin + c2 Ð (broken line in Fig. a and ᭹ in Fig. b) correlation functions for the antiferromagnetic state with λ = 0.1 and δ = 0.15 (a) on the KÐK bonds and in the form and for the singlet state with λ = 0.05, δ = 0.14, and h+, Ð = z ψ s2 ∝ (|…↑↑…↓↓…〉 |…↓↓…↑↑…〉 1.4h (b). l + , (7) ψ s2 ∝ (|…↑↓…↑↓…〉 + |…↓↑…↑↓…〉 t temperatures are determined according to the maxima ξ ξ on the I bonds. Excitations of this type do not lead to a of the correlation radii 2(T) and 4(T), which are change in the z component of the total spin (∆Sz = 0) shown in Fig. 2 for antiferromagnetic and singlet states, and do not contribute to the longitudinal susceptibility, according to the maximum change in the longitudinal so that the minimum in the temperature dependence component of the squared total spin 〈(Sz)2〉, i.e., accord- χ z 2 (T) for some temperature Tsi corresponds to filling of ing to the maxima of d〈(S ) 〉/dT and the extremal points the band of singlet spinon excitations. Excitations on of the temperature dependence q(t), presented in Fig. 3. the IÐK bonds change the z component of the spin On the basis of an analysis of the temperature behavior ∆Sz = 1 and are spinons, or spin waves; this gives rise of the susceptibility χ(T) (Fig. 4), the critical tempera- to a maximum in the temperature dependence of the tures were associated to the filling energy of triplet (χ = susceptibility at Tti. The filling of the singlet band of max) and singlet (χ = min) spin excitation bands. The excitations in the IÐI sublattice will lead to an increase filling of these bands forms three maxima in the tem- in the dimmer ordering parameter q (4), and in the KÐK perature dependence of the specific heat (Fig. 4). sublattice it will lead to a sharp decrease of the parameter q. We shall determine the splitting into subbands A qualitative estimate of the relations between these according to the magnitude of the temperature variation 2 2 temperatures, T Ð T ∝ λ ± 2δλ + δ , where the of the dimerization parameter qx, y, z and the near-range ti si minus sign corresponds to T and the plus sign to T , 〈 α β〉 s1 s2 correlation functions S0 S1 with respect to the longi- apparently, will also be valid for the energy gaps tudinal and transverse spin components. between these excitation bands. An even weaker effect The temperature dependences of the above-indi- is the splitting of the proposed spin bands by the sublat- cated characteristics are calculated for three interchain tice field, which occurs for δ > 0.1. The temperature at exchange parameters, λ = 0.05, 0.1, and 0.25, and the which the change in the correlation function between corresponding values of the exchange alternation, δ = nearest neighbors with respect to the longitudinal spin 0.05, 0.075, 0.12, 0.14, and 0.2; δ = 0.1, 0.15, 0.2, 0.3, components is much greater than this change with 0.45, 0.6; and, δ = 0.15, 0.3, 0.45, 0.6, 0.75. The critical respect to transverse components (this appears most

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d〈S+, zS –, z〉/dT 〈 s, + z, Ð〉 0 1 strikingly in the calculation of d S0 S1 /dT (inset in 0.3 Fig. 3)) refers to the excitation energy of the longitudi- qa nal spin mode. In the singlet state, the sublattice field at 0.2 a distance of the order of the correlation radius influ- 0.15 0.1 ences the spin excitations with wavelength ~π/ξ, and for δ ӷ δ the interaction of chains has no effect on the 0 c redistribution of the spin excitation density (as com- –0.1 λ ≤ 0.10 pared with the one-dimensional chain) for 0.01. It 0 0.1 0.2 is possible that each spin subband is characterized by a T/J π definite wave vector of the structure Qi < /a, which can found from the Fourier spectrum of the spin correlation 0.05 function S(q), determined in the singlet state at a dis- ξ tance k ~ 1/ 2. Thus, S(q) in the singlet state contains weak additional maxima at Qi, whose number increases 0 0.2 0.4 0.6 T/J with temperature. The Néél temperature was determined from the sub- Fig. 3. Temperature dependence of the dimmer ordering α lattice magnetization σ 0. In the range of bond parameter q (α = z (ᮀ), x, y = (᭺)) in an antiferromagnet with λ = 0.1 and δ = 0.15. Inset: Temperature dependence of alternations close to the critical value, two sharp 〈 α β〉 α β dropoffs appear clearly in the temperature behavior of the derivative d S0 S1 /dT with , = z (dotted line) and the magnetization, for example, for δ = 0.15 and λ = 0.1 with α = +, β = Ð (solid line). for Ts1/J1 = 0.06 and Tt/J1 = 0.11; they are associated with the filling of the triplet spin excitation band in the χJ/N temperature range Ts1 < T < Tt and in the spin-wave 0.4 band at temperatures T > Tt. The computed critical tem- (a) peratures for λ = 0.1 are presented in Fig. 5. The van- ishing of the long-range antiferromagnetic order can be 0.3 understood from this diagram. As exchange alternation increases, the spin-wave excitation density decreases 0.2 0.3 and vanishes at TN ~ Tt. When the sublattice fields are the same, h+, Ð = hz, the splitting into subbands vanishes,

N 0.2 /

J and for the singlet and paramagnetic states only two 0.1 χ 0.1 critical temperatures exist, above and below the spin- 0 Peierls transition temperature, which are shown in Fig. 5 0 0 0.2 0.4 0.6 0.8 1.0 by the dashed and dotted lines. From the standpoint of T/J symmetry, three phases can be distinguished in the tem- 0 0.2 0.4 0.6 0.8 1.0T/J peratureÐexchange alternation plane: a region with long-range antiferromagnetic order, a region where the C/kN thermodynamic value of the spin is zero, i.e., a singlet 0.5 state, and a region where 〈Sz〉 ~ H/k T, i.e., the paramag- (b) B 0.4 netic state. The phase diagrams (see Fig. 5b), calculated for the three parameters λ in normalized units, are the 0.3 same to within the computational error. 0.2 A variety of incomprehensible experimental data for the antiferromagnet CuWO with alternating exchange 0.1 4 [2] can be explained on the basis of the results obtained: 0 the existence of an energy gap at the center of the band of the spin excitation spectrum at ω = 1.4 meV and the 0 0.2 0.4 0.6 0.8 1.0 T/J different temperature dependences of the intensity of spin modes, one of which vanishes at T = 24 K and the Fig. 4. (a) Temperature dependence of the susceptibility, other (gapless) remains even at T = 36 K [2]. The tem- calculated in the sublattice ﬁeld h+, Ð = 0 (ᮀ) and h+, Ð ≠ 0 perature dependence of the susceptibility in the range (᭿) according to (3) in an antiferromagnet with λ = 0.1 and 40 < T < 70 K has a concave form [9]. Our computed δ = 0.15. Inset: χ(T) in the singlet state with λ = 0.05, δ = intrachain exchange parameters J = 11.6 meV and K = 0.14, and h+, Ð = 1.4hz. (b) Temperature dependence of the 8.9 meV agree well with neutron diffraction measure- speciﬁc heat in an antiferromagnet with alternating exchange and the parameters λ = 0.1 and δ = 0.15 (solid ments J = 11.56 and K = 9.25 meV [26]. The interchain line) and λ = 0.05 and δ = 0.14 (᭿). exchange J2 ~ 1 meV agrees fairly well with the aver-

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Tci/J 0.5 (a) 8 (b) δ Tc( )/Tc(0) 2.8 0.4 7 2.4 PM 6 0.3 2.0 5 1.6 0.2 1.2 4 SS 2 0.8 0.1 3 1 AF 0.4 δ δ δ 0 0.1 0.2 0.3 0.4 0 1 2 / c

Fig. 5. Critical temperatures, associated with the ﬁlling energy of singlet (1, 2, 7, 8) and triplet (3, 4, 5, 6) spinon bands as a function of the exchange alternation for h+, Ð ≠ hz (a). Phase diagram of an antiferromagnet (AF), singlet state (SS), and paramagnet (PM) in the plane normalized temperature Ð normalized exchange alternation for h+, Ð = hz (b).

≈ ω age value J2 1.7 meV [2]. According to the Monte For example, in (VO)2P2O7 two gaps were found at = Carlo calculations, the spin triplet band is filled in the 3.12 and 5.75 MeV [5]. It is possible that one other sin- temperature range 17 < T < 24 K. At T > 24 K the gap- glet mode with gap at band center at ω Ӎ 0.6 meV and less mode corresponds to spin-wave excitations, which with weak intensity exists in this compound. For λ Ӎ should vanish at T ≈ 40 K. The band of singlet excita- 0.02 [5] our computed exchange alternation δ ≈ 0.15(2) c δ δ tions, not contributing to the magnetic susceptibility, is lies in the range n = 0.12 and χ = 0.18, which were filled in the temperature range 52 < T < 86 K, which determined, respectively, in neutron resonance (n) gives rise to the inflection in the temperature behavior experiments [5] and from the temperature dependence of χ(T). It is possible that another singlet mode with gap of the susceptibility [27]. energy ω ~ 0.7 meV exists at temperatures T < 12 K. The compound CuGeO3 with a spin-Peierls transi- A series of energy gaps at the center of the band in tion was investigated in greatest detail. According to Bi2CuO4 [4, 7] with tetragonal symmetry P4/ncc and the neutron diffraction data, the ratio of the exchange alternating exchange in the [111] direction can be interactions is Jc : Jb : Ja = 100 : 10 : 1 [28] and, accord- δ δ Ӎ explained by the existence of singlet and triplet spin ing to our estimates, for > c 0.1 a magnet with such excitations. The temperature dependences of the reso- an exchange ratio can be in a singlet state. Inelastic nance field H0 and the anisotropy field Ha are non- neutron scattering data revealed three gaps at band cen- ω ± monotonic, and their derivatives dH0/dT and dHa/dT ter at i = 0.8 meV (1.9Ð2.1) meV [29] and 4( 1) meV possess several maxima, with different magnitudes, at [30] and a wide maximum in neutron scattering ω Ӎ the corresponding temperatures, T = 8, 12, 18, 26 K 6 meV [29], two values for which are close to the ci ω and Tci = 7, 11, 17, 26, 38 K [10]. For the intra- and results = 1.86 and 4.74 meV obtained from EPR mea- interchain exchange parameters chosen, J = 107 K and surements [14]. Light scattering at the boundary of the 1 ω δ Brillouin zone also leads to three energy gaps i = 2.2, J2 = 28 K, the exchange alternation is = 0.2, and our computed critical temperatures, related with the filling 3.6, and 5.8 meV [31, 32]. The magnetic thermal conductivity in a magnetic field up to H = 14 T has two MC Ӎ of the split singlet and triplet spinon bands at T ci 7, maxima at T = 5.5 and 22 K [16], and the width of the 11, 15, 25, 30, 35, agree well with the experimental EPR line diverges at T Ӎ 4 and 14 K [15]. In a magnetic ω results. It is possible that the energy gaps at s1 = 0.7 meV field applied in the direction of the c axis, the tempera- ω and s2 = 3.4 and 4 meV at the center of the Brillouin ture dependence of the magnetostriction constants has zone are due to singlet excitations of longitudinal and three maxima at T Ӎ 6, 13, and 26 K and in a field H || b at ω Ӎ transverse spinon modes, and at t = 1, 7, and 2.1 meV T 4, 11, and 20 K [17]. by triplet excitations. Several spin-excitation modes These results can all be explained well by choosing also exist in singlet magnets with alternating exchange. the two intrachain exchange parameters I = 145 K and

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 200 APLESNIN K = 109 K, which agree fairly well with the data which can be interpreted on the basis of the model of obtained based on the 1D Heisenberg model, in which additional spinon singlet and triplet excitations, were the triplet gap ω Ӎ 2 meV has been calculated [33] and found in an alternating antiferromagnet and in the sin- +, Ð z the spin-Peierls transition temperature is Tsp = 14 K. glet state. The sublattice self-consistent field (h ≠ h ) The observation of thermodynamic anomalies above splits the spinon bands into longitudinal and transverse and below Tsp, additional energy gaps, anisotropy of the excitation modes. The temperatures corresponding to critical magnetic fields, which are encountered in a the maxima of the derivatives of the resonance field and ~10% range, where the field Hc = 13.9 T is applied in the linewidths as a function of temperature in Bi2CuO4 the direction of the exchange alternation axis and the were calculated using dynamic scaling between the field Hb = 12.6 T is applied perpendicular to this axis relaxation time and the correlation radius. Possible [34], remains unexplained. As indicated above, at a dis- spinon excitation modes and energy gaps at the center of ξ Ӎ tance of the order of the correlation radius 2 10c, the the band in antiferromagnetic states of the compounds interchain interaction leads to splitting of the low- CuWO4 and Bi2CuO4 and in the singlet states of the com- energy band of singlet excitations with respect to its pounds (VO)2P2O7 and CuGeO3 were predicted. The tem- center into longitudinal and transverse excitation modes peratures corresponding to the maxima of the magnetic ω Ӎ ω Ӎ with gap energies b 0.5 meV and c 0.8 meV and thermal conductivity, the magnetostriction constants, and Ӎ Ӎ corresponding critical temperatures Tb 3.7 K and Tc the divergence of the EPR linewidth in CuGeO3 were 6 K, splitting of the triplet modes with gap energies calculated. A polarization dependence of light and neu- ω Ӎ ω Ӎ b 1.7 meV and c 2 meV and corresponding crit- tron scattering along the dimerization axis of a chain Ӎ Ӎ was predicted. ical temperatures Tb 14 K and Tc 18 K, and splitting of the high-energy longitudinal and transverse singlet ω Ӎ ω Ӎ modes with b 4.5 meV and c 5.8 meV and cor- ACKNOWLEDGMENTS Ӎ Ӎ responding temperatures Tb 30 K and Tc 39 K. However, at temperatures T > T small structural dis- This work was supported by INTAS (Grant sp no. 97-12124). tortions are observed in CuGeO3, which lead to a change in the magnitude of the exchange and its alternation. For this reason, in the temperature range 20 < T < REFERENCES 26 K one can talk about a qualitative agreement ∆ 1. D. Khomskii, W. Geertsma and M. Mostovoy, Czech. J. between the temperature interval Tex = 6 K and the Phys. 46, suppl., pt. 56, 3239 (1996). ∆ Monte Carlo computational results TMC = 9 K. It is 2. B. Lake, D. A. Tennant, R. A. Cowley, et al., J. Phys. possible that in CuGeO3 in the singlet state the quanti- Cond. Mat. 8, 8613 (1996). zation axis is directed along the b axis, and then the 3. K. Yamada, K. Takada, S. Hosoya, et al., J. Phys. Soc. anisotropy of the critical magnetic field is explained Jpn. 60, 2406 (1991). well. Thus, our calculations predict a polarization depen- 4. B. Roessli, P. Fischer, A. Furrer, et al., J. Appl. Phys. 73, dence of light scattering and inelastic neutron scattering 6448 (1993). along the dimerization axis of a chain. At low tempera- 5. A. W. Garrett, S. E. Nagler, D. A. Tennant, et al., Phys. tures the system is nonlinear, so that the definition of the Rev. Lett. 79, 745 (1997). 3 χ 6. M. Hirotaka and Y. Isao, J. Phys. Soc. Jpn. 66, 1908 nonlinear susceptibility Mγ = γ, α, β, δHβHαHδ must be (1997). used to calculate the resonance absorption frequencies. 7. G. A. Petrakovskii, K. A. Sablina, A. I. Pankrats, et al., It is possible that as a result of the nonlinear interaction J. Mag. Magnet. Mat. 140, 1991 (1995). of the field with the spin subsystem, a transition occurs 8. M. Ain, G. Dhalenne, O. Guiselin, et al., Phys. Rev. B from the singlet ground state into a singlet excited state, 47, 8167 (1993). as is observed in EPR measurements at frequency ω = 9. A. G. Anders, A. I. Zvyagin, M. I. Kobets, et al., Zh. 294 GHz [14], and transitions occur between the sub- Éksp. Teor. Fiz. 62, 1798 (1972) [Sov. Phys. JETP 35, bands ψs ψs at the frequency f = 34 GHz in a field 934 (1972)]. l t 10. A. I. Pankrats, G. A. Petrakovskii, and K. A. Sablina, H ≈ 12 kOe [15]. The intensity of both resonances has Solid State Commun. 91, 121 (1994). a maximum at T = 6 K and vanishes at T < 2 K; this 11. Yu. P. Gaidukov, V. N. Nikiforov, and N. N. Samarin, agrees well with our estimates for the energy of the sin- Fiz. Nizk. Temp. 22, 920 (1996) [Low Temp. Phys. 22, glet gap and the critical temperature. 705 (1996)]. Thus, the long-range antiferromagnetic order in the 12. L. N. Bulaevskii, A. I. Buzdin, and D. I. Khomskii, Solid quasi-low-dimensional antiferromagnet with alternat- State Comm. 27, 5 (1978). ing exchange remains for λ ≤ 0.52(3)δ. Alternation 13. G. S. Uhrig, Phys. Rev. Lett. 79, 163 (1997). gives rise to quantum spin reduction at a site σ/σ(0) Ð 1 = 14. T. M. Brill, J. P. Boucher, J. Voiron, et al., Phys. Rev. Lett. 73, 1545 (1994). δ2.5(2) σ λ 0.71(6) , where (0) = 1.9 . Several tempera- 15. A. I. Smirnov, V. N. Glaznov, L. I. Leonyuk, et al., tures, at which the correlation radii are maximum and Zh. Éksp. Teor. Fiz. 114, 1876 (1998) [JETP 87, 1019 the thermodynamic characteristics exhibit features, (1998)].

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16. Y. Ando, J. Takeya, D. L. Sisson, et al., Phys. Rev. B 58, 26. J. P. Doumerc, J. M. Dance, J. P. Chaminade, et al., R2913 (1998). Mater. Res. Bull. 16, 985 (1981). 17. G. A. Petrakovskiœ, A. M. Vorotynov, G. Shimchak, 27. G. Barnes, Phys. Rev. B 35, 219 (1987). et al., Fiz. Tverd. Tela 40, 1671 (1998) [Phys. Solid State 28. M. Nishi, O. Fugita, and J. Akimitsu, Technical Report 40, 1520 (1998)]. of ISSR, Ser. A 2759, 1 (1993). 18. L. D. Fadeev and L. A. Takhtajan, Phys. Lett. A 85, 375 29. M. Ain, J. E. Lorenzo, L. P. Regnault, et al., Phys. Rev. (1981). Lett. 78, 1560 (1997). 19. D. J. Scalapino, Y. Imry, and I. Pincus, Phys. Rev. B 11, 30. B. Roessli, P. Fischer, J. Schefer, et al., J. Phys.: Cond. 2042 (1975). Matt. 6, 8469 (1994). 31. A. Damascelli, van der Marel, F. Parmigiani, et al., Phys. 20. H. J. Schulz, Phys. Rev. Lett. 77, 2790 (1996). Rev. B 56, R11373 (1997). 21. H. Raedt and A. Lagendijk, Phys. Rep. 127, 233 (1985). 32. G. Els, M. van Loosdrecht, and P. Lemmens, Phys. Rev. 22. S. S. Aplesnin, Fiz. Tverd. Tela 38, 1868 (1996) [Phys. Lett. 79, 5138 (1997). Solid State 38, 1031 (1996)]. 33. N. Nishi, O. Fujita, J. Akimitsu, et al., Phys. Rev. B 52, R6959 (1995). 23. N. Kawashima and J. E. Gubernatis, Phys. Rev. Lett. 73, 1295 (1994). 34. H. Hori, M. Furusawa, S. Sugai, et al., Physica B 211, 180 (1995). 24. P. W. Anderson, Mater. Res. Bull. 8, 153 (1973). 25. M. C. Gross and D. S. Fisher, Phys. Rev. B 19, 402 (1979). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 202Ð216. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 227Ð242. Original Russian Text Copyright © 2000 by Ginzburg, Savitskaya. SOLIDS

Self-Organization of the Critical State in Granular Superconductors S. L. Ginzburg and N. E. Savitskaya* St. Petersburg Institute of Nuclear Physics, Gatchina, Leningrad oblast, 188350 Russia *e-mail: [email protected] Received June 29, 1999

Abstract—The critical state in granular superconductors is studied using two mathematical models: systems of differential equations for the gauge-invariant phase difference and a simpliﬁed model that is described by a system of coupled mappings and in many cases is equivalent to the standard models used for studying self-organized criticality. It is shown that the critical state of granular superconductors is self-organized in all cases studied. In addition, it is shown that the models employed are essentially equivalent, i.e., they demonstrate not only the same critical behavior, but they also lead to the same noncritical phenomena. The ﬁrst demonstration of the existence of self-organized criticality in a system of nonlinear differential equations and its equivalence to self- organized criticality in standard models is given in this paper. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION external parameters for its existence. Structurally, the critical state that arises is a set of metastable states, Interest in investigations of granular superconduc- transforming into one another by means of “ava- tors has increased substantially in the last few years pri- lanches,” which arise as a result of a local external per- marily in connection with the fact that most HTSC turbation. Such a critical state is said to be self-orga- materials can be obtained precisely in the form of a nized, and the criterion for the existence of self-organi- granular system. Specifically, there have appeared a zation in the system is a power-law dependence of the number of theoretical works [1Ð3] on the magnetic probability density of the sizes of the avalanches. properties of such systems where the granular superconductor was modeled by a system of superconduct- The concept of self-organized criticality can be used ing granules joined by an insulator, i.e., an ordered lat- to describe the behavior of a large number of dissipa- tice of Josephson junctions. It has been shown that the tive dynamical systems arising in various fields of mod- magnetic properties of such a system are similar to the ern science. The properties of granular superconductors properties of hard type-II superconductors; specifically, are such that self-organized criticality can also be a critical state, which is described well by Bean’s model observed in them. This not only gives a new represen- [4], can also arise in them. However, the properties of a tation of the nature and structure of the critical state in granular system (specifically, the magnetic-field profile these systems, but it also opens up the possibility of in the sample in the critical state) depends strongly on studying SOC experimentally. This is especially impor- 3 Φ tant, since thus far, despite the generality of the phe- its main parameter V ~ jca / 0 (jc is the critical current Φ nomenon, SOC has been studied only on model sys- density at the junction, a is the lattice constant, and 0 tems which are difficult to realize in practice, and is the quantum of magnetic flux). It has been shown in experimental investigations have been performed only [1Ð3] that when this parameter is large (V ӷ 1), each for a sandpile [6]. cell of such a system becomes a pinning center and is The possibility of self-organization of the critical capable of confining a large number of magnetic flux ӷ quanta. It has also been shown [3] that for V ӷ 1 the state in granular superconductors for V 1 has been system as a whole possesses a large number of metasta- considered in [7], where a simplified model of such a ble states. system was introduced and studied. In constructing this model, taking account of the physical characteristics of The capability of the system to arrive in the critical the behavior of granular superconductors for V ӷ 1, the state without precise adjustment of the external param- system of differential equations for the gauge-invariant eters and the presence of a large number of metastable phase difference on junctions that describes a super- states are characteristic features of systems in which conductor was replaced by a system of mappings self-organized criticality (SOC) is observed [5]. It con- (equations with discrete time) for the currents through sists in the fact that the dynamical system naturally the junctions. It was found that for a two-dimensional evolves to the critical state, which subsequently is self- multijunction SQUID, the system of mappings maintaining, not requiring accurate adjustment of the obtained is equivalent to the classical model for study-

SELF-ORGANIZATION OF THE CRITICAL STATE IN GRANULAR SUPERCONDUCTORS 203 ing SOC: an Abelian model of a sandpile [8]. The mag- ductors under various conditions, proceeding directly nitude of the current in the junction played the role of from the system of differential equations for the gauge- the pile height at a given node, and current injection invariant phase differences at junctions. The corre- was the analog of adding sand grains. In this manner, it sponding simplified models described by systems of was shown that the critical state in the simplified model mappings were studied in parallel. As a result, it was of a multijunction SQUID described by a system of shown that the critical state in granular superconduc- mappings is self-organized. tors is self-organized, and the behavior of the simplified Subsequent study of the simplified model of granu- model is completely equivalent to the behavior of the lar superconductors showed that, on account of the initial system described by a system of differential physical properties incorporated in it, SOC in this sys- equations. tem possesses a number of interesting properties, In Sec. 2, the one- and two-dimensional granular which have not been observed in the Abelian sandpile superconductors (multijunction SQUIDs) are studied, model. the systems of differential equations describing them Thus, in [9] the simplified model of a multijunction and the corresponding simplified models are presented, SQUID was studied with various methods of current and the basic characteristics of the system are also injection into the system. Specifically, the case of cur- introduced: the average current and voltage, whose sta- rent injected simultaneously into all nodes in small tistical properties in the critical state will be studied. increments, which is not realizable in the sandpile In Sec. 3, the results of computer simulation of the model, since a sand grain in the sandpile model pos- critical state of multicomponent SQUIDs under various sesses a finite size, was studied. It was found that the conditions are presented: for open boundary conditions probability density of the avalanche sizes for all injec- with various methods of current injection and for tion methods is a universal function, while the finer closed boundary conditions and an increasing external characteristics (interavalanche correlators) differ. magnetic field. For each case simulation was performed In addition, in [10] the conjecture advanced earlier for the system of differential equations and for simpli- in [7] that in the simplified model of a multijunction fied models, and the results were compared. SQUID placed in an increasing magnetic field SOC is The basic results of this work are formulated briefly realized even under closed boundary conditions was in Sec. 4. verified. This is impossible, in principle, to do in the sandpile model, since in this case the possibility for sand to leave the system is one of the main require- 2. ONE- AND TWO-DIMENSIONAL ments for the existence of SOC. In the case of a multi- MULTIJUNCTION SQUIDS contact SQUID, however, the process of current leaving from the system is replaced by a fundamentally dif- 2.1. Basic Equations ferent process, the annihilation of currents with A two-dimensional multicontact SQUID consists of opposite signs, and the SOC can also exist in a closed two superconducting plates connected by Josephson system. junctions located at the nodes of a N × M lattice. The Thus, the study of the simplified model makes it junction size l is much less than the lattice period a possible to infer that the critical state in granular super- (Fig. 1a). In the geometry described, Josephson current conductors is self-organized and possesses a number of flows along the z axis. Then the system of equations for ϕ interesting properties, which have not been observed in the gauge-invariant phase difference n, m on the junc- previously investigated models with SOC. tions with coordinates (na, ma) can be written on the However, no conclusion can be drawn about the basis of the equation for a large Josephson junction existence and properties of SOC in real superconduc- [11]. Using the resistive model of a Josephson junction tors, whose behavior is described by differential equa- and neglecting thermal fluctuations [12], we obtain in tions, on the basis of only the simplified model, since dimensionless form the following system of equations: the simplifications introduced could have led to a loss ϕ ϕ τd n, m ∆ (ϕ) π of some properties of the initial system and to the V sin n, m + ------= n, m + 2 Fn, m, appearance of new qualities which are uncharacteristic dt of the initial system. For this reason, in the first place, π Φ the problem of investigating the properties of the criti- 2 jc 0 V = ------, jϕ = ------2------, cal state, proceeding directly from a system of differen- jϕ 8πl λ tial equations for the phase difference, remains open; in L the second place, it is interesting to investigate the Φ jen, m τ 0 question of the degree to which the simplified model Fn, m = ------, = ------, (1) jϕ ρ jϕ reflects the real behavior of the initial system. 0

Thus, the main objective of the present work was to ∆ , (ϕ) = ϕ , + ϕ , study theoretically and using a computer the critical n m n + 1 m n Ð 1 m ϕ ϕ ϕ state of one- and two-dimensional granular supercon- + n, m Ð 1 + n, m + 1 Ð 4 n, m,

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204 GINZBURG, SAVITSKAYA

z to the existence of phenomena such as quantization of (a) the magnetic flux and hysteresis [12]. These properties remain for a multijunction SQUID and V ӷ 1. In what follows, we shall study multijunction SQUIDs precisely for V ӷ 1. We shall study, in parallel with the systems of differential equations (1) and (2), simplified models of mul- x l a tijunction SQUIDs, which have analogs among previ- y ous models used to study SOC, such as, the Abelian (b) sandpile model and the one-dimensional “sandpile” model studied in [13, 14]. The transition to simplified models described by systems of coupled mappings has been examined in detail in [7]. In what follows, we shall present this in a brief form.

(n, m) 2.2. Simplified Model of Two- and One-Dimensional Multijunction SQUIDs a x n = (N – 1)/2 + 1 The simplified models of multijunction SQUIDs are z formulated in terms of the dimensionless current zn, m (c) through a junction: ϕ τ d n, m z , = z sin(ϕ , ) + ------, (3) n m c n m 2π dt π where zc = V/2 . For the dimensionless current, equa- x tions (1) are rewritten in the form a 1 I I z , (ϕ) = ------∆ , (ϕ) + F , . (4) en en + 1 n m 2π n m n m Fig. 1. (a) Two-dimensional multijunction SQUID, (b) pro- The system (3) and (4) obtained for V ӷ 1 can be sim- jection of the two-dimensional multijunction SQUID on the plified using the properties of the solution of equation (3), (x, y) plane, and (c) one-dimensional multijunction SQUID. which is an equation for the phase for a single Joseph- son junction. It is obvious from this equation that if ≤ ϕ zn, m zc, then d n, m/dt = 0, i.e., the phase does not ӷ λ change. However, if zn, m > zc, then for V 1 and where jc is the critical current density, L is the London | − | Ӷ ϕ Φ ρ zn, m zc zc this equation has a solution n, m(t), penetration depth, 0 is the flux quantum, 0 is the surface resistivity of the junction, j is the injection cur- which at first varies very little in a time interval T and en, m then changes by 2π in a time τ , where τ Ӷ T. rent density, and ∆ is the two-dimensional discrete 0 0 n, m ∆ ϕ Laplacian. Next, we neglect in the term n, m( ) this weak (of the order of 1/V) change and we approximate the phase For a one-dimensional multijunction SQUID (Fig. 1c) by a step function [12], which consists of two superconducting plates, ϕ ≈ π π ⁄ which are infinite along the y axis and are connected n, m 2 qn, m + 2, (5) with one another by Josephson junctions at the points where q is an integer. In this approximation the sys- with coordinates x = an, n = 1, …, N, the system of n, m equations (1) can be written as follows [2]: tem (3) and (4) can be rewritten as ϕ τ d n, m ∂ϕ z , = z sin(ϕ , ) + ------, V sinϕ + τ------n = Ð 2ϕ + ϕ + ϕ + 2πF , n m c n m 2π dt n ∂t n n + 1 n Ð 1 n (2) z , = ∆ , (q) + F , , (6) 4πaS j 4π j aS 4πaS n m n m n m V = 2π------c , F = ------e--n------, τ = ------, Φ n Φ ρ ϕ 0 0  n, m 1 qn, m = Int------π---- + -- , where S is the area of a cell between two junctions. 2 4 The parameter V appearing in the equations is the where Int(x) is the integer part of x. main characteristic of the SQUID. Thus, for V ӷ 1 the The equations (6) form a closed system in which the energy of a single-junction SQUID has a large number phases on different junctions do not interact with one (of the order of V) metastable states, and this fact leads another.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 SELF-ORGANIZATION OF THE CRITICAL STATE IN GRANULAR SUPERCONDUCTORS 205

We shall describe the solution of such a simpliﬁed Then we have for zn, m system in terms of the dimensionless current. ( ) ( ) ∆ {θ[ ( ) ]} ξ ( ) First, we examine the change in the phase at a site zn, m k + 1 = zn, m k + n, m z k Ð zc + n, m k (n0, m0). We choose as initial conditions the situation ( ) θ[ ( ) ] θ[ ( ) ] = zn, m k + zn + 1, m k Ð zc + zn Ð 1, m k Ð zc where a current Fn, m = K = Int(zc) less than the critical current is injected into all junctions, and the phases on + θ[z , (k) Ð z ] + θ[z , (k) Ð z ] (12) all junctions are the same and are equal to n m + 1 c n m Ð 1 c

Ð 4θ[z , (k) Ð z ] + ξ , (k), 2πK π  2πK n m c n m ϕ , (0) = arcsin ------= -- Ð 2 1 Ð ------. (7) n m     ξ ( ) ( ) ( ) V 2 V n, m k = Fn, m k + 1 Ð Fn, m k . ∆ The qn, m = 0 and n, m(q) = 0. The mappings obtained are completely equivalent to We shall assume that the current is injected into the the algorithm describing the Abelian sandpile model. system in integral portions, i.e., Fn, m is an integer, so The quantity zn, m plays the role of the pile height at a that z is also an integer. We add at the site n , m one given site, and current injection, described by the func- n, m 0 0 ξ unit of current, i.e., F , = K + 1. Then tion n, m(k), plays here the role of the addition of a sand n0 m0 grain. z , z , + 1. (8) n0 m0 n0 m0 For a one-dimensional multicomponent SQUID the system of mappings can be written in the form Now, the current z , exceeds the critical current, n0 m0 q (k + 1) = q (k) + θ[z (k) Ð z ], and then dϕ , /dt > 0. From the first equation of our n n n c n0 m0 system (6) we find the solution for ϕ , . It is such that z (k) = ∆ (q(k)) + F (k), (13) n0 m0 n n n the phase is essentially unchanged in a time T = ∆ (q) = Ð 2q + q + q . πτ (( ) ⁄ ) n n n + 1 n Ð 1 2 0/ 2 K + 1 zc Ð 1 , after which it rapidly π τ changes by 2 in time 0. At the moment the phase Then, for zn we have changes by 2π the following changes occur: z (k + 1) = z (k) + ∆ {θ[z(k) Ð z ]} + ξ (k) ∆ ∆ n n n c n qn, m qn, m + 1, n, m n, m Ð 4, (9) ( ) θ[ ( ) ] θ[ ( ) ] ∆ ∆ ∆ ∆ = zn k + zn + 1 k Ð zc + zn Ð 1 k Ð zc n ± 1, m n ± 1, m + 1, n, m ± 1 n, m ± 1 + 1. (14) θ[ ( ) ] ξ ( ) Ð 2 zn k Ð zc + n k , Then zn , m changes according to the law 0 0 ξ ( ) ( ) ( ) n k = Fn k + 1 Ð Fn k . z , z , Ð 4, z ± , z ± , + 1, n0 m0 n0 m0 n0 1 m0 n0 1 m0 (10) The system of mappings (14) is equivalent to the rules z , ± z , ± + 1. n0 m0 1 n0 m0 1 for the change in the slopes in the one-dimensional sandpile model studied in [13]. We note that the rules (8) and (10) for z , to n0 m0 We note that equations (12) and (14) describing the change are identical to the rules for adding sand grains current dynamics can be briefly written in the form and for falling of sand grains in the Abelian sandpile ∆ {Ψ[ ]} ξ model. zn, m(k + 1) = zn, m(k) + n, m z(k) + n, m(k), (15) The following difficulty arises when switching to ( ) ( ) ∆ {Ψ[ ( )]} ξ ( ) the entire lattice. The problem is that the time T for a zn k + 1 = zn k + n z k + n k , (16) weak change in phase to occur is different at different Ψ( ( )) θ( ) sites and constantly changes because during this weak z k = z Ð zc . (17) change jumps in phase can occur at a neighboring site. The equations (6) take this change in T into account, but Let us consider the relation between the function Ψ they are difficult to analyze. We shall make one other (z(k)) and the electrodynamic system and its possible simplification. We shall assume that T is the same for modifications. To understand the physical meaning of Ψ all sites. Then, we can introduce into the system the dis- (z(k)), we shall examine the relation between the voltage U (t) on a junction and the phase. This relation is crete time tk = kT. In this case, the advance of the phases n, m expressed as follows: can be ignored, and only the change in qn, m and zn, m need be considered. The rules for a change in zn, m in Φ ∂ϕ 0 n, m this case can be written in the form U , = ------. (18) n m 2π ∂t ( ) ( ) θ[ ( ) ] qn, m k + 1 = qn, m k + zn, m k Ð zc , (11) If the current through the junction is greater than the ( ) ∆ ( ( )) ( ) ϕ zn, m k = n, m q k + Fn, m k . critical current, then in a time T the phase n, m changes

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 206 GINZBURG, SAVITSKAYA by 2π. Then the average voltage on the junction over and M + 1, and 1 ≤ n ≤ N. We shall assume that the crit- this time is ical current density on them is inﬁnite:

t j = ∞, k + 1 Φ c  1 1 0 U , k + -- = --- U , (t)dt = ------, (19) n = 0, N + 1; 1 ≤ m ≤ M; (24) n m 2 T ∫ n m T tk m = 0, M + 1; 1 ≤ n ≤ N. or, in a more general form, Then the current on junctions with n = 0 and N + 1, 1 ≤ m ≤ M, m = 0 and M + 1, and 1 ≤ n ≤ N can only Φ  1 0 increase, i.e., the current can leave the sublattice where U , k + -- = ------θ[z , (k) Ð z ] ≤ ≤ ≤ ≤ n m 2 T n m c 1 n N and 1 m M, which corresponds to open (20) boundary conditions. The condition (24) is equivalent Φ 0 to the condition = ------[q , (k + 1) Ð q , (k)]. T n m n m ϕ n, m = 0,

Then we can write zn, m in the form n = 0, N + 1; 1 ≤ m ≤ M; (25) ( ) ( ) m = 0, M + 1; 1 ≤ n ≤ N. zn, m k + 1 Ð zn, m k (21) The equations (1) for nodes at the boundary, for exam- T ∆  1 ξ ( ) ple, for n, m = 1 and n = 1, m ≠ 1, M, and taking account = -Φ----- n, m Uk + -- + n, m k , 0 2 of the condition (24), can be written as follows: ϕ i.e., d 1, 1 V sinϕ , + τ------= ϕ , + ϕ , Ð 4ϕ , + 2πF , , 1 1 dt 2 1 1 2 1 1 1 1 Φ  1 0 U , k + -- = ------Ψ[z , (k)]. (22) ϕ n m 2 T n m d 1, m V sinϕ , + τ------1 m dt Since z is the dimensionless current through a junc- n, m ϕ ϕ ϕ ϕ π Ψ = 2, m + 1, m Ð 1 + 1, m + 1 Ð 4 n, m + 2 F1, m. tion, Un, m(k + 1/2) and therefore (z) have a simple physical interpretation: They are the current-voltage For the simpliﬁed models (12) and (14), open characteristic (IVC) of a junction. boundary conditions mean We note that only one threshold is taken into z = ∞, account in the expression (15) for the IVC. The com- c plete IVC, taking account of both thresholds, is given n = 0, N + 1; 1 ≤ m ≤ M; (26) by the expression m = 0, M + 1; 1 ≤ n ≤ N, Ψ( ) θ( ) θ( ) z = z Ð zc Ð Ð z Ð zc . (23) which is equivalent to the condition

zn, m = 0, 2.3. Boundary Conditions n = 0, N + 1; 1 ≤ m ≤ M; (27) The behavior of our system is not completely deter- m = 0, M + 1; 1 ≤ n ≤ N, mined by equations (1) and (2) or the systems of map- which is usually studied in problems concerning sys- pings (12) and (14). In problems of studying the SOC, tems with SOC. the boundary conditions and the method of perturbation of the system play an important role. Usually, two types With these boundary conditions the mappings (12) of boundary conditions are considered in problems have the form with SOC: open and closed. ( ) ( ) θ[ ( ) ] z1, 1 k + 1 = z1, 1 k + z2, 1 k Ð zc In this section we shall examine what different + θ[z , (k) Ð z ] Ð 4θ[z , (k) Ð z ] + ξ , (k), boundary conditions signify for our system. 1 2 c 1 1 c 1 1 ( ) ( ) θ[ ( ) ] In our case, open boundary conditions can be real- z1, m k + 1 = z1, m k + z2, m k Ð zc (28) ized, if it is assumed that our SQUID is shunted by a θ[ ( ) ] θ[ ( ) ] normal superconductor, whose critical current is sev- + z1, m + 1 k Ð zc + z1, m Ð 1 k Ð zc eral orders of magnitude greater than the critical cur- Ð 4θ[z , (k) Ð z ] + ξ , (k). rent of Josephson junctions. This means that to a lattice 1 m c 1 m of junctions with 1 ≤ n ≤ N and 1 ≤ m ≤ M are added Close or reﬂective boundary conditions mean that junctions with n = 0 and N + 1, 1 ≤ m ≤ M and m = 0 current is conserved and cannot leave the system. In

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 SELF-ORGANIZATION OF THE CRITICAL STATE IN GRANULAR SUPERCONDUCTORS 207 this case, equations (1) for nodes on the boundary, for 1. Current injection in all junctions simultaneously example, for n, m = 1, have the form in small portions. For the system (1), this means that ϕ after arrival in the next metastable state the injection ϕ τd 1, 1 ϕ ϕ ϕ π current on all junctions changes as follows: V sin 1, 1 + ------= 2, 1 + 1, 2 Ð 2 1, 1 + 2 F1, 1. dt (29) F , (t + δt) = F , (t ) + δF, For the simplified model n m 0 n m 0 (35) δ δ Ӷ ( ) ( ) θ[ ( ) ] t 0, F 1. z1, 1 k + 1 = z1, 1 k + z2, 1 k Ð zc (30) + θ[z , (k) Ð z ] Ð 2θ[z , (k) Ð z ] + ξ , (k). For the simplified model this method of perturbation 1 2 c 1 1 c 1 1 means that after the system arrives in the next metastable state for all n, m the function 2.4. Perturbation of the System ξ , (k ) = p, p Ӷ 1. (36) When studying SOC in the sandpile model, the system n m 0 is usually considered in the following regime. Starting in a The next method of perturbation is closest to being metastable state, the system is perturbed by a prechosen realized in practice in an experiment. method, for example, a single sand grain is added at a randomly chosen node. The perturbation is followed by a 2. To study the critical state in superconductors a relaxation process (“avalanche”), which puts the sys- sample is placed in an external ac magnetic field and tem into the next metastable state, the perturbation is current in the junctions is excited not by direct injection repeated, and so on. but rather by induction by an external field. In what fol- Two methods of perturbation are ordinarily used in lows we shall assume that the magnetic field is directed problems studying SOC in the sandpile model. along the y axis (Fig. 1b). Then it can be shown (see [7] 1. Addition of a sand grain at a randomly chosen for a more detailed discussion) that the effect of an node. In our case, this is equivalent to injection of a unit external increasing magnetic field effectively reduces current in a randomly chosen junction, i.e., when the to injection of a positive current in all junctions on the ∂ϕ ∂ right-hand boundary of a SQUID and the same current SQUID is in a metastable state (all ( n, m/ t)(t0) = 0), the injection current on a randomly chosen junction in absolute magnitude but negative in sign in all junc- increases by one unit: tions on the left-hand boundary. Then the total current injected into the system is zero. This means that when F , (t + δt) = F , (t ) + 1, δt 0. (31) n0 m0 0 n0 m0 0 the system is in a metastable state the injection current changes as follows: Subsequently, the magnitude of the injection current remains unchanged in the course of the relaxation pro- F , (t + δt) = F , (t ) Ð F , cess. 1 m 0 1 m 0 ex ( δ ) ( ) For the simplified model, this method of perturba- FN, m t0 + t = FN, m t0 + Fex, (37) tion means that after the system arrives in the next 1 ≤ m ≤ M. metastable state (all zn, m(k0) < zc) the function at the randomly chosen site (n0, m0) For the simplified system this will be ξ ( ) n , m k0 = 1, (32) ξ ξ ≤ ≤ 0 0 1, m = ÐFex, N, m = Fex, 1 m M. (38) and ξ = 0 during relaxation at all sites. n, m We note that in contrast to previous methods for per- 2. Addition of a sand grain at the central node of turbing the system, here not only a positive but also a the lattice. For the system (1), this is equivalent to negative current is injected into the system. Therefore, injection of a unit current in the central junction, i.e., a negative threshold must also be taken into account in when the SQUID is in a metastable state, the injection the IVCs of junctions in the simplified model of a gran- current on the central junction increases by one unit: ular superconductor. It was immaterial in previous ( δ ) ( ) δ Fn , m t0 + t = Fn , m t0 + 1, t 0. (33) cases, since there the system was always near the posi- c c c c tive threshold. Thus, in the method described for excit- For the simplified model, this method of perturba- ing the system the simplified model will be described tion means that after the system arrives in the next by the system of mappings (16) with the IVC (23). metastable state the function in the central node (n , m ) c c The method for perturbing the system in a one- ξ , (k ) = 1. (34) nc mc 0 dimensional state also has its own specific features. To study multijunction SQUIDs, besides the pertur- 3. In the sandpile model the SOC does not occur in bation methods described above, other methods, which the one-dimensional case [5]. In [13, 14] it was shown are physically meaningless for the Abelian sandpile model that the SOC in the one-dimensional case can occur if but are natural for SQUIDs can also be considered. the method for adding a sand grain is different from the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 208 GINZBURG, SAVITSKAYA method ordinarily used in the Abelian sandpile model. the system, i.e., the system-averaged current at the For our case, it can be expressed as follows: moment the ith avalanche ends: ( δ ) ( ) ⁄ Fn t0 + t = Fn t0 + 1 2, (39) 1 st ( ) ( δ ) ( ) ⁄ zi = ------∑zn, m tei , Fn + 1 t0 + t = Fn + 1 t0 Ð 1 2, NM n, m (44) ≠ where n N is a randomly chosen node. st V z , = ------sinϕ , (t ). For the simplified system this means n m 2π n m ei ξ ⁄ ξ ⁄ ≠ n = 1 2, n + 1 = Ð1 2, n N. (40) In the Abelian sandpile model the total mass of the system was the analog of this quantity [15]. We note that in [13, 14] a one-dimensional system was studied with one open and one closed boundary, For the simplified model of a multijunction SQUID, and even though negative slopes appeared in the sys- analogous quantities will have the index R. By analogy tem, for them only one threshold was studied. We shall to the expression (41) we obtain for the simplified study in this case the simplified model, taking account model the following expression for the integral of the of both thresholds (23), in order to be able to compare voltage: with the case of differential equations. k = tei R 1 W = ------∑ ∑θ(z , (k) Ð z ) (45) i NM n m c , 2.5. Average Current and Integral of the Voltage k = tbi n m The mathematical criterion for the existence of self- and for the average current organization in the system is power-law behavior of the probability density of avalanche sizes. The size of an R 1 z = ------∑z , (k = t ). (46) avalanche can be defined as the total number of top- i NM n m ei pings over the avalanche time [15]. This quantity can be n, m expressed as follows:

k = tei 3. RESULTS OF COMPUTER SIMULATION 1 S = --- ∑ ∑θ(s , (k) Ð s ), (41) In the present section the results obtained by com- i L n m c k = t n, m puter simulation of the critical state of one- and two- bi dimensional multijunction SQUIDs directly from the where L is the number of nodes in the lattice, tbi and tei system of differential equations with V = 40 are pre- are the moments when the ith avalanche starts and ends, sented. The equations were solved using Euler’s sn, m is the pile height at a given site in the Abelian sand- scheme with a time step dt = 0.01. Decreasing this pile model or the slope of the pile in the model of [13], quantity further and using other integration schemes and sc is the critical value for sn, m. In our case the ana- did not change the physical results. Systems with vari- log of sn, m is the value of the current in a junction. ous boundary conditions and with various methods of When the current in a junction exceeds the critical perturbation were studied. value, a jump in phase by 2π occurs at the junction, and Unless stipulated specially, the system was studied according to Eq. (18) this leads to a voltage pulse. Thus, in the standard regime used for studying systems with in our case, the SQUID-averaged integral of the voltage SOC. Starting with “smooth” initial conditions, for ϕ over the avalanche time plays the role of the avalanche which all n, m = 0 and Fn, m = 0, we perturbed the sys- size: tem, changing the magnitude of the injection current by one of the methods described above. Then, the system tie 1 relaxed to the next metastable state. The magnitude of U = ------∑ U , (t)dt. (42) i NM ∫ n m the injection current remained unchanged during relax- n, m tib ation. When the system reached a metastable state, we once again perturbed the system by the same method, Taking account of Eq. (18), we have for Ui the follow- and so on. In the simulation it was assumed that the sys- ing expression: ∂ϕ ∂ Ð8 tem reached a metastable state if n, m/ t < 10 for all Φ junctions. 0 U = ------∑[ϕ , (t ) Ð ϕ , (t )]. (43) After a transient process, the system reached a sta- i 2πNM n m ei n m bi n, m tionary critical state consisting of a set of metastable critical states transforming into one another after the For convenience in making comparisons below, we Φ next perturbation. We shall term the process whereby shall consider the dimensionless quantity Ui/ 0. the system arrives in the next metastable state an “ava- Besides the integral of the voltage, we shall also lanche.” For each avalanche the system-averaged cur- consider the properties of a different characteristic of rent, zi (Eq. (44)), at the moment the avalanche termi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 SELF-ORGANIZATION OF THE CRITICAL STATE IN GRANULAR SUPERCONDUCTORS 209 nates and the system-averaged integral of the voltage, value z while on the right-hand side they are negative Φ c Ui/ 0 (Eq. (43)), were determined. and close to Ðzc (Fig. 2a). This picture resembles the A corresponding simplified model described by the critical state in Bean’s model [4], but Bean’s model π system of mappings with zc = V/2 was studied each there is only one metastable state, into which the system time in parallel with the system of differential equa- returns each time after the next successive perturbation. In tions. The simplified model was studied in the same our case, however, after a perturbation the system relaxes regime and with the same perturbation method as the to a different metastable state, in which the separation into initial system. For the simplified model, “smooth” “positive” and “negative” parts remains but the values of the currents at the junctions change somewhat. This is boundary conditions means that all zn, m = 0, and the metastable state is determined as the state in which all the behavior characteristic for systems with self-orga- | | nization. zn, m < zc. For the simplified model, for each avalanche the average current z R (Eq. (46)) and the integral of We note that in the case of a one-dimensional “sand- i pile” the separation of the system into positive and neg- R voltage Wi (Eq. (45)) were also determined for each ative parts does not arise [13, 14], even though both avalanche. positive and negative slopes do occur. In our case the separation is due to the fact that two threshold values of Next, in the simplified model and for the initial sys- ρ the currents are present in the system. An interesting tem, the probability density (x) was calculated for feature of our system is that the separation occurs even each quantity, and in some cases the interavalanche cor- though the positive and negative currents are injected in relation functions Dx(j) and the power spectra Sx(f) of randomly chosen junctions. these quantities were also studied [16]: In the numerical simulation the quantities U /Φ for ρ( ) 〈δ( )〉 i 0 x = x Ð xi , R system (2) and Wi for the simplified model were cal- ( ) 〈 〉 〈 〉 2 Dx j = xi xi + j Ð x , culated for each avalanche. (47) ∞ It should be noted that in calculating them, only the ( ) ( ) Ði2πfj Sx f = 2 ∑ Dx j e . positive parts of the systems were considered. This is j = Ð∞ because our systems are closed, i.e., one of the most 〈 〉 important conditions for the existence of self-organized Here … denotes averaging over the number of the criticality, the possibility of a sink, is not satisfied for R Φ R avalanche i, and x can be z, z , U/ 0, or W . them. However, a process replacing a sink exists sepa- The results obtained for the simplified models and rately for positive and negative parts of the system. It initial systems of differential equations were compared. consists in the fact that the excess positive current, which cannot leave the system through the boundary, flows into the negative part of the system where it anni- 3.1. One-Dimensional SQUID hilates with the negative current, and similarly the The one-dimensional multijunction SQUID has excess negative current is shed into the positive part. been studied in detail in [17]. We shall briefly describe Thus, it can be assumed that junctions on the boundary the basic results obtained. between the positive and negative subsystems act for A one-dimensional multijunction SQUID described each of them effectively as a reservoir for the current by the system of differential equations (2) was studied shed from the subsystems, i.e., the positive parts are with open boundary conditions systems with one open boundary and self-organization of the critical state is possible in them. ∂ϕ ϕ τ 1 ϕ ϕ π R V sin 1 + ------= Ð 1 + 2 + 2 Fe1, The probability densities ρ(U Φ ρ ∂t i/ 0) and (Wi ), dis- (48) played in Fig. 2b, were also calculated. It is evident in ∂ϕ V sinϕ + τ------N- = Ð ϕ + ϕ + 2πF the figure that ρ(U /Φ ) and ρ(W R ) demonstrate N ∂t N N Ð 1 N i 0 i power-law behavior, whence it follows that SOC is with the perturbation method (39). realized in the positive part of the system described by In parallel with the system of differential equations, a the differential equations (2) and in the simplified simplified one-dimensional model (16) with the IVC (23) model. was also studied in the same regime with closed boundary It is clear from Fig. 2b that the results obtained for conditions and with the perturbation method (40). the simplified model and in the description of the sys- In both cases, after a transient process, the systems tem with differential equations are essentially identical. reached a critical state, which consisted of a set of The agreement obtained shows that the simplified metastable states passing into one another in which the model accurately reflects the basic feature of the behav- values of the dimensionless currents on the left-hand ior of the initial system and belongs the same universal- side of the lattice are positive and close to the critical ity class and the simplying assumptions made do not

