A General Exact Solution of the Einstein–Dirac Equations with The

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 619Ð628. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 707Ð716. Original Russian Text Copyright © 2004 by Zhelnorovich. GRAVITATION, ASTROPHYSICS

A General Exact Solution of the EinsteinÐDirac Equations with the Cosmological Constant in Homogeneous Space V. A. Zhelnorovich Institute of Mechanics, Moscow State University, Vorob’evy gory, Moscow, 119992 Russia e-mail: [email protected] Received May 29, 2003

Abstract—We have obtained a general exact solution of the system of EinsteinÐDirac equations with the cosmological constant in homogeneous Riemannian space of the ﬁrst type according to the Bianchi classiﬁcation. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ant way. Using this gauge allows the number of unknown functions in the EinsteinÐDirac equations to We face two problems when integrating the Ein- be reduced by six, while keeping the equations invari- steinÐDirac equations. ant under transformation of the observer’s coordinate The first, purely technical problem stems from the system. fact that the EinsteinÐDirac equations constitute a complex system of nonlinear partial differential equations With the tetrad gauge used here, all of the equations of the second order for 24 unknown functions. Previ- can be written as equations of the first order only for ously, some of the particular exact solutions [1Ð6] to two invariants of the spinor field and the Ricci rotation the EinsteinÐDirac equations in homogeneous space symbols of the proper vector bases determined by the have been obtained only for diagonal metrics of the spinor field. The tetrad gauge transforms the Dirac Riemannian event space. equations to equations for the Ricci rotation symbols and for the spinor field invariants, with the Ricci rota- The second problem is fundamental in nature and tion symbols in the Dirac equations being linear and ψ stems from the fact that the spinor field functions in without derivatives. Therefore, the Dirac equations in the Riemannian event space can be determined only in homogeneous Riemannian space close the system of certain nonholonomic orthonormal bases (tetrads) Einstein equations for the Ricci rotation symbols with- which that must be specified, or a tetrad gauge is said to out using additional equations. In this case, we can first be needed. A large number of such gauges are known, integrate the first-order equations for the Ricci rotation and different authors have suggested various gauges. symbols and the spinor field invariants and then inte- All of these are either noninvariant under transforma- grate the first-order equations for the tetrad (Lamé) tion of the variables of the observer’s coordinate system coefficients. or are written in the form of differential equations, which complicates the initial system of equations. These two factors—the reduction in the number of unknown functions by six and the possibility of inte- Physically, all gauges are equivalent, because the grating the second-order equations in two steps — con- EinsteinÐDirac equations are invariant under the choice siderably simplify the EinsteinÐDirac equations. As a of tetrads. Mathematically, however, using a bad gauge result, it becomes possible to obtain new exact solu- (i.e., additional equations that close the EinsteinÐDirac tions of these equations. equations) can greatly complicate the equations, while using a good gauge can significantly simplify them. A general exact solution of the EinsteinÐDirac equations in homogeneous Riemannian space of the first In many respects, the problem of choosing a reason- type according to the Bianchi classification was able tetrad gauge stems from the fact that the solutions obtained in [7Ð9]. Recently, new papers in which the of the EinsteinÐDirac equations have previously been authors discuss models described by the Einstein equa- obtained only for diagonal metrics. Since the basis vec- tions with the cosmological constant (including those tors of a holonomic coordinate system for such metrics with the spinor fields) have appeared. In this paper, we are orthogonal, the tetrads associated with the orthogo- obtain a general exact solution to the system of Ein- nal holonomic basis of the Riemannian space can be steinÐDirac equations with the cosmological constant chosen naturally. in homogeneous Riemannian space in connection with In this paper, we use a new tetrad gauge [7] that is increasing interest in studying the role of the cosmolog- algebraic and, at the same time, is formed in an invari- ical constant.

620 ZHELNOROVICH 2. THE SPINOR FIELDS complex conjugation; the invariant spinor of the second IN FOUR-DIMENSIONAL RIEMANNIAN SPACE rank β is given by the equations Let V be the four-dimensional Riemannian space γ T βγ βÐ1 βú T β with metric signature (+, +, +, Ð) referred to a coordi- ú a ==Ð a , . (4) nate system with variables xi and with a holonomic vector basis Ji (i = 1, 2, 3, 4). We specify the metric tensor In general, the four-component spinor field ψ has of space V in basis Ji by the covariant components gij; two real invariants, ρ and η, that can be determined by the connectivity is defined by the Christoffel symbols using the equation Γs . In space V, we introduce a smooth field of ij + + 5 i ρexp()iη = ψ ψ + iψ γ ψ, (5) orthonormal bases (tetrads) ea(x ) (a = 1, 2, 3, 4) in the form where i a ea ==ha Ji, Ji hi ea, (1) γ 5 1 εabcdγ γ γ γ = ------a b c d, a i 24 where hi and ha are the scale factors. Below, we denote the indices of the tensor components calculated εabcd are the components of the four-dimensional LeviÐ in basis Ji by the Latin letters i, j, k, …, and the indices Civita pseudotensor, ε1234 = –1. Using the spinor field ψ of the tensor components calculated in the orthonormal and the conjugate spinor field ψ+ in Riemannian space bases ea by the first Latin letters a, b, c, d, e, and f. V, we can determine the proper orthonormal vector The differential of the vectors of the orthonormal basis eù of the spinor field: i a basis ea(x ) is defined by the Ricci rotation symbols b i i i i i de = ∆ , e dx , which can be expressed in terms of the π ξ σ a ia b ùe1 ====Ji, ùe2 Ji, eù 3 Ji, eù4 u Ji. scale factors The vector components πi, ξi, σi, and ui are specified by ∆ 1[ j ()∂ ∂ the relations [7, 10] iac, = --- hc ih ja Ð jhia 2 (2) j ()∂ ∂ b j s ()]∂ ∂ ρπi ==Im()ψT Eγ iψ , ρξi Re()ψT Eγ iψ , Ð ha ih jc Ð jhic + hi hahc jhsb Ð sh jb . (6) ρσi ψ+γ iγ 5ψ ρ i ψ+γ iψ ∂ ∂ ∂ i ==, u i , Here, i = / x . i Let us define the spinor field of the first rank, y(x ), in which the spin tensors γ i = hi γ a satisfy the equation in Riemannian space V specified by the contravariant a components ψA(xi) (A = 1, 2, 3, 4) in the orthonormal i j j i ij i γ γ γ γ bases ea(x ). The spinor indices are juggled by using the +2= g I. ψA ABψ ψ ψB formulas = e B and A = eAB . In these formulas, || || Ð1 || AB|| ù i E = eAB and E = e are the covariant and contra- Clearly, the scale factors ha that correspond to the variant components of the metric spinor, respectively, proper basis eù are defined by the matrix given by the equations a

γ T γ Ð1 T 1 1 1 1 a ==ÐE aE , E ÐE. (3) π ξ σ u 2 2 2 2 γ i π ξ σ Here, T is the transposition symbol; a are the four- ù u ha = . (7) dimensional Dirac matrices, which, by definition, sat- π3 ξ3 σ3 u3 isfy the equation π4 ξ4 σ4 u4 γ γ γ γ a b +2b a = gabI, γ β || || If the spintensors a, E, and are specified as where I is a unit four-dimensional matrix; and gab = diag(1, 1, 1, Ð1) are the covariant components of the metric tensor in the orthonormal basis ea. 000i 00 01 Let us also define the conjugate spinor field by the γ = 00i 0 , γ = 00Ð0 1 , ψ+ ψ+ 1 2 covariant components = A using the relation 0 Ð00i 01Ð00 ψ+ = ψú T β , in which the dot above the letter means Ð000i 10 00

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the covariant derivatives for the partial derivatives. 00i 0 00i 0 Eqs. (11) are identically valid by the deﬁnition of ρ, η,

γ 000Ði γ 000i and ∆ù , . 3 = , 4 = , (8) sij Ð000i i 000 0 i 00 0 i 00 3. THE EINSTEINÐDIRAC EQUATIONS WITH THE COSMOLOGICAL CONSTANT 0100 0010 Let us consider the system of equations E ==Ð0001 , β 0001 , γ a∇ ψ ψ 0001Ð 1000 a +0,m = 0010 0100 1 R Ð ---Rg + λg = κT , ab 2 ab ab ab then the components of the spinor ψ calculated in the (13) ù ρ η 1[∇ψ+γ ψ proper basis ea are defined by the invariants and as T ab = --- a b follows [7, 10]: 4 ()γ∇ ψ+ ψψ+γ ∇ ψ∇()γψ+ ψ] Ð b a + b a Ð a b . 1 2 1 i ψùù==0, ψ i ---ρexp ---η , 2 2 Here, ψ is the four-component spinor field in four- (9) dimensional Riemannian event space V specified in the 3 4 1 i λ κ ψùù==0, ψ i ---ρexp Ð---η . orthonormal basis ea; m, , and are constants; R =  ab ab 2 2 g Rab is the scalar curvature of space V, R are the Ricci tensor components calculated in the basis e ; ψ a The covariant derivatives of the spinor fields and gab = diag(1, 1, 1, Ð1); and T are the energyÐ ψ+ ab in Riemannian space are known to be given by the momentum tensor components for the spinor field in equations [11] the basis ea.

1 i j Since Eqs. (13) are invariant under an arbitrary ∇ ψ∂= ψ Ð ---∆ , γ γ ψ, s s 4 sij pseudoorthogonal transformation of the tetrads ea, the (10) equations that define the tetrad ea must be added to ∇ ψ+ ∂ ψ+ 1ψ+∆ γ iγ j s = s + --- sij, . close Eqs. (13). These additional equations are com- 4 monly called gauge conditions. As the gauge conditions for the tetrads ea, we assume that an arbitrary ea tetrad The following equations are valid for the covariant in Eqs. (13) is identical to the proper tetrad of the spinor ψ ψ+ derivatives of the spinor fields and : ψ i field , i.e., ea = eù a . For this gauge, the scale factors ha ù i ∇ ψ 1 ∂ ρ 1γ 5∂ η 1∆ù γ iγ j ψ in Eqs. (13) are identical to the coefficients ha defined s = ---I s ln Ð --- s Ð --- sij, , 2 2 4 by matrix (7). (11) In this case, the Dirac equations can be written as ∇ ψ+ ψ+1 ∂ ρ 1γ 5∂ η 1∆ù γ iγ j s = ---I s ln Ð --- s + --- sij, . ∆ù 2 2 4 equations for the Ricci rotation symbols abc, and the invariants of the spinor field [7, 10]: Here, the invariants of the spinor field are defined by b Eq. (5), and the Ricci rotation symbols, ∆ù , = ù∂ ρ∆ù σ η ijk a ln+ ba, = 2m ù a sin , ùùùb c∆ (14) h j hk ibc, , correspond to the proper bases of the spinor ù a 1 abcd a ∂ η +2---ε ∆ù , = mσù cosη. field and can be calculated by using the formula 2 bcd 1 ∆ù , = ---(ξπ ∇ π Ð π ∇ π + ∇ ξ Ð ξ ∇ ξ Here, sij 2 i s j j s i i s j j s i (12) σ ∇ σ σ ∇ σ ∇ ∇ ) ∂ùùi ∂ {}πi∂ ,,,ξi∂ σi∂ i∂ + i s j Ð j s i Ð ui su j + u j sui . a ==ha i i i i u i

Equations (11) in four-dimensional pseudo-Euclid- is the differentiation operator along the vectors of the ean space were obtained in [7, 12]. Eqs. (11) in Rie- σù σùa mannian space can be derived from the corresponding proper basis, and a = = (0, 0, 1, 0) are the compo- equations in pseudo-Euclidean space by substituting nents of the vector eù 3 in the proper basis.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 622 ZHELNOROVICH ∆ù The symbols abc, in Eqs. (14) are related to the Equations (14)Ð(20) in the given coordinate system ù i xi form a closed system of equations for the functions scale factors ha by π j ξ j σ j j ρ j η j i(x ), i(x ), i(x ), ui(x ), (x ), and (x ). The metric tensor components for Riemannian space are related to 1 j j ∆ùùùùùùùùùù[ ()∂ ù ∂ ()∂ ∂ π j ξ j σ j j abc, = --- ha bh jc Ð ch jb + hc ah jb + bh ja the functions i(x ), i(x ), i(x ), and ui(x ) by 2 (15) Ð hùùùj ()]∂ h + ∂ ùùh . ij ù i ù j ab πiπ j ξiξ j σiσ j i j b a jc c ja g ==hahbg ++ Ð u u . (21) Equations (14) are identical to the Dirac equations in the proper basis eù a . These equations can also be 4. THE GENERAL EXACT SOLUTION derived from the Dirac equations in system (13) by OF THE ENSTEINÐDIRAC EQUATION changing the derivatives in them using formula (11) and IN HOMOGENEOUS SPACE by subsequent algebraic transformations. Let us consider the four-dimensional Riemannian ù i ∆ù Substituting ha for abc, in Eqs. (14) leads to the space referred to a synchronous coordinate system with following system of invariant tensor equations [7, 10]: variables xi in which, by definition, the following equations hold: ∇ ρπi ∇ ρξi i ==0, i 0, 44 4α α ,, g44 = g = Ð 1, g4α = g = 0, = 1 2 3. (22) ∇ ρσi ρη∇ ρ i i ==2m sin, i u 0, (16) We will seek a solution of Eqs. (14)Ð(20) in the syn- ∇iη 1εijms(π ∇ π ξ ∇ ξ Ð --- j m s + j m s chronous coordinate system by assuming that all of the 2 sought-for functions depend only on the parameter x4 = t. Thus, the event space is assumed to be a homo- σ ∇ σ ∇ ) σi η + j m s Ð2u j mus = m cos . geneous space of the first type according to the Bianchi classification. It is convenient to write the Einstein equations in the In this case, Eqs. (16) can be written as proper basis eù a as ∂ ()Ðgρπ4 = ∂ ()Ðgρξ4 = ∂ ()Ðgρu4 = 0, ùùκ 1κ ρηλ 4 4 4 Rab = T ab + --- m cos + gab. (17) 2 ∂ ()ρσ4 ρη 4 Ðg = 2mgÐ sin , (23) ∂ η σ4 η To transform the Einstein equations to (17), we 4 = Ð2m cos . should take into account the fact that, in view of (13), the following equation holds: Equations (23) and the equation R ==ÐκT a + 4λ κmρηcos + 4λ. (18) a g44 ≡ π4π4 ++ξ4ξ4 σ4σ4 Ð u4u4 = Ð1 (24) An expression for the tetrad components of the π4 ξ4 σ4 4 energyÐmomentum tensor in the gauge ea = eù a was form a complete system for the functions , , , u , obtained in [10]: η, and ρ Ðg . The general solution of Eqs. (23) and (24) is [8, 9] ù 1ρ σ ∂ù ησ∂ù η T ab = --- Ð ù b a Ð ù a b --- 4 4 4 4 Cρ π ξ u 1 (19) ------= ------= ------= ------= ------, 1 cde cde ρ Cπ Cξ Cu 2 2 + ---σù ()∆ù , ε + ∆ù , ε . Ðg 1 + Cσ cos ()2mt + ϕ 2 e acd b bcd a ε ()ϕ σ4 Cσ sin 2mt + The tensor components Rù can be expressed in = ------, (25) ab 2 2 ()ϕ terms of the Ricci rotation symbols as 1 + Cσ cos 2mt + 1 + iCσ cos()2mt + ϕ ùùùù1 ∂ []()ùj ∆ c j ∆ c exp()iη = ε------, Rab = ------j Ðg hc ba, Ð hb ca, 2 2 Ðg (20) 1 + Cσ cos ()2mt + ϕ ∆ù c∆ù f ∆ù c∆ù f Ð fb, ca, + ca, fb, , ϕ ≥ where , Cπ, Cξ, Cσ, Cu 1, and Cρ > 0 are the integra- || || ε where g = det gij . tion constants; the coefficient can take on any of the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A GENERAL EXACT SOLUTION OF THE EINSTEINÐDIRAC EQUATIONS 623 two values: +1 or Ð1. In view of the synchronism con- In view of solution (25), the quantity ρ Ðgcosη on dition (24), the constants C are related by the right-hand side of the first equation in (31) is a constant: 2 2 2 2 Cπ ++Cξ Cσ Ð Cu = Ð 1. (26) ρ Ðgcosη = εCρ. (32) Let us denote h = hù 4 = (π4, ξ4, σ4, u4). We find from a a The first equation in (31) can be obtained by con- solution (25) that tracting the Einstein equations (17) with the components of the tensor δa + h ha in index a. The second {}h ,,h h c c 1 2 4 equation in (31) can be obtained by contracting the Ein- 1 {},, (27) stein equations (17) with the components of the tensor = ------Cπ Cξ Cu . ab a b 2 2 g + 2h h in indices a and b. 1 + Cσ cos ()2mt + ϕ Contracting the first equation in (31) with gab in Thus, we see that, in view of solution (25), the direc- indices a and b yields the equation tion of the three-dimensional vector with the components h1, h2, and h4 does not depend on the parameter t. 3 ∂ ∂ Ðg =3---κmεCρ + λ Ð,g (33) 4 4 2 It follows from definition (15) that the Ricci rotation ∆ symbols, a, bc, can be represented as that defines the quantity Ðg . ∆ù 1() λ abc, = --- hbsac Ð hcsab Ð haabc , (28) If the cosmological constant > 0, then the solution 2 of Eq. (33) is where, by definition, εκm Ðg = ------λ 2 (34) ùùi ∂ ù i ∂ ù sab ==sba ha 4hib + hb 4hia, × []()λ ()λ (29) Cρ Ð1 ++f 1 sinh 3 t f 2 cosh 3 t , ùùi ∂ ùùi ∂ aab ==Ðaba ha 4hib Ð hb 4hia. where f1 and f2 are arbitrary constants. Since the quantities ha in (28) are defined by solu- For λ < 0, we obtain tion (25) as functions of the parameter t, Eq. (28) ∆ù εκ expresses the 24 dependent functions abc, only in m {}[]λ() Ðg = ------Cρ Ð1 + f sin 3 ttÐ 0 , (35) terms of the 16 functions sab and aab. 2λ It follows from Eqs. (14) that the antisymmetric quantities a are defined by the equality where f and t0 are the integration constants. The case of ab λ = 0 was considered in [7Ð9]. []()ηεσ σ σc d η Since the direction of the three-dimensional vector aab = 4m ùùahb Ð bha sin Ð abcd ùh cos . (30) with the components h1, h2, and h4 does not depend on the parameter t, the components h1 and h2 can be made Using Eqs. (30) and definitions (19) and (20) for equal to zero by a constant Lorentz transformation of Tù and Rù , we can write the Einstein equations (17) ab ab the basis vectors eù 1 , eù 2 , and eù 4 . It is easy to see that as the equivalent system of equations the initial system of equations (14)Ð(20) is invariant under an arbitrary Lorentz transformation of the vectors ∂ () ()σc η ù ù ù 4 Ðgsab Ð 2mgÐ hasbc + hbsac ù sin of the basis e1 , e2 , and e4 that is independent of the variables xi. Therefore, it will suffice to consider the solution of Eqs. (14)Ð(20) only for h1 = h2 = 0. Under η 1κρ ()ε f ε f cσù e Ð2 Ðgmcos + --- cefasb + cefbsa h this condition, the first equation in (31) can be written 8 (31) in expanded form as = ()κmρ Ðgcosη + 2λ Ðg ()g + h h , ab a b ∂ ()σ4 η 4 s33 Ðg Ð 4m sin s33 Ðg ()a 2 ab ()κρ ηλ ()εκ λ ()4 2 sa Ð8sabs = mcos + . = mCρ + 2 Ðg u ,

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4 ∂ ()σ η 1 4 4 s23 Ðg Ð 2m sin s23 Ðg s = Ð---ρu Bcos()2()ζβ+ , 12 2 1 4 (37) Ð2mcosη + ---κρ u s Ð0,g = 1 4 2 4 13 s = ---ρ()u Acos()ζα+ , 13 2

4 ∂ ()σ η 1 4 2 4 s13 Ðg Ð 2m sin s13 Ðg s = ---ρ()u Asin()ζα+ , 23 2 1 4 +2mcosη + ---κρ u s Ð0,g = 4 23 2ρσ4()4 21 ∂ s34 = --- u N + ------4 Ðg , 3 CρCu ∂ ()η 1κρ 4 4 s11 Ðg + 22mcos + --- u s12 Ðg  1 4 4 4 s = ---ρu σ Acos()ζα+ , 14 2 = εκmCρ + 2λ Ð,g 1 4 4 s = ---ρu σ Asin()ζα+ . 24 2 ∂ ()η 1κρ 4 4 s22 Ðg Ð 22mcos + --- u s12 Ðg 4 Here, A, B, N, α, and β are arbitrary constants, Ðg in solution (37) is deﬁned by equalities (34) and (35), and = εκmCρ + 2λ Ð,g ζ (36) is given by 1 ∂ () η κρ 1 4 4 s12 Ðg Ð 2mcos + --- ζ = 2mcosη + ---κρ u dt 4 ∫4 × 4() (38) u s11 Ð s12 Ð0,g = tan()2mt + ϕ 1 = εarctan------+ ---κτ. 2 4 1 + Cσ ∂ ()η 1κρ 4 s14 Ðg + 2mcos + --- 4 The parameter τ in (38) depends on the cosmologi- λ × 4 4 η cal constant and is deﬁned by the integral u s24 Ð2g Ð0,mu sin s31 Ðg = τρ= ∫ u4dt. 1 ∂ ()s Ðg Ð 2mcosη + ---κρ In view of Eqs. (18) and (32), the scalar curvature R 4 24 4 of the Riemannian event space can be expressed in × 4 4 η terms of Ðg : u s14 Ð2g Ð0,mu sin s23 Ðg =

εκmCρ 4 4 λ ∂ ()s Ðg Ð 2msinησ()s + u s Ðg R = ------+ 4 . (39) 4 34 34 33 Ðg 4 4 = ()σεκmCρ + 2λ Ðg u , It thus follows that the points at which Ðg becomes ∂ ()4 η 4 s44 Ðg Ð 4mu sin s34 Ðg zero are the singular points of the curvature tensor. 2 4 Substituting the components sab from (37) into the = ()σεκmCρ + 2λ Ðg (). second equation of system (31) yields a relation The general solution of Eqs. (36) is between the integration constants A, B, N and f1, f2. For λ > 0, we obtain ρ 4 1 2 ∂ 1 ()()ζβ λ s11 = u Ð ---N ++------4 Ðg ---Bsin 2 + ,2 2 21 2 1 2 1 2 ≤ 3 3CρCu 2 f 2 Ð1f 1 = Ð ------Cu---A ++---B ---N 1. κ2m2 4 4 3 4 1 2 1 s = ρu Ð ---N +,------∂ Ðg Ð ---Bsin()2()ζβ+ 22 4 2 ≥ 2 3 3CρCu 2 If f 2 f 1 , then formula (34) for Ðg can be represented as 2ρ()4 31 ∂ s33 = --- u N + ------4 Ðg , εκm 3 CρCu Ðg = ------Cρ{}Ð1 + f cosh[]3λ()ttÐ , (40) 2λ 0 2 4 4 21 s = ---ρu ()σ N + ------∂ Ðg , 44 4 2 2 2 3 CρCu where t0 and f are arbitrary constants, f = f 2 Ð f 1 .

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The singular points of the solution are deﬁned by the general solution of Eqs. (14)Ð(20), it will now sufﬁce to equation integrate Eqs. (29), from which it follows that

[]λ() 1 a f cosh3 ttÐ 0 = 1. ∂ hù = ---()s + a hù . (44) 4 ib 2 ab ab i The solution has one singular point at f = 1 and two sin- ≤ ε a gular points for 0 < f < 1. If f 0 (in this case, = Ð1), Given deﬁnition (7) of hù and solutions (30) and then the solution has no singular points. i (37) for sab and aab, Eqs. (44) can be transformed to 2 2 If f 2 < f 1 , then the following equality is valid for d 4 4 Ðg : ----- ()u σ Ð σ u dτ j j εκm 1 Ðg = ------Cρ{}Ð1 + f sinh[]3λ()ttÐ , (41) = ---A[]π cos()ζα+ + ξ sin()ζα+ 2λ 0 4 j j

4 4 1 d 1 2 2 2 ()σ σ in which f =f Ð f . In this case, the solution has one + u j Ð u j ------Ðg + ---N , 1 2 3 Ðgdτ 3 singular point defined by the equation d f sinh[]3λ()ttÐ = 1. ----- ()ξξ + iπ = ()+ iπ (45) 0 dτ j j j j ± If f2 = f1 = f, then × 1 d 1 2m η ------Ðg Ð ---N Ð i------cos εκm 3 Ðgdτ 6 ρ Ðg = ------Cρ{}Ð 1 + f exp()± 3λt . (42) 2λ i Ð ---()ξ Ð iπ Bexp[]Ð2i()ζβ+ For λ < 0, we obtain the following formula for the 4 j j integration constant f in Eq. (35): i ()4σ σ4 []()ζα + ---Au j Ð u j exp Ði + . λ 4 2 21 2 1 2 1 2 ≥ f = 1 Ð ------Cu---A ++---B ---N 1. κ2m2 4 4 3 The system of equations (45) must be complemented with the synchronism condition In this case, there is an infinite number of singular ≡ σ σ α ,, points at which Ðg becomes zero. g4α 4 α Ð u4uα ==0, 123. (46) The positivity condition, Ðg > 0, imposes con- In view of Eqs. (25), we obtain straints on the possible values of the integration constants and the domain of existence of the solution. τρ4 dt ==∫ u dt CρCu∫------. Using definition (9) and solution (25) for ρ and η, let Ðg us write out the solution for the spinor field components in the proper basis: If the cosmological constant is positive, λ > 0, and κm ≠ 0, then using Eqs. (40) and (41), we obtain 0 λ τ 2 Cu dt ()ϕ = ------∫------(47) 1 + iCσ cos 2mt + εκm Ð 1 + f sinh[]3λ()ttÐ i εCρ------0 ψ ± 2 Ðg ù = , (43) and 0 ()ϕ 2λC dt ε 1 Ð iCσ cos 2mt + τ = ------u ------. (48) i Cρ------εκm ∫ []λ() 2 Ðg Ð 1 + f cosh 3 ttÐ 0 For λ < 0 (f 2 > 1), using (35), we find that where Ðg is given by (35) or (40)Ð(42), depending on the sign of λ. λ 2 Cu dt Equations (25), (30), and (37) completely define the τ = ------∫------. (49) εκm Ð 1 + f sin[]Ð3λ()ttÐ ∆ù 0 Ricci rotation symbols abc, by formula (28) and constitute the first integral of Eqs. (14)Ð(20). To find the Integrals (47)Ð(49) can be found in [13].

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Let us now change the unknown functions (πλ, ξλ, written as 0 0 σλ, uλ) ()πλ,,ξλ θλ in Eqs. (45):  ξ0 π0 ()α 1 τ λ + i λ = exp Ði ÐAHλ exp---N  2 0 0 1/3 1 ξλ +iπλ = ()ξλ + iπλ ()Ðg expÐ ---Nτ Ð iζ , 6 +Ð[][]κ + Bexp()2i()αβÐ Fλ + 4iQGλ cos()Qτ (53) 1/3 4 1 σλ = θλ()Ðg u exp Ð---Nτ , (50) 6  +Ð[][]κ + Bexp()2i()αβÐ Gλ Ð 4iQFλ sin()Qτ , 1/3 4 1  uλ = θλ()Ðg σ exp Ð---Nτ , 6 1 θλ = []κ Ð Bcos()2()αβÐ Hλ exp ---Nτ where ζ is given by Eq. (38), σ4 and u4 are defined by 2 solution (25), and Ðg is defined by equalities (35) or + AF[]λ cos()Qτ + Gλ sin()Qτ . (40)Ð(42). As a result, the synchronism condition (46) is satisfied identically, and the system of equations (45) Here, transforms to a system of linear equations with constant 1 2 2 2 Q = --- κ Ð A Ð,B coefficients: 4

d 0 0 i 0 0 Fλ, Gλ, and Hλ are the integration constants. ----- ()ξλ + iπλ = ---κξ()λ + iπλ dτ 4 For the spatial components of the metric tensor, we obtain the following expression using Eqs. (21) and (50): i 0 0 i Ð ---Bexp()ξÐ2iβ ()λ Ð iπλ + ---Aiexp()θÐ α λ, (51) 4 4  ()2/3 1 τ 2 ()τ d 1 0 0 1 gαβ = Ðg expÐ---N Hα HβZ exp N ------θλ = ---A()πλ cosαξ+ λ sinα + ---Nθλ. 3 dτ 4 2 

+ FαFβ[]SM++cos()2Qτ 8QNsin()2Qτ At j = 4, Eqs. (45) are satisﬁed identically in view of the conditions h1 = h2 = 0. + GαGβ(SMÐ cos()2Qτ Ð 8QNsin()2Qτ ]

The following characteristic equation for the eigen- + ()FαGβ + FβGα (54) value q corresponds to Eqs. (51) for each value of the × []M sin()2Qτ Ð 8QNcos()2Qτ index λ = 1, 2, 3: 1 + ()Fα Hβ + Fβ Hα 2κAexp ---Nτ cos()Qτ 2()N Ð 2q ()16q2 + κ2 Ð A2 Ð B2 2 (52) 2 + A []2NBÐ sin()2()αβÐ = 0.  ()κ 1 τ ()τ + Gα Hβ + Gβ Hα 2 Aexp---N sin Q , 2  In general, the solution of Eq. (52) is given by the Cardano formulas. A simple solution of this equation is α β α β where 2N = Bsin( Ð ), and we use the following nota- obtained, for example, for A = 0 or 2N = Bsin(2( Ð )). tion for the constants: The solution of Eqs. (51) for A = 0 was considered in [8, 9]. MA= 2 ++B2 κBcos2()αβÐ , Let us consider the case where the equation 2N = κκ[]()()αβ Bsin(2(α Ð β)) holds, A ≠ 0. In this case, the eigenvalues S = + Bcos 2 Ð , are Z2 = A2 + []κ Ð Bcos()2()αβÐ 2.

1 1 2 2 2 1 2 2 2 ---N, --- A +,B Ð κ Ð--- A +.B Ð κ The quantity Ðg on the right-hand side of equal- 2 4 4 ity (54) is deﬁned by Eqs. (35) or (40)–(41). Formula (54) deﬁnes the oscillatory approach to the singular points If 0 < A2 + B2 < κ2, then the solution of Eqs. (51) can be of the solution.

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The metric is particularly simple when the phases α Calculating the energyÐmomentum tensor of the and β satisfy the condition α Ð β = kπ, k = 0, ±1, ±2, …: spinor field for the solutions obtained yields the following relation for its component T44 in the synchronous 2/3 ()Ðg coordinate system: αβ g = ------2 16Q ε ρη mCρ T 44 ==m cos ------. (60) SM+ cos()2Qτ M sin()2Qτ 2κAQcos()τ Ðg × M sin()2Qτ SMÐ2cos()2Qτ κAQsin()τ . 2 Since, by definition, Cρ > 0, the above solutions with 2κAQcos()τ 2κAQsin()τ Z ε = Ð1 correspond to a negative energy density; for ε = 1, the energy density of the spinor field is positive. In this case, N = 0; A and B are arbitrary; and the Let us consider the transformation of the variables constants Fλ, Gλ, and Hλ are given by the equalities of the observer’s coordinate system (xα, t) (xα, τ) τ ρ 4 {}()κ2 2 Ð1/2,, defined by the equation d = u dt. The explicit depen- Fα = Ð B 00, dence of the function τ(t) is given by relations (47), {},,()κ2 2 Ð1/2 (55) (48), or (49). Our calculation shows that the following Gα = 0 Ð B 0 , equation holds in the coordinate system with variables α 2 2 Ð1/2 x and τ: Hα = {}00,,()κ Ð B .

2 2 κ2 ij d 4i If A + B > , then the trigonometric functions []∂ ()Ðgg ' ==----- ()Ðgg '0. (61) in (54) are substituted with hyperbolic functions. j dτ The case where the integration constants A and B in Eqs. (51) are equal to zero, A = B = 0, corresponds to a Thus, the coordinate system with variables xα and τ diagonal metric gij. In this case, is harmonic. It is easy to see that the parameters t and τ/Cρ, where the integration constant Cρ is defined by solution (25), have the same dimensions. ξ0 π0 κ()i κτ λ + i λ = Ð Fλ + iGλ exp--- , 4 After the transformation (xα, t) (xα, τ), the spatial part of the metric does not change. It should be θ κ 1 τ noted that the time dependence of the determinant γ = λ = Hλ exp---N , 2 det||gαβ|| of the spatial metric tensor components, which (56) governs the expansion or contraction of the Universe, is κ2()2/3 1 τ ()distinctly different in the synchronous and harmonic gαb = Ðg expÐ---N FαFβ + GαGβ 3 coordinate systems. This difference is responsible for the significant difference between the evolutionary sce- 2 τ + exp---N Hα Hβ . narios of the Universe in the synchronous and harmonic 3 coordinate systems. In conclusion, we note some of the qualitative dif- Here, Ðg takes the form of (35) or (40)Ð(42). ferences between our solutions of the Einstein equa- If we define Fλ, Gλ, and Hλ by the relations tions with the cosmological constant λ ≠ 0 in the synchronous coordinate system and the solutions of these Ð1 Ð1 Fα =={}κ ,,00, Gα {}0,,κ 0 , equations without any cosmological constant. (57) λ ≠ {},,κÐ1 (1) At 0, there are solutions without singular Hα = 00 , points (solution (40) for f > 1, ε = Ð1 and f < 0, ε = Ð1). then metric (56) is diagonal: (2) There are solutions with a horizontal asymptote for Ðg ; i.e., the Universe can expand only to a certain 2/3 ÐNτ/3 ÐNτ/3 2Nτ/3 gαβ = ()Ðg diag{}e ,,e e . (58) limit defined by the integration constants (solution (42)). For A = B = N = 0, metric (54) corresponds to an iso- (3) There are solutions with an infinite number of tropic Universe: singular points. In this case, the Universe passes through a closed cycle of evolution (solution (35)). 2 2/3 gαβ = κ ()Ðg ()FαFβ ++GαGβ Hα Hβ . (59) REFERENCES 2 Here, Ðg is given by formula (40) in which f = 1 for 1. C. J. Isham and J. E. Nelson, Phys. Rev. D 10, 3226 λ > 0; for λ < 0, Eq. (35) in which f 2 = 1 holds. (1974).

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2. C. J. Radford and A. N. Klotz, J. Phys. A: Math. Gen. 16, 9. V. A. Zhelnorovich, Gravit. Cosmol. 2, 109 (1996). 317 (1983). 10. V. A. Zhelnorovich, Dokl. Akad. Nauk SSSR 296, 571 3. P. Wils, J. Math. Phys. 32, 231 (1991). (1987) [Sov. Phys. Dokl. 32, 726 (1987)]. 4. Yu. P. Rybakov, B. Sakha, and G. N. Shikin, Izv. Vyssh. 11. V. Fock, Z. Phys. 57, 216 (1929). Uchebn. Zaved. Fiz., No. 7, 40 (1994). 12. V. A. Zhelnorovich, Dokl. Akad. Nauk SSSR 311, 590 5. V. G. Bagrov, A. D. Istomin, and V. V. Obukhov, Gravit. (1990) [Sov. Phys. Dokl. 35, 245 (1990)]. Cosmol. 2, 117 (1996). 6. G. Platania and R. Rosania, Europhys. Lett. 37, 585 13. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, (1997). Integrals and Series. Elementary Functions (Nauka, Moscow, 1981). 7. V. A. Zhelnorovich, Theory of Spinors and Its Applica- tions (Avgust-Print, Moscow, 2001). 8. V. A. Zhelnorovich, Dokl. Akad. Nauk 346, 21 (1996). Translated by V. Astakhov

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 629Ð642. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 717Ð733. Original Russian Text Copyright © 2004 by Vladimirskiœ, Bruns. GRAVITATION, ASTROPHYSICS

On the Presence of Fictitious Solar Neutrino Flux Variations in Radiochemical Experiments B. M. Vladimirskiœ* and A. V. Bruns Crimean Astrophysical Observatory, pos. Nauchnyœ, Crimea, 98409 Ukraine *e-mail: [email protected] Received April 4, 2003

Abstract—The currently available data on solar neutrino flux variation in radiochemical experiments and Cherenkov measurements have so far defied a simple interpretation. Some of the results concerning these variations are indicative of their relationship to processes on the solar surface. It may well be that a poorly understood, uncontrollable factor correlating with solar activity indices affects the neutrino flux measurements. This factor is assumed to modulate the detection efficiency on different detectors in different ways. To test this assumption, we have analyzed all available radiochemical measurements obtained with the Brookhaven (1970−1994, 108 runs), GALLEX (1991Ð1997, 65 runs), and SAGE (1989Ð2000, 80 runs) detectors for possible instability of the detection efficiency. We consider the heliophysical situation at the final stage of the run, the last 7Ð27 days, when the products of the neutrino reaction with the target material had already been accumulated. All of the main results obtained previously by other authors were found to be reproduced for chlorineÐ argon measurements. The neutrino flux anticorrelates with the sunspot numbers only for an odd solar cycle. A similar behavior is observed for the critical frequencies of the E-ionosphere. The neutrino flux probably correlates with the Ap magnetic activity index only for an even solar cycle. The predominance of a certain sign of the radial interplanetary magnetic field (IMF) in the last 14 (or 7) days of the run has the strongest effect on the recorded neutrino flux. The effect changes sign when the polarity of the general solar magnetic field is reversed and is most pronounced for the shortest runs (less than 50 days). The dependence of the flux on IMF polarity completely disappears if the corresponding index is taken for the first rather than the last days of the run. The IMF effect on the recorded neutrino flux was also found for short runs in the galliumÐgermanium experiment, but this effect for a given time interval in the SAGE measurements is opposite in sign to that detected with the Brookhaven and GALLEX detectors. The anticorrelation with the Ap activity index, which is absent in the SAGE measurements, contributes significantly to the flux variations on the GALLEX detector. If a magnetic storm with a sudden commencement occurs in the last 7 days of the run, then an effect of generally the same type, a significant increase in the variance, is observed for all three detectors. In all other indices, the flux variations on the Brookhaven and GALLEX detectors are the opposite of those on the SAGE detector. We have found that the GALLEX and SAGE measurements for the runs that ended simultaneously, to within about 10 days or less, anticorrelate, while the Brookhaven and GALLEX measurements correlate. We conclude that there are fictitious variations in the measurements under consideration that are attributable to the influence of geophysical factors (probably, very-low-frequency electromagnetic fields) controllable by solar activity on the physicalÐchemical kinetics of the target material. We discuss possible experiments to check whether the detected effects are real. If all of these effects are indeed real, then the neutrino flux was underestimated in the radiochemical measurements. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 1.1. The ChlorineÐArgon Experiment The question of possible solar neutrino variations in (1) There is a significant anticorrelation between the a chlorineÐargon experiment that was raised by neutrino flux and the global solar activity index, the G.T. Zatsepin back in the 1960s and initiated by Bazi- sunspot numbers R, only after 1977 [3, 4]. No convinc- levskaya et al. [1] has already been discussed for almost ing interpretation of the absence of such an anticorrela- two decades. Many authors have independently ana- tion for 1970Ð1976 has been offered. lyzed the accumulated experimental data by using var- (2) A similar anticorrelation is observed if the ious heliogeophysical indices. If we summarize the photospheric magnetic fields measured with a magne- results of this work, then the following picture emerges tograph are used as a heliophysical index. The anticor- (all of the necessary information about the experiment relation proves to be statistically significant only for itself and a bibliography of previous publications are low-heliolatitude data (the disk center [5, 6]). It is also given in the monograph [2]). more pronounced for 1977Ð1990, with the spring

630 VLADIMIRSKIŒ, BRUNS months of each year making an important contribution 1.2. The GalliumÐGermanium Experiments to such variations. For obvious reasons, the question of neutrino flux (3) Most of the authors agree that there is an annual variations in the GALLEX and SAGE experiments is period in the intensity variations [7Ð9]. This period is less clear. It seems that no correlation with the sunspot found when analyzing data irrespective of any helio- number is found in these measurements [25], although physical indices [10, 11] and when correlating the flux the annual period is probably present. Two specific Q with the sunspot numbers R [3, 8] and the coronal results deserve rapt attention. index (the intensity of the λ = 5303 Å line [12]). All of (1) The GALLEX measurements revealed a period the authors interpret the annual period as resulting from close to one of the solar rotation modes (28 days). the Earth’s spatial displacements relative to the helio- There may also be a period that coincides with the well- equator. known harmonic of the solar free inertial oscillations (about 157 days) [25]. A period close to 28 days may be (4) There is a weak positive correlation between the present in the Brookhaven experimental data [25]. neutrino flux and the magnetic activity level (the A p (2) There is a remarkable feature in the set of data index [13, 14], the aa index [6]). This correlation is usu- from both galliumÐgermanium experiments: the distri- ally interpreted as resulting from variations in solar bution of their results exhibits two peaks. The bimodal wind parameters. It seems to be more pronounced for pattern of the distribution could be indicative of the the wind density estimated in direct measurements existence of two discrete intensities [26]. The time (1973Ð1993 [15]). It is not quite clear how this correla- scale of the transition between the states was estimated tion matches with the anticorrelation between the neu- by the authors to lie within the range of 10Ð60 days. trino flux and the sunspot numbers. The situation described above must be complemented (5) There is a positive correlation between the neu- by data on the high-energy neutrinos observed with trino flux and the Galactic cosmic-ray intensity mea- Cherenkov detectors. According to KAMIOKANDE sured with neutron monitors. Most of the authors agree data [27], the neutrino flux shows no anticorrelation that this correlation is not causal, but results from the with the sunspot numbers for solar cycle 22 (a weak anticorrelation between Q and R. The correlation correlation is not ruled out). The absence of intensity between the neutrino flux and chromospheric flares has variations that correlate with solar activity in these and not been confirmed. subsequent measurements calls into question the reality of the neutrino flux variations in the radiochemical (6) A significant linear anticorrelation has been experiments. observed between the neutrino flux and the frequency shift of the free solar acoustic oscillations (P-modes, 1980Ð1991 [16]). It is not clear whether this anticorre- 2. THE EXPERIMENT lation is causal. Since the above shift anticorrelates AND THEORETICAL MODELS with the solar radius, the neutrino flux may be assumed The behavior patterns described above must be to positively correlate with the solar-radius variations. compared with the following three major types of mod- (7) A set of periods is found in the neutrino flux vari- els that describe the variations. ations. Various (more than ten) authors are unanimous (1) The neutrino source in the solar core is stable; that a quasi-biennial period (2.17 ± 0.03 years) is variations arise when the neutrino flux propagates observed. As regards other periods, the results slightly through the solar matter due to the transitions between disagree. The periods of 8Ð9 and 4.5Ð5 years are quoted various neutrino states, in particular, due to its interac- most commonly (see, e.g., [17Ð20]). Clearly, the peri- tion with solar subphotospheric magnetic field, as long odograms (power spectra) must exhibit a one-year as the neutrino is postulated to have a large magnetic period; it is actually present. At the same time, it also moment (for the physics of these phenomena, see [28]). often exhibits a period of 1.3 years. A big surprise was (2) The neutrino source in the core pulsates due to the the detection of one of the solar rotation modes excitation of special oscillations; these oscillations are (28.34 days) in the neutrino flux variations [21]. synchronized with the solar activity variations [29, 30]. (8) There is compelling evidence suggesting that (3) The neutrino flux is stable both in the solar core many authors have overestimated the statistical signifi- and near the Earth; variations arise from unnoticed vari- cance of their results concerning the neutrino flux variations in the detection efficiency in the instrument ations and the corresponding correlations [22Ð24]. itself; i.e., they are fictitious. These variations are con- However, we cannot agree with the opinion that all of jugate with the solar activity variations, because the the above results are attributable to statistical fluctua- solar activity itself affects the measuring technique tions. Most of them are consistent with the well-known through geophysical fields, acting as an uncontrollable patterns of solarÐterrestrial physics. For example, factor [31]. almost all of the variation periods (listed in (7)) are the Many publications are devoted to the first type of well-known cosmophysical periods. models. A familiarity with them shows that a large