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 210 GINZBURG, SAVITSKAYA ρ zst –1 10 10 (a) (b) ρ Φ (Ui/ 0) 8

10–2 ρ(WR) 6 i f(x) ~ x–1.76 4 10–3 2

0 10–4 –2

–4 10–5

–6

–8 10–6 0 20 40 60 80 100 120 1 10 100 Φ R n Ui/ 0, Wi

Fig. 2. Results of a computer simulation of the critical state of a one-dimensional multijunction SQUID for the system of differential equations (2) and for the simplified model (16) with the IVC (23). (a) One of the metastable states that comprise the self-organized critical state of a one-dimensional multijunction SQUID, obtained from a system of differential equations; (b) voltage probability ρ Φ ρ R density (Ui/ 0) for the system (2) and the probability density (Wi ) for the simplified model. lead to a loss of the basic properties of the initial system where l and j are random and independent of one by the model. another. This perturbation rule does not violate the main physical requirement that the total current injected into the system in a single step be zero. 3.2. Two-Dimensional Multijunction SQUID in an External Magnetic Field For such a perturbation method, after a certain transient period, the system reached a critical state, which In [10] the critical state in the simplified model of a consisted of a set of metastable states that transform two-dimensional multijunction SQUID placed in an ac into one another and in which the currents were positive magnetic field was studied by computer simulation on in the right-hand side of the system and negative in the an N × M lattice. In this case closed boundary condi- left-hand side. Just as for a one-dimensional SQUID, tions are realized in the system and the current in the SOC was realized in the positive and negative parts of junctions is excited not by the injection current but the system. This was indicated by the power-law prob- rather by an ac magnetic field, according to the rules ability density ρ(W R ). The realization of SOC in a (38). However, as noted in [10], for this method of per- i turbation SOC is not realized in the system. This is closed system is possible in this case also because of because the system degenerates into a set of one- annihilation of the currents at the boundary of the pos- dimensional strips in which the appearance of SOC is itive and negative subsystems for odd N at contacts with impossible for the perturbation method (38). For this rea- the coordinates n = (N Ð 1)/2 + 1 (Fig. 1b). It was shown son, a different perturbation method was used in [10]. in [10] that for odd N the properties of the critical state When the system reached a metastable state, a unit of cur- for the positive subsystem in a system with annihilation rent was added in a randomly chosen junction on the right- and in a system of corresponding size with one open hand boundary and a current Ð1 was added at a randomly boundary are completely equivalent. chosen contact on the left-hand boundary, i.e., In the present work the behavior of a two-dimensional multijunction SQUID was simulated by the sys- ξ ξ 1, l = Ð1, N, j = 1, (49) tem of differential equations (1) on a 31 × 15 lattice

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ρ S –1 0 (a) ρ Φ 10 (b) 10 (U / 0) i Φ SUi/ 0 ρ R –1 (Wi ) –2 10 10 S R f(x) ~ x–1.15 Wi –2 10 10–3 10–3 10–4 10–4 10–5 10–5 0.1 1 10 10–4 10–3 10–2 10–1 100 Φ R Ui/ 0, Wi f ρ S 0.07 (c) (d) –3 S ρ(z ) 10 zi 0.06 i ρ zR SzR 0.05 ( i ) 10–4 i 0.04 10–5 0.03 –6 0.02 10

0.01 10–7

5.10 5.15 5.20 5.25 5.30 5.35 5.40 10–4 10–3 10–2 10–1 100 R zi, zi f

Fig. 3. Results of a computer simulation of the critical state of a two-dimensional multijunction SQUID, placed in an external magnetic field, for the system of differential equations (1) and the simplified model (16) with the IVC (23). (a) Voltage probability density R ρ(U /Φ ) for the system (1) and the probability density ρ(W ) for the simplified model, (b) voltage power spectra S ⁄ Φ (f) for i 0 i Ui 0 ρ ρ R the system (1) and S R (f) for the simplified model, (c) probability density (zi) for the average current for the system (1) and (zi ) Wi

for the simplified model, and (d) power spectra of the average current S (f) for the system (1) and S R (f) for the simplified model. zi zi with closed boundary conditions. The system was per- Figure 3 also displays similar characteristics for the turbed by the rule (49), which for the system of differ- positive subsystem in the simplified model. ential equations means It is evident in Fig. 3a that the probability density ( δ ) ( ) ρ Φ F1, l t0 + t = F1, l t0 Ð 1, (Ui/ 0) of the voltage for the positive subsystem, just (50) ( δ ) ( ) as in the simplified system, possesses a power-law dis- FN, j t0 + t = FN, j t0 + 1, tribution, which indicates that SOC is realized in the where l and j are random and independent of one positive subsystem of the closed system. In addition, another. just as in the one-dimensional case, the results for the The structure of the critical state arising in the sys- simplified model and the initial system of differential tem is completely analogous to that of the critical state equations are essentially identical. Therefore, even in in the simplified model. Just as in the simplified model, this case both systems are in the same universality a positive subsystem in which for each avalanche in the class. stationary critical state the subsystem-averaged current We also compared the positive subsystem for the and integral of the voltage and the probability density case where the SQUID is described by differential and power spectra for these quantities were calculated equations with a 15 × 15 system with one open bound- for each avalanche. The results are displayed in Fig. 3. ary and injection of positive current on the boundary

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ρ S 100 (a) ρ Φ (b) (Ui/ 0) R SU /Φ –1 ρ(W ) i 0 10 i 10–4 ( ) ~ –1.29 S R f x x Wi 10–2

–3 10 10–5

10–4

–5 10 10–6 0.01 0.1 1 10–4 10–3 10–2 10–1 100 Φ R Ui/ 0, Wi f ρ S (c) (d) 0.05 ρ Sz (zi) 10–4 i ρ R SzR 0.04 (zi ) i 10–5 0.03 10–6 0.02 10–7 0.01 10–8 0 4.95 5.00 5.05 5.10 5.15 5.20 10–4 10–3 10–2 10–1 100 R zi, zi f

Fig. 4. Results of a computer simulation of the critical state of a two-dimensional multijunction SQUID with current injection into a randomly chosen junction for the system of differential equations (1) and the simpliﬁed model (12). (a) Voltage probability density R ρ(U /Φ ) for the system (1) and the probability density ρ(W ) for the simplified model; (b) voltage power spectra S ⁄ Φ ( f ) for i 0 i Ui 0 ρ ρ R the system (1) and S R ( f ) for the simpliﬁed model; (c) probability density (zi) of the average current for the system (1) and (zi ) Wi

for the simpliﬁed model; and, (d) power spectra of the average current S ( f ) for the system (1) and S R ( f ) for the simpliﬁed model. zi zi

opposite to the open boundary. Just as in the simplified The critical state arising in this case in the system, model [10], the basic characteristics of the system in just as in the simplified model, is a set of metastable the critical state were essentially the same. states can transform into one another. Just as in [9], we studied not only the probability density for the average Φ current zi and the voltage Ui/ 0, but also the interava- 3.3. Two-Dimensional Multijunction SQUID lanche correlators and the power spectra of these quan- with Open Boundary Conditions tities. The critical state in the simplified model of a two- We obtained the following results for various pertur- dimensional multijunction SQUID with open boundary bation methods. conditions has been studied in detail in [9] with various methods of current injection into the system. Here we 1. Current injection into a randomly chosen junction present the results of a study of the critical state of a (method (31) for (1) or (32) for the simplified model (12)). two-dimensional multijunction N × N (N = 21) SQUID Figure 4a shows the voltage probability densities in described by the system of differential equations (1) both cases. We can see that they demonstrate power- with open boundary conditions (25) and the same per- law behavior. This indicates that a self-organized criti- turbation methods as in [9]. cal state is realized in the system.

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ρ S 100 ρ Φ 10–1 (a) (Ui/ 0) (b) R –2 SU /Φ –1 ρ(W ) 10 i 0 10 i R –1.65 –3 S f(x) ~ x 10 Wi 10–2 10–4 10–3 10–5 10–4 10–6 10–7 10–5 10–8 0.01 0.1 1 10–4 10–3 10–2 10–1 100 Φ R Ui/ 0, Wi f ρ S (c) (d) 0.05 10–3 ρ (zi) Sz 0.04 ρ R –4 i (zi ) 10 Sz R i 0.03 10–5 0.02 10–6

0.01 10–7 0 4.9 5.0 5.1 5.2 5.3 10–4 10–3 10–2 10–1 100 R zi, zi f D (e) 0.0004 D zi 0.0002 DzR 0 i –0.0002 0 500 1000 1050 2000 2500 j

Fig. 5. Results of a computer simulation of the critical state of a two-dimensional multijunction SQUID with current injection into the central junction for the system of differential equations (1) and the simpliﬁed model (12) in the case of random initial conditions. ρ Φ ρ R (a) Voltage probability density (Ui/ 0) for the system (1) and the probability density (Wi ) for the simpliﬁed model; (b) voltage

probability spectra S ⁄ Φ ( f ) for the system (1) and S R ( f ) for the simpliﬁed model; (c) probability density of the average current Ui 0 Wi R ρ(z ) for the system (1) and ρ(z ) for the simplified model; (d) power spectra of the average current S ( f ) for the system (1) and i i zi

S R ( f ) for the simpliﬁed model; (e) correlation functions of the average current D ( j) for the system (1) and D R ( j) for the sim- zi zi zi pliﬁed model.

It is also clearly seen in Figs. 4aÐ4d that the charac- For smooth initial conditions, the results for both teristics considered are also essentially identical for the systems were essentially the same. For random initial simplified model and the initial system in the present conditions, a difference is observed only in the proba- case. This shows that both of these systems in this situ- bility density of the average current (Fig. 5). The ation fall into the same universality class. behavior of the probability density remains the same but the magnitude of the average current decreases in the 2. Current injection in the central junction (method simplified model and increases in the initial system (1). (33) for (1) and (34) for the simplified model (12)). This is because the mappings and differential equations respond differently to the introduction of nonintegral In this case, for the simplified model (12) and the initial conditions. initial system (1) both “smooth” and “random” initial Irrespective of the initial conditions, the voltage conditions, for which Fn, m(0) are random numbers and probability density for this method of perturbation, just ≤ ≤ π 0 Fn, m(0) V/2 , were used. as in the previous cases, shows power-law behavior,

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ρ S –2 100 ρ Φ 10 (a) (Ui/ 0) (b) SU /Φ –1 ρ R –3 i 0 10 (Wi ) 10 R SW f(x) ~ x–1.23 i 10–2 10–4 10–3 10–5 10–4 10–6 10–5 10–7 0.01 0.1 1 10–4 10–3 10–2 10–1 100 Φ R Ui/ 0, Wi f ρ S –2 0.030 (c) 10 (d) ρ(z ) –3 Sz 0.025 i 10 i ρ z R) Sz R ( i 10–4 i 0.020 10–5 0.015 10–6 0.010 10–7 0.005 10–8 0 10–9 4.8 4.9 5.0 5.1 5.2 5.3 5.4 10–4 10–3 10–2 10–1 100 R f zi, zi D (e) × –4 Dz (j), D(0) = 3 10 0.0001 i × –4 0 Dz R(j), D(0) = 2 10 –0.0001 i 0 500 1000 1500 j

Fig. 6. Results of a computer simulation of the critical state of a two-dimensional multijunction SQUID with current injection simultaneously into all junctions for the system (1) with δF = 1/N2 and the simplified model (12) with p = 1/N2. (a) Voltage probability R density ρ(U /Φ ) for the system (1) and ρ(W ) for the simplified model; (b) voltage power spectra S ⁄ Φ ( f ) for the system (1) i 0 i Ui 0 ρ ρ R and S R ( f ) for the simplified model; (c) probability density of the average current (zi) for the system (1) and (zi ) for the sim- Wi

pliﬁed model; (d) power spectra of the average current S ( f ) for the system (1) and S R ( f ) for the simpliﬁed model; (e) correlation zi zi

function of the average current D ( j) for the system (1) and D R ( j) for the simplified model. zi zi i.e., the system remains self-organized. However, the function for the current. Quasiperiodicity is clearly exponents for the cases of smooth and random initial seen in its behavior. conditions differ somewhat. This phenomenon was 3. Current injection simultaneously into the entire already noted in [18], where the sandpile model with lattice in small portions (method (35) for (1) or method sand grains added at the center and different initial con- (36) for the simplified model). ditions was studied. The difference in the exponents In this case, “random” initial conditions with a vari- could be due to breaking of the internal symmetry of ≤ ≤ the system. ance of the random quantities Fn, m, such that 0 Fn, m V/2π, were used to study the critical state in the simpli- At the same time, as shown previously in [9], in con- fied model and for the initial system (1). This is because trast to the case where current is injected into a ran- for smooth initial conditions the behavior of the system domly chosen junction, a peak at frequency f ~ 1/N2 is became purely periodic and SOC did not arise in it. noticeable in the power spectrum of the current. This We chose the quantity δF for the system of differen- indicates the presence of a quasiperiodic process in the tial equations or p for the simplified model in a manner system. Figure 5e shows the interavalanche correlation so that in one step a unit of current was injected into the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 SELF-ORGANIZATION OF THE CRITICAL STATE IN GRANULAR SUPERCONDUCTORS 215 system, i.e., N2δF = N2p ≈ 1. Figure 6 displays the prob- for study and computer simulation than the initial sys- ability density for the average current and the voltage, tem of differential equations. Thus, SOC in granular the power spectra of these quantities, and the interava- superconductors can be further studied using the sim- lanche correlation function of the average current for plified models introduced in [7] for various cases. the simplified system (12) and the system (1). It is clear In conclusion, we note once again that the physical from the figure that, just as in the preceding cases, the properties of granular superconductors make it possible results for both systems are essentially identical. A dif- to study a system under conditions which are unphysi- ference occurs only for the probability density of the cal in previously proposed models for studying SOC. currents. As a result of our study of the characteristic features of As has already been shown in [9], the voltage prob- SOC in SQUIDs, it was shown that for all boundary ability density possesses a power-law distribution. This conditions and methods for perturbing the system indicates the presence of SOC in the system. However, which were studied, the system remains self-organized while the probability density of the voltage shows in the generally accepted sense, i.e., the voltage proba- power-law behavior for all methods considered for per- bility densities have a power-law distribution. At the turbing the system, the power spectra for a given same time, interavalanche correlators, as characteris- method differ from the spectra obtained for the central tics which are more sensitive to the characteristic fea- and random injections. The spectra contain a distinct tures of the system, behave differently. A study of these peak attesting to the presence of a quasiperiodic pro- correlators showed that quasiperiodic processes can cess with period T ~ pÐ1 or T ~ δFÐ1 in the systems. exist in a system with SOC. Moreover, it was shown that in granular superconductors SOC can exist even As noted in [9], the relative magnitude of the peak with closed boundary conditions. In this case the exist- at frequency f ~ p or f ~ δF increases when the magni- ence of SOC is ensured not by the possibility of a sink, tude of the current injected into the system in one step as in all previously studied open systems, but rather by increases. In general, however, the periodicity in the a fundamentally different mechanism Ð annihilation of behavior of the system is observed not only with the currents with opposite signs. Thus, the discovery of increasing current injected into the system in one step SOC in granular superconductors gives new informa- but also with “smoothing” of initial conditions, i.e., a tion about this phenomenon. decrease in the initial variance of Fn, m, as well as when We are grateful to O.V. Gerashchenko and M.A. Pus- the current was injected not into all junctions but into a tovoit for an interesting discussion of the problem and definite group of junctions. for valuable remarks. This work was sponsored by the Russian Founda- 4. CONCLUSIONS tion for Basic Research (Grant no. 99-02-17545) and is supported by the Scientific Council on the direction In the present paper, the critical state of two- and “Superconductivity” of the program “Current Direc- one-dimensional multijunction SQUIDs with a large tions in Condensed-Media Physics” as part of project parameter V ӷ 1 was studied by computer simulation no. 96021 “Profile,” and also the subprogram “Statistical directly from a system of differential equations for the Physics” of the State Scientific and Technical Program phase differences on the junctions. It was shown that in “Physics of Quantum and Wave Processes” as part of this case the critical state in the system is self-orga- project VIII-3 and the State Program “Neutron Investiga- nized, i.e., it consists of a set of metastable states, tions of Matter.” One of us (N. S.) is grateful to the Rus- which pass into one another by means of avalanches sian Foundation for Basic Research for partial financial arising after a local external perturbation of the system. support (projects 96-15-96775 and 96-15-96764). Thus, a physical system with SOC that can be studied not only theoretically but also experimentally was described. REFERENCES In parallel with the initial systems described by dif- 1. D.-X. Chen, J. J. Moreno, and A. Hernando, Phys. Rev. ferential equations, simplified models of one- and two- B 53, 6579 (1996). junction SQUIDs, described by systems of mappings 2. D.-X. Chen, A. Sánchez, and A. Hernando, Phys. Rev. B and having analogs among standard models for study- 50, 10342 (1994). ing SOC, such as the belian sandpile model, were stud- 3. A. Majhofer and T. Wolf, Phys. Rev. B 47, 5383 (1993). ied. It was shown that for each of the boundary conditions and methods for perturbing the system which 4. C. P. Bean, Rev. Mod. Phys. 36, 31 (1964). were studied in this paper, the behavior of the basic 5. P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, quantities is the same for both systems, i.e., they fall 381 (1987). into the same universality class. Therefore the simplifications introduced do not change the basic physical 6. G. A. Held, et al., Phys. Rev. Lett. 65, 1120 (1990). properties of the system. This result is also important 7. S. L. Ginzburg, Zh. Éksp. Teor. Fiz. 106, 607 (1994) because the simplified model is much more convenient [JETP 79, 334 (1994)].

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8. D. Dhar, Phys. Rev. Lett. 64, 1613 (1990). 14. S. T. R. Pinho, C. P. C. Prado, and S. R. Salinas, Phys. 9. S. L. Ginzburg, M. A. Pustovoit, and N. E. Savitskaya, Rev. E 55, 2159 (1997). Phys. Rev. E 57, 1319 (1998). 15. S. S. Manna and J. Kertesz, Physica A 173, 49 (1991). 10. S .L. Ginzburg and N. E. Savitskaya, Pis’ma Zh. Éksp. 16. J. S. Bendat and A. G. Piersol, Random Data: Analysis Teor. Fiz. 69, 119 (1999) [JETP Lett. 69, 133 (1999)]. and Measurement Procedures (Mir, Moscow, 1989; Wiley, New York, 1986). 11. O. I. Kulik and I. K. Yanson, The Josephson Effect in Superconducting Structures (Nauka, Moscow, 1970). 17. S. L. Ginzburg and N. E. Savitskaya, Pis’ma Zh. Éksp. Teor. Fiz. 68, 688 (1998) [JETP Lett. 68, 719 (1998)]. 12. K. K. Likharev, Introduction to the Dynamics of Joseph- son Junctions and Circuits (Nauka, Moscow, 1985; Gor- 18. K. Wiesenfeld, J. Theiler, and B. McNamara, Phys. Rev. don and Breach, New York, 1986). Lett. 65, 949 (1990). 13. L. Kadanoff, S. R. Nagel, L. Wu, et al., Phys. Rev. A 39, 6524 (1989). Translation was provided by AIP

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 217Ð221. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 243Ð247. Original Russian Text Copyright © 2000 by Moskalenko, Perel’, Yassievich. SOLIDS

Effect of a Magnetic Field on Thermally Stimulated Ionization of Impurity Centers in Semiconductors by Submillimeter Radiation A. S. Moskalenko1, V. I. Perel’*1, and I. N. Yassievich**1, 2 1 Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia 2 Department of Theoretical Physics, Lund University, Lund, S-223 62 Sweden *e-mail: [email protected] **e-mail: [email protected] Received July 7, 1999

Abstract—The probability of electron tunneling from a bound state into a free state in crossed ac electric and dc magnetic fields is calculated in the quasiclassical approximation. It is shown that a magnetic field decreases the electron tunneling probability. This decreases the probability of thermally activated ionization of deep impurity centers by submillimeter radiation. The logarithm of the ionization probability is a linear function of the squared amplitude of the electric field and increases rapidly with the frequency of the electric field. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. QUASICLASSICAL WAVE FUNCTION OF A TUNNELING ELECTRON In the last few years it has been shown experimentally and theoretically that the main mechanism of ion- For exponential accuracy the wave function in the ization of impurity centers in semiconductors by an quasiclassical approximation is given by the expression electric field and by submillimeter radiation is thermally activated tunneling [1Ð4]. This process includes Ψ(), ∝ i ˜ (), electron tunneling and tunneling of a vibrational sys- rt exp ប--- S r t , (1) tem of the center. Recently, the effect of a magnetic field on this process in a dc electric field [5] was studied theoretically using the results of [6] for the electron where the action S˜ (r, t) satisfies the Hamilton–Jacobi tunneling probability. equations The present paper is devoted to the theory of ther- ∂S˜ mally activated tunneling ionization of centers by an ac ------==ÐᏴ ()Pr,,t,∇S˜P (2) electric field in the presence of a dc magnetic field per- ∂t pendicular to the electric field. The paper consists of two parts. In part one the method used in [7] is applied with the boundary condition S˜ (r, t) = Ðεt as r 0, to derive an expression for the probability of electron where ε is the electron energy on a center (ε < 0). In tunneling through a time-varying potential barrier. The equations (2) Ᏼ is the Hamiltonian and P is the gener- calculations are performed for the short-range potential alized momentum. In the present case of a center. In the case of a charged center, there is hope that an attractive Coulomb potential will increase the 2 1e tunneling probability but will not greatly change the Ᏼ()Pr,,t =------P Ð -- A ÐzF() t , linear dependence of the logarithm of the probability of 2mc (3) thermally activated tunneling ionization on the squared e amplitude of the electric field (equation (25); see dis- P= mrú+ -- A , cussion of a similar question in the case of a dc electric c field in the book [3, §10.5]. In part two thermally stim- Ω ulated tunneling is studied. We confine our attention to where F(t) = Fcos t is the force exerted on an electron exponential accuracy. In this approximation, the loga- by the electric field of the wave and F = eᏱ. It is rithm of the ionization probability is a linear function of assumed that the field Ᏹ is directed along the z axis, and the squared amplitude of the electric field. A magnetic the dc magnetic field H is directed along the x axis. In field decreases the ionization probability. what follows, we use the gauge Ax = Az = 0 and Ay = ÐHz.

218 MOSKALENKO et al.

The action S˜ satisfying the required conditions can Using the second equation of equations (2), we obtain be written in the form [7, 8] the condition

t ImP = 0. (10) ˜ ε ᏸ( , , ) From the first two equations of motion (6) and the sec- S = S0 Ð t0, S0 = ∫ r' rú' t' dt', (4) ond equation (3) it follows that Px and Py are integrals t0 of motion: P = mv and P = mv and, hence, v and ᏸ x 0x y 0y 0x where is the Lagrangian, v0y are real. Then, it follows from equation (9) that v0z is a purely imaginary quantity. It follows from equa- mrú'2 eH ᏸ(r', rú', t') = ------Ð z' ------yú' Ð F(t') . (5) tions (2) that for real values of t the imaginary part of 2 c the action does not depend on t in the region of space determining the result of tunneling. Therefore, in what The radius vector r'(t') satisfies the equations of motion follows, we can set t = 0. F We shall write out the solution of the equations of úxú' = 0, úyú' = ω zú', úzú' = --- cosΩt' Ð ω yú' (6) c m c motion (6) that satisfies the first of the conditions (7): ( ) ω x' = v 0x t' Ð t0 , (where c is the cyclotron frequency) under the conditions ( ω ω ) ( ω ω ) y' = a1 cos ct' Ð cos ct0 + a2 sin ct' Ð sin ct0 ( ) ( ) r' t0 = 0, r' t = r. (7) ω c ( Ω Ω ) Ð -----b sin t' Ð sin t0 , In equations (4) t0 should be taken to be a function of r Ω and t, determined by the condition ( ω ω ) ( ω ω ) z' = Ða1 sin ct' Ð sin ct0 + a2 cos ct' Ð cos ct0  dS˜  ( Ω Ω ) ------ = 0. (8) Ð b cos t' Ð cos t0 , dt0 r, t F We underscore that, according to the meaning of the b = ------. (11) (Ω2 ω 2) wave function, the quantities r and t are real, while r', m Ð c t', and t0 can also be complex. Using the equality [2] Here a1, a2, and v0x are, for the time being, arbitrary dS  constants, but they are expressed in terms of x, y, z, and ------0 = Ᏼ ,   t = t0 t by the second of equations (7). However, the condi- dt0 , 0 r t tions (10) are sufficient to determine these constants in equation (8) can be written in the form the region that is important for tunneling. These conditions can be rewritten (taking into account the fact that m 2 2 2 --- (v + v + v ) = ε, (9) z is real) in the form 2 0x 0y 0z dr' ú Im ------= 0. where v0 = r' (t0) is the velocity at the initial time t0.  dt' According to equation (9), this velocity is purely imag- t' = t ε inary, since < 0. This is natural, since the electron is Hence it follows that a1, a2, and v0x are real. located below the barrier. τ τ We set t0 = 0 + i e and we write out the condition for z to be real and v0z imaginary, using the last of equa- 3. PROBABILITY OF DIRECT TUNNELING tions (11) with t' = t and z' = z, and differentiating this OF AN ELECTRON equation with respect to t' and setting t' = t0. Then, we obtain Having solved the equation of motion (6) under the conditions (7), we can find v as a function of r, t, and Ωτ 0 ω τ ω τ sinh e Ωτ t . Then equation (9) determines t as a function of r a1 cos c 0 + a2 sin c 0 = b------ω------τ---sin 0 0 0 sinh c e ˜ (12) and t. As a result, the action S can be found as a func- Ω coshΩτ Ψ e Ωτ tion of r and t and therefore the wave function (r, t) = bω------ω------τ---sin 0. can also be found. c cosh c e However, to find the flux from the center (propor- It follows from the second equality in equation (12) that Ωτ tional to |Ψ|2) it is sufficient to find ImS˜ in the region sin 0 = 0, i.e., τ τ of space where this quantity is maximum, i.e., for val- 0 = 0, t0 = i e, a1 = 0 (13) ues of r such that Ωτ (strictly speaking, the condition sin 0 = 0 admits non- ∇ ˜ ImS = 0. zero solutions for t0, but it can be shown that they lead

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 EFFECT OF A MAGNETIC FIELD ON THERMALLY STIMULATED IONIZATION 219 to the same results). The fact taht x and y are real (the Thus, the ionization probability is given to exponential ε τ ε first two of equations (11) with t' = t) leads to the equal- accuracy by equation (17), where Se( ) and e( ) are ities determined by equations (16) and (18). Fω sinhΩτ We note the relation v = 0; a = ------c------e-. (14) 0x 2 2 2 ω τ ˜ ∂ mΩ(Ω Ð ω ) sinh c e dS  S0  dt0 c ------= ------Ð ε ------Ð t = Ðt , ε  ∂  dε 0 0 Thus, the solution of the equations of motion that satis- d t0 0 fies the first of the conditions (7), the condition (10), whence it follows that and the condition for x', y', and z' to be real at t' = t and ∂S z' to be imaginary at t' = t0, we have τ ប e e = Ð --∂---ε--. (20) x' = 0 τ ω Ωτ The quantity e can be called the electron tunneling F c sinh e ω y' = ------Ð sinΩt' + ------sinω t' , time. Without a magnetic field (for c = 0) equations Ω(Ω2 ω 2) sinhω τ c m Ð c c e (16)Ð(19) are identical to the result for the tunneling (15) ionization in an ac electric field [7] and for Ω = 0 they F ω sinhΩτ z' = ------c ------e-(cosω t' Ð coshω τ ) are identical with the result for tunneling ionization in (Ω2 ω 2) Ω sinhω τ c c e a dc electric field in the presence of a magnetic field [6]. m Ð c c e As the magnetic field increases, the ionization prob- ( Ω Ωτ ) ability decreases and the tunneling time increases. -Ð cos t' Ð cosh e . 4. THERMALLY ACTIVATED TUNNELING Hence it is easy to obtain v0, differentiating these τ IONIZATION OF A SHORT-RANGE CENTER expressions with respect to t' and setting t' = i e. Then τ the relation (9) can be used to determine e. This rela- The probability of tunneling ionization with the par- tion assumes the form ticipation of phonons can be regarded as being the result of three processes: thermal excitation of the sys- Ωω 2 ω 2 tem into a vibrational level E in an adiabatic potential 2 Ωτ  c   Ωτ c ω τ  1 sinh e 1 Ð ------coth e Ð ----- coth c e U (x), corresponding to an electron bound on a center Ω2 ω 2  Ω  1 Ð c (x is the vibrational coordinate), tunneling transition of Ω2 (16) the vibrational system to the adiabatic potential U2ε(x), = ------2m ε . corresponding to a free electron with a negative energy F2 ε, and a tunneling transition of an electron under the action of an electric field from a well into a free state The ionization probability Pe can be written in the form with negative energy ε. P ∝ exp[Ð2S (ε)], (17) In the quasiclassical approximation, the ionization e e ε probability for fixed and E1 is proportional to the ε ˜ ប ˜ where Se( ) = ImS / , and S is determined by the for- expression mulas (4) and (5) where r' is given by the formulas (15), τ  E1 t0 = i e and t = 0. Thus, [ ( )] [ (ε)] expÐ------ exp Ð2 S2ε Ð S1ε exp Ð2Se , (21) τ kT e m 2 2 ετ S (ε) = ------(zú' Ð yú' )dτ Ð ------e. (18) where the three cofactors give the probabilities of the e 2ប∫ ប three processes enumerated above, 0 x In deriving this expression, the last two terms in equa- cε 2M ( ) ε tion (5) were integrated by parts. The variable of inte- S1ε = -----ប------∫ U1 x Ð Edx, E = E1 Ð T , τ gration = Ðit' was introduced in equation (18), so that a 1 (22) Fω ω sinhΩτ xcε yú' = ------c------Ð coshΩτ + -----c ------e-coshω τ , 2M 2 2 c ( ) ( ) ( ) ε (Ω ω ) Ω sinhω τ S2ε = ------U2ε x Ð Edx, U2ε x = U2 x + m Ð c c e ប ∫ a ε Fω 2 zú' = ------c------(19) (the notations are illustrated in Fig. 1), and M is the (Ω2 ω 2) m Ð c mass associated with the vibrational mode playing the main role in the process. Ω ω sinhΩτ × ----- sinhΩτ Ð -----c ------e-sinhω τ i. The total ionization probability is obtained by inte- ω Ω ω τ c ε c sinh c e grating the expression (21) with respect to E1 and .

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(a) The integrals can be calculated by the method of steep- U2(x) est descent. This gives two equations for the optimal ε U1(x) quantities E1m and m: τ τ ប ⁄ τ τ 2ε = 1ε + 2kT, 2ε = e, (23) U2ε(x) τ where the “tunneling times” nε of the vibrational system are determined by the expression ε ∂ xcε E x Snε τ ε = Ðប------. n ∂E ε ε For not too strong fields, the quantity m is small and we T ε E1 can set = 0 in the first equation in equations (23). It determines the energy E1m, which does not depend on the electric field. In the expression (21) the quantity ε S2ε Ð S1ε can be expanded in powers of , retaining only (b) the first term in the expansion, U2(x) ετ U (x) 2 1 S2ε Ð S1ε = S2 Ð S1 + ---ប----, (24) τ τ where S2, S1, and 2 are the values of S2ε, S1ε, and 2ε at ε = 0. Then, using equation (18) with τ = τ , the ioniza- U ε(x) e 2 2 tion probability e(F) can be obtained as ε xcε E x 2 ( ) ( ) F e F = e 0 exp-----2, (25) Fc ε T where it is convenient to write E1 2 3mប Fc = ------, τ 3 2*

Fig. 1. Diagram of adiabatic potentials, illustrating the tun- ω 2 neling rearrangement of the vibrational system accompany- τ 3 3 c ing ionization for two possible arrangements of the poten- 2* = ------2 (Ω2 ω 2) tials (a and b; autolocalization occurs in the case b). Expla- Ð c nations are presented in the text. τ (26) 2 2  ω sinhΩτ τ*3 Ω τ*3 Ω ω τ3 ×  Ð coshΩτ + -----c ------2- coshω τ ( 2 ( , 0) – 2 ( , c))/ 2 ∫ Ω ω τ c  sinh c 2 2.0 0 Ωτ 2 = 4 2 Ω ω sinhΩτ  Ωτ c 2 ω τ  τ + ω----- sinh Ð -Ω------ω------τ--- sinh c d . 1.5 c sinh c 2  τ The dependence of 2* on the magnetic field is illus- Ωτ 1.0 2 = 2 trated in Fig. 2. ω Without a magnetic field (in the limit c 0) the Ωτ 2 = 1 expressions (25) and (26) are identical to the expression 0.5 for the probability of multiphonon thermally activated Ωτ = 0 tunneling under the action of an ac electric field [2]. In 2 the limit Ω 0, it follows from equation (26) that 3τ τ*3 ------2(ω τ ω τ ) 0 1 2 3 4 5 2 = 2 c 2 coth c 2 Ð 1 , (27) ω τ ω c 2 c which gives the result for the probability of this process τ*3 Ω τ*3 ω Ω τ3 ω τ Fig. 2. [ 2 (0, ) Ð 2 ( c, )]/ 2 versus c 2 for vari- under the action of a dc electric field in the presence of Ωτ ous values of the parameter 2. a magnetic field [5].

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We note that the applicability condition for equations 96392). I. N. Yassievich is grateful to the Swedish ӷ |ε | Ӷ ε (25) and (26) are the inequalities F Fc and m T, NSRC (Grant O-AH/KG 03996-322). |ε | τ τ where m is determined by equation (16) with e = 2. ε According to the first of equations (23) with = 0 REFERENCES τ τ ប ⁄ 2 = 1 + 2kT, (28) 1. V. Karpus and V. I. Perel’, Pis’ma Zh. Éksp. Teor. Fiz. 42, 403 (1985) [JETP Lett. 42, 497 (1985)]. τ whence 1 is essentially temperature-independent. For 2. S. D. Ganichev, E. Ziemann, Th. Gleim, et al., Phys. Rev. adiabatic potentials of the type shown in Fig. 1b Lett. 80, 2409 (1998). τ (autolocalization), we have 1 < 0. 3. V. N. Abakumov, V. I. Perel’, and I. N. Yassievich, Non- radiative Recombination in Semiconductors (Peterburg- skiœ Institut Yadernoœ Fiziki im. B. P. Konstantinova 5. CONCLUSIONS RAN, St. Petersburg, 1988); Modern Problems in Con- In summary, in the present paper an expression was densed Matter Sciences, Ed. by V. M. Agranovich and A. A. Maradudin (North-Holland, Amsterdam, 1991), obtained for the probability of a tunneling transition of Vol. 33. an electron from a bound state into a free state under the 4. S. D. Ganichev, I. N. Yassievich, and W. Prettl, Fiz. action of an ac electric field in the presence of a mag- Tverd. Tela 39, 1905 (1997) [Phys. Solid State 39, 1703 netic field oriented perpendicular to the electric field (1997)]. (equations (17)Ð(19)). 5. V. I. Perel’ and I. N. Yassievich, Pis’ma Zh. Éksp. Teor. The result was used to calculate the probability of Fiz. 68, 763 (1998) [JETP Lett. 68, 804 (1998)]. thermally activated tunneling ionization of deep impu- 6. V. S. Popov and A. V. Sergeev, Zh. Éksp. Teor. Fiz. 113, rity centers in semiconductors by submillimeter radia- 2047 (1998) [JETP 86, 1122 (1998)]. tion in a magnetic field (equations (25), (26), and (28)). 7. L. D. Landau and E. M. Lifzhitz, Quantum Mechanics: All results were obtained to exponential accuracy. Non-Relativistic Theory (Nauka, Moscow, 1989; Perga- mon, Oxford, 1977). 8. L. D. Landau and E. M. Lifzhitz, Mechanics, 3rd ed. ACKNOWLEDGMENTS (Nauka, Moscow, 1988; Pergamon, Oxford, 1976). This work was supported by the Russian Foundation for Basic Research (Grants 98-02-18268 and 96-15- Translation was provided by AIP

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 222Ð226. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 248Ð252. Original Russian Text Copyright © 2000 by Il’ichev. MISCELLANEOUS

Recombination Accompanied by Teleportation of “Quantum Entanglement”

L. V. Il’ichev Institute of Automatics and Electrometry, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia; e-mail: [email protected] Received May 13, 1999

Abstract—It is remarked that the recently proposed [2] and partially realized [3] teleportation of quantum states of microparticles is (in a certain modiﬁcation) a natural element of the process of cross recombination of radical spin pairs [4]. In the present paper the situation where singlet and triplet pairs are created in the medium and then diffuse and can recombine, if the particles encountering one another are in a singlet state, is examined in detail. As a result of cross recombination and teleportation, there arises a countable set of spin states of radical pairs. The distributions of the densities of radical pairs corresponding to this set are found under stationary and spatially uniform conditions. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION and Bob, each possess a pair of spin-1/2 particles. We denote Alice’s particles by the numbers 1 and 3 and The phenomenon of so-called quantum entangle- Bob’s particles by the numbers 2 and 4. We consider a ment (QE) has been known for a long time, but it has situation where the state of all four particles has the aroused great interest in the last few years in connec- form tion with its role in the quantum theory of information and the theory of a quantum computer (see, for exam- ρˆ ()ρ1234,,, =ˆ()ρ12, ⊗ˆ()34, . (2) ple, [1]). Quantum entanglement gives rise to specific correlations, which have no classical analog, between In this case, as one can easily see, the pairs of Alice’s subsystems of a single quantum system. In the present and Bob’s particles a priori are in the states paper the following general negative definition of QE is ρ used: A state (the density matrix) ˆ (1, 2) of the union ρ()pre (), ρ()ρ,,, ()⊗ ρ() ˆ 13 ==Tr24, ˆ1234 ˆ1ˆ3, of two subsystems S1 and S2 is said to be “nonentan- (3) ()pre gled” if it can be represented as a convex linear combi- ρˆ()24, ==Tr , ρˆ()ρ1234,,, ˆ()2⊗ ρˆ()4 nation of products of the states of the subsystems, spe- 13 cifically, and in accordance with equation (1) they do not possess the property of QE. Alice performs a measurement on her particles, checking to see whether or not her pair is ρˆ ()12, = pρˆ()1⊗ρˆ()2, (1) ∑ii i in a singlet state, i ˆ()Ψ, ≡|〉Ψ(), 〈|(), P013 013 013 , (4) where pi > 0 and ∑ipi = 1. We note that for a “pure” state ρˆ (1, 2) = |ψ(1, 2)〉〈ψ(1, 2)| this definition of “non- where entanglement” reduces to the well-known requirement |〉Ψ(), that the wave function be factorizable: |ψ(1, 2)〉 = |ψ(1)〉 ⊗ 013 |ψ(2)〉. 1 ()|〉ψψ ()⊗|〉ψ() |〉ψ()⊗|〉() = ------+ 1 Ð3ÐÐ1 +3, The existence of QE is the main attribute in realizing 2 the so-called quantum teleportation [2, 3]. We shall be interested in a modification of this phenomenon, specifi- |ψ±(i)〉 is the state of particle i with spin projection ±1/2 cally, quantum entanglement teleportation (QET). The crux of the QET process (in the form in which it in a certain fixed direction. We note that the state Pˆ0 (1, 3) appears below) is as follows. Two individuals, Alice refers to the so-called “maximally entangled” state, in

RECOMBINATION ACCOMPANIED BY TELEPORTATION 223 which the states of the particles 1 and 3 are maximally The view that QET is an exotic phenomenon is mixed: probably incorrect. In the present paper we study a simple model process of creation and recombination of 1 spin radicals in a medium that is accompanied by QET. ρˆ (1) = Tr Pˆ (1, 3) = --Iˆ(1), 0 3 0 2 As far as I know, the aspect associated with teleportation was ﬁrst brought up in [4]. The crux of the phe- 1 ρˆ (3) = Tr Pˆ (1, 3) = --Iˆ(3). nomenon is as follows. Let us assume that only singlet 0 1 0 2 pairs can be created and recombined, and the spin states of the correlated pairs are not subjected to any pertur- All information carried by such states is contained in bations and remain singlets in the intervals between the the interparticle correlations. creation and recombination events. As a result of diffusion, particles from different pairs can approach one For a positive outcome of an experiment, Alice’s another.1 Successful recombination is identical to a pair at Bob’s location are in the state positive outcome of Alice’s experiment. The photon

( ) emitted on recombination carries a classical message ρˆ post (2, 4) indicating success, and the remaining fragments of the two pairs are in a singlet state. The kinetics of this pro- Tr , [Pˆ (1, 3)(ρˆ (1, 2) ⊗ ρˆ (3, 4))Pˆ (1, 3)] (5) = ------1----3------0------0------cess, taking account of possible creation of triplet pairs Tr , , , [Pˆ 0(1, 3)(ρˆ (1, 2) ⊗ ρˆ (3, 4))Pˆ 0(1, 3)] also, is studied in Section 3, but ﬁrst we need to develop 1 2 3 4 a convenient language for quantitative description of in accordance with von Neumann’s projection postu- QET in our model. late. It is convenient to introduce the QET mapping τ 1, 3, acting on the set of all states of particles 1, 2, 3, and 4, of the form (2). The result of this mapping is the 2. GENERAL RELATIONS state (5): FOR QET ACCOMPANYING RECOMBINATION

τ ρ( , ) ⊗ ρ( , ) ρ(post)( , ) As already mentioned in the Introduction, QET in 1, 3: ˆ 1 2 ˆ 3 4 ˆ 2 4 . (6) the presence of recombination appears to be especially simple if all correlated pairs are singlets. In this case, In the next section it would be shown that in the pres- each recombination event annihilates two singlet pairs ence of an initial “entanglement” within the pairs 1, 2 and creates one new pair—“teleportation of singlet- ( ) and 3, 4 (the so-called QE resource) the state ρˆ post (2, 4), ness” occurs. The problem of this section is to deter- (pre) mine the quantitative characteristics of QET accompa- in contrast to ρˆ (2, 4), is “entangled.” nying recombination of fragments of two pairs mixed in an arbitrary manner. These characteristics are The fundamental condition, anticipating any use of required in the situation where triplet pairs appear in quantum correlations of the QE type which exist in the the medium for some reason. As will be shown below, ( ) state ρˆ post (2, 4), is Bob receiving a signal from Alice this is equivalent to a decrease in the quality of EPR indicating a positive outcome for her experiment. In the channels. We shall conﬁne our attention to the spatially literature, this circumstance is often interpreted as the isotropic situation. In this case, the most general states necessity of the existence of two types of channels for of pairs of particles i and j have the form transmitting information in realizing QET (and also standard quantum teleportation)—a classical channel, (α) 3α + 1 ρˆ (i, j) = ------Pˆ (i, j) along which Alice reports about the outcome of her 4 0 experiment, and the so-called EinsteinÐPodolskyÐ (7) Rosen (EPR) quantum channels, formed by the world 1 Ð α + ------( P ˆ 1 , Ð 1 ( i , j) + Pˆ 1, 0(i, j) + Pˆ 1, +1(i, j)), lines of “entangled” pairs of particles 1, 2 and 3, 4. It 4 should be noted that in our exposition the states of the pairs 1, 2 and 3, 4 appear as two equivalent EPR chan- where we have introduced the parameter α ∈ [Ð1/3, 1], nels. The pertinence of such a point of view is dictated ˆ ˆ ˆ by the speciﬁc nature of the problem considered below. and the operators P1, Ð1 (i, j), P1, 0 (i, j), and P1, +1 (i, j), Our symmetric interpretation differs from the conven- by analogy with equation (4), are projectors onto the tional one [2], where the EPR channel is only one pair, states for example, 1, 2, and this channel is used for telepor- |Ψ ( , )〉 |ψ ( )〉 ⊗ |ψ ( )〉 tation of the “entanglement” existing between the spin 1, Ð1 i j = Ð i Ð j , states of particles 3 and 4 into a new pair 2, 4. A more 1 This fundamentally distinguishes the model under consideration accurate term, used in the literature for the process from the process of geminal (cell) recombination [5]. The recom- under study, entanglement swapping, is the transport or bination of radicals from different pairs has been studied, for transfer of entanglement. example, in [6].

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 224 IL’ICHEV |ψ ( )〉 ⊗ |ψ ( )〉 |ψ ( )〉 ⊗ |ψ ( )〉 3. KINETIC EQUATIONS |Ψ ( , )〉 + i Ð j + Ð i + j 1, 0 i j = ------, AND THEIR SOLUTIONS 2 We shall now formulate the kinetic model. We shall consider an unbounded medium in which creation and |Ψ , (i, j)〉 = |ψ (i)〉 ⊗ |ψ ( j)〉, 1 +1 + + recombination of pairs of particles with spin 1/2 occur.2 respectively. The sum in parentheses in equation (7) is, For simplicity, we shall assume that the particles cre- to within a factor of 1/3, an isotropic triplet state. The ated are identical. We assume that only singlet pairs can parameter α, which gives the fraction of singletness recombine, while both singlet and triplet pairs can be and tripletness, is introduced in a manner so that α = 1 created. In the intervals between creation and recombi- corresponds to a purely singlet pair, α = 0 corresponds nation events, the particles diffuse with diffusion coefficient D, which does not depend on the state of the par- to an uncorrelated maximally mixed state of the parti- (n) cles i and j, and α = Ð1/3 corresponds to a purely triplet ticle. We introduce the set of functions f (r1, r2), two- state. coordinate spatial densities of pairs of particles in the ρ(n) It is not difficult to show that in the model under spin state ˆ from equation (9). The densities satisfy consideration the recombination of two particles from the following system of kinetic equations: different pairs occurs with probability 1/4 (this immedi- ∂ (n)( , ) ately determines the normalization factor in equation (5)) t f r1 r2 and, which is especially important, it dictates the unique ν ( ) (n)( , ) ν (n)( , ) ( ) multiplicative character of the teleportation mapping from = Ð f r1 f r1 r2 Ð f r1 r2 f r2 equation (6): n (11) ν (k)( , ) (n Ð k)( , ) 3 + ∑∫ f r1 r' f r' r2 d r' τ ρ(α)( , ) ⊗ ρ(β)( , ) ρ(αβ)( , ) 1, 3: ˆ 1 2 ˆ 3 4 ˆ 2 4 . (8) k = 0 ( ) ( ) α β + D(∆ + ∆ ) f n (r , r ) + γ n (r )δ(r Ð r ). Since the parameters and of the initial pairs do not r1 r2 1 2 1 1 2 exceed 1 in magnitude, we have Here the first two terms on the right-hand side corre- 0 ≤ αβ ≤ min( α , β ). spond to annihilation of pairs accompanying recombination of one of its particles, located at the points r1 This reflects the unavoidable degradation of the tele- and r2; ported state as compared with the states of the EPR ∞ channels 1Ð2, 3Ð4. (n) 3 f (r) = ∑ ∫ f (r, r')d r' If it is assumed that triplet pairs, just like singlet n = 0 pairs, arise as a result of a dissociation-type process, then it can be concluded from the multiplicative char- is the total density of the recombining pairs. The third acter of the mapping τ that only pairs whose parameter term corresponds to creation of a pair accompanying α is an integer power n = 0, 1, 2, … of the number –1/3, annihilation of two other pairs. The parameters of the corresponding to a triplet state, are present in the created pair are related with the parameters of the anni- medium: hilated pairs via the QET mapping (10). The last two terms are diffusion terms and the source of local pair (0) (1) n creation. We assume that only the quantities γ and γ ρ(n)( , ) 1  1 ρ(0)( , ) ˆ i j = -- 1 + 3Ð-- ˆ i j are different from zero. We note that in the case of sin- 4 3 glet and triplet pairs we use singular sources in equa- (9) n tion (11). At the same time, it is known that because of 3  1 (1) + -- 1 Ð Ð-- ρˆ (i, j). the statistics of spin-1/2 particles the coordinate density 4   3 matrix of a created triplet pair and together with it the γ(1) function (r1, r2) must approach zero as r1 r2. Here the state ρˆ is indexed by the power n and not by However, in our approach, where all spatial scales are the parameter α itself, as in equation (7). In the chosen much greater than the molecular scale (and this is the ( ) notation, ρˆ 0 (i, j) corresponds to a singlet state and characteristic size of the radical pair just created), we (1) are fully justified in replacing the true source of triplet ρˆ (i, j) corresponds to a triplet state. In the new nota- pairs by its singular approximation. tion the teleportation mapping becomes additive: We shall consider the spatially uniform and stationary situation where f(n)(r , r ) = f(n)(|r Ð r |), f(r) = f, τ ρ(n)( , ) ⊗ ρ(m)( , ) ρ(n + m)( , ) 1 2 1 2 1, 3: ˆ 1 2 ˆ 3 4 ˆ 2 4 . (10) 2 Spin must be understood in the generalized sense. It can be the The maximally mixed state must now be designated as standard spin and any dichotomous observable, for example, (∞) chirality. In the latter case, the states “spin up” and “spin down” ρˆ (i, j). must be identified with right-hand and left-hand enantiomers.

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(0) 3 f (r)l /f C1 = 1, n = 2, 3, …. The characteristic form of the dis- (0) 1.2 tribution f (r) is displayed in the figure. The plot was constructed for equal rates of creation 1.0 of triple and singlet pairs (γ(0) = γ(1)). An expression for the characteristic length l is presented in the Conclu- 0.8 sions (see equation (16)). We note the unique step character of the function f(0)(r). 0.6 4. CONCLUSIONS 0.4 We have described on the basis of a simple “mass 0.2 action” type law the kinetics of the creation, diffusion, and recombination of pairs of particles in “entangled” quantum spin states. A nontrivial feature of the model 0 0.1 0.2 0.3 r/l is that the process of QET in the recombination of particles is taken into account. The stationary uniform Stationary density of singlet pairs (see explanation in regime admits, as shown above, a simple analytical the text). description. We note that taking account of the evolution of the spin states between creation and recombination events should substantially change the results γ(n)(r) = γ(n). Then the equations for the Fourier trans- obtained. (n) | | forms of the functions f ( r1 Ð r2 ), As follows from equations (12) and the figure, as a (n) (n)( ) ( ⋅ ) 3 result of diffusion and the QET process, a substantial f q = ∫ f r exp Ðiq r d r, fraction of “entangled” spin states with characteristic size 0.1l can appear, where acquire the especially simple form 2 1/4 n  D  l = ------. (16) (ν 2) (n) ν (k) (n Ð k) γ (n)  (0) (1)  2 f + Dq f q = ∑ f q f q + . (12) ν(γ + γ ) k = 0 Nonlocal, nonclassical spin correlations arise in the ∞ ( ) The total stationary particle density f = ∑ f n is medium. This should have nontrivial thermodynamic n = 0 0 consequences. We shall examine as an example the determined by the same system of equations: fluctuations of the total projection of the spin of the particles located in a certain region Ω. For simplicity we γ (0) γ (1) f = ------+ ------. (13) shall assume that only singlet pairs are present. We ν ν present without derivation the result of calculation of the distribution function p(m) for the probability of (0) A quadratic equation is obtained for the density f q of finding the total spin projection m of all particles in a (0) chosen direction: singlet pairs (and only for f q ). The stable solution is ( ) (ξ) Ðξ p m = I 2m e . (17) 2 (0) (0) D 2  D 2 γ f q = f + ----q Ð f + ----q Ð ------. (14) ν  ν  ν Here In(z) is a modified Bessel function, and the parameter The solution of the system (12) for the Fourier trans- (n) forms f q of the densities for n = 1, 2, … has the form ξ 3 3 (0)( ) = ∫d r1∫d r2 f r1 Ð r2 (18) ( ) ( ) 1 n 2 0 1/2 Ð n Ω (n) γ   D 2 γ Ω f q = Cn-----ν--  f + -ν---q  Ð ---ν---- , (15) 2 is defined as the integral over the volume of the region Ω Ω ξ where the constants Cn are determined from the system and its exterior part . Physically, the parameter is of recurrence relations the average number of singlet pairs, one fragment of which lies inside Ω and the other outside Ω. It is obvi- n Ð 1 1 ous that only such pairs will contribute to the total spin C = -- ∑ C C , Ω ξ n 2 k n Ð k of the region . It is also obvious that depends not k = 1 only on the volume but also on the shape of the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 226 IL’ICHEV region Ω, especially if the characteristic size of Ω or its REFERENCES individual parts is of the order of l. 1. B. B. Kadomtsev, Dynamics and Information (Editorial The situation where a source of triplet pairs is ofﬁce of the journal Usp. Fiz. Nauk, Moscow, 1997). present is more complicated. This case, and the general 2. C. H. Bennett, G. Brassard, C. Crepeau, et al., Phys. Rev. problems of thermodynamics of a medium of spin-cor- Lett. 70, 1895 (1993). related pairs, require a separate analysis. 3. D. Bouwmeester, J.-W. Pan, K. Mattle, et al., Nature 390, 575 (1997). 4. B. Brocklehurst, Int. Rev. Phys. Chem. 4, 279 (1985). ACKNOWLEDGMENTS 5. A. L. Buchachenko, R. Z. Sagdeev, and K. M. Salikhov, Magnetic and Spin Effects in Chemical Reactions I am grateful to S.V. Anishchik, N.N. Lukzen, and (Nauka, Novosibirsk, 1978). P.L. Chapovskiœ for a fruitful discussion of the problem. This work was supported by the Russian Foun- 6. S. V. Anischik, O. M. Usov, O. A. Anisimov, et al., dation for Basic Research (grant no. 98-03-33124a) Radiat. Phys. Chem. 51, 31 (1998). and the Federal target program “Integration” (project no. 274). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 24Ð29. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 29Ð34. Original Russian Text Copyright © 2000 by Kovarskiœ, Prepelitsa. ATOMS, SPECTRA, RADIATION

Influence of a Polar Ambient Medium on Multiphoton Resonances in Dipole Molecules V. A. Kovarskiœ and O. B. Prepelitsa Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau, 277028 Moldova; e-mail: [email protected] Received May 26, 1999

Abstract—The influence of fluctuations of a polar ambient medium on multiphoton resonances in dipole molecules interacting with an external electromagnetic field is studied. It is shown that the electric fields created by the fluctuating dipole moments of the medium strongly influence the fulfillment of the multiphoton resonance conditions. Cases of slow and fast fluctuations of the polar ambient medium are considered separately. It is established that the temperature dependence of the multiphoton resonance probability is non-Arrhenius because of the influence of the fluctuations on the profile of the potential barrier. For fast fluctuations it is predicted that the rate of the multiphoton transition will decrease as the intensity of the fluctuations of the polar ambient medium increases. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION external electromagnetic field. One model [20–22] frequently used for simplicity only allows for direct inter- Multiphoton transitions in isolated molecules have action of the electromagnetic field with the electron and formed the subject of numerous theoretical and experi- neglects nonresonant interaction of the electromagnetic mental investigations (see, for example, the book by field with intramolecular vibrations and with a polar Kovarskiœ et al. [1]). Recently, increasing interest has ambient medium. We also shall adopt this model. been directed toward a new class of nonlinear optical effects involving the generation of higher harmonics If a molecule has a nonzero average electron dipole [2Ð4]. Investigating this problem revealed that the moment, this leads to strong interaction between this ambient medium may well play a role in high-order molecule and a high-intensity electromagnetic field. In multiphoton processes. The simplest task in the theory the simplest case of a two-level molecule in an external of nonlinear interaction of electromagnetic radiation electromagnetic field the system may be described by with atoms and molecules involves studying the mul- the Hamiltonian tiphoton absorption process and thus, in the present HH=++H H , paper we consider the influence of the ambient medium e vib int on this process and specifically analyze the influence of where a polar ambient medium on multiphoton resonance in a 1 1 dipole impurity molecule. The presence of fluctuating H =--- ()σε Ð ε +--- E ()d Ð d dipole moments of an ambient medium may bring about e 2 2 1 z20 22 11 significant changes in the multiphoton absorption and ×σ cos()Ωt + E d σ cos()Ωt scattering of light by an isolated molecule. This influ- z 0 21 x ence of a polar ambient medium on quantum nonradia- is the Hamiltonian of a two-level system whose unper- | 〉 ε tive transitions involving intramolecular vibrations has turbed states i have the energies i and the intrinsic already been examined in the literature [5Ð22]. For a dipole moment dii. Here i = 1, 2 denotes the ground and molecule having a dipole-active vibration, the ambient excited states of the system, d21 is the dipole moment of medium strongly influences the absorption process as a σ the transition between the ground and excited states, z result of dipoleÐdipole interaction. At the same time, σ and x are the Pauli matrices satisfying the familiar the establishment of multiphoton resonances in a mol- commutation relationships, and E and Ω are the ampli- ecule in many respects depends on the contribution of 0 electronÐvibrational interactions. This particularly applies tude and frequency of the applied linearly polarized to the influence of the electric fields created by fluctua- electromagnetic field. In addition, Hvib is the Hamilto- tions of the dipole moments of the ambient medium on the nian of the vibrational subsystem and Hint is the inter- dynamics of intramolecular vibrations and therefore on action Hamiltonian of the electronic and vibrational the establishment of multiphoton resonances. The dipole subsystems. moment of a molecule is formed by electronic and vibra- In the simple case of a multiphoton transition with- tional states and determines its interaction with an out any vibrations involved, the problem is in many

INFLUENCE OF A POLAR AMBIENT MEDIUM ON MULTIPHOTON RESONANCES 25 respects similar to that of a multiphoton transition to a over the vibrational states in the initial electronic state degenerate level of the hydrogen atom [23]. In this with the density matrix case, the transition rate W21 is described by the generat- τ ρ = AÐ1 exp(ÐβH ), ing function I21(t, ) of the multiphoton process: T 1 ∞ 1 A = Tr(exp(ÐβH )), β = ------, τ  i (ε ε )τ ( , τ) 1 k T W21 = 2∫d expប-- 2 Ð 1  I21 t , (1) 0 0 k0 is the Boltzmann constant, and T is the absolute tem- where the generating function perature. The value of S(τ) may be calculated exactly [1]: 2   E0d21 (τ) i (ε ε )τ (ωτ ϕ) I (t, τ) = ------S = exp-- 1 Ð 2 + zcos + , 21  ប  ប  × exp{i(ρ Ð ρ )[sin(Ωt) Ð sin(Ω(t Ð τ))]} 22 11 ( ) 1 (2) z = a n n + 1 , n = ------(--ប----ω------⁄------)------, (4) × cos(Ωt)cos(Ω(t Ð τ)), exp k0T Ð 1 i ρ E0d22 ρ E0d11 ϕ = ------. 22 = ---ប----Ω------, 11 = ---ប----Ω------, 1 + 2n is written in the lowest order of perturbation theory in The value of a is the so-called heat release constant, terms of the interaction (E d )σ cos(Ωt), which defined as the square of the displacement of the adia- 0 21 x batic potentials relative to each other: mixes states |1〉 and |2〉, while the interaction of the 2 intrinsic dipole moments dii, i = 1, 2 is taken into  V  ប a = ------. (5) account exactly [1]. បω mω The subsequent generalization of formula (1) corresponds to allowing for the contribution of the vibra- It is easy to show that by using formulas (3)Ð(5) the tional degrees of freedom to the multiphoton transition rate of a multiphoton process involving vibrational probability. We shall confine our analysis to the case of quanta may be reduced to the form a single vibrational mode q of frequency ω active in the 2 ∞ 2 ( ) (E d ) J (ρ) electronic transition. The adiabatic approximation is W a = 2π------0------21------∑ R (z)n2----n------21 ប p ρ2 usually used to describe electronÐvibrational interac- , ∞ tion. In accordance with this, we take the electronÐ n p = Ð × δ(ε ε បΩ បω) vibrational interaction in the electronic state |2〉 in the 2 Ð 1 Ð n + p , form   1  1 p ⁄ 2 (6) R (z) = exp Ða n + -- 1 + -- I (z), Hint = Vq. p   2  n p Here V is the matrix element of the coefficient function ρ ρ ρ of the electronÐvibrational interaction. The assumed = 22 Ð 11. ρ constraint of a linear term with respect to the coordinate Here Jn( ) and Ip(z) are Bessel functions of real and q in the Hamiltonian Hint and the constraint of the har- imaginary arguments, respectively. monic approximation for the Hamiltonian Hvib are con- We shall subsequently investigate the influence of a sistent with the “displaced parabola” model for the adi- polar ambient medium on multiphoton processes in abatic terms of the ground and excited electronic states. dipole molecules caused by their active interaction with In this case, calculations of the generating function fluctuations of the dipole moments of the ambient using standard methods in the theory of multiphoton medium. We shall generalize formula (6) to the case of a processes [1] yield the following formula: polar ambient medium and identify the conditions under ∞ which this medium has a stabilizing or destabilizing (a) τ ( , τ) (τ) influence on multiphoton resonances. For this we shall W21 = 2∫d I21 t S . (3) assume that under certain conditions (fast fluctuations) 0 the optical absorption bands undergo a familiar narrow- Here the generating function I (t, τ) is given by for- ing effect with increasing fluctuation intensity [24] and 21 the rates of the nonradiative processes decrease [21, 22]. mula (2), For the real energy levels of an impurity molecule, (τ) 〈 ( τ) ( τ)〉 S = exp iH2 exp ÐiH1 av , the resonance condition involving electromagnetic field photons and vibrational quanta is not always satisfied. where Hi are the Hamiltonians of the vibrational sub- Any resonance detuning may be compensated by means 〈 〉 system in states i = 1, 2 and … av denotes averaging of additional interactions introduced by the polar ambi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 26 KOVARSKIŒ, PREPELITSA ent medium, changes in the kinetic energy as a result of ω2 ( ) m 2 ( ) collisions, and other factors. However, we shall identify U2 q = ------q Ð Vq Ð F t q. the particular role which may be played by fluctuations 2 of a polar ambient medium since, as will be shown sub- The rate of the transition between electronic states 1 sequently, the intensity of these fluctuations can not and 2 of the impurity molecule under the action of an only increase the rate of the multiphoton transition but applied electromagnetic field in the presence of a also, which is particularly important, can reduce the dipole ambient medium is written in the form probability of multiphoton resonance. The reduction in ∞ the transition rate in this last case is occasioned by i τ  (ε' ε )τ ( , τ) broadening of the energy levels caused by the interac- W21 = 2∫d expប-- 2 Ð 1  I21 t tion between an impurity molecule and the polar ambi- 0 ent medium. × ϕ* Ψ , Ψ* , τ ϕ ∫dq1dq2 0 (q2)∑ p(q2 t) p (q1 t Ð ) 0(q1) . 2. INFLUENCE OF FLUCTUATIONS p OF A POLAR AMBIENT MEDIUM ε ε បω ϕ 〈 | 〉 ON THE MULTIPHOTON TRANSITION RATE Here we have 2' = 2 + a/2, 0(q) = q 0 is the initial (ground) state of the oscillator (here and subsequently We shall consider an impurity dipole molecule in an it is assumed that k T Ӷ បω so that the probability of external electromagnetic field interacting with a polar 0 direct thermal excitation of the oscillator is negligible) ambient medium. The proposed model is based on the | 〉 fact that the electromagnetic field interacts actively relative to the electronic state 1 and in addition Ψ ( , ) 〈 ( , ∞) ϕ 〉 with the impurity molecule and barely perturbs the p q t = q U t Ð p , polar ambient medium. If the ambient medium creating ϕ the fluctuations of the dipole moment is a plasma, this where p(q) are the oscillator wave functions of the implies that the electromagnetic field frequency Ω is far electronic state |2〉, U(t, Ð∞) is the evolution operator ω from resonance with the plasma frequencies p. Using determined by the vibrational Hamiltonian H2, and the the model for an impurity molecule described in the double angle brackets denote averaging over realiza- Introduction, i.e., intrinsic dipole moments exist in the tions of the random process F(t) [see formula (7)]. | | ≠ | | ground and excited electronic states, where d11 d22 , Using the representation of the Green function we shall use a GaussÐMarkov autocorrelation function ( , , , τ) to describe the fluctuations of the polar ambient K q2 t q12 t Ð medium: i 2 Ψ ( , )Ψ*( , τ) τ ≥ 〈 〈 ( ) ( )〉〉 ( γ ) = Ð-- ∑ p q2 t p q1 t Ð , 0, F t F t' = F0 exp Ð t Ð t' , (7) ប p 2 where F(t) is a random force, F0 is the intensity of the we write the transition probability as follows: fluctuations of the ambient medium which depends on ∞ 2 i ∝ γÐ1 ប τ  (ε' ε )τ temperature (F0 T) and is the characteristic cor- W21 = 2i ∫d expប-- 2 Ð 1  relation decay time. It is assumed that the random 0 forces act differently on the vibrational motion of the × ( , τ) 〈 〈 ( , , , τ)〉〉 nuclei in different electronic states. Thus, for simplicity I21 t K q2 t q1 t Ð , ≠ we can only assume F(t) 0 for vibrations in the and we write the Green function K(q , t, q , t Ð τ) in the excited electronic state. In addition, we can neglect any 2 1 direct change in the electronic state introduced by fluc- form of the functional integral: tuations of the ambient medium [20Ð22]. The action of ( , , , τ)  i ( , τ) the dipole ambient medium on the vibrational spectrum K q2 t q1 t Ð = ∫Dqexpប-- S t t Ð  , (8) of the molecule in the excited electronic state will be described by the interaction Hamiltonian F(t)q. Hence, where the vibrational Hamiltonians in electronic states 1 and 2 have the form t ( , τ) S t t Ð = ∫ dt1 ប2 ∂2 t Ð τ H , = Ð ------+ U , (q), 1 2 2m∂ 2 1 2 q ( ) 2 ω2 × mdq t1  m 2( ) ( ( )) ( ) ------Ð ------q t1 + V + F t1 q t1 2 2  dt  2 mω 2 1 U (q) = ------q , 1 2 is the classical action.

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The continuous integral (8) is calculated over the After differentiating twice, the integrodifferential trajectories q(t) satisfying the boundary conditions equation (9) is reduced to an ordinary fourth-order dif- τ q(t Ð ) = q1, q(t) = q2. ferential equation: We shall use the well-known property of a GaussÐ (4) (ω2 γ 2) (2) Markov process: q + Ð q 2 (12) t t  F 2 2 2 V 2 γ 0 γ ω γ  i  1 i  + 2i ------Ð  q + --- = 0 exp-- dt f (t )q(t ) = exp-- -- dt mប m ប ∫ 1 1 1  2ប ∫ 1 τ τ τ t Ð t Ð with the boundary conditions q(t Ð ) = q1 and q(t) = q2. t  The two additional integration constants are deter- τ × ( ) 〈 〈 ( ) ( )〉〉 ( ) mined from equation (9) at the points t1 = t Ð , t1 = t. In ∫ dt2q t1 F t2 F t1 q t2 ,  general, the solution of equation (12) is fairly cumber- t Ð τ some. Thus, at this point it is convenient to select the 〈〈 τ 〉〉 and then the averaged Green function K(q2, t, q1, t Ð ) limiting cases of fast and slow fluctuations of the ambi- can be written as ent medium. In the case of slow fluctuations b/Ꭽ ӷ 1, where b =  i  〈 〈K(q , t, q , t Ð τ)〉〉 = Dqexp -- S (t, t Ð τ) . 2 បω3 Ꭽ γ ω 2 1 ∫ ប eff  F0 /m , = / , equation (12) gives an expression for the multiphoton transition similar to formula (6) The effective action has the form where, however, the presence of the ambient medium is manifest in the appearance of an additional force in the t ( ) 2 ω2 heat release constant (5). This constant should be mdq t1  m 2( ) ( ) Seff = ∫ dt1 --- ------ Ð ------q t1 + Vq t1 ∫ replaced by 2 dt1 2 t Ð τ V + F 2 ប a˜ = ------. 2 t  បω  mω iF0 ( γ ) ( ) ( ) + ------∫ dt2 exp Ð t2 Ð t1 q t2 q t1 . 2ប In this case, the transition rate W should be averaged τ 12 t Ð over slow fluctuations of F with the distribution func- The extremal trajectory which minimizes the action tion satisfies the equation  2  2 1 F ( ) iF ------expÐ------ . 2 ( ) ω2 ( ) 0 2 q t1 + q t1 = ------πF  2F  mប 0 0 t (9) V Hence, the unknown transition rate is written as fol- × dt exp(Ðγ t Ð t )q(t ) + --- . lows: ∫ 2 2 1 2 m t Ð τ ∞  2  1  F  ( ) The effective action on the extremal trajectory has the W21 = ------∫dFexp Ð------W21 a˜ , (13) 2 πF  2F2 form 0 0 0 t (cl) m ( ) t V where W21(a˜ ) is determined by formula (6) in which S (t, t Ð τ) = --- q(t )dq t1 + --- dt q(t ). (10) eff 2 1 ------2 ∫ 1 1 the constant a should be replaced by a˜ . dt1 τ t Ð t Ð τ We shall confine ourselves to an analysis of pro- The continuous integral (8) may be written as cesses involving photon emission. We calculate the integral (13) by the method of steepest descents, as a ∂2 (cl) ( , , , τ) 1 Seff result of which we obtain with exponential accuracy K q2 t tq1 t Ð = Ð----π---ប---- ∂------∂------2 i q2 q1 2 (11) ∼ ab 2n0 2n0   (a) W21 exp------Ð ln------+ 1  W21 , (14) ×  i (cl)( , τ) 2 a a expប-- Seff t t Ð  . where n0 is the photonicity of the process which makes Note that in general, expression (11) is exponen- the main contribution to the probability of a transition tially exact but since the extremal trajectory q(t) is a lin- accompanied by phonon emission. It follows from this ear profile along q and q , the pre-exponential factor in expression that the fluctuations of the ambient medium in 2 1 Ӷ (11) does not depend on q2 and q1 so that in this case the case 2n0/a 1 and bn0/a > 1 appreciably increase the expression (11) is also exact. probability of a multiphoton transition. For these rela-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 28 KOVARSKIŒ, PREPELITSA tionships between the parameters expression (14) may tion rates [24]. In this case, the probability of a mul- be simplified and reduced to the form tiphoton transition decreases as the fluctuation intensity ε ε បΩ បω Ӷ ប Γ 2 increases. In fact, for 2' Ð 1 Ð n0 + ( + D) bn  (a) W ∼ exp ------0 W . and Γ Ӷ D expression (15) may be written in the form 21  a  21 2 eE d  2 2 1 2 W ≈ 2π ------0------21-- n J (ρ)----. Since b ~ F0 ~ T, it follows from this last expression 21  បρ  0 n0 D that for transitions accompanied by phonon emission, the transition probability has an “anomalous” tempera- From this it follows that as the fluctuation intensity ture dependence W21 ~ exp(T/A), where A is a constant. increases, the multiphoton transition probability Ӷ (Here it is assumed that because of the condition k0T decreases. Moreover, since D ~ b and b ~ T, the optical បω the probability of transitions accompanied by transition probability decreases with increasing tem- phonon emission in the absence of an ambient medium perature. does not depend on T.) The characteristic value of the parameter A for some biomolecules reaches 43 K [22]. 3. CONCLUSIONS We shall now consider the opposite case of fast fluctuations of the ambient medium. If the decay time for This analysis has shown that the fluctuations of a the correlations of a polar ambient medium is fairly polar ambient medium lead to changes in the probabil- short compared with the period of the molecular vibra- ity of multiphoton resonances. This influence is deter- tions, the fluctuations of the ambient medium have little mined to a considerable extent by the relationship influence on the quantum transition processes. In the between the correlation decay time and the frequency asymptotic limit Ꭽ ∞ the influence of the ambient of the molecular vibration active in the optical transi- medium on the multiquantum transition process is tion. The conclusion that the multiphoton transition “switched off”, i.e., the transition probability ceases to probability does not have an Arrhenius temperature depend on the fluctuations of the medium. The case dependence for media with slow correlation decay is Ꭽ ≥ 1 for b/Ꭽ Ӷ 1 is more interesting. In this case, the fundamentally new. This effect is entirely attributed to extremal trajectory, being a solution of equation (12) a change in the profile of the potential barrier caused by with corresponding boundary conditions, has the form fluctuations of the ambient medium. In the other limiting case of fast fluctuations the theory predicts a ( ) 1 decrease in the multiphoton transition probability with q t1 = ------(---Ω------τ---)- sin f increasing fluctuation intensity. It is readily established that the characteristics attributable to the ambient × [ (Ω ( τ)) (Ω ( ))] q2 sin f t1 Ð t + Ð q1 sin f t1 Ð t medium are also found for the multiphoton scattering of coherent radiation by impurity dipole molecules. V sin(Ω (t Ð t)) Ð sinΩ (t Ð t + τ) + ------1 + ------f------1------f------1------, Ω2 sin(Ω τ) m f f REFERENCES Ω ω( ) ⁄ Ꭽ f = 1 Ð iD , D = b . 1. V. A. Kovarskiœ, N. F. Perel’man, and I. Sh. Averbukh, Multiquantum Processes (Énergoatomizdat, Moscow, Using this solution, we find after various transfor- 1985). mations 2. M. Ferray, A. Huillier, X. Li, et al., J. Phys. B 21, L31 2 (1988). πeE0d21 2 2(ρ) W21 = 2 ------ρ------ ∑n Jn 3. J. H. Eberly, Q. Su, and J. Javanainen, Phys. Rev. Lett. n 62, 881, 1989 (1989). Γ + D 4. T. P. Platonenko, Laser Phys. 6, 1173 (1996). × ------. (ε ε បΩ បω)2 ប2(Γ )2 5. R. A. Marcus, J. Phys. Chem. 90, 3460 (1986). 2' Ð 1 Ð n + + + D 6. A. Warshel and S.T. Russell, Quarterly Rev. Biophys. 17, Here Γ is the width of the electronic state in the absence 283 (1984). of a polar ambient medium. 7. H. Sumi and R. A. Marcus, J. Chem. Phys. 84, 4894 ε ε បω បΩ In the particular case 2' Ð 1 + ~ n0 where the (1986). contribution of the resonant term (containing n0) to the 8. A. I. Burshtein and A. G. Kofman, Chem. Phys. 40, 289 transition probability is decisive, the dependence of the (1979). multiphoton transition rate on the fluctuation intensity (in terms of the parameter D) is similar to that obtained 9. L. D. Zusman, Tekh. Éksp. Khim. 15, 227 (1979). in the case of dynamic narrowing of the optical transi- 10. L. D. Zusman, Chem. Phys. 49, 295 (1980).

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11. I. V. Aleksandrov, Tekh. Éksp. Khim. 16, 435 (1980). 20. I. Sh. Averbukh, V. A. Kovarskiœ, A. A. Mosyak, et al., 12. A. B. Gel’man, Tekh. Éksp. Khim. 19, 281 (1983). Teor. Mat. Fiz. 81, 271 (1989). 13. A. B. Helman, Chem. Phys. 65, 271 (1982). 21. I. Sh. Averbukh, V. A. Kovarskiœ, A. A. Mosyak, et al., Kinetic Effects in Molecules and Solid-State Systems in 14. E. M. Kosower and D. Huppert, Chem. Phys. Lett. 96, External Fields, Ed. by V. A. Kovarskiœ (Shtinitsa, Ki- 433 (1983). shinev, 1990). 15. J. N. Onuchic and D. N. Beratan, J. Chem. Phys. 16, 22. A. V. Belousov and V. A. Kovarskiœ, Zh. Éksp. Teor. Fiz. 3707 (1986). 114, 1944 (1998). 16. A. Gard, J. N. Onuchic, and V. J. Ambegankar, Chem. Phys. 83, 4491 (1985). 23. V. A. Kovarskiœ, Zh. Éksp. Teor. Fiz. 57, 1217 (1969). 17. I. Rips and J J. Jortner, Chem. Phys. 87, 4491 (1987). 24. S. A. Akhmanov, Yu. E. D’yakov, and A. S. Chirkin, Introduction to Statistical Radiophsics and Optics 18. G. W. Lohson and D. W. Oxtby, Chem. Phys. 87, 781 (Nauka, Moscow, 1989). (1987). 19. A. Warshel and J.-R.Hwang, J. Chem. Phys. 84, 4938 (1986). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 30Ð44. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 35Ð50. Original Russian Text Copyright © 2000 by Pozdneev. ATOMS, SPECTRA, RADIATION

Collisions of Electrons with Molecules in Excited VibrationalÐRotational States S. A. Pozdneev Lebedev Physics Institute, Russian Academy of Sciences, Moscow, 117924 Russia; e-mail: [email protected] Received December 4, 1998

Abstract—Results are presented of calculations of cross sections for scattering of electrons by diatomic molecules in speciﬁc excited vibrationalÐrotational states. The calculations were made using an approximation based on a quantum theory of scattering in a system of several bodies which can be applied to calculations of direct reactions and reactions involving the formation of an intermediate transition complex. Results of calculations of cross sections for collisions of electrons with hydrogen, nitrogen, lithium, sodium, and hydrogen halide molecules are compared with existing experimental data and the results of calculations made by other authors. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION hydrogen, nitrogen, sodium, and lithium molecules in specific vibrational–rotational states: Various electronÐmolecule collision processes e + AB(v = n, J = N) AÐ + B. , e+ AB()v1 J1 Ð Various series of experiments[2, 6, 11Ð14] have shown that the cross sections of these processes depend on elastic scattering processes the degree of excitation of the vibrationalÐrotational state , e + AB()v2 J2 Ð of the target molecule. A similar situation is observed for hydrogen [10], nitrogen [11], and other molecules [1, 2]. vibrationalÐrotational excitation In addition, the dissociative attachment of an elec- , Ð Ð e +AB()v1 J1 A ()BeA +,AB+()eB Ð tron to a lithium molecule has the following characteristic features [12]: dissociative attachment of an (1) a fairly high rate of formation of negative lithium electron to a molecule ions by dissociative attachment to molecules in highly excited vibrational states; e ++AB Ð (1.1) (2) a negative lithium ion will most probably play molecular dissociation the same role in the future as a negative hydrogen ion because of the similarity between the electronic proper- play an important role in negative ion production pro- ties of these molecules; cesses, fast generation of fluorine and chlorine atoms from halogen-containing molecules in excimer lasers, (3) since the lithium molecule has the same valence gas-discharge plasmas, and so on [1Ð4]. as a hydrogen molecule, methods used to interpret the experimental data on collisions between electrons and In addition, dissociative attachment is the simplest hydrogen molecules can be used to study collisions chemical reaction induced by electrons and has thus between electrons and lithium molecules. been studied on a considerable number of occasions The well-known methods of investigating dissociative [1Ð10]. However, studies dealing with collision pro- attachment [1Ð8] (the boomerang method, the R-matrix cesses between electrons and molecules in predefined method, the time evolution of the wave function, the Fes- excited vibrationalÐrotational states have only recently hbach operator method, and so on) are based on the begun to appear [9Ð14]. This is because of the major treatment of this process as a multistage process: difficulties involved in the experimental preparation of (1) the first stage of the process involves the capture these excited states and the overall realization of the of an electron by the molecule with the consequent for- experiment. mation of a negative molecular ion; Hence the present paper is devoted to a theoretical (2) the second stage involves the decay (evolution) study of the dissociative attachment of electrons to of this state into various states of the decay products: a

1063-7761/00/9001-0030 $20.00 © 2000 MAIK “Nauka/Interperiodica” COLLISIONS OF ELECTRONS WITH MOLECULES 31 negative ion and a neutral or excited atom (dissociative In the present study, an approach based on the quan- attachment); two neutral or excited atoms and electrons tum theory of scattering in a system of several bodies is (molecular dissociation); an excited molecule and an used to calculate cross sections for collision processes electron (electron impact molecular excitation). between electrons and diatomic hydrogen, nitrogen, The basis of this formalism, i.e., the formation of an lithium, sodium, and hydrogen halide molecules in pre- intermediate state of a molecular negative ion, is not deﬁned vibrationalÐrotational states. always justiﬁed from the physical point of view. For In this approach the main approximation is that the instance, in the case of dissociative attachment of an interaction of an impinging electron with electrons and electron to a hydrogen molecule the lifetime of this nuclei of the target molecule is replaced by interaction complex is comparable with the time taken for free drift of an impinging electron with each atom as a whole, of the electron over a distance equal to the diameter of considering the atom to be a force center. Hence, the the hydrogen molecule. A similar situation arises in the complex multiparticle problem needed to calculate case of a molecular reaction [15] without the formation cross sections for scattering of an electron by diatomic of an intermediate complex, molecules is reduced to a collision problem in a three- body system which can be solved using the quantum O(3P) + CS(X1Σ+) CO(X1Σ+) + S(3P), scattering problem in a three-body system. when an appreciable fraction of the translational energy This approximation is reasonable at impinging elec- (in accordance with the momentum limit Ev/Et ~ 0.88 tron energies lower than the electronic excitation [14, 15]) is converted into vibrational energy of the CO energy of the molecule [9]. The initial data in this for- molecule. mulation of the problem are the interaction pair poten- Finally there are many examples where a long-lived tials, and the masses and energies of the colliding par- intermediate complex forms in the reaction process (for ticles. For the pair potentials of the interaction of elec- further details see [1Ð8, 10Ð14]), although the pro- trons with molecular atoms we used the zero-range cesses described above also exist, and these show that a potentials [8] and potentials having the form preliminary analysis of the experimental data on the λ ( β ) ⁄ process is required for a theoretical analysis of various V(r) = exp Ð r r, (1.2) collisions. The absence of such an analysis fairly fre- whose parameters were determined using the electron quently leads to erroneous interpretations of the exper- binding energy in the negative ion, the scattering lengths, imental data, as in the case of the dissociative attach- and the effective radius where the spin (in the case of ment of an electron to a hydrogen molecule [4]. In this homonuclear molecules) is taken into account as fol- case, the dissociative attachment cross section calcu- lows. For the scattering length we used the quantity [19] lated using the intermediate state model is ten times higher than the experimental data [16]. 1 1 1 1 3 1  -- = ---- = ---- = ----- + ---- , A similar situation is now encountered in quantum a a1 a2 4 at as chemistry where “the potential energy surface plays a central role in the numerical modeling of chemical reactions” where at and as are the triplet and singlet scattering [17]. However, there are many examples, one of which is lengths, respectively. given above, where an intermediate complex does not The pair potentials of the interaction between atoms appear during the reaction process and consequently , the in molecules were simulated by the Morse potentials existence of a potential energy surface presents problems. V(r) = D(1 Ð exp[Ðα(r Ð r )]), (1.3) Thus, in atomic and chemical physics there is a class 0 of processes, described as direct by analogy with whose parameters were determined using spectro- nuclear physics, whose basic characteristic is that no scopic data [20]. intermediate long-lived complex forms during the scattering process. Consequently, in order to interpret these direct pro- 2. FUNDAMENTAL APPROXIMATIONS cesses we need to develop suitable theoretical tools and AND METHODS OF CALCULATION methods, including the well-known methods to The quantum theory of scattering in a system of sev- describe reactions involving the formation of an inter- eral particles (in our case reaction (1.1) and, in the mediate complex. approximation studied, three particles) is formulated These methods were developed by L.D. Faddeev for the three parts which make up the complete wave and O.A. Yakubovskiœ (see [18]) who presented a quan- function of the three-body system tum theory of scattering in systems of several particles, 3 free from model assumptions on the formation of an Ψ = ∑ Ψ , (2.1) intermediate complex in the collision process. This i method can be used to describe both direct processes i = 1 and those involving the formation of intermediate long- each part corresponding to all possible ways of dividing lived states. the three-particle system into noninteracting subgroups.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 32 POZDNEEV These equations in momentum space for the scatter- variables of integration in (2.2). For example, in the Ψ ing of particle 1 at the bound pair (2,3) have the follow- integral corresponding to the expression G0T1 2 it is ing form [18]: most convenient to take k2 and p2 as the variables of Ψ = Φ δ Ð G (Z)T (Ψ + Ψ ), integration. In this case, the variables k1 and p1 on i i i1 0 i j k (2.2) which the kernel of the operator T depends, should be , , , , , , , , 1 i j k = 1 2 3; 3 1 2; 2 1 3, expressed in terms of the variables k2 and p2. In some Φ cases, in this situation it is convenient to take p and p where 1 describes the initial state of the three-body 1 2 system: the free motion of particle 1 and the bound state as the variables of integration. of the pair (2, 3): G (Z) = (H Ð Z)Ð1, Z = E + i0, H is 0 0 0 The pair T-matrices t (k , k' ; Z) appearing in the ker- the free motion operator of the three particles, E is the i i i total energy of the three-body system which is equal to nels of the equations (2.2) have singularities in terms of the sum of the kinetic energy of the impinging particle 1 the variable Z: poles corresponding to a discrete spec- and the binding energy of the pair (2, 3), and Ti is the trum of pair subsystems and a branch cut across the pair T-matrix which is uniquely determined by the interac- positive part of the real axis generated by the spectrum tion pair potentials Vi using the LimpanÐSchwinger equa- of the two-body problem. The explicit form of these tions [18] singularities gives the spectral representation of the T-matrix [18]: T i = V i + V iGiT i. (2.3) ψ ψ In order to describe the motion of three particles in a i(k) i(k') t(k, k'; Z ) = V ( k Ð k ' ) + ∑ ------center-of-inertia system we use the usual Jacobi coor- κ2 + Z dinates which are deﬁned as follows: i i 2 2 dx , ± , ± m3q2 Ð m2q3 + ∫t(k x; x i 0 )t(k' x; x i 0 )------2------, k1 = ------, Z Ð x ⁄ 2m m2 + m3 (2.4) (m + m )q ψ κ2 2 ϕ κ2 ϕ 1 2 3 where i(k) = ( i + k ) i(k), and i and i(k) are the p1 = m3(q1 + q2) Ð ------, m1 + m2 + m3 energy and wave function of the bound state of the two particles in the momentum representation. where mi and qi are the mass and momentum of each particle. The coordinates k2, p2, k3, and p3 are deﬁned The poles of the T-matrix corresponding to the dis- similarly. crete spectrum generate singularities in the components Ψ In these variables of the wave function i, and by separating these out we obtain the following representation: Φ , ϕ δ 0 1(k1 p1) = (k1) (p1 Ð p1), Ψ , 0 where ϕ is the wave function of the initial state of the i(ki pi; p i ) κ2 0 system (2, 3) with the binding energy 1 , p1 is the 0 , momentum of the impinging particle 1, ϕ δ 0 Bi(ki pi; pi ; Z) = (ki) (pI Ð pi ) Ð ------, p2 ⁄ 2n + k 2 ⁄ 2m Ð Z (2.5) 02 κ2 i i i jk p1 1 Z = ------+ ------+ i0, 0 B (k , p ; p ; Z ) 2n1 2m23 i i i i

m1(m2 + m3) m2m3 3 ϕ 0 n1 = ------, m23 = ------. 0 j(p j)R ji(k j; p i ; Z ) m + m + m m + m = Ð ∑ Q (k , p ; p ; Z ) Ð ------, 1 2 3 2 3 j i i i 2 ⁄ κ j = 1 p j 2n j Ð j Ð Z The operators G0 and Ti are integrals with kernels having the form: Qj, Rji are smooth functions of their variables. This sep- 2 aration of the singularities occurs naturally when the , , ,  , pi  δ Ti(ki ki' pi pi'; Z ) = tiki k'i; Z Ð ------ (pi Ð p'i), integral equations (2.2) are solved numerically. To 2ni obtain a unique deﬁnition of the functions Qj, Rji we can proceed as follows: we substitute Ψ in the form δ δ i (ki Ð k'i) (pi Ð p'i) (2.5) into the equations (2.2) and equate coefﬁcients at , , , G0(ki ki' pi pi'; Z ) = ------. k2 ⁄ 2m + p2 ⁄ 2n Ð Z the same singularities. Thus, we obtain equations for i jk i i these functions in terms of which all the main charac- It should be borne in mind that any system of vari- teristics of the three-body problem can be expressed ables which is the most convenient may be taken as the explicitly, i.e., the wave function, the elements of the

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S-matrix, and the amplitudes and cross sections of the The solution of the equations (2.3) with the potential various processes taking place in the three-body system: (2.10) has the form 1 + (2, 3) n ˜t(k, k', Z) = ∑C (Z)V(k, s )V(k', s ), (2.11) 1 + (2, 3)—elastic scattering processes ij i j i, j ( , )∗ 1 + 2 3 —excitation processes, (2.6) where ( , )∗ ( , )∗ 3 + 1 2 , 2 + 1 3 —rearrangement , Cij(Z) = V(si s j) processes with excitation, ∞ ∞ , , 2 1 + 2 + 3—ionization processes. π V(k si)V(k' s j)k dkdk' + 8 mij∫∫------. k2 ⁄ 2m Ð Z The cross section of the elastic scattering process 0 0 ij has the form It should be noted that this method of constructing the separable T-matrix (2.11) can be used for any local dσ ------11-- = (2π)4n R 2, (2.7) potential. In addition, this method appreciably simpli- dθ 1 11 fies the numerical solution of the system of integral equations (2.2) and in some cases an analytic solution the cross section of the rearrangement process is can be obtained. 2 The integral equations (2.2) only possess good dσ n p R ------1--i = (2π)4 ----i-----f------1--i---, (2.8) properties from the mathematical point of view (such as dθ 0 p1 Fredholm properties, unique solubility, and so on) under certain conditions for two-particle data [18]: and the cross section of the breakup process (1) the pair potentials, which are generally nonlocal, 2 Vi(k, k') are smooth functions of k, k' and satisfy the dσ → n p B ------1------3 = (2π)4 ----i-----f------0--i---, (2.9) condition θ 0 d dp p ⑀ 1 , ≤ ( )1 Ð ⑀ > V i(k k') 1 Ð k Ð k' , 0; 02 κ 2 κ2 where pf = 2ni( pi /2ni Ð 1 Ð i ). (2) the point Z = 0 is not a singular point for the equations (2.3), i.e., all three scattering lengths in the It should be noted that the potentials do not partici- pair channels are finite; pate in explicit form in equations (2.2), but contain the more general characteristic of the T-matrix and they are (3) the positive two-particle spectrum is continuous. related to the potentials by equations (2.3). Hence, This condition is important for nonlocal potentials although this method formally uses the potentials, essen- since positive eigenvalues can only appear in this case, tially the T-matrices are modeled, these being constructed and it is satisfied for almost all real physical potentials. using the Beteman method [9, 15, 16, 18Ð24]. The Coulomb potentials and the hard shell poten- According to this method [24], the partial harmonic tials do not satisfy the first condition, and the Coulomb of the local potential in momentum space, potentials lead to singularities in T-matrices of the type |k Ð k'|Ð2 while the hard shell potentials lead to a slow π decay of the T-matrix at high momenta. When the sec- V (k, k') = -- r2 j (kr)V(r) j (k'r)dr l 2∫ l l ond condition is violated, the equations (2.1) cease to be Fredholm equations for Z = 0, which leads to the Efi- is approximated by the expression mov effect [19] whereby at some critical value of the coupling constant, when the scattering length first goes n , , to infinity, an infinite discrete spectrum may appear in ˜ , V l(k si)V l(k' s j) the three-particle system under certain conditions. V l(k k') = ∑------,------, (2.10) V l(si s j) i, j The Faddeev integral equations (2.2) are exact equations to describe the dynamics of three pairwise inter- and the parameters si, sj are determined from the acting structureless particles, for which it is natural to expression for the minimum of the functional assume pair interaction between the particles in the three-particle system since all the characteristics of the χ , , , …, (s1 s2 s3 sn) processes in this system will primarily be determined , , , , , , …, by the pair interaction. This is completely confirmed by ∫∫ V l(k k') Ð V l(k k' s1 s2 s3 sn) dkdk' the experimental results in direct reactions in nuclear = ------. , and atomic physics where the pair potentials determine ∫∫V l(k k')dkdk' the entire dynamics of the reaction [8, 9, 21].

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 34 POZDNEEV At this point, particular mention should be made of to describe the system [9, 18]. Hence, methods of con- the fact that the classical trajectory method cannot gen- structing analytic solutions of the system of integral erally be applied to calculate the collision between an equations (2.2) are of particular importance. This can atom and a diatomic molecule based on the surface be achieved when systems of several bodies contain potential energy, i.e., the potential energy surfaces can additional integrals of motion associated with specific only appear when the chemical reaction involves an features of the pair interaction or constraints are intermediate complex. The method of the quantum scat- imposed on the system [9, 10]. tering problem can be applied to any chemical reactions The importance of constructing analytic solutions of for which the above conditions are satisfied. the integral equations (2.2) describing various dynamic In molecular reactions, the particles taking part in processes in systems of three or more particles arises the reaction are intricate complexes (atoms with an because in the case of three particles, for example, new electron shell) and thus their internal structure may qualitative characteristics appear, completely unchar- play a significant role. As a result, the interaction poten- acteristic of the two-particle model: the Efimov effect tial in this system, which is generally nonlocal, will [9, 18, 20, 24] described above, the use of T-matrices include many-particle potentials as well as pair poten- instead of the potentials in the equations, which include tials and in [9Ð17, 22] it was shown how these many- several new interaction characteristics not observed in particles potentials appear and in what form, in a three- the two-body problem [9, 18, 24]. particle system consisting of three like atoms interact- The large ratio of the proton mass to the electron ing using the potentials mass is a favorable circumstance which can be used to 1 1 1χ 3 3 3χ obtain analytic solutions of the system (2.2) and conse- V i(ri) = V i (ri) i + V i (ri) i, quently cross sections of the processes (2.6)Ð(2.10) where which can be expressed in terms of elementary and special functions [9Ð11, 15, 16]. Hence, we shall consider 1χ 3χ i = 1/4 + s jsk, i = 3/4 Ð s jsk, various particular cases when the systems of integral equations (2.2) can be solved analytically [9, 15, 16]. si is the spin of the ith atom, and the entire interaction ≡ We consider the case of a single light particle m1 m has the form ≡ and two like heavy particles m2 = m3 M. A similar situation is encountered in problems of proton scattering V = ∑V i. (2.12) at hydrogen atoms or collision between an electron and i a diatomic molecule consisting of like atoms where, in this last case, the condition m/M 0 is the basis of Mention should be made of [22] in which similar rea- the adiabatic approximation. soning was used to construct the potential energy surface of diatomic LiF and LiBr molecules. We set [21] Ψ Ψ By directly solving the equations (2.2) after separat- 2 = G0F2, 3 = G0F3. (2.13) ing the angular variables and expanding in terms of partial waves in momentum space, we also obtain a many- Then using (2.2) we obtain the following system of particle nonlocal potential [9, 11, 15, 16] for which the equations for F2 and F3 first terms of its Fourier transformation in the BornÐ ( ) Oppenheimer approximation are the same as (2.12). 1 Ð T 2G0T 1G0 F2 However, unlike (2.12), this potential has the asymp- ( ) Φ + T 2G0 Ð T 2G0T 1G0 F3 = T 2 , ρ2 ρ ∞ ρ (2.14) totic form ~C/ for where is the distance ( ) between the impinging particle and the center of mass 1 Ð T 3G0T 1G0 F3 of the pair. This property of the interaction potential + (T G Ð T G T G )F = T Φ, leads to threshold characteristics of the amplitudes and 3 0 3 0 1 0 2 3 scattering cross sections which take the form of oscilla- and the following expression for the complete wave tions at the threshold for breakup into three particles [9, function: 16, 23]. Note that these characteristics do not depend Ψ Φ Φ on the specific type of pair forces and are determined by = Ð G0T 1G0(F2 + F3) Ð G0T 3( + G0F2) (2.15) the presence of resonances in these and can thus exist Ð G T (Φ + G F ) Ð G (T + T )G T (F + F ). in any system of three particles. 0 2 0 3 0 3 2 0 1 2 3 The equations of motion similar to (2.2) for three or The meaning of the various terms of the expansion more particles interacting using arbitrary pair poten- becomes clear if we assume that the residues of expres- tials are generally nonintegrable and thus neither the sions of the type G0TiFi at the poles Ti are (apart from a bound states nor the scattering states in these systems constant) the amplitudes of one of the possible particle can be expressed in the form of quadratures. This is excitation or redistribution processes as a result of col- because the number of existing integrals of motion is lisions, and the residue at the pole G0 gives the breakup smaller than the number of dynamic variables required amplitude.