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ON THE PRESENCE OF FICTITIOUS SOLAR NEUTRINO FLUX VARIATIONS 631 number of various fitting assumptions are made when In the model being described, the correlation with comparing experimental data with specific computa- solar activity arises, because the state of the target liq- tional models. Therefore, the popular belief that the uid depends on the parameters of the background geo- problem of neutrino flux variations has been solved physical fields, in particular, very-low-frequency elec- almost completely is a gross exaggeration. tromagnetic fields controllable by solar activity. When Ar37 is trapped into a clathrate structure, for example, In contrast, the idea of solar core oscillations the number density of these structures in the target receives little support. These oscillations, if they actu- depends on the liquid run in the fields mentioned above. ally exist, must seemingly be synchronized with the natural timer—the oscillations attributable to a motion In the past, this kind of reasoning could not be con- of the Sun with respect to the system’s barycenter. The vincingly justified and was perceived as speculation. In periods found for the parameters of this motion (in the last decade, the situation has changed significantly. years)—2.41, 3.51, 4.26, … [32]—are in poor agree- The following arguments now force us to treat the idea ment with the periods found in the chlorineÐargon of modulation of the detection efficiency by solar activ- experiment (see Section 1.7). ity in earnest. First, the pattern of the dependence of neutrino flux To most researchers, the third type of models seems variations on heliogeophysical indices closely follows to be completely implausible. In this case, the main idea 37 71 the patterns attributable to the surface effects of solar is that the reaction products (radioactive Ar and Ge ) activity that are known from solarÐterrestrial physics. in the target liquid may prove to be bonded somehow Significantly, the 28-day period with its ghosts found with a certain molecular structure and do not always by Sturrock et al. [21] is precisely the period that was fall into the counting system. Of course, such bonding observed in the sunspot numbers at the epoch under is possible in principle at a certain stage of the complex study [25]. Further, the heliophysical indices taken for technique of extracting a small number of product the central zone of the solar disk are known to correlate atoms from a large number of target atoms. This possi- better with the parameters of purely terrestrial pro- bility seems to be ruled out, because the extraction effi- cesses than the same disk-averaged indices do. Charac- ciency is controlled in real time by adding a known teristically, the neutrino flux shows a correlation both number of atoms of the corresponding stable isotope to with the global solar indices (pertaining to the entire the target material. However, this procedure is not quite 36 disk) and with the solar-wind (magnetic activity) correct: neutral Ar atoms are added for control, while parameters that reflect the processes in a narrow zonal the neutrino reaction product appears in the form of a region for a given heliolatitude. In solarÐterrestrial fast ion. It may well be that the bonding into a structure physics, this applies to two different connection chan- will be different in these cases. This well-known prob- nels through the short-wavelength radiation (iono- lem of a hot ion can be solved by adding an appropriate sphere) and the solar wind (magnetosphere). radioactive isotope to the target material, where the product being extracted also appears in hot form. In this Second, direct evidence for the instability of the case, however, the uncertainty is not completely elimi- detection efficiency in the chlorineÐargon experiment nated either: this experiment only simulates the real sit- has been obtained. For example, for some reason, there uation, without replicating it in all details. In addition, is a strange anticorrelation between the flux Q and the these changes are isolated episodes in nature. Whether run time [4, 24]. If this anticorrelation is real, then it the extraction efficiency changes from session to could correspond to continuous bonding of the reaction session in normal working conditions is still an open product, its fictitious destruction, in certain time inter- question. vals. The anticorrelation between the flux Q and the background correction [24] appears no less strange. +37 +71 The bonding of Ar or Ge ions can include var- Third, a number of examples where the effects of ious reactions. Until now, only one type of this process solar activity on the processes in condensed phases has been discussed: falling into a molecular cavity trap. were found with confidence have been accumulated to In particular, three possible “containers” for the neu- date. The directly acting agent—the very-low-fre- trino reaction product have been proposed for Ar+37: quency electromagnetic background disturbances— Jacobs [33] assumed that the globules produced by per- acts in all these cases as an uncontrollable factor in the chloroethylene polymerization could play an important given experiment. The widely known example is a reg- role in the possible argon bonding. Subsequently, argon ular increase in the measured biochemical reaction during the cooling may prove to be bonded in the struc- rates during each solar minimum over three 11-year ture formed in a microbubble (babstone) [34]. If there cycles [35]. The above studies (of the so-called macro- is water in the form of an admixture in perchloroethyl- scopic fluctuations) based on a comparison of histo- ene, then the formation of a gas hydrate of the second grams have revealed characteristic periods of about type—an A á 2B á 17H2O structure, where A and B are 27 days and about a year for normal measurements of the cavities populated by various molecules, radicals, the count rate of radioactive standards [36]. A behavior and ions—is probable. Cavity B can be a cavitand for of the same type in an 11-year cycle was also discov- Ar37 [31]. ered when analyzing the gravitational constant mea-

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632 VLADIMIRSKIŒ, BRUNS

Table 1 were used as the source data. We took unweighted (best-fit) values. The heliogeophysical indices—the Revealed interval Runs sunspot numbers R and the Ap magnetic activity indi- Even cycle 20, 1970Ð1976 18Ð46 ces—were taken from the SolarÐGeophysical Data. The index I of the mid-latitude E-ionosphere was tabu- Odd cycle 21, 1977Ð1986 47Ð91 lated in [42]. The IMF signs restored from geophysical Even cycle 22, 1987Ð1994 92Ð133 data were taken from the catalog of laboratory polar Plus at solar north pole 18Ð64; 109Ð133 measurements (Institute of Terrestrial Magnetism, Iono- sphere and Radio-Wave Propagation, IZMIRAN) [43]. Minus at solar north pole 65Ð108 It is clear from general considerations that an optimal time interval for which the effect of heliogeophysical variations on the target material is most pronounced surements using a torsion pendulum [37]. In this case, must exist. Since this interval is not known, we used the point of action of the uncontrollable factor is the three different intervals in many cases: 27, 14, and 7 torsion pendulum suspension filament. This conclusion days. They were counted backward from the comple- was reached after studying the dependence of the mea- tion date, which was the main reference date of our surements with this detector on the index used analysis. As is customary in solarÐterrestrial physics, below—the sign of the radial interplanetary magnetic the evenÐodd solar cycles are considered separately. It field (IMF) [38]. Semiannual variations of seemingly is of no less importance to distinguish the epochs at the same nature [39] (for an overview of the solar which the general solar magnetic field changes sign. activity effects on the “technosphere,” see the mono- Table 1 presents the distribution of runs (their numbers) graph [40, Chapter 7]) are now known even in neutrino for the selected time intervals. The sign of the field in mass measurements. In general, as long as the solar polar solar regions is known to be established relatively activity effects, which apparently modulate the detec- slowly. Therefore, referring a certain run to the interval tion efficiency in many cases, do not seem absurd, a of a particular polarity at such transitional epochs is special search for similar effects in radiochemical controversial. experiments is quite justifiable. Although the targets in The diagrams presented in the figures were con- these experiments are placed at relatively large depths, structed by using the same standard method: the helio- they are nevertheless within the reach of electromag- physical index obtained for a given run (its final part) netic disturbances. For typical electrical conductivities allowed it to be placed in a certain bin of this index. The of rocks, the skin depth for a frequency of 8 Hz is sev- Ð1 eral kilometers. In addition, there is a significant com- fluxes Q (in atoms day ) were averaged within this bin. ponent of lithospheric origin in the variations of these The indices R (sunspot numbers) and Ap (magnetic geophysical fields. This component of the electromag- activity) were used for comparison with the corre- netic background also varies synchronously with solar sponding data by other authors. The ionospheric and activity. IMF data are physically more meaningful. If the neutrino flux variations in the radiochemical The neutrino flux Q is plotted against the sunspot experiments are attributable not to variations in the numbers in the last 27 days of the run in Fig. 1 (Q was number of reaction product atoms, but to variations in averaged for each R bin). The distinct anticorrelation the number of these atoms extracted from the target for the odd solar cycle corresponds to the anticorrela- material, then the heliogeophysical situation at the end tion found for approximately the same time interval by of the run is of great importance in analyzing the other authors who used the mean R for the entire run results. In the following sections, we present some of (see, e.g., [4]). For the even cycles, we clearly see a the results of our analysis of data from the chlorineÐ weak opposite tendency. The ionospheric index I exhib- argon experiment and the GALLEX and SAGE experi- its a similar pattern (Fig. 2) in the last 27 days of the ments in connection with the solar activity and mag- run; I has the meaning of a corrected daily mean critical netic disturbance variations when the corresponding frequency. The similarly constructed data for the Ap indices were calculated for the final stage of the run index are shown in Fig. 3. The index was averaged over (less than 25% of the session duration). the last 7 days of the run. In this case, we see a tendency

for Q to increase with Ap for the even cycle (this ten- 3. CORRELATION OF THE NEUTRINO FLUX dency for the entire data set was also found by other IN THE RADIOCHEMICAL EXPERIMENTS authors who used the mean indices for the entire run WITH HELIOGEOPHYSICAL INDICES time). AT THE END OF THE RUN Finally, Figs. 4Ð6 show plots of Q against IMF sign 3.1. The ChlorineÐArgon Detector in the last 14 days of the run (the sum of the numbers of The measurements processed by the maximum-like- days of positive, negative, and mixed polarities divided lihood method [41] (a total of 108 runs, 1970Ð1994) by 14). We clearly see a tendency for Q to decrease

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ON THE PRESENCE OF FICTITIOUS SOLAR NEUTRINO FLUX VARIATIONS 633 with increasing dominance of the positive polarity in Q, Ar37 atoms dayÐ1 Fig. 4a and the change in sign of the effect when the 0.9 polarity of the general solar magnetic ﬁeld is reversed. n = 108 2 In general, this pattern does not change if the IMF signs 9 0.7 6 are taken in the last 7 and 27 days of the run, but for 14 6 5 7 days it is move distinct. Figure 5 shows data convolu- 5 10 0.5 13 9 tion for the entire data set (Fig. 4b was inverted relative 17 to the zero line). According to the MannÐWhitney test, 0.3 10 7 the general means for different polarities differ at the Ð2 10 signiﬁcance level. We can see from Fig. 6 that the 0.1 2 effect is systematic in nature; in the means, it manifests itself in a redistribution of extreme values: 80% of the 0 30 60 90 120 150 180 210 values close to zero belong to one IMF sign, and all of R (27 days)

Q > 1.0 belong to the other IMF sign. Fig. 1. Mean neutrino flux versus sunspot numbers R averaged over the last 27 days of the run for the even (thin solid It is easy to verify by direct comparison that the and dash-dotted lines) and odd (heavy solid and dashed dependence of Q on the IMF sign contributes apprecia- lines) solar cycles. Q was calculated for all of the runs that bly to the annual variation. According to the standard fell within a given R bin. The numbers near the dots give the number of such runs. The error is a typical rms scatter; the ColemanÐRosenberg law, the phase of this variation is straight lines were drawn by least squares; n is the number reversed when the polarity of the general solar mag- of runs. netic field is reversed. It is important to emphasize that the dependence Q, Ar37 atoms dayÐ1 Q (R), which is distinct for the odd solar cycle (see 0.8 n = 99 Fig. 1), holds irrespective of the IMF sign. If we con- 3 struct such a dependence separately for all cases of dif- 12 0.6 5 ferent IMF polarities, then it will remain valid. Thus, 11 the dependences Q (R) and Q (IMF) cannot be reduced 5 to each other and represent different effects. It is partic- 0.4 ularly important to take this into account when analyz- 8 ing the dependence of Q on the run time found in [4]. 0.2 It is shown in the form adopted here in Fig. 7a and holds irrespective of the solar activity level (R) and the IMF sign. There is no such general anticorrelation for the 160 200 240 280 320 360 400 even cycle. However, it has emerged that this kind of I (27 days) relationship exists for different IMF polarities (Fig. 7b). As we see, the tendencies are opposite for dif- Fig. 2. Same as Fig. 1 for the index I of the mid-latitude ferent IMF signs. Note that the dependences in Fig. 7b E-ionosphere at the Moscow station. The even solar cycle is 37 represented by the thin solid line; the odd solar cycle is rep- cannot be explained by the Ar bonding with a constant resented by the heavy solid and dashed lines. rate (the aging of the solution), as should seemingly be done for the case of Fig. 7a. The curves in Fig. 7b were constructed in such a way that, most likely, there is no bonding for one of the IMF signs at short runs. The Ð1 n = 108 effect in a weakened form for longer runs proves 0.9 7 noticeable because of the inertia of the index used. If the IMF sign is taken for the final 7 days of the run 0.7

(rather than 14 days, as in Fig. 7b), then this difference atoms day 7 3 in ﬂuxes Q disappears abruptly at a run time of L = 37 0.5 17 Ar 60 days. In general, the IMF sign index differs only , 16 14 slightly from one 27-day interval to another. Therefore, Q 0.3 when this interval is used, the effect for L < 80 days dis- 5 10 15 20 25 30 35 appears completely. However, when the IMF signs are Ap (7 days) taken for 7 and 14 days at L ≤ 60 days, the changes in − ≈ Fig. 3. Same as Fig. 1 for the Ap magnetic activity index in the means are the same: Q ( )/Q (+) 3 (in all cases, the last 7 days of the run. The even solar cycle is represented when the sign of the general solar magnetic ﬁeld by the heavy solid and dashed lines; the odd solar cycle is changed, the IMF polarities also changed in accordance represented by the thin solid line.

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27 = 62 (a) 109 49 n

47 1.0 119 50 37 51 Ð1 56 129 58 130 112 29 122 114 54 atoms day 118 116 120

37 115 36 55 62 64 39 42 Ar 41 125 , 0.5 46 18 Q 128 124 30 48 24 19 40 111 45 110 32 33 27 22 59 20 126 113 38 53 30 121 63 27 60 21 43 61 0 62 35 127 44 –0.8 –0.6 –0.4 –0.2 0 +0.2 +0.4 +0.6 +0.8 IMF (14 days)

71 (b) 86 n = 42

1.0 106

Ð1 100 99 91 80 98 97 94 78 atoms day 95 72 89

37 85 102 84 79 92 Ar 75 105 , 0.5 82 69 Q 107 74 76 77 66 68 103 81 96 65 67 88 108 83

101 73 87 104 70 0 –0.8 –0.6 –0.4 –0.2 0 +0.2 +0.4 +0.6 +0.8 IMF (14 days) Fig. 4. “Neutrino flux Q—mean IMF polarity sign in the last 14 days of the run” scatter diagram: (a) the general solar magnetic field at the north pole is positive; (b) the general solar magnetic field at the north pole is negative. The numbers near the dots give the number of runs. with the pattern shown in Fig. 4). The Q difference is ces for the final and initial run intervals may be shown statistically significant at the 1.7 × 10Ð3 level for an to be uncorrelated. Therefore, we once again con- interval of 30Ð50 days and at the 4.8 × 10Ð2 level for an structed a convolution of the dependence of Q on the interval of 50Ð60 days. IMF sign, but for the initial 14 days of the run. We found no correlation between Q and the IMF sign: the The effect of the IMF sign in the last days of the run corresponding values were Q (Ð) = 0.476 ± 0.041 and on the recorded neutrino flux Q , which is important for Q (+) = 0.479 ± 0.048 (see Fig. 5, where the corre- the subsequent analysis, can be tested independently. sponding means differ by a factor of 1.6; the standard For the set of runs used in the experiment under consid- deviations pertain to the scatter of points with different eration (a median value of about 70 days), the IMF indi- signs).

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3.2 The Gallium–Germanium Detectors— Q, Ar37 atoms dayÐ1 the GALLEX and SAGE Experiments 1.0 The well-known published data (GALLEX Coll., n = 104 1993, 1994, 1996, and 1999 [44Ð48], 65 runs in 1991Ð 1997) served as the source data for our analysis of the GALLEX experiment. For a comparison with magnetic 0.5 activity, we also used 19 runs of the continuation of these measurements (GNO, 1998Ð1999). The SAGE measurements were received from the authors of the 0.557 ± 0.042 0.341 ± 0.056 0 experiment (1989Ð2000, 80 runs). The method of anal- –0.8 –0.4 0 0.4 0.8 ysis was similar to that used above. All of the cosmophysical data were counted for the last 7 days of the IMF (14 days) run. Since the IMF polarity estimate for a 7-day interval Fig. 5. Convolution of the data shown in Figs. 4a and 4b may contain a significant error due to the erroneous after the inversion of the data in Fig. 4b relative to the zero determination of the field sign for a single day in the IMF line. The difference between the means, 0.557 ± 0.042 IZMIRAN catalog, the data were checked by using and 0.341 ± 0.056, for the negative and positive IMF signs, independent direct measurements (the OMNI data- respectively, is statistically significant at the 10Ð2 level. base). The gaps in these data were filled by interpolation. The IMF sign was assumed to have been determined reliably only when it was the same in both types 30 of data. For the intervals of the GALLEX and SAGE 1 n = 104 measurements, the sign proved to be uncertain for 25% 20 2

and 10% of the runs, respectively (a trivial factor, geo- , % magnetic disturbances, is responsible for the discrep- n 10 ancy between the data in the catalogs mentioned above in half of the cases). Although, at first glance, the run statistics for the galliumÐgermanium experiments is 0 0.2 0.4 0.6 0.8 1.0 1.2 comparable to that for the chlorineÐargon measure- 37 Ð1 ments, analysis of the GALLEX and SAGE data runs Q, Ar atoms day into additional difficulties. Since the transition from Fig. 6. The distribution (frequency of occurrence) of mea- (even) solar cycle 22 to odd solar cycle 23 occurred in sured Q for the positive (1) and negative (2) IMF polarities. the spring of 1996, these measurements do not completely cover a cycle of this type. The possible variations through the two channels of solarÐterrestrial rela- in the same notion in Fig. 8. The corresponding data are tionships mentioned above are largely independent. presented in Table 2. As we see, the GALLEX data for Therefore, for a correlation between the neutrino flux the measurement interval under consideration behave and a given cosmophysical index to be found, we must in exactly the same way as the Brookhaven data: the consider only those data that were obtained at constant intensity is higher for the negative IMF polarities for values of another index. This approach cannot be short runs and is close to the mean for longer runs (Fig. consistently implemented for the accumulated body of data. 8a). For the SAGE data, the effect is less distinct and is opposite in sign (Fig. 8b). It is all the more surprising that the very important result on the intensity variations of the chlorineÐargon On average, the intensity is generally slightly higher experiment (see Fig. 7b) is also reproduced for the gal- for short runs for the two experiments under consider- liumÐgermanium measurements. This result is shown ation (as in the chlorineÐargon experiments, about

Table 2

Experiment Run “+” IMF n “–” IMF nP(U*)

GALLEX about 21 days 42.4 ± 69.5 7 94.4 ± 45.7 10 0.13 about 28 days 73.3 ± 39.9 18 75.6 ± 43.4 10 Ð SAGE 35 days 84.6 ± 60.9 20 63.1 ± 49.9 15 0.24 about 45 days 64.4 ± 35.4 10 70.9 ± 50.3 14 Ð

Note: n is the number of runs, and P(U*) is the signiﬁcance level.

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Q, Ar37 atoms dayÐ1 Q, Ar37 atoms dayÐ1 6 0.7 n = 43(+1) (a) 1 0.7 (b) 14 12 11 12 0.5 8 12 0.5 11 n = 40(+ 4) 10 0.3 10 9 2 9 0.3 n = 59(+ 2) 3 0.1 6 0.1 30 50 7090 110 30 50 70 90 110 L, days L, days

Fig. 7. (a) The relationship between mean neutrino flux Q and run time L found in [4]. The data pertain to the odd solar cycle 21. (b) The neutrino flux Q for the given run time L for the positive (curve 1) and negative (curve 2) IMF signs in the last 14 days of the run. The numbers near the dots give the number of runs in the given interval L. A typical rms scatter is shown for an interval of 70Ð80 days. Six points from the entire data set were excluded from our analysis (the IMF polarity is equal to zero, L > 110). Accord- ing to the MannÐWhitney test, the difference between the mean Q for an interval of 30–50 days is statistically significant at the 1.7 × 10Ð3 level.

100 94.4 ± 45.7(10) (a) 100 2 (b) 84.6 ± 60.9(20) 75.6 ± 43.4(10) 2 70.9 ± 50.3(14) 73.3 ± 39.9(18) 63.1 ± 49.9(15) 1 64.4 ± 35.4(10) Intensity, SNU Intensity, 50 SNU Intensity, 50 1 42.4 ± 69.5(7)

0 1020 30 0 3040 50 L, days L, days Fig. 8. The neutrino ﬂux (SNU) for the run time under consideration for the positive (curve 1) and negative (curve 2) IMF signs in the last 7 days of the run. The numbers are the corresponding means with their standard deviations; the number of runs is given in parentheses. The same as Fig. 7b for GALLEX (a) and SAGE (b).

20%). More importantly, a significant (by a factor The values of Ap > 25 correspond with a high prob- of 1.7) increase in the variance is characteristic of short ability to the falling of a magnetic storm with a sudden runs. This increase may be considered as evidence for a commencement within the seven-day interval in ques- higher sensitivity of the measurements to the influence tion. This is a special type of global electromagnetic of the uncontrollable factor precisely for short runs. disturbance. Such events should be considered separately. The corresponding counts are summarized in Figure 9 shows an “A p (7 days) – intensity” scatter dia- Table 4. (No magnetic storms appear in the right-hand gram for the GALLEX measurements. There is a weak part of Table 3 and Fig. 9). tendency for an anticorrelation (Ð0.22 ± 0.09), which is To facilitate our comparison of the various experi- also observed separately for the 19 GNO runs. It is ments, the last column in Table 4 gives the data normal- more distinct for short runs and is definitely absent for ized to the corresponding means. In all cases, an effect the SAGE data. This can be seen from Table 3, which of the same type is observed (in all cases, the mean A gives the means with their rms deviations when the p (7 days) > 30). If the revealed tendencies are real, then entire data set is broken down into groups for quiet and it makes sense to return to Table 2 to check the sensitiv- disturbed conditions. The conditional boundary corre- ity of the measurements to the change of IMF sign for sponds to A p (7 days) = 12.5. the most favorable conditions—short runs and a fixed

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200 64 62 n = 74 57 59 13 18 24 54 9 6 17 10 + 49 56 + 7 23 14 58 37 + 38 65 53 + 16 61 41 15 3 20 100 39 + 13 + 35 11 1 19 18 22 26 + 27 63 + 64 12 4 55 21 28 46 50 + 11 36 51 + 16 6 47 25 10 + 34 43 33 + + + 30 2 19 17 15 12 45 48 + + 52

Intensity, SNU Intensity, 0 5 10 15 20 25 44 5 Ap (7 days) 8 5 + + 60 42 –100 100.0 ± 48.2(12) 79.3 ± 48.0(28) 46.5 ± 54.0(14) 60.0 ± 49.4(20)

Fig. 9. “Intensity (SNU)—Ap index in the last 7 days of the run” scatter diagram for the GALLEX (circles) and GNO (crosses) measurements. The numbers near the dots give the number of runs. The means shown at the bottom of the figure correspond to the dashed histogram; the errors are standard deviations; the number of runs in the interval under consideration is given in parentheses. The correlation coefficient is r = 0.22 ± 0.09. level of geomagnetic activity. Of course, the number of that ended when the conditions were geomagnetically corresponding runs decreases. The result can be seen quiet (Ap < 12.5), we may attempt to find the possible from Table 5. influence of sunspot number variations on the results of the measurements. The picture obtained is clear from Note that the variations in the experiments being Table 6. The conditional boundary between the high analyzed show opposite tendencies: in the GALLEX and low levels of solar activity was adopted for R = and SAGE measurements, the intensity is enhanced 60.0. As follows from the data obtained in Section 3.1 when, respectively, the negative and positive IMF for the chlorineÐargon measurements, the even and odd polarities dominate at the end of the run. As the geo- solar cycles should be considered separately. magnetic disturbance level increases, the intensity on GALLEX decreases, while the intensity on SAGE As we see from an examination of Table 6, the dif- increases (the last column in Table 5—the significance ferences are marginally significant, but we obtain a of the differences between the means for different signs coherent picture: the GALLEXÐGNO data reveal a cor- in quiet conditions). Finally, having selected the runs relation for the even solar cycle 22 and an anticorrela-

Table 3

Conditions Experiment Run All data P(U*) quiet, Ap < 12.5 disturbed, Ap > 12.5 GALLEX Less than 21 days 82.5 ± 75.5 86.1 ± 62.0 41.4 ± 75.6 0.18 n = 28 n = 13 n = 7 About 28 days 74.6 ± 44.9 89.1 ± 46.7 67.1 ± 38.6 0.20 n = 37 n = 16 n = 19 SAGE 35 days 86.1 ± 75.4 73.1 ± 55.8 85.3 ± 74.6 Ð n = 51 n = 30 n = 15 About 45 days 70.4 ± 43.3 72.4 ± 43.4 62.1 ± 40.2 Ð n = 29 n = 17 n = 11

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Table 4 search for synchronous runs for extreme intensities on a particular detector. The corresponding examination Experiment n Intensity, SNU Normalized values shows that through the extremely large values obtained Brookhaven 18 3.09 ± 1.49 1.19 ± 0.58 on SAGE (more than 150 SNU), we can find seven “twins” in the GALLEX–GNO data if the requirement GALLEX 10 95.9 ± 79.1 1.30 ± 1.08 for synchronism is relaxed to |∆| ≤ 10 days. The result SAGE 7 136.6 ± 115.7 1.77 ± 1.50 of our comparison by the epoch-folding technique is shown in Fig. 10. As we see, extremely low values for GALLEXÐGNO correspond to anomalously high val- tion for the odd solar cycle 23. This result is in close ues for SAGE. The mean mismatch between the runs |∆| ± Ð3 agreement with the results obtained above for the for this sample is = 6.7 2.7 days, P(U*) < 10 . Brookhaven data. The variations for the SAGE mea- Unfortunately, the number of corresponding syn- surements are opposite: an anticorrelation for the even chronous runs is too small for anomalous values of the cycle and a correlation for the odd cycle. It follows opposite sign. The aforesaid suggests that the annual from these data that the GALLEX and SAGE results neutrino intensity variations must be similar in the show an opposite behavior for all of the cosmophysical GALLEX and Brookhaven measurements. In contrast, indices under consideration: for the IMF sign (Table 2), the annual variations in the GALLEX and SAGE mea- the Ap magnetic activity indices Ap (Table 5), and the surements must be different. Indeed, if we construct the sunspot numbers (Table 6). If the above tendencies are annual variations for the even solar cycle, then we will real, then the data obtained with these detectors must observe the characteristic minima for the GALLEX generally anticorrelate, while the Brookhaven and data in April and November and the maximum in Sep- GALLEX data must correlate. tember that have long been known for the chlorineÐ argon measurements. The SAGE data do not contain these features and even show a tendency for anticorre- 3.3. Comparison of the Data lation with the profile mentioned above (it is pertinent to from Different Detectors recall that the annual variation in this case was con- To test the above prediction, we must choose the structed for the completion date of the run). runs from the corresponding sets of measurements that would end simultaneously, with a small mismatch. As this mismatch, we chose |∆| ≤ 5 days for the galliumÐ 4. DISCUSSION germanium measurements and ∆ ≤ 10 days for the Thus, we may summarize the main results presented Brookhaven and GALLEX detectors. For the above as follows: the measured solar neutrino intensity 12 synchronous runs, the actual value was |∆| = 3.2 ± in all three radiochemical experiments depends on the ∆ ± heliophysical indices pertaining to the last run interval. 1.6 days in the former case and = 4.5 3.1 days in At the same time, all of the previously known main the latter case. Next, we may choose lower (higher) results, with the important refinement that an anticorre- intensities from the catalogs compiled in this way for lation between the neutrino flux and the sunspot num- the entire data set and calculate the mean of the syn- bers is only observed for the odd solar cycle, while the chronous runs for another detector. The inverse proce- IMF sign has the strongest effect on the recorded flux, dure can serve as a check. The data normalized to the are reproduced in the chlorineÐargon measurements. corresponding means are presented in Tables 7 and 8. This effect for shorter runs is also found for the gal- The large variance at the first point in the right-hand liumÐgermanium measurements. For all three detec- column in Table 7 is attributable to the presence of two tors, we found a magnetic storm effect of the same type: extremely large values. If they are disregarded, then the a sharp increase in the mean scatter of results. Evidence inequality remains valid. Another check of the suggests that the variations on the Brookhaven and GALLEX and SAGE data for anticorrelation is to GALLEX detectors are of the same type, while those

Table 5

Quiet conditions, Ap < 12.5 Disturbances, Ap > 12.5 Experiment P(U*) “+” IMF “–” IMF “+” IMF “–” IMF GALLEX 76.3 ± 20.5 103.1 ± 40.0 17.0 ± 81.5 74.0 ± 51.3 0.19 n = 3 n = 7 n = 4 n = 3 SAGE 79.6 ± 61.9 48.8 ± 23.1 96.2 ± 56.8 84.7 ± 68.1 0.23 n = 14 n = 9 n = 6 n = 6

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Table 6 search for dark matter, etc.), the extreme points are arranged in a similar fashion. Finally, note that an Experi- Solar activity examination of the GALLEX calibration measure- Cycle P(U*) 51 ment low, R < 60 high, R ≥ 60 ments using Cr (seven runs, JuneÐJuly 1994 [48]) makes it possible to understand why this procedure GALLEX 22, even 62.4 ± 50.1 82.6 ± 40.9 0.16 does not completely solve the question of flux varia- n = 14 n = 5 tions either: the runs mentioned above were taken for various values of the cosmophysical indices used here. CNO 23, odd 125.8 ± 40.3 74.9 ± 28.7 0.16 What level of the reconstructed activity of the source n = 11 n = 10 should be taken—at a low (60.7 ± 3.9) or enhanced ± SAGE 22, even 77.4 ± 45.8 60.8 ± 73.4 0.29 (72.2 2.1) geomagnetic disturbance level? n = 13 n = 10 Of course, by no means all of the empirical data pre- ± ± sented above can be understood. Even in those cases 23, odd 61.1 36.6 84.6 41.7 0.01 where a simple explanation can be found for the effect n = 12 n = 14 in question, it is most often ambiguous or not compre- hensive. We have failed to find any idea that would explain why the measured flux for all three experiments on the SAGE detector are opposite. For the runs that is, on average, higher for shorter runs. The even and ended simultaneously, to within several days, the odd solar cycles are generally known to differ in vari- GALLEX and SAGE data probably anticorrelate, while ous properties, but it is not clear what these differences the Brookhaven and GALLEX data correlate. specifically in parameters of the acting electromagnetic In many cases, the statistical significance of the fields are. The anticorrelation between the ionospheric results obtained above is low (10Ð2), and, strictly speak- index and the neutrino flux could in principle be an ing, they need to be tested. Such a test can be made only important argument for our hypothesis that very-low- when additional data will be accumulated. In our case, frequency emissions affect the target material. How- the statistical reliability is determined by random fluc- ever, the conclusion that the anticorrelation between the tuations (not by systematic errors). Therefore, circum- ionospheric index and the flux for the odd solar cycle stantial evidence is also important for assessing (see Fig. 2) is causal cannot be considered to have been whether the tendencies under consideration are real. In proved. This correlation may just arise from the corre- particular, it is important to note that the results lation between I and R. obtained fit into a self-consistent picture. Note, in par- In contrast, the influence of the sign of the radial ticular, that the correlations of the neutrino flux with the IMF on the results of the measurements can be inter- IMF sign, R, and Ap necessarily give rise to the already preted unambiguously in our case. All of the parame- detected about 27-day periodicity. These correlations ters of geomagnetic micropulsations and low-fre- give a qualitative insight into the bimodal distribution quency emissions in the IMF sectors of different signs of the results from [26]: the two stated found by the have long been known to vary significantly. It has been authors of [26] may correspond to the two IMF signs firmly established that some physicochemical systems and the corresponding quasi-periodic Ap variations. respond to the change in sign of the IMF sector (in par- Some of the periods found in the neutrino flux varia- ticular, this has been established for the rest reaction tions (see point 7 in the Introduction) are equal to the that reveals an 11-year solar cycle; see [49]). Clearly, periods found in the IMF variations (1.3, about 3, and this dependence must change sign with polarity rever- about 4.5 years). The annual period is of greatest inter- sal of the general solar magnetic field (although the est. As was noted above, the profile of the annual vari- dynamics of such phenomena has been studied inade- ations for the even solar cycle for the Brookhaven and quately). It should be emphasized that the change in GALLEX detectors share common characteristic fea- IMF sign and the accompanying change in the excita- tures. Interestingly, in the various precision measure- tion regime of geomagnetic micropulsations and kilo- ments that also reveal the annual period of unknown hertz magnetospheric emissions is a geophysical effect. origin (the frequency drift of the atomic standards, the Therefore, large variations in neutrino flux Q for differ-

Table 7

GALLEXÐGNO, initial sample smaller than M = 78.0 SAGE, mean of synchronous runs P(U*)

n = 6; 0.43 ± 0.35 n = 6; 1.86 ± 1.79 5 × 10Ð2 SAGE, initial sample smaller than M = 77.0 GALLEXÐGNO, mean of synchronous runs

n = 7; 0.56 ± 0.25 n = 7; 1.05 ± 0.41 3 × 10Ð2

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R > (<) 90; I > (<) 220; and Ap > (<) 12.5. Choosing the intervals of ﬂux-stimulating IMF polarities and requir- 1 200 ing the satisfaction of the inequalities written above, we obtain for the odd cycle,

Qmax = 0.726± 0.338 ()n = 9 ,

Qmin = 0.431± 0.313 ()n = 35 ; 100 2

Intensity, SNU Intensity, for the even cycle,

Qmax = 0.690± 0.226 ()n = 17 ,

Qmin = 0.400± 0.267 ()n = 47 ; 0 and for the entire data set, –4 –2 0 2 4 Qmax = 0.702± 0.293 ()n = 26 , Fig 10. Intensity (SNU) versus run sample on SAGE (1) and GALLEX (2). The zero interval—synchronous runs (the mismatch between their ends is no more than 10 days). The Qmin = 0.413± 0.228 ()n = 82 initial SAGE sample—extremely large values (>150 SNU). The corresponding runs adjacent to the synchronous runs (all of the errors are standard deviations). are to the left and to the right of zero (the minus and plus correspond to the preceding and succeeding runs, respec- Finally, if we take into account the signiﬁcant effect tively). of the IMF sign (see Fig. 7b) and exclude the long (∆ >

70 days) runs from the list for Qmax , then we can obtain ent IMF signs at the end of the run is a decisive argu- the extreme value, Qmax = 0.768 ± 0.289 (n = 17; 4.5 ± ment that there are fictitious variations in the radio- 1.5 in SNU). Clearly, if this flux is taken as the actual chemical experiments that were not caused by real vari- value, then the problem of its deficit becomes less ations. acute. Unfortunately, such an estimate for the The model of the variations under consideration GALLEX and SAGE data cannot be obtained because assumes that the action on the physicochemical kinetics of the poor statistics. However, recall the result of ana- through both channels (through the short-wavelength lyzing the data distribution (histograms) in the galliumÐ solar radiation, the ionosphere, and through the solar germanium measurements [26]. The two states found in wind, the magnetosphere) always only reduces the the above paper may correspond to the two IMF signs, measured flux, differently at different epochs and on with the higher state approaching the theoretical flux. different detectors. Therefore, the measured fluxes Without touching upon the complicated question of must be lower than the actual fluxes. The actual the influence of weak electromagnetic fields on the recorded flux can be estimated by assuming that all of physicalÐchemical kinetics of solutions (where the the correlations under consideration are real. For the most important events develop at the mesolevel, e.g., chlorineÐargon detector, these are the correlations the drawing of ions or atoms into nanotubes), we will (anticorrelations) with the sunspot numbers, the critical briefly list the possible tests of the ideas presented above. Most of them are simple and quite realizable. ionospheric frequencies, and the Ap index. By analyzing the distribution of these indices, we can obtain the (1) It would be instructive to carry out a series of following limiting values for a high (low) activity level: direct measurements of the variations in electromagnetic fields of low and very low frequencies near the detector, including the fundamental mode frequencies Table 8 of the Schumann ionospheric resonance and Pc3-type geomagnetic pulsations. GALLEXÐGNO, initial Brookhaven, mean of sample larger than M = 78.0 synchronous runs (2) It seems of considerable interest to carry out a series of measurements on all of the radiochemical n = 7; 1.76 ± 0.84 1.27 ± 0.46 detectors (e.g, for a year) in such a way that the runs end simultaneously. Brookhaven, initial sample GALLEX, mean of (3) It would be appropriate to measure the detection larger than M = 0.478 synchronous runs efficiency of a suitable radioactive standard near the n = 9; 1.40 ± 0.29 1.37 ± 0.96 detector by using standard measuring devices, prefera- bly simultaneously through several channels (scintilla-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE PRESENCE OF FICTITIOUS SOLAR NEUTRINO FLUX VARIATIONS 641 tors, Geiger counters, semiconductor detectors). It is ACKNOWLEDGMENTS quite probable that the count-rate measurements in this We are grateful to G.S. Ivanov-Kholodnyœ for pro- case could correlate (anticorrelate) with the recorded viding unpublished ionospheric data and helpful neutrino flux variations. remarks; to V.I. Odintsov for providing IMF data; and to (4) It seems of considerable importance to automat- S.E. Shnol and A.A. Konradov for helpful discussion. ically monitor the parameters of the target liquid (e.g., perchloroethylene) in stable laboratory conditions. The various parameters of the liquid that could be systemat- REFERENCES ically measured, such as the electrical conductivity, the 1. G. A. Bazilevskaya, Yu. I. Stozhkov, and T. N. Chara- dielectric loss tangent, the heat conductivity, etc., are khch’yan, Pis’ma Zh. Éksp. Teor. 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Dorman, Yad. Fiz. 63, 1064 (2000) [Phys. At. Nucl. 27-day periods could be easiest to find at the epochs of 63, 984 (2000)]. the Earth’s high heliolatitudes. These variations could 12. S. Massetti and M. Storini, Astrophys. J. 472, 827 be detected and studied in detail in laboratory condi- (1996). tions by measuring the beta activity of a suitable radioactive standard with Cherenkov water detectors. 13. D. Basu, Sol. Phys. 81, 363 (1982). 14. R. Wilson, Sol. Phys. 149, 391 (1994). 15. R. L. McNutt, Science 270, 1635 (1995). 5. CONCLUSIONS 16. Ph. Delache, V. Gavryusev, E. Gavryuseva, et al., Astro- Our analysis has led us to the following conclusions. phys. J. 407, 801 (1993). (1) The neutrino flux in all of the radiochemical 17. H. J. Haubold and E. Gerh, Astron. Nachr. 306, 203 experiments depends on the heliogeophysical situation (1985). at the very end of the run, when the neutrino reaction 18. H. J. Haubold and E. Gerh, Sol. 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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 643Ð650. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 734Ð743. Original Russian Text Copyright © 2004 by Grigorov, Tolstaya. GRAVITATION, ASTROPHYSICS

The Spectrum of Cosmic-Ray Particles and Their Origin N. L. Grigorov and E. D. Tolstaya* Skobeltsyn Research Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119992 Russia *e-mail: [email protected] Received May 21, 2003

Abstract—An analysis of all the direct measurements of the spectrum for all cosmic-ray particles over the energy range 0.1Ð10 TeV reveals an anomaly in the spectrum in the form of a step if the spectrum is represented β as E I0(E). The pattern of the anomaly unequivocally implies a proton spectrum with a knee at energy close to 1 TeV. The qualitative difference between the spectra of protons and nuclei with Z ≥ 2 (the latter have a purely power-law spectrum over a wide energy range) leads us to conclude that the acceleration conditions for protons and nuclei are different. We consider the process characteristic only of protons that may be responsible for the emergence of a knee in the proton spectrum. © 2004 MAIK “Nauka/Interperiodica”.

1. DIRECT MEASUREMENTS spectra taken by the authors of [2] were published in a OF ALL GALACTIC COSMIC-RAY PARTICLES tabulated form [3].) OVER THE ENERGY RANGE 0.1Ð100 TeV The discovered anomaly had been neither confirmed Historically, there have been very few direct mea- nor disproved over the whole 25 years. The spectrum of surements of the spectrum for all galactic cosmic-ray all particles was remeasured with a thin ionization cal- (GCR) particles. In general, information about the orimeter (TIC) flown on a balloon only in 1997 [4]. The spectrum of all particles, I0(E), have been obtained by TIC measured the energy release of all the particles that adding the spectra of the individual components. fell on the instrument in any direction. As the authors of [4] showed, the energy release spectrum revealed the Since the spectra of the individual components were same anomaly in the spectrum of all particles that was measured by different methods (using electronic observed in the Proton satellite measurements [2]. devices at energies E < 1 TeV and, as a rule, X-ray When the energy release was recalculated to the parti- emulsion chambers (XEC) with a high detection cle energy, as was done in [5], the TIC energy spectrum threshold at E > 1 TeV (5Ð10 TeV)), the proton spec- was quantitatively identical to the spectrum obtained by trum contained a range from about 1 to 5Ð10 TeV in the authors of [2]. The TIC, SEZ-14, and SEZ-15 mea- which virtually no direct measurements were carried surements are shown in Fig. 1. The solid line in Fig. 1 out. In the spectrum of all particles obtained by adding indicates the best fit Φ(E) to the experimental spectrum the individual components, the energy range 1 < E < 5Ð of all particles at a = 0.4 TeV: 10 TeV was not covered by direct measurements. This energy range in the spectrum of all particles is usually 2.6 1.1 drawn by interpolation based on the confidence that the Φ() () E ==E I0 E ------0.2- proton spectrum is similar to the spectrum of nuclei. []1 + ()E/a 3 This spectrum is commonly considered as the spectrum (1) 3 of all GCR particles I (E) [1]. ()E/a Ð2 Ð1 Ð1 1.6 0 × 1+ 0.37------+ 0.130 m s sr TeV . Direct information about the spectrum of all parti- 1 + ()E/a 3 cles in the energy range 1 to 5Ð10 TeV can be obtained by using direct measurements of the spectrum for all We see from Fig. 1 that the anomaly in the spectrum of particles with electronic equipment that covers a wide all particles, if it is represented as EβI (E), appears as a energy range containing particles both to the left and to 0 the right of the narrow interval 1Ð5 TeV to be measured. step. Such information was first obtained in 1972 by measur- Figure 1 shows that the anomaly in the spectrum of ing the spectrum of all particles with the SEZ-14 instru- all particles is revealed irrespective of the thickness of ment onboard the Proton-1,2,3 satellites over the the ionization calorimeter (IC) in the instrument (it is λ λ λ energy range 0.07Ð17 TeV and with the SEZ-15 instru- ~1 p in TIC, ~1.71 p in SEZ-14, and ~3 p in SEZ-15). ment onboard the Proton-4 satellite over the energy Therefore, we decided to look for such an anomaly in range 0.19Ð103 TeV [2]. These measurements first the spectrum of all particles measured with the BFB-S revealed an anomaly in the spectrum of all particles in instrument, in which the IC had a mean thickness λ the energy range 1Ð10 TeV. (Subsequently, the energy of ~0.7 p.