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Thus, the second term in (2.15) gives the amplitude particle collisions. Neglecting the operator G0T1G0 is Ψ Φ of elastic scattering or excitation, and the third and equivalent to replacing 1 by in the equations (2.1). fourth terms give the amplitudes of the rearrangement From this it follows that the system of equations (2.17) reactions in the three-body system. The last term makes describes the scattering of particle 1 as if the momen- contributions to all the amplitudes and thereby allows tum distribution of particles 2 and 3 conserved its initial ψ for the mutual influence of these processes. value 0 during the scattering process. This approxi- It can be shown [9, 21] that the action of the operator mate description of the scattering process forms the G0TiG0 in (2.15) on an arbitrary fairly smooth function basis of the well-known momentum approximation, κ F vanishes in the limit m/M 0 provided that n is which is based on the following assumptions. fixed. Thus, the equations (2.14) in the limit m/M 0 have the following form: (1) The effective radius of interaction of an impinging particle with the particles of a complex system is Φ F2 + T 2G0F3 = T 2 , much smaller than the average distance between the (2.16) Φ particles in the system, i.e., the interaction of the imping- F3 + T 3G0F2 = T 3 . ing particle and a system particle is considerably stron- This approximation consists in determining F and ger than the interaction between the particles in a bound 2 system and consequently, the interaction potential in this F3 from the system (2.16) where m/M 0 for finite, κ system can be considered to be a pair potential. small values of m/M. Here it should be noted that if n is fixed and M ∞, all the energy levels of the cou- (2) The pair interaction time is much shorter than pled system (2, 3) tend to zero. However for finite M the the characteristic time for the system of scatterers. energy of the coupled system (2, 3) is nonzero. Conse- quently, (2.16) will only be a reasonable approximation Thus, it has been shown [9, 21] that in the three- when the existence of coupling in the system (2, 3) body problem for the scattering of a light particle by a plays no significant role in the elastic scattering of par- system of two heavy particles, the adiabatic and momen- ticle 1. However, this can only be achieved when the tum approximations are related as the Fourier transform energy of the impinging particle is appreciably higher and the inverse transform. than the binding energy of the pair. The operator solu- A similar result is obtained in the quasiclassical tion of the system (2.16) can be expressed in the form description of a particle collision with a bound pair in Φ T 2(1 Ð G0T 3) which the scattering amplitude is also expressed in F2 = ------, terms of the pair amplitudes [3, 9, 21]. It can be seen 1 Ð T2G0T 3G0 (2.17) from these examples that the equations of quantum T (1 Ð G T )Φ scattering theory in several-particle systems may serve F = ----3------0------2------3 1 Ð T G T G as a universal mathematical basis for deriving these 3 0 2 0 approximations, and the iterations of these equations and agrees with the expressions obtained for multiple form a representation for the scattering amplitudes in the scattering by two intersecting systems of centers [9, 15, form of a series (2.17) which contains the product of the 16, 21], i.e., in the adiabatic approximation; all the pair amplitudes, and the number of cofactors in the terms characteristics of the amplitudes and the cross sections determines the order of the multiple scattering. Multiple of the processes in the three-body system are deter- scattering approximations can be obtained by using the mined in terms of multiple scattering. corresponding approximations for the T-matrices (for Taking into account (2.15) and (2.17) for the com- example, the separable approximation) in the iterations of plete wave function of the three-body system we have the equations (2.2). In addition, pair single and double the following representation: multiple scatterings of colliding particles are responsible for the appearance of nonintegrable δ-functions and pole Ψ ∼ ( )Φ 1 Ð G0T 3 Ð G0T 2 singularities in the kernels of the scattering matrix for Ð G T G T Φ Ð G T G T Φ Ð … breakup processes which correspond to the free terms 0 3 0 2 0 2 0 3 and the first iterations of the equations (2.2). In this approximate expression, the second and third terms correspond to the single scattering approxima- Similar expressions can also be obtained for scatter- tion, while the fourth and fifth terms correspond to the ing in systems of charged particles [18, 25]. In this double scattering approximation. Thus, in this approx- case, the elements of the scattering matrix are determined by the resolvent imation the terms G0T3G0F2 and G0T2G0F3 in (2.15) allow for multiple scattering at the potentials V2 and V3, i.e., all the characteristics of the amplitudes and the ˜ cross sections are determined by multiple scattering. R00 = Fc + ∑Fα + ∑ Fα, β + F0, Hence, in this approximation the scattering in a α α ≠ β three-body system (electron and atoms of a molecule) ⁄ 5 + 2iη may be considered to be the result of successive two- Fc = Ac p Ð p' ,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 36 POZDNEEV whose singularities are determined by the residual Cou- of the Hamiltonian hα; A = (α, i), i = 1, 2, 3, …, N; lomb interaction, ψα c are the wave functions describing scattering in χ 0 nα(1 Ð α) α | | α V α = ------, the Coulomb field V c = nαα/ yα ; and U0 = xα ψ ψα , α(xα, kα) c (yα pα) is the wave function of the con- single scattering, tinuous spectrum. , It has therefore been shown that in general the adia- f α(kα kα' ) batic approximation in momentum space is equivalent Fα = Aα------3--Ð---i--a--, p Ð p' α to the momentum approximation and may be expressed in terms of multiple scattering. This approximation in double multiple scattering, configuration space corresponds to dividing the system into fast and slow subsystems in particular regions of , , f (kα kα' β) f β(kβ' kα' β) the configuration space, where this division is not uni- Fαβ = Aαβ------, 1 Ð iaαβ versal and has a dynamic nature. Thus, in various stud- (kα' β Ð kβ' Ð i0) ies [18, 23, 25] attempts were made to use methods of where fα are the two-particle scattering amplitudes for differential geometry and algebraic topology to make the operators hα and the form of the functions Aα, Aαβ such a division and in this approach the nontrivial cova- and the parameters aα, aαβ is given in [5]. riant derivative and the differential topological invari- ants can be expressed in terms of the spectral geometry, At this point it should be noted that the singularities i.e., in terms of the spectral properties of the fast Hamil- of the S-matrix for the breakup processes most typical tonian. of a three-body system are concentrated in the same directions of configuration space as for neutral parti- In momentum space the system of equations (2.16) cles. However, unlike the case of breakup processes in has the form [21] systems of neutral particles, characterized by singular- , , , t3(p k Ð k'; Z ) F 2 (k Ð k ' k ') δ δ F3(k p) = 2m1∫------dk' ities of the type (p Ð p') (pα Ð pα' ), in the case of charged ( )2 2 k Ð k' Ð p0 Ð i0 Ð5 Ð 2iη particles the generalized functions |pα Ð pα' | and m Ð 3 Ð iaα 2 2  3  Ð1 , ϕ |pα Ð pα' | appear and the pole (kα' β Ð kβ' Ð i0) is = t3(p p0; Z ) 0  k Ð ------p0 , m2 + m3 2 2 )Ð 1 Ð iaαβ (2.18) replaced by the distorted pole (kα' β Ð kβ' Ð i0 . t (p, k Ð k'; Z ) F ( k Ð k ', k ') F (k, p) = 2m --2------3------dk' The physical reason for these singularities is that the 2 1∫ ( )2 2 region where pair multiple scattering of free particles k Ð k' Ð p0 Ð i0 can take place is unbounded [18, 25]. In elastic scatter- m ing, rearrangement, and breakup processes, all the sin- , ϕ  2  = t2(p p0; Z ) 0  k + ------p0 gularities are determined by two-particle collisions in a m2 + m3 system of two Coulomb centers. and can be solved analytically when the T-matrices can For the case of elastic scattering we can explicitly be represented in the separable form (2.10), (2.11). obtain the leading terms of the amplitudes by assuming α β In the approximation of zero-range potentials, Rαβ Ӎ R or R and replacing the Green functions of the which are a particular case of separable potentials, the intermediate states by their eikonal approximation [25]. T-matrices do not depend on k and have the following In this case, we have form [8, 9, 18]: ˜ Ð1 f AA = f c + f AA, [( π)2 (α )] ti(Z) = 2 mij i + i 2mijZ , ˜ ∼ 1 ˆ ( ˆ ˆ ˆ ˆ α f AA U AV 1 R2V 2 + R3V 3 where the values of i are the reciprocal scattering ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) 1 lengths aÐ1 = α . Thus the functions F and F in the + R2V 2 R3V 3 + R3V 3 R2V 2 U A . i i 2 3 approximation of zero-range potentials satisfying the For breakup processes we have the following system of equations (2.18) will depend on one rather expression: than two vector variables. The limiting expressions of ti ˜f ∼ U1Vˆ (Rˆ Vˆ + Rˆ Vˆ for m/M 0 which should be substituted into (2.17) 0A 0 1 2 2 3 3 have the form 1 + Rˆ Vˆ Rˆ Vˆ + Rˆ Vˆ Rˆ Vˆ )U , Ð1 2 2 3 3 3 3 2 2 A [( π)2 (α )] t3 = 2 m1 3 + i p0 , α , ψ ψα , where U A(X pα) = A(xα) c (yα pα) is the wave func- Ð1 [( π)2 (α )] tion of the discrete spectrum, ψα are the eigenfunctions t2 = 2 m1 2 + i p0 .

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–22 2 For F2 and F3 we then obtain the following system σ, 10 cm of equations: 5

F2(k') F3(k) = 2t3m1∫------2------2------dk' – ( ) H /H2 k Ð k' Ð p0 Ð i0

ϕ  m3  = t3 0k Ð ------p0 , m2 + m3 0 0.5 F3(k') F2(k) = 2t2m1∫------dk' (k Ð k')2 Ð p 2 Ð i0 0 D–/HD ϕ  m2  = t2 0k + ------p0 . m2 + m3 0 It should be noted that outside the limits of the adiabatic approximation it is impossible to use zero-range potentials [8, 9, 18, 21] because in a system of three 0.005 particles 1, 2, 3 of which two, say 2 and 3, interact with – 1 using zero-range potentials, there is an attraction D /D2 between particles 2 and 3 proportional to ~C/r2, where the coefﬁcient C depends on the mass ratio. This interaction leads to unique features in the behavior of a real physical system of three particles and, speciﬁcally, to coalescing to a central point. The nonuniqueness of the 0 solution of the integral equation describing scattering 3.5 4.5 5.5 in this system is related to this factor since it is found E, eV that the corresponding homogeneous equation has a nontrivial solution [9, 21]. Fig. 1. Dissociative attachment of electrons to hydrogen molecules and their isotopically substituted analogs: cir- However, the homogeneous equations obtained by cles—experimental data [28], curves—results of the present discarding the right-hand sides in the system (2.18) calculations. have no nonzero solutions because t2 and t3 are complex numbers. Hence, in the adiabatic approximation the features of the cross section on the impinging electron energy is described above do not appear. This formally demon- nonmonotonic. strates that zero-range potentials can be used in this approximation. The system of integral equations (2.18) in the Calculations of this effect for the dissociative approximation of zero-range potentials can be solved attachment reaction of a hydrogen molecules are plot- exactly by using a Fourier transformation and its solu- ted in Fig. 1 where it can be seen that the dependence tion has the following form:

exp(ip ⋅ r + iκp ⋅ r) (α + i p ) exp(Ðγ p ⋅ r) Ð ------0------0------ϕ(r) 2 0 0 r F (r) = ------, 3 ( π)2 [(α ) α ( ⋅ ) ⁄ 2] 2 m1 2 + i p0 ( 3 + i p0) Ð exp 2ip0 r r

exp(ip ⋅ r + iγ p ⋅ r) (α + i p ) exp(Ðκp ⋅ r) Ð ------0------0------ϕ(r) 3 0 0 r F (r) = ------, 2 ( π)2 [(α ) α ( ⋅ ) ⁄ 2] 2 m1 2 + i p0 ( 3 + i p0) Ð exp 2ip0 r r

m m γ = ------3------, κ = ------2------. m2 + m3 m2 + m3

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 38 POZDNEEV Similarly, we can construct analytic solutions of the which in coordinate space correspond to the local equations by using the approximation (2.10) with the potential form factors λexp(Ðβ r) V (r) = ------i-----. i r 1 V (k, β ) = ------, In this case the solution of the system has the fol- i i 2 β2 k + i lowing form [3, 9, 10]:

, Λ , β Fi(r r) = i(r i)Ci(r), Ð1 λ V (x)x2dx Λ = ------i- 1 + 4πλ ------i------, i m i∫ 2 jk 2m jk Zi Ð x V (p )exp(iκp ⋅ r) Ð 2m Λ V (p )Φ(r)exp(Ðiγ p ⋅ r)ϕ(r) C (r) = -----2------0------0------1------2------3------0------0------, 2 Λ Λ [ Φ ]2 1 Ð 2 3 2m1 (r) V (p )exp(iγ p ⋅ r) Ð 2m Λ V (p )Φ(r)exp(Ðiκp ⋅ r)ϕ(r) C (r) = -----3------0------0------1------3------2------0------0------, 3 Λ Λ [ Φ ]2 1 Ð 2 3 2m1 (r) 2π exp(i p r) Φ  0 (r) = ------2- r ( 2 β2) p0 + ( γ ) 2 2 r exp Ði p r  Ð 1 Ð ( p + β )------0----- , 0 β 2 2 ( 2 β2)  p0 + where ρ is the distance between the impinging particle formulas (2.7)Ð(2.9) or from expression (2.18) for the and the center of gravity of the bound particle pair. complete wave function. In this last case, the amplitude The amplitudes of the various processes (2.6) taking is determined by the following scheme [9, 21] Near the place in a three-particle system are determined using pole corresponding to the bound state of the system we have

σ, cm2  k2   k'2  –18 → – ------Ð ε ------Ð ε  10 e + H2(v, J) H + H  n n 2 2m jk 2m jk  t (k, k'; Z Ð p ) = ------i 2 ε Z Ð p Ð n ϕ ϕ + n(k) n(k') + smooth part of function, ϕ where n is the wave function of the bound state of the ε particle pair (j, k) and n is the binding energy of this 10–20 pair. Consequently, the expression G0TiG0Fi is factorized (2, 0) as the product of three factors: (1, 8) p2 Ð1 G T G F = ϕ (k) ----- + ε Ð Z M. (0, 11) 0 i 0 i n 2n n (0, 0) The middle factor describes the free motion of particle 10–22 2.5 3.5 4.5 E, eV i relative to the bound system (j, k) where in coordinate space this corresponds to a diverging wave. From this it Fig. 2. Dissociative attachment of electrons to hydrogen is clear that the factor M, expressed in terms of the inte- molecules initially in excited vibrationalÐrotational states: ϕ dashed curves—results of calculations from [10], solid gral of the product of the functions n and Fi, is the curves—results of the present calculations. scattering amplitude.

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Thus, the amplitudes and cross sections of the elas- agrΓ(Ð2x) x = i 2εm ⁄ a, δ(x) = ------= Ðδ(Ðx), tic scattering and excitation processes are given by the 23 Γ(1/2 Ð x Ð l/2) expressions ( ⁄ πε)1/4 Aε = m23 8 , M = (2π)2m ϕ*(r)[exp(iγ p ⋅ r)F (r, ρ) if 1∫ i 3 ε is the kinetic energy of the expanding nuclei and in ( κ ⋅ ) , ρ ]ϕ order to calculate the transitions to the continuous spec- + exp Ði p0 r F2(r ) f (r)drr, trum from high vibrational states of the molecule we dσ can use their following asymptotic form [9, 27]: ------i-f = M 2, dΩ if ψ ∼ L 1 ϕ ϕ ε(r) 2Aε sin 2m23(r Ð r0) Ð ln--- -- . where i and f are the vibrationalÐrotational wave a r functions which have the following form in the adiabatic representation [8, 9, 21] For like particles m2 = m3, the amplitude of elastic scattering and excitation is given by ϕ (r) = ψ (r)Y (θ, ξ), ϕ (r) = ψ (r)Y (θ, ξ), i v JM f v J'M' cos( pr ⁄ 2)cos( p r ⁄ 2) ϕ* 0 Y (θ, Ω) are spherical functions [26], ψ are the vibra- Mif = 2∫ f (r) -α------(------)---⁄---- JM v + i p0 + exp i p0r r tional wave functions of the Morse potential, (2.19) ( ⁄ ) ( ⁄ ) sin pr 2 sin p0r 2 xv + ------ϕ (r)dr.  y y α ( ) ⁄ i ψ (r) = A F (Ðv, 1 + 2x , y)exp Ð------, + i p0 + exp i p0r r v v 1 v  2 r The total cross sections of all the processes including dissociation and all possible reactions taking place in a [ ( )] L Ð 1 y = Lexp Ða r Ð r0 , xv = ------Ð v , three-particle system may be obtained using the optical 2 theorem [9, 21, 24]. As a result, we have a(2x + 1)…(2x + v ) L = 2 m D ⁄ a, A = ------v------v------, 23 v Γ 4π 2 1 (2xv)v! σ ϕ tot = ------∫ 0(r) ------2  αr + cos( pr)  2 p0 ------F1(a, b, x) is a degenerate hypergeometric function [26]. 1 +  ( ) p0r + sin p0r The amplitude of the rearrangement process has the (2.20) form 1 m + ------dr, ( π)2 ϕ*  23 ,  αr + cos( pr)  2 MR = 2 m23∫ 12, 13 expi ------ F2, 3(r r)drdr, 1 + ------m1  ( ) p0r Ð sin p0r where ϕ is the wave function of the bound system 12, 13 which is the same as the expressions obtained in [3Ð8]. formed in the reaction process. The expression for the The expressions (2.19) and (2.20) reproduce the cross section of the rearrangement process is as follows: main characteristics in the cross sections of the processes, especially processes involving collisions σ between electrons and diatomic molecules. For exam- d R m12 2 -----θ---- = ------MR . ple, for collisions between electrons and homonuclear d m23 (m2 = m3) molecules, in the denominators of the func- For processes of molecular dissociation by elec- tions fi for fixed values of r and varying momentum p0 trons, the final state is the state of the continuous spec- it is possible to have cases when trum and thus functions having the following form [9, 27] [ ] α ± ( ) ⁄ Re J±(p0) = Ð i p0 exp i p0r r = 0. should be used as the wave function ψv These points correspond to resonant maxima of the ψ  y  functions Fi which lead to the appearance of maxima in ε(r) = Aε expÐ----- σ 2r the cross sections of all the processes, and also in tot. ⁄  In addition, the presence of the ratio m23 m1 in the × [ δ ] 1 L, ,  x exp i (x) F1-- + x Ð --- 1 + 2x y y argument of the exponential function for the reaction  2 2 amplitudes results in an appreciable isotope effect, first predicted by Demkov [5] and clearly observed experi-  mentally [28]. [ δ ] 1 L, ,  1 + exp i (x) F1-- Ð x Ð --- 1 Ð 2x y ---- , 2 2 x  It has been noted that all the characteristics of the y cross sections of the processes (2.6) are expressed in

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σ, 10–16 cm2 terms of multiple scattering whose action is character- 0.6 ized by the term exp(ip0r)/r in the functions Fi. 1Σ+ v → 1Σ+ v e + H2(X g, = 0, J = 0) e + H2(X g, = n, J = 0) However, it has also been noted that the traditional 0.4 interpretation of resonant phenomena accompanying the interaction of electrons or atoms with molecules is 0.2 n = 2 based on the assumption that transition complexes are formed during the collision process. The relationship 0.6 between the multiple scattering approximation and the theory of a transition state can be seen immediately if 0.4 we bear in mind that the imaginary zeros of the function J±(p0) determine the quasi-steady-state terms of the 0.2 n = 3 transition state [9, 21] and the complex zeros correspond to the continuations of the terms into the region of quasi-steady states. 0.6 Another method [9] of obtaining an analytic solution of the system of integral equations (2.2) involves 0.4 first going to the limit at m/M 0 directly in the equations (2.2) and then using the approximation of 0.2 n = 1 separable variables. In this case it is convenient to take p2 and p3 as independent variables. 0 2 4 6 E, eV Similarly we can obtain an analytic solution of the ≡ equations (2.2) for the case of one heavy particle, m3 M Fig. 3. Vibrational excitation of hydrogen molecules by and two identical light particles m = m ≡ M, for exam- electrons: circles—experimental data [28], solid curves— 1 2 results of the present calculations, dashed curves—calcula- ple, for the scattering of an electron at a hydrogen atom. tions using quasiclassical approximation [3]. In this case, the momenta of the light particles k1 and k2 can be conveniently assumed to be independent variables. σ –14 2 , 10 cm It should be noted that the smallness of the mass 1.0 ratio and specific models for the T-matrices or Green → e + D2(v = n) e + D + D functions were used to obtain analytic solutions of the systems of equations (2.2), (2.14). This is undoubtedly the main advantage of the formulation of these equa- n = 24 tions which allows us to explicitly isolate objects for 0.5 which approximate expressions need to be constructed. In addition, in analyses of various limiting cases n = 23 such as a single light particle and two identical heavy ones or one heavy particle and two identical light ones, n = 22 a preferred system of coordinates is obtained, which can be used to express the characteristics of all the 0 → dynamic processes taking place in these systems. 0.2 e + H2(v = n) e + H + H n = 15 n = 14 3. RESULTS AND DISCUSSION Figures 1Ð10 give the results of calculations of the cross sections for scattering of electrons by hydrogen 0.1 molecules and their isotopically substituted analogs (Figs. 1Ð4), hydrogen halides (Figs. 5, 6), nitrogen = 16 (Figs. 7, 8), lithium (Fig. 9), and sodium (Fig. 10), in both n the ground and excited vibrationalÐrotational states. The calculations were made using the approximation of zero- range potentials and in the more general case of separa- 0 0.2 0.4 0.6 E, a–1 ble variables. It can be seen that the cross sections of 0 these processes are accurately reproduced by formulas (2.7)Ð(2.9), which give a quantitative description of the Fig. 4. Dissociation of vibrationally excited hydrogen and deuterium molecules by electrons: dashed curves—results of processes (1.1), (2.6) and a qualitative picture of the calculations using the approximation of zero-range potentials effect whereby it is sufficient to allow only for the [27], solid curves—results of the present calculations. action of multiple scattering in this approximation.

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σ, cm2 dσ(E, 120°)/dΩ, 10–16 cm2/sr –14 10 e + HBr(v = 0) → e + HBr(v = 1) 3

2 e + HCl(v, J) → H + Cl– 1 10–15 0 e + HCl(v = 0) → e + HCl(v = 1) 1.5

10–16 1.0 0.5 0 (v = 2, J = 0) v → v 10–17 0.6 e + HF( = 0) e + HF( = 1) (v = 1, J = 0) 0.4 (v = 0, J = 15) 0.2 (v = 0, J = 0) 10–18 0.4 0.8 1.2 E, eV 0 1 2 3 E, eV Fig. 5. Dissociative attachment of electrons to HCl molecules initially in excited vibrationalÐrotational states: solid Fig. 6. Vibrational excitation of hydrogen halide molecules curves—results of calculations from [10], dashed curves— (HBr, HCl, HF) by electrons: circles—experimental data results of the present calculations. [28], curves—results of the present calculations.

In this approximation an isotope effect is observed, Note that various theories have now been put for- which was first predicted by Demkov [5] (Figs. 1, 2). ward to calculate the cross sections of processes We note that the results of calculations of the cross accompanying the interaction between electrons and sections of the processes (2.6) using the single scatter- molecules [1Ð8, 10Ð14] for which the terms of molec- ing approximation [9] differ substantially from the ular negative ions and various other empirical parame- experimental data since the single scattering approxi- ters are required as initial data. However, in many cases mation is inappropriate and multiple scattering must be it is impossible to make specific calculations according taken into account. to these theories, even for diatomic molecules, because for the vast majority of molecules the data are lacking. Figure 1 gives the results of calculating the cross sections for dissociative attachment of an electron to In addition, there is a large discrepancy between the hydrogen molecules and their isotopically substituted experimental results and the results of these calculations, modifications made by solving the system of equations as is demonstrated clearly by the calculations made (2.2) numerically using a method described in [9, 29]. using the FirsovÐSmirnov model [4] which uses approx- Ð The energy dependences of the cross sections of the imate values of the energy and the width of the H2 dissociative attachment processes are nonmonotonic. quasi-steady-state term obtained using the approxima- This indicates that the cross sections of the reactions tion of zero-range potentials. At energies below the dis- taking place in a three-particle system have threshold sociation threshold the cross sections were too high, characteristics which are observed most clearly in sys- whereas at energies above the threshold the agreement tems consisting of two like heavy particles and one between the experiment and our calculations was con- light particle. siderably better. A comparison between these calculations [9, 23] and the experimental data [28] shows that modeling the Thus, this particular case clearly reveals the advan- interaction of an electron with each of the atoms in a tages of an approach based on quantum scattering the- molecule using the potentials given above in the multi- ory in a system of several particles, in which the ple scattering approximation can give satisfactory required initial data are the pair T-matrices, which can agreement with the experiment (the orders of the cross be constructed using the pair interaction potentials and sections agree, including the isotope effects and the wave functions corresponding to the initial state of the threshold characteristics). scattering process.

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σ –16 2 σ –20 2 , 10 cm , 10 cm → – e + N2(v = n) N + N 2 → e + N2 e + N2(v = n) 2

1 n = 1 1 n = 1 0

2 0 1 n = 2 1 n = 2 0

3 0

2 1

n = 3 1 n = 3

0 0 1 2 3 4 9 10 11 12 E, eV E, eV

Fig. 7. Vibrational excitation of nitrogen molecules by elec- Fig. 8. Dissociative attachment of electrons to nitrogen trons: solid curves—experimental data [28], dot-dash molecules initially in excited vibrational states: solid curves—results of calculations using quasiclassical approxi- curves—results of calculations from [11], dashed mation [3], dashed curves—results of the present calculations. curves—results of the present calculations.

Figures 7 and 8 give the results of calculations of the [30]. No experimental data or theoretical estimates are cross sections for electron collisions with nitrogen mol- available for the effective radius. Thus, the absolute ecules based on the quantum scattering theory in a value of the cross section for dissociative attachment of three-body system, experimental data from [28], results an electron to a nitrogen molecule was used to determine of quasiclassical calculations [3], and calculations the effective radius and the other parameters required made using the strong coupling approximation [9, 11]. [9, 11]. A distinguishing feature of the dissociative attach- The interaction potential between the atoms of the ment of an electron to a nitrogen molecule is that as a molecule was simulated by the Morse potential, In result of the reaction a negative nitrogen ion forms in order to construct the separable T-matrix from this the p-state: potential we used the Beteman method described above e + N N(4S) + NÐ(3P). and the method described in [9, 29] to obtain a numer- 2 ical solution of the system of integral equations in this Consequently, in order to describe this reaction in approximation. the separable approximation we need to use the potentials (2.10) with the form factors [11] A comparison between the results of the calculations and the experimental data [11, 28] shows that g(k) = [1 + µkY (ρ)] ⁄ (k2 + β2). modeling the interaction of an electron with the atoms 10 forming a molecule using the nonlocal separable poten- Formulas linking the parameters λ, µ, β with the tials (2.11), (2.12) can give satisfactory agreement with binding energy, scattering length, and effective radius the experimental data on average, which is a natural are given in [8]. consequence of the properties of the nonlocal poten- Unfortunately, only the binding energies have been tials. It should also be noted that by selecting the determined experimentally so far [7, 28] and only theo- parameters of the separable potentials using the exper- retical estimates are available for the scattering length imental data [9, 11, 28] it is possible to calculate all

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σ 2 , a0 possible electronÐmolecule collision processes, which → – e + Li2(v, J) Li + Li conﬁrms the self-consistency of the proposed model. In cases of dissociative attachment of electrons to 100 vibrationalÐrotationally excited lithium, sodium, nitrogen, and hydrogen halide molecules (Figs. 5, 8Ð10) at speciﬁc impinging electron energies the calculations show that the vibrational excitation of the molecules 10–2 predominates over their rotational excitation.

ACKNOWLEDGMENTS 10–4 This work was supported by the Russian Foundation for Basic Research (project 98-002-17266) and by the 7 Academy of Sciences of Taiwan (project NCS-85- 2112-M-007-009). 10–6 8 9 1 2 3 REFERENCES 4 1. L. G. Christophorou, Electron Molecule Interaction and 10–8 their Application (Academic, New York, 1984). 5 2. A. Chutjian, A. Garscadden, and J. M. Wadehra, Phys. 6 Rep. 264, 393 (1995). 3. A. K. Kazanskiœ and I. I. Fabrikant, Usp. Fiz. Nauk 143, 0.4 0.5 0.6 0.7 E, eV 602 (1984). 4. E. Illenberger and B. M. Smirnov, Usp. Fiz. Nauk 168, Fig. 9. Dissociative attachment of electrons to lithium mol- 731 (1998). ecules initially in excited vibrationalÐrotational states: 1, 2, 5. Yu. N. Demkov, Phys. Lett. 15, 235 (1965). 3—results of calculations from [12], 4, 5, 6—results of the present calculations using zero-range potential approxima- 6. W. Domcke, Phys. Rep. 208, 98 (1991). tion, 7, 8, 9—results of the present calculations using separable potential approximation (1.2). 7. A. Herzenberg, Electron-Molecular Collision (Plenum, New York, 1984, 191). 8. Yu. N. Demkov and V. N. Ostrovskiœ, Zero-Range Poten- tials and Their Application in Atomic Physics (Plenum, σ 2 , a0 New York, 1988). 103 → – 9. S. Pozdneev, Nova Sci. Publ. 212, 99 (1996). e + Na2(v, J = 9) Na + Na 10. J. M. Wadehra and J. N. Bardsley, Phys. Rev. Lett. 41, 1791 (1978); Phys. Rev. Lett. 41, 1795 (1978); S. Pozd- 102 neev and G. F. Drukharev, J. Phys. B: Atom Mol. Phys. 13, 2611 (1980). 11. A. Huetz, F. Grestean, J. Mazeau, et al., J. Chem. Phys. 72, 5297 (1980); S. Pozdneev, J. Phys. B: Atom Mol. 101 8 Phys. 16, 867 (1983). 12. J. M. Wadehra, Phys. Rev. A 41, 3607 (1990). 6 13. M. Kulz, M. Keil, B. Kortyna, et al., Phys. Rev. A 53, 0 10 4 3324 (1996). 14. E. E. Nikitin, Theory of Elementary Atomic and Molecu- v = 2 lar Processes in Gases (Khimiya, Moscow, 1970; Claren- don, Oxford, 1974). 10–1 0 0.1 0.2 0.3 0.4 0.5 15. S. A. Pozdneev, Khim. Vys. Énerg. 18, 290 (1984). E, eV 16. S. A. Pozdneev, Zh. Éksp. Teor. Fiz. 77, 38 (1979).

Fig. 10. Dissociative attachment of electrons to sodium 17. N. F. Stepanov and V. I. Pupyshev, Molecular Quantum molecules initially in excited vibrationalÐrotational states: Mechanics and Quantum Chemistry (Mosk. Gos. Univ., solid curves—results of calculations from [13], dashed Moscow, 1991); N. F. Stepanov, Sorosov. Obraz. Zh., curves—results of the present calculations using separable No. 10, 33 (1996); A. I. Nemukhin, Sorosov. Obraz. Zh., potential approximation (1.2). No. 6, 48 (1998).

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18. S. P. Merkur’ev and L. D. Faddeev, Quantum Scattering 26. Handbook of Mathematical Functions, Ed. by Theory for Systems of Several Particles (Nauka, Mos- M. Abramowitz and I. A. Stegun (Nauka, Moscow, cow, 1985). 1979; Dover, New York, 1965). 19. V. Eﬁmov, Nucl. Phys. A 362, 1024 (1981). 27. R. I. Boœkova, Vestn. Leningr. Gos. Univ., Ser. Fiz. 20. K. P. Hube and G. Gerzberg, Constants of Diatomic Mol- Khim., No. 10, 31 (1977). ecules (Springer-Verlag, New York, 1979). 28. G. J. Schultz, Rev. Mod. Phys. 45, 423 (1973). 21. G. F. Drukarev, Zh. Éksp. Teor. Fiz. 67, 38 (1974). 22. D. A. Micha, Adv. Chem. Phys. 30, 7 (1975). 29. N. M. Larson and J. H. Hetherington, Phys. Rev. C 9, 699 (1974). 23. S. A. Pozdneev, Kratk. Soobshch. Fiz., No. 6, 61 (1987). 24. V. B. Belyaev, Lectures on the Theory of Few-Particle 30. N. B. Berezina and Yu. N. Demkov, Zh. Éksp. Teor. Fiz. Systems (Énergoatomizdat, Moscow, 1986). 68, 848 (1975). 25. A. M. Veselova, S. P. Merkur’ev, and L. D. Faddeev, Dif- fraction Interaction of Hadrons with Nuclei (Naukova Dumka, Kiev, 1987), p. 107. Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 45Ð49. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 51Ð56. Original English Text Copyright © 2000 by Kravtsov, Mikhaœlov. ATOMS, SPECTRA, RADIATION

Temperature Dependence of the Formation Rates of Hydrogen-Helium Mesic Molecules in Collisions of Slow Hydrogen Atoms with Helium¦ A. V. Kravtsov* and A. I. Mikhaœlov** St. Petersburg Nuclear Physics Institute Gatchina, Leningradskaya oblast, 188350 Russia *e-mail: [email protected] **e-mail: [email protected] Received February 8, 1999

Abstract—The rates of molecular muon transfer from the ground-state muonic hydrogen to helium isotopes are calculated in an improved adiabatic approximation. The results obtained by us at various temperatures are compared with the available experimental data. © 2000 MAIK “Nauka/Interperiodica”.

Muon transfer from the ground state of the muonic The rate of the formation of the muonic molecule1 hydrogen to nuclei with Z > 1 is one of the important prob- is determined by a dipole transition with the conversion lems of mesic-atom physics for more than 30 years. The of the atomic electron and is given by [6] transfer to helium is of special interest, since it is con- 2 nected with the problem of muon-catalyzed fusion in the 16π 3 5 Ð 1 2 2 dλ =------N a ξ q I ()q 〈〉d ν, (1) deuteriumÐtritium mixture. Helium nuclei are unavoid- 3 0 e e ably accumulated in the mixture due to the nuclear fusion ν 4 ប3 × 16 Ð1 reaction and tritium decay. Muon transfer in collisions where e = mee / = 4.134 10 s is an atomic fre- ប2 2 × Ð8 of muonic hydrogen atoms with helium stops the cycle quency unit, ae = /mee = 0.529 10 cm is the Bohr of the catalysis; therefore, the transfer rate is an impor- ξ radius of the hydrogen atom, = me/m (here, me is the tant characteristic of the muon catalyzed fusion. electron mass and m is the reduced mass of mesic The rate of the direct muon transfer to helium is Ð1 Ð1 Ð1 rather low (~106 sÐ1) [1, 2], because the crossing point hydrogen; m = mµ + M1 , where mµ is the muon of the lowest terms of the system (2pσ and 1sσ), which mass, and M1 is the mass of the nucleus of the hydrogen × 22 Ð3 corresponds to the initial and ﬁnal states of the reaction, isotope), and N0 = 4.25 10 cm is a liquid-hydrogen turns out to be deep under the barrier at energies ~1 eV. density. The quantity qÐ1I2(q) is calculated in atomic Another possible mechanism for muon transfer with units and determines the rate of the electron transition the formation of the intermediate molecular state via from the bound 1s-state of the helium atom to the con- the conversion of the atomic electron was proposed in tinuum. The quantity 〈d〉2 is calculated in mesic-atom [3]. The resulting molecular ion is in an excited state (in units (see below) and determines the rate of the dipole the muon motion) and undergoes deexcitation to the transition of the system of three bodies from the contin- lower term 1sσ in a time of ~10Ð12 s. As a result, the uum to the bound state on the term 2pσ. muon turns out to be bound by a helium nucleus, form- The momentum q and the energy ε of the conver- ++µ e ing a muonic helium atom in the ground state (He )1s. sion electron are given by Since the rate of formation of the muonic mole- ε ε εε cule (~108 sÐ1) is much lower than that of its decay q==2me e, e +JvÐIe, (2) 12 Ð1 (~10 s ), the muon-transfer rate nearly coincides ε ε with the formation rate. The latter was calculated in a where is the collision energy, Jv is the energy of the number of papers [3Ð6] in various approximations. mesic molecule in the rotationalÐvibrational state (J, v) In this paper, we pay special attention to the con- (here, J and v are the rotational and the vibrational struction of the effective potential in accordance with the quantum number, respectively), and Ie is the binding prescription given in [7] (“simple-approach approxima- energy of the 1s-electron in the helium atom. tion”). We calculated the muon-transfer rates in a low The integral I(g) is the matrix element of the vari- energy collision of hydrogen isotopes with helium iso- able 1/ρ2 (ρ is the distance between the electron and the topes and compared them with new experimental data. 1 For brevity, we shall call a molecular ion (HµHe)++, H ≡ p, d or t, ¦This article was submitted by the authors in English. mesic molecule.

46 KRAVTSOV, MIKHAŒLOV

Table 1 ∞ 1 χ ( )χ ( ) ε Ð1 2 I1 = --∫ f R i R RdR, e, eV q I(q) q I (q) k 0 4 0.542 0.491 0.445 ∞ (7) ⋅ 8 0.767 0.607 0.481 1 χ ( )χ ( ) φ2 ( , )R r I2 = --∫ f R i R dR∫ 2 pσ R r ------dr. 12 0.939 0.712 0.540 k R 0 16 1.084 0.805 0.598 M is the mass of the helium isotope, φ σ(R, r) is the 20 1.212 0.887 0.649 2 2p wave function of the muon in the 2pσ state in the ﬁeld 24 1.328 0.964 0.700 of the two Coulomb centers, R is the internuclear dis- 28 1.435 1.030 0.739 tance, and r is the muon coordinate calculated from the 32 1.534 1.089 0.774 middle of the internuclear distance. The radial wave χ χ 36 1.627 1.144 0.805 functions i(R) and f(R) describe the relative motion of the nuclei in the initial and ﬁnal states, respectively. In 40 1.715 1.196 0.834 the one-channel approximation they are obtained from 44 1.798 1.248 0.866 the Schrödinger equation 48 1.878 1.300 0.899 d2 J(J + 1) 52 1.955 1.352 0.934 (ε ) χ( ) ------2 + 2M Ð W Ð ------2------R = 0. (8) 56 2.029 1.403 0.971 dR R 60 2.100 1.455 1.008 Here J is the total orbital angular momentum of the system of three particles, M is the reduced mass of the nuclei, ε = k2/2M is the collision energy, and k is the asymptotic center of mass of the system) calculated with the elec- momentum of the relative nuclear motion in the effec- tron wave functions tive potential W [the same momentum enters formulas ∞ (7)]. All the quantities in Eq. (8) are used in the mesic atom units e = ប = m = 1, where m = M mµ/(M + mµ). In these ( ) ρ (ρ) (ρ) 1 1 I q = ∫d Rq1 R1s , (3) units, 0 λ λ ⁄ mµ = 1 + 1, 1 = mµ M1. (9) ρ where R1s( ) is the radial ground-state wave function of the electron in the helium atom and R (ρ) is the radial The effective potential W consists of the term E, the q1 adiabatic corrections, and the energy of the Coulomb wave function of the p-wave electron (l = 1) in the con- repulsion of the nuclei [8]: tinuum. ρ 2 1 + Ð The functions R( ) are normalized by the conditions W = E + --- + ------[H Ð H* + κ(H Ð 2H*)], (10) ∞ R 2M R2 (ρ)ρ2dρ = 1, M M M Ð M ∫ 1s M = ------2------1---, κ = ------2------1-. (11) 0 M2 + M1 M2 + M1 ∞ (4) ∞ + (ρ) (ρ)ρ2 ρ δ( ) For R , we have E2pσ Ð1/2, H2 pσ 1/4, ∫ Rq'1 Rq1 d = q' Ð q . * Ð 0 H2 pσ 1/4, H2 pσ 1/2, so that The HartreeÐFock wave functions were used as 1 ρ W σ Ð -- = E (Hµ). (12) R( ). The wave function of the emitted electron was 2 p 2 1s calculated in the frozen core model (in the model with 2 the core reconstruction, the results for I (q) are about The asymptotic value of W2pσ coincides with the energy 10% smaller [6]). The values of the integral I(q) for sev- of the Hµ-atom in the ground state (which corresponds ε eral energies e are given in Table 1. to the initial conditions of the collision). However, the The dipole matrix element calculated with the wave reduced mass M in Eq. (11) (and hence the asymptotic functions of the system of three bodies has the form [5] momentum k), which enters Eq. (8) differs from the true reduced mass of the system Hµ + He, which is 〈 〉 d = aI1 + bI2, (5) ( ) λ ˜ M2 M1 + mµ 1 + 1 M = ------= M------λ---, M2 Ð M1 1 2mµ M2 + M1 + mµ 1 + a = ------Ð --, b = 1 + ------, (13) Mt 2 Mt (6) mµ λ = ------. Mt = M1 + M2 + Mµ, M2 + M1

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˜ If one makes the replacement M M in Eq. (8), the χ (R) → ∞ = sin(kR Ð Jπ ⁄ 2 + δ ), k = 2M˜ ε, (19) ε i R J calculated energy levels of the mesic molecule Jv [9] turn out to be in better agreement with accurate calcu- ˜ 1/2 χ (R) → ∞ = exp[Ð(2M ε v ) R]. lations2 [10Ð12] than those obtained earlier [3] with the f R J mass M. Even better agreement can be obtained if one The energy of the rotational-vibrational state (J, v) of replaces κ in Eq. (10) by κ˜ , which gives the correct the molecular ion (HµHe)++ is obtained together with χ asymptotic value of the effective potential in the 1sσ the wave function f(R) when solving the Schrödinger channel. We are solving a one-level problem; however, equation. For slow collisions, it is enough to consider the 1sσ channel is the second open channel in the prob- Ji = 0 for initial and Jf = 1 for final states (the molecule lem of slow collisions of mesic hydrogen with helium, is formed via the dipole transition). There exist only σ and its influence can be taken into account indirectly3 three bound states in the 2p channel, which have v = 0 and J = 0, 1, 2. The level energies ε and ε obtained by the replacement κ κ˜ . In order to obtain κ˜ , let us 00 10 σ in this paper are given in Table 2 together with the write the asymptotics of the potentials in the 1s chan- results of other papers. ∞ + nel: for R , we have E1sσ Ð2, H1sσ 1, As can seen from the table, the level energies calcu- Ð lated in this paper are very close to the high-accuracy H* σ 1, H σ Ð2, so that 1s 1s calculations with large number of basis functions [10Ð 12]. For this reason, when calculating the rates of the 1 [ κ˜ ( )] W1sσ Ð2 + ------1 Ð 1 + Ð 2 Ð 2 formation of mesic molecules, we use Eq. (17), where 2M˜ (14) M˜ and κ˜ are defined by Eqs. (13) and (16).  κ˜  = Ð 21 + ---- . When calculating the wave function of the initial state ˜ χ M i(R), the electron screening was taken into account by Actual asymptotic value of the potential in the 1sσ an additional term in the potential of Eq. (17). The influ- channel should coincide with the ground-state energy ence of the electron shell of the helium atom on the of the Heµ-atom: final (bound) state of the molecule is negligible because of the short length of the corresponding wave function ( µ) E1sσ He = Ð2m', (see Refs. [4Ð6] for more details). Table 3 shows the rates of the molecule formation mµM 1 + λ mµ (15) 2 1 λ λ (108 sÐ1) averaged over the Maxwellian energy distri- m' = ------µ-- = ------λ----, 2 = ------. M2 + 1 + 2 M2 bution. The electron screening is taken into account. Recently new experimental data of λ have been Let us choose such κ˜ that W σ E σ(Heµ). Then, 1s 1s obtained. In the experiment, the total muon transfer rate (1 + λ )2 from the ground-state muonic hydrogen to helium, κ˜ = κ------1------. (16) (1 + λ)(1 + λ ) λ λ λdir 2 pHe = pµHe + pHe, (20) χ So, the wave functions i, f (R) are solutions to the λdir equation is measured, where pHe is the rate of the direct muon transfer without the molecule formation. This rate was cal- 2 ( ) d Ji, f Ji, f + 1 culated in [1] for the systems pµ + 3, 4He and dµ + 3, 4He. ------+ 2M˜ (ε Ð W˜ σ) Ð ------2 2 p 2 ε ≤ dR R (17) For collision energies 0.1 eV, the rate of the direct muon transfer does not depend on energy and × χ ( ) i, f R = 0, amounts to where λdir ≈ × 8 Ð1 λdir ≈ 6 Ð1 pHe 0.06 10 s and dHe 10 s . (21) 2 W˜ σ = E σ + --- 2 p 2 p R As was mentioned, the rate of molecular transfer (18) coincides with the rate of the molecule formation (see 1 [ + * κ( Ð * )] Table 3). So, when comparing experimental rates with + ------H2 pσ Ð H2 pσ + ˜ H2 pσ Ð 2H2 pσ 2M˜ theoretical ones (Table 4), the latter were enlarged according to Eqs. (20) and (21). with the boundary conditions As a matter of fact, the genuine muon transfer to the χ (0) = χ (0) = 0, helium nucleus occurs when the molecular ion decays i f into hydrogen and muonic helium. Such a decay may 2 These calculations make use of about 300 to 3000 basis functions. be radiative (or via the electron conversion), as well as via 3 A similar procedure for solving the two-channel problem was predissociation. The latter channel was not considered in proposed earlier in [7] (“simple approach” of “improved two- the first papers on the molecular charge exchange. This led channel approximation”). to large discrepancies when comparing the calculations

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Table 2. The binding energies (eV) the hydrogen-helium muonic molecules

Energy Reference pµ3He pµ4He dµ3He dµ4He tµ3He tµ4He

[3]a 67.2 73.9 69.5 77.6 71.6 80.5 [9]b 69.0 75.4 70.6 78.7 72.3 81.3 This paperc 73.2 80.8 71.0 79.4 72.3 81.4 ε Ð 00 [10] 70.7 [13] 67.7 74.4 70.0 78.0 71.9 80.8 [14] 72.8 80.6 69.4 77.5 [3]a 34.9 41.6 46.5 55.9 52.3 62.9 [9]b 38.1 45.4 48.2 57.6 53.4 63.9 This paperc 41.5 50.0 48.5 58.3 53.4 64.0 [10] 50.0 47.9 57.8 ε Ð 10 [11] 48.4 58.2 [12] 48.4 58.2 [13] 33.8 41.2 46.8 56.1 52.7 63.1 [14] 38.8 47.4 46.3 55.7

Notes: a One-channel approximation with M and κ. d One-channel approximation with M˜ and κ. c One-channel approximation with M˜ and κ˜ .

Table 3. The rates of the molecule formation λ (108 sÐ1) averaged over the Maxwellian energy distribution. The electron screening is taken into account

T, K pµ3He pµ4He dµ3He dµ4He tµ3He tµ4He

15 0.52 0.33 2.40 12.7 51.2 1.89 20 0.52 0.33 2.34 11.8 45.5 1.86 25 0.51 0.32 2.28 11.0 41.1 1.84 30 0.51 0.32 2.24 10.4 37.6 1.82 35 0.51 0.32 2.20 9.8 34.7 1.79 40 0.50 0.31 2.16 9.3 32.3 1.77 50 0.50 0.31 2.09 8.5 28.6 1.74 100 0.47 0.29 1.85 6.2 18.8 1.60 150 0.46 0.28 1.70 5.1 14.4 1.51 200 0.45 0.27 1.60 4.4 11.9 1.43 250 0.44 0.27 1.52 3.8 10.2 1.37 300 0.43 0.26 1.45 3.5 9.0 1.31 350 0.42 0.25 1.39 3.2 8.1 1.27 400 0.42 0.25 1.34 2.9 7.3 1.22 450 0.41 0.25 1.30 2.7 6.7 1.19 500 0.40 0.24 1.26 2.6 6.2 1.15

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Table 4. Muon transfer rates (108 sÐ1) for various isotopes 6. V. K. Ivanov, A. V. Kravtsov, A. I. Mikhaœlov et al., Zh. and temperatures Éksp. Teor. Fiz. 91, 358 (1986) [Sov. Phys. JETP 64, 210 (1986)]. Experiment Theory 7. L. I. Ponomarev, L. N. Somov, and M. P. Faœfman, Yad. λ(p3He, 30 K) 0.46 ± 0.15 [15] 0.57 Fiz. 29, 133 (1979) [Sov. J. Nucl. Phys. 29, 67 (1979)]. λ 4 ± 8. S. I. Vinitsky and L.I. Ponomarev, Fiz. Elem. Chastits At. (p He, 30 K) 0.42 0.07 [15] 0.38 Yadra 13, 1336 (1982) [Sov. J. Part. Nucl. 13, 557 λ(p4He, 300 K) 0.36 ± 0.10 [16] 0.32 (1982)]. λ(p4He, 300 K) 0.44 ± 0.20 [17, 9] 0.32 9. A. V. Kravtsov, A. I. Mikhaœlov, and V. I. Savichev, λ(p4He, 300 K) 0.5 ± 0.1 [18, 19] 0.32 Hyperfine Interact. 82, 1205 (1993). 10. S. Hara and T. Ishihara, Phys. Rev. A 39, 5633 (1989). λ(d3He, 30 K) 1.86 ± 0.08 [20] 2.25 λ 3 ± 11. Y. Kino and M. Kamimura, Hyperfine Interact. 82, 195 (d He, 40 K) 2.25 0.15 [21] 2.17 (1993). λ(d3He, 300 K) 1.24 ± 0.05 [22] 1.46 12. V. I. Korobov, Hyperfine Interact. 101/102, 329 (1996). λ(d4He, 30 K) 10.50 ± 0.21 [20] 10.4 13. S. S. Gershteœn and V. V. Gusev, Hyperfine Interact. 82, λ(d4He, 300 K) 3.68 ± 0.18 [23] 3.48 185 (1993). λ(t3He, 15 K) 46 ± 4 [24] 51 14. V. B. Belyaev, O. I. Kartavtsev, V. I. Kochkin, et al., Phys. Rev. A 52, 1765 (1995). 15. S. Tresch, F. Mulhauser, C. Piller, et al., Phys. Rev. A 58, with the experiments, in which the probability of the 3528 (1998). transfer was obtained by the measurement of the X-ray 16. V. M. Bystritsky, V. P. Dzhelepov, V. I. Petrukhin, et al., yield. These discrepancies were removed when the pre- Zh. Éksp. Teor. Fiz. 84, 1257 (1983) [Sov. Phys. JETP dissociation channel was pointed out [13, 9] and taken 57, 728 (1983)]. into account. 17. H. P. von Arb, F. Dittus, H. Hofer, et al., Muon Catal. Fusion 4, 61 (1989). Comparing the experimental and theoretical values for muon transfer from ground-state muonic hydrogen 18. C. Piller et al., Helv. Phys. Acta 67, 779 (1994). to helium, one can see that they are in reasonable agree- 19. S. Tresch, P. Ackerbauer, W. H. Breunlich, et al., Hyper- ment. This means that the main features of the process fine Interact. 101/102, 221 (1996). are understood correctly. 20. B. Gartner, P. Ackerbauer, M. Augsburger, et al., in Pro- ceedings of Workshop on Exotic Atoms, Molecules and The authors are grateful to S. Tresch and B. Gartner Muon Catalyzed Fusion, Ascona, 1998 (submitted to for sending the experimental results prior to publica- Hyperfine Interact.). tion. 21. D. V. Balin, T. Case, K. M. Crowe, et al., in Proceedings of Workshop on Exotic Atoms, Molecules and Muon Cat- REFERENCES alyzed Fusion, Ascona, 1998 (submitted to Hyperfine Interact.). 1. A. V. Matveenko and L. I. Ponomarev, Zh. Éksp. Teor. 22. D. V. Balin, V. N. Baturin, Yu. A. Ghestnov et al., in Fiz. 63, 48 (1972) [Sov. Phys. JETP 36, 24 (1972)]. Muonic Atoms and Molecules, Ed. by L. A. Schaller and 2. S. S. Gershteœn, Zh. Éksp. Teor. Fiz. 43, 706 (1962) [Sov. C. Petitjean (Birkhäuser Verlag, Basel, 1993), p. 25. Phys. JETP 16, 501 (1963)]. 23. D. V. Balin, A. A. Vorobyov, An. A. Vorobyov, et al., 3. Yu. A. Aristov, A. V. Kravtsov, N. P. Popov, et al., Yad. Pis’ma Zh. Éksp. Teor. Fiz. 42, 236 (1985) [JETP Lett. Fiz. 33, 1066 (1981) [Sov. J. Nucl. Phys. 33, 564 (1981)]. 42, 293 (1985)]. 4. A. V. Kravtsov, A. I. Mikhaœlov, and N. P. Popov, J. Phys. B: 24. T. Matsuzaki, K. Nagamine, K. Ishida, et al., in Proceed- At. Mol. Phys. 19, 1323 (1986). ings Workshop on Exotic Atoms, Molecules and Muon 5. A. V. Kravtsov, A. I. Mikhaœlov, and N. P. Popov, J. Phys. B: Catalyzed Fusion, Ascona, 1998 (submitted to Hyper- At. Mol. Phys. 19, 2579 (1986). fine Interact).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 50Ð58. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 57Ð66. Original Russian Text Copyright © 2000 by Panteleev, Rerikh, Starostin. ATOMS, SPECTRA, RADIATION

Influence of Interference Effects of Spontaneous Emission on the Behavior and Spectra of Resonance Fluorescence of a Three-Level Atom in a Strong Wave Field A. A. Panteleev*, Vl. K. Rerikh**, and A. N. Starostin Russian State Scientiﬁc Center, Troitsk Institute of Innovative and Thermonuclear Research (TRINITI), Troitsk, Moscow oblast, 142092 Russia *e-mail: [email protected] **e-mail: vroerieh@ﬂy.triniti.troitsk.ru Received June 7, 1999

Abstract—An investigation is made of the influence of quantum interference processes accompanying radiative relaxation of excited states on the population dynamics, total intensity, and spectra of the resonance fluorescence of three-level V-type atoms. Analytic expressions are obtained for the total intensity and spectra of the resonance fluorescence taking into account the off-diagonal nature of the radiative relaxation operator. It is shown that quantum interference process can substantially alter the total spontaneous emission intensity of the atoms and the population dynamics of the atomic levels, as well as the resonance fluorescence spectra. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The simplest system in which interference effects are observed in one-photon processes contains three Interference effects accompanying the spontaneous levels since a minimum of two correlated paths is emission of atoms have recently formed the subject of required. Three-level systems satisfying this condition numerous studies in connection with the development Λ of lasers with no population inversion, the generation of may have three configurations: stepped, -type, and squeezed states, studies of laser-induced transparency, V-type. Interference effects only influence the sponta- suppression of absorption and emission, and other neous relaxation rate for transitions having the same effects. Interference effects accompanying the sponta- final states. Hence, interference terms influencing the neous emission of atoms have been known for some rate of radiative relaxation can only occur for the V-con- time: modification of the polarization structure of spon- figuration. In the present paper we study the influence taneous emission in a magnetic field (Hanle effect), of interference effects accompanying the spontaneous level crossing, quantum resonances, transient quantum emission of a three-level V-type atom in a laser field on beats, and so on [1]. These effects may be classified as its population dynamics and also on its absorption and linear. In addition, nonlinear effects may be observed as emission spectra. Note that the resonance fluorescence a result of nonlinear parametric processes (the Stark properties of a doublet-type three-level atom were effect, the dynamic Stark effect, two- and multiphoton investigated experimentally and theoretically in [14] scattering, saturation effects, multiple harmonic gener- where it was shown that these spectra can contain up to ation, bleaching under multiphoton resonance, and so seven peaks. This is because of the high-frequency on [2Ð10]). Interference effects do not usually influence Stark effect: the presence of a strong field binding the the rate of radiative relaxation of the atomic states. N levels leads to the formation of quasienergy states From the mathematical point of view their contribution and the transitions between these are responsible for becomes zero either when integrated over angles (for the multiplet structure of the spectra. In this case, the maximum number of peaks is determined by the rela- polarization effects) or when integrated over the spec- 3 trum (for nonlinear interference effects). tionship [15] (N + 1)/(N + 1). Note that we have already investigated the characteristics of quantum The influence of interference effects on the rate of interference accompanying the spontaneous emission radiative decay was taken into account in studies of the of a three-level atom in problems involving the emis- spontaneous emission spectra [7, 11], in analyses of the sion of an atom in a cavity and also in analyses of the dynamics of molecular quasienergy states [12], and in characteristics of scattered radiation [16]. We investi- studies of the spontaneous Raman scattering spectra of gated the influence of interference effects on scattered two adjacent states [13]. Note that in these studies the radiation transport processes in [17]. off-diagonal terms in the radiative relaxation operator corresponding to interference effects were introduced The present paper is constructed as follows. In the formally. second section we give the equations of motion for a

INFLUENCE OF INTERFERENCE EFFECTS OF SPONTANEOUS EMISSION 51 three-level V-type atom in a monochromatic electro- where α = +1 or –1 depending on the specific properties magnetic field and we analyze the influence of interfer- of the atomic system. The factor p determines the cor- ence effects on the radiative relaxation operator. In the relation between the interference contributions of dif- third section we investigate the dynamics of the atomic ferently polarized photons. In [7Ð12] p was assumed to states and the total intensity of the resonance fluores- be unity, whereas in descriptions of the Raman scatter- cence. Finally, in the fourth section we study the influ- ing spectra in [13] p varied between 0 and 1 depending ence of interference effects on the emission spectra and on the direction of observation of the scattered quanta. discuss the results. It should be stressed that this approach is incorrect: the rate of the relaxation processes of the atomic system does not depend on the direction of observation of the 2. EQUATIONS OF MOTION scattered radiation. In the present paper we investigate FOR A THREE-LEVEL ATOM a model three-level system neglecting any degeneracy IN AN ELECTROMAGNETIC FIELD of the magnetic moment. Thus, p will be taken as a model parameter. Expressions for the elements of the The behavior of the atomic system will be described Γ using the equation for the density matrix radiative relaxation operator js for real systems are given in the Appendix. It should be stressed that for iបρú = [Hˆ 0 + Vˆ , ρ] + iRˆ , (1) most real systems the parameter is p = 0 because of the phase quenching of the contributions from photons of where Hˆ 0 is the Hamiltonian of the free atom: different polarizations [see (A.2)]. However, in some cases, for example when the density of states of the electromagnetic field varies, the ˆ ប ∆ ˆ H0 = ∑ jS jj, (2) quenching is destroyed and the value of p will be non- j = 2, 3 zero. In this case, we need to bear in mind the change in the diagonal elements Γ . Since the radiative relax- ˆ jj and V is the Hamiltonian of the interaction between the ation constants will also be taken to be model parame- atom and a monochromatic electromagnetic wave hav- ters in the following investigation, we shall assume that ω ing the field strength EL and the frequency L (in the the interference change in the diagonal elements is rotating wave approximation): already taken into account. The proposed model may be used for a qualitative description of the spontaneous ˆ ប ( ˆ Ð ˆ + *) luminescence on the D-line of alkali metal vapor in the V = ∑ V jS j + S j V j . (3) field of a linearly polarized laser wave near the metal j = 2, 3 surface. A detailed analysis of the resonance fluores- µ ប µ cence of the sodium D-line in the field of a high-inten- Here Vj = Ð j1EL/2 , j1 is the matrix element of the Ð + sity electromagnetic wave in a formulation of the prob- dipole moment, Sˆ jm = |m〉〈j|, Sˆ j = |1〉〈j|, Sˆ j = |j〉〈1|, |j〉 lem allowing for the complete angular momentum of are the projection operators of the jth state [18] (j = 1 cor- the states requires a separate analysis. responds to the ground state, and j = 2, 3 corresponds to From expressions (1)Ð(4) we obtain equations for the excited states), ∆ = ω Ð ω (ω is the frequency cor- ρ j j1 L j1 the components of the atomic density matrix jm = responding to the |j〉 |1〉 transition). We express the 〈j|ρ|m〉: ˆ radiative relaxation operator R in the form [18] ρ (ω ω )ρ i ú jm = j1 Ð m1 jm ប Rˆ = -- [(Γ + Γ+ )SÐ ρS+ ∑ jm mj m j i (Γ ρ ρ Γ ) ρ ρ 2 (4) Ð -- ∑ js sm + js sm + V j1 1m Ð j1V 1m, m, j = 2, 3 2 s = 2, 3 Ð Γ ρS+ SÐ Ð S+SÐ ρΓ+ ], jm m j j m mj ρ (ω ω )ρ i ú j1 = j1 Ð L j1 (6) Γ γ ∆L γ ∆L where for j = m we have jj = j + i j , and j and j i determine the radiative relaxation rate and the Lamb Ð -- ∑ Γ ρ + V ρ Ð ∑ ρ V , shift corresponding to the transition |j〉 |1〉. The first 2 js s1 j1 11 jr rj term in expression (4) describes the radiative influx to s = 2, 3 r = 2, 3 the ground state, and the next two terms describe radia- ρ = ρ* , j, m = 2, 3, tive damping and the level shift. A distinguishing feature 1 j j1 of the last terms is that they are off-diagonal. We postu- subject to the conditions ρ + ρ + ρ = 1. It follows δ ω ω Ӷ ω 11 22 33 late that the excited states are adjacent, = 2 Ð 3 2 from the equations (6) that the total intensity of the Γ ≠ whereas for jm where j m, we have scattered radiation is given by Γ [ γ γ α ∆L∆L 1/2] ∝ បω(γ ρ γ ρ Γ' ρ ) jm = p j m + i j m , (5) Irad 2 22 + 3 33 + 2 32Re 23 . (7)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 52 PANTELEEV et al.

|2〉 state. We checked the accuracy of equations (4) and (6) using the Keldysh diagram technique for the nonequi- 〉 librium Green functions [19] and also the atomÐphoton |3 density matrix [20Ð22] using an approach developed in [23]. Note that some aspects associated with describing the spectral relaxation characteristics of the excited ω ω γ γ ω 2 L 2 3 3 states were taken into account in [11, 12] using the technique of probability amplitudes.

3. INVESTIGATION OF THE DYNAMICS OF ATOMIC STATES |1〉 AND THE TOTAL RESONANCE FLUORESCENCE INTENSITY Fig. 1. Diagram of a V-type three-level atom. In this section we consider the influence of quantum interference effects on the total spontaneous emission intensity of the atoms and the populations of the upper Γ Γ Γ Here we assume that jm = 'jm + i ''jm . Note that levels. As we noted in the previous section, the off-diag- expression (7) derives from the off-diagonal structure onal nature of the radiative relaxation operator leads to Γ ρ of the relaxation operator (4). In the standard descrip- the appearance of an additional term 2 32' Re 23 in the tion [2, 18] the drift terms have a diagonal structure: standard expression for the total intensity and allow- ( ) ance for this term may be significant. R 1 = ប(γ + γ )ρ ⁄ 2 (8) jk j k jk Figures 2 and 3 give the total intensity of the reso- and correspond to the usual expression for the total nance fluorescence as a function of the laser field scattered radiation intensity: strength for various relationships between the radiative relaxation times and various intensities of the interfer- ∝ បω(γ ρ γ ρ ) Irad 2 22 + 3 33 (9) ence effects (various values of the parameter p). It can be seen from these graphs that interference effects may ω ≈ ω ≈ ω [here and in (7) 32 21]. We shall subsequently have a substantial influence on the emission properties normalize the total intensity of the spontaneous emis- of the atom, which for specific values of the parameters ∝ បωγ sion Irad to I0 2 which is proportional to the total may lead to a sharp drop (as far as zero) in the intensity luminous intensity of atoms excited only to the second (see Figs. 2b and 3b).

ρ ρ Irad/I0, 22, 33 0.8 0.9 (a) (b)

0.6 0.6

0.4

0.3 0.2

0 150 300 0 150 300 γ V/ 2

Fig. 2. Dependences of the upper level populations and the total intensity of the resonance ﬂuorescence on the laser ﬁeld strength γ γ ∆ γ ∆ γ បωγ ∆L for the parameters 2 = 3, 2/ 2 = 10, 3/ 2 = Ð60, V = V21 = 2 V31, I0 = 2, and the Lamb shift j = 0, p = 0 (a), p = 1 (b): ρ ρ 22—dashed curve, 33—dot-dash curve, Irad—solid curve.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INFLUENCE OF INTERFERENCE EFFECTS OF SPONTANEOUS EMISSION 53

ρ ρ Irad/I0, 22, 33 0.6 0.9 (a) (b)

0.4 0.6

0.2 0.3

0 150 300 0 150 300 γ V/ 2

Fig. 3. Dependences of the upper level populations and the total intensity of the resonance ﬂuorescence on the laser ﬁeld strength γ γ for the parameters 3/ 2 = 0.5 (the values of the other parameters and the notation are the same as in Fig. 2).

We can identify several main types of behavior of cence on the Lamb component of the off-diagonal the total resonance fluorescence intensity as a function relaxation constant Γ'' . We established that in quan- of the pump wave intensity. jm tum interference effects the Lamb shift has little influ- γ γ µ µ 2 When the relationship 2/ 3 > ( 21/ 31) is satisfied, ence on the populations and the total intensity. How- the dependence of the fluorescence intensity on the ever, as will be shown later, the Lamb shift can substan- wave intensity does not change drastically as the con- tially influence the form of the spectra in regions where tribution of the interference effects increases, because the intensity of the atomic spontaneous emission is low. of the slight changes in the populations of the upper levels and the value of Reρ . As a result, we observe a 23 I /I , ρ , ρ linear dependence of the intensity on the parameter p rad 0 22 33 0.8 (see expressions (7) and (5)) whose behavior for a given laser field strength is determined by the magni- ρ tude and sign of Re 23. γ γ We may observe similar behavior for the case 2/ 3 < 0.6 µ µ 2 ӷ ∆ Ӷ ∆ ( 21/ 31) at high (Vj1 j1) or low (Vj1 j1) pump ∆ wave intensities. However, for Vj1 ~ j1 the influence of quantum interference effects may lead to substantial 0.4 changes in the upper level populations and the coher- ence between them (see Fig. 4) which gives rise to a sharp drop in the luminescence of the atoms in this range of parameters. Formally for p = 1 the total inten- 0.2 sity of the resonance fluorescence may even have zero values (see Fig. 2b). The reduction in the total intensity of the resonance fluorescence is observed particularly γ γ µ µ 2 clearly under the condition 2/ 3 = ( 21/ 31) and p = 1 (see Fig. 3b). This case should be an isolated one 0 0.2 0.4 0.6 0.8 1.0 because, despite an increase in the laser field and total p ρ ρ ρ population inversion 11 < 22 < 33, the intensity of the atomic spontaneous emission decreases. Fig. 4. Dependences of the upper level populations and the total intensity of the resonance fluorescence on the parame- γ γ We also investigated the dependences of the popula- ter p for 2 = 3, V = V21 = 140, p = 1 (the values of the other tions and the total intensity of the resonance fluores- parameters and the notation as the same as in Fig. 2).

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4. INVESTIGATION OF THE SPECTRA Following [15], the coupling constants gj can be related In this section we investigate the resonance fluores- to the elements of the radiative relaxation operator and cence spectra taking into account interference effects the angular distribution function of the spontaneous and the angular distribution of the spontaneous emis- emission as follows: sion. In order to construct the spectra we use the tech- Γ * nique of the atomÐphoton density matrix [20Ð23]. Fol- jk = ∫g jgk a jldk lowing the general procedure, we add the equation for (13) the lower state ρ Γ Γ Φ (φ, θ) 11 = ∫ jkdO = ∫ jk jk dO. ρ d 11 i Here k is the wave vector of the vacuum mode, a is the ------= -- ∑ (Γ ρ + ρ Γ ) jk dt 2 js sj js js complex line profile, dO is an element of the solid j, s = 2, 3 Φ φ θ θ θ φ (10) angle, and the quantity jk( , )sin( )d d determines the resonance fluorescence intensity in the angle dφdθ ( ρ ρ ) + ∑ V 1 j j1 Ð 1 jV j1 in spherical coordinates and possesses the normaliza- j = 2, 3 tion to the system (6) and rewrite it in the following form: Φ (φ, θ) ∫ jk dO = 1. dρ i------= Mρ, (11) dt Generally in this model the angular distribution func- Φ φ θ tion of the spontaneous emission jk( , ) cannot be where obtained from expression (13). In order to determine ρ = (ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ )T , this, we need to make a complete analysis which takes 11 22 33 21 12 31 13 32 23 into account the magnetic sublevels and the polariza- and the matrix M is determined by the system of equation composition of the spontaneous emission and the Φ φ θ tions (6) and (10). pump wave. The expression for the function jk( , ) The system of equations for the atomÐphoton den- obtained in this analysis is given in the Appendix [see ν (A.3)]. In this model the function Φ (φ, θ) is introduced sity matrix ρ has the form jk phenomenologically. ν dρ ν The general expression for the fluorescence spec- i------= (M Ð ν)ρ + g*F + g*F . (12) dt 2 2 3 3 trum for observations in a certain direction has the form

Here the vector   2 ν ν dν ν ν ν ν ν ν ν ν ν ν T A(ν)dνdO = Im ∑ g ρ  ------dO. (14) ρ = (ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ , ρ ) ,  j j1 3 11 22 33 21 12 31 13 32 23 j = 2, 3 c ρν ρν js = j s Using (13), we can reduce this to the simpler form: are the projections of the atomÐphoton density matrix   on the atomic variables, ν = ω Ð ω , ω is the frequency Ð1 L A(ν) = Im ∑ Γ jk(M F j)k1 , (15) of a scattered quantum), g is the coupling constant of   j j, k = 2, 3 the scattered radiation with the transition |j〉 |1〉: Ð1 Ð1 2πω  1/2  ν  where (M F j )k1 are the elements of the vector M F j g = µ ------L- 1 + O ------ρ j 1 j ប   ω  whose positions correspond to those of the elements k1 W L of the vector ρ . The explicit form of Γ jk depends on (W is the quantization volume). The expression for the the geometry of the problem. For most problems where coupling constant gj for real physical systems is given we are not interested in the angular distribution of the in the Appendix (see (A.1)). The atomic density matrix Φ φ θ Γ is given by scattered radiation, we assume jk( , ) = 1 and jk = γ δ (δ is the Kronecker delta). In the present paper we ν j jk jk ρ = Sp(ρ ) ph, shall assume that the system has a preferred axis and then, using expression (A.3), we can show that the val- where the subscript ph denotes the photon variables. ues of Γ jk have the form The vectors F2 and F3 are expressed in terms of the elements of the atomic density matrix as follows: Γ Γ ( ( θ)) jk = jk c jk + b jk cos 2 . (16) (ρ , , , ρ , , ρ , , , )T F2 = 12 0 0 22 0 32 0 0 0 , The ﬁrst term in (16) describes the contribution of the standard spontaneous emission and the second describes (ρ , , , ρ , , ρ , , , )T F3 = 13 0 0 23 0 33 0 0 0 . the polarization effects whose contribution vanishes

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INFLUENCE OF INTERFERENCE EFFECTS OF SPONTANEOUS EMISSION 55 when integration is performed over angles. In this case, main changes to the spectrum are associated with the the expression for the spectrum has the form term 8 A(ν) Im( f )γ ∑ ------n------n------, (ν Ω )2 γ 2   (17) Ð n + n = Im ∑ Γ (c + b cos(2θ))(MÐ1F )  . n = 1 ik jk jk j k1 Ω   and since the value of j remains unchanged for con- j, k = 2, 3 δ stant EL and j, as p increases the main changes take γ In order to interpret the spectra of a three-level atom, place for Im(fn) and n. formula (17) can be transformed to give It can be seen from the spectra that the changes in γ 9 Im(fn) and n may not be the same for all the peaks. (ν) f n Moreover, this is difficult to investigate because of the A = ∑ -ν------Ω------γ--- + c.c., (18) Ð n + i n large number of parameters which are generally inter- n = 1 related for real systems (for example, the constants cjk, where Ω Ð iγ are the eigenvalues of the matrix M, and n n bjk, and p). However, some general dependences can be fn is a complex quantity which determines the weight of identified. the corresponding term. The values of Ω have three- n ∆L fold degeneracy near zero while the remaining six For p =0 and j = 0 the system has a symmetric ±Ω ±Ω ±Ω ν γ ∆ ( 1, 2, 3) are symmetric relative to the axis. spectrum regardless of the relationships between j, j, These are the Rabi frequencies of the three-level sys- and Vj1 (see Fig. 5a). This agrees with the well-known tem and describes the high-frequency Stark effect. In result obtained by Mollow for the spectra of a two-level addition, one of the eigenvalues of the matrix M is zero, atom [26]. It should be noted that the asymmetry of the Ω γ i.e., for one of the degenerate n = 0 and also for n = 0, spectra obtained in [14] using correct overall method- this corresponds to elastic Rayleigh scattering taking ology is attributed to the incomplete averaging over place without any change in frequency. Bearing this in g g* (see (13) and (14)). mind and using the Sokhotskiœ formula, we can isolate j k the unshifted Rayleigh scattering component in (18): Another interesting result concerns the influence of quantum interference on the form of the spectrum. In most cases, as p increases the intensity of the outer lines el Γ ρ ρ δ(ν) A = ∑ jk 1 j k1 . (19) decreases relative to the central ones while the symme- j, k = 2, 3 try of the spectrum is conserved (see Figs. 5b and 5c) (an exception is the region where the total intensity of Here δ(ν) is the Dirac delta function. Note that in strong the resonance fluorescence is almost zero where the fields, and also in the presence of collisions, the contri- dispersion term (20) makes an important contribution). bution of the elastic component is small so that we shall An investigation of the influence of the Lamb shift neglect this and rewrite formula (18) in the form on the spectra shows that this does not lead to any sig- 8 nificant changes at high spontaneous emission intensi- Re( f )(ν Ð Ω ) + Im( f )γ A(ν) ≈ ∑ ------n------n------n------n- + Aelδ(ν). (20) ties. However, in regions of low total intensity (see Sec- (ν Ω )2 γ 2 Ð n + n tion 3) allowance for the Lamb shift may lead to sub- j = 1 stantial modification of the spectrum (see Fig. 5d). Ω In expression (20) the central peak with 9 = 0 is dou- Dependences of the spectra on the angular distribu- bly degenerate and is the sum of two Lorenzians of dif- tion function [see (13)Ð(17)] are plotted in Fig. 6. It can ferent widths. Consequently, in the resonance fluores- be seen from the figures that these characteristics may cence spectrum we predict that up to seven components have a substantial influence on the form of the spectra will be present. and in certain directions, they may lead to changes in The dispersion term the total intensity of the resonance fluorescence and to degeneracy or, conversely, to strong amplification of 8 Re( f )(ν Ð Ω ) isolated lines in the spectrum. ------n------n--- ∑ 2 2 It should be reemphasized that this investigation is (ν Ð Ω ) + γ n = 1 n n to a certain extent illustrative. The explicit form of the Φ φ θ may lead to asymmetry of the peaks and reduce their angular distribution jk( , ) depends strongly on the intensity to zero for some frequencies. This possibility geometry of the problem which, like the parameter p, was investigated in [11] for the spontaneous emission depends on the quantum interference effects. Thus, a spectra. It can be seen from the graphs that the disper- comprehensive analysis of real systems involves allow- sion component only makes a significant contribution ing for the degeneracy of the angular momentum and for p = 1 in regions where the total intensity of the res- the relationships between the constants cjk, bjk, and p. onance fluorescence is low, which can be seen from the We have therefore shown that quantum interference slight asymmetry of the peaks (see Fig. 5c). Thus, the processes accompanying the radiative relaxation of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 56 PANTELEEV et al.

A, arb. units 0.3 0.3 (a) (b)

0.2 0.2

0.1 0.1

0 0 0.004 0.006 (c) (d)

0.004

0.002

0 0 –200 0 200 –200 0 200 ν γ / 2

γ γ ∆ γ ∆ γ ∆L γ Fig. 5. Resonance ﬂuorescence spectra for 2 = 3, 2/ 2 = 10, 3/ 2 = Ð60, V = V21 = 2 V31 = 60, bjk = 0, ajk = 1: (a) p = 0, j / 2 = 0; ∆L γ ∆L γ ∆L γ (b) p = 0.5, j / 2 = 0; (c) p = 1, j / 2 = 0; (d) p = 1, j / 2 = 3.

A, arb. units 0.3 0.3

(a) (b)

0.2 0.2

0.1 0.1

0 –200 0 200 –200 0 200 ν γ / 2

θ ∆L Fig. 6. Resonance ﬂuorescence spectra for p = 0.8, bjj = 0.2, = 0, j = 0, ajk = 1, b23 = b32 = 0 (a), b23 = b32 = 0.5 (b) (the values of the other parameters are as in Fig. 5).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 INFLUENCE OF INTERFERENCE EFFECTS OF SPONTANEOUS EMISSION 57 excited states may substantially alter the total intensity wave vector k (λ is the polarization index in rotated of the atomic spontaneous emission and its angular dis- 1 Eulerian coordinates), Dλσ is the Wigner function, and tribution, as well as the dynamics of the atomic level ω σ σ populations and the resonance fluorescence spectra. In ajs = ajs( , , ') is the complex line profile. addition, we have derived analytic expressions for the Expression (A.2) can be used to isolate all the main total intensity and the spectra of the resonance fluores- spontaneous emission characteristics and is obtained in cence taking into account the off-diagonal nature of the energy balance with the equation for the photon field radiative relaxation operator and the angular distribu- dynamics (see, for example, [15]). This expression can tion of the spontaneous emission. These results may be be used to describe the angular structure of the scat- used for a qualitative description of the D-line sponta- tered radiation [24] and polarization effects. Using neous luminescence of alkali metal vapor in the field of (A.2) we can write the angular distribution function of a linearly polarized laser wave near a metal surface. the spontaneous emission This work was partly financed by the Russian Foun- σ σ ΦM j Ms '(φ, θ) dation for Basic Research (Grants nos. 99-02-17063, js 96-15-96446, and 99-02-18176).

J j M j Js Ms = ∑ C σC σ U σU σ (A.3) J1 M11 J1 M11 ' k k ' APPENDIX , λ M1 Structure of Interference Terms × 1 (φ, θ, )( 1 (φ, θ, )) in the Radiative Relaxation Operator Dλσ' 0 Dλσ 0 *. for Real Atomic Systems When analyzing (A.2), we need to bear in mind the Expression (5) for the radiative relaxation operator following properties of the sums [25]: allows for the influence of interference effects in the spontaneous emission of atoms but it describes a model 1, σ ≠ σ', J j M j Js Ms nondegenerate atomic subsystem. We shall consider the ∑CJ M 1σCJ M 1σ' =  (A.4) 1 1 1 1 δ , , σ = σ',  J j Js role of these effects for a real system allowing for angu- M1 lar momentum degeneracy. The radiative lifetime, or the rate of radiative relaxation, of the excited state of an and also the orthogonal property of the Wigner func- atomic system is determined using the standard tions: WignerÐWeisskopf procedure using second-order per- 1 ( 1 ) δ turbation theory in terms of the coupling constant in the ∫Dλσ' Dλσ *dO = σ'σ. (A.5) resonance approximation. We assume that adjacent excited states have the angular momentum J2 and J3 and It can be seen from (A.4) and (A.5) that for most real then the explicit expression for the coupling constant atomic systems no interference damping occurs, i.e., Γ ≠ may be given in the form [15] js = 0, where j s. The only exceptions are adjacent states having the same values of the angular momen- kσ( , ) ( )σ 1 Ð σ g j1 M j M1 = i Ð1 CJ M J Ð M tum. Our analysis for atomic systems, including hyper- j j 1 1 fine splitting, confirms this result. However, we note µ πω (A.1) that in many cases where J ≠ J the coefficient p in (5) × j1 ( ) 2 k 2 3 ------Ukσ r ------. may be nonzero. 3 បW (1) If the atoms are located in a high-Q resonator Using the analog of the general expression (13) for the which changes the density of states of the electromag- elements of the radiative relaxation operator, we have netic field and the resonator axis defines a specific direction (quantization axis). The density of states for Γ kσ , kσ' , dk photons of different polarization may differ substan- js(M j Ms) = ∑∫g j1 (M j M1)g1s (M1 Ms)a js------(2π)3 tially and in the expression (A.2) interference quench- M1 ing disappears. Naturally, the Parcel effect must be π µ µ taken into account here, i.e., variation of the ordinary 2 j1 s1 J j M j Js Ms (A.2) γ = ------∑ C σC σ rates of radiative relaxation j in the resonator. ប ( )( ) J1 M11 J1 M11 ' 2J j + 1 2Js + 1 , λ, σ, σ M1 ' (2) If the atoms are in a highly dispersive medium having different values of ε(ω, σ). In this case, the value 1 1 dk × U σU σ Dλσ (Dλσ)*ωa ------. of Γ is a function of the projection M of the angular ∫ k k ' ' js( π)3 23 2 momenta. Then, for example, for sodium D-lines excited by linearly polarized radiation we have for Γ (1/2, 1/2) J j M j Js Ms 1 Ð σ 23 In (A.1) and (A.2) C σ , C σ , and C J1 M11 J1 M11 J j M j J1 Ð M1 2 are the ClebschÐGordan coefficients, Ukσ(r) is the spa- p = ------(φ(0) Ð φ(+1)) (A.6) tial mode factor of σ-polarized photons having the 3

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 58 PANTELEEV et al. Γ and for 23(Ð1/2, Ð1/2) 6. E. G. Pestov, Tr. Fiz. Inst. Akad. Nauk SSSR 187, 60 (1988). 2 p = ------(φ(Ð1) Ð φ(0)). (A.7) 7. D. Agassi, Phys. Rev. A 30, 2449 (1984). 3 8. E. A. Manykin and A. M. Afanas’ev, Zh. Éksp. Teor. Fiz. 48, 931 (1965). The quantity φ(σ) is determined by the convolution 9. E. A. Manykin and A. M. Afanas’ev, Zh. Éksp. Teor. Fiz. dω 52, 1246 (1967). φ(σ) = ε(ω, σ)a (ω, σ)------, (A.8) ∫ 23 2π 10. Ce Chen, Yi-Yian Yin, and D. S. Elliot, Phys. Rev. Lett. 64, 507 (1990). where σ = 0, ±1. If this transition makes a significant 11. S. Y. Zhu, R. C. F. Chan, and C. P. Lee, Phys. Rev. A 52, contribution to the permittivity of the medium, its value 710 (1995). should be calculated self-consistently. If this contribu- 12. A. E. Devdariani, V. N. Ostrovskiœ, and Yu. N. Sebyakin, tion is small and ε(ω, σ) is determined by a different Zh. Éksp. Teor. Fiz. 71, 909 (1976). type of particle in the medium and varies only slightly 13. S. Y. Zhu and M. O. Scully, Phys. Rev. Lett. 76, 388 ε ω σ within the line profile of these transitions, ( 0, ) can (1996). be removed from the integral in (A.6). 14. A. G. Leonov, A. A. Panteleev, A. N. Starostin, et al., 3. If the atoms are in a strong electromagnetic field Zh. Éksp. Teor. Fiz. 105, 1536 (1994). where nonlinear dynamic relaxation effects become 15. A. A. Panteleev, Zh. Éksp. Teor. Fiz. 111, 440 (1997). substantial and there is an appreciable difference 16. V. Savchenko, A. A. Panteleev, and A. N. Starostin, in between the line profiles for photons of different polar- Proc. 13th Intern. Conf. of Spectral Line Change ization. In this case, we have the following estimate for (Firenze, 1996), p. 132. the parameter p: 17. V. I. Savchenko, N. J. Fisch, A. A. Panteleev, et al., Phys. Rev. A 59, 708 (1999). ∝ V 18. G. S. Agarwal, in Progress in Optics, Ed. E. Wolf, p ω-----. (A.9) 0 (North-Holland, Amsterdam, 1973), Vol. 11, p. 2. 19. L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1515 (1964). Here we have V ~ Vj1. We note that in all three cases discussed above the parameter p may be either positive or 20. M. O. Scully and W. E. Lamb, Jr., Phys. Rev. 159, 208 negative. The specific sign of p depends on the physical (1967). conditions. 21. S. Stenholm, D. A. Holm, and M. Sargent, III, Phys. Rev. A 31, 3124 (1985). 22. M. Sargent, III, D. A. Holm, and M. S. Zubary, Phys. REFERENCES Rev. A 31, 3112 (1985). 1. E. B. Aleksandrov, G. I. Khvostenko, and M. P. Chaœka, 23. A. A. Panteleev and A. N. Starostin, Zh. Éksp. Teor. Fiz. Interference of Atomic States (Nauka, Moscow, 1991). 106, 1606 (1994). 2. S. G. Rautian, G. I. Smirnov, and A. I. Shalagin, Nonlin- 24. L. D. Landau and E. M. Lifshits, Electrodynamics of ear Resonances in Atomic and Molecular Spectra Continuous Media (Pergamon, Oxford, 1984; Nauka, (Nauka, Novosbirsk, 1990). Moscow, 1982). 3. S. Stenholm, Foundations of Laser Spectroscopy (Mir, 25. D. A. Varshalovich, A. P. Moskalev, and V. K. Kherson- Moscow, 1987; Wiley, New York, 1984). skiœ, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988; Nauka, Leningrad, 1975). 4. S. G. Rautian, Pis’ma Zh. Éksp. Teor. Fiz. 61, 461 (1995). 26. B. R. Mollow, Phys. Rev. 188, 1969 (1969). 5. F. A. Lomaya and A. A. Panteleev, Zh. Éksp. Teor. Fiz. 103, 1970 (1993). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 59Ð65. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 67Ð74. Original Russian Text Copyright © 2000 by Karshenboœm, Ivanov, Shabaev. ATOMS, SPECTRA, RADIATION

Polarization of Vacuum in a Hydrogen-Like Relativistic Atom: Hyperfine Structure S. G. Karshenboœm*, V. G. Ivanov**, and V. M. Shabaev*** *State Science Center, Mendeleev All-Russia Research Institute of Metrology, St. Petersburg, 198005 Russia; e-mail: [email protected] **Main Astronomical Observatory, Russian Academy of Sciences, St. Petersburg, 196140 Russia ***St. Petersburg State University, St. Petersburg, 198904 Russia Received July 19, 1999

Abstract—An analysis is made of the contribution of the polarization of vacuum to the hyperﬁne splitting of the ground state of a hydrogen-like atom. An expression for the correction to the energy is obtained as an explicit function of the parameter Zα. The ﬁnal expression derived in terms of generalized hypergeometric functions and their derivatives is a function of the particle mass ratio in orbit and in a vacuum loop, and is therefore valid for both ordinary and muonic atoms. Various asymptotic forms are also given. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION A nonrelativistic result was recently published for a muonic atom [9] where the single-potential contribu- Vacuum polarization effects influence the energy tion was obtained in analytic form and the double- levels in ordinary (electronic) and muonic atoms. The potential contribution was obtained as a single integral. most important is the Lamb shift which occurs in muonic atoms in the principal approximation as a result It should be noted that all these calculations, apart of these effects (Fig. 1a). An analytic calculation of this from the numerical ones [7, 8], were made for a point correction was presented in [1]. The present paper is nucleus. Obviously charge distribution and magnetic concerned with the contribution to the hyperfine split- moment effects become important as Z increases. How- ting of the ground state of a hydrogen-like atom and is ever, it should be noted that similar calculations can a direct continuation of [1]. also be made for more highly excited levels for which these effects are not so important. For electronic atoms Calculations of the leading polarization correction the most interesting are the 1s and 2s levels for which to the hyperfine splitting are more complex than those we found new terms of the expansion for low values of for the case of the Lamb shift. Contributions are made the parameter Zα relevant to muonium and hydrogen. by the single-potential (Fig. 1b) and the double-poten- For muonic atoms the calculations can be made for tial diagrams (Fig. 1c): higher levels since hyperfine structure effects in muonic atoms are more important than in ordinary ∆ Ehfs = ETU + EUT. (1) atoms. The single-potential contribution is exactly the same as The present study is organized as follows: we first the contribution to the Lamb shift analyzed in [1, 2] discuss a general expression for the single-potential whereas the contribution of the diagrams in Fig. 1c con- contribution and then one for the double-potential containing the Uehling potential and the magnetic interac- tribution. We then analyze various asymptotic forms. tion, with the Coulomb Green function connecting Finally we discuss the results and compare them with them is far more complex [3]. Nevertheless it can be the numerical calculations. calculated in closed analytic form. In the present study we use exactly the same nota- We recall the main results for the hyperfine splitting tion as in [1] and, in particular of the ground state of hydrogen-like electronic and ε ()α2 κα⁄ muonic atoms. In the electronic case, the first terms of ==11ÐÐ,Z Zmml, the expansion in terms of the small parameter Zα are known [4Ð6] while numerical calculations are available where ml is the particle mass in the loop, m is the mass for high values of the nuclear charge [7, 8]. of a bound particle, and ប = c = 1.

60 KARSHENBOŒM et al. µ (a) (b) N is the nuclear magneton, I is the nuclear spin, and mp m is the proton mass. In general, the dipole magnetic interaction contains ml spinÐspin and spinÐorbit terms. For s-states the second term is obviously zero which appreciably simpliﬁes the expression for the correction

∞ (c) λ ∫ drr2 f (r)g(r)(∂ ⁄ ∂r)(eÐ r ⁄ r) ᏾ (λ) 0 TU = ---∞------. (8) + ∫ drr2 f (r)g(r)(∂ ⁄ ∂r)(1 ⁄ r) 0 Fig. 1. Contribution of the vacuum polarization to the The radial integral is easily taken energy. The bound particle has the mass m and the particle in the polarization loop has the mass ml . The thick line gives ∞ the reduced Green function of a particle in the Coulomb 2 Ð2γr Ð2ε Ðλr ﬁeld of the nucleus. ∫ drr e r (∂ ⁄ ∂r)(e ⁄ r) ᏾ (λ) 0 TU = ---∞------2. GENERAL EXPRESSION γ ε ∫ drr2eÐ2 rrÐ2 (∂ ⁄ ∂r)(1 ⁄ r) (9) 2.1. Single-Potential Contribution 0 It is convenient to begin our calculations of the sin-  2γ  1 Ð 2ε λ gle-potential contribution (Fig. 1b) with a substitution = ------1 + (1 Ð 2ε)------, in the momentum representation, which is equivalent to 2γ + λ 2γ + λ inserting the free polarization of vacuum (cf. (4) in [1]): and the ensuing single integral 1 µ 2 2 µ qb c α v (1 Ð v ⁄ 3) qb c 1 ε ------dv ------ε ------, (2) 2 2 abc 2 π∫ 2 abc 2 2 v (1 Ð v ⁄ 3) κ  q 1 Ð v q + λ R (κ) = dv ------0 TU ∫   1 Ð v 2 1 + κ 1 Ð v 2 where µ is the magnetic moment of the nucleus, 0 (10) Ð2ε 2m  κ 1 Ð v 2  1 λ(v ) = ------l----, (3) × ------ 1 + (1 Ð 2ε)------2  2 2 1 Ð v 1 + κ 1 Ð v 1 + κ 1 Ð v and a loop particle has the mass ml. In the coordinate is easily expressed in terms of the base integrals representation the substitution has the form (see (16) in [1]) 1 α 2( 2 ⁄ ) Ðλr α E ε ∇ µ 1 v 1 Ð v 3 ε ∇ µ e E (1s) = ------F------abc b c-- ---∫dv ------abc b c------. (4) TU π r π 1 Ð v 2 r 2[1 Ð (Zα)2] Ð 1 Ð (Zα)2 0 (11) The single-potential contribution is determined by  1  1 Ð 2ε 1   ×  I Ð --I + ------I Ð --I . α  121 221 κ  132 232 (κ)  3 3  ETU = -π--ET RTU . (5) The analyticity of the result is determined by the fac- Here ET is the matrix element of the magnetic exchange tor (6): without any radiation correction, which was obtained for the ground state in [10] (Zα) < 3 ⁄ 2, or Z < 118.676… . (12) ( ) EF ET 1s = ------, (6) 2 2 2[1 Ð (Zα) ] Ð 1 Ð (Zα) 2.2. Double-Potential Contribution where the Fermi energy EF corresponds to the nonrela- We shall now analyze the contribution of the diagrams tivistic interaction of two magnetic moments: in Fig. 1c. The difference between this contribution and µ the single-potential one consists in the presence of a Cou- 4α( α)3 m 2I + 1 lomb Green function connecting the potentials. However, EF = -- Z µ------m, (7) 3 N mp 2I the contribution may be rewritten in the form of some

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 POLARIZATION OF VACUUM IN A HYDROGEN-LIKE RELATIVISTIC ATOM 61 matrix element of the Uehling potential examined by us 3.1. Single-Potential Contribution in [1] in terms of the wave function perturbed by the In the Schrödinger approximation the result is magnetic ﬁeld. This function was obtained explicitly in known analytically [9]: [11]. Using the explicit expression for the perturbed wave function (see the Appendix) we obtain (cf. [1]) 2 Ð κ2 + 2κ4 2 π 6 + κ2 r (κ) = ------Ꮽ(κ) Ð ------+ ------, (19) α TU κ3 κ3 2 κ2 (κ) 4 3 9 EUT = ---ET RUT , (13) π where where (κ) (κ κ2 ) Ꮽ(κ) arccos ln + Ð 1 1 = ------= ------, (20) 2( 2 ⁄ ) κ2 κ2 (κ) v 1 Ð v 3 ᏾ (λ( )) 1 Ð Ð 1 RUT = ∫dv ------UT v , (14) 1 Ð v 2 and by analogy with expression (18) in [1] we deter- 0 mine ∞ 2 λ (κ) (κ) ε (κ) ᏻ(ε ) ( )[( α) Ð r ⁄ ] RTU = rTU Ð 2 pTU + . (21) ∫ f 1s X1s + g1sY1s Z e r dr ᏾ (λ) 0 Further expansion in the nonrelativistic approximation UT = ------∞------, (15) yields the result for small κ: f g dr π π ∫ 1s 1s 3 κ 4κ2 5 κ3 RTU = ------Ð -- + ------0 8 5 24 (22) and explicit expressions for the radial components of 3π κ π  2 Ð ------ln-- + ----- κ(Zα) . the wave functions are given in the Appendix.  8 2 32 It is not difﬁcult to calculate the radial integrals in (15) and thus obtain Substituting the expanded normalization factor

3 2 ε ε2 ε3 E = E 1 + --(Zα) , (23) (κ) 3 Ð 8 + 8 Ð 2 (κ) T F RUT = ------J20 2 (1 Ð ε)2 (16) for the electronic atom (κ = Zα) we obtain the result (ε ) 2 (κ) 2 + 1 (κ) ( ε) (κ) Ð ------ε--J21 + ------ε----J20 Ð 2 1 Ð J30 , 1 Ð 1 Ð 2 α 3π 4 2  ( α) ( α) ETU = -π--EF ------Z Ð -- Z where the J-integrals  8 5 (24) 1 2 2 ( ⁄ ) 3π Zα 71π 3  (κ) v 1 Ð v 3 + Ð------ln------+ ------(Zα) , Jmn = ∫dv ------1 Ð v 2 8 2 96  0 (17) m Ð 2ε  κy  n κy which confirms all the previously known coefficients × ------ln ------1 + κy 1 + κy (see [4, 6]) and contains a single new one, denoted by C30. are expressed in terms of the derivatives of the integrals Iabc (see (16) in [1]) [3]: 3.2. Double-Potential Contribution ∂n 1 Unlike the single-potential contribution, even in the J = ------I µ Ð --I µ . (18) leading nonrelativistic approximation it is impossible to mn n 12 3 22 ∂µ µ = m obtain a fairly compact expression for the double-potential contribution. In particular, the integral J cannot be As for the single-potential term, the analyticity of Z 21 simplified substantially [9]: is determined by condition (12). κ2 κ4 κ2 (κ) π Ð 2 5 Ð 8 + 6 RUT = ------3---- + ------2------2----- 3. CORRECTION TO THE HYPERFINE 2κ 3κ (1 Ð κ ) α Ӷ (25) SPLITTING FOR Z 1 κ6 κ4 κ2 2 Ð 3 + 4 Ð 2Ꮽ(κ) + ------Ð 2J21. For large Z the finite dimensions of the nucleus play κ3(κ2 Ð 1) a significant role, so these formulas are most topical in the nonrelativistic limit (Zα Ӷ 1). We shall begin our In [9] asymptotic forms of this expression were also analysis with the single-potential contribution. obtained for large and small values of the parameter κ.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 62 KARSHENBOŒM et al. A further nonrelativistic expansion for small κ where (see (34) in [1]) yields the result (1) 2 b Ð 2 --κ ( ε, ) π Ibc = B1 Ð δ 2 Ð b + c Ð 2 b Ð 2 , (κ) 3 κ 214 8 ( κ) κ2 3 RUT = ------+ ------Ð ----- ln 2  (32) 8 225 15 1 (26) δ = ------Ӷ 1 π π κ π π κ 1 + κ  43 5  κ3 23 3  κ( α)2 + Ð ------+ ------ln-- + ------Ð ------ln-- Z . 288 24 2 32 8 2 and κ α ( ε , ) For electronic atoms, assuming = Z we ﬁnd [12] B1 Ð δ c Ð 2 Ð 1 1 1 1 c Ð 2ε  1  (33) 3π(Zα) 214 8  2 = ------Ð -- + ------+ ᏻ ----- . R (κ) = ------+ ------Ð ----- ln(2Zα) (Zα) ε κ 2  2 UT 8 225 15  c Ð 2 Ð 1 2κ κ (27) π π α The second term is given by 41 Z  ( α)3 + ------Ð -- ln------ Z 72 6 2 π ε (2)  2 c Ð 2 ᏻ 1  I3c = Ð -- Ð -- + ------+ ----- , (34) or 4 3 6κ κ2 and ultimately we obtain α  π α 3 Z 214 8 ( α) ( α)2 EUT = ---EF------+ ------Ð ----- ln 2Z Z π  8 225 15 1 2 1 π1 c Ð 2ε ᏻ 1  -- I3c = ------Ð -- -- + ------+ ----- . (35) (28) κ 3c Ð 2ε Ð 1 4κ 2κ2 κ3 163π π Zα 3  + ------Ð -- ln------(Zα) . Finally the single-potential correction for the hyper- 6 144 2  ﬁne splitting in a muonic atom is given by

As in the single-potential case (24), this expression con- 2 2 1 R (κ) = -- ln(2κ) + --[ψ(1) Ð ψ(1 Ð 2ε)] + -- firms all the expansion coefficients known so far [4, 6] and TU 3 3 9 also gives the result for one new one (C30). (36) 2 5 5 2 1  1  + -- Ð --ε Ð --ε ----- + ᏻ ----- . 3 6 3 κ2 κ3 4. HYPERFINE SPLITTING IN MUONIC ATOMS The expression contains elementary functions and a ψ-function which is a logarithmic derivative of a 4.1. Single-Potential Contribution gamma function. The result agrees with the double α Ӷ κ ӷ For nonrelativistic muonic atoms the asymptotic asymptotic form (Z 1 and 1) given in (29). form for small Zα and large κ is more relevant: 4.2. Double-Potential Contribution   (κ) 2 ( κ) 1 RTU = -- ln 2 + --  We first introduce the double limit Zα Ӷ 1 and κ ӷ 1: 3 9  (29) 2 2 2π π 2 4  1  R (κ) = 2ln(2κ) Ð 3 + ------+ -----(Zα) + ᏻ((Zα) ) + ᏻ ----- . UT 9 9 κ2 π2  ( κ) 4ψ ( ) 4  ( α)2 In the more general case we can obtain the expres- + 2ln 2 Ð -- '' 2 Ð 3 + ------ Z (37) sion 3 9

1 4  1  (κ) ( ε)-- + ᏻ((Zα) ) + ᏻ ----- . RTU = I21 + 1 Ð 2 κI32, (30) κ2 κ ӷ α where the integrals Ibc are determined in (32) in [1]. In the more general case ( 1 for arbitrary Z ), deriv- ψ One of these (I21) was obtained in [1] (see (37)) while atives of the -function appear in the expression: the other (I ) is known only partly: 32 ε ε2 ε3 (κ) 3 Ð 6 + 4 Ð 2 (1) (2) RUT = ------2------I32 = I32 + I32 , (31) (1 Ð ε) (1 Ð 2ε)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 POLARIZATION OF VACUUM IN A HYDROGEN-LIKE RELATIVISTIC ATOM 63

ML Ð Ue EL Ð Ue Analytic results of the present study and numerical results from [8] (denoted in [8] as EVP and EVP , respectively

Single-potential contribution Double-potential contribution F F , Z 0 Numerical F F , Z 0 Numerical Z TU TU TU TU (11) (24) value [8] (16) (28) value [8] 1 0.0085578 0.0085579 0.0085578 0.0087703 0.0087702 0.0087691 3 0.0254869 0.0254877 0.025487 0.0271133 0.0271086 0.027112 5 0.0422566 0.0422619 0.042257 0.0464153 0.0463843 0.046414 7 0.0589599 0.0589771 0.058960 0.0666723 0.0665652 0.066671 10 0.0840724 0.0841284 0.084072 0.0989572 0.0985560 0.098955 25 0.2192738 0.2195350 Ð 0.3068919 0.2936806 Ð

2 2 1 and × -- ln(2κ) + --[ψ(1) Ð ψ(1 Ð 2ε)] + -- 3 3 9 ( ) α (38) ∆E 4 = ---E (Zα)4 4 2 3 Ð ε UT π F + ------ψ'(2 Ð 2ε) Ð ------3(1 Ð ε) 3(1 Ð ε)(1 Ð 2ε) 2  8 2 5336 767881 2π  × ----- ln (2Zα) Ð ------ln(2Zα) + ------Ð ------. 2 15 1575 165375 45  3 Ð 2ε + ε ᏻ 1  + ------2------+ -----3 . κ κ It has been noted that numerical results are also available for calculations of this correction [7, 8]. For comparison we ﬁrst note that a slightly different param- 5. DISCUSSION OF RESULTS etrization of the result is frequently used for numerical and analytic calculations in electronic atoms (κ = Zα) We shall brieﬂy discuss these results. For the case of ordinary (electronic) atoms we obtained an expansion α α ( ) ( α) for small Z Ehfs 1s = -π--EFF Z . (40)

α  π ∆ 3 α 34 8 ( α) ( α)2 Results for the single-potential and double-potential Ehfs = ---EF------Z + ------Ð ----- ln 2Z Z π  4 225 15 contributions to the total expression F (see (11) and (16)), the expansions (see (24) and (28)), and the (39) numerical results are compared in table. We only make 539π 13π Zα 3  ≤ + ------Ð ------ln------(Zα) , a comparison with the results [8] for Z 10 since for 288 24 2  larger Z we used a nonpoint nucleus in this study, as in the calculations made in [7]. which contains one new coefﬁcient in addition to the We also give the asymptotic forms obtained above known ones. The possibility of expanding as far as arbi- for muonic atoms trary order of smallness is important for the polarization correction and for studying the structure of the α  ε ε2 ε3 α ∆ 4 Ð 10 + 9 Ð 4 series in terms of the parameter Z in general. The Ehfs = ---ET ------appearance of the square of the logarithm of this π  (1 Ð ε)2(1 Ð 2ε) parameter in various corrections [13, 6] makes accurate calculations preferable. This result together with calcu- 2 2 1 lations of the Uehling correction was ﬁrst obtained in a × -- ln(2κ) + --[ψ(1) Ð ψ(1 Ð 2ε)] + -- closed analytic form as an explicit function of the 3 3 9 parameter Zα. 4 2 3 Ð ε It is easy to obtain expansions of higher orders as + ------ψ'(2 Ð 2ε) Ð ------high as necessary. For example, in the fourth order of 3(1 Ð ε) 3(1 Ð ε)(1 Ð 2ε) the parameter Zα we obtain

11 17 2 2 1  1   (4) α 44 303 -----ε --ε ----- ᏻ -----  ∆ ( α) ( α) + ----- Ð Ð  2 +  3 , ETU = ---EF Z -- ln 2Z Ð ------3 3 3 κ κ π 5 175 

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 64 KARSHENBOŒM et al.