644 GRIGOROV, TOLSTAYA

its scientiﬁc signiﬁcance and was not published. Now, E ]

1.6 2.0 we are interested in the presence (or absence) of an log d

/ irregularity in a narrow spectral range that cannot arise 1.5 ε TeV dN from the nonlinear relationship between and E. There-

–1 –0.5 1.0 1.61 fore, we recalculated the measured energy release to the

sr 0.1 1 10 E β

–1 spectrum of all particles I (E) and plotted E I (E) in –0.6 E, TeV 0 0 s Fig. 1 (crosses). We see that the measured BFB-S spec- –2 –0.7 trum also exhibits an irregularity in the form of a step

), [m in the spectrum of all particles in the same energy range 0

I –0.8 in which it is recorded by other instruments. The

2.6 smaller step height is a natural result of the IC thinness E –0.9 (protons—the culprits of the step—make a small con-

log( –1.0 tribution to the number of recorded particles). 0.1 1 10 100 E, TeV A preliminary result of the ATIC measurement of the spectrum for all particles was published at the 27th 2.6 Fig. 1. E I0 versus E, as measured with different instru- International Conference on Cosmic Rays [7]. The ments: SEZ-14 (᭹) [3], SEZ-15 (+) [3], TIC (᭺) [5], and spectrum of energy release in the calorimeter of the BFB-S (×) (this paper). The ATIC data are shown in the instrument presented in this paper was thoroughly mea- upper right corner; arbitrary units are along the vertical axis, sured in [8]. As a result, the authors obtained a depen- and energy release in the calorimeter is along the horizontal β axis. dence of E (dN/d logE ) on logE in different ranges of energy release. This dependence is shown in the upper right corner of Fig. 1. It convincingly demonstrates that The BFB-S was installed on the Intercosmos-6 sat- there is also the same step as that recorded by previous ellite and described in [6]. The ionization calorimeter of instruments in the ATIC energy release spectrum for all this instrument consisted of two identical sections. particles. Each section consisted of eight 1.5-cm-thick lead plates. A 0.5-cm-thick scintillator was placed under Thus, we have measurements of the spectrum for all each odd plate. All four scintillators were viewed by particles with five different instruments: SEZ-14, SEZ- two photomultipliers from their ends, one from each 15, TIC, BFB-S, and ATIC. They all measured the spec- side. The signals from both photomultipliers were trum over a wide energy range that contained the inter- added and fed to a pulse-height analyzer. It recorded the val 1Ð10 TeV concerned, and they all revealed a similar energy release spectrum without being connected with anomaly in the spectrum in the form of a step: with dif- the trigger that controlled the operation of the instru- ferent spectral indices in different energy ranges. β ment. Accordingly, the values of E I0(E) before and after the step are also different. The five qualitatively identical Since the photomultiplier photocathodes were results suggest that the step in the spectrum of all parti- strongly diaphragmed, the particles emerging from the cles is an objective reality that has certain quantitative IC affected the signal from the photomultiplier. As a characteristics. These characteristics include the fol- result, the relationship between the measured energy β lowing: the spectral index at energies up to 1 TeV ( 1), release ε and the particle energy E turned out to be non- β the spectral index in the energy range 1Ð5 TeV ( 2), the linear, E = εα (at α = 0.78). Therefore, the spectrum lost ≥ β spectral index at E 10 TeV ( 3), and the mean values β ≥ of E I0(E) at E < 1 TeV and E 5 TeV. We determined Table 1 all of these characteristics from the results of each experiment and gathered them together in Tables 1 β β β and 2. Instrument 1 2 3 Source SEZ-14 2.59 3.00 Ð [3] The differences between the mean spectral indices in different energy ranges are SEZ-15* Ð 2.94 2.63 [3] TIC Ð 2.80 2.65 [4, 5] 〈〉β 〈〉 β ± 〈〉β 〈〉 β ± 2 Ð1 = 0.28 0.04, 2 Ð 3 = 0.23 0.04. BFB-S 2.59 2.78 2.66 This paper These values allow us to formulate the first characteris- ATIC 2.61 2.87 Ð [8] tic of the anomaly in the spectrum of all particles: the Literature 2.62 Ð 2.67 [1, 3, 9] spectral indices at energies E < 1 TeV and E > 5 TeV are Mean 2.60 ± 0.01 2.88 ± 0.04 2.65 ± 0.01 almost equal and close to 2.6. The spectral index in the energy range 1Ð5 TeV is larger than that outside this * In some of the publications, SEZ-15 is called IC-15. range by 0.2Ð0.25.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 THE SPECTRUM OF COSMIC-RAY PARTICLES AND THEIR ORIGIN 645

If the spectrum of all particles is represented as Table 2

Φ() β () β Φ , mÐ2 sÐ1 srÐ1 Φ , mÐ2 sÐ1 srÐ1 E ==E I0 E , 2.6, Instru- 1 2 1.6 1.6 K = Φ /Φ ment TeV TeV 1 2 then in the energy regions where I0(E) is described by a (E < 1 TeV) (E > 5 TeV) Φ power function with a spectral index of 2.6, (E) will ± ± be constant over the entire energy range. This implies SEZ-14 0.247 0.009 Ð 1.66 0.07 Φ Φ Φ that must have some constant values 1 and 2 in the SEZ-15 Ð 0.149 ± 0.003 Ð spectrum of all particles at E < 1 TeV and E > 5 TeV, TIC 0.240 ± 0.018 0.134 ± 0.008 1.79 ± 0.17 respectively. The values of Φ and Φ obtained from 1 2 ± ± ± each experiment are given in Table 2. BFB-S 0.237 0.012 0.198 0.007 1.20 0.07 The second quantitative characteristic of the anom- ATIC [8] Ð Ð 1.49 ± 0.08 aly in the spectrum of all particles is the ratio of the step Literature 0.270 [1] 0.160 ± 0.007 [9] 1.69 ± 0.07 height to the flux of all particles before the step, i.e., Φ Φ Φ Mean 0.249 ± 0.007 0.148 ± 0.008 1.66 ± 0.06 ( 1 Ð 2)/ 1. This parameter is close to the ratio of the proton flux to the total flux of all GCR particles at equal Note: The mean values do not include the BFB-S data. The errors energy per particle. of the mean are the rms deviations from the mean. The first Φ Φ row in the column K = 1/ 2 was obtained from SEZ-14 The following two remarks should be made regard- and SEZ-15 data. ing the results presented in Table 2. First, if we take data from the All-Union State Stan- dard (GOST) [10] for energies E < 1 TeV, then we The fact that these two values are almost equal suggests obtain the sum that there are very few protons in the flux of all particles at E > 5 TeV. 28 2.6 ± Ð2 Ð1 Ð1 1.6 Thus, Table 2 and the remarks to it lead us to con- ∑ E IZ = 0.258 0.005 m s sr TeV clude that the step in the spectrum of all particles is pro- Z = 1 duced by protons. which is almost the same as that obtained by directly measuring the spectrum of all particles with the instru- 2. PARAMETERS OF THE PROTON SPECTRUM ments listed in Table 2. This implies that there are no (FROM THE SPECTRUM OF ALL PARTICLES) significant systematic errors in these measurements. We obtain the proton spectrum Ip(E) from the obvi- The above sum consists of the sum of two quanti- ous equality ties: one refers to protons and is equal to () () () I0 E = I p E + IZ E , 2.6 Ð2 Ð1 Ð1 1.6 E I p = 0.120 m s sr TeV , where IZ(E) is the spectrum of the sum of all nuclear ≥ components with Z ≥ 2. Multiplying all terms of this and the other refers to all nuclei with Z 2 and is equal 2.6 to equality by E and interchanging I0 and Ip yields 2.6 2.6 2.6 β () () () ± Ð2 Ð1 Ð1 1.6 E I p E = E I0 E Ð E IZ E . (2) E IZ = 0.138 0.005 m s sr TeV . Since In other words, the protons at E < 1 TeV account for 0.120/0.258 = 0.46 of the total particle flux at equal E2.6 I ()E ==const Φ , energy per particle (this is a well-known result). Z Z Second, it is well known that for the nuclei, over a wide energy range, equality (2) may be rewritten as E2.6 I = const Z 2.6 () 2.6 Φ E I p E = E I0 Ð Z. over a wide energy range of several orders of magnitude. At E < 1 TeV, Since 2.6 () Φ 28 E I0 E ==const 1 2.6 ± Ð2 Ð1 Ð1 1.6 ∑ E IZ = 0.138 0.005 m s sr TeV , at E < 1 TeV, Z = 2 2.6 Φ Φ E I p ==1 ÐZ const and at E > 5 TeV for the flux of all particles I0, = 0.11± 0.01 mÐ2 sÐ1 srÐ1 TeV1.6 2.6 ± Ð2 Ð1 Ð1 1.6 E I0 = 0.148 0.008 m s sr TeV . in this energy range.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 646 GRIGOROV, TOLSTAYA Φ ± Hence, we obtain for E < 1 TeV If we use the mean value of 1 = 0.249 0.007 from β ± Φ Table 2, Ð 2.6 = 0.24 0.04 from Table 1, and Z = () ()± Ð2.6 Ð2 Ð1 Ð1 Ð1 ± I p E = 0.11 0.01 E m s sr TeV . 0.138 0.005, then we obtain β ()± ⋅ ()± ± The proton spectrum at E > 1 TeV must decrease p Ð2.6 = 0.24 0.04 2.26 0.18 = 0.54 0.09, Ð2.6 β faster than E ; i.e., it must have a spectral index p > β whence β = 3.14 ± 0.09 at E > 1 TeV. 2.6. The value of p can be determined from the data in p Tables 1 and 2. To this end, we represent the proton Thus, the step in the spectrum of all particles inevi- spectrum in a slightly simplified form: tably leads us to conclude that the proton spectrum has a knee at energy close to 1 TeV. The proton spectral EÐ2.6, EE< , index is β = 2.6 before the knee and β = 3.14 ± 0.09 ()∝ c p p I p E  β after the knee.  Ð p > E , EEc. The shape of the proton spectrum can be obtained in more detail from the same equality (2) if we subtract For this proton spectrum, the spectrum of all particles is Φ Φ the contribution of nuclei Z from the function (E) ()β that describes the spectrum of all particles. If expres- 2.6 () Ð p Ð 2.6 Φ E I0 E = BE + Z. sion (1) is taken as Φ(E), then the proton spectrum takes the form At E = Ec, 2.6 0.11 ()β Ð 2.6 E I ()E = ------2.6 Φ ()Φ Φ p p 3 0.2 E I0 ==1, B 1 Ð Z Ec . []1 + ()E/a (3) Therefore, 3 ()E/a Ð2 Ð1 Ð1 2.6 ×  1+ 0.37------3 m s sr TeV . Ð()β Ð 2.6 () 2.6 ()() Φ Φ ()p Φ 1 + E/a E I0 E = 1 Ð Z E/Ec + Z. The sum of two power functions, The coefficient a should be determined by comparing (3) with the experimental proton spectrum (see Ðγ Ðγ below). BE 1 + CE 2 , It is important to emphasize that we obtained the may be substituted with a good accuracy by one power proton spectrum (3) with a knee at E = a from the spec- function DEÐγ, where trum of all particles, i.e., from the experiments to which the reverse particle current from the ionization calorim- B C eter (which frighten many experimenters) bears no rela- γ = ------γ + ------γ CB+ 1 CB+ 2 tion whatsoever. Let us now consider the direct measurements of the (see [11]). In our case, proton spectra. They all refer to energies E > 4Ð10 TeV, and most of them were carried out by the XEC method Φ Φ γ β 1 Ð Z without experimental chamber calibration. Therefore, 1 ==p Ð 2.6, B ------Φ , caution should be exercised when dealing with the 1 absolute fluxes determined by this method. However, β Φ this remark does not apply to the spectral index p con- γ Z 2 ==0, C ------Φ-. cerned. 1 β Table 3 lists the values of p obtained by different Therefore, the power-law index of the sum of the spec- authors. tra is Its columns give the following data: (1) the author and the measurement method, (2) the minimum proton Φ Ð Φ β 1 Z ()β energy in the spectrum (in TeV), (3) the value of p with ------Φ p Ð 2.6 . 1 its error, and (4) the number of protons N0 used to construct the spectrum. The number N0 without an asterisk At E > Ec, the spectral index of the spectrum for all par- is given in the paper; the number with an asterisk was β 2.6 ticles is equal to . Therefore, E I0(E) is a power func- estimated by us from the statistical errors. tion with an index β Ð 2.6. As a result, β The mean p of the five measurements listed in Φ Φ Table 3 is 1 Ð Z ()ββ ------Φ p Ð 2.6 = Ð 2.6. 〈〉β ± 1 p = 2.94 0.07.

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The rms deviation σ of the individual result from the E2.6 I , [m–2 s–1 sr–1 TeV1.6] mean is 0.14, i.e., of the same order of magnitude as the p error of the individual measurements. This circum- β stance suggests that the scatter of numerical p values 0.10 β is purely statistical in nature, and the individual p values can differ significantly from the mean at the statistical errors characteristic of these experiments. We see that the direct measurements of the proton spectrum yield β = 2.94 ± 0.07 for energies 5Ð20 TeV, β ± which agrees with the above value of p = 3.14 0.09 for energies 1Ð5 TeV (the region of the step in the spectrum of all particles). 0.01 In [12, 15], the XEC had targets. As was noted 0.1 1 10 E, TeV in [16, 17], protons with energies close to the detection Fig. 2. Measured SEZ-14 proton spectrum [11]. The solid threshold are detected with a low efficiency in such β curve represents the fit to the difference between the spec- chambers, which may cause p to decrease. Therefore, trum of all particles and the spectrum of nuclei with Z ≥ 2 β ≥ (formula (3)). we determined p for E 20 TeV, farther from the threshold energy in the spectra of [14, 15]. It turned out β ± ± that p = 3.17 0.19 and 3.05 0.19 in these spectra in β ± cannot be acquired during the propagation of particles the above energy range. The mean value is p = 3.11 in the Galaxy. In this case, particles of identical rigidity 0.14 (see [18]). undergo identical disturbances, irrespective of their β Within the error limits, all three values of p (3.14, charge and mass. Therefore, changes in the spectrum, if 2.94, and 3.11) refer to the same spectral index that they were caused by the propagation processes, would characterizes the proton spectrum in an energy range have the same effect both on protons and on nuclei. In from approximately 1 to 40Ð50 TeV. The weighted other words, the observed difference is acquired in the β ± mean p of these three values is 3.02 0.05. sources. Consequently, the protons and the nuclei have There is only one experimental proton spectrum in different sources; the sources of nuclei give particles an energy range of about 0.1Ð10 TeV. It was obtained with a purely power-law spectrum, while the sources of more than 30 years ago from the Proton-2 and 3 satel- protons give particles with a knee in their spectrum. lites; this spectrum was published in integral and differ- A characteristic feature of the knee in the proton β ential forms in [19] and [11], respectively. We show it spectrum is that the energy range in which p changes in Fig. 2 (from [11]) together with the proton spec- by 0.5Ð0.6 is narrow. This circumstance is indicative of trum obtained from the spectrum of all particles: a universality of the knee formation that depends expression (3) at a = 0.8 TeV. We see from Fig. 2 that weakly on the specific properties of the source. the experimental proton spectrum matches the spectrum obtained from the spectrum of all particles; as was When explaining the formation of the knee in the shown above, the latter matches the direct measure- proton spectrum, we should also point out the cause of ments in an energy range of about 5Ð40 TeV. the formation of a knee at energy near 1 TeV, and not at some other energy that differs greatly from 1 TeV. Thus, we may assert that all of the direct measurements yield the same result: the proton spectrum has a We believe that the particles themselves rather than β knee at energy near 1 TeV. The spectral index is p = 2.6 the sources are mainly responsible for the difference in (or possibly 2.7) before the knee and is larger by 0.5Ð the spectra of protons and nuclei. What are the differ- 0.6, i.e., 3.0Ð3.1, after the knee. It should be added to this conclusion that all of the indirect measurements of high-energy secondary parti- Table 3 cles in the Earth’s atmosphere (hadrons, muons, γ pho- Method and E , tons) lead us to conclude that β = 3.0 in the TeV energy min Spectral index N p reference TeV 0 range [11]. β ± XEC, [12] 5 p Ð 1 = 1.82 0.13 90* β ± 3. THE ORIGIN Calorimeter, [9] 4 p Ð 1 = 2.11 0.15 90* OF GALACTIC COSMIC-RAY PROTONS β ± Calorimeter, [13] 5 p = 2.85 0.14 160* Even a fleeting glimpse of the proton and nuclear XEC, [14] 10 β = 3.14 ± 0.08 602 spectra reveals a significant difference between them: p β ± 1 the nuclei have a purely power-law spectrum over a XEC, [15] 6 p = 2.80 0.04 656 wide energy range, while the protons have a power-law 1 The error of 0.04 given in [15] has no physical meaning, because spectrum, but with a knee near 1 TeV. This difference it is smaller than the statistical error of 0.07.

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σ pp, mb bility.) Having a speed close to the speed of light, the 200 protons will rapidly escape from the acceleration region and, moving through the stellar envelope, will escape from it. Clearly, during this escape from the exploded supernova, the protons will inevitably have to traverse a significant thickness of material of several hundreds or thousands of g cmÐ2. What will happen to the original spectrum in this case? 100 Suppose that the protons in the acceleration region Total acquire a power-law spectrum of the form σin = σ (1 + bln(E/E )) 0 0 () Ðβ IE = I0E . Inelastic The equation that describes the passage of protons 0246810 through matter is log(p ), [GeV] ∞ beam ∂ (), (), (), IEx IEx IE' x (), Fig. 3. Effective cross section for pÐp interaction versus ------∂ - = Ð ------λ - + ∫------λ -PE' E dE'. (4) energy [23]. x E If the effective cross section for inelastic interaction ences between the protons and the nuclei that can have of protons with matter, σin, does not depend on energy, a crucial effect on their acceleration and escape from then the mean free path before inelastic interaction is the sources? There is one qualitative difference λ 0 = const. In this case, the solution of Eq. (4) is known between these types of particles: as they accelerate and to be escape from the sources, the protons can undergo an β () infinite number of inelastic collisions while remaining (), Ð Ð x/L0 nucleons. In contrast, the nuclei are too fragile: after IEx = I0E e , several inelastic collisions, they break up into their con- where the absorption mean free path L0 is related to the stituent nucleons and cease to exist as nuclei. As a result λ of this difference, the protons can accelerate in a suffi- mean free path before interaction 0 by ciently dense medium and traverse a significant thick- 〈〉β Ð 1 ness of material (~102Ð103 g cmÐ2). In contrast, the 1 1 Ð u ----- = ------λ -, nuclei can accelerate only in a low-density medium, L0 0 i.e., in a greatly expanded supernova envelope. 1 The possibility of particle acceleration to high ener- 〈〉β Ð 1 β Ð 1 () gies at the initial phase of a supernova explosion was u = ∫u Pudu. considered by Colgate and Johnson [20]. They showed 0 that acceleration was possible even in dense stellar layers; in this case, a power-law spectrum of the acceler- In other words, in the case under consideration, a ated particles can be formed. proton beam with the same power-law energy distribution as that formed in the acceleration region, but Without going into the details of the acceleration with a lower intensity, would emerge from the stellar process, let us consider what particles will escape from envelope. such a source and what spectrum they will have. In reality, however, the effective cross section for A supernova explosion is the final phase of stellar inelastic interaction depends on energy, as shown in evolution. Therefore, old stars, red giants and super- Fig. 3. In the first approximation, this dependence at giants, explode. In these stars, hydrogen has long E > E0 may be fitted by the function burned out, and the shells are composed of complex σin() σ()() nuclei heavier than hydrogen. The explosion energy is E = 0 1 + bEln /E0 . (5) released in the stellar core and the adjacent shell regions. Therefore, particle acceleration can begin in For a hydrogen medium, as shown in Fig. 3, b = 0.08. sufficiently dense shell layers. The nuclei will acceler- For the Earth’s atmosphere, b = 0.04Ð0.05. For this ate, because the shell consists of nuclei. However, the energy dependence of the cross section, the situation accelerated nuclei in a dense medium will undergo must change. inelastic collisions and continuously fragment into Indeed, the probability of an inelastic collision lighter parts. Therefore, the accelerated nuclei will rap- between a nucleon and nuclei increases with E. There- idly turn into a beam of energetic protons. (The neu- fore, a higher-energy nucleon undergoes a larger num- trons will also turn into protons because of their insta- ber of collisions in a given layer of material than does a

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 THE SPECTRUM OF COSMIC-RAY PARTICLES AND THEIR ORIGIN 649 lower-energy nucleon. Therefore, a higher-energy E 2.6 Iex/L0, arb. units nucleon loses a larger fraction of its initial energy than 103 does a lower-energy nucleon. Hence, as it emerges from the absorbing layer, a higher-energy nucleon is shifted on the energy scale toward the lower energies by a larger interval than a nucleon with a lower initial 1 energy. Consequently, the initial power-law spectrum emerging from an absorber will be softer than the orig- 2 inal spectrum; i.e., the spectral index will be larger than 3 that for the original spectrum. One of us explained the softer spectrum of high-energy hadrons deep in the Earth’s atmosphere than the spectrum of primary GCRs 4 by this effect back in 1965 [21]. An approximate solution of Eq. (4) with ﬁt (5) was found in [22]: (), Ðβ Ðx/LE() IEx = I0E e , where 102 1011 1012 1013 L LE()= ------0 -, E, eV 1 + bEln()/E 0 Fig. 4. Proton spectra at various depths x of a hydrogen atmo- or, in a different form, sphere: x = 100 (1), 200 (2), 300 (3), and 450 (4) g cmÐ2.

()βδ (), Ð + Ðx/L0 IEx = I0E e , To find out how wide the region in which the spec- β β δ where tral index changes from 0 to 0 + is and how this region changes with the amount of material traversed, δ = bx/L0. we performed Monte Carlo numerical simulations of the passage of a nucleon beam through different thick- This approximate solution refers to particles with E > nesses of material using fit (5). The results of these sim- Ðβ E0 under the boundary conditions I(E, x = 0) = I0E . It ulations are shown in Fig. 4. yields an error in L(E) of only 2% for a layer of absorb- Using Fig. 4, we may relate the energy E at which Ð2 k ing material x = 700 g cm and b = 0.04. the knee in the spectrum occurs to the amount of tra- σin We see from Fig. 3 that = const at E < E0. There- versed material x by the empirical relation Ӷ fore, for particles with E E0, the solution of Eq. (4) must correspond to σin = const; i.e., it must be ()Ð0.8 Ek = 3.4 x/L0 TeV. β (), Ð 0 Ðx/L0 IEx = I0E e . As we see, the location of the knee in the proton spectrum depends weakly on the amount of material tra- Let us apply these solutions to the acceleration of versed. Therefore, the knee region in the observed spec- particles at the early phase of a supernova explosion, trum, which is the sum of the spectra from many i.e., in sufficiently deep regions of its envelope. sources in which the protons traverse different amounts Suppose that the accelerated particles initially had a of material, will be smeared only slightly; i.e., the power-law spectrum of the form observed location of the knee will be close to E0 in Ðβ dependence (5), as observed in the experiment. IE()= I E 0. 0 A crucial factor in considering various cosmic-ray acceleration mechanisms is usually the spectral shape From the place of acceleration to the escape of particles β from the star, they had to traverse a significant thickness of the accelerated particles and the spectral index in of envelope material. Therefore, the spectrum emerging this spectrum. However, a different approach to the for- from the envelope will have different spectral indices in mation of the observed GCR spectrum is also possible. β β different spectral ranges. The spectral index is = 0 + According to this approach, particles are generated δ δ , where = bx/L0 (x is the amount of traversed mate- in the sources with a spectrum far from the power law β β Ӷ Ðβ rial) at E > E0 and is constant and equal to = 0 at E I ∝ E with β = 2Ð2.6. If there is a characteristic E0. Thus, the initially power-law spectrum of the accel- parameter ξ in the generated spectrum that determines erated protons as they emerge from the supernova will the spectral shape and if the sources themselves have a ξ ∝ be a power-law spectrum with a knee. power-law distribution of this parameter, i.e., Ik( )

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 650 GRIGOROV, TOLSTAYA ξ−β , where Ik is the intensity of source k, then the total supernovas in which the protons were accelerated, but spectrum can be a power law with an index equal to β. at a later phase: ﬁrst, the protons accelerate, and then, A glaring example of the formation of a power-law after a lapse of time, the nuclei accelerate. spectrum in this way is the spectrum of energetic γ-ray π0 photons in the Earth’s atmosphere from -meson REFERENCES decay. If the π0 mesons (sources) have a power-law distribution in the Lorentz factor (or, equivalently, in 1. T. Shibata, Nuovo Cimento C 19, 713 (1996). γ Emax—the maximum energy of the generated -ray pho- 2. N. L. Grigorov, V. E. Nesterov, I. D. Rappoport, et al., in tons), then the total γ-ray spectrum will be a power law Space Research (Akademie, Berlin, 1972), Vol. 12, p. 1617. with an index that determines the distribution in Emax. At the same time, the γ-ray photons in the sources them- 3. N. L. Grigorov, Kosm. Issled. 33, 339 (1995). selves (the rest frame of π0 mesons) are monoenergetic; 4. G. Adams, V. I. Zatsepin, M. I. Panasyuk, et al., Izv. i.e., their distribution is very far from a power law. Ross. Akad. Nauk, Ser. Fiz. 61, 1181 (1997). If we take into account the fact that the power-law 5. N. Grigorov and E. Tolstaya, in Proceedings of the 27th distribution of a particular property of the various phys- ICRC (Hamburg, Germany, 2001), p. 1647. ical quantities in nature is widespread, then the forma- 6. A. Shomodi, S. Sugar, B. Chadraa, et al., Yad. Fiz. 28, tion of the observed proton spectrum by the process 445 (1978) [Sov. J. Nucl. Phys. 28, 225 (1978)]. under consideration also seems likely. In this case, the 7. J. Wefel (for the ATIC Collaboration), in Proceedings of β β observed value of is the mean of many i values for the 27th ICRC (Hamburg, Germany, 2001), p. 2111. the spectra from the individual sources. The value of 8. Yu. I. Stozhkov, Kratk. Soobshch. Fiz. (in press). δ 〈 〉 = b x/L0 is also the mean of many individual thick- 9. N. L. Grigorov, Yad. Fiz. 51, 157 (1990) [Sov. J. Nucl. nesses of traversed material in individual supernovas. Phys. 51, 99 (1990)]. A peculiar feature of this formation of the observed 10. GOST (State Standard) 25645.122-85 (1985); GOST spectrum is its ﬂattening with increasing energy. This is (State Standard) 25645.125-85 (1985); GOST (State because the observed spectrum consists of a set of spec- Standard) 25645.144-88 (1988). β tra with different i. As E increases, the contribution of 11. N. L. Grigorov and E. D. Tolstaya, Pis’ma Zh. Éksp. β the components with larger i will decrease, and, Teor. Fiz. 74, 147 (2001) [JETP Lett. 74, 129 (2001)]. accordingly, the contribution of the components with 12. Ya. Kawamura, H. Matsutani, and H. Najio, Phys. Rev. β smaller i will increase. This effect is experimentally D 40, 729 (1989). observable. 13. I. P. Ivanenko, V. Ya. Shestoperov, I. D. Rappoport, et al., To conclude the discussion of the proton spectrum, in Proceedings of the 23rd ICRC (Calgary, Canada, we emphasize that the existence of a knee in the proton 1993), Vol. 2, p. 17. spectrum at Ek ~ 1 TeV is important evidence that the 14. V. I. Zatsepin, T. V. Lazareva, G. P. Sazhina, et al., Yad. cosmic-ray protons are generated in dense objects in Fiz. 57, 684 (1994) [Phys. At. Nucl. 57, 645 (1994)]. which they traverse hundreds of g cmÐ2 of material. 15. M. L. Cherry (for the JACEE Collaboration), in Proceed- This circumstance can be important evidence for the ings of the 25th ICRC (Rome, 1997), Vol. 4, p. 1. Galactic origin of the cosmic-ray proton component. 16. N. S. Konovalova, Candidate’s Dissertation in Physics The cosmic-ray particles discussed above constitute and Mathematics (Physical Inst., Russian Academy of the bulk of the beam. Their energies are within the so- Sciences, Moscow, 1996). called knee in the spectrum of all particles at E ≈ (3Ð 17. RUNJOB Collaboration, Astropart. Phys. 16, 13 (2001). 5) × 1015 eV. In general, they are investigated by direct 18. N. Grigorov and E. Tolstaya, in Proceedings of the 26th methods in experiments on balloons and Earth satel- ICRC (Salt Lake City, USA, 1999), Vol. 3, p. 183. lites. According to the popular view, their sources are 19. N. L. Grigorov, V. E. Nesterov, I. D. Rappoport, et al., supernovas of our Galaxy, and their acceleration mech- Yad. Fiz. 11, 1058 (1970) [Sov. J. Nucl. Phys. 11, 588 anism involves the shock waves of the expanding (1970)]. supernovas envelope. The range of ultrahigh-energy 20. S. A. Colgate and M. H. Johnson, Phys. Rev. Lett. 5, 235 cosmic-ray particles extends from the knee up to the (1960). 20 measured end of the spectrum (~10 eV). This range 21. N. L. Grigorov, I. D. Rappoport, I. A. Savenko, et al., Izv. has been investigated only by indirect methods. In this Akad. Nauk SSSR, Ser. Fiz. 29, 1656 (1965). range, there are interesting problems far from those 22. N. L. Grigorov, Yad. Fiz. 25, 788 (1977) [Sov. J. Nucl. considered above (see review of current status in [24]). Phys. 25, 419 (1977)]. The formation of a proton spectrum with a knee at 23. T. Gaisser, Cosmic Rays and Particle Physics (Cam- energy of ~1 TeV considered here does not affect exist- bridge Univ. Press, Cambridge, 1990). ing models for the acceleration of nuclei in any way. 24. E. Roulet, astro-ph/0310367. Moreover, since the acceleration of nuclei is possible only in a low-density medium, the nuclei can be accelerated by shock waves in the envelopes of the same Translated by V. Astakhov

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 651Ð660. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 744Ð754. Original Russian Text Copyright © 2004 by Fedorov, Yurkov. NUCLEI, PARTICLES, AND THEIR INTERACTION

QED with Vector and Axial Vector Types of Interaction and Dynamical Generation of Particle Masses D. K. Fedorov* and A. S. Yurkov** *e-mail: [email protected] **e-mail: ﬁ[email protected] Received August 18, 2003

µ Abstract—We consider a model of electrodynamics with two types of interaction, the vector (eψ (γ Aµ)ψ) and ψ γ µγ5 ψ axial vector (eA ( Bµ) ) interactions, i.e., with two types of vector gauge fields, which corresponds to the ⊗ local nature of the complete massless-fermion symmetry group U(1) UA(1). We present a phenomenological model with spontaneous symmetry breaking through which the fermion and the axial vector field Bµ acquire masses. Based on an approximate solution of the Dyson equation for the fermion mass operator, we demonstrate the phenomenon of dynamical chiral symmetry breaking when the field Bµ has mass. We show the possibility of eliminating the axial anomalies in the model under consideration when introducing other types of fermions (quarks) within the standard-model fermion generations. We consider the polarization operator for the field Bµ and the procedure for removing divergences when calculating it. We demonstrate the emergence of a mass pole in the propagator of the particles that correspond to the field Bµ when chiral symmetry is broken and consider the problems of regularizing closed fermion loops with axial vector vertices in connection with chiral symmetry breaking. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In this form, the interaction Lagrangian is actually a local simplification of the interaction between the two In the universally accepted quantum electrodynam- fermion currents through the nonlocal interaction that ics (QED), the fermion masses are assumed to be given, describes the vector field photon exchange in a phe- and the question about the origin of these masses is not nomenological form. DCSB with the generation of fer- considered. At the same time, the idea that the electron mion mass m and vacuum condensate 〈〉0 ψψ 0 is mass is generated by electromagnetic interactions dates back to the end of the 19th centuryÐbeginning of the observed in the NambuÐJona-Lasinio model with 20th century. Despite such a long history, this idea is massless fermions if the coupling constant G exceeds a also embodied in present-day studies [1Ð4], now on the critical value, basis of quantum theory. ≥ π2 Λ2 GGcrit = 2 / , It should be noted that the symmetry of the Lagrangian for massless fermions is higher than the where Λ2 is the invariant cutoff parameter. symmetry of the Lagrangian for massive fermions. In the quantum electrodynamics of massless fermi- Massless fermions have a chiral symmetry, while mas- α α ons, DCSB also exists for strong coupling > c, sive fermions have no such symmetry. Thus, if the fer- α where c is the critical coupling constant, and is absent mion masses are generated dynamically, through inter- α α α α for < c [1Ð4]. If > c and if there is an invariant actions, then dynamical chiral symmetry breaking cutoff parameter Λ, then a nontrivial solution to the (DCSB) takes place. Dyson (SchwingerÐDyson) equation exists for the One of the first quantum-field models that described dynamical fermion mass function β(k2), 1/(pˆ Ð β(k2)) is DCSB was the so-called NambuÐJona-Lasinio model [5] the fermion Green function in momentum space. with the local interaction Lagrangian At the same time, it should be noted that the leading int []()ψψ 2 ()ψ γ 5ψ 2 principle of almost any current quantum-field interac- LNJL = G + i , tion theory is the idea of gauge (local) invariance. In this case, the symmetry of the Lagrangian for free (gen- where G is the coupling constant. In a more general erally fermion) fields uniquely determines the form, the interaction Lagrangian for this model appears Lagrangian for the interaction of these fields with the as the product of two fermion currents, additional vector fields that ensure the satisfaction of the local (gauge) invariance conditions and that are int ()ψγψγµψ ()ψ LNJL = G µ . interaction carriers.

652 FEDOROV, YURKOV The standard quantum electrodynamics is nothing is symmetric relative to the gauge transformations but the gauge U(1) theory. However, in contrast to the U(1): case of a nonzero mass, the symmetry group of the ⊗ ψ ()ψ1 + iΘ()x , Lagrangian for massless fermions is U(1) UA(1), where the group U(1) corresponds to the ordinary phase 1 (2) Aµ Aµ + ---∂µΘ()x , transformations of the field function, and the group e UA(1) corresponds to the chiral transformations. In this case, it seems more natural to consider both U(1) and which are represented in (2) in an infinitesimal form. However, there is also a symmetry of Lagrangian (1) UA(1) as equivalent local transformations and to con- ⊗ relative to the global chiral transformations U (1): struct the interaction from the group U(1) UA(1). A Clearly, new particles (axial photons) will corre- ψ ()ψ1 + iγ 5Θ . (3) spond to the additional gauge field, but they acquire a A mass through chiral symmetry breaking; their mass The electromagnetic field described by the potentials may prove to be very large, or, in a sense, the field of Aµ corresponds to the interaction of charge particles. these particles may be considered as a kind of a regula- In this formulation of electrodynamics, the symme- tor field of the theory. try relative to transformations (2) and (3) is a gauge and In this paper, we consider a model of electrodynam- nongauge one, respectively, which is connected with an ics with two types of interaction, the vector imbalance in the virtually equivalent symmetries of ψ γµ ψ ψ γ µγ5 ψ ((e Aµ) ) and axial vector (eA ( Bµ) ) interac- Lagrangian (1). tions, i.e., with two types of interaction-carrying vector Let us introduce the second gauge field Bµ that cor- gauge fields, which corresponds to the local nature of responds to the chiral symmetry transformation U (1). the complete massless-fermion symmetry group A ⊗ σ Lagrangian (1) takes the form U(1) UA(1). A phenomenological ( -type) model with spontaneous symmetry breaking through which 1 µν 1 µν L = Ð ---F Fµν Ð ---G Gµν the fermion and the axial vector field Bµ acquire masses 4 4 is presented in Section 2. In Section 3, based on an ψ()ψγ µ∂ γ µ γ µγ 5 (4) approximate solution of the Dyson equation for the fer- + i µ ++e Aµ eA Bµ , mion mass operator, we demonstrate the phenomenon Gµν = ∂µ Bν Ð ∂ν Bµ; of DCSB in a model with two gauge fields when the field Bµ has a mass. In Section 4, we show the possibil- it is invariant relative to the gauge transformations ity of eliminating the axial anomalies in the model U(1), under consideration when introducing other types of ψ ()ψΘ() fermions (quarks) within the standard-model fermion 1 + i x , generations for an appropriate choice of axial coupling 1 (5a) α α Aµ Aµ + ---∂µΘ()x , constants eA (axial charges) A = for each type of fer- e mions. In Section 5, we consider the polarization operator for the field Bµ and the procedure for removing and UA(1), divergences when calculating it. We demonstrate the ψ ()ψγ 5Θ () emergence of a mass pole in the propagator of the par- 1 + i A x , ticles that correspond to the field Bµ when chiral sym- (5b) 1 ∂ Θ () metry is broken. This confirms that the solution of the Bµ Bµ + ----- µ A x . e Dyson equation for the fermion mass operator is ade- A quate in a situation with DCSB. We also consider the The symmetry transformation UA(1) becomes local problems of regularizing closed fermion loops with Θ with the local parameter A(x), while the field Bµ is a axial vector vertices in connection with chiral symme- pseudovector. The coupling constant e , the axial fer- try breaking. A mion charge, is introduced to couple the field Bµ with the fermion field ψ. 2. THE PSEUDOVECTOR INTERACTION If we naively described the electrodynamics using IN QED AND SPONTANEOUS CHIRAL Lagrangian (4), then we would have two massless SYMMETRY BREAKING gauge fields, Aµ and Bµ; i.e., we would have two types The standard quantum electrodynamics with mass- of massless photons corresponding to Aµ and Bµ. How- less fermions described by the Lagrangian ever, only the Aµ photons, i.e., ordinary photons, are observable massless photons, so we should take into 1 µν µ µ L = Ð---F Fµν + ψ()ψiγ ∂µ + eγ Aµ , account the UA(1) symmetry breaking. 4 (1) The actual charged fermions have observable finite ∂ ∂ Fµν = µ Aν Ð ν Aµ, masses; i.e., the UA(1) symmetry is broken. As a result

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 QED WITH VECTOR AND AXIAL VECTOR TYPES OF INTERACTION 653 σ σ σ σ of symmetry breaking, the charged fermions acquire a broken symmetry. By substituting ' = Ð 0 for the mass [1Ð3, 6]. field, we can obtain the fermion mass term in By analogy with the σ-models used to describe Lagrangian (6), strong interactions (see, e.g., [5]), in which the fermi- mψψ, mg= σ , (8) ons acquire a mass when the symmetry is spontane- 0 ously broken, we write the Lagrangian that clearly and the mass term for the pseudovector boson Bµ, shows spontaneous chiral symmetry breaking in our case as 1 2 µ 2 2 2 ---M B Bµ, M = 4e σ . (9) 2 A 0 1 µν 1 µν L = Ð ---F Fµν Ð ---G Gµν 4 4 Thus, the fermion and the gauge field of the pseudovector boson acquire masses. The σ field µ µ µ 5 ψ()ψγ ∂ γ γ γ 2 2 + i µ ++e Aµ eA Bµ λσ π remains massive with the mass Mσ = 2 0 , but the σ T ← field remains massless, showing the emergence of 1()∂ τ + --- µ + i 22eA Bµ (6) massless Goldstone modes (excitations) in the case of 2 π spontaneous symmetry breaking. As might be expected, the electromagnetic field Aµ acquires no →µ µ σ ×∂()Ð iτ 2e B mass. 2 A π When the σ models for strong interactions are ana- λ lyzed, π-like fields are associated with low-mass π ()σ2 π2 σ2 2 ψσ()ψγ 5π Ð --- + Ð 0 Ð g + i . mesons, but when the electrodynamics is considered, 4 the existence of a physical massless pseudoscalar field This Lagrangian is invariant relative to the gauge trans- is unacceptable. To eliminate the massless pseudoscalar formations U(1), field π, we should take into account the fact that the theory under consideration is a gauge one. In accordance 1 ψ ()ψ1 + iΘ()x , Aµ Aµ + ---∂µΘ()x , (7a) with the standard scheme for demonstrating spontane- e ous symmetry breaking in theories with gauge fields, and U (1), we may choose a (unitary) gauge of the field Bµ such A that the π field vanishes. ψ ()ψγ 5Θ () 1 + i A x , Thus, the explicit scheme of spontaneous chiral symmetry breaking with the pseudovector gauge field β 1 ∂ Θ () Bµ and a set of Higgs bosons (the phenomenological µ Bµ + ----- µ A x , (7b) eA scalar and the pseudoscalar) show that the fermion and σσΘ ()π ππΘ ()σ pseudoscalar gauge fields acquire masses and that there + 2 A x , Ð 2 A x are no physical fields corresponding to massless Gold- or stone modes. σ σ In calculations in the theory described by  ()τ Θ () Lagrangian (6), the propagator of the massive field Bµ  1 + i 22 A x , (7c) π π in the unitary Bµ gauge is where 5 1 kµkν µν() ------µν D k = 2 2 g Ð ------2 ;  k Ð M + i0 M τ 0 Ði 2 = . i 0 its use causes difficulties with the removal of divergences and renormalization. However, we may use In (6), the Higgs σ and π bosons (scalar and pseudo- other gauges with a gauge-fixing term of the form scalar, respectively) are the phenomenological fields 1 ()∂µ 2 that are coupled with the fermion field by the coupling LGF = Ð-----ξ- Bµ . constant g; they have no electric charge e, but have an 2 axial charge 2eA equal to twice the fermion axial The following propagator of the field Bµ corresponds to charge. these gauges: The potential 2 1 Dµν()k = ------()σπ, ()σ λ()2 π2 σ2 2 2 2 V = /4 + Ð 0 k Ð M + i0 (10) formed from the Higgs fields have minima at σ2 + π2 = kµkν × gµν Ð ()1 Ð ξ ------; σ2 2 ξ 2 0 that correspond to a vacuum with a spontaneously k Ð M

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 654 FEDOROV, YURKOV a special case of it is the propagator in the transverse In the initial approximation, we use the fits for the fer- Landau gauge: mion Green function β(k2) ≈ m m is the electromagnetic fermion mass) 5 1 kµkν Dµν()k = ------gµν Ð ------. (11) 2 2 2 ()≈ 1 1 []Σ() k Ð M + i0 k Gint k ------, m = ---tr 0 , (13) kˆ Ð m + i0 4 Renormalizability is restored in these gauges, but the unphysical π field remains in the theory and inter- which are similar to those used in demonstrating DCSB acts with the field Bµ via an interaction term of the form in the NambuÐJona-Lasinio model [5]. µ B ∂µπ. When using propagator (11), this interaction is The following approximation may be used for the µ ineffective, or the field ∂ Bµ is said to be nonpropagating. vertex function of the electromagnetic field: µ Γ ()γ,, ≈ An interaction of the form B ∂µπ does not emerge µ kpkp+ µ. (14) explicitly in the t’Hooft gauges with a gauge-fixing term of the form Since Lagrangian (4) is symmetric relative to the gauge transformations (5), the Ward axial identity may µ 2 5 1 ()∂ ξ π be written for the vertex function Γµ (p + k, k, p). In LGF = Ð------Bµ + M . 2ξ momentum representation, it appears as Propagator (10) also corresponds to the family of µ 5 Ð1 5 5 Ð1 k Γµ()pkkp+ ,, = G ()γpk+ + γ G ()p . (15) t’Hooft gauges. int int When ξ ∞ (physical limit), this propagator cor- When using fit (13) for the complete fermion prop- responds to the propagator of a massive vector field. agator, we can find from Eq. (15) that π The unphysical fields are present in these gauges at a 2 ξ 5 5 5 β()β()pk+ + ()p finite . However, the physical results should not Γµ()γpkkp+ ,, = µγ Ð γ kµ------depend on the gauge, i.e., on the parameter ξ. k2 (16) 5 52mkµ ≈γµγ Ð γ ------, when k µ 0. 3. THE MASS OF THE FIELD Bµ k2 AND DCSB β 2 We will use the Dyson equation [1Ð3, 6] to demon- If we expect (p ) and m to be nonzero and to result strate the DCSB in the model with Lagrangian (4). For from DCSB, then the second (pole) term in (16) does the two types of interaction in momentum representa- not vanish and the emergence of a pole at the vertex 5 2 tion, this equation may be written as Γµ (p + k, k, p) for k = 0 corresponds to the generation of massless Goldstone states in the theory. Σ() Ð1() Ð1() k = G k Ð Gint k However, if we use the propagator of the field Bµ in 4 2 νµ d p a transverse gauge similar to (11), then this pole term = Ðie γ νG ()Γkp+ µ()kppk+ ,, D ()p ------∫ int int ()π 4 makes no contribution because of the transverse tensor 2 structure of propagator (11), much as the interaction (12) µ 4 B ∂µπ of the field Bµ with the Goldstone π fields is inef- 2 γ γ 5 ()Γ5 (),, νµ ()d p Ð ieA∫ ν Gint kp+ µ kppk+ D5, int p ------, fective in the model with Lagrangian (6). ()2π 4 Based on what we have said above about the pro- Σ where (k) is the fermion mass operator; Gint(k) is the pagator of the field Bµ, it would be appropriate to use complete fermion Green function (propagator); G(k) is the fit νµ νµ the propagator of a free fermion; D ()k and D , ()k int 5 int νµ () 1 D5, int k = ------are the Green functions for the electromagnetic field ()k2 + i0 ()1 Ð P5()k2 /k2 5 (photon) and the field Bµ; and Γµ(k + p, p, k) and Γµ (k + ν µ p, p, k) are the vector and axial vector vertex functions, × νµ k k 1 g Ð ------(17) respectively. k2 k2 Ð M2 + i0

The complete fermion propagator Gint(k) may be ν µ νµ k k 5 2 2 represented as × , P ()≈ g Ð ------2 k M , () []ˆ Σ() Σ() α()2 ˆ β()2 k Gint k ==1/ k Ð k , k k k + k , where P5(k2) is the polarization operator of the particle 2 1 µ (see Section 5) that corresponds to the ﬁeld Bµ and M is β()k ==---tr[]Σ()k , kˆ γ kµ. 4 the mass of the ﬁeld Bµ generated by DCSB. Accord-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 QED WITH VECTOR AND AXIAL VECTOR TYPES OF INTERACTION 655 ingly, we may use a fit similar to (14) for the vertex Thus, a nontrivial solution of Eq. (12) for the mass 5 operator Σ(p = 0) = m that corresponds to dynamical Γµ (p + k, k, p), chiral symmetry breaking is possible for a nonzero 5 5 mass of the vector field Bµ. In this case, calculating the Γµ()γpkkp+ ,, µγ , (18) integral in (20) does not require the introduction of an and the Landau gauge for the Green function of the ultraviolet cutoff (as would be the case, for example, in electromagnetic field, the NambuÐJona-Lasinio model [5]), and a nontrivial ν µ solution for Σ(p = 0) = m at a nonzero value of M exists νµ() 1 νµ k k for any value of the coupling constant Dint k = ------g Ð ------()k2 + i0 ()1 Ð Pk()2 /k2 k2 α 2 π 2 πα ν µ (19) ===e /4 eA/4 A. 1 νµ k k ------g Ð ------, k2 + i0 k2 Whereas DCSB arises in the NambuÐJona-Lasinio model at a coupling constant of the model where P(k2) is the photon polarization operator. ≥ π2 Λ2 Setting pµ = 0 in the Dyson equation (12) and sub- GGcrit = 2 / , stituting fits (13)–(19) for the Green and vertex functions into it, we obtain with the corresponding invariant cutoff parameter Λ2, such a natural parameter as the mass of the field Bµ 1 2 ---tr[]Σ()p p = 0 m plays a role similar to that of Λ in the model considered 4 here.