(a) (b) (c) APPENDIX Perturbation of the Dirac Coulomb Wave Function + by the Magnetic Field of the Nucleus The first-order correction for the hyperfine interaction to the Dirac Coulomb wave function may be Fig. 2. Skeletal diagrams. obtained analytically using the method of generalized virial relations developed in [15]. An explicit solution for the ground state in a convenient form for practical which for low nuclear charge have the form calculations was given in [11] (in the second line of ε equation (18) in this article the sign before nκ should be changed). Below we follow the notation used in α  π2 ∆ 8 ( κ) 26 2 11 [16]. The Dirac wave function is defined by us as fol- Ehfs = ---EF-- ln 2 Ð ----- + ------+ ------π 3 9 9 3κ2 lows: f (r)Ω (n) Ψ  njl jlm  2 njlm = , (A.1)  4 22 8π 103  2  ( ⋅ ) ( )Ω ( ) + 6ln(2κ) Ð --ψ''(2) Ð ----- + ------+ ------(Zα) Ði s n gnjl r jlm n  3 3 9 κ2 24 where  ( α)3 3( ε) ᏻ(( α)4) ᏻ 1  Z m 2 Ð ( α )Ðε ÐZαmr + Z + ----- . f 1s = 2 ------2Z mr e , (A.2) κ3  Γ(3 Ð 2ε) ( α)5 3 Z m ( α )Ðε ÐZαmr Both these asymptotic forms contain a large loga- g1s = Ð2 ------2Z mr e (A.3) rithm (ln(2κ)) which can be determined using the run- (2 Ð ε)Γ(3 Ð 2ε) ning coupling constant and the known matrix elements are the components of the pure Coulomb function for the skeletal diagrams (Fig. 2) [12], in particular (see [16]). Sc ( ) ( ) The linear correction to the ground state for the ETU 1s = ET 1s and magnetic moment of the nucleus is a complex expres- ∂ ( ) 2 3 sion containing contributions of s- and d-waves. We are Sc E 1s 3 Ð 6ε + 4ε Ð 2ε E (1s) = Zα------T------= ------E (1s). interested in the matrix element of the central potential UT ∂( α) 2 T Z (1 Ð ε) (1 Ð 2ε) (see (15)), so we merely give the first of these two contributions here The formula for calculating the logarithmic terms µ involves multiplying the skeletal matrix elements by (δΨ )s 2α 1 1, 1/2, 0, m = Ð-- µ------3 N mp 2α qat ----- ln-----, F(F + 1) Ð I(I + 1) Ð 3 ⁄ 4 3 π m × ------(A.4) l 2I which is obtained from the known expression for the ( )Ω ( )  X1s r 1/2, 0, m n  running coupling constant (see [1]). The characteristic × , α κ  ( ⋅ ) ( )Ω ( ) atomic momentum is qat = Z m = ml. Finally we Ði s n Y1s r 1/2, 0, m n obtain the logarithmic contributions contained in (36) where the total moment of the system F may have values and (38) which serves as further confirmation of these of I ± 1/2 and the radial components have the form [11] results. ( α)3 To conclude, we note that similar calculations may 1 2 Z m 3 X1s = ------------Ð -- f 1s be made for the hyperfine splitting of any level. The 1 Ð 4(1 Ð ε)2 1 Ð ε r expression for the relativistic energy E is known [14] (A.5) T  2Zα 2Zα(3 Ð 2ε)m as is the method of calculating the correction to the + 3(2 Ð ε)m Ð ------g Ð ------F , wave function perturbed by the magnetic field of the  r  1s r 1s nucleus [11, 15]. 1 2(Zα)3m 3 This work was supported by the program “Funda- Y = ------+ -- g 1s 2  ε  1s mental Metrology” and also by the Russian Foundation 1 Ð 4(1 Ð ε) 1 Ð r (A.6) for Basic Research (project no. 98-02-18350) and the pro- α α( ε)  ε 6Z  2Z 3 Ð 2 m gram “Universities of Russia. Fundamental Research” + Ð 3 m + ------ f 1s Ð ------G1s , (project no. 3930). r r

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 POLARIZATION OF VACUUM IN A HYDROGEN-LIKE RELATIVISTIC ATOM 65

α ( ε) α ε 4. S. L. Brodsky and G. W. Erickson, Phys. Rev. 148, 26 ( ) Z m 2 Ð ÐZ mr( α )1 Ð (1966). F1s r = ---Γ----(------ε---)---e 2Z mr 3 Ð 2 5. S. M. Schneider, W. Greiner, and G. Soff, Phys. Rev. A 50, 118 (1994). ψ( ε) 1 × 3 Ð 2 ε 6. S. G. Karshenboœm, Z. Phys. D 36, 11 (1996). ------ε------+ 2 Ð Ð ----(------ε---)- (A.7) 1 Ð 2 1 Ð 7. V. M. Shabaev, M. Tomaselli, T. Kühl, et al., Phys. Rev. A 56, 252 (1997). 1 -Ð Zαmr Ð ------ln(2Zαmr) , 8. P. Sunnergren, H. Persson, S. Salomonson, et al., Phys. 1 Ð ε Rev. A 58, 1055 (1998). 9. S. G. Karshenboœm, U. Jentschura, G. Soff, et al., Eur. J. 3 Phys. D 2, 209 (1998). (Zα) m ÐZαmr 1 Ð ε G (r) = Ð ------e (2Zαmr) 1s (2 Ð ε)Γ(3 Ð 2ε) 10. G. Breit, Phys. Rev. 35, 1477 (1930). 11. M. B. Shabaeva and V. M. Shabaev, Phys. Rev. A 52, ψ(3 Ð 2ε) 1 2811 (1995). × ------+ 2 Ð ε + ------(A.8) 1 Ð ε 2(1 Ð ε) 12. S. G. Karshenboœm, V. G. Ivanov, and V. M. Shabaev, Physica Scripta 80, 491 (1999). 13. S. G. Karshenboœm, Zh. Éksp. Teor. Fiz. 103, 1105 α 1 ( α ) (1993). --Ð Z mr Ð ------ε-- ln 2Z mr . 1 Ð 14. P. Pyykko, E. Pajanne, and M. Inokuti, Int. J. Quantum Chem. 7, 785 (1973). 15. V. M. Shabaev, J. Phys. B 24, 4479 (1991). REFERENCES 16. V. B. Berestetskiœ, E. M. Lifshits, and L. P. Pitaevskiœ, 1. S. G. Karshenboœm, Zh. Éksp. Teor. Fiz. 116 (11) (1999). Quantum Electrodynamics (Pergamon, Oxford, 1982; 2. S. G. Karshenboœm, Can. J. Phys. 76, 168 (1998). Nauka, Moscow, 1980). 3. S. G. Karshenboœm, V. G. Ivanov, and V. M. Shabaev, Can. J. Phys. 76, 503 (1998). Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 66Ð78. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 75Ð89. Original Russian Text Copyright © 2000 by Lee, Milsteœn, Strakhovenko. ATOMS, SPECTRA, RADIATION

Quasiclassical Green Function in an External Field and Small-Angle Scattering Processes R. N. Lee, A. I. Milsteœn*, and V. M. Strakhovenko Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia *e-mail: [email protected] Received July 22, 1999

Abstract—A representation is obtained for the quasiclassical Green functions of the Dirac and Klein–Gor- don equations allowing for the ﬁrst nonvanishing correction in an arbitrary localized potential which generally possesses no spherical symmetry. This is used to obtain a solution of these equations in an approximation similar to the FurryÐSommerfeldÐMaue approximation. It is shown that the quasiclassical Green function does not reduce to the Green function obtained in the eikonal approximation and has a wider range of validity. This is illustrated by calculating the amplitude of small-angle scattering of a charged particle and the amplitude of Delbrück forward scattering. A correction proportional to the scattering angle was obtained for the amplitude of charged particle scattering in a potential possessing no spherical symmetry. The real part of the Delbrück forward scattering amplitude was calculated in a screened Coulomb potential. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tions generalize the results obtained in the FurryÐSom- merfeldÐMaue approximation [8Ð10] to the case of an As we are well aware, quantum electrodynamics arbitrary localized potential. processes in external fields can be conveniently analyzed using wave functions which are a solution of the The eikonal approximation is frequently used to cal- Dirac equation in a given field, and the corresponding culate the amplitudes of processes at high energies. The Green functions. In cases where the particle energy is corresponding wave functions and Green functions high compared with the mass and the scattering angles generally differ from the quasiclassical ones and have a are small when the characteristic angular momenta in narrower range of validity. Thus, using the eikonal the process are large, the quasiclassical approximation approximation without any special justification may can be applied. Moreover, using the quasiclassical yield incorrect results. In Section 4 we demonstrate this Green functions considerably simplifies the calcula- for the small-angle scattering of a charged particle in an tions. external field. In this Section we put forward a systematic derivation of an expression for the amplitude of the |ε The quasiclassical electron Green function G(r2, r1 ) process using quasiclassical wave functions. In partic- was first obtained for a Coulomb potential in [1, 2] ular, we find that for the case of a screened Coulomb using the exact Green function of the Dirac equation potential the quasiclassical wave function rather than [3]. A more convenient representation of the quasiclas- the eikonal one must be used to obtain a correct sical Coulomb Green function for almost collinear vec- description in cases of momentum transfer larger than tors r1 and r2 was obtained in [4, 5]. The same geometry the reciprocal screening radius (when screening can be was used to obtain the quasiclassical Green function for neglected). an arbitrary spherically symmetric decreasing potential In Section 5 the expressions obtained for the quasi- in [6, 7]. classical Green function are used to calculate the In Section 2 of the present study, expressions for the amplitude of elastic forward scattering of a photon in quasiclassical Green functions of the Dirac and KleinÐ the electric field of an atom (Delbrück scattering). It has Gordon equations are derived for the case of a localized been shown [2] that corrections to the quasiclassical potential which does not generally possess spherical Green function must be taken into account to calculate symmetry. By localized we understand a potential this amplitude. These corrections are significant in the which has a maximum at a certain point and decreases range of validity of the eikonal approximation whereas fairly rapidly with increasing distance. The Green func- the contribution of higher orders of perturbation theory tions are obtained allowing for the first correction. in terms of the external field (Coulomb corrections) is These are then used to find the quasiclassical wave determined by the range of variables where the eikonal functions of the Dirac and KleinÐGordon equations and Green function cannot be used. The real part of the Del- the corrections to them (Section 3). These wave func- brück scattering amplitude is obtained for the first time

QUASICLASSICAL GREEN FUNCTION IN AN EXTERNAL FIELD 67 ρ Γ for a screened Coulomb potential. For scattering at parameter * of the rectilinear trajectory linking the comparatively low photon energies the real parts of the points r and r as Delbrück amplitude and the amplitude of Compton 1 2 scattering at atomic electrons are of the same order. φ(x) This factor may prove important in descriptions of pho- ρ = minlim ------. (7) ton propagation in matter. * x ∈ Γ ∇⊥φ(x)

Let us assume that this minimum is achieved at a cer- 2. GREEN FUNCTION tain point x = r*. A necessary condition for the validity It has been shown [4, 6] that in order to calculate the κρ ӷ amplitudes of various quantum electrodynamic proof the quasiclassical approximation is that * 1, cesses it is convenient to use the Green function of the which we assume to be satisfied. We introduce the nota- “squared” Dirac equation D(r , r |ε), which is related to | | 2 1 tion a1, 2 = r1, 2 Ð r* . We consider the two overlapping the ordinary Green function G(r , r |ε) by 2 1 regions: , ε [γ 0 ε ] , ε G(r2 r1 ) = ( Ð V(r2)) Ð gp2 + m D(r2 r1 ), 2 (1) , Ӷ κ ρ , ӷ ρ 1) min(a1 a2) *, 2) min(a1 a2) *. (8) where γµ are the Dirac matrices, p = Ði∇ is the momentum operator, and V(r) is the potential energy. In the In the first region the right-hand side of equation (6) is |ε quasiclassical approximation the function D(r2, r1 ) small. The solution of this equation with zero right- may be expressed in the form hand side has the form D(r , r ε) 2 1 1 ( ) (2) φ i ( ⋅ (∇ ∇ )) 0 , ε F0 = exp Ðir∫ (r1 + xr)dx , (9) = 1 Ð ----ε- a 1 + 2 D (r2 r1 ), 2 0 where which corresponds to the eikonal approximation. In (0) , ε 1 order to calculate the first correction to F0, we shall D (r2 r1 ) = r2 r1 , ----2------κ Ð H + i0 (3) seek a solution of equation (6) in the form F = F0(1 + g) and we shall neglect g on the right-hand side of this 2 ε 2 H = p + 2 V(r) Ð V (r), equation. As a result, we obtain the equation for g: κ2 = ε2 Ð m2. Thus the problem is reduced to calculating 1 the quasiclassical Green function D(0) of the KleinÐ ∂ κ φ2 2∆ φ Gordon equation with the potential V(r) (the 2i g = (r1 + r) + ir dxx 1 (r1 + xr) ∂r ∫ Schrödinger equation with the Hamiltonian H). 0 (0) |ε (10) In the function D (r2, r1 ) we now convert from 1 2 the variables r1 and r2 to the variables r1 and r = r2 Ð r1. 2 ∇ φ + r ∫dxx 1⊥ (r1 + xr) , In these variables the function D0 satisfies the equation 0 [κ2 κφ 2] (0) , ε δ Ð 2 (r1 + r) Ð p D (r + r1 r1 ) = (r), (4) where the index 1 in the derivatives denotes differenti- φ λ 2 κ λ ε κ ∇ ∇ where = V Ð V /2 , = / , and p = Ði r. In the ation with respect to r1, 1⊥ is the component of the ultrarelativistic case we have λ = +1 for ε > 0 and λ = ∆ ∇2 −1 for ε < 0. gradient perpendicular to r, and 1 = 1 . After integrat- We shall seek a solution of this equation in the form ing over r, we obtain the expression for g:

(0) exp(iκr) 1 D (r + r , r ε) = Ð------F(r, r ). (5) 1 1 π 1 1 2 4 r g = ------r dxx(1 Ð x)∆ φ(r + xr) 2κ ∫ 1 1 Note that the factor at F in (5) is the Green function of 0 equation (4) for φ = 0. The function F satisﬁes the equa- 1 tion φ2 Ð ir∫dx (r1 + xr) (11) ∂ φ 1 ∆ ⁄ 0 i Ð (r1 + r) F = Ð------r (F r) (6) ∂r 2κ 1 x 3 Ð 2ir dx(1 Ð x)∇ ⊥φ(r + xr) dyy∇ ⊥φ(r + yr) . with the boundary condition F(r = 0, r1) = 1. For the ∫ 1 1 ∫ 1 1 following reasoning we deﬁne the effective impact 0 0

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Finally, taking into account the correction, the Substituting (15) into (14) we obtain an equation for Ᏺ Green function D(0) in the first region has the form ∂ 1 φ˜ Ᏺ i∂ Ð (0) , ε 1 κ λ r D (r2 r1 ) = Ð------exp i r Ð i r∫V(r1 + xr)dx (16) 4πr 2 2 2 0 1 ∂ iL  ∂ 1 iL  = Ð------Ð ------Ð -- Ð ------Ᏺ, κ 2 2∂ 2  1 2 ∂r κr r r 4κr × 1 λ 2 ∆ 1 + ------r ∫dxx(1 Ð x) 1V(r1 + xr) (12)  2κ where 0 ÐiA iA 1 x φ˜ φ  = e (r1 + r)e . (17) 3 ∇ ∇ Ð 2ir ∫dx(1 Ð x) 1⊥V(r1 + xr)∫dyy 1⊥V(r1 + yr) .  We shall seek a solution of this equation taking into 0 0 account the first correction in terms of the parameter φ κρ φ˜ In this expression we used the relationship between 1/ *. For this purpose in the expansion of in terms ε ӷ λ2 and V and assumed that m (we set = 1). Substi- of commutators of the operator A with φ it is sufficient tuting (12) into (2), we obtain an expression for the to retain only the first two terms: function D in the first region:

1 φ˜ ≈ φ Ð i[A, φ]. , ε 1 κ λ D(r2 r1 ) = Ð----π---- exp i r Ð i r∫V(r1 + xr)dx In the principal approximation in terms of the parame- 4 r κρ 0 ter 1/ * we can neglect the right-hand side of (16) and 1 replace φ˜ by φ. In this approximation the function Ᏺ is  r ×  ∇ the same as the eikonal function F [see (9)]. In order to 1 Ð ----κ--∫dxa 1V(r1 + xr) 0  2 find the first correction, we express Ᏺ in the form 0 (13) 1 2 Ᏺ λ = F0(1 + g1). r ∆ + ------∫dxx(1 Ð x) 1V(r1 + xr) 2κ With the necessary accuracy we obtain the following 0 equation for g1: 1 x 3  ir ∇ ∇ ∂ 2 1 1  Ð ------∫dx(1 Ð x) 1⊥V(r1 + xr)∫dyy 1⊥V(r1 + yr) . 2iκ g = φ (r + r) Ð -- Ð ---- κ  ∂r 1 1  r a  0 0 1 1 (18) We shall now analyze the second region. In this case × 2φ φ φ the right-hand side of equation (6) is not small. Using iL (r1 + r) + 2r(L (r1 + r))∫dxL (r1 + xr) . spherical coordinates we can rewrite this in the form: 0 2 ∂ L2 1 ∂ Integrating this over r, we find i Ð φ(r + r) Ð ------F = Ð------F, (14) ∂ 1 2 κ 2 r 2κr 2 ∂r 1 ir 2 2 g = Ð------dxφ (r + xr) where L is the square of the angular momentum oper- 1 2κ ∫ 1 ator. In this equation r is an independent variable. We 0 are interested in the value of the function F(r, r ) for 1 1 r = r2 Ð r1. The polar axis of the spherical coordinates  r  + ir dxx 1 Ð ----x ∆ ⊥φ(r + xr) is for convenience directed along r2 Ð r1. The term on ∫  a  1 1 2 1 the left-hand side of the equation containing L is sig- 0 (19) ≥ 2 1 nificant for r a1. This becomes clear if the operator L 2  r  is made to act on the eikonal function (9). Then in order + 2r dx 1 Ð ----x (∇ ⊥φ(r + xr)) ∫  a  1 1 to calculate the function F(r2 Ð r1, r1) it is sufficient to 1 analyze a narrow region of polar angles of the vector r 0 θ ρ Ӷ x of the order of ~ */a1 1. The right-hand side of × ∇ φ equation (14) is a small correction. We shall seek a ∫dyy 1⊥ (r1 + yr) . solution of this equation in the form 0 2 iA 1 1  L 2 Ᏺ Here we have ∆ ⊥ = ∇ ⊥ . Using an expansion in terms F = e , A = -- Ð ---- ----κ--. (15) 1 1 r a1 2 of spherical functions we can show that when β Ӷ 1 for

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 QUASICLASSICAL GREEN FUNCTION IN AN EXTERNAL FIELD 69 an arbitrary function g(r), the following relationship is inclusive. Thus, we can write a formula which is valid satisﬁed with the required accuracy κρ ӷ in both regions subject to the condition * 1:

2 2 dq iq2 1 exp[Ðiβ L ]g(r) ≈ ----- e g(r + 2βrq), (20) iκr ∫ iπ (0) , ε ie 2 λ D (r2 r1 ) = ------∫dqexp iq Ð i r∫dxV(Rx) 4π2r where q is a two-dimensional vector perpendicular to r. 0 Using (15), (19), and (20) we obtain the following 1 (0)  λ expression for the Green function D in the second × 1 Ð ------2∫dxV(Rx) Ð V(r1) Ð V(r2) region allowing for the correction:  2κ 0 (24) 1 1 x iκr 3 (0) ie 2 D (r , r ε) = ------dqexp iq Ð iλr dxV(r ) ir [ ( ) ] 2 1 2 ∫ ∫ x + ------∫dx∫dy x(1 Ð x)y(1 Ð y) Ð 1 Ð x y 4π r κ 0 0 0

1  ir  r   ×  λ ∆ × (∇ ⊥V(R ))(∇ ⊥V(R )) , 1 Ð ----κ-- i r∫dxx1 Ð ----x 1⊥V(rx) (21) 1 x 1 y  2 a1  0 where the value of R is determined in (22). In fact, in 1 x  x 2  r  (∇ ) ∇ the ﬁrst region we can expand V(Rx) in terms of q as far + 2r ∫dx1 Ð ----x 1⊥V(rx) ∫dyy 1⊥V(ry) , a1  as quadratic terms and integrate over dq which directly 0 0 yields formula (12). In the second region we need to bear in mind that, as we have already noted, the main ⁄ (κ ) where rx = r1 + xr + qx 2r(r Ð a1) a1 and r = r2 Ð contribution to the integrals is made by a narrow region of x and y near a /r of width δx ~ δy ~ ρ /r. Thus we r1. Note that the gradients with respect to r1 in this for- 1 * mula are taken for ﬁxed r. obtain

We note that in r the term proportional to q need 2 x | | 2[ x(1 Ð x)y(1 Ð y) Ð (1 Ð x)y] = x Ð y Ð (x Ð y) only be taken into account in the narrow region x Ð a1/r ~ ρ Ӷ 2 */r 1. Utilizing this fact, we rewrite formula (21) in ( ) ≈ a form which does not depend explicitly on a . For this Ð x(1 Ð x) Ð y(1 Ð y) x Ð y δ 1 we write rx = Rx + rx, where ρ to within quadratic terms with respect to * /r. In addi- ⁄ κ tion, the linear terms with respect to V in the preexpo- Rx = r1 + xr + q 2x(1 Ð x)r , (22) nential factor also make a negligible contribution in δ ⁄ κ( ⁄ ( ) ) terms of the parameter ρ /r. Hence, in the second rx = 2r x r a1 Ð 1 Ð x(1 Ð x) q. * region (24) yields (21). We expand V(r ) in phase in terms of δr as far as the x x Using (2) we obtain a ﬁnal expression for the Green ﬁrst term, we replace rx with Rx in the preexponential function D(r , r |ε) factor, and integrate by parts over q. As a result we 2 1 1 obtain iκr ie 2 , ε ------λ 1 D(r2 r1 ) = 2 ∫dqexp iq Ð i r∫dxV(Rx) iκr 4π r (0) , ε ie 2 λ 0 D (r2 r1 ) = ------∫dqexp iq Ð i r∫dxV(Rx) 4π2r 0 1 (23)  r 1 x ×  ∇  3  1 Ð ----κ--∫dxa 1V(Rx) × ir (∇ )(∇ )  2 1 + ------∫dx∫dy(x Ð y) 1⊥V(Rx) 1⊥V(Ry) . 0  2κ  0 0 1 λ As we have already discussed, the ranges of validity of Ð ----κ-- 2∫dxV(Rx) Ð V(r1) Ð V(r2) (25) ρ Ӷ 2 formulas (12) and (23) overlap. In fact for * 0 2 min(a , a ) Ӷ κρ we can expand V(R )in terms of q 1 x 1 2 * x 3 as far as quadratic terms and integrate over dq. Then ir [ ( ) ] + --κ----∫dx∫dy x(1 Ð x)y(1 Ð y) Ð 1 Ð x y formula (23) yields (12) to within terms O(1/κρ ) * 0 0

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 70 LEE et al. θ π θ  If the angle ' = Ð between the vectors r1 and r2 × (∇ )(∇ ) is small, we find 1⊥V(Rx) 1⊥V(Ry) .  (0) , ε 1 { κ D (r1 r2 ) = Ð----π------exp i r1 Ð r2 An advantage of the representations (24) and (25) is 4 r1 Ð r2 (27) that they retain their form in any coordinate system. λ (Φ Φ ) } + i sign(r1 Ð r2) (r1) Ð (r2) . Integration over the variable q in formulas (24) and For the following transformations it is convenient to (25) may be interpreted as allowing for quantum fluctu- rewrite formula (26) in a different form, using the iden- ations near the rectilinear trajectory between the vec- tity tors r1 and r2. The main contribution to the integral over q is made by the region q ~ 1. Quantum fluctuations 2 1 2 dllJ (lθ)g(l ) = ------dqexp(iq ⋅ q)g(q ), may be neglected if ∫ 0 2π∫ 1 where g(x) is an arbitrary function and q is a two- ⁄ κ ∇ ∫dx x(1 Ð x)r 1⊥V(r1 + xr) dimensional vector. We substitute this relation into (26) and change the variables 0 1 κ κ 2 r1r2 r1r2 Ӷ dx V(r + xr) , q ------q Ð ------q. ∫ 1 r1 + r2 r1 + r2 0 We define the impact parameter r by which is in fact equivalent to the first condition in for- r × [r × r ] mula (8) which ensures the validity of the eikonal r = ------1------2---, approximation. r3 In formulas (24) and (25), the terms in the preexpo- θ Ӷ where r = r2 Ð r1. Bearing in mind that 1, the nential factor containing the potential determine the ≈ ρ Ӷ correction to the quasiclassical Green function. The impact parameter is r qr1r2/(r1 + r2) and r we condition for the validity of these expressions is that can reduce formula (26) to the form these corrections are small. Our results were obtained iκr (0) , ε ie for the case of a localized potential. In fact, when these D (r2 r1 ) = ------2---- corrections are small, they can also be used for poten- 4π r (28) tials which are not localized, for example, for a super- 1 position of localized potentials. × 2 λ ( ⁄ κ ) ∫dqexp iq Ð i r∫dxV r1 + xr + q 2r1r2 r . For a spherically symmetric potential the quasiclassical Green function without corrections can also be 0 obtained from the results of [6, 7] in which this function Here q is a two-dimensional vector lying in the plane was calculated for the case of almost collinear vectors perpendicular to the vector r. This formula was θ r1 and r2. In these studies quasiclassical radial wave obtained for small angles . However, it is also valid for functions were used to obtain the following expression θ ~ 1 since in this case we can neglect the term propor- (0) |ε θ for the function D (r2, r1 ) for a small angle tional to q in the argument of the potential and the between the vectors r2 and Ðr1: Green function (28) becomes the eikonal function. In particular, for θ' Ӷ 1 it is the same as (27). If the correc- κ ∞ i (r1 + r2) 2 tion is omitted in formula (24), it agrees with (28) since (0) ie  l (r + r ) , ε θ  1 2 D (r2 r1 ) = ----π----κ------∫dllJ0(l )exp i ------κ------in situations where the term containing q needs to be 4 r1r2  2 r1r2 retained in the argument of the potential, we can 0 (26) ⁄ 2  replace x(1 Ð x) with r1r2 r . λδ l  λ(Φ Φ ) + 2 -- + (r1) + (r2) . κ  3. WAVE FUNCTIONS In this formula we have IN THE QUASICLASSICAL APPROXIMATION ∞ The expressions obtained for the quasiclassical Φ ζ ζ Green function can be used to find the quasiclassical (r) = ∫V( )d , wave functions taking into account the corrections. In r earlier studies [8, 9] the quasiclassical wave functions ∞ were obtained for a Coulomb field (FurryÐSommer- feldÐMaue functions) and in [10] they were obtained δ ρ ζ2 ρ2 ζ λ ε ⁄ κ ( ) = Ð∫V( + )d , = . for an arbitrary decreasing central field. These wave 0 functions were calculated in the principal approxima-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 QUASICLASSICAL GREEN FUNCTION IN AN EXTERNAL FIELD 71 tion. In [11] the wave functions and corrections to them 4. SCATTERING OF CHARGED PARTICLES were determined for an arbitrary potential in the eikonal approximation. In this section we use the quasiclassical Green func- In order to calculate the wave functions we use the tion to calculate the amplitude of small-angle scattering well-known relations (see, for example [12]) of high-energy charged particles in a localized potential. For a spin zero particle the scattering amplitude has [ κ ] (0) , ε exp i r2 ψ(Ð)* the form lim D (r2 r1 ) = Ð------p (r1), → ∞ π 2 r2 4 r2 (29) κ ( ) ⋅ [ κ ] Ð * ip1 r (0) exp i r1 (+) f (p , p ) = Ð------drψ (r)φ(r)e . (32) , ε ψ 2 1 ∫ p2 lim D (r2 r1 ) = Ð------p (r2). 2π → ∞ π 1 r1 4 r1 φ λ 2 κ κ κ ψ(+) (ψ(Ð) ) We note that = V Ð V /2 . Here p1 = Ð r1/r1, p2 = Ð r2/r2, and p (r) p (r) denotes the solution of the KleinÐGordon equation con- For convenience, we position the origin at the point taining at inﬁnity a plane wave having the momentum of maximum potential and convert to cylindrical coor- p and a diverging (converging) spherical wave. From dinates with the z axis parallel to the vector p2. Substi- (24) and (29) we obtain the quasiclassical wave func- tuting (30) into (32) we obtain, allowing for the ﬁrst tions for the KleinÐGordon equation allowing for the correction, correction:

∞ ∞ iκ ψ( ±) ± dq ⋅ ± 2 +− λ f = ------∫ dz∫dr∫dq p (r) = ∫---π-- exp ip r iq i ∫dxV(rx) π2 i 2 ∞ 0 Ð  λ ∞ × 1 + ------V(r) (30) × ⋅ 2 λ ,  2κ exp Ð iQzz Ð iQ⊥ r + iq Ð i ∫dxV(x + z rx) 0 ∞ x (33) i  ∞ x ± [ ](∇ ) ∇  i κ-- ∫dx∫dy xy Ð y ⊥V(rx) ( ⊥V(ry)) , × λ , [ ]  V(z r) 1 Ð κ-- ∫dx∫dy xy Ð y 0 0 0 0 − ⁄ κ ⁄ κ κ rx = r + px + q 2x , = p . × (∇ , ) ∇ , Here q is the two-dimensional vector perpendicular to ρV(x + z rx) ( ρV(y + z ry)) , p and ∇⊥ is the component of the gradient perpendicular to p. Similarly for the Dirac equation the quasiclassical wave functions with the correction are obtained ⁄ κ from (25): where rx = r + 2x q , Q = p2 Ð p1. ∞ We first demonstrate the need to use quasiclassical Ψ( ±) ± dq ⋅ ± 2 − λ rather than eikonal wave functions in the scattering p (r) = ∫----- exp ip r iq + i ∫dxV(rx) iπ problem. To do this, we calculate the amplitude in the 0 principal approximation which corresponds to replac- ∞  λ ing the brackets in the pre-exponential factor in formula × − 1 ⋅ ∇ 1 + ------∫dxa V(rx) + ------V(r) (31) (33) by one. We divide the region of integration over z  2κ 2κ ∞ ∞ 0 into two: (Ð , 0) and (0, ). In the region between zero and infinity we can replace r by r (neglecting quan- ∞ x x  tum fluctuations) and integrate over q. The contribution ± i ( )(∇ ) ∇ λ -- ∫dx∫dy xy Ð y ⊥V(rx) ( ⊥V(ry)) up, of this region then has the form κ  0 0 ∞ λ λ λ κ where up are free positive- ( = 1) and negative- ( = Ð1) f = Ð------dz dr + 2π ∫ ∫ frequency Dirac spinors having the momentum p. Ð∞ When V(rx) can be expanded in terms of q, formula ∞ (34) (31) yields the wave function in the eikonal approxima- × ⋅ λ , λ , tion with corrections and agrees with the result exp Ð iQzz Ð iQ⊥ r Ð i ∫dxV(x + z r) V(z r). obtained in [11]. 0

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Now integrating by parts over z using the relation Adding f+ and fÐ we obtain the expression familiar to us as ∞ the scattering amplitude in the eikonal approximation: λ , λ , iκ V(z r)exp Ði ∫dxV(x + z r) f = Ð------drexp[ÐiQ⊥ ⋅ r ] 2π∫ 0 (35) ∞ ∞   (39) ∂   λ , × exp Ðiλ ∫ dxV(x, r) Ð 1 . = Ð i∂ exp Ði ∫dxV(x + z r) Ð 1,   z  Ð∞ 0 This result is usually obtained using the eikonal wave we obtain function over the entire range of integration over z and ∞ neglecting the term Qzz in the argument of the exponen- κ  tial function. It follows from our calculations that, for i [ ⋅ ] λ , f + = Ð------drexp ÐiQ⊥ r exp Ði dxV(x r) arbitrary transfers of momentum, both these approxi- 2π∫  ∫ 0 mations are generally inaccurate. To illustrate this we ∞ ∞ consider the screened Coulomb potential with the    screening radius r . The main contribution to the ampli- [ ]  λ ,  c Ð 1 Ð iQz∫dzexp ÐiQzz exp Ði ∫dxV(x r) Ð 1 . ρ    tude is made by the region of integration z ~ rc, ~ 1/Q⊥. 0 0 2 If the value of Q z ~ Q⊥ r /κ in this region is small com- (36) z c pared with unity, the term containing q must be

The main contribution to the integral over z in this for- neglected in r since ρ ~ 1/Q⊥ ≤ r ⁄ κ . Thus, it is not ρ x c mula is made by the region z ~ . The relative value of the permissible to use the eikonal wave function. At the contribution proportional to this integral compared with ρ Ӷ same time, the quantity Qzz in the argument of the the term outside the integral is Qz ~ Qz/Q⊥ 1 so that in exponential function cannot be neglected. If this term is the principal approximation this contribution can be retained but the eikonal wave function is used, under neglected. In the region between Ð∞ and zero in the 2 | | the condition Q⊥ r /κ ≥ 1 the result for the scattering principal approximation x may be replaced by z in rx. c ⁄ κ amplitude is incorrect. In particular, if this procedure is We then make the change r r Ð q 2 z and adopted, the well-known result (Rutherford formula) is obtain ∞ not obtained in the limit rc (unscreened Cou- 0 lomb potential). Hence, formula (39) is valid for any iκ Q⊥ Ӷ κ, but it cannot be derived correctly without ------f Ð = 2 ∫ dz∫dr∫dq using the quasiclassical wave function. 2π Ð∞ We shall now calculate the scattering amplitude

2 2 allowing for the correction. For convenience integra-  Q⊥  z  tion over z will be performed between ÐL and +∞ and × exp Ð i Q Ð ------z Ð iQ⊥ ⋅ r + i q + Q⊥ ------(37)  z 2κ  2κ we then make L tend to infinity. Using the identity ∞ ∞ ∂ λ , λ dx ⋅ ∇ ,  2 z  V(z r) + i + ∫------q ρV(z + x rx) Ð iλ dxV(x + z, r) λV z, r Ð q ------. ∂z 2κx ∫  κ  0 0 ∞ (40) × λ , We note that for κρ ӷ 1 the condition 2 z ⁄ κ Ӷ exp Ði ∫dxV(x + z rx) = 0, max(|z|, ρ)is satisfied for any z which means that the 0 term proportional to q can be neglected in the argument we can integrate by parts over z in the first term in of the potential. In the small-angle approximation we brackets in formula (33). As a result, the expression for 2 ⁄ ( κ) f has the form have Qz = Q⊥ 2 , i.e., the coefficient of z in the argument of the exponential function becomes zero. f = f0 + f1, Calculating the integral over q and then integrating where over z using the identity (35), we find 2 κ Q⊥ κ ⋅ i f 0 = Ð------lim ∫dr∫dqexp i------L Ð iQ⊥ r f = Ð------drexp[ÐiQ⊥ ⋅ r ] π2 L → ∞ 2κ Ð 2π∫ 2 (41)  ∞ ∞  (38) ∞ × exp Ðiλ dxV(x, r) Ð exp Ðiλ dxV(x, r)  . + iq2 Ð iλ dxV(x, r ) ,  ∫ ∫  ∫ x + L Ð∞ 0 ÐL

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∞ potential. Then the expression for f is transformed to iκ 1 f = ------dz dr dq give 1 π2 ∫ ∫ ∫ 2 ∞ ∞ Ð dq f = dz dr ----- ∞ 1 ∫ ∫ ∫ π 2 i Q⊥ 2 Ð∞ (44) × exp Ði------z Ð iQ⊥ ⋅ r + iq Ð iλ dxV(x + z, r ) 2κ ∫ x 2 Q⊥ 2 0 × exp i------z Ð iQ⊥ ⋅ r + iq g(z, r + 2 z ⁄ κq) 2κ ∞ x iλ with a certain function q. After making the transposi- × ----V(z, r) dx dy( xy Ð y) (42) κ ∫ ∫ tion r r Ð q 2 z ⁄ κ , the integral over q becomes 0 0 elementary. As a result we obtain × (∇ , )(∇ , ) ∞ ρV(x + z rx) ρV(y + z ry) [ ⋅ ] , f 1 = ∫ dz∫drexp ÐiQ⊥ r g(z r). (45) ∞ 2 Ð∞ Q⊥ λ dx ( ⋅ ∇ , ρ ) + ------Ð ∫------q ρV(z + x x) . 2κ 2κx Following this procedure we obtain the following 0 expression for f1: ∞ ∞ In the expression for f0, the term which does not depend 1 ≠ f = Ð------dr dzexp Ð iQ⊥ ⋅ r Ð iλ dxV(x + z, r) on the potential and vanishes for Q⊥ 0 is omitted. We 1 2π∫ ∫ ∫ taken this into account explicitly in the final expression Ð∞ 0 for the amplitude. In order to find the limit in the func- ∞ x ⁄ κ × λ , tion f0, we make the transposition r r Ð q 2L i V(z r)∫dx∫dy( xy Ð y) (46) 0 0 and q q Ð Q⊥ L ⁄ (2κ) . Thereafter, calculation of × (∇ , )(∇ , ) the limit L ∞ and integration over q become ele- ρV(x + z r) ρV(y + z r) mentary. Taking into account the first correction with ∞ ∞ respect to Q⊥/κ we obtain 1 +-- dx dy(1 Ð x ⁄ y)(∇ρV(x + z, r))(∇ρV(y + z, r)) . 2∫ ∫ ∞ 0 0 iκ ⋅ λ , When deriving this formula we integrated by parts over f 0 = Ð----π--∫drexp Ð iQ⊥ r Ð i ∫ dxV(x r) 2 2 Ð∞ r the term proportional to Q⊥ in the preexponential ∞ (43) λ factor. We can check that the expression in the inte- × i ⋅ ∇ , grand of (46) is a total derivative with respect to z so 1 + ------∫ dxxQ⊥ ρV(x r) . 2κ that integration over z is trivial. Combining these Ð∞ expressions with (43) and integrating by parts over r We now calculate f . The integral over r in (42) con- the term proportional to Q⊥, we finally obtain the scat- 1 tering amplitude allowing for the first correction: verges in the region ρ ~ 1/Q⊥. Quantum fluctuations are ∞ only important for z < 0 and x near Ðz. Since f1 is a cor- iκ  [ ⋅ ] λ , rection, in formula (42) we can replace with | |. In f = Ð------∫drexp ÐiQ⊥ r exp Ði ∫ dxV(x r) rx r z 2π  addition V(z, r) in the pre-exponential factor may be Ð∞ replaced by V(z, r| |) since the difference between them ∞ ∞ z λ is small for any z. The integral over z between zero and λ , ∆ , Ð 1 + exp Ði ∫ dxV(x r) ----κ-- ∫ dxx ρV(x r) (47) infinity converges in the range z ≤ ρ. We then have 2 Ð∞ Ð∞

2 2 ∞ x zQ⊥ /2κ ~ ρQ⊥ /κ ~ Q⊥/κ Ӷ 1.  i (∇ , )(∇ , ) Ð -- ∫ dx ∫ dyy ρV(x r) ρV(y r) . κ  Hence, in the exponential function we can replace Ð∞ Ð∞ 2 2 κ | | κ 2 zQ⊥ /2 with Ð z Q⊥ /2 . We integrate the term propor- Here we have ∆ρ = ∇ρ . Using the wave function (31), tional to q by parts over q so that in the preexponential we can show (see, for example [13]) that for small- factor this vector only appears in the argument of the angle scattering the amplitude for spin 1/2 particles is

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 74 LEE et al. the same as the amplitude (47) for spin zero particles to light in the medium we also need to know the real part within terms Q⊥/κ inclusive. of the scattering amplitude. We now show that this amplitude has regular prop- In order to calculate the amplitudes of Delbrück erties relative to shifts of the origin. It follows from scattering using Green functions, we need to use vari- formula (33) that when the potential is transposed ous approximations for these functions for various ∆ V(r) V(r + r ), the amplitude f(p , p ) becomes transfers of momentum = k2 Ð k1 (k1 and k2 are the 0 2 1 ∆ ω f(p2, p1)exp[iQ á r0]. The amplitude (47) clearly has momenta of the initial and final photons). For ~ the this property for a shift perpendicular to p2 along which quasiclassical approximation cannot be used since the the z axis is directed. We now analyze the shift along main contribution to the amplitude is made by the ω ∆ the z axis: V(z, r) V(z + z0, r). Transposing the orbital moments l ~ / ~ 1. The amplitude for these variables x x Ð z0 and y y Ð z0 in the integrals transfers of momentum was calculated in [15]. In the over x and y, we obtain a correction to the amplitude region ω ӷ ∆ ӷ m2/ω we can use the quasiclassical proportional to z0: Green function without any correction [1, 16]. Calcula- tion of the Delbrück scattering amplitude in a Coulomb ∞ field for ∆ ≤ m2/ω necessitated a special analysis (see δ iz0 ⋅ λ , f = ----π--∫drexp Ð iQ⊥ r Ð i ∫ dxV(x r) [2, 17]). In this case a contribution to the amplitude is 4 2 Ð∞ made by the impact parameters ρ as far as ρ ~ ω/m . For these impact parameters we need to allow for a correc- ∞  ∞  2 tion to the quasiclassical Green function [2]. For a 2 × λ dx∆ρV(x, r) Ð i dx∇ρV(x, r) screened Coulomb potential where ω/m ӷ r (r is the ∫  ∫  c c Ð∞ Ð∞ (48) screening radius, in the ThomasÐFermi model rc ~ αÐ1 Ð1/3 ӷ z (m ) Z 1/m) a contribution to the amplitude is 0 [ ⋅ ] ρ ≤ Ӷ ω 2 = Ð----π--∫drexp ÐiQ⊥ r made by the impact parameters rc /m . In this 4 case we can use the quasiclassical Green function with- ∞ ∆ Ӷ ω ∆ Ð1 ≤   out any correction for arbitrary [6, 7]. If , rc × ∆ρexp Ðiλ dxV(x, r) Ð 1 . m2/ω, the correction must be taken into account. More-  ∫  Ð∞ over the expression for the forward scattering amplitude (∆ = 0) obtained using the Green quasiclassical 2 Integrating by parts over r and replacing Q⊥ /2κ with function without any correction is indeterminate. Sub- δ ≈ sequently we obtain the Delbrück scattering amplitude Qz, we find that f + f f exp(iQzz0) provided that ω 2 Ӷ for an arbitrary relation between rc and /m . Qzz0 1. Thus, the regular law for transformation of the scattering amplitude in translation is satisfied with It was shown in [6] that the Delbrück scattering the same accuracy as formula (47). amplitude at high photon energies may be expressed in the form For a potential satisfying the condition V(z, r) = V(Ðz, r), expression (47) agrees with that obtained in ω [11] in the eikonal approximation allowing for the cor- α ε [ ⋅ ( )] M = i ∫d ∫dr1dr2 exp ik r1 Ð r2 rection. However, as we have already discussed, formula (47) may also be valid for a relationship between 0 (49) × [( ∗ ⋅ ∗ˆ ) , ω ε ] momentum transfers and potential parameters for Sp 2e p2 Ð eˆ k D(r2 r1 Ð ) which the eikonal approximation cannot be applied. × [( ⋅ ˆ ) , ε ] 2e p1 + eˆk D(r1 r2 Ð ) , 5. DELBRÜCK SCATTERING where e and k are the photon polarization and four- ∇ momentum, p1, 2 = Ði 1, 2. In this formula it is implied The coherent scattering of photons in the electric that its value in a zero external field is subtracted from field of atoms via virtual electronÐpositron pairs (Del- the expression in the integrand. Since in a central field brück scattering) has been studied intensively both the- the scattering amplitude does not depend on the photon oretically and experimentally (see, for example, the polarization, in formula (49) it is convenient to make review [14]). In this section we analyze the amplitude δ ω2 of Delbrück scattering in a screened Coulomb field as the substitution ei*e j ( ij Ð kikj/ )/2. another example of using this quasiclassical Green In (49) we convert from the variables r to function. The photon energy ω is assumed to be large 1, 2 × [ × ] ⋅ compared with the electron mass m. In accordance with r r1 r2 r r1 r = r2 Ð r1, r = ------, z = Ð------. the optical theorem, the imaginary part of this ampli- r3 r2 tude is expressed in terms of the cross section for creation of electronÐpositron pairs by a photon in an Since r á r = 0 integration over r is performed in the atomic field. In order to describe the propagation of plane perpendicular to the vector r. The main contribu-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 QUASICLASSICAL GREEN FUNCTION IN AN EXTERNAL FIELD 75 tion to the amplitude is made by the region of integra- 2 × εκ[ ⋅ 2] tion in which r ~ ω/m , |z| ~ 1 and the angles between 2 q1 q2 Ð z(1 Ð z)qr θ ω Ӷ the vector r and k are of the order r ~ m/ 1. As a result of the smallness of the angles θ we can assume r ω ω that the vector r is perpendicular to k. Quite clearly the + ------(εq Ð κq ) ⋅ (q Ð q ) Ð i--- , ρ ≤ 2 ω 1 2 1 2 main contribution is made by 1, m rc/ . 4(z(1 Ð z)) r ρ We divide the region of integration over into two: κ ω ε ρ ρ ∞ ω Ӷ where = Ð , and qr is a two-dimensional vector in between 0 and 0 and between 0 and , where m/ κ ρ Ӷ 2 ω ρ ρ the plane perpendicular to (like the vectors q1, 2). We 0 1, m rc / . In the first region (where < 0) we can use the following representation for the Green function draw attention to the fact that the integral over z in this κ formula is taken between zero and one. This is because iei r a ⋅ q κr outside this interval the Green functions are eikonal and D(r , r ε) = ------dq 1 + ------2 1 2 ∫ ε the potential-dependent phases are reduced. For ρ < 4π r 2r1r2 ρ Ӷ 1 this leads to a smaller contribution of the region (50) 0 1 of integration outside the interval 0 ≤ z ≤ 1 compared × 2 λ ⁄ κ exp iq Ð i r∫dxV(r1 + xr + q 2r1r2 r) , with that made by this interval. 0 We now integrate over qr, convert from the variables which is obtained by substituting (28) into (2). In this q1, 2 to Q = (q1 + q2)/2 and q = (q1 Ð q2)/2, and make the case in the term containing a we omitted the longitudi- transposition r r + Q. Then the integral over r has nal components of the gradient which have additional ρ the form smallness compared with the transverse ones, and we 2iZα integrated by parts over q. Moreover, because of the  r Ð q  ρ J = ∫ dr Re ------Ð 1 . (52) definition of 0 we can neglect screening in this region  r + q  < ρ and replace V(r) by the Coulomb potential Vc(r) = r + Q 0 − α Z /r after which the integral over x is easily taken in The main contribution to the amplitude (51) is made by phase. ω Ӷ ρ Q, q ~ m/ 0 so that in (52) we can omit Q in the Screening is only significant in the second region limits of integration. We use the following procedure to where the representation (13) can be used, i.e., the calculate the integral (52). We subtract and add to the eikonal Green function with a correction. By substitut- expression in the integrand the function Ð2(Zα)2[2r á ing (13) into (49), the phases of the Green functions ρ2 2 2 which depend on the potential are reduced. Hence the q/( + q )] which can easily be integrated: contribution of the second region is quadratic in terms 2 2 2r ⋅ q  of the potential, i.e., it corresponds to the first Born J = dr Ð2(Zα) ------approximation. In addition, the reduction of the phases 1 ∫ ρ2 2 ρ < ρ + q means that it is necessary to allow for the correction to 0 (53) the Green function. In this region the contributions to  ρ 2  the amplitude from the correction and the principal = Ð4π(Zα)2q2ln----0- Ð 1 . term in the Green function are of the same order.  q2  We shall now calculate the contribution M1 of the ρ first region to the amplitude of the process. We substi- Then in the integral of the difference, we can replace 0 tute into (49) the Green function (50) for the case of a by infinity. To calculate this integral it is convenient to Coulomb potential, differentiate, and calculate the trace multiply the expression in the integrand by of the γ-matrices. Using the smallness of the angle 1 between the vectors r and k and the relationship ⋅ ≡ δ 2r q  2 2 2 1 ∫ dy y Ð ------ ε Ð m ≈ |ε| Ð m /2|ε| we obtain ρ2 + q2 Ð1 ω (54) 1 1 iα 5 dz M = Ð------dεεκ drr dq dr ------(ρ2 2) dyδ ( ⁄ )2 2 ⁄ 2 1 ( π)4∫ ∫ r ∫ ∫( )3 = + q ∫ ----- ( r Ð q y Ð q (1 y Ð 1)) 2 ρ < ρ z(1 Ð z) y 0 0 0 Ð1

2iZα r Ð q and change the order of integration over r and y. As a ×  1  result, the integral over r becomes elementary and we ∫dq1dq2 Re------ Ð 1 r Ð q2 obtain

1 iZα 2 2 2  ω ε κ  2 dy 1 Ð y 2 2 × r ω 2 m q1 + q2 π ( α) expi-- qr Ð ------+ ------ (51) J2 = 4 q ∫----- Re------ Ð 1 + 2 Z y . (55)  2 εκ z(1 Ð z)  y3 1 + y 0

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1 After replacing y = tanhτ , the integral over τ becomes 2 4 2 ω tabular and for J = J + J we obtain × dy(1 Ð y) --(1 Ð y) + 2y Ð ----- (59) 1 2 ∫ 3 εκ 0 J = 8πq2(Zα)2 × [∇ρV(R )] ⋅ [∇ρV(R )]. 2q (56) z z Ð y × ln----- Ð 1 + Reψ(1 + iZα) + C , ρ Ӷ ≈ α 0 For r rc the potential is V(r) ÐZ /r. It is therefore convenient to express the contribution (59) in the form where C = 0.577… is the Euler constant, ψ(x) = c δ c dlnΓ(x)/dx. M2 = M2 + M, where M2 is the value of M2 for V(r) = V (r) = ÐZα/r. After calculating the integral over r in (51), the c ε remaining integrals can be conveniently taken in the For V = Vc the integrals over r and are easily order: over Q, q, r, z, and ε. Finally the contribution of obtained and we find the first region is given by ∞ ∞ 2 2 c 2iα(Zα) 3 28α(Zα) ω M = Ð------dρρ dz M = i------2 2 ∫ ∫ 1 2 m 9m ρ Ð∞ (57) 0 (60) 1 ωρ 2 0 π 47 ( ) ⁄ ⁄ × ln------Ð i-- Ð Reψ(1 + iZα) Ð C Ð ----- . × ( ) 2 1 Ð y 9 + y 3 Ð 1 ∫dy 1 Ð y ------3/2----. m 2 42 2 2 3/2 2 2 0 [z + ρ ] [(z Ð y) + ρ ] The contribution of higher orders of perturbation theory with respect to the external field (Coulomb correc- Using the Feynman parametrization of the denomina- tions) corresponds to the term Reψ(1 + iZα) Ð C in (57) tors and agrees with the well-known result [17]. Thus, the 1 Coulomb corrections are completely determined by the 1 8 v(1 Ð v) ------= -- dv ------, first region in which the quasiclassical Green function 3/2 π∫ 3 (AB) [Av + B(1 Ð v)] is not reduced to the eikonal one. 0 We shall now calculate the contribution M2 of the we take the integrals over ρ and z, and then over y and second region. After differentiating over r1, 2, calculat- v bearing in mind that ρ Ӷ 1. Finally for this (Cou- ing the trace of the γ-matrices, and taking the integral 0 lomb) contribution we obtain over qr, we arrive at the following representation for M2: 2 ρ c 28α(Zα) ω 0 31 ω ∞ M2 = Ði------2------ln----- + ----- . (61) α ω 2 9m 2 21 ε 2 rm M2 = ----π---ω---∫d ∫drr exp Ði------ε---κ---- ∫ dr ∫ dz 2 2 The sum of the contributions (57) and (61) gives the 0 ρ > ρ Ð∞ 0 well-known expression for a Coulomb field [17]: 1 1 ω2 α( α)2ω × dxdy 2y(1 Ð x)[2ϑ(x Ð y) + 1] Ð ------(58) 28 Z ∫∫ εκ Mc = i------2 9m2 0 0 (62) × [∇ ] ⋅ [∇ ] 2ω π 109 ρV(Rz Ð x) ρV(Rz Ð y) , × ln------Ð i-- Ð Reψ(1 + iZα) Ð C Ð ------. m 2 42 2 ρ2 where Rs = r s + . In this formula we omit terms In order to calculate the correction δM caused by the which are antisymmetric relative to the substitution presence of screening, by subtracting from the expres- ε ω Ð ε, z 1 Ð z whose integral is zero. Note sion in the integrand in (59) its value for V = V we can ρ c ρ that unlike the first region, in the second integration replace 0 in the lower limit of integration over by over z is performed within infinite limits. Expression zero. For the following calculations it is convenient to (58) is simplified if in the third integral over x, y, z we use the momentum representation for the potentials: transpose the variables z z + x, y y + x and integrate over x. We obtain dp ip ⋅ r ˜ V(r) = ∫------3e V(p). ω ∞ ( π) 2 2 α 2 ωrm M = ------dε drr exp Ði------dr dz 2 2πω∫ ∫ 2εκ ∫ ∫ Using this representation we take the integrals over r ρ > ρ ∞ 0 0 Ð and z. The result of this integration is proportional to

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 QUASICLASSICAL GREEN FUNCTION IN AN EXTERNAL FIELD 77 δ (p1 Ð p2). Integrating over p2 and the angles of the vec- all the integrals in (65) are taken: tor p1, we have 2 4iα(Zα) ω 2 3 4 ω δM = ------33 Ð 13τ + --τ α ωrm2 9m2 2 δM = ------∫dε∫drrexp Ði------2π3ω 2εκ 0 1 2 4 3 2 6 2 + --τ(24 Ð 13τ + 3τ )L + --(8 Ð 9τ + τ )L , (66) 1 2 8 2 4 2 ω × dy(1 Ð y) --(1 Ð y) + 2y Ð ----- (63) ∫ εκ 2 3 2im τ Ð 1 0 τ = 1 + ---ω----β----, L = lnτ------. ∞ + 1 4 2 2  sinζ cosζ × ∫dp[ p V˜ (p) Ð (4πZα) ] ------Ð ------, More realistic is the Moliére potential [18] for  ζ3 ζ2  which 0 3 where ζ = rpy. Changing from the variable r to ζ, we α ˜ π α ------n------integrate by parts over p and take the integral over ζ. V(p) = Ð4 Z ∑ 2 2, (67) p + β Making the transposition ε ωε, we obtain n = 1 n 1 1 α α α β β αω 1 = 0.1, 2 = 0.55, 3 = 0.35, n = 0bn, δ εε ε 1  M = ------∫d (1 Ð )∫dy-- Ð 1 2π3m2 y mZ1/3 0 0 b = 6, b = 1.2, b = 0.3, β = ------. 1 2 3 0 121 ∞ 4 2 1 4 2 × --(1 Ð y) + 2y Ð ------dp(∂ p V˜ (p)) (64) In this case we have 3 ε(1 Ð ε) ∫ p 1/2 0 2 4iα(Zα) ω dy 2 δM = Ð------(6y + 7y + 7) 1 Ð 2y 1 1 1   1 + η  2 ∫ y × --- + -- 1 Ð ----- ln ------, 9m η 2 η2 1 Ð η Ð i0 0

η ωε ε 2 2 1 + 1 Ð 2y where = 2 (1 Ð )py/m . We then transpose the vari- + 3y(2y Ð 3y Ð 3)ln------ ables y y/(2ε(1 Ð ε)), change the order of integra- 1 Ð 1 Ð 2y tion over ε and y, and take the integral over ε. Finally (68)  γ 2i 2  n + i the contribution to the amplitude of Delbrück scattering ×  α α 1 Ð ------Ð ------ln ------∑ n k γ γ 2 2 γ  associated with the screening is given by  n + k γ γ k + i n ≠ k n Ð k ∞ αω δ (∂ 4 ˜ 2 ) M = Ð------∫dp p p V (p) γ  π3 2 α2 n  18 m + ∑ nγ------, 0 n + i  n 1/2 dy 1 1 1   1 + η  γ ωβ 2 × ------+ -- 1 Ð ----- ln ------where n = ny/m . ∫ η  2  η  y 2 η 1 Ð Ð i0 We know that the imaginary part of the Delbrück 0 (65) scattering amplitude is related to the total cross section × (6y2 + 7y + 7) 1 Ð 2y for photon creation of electronÐpositron pairs in an external ﬁeld by σ = ImM/ω. Using (62) and (68), we obtain the cross section for creation of electronÐ ( 2 ) 1 + 1 Ð 2y positron pairs for the Moliére potential: + 3y 2y Ð 3y Ð 3 ln------ , 1 Ð 1 Ð 2y α( α)2  ω σ 28 Z 2 ψ α 109 η ω 2 = ------ln------Ð Re (1 + iZ ) Ð C Ð ------where = py/m . This formula holds for an arbitrary 9m2  m 42 type of screened Coulomb potential. In particular cases, this can be simpliﬁed substantially. For instance, for the 1/2 potential dy 6 2  Ð ∫ ----- --y + y + 1 1 Ð 2y V(r) = ÐZαexp(Ðβr) ⁄ r, y 7 0 (69) which corresponds to 3 ( 2 ) 1 + 1 Ð 2y 2 2 + --y 2y Ð 3y Ð 3 ln------ V˜ (p) = Ð4πZα ⁄ ( p + β ), 7 1 Ð 1 Ð 2y

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Re M/|MComp| tudes of Compton and Delbrück scattering even at com- 1.2 paratively low energies. For the case of Delbrück scattering we reafﬁrm that 1.0 using the eikonal approximation to describe small- 0.8 angle scattering processes at high energies without due substantiation may lead to incorrect results. For 0.6 instance, if the Delbrück scattering amplitude is calculated using the eikonal Green function, the expression 0.4 for the imaginary part of the amplitude proportional to 0.2 the cross section for creation of electronÐpositron pairs will contain no Coulomb corrections. 0 2 4 6 8 10 ω, GeV REFERENCES

Real part of the Delbrück scattering amplitude as a function 1. A. I. Milstein and V. M. Strakhovenko, Phys. Lett. A 95, 135 (1983). of ω for the potential V(r) = ÐZαexp(ÐmαZ1/3r)/r in units of 4πZα/m, Z = 82. 2. A. I. Milstein and V. M. Strakhovenko, Zh. Éksp. Teor. Fiz. 85, 14 (1983). 3. A. I. Milstein and V. M. Strakhovenko, Phys. Lett. A 90, 1 + γ 2 γ 2  447 (1982). × α α 1  n α2 n ∑ n k 1 Ð ------ln ------+ ∑ n------, 4. R. N. Lee, A. I. Milstein, and V. M. Strakhovenko, γ 2 γ 2  γ 2 γ 2  n ≠ k n Ð k 1 + k n 1 + n Zh. Éksp. Teor. Fiz. 112, 1921 (1997). 5. R. N. Lee, A. I. Milstein, and V. M. Strakhovenko, Phys. which agrees with the well-known expressions in the Rev. A 57, 2325 (1998). literature (see, for example [19]). 6. R. N. Lee and A. I. Milstein, Phys. Lett. A 198, 217 We now discuss the dependence of the real part of (1995). the Delbrück amplitude on the photon energy ω. As 7. R. N. Lee and A. I. Milsteœn, Zh. Éksp. Teor. Fiz. 107, ω 2 Ӷ long as /m rc, screening can be neglected and 1393 (1995). πα α 2ω ⁄ 2 8. W. Furry, Phys. Rev. 46, 391 (1934). ReM = ReMc = 14 (Z ) 9m . 9. A. Sommerfeld and A. Maue, Ann. Phys. 22, 629 (1935). As ω increases, the real part increases more slowly and 10. H. Olsen, L. C. Maximon, and H. Wergeland, Phys. Rev. ω 2 ӷ 106, 27 (1957). for /m rc we obtain from (62) and (65) 11. A. I. Akhiezer, V. F. Boldyshev, and N. F. Shul’ga, Teor. ∞ Mat. Fiz. 23, 11 (1975). α  ω  dp 4 2 ReM ≈ ------ln2 ------(∂ p V˜ (p)). (70) 12. A. I. Baz’, Ya. B. Zel’dovich, and A. M. Perelomov, Scat- π3  2  ∫ p p 2 m rc tering, Reactions and Decay in Nonrelativistic Quantum 0 Mechanics (Israel Program for Scientific Translations, In this asymptotic form we only retained the term con- Jerusalem, 1966; Nauka, Moscow, 1971) taining the leading power of the large logarithm. For 13. V. N. Baœer and V. M. Katkov, Dokl. Akad. Nauk SSSR illustration figure gives the real part of the Delbrück 227, 325 (1976). scattering amplitude as a function of ω for Z = 82 and 14. A. I. Milstein and M. Schumacher, Phys. Rep. 243, 183 the potential (1994). 1/3 15. A. I. Milstein and R. Zh. Shaisultanov, J. Phys. A 21, V(r) = ÐZαexp(ÐmαZ r) ⁄ r. 2941 (1988). One of the main mechanisms for elastic scattering 16. R. N. Lee, A. I. Milsteœn, and V. M. Strakhovenko, Zh. of a photon is Compton scattering at atomic electrons. Éksp. Teor. Fiz. 116, 78 (1999). For forward scattering the amplitude of this process is 17. M. Cheng and T. T. Wu, Phys. Rev. D 2, 2444 (1970). a real quantity and does not depend on ω: 18. G. Z. Moliére, Z. Naturforsch. A 2, 133 (1947). 19. H. Davies, H. A. Bethe, and L. C. Maximon, Phys. Rev. π α ⁄ MComp = Ð4 Z m. 93, 788 (1954). It can be seen from figure that for photon scattering we need to allow for interference between the ampli- Translation was provided by AIP

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000

Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 79Ð92. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 90Ð104. Original Russian Text Copyright © 2000 by Vagin, Silin, Uryupin. PLASMA, GASES

Stimulated Brillouin Scattering in a Plasma with Ion-Acoustic Turbulence K. Yu. Vagin*, V. P. Silin**, and S. A. Uryupin Lebedev Physics Institute, Russian Academy of Sciences, Moscow, 117924 Russia *e-mail: [email protected] **e-mail: [email protected] Received June 18, 1999

Abstract—A theory of stimulated Brillouin scattering (STBS) in a plasma with ion-acoustic turbulence is developed using concepts of parametric instability under conditions when equations of two-temperature hydrodynamics can be used to describe ion-acoustic perturbations of the electron density. The temporal growth rate of the absolute instability and the spatial gain of the scattered wave are determined. The dependence of the threshold density of the radiation ﬂux on the angle between the scattering wave vector and the direction of anisotropy of the turbulent noise is described. A new effect of STBS forbiddenness caused by anomalous turbulent heating of the ions is predicted for a plasma with a high level of turbulent noise. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION characteristics determining the heat transfer process has led to the observation of qualitatively new scatter- The phenomenon of stimulated Brillouin scattering ing laws in a plasma with ion-acoustic turbulence [14]. has formed the subject of numerous experimental (see, In this type of plasma the electron heat transfer behav- for example, [1Ð5]) and theoretical [6Ð9] investiga- ior is determined to a considerable extent by electron tions. In general, the theoretical description of this phe- scattering at ion-acoustic oscillations of the charge den- nomenon is based on a joint analysis of equations for sity. In [14] Ovchinnikov et al. put forward a theory of θ the fields of the incident and scattered waves and an STBS for the case of fairly large scattering angles 0 equation for the perturbation of the electron density when the condition created by the acoustic wave field. These equations 2 2()θ ⁄ Ӷ have been studied in the greatest detail for a laminar 4Zk0ltleisin 0 2 1, (1.1) plasma, where the electron and ion scattering processes are determined by pair collisions of charged particles. is violated, where Z is the degree of ionization of the Two STBS regimes are usually distinguished. In one ions, k0 is the wave number of the fundamental-fre- the electron density perturbation is determined by the quency wave, lt and lei are the electron mean free paths ponderomotive action of a bilinear combination of inci- for scattering at ion-acoustic oscillations of the charge dent and scattered wave fields. In the other the density density and ions. Under conditions when the inequality perturbation is caused by competition between two opposite to (1.1) is satisfied, a correct description of physical phenomena: nonuniform heating of the elec- scattering is obtained using concepts of nonlocal heat trons as a result of electron–ion collisions in the fields transfer in a turbulent plasma. of the incident and scattered waves, and electron heat However, scattering at small angles when condition transfer. In this case we talk of the thermal mechanism (1.1) is satisfied is also of interest. A theory of small- for STBS. The perturbation of the electron density then angle STBS is put forward in the present study. This depends strongly on the particular characteristics of theory differs substantially from that put forward in heat transfer in a particular plasma state. The theory of [14], first because of the possibility of using a detailed thermal STBS has been developed in the greatest detail theory of local transport in a turbulent plasma and sec- under conditions when the electron mean free paths are ond, because the change in the turbulence noise as a so short compared with the wavelength of the ion- result of long-wavelength ion-acoustic perturbations of acoustic waves that classical heat transfer theory can be the temperatures and densities of the plasma particles used to describe the heat transfer [10]. Comparatively can be taken into account self-consistently. This last recently, STBS theory has been developed under condi- factor in particular not only allowed us to give a sys- tions of nonlocal electron heat transfer in a laminar tematic description of STBS as a parametric instability plasma [11Ð13]. The theoretically established strong but also revealed conditions for its forbiddenness as a dependence of thermal STBS on the electron scattering result of anomalously effective turbulent ion heating.

1063-7761/00/9001-0079 $20.00 © 2000 MAIK “Nauka/Interperiodica” 80 VAGIN et al. The frequency difference ∆ω as a result of scattering is The spatial gain of the scattered wave is determined. In ω determined by the acoustic frequency s, Section 5 we give estimates to demonstrate the possi- θ bility of observing small-angle STBS experimentally. ∆ω = ω = 2k v sin----0, (1.2) s 0 s 2 2. HYDRODYNAMIC TRANSPORT EQUATIONS where vs is the velocity of sound. Since the condition IN A TURBULENT PLASMA 2 4Zk0ltlei > 1 is usually satisfied, the equation (1.2) and We consider a nonisothermal plasma with ion- the inequality (1.1) yield the constraint on the relative acoustic turbulence in an rf field frequency difference: 1 --E(r, t)exp(Ðiω t) + c.c., (2.1) ∆ω ν 2 0 Ӷ m ei lei Ӷ -ω------ω------1 , (1.3) 0 mi 0 lt where the function E(r, t) varies weakly with time over π ω ω the period of the field 2 / 0. We shall assume that the where 0 is the pump wave frequency, m and mi are the electron and ion masses, respectively, and ν is the fundamental frequency is much higher than the plasma ei frequency of the electrons ω . This means that we can electronÐion collision frequency. The inequality (1.3) Le implies that in order to observe small-angle scattering neglect the influence of low-frequency turbulent fluctuations of the charge density on the rapidly varying experimentally we need to use lasers having a fairly ω narrow emission line width. At the same time, by virtue motion of the electrons of frequency 0 (see, for exam- Ӷ ple, [15]). At the same time, the slow macroscopic of the inequality lt lei, the conditions (1.1) and (1.3) are established more easily in a turbulent plasma com- motion of the electrons and ions is determined to a con- pared with similar conditions in a laminar plasma. siderable degree by the ion-acoustic turbulence, and to Hence, the constraint on the range of scattering angles describe this we use a system of hydrodynamic equa- in a turbulent plasma considered subsequently is less tions for the electron density n, the plasma velocity u, stringent than that in a laminar plasma. and the temperatures of the electrons T and the ions Ti [16Ð18]: In Section 2 we give the necessary information on the hydrodynamic equations for a turbulent plasma in ∂n ----- + div(nu) = 0, (2.2) an rf electromagnetic field. In Section 3 we use these ∂t equations as the basis to describe the damping of ion- acoustic oscillations having wavelengths much greater  ∂ ∂  Z ∂ Zη  1 ∂  + u u = Ð ------p + ------∆u + -- divu than the characteristic scales of the turbulent fluctua- ∂t ∂r k nm ∂r nm  k 3∂r  tions of the charge density. For a plasma without an rf i k i k (2.3) field we specify the conditions under which, as a result 2 Ze  ∂ 2 ∂  of the relatively strong ion damping, acoustic perturba- Ð ------n E Ð [n(E E* + E * E )]  , ω2 ∂r ∂r k j k j tions of the hydrodynamic quantities do not result in the 4nmmi 0 k j  buildup of instability. The STBS effect is considered in Section 4. A description of the scattering in the limit  2   ∂ ∂  e 2 (1.1) is obtained using the field equations supple- + u T + ------E  ∂t ∂r ω2 mented with a system of hydrodynamic equations to  6kBm 0  find the perturbations of the electron density in a plasma (2.4) 2 with ion-acoustic turbulence. We discuss the conditions 2 2 e 2 + ------divq--Tdivu = ν ------E Ð ν T, under which a quasi-steady-state turbulence spectrum 3nk 3 ei ω2 T and electron fluxes can be established within the charac- B 3kBm 0 teristic time of variation of the ion-acoustic density per- ∂ ∂ turbations. A systematic description of STBS is given   2 ν  + u  Ti + --Tidivu = Z T T. (2.5) under these conditions when the buildup of hydrody- ∂r ∂r 3 namic instability of the turbulent plasma is suppressed. In these equations e is the electron charge, η = We determine the dependence of the radiation flux den- ν nkBTi/Z ii is the coefficient of viscosity, kB is the Boltz- sity corresponding to the STBS threshold on the angle ν θ between the scattering wave vector and the anisot- mann constant, ii is the ionÐion collision frequency, ropy axis of the turbulent noise. We describe the depen- p = nkBT + nkBTi/Z is the pressure, dence of the STBS threshold on the degree of noniso- π Λ 4 thermicity of the plasma and the turbulent noise distri- ν 4 2 Z e n ei = ------(2.6) bution in wave vector space, and we show that in a 3 m(k T)3/2 plasma with a high level of turbulent noise, there are B directions in which STBS is forbidden. This qualita- is the electronÐion collision frequency, Λ is the Cou- ν tively new effect of STBS forbiddenness is caused by lomb logarithm, T is the turbulent temperature relax- anomalously effective turbulent heating of the ions. ation frequency, and q is the electron thermal flux.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 STIMULATED BRILLOUIN SCATTERING IN A PLASMA 81 Ӷ Equations (2.2)Ð(2.5) were written assuming that the where for R RNL we have aT = 1, a* = [1 + electric current density j is zero and the electroneutral- π β β Ð1 ӷ ity condition n = Zn is satisfied, where n is the ion den- (3 /32 ||)(1 Ð ||)] = 0.42, and for R RNL we have i i ν sity. The condition j = 0 is frequently used to analyze a* = aT = 1.3. Note that in the last expression for T in Ӷ the hydrodynamics of a laser plasma. When a quasi- (2.13) in the limit R RNL only the leading terms with steady state of ion-acoustic turbulence is established as respect to the small parameter (2.11) are retained. a result of competition between Cerenkov radiation Allowing for the relations (2.6)Ð(2.13), the system of from waves at electrons and induced scattering of equations (2.2)Ð(2.5) for a plasma in a given pump field waves by thermal ions, the explicit form of the heat flux (2.1) is closed and can be used to study the hydrody- q in (2.4) and the turbulent frequency ν in (2.4), (2.5) T namic motion of a current-free plasma. depend on the relationship between the density of the turbulence-exciting force R (A.1) and the quantity [19, 20] The order-of-magnitude turbulent frequency of tem- 2 perature relaxation (2.13) can be written in the follow- 1 r R = ------mnv ω ---D----e, (2.7) ing form: NL 6π s Li 2 rDi ω ν ∼ Zmν R where vs = LirDe is the velocity of sound, rDe(i) is the T ------t 1 + ------, ω mi RNL Debye length of the electrons (ions), and Li is the ion Langmuir frequency. Using the condition for the where ν is the turbulent frequency of relaxation of the absence of current j = 0 and the relations (A.2), (A.4) t we eliminate from R (A.1) the unknown quasi-steady- electron momentum: state electric field. In this case, the density of the force R and the electron temperature gradient ∇T are 9π R ν = ------, R Ӷ R , (2.14) directed along the unit vector n = ∇T/|∇T|. From (A.5) t 8 nmv NL ӷ s for R RNL we then find π 3 ∇ ν 9 RRNL ӷ R = -- p lnT , (2.8) t = ------, R R NL. (2.15) 2 8 nmv s

128 R Under anomalous transport conditions we have ν ӷ ν . q = Ð------β|| pv ------n ≡ Ðnq, (2.9) t ei π s ν 3 RNL Hence the frequency T appreciably exceeds the tem- ν β Ӷ perature relaxation frequency in a laminar plasma ε ~ where || = 0.25. If R RNL, the thermal ﬂux (A.3) has ν the form (m/mi) ei. The inequality

ν ӷ ν q = Ðpv s T ε (2.16)  α  (2.10) × 25 β α  128β α ≡ means that we can neglect the contribution of electronÐ -----1 Ð ||( ) + ----- + ------||( )n Ðnq,  4 12 3π  ion collisions to the heat exchange between electrons and ions in equations (2.4) and (2.5). β α β β α where ||( ) = || + Cβα, || = 0.177, Cβ = 0.062, and In order to describe STBS in a turbulent plasma as a is a small parameter parametric instability we shall now determine the ln2 ground state. We shall assume that in the ground state a α Ӷ = ------(------π------⁄------)- 1. (2.11) plasma in an rf ﬁ eld (2.1) is characterized by the elec- ln 3 RNL 8R tron density n0 and also by the electron temperature T0 The force density R in expressions (2.10) and (2.11) is and the ion temperature Ti0. Let us assume that LT is the related to the modulus of the electron temperature gra- characteristic scale of nonuniformity of the electron dient by temperature in the ground state, as given by ∇ π α p lnT 2  β α  ------= -- + ------1 Ð ||( ) + ----- . (2.12) 1 ∇ R 3 16β||(α) 12 ----- = lnT0 . (2.17) LT In these limiting cases for the turbulent frequency of temperature relaxation taking into account (2.8) and We shall use the system of equations (2.2)Ð(2.5) to (2.12) we have describe the interaction between rf radiation and a nonisothermal plasma whose hydrodynamic motion 2 R ν = --a v --- = a v ∇lnT , (2.13) can be neglected, u = 0. The plasma density can then be T 3 T s p * s assumed to be independent of time and the evolution of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 82 VAGIN et al. the electron and ion temperatures in this ground state is ∂ ∂ δ 2 δ 2T δ described by the equations ∂ T + ------div q Ð ------∂ n t 3nkB 3n t (2.21) ∂ 2 2δ 2 T 0 + ------divq0 ν 2e E δ[ν ] ∂t 3n k = ei------Ð T T , 0 B ω2 (2.18) 3kBm 0 2 v (ν ) e 2 s0 = ei 0------E0 Ð a ------T0, ∂ ∂ ω2 * L 2T i 3kBm 0 T δT Ð ------δn = Zδ[ν T]. (2.22) ∂t i 3n ∂t T

∂ v S0 T = Za ------T , (2.19) δ| |2 δ ∂ i0 0 Here E and (Ek E*j + Ek*E j) are the perturbations t * LT of the bilinear combinations of rf field components, δq which follow directly from (2.4), (2.5), (2.13), and is the perturbation of the electron thermal flux density: (2.17). In equations (2.18)Ð(2.19) the values of q0, ν ( ei)0, and vS0 are determined by the plasma parameters ∇δT 1∇T(∇δT ⋅ ∇T) δ  ------in the ground state and the amplitude of the pump field q = Ðq ------Ð 3  ∇T 2 ∇T E0 is assumed to be given. It can be seen from equation (2.19) that in a nonisothermal plasma with a quasi- (2.23) δ  steady-state spectrum of ion-acoustic turbulence the ions 3δn 2 Ti ∇T  ӷ + ------- + ------ --∇------, R R NL, undergo comparatively fast heating. In accordance with 4 n 3 T i T  equation (2.19), the characteristic time for doubling of |∂ ∂ |Ð1 the ion temperature is lnTi0/ t ~ (Ti0/ZT0)LT/vS0. As ∇T × [∇δT × ∇T] δn ∇T  the ions are heated, the degree of nonisothermicity of δq = Ðq------+ ------ 3 ∇ the plasma ZT0/Ti0 decreases and at times of the order  ∇T n T  LT/vS0 the ion temperature Ti0 becomes comparable δ (2.24) with ZT0. When describing the state of a plasma with b 2 ∇T(∇δT ⋅ ∇T) Ti ∇T Ð pv ------α ------+ ------, ion-acoustic turbulence for such long times, we need to s ln2 ∇T T ∇T take into account the comparatively strong Cerenkov i Ӷ attenuation of the sound by the ions, which leads to a R R NL, substantial change in the electron heat transfer and a difference in the heat flux from that described by where the numerical coefficient is b = (25/4)(1/12 Ð Cβ) + expressions (2.9) and (2.10). Subsequently, without (128/3π)Cβ = 0.97 and analyzing time intervals greater than LT/vS0, we shall ∇T ⋅ ∇δT confine our study to STBS in a strongly nonisothermal δ[ν ] v ------T T = a* s ∇ (2.25) plasma when Cerenkov attenuation of sound at ions can T be neglected. is the perturbation of the electronÐion heat exchange. In addition to equations (2.18) and (2.19) describing Note that expression (2.24) is written to within terms of the plasma ground state, in order to study STBS we also the second order with respect to the small parameter α, require equations for the small perturbations of the den- (2.11) as is required for the following calculations. sity δn = n Ð n , and the electron δT = T Ð T and ion δ 0 0 temperatures Ti = Ti Ð Ti0, which vary on spatial scales much smaller than the scales of nonuniformity of the 3. LONG-WAVELENGTH ground state. In our subsequent analysis of STBS, in ION-ACOUSTIC OSCILLATIONS order to avoid complicating the formulas we shall omit IN A PLASMA WITH SHORT-WAVELENGTH the subscript 0 from n0, T0, and Ti0 which characterize TURBULENCE the plasma ground state. Then for small-scale perturbations of the plasma hydrodynamic parameters we can In this section we shall analyze long-wavelength write the following equations which follow directly sound in a plasma with ion-acoustic turbulence. We shall from the initial equations (2.2)Ð(2.5), (2.9), (2.10), and denote by k the wave vector of the long-wavelength ion- (2.13): acoustic oscillations. We assume that the wave vector of the acoustic oscillations satisfies the condition 2  ∂ 4 Zη ∂ 2  Znk  δT   Ð ------∆ Ð v ∆δn = ------B-∆δT + ------i  Zk2l l Ӷ 1, (3.1) 2 ∂ s t ei ∂t 3min t  mi  Z  ν (2.20) where lei = vT/ ei and lt = vT/vt are the mean free paths 2 2 Ze  2 ∂  of electrons having the thermal velocity vT for electronÐ + ------∆δ E Ð ------δ(E E* + E*E ), 2 ∂ ∂ k j k j ion collisions and scattering of electrons by ion-acous- ω  rk r j  2mim 0 tic oscillations of the charge density, respectively.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 STIMULATED BRILLOUIN SCATTERING IN A PLASMA 83

We know (see, for example, [19]) that turbulent In accordance with equations (2.2)Ð(2.5), perturba- noise is concentrated in a region of comparatively high tion of the density is accompanied by perturbations of wave numbers for which the remaining hydrodynamic quantities: ν ω r2 1 k > ----i-i------L--e-----D---i . (3.2) --δuexp(Ð iωt + ik ⋅ r) + c.c., t ω2 3 2 LirDe (3.8) 1 Instead of the vector k for the wave vectors of the turbulent --δT( ) exp(Ð iωt + ik ⋅ r) + c.c. 2 i charge density fluctuations in (3.2) we use the notation kt. It can be seen from a comparison of inequalities (3.1) and Taking into account inequality (3.5) for small perturba- (3.2) that in the most interesting case when δ δ δ tions n, T(i), and u we have from (2.2)Ð(2.5) the lin- ω 1/2 1/2 earized system of equations LeZT Zlt > -ω------------ ------ 1, (3.3) Li Ti lei ωδn = n(k ⋅ δu), (3.9) the long-wavelength perturbations being analyzed do 2 k kB not fall within the turbulence region since k < kt. It ωk ⋅ δu = ------{Zδ(nT) + δ(nT )} nm i should be stressed that turbulent noise with the wave i (3.10) vectors kt ~ 1/rDe from the interval (3.2) determines the 4i Zη 2 Ð ------k k ⋅ δu, transport processes in the plasma [19] and thereby 3 nm strongly influences the long-wavelength perturbations i of the hydrodynamic quantities. ωδ 2 ⋅ δ 2 ⋅ δ δ[ν ] In order to analyze long-wavelength sound we shall T Ð ------k q Ð --Tk u = Ði T T , (3.11) use equations (2.2)–(2.5), in which there is no rf field. 3nkB 3 We shall assume that the long-wavelength perturbation 2 of the electron density has the form ωδT Ð --T k ⋅ δu = iZδ[ν T]. (3.12) i 3 i T 1 --δnexp(Ð iωt + ik ⋅ r) + c.c. (3.4) 2 In equations (3.11) and (3.12) the explicit form of the heat flux perturbation δq and the perturbation of the We shall subsequently analyze those perturbations for energy fraction δ[ν T] transferred from the electrons to which the wave vector k satisfies the condition T the ions depends on the ratio of R to RNL (2.7). Using ӷ ӷ kL 1, (3.5) (2.23) for R RNL we find where L is the smallest scale of nonuniformity of the ⋅ δ 2χ θ δ hydrodynamic quantities n, T, and Ti in the ground state k q = Ðik t( ) T and the period 2π/ω is much smaller than the character- 32 3 p δ 2δT  (3.13) istic time of variation of these quantities in the plasma β v θ n i Ð --π--- || p skcos ----------- + ------ , ground state. In addition, we shall assume that the fre- 2RNL LT n 3T i quency of the perturbations is much lower than the turbulent relaxation frequency of the electron momentum where θ is the angle between the wave vector of the per- ω Ӷ ν and 1/t , which is the reciprocal time taken to turbations k and the unit vector ∇T/|∇T| along which t s χ θ establish a quasi-steady-state turbulence spectrum in the heat flux (2.9) propagates, and t( ) is the effective the region kt ~ 1/rDe: thermal conductivity ω2 ⁄ Ð1 Li R RNL 32 2 p 2 ω Ӷ ∼ χ (θ) = -----β||nk v ------L (1 + sin θ). (3.14) t s ------. (3.6) t π B s T ω ⁄ 3RNL Le 1 + R RNL Ӷ Under the conditions (3.1), (3.5), and (3.6), first the dis- Then, taking into account expression (2.24) for R RNL tribution of turbulent noise with kt ~ 1/rDe which deter- we have mines the electron fluxes and the energy relaxation fre- ⋅ δ 2χ θ δ quency can follow slow changes in the hydrodynamic k q = Ðik t( ) T quantities. Second, in order to describe transport pro- δ δ (3.15) cesses we can use expressions for the steady-state θ (α α) n b α2 T i Ð pv sk cos + b ----- + ------, charge and heat fluxes (A.2)Ð(A.5). In connection with n ln 2 T i the use of hydrodynamic equations for the ions, we note χ θ that this approach is meaningful if t( ) = nkBv s LT ν ӷ ω, ii k v T i , (3.7) 2 b 2 2 (3.16) × (a + bα)sin θ + ------α cos θ , where vTi is the ion thermal velocity. ln2

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 84 VAGIN et al. where α is determined by formula (2.11), and a is a electrons and adiabatic for the ions [16], and the ion numerical coefficient: ⁄ acoustic velocity vs = ZkBT mi is small and will sub- 25( β ) 128β sequently be neglected. a = ----- 1 Ð || + -----π--- || = 7.55. (3.17) 4 3 ӷ In the limit R RNL, inequality (3.21) ensures that In these limiting cases, to within small corrections of equation (3.19) is satisfied for all angles θ. In the oppo- 2 Ӷ θ Ӷ α the order 1/kL Ӷ 1 or α Ӷ 1 for the perturbation of the site limit R RNL, and for small angles equation term describing energy exchange between electrons (3.19) is satisfied if and ions (2.25), taking into account (2.13) we have α2 ( , ⁄ ) ӷ δ[ν ] δ max kLT ZT T i 1. (3.23) T T T θ -----ν------= i------kLT cos . (3.18) T T T In the same limit, but for θ ӷ α, the condition for valid- In accordance with (3.18), the small-scale (3.5) hydro- ity of (3.19) is less stringent dynamic perturbations (3.4), (3.8) increase the relative α2 ⁄ 2θ ӷ change in the energy fraction transferred from the elec- ZT T i + kLT sin 1. (3.24) | θ| Ӷ trons to the ions by a factor of kLT cos 1 for angles θ far from π/2. It can be seen from the definition (2.13) The dispersion equation (3.19) has solutions describ- and relations (2.8) and (2.12) that the additional param- ing acoustic oscillations in a plasma at frequency ω which θ ω eter kLTcos appears in (3.18) because the projection differs from the acoustic frequency s by a small imagi- ω Ӎ ω γ |γ| Ӷ ω of the scale of nonuniformity of the perturbations on nary correction s + i , s. Neglecting the the direction of nonuniformity in the unperturbed state small variation in the ion-acoustic oscillation frequency θ is determined by the value of 1/kcos rather than by LT. in the following analysis, we find from (3.19) From an analysis of equations (3.9)Ð(3.18) we can γ γ θ ≡ γ γ θ obtain the following dispersion equation linking the com- = Ð s( ) Ð i Ð e( ), (3.25) plex frequency ω with the perturbation wave vector k: γ θ where the function e( ) characterizing the interaction  ν between the ion-acoustic waves and the electrons, has ω2 ωγ 2 2 ω2 T θ  + 2i i Ð k v sT Ð s 1 Ð -----kLT cos  the form  ω

2 ν ω2  q  k γ θ t Li ψ θ × ω Ð ω a ------cosθ i------χ (θ) (3.19) e( ) = ------( )  s q pv  nk t 2 ω2 s B Le (3.26) ν ω Ð1   q  ZT T s ∂q  2 × ( θ) ------θ Ð ------kL ------cos θ  = 0, 1 Ð a*cos 1 Ð aq cos  . T ω ∂  pv s T i pv s lnR  The function ψ(θ) which determines γ (θ) (3.26) where ω = kv is the frequency of ion-acoustic oscilla- e s s depends strongly on the thermal conductivity in the tur- tions of wavelength much greater than the electron χ θ Ӷ bulent plasma t( ) (3.14), (3.16) and is positive for all Debye length, aq = 1 for R RNL and aq = 13/12 for R ӷ R , angles θ. The explicit form of the function ψ(θ) in NL Ӷ (3.26) depends on the ratio R/RNL. For R RNL γ 2 2 Z η i = --k ------(3.20) 3 nmi 4 2 2 b 2 2 ψ(θ) = ------asin θ + ------α cos θ π ln2 is the sound attenuation decrement at the ions. When 9 a* deriving equation (3.19), we assumed that  ӷ 2 kLT 1, (3.21)  2 b 2 2 ×  asin θ + ------α cos θ ⁄ ӷ ln2 ZT T i 1. (3.22)  (3.27)  Inequality (3.22) corresponds to the conditions for the existence of ion-acoustic turbulence [19, 20]. It should Ð1 2  be noted that under the condition (3.22), the difference ZT b 2 2  --a---*------α θ  > between the expression for the velocity of sound vsT = + cos 0, kLT Ti ln2  ⁄ ( ⁄ ) kB mi ZT + 5T i 3 , which is isothermal for the 

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 STIMULATED BRILLOUIN SCATTERING IN A PLASMA 85 ӷ and in the opposite limit R RNL where c is the velocity of light in vacuum. The frequencies and wave vectors of the incident and scattered 2π 2 ω ψ(θ) = ------(2 Ð cos θ) waves differ by the frequency and the wave vector k 48β|| of the electron density perturbations

Ð1 ω ω ω 2 (3.28) Ð1 = 0 Ð , kÐ1 = k0 Ð k. (4.4) 2   × ( 2θ) a ZT 2θ > 2 Ð cos + -----*------cos  0. The density perturbation itself has the form (3.4). kLT Ti  Assuming that the field E of the fundamental-fre- γ θ 0 In the range of angles where e( ) > 0, expression (3.26) quency wave is given, we use the following equation to describes the attenuation of ion-acoustic waves at elec- determine the amplitude E of the scattered wave field γ θ Ð1 trons, and for those angles where e( ) < 0, wave generation takes place. ν δ ω2 2 2 ω2  ei  * nω2 * Ð1 Ð kÐ1c Ð Le1 + iω------ EÐ1 = ----- LeE0 , (4.5) As has already been noted [18], the existence of Ð1 2n anomalous transport means that it is necessary to analyze the stability of the resulting hydrodynamic struc- which contains the amplitude of the electron density tures. It can be seen from expression (3.25) that a perturbations to be determined self-consistently. The plasma with advanced ion-acoustic turbulence is stable incident E0 and scattered EÐ1 waves are plane trans- with respect to the small perturbations (3.4), (3.8) verse electromagnetic waves for which the conditions under study if the following inequality is satisfied k0 á E0 = kÐ1 á EÐ1 are satisfied and as can be seen from γ > max(Ðγ (θ)). (3.29) (4.5) the directions of the vectors E0 and EÐ1 are the i e δ Ӷ same. In order to search for n we shall use the system In the limit R RNL, (3.29) may be expressed in the of hydrodynamic equations (2.20)Ð(2.22) given in Sec- form tion 2 which, for the perturbations (3.4) and (3.8), may Λ 5/2 be written in the form  kLT  1.3 iZT Zm 2 min 1, ------< k l l ,  ⁄  2 Λ   t ei 0.84ZT T i α Ti mi {ω2 ωγ ω2}δ (3.30) + 2i i Ð s n and conversely, when R ӷ R condition (3.29) corre- NL 2 ⋅ * (4.6) Znk 2 δT  Z 2e E E sponds to the relationship = ------B-k δT + ------i  + ------k ------0------Ð---1 , m Z 2m ω2 Λ 5/2 i   i m 0 0.04 i R ZT Zm < 2 ------Λ------------ ------k ltlei, (3.31) 0.1 ZT RNL Ti mi δ 1 + ------ω δ 2ω n 2 ⋅ δ kLT Ti s T Ð -- sT----- Ð ------k q 3 n 3nkB Λ where i is the ion Coulomb logarithm. When analyz- (4.7) ing inequalities (3.30) and (3.31), we must bear in 2e2E ⋅ E* = iν ------0------Ð--1 + a kv δT cosθ, mind that these should be satisfied together with con- ei ω 2 * s dition (3.1). 3kBm 0 δ ω δ 2ω n δ θ 4. STIMULATED BRILLOUIN SCATTERING s T i Ð -- sT i----- = Ða kv sZ T cos . (4.8) 3 n * Returning to our analysis of the scattering of electromagnetic radiation in a plasma with ion-acoustic tur- In equations (4.7) and (4.8) we neglect the small differ- bulence, we express the rf field (2.1) as the sum of the ence between the frequency ω and the acoustic fre- ω pump field quency s. Unlike the system (3.9)Ð(3.12) which 1 describes long-wavelength acoustic perturbations in a --E exp(Ð iω t + ik ⋅ r) + c.c. (4.1) turbulent plasma, the system (4.6)Ð(4.8) contains terms 2 0 0 0 proportional to E* and together with equation (4.5), and the scattered wave Ð1 describes the scattering of the pump wave (4.1) at low- 1 frequency long-wavelength acoustic perturbations of --E exp(Ð iω t + ik ⋅ r) + c.c., (4.2) 2 Ð1 Ð1 Ð1 the charge density. ω ω whose frequencies 0 and Ð1 are related to the wave We consider the contribution to the pressure pertur- vectors k0 and kÐ1 by bation on the right-hand side of equation (4.6) ω 2 ω2 2 2 ω 2 ω 2 2 2 δ δ ⁄ 0 = Le + k0 c , Ð1 = Le + kÐ1 c , (4.3) kBn( T + T i Z), (4.9)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 86 VAGIN et al. which is attributable to perturbations of the electron δT and (3.26)Ð(3.28) we obtain the electron temperature δ and ion temperatures Ti. From equation (4.8) we perturbation from (4.7) obtain for the ion temperature perturbation 2 ⋅ * ψ θ γ θ δ 2 δn e E0 EÐ1 ( ) e( ) n δ ------δ θ δT = ------Ð 2i------T-----. (4.13) T i = Ti Ð a*Z T cos . (4.10) ω 2 2 ω n 3 n kBm 0 k ltlei s The first term on the right-hand side (4.10) is responsi- When writing (4.13) we neglected small quantities of ble for the adiabatic contribution of the ions to the the order 1/kLT which make small corrections to the real velocity of sound, well known in the hydrodynamics of ω frequency s. The first term on the right-hand side of a laminar plasma. As we showed in the previous sec- (4.13) describes the perturbation of the electron tem- tion, in the limits of condition (3.22) this contribution perature δT caused by mixing of the scattered wave to the velocity of sound is small compared with the isothermal contribution of the electrons. We shall subse- field EÐ*1 with the pump field E0 and corresponds to the quently neglect this contribution. Unlike a laminar balance of the nonuniform heating of the electrons by plasma, in a turbulent plasma the ion temperature incre- inverse bremsstrahlung absorption of this interference ment as a result of energy exchange with electrons field and the electron heat transfer. The second term on described by the second term on the right-hand side of the right-hand side of (4.13) corresponds to the small (4.10) is by no means small. This substantial increase in dissipative contribution of the electrons to the ion the contribution of electronÐion heat exchange to δT is γ θ i sound increment s( ). caused first by an increase in the temperature relaxation δ δ frequency in a turbulent plasma (2.16) compared with a Eliminating T and Ti using (4.10) and (4.13), laminar plasma and second, by an increase in the rela- instead of the system (4.6)Ð(4.8) we write a single tive variation of the energy fraction (3.18) transferred equation to determine the density perturbation created | θ| ӷ by the bilinear combination of the fields of the incident from the electrons to the ions when kLT cos 1. Tak- ing into account (4.10) and these observations we and scattered waves: obtain for the pressure perturbation ω 2 2 2 δn 1 e [ω ω ωγ θ ] ⋅ *  s  δ δ ⁄ θ δ Ð s + 2i s( ) ----- = --E0 EÐ1-----ω------ kBn( T + Ti Z) = kBn(1 Ð a cos ) T. (4.11) n 2 m 0v T * (4.14) ψ(θ) For R > RNL when a = 1.3 it follows from the right- × 1 + 2(1 Ð a cosθ)------. * * 2 hand side of (4.11) that angles θ exist for which the k ltlei contribution to the pressure perturbation (4.9) is in antiphase with the electron temperature perturbation The terms in the brackets on the right-hand side of for- δT. This occurs first when the modulus of the contribu- mula (4.14) are of a different nature. The first term, one, δ occurs as a result of the ponderomotive action of the tion to the ion temperature perturbation Ti caused by | δ | fields. The second, which contains the small parameter heat exchange with electrons exceeds Z T and second, 2 Ӷ is in antiphase with δT. k ltlei 1 in the denominator [see (3.1)] describes the density perturbation as a result of the contribution to The contribution to the last term on the left-hand the pressure perturbation (4.11) generated by the bilin- side of equation (4.7) made by the electron thermal ⋅ conductivity (3.14), (3.16) which appears in the diver- ear interference combination of the fields E0 EÐ*1 . gence of the electron thermal flux (3.13), (3.15) may be By jointly analyzing equations (4.5) and (4.14) we written (in order of magnitude) in the form can obtain the following dispersion equation which 2 describes parametric instability and links the frequency k χ θ δ ∼ δ ω ------t( ) T kv eff (k) T. (4.12) , which is generally complex, with the perturbation nkB wave vector k Here v (k) ~ (kL ) 1 + R ⁄ R v characterizes the (ω ∆ γ )[ω2 ω 2 ωγ θ ] eff T NL s Ð Ð + i E Ð s + 2i s( ) effective rate of heat removal by electron heat transfer ω 2 (4.15) from the region of action of the heating field. The first 2 ψ(θ) = ------s------v 1 + 2(1 Ð a cosθ)------, term on the left-hand side of (4.7), corresponding to the ω 2 E * 2 δ δ 8 0rDe k ltlei time variation of T, is given by kvs T. Allowing for (3.21), this is small compared with (4.12). Then, the | ω | where vE = eE0/m 0 is the amplitude of the electron last term on the right-hand side of (4.7) describing oscillation velocity in the field of the fundamental-fre- energy exchange between electrons and ions is also δ quency wave, given in order of magnitude by the expression ~kvs T and also makes a small contribution to the electron heat 2 ω2 c 2 balance compared with (4.12). Bearing all this in mind, ∆ = ------(2k ⋅ k Ð k ), γ = ν -----L---e-. (4.16) 2ω 0 E ei ω2 substituting (4.10) into (4.7), and using (3.13)Ð(3.16) 0 2 0

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 STIMULATED BRILLOUIN SCATTERING IN A PLASMA 87 |γ θ | When deriving equation (4.15) and also when analyzing the value of s( ) may be either larger or smaller than the acoustic perturbations in the absence of the pump field, 2π/τ which is the reciprocal duration of the laser pulses we assumed that the conditions (3.21)Ð(3.24) were satis- used to investigate STBS. Consequently, it is interest- fied. In addition to these inequalities, we used the fol- ing to discuss the two possibilities. We shall first ana- lowing constraint to derive equation (4.15) lyze the situation when the pump pulse duration τ is ω Ӷ ω long and the following inequality holds 0, (4.17) |γ θ |τ ӷ π ω ω Ӷ ω s( ) 2 . (4.20) which is known to be satisfied for ~ s 0. In order to complete the picture, we note that non- We shall consider the conditions under which the Ӷ summed sound attenuation decrement at the ions and relativistic motion of the plasma at the velocity u c γ θ θ leads to a Doppler shift of the frequency ω and is taken electrons s( ) is positive for all values of the angle , into account by replacing ω with ω' = ω Ð k á u in equa- i.e., inequality (3.29) is satisfied and no hydrodynamic tion (4.15). instability develops in the plasma. The square root of ω Ӎ ω γ |γ| Ӷ ω equation (4.18) corresponding to the possibility of We shall also assume that s + i where s STBS then has the form (not to be confused with γ from Section 3). We shall analyze the conditions for which the following inequal- 1 γ = Ð--[γ + γ (θ)] ity holds for the resonance detuning ∆ 2 E s ∆ ω (4.21) = s 1 2 + -- [γ Ð γ (θ)] + W (θ). and the STBS process takes place more efficiently. 2 E s E γ Then, to determine the imaginary correction we have In accordance with (4.21), the threshold radiation from (4.15) intensity for STBS is obtained from 1 θ γ γ θ > (γ + γ )[γ + γ (θ)] = --W (θ), (4.18) W E( ) = 4 E s( ) 0. (4.22) E s 4 E Estimates show that for most laser plasmas the condi- where γ (θ) is described by relations (3.25)Ð(3.28), and |γ θ | Ӷ γ s tion s( ) E holds and the solution of (4.21) near the 2 threshold may be expressed in the following simple ω v ψ(θ) W (θ) = ------s------E-- 1 + 2(1 Ð a cosθ)------. (4.19) form: E 4ω 2 * 2 0 rDe k ltlei W (θ) γ Ӎ γ θ E Ð s ( ) + ------γ------. (4.23) As we noted in Section 3, the function ψ(θ) determin- 4 E ing W (θ) (3.27), (3.28) depends strongly on the ther- E Then using relations (3.20), (3.25), (3.27), (3.28), and mal conductivity in the turbulent plasma χ (θ) (3.14), t (4.16) we express condition (4.22) in the form (3.16) and is positive for all values of the angle θ. The expression (4.19) describes the nonlinear inter- (v 2 ) ν γ + γ (θ) ------E-----t--h = 8----e--i ----i------e------action of the scattered (4.2) and acoustic (3.4) waves 2 ω ω v T 0 s and its sign characterizes the possibility of STBS. When (4.24) (4.19) is positive, STBS may occur, whereas when it is 1 × ------> 0. negative STBS is impossible. Since under the conditions θ ψ θ ⁄ 2 2 Ӷ 1 + 2(1 Ð a cos ) ( ) k ltlei being discussed (3.1) the parameter is k ltlei 1, the * main contribution to the nonlinear interaction is made by We shall discuss expression (4.24) which determines the pressure perturbation (4.11) which corresponds to the the pump radiation intensity at which STBS occurs. As thermal mechanism of STBS. Thus, for R > RNL for angles usual in STBS theory, the presence of the two small θ for which the contribution to the pressure perturbation ν ω γ θ ω parameters ei/ 0 and s( )/ s on the right-hand side of ⋅ and the bilinear combination E0 EÐ*1 generating this are (4.24) indicates that STBS occurs at a comparatively phase-shifted by 180°, i.e., (1 Ð a cosθ) < 0, the sign of low radiation intensity when the amplitude of the elec- * tron oscillation velocity in the fundamental-frequency the nonlinear interaction is negative and STBS cannot wave field is considerably lower than the electron ther- develop. It should be noted that for a fairly high degree mal velocity. When discussing the relation (4.24), it is of nonisothermicity of the plasma when the condition θ convenient to introduce the scattering angle 0 between ZT/T > 0.5L (l l )Ð1/2 > 1 is satisfied, even in the limits i T t ei the wave vectors of the incident k0 and scattered waves (3.1) STBS is determined by the ponderomotive mech- kÐ1, which has already been used in formulas (1.1) and anism and it is impossible for it to be forbidden. (1.2). For the wave vector k of the acoustic waves In equation (4.18) the absolute value of the decre- which determines the scattering we then have |γ θ | ω ment s( ) is small compared with the frequency s, θ which is itself low in this particular limit (3.1) of fairly k = 2k sin----0. long wavelengths. For these laser plasma parameters 0 2

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I It follows from (4.25) that as the significant dependence corresponding to the threshold of the radiation 0.4 flux density as a function of the angle θ decreases, 0.3 where θ is the angle between the scattering vector and 1 0.2 the anisotropy axis of the turbulent noise, the STBS forbiddenness is lifted. This dependence is observed par- 2 0.1 ticularly clearly in the limit R > RNL when, as has –1.0 –0.5 0 0.5 1.0 already been noted in the discussion of equation (4.14) cosθ and relation (4.19), the phase shift between the contribution to the pressure perturbation (4.9) and the bilin- ⋅ ° Fig. 1. Dependence of the STBS threshold on the angle ear field combination E0 EÐ*1 changes by 180 . The between the vector of the ion acoustic waves and the force θ density vector R. The curves I(θ) were obtained for α = 0.3, change in the phase matching for small 0 has the result ν γ (ω2 ⁄ ω2 ) that instead of a buildup of ion-acoustic oscillations of ZT/Ti = 10, kLT = 50, (( t/2 i) Li Le = 0.02, and the density, these are suppressed. As a result of this sup- 2 k ltlei = 0.1 (1) and 0.01 (2). pression of the ion-acoustic oscillations STBS is forbidden for those angles θ where expression (4.25) is negative. The functions I 2.0 (v 2 ) ω ω I(θ) = ------E-----t--h ------0------s 1 1.5 2 8ν γ v T ei i 1.0 θ 2 0.5 over the entire range of angles are plotted in Figs. 1Ð3

...... (ν ⁄ γ )(ω2 ⁄ ω2 ) –1.0 –0.5 0 0.5 1.0 for values of the parameter t 2 i Li Le satisfy- cosθ ing condition (3.29). In accordance with Fig. 1, when a Ӷ small force acts on the electrons where R RNL and Fig. 2. Form of the function I(θ) for R = 16R , kL = 10, (ν ⁄ γ )(ω2 ⁄ ω2 ) NL T t 2 i Li Le = 0.02, the STBS threshold is low- ν γ (ω2 ⁄ ω2 ) ( t/2 i) Li Le = 2. The other parameters are the same est for those electron density perturbations whose wave as in Fig. 1. vector k is directed along R. For vectors k oriented in the transverse directions to R the STBS threshold is ӷ considerably higher. For example, for R RNL and I (ν ⁄ γ )(ω2 ⁄ ω2 ) t 2 i Li Le = 2, and comparatively weak 1 1.2 nonisothermicity when, for example kLT = ZT/Ti = 10, the STBS threshold is lowest for vectors k directed at 0.8 large angles to R (see Fig. 2). For directions k close to 2 0.4 R the STBS threshold increases sharply. This increase 2 Ӷ is particularly appreciable for k ltlei 1 and as follows –1.0 –0.5 0 0.5 1.0 from (4.19), (4.24), (4.25), and Fig. 2, may lead to cosθ θ STBS forbiddenness for those angles where WE( ) < 0. It should be stressed that formula (4.25) is only mean- θ Fig. 3. Form of the function I( ) for R = 16RNL. Unlike Fig. 2, ingful for those angles θ where its denominator is pos- the nonisothermicity of the plasma is five times higher, itive. The condition that the denominator of (4.25) van- ZT/Ti = 50. The other parameters are the same. θ ishes corresponds to the limiting angle * which separates the θ angle space into the region where STBS can develop and the region where STBS is forbidden. It fol- Bearing in mind this definition and considering the case lows from (4.23) that for vectors k falling within the when conditions (3.30) and (3.31) are satisfied, we have θ γ range of angles for which WE( ) < 0, the decrement is approximately from (4.24) negative and no perturbations grow. The dependences 2 plotted in Fig. 2 change significantly as the plasma (v ) 32 ------E------t-h = ----- nonisothermicity increases. It can be seen from Fig. 3 2 3 that for the selected plasma parameters, an increase in v T (4.25) the nonisothermicity by a factor of 5 lifts the STBS for- (T ⁄ ZT)(ν ⁄ ν )(k v ⁄ ω )sin(θ ⁄ 2) biddenness for k oriented along R and the lowest values × ------i------e--i------i-i------0------s------0------0------> 0. 1 + (1 Ð a cosθ)ψ(θ) ⁄ (2k2l l sin 2(θ ⁄ 2)) of the STBS threshold are obtained for k directed at * 0 t ei 0 right angles to R.