2 By m in (13)Ð(22) we may also mean the sum of the ie γ ν()γkˆ + m µ = Ð------tr ------seed mass m0 inducing symmetry breaking and the 4 ∫ 2 2 2 ()k Ð m + i0 ()k + i0 variable mass mcb reﬂecting the degree of symmetry breaking: ν µ 4 × νµ k k d k g Ð ------(20) mm= 0 + mcb. k2 ()2π 4 By analogy, we obtain for the mass of the ﬁeld Bµ 2 γ γ 5()γˆ γ 5 ieA ν k + m µ Ð ------tr ∫------MM= + M ; 4 ()k2 Ð m2 + i0 ()k2 Ð M2 + i0 0 cb the parameters that induce symmetry breaking, m and ν µ 4 0 νµ × k k d k M0, are independent and tend to zero at the end of the g Ð ------. k2 ()2π 4 calculations (in accordance with the Bogolyubov method when analyzing the DCSB; see, e.g., [6]): The gauge U(1) and UA(1) symmetries for m0 0 and M0 0. If mcb = m and Mcb = M are Lagrangian (4) are equivalent. It would also be natural nonzero when passing to a zero limit of the seed param- to assume for the ordinary charge e characterizing the eters, then dynamical chiral symmetry breaking may be vector interaction and the charge eA characterizing the said to be present. axial vector interaction that ± eA = e. (21) 4. AXIAL ANOMALY It should also be noted that relation (21) corresponds It follows from the symmetry of Lagrangian (4) to the condition for eliminating the axial anomalies that (or (6)) relative to transformations (5) (or (7)) that the will be present in the model under consideration as in corresponding axial current is conserved, any theory with the pseudovector interactions µ 5 e ψ (γ γ Bµ)ψ (see also formulas (25) and (26) in Sec- µ 5 A ∂ Jµ = 0 , (23) tion 4). The sum of the integrals in (20) will converge under and that the corresponding Ward identity (15) is valid. condition (21). Integration in (20) yields an equation However, when the theory is quantized, equality (23) that relates the masses m and M: breaks down due to the triangular loop diagrams with 2 2 2 ψ γ µγ5 ψ 3α x 2 2 M e the axial vertices eA ( Bµ) , with three axial verti- mm===------lnx , x ------, α ------. (22) 4π x2 Ð 1 m2 4π ces (loop BBB in Fig. 1), and with one axial vertex (loop AAB in Fig. 2). The anomaly emerges, because Apart from the trivial solution m = 0, Eq. (22) has a the divergences cannot be removed from these dia- nonzero solution for m at a nonzero value of M. grams without explicit symmetry breaking, and exp-

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B Within the ﬁrst generation of charged fermions of the k standard model (e, u, d) (i.e., with an electron and two quarks) with charges γσ γ5 2 1 Qe= Ð1,,--- Ð--- , (25) 3 3 the anomalies can be eliminated if the following relation is valid for the coupling constants at axial vertices:

µ 2 1 γ γ5 γυ γ5 Q = e ±1,,+−--- +−--- . (26) A 3 3 p q Given the three color degrees of freedom of the B B quarks (Nc = 3), the anomalous components of the amplitudes that correspond to both loop BBB (Fig. 1) Fig. 1. and loop AAB (Fig. 2) for all charged fermions vanish (in the massless case, the sum of the diagrams for all fermions with the quark colors becomes equal to zero). B This anomaly elimination condition corresponds to k assumption (21). Clearly, all of the aforesaid also applies to other generations. The neutrinos as particles that have no charge, but that are members of the standard-model generations, are disregarded; it would be γσ γ5 natural to set eA = 0 for all neutrinos. It should also be noted that the anomalies are eliminated precisely within the standard-model generations.

5. THE POLARIZATION OPERATOR OF THE FIELD Bµ µ γυ γ AND CHIRAL SYMMETRY BREAKING

νµ The complete Green function D , ()k for the ﬁeld p q 5 int Bµ and the propagator for free particles (in the trans- A A verse Landau gauge) ν µ Fig. 2. νµ() 1 νµ k k D5 k = ------g Ð ------k2 + i0 k2 ression (23) for the model under consideration will can be related by appear as νµ () νµ() νθ() 5 () ρµ () D5, int k = D5 k + D5 k Pθρ k D5, int k , 2 2 µ 5 e µνρσ eA µνρσ ν µ ∂ Jµ = ------ε FµνFρσ + ------ε GµνGρσ. (24) νµ νµ (27) 2 2 () ()2 k k 16π 48π D5, int k = D5 k g Ð ------, k2 The emergence of anomalies leads to a number of 5 problems with the Bµ quantization and the renormaliz- where Pµν()k is the (axial) polarization operator. The ability of the theory [7]. In the model under consider- tensor structure of the axial polarization operator (in ation, the axial interaction is not an external object and any case, for the gauge used) may also be assumed to the anomaly causes the self-consistency of the theory to be transverse (i.e., we may consider only the transverse be broken. 5 projection of the operator Pµν()k ; the longitudinal Including other types of fermions in the theory parts, if they exist, make no contribution to the com- allows the anomalous components that emerge in the plete Green function in this gauge): loops in Figs. 1 (loop BBB) and 2 (loop AAB) to be eliminated. It would be appropriate to treat the fermions ν µ 5 () 5()νµ k k included in the model under consideration in accor- Pµν k = P k g Ð ------. (28) dance with the standard-model fermion generations. k2

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From (27), we may derive invariance, the corresponding axial current is conserved (for the time being, we disregard the axial anomalies), 2 1 1 5 2 2 5 () ------() () and the tensor structure of the operator Pµν()k must D5 k = ------2 - + 2 P k D5 k , k + i0 k + i0 have only transverse components. However, the com- (29) monly used regularization methods (e.g., cutoff regu- 2 1 D ()k = ------. 5 2 5 2 2 larization) themselves distort the tensor structure of ()k + i0 ()1 Ð P ()k /k 5 Pµν()k even in the presence of gauge invariance. The following Dyson equation similar to (12) is valid For the two normalization conditions, (31) and (35), 5 for the operator Pµν()k : to be satisfied, the scalar part of the transverse projection of the polarization operator P5(k2) after regulariza- 5 tion must be Pµν()k P5()k2 = c M2 ++k2()c + Fk()2 M2F ()k2 , (36) 2 []γ γ 5 ()Γ5(),, () 1 2 1 = ieA∫tr µ Gint pk+ ν pkkp+ Gint p (30) where c1 and c2 are constants that generally contain 2 d4 p divergences when removing regularization, and F(k ) × ------. 2 ()π 4 and F1(k ) are finite functions. If there were gauge 2 invariance and if the regularization procedure did not 5 Since divergences of the loop integrals arise when cal- distort the tensor structure of Pµν()k , then the constant 5 2 culating Pµν()k in (30), it would be appropriate to c1 (and the function F1(k )) would be equal to zero, determine the normalization conditions. Assuming that 5()2 2()()2 the complete propagator for DCSB must describe the P k = k c2 + Kk , propagation of a massive vector particle of the field Bµ, the first normalization condition is and it would be impossible to simultaneously satisfy both conditions, (31) and (35). Physically, this would 5 2 2 P ()M = M , (31) imply that the Bµ photons remain massless. Introducing the mass m (or the Bogolyubov seed which corresponds to the fit of the complete Green func- mass m , because m and M are generally independent tion using in (17) when solving Dyson equation (12). 0 parameters) in the fermion propagator, we obtain UA(1) We redefine the scalar part of the polarization oper- symmetry and gauge invariance breaking. In this case, ator as follows: 5 2 the operator Pµν()k loses its transverse structure, but 5 P5()k2 = M2 + ()k2 Ð M2 P˜ ()k2 , (32) this loss of transversality will be controllable by the breaking parameter m, while P5(k2) after the separation where M is the mass acquired by the field Bµ due to of the transverse part may be represented as (36). Since chiral symmetry breaking. The complete Green func- the regularization procedure can also distort the tensor 5 tion appears as structure of Pµν()k , we impose a condition on the con-

5, int 1 stant c1, Dµν = ------5 - ()k2 Ð M2 + i0 ()1 Ð P˜ ()k2 lim c1 0, (37a) (33) m → 0 kνkµ × gνµ Ð ------. 2 2 k and, similarly for the function F1(k ), The second normalization condition related to the nor- ()2 malization of the axial charge eA is lim F1 k 0; (37b) m → 0

5 ˜ () 2 P 0 = 0, (34) i.e., the components of c1 (F1(k )) that do not satisfy this condition arise in the calculations from the regulariza- or, equivalently, tion procedure. 5() 5()2 2 In general, using the standard regularization meth- P 0 ==P M M . (35) 5 ods that preserve the tensor structure of Pµν()k (the Calculating a loop integral similar to (30) requires dimensional regularization method or the PauliÐVillars using regularization. In the presence of gauge U(1) method for fermion loops) is not logically consistent.

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p + k and c1'' is a constant that generally contains divergences). In the massless case, the expressions for loops υ γµ γ5 γ γ5 with vector and axial vector vertices are identical. This identity reflects the facts that there is a symmetry rela- k tive to both gauge transformations (5), that the vector 5 and axial vector currents are conserved (∂µJµ = ∂µ Jµ = p 0, m = 0), and that both fields Aµ and Bµ are massless. p + k Applying dimensional regularization to the integrals on the right-hand side of the equality in Fig. 3 yields γµ γυ 5 = Pµν()k k d 4 Ð d 2 pˆ ++kˆ m pˆ + m d p µ γ ------γ ------p = i eA∫tr µ 2 2 ν 2 2 d (39) ()pk+ Ð m p Ð m ()2π 2 µυ 2 – 8im g (c1'' + f(k )) d µ4 Ð d 2 2 gµν d p Ð8i m eA∫------, Fig. 3. ()()pk+ 2 Ð m2 ()p2 Ð m2 ()2π d where µ is the mass operator that restores the correct The dimensional regularization method runs into prob- dimensions of the polarization operator; regularization γ5 lems when determining the matrix . The PauliÐVillars is removed for d 4. The first integral (vector loop) method for fermion loops requires introducing addi- on the right-hand side of equality (39) has a transverse tional regularizing fermion loops with masses that are tensor structure owing to the properties of the dimen- regularization parameters that explicitly break the axial sional regularization, and expression (39) may be writ- gauge UA(1) invariance, which, in our case, makes it ten as identical to simpler methods like the cutoff regularization method. 5 Pµν()k Although using the dimensional regularization method is not quite consistent, we will use it to show 2 2 eA 2kµkν k how the terms corresponding to DCSB emerge in the ------µν ' ------, = 2k g Ð ------2 c2 + I2 0 5 12π k 4m operator Pµν()k . To this end, we note that, in fermion loops with an even number of axial vertices, all of the 2 2 matrices γ5 in integral expression (30) may be rear- 2 eA k , + gµνm ------c1' + I1------0 , (40) ranged in such a way that they will be close and, as a 12π2 4m2 result, will disappear: 1 5 Pµν()k 14Ð x()1 Ð x t Itu(), = 6 x()1 Ð x ln ------dx, ∫ 14Ð x()1 Ð x u ˆ 4 ∼ 2 γ γ 5 pˆ ++k m γ γ 5 pˆ + m d p 0 eA∫tr µ ------2 2 ν ------2 2 ------4 ()pk+ Ð m p Ð m ()2π 1 () (), 14Ð x 1 Ð x t ˆ 4 I1 tu = 6∫ ln------dx, 2 γ pˆ ++k m γ pˆ Ð m d p 14Ð x()1 Ð x u = eA∫tr µ------ν------(38) 0 ()pk+ 2 Ð m2 p2 Ð m2 ()2π 4

4 where c1' and c2' are constants that contain divergences 2 pˆ ++kˆ m pˆ + m d p γ γ when removing regularization (d 4), and we have = eA∫tr µ------2 2 ν------2 2 ------4 ()pk+ Ð m p Ð m ()2π for the P5(k2) projection 4 2 2 gµν d p 2 2 Ð8m e ------. 5 2 2 eA k A∫ 2 2 2 2 4 P ()k = m ------c' + I ------, 0 ()()pk+ Ð m ()p Ð m ()2π 21 12 12π 4m In expression (38), the integral that formally corre- (41) e 2 2 sponds to an ordinary vector fermion loop and the addi- A 2' k , 2 + ------k c2 + I------0 . tion to it proportional to m appear in the last part of the 12π2 4m2 equality. This equality may be represented in diagram form, as shown in Fig. 3 (f(k2) is a ﬁnite scalar function, According to (36), it allows both normalization condi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 QED WITH VECTOR AND AXIAL VECTOR TYPES OF INTERACTION 659 tions, (31) and (35), to be satisﬁed: the transverse part proportional to P5(k2), it gives no contribution to the propagator of the ﬁeld Bµ. 2 2 ∼ 2 eA  2 γ m In accordance with conditions (31) and (35), we c1 m ------Ð + ln ------π2 24 Ð d µ2 5 2 M may obtain for the transverse projection of Pµν()k

λ2 5 2 2 1 ,  P ()k = M + ---I1------0  , (42) 6 4m2 k2 M2  I ------, ------2 2 2 α  2 2 2 2 e k λ 2 2 2 4m 4m F ()k ∼ m ------A I ------, ------, + ()k Ð M ------A k ------1 2 2 12 2 π 2 2 12π M 4m 4m 3  ()k Ð M  for d 4, γ is the Euler constant, and λ is the normalization point. Condition (37) is also satisfied. 2 2 2 M , 2 k ,  k I1------0 Ð M I1------0 The expansion of the axial loop presented as a dia- m2 4m2 4m2  gram in Fig. 3 is convenient for separating out the com- Ð ------ , ponents related to the breaking of gauge U (1) invari- M2 ()k2 Ð M2  A  ance (5) and to DCSB for any regularization method. The first term in Fig. 3 formally corresponds to the 2 2 polarization vector of the electromagnetic field Aµ, and k M  I ------, ------the vector current is conserved. Therefore, the final α  2 2 5 2 2 4m 4m result after the separation of divergences and the P˜ ()k = ------A k ------(43) π 2 2 removal of regularization for the first term in Fig. 3 3  ()k Ð M must be transverse and have the structure 

2()()2 ()2 2 2 k c2 + Fk gµν Ð kµkν/k , 2 M , 2 k ,  k I1------0 Ð M I1------0 m2 4m2 4m2  where F(k2) is a ﬁnite function, and the components Ð ------ , M2 ()k2 Ð M2  corresponding to DCSB with a gµν-type tensor structure  remain in the second term, which clearly corresponds to condition (37). 1 Thus, for an arbitrary regularization procedure, the 14Ð x()1 Ð x t Itu(), = 6 x()1 Ð x ln ------dx, terms without the above structure in the vector loop of ∫ 14Ð x()1 Ð x u the ﬁrst expansion term in Fig. 3 are assumed to be 0 unphysical and introduced by the regularization proce- 1 dure. They introduce no problems in the theory and in 14Ð x()1 Ð x t the corresponding divergences and can be removed by I ()tu, = 6 ln ------dx. 1 ∫ 14Ð x()1 Ð x u introducing appropriate counterterms. However, in 0 contrast to the addition related to the second term on the right-hand side of the expansion in Fig. 3, these terms For the proper energy operator of the Bµ photon have no physical meaning. When analyzing DCSB, one should use an expres- Π5 () 2 ()5 ()5() µν xxÐ ' = ieA 0 TJµ x Jν x' 0 sion of form (16) with the following pole term of the Goldstone state in (30) for the vertex function: in momentum representation, we have

Γ5 ()γ,, ≈ γ 5 γ 5 2 νµ () νµ() νθ()Π5 () ρµ() µ pkkp+ µ Ð 2mkµ/k . D5, int k = D5 k + D5 k θp k D5 k , ν µ νµ 1 νµ k k (44) Γ5 () ------This nontrivial addition to the vertex µ (p + k, k, p) dis- D5 k = 2 g Ð ------2 . rupts the transverse tensor structure of the operator k + i0 k 5 2 Pµν()k , and the integral for the Goldstone addition By analogy with the polarization operator, we may 2 5 2mkµ/k diverges when regularization is removed. How- consider only the transverse projection for Πµν()k , 5 2 ever, this addition to the polarization operator Pµν()k 5 5 2 2 has a purely longitudinal structure; when separating out Πµν()k = Π()k ()gµν Ð kµkν/k .

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We obtain for it Bµ itself is the condition for the existence of a nontrivial

2 5 2 solution to the Dyson equation that corresponds to 5 2 k P ()k Π ()k = ------, dynamical chiral symmetry breaking. 2 5 2 k + i0 Ð P ()k If the Bµ photons are assumed to be physical parti- (45) 2 2 2 2 5 2 cles, then they must be assumed to very massive in 5 2 k ()M + ()k Ð M P˜ ()k Π ()k = ------; order not to come into conflict with the actual experi- 2 5 ӷ ()k Ð M2 + i0 ()1 Ð P˜ ()k2 mental situation: according to (22), we have for M m π2 Π5 () M ∝ 8 i.e., µν k also has a pole that corresponds to the mas------exp------. m 3e2 sive Bµ photon exchange in momentum representation. At the same time, it should be noted that not only electromagnetic, but also other types of interactions can 6. CONCLUSIONS significantly affect the masses of real particles. Thus, Thus, based on our model for the electrodynamics actual physical numerical mass estimates for the Bµ of massless fermions, which, apart from the gauge field photons require further studies. Aµ corresponding to U(1) symmetry, also includes the field Bµ corresponding to chiral gauge UA(1) symmetry, REFERENCES we have shown that both the fermions and the pseudovector gauge field Bµ can acquire masses. We 1. P. I. Fomin, V. P. Gusynin, V. A. Miransky, and have demonstrated that the Dyson equation for the fer- Yu. A. Sitenko, Riv. Nuovo Cimento 6, 1 (1983). mion mass corresponding to DCSB when the mass of 2. T. Maskawa and H. Nakajima, Prog. Theor. Phys. 52, 1326 (1974). the gauge field Bµ is generated can have a nontrivial solution, and, when the fermion mass is generated 3. V. A. Miransky, Phys. Lett. B 91B, 421 (1980). through DCSB, the field Bµ can acquire mass. 4. V. P. Gusynin and V. A. Kushnir, Preprint No. 89-81E, ITF AN SSSR (Inst. for Theoretical Physics, USSR The massiveness of the field Bµ that results from Academy of Sciences, Moscow, 1990). DCSB is consistent with the existence of only one type 5. W. Weise, in Quarks and Nuclei, International Review of of massless particles—the electromagnetic interaction Nuclear Physics (World Sci., Singapore, 1984), Vol. 1, carriers (photons). The removal of anomalies in this p. 57; Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, QED model, especially within each standard-model 345 (1961). fermion generation, also places it in the class of models 6. V. E. Rochev, Preprint No. 2001-40, IFVÉ (Inst. for High that can be physically adequate. Energy Physics, Protvino, Moscow oblast, 2001). Condition (21) for the vector and axial vector cou- 7. S. Adler, Phys. Rev. 117, 2426 (1969); S. Bell and pling constants, which corresponds to the condition for R. Jakiw, Nuovo Cimento A 60, 47 (1969); L. D. Fad- eliminating the axial anomalies in the model under con- deev and A. A. Slavnov, Gauge Fields: Introduction to sideration, allows us to dispense with the ultraviolet Quantum Theory, 2nd ed. (Nauka, Moscow, 1988; Add- cutoff when working with the Dyson equation and ison-Wesley, Redwood City, Calif., 1990). when analyzing its solutions. The mass of the field Bµ acts as a cutoff factor, while the massiveness of the field Translated by V. Astakhov

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 661Ð666. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 755Ð760. Original English Text Copyright © 2004 by Soa, Long, Binh, Khoi. NUCLEI, PARTICLES, AND THEIR INTERACTION

Single Z' Production at Compact Linear Collider Based on eÐg Collisions¦ D. V. Soaa, H. N. Longb, D. T. Binhc, and D. P. Khoic,d aDepartment of Physics, Hanoi University of Education, Hanoi, Vietnam bPhysics Division, NCTS, National Tsing Hua University, Hsinchu, Taiwan; e-mail: [email protected]; on leave from Institute of Physics, NCST, Hanoi, Vietnam cInstitute of Physics, NCST, Hanoi, Vietnam dDepartment of Physics, Vinh University, Vinh, Vietnam Received October 10, 2003

Abstract—We analyze the potential of the compact linear collider (CLIC) based on eÐγ collisions to search for γ ⊗ ⊗ the new Z' gauge boson. Single Z' production on eÐ colliders in two SU(3)C SU(3)L U(1)N models, the minimal model and the model with right-handed neutrinos is studied in detail. The results show that new Z' gauge bosons can be observed on the CLIC and that the cross sections in the model with right-handed neutrinos are bigger than those in the minimal one. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION CERN large hadron collider [12] or the next linear col- Neutral gauge structures beyond the photon and the lider [13, 14]. In [15], two of us have considered single Z boson have long been considered one of the best production of the bilepton and shown that several thou- motivated extensions of the standard model of elec- sand events are expected at the integrated luminosity ≈ × 4 Ð1 troweak interactions. They have been predicted in many L 9 10 fb . In this work, single production of the models going beyond the standard one, including the new Z' gauge boson in the 3Ð3Ð1 models is considered. ⊗ ⊗ The paper is organized as follows. In Section 2, we give models based on the SU(3)C SU(3)L U(1)N (3Ð3Ð1) gauge group [1Ð5]. These models have some interesting a brief review of two models: the relation among real characteristics. First, they predict three families of physical bosons and constraints on their masses. Sec- tion 3 is devoted to single production of the Z' boson in quarks and leptons if the QCD asymptotic freedom is γ imposed. Second, the PecceiÐQuinn symmetry natu- the eÐ collisions. Discussion is presented in Section. 4. rally occurs in these models [6]. Finally, it is characteristic of these models that one generation of quarks is 2. REVIEW OF 3Ð3Ð1 MODELS treated differently from the other two. This might lead to a natural explanation for the unbalancing heavy top To put the context in a proper frame, it is appropriate quark. to briefly recall some relevant features of the two types The Z' gauge boson is a necessary element of vari- of 3Ð3Ð1 models: the minimal model proposed by ous models that extend the Standard Model. In general, Pisano, Pleitez, and Frampton [1, 2], and the model the extra Z' boson may not couple in a universal way. with right-handed neutrinos [4, 5]. There are, however, strong constraints from flavor- changing neutral current processes that specifically limit the nonuniversality between the first two genera- 2.1. The Minimal 3Ð3Ð1 Model tions. Lower bounds on the mass of Z' following from The model treats the leptons as the SU(3)L antitrip- analysis of a variety of popular models are found to be lets [1, 2, 16],1 in the energy range 500Ð2000 GeV [7, 8]. It was suggested recently that the 3Ð3Ð1 models eaL  arise naturally from the gauge theories in spacetime with f = Ðν ≈ ()130,, , (1) extra dimensions [9] where the scalar fields are the com- aL aL c ponents in additional dimensions [10]. A few different ()e a versions of the 3Ð3Ð1 model have been proposed [11]. where a = 1, 2, 3 is the generation index. Two of the Recent investigations have indicated that signals of three quark generations transform as triplets, and the new gauge bosons in models may be observed on the 1 The leptons may be assigned to a triplet as in [1]; the two models ¦This article was submitted by the authors in English. are mathematically identical, however.

662 SOA et al. third generation is treated differently. It belongs to an Symmetry breaking and fermion mass generation Φ ∆ ∆ antitriplet, can be achieved by three scalar SU(3)L triplets , , ' and a sextet η,

uiL 1 φ++ Q = d ≈ 33,,Ð--- , (2) iL iL 3   Φ = φ+ ≈ ()131,, , DiL  φ,0 ≈ (),, ≈ (),, uiR 312/3, diR 31Ð 1/3, ≈ (),, , + DiR 31Ð 1/3, i = 12, ∆ 1 ∆ = ∆0 ≈ ()130,, , d3L  ∆Ð ≈ (),, 2 Q3L = Ðu3L 332/3, (3)  T L ∆'0  u ≈ ()312/3,, , d ≈ ()31,,Ð 1/3, ∆' = ∆'Ð ≈ ()13,,Ð 1, 3R 3R  ≈ (),, ÐÐ T R 312/3. ∆'

a The nine gauge bosons W (a = 1, 2, …, 8) and B of  η++ η+ η0 SU(3)L and U(1)N are split into four light gauge bosons 1 1 /2 /2 and ﬁve heavy gauge bosons after SU(3) ⊗ U(1) has  L N η = η+ η 0 ηÐ ≈ ()160,, . 1 /2 ' 2/2 been broken into U(1)Q. The light gauge bosons are  those of the Standard Model: the photon (A), Z , and η0 ηÐ ηÐÐ 1 /2 2/2 2 W±. The remaining ﬁve correspond to new heavy gauge ± ±± bosons Z2, Y and doubly charged bileptons X . They η a The sextet is necessary to give masses to charged lep- are expressed in terms of W and B as [16] tons [3, 16]. The vacuum expectation value

+ 1 2 + 6 7 T 2Wµ ==Wµ Ð iWµ,2Yµ Wµ Ð iWµ, 〈〉Φ = ()00,,u/2 (4) ++ 4 5 2Xµ = Wµ Ð iWµ, yields masses for exotic quarks, the heavy neutral gauge boson Z', and two new charged gauge bosons ++ + 3 ()8 2 X , Y . The masses of the standard gauge bosons and Aµ = sW Wµ + cW 3tW Wµ + 13Ð tW Bµ , the ordinary fermions are related to the vacuum expectation values of the other scalar ﬁelds, 3 8 2 Zµ = c Wµ Ð s ()3t Wµ + 13Ð t Bµ , (5) W W W W 〈〉∆0 ==v /2, 〈〉∆'0 v'/ 2,

2 8 0 0 Zµ' = Ð 13Ð tW Wµ + 3tW Bµ, 〈〉η == ω/2, 〈〉η' 0. where we use the notation For consistency with low-energy phenomenology, the ⊗ mass scale of SU(3)L U(1)N breaking must be much ӷ v v c ≡ cosθ , s ≡ sinθ , t ≡ tanθ . larger than that of the electroweak scale, i.e, u , ', W W W W W W ω. The masses of gauge bosons are explicitly given by

2 1 2 2 2 2 The physical states are a mixture of Z and Z', m = ---g ()v ++v' ω , W 4 φ φ Z1 = Z cos Ð Z 'sin , 2 1 2()2 2 ω2 MY = ---g u ++v , (6) φ φ 4 Z2 = Z sin + Z 'cos , 2 1 2 2 2 2 M = ---g ()u ++v' 4ω , where φ is the mixing angle. X 4

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 SINGLE Z' PRODUCTION AT COMPACT LINEAR COLLIDER 663 and u3L  2 2 Q = d ≈ ()331/3,, , (11) 2 g 2 2 2 m 3L 3L m ==------()v ++v' ω ------W-, Z 2 2 T 4cW cW L ≈ (),, ≈ (),, 2 2 u3R 312/3, d3R 31Ð 1/3, 2 g c 2 M = ------W -u (7) ≈ (),, Z' 3 2 T R 312/3. 14Ð sW The doubly charged bileptons of the minimal model are 2 2 here replaced by complex neutral ones as 14Ð sW 2 2 2 3sW 2 () v v ω v + ------2 - ++ ' + ------2- ' . + 1 2 Ð 6 7 4c 14Ð s 2Wµ ==Wµ Ð iWµ,2Yµ Wµ Ð iWµ, W W (12) 0 4 5 Expressions in (6) yield a splitting between the 2Xµ = Wµ Ð iWµ. bilepton masses [17], The physical neutral gauge bosons are again related φ 2 2 ≤ 2 to Z and Z' through the mixing angle . Together with MX Ð3MY mW . (8) the photon, they are deﬁned as [5] Combining the constraints from direct searches and t t 2 neutral currents, we obtain the range for the mixing 3 W 8 W angle [16] as Aµ = sW Wµ + cW Ð------Wµ + 1 Ð ----- Bµ , 3 3 Ð101.6 × Ð2 ≤≤φ 710× Ð4 2 3 tW 8 tW and a lower bound on MZ Zµ = cW Wµ Ð sW Ð------Wµ + 1 Ð ----- Bµ , (13) 2 3 3 M ≥ 1.3 TeV. Z2 2 tW 8 tW Such a small mixing angle can be safely neglected. In Zµ' = 1 Ð -----Wµ + ------Bµ. 3 that case, Z1 and Z2 are the Z boson in the Standard 3 Model and the extra Z ' gauge boson, respectively. With Symmetry breaking can be achieved with just three the new atomic parity violation in cesium, we obtain a SU(3)L triplets, lower bound for the Z2 mass [18]: 0 M > 1.2 TeV. χ Z2  χ = χÐ ≈ ()13,,Ð 1/3, (14)  2.2. The Model with Right-Handed Neutrinos χ,0 In this model, the leptons are in triplets and the third member is a right-handed neutrino [4, 5], ρ+  ρ 0 ≈ (),, ν = ρ 132/3, (15) aL  ρ,+ f = eaL ≈ ()13,,Ð 1/3, aL  (9) ()νc L a η0  ≈ (),, Ð eaR 11Ð 1. η = η ≈ ()13,,Ð 1/3. (16)  The ﬁrst two generations of quarks are in antitriplets η,0 and the third one is in a triplet, The necessary vacuum expectation values are diL ≈ (),, 〈〉χ T ==()00,,ω/2, 〈〉ρ T ()0,,u/20, QiL = ÐuiL 330, (10) (17)  〈〉η T (),, DiL = v / 200. 〈χ〉 u ≈ ()312/3,, , d ≈ ()31,,Ð 1/3, The vacuum expectation value generates masses iR iR for the exotic 2/3 and Ð1/3 quarks, while the values 〈ρ〉 ≈ (),, , 〈η〉 DiR 31Ð 1/3, i = 12, and generate masses for all ordinary leptons and

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γ Z' γ Z' two Feynman diagrams contributing to reaction (22), e– depicted in Fig. 1. The s-channel amplitude is given by – Z ' ieg e M = ------⑀µ()p ⑀ν()k uk() – – – – s 2 2 2 1 e e e e 2cW qs (23) ×γν[]() ()γ γµ () Fig. 1. Feynman diagrams for the reaction eÐγ Z'eÐ. g2V e Ð g2A e 5 qs up1 ,

where qs = p1 + p2. The u-channel amplitude is quarks. After symmetry breaking, the gauge bosons Z ' ieg ⑀ ()⑀ ()()γν gain masses as Mu = ------2 µ k2 ν p2 uk1 qu 2cW qu (24) 2 1 2()2 2 2 1 2()2 ω2 µ mW ==---g u + v , MY ---g v + , ×γ []() ()γ () 4 4 g2V e Ð g2A e 5 up1 , (18) ⑀ ⑀ ⑀ ⑀ 2 1 2 2 2 where qu = p1 Ð k2; µ(p2), ν(p2) and ν(k2), µ(k2) are M = ---g ()u + ω , X 4 the respective polarization vectors of the photon γ and the Z' boson, and g2V(e), g2A(e) are the coupling con- and stants of Z' to the electron e. In the minimal model, they 2 2 are given by [16] 2 g ()2 2 mW mZ ==------u + v ------, (19) 2 2 3 2 4cW cW g ()e = ------14Ð,s 2V 2 W (25) 2 2 2()2 2 2 g 2 u v 12Ð sW () 1 2 M = ------4ω ++------.(20) g2A e = Ð------14Ð,sW Z' ()2 2 2 23 43Ð 4sW cW cW For consistency with low-energy phenomenology, and in the model with right-handed neutrinos [5], 〈χ〉 ӷ 〈ρ〉 〈η〉 Ӷ we have to assume that , , such that mW 1 2 1 g ()e = Ð2--- + s ------, MX, MY. 2V W 2 34Ð s2 The symmetry-breaking hierarchy gives us a split- W (26) ting between the bilepton masses [19] () 1 g2A e = ------. 2 2 ≤ 2 2 MX Ð MY mW . (21) 23Ð 4sW ≈ It is therefore acceptable to set MX MY. From Eqs. (25) and (26), we see that because of the The constraint on the Z Ð Z' mixing based on the Z factor decay is given in [5], 2 Ӷ 14Ð sW 1, × Ð3 ≤≤φ × Ð4 Ð102.8 1.8 10 ; the cross sections in the minimal model are smaller than in this model, we do not have a limit for sin2θ . With those in the model with right-handed neutrinos. We W work in the center-of-mass frame and let θ denote the this small mixing angle, Z1 and Z2 are the Z boson in the Standard Model and the extra Z' gauge boson, respec- scattering angle (the angle between the momenta of the initial electron and the ﬁnal one). We have evaluated the tively. From the data on parity violation in the cesium θ atom, we obtain a lower bound on the Z mass in the dependence of the differential cross section 2 dσ/dcosθ, the energy, and the Z' boson mass depen- range between 1.4 TeV and 2.6 TeV [18]. Data on the σ ∆ ≤ dence of the total cross section . kaon mass difference mK gives the bound M Z2 1) In Fig. 2, we plot dσ/dcosθ for the minimal 1.02 TeV [8]. model as a function of cosθ for the collision energy at CLIC s = 2733 GeV [20] and the relatively low value γ 3. Z' PRODUCTION IN eÐ COLLISIONS of mass mZ' = 800 GeV. From Fig. 2, we see that σ θ Now we are interested in the single production of d /dcos is peaked in the backward direction (this is new neutral gauge bosons Z ' in eÐγ collisions, due to the eÐ pole term in the u-channel) but is ﬂat in the forward direction. We note that the behavior of σ θ Ð()γ, λ (), λ Ð(), τ (), τ d /dcos for the model with right-handed neutrinos is e p1 + p2 ' e k1 + Z' k2 ' , (22) similar at other values of s . λ λ τ τ where pi and ki are the momenta and , ', , and ' are 2) The energy dependence of the cross section for the helicities of the particles. At the tree level, there are the minimal model is shown in Fig. 3. The same value

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Cross section, pb Cross section, pb 0.07 0.18 0.16 0.06 0.14 0.05 0.12 1 0.04 0.10 0.03 0.08 0.06 4 0.02 0.04 2 0.01 0.02 3 0 0 –1.0 –0.5 0 0.5 1.0 1200 1600 2000 2400 2800 cos θ s, GeV Fig. 3. Cross section σ(eÐγ Z'eÐ) of the minimal model as a function of s : 1—total cross section, 2—cross Fig. 2. Differential cross section of the minimal model, section in the u-channel, 3—cross section in the s-channel, s = 2733 GeV, mZ' = 800 GeV. 4—cross section in Standard Model; mZ' = 800 GeV.

≈ Ð1 of the mass as in the first case, mZ' = 800 GeV, is cho- ity L 100 fb , the number of events can be several sen. The energy range is thousand. ≤≤ In the final state, Z' decays into leptons and quarks. 1200 s 3000 GeV. Its partial decay width is equal to [21] Curve 1 is the total cross section for the minimal model, Γ() and curves 2 and 3 represent the respective cross sec- Z' ff tions of the u- and s-channels. Curve 4 is the cross sec- 2 tion for the Standard Model, reduced three times. The GFmZ ' F[]()f 2 f ()f 2 f = ------Nc g2A RA + g2V RV u-channel, curve 2, rapidly decreases with s , while 62π the s-channel has a zero point at s = mZ' and then 6.4 GeV for minimal model, slowly increases. In the high-energy limit, the s-chan- =  nel makes the main contribution to the total cross sec- 11.8 GeV for right-handed neutrinos model. tion. In Fig. 3, the cross section of the Standard Model Because of the coupling constants, the lifetime of Z' in reaches 0.18 pb and then slowly decreases to 0.05 pb, while the cross section of the minimal model is only 0.14 pb at s = 800 GeV and 0.05 pb at s = Cross section, pb 2733 GeV. The same situation occurs in the model with 0.35 right-handed neutrinos. In this model, we fix mZ' = 800 GeV and illustrate the energy dependence of the 0.30 cross section in Fig. 4. The energy range is the same as 0.25 in Fig. 3, 1200 ≤ s ≤ 3000 GeV. We see from Fig. 4 that the cross section σ decreases with s , from σ = 0.20 0.35 pb to σ = 0.08 pb. 3) We have plotted the boson mass dependence of 0.15 the number of events in the three models in Fig. 5. The 0.10 energy is fixed as s = 2733 GeV and the mass range is 800 ≤ m ≤ 2000 GeV. As we mentioned above, due Z' 0.05 to the coupling constant, the order of the lines of num- 1200 1600 2000 2400 2800 ber of events, from bottom to top, is as follows: the min- s, GeV imal model, the Standard Model, and the model with right-handed neutrinos. The smallest number of events Fig. 4. Cross section σ(eÐγ Z'eÐ) of the model with is for the minimal model. With the integrated luminos- right-handed neutrinos as a function of s ; mZ' = 800 GeV.