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We shall now analyze the STBS spatial gain which higher than those required for long pulses. As for the case is of interest for various applications. For this purpose, of long pump pulses, the threshold value of (4.29) depends when analyzing the dispersion equation (4.15), we shall strongly on the angle between the vectors n and k. As a ω ω ω ∆ ω assume that the frequency is real and = s, = s result of (3.29) and (4.27) the corresponding dependences and the wave vector has a small imaginary part ÐiG, are similar to those plotted in Figs. 1Ð3. G Ӷ k. Then for the spatial gain in the direction of prop- To conclude this Section, we note that in the limit of agation of the scattered wave GÐ1 = kÐ1 á G/kÐ1 we find short pump field pulses (4.27) formula (4.26) holds for ( 2 ) ν ω 2  2  the spatial STBS gain near the threshold where v E th ei Le v E GÐ1 = ------------Ð 1, (4.26) ( 2 ) ω 2 ( 2 ) should be replaced by v E th, τ as given by (4.29). 2 0k0c  v E th  ( 2 ) where the dependence of the function v E th on the 5. DISCUSSION θ angle is described by expression (4.24). In accor- We shall discuss the conditions under which STBS dance with (4.26), the spatial gain of the scattered wave θ may be observed in a plasma with ion-acoustic turbu- is most efficient for those angles for which the largest lence. According to inequality (3.5), the characteristic excess over the threshold is achieved. At the same time, Ӷ Ӷ scale of the hydrodynamic quantities determining the since we assumed that GÐ1 kÐ1 (or G k) to derive perturbation should be smaller than the spatial scale of relation (4.26), under the conditions for its validity the variation of these quantities. This implies that the scat- excess over the STBS threshold should not be anoma- tering angle θ should not be too small, lously large. 0 We shall now discuss STBS in the limit opposite to θ Ð2 × 15 0 ≈ θ ӷ × Ð 3 10 2 10 (4.20) when the pulse duration of the pump field (4.1) 2sin---- 0 1.5 10 ------, (5.1) 2 L [cm] ω [ Ð1] is short compared with the characteristic time of varia- T 0 s tion of the amplitudes of the hydrodynamic perturbations in the plasma, where it is assumed that L ~ LT. Generally, another lower constraint on θ is imposed by inequalities (3.30) γ θ τ Ӷ π 0 s( ) 2 . (4.27) and (3.31). However, in the most interesting case of |γ θ |τ Ӷ π We shall determine the threshold pump field intensity, short laser pulses when s( ) 2 no such constraint as is usually assumed for short pulses, using the rela- exists. At the same time, using the hydrodynamic tion description of the perturbations presupposes that the 2 parameter Zk ltlei (3.1) is small compared with unity, γτ = 2π, (4.28) which gives where γ is given by formula (4.21). We then obtain the 1/2 Λ 10Ð2 1/4 2 × 1015 following expression for the threshold intensity of the θ < 0.1 ------pump field: 0  5 L [cm] ω [ Ð1] T 0 s ( 2 ) ν (5.2) v E th, τ ei 2π 2π 1/4 5/8 ------= 8------1 + ------ZT R 1/4 100 9/8 n[cmÐ3] 2 ω ω τ γ τ × ------L ------. v T 0 s E T [ ] 20 (4.29) 10Ti p T eV 10 ψ(θ) Ð1 × 1 + 2(1 Ð a cosθ)------> 0. * 2 Another constraint of the theory put forward above k ltlei involves the assumption that the characteristic time for As in the case (4.24), it follows from formula (4.29), the establishment of a quasi-steady-state spectrum of the conditions |γ| Ӷ ω , and (4.28) that the amplitude of turbulent noise is shorter than the time of variation of s the perturbations of the hydrodynamic quantities (see the electron oscillation velocity corresponding to the (3.6)). In the limits of inequalities (3.1) and (3.6), the threshold pump field intensity is also low compared spectrum of turbulent noise and the electron fluxes can with the electron thermal velocity. It should be noted follow the comparatively slow variation of the pertur- that the comparatively low threshold energy flux den- bations. Inequality (3.6) imposes another upper con- sity (4.29) of a pump field having the pulse duration straint on the angle θ : (4.27) exceeds the corresponding value (4.24) for a 0 long pulse 10Ð2 1/2 2 × 1015 θ < 0.2 ------( 2 ) π π 0 [ ] Ð1 v E th, τ 2 2 LT cm ω [s ] ------= ------1 + ------ӷ 1 (4.30) 0 2 γ θ τ γ τ (v ) max s( ) E (5.3) E th 1/2 1/4 10T R 1/2 100 1/4 n[cmÐ3] times. Thus, in order for STBS to develop in short laser × ------i ---L ------. T [ ] 20 pulses, the energy flux densities of these fields must be ZT p T eV 10

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2 Ӷ k0 for Zk ltlei 1 or (5.2), STBS forbiddenness occurs in k –...... 1 the limit of high values of the effective density of the ϕ turbulence-exciting force R when R ӷ R . In this case, ...... NL forbiddenness occurs when the angle θ between the direction of the unit vector n = ∇T/|∇T| (which determines the axis of anisotropy of the ion-acoustic turbu- ...... n lence) and the difference k0 Ð kÐ1 (the wave vectors of the pump field k and the scattered wave k is equal to θ 0 Ð1 0 θ or smaller than θ . The angle θ is determined from the p * * condition that the denominator of formula (4.25) vanishes and separates the region of angles θ into two Fig. 4. Volume scattering diagram. θ θ π parts, where STBS is allowed when * < < and for- ≤ θ ≤ θ θ bidden when 0 *. Let us assume that p is the We shall consider the constraints (5.1)Ð(5.3) for an iron angle between the vectors n and k0 which we can call laser plasma under conditions typical of experiments the angle of incidence. We select the polar axis of the 20 Ð3 [21, 22] where T = 200 eV, n = 10 cm , Z = 5, LT = spherical coordinates in the direction of the vector k0. µ 100 m, and ZT/Ti = 20. For these plasma parameters Then the direction of the vector kÐ1 which defines the the inequalities (5.1) and (5.2) impose the following direction of observation of the scattered wave will be θ constraints on the scattering angle in degrees: determined by the polar angle 0 (the scattering angle) ° θ ° and the azimuthal angle ϕ measured from the projec- 0.2 < 0 < 4 , (5.4) tion of the vector n on the plane perpendicular to k0 (see and inequality (5.3) is weaker than (5.2). Note that for Fig. 4). In order to observe STBS forbiddenness exper- ≈ θ k k0 0 the inequalities (3.7) are also satisfied, which imentally, the following condition must be satisfied: means that the ions can be described hydrodynamically. Consequently, the anisotropy characteristics of the cosθ sin(θ ⁄ 2) Ð cosθ ϕ ≤ p 0 * STBS threshold in a turbulent plasma observed above cos ------θ------(--θ------⁄------)------, (5.6) should be observed for comparatively small scattering sin p cos 0 2 θ angles. For such small angles 0 the scattering wave which ensures that the inequality 0 ≤ θ ≤ θ is satisfied. vector k is almost perpendicular to the wave vector of * Since the STBS theory constructed by us for (3.1) is the incident wave k0. This implies that for waves propagating along n, which is the direction of anisotropy of valid for small scattering angles (5.2), in the zeroth approximation with respect to the small contribution of the ion-acoustic turbulence, the scattering characteristics θ Ӷ will be determined by the vectors k almost orthogonal to quantities of the order 0 1, condition (5.6) can be ⊥ written in the form n. Conversely, if k0 n, the small-angle scattering will be determined by the vectors k oriented along n.  cosθ   cosθ  To complete the picture we note the following. Out- π  * ≤ ϕ ≤ π  * Ð arccos ------θ----- + arccos ------θ----- , (5.7) side the range of angles (5.4) the scattering wave vector  sin p   sin p  is so large that the following inequality is satisfied 2 ӷ where the angle of incidence θ is in the range Zk ltlei 1. (5.5) p π ⁄ θ ≤ θ ≤ π ⁄ θ In order to describe scattering using relation (5.5), we 2 Ð p 2 + . (5.8) need to allow for the nonlocal nature of the electron * * heat transport. A theory of STBS under the conditions In particular, under the conditions of thermal STBS (5.5) was given in a recent publication [14] for the case being discussed when the turbulent noise distribution is defined. 1 To conclude our discussion of the characteristics of θ ----- Ӎ ° * = arccos 40 , (5.9) small-angle scattering we also note that, as can be a* seen from relation (4.29), for the plasma parameters π γ Ӷ and inequality (5.8) has the form given in this section and radiation pulses with 2 / E τ Ӷ π|γ | 2 / s , the STBS threshold does not exceed 50° ≤ θ ≤ 130°. (5.10) 10− 11 π |γ |τ 2 γ 8 Ð1 p ~10 10 (2 / s ) W/cm . In particular, for s ~ 10 s and τ ~ 1 ns the threshold energy flux density is In Fig. 5 the shaded region illustrates the range of azi- 12 13 2 ϕ ~10 Ð10 W/cm . muthal angles of the vector kÐ1 in which (5.7) is sat- The conditions for STBS forbiddenness have not isfied and STBS is forbidden. Here kÐ1⊥ and n⊥ are the been discussed so far. We shall make a special analysis components of the vectors kÐ1 and n perpendicular to of this qualitatively new effect. It has been shown that k0. It follows from (5.7) that the maximum width of the

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depend on the relationship between the effective density of the turbulence-exciting force k–1⊥ ∂ ϕ R = enE Ð p (A.1) q ∂r k n⊥ 0 and the force density RNL characterizing the influence of the induced scattering of waves by ions (see (2.7)) where Eq is the quasi-steady state electric field in the Ӷ plasma. When R RNL, the density of the current and the thermal electron flux are described by [19] Fig. 5. Diagram of scattering in the plane perpendicular to ϕ the vector k0. In the shaded area of azimuthal angles of the  3 α 16 vector k STBS is forbidden.  β α  β α Ð1 j = env s --1 Ð ||( ) + ----- + ----- ||( ) n  2 12 π (A.2) region of azimuthal angles ϕ of the vector k in which Ð1 24  STBS is forbidden is Ð -----(β||(α)n ⋅ zn + β⊥n × [z × n]) , π  π θ ≤ ϕ ≤ π θ Ð * + * (5.11)  α 15 β α  64β α and is achieved when the angle of incidence is θ Ӎ π/2, q = pv s -----1 Ð ||( ) + ----- + ----- ||( ) n p  4 12 π i.e., the pump wave is incident almost perpendicular to the axis of anisotropy of the turbulence. Under the con- (A.3) ditions of thermal STBS inequalities (5.11) have the 160  Ð ------(β||(α)n ⋅ zn + β⊥n × [z × n]) , form π  140° ≤ ϕ ≤ 220°. where n = R/R, z = (p/R)∂lnT/∂r, β⊥ = 0.02. In the ӷ By determining experimentally the limiting values of opposite limiting case when R RNL, the electron θ ϕ fluxes have the form [19] the angles p and for which STBS forbiddenness θ occurs, we can find and check, not only the numer- 16 R * j = -----env ------ical value of the constant a but also, most impor- π s * RNL tantly, we can check experimentally the recent state- (A.4) ment made in the literature that the ion heating in a  3  × β||n Ð --(β||n ⋅ zn + β⊥n × [z × n]), plasma with ion acoustic turbulence is anomalously  2  effective. This work was supported financially by the Russian 64 v R Foundation for Basic Research (projects nos. 96-02- q = --π--- p s ------RNL 16779, 99-02-18075), the program for State Support of (A.5) Leading Scientific Schools (project no. 96-15-96750) and  5  the Federal Target Scientific-Technical Program × β||n Ð --(β||n ⋅ zn + β⊥n × [z × n]), “Research and Development of Priority Directions in the  2  Development of Science and Technology for Civilian Pur- β poses,” subprogram “Physics of Quantum and Wave Pro- where ⊥ = 0.80. Expressions (A.2)Ð(A.5) can be used cesses.” to analyze electron charge and heat fluxes parallel and perpendicular to the axis of anisotropy of the turbulent noise.

APPENDIX In a highly nonisothermal plasma, the number of ions REFERENCES interacting with ion-acoustic waves by the Cerenkov scat- 1. H. A. Baldis and C. J. Walsh, Phys. Fluids 26, 3426 tering mechanism is comparatively small and their influ- (1983). ence on the turbulence spectrum can be neglected. In this 2. K. Tanaka, L. M. Goldman, W. Seka, et al., Phys. Fluids plasma, a quasi-steady state of ion acoustic turbulence is 27, 2960 (1984). established under the influence of various processes 3. P. E. Young, K. G. Estabrook, et al., Phys. Fluids B 2, 1907 such as Cerenkov emission of waves by electrons and (1990). induced scattering of waves at thermal ions. Under 4. G. P. Banfi, K. Eidmann, and R. Sigel, Opt. Commun. these conditions the laws governing electron transport 52, 35 (1984).

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5. J. Handke, S. A. H. Rizvi, and B. Kronast, Appl. Phys. 15. V. P. Silin and S. A. Uryupin, Zh. Éksp. Teor. Fiz. 98, 117 25, 109 (1981). (1990). 6. S. J. Karttunen and R. R. E. Salomaa, Phys. Lett. A 88, 16. A. V. Maksimov, V. P. Silin, and M. V. Chegotov, Fiz. 350 (1982). Plazmy 16, 575 (1990). 7. J. F. Drake, P. K. Kaw, Y. C. Lee, et al., Phys. Fluids 17, 17. V. P. Silin and S. A. Uryupin, Fiz. Plazmy 22, 790 778 (1974). (1996). 8. W. M. Manheimer and D. G. Colombant, Phys. Fluids 18. V. Yu. Bychenkov and V. P. Silin, Fiz. Plazmy 13, 1097 24, 2319 (1981). (1987). 19. V. Yu. Bychenkov, V. P. Silin, and S. A. Uryupin, Phys. 9. W. L. Kruer, Phys. Fluids 23, 1273 (1980). Rep. 164, 119 (1988). 10. L. Spitzer, Jr. and R. Harm, Phys. Rev. 89, 977 (1953). 20. V. Yu. Bychenkov, V. P. Silin, and S. A. Uryupin, Com- 11. R. W. Short and E. M. Epperlein, Phys. Rev. Lett. 68, ments Plasma Phys. Contr. Fusion 13, 239 (1990). 3307 (1992). 21. L. L. Losev and V. I. Soskov, Opt. Commun. 135, 71 12. P. K. Shukla, Phys. Fluids B 5, 4253 (1993). (1997). 13. A. V. Maximov and V. P. Silin, Phys. Lett. A 192, 67 22. A. A. Antipov, A. Z. Grasyuk, S. V. Eﬁmovskiœ, et al., (1994). Kvantovaya Élektron. 25, 31 (1998). 14. K. N. Ovchinnikov, V. P. Silin, and S. A. Uryupin, Zh. Éksp. Teor. Fiz. 113, 629 (1998). Translation was provided by AIP

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Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 93Ð101. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 105Ð114. Original Russian Text Copyright © 2000 by Belov, A. Ivanov, D. Ivanov, Pal’, Starostin, Filippov, Dem’yanov, Petrushevich. PLASMA, GASES

Coagulation of Charged Particles in a Dusty Plasma I. A. Belov*, A. S. Ivanov*, D. A. Ivanov*, A. F. Pal’**, A. N. Starostin**, A. V. Filippov**, A. V. Dem’yanov**, and Yu. V. Petrushevich** *Kurchatov Institute Russian Science Center, Moscow, 123182 Russia **Russian State Science Center, Troitsk Institute of Innovative and Thermonuclear Research (TRINITI), Troitsk, Moscow oblast, 142092 Russia; e-mail: afpal@ﬂy.triniti.ru, [email protected] Received June 3, 1999

Abstract—An investigation is made of characteristic features in the behavior of small particles in a dusty plasma attributable partly to the suppression of coagulation as a result of monopolar charging for particle sizes smaller than the Debye shielding length and partly to the reduction in the effect of charging for larger particles. Similarity relations linking the plasma composition and particle charge with the parameters of the dust component are used to determine the range of parameters for which the linear approximation of the particle charge as a function of their sizes holds. A modiﬁed classical theory of coagulation in the diffusion approximation is used to study some anomalies in the behavior of the particle size distribution. It is established that unlike an ordinary aerosol, in a dusty plasma the dispersion of the distribution and the average particle size may decrease with time. It is shown for the ﬁrst time that a long-lived “quasi-liquid” state of a dusty plasma may be established as a result of the anomalous behavior of the size distribution function of coagulating charged particles. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION observed, sometimes displaced by the inclusion of various particle charging mechanisms. In particular, if diffu- The problem of small particles in a dusty plasma has sion charging takes place in a weakly ionized atmosphere, attracted interest because of its fairly unique and not an average, negligible, almost symmetric charging is always understandable behavior in laboratory experi- observed with a Boltzmann particle charge distribution. ments. Particular mention should be made of the obser- This is attributable to the low concentration of electrons in vation of quasi-crystalline, levitating, dusty structures an atmosphere containing electronegative gases. In this whose appearance is usually attributed to the strongly case, the charging is accomplished by fluxes of positive nonideal properties of a dusty plasma and the establish- and negative ions having similar mobilities. The incor- ment of long-range particle interaction under these con- poration of other mechanisms such as thermionic or ditions [1]. Since a dusty plasma forms spontaneously photoemission, or electrification accompanying disper- in many cases during the dissociation and subsequent sion, naturally shifts the charge distribution in a partic- nucleation of the initial material in a gas discharge, par- ular direction. ticle coagulation is important for the self-organization processes of a dusty plasma. Studies of particle coagu- Particle charging in a plasma having a fairly high lation in a dusty plasma are also topical in relation to electron density has some important differences. As a material transport and surface contamination during the result of the appreciable difference between the mobil- fabrication of semiconductor devices [2]. Similar top- ities of the electrons and the positive ions, monopolar ics arise in problems associated with plasma-chemical charging takes place where the dispersed particles have technologies for fabricating powder materials [3]. In extremely large negative charges [1]. this last case, coagulation is one of the main mecha- The subject of the present investigation is a plasma nisms for dust particle growth. Depending on the exper- created by the ionization of gases (such as xenon) by imental conditions, a wide range of different objects decay products of radioactive dust particles having the have been observed during the coagulation process, following characteristic parameters: rate of creation of ranging from particles having a complex “cauliflower” electronÐion pairs 1015Ð1016 cmÐ3 sÐ1, pressure ~1 atm, structure to fractal clusters having branching dendritic electron density ~1010Ð1011 cmÐ3, particle charging structures [4, 5]. times 10Ð5Ð10Ð6 s, dust particle sizes up to 100 µm, and The phenomenon of coagulation has been studied in concentrations up to 107 cmÐ3. This plasma may be of fairly great detail in investigations of the behavior of interest for the development of a nuclear battery using various types of aerosols, including charged ones [6]. radioactive waste [7]. In this case, the particle charge In situations encountered in practice, such as, for exam- distribution is closely related to their sizes. Hence, the ple, atmospheric clouds, bipolar charging is generally particle size distribution function is one of the most

94 BELOV et al. important characteristics describing the behavior and ing time determined by the ion diffusion is consider- properties of a dusty plasma. The distribution function ably greater than these figures, but still considerably is formed mainly as a result of the coagulation and dep- shorter than the characteristic times of variation of the osition of dust particles. In some studies [8, 9] the particle size distribution function as a result of coagu- authors have postulated that a quasi-steady-state parti- lation over a wide range of particle sizes and concentra- cle size distribution is established. When this state is tions. In view of this, the problems of particle charging attained, the growth of particles of a particular size is and the formation of the distribution function may be compensated by their losses as a result of coagulation separated and solved systematically. Estimates of parti- and deposition. For very small particles the deposition cle charging in plasmas have been made in various losses are negligible whereas for comparatively large studies [7, 11]. The steady-state charge is determined ones coagulation can be neglected. from the condition that the electron and ion fluxes to a We know [10] that the rate of coagulation may particle are equal. It is easy to show that the particle increase or decrease depending on the polarities and charge q for low dust particle concentrations Nd is a lin- magnitudes of the charges at the particles. The ratio of ear function of the radius: the coagulation constants for charged and uncharged eq = r U, particles was first calculated by Fuks [6]. It was shown d that for a weakly charged bipolar aerosol the increase in where U is the floating potential, rd is the particle coagulation as a result of attraction is compensated to a radius, and e is the elementary charge. However, as Nd considerable extent by the decrease caused by repul- increases, this relationship is destroyed. Since the sion. At the same time, for a strongly charged bipolar dependence of the charge on the particle radius is basic aerosol the increase in coagulation as a result of attrac- for studying the coagulation process, we need to make tion is considerably greater than its decrease as a result a more detailed study of this dependence and estimate of repulsion and the rate of coagulation increases. the range of parameters where it is valid. According to [10], for a monopolar charged aerosol, To be specific, we shall assume the number of ion electrostatic repulsion leads to the separation of like and electron pairs formed per unit plasma volume per charged particles, lowers the rate of coagulation, and unit time is Q. This plasma is created by an external causes a drop in the aerosol concentration. Note that polarizer such as an electron beam or radioactive dust this statement only holds completely when the integral particles. We place aerosol particles in the plasma and electroneutrality of the system is destroyed. estimate their charge in the diffusion approximation. Characteristics of the particle behavior in a dusty We shall assume that the mean free path of the ions and plasma associated, on the one hand, with the suppres- electrons is much smaller than the particle sizes. For sion of coagulation as a result of monopolar charging atmospheric pressure this condition is satisfied for par- for particle sizes smaller than the Debye shielding ticle sizes ≥10Ð4 cm. length Rd and, on the other hand, with the reduction of The plasma composition is determined by the charging for particle sizes larger than Rd, result in these charge conservation equation (the plasma quasineutral- dependences being impaired. ity condition) and the equation of continuity for the In the present paper we investigate the influence of electrons and ions: various anomalies in the coagulation process on the ∂ NÐ β particle size distribution function in a dusty plasma. We ---∂------+ divjÐ = Q Ð NÐ N+, show for the first time that a long-lived “quasi-liquid” t dusty plasma state may be achieved as a result of the ∂ N+ β anomalous behavior of the size distribution function of ---∂------+ divj+ = Q Ð NÐ N+, the coagulating charged particles. t

∑q n + N = N , 2. PLASMA COMPOSITION AND CHARGING k dk + Ð (1) OF SMALL PARTICLES k First, we need to analyze the charge state of the rotE = 0, small particles and establish a relationship between the j = Ð D ∇N Ð K N E, magnitude of the charge, the particle size, and the Ð Ð Ð Ð Ð plasma composition for given conditions (for example, j = Ð D ∇N + K N E. for a given ionization source power). In the present study + + + + + we consider a plasma without negative ions. As has been Here N±, j±, D±, and K± are the concentrations, current noted, when dispersed particles enter a plasma, they densities, diffusion coefficients, and mobilities of the become negatively charged, mainly because the electron ions (“+”) and the electrons (“–”), E is the electric field mobility is substantially higher than the ion mobility. strength, β is the electronÐion recombination coeffi- Ð5 Ð6 The characteristic particle charging time is 10 Ð10 s cient, and ndk = nd(rk) is the concentration of particles [10]. It should be noted that the characteristic discharg- having sizes between rk and rk + 1 (see below). The evo-

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COAGULATION OF CHARGED PARTICLES IN A DUSTY PLASMA 95 lution of the particle size distribution function is con- This expression agrees with the expression to deter- sidered in the following section. mine the charge of a dust particle obtained in the In the steady-state and spherically symmetric case, bounded orbital approximation (see [1]) where the ratio the currents J± are given by [11]: of the ion and electron thermal velocities is naturally replaced by the ratio of the drift velocities or, ultimately, 2 ∂N±  the mobilities in the diffusion approximation. Bearing in J± = 4πer ± D±------Ð K±EN± . (2)  ∂r  ӷ ӷ ∞ ≈ ∞ mind that TÐ T+ and KÐ K+ for the case N+ NÐ , The mobilities K± and diffusion coefficients D± of the we have from (7) ions and electrons are related by the Einstein relationships kT Ðrd  KÐ q = Ð------ln 1 + ------. (8) 2  K  kT ±K± = eD±, (3) e + where k is the Boltzmann constant, and T± is the tem- Thus, the particle charge is negative and is determined perature. by the electron temperature. In particular, for xenon Ð4 We assume that in the region of greatest variation of having the radius rd = 10 cm and kTÐ = 2 eV, we have the ion and electron density the particle charge is not |q| ≈ 104. The mobility ratio in xenon, according to shielded by the plasma, and the electric field in this [12, 13], is K /K ≈ 103 under these conditions. region is determined by the Coulomb law. This assump- Ð + tion is mainly valid when the condition The condition for equal ion and electron densities at L Ӷ a , (4) infinity is satisfied for comparatively low aerosol parti- + d cle concentrations and fairly high-intensity ionization is satisfied where of the gas when the electron density is considerably ∞ higher than the total charge collected at the dust parti- N+ cles per unit volume. However, according to our estimate L+ = --(--∂------⁄--∂------)------of the charge this condition is not satisfied even for dust N+ r r = rd 7 Ð3 11 Ð3 concentrations Nd ~ 10 cm and NÐ ~ 10 cm . is the characteristic size of the region of greatest variation of the ion density and We shall attempt to estimate the influence of changes in the electron density on the charging of dust 1 particles. In the proposed approach this problem can be ad = ------3 ( ⁄ )π solved by redefining the electron and ion density at 4 3 Nd infinity. If condition (4) is satisfied, we can assume that is the mean interparticle distance. Note that condition these quantities are the same as the averages over the (4) is known to be satisfied if the simpler condition for Ӷ plasma volume N± . The plasma composition is deter- estimates: rd ad is satisfied. If the particle charge is | | 2 Ð4 ≈ mined by the equation of quasineutrality and the bal- q ~ 10 and the radius is rd ~ 10 cm, the field is E Ð1 ance equation for the rates of generation and loss of 1500 V cm . The reduced electric field strength at atmo- electrons and ions: ≈ × Ð17 2 spheric pressure is fairly high E/N0 5 10 V cm (N0 is the number of atoms per unit volume) and an N+ = NÐ Ð qNd, electron temperature difference occurs, which may (9) β ⁄ reach 1Ð5 eV in rare gases. Q = N+ NÐ + J+ Nd e. In the steady state when condition (4) is satisfied, the following relation holds for the currents Having supplemented these equations with relation ≈ (7), which links the electron and ion densities far from J+ = ÐJÐ const. (5) the particle, after simple transformations we obtain a By integrating relation (5) for the case of a monodis- system of equations to determine the charge q and the perse aerosol ndk = Nd, qk = q, rk = rd for all particles densities N+ and NÐ : with the boundary conditions ∞ N+ = NÐ Ð qNd, N± ∞ = N± , N± = 0, (6) r = r = rd 2 we obtain an expression linking the ion and electron Q = (β Ð β )N N + β N , (10) densities far from the dust particle: id + Ð id + ∞ { 2 ⁄ } { 2 ⁄ } N K+ 1 Ð exp Ðqe kT Ðrd N K 1 Ð exp Ðqe r kT Ð ------Ð + d Ð -----∞-- = 2 . (7) ------= ------2------, KÐ { ⁄ } K { ⁄ } N+ exp qe kT +rd Ð 1 N+ Ð exp qe rdkT + Ð 1

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4 –1 this ratio q/r is a function of the product N r = ξ. In q/rd, 10 cm d d d –1600 this context the following similarity relations hold: (ξ, ) NÐ = f 1 Q , –1200 1 2 (ξ, ) 3 N+ = f 2 Q , (13) –800 4 ⁄ (ξ, ) 5 q rd = f 3 Q . ξ ξ If we assume that q ~rd, i.e., f3( , Q) = const, then is –400 proportional to the total charge collected on the aerosol particles. Then the relations (13) have the fairly simple

0 1 3 5 meaning that the plasma composition is determined 10–1 10 10 10 only by the source power and the total charge collected ξ, –2 ≠ cm by the aerosol dispersed in it. Note that if f3 const, the parameter ξ no longer characterizes the total charge and Fig. 1. Ratio of charge to radius of dust particles as a func- it is clearly difﬁcult to obtain a simple interpretation of tion of the parameter ξ for various ionization rates. Curves 13 17 Ð3 Ð1 the relations (13). At the same time, the similarity rela- 1Ð5 correspond to ionization rates of 10 Ð10 cm s . tions hold and may be used to analyze the experimental results. Note that the similarity relations for the solu- N , cm–3 tion of the system of equations (10) may be expressed + in the simpler form: 1012 ⁄ ( ⁄ ξ) NÐ Q = f 4 Q , 10 10 ⁄ ( ⁄ ξ) N+ Q = f 5 Q , (14)

8 q ⁄ r = f ( Q ⁄ ξ). 10 1 d 6 2 Figures 1Ð3 give numerical solutions of the sys- 3 tem (10) for various rates of ionization of xenon at 106 4 5 atmospheric pressure. The electron temperature is taken to be 0.25 eV and the ion temperature 300 K. The 4 mobilities reduced to atmospheric pressure are: K = 10 1 3 5 Ð 10–1 10 10 10 3000 cm2 VÐ1 sÐ1 [12] and K = 0.55 cm2 VÐ1 sÐ1 [13]. The ξ, –2 + cm + coefficient of dissociative recombination of Xe2 (the Fig. 2. Average electron density N (ξ) for various ioniza- dominant ions in a xenon plasma at atmospheric pres- Ð sure) at the electron temperature given above is β = 0.9 × tion rates. The correspondence between the curves and the Ð6 3 Ð1 ionization rates is as in Fig. 1. 10 cm s [14]. ξ According to these calculations, the curve q( )/rd β (Fig. 1) has the form of a slightly broadened Fermi step. where id is the effective coefficient of ion recombina- The floating potential of the particles U = eq/rd remains tion at dust particles, which is given by almost constant over a wide range of parameters of the plasma dust component in the range ξ ≤ ξ . The value 4πeK 0 β + of the parameter ξ is determined by the ionization id = ------2------. (11) 0 { ⁄ } 15 Ð3 Ð1 1 Ð exp qe rdkT + source power. In particular, for Q = 10 cm s this state- ξ ξ ≈ 3 Ð2 β ment holds fairly accurately as far as = 0 10 cm . On When condition (4) is satisfied, the expression for id is the plane (r , N ) the region ξ ≤ ξ is positioned simplified and has the form d d 0 between the coordinate axes and the hyperbola Nd = ξ β = 4πeK . (12) 0/rd (see Fig. 4). The particle floating potential id + remains constant provided that the rate of volume loss The expression for ionÐion recombination in the Lan- of electrons and ions as a result of electronÐion recom- gevin theory [13] has the same form when one of the bination considerably exceeds their rate of loss at the ions (in our case a negatively charged dust particle) has surface of the dust particles. From expressions (10) and almost zero mobility. (12) we find that this condition is satisfied in the range of parameters of the dust component where It is easy to see that the dependence of N and N + Ð 2 ⁄ β on the parameters of the plasma dust component is ξ Ӷ e Q ------(------⁄------)-. (15) determined by the ratio q/rd. Thus, according to (10), T Ð ln 1 + KÐ K+

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It is interesting to note that if the initial concentra- –3 N+, cm tions and sizes of the aerosol particles satisfy the con- 12 ξ ≤ ξ 10 dition 0, during the coagulation process, which is accompanied by a drop in the particle concentration and an increase in their sizes, the inequality ξ still 1011 holds. This allows us to investigate the coagulation of dust particles under the condition U = const, deﬁning 10 the initial value of the parameter ξ to be inside the 10 shaded region (Fig. 4). In this case, as can be seen from 1 the system of equations (1), the charging of a polydis- 109 2 perse system of dust particles having the size distribu- 3 tion function ndk will be the same as that for a monodis- 4 108 5 perse system with the parameters N = n and r = 1 3 5 d ∑k dk d 10–1 10 10 10 ξ, cm–2 ∑k ndk rk /Nd. Thus, the results obtained in this section for monodisperse particles can also be applied to poly- ξ ξ ≤ ξ Fig. 3. Average ion density N+ ( ) for various rates of ion- disperse ones provided that the condition 0 is sat- ization. The correspondence between the curves and the ion- isﬁed. ization rates is as in Fig. 1.

3. PARTICLE SIZE DISTRIBUTION FUNCTION 7 –3 nd, 10 cm We shall now investigate the coagulation of particles 3.0 in a dusty plasma. The theory of aerosol coagulation usually starts from the assumption that the particles 2.5 coagulate, i.e., merge or coalesce, on each contact. This can be confirmed in numerous experiments [10]. How- 2.0 ever, if the particles have fairly high, like charges, this assumption is not completely justified because of elec- 1.5 trostatic repulsion. However, it should be borne in mind ξ = ξ0 that aerosol particles are not points. In many cases of practical interest the particles possess fairly high elec- 1.0 trical conductivity so that at distances comparable with the particle sizes polarization plays an important role in 0.5 their interaction. The polarization effect may be so appreciable that it leads to the attraction of like charged dust particles. In addition, shielding of the charges in 0 0.5 1.0 1.5 2.0 2.5 3.0 the plasma also plays an important role in the interac- r , 10–4 cm tion of charged dust particles. Thus, in order to solve d ξ ≤ ξ the coagulation problem systematically, we generally Fig. 4. Region where the condition 0 is satisfied on the need to make self-consistent calculations of the interac- plane (rd, Nd) where the particle floating potential remains tion of at least two nonpoint particles in an ionized gas. almost constant. In the most primitive model of the coagulation of a monodisperse aerosol, to simplify the problem it is assumed that one of the particles is fixed and the fre- where D is the diffusion coefficient of the dust. For quency of contact between this particle and others example, in air the coagulation constant for particles of undergoing thermal motion is determined. The shape 10Ð4 cm is 3.44 × 10Ð10 cm3 sÐ1 [6]. According to for- and size of the fixed particle are assumed to be mula (14), the time taken for the aerosol concentration unchanged. This approach can be used to estimate the to fall to half of N = 3 × 108 cmÐ3 is t = 1/GN ≈ 10 s. change in the concentration of aerosol particles with d 1/2 d time. Assuming that each contact reduces the number The description of the coagulation of a polydisperse of particles by one, we write the fundamental coagula- system having an arbitrary initial particle size distribution tion equation in the form [6] is based on solving the system of transport equations ∂ ( , ) nd rk t dNd 2 ------= ÐGNd . (16) ∂t dt The coagulation constant G is given by = Ð∑(1 Ð (1 ⁄ 2)δ )G(r , r )n (r , t)n (r , t) (18) π ik i k d i d k G = 8 rd D, (17) i

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+ (1 ⁄ 2)∑∑G(r , r )θ k n (r , t)n (r , t), Taking into account the spherical symmetry of the i j ij d i d j problem we find i j ∂ 2 nd  where nd(rk, t) is the concentration of particles having 4πr D------Ð BFn = J = const. (22)  ∂r d radii between rk and rk + 1 at time t, G(ri, rj) are the coagulation constants for particles having radii ri and rj, Note that this equation is quite similar to relation (2) δ θ k used above. The first term on the left-hand side is equal respectively, ik is the Kronecker delta, ij is the inter- polation coefficient (i, j = 1 Ð N), where (V is the vol- to the number of particles passing across the spherical i surface per unit time as a result of diffusion and the sec- ume of a particle of radius ri) [15]: ond term gives the number passing across this surface V Ð (V + V ) as a result of ordered motion in an electric field. In sum θ k = -----k--+---1------i------j--, if (V + V ) ∈ [V , V ], these give the rate of particle deposition on the sphere. ij V Ð V i j k k + 1 k + 1 k The function nd(r) should satisfy the boundary condi- ∞ ( ) tions nd = nd0 for r = and nd = 0 for r = ri + rk, where θ k V k Ð 1 Ð V i + V j n is the initial concentration of dust particles. Using ij = ------, d0 V k Ð 1 Ð V k (19) the boundary conditions and taking into account the if (V + V ) ∈ [V , V ], Einstein relation D = BkT, we find the ratio of the coag- i j k Ð 1 k ulation constants of the charged and uncharged particles (γ = J/J ): θ k 0 ij = 0 otherwise. 1 Ð1   r + r   For the case of diffusive coagulation the constants are γ  1 Ψ  i k  ik = ∫ exp------ik------ dx , given by [6]  kT x   (23) 0 Ψ ( ) ϕ ( ) ( , ) kT ( ) 1 1 ik r = eqk i r , G ri r j = ----η-- ri + r j --- + --- , (20) 3 ri r j ϕ where i is the electrostatic potential and x = (ri + rk)/r. where η is the viscosity of the gas. In the case of monopolar charging of aerosol particles, the polarization effect may play a significant role The system of equations (18) can be solved numer- particularly if the difference in the charges is large. ically using a range of computer programs, in particular However, in the present study we neglect this influence the NAUA code [15]. The NAUA program can analyze in the calculations and only take into account the the main parameters of a coagulating aerosol and study shielded Coulomb contribution its deposition in a given geometry. However, particle charging has not yet been included in the calculations   ϕ ( ) eqi r of this program. Generally, modifying the program to i r = ------expÐ----- , (24) make systematic allowance for the charging of particles r  Rd  in a plasma is fairly complex. However, preliminary estimates of the degree of influence of particle charging where Rd is the Debye shielding radius. In the absence ∞ on the coagulation process and the deformation of the of shielding, i.e., in the limiting case Rd , relation particle size distribution function were made subject to (23) is easily integrated analytically: various simplifying assumptions. λ 2 We shall estimate the coagulation constant of spher- γ ik λ qiqke ik = ---λ------, ik = ------. (25) ik (r + r )kT ical particles having radii ri and rk and charges qi and qk e Ð 1 i k following the logic put forward in [6]. Clearly we shall γ not incur a large error by using the method [6] for par- We shall estimate the value of ik for the case qi = × Ð5 1 ticles of various sizes since the rate constant for diffu- qk = 10, ri = rk = 5 10 cm, T = 300 K. In this case sive coagulation in this range depends weakly on the we have γ ~ 1.5 × 10Ð2. If the particle charge increases particle sizes [10]. We denote the force of the electro- to 50, then γ ~ 10Ð63. In this last case, coagulation is evi- static particle interaction by F(r) where r is the distance dently almost completely suppressed. According to between their centers. In the coagulation process we are these formulas, the ratio of the coagulation constants dealing with the diffusion of Brownian particles toward under the condition ri = rk (in the region of practical an absorbing medium in the presence of a radial electrostatic force under whose action the charged particles 1 Note that the range of variation of the particle sizes is broadened acquire the velocity V = BF, where B is the mobility of appreciably toward smaller sizes by introducing the Cunningham the charged dust particles. In the steady-state regime we correction [6]. Allowance for this correction reduces the coagulation constants of small particles. However, the decrease in coagu- have lation as a result of electrostatic repulsion of the charged particles ∆ ( ) is several orders of magnitude greater than this effect and its con- D nd = Bdiv Fnd . (21) tribution in this range of sizes leads to small quantitative changes.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 1 2000 COAGULATION OF CHARGED PARTICLES IN A DUSTY PLASMA 99 λ interest ik > 1) is a monotonically decreasing function rk, cm of particle size. However, allowance for shielding leads 0.002 to qualitatively different behavior of the coagulation 0.004 constant for the charged particles. We shall analyze in 0.006 greater detail the situation where U = const, i.e., eq = r U i i G, cm3 s–1 and eq = r U, and then k k 3 × 10–9

r + r  2 r r  r + r  Ψ ---i------k = xU ------i----k--- expÐ---i------k . (26) ik  × –9 x ri + rk  xRd  2 10

Accordingly the matrix of the coagulation constants for 1 × 10–9 charged particles having radii ri and rk is described by (r + r )2 ( , ) kT i k 0 G ri rk = ----η------3 rirk (27) 0.002 1   Ð1 × 1 Ψ ri + rk 0.004 ∫ exp------ik------ dx . kT x  0 0.006 r , cm The results of a numerical integration of relations j × Ð4 (27) are plotted in Fig. 5 (U = 0.15 V, Rd = 8.4 10 cm, Fig. 5. Matrix of coagulation constants of a polydisperse T = 300 K). According to these results the behavior of aerosol in a dusty plasma as a function of the colliding par- the coagulation constants in a dusty plasma changes ticle radii. radically. The rate of coagulation of very small charged particles with larger ones is extremely high. This is ﬁrst because of the weak charge at these particles and sec- equation (18) in accordance with relations (26) and (27). ond because of the comparatively high Brownian In the calculation scheme the range of variation of the mobility of these particles. As the size increases, the particle sizes (rmin, rmax) is divided in accordance with particle coagulation constants decrease sharply as a (m Ð 1) ln(r ⁄ r ) = ln(r ⁄ r ), result of the electrostatic repulsion of the charges. k k Ð 1 max min However, if the size of at least one of the two colliding where m is the number of points (m = 60). In order to particles exceeds the Debye length, charge shielding ensure that the largest particles do not build up in the begins to play an important role. In this case, the func- system and distort the distribution function, we intro- tion G(ri, rk) passes through a minimum with increas- duced their deposition. ing particle size, forming a deep dip at whose outer edge the coagulation rate constant approaches the value Our calculations of the dependences of the concentra- typical of an uncharged aerosol. The characteristic tions on the particle sizes taking this factor into account sizes and shape of the dip are determined by the plasma are plotted in Figs. 6Ð8. Graphs of nd(rk, t) for an parameters and thus in real situations their spectrum of uncharged aerosol at various times are plotted for compar- variation is extremely extensive. This leads to various ison (Figs. 6aÐ8a). The initial distribution in Figs. 6a and effects in the behavior of dusty structures during their 6b was taken to be a fairly wide log-normal particle size self-organization process. distribution with a standard geometric deviation of 2, and Ð4 It should be noted that an additional stimulus to the mean radius rk = 10 cm (the initial total concentration of combining of large particles with small ones is pro- dust particles is 108 cmÐ3). The calculations were made for × Ð4 vided by the polarization effect noted above, which a plasma having the Debye length Rd = 8.4 10 cm and leads to the attraction of like-charged particles at dis- ﬂoating potential U = 0.15 V. The range of variation of the Ð6 × Ð3 tances of the order of the radius of the larger particle. If particle sizes is (rmin = 10 cm, rmax = 3 10 cm). As the Debye length is of the same order of magnitude, the was predicted for the uncharged aerosol (Fig. 6a), the electrostatic interaction between the particles may particle concentration initially decreases rapidly and change sign, leading to an increase in the rate of coag- their average size increases. Figure 6b shows curves ulation even compared with its value in an uncharged describing the distribution of like-charged particles in a aerosol. The nonmonotonic dependence of the rate con- dusty plasma. A comparison of Figs. 6a and 6b shows stant on the particle size substantially changes the that charging of the particles leads to a general slowing behavior of the particle size distribution function. of the coagulation process. At the same time, the behav- The formulas obtained can be used to modify the ior of the distribution function also changes qualita- NAUA module describing aerosol coagulation fairly tively. Initially a small fraction which undergoes substan- simply by renormalizing the coagulation constant in tially less slowing of the coagulation than the larger parti-

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6 –3 6 –3 ndk, 10 cm ndk, 10 cm 3.0 3.0 (a) 2.5 0 s 2.5 0 s (a) 10 10 2.0 100 2.0 100 1000 1.5 1000 1.5 1.0 1.0 0.5 0.5 0 0 3.0 3.0 (b) 2.5 0 s 2.5 0 s (b) *** 100 100000 * ** 2.0 2.0 * * 1000 * * 1.5 * 10000 1.5 * * * 100000 1.0 * 1.0 * * 0.5 * 0.5 * * * 0 ********* * * 0 ******* ** ****************** 0.1 1.0 10 0.01 0.10 1.00 10.0 100 –4 –4 rk, 10 cm rk, 10 cm Fig. 7. Dynamics of variation of a narrow particle size distribution in an uncharged polydisperse aerosol (a) and in a × Ð4 dusty plasma (b) with rk = 1 10 cm. The distribution Fig. 6. Dynamics of variation of a wide particle size distri- functions for t = 0 and 105 s are almost the same hence the bution in an uncharged polydisperse aerosol (a) and in a last curve is shifted by one division upward for convenience × Ð4 dusty plasma (b) with rk = 1 10 cm. of representation.

Ð4 cles is rapidly “eaten away.” As a result, clusters with a rk = 10 cm substantially smaller than the Debye length × Ð4 “cauliflower” structure appear, consisting of numerous (Rd = 8.45 10 cm). Here and subsequently the stan- small particles. Collisions between the larger particles are dard geometric deviation of the initial distributions was impeded by the forming Coulomb barrier. As it moves taken to be 0.5 and the range of variation of the particle toward increasing particle size, the distribution function, Ð6 × Ð2 sizes (rmin = 10 cm, rmax = 3 10 cm). Figure 7a corre- as it were, “encounters” the Coulomb barrier. However, sponds to an uncharged aerosol and Fig. 7b to a charged for particles having sizes comparable with or greater than aerosol in the absence of deposition. The calculations the Debye length the electrostatic repulsion effect is show that in a dusty plasma (Fig. 7b) no appreciable reduced and their coagulation becomes quite probable. changes in the distribution profile occur for time inter- The distribution becomes narrower and the dispersion vals up to 107 s. For the uncharged aerosol the concen- decreases. The deposition of larger particles plays a tration decreases by an order of magnitude within decisive role here. In addition, the form of the distribu- ~103 s (Fig. 7a) as a result of coagulation. Figure 8 shows tion more closely resembles a normal one. Another × Ð3 interesting feature of the behavior of the distribution graphs of particle size distributions with rk = 2 10 cm function under these conditions is the possibility of a in an uncharged aerosol (Fig. 8a) and in a dusty plasma × Ð4 decrease in the average particle size with time. with Rd = 8.45 10 cm (Fig. 8b). In this case (rk > Rd) the influence of charging on the rate of coagulation of We have so far analyzed the behavior of a particle the dust particles is not as appreciable as in the previous size distribution with a comparatively broad spectrum. case. However, attention is drawn to the fact that in the If a fairly narrow distribution function with rk < Rd is absence of deposition during the coagulation process taken as the initial distribution and deposition caused the graph of nd(rk) has an additional maximum in the mainly by contact with the walls and by gravity is elim- region of larger sizes (Fig. 8b). inated, a highly charged aerosol may exist for a long Thus, the behavior of the particle size distribution in time without any perceptible changes in the particle a dusty plasma is anomalous and has various interesting concentration. In order to illustrate this, Figs. 7a and 7b features which may lead to some unusual physical show particle size distributions at various times for effects such as the levitation of a long-lived quasi-liq-

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6 –3 effects on structural rearrangement processes in cosmic ndk, 10 cm 3.0 dust formations is of particular interest. However, this (a) aspect requires special study. 2.5 0 s 10 2.0 100 ACKNOWLEDGMENTS 1000 1.5 The authors are deeply grateful to V.Yu. Baranov, V.E. Fortov, and H. Hora for their interest in the work 1.0 and useful discussions. 0.5 REFERENCES 0 1. V. N. Tsytovich, Usp. Fiz. Nauk 167, 57 (1997). 3.0 2. S. J. Choi and M. J. Kushner, J. Appl. Phys. 74, 853 (b) 0 s (1993). 2.5 100 3. S. A. Krapivina, Plasma-Chemical Technological Pro- cesses (Khimiya, Leningrad, 1981). 2.0 1000 4. S. R. Forrest and T. A. Witten, J. Phys. A: Math. Gen. 12, 1.5 L109 (1979). 5. F. J. Huang and M. J. Kushner, J. Appl. Phys. 81, 5960 1.0 (1997). 6. N. A. Fuks, Mechanics of Aerosols (Md., Army Chemi- 0.5 cal Center, 1958; Akad. Nauk SSSR, Moscow, 1955). 0 7. V. Yu. Baranov, I. A. Belov, A. V. Dem’yanov, et al.,. Pre- 1 10 100 1000 print No. IAE-6105/6 (Moscow, Institute of Atomic –4 rk, 10 cm Energy, 1998); I. A. Belov, A. S. Ivanov, D. A. Ivanov, et al., Pis’ma Zh. Tekh. Fiz. 25, 89 (1999). Fig. 8. Dynamics of variation of a narrow particle size dis- 8. G. M. Hidy and J. R. Brock, J. Coll. Sci. 20, 123 (1965). tribution in an uncharged polydisperse aerosol (a) and in a 9. G. M. Hidy and J. R. Brock, The Dynamics of Aerocol- × Ð3 dusty plasma (b) with rk = 2 10 cm. loidal Systems (Pergamon, Oxford, 1970). 10. P. C. Reist, Introduction to Aerosol Science (Mir, Mos- cow, 1987; Macmillan, London, 1984). uid dusty structure in an electrostatic trap. The lifetime 11. B. M. Smirnov, Aerosols in Gases and Plasmas (Inst. of an uncharged aerosol having a particle concentration Vysokh. Temp. Akad. Nauk SSSR, Moscow, 1990). higher than 109 cmÐ3 is extremely limited and its prop- 12. J. J. Dutton, J. Phys. Chem. Ref. Data 4, 557 (1975). erties have been little studied. Thus, the creation of a 13. B. M. Smirnov, Ions and Excited Atoms in Plasmas stable aerosol with such a high particle density is not (Atomizdat, Moscow, 1974). only of applied interest but also of purely scientific 14. J. B. A. Mitchell, Phys. Rep. 186, 215 (1990). interest. 15. H. Bunz, M. Kouro, and W. Schock, NAUA-MOD4ÐA To conclude, it should be noted that this investiga- Code for Calculating Aerosol Behaviour in LWR Core Melt tion was carried out in the diffusion approximation. Accidents, Code Description and Users Manual, KfK-3554 However, qualitatively similar results are also obtained (Kernforschungszentrum, Karlsruhe, 1983). for low pressures and smaller particles, i.e. in cases of molecular flow. In this context, the influence of these Translation was provided by AIP

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