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Number of events NCTS under a grant from National Science Council of 9000 Taiwan. This work was supported in part by the National Council for Natural Sciences of Vietnam. 2 7000 3 REFERENCES 1. F. Pisano and V. Pleitez, Phys. Rev. D 46, 140 (1992). 2. P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). 5000 3. R. Foot, O. F. Hernandez, F. Pisano, and V. Pleitez, Phys. Rev. D 47, 4158 (1993). 4. R. Foot, H. N. Long, and Tuan A. Tran, Phys. Rev. D 50, 3000 R34 (1994). 5. H. N. Long, Phys. Rev. D 53, 437 (1996); Phys. Rev. D 1 54, 4691 (1996). 1000 6. P. B. Pal, Phys. Rev. D 52, 1659 (1995). 800 1000 1200 1400 1600 1800 7. C. Geweriger et al. (LEPEWWG ffÐ Subgroup), Preprint m, GeV LEP2FF/00-03. Fig. 5. Number of events of three models: 1—minimal 8. H. N. Long and V. T. Van, J. Phys. G 25, 2319 (1999). model, 2—right-handed neutrinos model, 3—Standard 9. I. Antoniadis, K. Benakli, and M. Quiros, Preprints Model. ETH-TH/01-10, CERN-TH/2001-202; New J. Phys. 3, 20 (2002); T. Li and L. Wei, Phys. Lett. B 545, 147 (2002); S. Dimopoulos, D. E. Kaplan, and N. Weiner, the minimal model is longer than that in the model with Phys. Lett. B 534, 124 (2002); I. Gogoladze, Y. Mimura, right-handed neutrinos. and S. Nandi, Phys. Lett. B 554, 81 (2003). 10. H. Hatanaka, T. Inami, and C. S. Lim, Mod. Phys. Lett. A 13, 2601 (1998); H. Hatanaka, Prog. Theor. Phys. 102, 4. CONCLUSIONS 407 (1999); G. Dvali, S. Randjibar-Daemi, and R. Tab- In this paper, we have considered the production of bash, Phys. Rev. D 65, 064021 (2002); N. Arkani- a single Z ' boson in the eÐγ reaction in the framework Hamed, A. G. Cohen, and H. Georgi, Phys. Lett. B 513, of the 3Ð3Ð1 models. We see that with this process, the 232 (2001). reaction mainly occurs at small scattering angles. The 11. T. Kitabayshi and M. Yasue, Phys. Rev. D 63, 095002 results show that if the mass of the boson is in the range (2001); Phys. Lett. B 508, 85 (2001); Nucl. Phys. B 609, γ 61 (2001); Phys. Lett. B 524, 308 (2002); L. A. Sanchez, of 800 GeV, then single boson production in eÐ colli- W. A. Ponce, and R. Martinez, Phys. Rev. D 64, 075013 sions may give observable values at moderately high γ (2001); W. A. Ponce, Y. Giraldo, and L. A. Sanchez, energies. On a CLIC based on eÐ colliders, with the Phys. Rev. D 67, 075001 (2003). ≈ Ð1 integrated luminosity L 100 fb , we expect observ- 12. B. Dion, T. Gregoire, D. London, et al., Phys. Rev. D 59, able experiments in future colliders. Because of the val- 075006 (1999). ues of the coupling constants, cross sections in the 13. P. Frampton and A. Rasin, Phys. Lett. B 482, 129 (2000). model with right-handed neutrinos are bigger than in 14. H. N. Long and D. V. Soa, Nucl. Phys. B 601, 361 the minimal model. (2001). In conclusion, we have pointed out the usefulness of 15. D. V. Soa, T. Inami, and H. N. Long, hep-ph/0304300. electronÐphoton colliders in testing the 3Ð3Ð1 models 16. D. Ng, Phys. Rev. D 49, 4805 (1994). at high energies via the reaction 17. J. T. Liu and D. Ng, Z. Phys. C 62, 693 (1994); N. A. Ky, eÐγ eÐZ '. H. N. Long, and D. V. Soa, Phys. Lett. B 486, 140 (2000). If the Z' boson is not very heavy, this reaction offers a 18. H. N. Long and L. P. Trung, Phys. Lett. B 502, 63 (2001). much better discovery reach for Z' than the pair produc- + Ð Ð Ð 19. H. N. Long and T. Inami, Phys. Rev. D 61, 075002 tion in e Ðe or e Ðe collisions. (2000). One of the authors (H. N. L) would like to thank the 20. Z. Z. Aydin et al., hep-ph/0208041; R. W. Assman et al., members of the Physics Department, National Center CERN 2000-008 (2000), p. 6; J. Ellis, E. Keil, and for Theoretical Sciences (NCTS), Hsinchu, Taiwan, G. Rolandi, CERN-EP/98-03, CERN-SL/98-004 (AP), where this work was completed, for warm hospitality CERN-TH/98-33. during his visit. His work has been supported by the 21. Particle Data Group, Phys. Rev. D 66, 010001 (2002).

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 667Ð677. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 761Ð773. Original Russian Text Copyright © 2004 by Melentiev, Borisov, Balykin. ATOMS, SPECTRA, RADIATION

Zeeman Laser Cooling of 85Rb Atoms in Transverse Magnetic Field P. N. Melentiev, P. A. Borisov, and V. I. Balykin Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia e-mail: [email protected] Received August 7, 2003

Abstract—The process of Zeeman laser cooling of 85Rb atoms in a new scheme employing a transverse magnetic ﬁeld has been experimentally studied. Upon cooling, the average velocity of atoms was 12 m/s at a beam intensity of 7.2 × 1012 sÐ1 and an atomic density of 4.7 × 1010 cmÐ3. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION field [12]. Application of the transverse magnetic field allowed us to obtain the optimum distribution of mag- Cold atomic beams characterized by a small average netic field along the beam axis, which is necessary for velocity of atoms at a high intensity and high phase space density are widely used in various experiments in atomic effective cooling. Using the scheme with transverse beam optics, interferometry, and lithography [1, 2]. magnetic field, we succeeded in creating a compact and Low-energy atomic beams can be obtained by method effective Zeeman slower ensuring the formation of of laser cooling, which is known in several variants intense beams of atoms with an average velocity as low employing the Zeeman effect [3, 4], frequency-chirped as 10 m/s. laser radiation [5], isotropic light [6], and wideband laser radiation [7]. Unfortunately, the process of cooling by all these techniques is accompanied by unavoid- 2. ZEEMAN LASER COOLING able increase in the transverse temperature of atoms and, hence, by a decrease in the beam brightness and 2.1. Zeeman Slowing phase space density. in Longitudinal Magnetic Field An effective means of solving this problem is According to the method of laser cooling, an atomic offered by schemes employing a two-dimensional mag- beam interacts with the counterpropagating beam of neto-optical trap (2D-MOT) ensuring both transverse laser radiation with a frequency tuned in resonance compression of the atomic beam and a decrease in the with that of a given atomic transition. In the course of transverse velocity of atoms [8Ð10]. The degree of deceleration, the absorption frequency exhibits a Dop- compression and cooling in 2D-MOT is usually limited pler shift relative to the laser radiation frequency and, by a finite time of flight of atoms through the apparatus. hence, the efficiency of the process tends to decrease. Effective use of 2D-MOTs requires that atoms in a The Doppler shift can be compensated using the linear beam would possess a sufficiently low longitudinal Zeeman effect. The scheme of atomic beam cooling by velocity. laser radiation in a magnetic field, called Zeeman slow- An alternative method for obtaining atomic beams ing, is now most widely used for obtaining slow atomic of high brightness and high phase space density is beams. based on the extraction of atoms from a three-dimensional MOT (3D-MOT) [11], an advantage of this An experimental setup for Zeeman slowing com- device being a relatively high phase space density of prises a source of neutral atoms and a Zeeman slower atoms. However, the use of this technique for obtaining creating the required magnetic field distribution in the continuous atomic beams of high intensity is limited by zone of interaction between atoms and laser radiation. the large time required for the accumulation of atoms. The cooling laser radiation tuned in resonance with a given atomic transition propagates in the direction We have developed a new method for obtaining cold opposite to that of the atomic beam. In most Zeeman atomic beams of high intensity (7.2 × 1012 sÐ1) and a slower schemes, atoms are decelerated only inside the small average velocity of atoms (12 m/s) and have stud- apparatus and do not interact with the laser radiation ied this method in application to a beam of 85Rb atoms. outside. The required magnetic field configuration in The proposed technique employs the Zeeman laser the Zeeman slower is created using a magnetic solenoid cooling of thermal 85Rb atoms in transverse magnetic with the distance between turns varied so as to provide

668 MELENTIEV et al.

B The acceleration is determined by the expression Atomic Magnetic coil beam ប Γ Laser radiation, k G σ+ a = ------, (2) -polarization 2M 1 ++G ()∆ + kV Ð αB 2/γ 2 k B σ+ (a) where 2γ is the natural width of the given atomic tran- Two current-carrying sition, M is the atomic mass, G = I/Isat is the parameter Atomic wires E of saturation of the atomic transition, I is the laser radi- beam I ation intensity, and Isat is the laser saturation intensity. Laser radiation The latter quantity is given by the formula E I k បω B (b) I = ------0, (3) sat 2τσ Fig. 1. Schematic diagrams illustrating Zeeman laser cool- τ γ ω ing of an atomic beam in (a) longitudinal and (b) transverse where = 1/2 is the time of spontaneous decay and 0 magnetic fields. is the atomic transition frequency, and σ is the absorption cross section. for the optimum distribution of the magnetic field in the Let the laser frequency be tuned precisely in reso- axial direction. The solenoid axis coincides with the nance to that of the given atomic transition. Using con- axes of atomic and laser beams (Fig. 1a). It is possible dition (1), we obtain an expression for the required field to use ring permanent magnets instead of the magnetic profile: coil. In both cases, the magnetic field vector in the Zee- man slower is collinear with the wavevector of laser 2az Bz()= B 1 Ð.------σ+ 0 2 radiation. The laser radiation possesses a polariza- V tion and excites transitions between the Zeeman sub- 0 levels corresponding to a change in the magnetic quan- The existence of a maximum possible acceleration for tum number ∆m = +1. ប Γ a given atom in a magnetic field, amax = 2 k /M (for ӷ The Zeeman shift of the atomic transition frequency I Isat) poses a limitation on the maximum magnetic is proportional to the magnetic field strength (magnetic field gradient [3], ∆ω α α induction) B: Zeeman = B, where is a constant determined by the Zeeman effect. The resonance inter- dB ≤ dB amax ------------ω------λ , (4) action between atoms and laser radiation in the Zeeman dz max d V slower is determined by the condition where dB/dω = αÐ1. When the magnetic field gradient is ∆ +0,kV Ð αB = (1) below maximum, the laser radiation intensity is limited by the condition ∆ ω ω where = laser Ð 0 is the detuning of the laser radia- ω 1 tion frequency laser from the atomic transition fre- G ≥ ------. (5) ω a dB dz quency 0 in a zero magnetic field, V is the atomic 1 + ------max ------velocity, and k = 2π/λ is the wavevector. If the magnetic Vλ dωdB field B varies in space so that condition (1) is valid at all points of the atomic trajectory in the course of deceler- The minimum temperature TD of atoms that can be ation, then atoms occur in resonance with the laser radi- achieved by means of Zeeman slowing, called the Dop- ation. pler cooling limit, is determined by the formula [13]

The required magnetic field distribution in the Zee- បγ man slower can be readily determined as follows. When TD = ------. (6) condition (1) is fulfilled at all points of the atomic tra- 2kB jectory, the force of light pressure imparts a constant 85 µ acceleration a to an atom so that its velocity decreases For Rb atoms, TD = 141 K and the corresponding according to the law minimum atomic velocity is VD = 0.12 m/s. However, there are several other limiting factors that hinder obtaining such a low velocity by Zeeman slowing. The () 2 Vz = V 0 Ð.2az main difficulty is encountered in extracting low-energy

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ZEEMAN LASER COOLING OF 85Rb ATOMS IN TRANSVERSE MAGNETIC FIELD 669 atoms from the Zeeman slower [4]: at a low atomic F' = 4 (100.2 MHz) velocity (~10 m/s), the length of interaction between an atom and the laser field on which the velocity is 120.7 MHz 5P reversed is as small as a fraction of a millimeter. For 3/2 F' = 3 (–20.5 MHz) this reason, intense atomic beams with particle veloci- 63.4 MHz F' = 2 (–83.8 MHz) ties below 50 m/s could not be obtained by means of 29.3 MHz Zeeman slowing. F' = 1 (–113.1 MHz) 780 nm 2.2. Zeeman Slowing ~ in a Transverse Magnetic Field F = 3 (1.265 GHz) The principal difference between the Zeeman slow- 5S ing in a transverse magnetic field and the scheme con- 1/2 3.0357 GHz sidered above is that the magnetic field vector is perpendicular to the wavevector of the cooling laser radia- F = 2 (–1.771 GHz) tion (in the conventional scheme, these vectors are collinear). This mutual orientation of the magnetic Fig. 2. Schematic diagram of energy levels of the D2 line of field B and the wavevector k determines, in turn, the 85Rb atom. required polarization of the laser radiation, and the electric field vector is perpendicular to the magnetic field vector B (Fig. 1b). In this configuration, the laser field) is tuned in resonance to the F = 3 F' = 4 tran- radiation can induce atomic transitions with a change in sition. The second (auxiliary) mode is tuned in reso- the magnetic quantum number ∆m = +1 or ∆m = Ð1. nance to the F = 2 F ' = 3 transition so as to provide for the optical pumping of the ground state sublevel The method of Zeeman slowing in the transverse with F = 3. magnetic field offers two important advantages over the For 85Rb atoms possessing thermal velocities, a conventional scheme: (i) simpler realization of the Doppler frequency shift is greater than the distance required magnetic field distribution in the Zeeman between sublevels of the hyperfine structure of an slower and (ii) higher accuracy of controlling the length excited state. For this reason, the positions of energy of interaction between atoms and laser radiation, facil- levels in a magnetic field should be calculated for the itating the extraction of low-energy atoms from the cases of weak and strong magnetic fields. In a weak Zeeman slower. Let us consider application of the new magnetic field, each component of the hyperfine struc- scheme to cooling 85Rb atoms. ture of the ground and excited states of 85Rb atom splits into 2F + 1 Zeeman sublevels characterized by the The existence of hyperfine splitting of the ground magnetic quantum number m , and excited states in alkali metal atoms leads to transi- F tions between various sublevels of the hyperfine struc- U = µ g m B, (7) ture. Excitation with single-mode laser radiation leads mF B F F to optical pumping of atoms to one sublevel of the µ × Ð24 hyperfine structure of the ground state and drives these where B = 9.27 10 J/T is the Bohr magneton and atoms out of resonance with the laser radiation. For gF is the Lande factor. 85Rb atom (Fig. 2), a transition from the ground state In strong magnetic fields such that the energy of an with F = 3 to an excited state with F' = 4 is a cyclic tran- atom in the magnetic field is greater than the energy of sition and the F' = 4 F = 2 transition is forbidden electron interaction with the nucleus, the character of in the dipole approximation. Therefore, the former tran- splitting changes significantly. A level characterized sition can be used for Zeeman slowing of 85Rb atoms. by the magnetic quantum number J splits into (2J + 1)(2I + 1) sublevels, determined by the quantum num- Ð4 However, there is a small probability (6 × 10 for a bers mI and mJ, with the energies laser intensity on a saturation level) of a transition to the sublevel with F' = 3 of the excited state. From this state, U = µ g m BAm+ m , (8) the atom can equiprobably pass either to the ground mJ mI B J J I J state sublevel with F = 3 or to an excited sublevel with F = 2 spaced at 3 GHz from the sublevel with F = 3, where A is the hyperfine splitting constant. For the 5P3/2 which will break cyclic interaction with the laser radia- level of 85Rb atom, A = 25 MHz. tion. A commonly accepted straightforward solution of Figure 3 shows the pattern of splitting of the mag- this problem consists in using two-frequency laser radi- netic sublevels of the ground (5S1/2) and excited (5P3/2) ation. The dominant (cooling) laser mode (decelerating states of 85Rb atom in a magnetic field B. As can be

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Energy mJ mI 5/2 3/2 3/2 1/2 –1/2 –3/2 –5/2 5/2 1/2 3/2 5P3/2 1/2 –1/2 F' = 4 –3/2 120.7 MHz –5/2 F' = 3 –1/2 –5/2 –3/2 –3/2 –5/2

~ m F 3 2 5S field Weak Strong field 1 1/2 0 –1 F = 3 –2 –3 3 GHz –2 –1 F = 2 0 1 0 20 40 60 80 100 130 140 2 Magnetic field strength, G

85 Fig. 3. The hyperfine structure of energy levels of the ground (5S1/2) and excited (5P3/2) states (the levels with F' = 3, 4) of Rb atom in a magnetic field. seen, the weak magnetic field approximation is valid for the optical pumping of the atom to the sublevels for most of the sublevels under consideration in a field with F = 3, mF = 3 and F = 2, mF = 2. Therefore, an anal- with B < 20 G, while the strong magnetic field approx- ysis of the Zeeman slowing can be restricted to the imation is applicable when B > 80 G. F = 3, mF = 3 F' = 4, mF = 4 and F = 2, mF = 2 For the ground state level with F = 2, the Lande fac- F' = 3, mF = 3 transitions. tor is negative, while for the level with F = 3 this factor is positive. As a result, the Zeeman sublevels of the The number of cooled atoms at the output of the ground state levels with the same magnetic moment Zeeman slower is, together with the average velocity of atoms, among the most important parameters charac- projection mF behave differently in response to increasing magnetic field strength. As can be seen from Fig. 3, terizing the cooling efficiency. This number is deter- the field dependences of the frequencies of transitions mined primarily by two factors: (i) the fraction of the between magnetic sublevels with F = 3 and F' = 4 sig- initial velocity distribution of atoms cooled by laser nificantly differ from the analogous dependences for radiation in the Zeeman slower and (ii) the fraction of the states with F = 2 and F' = 3. Since the Doppler shift the primary atomic flux injected into the Zeeman for these levels is the same, the complicated behavior of slower. The former circumstance dictates the need for magnetic sublevels in the applied magnetic field will increasing the velocity interval of atoms subjected to lead to a loss in efficiency of excitation for the second cooling. However, this usually leads to a considerable mode of the two frequency laser radiation in the course increase in the length of the Zeeman slower and, of Zeeman slowing. The efficiency in interaction accordingly, to a decrease in the flux of thermal atoms between atoms and laser radiation can be increased at injected into the system. An analysis shows that the the expense of the field-induced broadening. According shorter the Zeeman slower, the greater the output flux to our calculations, the parameters G1 and G2 of the of cold atoms. In selecting the optimum Zeeman slower atomic transition saturation for the dominant (cooling) length, it is also necessary to take into account the fact laser mode and the second (auxiliary) mode, respec- that the real atomic velocity distribution in a beam is tively, must obey the condition G ӷ 0.1G . 2 1 depleted of the low-velocity fraction because of atomic When 85Rb atoms interact with a two frequency collisions in the beam [14]. With allowance for this fact, laser radiation in the presence of a magnetic field, sev- we selected a magnetic field configuration in the Zee- eral photon absorptionÐreemission cycles are sufficient man slower such that atoms are decelerated beginning

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30 cm Probing laser Photomultiplier

Zeeman slower

Photomultiplier Electronic gate Nonlinear AOMs resonances

Oscillograph Frequency stabilization

Cooling laser

Fig. 4. Schematic diagram of the experimental setup. with a velocity equal to half of the most probable value 9 MHz/h. The probing laser frequency was chirped in for atoms in the beam. the vicinity of the frequency of the F = 3 F' = 4 transition in 85Rb. The atomic velocity distribution was determined using the signal of fluorescence detected by 3. EXPERIMENTAL SETUP a photomultiplier. The experimental setup for studying the Zeeman In order to eliminate the influence of the cooling slowing of atoms is schematically depicted in Fig. 4. laser radiation during the atomic velocity measure- The radiation sources were semiconductor lasers ments, the cooling laser was switched off by means of employing the Littrov scheme. Both lasers operated in an acoustooptical modulator (AOM). The fluorescence a two frequency lasing regime ensured by resonance signal was measured using a BoxCar electronic gate excitation of the relaxation oscillations due to micro- (Fig. 5). In order to determine a stationary distribution wave modulation of the injection current [15]. The of the atomic velocities, the time for which the cooling microwave modulation frequency was equal to the dif- laser was switched on was selected sufficiently large, so ference in the frequencies of F = 3 F' = 4 and that slow atoms leaving the slower could reach the F = 2 F' = 3 transitions (2916 MHz). The micro- detector before the laser was switched off. In our exper- wave generator power was selected such that the inten- imental configuration, this time was 6 ms. With this sity at the fundamental frequency would be four times time delay, we measured the stationary distribution of that of the first side band. The dominant laser mode was 85 atomic velocities—the same as that established for the used to excite the F = 3 F' = 4 transition in Rb constant laser irradiation of the atomic beam. In order atoms, while the one at the side band frequency excited to reduce the mechanical action upon atoms from the the F = 2 F' = 3 transition. The maximum laser outside probe laser, the probing laser radiation was put power was 15 mW. The laser beam diameter at the switched on by an AOM only during the fluorescence Zeeman slower entrance and exit was 3.5 and 5 mm, measurements. respectively. The cooling lasers were operating in the regime of active frequency stabilization with respect to Since we employed the low-velocity part of the ini- the absorption signal in a cell placed in the magnetic tial atomic velocity distribution for laser cooling, spe- field [16]. The short-term laser frequency stability was cial measures were taken to obtain a beam of thermal 3 MHz, while the long-term frequency drift was within atoms with undepleted low-energy fraction of the total

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B Magnetic field calculated value). Investigation of the primary atomic 2-s field off beam characteristics showed that this source exhibited 2-s field on no depletion of the low-energy part of the velocity distribution as a result of the beam scattering from vapor in the vicinity of the exit diaphragm of the atomic oven. I 1.5 ms Cooling laser On c The magnetic field profile along the Zeeman slower axis was calculated using the formula 6 ms Off Delay Box Car gate On () amaxz Bz = B0 1 Ð,------2 - V 0 50 µs Off × 5 2 where amax = 1.07 10 m/s and the initial velocity is t V0 = 150 m/s. The Zeeman slower comprised two Fig. 5. Time series of the switching of magnetic field B(t), 22-cm-long aluminum stripe combs cut to various cooling laser field Ic(t), and BoxCar gate for monitoring the depths (Fig. 6). The variable cut depth allowed the atomic velocity distribution during Zeeman slowing. required field profile along the axis to be obtained because the current passed only via a continuous part of the metal stripe. The comb configuration provided for velocity distribution. This was achieved by using a an increase in the effective mass and the surface area, source of 85Rb atoms analogous to that described thus increasing the heat exchange rate under ultrahigh in [17]. The atomic source temperature could be varied vacuum conditions. from 20 to 500°C. The intensity of the atomic beam Figure 7 compares the calculated and experimen- formed with a 4-mm aperture at a source temperature of tally measured field profiles along the Zeeman slower T = 250°C was 4.5 × 1013 sÐ1 (approximately half of the axis for a current of I = 170 A. As can be seen, the max-

(a)

I Laser Rb atoms radiation V B I

(b)

Fig. 6. The Zeeman slower: (a) schematic diagram; (b) general view.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ZEEMAN LASER COOLING OF 85Rb ATOMS IN TRANSVERSE MAGNETIC FIELD 673 imum deviation of the measured magnetic field strength 160 from the calculated profile amounts to ∆B = 15 G. This deviation may lead to a decrease in the Zeeman slowing 140 efficiency as a result of the laser radiation being out of , G 120 resonance with the atomic transition frequency. The compensation of the deviation of the magnetic field 100 strength from that required for the effective cooling was achieved by increasing the laser radiation intensity. To 80 this end, the parameter of the atomic transition satura- 60 tion must be not less than 40

≈ Magnetic ﬁeld strength G 14. 20

The electric resistance of the Zeeman slower 0 5 10 15 20 25 (together with connecting leads in the vacuum cham- Length, cm ber) was R = 3 × 10Ð3 Ω. For a current of 170 A passing Fig. 7. Magnetic field distribution along the Zeeman slower through the device, the electric power converted into axis. Solid curve shows the ideal calculated profile; squares heat amounted to 90 W. In order to reduce the influence show the experimental data. of Joule heating of the Zeeman slower on the residual gas pressure in the vacuum chamber, the current was Number of atoms supplied in a quasiperiodic regime, with the current switched on for 2 s and off for 8 s. The residual gas V = 10 m/s pressure in the vacuum chamber was 3 × 10−7 Torr. For a correct analysis of the measured atomic veloc- (a) ity distributions, it is very important to know the position of zero velocity. For this purpose, a part of the probing laser radiation was introduced perpendicularly V = 12 m/s to the atomic beam and the corresponding fluorescence component was measured by a separate photomultiplier. Since this scheme eliminates the Doppler broad- (b) ening, the resonance between the atomic fluorescence signal and the probing radiation indicated the position V = 30 m/s of the exact resonance corresponding to the F = 3 F' = 4 transition, thus determining the atoms with a zero velocity. It should be noted that deviation of the angle (c) between the probing laser radiation and the atomic beam from 90° leads to an error in the zero velocity determination. In order to minimize this error, the prob- V = 48 m/s ing laser radiation was adjusted to cross the atomic ° beam at an angle of about 89 and reflected back by a (d) mirror. This resulted in the appearance of two peaks equally shifted from the zero velocity position. The accuracy of zero velocity calibration in this scheme is V = 64 m/s determined by the uncertainty of matching of the forward and reflected laser beams. In our experiments, this uncertainty led to an error below 2 m/s in the velocity (e) determination. We have also used an alternative technique for the zero velocity calibration based on the –50 0 50 100 150 200 250 300 monitoring of nonlinear absorption resonances in a cell Velocity, m/s with Rb vapor. Fig. 8. The velocity distribution of 85Rb atoms in a beam upon Zeeman slowing for various detunings of the cooling laser frequency ∆ (MHz): (a) Ð39, (b) Ð46, (c) Ð54, (d) Ð66, 4. EXPERIMENTAL RESULTS (e) Ð77. Figure 8 shows the atomic velocity distributions of a beam of 85Rb atoms upon Zeeman slowing at various atoms decelerated as a result of the Zeeman slowing. As detunings ∆ of the laser radiation frequency. The can be seen from these data, the average velocity of atomic source temperature was T = 250°C. A peak in atoms in this peak, as well as the peak amplitude, the low-velocity part of the distribution corresponds to depend on the laser frequency detuning. The closer the

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Number of atoms (and higher than the Doppler limit) temperature of the atomic beam. According to the results of our calcula- I = 1.52Iopt tions, spatial inhomogeneity of the laser radiation intensity leads to a finite width of the atomic velocity distribution on a level of 6 m/s, spatial inhomogeneity (a) of the magnetic field leads to additional broadening of about 3 m/s, and a contribution due to the impulse dif- I = 1.24Iopt fusion amounts to 0.5 m/s. The total calculated width of the low-velocity peak is ∆V ≈ 10 m/s, which corresponds to a temperature of ∆T ≈ 1 K. This estimate is (b) lower than the value observed in experiment. The dis- crepancy is related to a finite length of the detection I = I zone, in which atoms continue to interact with the cool- opt ing laser radiation. As a result, the average velocity of atoms at various points of the detection zone exhibits (c) various values. In order to study the dependence of the slowing effi- I = 0.761Iopt ciency on the magnetic field strength in the Zeeman slower, we varied the current passing through the system, all other parameters being fixed. The correspond- (d) ing velocity spectra are presented in Fig. 9. As can be seen, the curves measured for the current exceeding –100 0 100 200 300 400 Iopt = 170 A exhibit two peaks. This is related to the fact Velocity, m/s that the process at high currents does not obey the condition (4) for the maximum possible magnetic field gra- Fig. 9. The effect of magnetic field strength in the Zeeman slower on the atomic velocity distribution. The maximum dient during Zeeman slowing, which breaks the interac- cooling efficiency is observed for the magnetic field tion between atoms and laser radiation. For currents induced by the current I = Iopt = 170 A. below the optimum value, the magnetic field gradient drops and, hence, the cooling efficiency decreases. cooling laser frequency to that of the atomic transition, We have experimentally investigated the depen- the lower the average velocity of cooled atoms. dence of the flux of cold atoms on the laser radiation intensity Ilaser. In our experiments, the maximum laser The data in Fig. 8 show that the amplitude of the radiation intensity corresponded to a saturation param- peak of cold atoms remains virtually unchanged for the eter of G = 30. A twofold decrease in this value reduces laser frequency detunings corresponding to the average the detected flux of cold atoms approximately by half, velocities of cooled atoms above V = 12 m/s. The peak while the average velocity of cooled atoms remains of cold atoms accounts for about 7% of the total num- unchanged. A fourfold decrease in the value of Ilaser ber of atoms in the initial velocity distribution, which leads to a significant decrease in the number of cold agrees with theoretical estimates. For the average atoms, while the average velocity of these atoms velocity below 12 m/s, the peak amplitude drops with increases by a factor of about 1.4. This behavior is decreasing detuning ∆. A minimum average velocity explained by the fact that the magnetic field profile in for which the peak contains a significant fraction of the Zeeman slower deviated from the ideal shape. By atoms from the initial distribution is 10 m/s (∆ = increasing the laser radiation intensity, it was possible −39 MHz). This decrease in the low-velocity peak to compensate for the nonideal magnetic field distribu- amplitude is explained by a decrease in the efficiency of tion by means of field-induced broadening, which led detection of low-energy atoms, which is caused by a to a decrease in the average velocity and an increase in large divergence of the beam of atoms with longitudinal the flux of cold atoms. According to the estimates pre- velocities below 15 m/s. Good agreement of the exper- sented above, the imperfect magnetic field distribution imentally observed numbers of cooled atoms with the can be compensated provided that the atomic transition results of calculations showed that the proposed saturation parameter is G = 14. When the saturation scheme provides for the effective extraction of cold parameter for the dominant (cooling) laser mode in our atoms from the Zeeman slower. experiments was G1 = 14, the corresponding value for the second laser mode was as small as G2 = 3.5 that was In the spectra of Fig. 8, a full width at half maximum insufficient for the effective Zeeman slowing. (FWHM) of the peak of cold atoms amounts to ∆V = 28 ± 2 m/s. This value is significantly greater than the We have also studied the dependence of the Zeeman minimum width determined by the Doppler cooling slowing efficiency on the laser radiation polarization limit. There are several factors responsible for the final and on the mutual orientation of electric vector E and

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ZEEMAN LASER COOLING OF 85Rb ATOMS IN TRANSVERSE MAGNETIC FIELD 675 magnetic field B. As expected, any deviations of the y laser radiation polarization from the optimum (linear) reduced the efficiency of Zeeman slowing. These devi- Eeff (E = E cos θ) ations lead to a decrease in intensity of the laser radia- E θ eff tion component exciting the atomic transitions with k ∆m = +1 and, hence, to a drop in the cooling efficiency z (Fig. 10). B x 5. PARAMETERS OF A COLD ATOMIC BEAM Fig. 10. Schematic diagram illustrating the influence of the mutual orientation of the magnetic induction B and the elec- We have determined the main parameters of the cold tric vector E of the cooling laser radiation on the Zeeman atomic beam, including the intensity, density, diver- slowing efficiency. gence, brightness, and phase space density. The intensity of a beam of cold atoms with an average velocity of 12 m/s (at an atomic source temperature of 250°C) was 1013 3 × 1012 sÐ1. We studied the possibility of improving this This study parameter by increasing the source temperature. As the 1012 source temperature was increased to 400°C, the cold atomic beam intensity exhibited a growth by a factor of 1011 2.4, but further increase in the source temperature led to Ð1 FOM PTB Bonn Univ. a decrease in the beam intensity. This is related to the 1010 primary beam depletion of the low-velocity atoms as a Australian Univ. Colorado Univ. result of their scattering from fast atoms. Thus, the 109 maximum intensity of the beam of cold atoms in our experiments was I = 7.2 × 1012 sÐ1. The correspond- Atomic flux, s Stanford Univ. max 108 ≈ × ing density of cold atoms was nmax = Imax/SV 4.7 Hannover Univ. 1010 cmÐ3. 107 0 20 40 60 80 100 The brightness of an atomic beam is defined as Velocity, m/s Fig. 11. The flux and average velocity of cooled atoms I R = ------, obtained in this study in comparison to the results obtained π∆()2∆Ω by other researchers using cooled atomic beams (᭿) and x⊥ atomic beams from 3D-MOT (᭹): This study; Hannover Univ. [8]; PTB [20]; Australia Univ. [21]; Bonn Univ. [19]; Stanford Univ. [9]; FOM [23];Colorado Univ. [11]. where ∆x⊥ is the transverse size of the beam and ∆Ω is the solid angle in which atoms are confined. The latter 2 solid angle is determined as ∆Ω = π(∆V⊥/)V || , where atomic beam is given by the expression ∆V⊥ is the width of the transverse velocity component π distribution and V || is the average value of the longitu- Λ˜ = B ------h3. r 3 4 dinal velocity component. The maximum transverse m V || velocity of atoms was determined by the beam-forming ≈ diaphragms and amounted to V⊥ 4.5 m/s. Therefore, The maximum phase space density observed in our for an average longitudinal velocity of V || = 12 m/s, experiments was Λ˜ = 2.4 × 10Ð11. cold atoms move within a solid angle of ∆Ω = 0.14π. × 12 Ð1 Figure 11 shows a comparison of the parameters For the maximum atomic flux of Imax = 7.2 10 s , the brightness amounts to R = 1.3 × 1018 (sr m2 s)Ð1. The (plotted as the beam intensity versus average atomic spectral brightness of an atomic beam is defined as velocity) of cold beams obtained in various research centers. As can be seen from these data, the flux of cold atoms achieved in this study is more than two orders of p|| magnitude higher than the beam intensities reported by Br = R------. ∆ p|| other researchers. This increase in the total flux of cold atoms has become possible for two reasons: first, due to the implemented scheme with transverse magnetic In our experiments, the spectral brightness was Br = field, which allowed the length of the cooling tract to be 1.7 × 1018 (sr m2 s)Ð1. The phase space density of an significantly reduced; second, due to the use of an

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1025 cation of the 2D-MOT technique in our case will allow the brightness and phase space density to be increased B by a factor of 3 × 104 and 1.5 × 106, respectively. 1023 LCV Figure 12 presents a summary of data on the bright- Ð1 Bonn Univ. A ness and phase space density of cooled atomic beams s) 2 1021 Australian Univ. Hannover Univ. obtained in various research centers using the 2D-MOT Colorado Univ. technique. For the comparison, we have also plotted the brightness and phase space density of the atomic beam 19 10 FOM obtained in this study, as well as expected values of the This study phase space density that can be achieved using a 1017 2D-MOT technique in the case of Doppler (point A) Brightness, (sr m and sub-Doppler cooling (point B). As can be seen, our Stanford Univ. cold beam parameters obtained even without using the 15 10 –12 –10 –8 –6 –4 2D-MOT technique are comparable to the analogous 10 10 10 10 10 parameters achieved due to 2D-MOT. The arrows in Phase space density Fig. 12 show the calculated values of brightness and Fig. 12. The brightness and phase space density of cooled phase space density of a cold atomic beam obtained by atomic beams obtained in this study in comparison to the the proposed method in combination with 2D-MOT. results obtained by other researchers: This study; Australia According to Figs. 11 and 12, the proposed method of Univ. [21]; LCV Univ. [22]; Bonn Univ. [19]; Colorado obtaining cooled atomic beams significantly improves Univ. [11]; Stanford Univ. [9]; FOM [23]; Hannover Univ. [8]. the phase space density of a beam. Arrows show the values of brightness and phase space density of a cold atomic beam predicted for the proposed method in combination with 2D-MOT cooling: (A) for the transverse Doppler cooling; (B) for transverse sub-Doppler 6. CONCLUSIONS cooling. Using the method of Zeeman laser cooling in a transverse magnetic field, we obtained a source of cold 85 × 12 Ð1 atomic source providing for an intense primary thermal Rb atoms with a beam intensity of 7.2 10 s at an beam with undepleted low-velocity fraction of the total average atomic velocity of 12 m/s. The density of cold × 10 Ð3 velocity distribution. atoms in the source was 4.7 10 cm . We have estimated the possibility to further increase the level of brightness and phase space density of a cold ACKNOWLEDGMENTS atomic beam achieved in our experiments. This can be The authors are grateful to A.P. Cherkun, I.V. Moro- provided by the two-dimensional magnetooptical trap zov, and D.V. Serebryakov for their help in preparing to technique (2D-MOT). The density of atoms in a experiments and to M.V. Subbotin for his active partic- 2D-MOT is limited by the following physical factors: ipation in the initial stage of experiments. (i) dipoleÐdipole interaction between atoms; (ii) repul- This study was supported in part by the Russian Foun- sive potential created by scattered laser radiation; and dation for Basic Research (project nos. 01-02-16337 and (iii) attractive potential related to the absorption of laser 02-02-17014), the Presidential Grant for Support of Lead- radiation [18]. The large intensity of a cold beam ing Scientific Schools (project no. NSh-1772.2003.2), achieved in our case suggests that the most important and INTAS (grant no. 479). factor determining the transverse size and temperature of the beam in the course of transverse laser cooling is reabsorption of photons inside the atomic ensemble. REFERENCES The multiple reabsorption of photons leads to the heat- 1. W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002). ing of atoms and to a decrease in the compressive force, 2. J. T. M. Walraven, in Quantum Dynamics of Simple Sys- so that the maximum possible density of atoms in the tems, Ed. by G. L. Oppo and S. M. Barnett (Inst. of beam is restricted to a value on the order of nmax = Phys., London, 1996), p. 315. 1012 cmÐ3 [18]. Taking into account this limitation and 3. W. D. Phillips, J. V. Prodan, and H. J. Metcalf, J. Opt. considering the maximum cold beam intensity and the Soc. Am. B 2, 1751 (1985). average atomic velocity obtained in our experiments, a 4. T. E. Barrett, S. W. Dapore-Schwartz, M. D. Ray, et al., minimum possible transverse size that can be achieved Phys. Rev. Lett. 67, 3483 (1991). ∆ ≈ µ by means of the 2D-MOT technique is x⊥ 430 m. 5. W. Ertmer, R. Blatt, J. L. Hall, et al., Phys. Rev. Lett. 54, Upon cooling atoms in a 2D-MOT down to the Doppler 996 (1985). limit of laser cooling, the angular divergence of the 6. W. Ketterle, A. Martin, M. A. Joffe, et al., Phys. Rev. atomic beam is 2 × 10Ð2 rad, while reaching sub-Dop- Lett. 69, 2483 (1992). pler temperatures of about 3 µK allows the divergence 7. M. Zhu, C. W. Oates, and J. S. Hall, Phys. Rev. Lett. 67, to be reduced down to 2.7 × 10−3 rad. Therefore, appli- 46 (1991).

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8. M. Schiffer, M. Christ, G. Wokurka, et al., Opt. Com- 16. K. L. Corwin, Z. T. Lu, C. F. Hand, et al., Appl. Opt. 37, mun. 134, 423 (1997). 3295 (1998). 9. E. Riis, D. S. Weiss, K. A. Moler, et al., Phys. Rev. Lett. 17. R. D. Swenumson and U. Even, Rev. Sci. Instrum. 52, 64, 1658 (1990). 559 (1981). 10. J. Nellessen, J. Werner, and W. Ertmer, Opt. Commun. 18. V. I. Balykin and V. G. Minogin, Zh. Éksp. Teor. Fiz. 78, 300 (1990). 123, 13 (2003) [JETP 96, 8 (2003)]. 19. F. Lison, P. Schuh, D. Haubrich, et al., Phys. Rev. A 61, 11. Z. T. Lu, K. L. Corwin, M. J. Renn, et al., Phys. Rev. 13405 (2000). Lett. 77, 3331 (1996). 20. A. Witte, T. Kisters, F. Riehle, et al., J. Opt. Soc. Am. B 12. S. N. Bagayev, V. I. Baraulia, A. E. Bonert, et al., Laser 9, 1030 (1992). Phys. 11, 1178 (2001). 21. M. D. Hoogerland, D. Milic, W. Lu, et al., Aust. J. Phys. 13. V. S. Letokhov, V. G. Minogin, and B. D. Pavlik, Zh. 49, 567 (1996). Éksp. Teor. Fiz. 72, 1328 (1977) [Sov. Phys. JETP 45, 22. W. Rooijakkers, W. Hogervorst, and W. Vassen, Opt. 698 (1977)]. Commun. 123, 321 (1996). 14. N. F. Ramsey, Molecular Beams (Clarendon Press, 23. K. Dieckmann, R. J. C. Spreeuw, M. Weidenmuller, Oxford, 1956; Inostrannaya Literatura, Moscow, 1960). et al., Phys. Rev. A 58, 3891 (1998). 15. P. N. Melentiev, M. V. Subbotin, and V. I. Balykin, Laser Phys. 11, 891 (2001). Translated by P. Pozdeev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 678Ð686. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 774Ð784. Original Russian Text Copyright © 2004 by Taranukhin. ATOMS, SPECTRA, RADIATION

High-Order Harmonic Generation by Atoms in a Two-Color Laser Field: Phase Control of Recombination Radiation Spectrum and Duration V. D. Taranukhin Faculty of Physics, Moscow State University, Vorob’evy gory, Moscow, 119992 Russia e-mail: [email protected] Received April 21, 2003

Abstract—The feasibility of phase control over above-threshold tunnel ionization and subsequent recombination emission in two-frequency laser fields is studied. It is shown that, in such fields, we can control the instants of ionization t0 (within optical cycle T) and recombination tk. The conditions that minimize the characteristic δ Ӷ δ Ӷ times t0 T and tk T, within which effective ionization and recombination occur, were found. Phase control allows recombination radiation to be generated with the selection of a narrow spectral range, while additional high-frequency “background illumination” sets up high harmonic “amplification” conditions. It was shown that special two-frequency pumping with elliptically polarized radiation can generate coherent electromagnetic pulses of attosecond width. The width of the pulses decreases as the intensity of pumping increases and can reach subattosecond values. Experimental generation of such pulses may lead to a breakthrough in the development of new methods for femto- and attosecond diagnostics of fast processes. © 2004 MAIK “Nauka/Inter- periodica”.

1. INTRODUCTION tively, and c is the speed of light), which corresponds to a value on the order of one femtosecond. The feasibility of generating extremely narrow coherent radiation pulses 10Ð15Ð10Ð17 s in width has Simultaneously, methods for measuring the width of been extensively studied to develop effective methods such short pulses have been under development [1, 5]. of femto- and attosecond metrology [1]. Current sug- These methods use the idea of the phase control of gestions for generating femto- and attosecond pulses above-threshold tunnel ionization, that is, the control of are related to the generation of high harmonics in the the ionization instant (the instant of electron transfer above-threshold tunnel ionization of atoms in strong into the continuum) within the optical radiation period. laser fields [1, 2]. Generally, the spectrum of high har- Phase control is possible because an electron produced in tunnel ionization begins its motion in the continuum monics is a broad slowly sloping plateau that extends ω (after the subbarrier evolution stage) at a zero velocity from the pumping frequency 0 to the “cutoff” fre- Ω ≈ along the instantaneous direction of the electric pump- quency Ui + 3.17Up, where Ui is the ionization ing field. The motion of the electron in the continuum potential of the atom and Up is the ponderomotive includes both oscillatory and drift components. The potential of pumping radiation. It should be noted from drift velocity of the electron contains information about the outset that a certain part of this spectrum is only the instant of atom ionization [5] and is recorded by a needed for applications. Subfemtosecond pulses can be detector after the photoelectron ceases to interact with generated if the phase matching condition is satisfied radiation. In this work, we study the feasibility of the for a group of harmonics from the plateau [3] or by phase control over the generation of high harmonics in applying a polarization gate [4]. The latter technique two-frequency fields. It is shown below that the use of uses pumping radiation with time-dependent ellipticity. two-frequency radiation allows us: This allows the return of a photoelectron to the parent ion (and, accordingly, the duration of recombination (1) to control the spectrum of high harmonic gener- radiation) to be controlled, because such a return is only ation, for instance, generate recombination radiation in a selected narrow spectral range rather than as a broad possible at the instants when the pumping radiation is plateau; linearly polarized. In both cases, the duration of recom- τ Շ λ λ bination radiation is g T = /c (T and are the opti- (2) to control the duration of recombination radia- cal cycle duration and the pumping wavelength, respec- tion, which opens up a real possibility of generating

HIGH-ORDER HARMONIC GENERATION BY ATOMS IN A TWO-COLOR LASER FIELD 679 τ coherent pulses of width g ~ 1Ð10 attoseconds (1 as = contributions of all discrete states except the ground 10Ð18 s); state |0〉 to the wave function Ψ(r, t), (3) to create conditions for the “amplification” of Ψ(), ()|〉 Ψ Ψ ()|〉 high harmonics. r t ==at 0 + c, c ∫dpb p t p , (1) The recombination radiation that arises in the tunnel ionization of atoms followed by electron recombination where a(t) and bp(t) are the amplitudes of the ground with the parent ion is described by the wave equation atomic state and continuum states |p〉 that correspond to with a source, which is the second derivative of the electron states with momentum p. On the other hand, if field-induced dipole moment D(t) with respect to time. the intensity of radiation is insufficient for effecting The moment D is calculated by the Schrödinger equa- above-the-barrier ionization, and the electron begins its tion, which, in particular, determines its dependence on evolution in the continuum fairly far from the parent ion time t. This dependence (along with propagation (at the far boundary of a fairly broad potential barrier), effects) controls the duration of recombination radia- then the influence of Coulomb forces on the motion of tion. In this work, propagation effects, which are the electron in the continuum can be ignored. We also described by the wave equation and can both broaden ignore bremsstrahlung against the background of and narrow recombination pulses, are not considered. recombination radiation. The introduced approxima- We concentrate on single-atom response D(t) calcu- tions allow the Schrödinger equation to be solved (the lations. procedure for solving this equation is described in [6]). The amplitude of the p state of the continuum can be represented in the form 2. A QUANTUM MODEL OF HIGH-ORDER HARMONIC GENERATION t () IN TWO-FREQUENCY FIELDS () ()()A t0 b p t = it∫d 0at0 E t0 dpÐ ------c (2) The quantum model of high-order harmonic gener- 0 ation by one atom in a monochromatic field described × exp[]ÐiS()p,,tt , in [6, 7] can be generalized to more complex pumping, 0 for instance, pumping that includes radiation at two dif- where d(p) = 〈p|r|0〉 is the matrix element of the dipole ferent frequencies. We will use the same approximation moment of the transition from the ground state of the as in [6, 7], which corresponds to the tunnel conditions atom to the p state of the continuum and E(t ) and A(t ) of the ionization of atoms. Under these conditions, 0 0 high-order harmonic generation admits a quasi-classi- are the amplitudes of the pumping radiation electric cal description and clear physical interpretation as a field and vector-potential at the instant of ionization t0, process including three stages [8], namely, the ioniza- t tion of an atom proper (photoelectron transfer into the ()2 ε (),, 1 A t' continuum), kinetic energy gain by the electron to ~ S p tt0 = ∫dt'Ui + --- p Ð ------. (3) 2 c Up (the above-threshold ionization stage), and the t recombination of the electron with the parent ion with 0 the emission of a photon at the frequency ω = kU + U . k p i Note that the contribution to the bp amplitude is formed The k factor varies in the range 0Ð3.17 depending on during the whole effective ionization time [the integral the phase (instant) of ionization ϕ, which results in the over ionization instants t0 in (2)], which causes the for- generation of a broad spectrum of high harmonics. The mation of a longitudinal structure of the wave packet in maximum generation frequency Ω corresponds to the ϕ ≈ π the continuum. Substituting (2) into (1) allows us to ionization phase /10 counted from the pumping write the wave function Ψ in the form field maximum. Also note that, because the generation c of high harmonics is an essentially nonlinear process, t the “fine” structure of the spectrum of high harmonic Ψ (), (), (),, generation is complex, and, if narrow pumping pulses c rt = it∫d 0 B p0 t0 f r tt0 (4) are used, the spectrum is continuous. 0 × [](),, ⋅ As in [6, 7], we consider pumping radiation of a expÐ iS p0 tt0 + ip0 r , fairly high intensity I at which atoms experience tunnel ionization. This allows us to ignore the influence on where ground state ionization of intermediate resonances, which correspond to the transition to the continuum A()t from all the other bound states [9]. Also note that the (), ()()0 B p0 t0 = at0 E t0 dp0 Ð ------probability of recombination into the ground atomic c state is much higher than that of recombination into any other discrete state [10]. We can therefore ignore the is the amplitude of the electron wave packet at the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 680 TARANUKHIN instant t0 and the function spreading) and the evolution of its center. In strong (but not relativistic) fields, the electron path in the contin- ()() σ (),, dpÐ At0 /c uum L is much larger than the size of its wave packet ; f r tt0 = ∫dp------()() - at the same time, L Ӷ λ. We can then use the quasi-clas- dp0 Ð A t0 /c (5) sical approach, and the evolution of the center of the × [](),, (),, ⋅ |〉 exp Ð iS p tt0 + iS p0 tt0 Ð ip0 r p wave packet in the continuum can be described by a classical equation of motion; that is, classical trajecto- describes the shape and phase of the packet (packet ries for the wave packet center can be considered. For spreading effects). In these equations, p0(t, t0) is the the Coulomb potential (when the electron is close to the momentum that makes the major contribution to inte- parent ion), the applicability of such an approximation gral (5) and corresponds to the stationary phase deter- is not obvious. It can, however, be used because the mined by the condition ∂S/∂p = 0. The stationary phase electron largely gains kinetic energy (which is essential 2 ∝ can be used because the characteristic scale p 1/(t Ð t0) to high harmonic generation) far from the parent ion. In of changes in action S at times on the order of the opti- addition, the first return of the photoelectron to the par- 2 ∝ cal period is much smaller than the p Ui scale of ent ion makes the major contribution to high harmonic changes in the matrix element d. Further, the explicit generation, which allows us to assume that the shape of its wave packet does not change significantly during its form of the packet f(r, t, t0) will be of no interest to us. It is, however, important that its center, which corre- evolution. At the same time, the effect of wave packet spreading is of importance. It can be described as the sponds to momentum p0, moves along a classical trajectory (this directly follows from the condition spreading of a free particle packet but at a Vsp rate deter- ∂S/∂p = 0). The packet width will be described by a mined by tunnel ionization [13], fairly strict equation [see Eq. (7) below] valid for the tunnel ionization conditions. 2 2 2 σ()t = σ()t +,V ()ttÐ Taking (4) into account, the dipole moment 0 sp 0 (7) ≈ 1/2()Ð1/4 V sp Fi 2Ui , D()t = 〈| Ψ*()r, t r|〉Ψ()r, t

where Fi = F(t0) is the pumping field at the instant of is obtained in the form ≈ ionization. We suggest to use the Vsp 1 nm/fs value (at t pumping radiation intensities I ≈ 1014Ð107 W/cm2), () () (), which closely agrees with the experimental value D t = a* t ∫dt0 B p0 t0 (6) from [4]. 0 × 〈|0 r|〉f ()r,,tt exp[]Ð iS()p ,,tt + ip ⋅ r + c.c. During its evolution in the continuum, the electron 0 0 0 0 gains the energy ε = ∂S/∂t from the field. After returning to the parent ion, it can recombine and emit a photon Equation (6) describes the quasi-classical evolution of ω ε 〈 | | 〉 the photoelectron during above-threshold tunnel ion- with the energy k = Ui + . The 0 r … matrix element ization and admits a clear physical interpretation. At the in (6) determines both the probability of recombination first stage of high harmonic generation, the electron and the duration of recombination radiation from a sin- wave packet is formed in the continuum. The packet gle atom, because this matrix element is only nonzero ∝ when the electron closely approaches the parent ion amplitude B(p0, t0) a(t0) is determined by the probability W of tunnel ionization at time t and takes into (when the electron wave packet and the wave function i 0 of the ground state of the atom noticeably overlap). account ionization saturation (depletion of atomic Lastly, note that the integral in t in (6) describes the states). Analytic results for dipole moment (6) were 0 obtained in [6] in the approximation of a comparatively coherent contribution to high harmonic generation of low pumping radiation intensity, at which the probabil- electrons released from an atom at various times t0, that ity of ionization was small. This led to the Keldysh for- is, the longitudinal structure of the wave packet. mula for the dependence of the probability of tunnel Such a clear physical interpretation of high har- ionization on the instantaneous pumping field value F. monic generation, which is possible because the pro- The Wi(F) dependence can, however, be substantially cess is quasi-classical, allows us to extend the applica- different in strong laser fields. The modification of the bility of the high harmonic generation model suggested ionization probability Wi in strong fields is discussed in in [6] to fairly high pumping radiation intensities [11], detail in [11] (also see [12]). Independent ionization take into account the three-dimensional character of the probability calculations can be helpful in applying (6). evolution of the electron in the continuum, and consider At the next stage of tunnel above-threshold ioniza- pumping fields of arbitrary configurations [14]. In the tion, the wave packet experiences evolution in the con- next sections, we discuss pumping by combined radia- tinuum under the action of pumping radiation. The tion whose components have different frequencies and f(r, t, t0) function describes the shape of the packet (its are differently polarized.

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3. GENERATION where OF RECOMBINATION RADIATION IN A TWO-FREQUENCY FIELD β 1 Ð γ 2 β γ β ==------, β ------. WITH DIFFERENT COMPONENT 1 2 2 α POLARIZATIONS: GENERATION 2 1 Ð α 2 OF ATTOSECOND PULSES Here and throughout this section, we use dimensionless It is known that usual circularly polarized pumping variables, namely, time is normalized with respect to ω ω2 does not cause harmonic generation [4]. This is so 1/ ; coordinates, with respect to (eF0/m ); velocity, ω because, after ionization, the photoelectron goes far with respect to (eF0/m ); and energy, with respect to away from the parent ion and never returns to it; that is, the ponderomotive energy of the high-frequency pump- at all ionization instants, there are no collisional elec- ing component U = e2F 2 /4mω2, where m and e are the tron trajectories. Conversely, collisional trajectories p 0 exist at any ionization instant if pumping radiation is mass and charge of the electron. For collisional tra- linearly polarized. Let us show that, under special two- jectories, both x and y tend to zero simultaneously at time t . frequency pumping conditions (including circularly k polarized fields), high harmonic generation is possible It follows from (10) and (11) that, in contrast to one- and it becomes feasible to control generation parame- frequency pumping, collisional electron trajectories ters, recombination radiation duration in particular. (see Fig. 1) only exist for certain sets of the α, β, and γ parameters and certain ionization instants t0. It is desir- Consider two-frequency pumping, which is a com- able that the following requirements be met when ε bination of a high-frequency elliptically polarized field selecting these parameters: The electron energy k at the and a low-frequency linearly polarized field, recombination instant should be higher (for generating harmonics of higher orders), the time spent by the elec- ()α2 ()ω αω() tron in the continuum should be shorter (this decreases F = F0 xˆ 1 Ð cos t + yˆ sin t + Fdc, (8) wave packet spreading and increases recombination effectiveness), the ratio between the amplitudes of the β ()γ 2 γ low- and high-frequency fields should be smaller Fdc = Ð F0 xˆ 1 Ð + yˆ , (9) (because of the difference in the power of 1 and 10 µm lasers), and a single collisional trajectory should exist ω α where F0, , and are the amplitude, frequency, and during the whole period of pumping (for generating a ellipticity of the high-frequency pumping component; single recombination radiation pulse). An analysis xˆ and yˆ are the unit vectors; and β and γ are the param- of (10) and (11), however, shows that there is no unam- eters that determine the relative amplitude and direction biguous solution to this problem, namely, there exist many (generally, infinitely many) various sets of the α, of the linearly polarized low-frequency pumping com- β, γ, and t parameters that lead to collisional trajecto- ponent F . The role of the low-frequency component 0 dc ries. Of the greatest importance for generating the can be played by CO2 laser radiation synchronized with shortest recombination radiation pulses is to minimize the high-frequency field. During one optical cycle of the time during which the electron occurs close to the λ µ high-frequency radiation at ~ 1 m, the CO2 laser parent ion, that is, in the region field can be considered constant. For simplicity, we ∆ ≤ σ() only consider collinear propagation of the low-fre- r tk , (12) quency and high-frequency fields. As has been mentioned above, the evolution of the photoelectron in the where dipole moment (6) is nonzero. This is attained by continuum (in tunnel ionization) can be described by increasing both the kinetic energy of the electron at the ε classical trajectories of the center of the electron wave instant of recombination k (this energy is limited by the packet. Solving the classical equations of motion in intensity of radiation used for pumping) and the rate of field (8) with zero initial conditions for the coordinates the “transition” from collisional to noncollisional elec- and velocity of the electron [7] leads to the following tron trajectories. The latter factor is in effect when the trajectories: 2 2 dr(tk)/dt0 derivative, where r = x + y , is fairly large, because, at given α, β, and γ parameters, the ionization xt()= Ð 1 Ð α2 instant t determines whether or not the electron trajec- (10) 0 × []() β ()2 tory is collisional. The time of electron interaction with costtÐ cos 0 ++ttÐ 0 sint0 1 ttÐ 0 , the parent ion can be substantially decreased by selecting the pumping two-component field (8) parameters yt() that determine the spatial orientation of the electron (11) wave packet at the instant of its return to the nucleus. α[]() β ()2 = Ð sinttÐ sin 0 Ð ttÐ 0 cost0 + 2 ttÐ 0 , The packet can be oriented in such a way that its over-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 682 TARANUKHIN the condition β τ costk Ð2cost0 = Ð 2 . (16) y Neither (14) nor (16) determines a unique set of the α, β γ , , and t0 parameters at which collisional trajectories exist. They, however, considerably facilitate seeking x such sets with a computer. The above reasoning suggests the following procedure. Equation (14) or (16) is used to determine a suitable set of the α, β, and γ pumping parameters and ionization instant t0. Given these pumping parameters, t0 tk t computer simulation of electronic trajectories (10), (11) is performed for various ionization instants in the Fig. 1. Time dependence of photoelectron coordinates in the vicinity of t0 to determine the maximum deviation of α β ≈ δ continuum [see (10), (11)] for case Y at = 1/2 , the ionization instant t0 (and the corresponding maxi- −0.276, γ ≈ Ð0.96, and t = π/10. δ 0 mum deviation of the recombination instant tk) at which recombination condition (12) is still satisfied; lap with the parent ion region (and, therefore, recombi- that is, at which the deviation of the electronic trajec- nation) would only involve a small longitudinal packet tory from the parent ion is smaller than the electron wave packet width. Photoelectrons that appear in the part that corresponds to a short interval of ionization δ δ Ӷ π continuum outside the t0 interval do not collide with instants t0 2 . The time of interaction with the ion (and the duration of recombination) is then determined the parent ion and do not contribute to recombination by the cross size of packet (7). radiation. This procedure is used to determine the “elementary” duration of recombination radiation (radia- We found two families of solutions to (10) and (11) tion of a single atom) in the process of high harmonic τ ≈ δ that give large dr(tk)/dt0 derivative values. We denote generation, namely, g tk. them by X and Y. Case Y is shown in Fig. 1. It corre- Below, we present the results of numerical experi- sponds to electronÐion collision instants (recombina- ments performed for cases X and Y to determine the tion instants) t that are found from (10), according to τ k duration of recombination radiation g for high har- which x(tk) = 0. We then have monic generation under two-frequency pumping conditions (8), (9). Let the high-frequency component be () λ µ dx tk ≡ laser radiation with a wavelength of = 1Ð0.8 m and ------0, 15 17 2 dt0 intensity of IHF = 10 Ð10 W/cm . Such radiation (13) () ∂ ∂ ∂ intensities have already been used in experimental and dy tk ≡∞y y tk computational studies of above-threshold ionization ------∂ - + ------∂ -∂------= . dt0 t0 tk t0 and high harmonic generation on ions (the use of ions ensures tunnel ionization, which is essential for high The first equation in (13) is a corollary to the x(tk) = 0 harmonic generation). For instance, the emission of + equation, which determines the tk(t0) function, and the high harmonics by He ions under pumping by radia- 17 2 second equation ensures a large dr(tk)/dt0 derivative tion of intensity I = 10 W/cm was calculated in [15]. value. The condition that must be satisfied by the α, β, In [16], lasers with radiation intensity I ~ 1016Ð γ 17 2 and pumping parameters and the ionization instant t0 10 W/cm were used to effect the tunnel ionization of for the collisional trajectory to exist and be character- noble gases (Ar, Kr, Ne, and Xe). High by charged ions ized by the shortest time during which the electron (with charges up to Z = 8) were observed. High harmonic occurs close to the parent ion is determined by (13) and generation by such ions under pumping with intensity (10), (11), I ~ 1016Ð1019 W/cm2 was calculated in [17, 18]. Case X. The set of parameters that satisfied condi- sint Ð2sint = β τ, (14) k 0 1 tion (16) for pumping (8), (9) with intensity IHF = τ 1017 W/cm2 was α = 1/2 (this corresponded to circu- where = tk Ð t0. larly polarized high-frequency radiation), β ≈ 0.4, γ ≈ Case X corresponds to the opposite situation, 0.65. The only ionization instant (during the optical cycle of the high-frequency field) was t ≈ Ðπ/5 (t was dy() t dx() t 0 0 ------k ≡∞0, ------k = . (15) counted from the high-frequency field maximum). dt0 dt0 Numerical examination of electron trajectories (10), (11) with such parameters gave the recombination α β γ ≈ π The , , , and t0 parameters must then satisfy instant tk 1.2 and the kinetic energy of the electron

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 HIGH-ORDER HARMONIC GENERATION BY ATOMS IN A TWO-COLOR LASER FIELD 683 ε ≈ at the recombination instant k 3.6. The corresponding deviations of t0 and tk [at which recombination con- δ ≈ δ ≈ dition (12) was still satisfied] were t0 tk 0.02. δ Such a tk value corresponded to the dimensional dura- τ ≈ tion of recombination radiation by a single atom g 10 as. Note once more that the obtained set of the α, β, γ , and t0 parameters, which ensures the generation of t such short pulses, is neither unique nor optimal. It fol- t t τ 0 k lows that the actual g (at the selected high-frequency radiation parameters) can be still shorter. Also note that 17 a consideration of pumping with intensity IHF ~ 10 Ð 18 2 Fig. 2. Example of an extremely narrow high-frequency 10 W/cm generally requires taking into account the pumping pulse for generating a single attosecond pulse influence of the magnetic field of radiation on electron [see (17)]. trajectories [11], which, however, does not change the τ g value substantially, although the exact parameter values [and condition (16)] can change. frequency pumping pulse with peak intensity IHF = 1018 W/cm2 and field envelope Case Y. Equation (14) gave the following set of 15 2 α 2 parameters for pumping with IHF = 10 W/cm : = ()t Ð 7 F ()t ∼ exp Ð,------(17) β ≈ γ ≈ 0 1/2 , Ð0.22, Ð1, and t0 = 0. The simulation of 4 electron trajectories (10), (11) with these parameters ≈ π ε ≈ δ ≈ δ ≈ which describes the pulse shown in Fig. 2. The proce- gave tk 2 , k 4, and t0 tk 0.25, which corre- α β ≈ γ ≈ sponded to recombination radiation duration τ ≈ dure suggested above gives = 0.708, Ð0.275, g Ð0.96, and t = 7.3575 for such a pumping pulse. This 100 as. Increasing the I intensity decreases τ . For 0 HF g corresponds to a single collisional trajectory (t ≈ 12) × 17 2 k instance, the parameters found for IHF = 5 10 W/cm during the whole pulse (17). Varying the t0 parameter α β ≈ γ ≈ π δ ≈ δ ≈ were = 1/2 , Ð0.276, Ð0.96, and t0 = /10. gives t0 0.02 and tk 0.06; that is, the width of a sin- ≈ π ε ≈ δ ≈ τ ≈ These parameters gave tk 1.9 , k 4.5, t0 0.01, gle recombination radiation pulse is g 30 as. δ ≈ and tk 0.002, which corresponded to the recombina- These simulations lead us to conclude that the sug- τ ≈ tion radiation duration g 0.9 as. Obtaining such gested approach opens up the possibility in principle of pulses experimentally would mean a breakthrough in generating 1Ð10 as coherent electromagnetic radiation the development of new methods for femto- and pulses and even of overcoming the subattosecond bar- attosecond diagnostics of fast processes [1]. Note that rier. In essence, this approach is a variety of the method τ the g value found above is more than three orders of for phase control of tunnel ionization, because it is magnitude smaller than the optical period T of the high- based on the selection of photoelectrons (within the δ frequency field. Such a pulse, however, contains about ionization interval t0) that contribute to recombination 50 recombination radiation optical cycles (whose dura- radiation generation. Let us shortly consider the effi- ε tion is determined by the k parameter) and is almost ciency of this generation. As with usual pumping, the monochromatic in this sense. efficiency is determined by three factors, namely, the probability of ionization of an atom (ion), the rate of The above estimates are valid for each high-fre- electron wave packet spreading in the continuum, and quency field optical cycle. In each cycle, we observe a the probability of electronÐion recombination. The use τ recombination radiation burst of duration g. If the of combined pumping (including circularly polarized high-frequency radiation pulse is long, this results in waves) does not charge the probability of tunnel ioniza- the generation of a train of attosecond pulses with the tion to any significant extent, because this probability is repetition frequency ~ω. Such trains can, for instance, determined by instantaneous field strength. Wave packet be used in petahertz spectroscopy with attosecond time spreading and recombination probabilities per photo- resolution. The isolation of one pulse from such a train electron are approximately the same as under usual high is, however, a difficult problem. harmonic generation conditions. Pumping (8), (9), however, selects only a small part of all photoelectrons that δ With extremely short high-frequency radiation recombine with the parent ion ( t0/T ~ 0.1Ð0.01). This pulses (when the ionization of an atom effectively results in both the generation of short recombination occurs only during one optical cycle), the use of two- radiation pulses and the selection of a narrow region δω frequency pumping (8), (9) opens up the possibility of g in a wide plateau of the spectrum of high harmonic generating a single attosecond recombination radiation generation. Note that there is no contradiction between pulse automatically (without additional experimental the generation of a short recombination radiation pulse τ δω efforts). For instance, consider an extremely short high- (of duration g) and a relatively narrow ( g) genera-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 684 TARANUKHIN δω τ ӷ tion spectrum; indeed, g g 1 thanks to the high fre- respectively. Let both fields be polarized linearly and quency of recombination radiation. It follows that, parallel to each other (along axis x) and synchronized in when we use pumping (8), (9), the total energy of high such a way that the high-frequency pulse corresponds δ ϕ harmonic generation decreases (proportionally to t0/T to a certain phase of the optical cycle of the low-fre- δω Ω or g/ ). The intensity (recombination radiation har- quency field. If the FL amplitude is insufficient for tun- monics), however, remains approximately the same as neling an electron from an atom, the ionization of the δω under usual pumping conditions within the g spectral atom is determined by the short high-frequency radia- range. tion pulse. Conversely, the above-threshold ionization stage is determined by the low-frequency field. Note that such a scheme of high harmonic generation is fully 4. ACTIVE PHASE CONTROL equivalent to the scheme used in [5] to measure the OF TUNNEL IONIZATION pulse duration in the subfemtosecond range. AND THE GENERATION OF RECOMBINATION Further calculations and estimates refer to the fol- RADIATION WITH THE SELECTION lowing experimental situation: The low-frequency OF A NARROW SPECTRAL RANGE × component is CO2 laser radiation of intensity IL = 6 In the preceding section, we considered passive 1013 W/cm2 (the tunnel ionization, for instance, of phase control of tunnel ionization, when the ionization helium atoms by this radiation can be ignored, but its of an atom occurred at all time instants during optical ≈ ponderomotive energy is substantial, Up 500 eV). The cycle T, whereas the selection of photoelectrons ω high-frequency component is radiation at an H ~ occurred at the stage of their recombination with the τ 25 eV frequency of duration H ~ 1Ð3 fs and intensity parent ion. The two-frequency pumping parameters are 10 2 selected such that only those electrons recombine that IH ~ 10 W/cm . Such parameters are attained in mod- “go away” from the atom during a fairly short time ern high-harmonic generation experiments [2]. δ Ӷ interval t0/T 1. The short duration of recombination If the high-frequency field satisfies the inequality ω տ radiation is then attained because of the high sensitivity H Ui, the velocity of the photoelectron at the exit to ≈ of recombination condition (12) to ionization instant t0 the continuum is V0 0 or, more exactly, 0 < V0 < V0m. variations. Using two-frequency radiation, we can also The maximum initial velocity of the photoelectron in exercise active phase control, that is, the selection of the absence of phase modulation of the high-frequency photoelectrons already at the atom ionization stage ប τ 1/2 τ pulse, V0m = (2 /m H) , is determined by its width H. (when ionization only occurs during a small fraction of The probability of ionization is then proportional to the optical period T). The overall degree of medium ioniza- intensity IH and the threshold cross section of one-pho- tion is then much lower than under passive control con- ton ionization. The further evolution of the photoelec- ditions. A substantial concentration of free electrons in tron wave packet largely occurs under the action of the a medium is known to cause radiation defocusing, ӷ low-frequency field (because of the condition FL FH) which imposes limitations on increasing the effective- and, as has been shown above, can be described by the ness of high harmonic generation [2]. It follows that the classical trajectory x(t) of the center of the packet. If selection of ionization instants decreases pumping radi- Coulomb attraction to the parent ion is ignored (which ation defocusing and allows the phase matching length can be done if U ӷ U ), the equation for this trajectory and, therefore, the effectiveness of high harmonic gen- p i eration to be substantially increased. While remaining has the simple form within the framework of analyzing the single-atom 2 response, let us show that two-frequency pumping of a d x e ()ω ϕ ------= ----FL cos Lt + . (19) special kind also allows the width of the recombination dt2 m spectrum to be substantially decreased and thereby the effectiveness of using it to be increased. Solving (19) with the initial conditions x(t = 0) = 0 and Let pumping be performed by the following two- V(t = 0) = V0 allows calculation of the kinetic energy of ε component field: a strong low-frequency field FL the electron k at the instant of its return to the parent (which by itself does not cause the tunnel ionization of ion (that is, at the x ≈ 0 point where it recombines with an atom) and an extremely short high-frequency pulse the parent ion). Wave packet spreading (7) then only ω field H > Ui, capable, for instance, of inducing the influences the effectiveness of electron recombination ε one-photon ionization of an atom. We assume that the and has no significant effect on its kinetic energy k. ε high-frequency pulse duration is substantially smaller The k energy depends on the initial photoelectron than the optical period T of the low-frequency field. ϕ L velocity V0 and ionization phase (Fig. 3). The latter is We can write ϕ determined by the high-frequency field (the phases 1, 2 FF= cos()ω T + F cos()ω t + ϕ , (18) in Fig. 3 correspond to the high-frequency pulse start L L H H and end points), which allows us to control the recom- ӷ ω Ӷ ω where FL FH are the amplitudes and L H are the bination radiation spectrum. Note that the return of the frequencies of the low- and high-frequency fields, photoelectron to its parent ion and its recombination

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 HIGH-ORDER HARMONIC GENERATION BY ATOMS IN A TWO-COLOR LASER FIELD 685 ε with it only occur when the direction of velocity V0 is k opposite to that of the low-frequency field. As one-pho- Ω ∆Ω ton ionization generates a symmetrical two-lobe wave 1 ± packet ( V0), half the total number of the photoelec- 3 trons do not contribute to recombination radiation. V0 = 0 ε The electron energies k (actually, the recombina- 2 tion spectrum ω = ε + U ) calculated by (19) are plot- k k i V0 = V0m ted in Fig. 3, which shows that the high-frequency pulse 1 gates ionization phases (ϕ < ϕ < ϕ ) and causes gener- 1 2 2τ /T ation only in a narrow spectral range ∆Ω. The spectral H L ε ϕ 0 components that lie between the k( ) curves for V0 = 0 ϕ πϕπϕπ 1/ 2/ 0.2 0.4 / and V0 = V0m and correspond to the ionization phases ϕ ϕ ϕ ∆Ω 1 < < 2 only contribute to . As is shown in Fig. 3. Dependence of recombination radiation frequency τ ϕ on ionization phase (counted from the low-frequency field Fig. 3, a high-frequency pulse of width H = ( 2 Ð ϕ π ≈ ϕ ≈ π maximum) at various initial photoelectron velocities; the 1)TL/2 1.2 fs close to the relative phase /10 gating of the recombination spectrum by a high-frequency generates recombination radiation in the narrow fre- pulse of width τ is shown. ∆Ω ≈ Ω Ω Ω H quency range 0.12 1, where 1 = Ð Ui. The central frequency of the recombination spectrum can be retuned by changing the amplitude of the low-fre- ∆Ω /Ω ∝ ΩÐ2/3 , and can be noticeably smaller Ω ∝ min 1 1 quency field, because 1 IL. The V0m value used in than 1%. Fig. 3 corresponds to a high-frequency pulse with certain phase modulation. For a high-frequency pulse of the To summarize, the generation of an extremely nar- τ row recombination spectrum requires the use of high- same width ( H = 1.2 fs) but without phase modulation, ∆Ω frequency pulses of optimum width without phase the width of the spectrum is smaller by about 30%. modulation. These pulses should be synchronized with A decrease in the width of the high-frequency pulse the low-frequency field near the optimum phase ϕ Շ π increases the initial electron velocity V0m. At fairly /10 (the higher the initial electron velocity V0m, the τ small H, this can broaden the recombination spectrum closer the high-frequency pulse must be to the low-fre- rather than narrow it. It follows that there exists an opti- quency field maximum). Also note that a single high- mal high-frequency pulse width at which the recombi- frequency pulse causes a single event of recombination ε ϕ nation spectrum width is minimum. If the k( ) depen- radiation generation. This results in a continuous gen- dence is close to parabolic at the top (Fig. 3) and there is eration spectrum and decreases generation effective- no phase modulation of the high-frequency pulse, (19) ness. The effectiveness of generation can be increased yields if a train of high-frequency pulses is used. The synchro- nization of two lasers is then, however, a more complex 2 experimental task. ∆Ω B ------= A τ + ------. (20) Note in conclusion that the mechanism of ionization Ω H τ1/2 1 H of an atom by two-frequency field (18) considered in this work also explains the absorption of recombination τ Ω If TL and H are in femtoseconds and 1 is in electron radiation (harmonics) in the usual (one-component) ≈ 2 ≈ Ω1/2 scheme for high harmonic generation. Atoms under the volts, then A 55.8/T L and B 0.17TL/1 . It follows action of pumping radiation also experience the influ- from (20) that the smallest recombination spectrum ence of weak recombination radiation. Recombination ∆Ω width min is observed at the high-frequency pulse radiation photons change the effective ionization poten- 2/3 Ð1/3 ω τ 2/3 ∝ Ω tial of the atom, Ui eff = Ui Ð H [19], and can therefore width 0 = (B/2) T L 1 and is given by the equation participate in the tunnel ionization of an atom by the low-frequency pumping field. Because of the exponen- ∆Ω tial dependence of the probability of tunnel ionization min τ2 on the ionization potential, this causes a sharp (by sev------Ω = 9 A 0. (21) 1 eral orders of magnitude) increase in the rate of ionization and, therefore, effective absorption of recombina- ≈ For TL 35 fs (CO2 laser low-frequency radiation), the tion radiation. This effect limits the effectiveness of τ ≈ following estimates were obtained: 0 0.2 fs and high harmonic generation when the recombination ∆Ω Ω ≈ radiation absorption length is shorter than the active min/ 1 0.016. The absolute recombination spectrum width increases as the intensity of low-frequency medium length or coherence length. When additional radiation (high-frequency background illumination) is ∆Ω ∝ Ω1/3 Ω ∝ radiation grows, min 1 , 1 IL. However used, this effect can, however, be inverted and used to simultaneously, the relative spectrum width decreases, increase the effectiveness of high harmonic generation.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 686 TARANUKHIN

For instance, if the main pumping is accompanied by (2002); A. Baltuska, Th. Udem, M. Uiberacker, et al., comparatively weak radiation at the frequency of one of Nature 421, 611 (2003); P. M. Paul, E. S. Toma, the harmonics, the rate of the tunnel ionization of atoms P. Breger, et al., Science 292, 1689 (2001); R. Kien- considerably increases. This in turn causes an increase berger, M. Hentschel, M. Uiberacker, et al., Science 297, in the intensity of all recombination spectrum compo- 1144 (2002). nents. There is no real radiation amplification at the fre- 2. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 quency of the harmonic used for additional (back- (2000). ground) pumping because of a fairly low efficiency of 3. Ph. Antoine, A. L’Huillier, and M. Lewenstein, Phys. high harmonic generation. The intensity of the other Rev. Lett. 77, 1234 (1996). components of the high harmonic spectrum, however, 4. P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, Opt. Lett. noticeably increases. 19, 1870 (1994). 5. E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Cor- kum, Phys. Rev. A 56, 3870 (1997). 5. CONCLUSIONS 6. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, et al., Phys. To summarize, two-frequency fields allow us to Rev. A 49, 2117 (1994). exercise phase control over the above-threshold tunnel 7. R. V. Kulyagin and V. D. Taranukhin, Kvantovaya Élek- ionization process. A consequence of such control is tron. (Moscow) 23, 866 (1996) [Quantum Electron. 26, the unique possibility of controlling recombination 866 (1996)]. radiation parameters. In this work, we showed the fea- 8. P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). sibility of actively controlling the recombination radia- 9. E. Mevel, P. Breger, R. Trainham, et al., Phys. Rev. Lett. tion spectrum (its central frequency and width, which 70, 406 (1993). can be decreased by two orders of magnitude) for the 10. J. B. Watson, A. Sanpera, and K. Burnett, Phys. Rev. A example of single atom radiation. We also showed the 51, 1458 (1995). possibility in principle of generating coherent pulses of 11. V. D. Taranukhin, Laser Phys. 10, 330 (2000). τ 12. A. Scrinzi, M. Geissler, and T. Brabec, Phys. Rev. Lett. width g = 1Ð10 as. Obtaining such pulses in experiments would signify a fundamentally new step in the 83, 706 (1999). development of methods for diagnostics of fast pro- 13. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. cesses. It should especially be noted that conditions (14), Éksp. Teor. Fiz. 51, 309 (1966) [Sov. Phys. JETP 24, 207 (16) for attaining such pulse widths admit further selec- (1966)]. tion of two-frequency pumping parameters. Additional 14. V. D. Taranukhin and N. Yu. Shubin, J. Opt. Soc. Am. B efforts (experimental or in the field of numerical simu- 17, 1509 (2000). lations) will allow us to obtain coherent radiation 15. M. W. Walser, C. H. Keitel, A. Scrinzi, and T. Brabec, pulses of width actually smaller than one attosecond. Phys. Rev. Lett. 85, 5082 (2000). 16. S. Augst, D. Strickland, D. D. Meyerhofer, et al., Phys. Because of propagation effects and spatial inhomo- Rev. Lett. 63, 2212 (1989). geneity of laser radiation, the calculation of recombina- 17. D. B. Milosevic, S. Hu, and W. Becker, Phys. Rev. A 63, tion radiation can change (increase or decrease), which 011403R (2001). requires a separate study [20]. Of primary importance 18. V. D. Taranukhin and N. Yu. Shubin, J. Opt. Soc. Am. B for any such study are, however, the single-atom 19, 1132 (2002). response calculations performed in this work. 19. N. B. Delone, N. L. Manakov, and A. G. Faœnshteœn, Zh. Éksp. Teor. Fiz. 86, 906 (1984) [Sov. Phys. JETP 59, 529 REFERENCES (1984)]. 20. N. Milosevic, A. Scrinzi, and T. Brabec, Phys. Rev. Lett. 1. M. Hentschel, R. Kienberger, Ch. Spielmann, et al., 88, 093905 (2002). Nature 414, 509 (2001); M. Drescher, M. Hentschel, R. Kienberger, et al., Science 291, 1923 (2001); H. Niikura, F. Legare, R. Hasbani, et al., Nature 417, 917 Translated by V. Sipachev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 687Ð690. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 785Ð788. Original Russian Text Copyright © 2004 by Milstein, Terekhov. ATOMS, SPECTRA, RADIATION

Scattering of a Low-Energy Electron in a Strong Coulomb Field A. I. Milstein and I. S. Terekhov Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia e-mail: [email protected] Received October 28, 2003

Abstract—The low-energy electron scattering cross section in a strong Coulomb field is analyzed theoretically. It is shown that the exact cross section in a wide energy range significantly differs from the results obtained in the first Born approximation and in the nonrelativistic approximation. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION () exp ipr1 ψ()+ () = Ð------π - ∑ λp r2 uλp, (1) 4 r1 An explicit expression for the scattering cross sec- λ = 12, tion of an electron with an arbitrary energy in a strong

Coulomb field was derived long ago by Mott [1] and ϕλ contained an infinite series in Legendre polynomials. ε  uλp = + m sr⋅ , Although the methods of summation of this series were ------φ developed in a number of publications, the numerical ε + m λ calculation of the cross section remains a complicated problem. A detailed review of publications devoted to ε2 2 ψ()+ () this problem can be found in monograph [2]. The elec- where p = Ð m ; m is the electron mass; λp r tron scattering cross sections in a Coulomb field were denotes the solution to the Dirac equation, containing at calculated in [3Ð5] for various scattering angles, infinity a plane wave with momentum p = Ðpn1 (n1, 2 = λ ± nuclear charges Z, and electron energies above r1, 2/r1, 2) and a diverging spherical wave; = 1 0.023 MeV. In an analysis of backward scattering for denotes two independent spinors φλ; and ប = c = 1. In Z = 80, it was proved that the ratio of the exact relativ- the Coulomb field, the right-hand side of expression (1) istic cross section to the nonrelativistic (Rutherford) iq αε contains the additional factor (2pr1) , where q = Z /p cross sections increases from 0.15 to 2.35 upon a α decrease in the electron energy from 1.675 to 0.023 MeV. and = 1/137 is the fine-structure constant. A conve- In view of such a large difference between the exact nient integral representation of the electron Green result and that obtained in the nonrelativistic approxi- function in a Coulomb field was derived in [6]. Using mation, it would be interesting to study the behavior of equalities (19)Ð(22) from [6], we obtain the exact scattering cross section for slow electrons in a strong Coulomb field. Here, we obtain the answer to ∞ f 1 ()+  this question by calculating the asymptotic form of the ψ () ε ⋅ λp r2 = + m∑ ps n2 , cross section for an arbitrary Z and a low electron ------f 2 l = 1 ε + m energy.

mZα − f , = R Ai+ ------R B M + iR BM φλ, 2. ELECTRON SCATTERING CROSS SECTION 12 1 p 2 1 2 2 IN A COULOMB FIELD ψ d ()() () The electron wave function λp(r) in an external Al= ------Pl y + Pl Ð 1 y , |ε dy ﬁeld can be derived from the Green function G(r2, r1 ) of the Dirac equation in this ﬁeld. We will use the famil- d iar relation B = ------()P ()y Ð P ()y , (2) dy l l Ð 1 (), ε lim G r2 r1 → ∞ ⋅ +− ()⋅ ()⋅ r1 y ==n1 n2, R12, 1 s n2 s n1 ,

∞ ()πν strong Coulomb field. We consider the ratio of the exact exp ipr2 Ð i ()+−12Ð iq α M , = i------t cross section (in parameter Z ) to the cross section 12 pr ∫ 2 2 2 0 dσ q p ------B = ------1 Ð ------()1 Ð x (5) × ()2 () dΩ p2()1 Ð x 2 2ε2 exp it J2ν 2t 2 pr2 dt. obtained in the first Born approximation. Using the Here, P (x) are Legendre polynomials, J ν are Bessel l 2 asymptotic form of the Γ function for large values of 2 2 σ σ functions, and ν = l Ð ()Zα . The integrals in func- the argument, the expression for S = d /d B can be reduced to the form tions M1, 2 can be expressed in terms of degenerate hypergeometric functions. Result (2) is in accordance with the well-known solution to the Dirac equation in a 1 Ð x  1 Ð x S = 1 + ------ImexpÐiqln------Coulomb field. q  2 To find the scattering amplitude, we must calculate ∞ coefficient Wλ of the diverging spherical wave × ()l[]() ()[]()πν exp[ipr2 + iqln(2pr2)]/r2 in the asymptotic form of ∑ l Ð1 Pl x Ð Pl Ð 1 x expÐ2i Ð 1 (6) () ψ + () ∞ l = 1 function λp r2 (r2 ). A nonzero contribution to Wλ comes from function il2  M (see Eqs. (2)), the required asymptotic form of this × exp ----- . 1 q function being determined by the integration domain  Ӷ t 1. We have For 1 + x ӷ 1/q, factor exp(il2/q) in this formula can be replaced by unity. This gives ∞ f ε ⋅ Wλ = + m∑ ps n2 , 1 Ð x  1 Ð x ------f  l = 1 ε S = 1 + ------Re expÐiqln------+ m (3) q  2 ()Γνπν () α iiexp Ð Ð iq mZ φ f = ------R1 Ai+ ------R2 B λ. 22 2 4 1 + x 2 pΓν()++1 iq p ×π()Zα ------1Ð Ð iπ ()Zα ln ------1 + x 2 Calculating the flux of scattered particles, averaging it ∞ (7) over spins, and dividing by the incident current density, ()l[]() () we obtain the scattering cross section: Ð il∑ Ð1 Pl x Ð Pl Ð 1 x l = 1 dσ 1 + ------= ------∑ Wλ()a ⋅ n Wλ dΩ 4 p 2 iπ()Zα 2 π2()Zα 4  λ , × ()πν = 12 expÐ2i Ð 1 Ð ------+ ------. l 2l2  2 2 2 2 mZα F () ӷ = -----2 1 + x F ' + ------, Thus, for 1 + x 1/q, the correction to function S is p p 1 Ð x proportional to 1/q. It should be noted that the sum over l in expression (7) converges very rapidly for an arbi- ∞ i Γν()Ð iq trary x. If 1 + x ~ 1/q (backward scattering), the main F = Ð---∑ liexp[]π()l Ð ν ------(4) 2 Γν()++1 iq contribution to sum (6) comes from moments l ~ q ӷ l = 1 1. Using the asymptotic form of the Legendre polyno- × []P ()x Ð P ()x , mials for x Ð1 and replacing summation by integra- l l Ð 1 tion, we obtain dF ϑ ⋅ π3/2()α 2 π () F ' ===------, x cos Ðn1 n2. ()Z q 1 + x dx S = 11+ Ð x ------cos--- + ------q 4 4 This result coincides with that obtained earlier in [1]. (8) q()1 + x × J ------. 04 3. SCATTERING CROSS SECTION IN THE LOW-ENERGY LIMIT It can be seen that the correction to function S for 1 + Let us calculate the scattering cross section (4) for x ~ 1/q is proportional to 1/q ∝ p/ε . Consequently, q = Zαε/p ӷ 1 and Zα ~ 1. This range of parameters the value of S tends very slowly to unity as v 0. The corresponds to scattering of a low-energy electron in a exact (in Zα) backward scattering cross section signifi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 SCATTERING OF A LOW-ENERGY ELECTRON IN A STRONG COULOMB FIELD 689

S for x = Ð1 and for several values of Z. The same ﬁgure 8 shows the low-energy asymptotic form (8) for x = Ð1: 3/2 3/2 7 3 S = 12+ π ()Zα v . (10) 6 It can be seen that, for backward scattering, the exact 5 α 2 (in Z ) cross section signiﬁcantly differs from the Born 4 cross section even for small velocities v, the difference 3 1 decreasing very slowly (in proportion to ∝ v ). It can 2 be seen from Fig. 1 that the exact result virtually coin- 1 cides with the asymptotic form for v < 0.2. It is well known that the Born cross section is trans- 0 0.2 0.4 0.6 0.8 1.0 formed to the Rutherford cross section for v Ӷ 1: v dσ Zα 2 ------R = ------. (11) Fig. 1. Dependence of function S on v for x = Ð1 and Zα = dΩ 2() 0.6 (1), 0.7 (2), and 0.8 (3). Solid curves correspond to exact mv 1 Ð x results and dotted curves, to the asymptotic form. σ σ It is interesting to analyze the ratio S1 = d /d R of the exact scattering cross section to the Rutherford cross cantly differs from dσ /dΩ even for comparatively low section. Figure 2 shows the dependence of S1 on q = B Zα/v for x = Ð1 and for several values of parameter energies. α Z . It can be seen that the ratio S1 of the cross sections For x = Ð1 and for arbitrary values of q, the exact increases with decreasing energy, attains its maximal expression for function S (see relations (4)) has a sim- value, and then slowly tends to unity (S Ð 1 ∝ ple form: 1 1/q ∝ v ). ∞ 2 Γν()Ð iq S = 4 ∑ liexp()Ð πν ------. (9) In classical electrodynamics, for scattering with Γν()++1 iq impact parameters ρ satisfying the relation Zα/mvρ տ 1, l = 1 a particle attains velocities of vmax ~ 1 at minimal dis- Figure 1 shows the dependence of function S on v = p/ε tance from a Coulomb center and relativistic effects

S1 5 (a) (b) 4

0 5

4 (c) (d)

0 5 10 15 20 0 5 10 15 20 q q

α α Fig. 2. Dependence of function S1 on variable q = Z /v for Z = 0.5 (a), 0.6 (b), 0.7 (c), and 0.8 (d). Solid curves correspond to the exact result and dashed curves describe the asymptotic form.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 690 MILSTEIN, TEREKHOV become signiﬁcant. In relativistic classical mechanics, This study was ﬁnanced by the Russian Foundation for ρ < (Zα)/mv (we consider the case with v Ӷ 1 as for Basic Research (project no. 03-02-16510). before), the phenomenon of falling to the center takes place. In addition, the cross section of scattering through angles close to π is singular. For 1 + x Ӷ v4/3 Ӷ 1, the for- REFERENCES mulas given in [7] readily give 1. N. F. Mott, Proc. R. Soc. London, Ser. A 124, 426 (1929). dσ Zα 2π 1/3 1 ------cl- = ------(12) 2. H. Überall, Electron Scattering from Complex Nuclei Ω 4 d mv 2v 621()+ x (Academic, New York, 1971). in this region. However, in the quantum-mechanical 3. J. H. Bartlett and R. E. Watson, Proc. Am. Acad. Arts Sci. case, for Zα ~ 1 and v 0, the backward scattering 74, 53 (1940). cross section (small values of ρ) tends to the nonrelativ- 4. J. A. Doggett and L. V. Spencer, Phys. Rev. 103, 1597 istic limit. This is due to the fact that, for a given ρ, (1956). indeterminacy ∆φ ~ 1/mvρ in the scattering angle 5. N. Sherman, Phys. Rev. 103, 1601 (1956). ρ Շ appears, which becomes of the order of unity for 6. A. I. Milstein and V. M. Strakhovenko, Phys. Lett. A Zα/mv. 90A, 447 (1982). 7. L. D. Landau and E. M. Lifshitz, The Classical Theory ACKNOWLEDGMENTS of Fields, 6th ed. (Nauka, Moscow, 1973; Pergamon Press, Oxford, 1975). The authors are grateful to R.N. Li and V.M. Stra- khovenko for fruitful discussions and for their interest in this research. Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 691Ð704. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 789Ð804. Original Russian Text Copyright © 2004 by Perminov, Rautian, Safonov. ATOMS, SPECTRA, RADIATION

On the Theory of Optical Properties of Fractal Clusters S. V. Perminova, S. G. Rautianb, and V. P. Safonovc aInstitute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia bLebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia cInstitute of Automation and Electrometry, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: [email protected]; [email protected] Received September 24, 2003

Abstract—A theory of optical properties of clusters of spherical metal nanoparticles characterized by an arbitrary size distribution is developed in a quasi-static dipole approximation. The equations for coupled dipoles and general relations are formulated in terms of reduced dipole moments. It is shown that the dipole resonant frequencies and amplitudes, the absorbed power, and the acting-ﬁeld magnitudes strongly depend on the ratios of particle radii in a cluster. Properties of linear, planar, and three-dimensional systems are examined. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION that the dominant role is played by the nearest neighbors of a particle, whereas the contribution of a more Optical properties of disordered fractal clusters of distant environment to local field characteristics is not spherical metal nanoparticles have been studied in a important. The interaction indicated above determines vast literature, including a number of reviews and the structure of field fluctuations and explains the monographs (e.g., see [1Ð6]). A variety of new linear essential difference of the optical properties of media and nonlinear effects have been discovered: inhomoge- consisting of fractal clusters from those of “normal” neous broadening of the plasmon band, giant field fluc- (weakly inhomogeneous) media. tuations with correlation radii much smaller than the wavelengths, surface-enhanced Raman scattering, pho- The observations enumerated above suggest that a tomodification of clusters, fast-response highly nonlin- collection of fractal clusters should be considered as an ear behavior, and nonlinear optical activity. Fine tech- unconventional medium that has unusual optical prop- niques have been developed for producing clusters with erties and is different from a gas, plasma, liquid, or controlled values of basic parameters. It was proved solid. that many natural systems with interesting properties The existing theories rely on the assumption of have fractal structure [6, 7]. The key role played by the equal particle size (an interaction parameter of primary fractal structure of clusters with Hausdorff dimension importance). The actual polydispersity is taken into substantially lower than three (D ~ 1.5Ð1.8) was dem- account by averaging the results over some size distri- onstrated both experimentally and theoretically. In bution (e.g., see [6]). This approach is not self-consis- recent years, fractal aggregates of metal nanoparticles tent. Indeed, if particles of difference size exist, then were found to be increasingly useful for various appli- nearest neighbors may have different diameters. How- cations. ever, electrodynamic interaction between particles of In the pioneering studies [8, 9], it was already equal size is characterized by a certain symmetry, revealed that optical properties of fractal clusters are which implies a corresponding degeneracy in the vibra- mainly determined by electrodynamic interactions tional spectrum and certain selection rules. For exam- between particles, which is manifested in both linear ple, in a system of two interacting particles (dimer), and nonlinear effects. In particular, the interaction is dipole oscillations of certain types cannot be excited by directly responsible for the strong plasmon-band split- an external field because of its simple structure. In this ting characteristic of an isolated particle and for a wide, respect, a dimer is analogous to a molecule consisting inhomogeneously broadened tails in the absorption of two identical atoms, for which infrared absorption is spectra of clusters. Numerical simulations showed that prohibited. The analysis presented below shows that the splitting interval is independent of the number N of particle-size variability has fundamental consequences particles (when N is sufficiently large) and of the opti- for spectroscopy of fractal clusters, nonlinear effects, cal properties of the particle material (when measured properties of field fluctuations, etc. Thus, the existing in certain units) [10, 11]. This fundamental fact implies theories of optical properties of clusters are essentially

692 PERMINOV et al.

r inadequate. Therefore, both foundations of the theory Equations (2.1) and (2.2) are rewritten in terms of di as and its applications to real systems must be revised. This problem is addressed in the present study. κ r ξ []r ψ ()ϕ⋅ r di + ∑ ij d j ij Ð 3nij nij d j ij ji≠ (2.6) 2. GENERAL RELATIONS 3/2ε () = ai hE0 ri . Consider a system of N spherical particles (mono- ε r mers) in a medium with a dielectric constant h. The The use of the reduced dipole moment di facilitates particle material is characterized by a dielectric con- further analysis. Since its square has the dimension of ε stant . Suppose that each particle radius ai (i = 1, 2, …, energy, both energy and amplitude relations are most N) is sufficiently small as compared to the wavelength, r so that the dipole approximation is valid. The dipoleÐ conveniently formulated in terms of di . Define column dipole interactions between particles and their interac- vectors dr and Er with the components tion with a monochromatic external field E0(r) of fre- ω r 3/2 r 3/2ε () quency are described by a system of equations for diα ==diα/ai , Eiα ai hE0α ri . (2.7) their dipole moments di [10, 11]: εε Following [10], rewrite Eqs. (2.6) in operator form: α ε α 3 Ð h di ==i hEi, i ai ------ε ε-, (2.1) + 2 h ()κ + U dr = Er, (2.8) U α, β = ξ []δαβψ Ð 3n αn βϕ , ij≠ . () Eij i j ij ij ij ij ij Ei = E0 ri + ∑------ε , ≠ h ji (2.2) The matrix elements Uiα, jβ of the interaction operator U 3n ()ϕn ⋅ d Ð d ψ are symmetric under the permutation iα jβ of the ij ij j ij j ij αβ Eij = ------3 , particle and coordinate indices, ij and , and are com- rij plex in the general case. As in [10, 11], the operator U can be diagonalized, and the eigenvalues um and eigen- rij ===ri Ð r j, rij rij , nij rij/rij, (2.3) states defined by the relation

ϕ []()2 () |〉 |〉 ij = 1 Ð ikrij Ð krij /3 exp ikrij , Um = um m (2.4) ψ []()2 () ij = 1 Ð ikrij Ð krij exp ikrij . can be used to represent the solution to Eqs. (2.6) as

The field Ei acting on the ith dipole is the superposition r 〈〉κα ()Ð1 〈〉β r ε diα = ∑ i m + um mj E jβ. (2.9) of E0(ri) and the fields Eij/ h generated by all dipoles (indexed by j) at the point ri where the ith dipole is located.1 The retardation effects due to the nonzero dis- Formally, both Eq. (2.8) and its solution (2.9) are simi- tance between the ith and jth particles are represented lar to those describing systems of identical particles. ϕ ψ Moreover, the proposed model (2.7)Ð(2.9) has the addi- by ij and ij. The general analysis of system (2.1), ξ κ (2.2) presented in [10, 11] was entirely based on the tional advantage that U, ij, , and um are dimensionless equality of all particle radii. Since this assumption was quantities, whereas their counterparts in [10, 11] are inherent in the theory developed in [10, 11], the particle measured in cmÐ3. Note that Eqs. (2.6) and (2.8) can be diameter was used as the unit of length. System (2.1), separated with respect to parametric dependence (when κ (2.2) is free of this restriction, but it can be transformed krij = 0): the parameter depends only on the dielectric into the one that was analyzed in [10, 11] by changing properties of particles and medium, while the operator the variables. Define U and its eigenvalues are determined by the system’s r ∝ 3/2 geometry. Note also that E jβ a j in (2.9), whereas d ()a a 3/2 dr ==------i , ξ ------i j , the column vector on the right-hand side of the equation i 3/2 ij 3 analogous to (2.8) in the standard theory [10, 11] ai rij (2.5) depends on j only via rj in E0α(rj). A detailed analysis is 3 ε ε easier to perform in terms of κ = Ðu . κ ai + 2 h m m ==----α------εε-. i Ð h It is obvious that the variability of ai is essential in r ∝ 3/2 various respects. Since E jβ a j , the amplitude of the 1 The near field of a dipole is inversely proportional to ε , and this h κ κ Ð1 dependence is factored out in (2.2). The polarizability of a ball is resonance ( Ð m) depends on the relations between α ε ε i h (see (2.1)), and the factors h cancel out. particle radii. In particular, certain amplitudes that van-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE THEORY OF OPTICAL PROPERTIES OF FRACTAL CLUSTERS 693 ≡ ish when ai a [5] are finite in the general case. Fur- The total work A done on the dipoles di by the exter- thermore, the coupling parameters nal field E0 is the sum of contributions due to the acting fields Ei. Indeed, the sum over i and j on the right-hand ξ ()3/2 3 side in the relation ij = aia j /rij ⋅ () ⋅ depend on the ratios of particle radii via both aiaj and rij Re i∑di E0* ri = Re i∑di Ei* ≥ (rij ai + aj). Both eigenvalues um and eigenfunctions of i i U are modified accordingly. In particular, degenerate eigenstates split since the symmetry associated with  ⋅ []() ⋅ * equality of particle diameters is broken when the ratios +Rei∑di E0* ri Ð Ei* = Re i∑di Ei  ai/aj are arbitrary. A detailed analysis of the effects due to i i particle-size variability is presented in Sections 3 to 5.  εÐ1 []⋅ ()⋅ ()⋅ 3 In the quasi-static approximation adopted in what +Rei h ∑ di d*j Ð 3 nij di nij d*j /rij ϕ ψ  follows (krij 0), ij = ij = 1. In this case of highest ij, ≠ i practical importance, both operator U and eigenvalues is obviously real, and the corresponding term vanishes. um are real. By virtue of (2.1), E* can be replaced by d* /ε α* , It is assumed above that all particles consist of the i i h i ε same material and are characterized by equal permittiv- with h assumed to be real: ities ε. The theory can be naturally extended to particles 2 ε ω di of different size (ai) and material ( i). This may be Q = Ð----Re i ------a ∑α ε important for analyzing the effects of particle size and 2 i* h ε i temperature T on i. It is reasonable to define the fol- (2.12) lowing quantities and introduce a relation between ω 2 1 them: = Ð ------ε-∑ di Imα----- . 2 h i i d ε Ð ε d' =,------i- α = a 3------i h-, Expression (2.12) can also be derived from a standard i α1/2 i i ε + 2ε i i h expression for the absorbed power per unit volume q [13]: 2 ()1 + V d' = Ᏹ, q ==ωε'' E /8π, ε'' Imε. ()α α 1/2 (2.10) To calculate the power Qai absorbed by the ith particle, i j []δ ψ ϕ this expression must be multiplied by its volume, with V iα, jβ = ------3 - αβ ij Ð 3nijαnijβ ij , rij E interpreted as the field strength Ei, in inside the particle related to the acting field Ei [13]: diα Ᏹ α1/2ε () di'α ==------, iα i hE0α ri . E 3ε 3d α1/2 i h ------i - i Ei, in ==------ε ε- α ()ε ε . + 2 h i + 2 h The operator V is also invariant under the permutation As a result, we have iα jβ, and all general conclusions made above apply to Eq. (2.10). ω 2 3 Q = ----3ε'' di a . (2.13) ai 2 α------()ε ε - i In the model adopted here, the work A done on the i + 2 h dipoles di by the external field is consumed to produce scattered radiation and to heat the particles [12, 13]: It is easy to show that expression (2.13) is equivalent to the ith summand in (2.12). The inverse specific susceptibility κ represented as ()ω ⋅ () A ==Ð /2 Re i∑di E0* ri Qs + Qa. (2.11) follows [10]: i κ = Ð X Ð iδ, (2.14) where the sign of the real part X is consistent with the opti- An estimate obtained below (see Eq. (2.16)) shows that ε the scattered power Q is much lower than the absorbed cal properties of metals ( ' < 0). Now, expression (2.12) s has a particularly simple form: power Qa for systems of interest here, and Qs is 2 2 neglected in the analysis that follows. ωδ d ωδ 2 Q ===Q , Q ------i ------dr . (2.15) a ∑ ai ai ε 3 ε i 2 2 h 2 h Following [14], we can take into account the effect of scattering i ai on relaxation characteristics. However, we focus here on the fundamental role played by the variability of particle size and do not Expressions (2.12) and (2.15) extend the results expand the model to allow for corrections of this kind. of [10, 11] to the case of an arbitrary size distribution.

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Formula (2.15) demonstrates that Qai is determined by description of these effects must take into account the the squared absolute value of the reduced dipole variability of particle size, which is suitably character- moment. Expressions (2.12) and (2.15) determine the ized in terms of dr . power absorbed by each particular particle, i.e., the dis- i tribution of power over particles parameterized by their Analyses of fractal clusters of metal nanoparticles size, ﬁeld frequency, properties of surrounding parti- commonly rely on the dielectric constant given by cles, and other characteristics. It is obvious that the Drude’s formula quantity {Ð(ω/2)Re[id á E* (r )]} in (2.11) cannot be εεω2 ωω()Γ i 0 i = 0 Ð p/ + 2i , (2.21) interpreted as the absorption by the ith particle. In par- ω Γ ticular, the contributions of some individual summands where p is the plasma frequency, is the damping in (2.11) to the work A done by the ﬁeld are negative. constant (generally, a function of particle size), and ε − 0 1 is the contribution of interband transitions. Set- It is well known that the power scattered by a parti- ε ≈ ε ting h 0 for simplicity, we obtain cle characterized by a dipole moment di is Qsi = ω4|d |2/3c3. For a ≈ 10 nm, 2πc/ω ≈ 103 nm, and δ ≈ i i ω2 ω2 δ ωΓ ω2 ω 2 ω2 ε 0.1Ð0.01 characteristic of metal nanoparticles, X = / p Ð1, = 2 / p, p = p/3 h. (2.22)

ε ω 3 For example, silver is characterized by ω = 2.5 × Qsi 2 hai Ӷ p ------= ------δ-------1. (2.16) 4 Ð1 Γ ≈ Ð1 Qai 3 c 10 cm and 500 cm . According to (2.22), the κ ω2 equation X = Ð m yields resonant values of , which Therefore, scattering is negligible as compared to are equivalent to absorption. This conclusion holds even if interference ω ω ()κ 1/2 of the fields scattered by different monomers is taken / p = 1 Ð m (2.23) into account. ω κ on the scale. When m is positive or negative, the cor- It is well known that fluctuations and nonlinear phe- responding resonance is shifted toward the low- or nomena depend on the acting field strength [3]. This high-frequency end of the spectrum, respectively. Sub- quantity of special importance for fractal clusters can stituting the expression for Qai in (2.15) into (2.22), we r be calculated by using solutions di to Eqs. (2.6): obtain

r 2 κ ω Ð1 r 2 di di Q = ------ε Γ d . (2.24) Ei ==------. (2.17) ai ω h i α ε 3/2ε p i h ai h Its absolute value squared is proportional to the Since it is obvious that absorbed power per unit volume: r 2 ∝∝δÐ2 ωÐ2 di ()κ3 2 2 Qai/ai 2 Ei = ------ε ωδ . (2.18) at resonant frequencies, the factor ω in (2.24) is can- 2 h celed out in resonant values of Qai. The model of irregular fractal clusters of identical According to (2.17) and (2.18), the value of Ei strongly depends on the monomer radius. particles involves the random parameters rij (spacing The energy of a dipole placed in an external field is between particles), nij (relative position of particles), and a (particle radius). In the generalized model con- 2 d structed here, the ratio ai/aj is an additional random 1 ()⋅ * 1 i Ui ==Ð---Re di E0 Ð---Reα------ε- parameter. An exact analysis must rely on an N-dimen- 2 2 * h (2.19) sional size distribution P(a1, a2, …, aN). Spectral prop- r 2 erties of clusters depend primarily on the interaction 1 di κ = Ð------ε Re . between monomers that are relatively close to each 2 h other. Therefore, the minimal required dimension of The ponderomotive force exerted by the external field P(a1, a2, …) may be relatively low. This means that a on the ith dipole and other dipoles is much greater number of random realizations must be computed in numerical experiments of the kind ∇ Fi = Ð Ui. (2.20) reported in [10, 11]. After this brief discussion of statistical aspects, we According to [15], the motion of a particle driven by focus below on dynamics of clusters. Some dynamical the force given by (2.20) changes its kinematic proper- aspects are directly related to experiment, as an analysis ties and leads to certain nonlinear effects. An adequate of photomodification of clusters based on electron dif-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE THEORY OF OPTICAL PROPERTIES OF FRACTAL CLUSTERS 695 fraction patterns visualizing individual monomers and dipole moment d = d1 + d2 of a dimer are aggregates. ε ()3 3 () dn = Ð hE0n a1 + a2 f n X , 3. DIMER 1 Ð C C (3.7) f ()X = ------+ ------, n X ++2ξ iδ X Ð 2ξ + iδ Consider the case of two particles (a dimer) as an illustration of the principal optical properties of clusters ε ()3 3 () due to electrodynamic interaction between particles [8, d⊥ = Ð hE0⊥ a1 + a2 f ⊥ X , ≤ 16]. We suppose that a2 a1 and drop the subscripts ij 1 Ð C C (3.8) r f ⊥()X = ------+ ------, in nij and rij as unnecessary. For the projections of din X Ð ξ + iδ X ++ξ iδ r and di⊥ on n and the plane perpendicular to it, respec- 3/2 tively, Eqs. (2.6) reduce to 1 2()a a C 2 C ==--- 1 Ð,------1 2 ------B . (3.9) 2 3 3 1 Ð C ()κ ξ ()r r ()ε3/2 3/2 a1 + a2 Ð 2 d1n + d2n = a1 + a2 hE0n, (3.1) ()κ ξ ()r r ()ε3/2 3/2 ± ξ ±ξ + 2 d1n Ð d2n = a1 Ð a2 hE0n, The resonance shifts 2 and in (3.7) and (3.8) depend on the distance r between the particles, and ()κξ()r r ()ε3/2 3/2 absorption in different spectral ranges is due to dimers + d1⊥ + d2⊥ = a1 + a2 hE0⊥, (3.2) with different values of r. This theoretical conclusion is r r 3/2 3/2 generally used as a basis for interpreting inhomoge- ()κξÐ ()d ⊥ Ð d ⊥ = ()εa Ð a E ⊥, 1 2 1 2 h 0 neous broadening of an absorption band. This interpre- ≠ tation holds for a1 a2, but is more complicated and ξ ==()a/r 3, aa()a 1/2, 1 2 (3.3) entails additional properties. In particular, formulas (3.6) E ==nE⋅ E cosθ, E ⊥ =E sinθ. and (3.9) yield B = 0 and C = 0 for a1 = a2; i.e., two res- 0n 0 0 0 0 onant amplitudes vanish. The remaining two reso- For the n-projections, κ = 2ξ and κ = Ð2ξ correspond nances correspond to oscillations that are parallel and n n ξ to the sum- and difference-frequency oscillations (the perpendicular to n and are characterized by Xn = Ð2 ξ ≠ ≠ latter mode was called antisymmetric in [14]). For the and X⊥ = , respectively. If a1 a2, then C 0, oscilla- ⊥-projections, the sum- and difference-frequency tion amplitudes change, and the band proﬁle is differ- modes are characterized by κ⊥ = Ðξ and κ⊥ = ξ, respec- ent. The difference-frequency resonant amplitudes van- r ish for a1 = a2 in the approximation of kr = 0 adopted tively.The oscillations of di⊥ are twice degenerate by here. However, the difference-frequency resonant virtue of axial symmetry. It should be noted that the amplitude does not vanish because of retardation normal modes described by (3.1) and (3.2) are effects [14] even if a1 = a2. In this case, the amplitude r λ ≈ 3 ≈ expressed in terms of reduced dipole moments diα . is proportional to 1 Ð cos(kr). For 10 nm and r ≈ ≈ Ð1 ≈ −2 This observation applies to systems consisting of a 2a1, 2 20 nm, we have kr 10 and 1 Ð cos(kr) 10 . greater number of monomers as well. An amplitude of similar order of magnitude is obtained when the particle radii differ by about 10%.3 The solutions to Eqs. (3.1) and (3.2) can be written 2 as The dependence of B on a2/a1 is illustrated by 2 curve 1 in Fig. 1. When a2/a1 is small, the coefﬁcient B 3/2 3/2 3/2 2 r a a Ð a rapidly decreases to B ≈ 0.5 at a /a = 0.3. The ratio of d = Ðε E ------+ ------i , 2 1 in h 0nX ++2ξ iδ X Ð 2ξ + iδ resonant amplitudes (3.4) and (3.5) in dipole oscilla- (3.4) tions is B, and its dependence on the difference in radii a3/2 + a3/2 is stronger. In (3.7) and (3.8), we expose the factors a3/2 = ------1 2 , 2 3 3 (a1 + a2 ) proportional to the total volume of the interacting particles. Thus, the functions f (X) and f⊥(X) are 3/2 3/2 3/2 n r ε a ai Ð a normalized to the unit volume of the particle material. di⊥ = Ð hE0⊥------+ ------, (3.5) X Ð ξ + iδ X ++ξ iδ The sums of resonant amplitudes in fn(X) and f⊥(X) are constant. Therefore, the appearance of doublet compo- 3/2 3/2 2 ≠ a Ð a 1 Ð t a 3/2 nents proportional to B (when a1 a2) is related to a i ± ------2 ------3/2 - ==B, B , t = . (3.6) decrease in absorption in stronger components. a 1 + t a1 3 In [5, p. 158], it was claimed that nonvanishing different-fre- In (3.6), plus and minus correspond to i = 1 and i = 2, quency resonance amplitudes should be explained solely by retar- respectively. The components of the total induced dation effects. This assertion obviously disagrees with reality.

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B2, ξ /ξ polarizability of the smaller particle. Indeed, the value max 0 ξ 1.0 max corresponding to the minimal distance rmin = a1 + a2, 0.8 2 ()a a 1/2 3 2()a a 1/2 3 ξ ==------1 2 ------1 2 - ξ , max r a + a 0 (3.10) 0.6 min 1 2 ξ 0 = 1/8, 0.4 reaches its maximum when a1 = a2. Curve 2 in Fig. 1 ξ ξ represents max/ 0 as a function of a2/a1. It demonstrates 0.2 1 that the “new” resonances already have appreciable amplitudes (curve 1) at a2/a1 = 0.25Ð0.50, while the splitting interval remains relatively wide (curve 2). 0 0.2 0.4 0.6 0.8 1.0 Equations (2.1), (2.2), and (2.6) are valid when the a2/a1 field generated by a dipole changes insignificantly over ≈ Fig. 1. Resonant-amplitude ratio (curve 1) and largest shift the length of a neighboring particle. When r a1 + a2, (curve 2) versus ratio of particle radii in a dimer. this condition is violated. A detailed analysis shows that the change in the dipole’s field is equivalent to an increase in 2a/r by a factor of (6/π)1/3 [3, 17, 18]. Qual- 1 δ itative considerations suggest that a similar renormal- 1Im(0.8 fn) ≠ (a) ization of (a1 + a2)/r applies when a1 a2 [3]. The 0.6 0 ξ 0.5 renormalization changes the value of 0 by a factor of 0.4 almost 2: 0.3 ξ ξ π 0.2 0 ==1/8 0 3/4 . 0.1 0.05 The imaginary parts of fn(X) and f⊥(X) determine the spectral profile of absorption on the X scale. Figures 2a and 2b present, respectively, the resonance profiles δ δ Imfn(X) and Imf⊥(X) calculated for several values of a2/a1 and a constant difference (a1 + a2) Ð r between the particle surfaces. Figures 2a and 2b show that the n- and ⊥ -polarization resonances “prohibited” for a2 = a1 can contribute substantially to the high- and low-frequency Im(δ ⊥) 1 0.8 1 f (b) 0.05 tails of the band, respectively. Note also that the reso- 0 0.5 0.6 nances shift as a2/a1 varies from 1 to 0.1 and the doublet 0.4 components merge as a2/a1 0. The physical explanation of this fact is obvious: as a2/a1 0, a dimer becomes a monomer. In summary, there exist two 0.1 mechanisms of inhomogeneous broadening of the 0.2 absorption band: variability of the distance r and poly- 0.3 dispersity of particles. The integral absorption (over X) includes a partial 3 contribution (a2/a1) due to the smaller particles, whereas their contributions to the resonance shifts are 3/2 Ӷ –0.15 –0.10 –0.05 0 0.05 0.10 0.15 proportional to (a2/a1) for a2/a1 1, i.e., much X greater. For example, when a2/a1 = 1/5, the contribution of the smaller particles to the integral absorption is less Fig. 2. Normalized absorption coefficient for a dimer ori- ξ ≈ ented (a) parallel and (b) perpendicular to the electric field than one per cent, while max 0.5, which amounts to ξ ξ vector for the values of a2/a1 shown at the curves, = 40% of the highest value of 0 (1/8). For sufficiently ξ δ max/2, and = 0.01. small δ, shifted resonances give rise to appreciable inhomogeneous broadening. Therefore, relatively small particles can substantially modify the absorption At first glance, it may seem that a decrease in a2 profile without contributing to the integral absorption. implies a smaller minimal distance between particles In the conventional model of identical spheres, a and a wider splitting interval. However, this factor is change in the central portion of an absorption band is weaker than the concomitant decrease in volume and explained by interaction between relatively distant par-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE THEORY OF OPTICAL PROPERTIES OF FRACTAL CLUSTERS 697 ticles, which implies cooperative effects [10]. This tal cluster consisting of many spherical particles [10, 11]. explanation is not complete: when a2/a1 is sufficiently This implies that the nearest neighbors of a particle play small, the resonance shifts are smaller than their half- a dominant role, whereas distant particles do not con- width, tribute substantially to local field characteristics. In the units used here, the width of the range in question is ξ ≈ ()3/2 < δ ξ max a2/a1 . (3.11) approximately 8 0, whereas the largest width of the ± ξ In this situation, the central portion of the absorption eigenfrequency splitting interval for a dimer is 2 0. band results from the combined contributions of distant Therefore, the dimer model cannot be used to explain particles of nearly equal size and dimers with particles optical properties of real fractal clusters. It would be interesting to identify the microscopic systems respon- having disparate sizes. For example, condition (3.11) is ± ξ δ sible for a splitting width of 4 0 and find out if the satisfied for = 0.1 when a2/a1 = 0.215. These trends are demonstrated in graph form by the curves corre- width varies with the ratio of monomer radii. It should sponding to a /a = 0.05 and δ = 0.01 in Figs. 2a and 2b. also be expected that the equal of power absorbed by 2 1 different monomers is specific to dimers, whereas Now, let us discuss the individual properties of the many-particle systems are characterized by nonuni- dipoles d1 and d2. For the projections d1n and d1⊥ of the form distributions of Qai over particles. These questions larger particle’s dipole, the amplitudes of both reso- are addressed in Sections 4 and 5. 3/2 3/2 nances have like signs (a1 > a ); for d2n and d2⊥, the Absorption spectra for systems of several mono- signs are opposite. Moreover, it is easy to show that mers in the approximation of dipoleÐdipole interaction both Imd2n and Imd2⊥ can be negative. were calculated in numerous studies (see [5] and references therein). Detailed results of eigenvalue calculations Consider the power Qain absorbed by individual particles. According to expressions (3.4) and (3.5), their for linear, planar, and three-dimensional systems with N varying from 2 to infinity were presented in [20, 21]. respective Lorentzian profiles do not overlap and the |κ ξ | ≈ resonant amplitudes corresponding to particles 1 and 2 It was established in [21] that m/ 0 4 are character- are equal. Therefore, istic of various configurations with N = 6 and 7. However, since all studies of this kind were conducted for particles of equal size, the problem should be reex- Qa1n ==Qa2n, Qa1⊥ Qa2⊥ (3.12) amined. near the resonance points. Note also that the ratio of the System (2.6) written for N interacting dipoles is on 2 resonant amplitudes in Qain is B , as in Imfn(X) and the order of 3N, and analytical results are difficult to Imf⊥(X). obtain and understand. The present analysis is restricted to several simple configurations, but its The quantity Qain is the total power absorbed by the ith particle. Approximate equality of Q and Q results provide a basis for some general conclusions a1n a2n that answer the questions posed in this study. implies that the smaller particle absorbs a higher power per unit volume. Therefore, the smaller particle is First, we consider a linear aggregate of N particles. heated by radiation to a higher degree. This can be man- The corresponding system (2.6) breaks up into equa- r r ifested in the temperature dependence of dielectric con- tions for the projections din and di⊥ on the chain axis stant and in photomodification of clusters [4, 19], for and the plane perpendicular to it: example, when particles melt or coalesce. Additional κ r ξ r 3/2ε factors include the rate of particle cooling, the duration din Ð 2∑ ijd jn = ai hE0n, of the irradiating laser pulse, thermal properties of the ij≠ (4.1) particle material, etc. The problems related to photo- ξ ()3/2 3 , ,,… , modification of clusters require a special analysis. ij ==aia j /rij, ij 12 N,

Many optical properties of fractal clusters strongly r r 3/2 κd ⊥ + ξ d ⊥ = a ε E ⊥. (4.2) depend on the local acting field Ei. Since Eq. (2.18) i ∑ ij j i h 0 entails ij≠ ∝ ()3 1/2 System (4.2) can be obtained from (4.1) by performing Ei Qai/ai , the change the field acting on (smaller) particle 2 is stronger than 2ξ Ðξ , E E ⊥. 3/2 ij ij 0n 0 that acting on particle 1 by a factor of 1/t = (a1/a2) by virtue of (3.12). The analysis presented here is mainly performed for the simpler system (4.2). Since ξ is a cubic function of 1/r , the interactions 4. EIGENFREQUENCIES ij ij OF MANY-PARTICLE SYSTEMS between neighboring particles play the dominant role. As a first approximation, we set The eigenfrequency range of the interaction opera- ξ ξ δ tor is independent of the number N of particles in a frac- ij = ii ± 1 i ± 1 j. (4.3)

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Table

N 123456789 κ ξ ± ± ± ± m/ 0 10 0.618 0 0.445 0 0.351 0 ±21/2 ±1.618 ±1 ±1.245 ±0.766 ±1 ±0.618 ±31/2 ±1.800 ±1.414 ±1.523 ±1.176 ±1.845 ±1.882 ±1.618 ±1.902

κ ξ Matrices of this form (with as diagonal entries) are a The lowest-order corrections (linear in ij with special case of Jacobian matrices [22]. We denote the |i − j| > 1) amount to about 10% and break the symme- ∆ determinant of a matrix of order N by N and recall a try about κ = 0, in particular, by shifting the zero root. ∆ m number of properties of N important for this study [22]. To illustrate the error due to approximation (4.3), we ∆ κ All roots of N are real and distinct. The sum of the present the exact values of max( m) obtained for N = 6 ∆ ∆ roots of N is zero. If N is even (odd), then N contains and for an infinite chain of contacting spheres [20, 21]: only even (odd) powers of κ. The interval between two ()κ ξ ∆ max m/ ==2.008, N 6, consecutive roots of N contains exactly one root of ∆ ()κ ξ ∞ N − 1. Both the largest positive root and the largest abso- max m/ = 2.40, N . lute value of a negative root increase with N. The smallest root min(κ ) of ∆ is a decreasing function of ξ . Its The corresponding approximate values given by (4.5) m N ij are 1.800 and 2.000, respectively; i.e., the error largest root max(κ ) is an increasing function of ξ . The m ij increases with N, reaching 10Ð15% at N ≈ 10. minimal and maxima roots satisfy the inequalities Let us use (4.3) and (4.5) to examine the influence ()ξ [] π()≤ ()κ of difference in radii on κ . For a chain of alternating Ð2max ij cos / N + 1 min m m particles of radius a1 and a2, ≤ ()ξ [] π() Ð2min ij cos / N + 1 , ()1/2 3 (4.4) a1a2 ()ξ [] π()≤ ()κ ξ ==ξ ------. (4.6) 2min ij cos/ N + 1 max m ij a1 + a2 ≤ ()ξ [] π() κ ≠ 2max ij cos / N + 1 . In this case, the ratio of m corresponding to a1 a2 and a1 = a2 is equal to that for dimers: The equalities in (4.4) are valid for ξ = ξ. In this case, ij ()1/2 3 ξ 2 a1a2 κ ξπ[]() ----- = ------< 1 m = Ð2 cos m/ N + 1 , ξ a + a (4.5) 0 1 2 ,,… ,ξ ξ m ==12 N, ij . (see (3.10) and curve 2 in Fig. 1). κ ξ Somewhat similar results are obtained by analyzing The table shows the numerical values of m/ given 4 κ two- and three-dimensional configurations. In particu- by (4.5). It demonstrates that the distribution of m cal- lar, for the “seven-leafed rosette” consisting of six par- culated in approximation (4.3) is symmetric about the ticles of radius a (j = 1, 2, …, 6) located equidistantly κ 1 point m = 0 and is nearly uniform over the interval on a circumference of radius r and a particle of radius bounded by ±κ (becoming slightly “condensed” N a2 at the center, toward max|κ |. The largest root κ almost reaches the N N ξ ()3 ξ limit 2ξ when N = 6. With further increase in N, both the jj + 1 ==a1/r , largest and smallest roots remain nearly constant. Since ξ ξ ()3/2 ξη the number of roots increases, so does their density in jj + 2 ===/3 0.192 , the interval bounded by ±2ξ. Recall that these results ξ ξ ζ ξ []()1/2 3 ξ are valid for dipole polarization perpendicular to the jj + 3 ==/8 , j7 =a1a2 /r =1, (4.7) ӷ chain axis. For axially polarized dipoles with N 1, the ()< |κ | ξ r = 2a1 a2/a1 1 , interval occupied by the roots corresponds to N / = 4, ()> which agrees with the numerical results of [10, 11]; i.e., ra= 1 + a2 a2/a1 1 . approximation (4.3) adequately describes linear aggregates in this case. When a2/a1 > 1, the outer particles are in contact with the central one. When a2/a1 < 1, the outer particles are 4 Approximation (4.3) applies to many physical models, e.g., the in contact with one another. In the case of polarization linear crystal [23, 24]. perpendicular to the rosette’s plane, the seventh-order

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE THEORY OF OPTICAL PROPERTIES OF FRACTAL CLUSTERS 699 κ system of equations breaks up into five first-order equa- exhibit a substantial and intricate dependence of m tions and a second-order system, which yield on a2/a1. κ ()ξξη ξ To illustrate the dependence of roots on particle size 1 ==2 Ð + 1.741 , for three-dimensional systems, we now consider a clus- κ ξηζ ξ () 23, ==+Ð 1.067 2, ter with six spherical particles of radius a1 (j = 1, 2, …, 6) lying on the coordinate axes at equal distances from κ ξ ηζ ξ () 45, ==Ð + +Ð 0.683 2, a central spherical particle of radius a (an endohedrally (4.8) 2 r κ , = Ð ξ Ð η Ð ζ/2 67 doped octahedron). The system of equations for diα breaks up into third-, second-, and first-order systems. ± []ξ2 ()ξηζ 2 1/2 6 1 + ++/2 The sum-frequency roots are

3 1/2 {}ξ± []() κ , ()ηζ = Ð 1.255 6 a2/a1 + 1.575 . 13 = Ð Ð /2 1/2 (The numbers in parentheses are the multiplicities of ± []12ξ2 ++2()ηζ+ 2 ()ηζÐ /2 2 ()3, (4.10) κ κ degeneracy.) The roots 1, …, 5 correspond to differ- κ () ence-frequency dipole oscillations of outer particles 2 = 0 3 , and depend only on the parameters of their interaction ≤ κ and the difference-frequency roots are (for a2/a1 1). The roots 6, 7 are associated with the κ η ζ () interaction of the central dipole with the resultant 4 = 3 + 2 2, peripheral dipole, varying with a /a . For a /a = 1/2 2 1 2 1 κ ηζ() 5 = 3 + 3, and 1 (r = 2a1), we obtain κ ξ κ ηζ 6/ = Ð2.780,Ð 4.007, 6 = 2 Ð (3), (4.11) κ ξ 7/ = 0.270, 1.497; κ η ζ 7 = Ð 3 + (3), i.e., the roots strongly depend on a /a . When a /a > 1, κ η ζ 2 1 2 1 8 = Ð26 + . the distance r increases with a2 as 1 + a2/a1 and the fre- κ κ Ð3 quencies 1, …, 5 scale with (1 + a2/a1) ; i.e., they Since the particles lying in each coordinate plane con- vary rapidly. stitute a five-leafed rosette similar to that considered above, the parameters ξ, η, and ζ are given by (4.9). For Analogous results are obtained for the five-leafed α α t = 2/ 1 = 1/2, 1, and 2, we obtain rosette consisting of four particles of radius a1 located equidistantly on a circumference and of a particle of κ ξ 1/ 0 = 2.698, 3.251, 2.825; radius a2 at the center. The interaction between the reduced central and resultant peripheral dipoles polar- κ /ξ = Ð4.078 , Ð3.833 , Ð;2.998 ized perpendicular to the rosette’s plane is character- 3 0 ized by the roots κ ξ 4/ 0 = 3.107, 1.311, 0.388; κ η ζ ± []ξ2 ()ηξ 2 1/2 κ ηζ 12, = Ð Ð4/2 + + /2 , 3 = 2 Ð . κ ξ 5/ 0 = 2.810, 1.186, 0.351; The difference-frequency root is doubly degenerate: κ ξ κ ζ 6/ 0 = 1.380, 0.582, 0.172; 45, = . κ ξ In the case of outer particles in contact with the central 7/ 0 = Ð2.218 , Ð0.936 , Ð;0.277 one, κ /ξ = Ð4.436 , Ð1.871 , Ð.0.554 1/2 3 1/2 3 8 0 ξ 2t ξ η 2 ξ ==------0, ------0, κ 1 + t 1 + t (4.9) The sum frequencies 1, 3 (associated with interaction of the peripheral particles with the central one) vary rel- ζ ()Ð3ξ ≥ 1/2 ==1 + t 0, ta2/a1 2 Ð 1. atively slowly, because an increase in a2 affects both the interparticle distance and the central particle’s pola- The numerical values for t = 1/2, 1, and 2, κ rizability. Accordingly, 1, 3 varies approximately as κ ξ 1/2 3 45, / 0 = 0.296, 0.125, 0.037; [2t /(1 + t)] . Since the difference modes are deter- κ ξ mined only by the interactions between peripheral par- 3/ 0 = 1.380, 0.582, 0.173; κ κ Ð3 ticles, the frequencies 4, …, 8 vary as (1 + t) with κ ξ increasing a . These trends are analogous to those char- 2/ 0 = Ð2.931,Ð 2.459,Ð 1.804; 2 acteristic of the seven- and five-particle two-dimen- κ ξ 1/ 0 = 0.959, 1.627, 1.557, sional systems considered above. Note also that

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 700 PERMINOV et al. |κ | ξ max m almost reaches 4 0 for the endohedrally doped The analysis that follows is focused on the distribu- octahedron. ||r ||r 3/2 tions of di and di /ai over particles near the stron- The difference modes strongly depending on the rel- gest resonances and (as in Section 4) is restricted to ative particle sizes are not excited in the symmetric sys- simple cases, but the results provide a basis for some tems discussed here, and the corresponding linear phe- general conclusions. For longitudinally and trans- nomena cannot be manifested in optical spectra. It is versely polarized linear aggregates, the strongest reso- κ κ obvious that these difference modes must be excited nances correspond to the largest m and Ð m, respec- when the symmetry is broken (as in dimers), and their tively. Consider a symmetric linear trimer with contact- frequency will be sensitive to ratios of radii. ing particles and a1 = a3. Using approximation (4.3) ξ ξ ӷ δ The specific cases analyzed here illustrate the gen- and assuming that 12 = 23 , we find that eral considerations about the influence of relative parti- r κ d2 1/2 Qa2 r r cle size on um and m presented in Section 1. When the ------2===, ------2, d d . (5.1) κ r Q 3 1 eigenfrequency m has an extremum for a2 = a1, it var- d1 a1 ies slowly. For other types of oscillations or geometries, κ may strongly depend a /a . For this reason, mono- These results are independent of polarization and the m 2 1 value of a /a . The ratios of the acting-field magnitudes mer polydispersity should be taken into account in 2 1 κ and absorbed powers per unit volume are numerical m calculations analogous to [10, 11]. The 3/2 results presented in [10, 11] are of fundamental impor- E2 1/2a1 tance for understanding optical properties of fractal ------= 2 ----- , E1 a2 clusters, but have a limited scope as applied to real sys- (5.2) tems. In particular, the range of eigenvalues obtained Q a 3 a 3 ------a2------1 = 2 -----1 . in [10, 11] would but slightly depend on size distri- Q a a bution, because its boundaries are determined by inter- a1 2 2 action between particle of equal size. However, allow- Therefore, the fields acting on the peripheral particles is ance for size variability would obviously result in a stronger than that acting on the central one if a1/a2 < slower variation of the eigenfrequency spectrum at the 2−1/3 = 0.794. This phenomenon is analogous to the boundaries. increase in field strength at a metal needlepoint. Under this condition, the peripheral particles are heated stronger than the central one. When the interaction between 5. ABSORBED POWER AND ACTING FIELD particles 1 and 3 is taken into account, the resulting cor- FOR MANY-PARTICLE AGGREGATES rection to (5.1) depends on a2/a1 and amounts to about ||r ||r In many phenomena, including nonlinear effects 10%. In particular, d1 slightly increases, while d2 and photomodification of clusters, an essential role is slightly decreases; i.e., the “needlepoint effect” played by the local fields Ei acting on individual parti- becomes more pronounced. cles. According to one hypothesis, photomodification For a symmetric tetramer with contacting particles of clusters may be due to melting, evaporation, coales- (a = a , a = a ), we obtain cence, and other changes in particles strongly heated by 1 4 2 3 r absorbed radiation. If the pulse duration is too short for d p 2 1/2 p heat exchange between particles or heat transfer to the ------2-1= + --- + ---, r 2 2 environment to occur, then the initial temperature d1 change is determined by the energy per unit volume (5.3) 2 1/2 absorbed by a particle: Qa2 p p ------1= + p1 + --- + --- , Qa1 2 2 τ Q r 2 p ai ∝ τ di ------p , r r r r 3 ------3/2- a d4 ==d1 , d3 d2 , i ai 3 1/2 1/2 τ 1 a2 a1 where p is the effective pulse duration. According p = --- ----- + ----- . to (2.18), the square root of this quantity determines the 2 a1 a2 magnitude of the acting field E . i ≥ Since p 1 (p = 1 when a2 = a1), it holds that For transversely and longitudinally polarized r dimers with different particle radii, it was shown in Sec- d 15+ 1/2 Q ξ ӷ δ ------2- ≥ ------= 1.618, ------a2- ≥ 2.618. (5.4) tion 3 that Qa1 = Qa2 near the resonances when . In r 2 Q d1 a1 contrast, the distribution of Qai over particles for linear, planar, and bulk many-particle systems depends on Thus, the difference in absorbed power between outer many factors, including ratios of particle sizes. and inner particles is slightly greater than in the case of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE THEORY OF OPTICAL PROPERTIES OF FRACTAL CLUSTERS 701 | | | | | | | | a trimer. The ratio of the acting-ﬁeld magnitudes is The values of a1/a2 for which E2 / E1 = 1 and E3 / E1 = 1, derived from (5.3): respectively, are

3/2 r E2 a1 d2 2/3 ------= ------a1/a2 ===0.676, a1/a2 2 Ð1 0.588. (5.9) E a r 1 2 d1 1/2 (5.5) Thus, a stronger field acting on an outer particle (as 6 3 3 compared to inner ones) is associated with an even Ð4 a1 8a1 a1 = 2 1 + ----- + 2 ----- + 1 + ----- . smaller size of outer particles in a pentamer as com- a a a 2 2 2 pared to a tetramer or trimer. Otherwise, the fields act- | | | | ≥ ing on the inner particles are stronger. A simple calculation shows that E1 / E2 / 1 if These particular cases suggest the following general ≤ a1/a2 0.717. (5.6) ||r rule. The degree of nonuniformity of a di or Qai distri- Therefore, the needlepoint effect manifests itself at a bution over particles increases with the number of smaller ratio a /a in a tetramer as compared to a trimer. 1 2 monomers in a chain, and ||dr is increases toward the For the strongest resonance in a symmetric linear i center of a chain. Therefore, nonlinear effects in an system of five contacting monomers (a1 = a5, a2 = a4), we obtain aggregate of identical particles are localized at its center. With decreasing outer-particle radius, the localiza- r r d ξ 2 1/2 d 2ξ tion of nonlinear effects shifts to the corresponding end ------2-2==------23- + 1 , ------3------23-, r ξ r ξ of a chain. For example, melting must begin either at d1 12 d1 12 the center of a chain of identical particles or at the ends r r r r of a nonuniform chain with sufficiently small outer par- d5 ==d1 , d4 d2 , ticles. This rule applies to two- and three-dimensional 3/2 systems as well. 1/2 ()a a (5.7) κ ==Ð()ξ2 + 2ξ2 , ξ ------1 2 , Let us discuss this rule as applied to the symmetric 1 12 23 12 ()3 a1 + a2 five-leafed rosette considered in Section 4. For polarization perpendicular to the rosette’s plane, we obtain ()a a 3/2 ξ ------2 3 23 = 3. r ()a + a d 4ξ 2 2 3 ------2- ==Ð------, r κ 2 1/2 If ai = a, then d1 1 ()1 + q + q r r d2 1/2 d3 1/2 3/2 3/2 ------3==1.732, ------=2. 12 a1 a1 dr dr q ==---1 + ------------0.208----- , (5.10) 1 1 2 8 2a2 a2 ||r ||r According to (5.7), d2 / d1 > 1 for any ai/aj, and E2 2 r r ------= ------. |||| 3 1/2 d3 / d2 can be both greater and smaller than unity. E1 []() a2/a1 + 0.0433 + 0.208 Setting a2 = a3 for simplicity, we obtain ξ ()3/2 ξ 2 1/2 Therefore, if a2 = a1, then 12 8 a1a2 ≤ξκ 12 ------==------1, 1 Ð 23 2 + ------, ξ ()3 ξ r 23 a1 + a2 23 d ------2-= 1.627, ξ r 23 = 1/8. d1 Furthermore, which is close to the values obtained for linear systems. 6 3 1/2 E2 Ð5a1 a1 ||r ||r | | | | ------= 2 1 + ----- + ----- , Furthermore, both d2 / d1 and E2 / E1 are monotoni- E a a 1 2 2 ||r ||r ≥ cally decreasing functions of a2/a1, and d2 / d1 1 E ()1 + a /a 3 and |E |/|E | ≤ 1, respectively, when ------3 = ------1 2 , (5.8) 2 1 E 4 1 ≥ ≤ a2/a1 0.425, a1/a2 0.681. (5.11) E ξ 2 1/2 2 1/2 ------3 = 2 / 2 + ------12- ≥ --- . E ξ 3 Thus, when the radius of the central particle 2 is not 2 23 sufficiently small, its reduced dipole moment is greater | | | | | | | | Both E2 / E1 and E3 / E1 monotonically increase with than that of the outer particles. Owing to the needle- | | | | a1/a2, and E3 / E2 reaches a minimum when a2 = a1. point effect, the acting field and absorbed power per

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 702 PERMINOV et al. unit volume are greater for the outer particles if their an aggregate. However, nonlinear effects can be more radius is sufficiently small. pronounced either for inner or for outer particles, Analogous results are obtained for the seven-leafed depending on their relative radii. A sufficiently small rosette considered in Section 4: outer particle can be the “locus” of a nonlinear effect. To perform a more detailed calculation, one must spec- r ify the cluster geometry and state a physical problem to d 6 a 3/2 ------2- ==------, q 1.255 -----1 , be solved. r 2 1/2 a d1 ()6 + q + q 2 6. DISCUSSION r d2 () In the 1980s and 1990s, when the principal objective ------r-= 1.497 a1 = a2 , (5.12) d1 was to highlight the properties of fractal clusters as a special optical medium, research was focused on large E 6 systems as whole entities, e.g., on the absorption spec------2 = ------. tra of N-particle aggregates with N ~ 104 and higher. 3 1/2 E1 []() 6 a2/a1 + 1.575 + 1.255 However, disordered, strongly bonded fractal clusters always contain subsystems of closely spaced, nearly Instead of (5.11), we obtain contacting particles, whereas the density of distantly spaced particles is relatively low. This feature distin- ≤≤ a2/a1 0.632, a1/a2 0.659. (5.13) guishes fractal clusters from uniform media. In particular, closely spaced particles determine the width of the Now, consider the endohedrally doped octahedral dipole eigenfrequency range and, therefore, the width cluster with a central particle of radius a2 and identical of the inhomogeneously broadened tail of the plasmon peripheral particles of radius a . If the external field is band and the spectral range where resonant nonlinear 1 effects can be observed. It is shown in Section 4 that parallel to the z axis, then the four particles lying in the |κ | ξ ≈ ≈ xy plane are geometrically equivalent and are called max( m / 0) 4 is reached for N 7 in linear, planar, particles 1, the particle at the origin is referred to as par- and bulk systems; i.e., the number of particles is more ticle 2, and those on the z axis are called particles 3. For important than the aggregate’s geometry. This result this symmetric system, we obtain highlights the dominant role played by the dynamics of dipoleÐdipole interaction of a relatively small number r of monomers, as compared to statistical aspects of the d 12 ------2z = ------() 2.505 , problem. The approach adopted above must be justified r []()3 1/2 ()3/2 with regard to various nonlinear problems. For exam- d1z 12+ 0.543 a1/a2 + 1.248 a1/a2 ple, photomodification of clusters is studied by using electron diffraction patterns to analyze changes in dr ------3z = 2 aggregate structure, and this analysis is not amenable to r statistical averaging of any kind. d1z (5.14) In a widely applied model [1Ð11], the particle radius 2.871 () was assumed to be equal for all monomers in a cluster. Ð ------1/2 1.401 , []12+ 0.543()a /a 3 + 1.248()a /a 3/2 In Sections 3 and 4, it is shown that the eigenfrequen- 1 2 1 2 κ cies m strongly depend on the ratio of radii and the |κ | largest max m corresponds to approximately equal E2z 12 κ ------= ------() 2.505 . radii. Accordingly, the spectrum width (the range of m) 3 1/2 E1z []() 12 a1/a2 + 0.543 + 1.248 is determined by interaction between almost identical contacting particles. The physical explanation of this The numbers in parentheses are the corresponding result lies in the fact that the polarizability of the ratios for a = a . Again, we see that the dipole moment smaller particle decreases as its radius cubed and this 1 2 trend has a stronger effect than the possibility of a of particle 2 is greater than that of particles 1. There- | | | | decrease in interparticle distance. However, the fre- fore, E1z / E2z > 1 only if a1 is sufficiently small as com- κ quency m can vary drastically (as radius cubed) within pared to a2 (cf. (5.11) and (5.13)): κ κ κ the interval (min m, max m). This behavior of m spec- ≤ trum is independent of the aggregate geometry. Thus, a1/a2 0.471. (5.15) inhomogeneous broadening of absorption spectra is explained by variation of both interparticle distance and Thus, the distributions of absorbed power and act- particle radii. ing-field magnitude strongly depend on the number of particles in a cluster, its geometry, and the relative radii A difference in radii has an even stronger effect on of the constituent monomers. A general rule is that the resonant amplitudes, particularly those corresponding r r reduced dipole moment increases toward the center of to difference modes. According to (2.7), di Ð d j is pro-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 ON THE THEORY OF OPTICAL PROPERTIES OF FRACTAL CLUSTERS 703

3/2 3/2 portional to ai Ð a j . Therefore, difference reso- The diversity of factors that affect the eigenvalue nance amplitudes strongly depend on ai/aj, and the res- distributions and resonant amplitudes for disordered clusters and the need for statistical averaging over var- onant modes prohibited when ai = aj can be excited by an external field. Figure 2 illustrates this effect in the ious random parameters suggest that absorption and simplest case of a dimer. refraction spectra must be insensitive to the details of models. Wide and featureless inhomogeneously broad- The distributions of absorbed power Qai, absorbed ened tails of an absorption band merely imply that the power per unit volume Q /a3 , and acting-field mag- particles are strongly bonded in a fractal cluster. To val- ai i idate a model, one should consider experiments that nitude yield “more local” characteristics. Valuable information of this kind may be extracted from nonlinear 1/2 E ∝ []Q /a3 effects, such as creation of “holes” in absorption spec- i ai i tra, experimental methods based on near-field optics, or experiments on a relatively small number of particles. are of primary interest for analysis of nonlinear phenomena. According to the general rule stated in Section 5, the value of Qai increases toward the center of an aggregate ACKNOWLEDGMENTS of monomers of equal size. Moreover, Qai may change We thank R.V. Markov, A.M. Shalagin, and by several times within an aggregate. If the radii of D.A. Shapiro for helpful discussions of a number of peripheral particles are sufficiently small, then the field issues addressed in this paper. This work was supported acting on them is stronger than that acting on “inner” by the Russian Foundation for Basic Research, project particles. This effect is qualitatively consistent with the nos. 02-02-17885, 03-02-06600, and NSh-439.2003.2). electrostatic needlepoint effect. Thus, nonlinear phenomena can be more pronounced either for inner or for outer particles in an aggregate, depending on their rela- REFERENCES tive radii. 1. V. M. Shalaev, Nonlinear Optics of Random Media: | | Fractal Composites and MetalÐDielectric Films The distribution of Ei is essential for scattering that (Springer, Berlin, 2000). is not degenerate with respect to frequency. For exam- 2. V. M. Shalaev, Phys. Rep. 272, 61 (1996). ple, Raman scattering by molecules adsorbed on a | |2| |2 3. V. M. Shalaev, in Optical Properties of Nanostructured nanoparticle surface depends on the product Ei EiR , Random Media, Ed. by V. M. Shalaev (Springer, Berlin, where EiR is the local field at the Stokes frequency [1]. 2002), p. 93. Thus, the strongest effect is attained when both pump 4. V. P. Drachev, S. V. Perminov, S. G. Rautian, and and scattered fields are strong at the point where the V. P. Safonov, in Optical Properties of Nanostructured scattering molecule is located. Since local fields reach Random Media, Ed. by V. M. Shalaev (Springer, Berlin, maximum values at resonant frequencies, the distribu- 2002), p. 113. tion of modes over the particles that make up a cluster 5. U. Kreibig and M. Vollmer, Optical Properties of Metal has to be calculated in the theory of giant scattering. A Clusters (Springer, Berlin, 1995). similar problem arises in studies of nonlinear scatter- 6. S. V. Karpov and V. V. Slabko, Optical and Photophysi- ing. According to Section 3, a dimer with a1 = a2 is cal Properties of Fractal-Structured Sols of Metals (Sib. characterized by a single resonance in the long-wave- Otd. Ross. Akad. Nauk, Novosibirsk, 2003). length absorption tail, whereas another resonance 7. B. M. Smirnov, Physics of the Fractal Clusters (Nauka, appears for particles of different size; i.e., high local Moscow, 1991).

field magnitudes can be obtained simultaneously at two 8. V. M. Shalaev and M. I. Shtokman, Zh. Éksp. Teor. Fiz. frequencies. This example demonstrates that the use of 92, 509 (1987) [Sov. Phys. JETP 65, 287 (1987)]. aggregates of particles of different size offers new pros- pects in nonlinear optics. In other words, particle-size 9. A. V. Butenko, V. M. Shalaev, and M. I. Shtokman, Zh. Éksp. Teor. Fiz. 94, 107 (1988) [Sov. Phys. JETP 67, 60 variability may be introduced into a model as an ele- (1988)]. ment of practical importance rather than a complicating factor that simply cannot be ignored. 10. V. A. Markel’, L. S. Muratov, and M. I. Shtokman, Zh. Éksp. Teor. Fiz. 98, 819 (1990) [Sov. Phys. JETP 71, 455 Thus, optical properties of a cluster strongly depend (1990)]. on the relative sizes of its constituent particles. This 11. V. A. Markel, V. M. Shalaev, E. B. Stechel, et al., Phys. conclusion is supported both by the general analysis Rev. B 53, 2425 (1996). presented in Section 2 and by illustrative numerical cal- 12. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, culations. Effects due to difference in particle size are New York, 1941; Gostekhizdat, Moscow, 1948). so strong that they may change the overall physical pic- 13. L. D. Landau and E. M. Lifshitz, Course of Theoretical ture of optical phenomena in cluster systems, particu- Physics, Vol. 8: Electrodynamics of Continuous Media, larly those involving nonlinear optical properties and 2nd ed. (Nauka, Moscow, 1982; Pergamon Press, local field fluctuations. Oxford, 1984).

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14. V. A. Markel, J. Opt. Soc. Am. B 12, 1783 (1995). 20. P. Clippe, R. Evrard, and A. A. Lucas, Phys. Rev. B 14, 1715 (1976). 15. V. P. Drachev, S. V. Perminov, S. G. Rautian, et al., Zh. Éksp. Teor. Fiz. 121, 1051 (2002) [JETP 94, 901 (2002)]. 21. M. Ausloos, R. Clippe, and A. A. Lucas, Phys. Rev. B 18, 7176 (1978). 16. S. V. Perminov, S. G. Rautian, and V. P. Safonov, Opt. 22. F. R. Gantmakher, Oscillation Matrices (Gostekhizdat, Spektrosk. 95, 447 (2003) [Opt. Spectrosc. 95, 416 Moscow, 1950). (2003)]. 23. E. Fermi, Molecole e Cristalli (Zanichelli, Bologna, 17. E. M. Purcell and C. R. Pennipacker, Astrophys. J. 186, 1934; Barth, Leipzig, 1938; Inostrannaya Literatura, 705 (1973). Moscow, 1947). 18. B. T. Draine, Astrophys. J. 333, 848 (1988). 24. C. Kittel, Introduction to Solid State Physics, 4th ed. (Wiley, New York, 1971; Fizmatgiz, Moscow, 1963). 19. S. V. Karpov, A. K. Popov, S. G. Rautian, et al., Pis’ma Zh. Éksp. Teor. Fiz. 48, 528 (1988) [JETP Lett. 48, 571 (1988)]. Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 705Ð706. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 805Ð807. Original Russian Text Copyright © 2004 by Stanislavsky. ATOMS, SPECTRA, RADIATION

Beam Propagation in a Randomly Inhomogeneous Medium A. A. Stanislavsky Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, 61002 Ukraine e-mail: [email protected] Received October 8, 2003

Abstract—An integrodifferential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the “3/2 law.” © 2004 MAIK “Nauka/Interperiodica”.

Fluctuations of the beam propagation direction in a bility density. In contrast to rotational Brownian randomly inhomogeneous medium are frequently motion, the random walks analyzed here consist of ran- ∆θ observed in nature. Examples include random refrac- dom angle jumps i at points separated by segments of ∆σ tion of radio waves in the ionosphere and solar corona, random length i. Exact knowledge of the distribu- stellar scintillation due to atmospheric inhomogene- tions of these random variables is not required. It is suf- ities, and other phenomena. The propagation of a beam ficient to assume that the angle jumps are independent (of light, radio waves, or sound) in such media can be random variables belonging to the domain of attraction described as a normal diffusion process [1, 2]. One of Gaussian probability distributions. The random seg- extraordinary property predicted for random media— ment lengths ∆σ are also identically distributed inde- and later revealed—is the Anderson localization [3], i pendent random variables, with distribution is charac- which brings normal diffusion to a complete halt. In the α ∆σ context of wave propagation in a random medium, the terized by an exponent . Since i is a nonnegative Anderson localization is caused by interference of quantity, this distribution is totally asymmetric, and 0 < waves resulting from multiple scattering [4]. When two α ≤ 1. Recall that a random variable characterized by a waves propagating in opposite directions along a closed probability distribution f(x) of this kind is described by path are in phase, the resultant wave is more likely to the Laplace transform return to the starting point than propagate in other directions. The properties of a randomly inhomoge- ∞ neous medium vary not only from point to point, but φ() ()() {}()α also with time. Consequently, localization may take s ==∫ exp Ðsx d fx exp Ð Ax , place both at random locations and at random times. 0 Random localization affects diffusive light propagation in a random medium. An approach to describing this where x ≥ 0 and A > 0 [5]. The total path length is the effect is developed in this paper. ∆σ ∆σ ∆θ sum of all i. Both i and i are Markov processes. Suppose that the medium is statistically homoge- However, since the former is the master process with neous and isotropic. Then, a beam propagating through respect to the latter, the resultant process may not pre- the medium is deflected at random. Localization serve the Markov property [6]. Convergence of distri- implies that the beam is trapped in some region. Since butions ensures passing to a continuous limit [7]. This the trapped beam returns to the point where it was leads to the diffusion equation trapped, its propagation is “frozen” for some time. After that, a randomly deflected beam leaves the region and propagates further until it is trapped in another Wα()θσ, Ð Wα()θ, 0 region (or at a point), and the localization cycle repeats. σ ∂ ()θσ, The randomly winding beam path due to inhomogene- D ∂ Wα ' α Ð 1 = ------sinθ------()σσÐ ' dσ', ities is responsible for the random refraction analyzed ∫Γα()sin θ∂θ∂θ in this study. 0 The angle θ of deviation of a beam from its initial direction is characterized by a probability density where D is a diffusion coefficient and Γ(x) is the Wα(θ, σ), where σ is the path traveled by the beam. Let gamma function. The solution to this equation can be us derive an integrodifferential equation for the proba- expressed as an integral transform of the probability

706 STANISLAVSKY distribution associated with rotational Brownian Now, the mean square deviation of the beam from its motion: initial direction can be calculated by combining (1) with (3): ∞ α α ()θσ, () ()θσ, 2 2 2 2σ 1 α Wα = ∫Fα z W1 z dz, ρ ==r Ð z ------Ð ------[]1 Ð Eα()Ð2Dα 3DΓα()+ 1 2D2 0 (5) 1 α where + ------[]1 Ð Eα()Ð6Dσ . 18D2 ∞ ()Ðz k If Dσ is small, then a generalized 3/2 law [8] is obtained: Fα()z = ∑ ------. k!Γ()1 Ð α Ð kα 2 22 1/2 3α/2 k = 0 ρ ≈ ------D σ . (6) Γ()3α + 1 The diffusion equation yields the mean The mean squares given by (1), (3), and (5) increase as α σα σ α cosθ = Eα()Ð2Dσ , at large . The case of = 1 corresponds to normal diffusion without wave localization. Thus, the approach where developed here subsumes classical results of the theory of beam propagation in a randomly inhomogeneous ∞ medium [1, 2]. ()Ðx n Eα()Ðx = ∑ ------Finally, it should be recalled that considerable Γ()1 + nα n = 0 experimental deviations from the 3/2 law (more precisely, from an exponent of 3/2 in the classical power is the Mittag-Lefﬂer function. At large σ, all beam law) were mentioned in [9]. However, they were attrib- directions are equiprobable. However, in contrast to uted to systematic measurement errors, probably normal diffusion (α = 1), a beam has to travel a longer because of the lack of plausible interpretation. This path σ to reach this state. This process is somewhat problem can be revisited in view of the results obtained analogous to “anomalously slow” relaxation. in this study. Moreover, new accurate experimental Following the method developed in [8], one can ﬁnd studies of beam propagation in suitable randomly inho- the mean square of the distance r from the starting point mogeneous media would be extremely useful for veri- to the observation point reached by the beam that has fying the model proposed here. traveled an intricate path of length σ through the medium: REFERENCES α 2 σ 1 α 1. S. M. Rytov, Introduction to Statistical Radiophysics r = ------Ð ------[]1 Ð Eα()Ð2Dσ . (1) (Nauka, Moscow, 1966). DΓα()+ 1 2 2D 2. L. A. Chernov, Wave Propagation in a Random Medium If Dσ Ӷ 1, then (Nauka, Moscow, 1975; McGraw-Hill, New York, 1960). α 2 2α 1 2Dσ 3. P. W. Anderson, Phys. Rev. 109, 1492 (1958). r ≈ 2σ ------Ð.------(2) Γ()2α + 1 Γ()3α + 1 4. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997). If the z axis of a polar coordinate system is aligned with 5. V. M. Zolotarev, One-Dimensional Stable Distributions the initial beam direction, then the mean square of the (Nauka, Moscow, 1983; Am. Math. Soc., Providence, distance passed by the beam along this axis is given by RI, 1986). the formula 6. W. Feller, An Introduction to Probability Theory and Its α Applications, 3rd ed. (Wiley, New York, 1967; Mir, Mos- 2 1 σ 1 α cow, 1984). z = ------Ð.------()1 Ð Eα()Ð6Dσ (3) 3D Γα()+ 1 6D 7. A. Stanislavsky, Phys. Scr. 67, 265 (2003). If Dσ is small, then 8. L. A. Chernov, Zh. Éksp. Teor. Fiz. 24, 210 (1953). 9. I. G. Kolchinskiœ, Astron. Zh. 29, 350 (1952). α 2 2α 1 6Dσ z ≈ 2σ ------Ð.------(4) Γ()2α + 1 Γ()3α + 1 Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 707Ð718. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 808Ð820. Original Russian Text Copyright © 2004 by Babich. PLASMA, GASES

Collision Operator for Relativistic Electrons in a Cold Gas of Atomic Particles L. P. Babich Russian Federal Nuclear Center, Institute of Experimental Physics, Sarov, Nizhegorodskaya oblast, 607188 Russia e-mail: [email protected] Received September 18, 2003

Abstract—The collision operator of relativistic electrons with a cold gas of atomic particles is derived consistently taking into account elastic interactions, excitation of electron shells, and ionization. The creation of secondary electrons is described accurately. In the range of energies exceeding the binding energy of atomic electrons, the operator implicates only the angular scattering by nuclei and the ionization integral that automatically allows for scattering by atomic electrons. The collision operator used earlier for studying the kinetics of avalanches of relativistic runaway electrons is analyzed. A more exact operator derived in the present study is simpler in form and saves time in computer calculations. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Operators (1.2) and (1.3) describe the flux in the momentum space and the angular diffusion due to scat- In collision operators of the kinetic equation (KE), tering from atomic nuclei (the component containing which are reduced to the forms convenient for numeri- factor Z /2 in Eq. (1.3)) and electrons. The creation of cal calculations of transport of relativistic electrons in a mol high-energy electrons is described by ionization inte- substance, infrequent events of creation of high-energy µ electrons are usually ignored. However, Gurevich et al. gral (1.4). Here, f(t, p, ) is the electron distribution [1, 2] proved that such events may change the course of function (EDF) over the momentum modulus p and the an ionization process in the presence of electric field, cosine of the angle between p and unit vector e = ÐE/E in the direction of the electric force, where E is the elec- leading to the development of relativistic runaway elec- ε tron avalanches (RREAs) and to gas breakdown in tric field vector; e is the elementary charge; ion is the weaker fields than those required for the conventional ionization threshold; Nmol is the molecule concentra- α breakdown. The theory of breakdown in air and the tion; Zmol is the number of electrons per molecule; is mechanisms of ascending atmospheric discharges the azimuth angle (see Fig. 1); involving RREAs was developed in terms of the kinetic ∂ ∂ equation [3, 4] ˆ ()µ2 Lµ = ∂µ 1 Ð ∂µ ∂f 1 Ð µ2 ∂ ∂ ----- Ð ------f + µ f eE ∂t p ∂µ ∂ p (1.1) is the angular part of the Laplace operator in the spher- γ β2 β = Stfr ++Stsc Stion ical system of coordinates; = 1/1 Ð ; = v/c; F(p) is the drag force describing the average energy losses of with the following components of the electronÐmole- an electron [5, 6]; σ (ε', ε) is the differential ionization cule collision operator: ion cross section; and ε = (γ Ð 1)mc2 is the kinetic energy. 1 ∂ 2 Primes denote the values of the quantities prior to inter------[],,µ Stfr = 2 p Fp()ftp(), (1.2) action events. p ∂ p The dependence of the characteristic time t of () e Zmol/2+ 1 Fp() St = ------Lˆ µ ftp(),,µ , (1.3) RREA enhancement on eE/Fmin was calculated in [3, 4] sc 4γp by numerically solving Eq. (1.1)Ð(1.4). The discrep- ∞ ancy with the values of te obtained by the Monte Carlo γ'2 Ð 1 method [7, 8] was partly eliminated in subsequent pub- St = N βc dε' ------ion mol ∫ γ 2 lications [8Ð12], where the procedure for solving εε Ð 1 Eq. (1.1)Ð(1.4) was refined. Satisfactory agreement 2 + ion π (1.4) was reached in [10Ð12], where the ionization process was described in the KE in 3D geometry, contrary to 2D ×σ εε, 1 ,,ε µ α ion()' π---∫ ft()' ' d . geometry used in [3, 4], and kinetic equation (1.1)Ð(1.4) 0 was written in divergent form. The remaining discrep-

708 BABICH

i ing interactions is the same as in [3, 4]. In the latter pub- ε e lications, the explicit dependence of cross section ion α on ε' and ε as well as the procedure of its factorization p' were used, although neither of these are necessary. In the gas of molecules consisting of n identical θ atoms, molecular quantities N and Z are connected θ' mol mol p'sinψ ψ with atomic quantities Nat and Zat via the relations N =,nN Z = Z /n. (1.5) k p at mol at mol The operator components describing the excitation and p'sinψsinα ionization contain product NZ, which, in accordance j with relations (1.5), is the same for atoms and molecules. The component responsible for scattering from a nucleus contains a factor with number n if calculations Fig. 1. Scattering geometry: vector e defines the direction of are carried out in terms of molecular quantities (see the electric force, p' and p are the electron momenta before and after the interaction, and ψ is the scattering angle. Section 7): 2 2 NatZat = NmolZmol/n. (1.6) ancy is probably due to the approximations made in [3, 4] in deriving operators (1.2)Ð(1.4). 2. GEOMETRY OF SCATTERING PROCESSES Here, we expound on a consistent derivation of the collision operator in a medium with a uniform external Figure 1 shows the scattering geometry in the coor- electric field, specifying a preferred direction, and dinate system defined by unit vectors describe all approximations made. We assume that 2 2 2 interactions with neutral atoms (molecules) dominate ipep==× []× / p sinθ ()e p Ð pp()⋅ e / p sinθ, and atoms are immobile, so that their energy distribution is insignificant. We use the procedure developed by jpe==[]× / psinθ, k p/ p, Holstein [13], who obtained an exact nonrelativistic operator consisting of rigorous balance components where p(p, θ, ϕ) is the electron momentum vector after responsible for elastic collisions, excitation, and ioniza- scattering [13]. In this system, k is the polar axis and tion. In contrast to [13], in reducing the exact operator scattering angle ψ ∈ [0, π] becomes the polar angle, to the form convenient for calculations, we will not use while angle α ∈ [0, 2π] between j and the momentum the Lorentz approximation, which is valid for weakly projection p'(p', θ', ϕ') onto plane p = 0 before scatter- anisotropic EDFs, but will take advantage of the fact ing is the azimuth angle. Formula (19b) from [13], that interactions with small variations of the direction and magnitude of the momentum dominate in the high- cosθ' = cosθψcos + sin θψαsin cos , (2.1) energy range. For this reason, the parts of the operator responsible for elastic interactions and excitation of which connects angles α and ψ with angles θ' and θ atomic particles can be reduced to differential form. between vector e and the directions of the electron The ionization integral describing the population of an momenta before (p'(p', θ', ϕ')) and after (p(p, θ, ϕ)) element of the phase volume remains nonreduced with scattering, was derived under the assumption that p' = the lower integration limit equal to the accurate value of p, which is violated in inelastic collisions and which is ε + ε as in [13] and not to 2ε + ε as in relation (1.4). ion ion actually superfluous. If we form the scalar product of In contrast to [13], this integral takes into account the the regular decomposition of p' in unit vectors i, j, k, relation between the scattering angle and the electron energies as well as the exact relation between the angles p pe× formed by the electron momentum vectors participat- p' = --- p'cosψ + ------p'sinψαsin ing in ionization event with direction e. p psinθ (2.2) To find the differences with relations (1.2)Ð(1.4), we pep× []× + ------p'cosψαcos , reduced the collision operator to a form similar 2 p sinθ to (1.2)Ð(1.4). For this purpose, we divide the ionization integral into two parts, one of which describes “weak” interactions and the other, “strong” interactions and vector e, result (2.1) is obtained directly without the with an energy change of an impinging electron, such assumption that p' = p. that both electrons after ionization event enter the high- The change in p' is especially large in ionizing inter- energy range (in particular, both electrons can become actions since the energy ε' of a primary electron not ε ε ε runaway electrons). The criterion adopted for separat- only decreases by ion, but the remaining energy ' Ð ion

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 COLLISION OPERATOR FOR RELATIVISTIC ELECTRONS IN A COLD GAS 709 is divided between two free electrons. The allowance p θ θ z for the exact relation between angles ' and is very du important in the problem of multiplication of runaway p electrons since the runaway energy threshold depends on the angle at which an electron moves relative to direction e. This relation determines whether both elec- ' trons (primary and secondary) are in the runway mode du p' ψ or only one of them will be a runway electron. dω dω' 0 py 3. ELASTIC COLLISIONS

In the approximation of symmetry relative vector e, px the EDF depends only on the modulus p of the momentum and angle θ between the directions of p and e. In Fig. 2. Scattering geometry: du' and du are the volume ele- this section, we denote by ε the total relativistic energy. ments in the momentum space before and after the interac- The electron elastic scattering probability per unit time tion, p' and p are the electron momenta before and after the from the phase volume element du'dV to element dudV, interaction, and ψ is the scattering angle. whose element du = p2dpdω is seen from the coordi- ω nate origin in the momentum space at solid angle d other elements du'dV is given by (Fig. 2), is defined as , ,,,ψ ,ω ∫ f ()p' t T el()p' p dp d dVu'd ,,,ψ , ω T el()p' p dp d (3.1) ,,µ ,,,ψ , ω du' v σ , ψ ωδ ϕ εε,,ψ v = dV∫ f() p' ' t Tel()p' p dp d ------du = Nat '()el()p' d ()el()' dp, du (3.5) ,,µ ()σ , ψ ()2 where = dVduNatv ∫ fp()' ' t el()p' p'/p ×δ() ϕ εε,,ψ ω σ σ , ψ ω el()' v'd 'dp'. d el = el()p' d ' µ θ δ ϕ Here, ' = cos '. Function ( el) satisfies the formal is the differential cross section of elastic scattering. relation [14] Delta function δ(ϕ ) takes into account the energy and el δ() momentum conservation laws: p' Ð p1 δϕ()()εε,,' ψ = ------el ∂g ()ε', ψ ε ϕ εε,,ψ ε ε , ψ el d ' el()' ==Ð0.gel()' (3.2) ------∂ε' dp' p' = p 1 (3.6) The formulas describing the scattering of a particle δ()p' Ð p = ------1 -, of mass m by an immobile particle of mass M in the lab- ∂g ()ε', ψ oratory reference frame [14] can be used to derive the el ------∂ε -v' exact expression for the energy transferred to the sec- ' p' = p1 ond particle. This expression is simplified for m Ӷ M as where p is the solution to Eq. (3.2). Evaluating the follows: 1 derivative of gel, we obtain the expression δ()p' Ð p 2 ()ε1 + ξ ' + M δϕ 1 ε' Ð ε = p' ()1 Ð ξ ------. (3.3) ()el = ------2 -, (3.7) ()ε' + M 2 Ð ξ2 p'2 ()p/ p' v' which makes it possible to integrate in Eq. (3.5) with Here, ξ = cosψ and the energy is measured in units of respect to momentum modulus p', the electron rest energy mc2. Since ξ2m2 Ӷ M2, we can v ,,µ ()σ , ψ ()2 replace p'2 in the denominator by ε'2 and simplify rela- dV du Nat ∫ fp()' ' t el()p' p'/p tion (3.3) so that law (3.2) can be written explicitly in δ()p' Ð p the form × ------1-v'dω'dp' ()p/ p' 2v' (3.8) 2 m ()ε' Ð 1 -----()1 Ð ξ M 4 ε ==g ()ε', ψ ε' Ð ------. (3.4) = dV du N v fp()',,µ' t ()σ ()p', ψ ()p'/p dω', el m at ∫ el 1 + ε'-----()1 Ð ξ ω M ' where p' = p1. Using relation (3.3), we obtain a cumber- The total number of transitions to dudV from all some expression for the derivative of gel. A disadvan-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 710 BABICH tage of relation (3.4), which is insigniﬁcant as long as angle θ in a single collision event is small. Expanding ε' Ӷ M, is the independence of the condition for the the right-hand side of operator (3.12) into a series in smallness of the transferred energy, ∆µ, we obtain the following diffusion approximation: ε ε ' Ð Ӷ 1 2π ------ε 1 , ' Ð m ξα Stel() 1 = Natv ∫ d ∫ d on the energy itself: Ð1 0 (3.14) ()m/M ()1 Ð ξ Ӷ 1. ∂f ∂2 f ()∆µ 2 × ------∆µ + ------σ ().p, ψ dµ ∂µ2 2 el Subtracting from relation (3.8) the total number of transitions from dudV to other elements, we obtain the Expanding the left-hand side of relation (2.1) into a following expression for the elastic collision operator: series, we reduce it to the form [ ,,µ ()4()σ , ψ Stel = Natv ∫ fp()' ' t p'/p el()p' 2 2 (3.9) ∆µ= Ð µ()11 Ð ξ + Ð1µ Ð ξ cosα. (3.15) ,,µ ()]σ , ψ ω Ð fp()t el()p d '. It can be seen that small values of ∆µ are realized for Taking into account the smallness of ∆p = p' Ð p as ξ 1. The quadratic form of relation (3.15) is as compared to p', we expand the part of the integrand in follows: expression (3.9) responsible for the population of element dudV, into a series: ()∆µ 2 = µ2()1 Ð ξ 2 Ð 2µ()11 Ð ξ Ð1µ2 Ð ξ2 cosα 4 fp()',,µ' t ()p'/p ()σ ()p', ψ (3.16) el +1()Ð µ2 ()α1 Ð ξ2 cos2 . = fp(),,µ' t ()σ ()p, ψ el (3.10) µ σ ψ α ∈ 1 ∂ 4 Since f( , p, t) and el(p, ) are independent of + ----- fp()',,µ' t ()p' ()σ ()p', ψ ∆p. π 4 ∂ p' el [0, 2 ], we can integrate relations (3.15) and (3.16) p p' = p with respect to this variable: ∆ ∆ε ψ ∆ Expressing p in terms of ( ) = v p and then, in 2π accordance with relation (3.4), ∆ε(ψ) in terms of ε, ∫ ∆µd α = Ð2πµ()1 Ð ξ , (3.17) 2 2 4 m ()ε Ð m c -----()1 Ð ξ 0 M ∆ε() ψ = ------, (3.11) 2π 2 m mc Ð ε-----()1 Ð ξ ()∆µ 2 α πµ2()πξ2 ()µ2 ()ξ2 M ∫ d = 2 1 Ð + 1 Ð 1 Ð (3.18) we transform relation (3.9) as follows: 0 ≈ 2πµ[]2()1 Ð ξ 2 + ()1 Ð µ2 ()1 Ð ξ . ()σ,,µ ,,µ ()ω, ψ Stel = Natv ∫ fp()' t Ð fp()t el()p d ' Substituting these relations into operator (3.14), we obtain ω'

2 2 4 m ()ε Ð m c -----()1 Ð ξ v ∂2 N σ ˆ σ µ2 f at M (3.12) Stel() 1 = Nat---- tr()p Lµ f +,()p ------2 (3.19) + ------4-∫ ------2 ∂µ 2 m p ω mc Ð ε-----()1 Ð ξ ' M where ∂ × ,,µ 4 σ , ψ ω ∂ fp()' t p ()el()p' d '. 1 p σ π ()σξ , ξ ξ tr()p = 2 ∫ 1 Ð el()p d , (3.20) In the coordinates depicted in Fig. 1, an element of the solid angle is given by [13] Ð1 1 dω' ==sinψdψdα Ðdξdα. (3.13) σ π ()ξ 2σ , ξ ξ ()p = 2 ∫ 1 Ð el()p d . (3.21) In the nonrelativistic limit, relation (3.12) can be Ð1 reduced to formula (11) from [13]. Since transport cross section (3.20) is much larger than Further, we will use the conventional procedure cross section (3.21) because ξ 1, we can disregard developed for high energies, when the variation of the second term in Eq. (3.19), which makes it possible

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 COLLISION OPERATOR FOR RELATIVISTIC ELECTRONS IN A COLD GAS 711 to satisfy the condition of conservation of the number The total number of transitions to dudV from all of particles in elastic collisions: other elements du'dV is 1 , ()i , ()i du' µ ∑∫ f ()p' t T ex dVu'd = dV∑∫ f()p' t T ex ------du ∫ Stel() 1 d = 0. (3.22) du i i Ð1 () We continue to expand the second term in rela- = dVduN ∑ fp()',,µ' t v'σ i ()p', ψ dω (4.4) tion (3.12) in ∆µ, neglecting the terms quadratic in at ∫ ex (m/M)(1 Ð ξ)ε as compared to mc2 and taking into i 2 ω account relation (3.17): ×δ() ϕ()i p' dp'd ' ex vdp------. 2 2 ω N m p ∂ 4 p dpd St ≈ ------at------p el() 2 4 ∂ p M m p ∂ ()i ∂ε (3.23) Since gex / ' = 1, the analog of relation (3.7) has the ∂ form × µ,,σ µ f σ f ()pt tr()p Ð.∂µ------()p () δ()p' Ð p δϕ()i ()εε,,' ψ =,------i- (4.5) This operator satisﬁes the condition of conservation of ex v' the number of electrons, so that where pi is the solution to Eq. (4.1), and relation (4.4) ∞ can be reduced to the integral 2 ∫Stel() 2 p dp = 0 (3.24) γ dVd Natv 0 µ ∞ () (4.6) due to the fact that f(p, , t) 0 for p . Since × fp()',,µ' t σ i ()p', ψ ()p'/p 2dω', values of ξ ~ 1 dominate, we disregard the second term ∑∫ ex in relation (3.23) and obtain the following expression i for the total operator of elastic collisions: in which p' = pi. Subtracting the number of transitions ∆ v from dudV and expanding into a power series in p = ≈ σ ˆ µ,, ()i Stel Nat ---- tr()p Lµ f ()pt ∆ε ε 2 /v = ex /v, we obtain the following expression (3.25) for St : 2 ex 1 m p ∂ 4 + ------p σ ()p f ().µ,,pt () 4 ∂ tr [ ,,µ σ i , ψ ()2 p M m p Stex = Natv ∑∫ fp()' t ex ()p' p'/p i ,,µ σ()i , ψ ] ω 4. INELASTIC INTERACTIONS Ð fp()t ex ()p d ' (BOUND-BOUND TRANSITIONS)  (4.7) []σ,,µ ,,µ ()i , ψ ω Let us derive operator Stex responsible for excitation = Natv ∑∫ fp()' t Ð fp()t ex ()p d '  of atomic particles to state (i) with excitation energy i () ε i . In this case, the energy conservation law can be ex ∆ ∂  p 2 ,,µ σ()i , ψ ω written in the form + ------p fp()' t ex ()p d ' . p2 ∂ p  ϕ()i εε,,ψ ε ()i ε , ψ ex ()' ==Ð0,gex ()' (4.1) where Repeating procedure (3.14)–(3.18) for the ﬁrst part of Stex , we obtain ()i ε , ψ ε ε()i gex ()' = ' Ð ex . (4.2) v St = N ----σ , ()p Lµ fp(),,,µ t (4.8) Analogously to relation (3.1), the probability of ex() 1 at 2 ex tr scattering per unit time from du'dV to dudV is given by where we have omitted the term quadratic in (1 Ð ξ) and ()i ,,,ψ , ω T ex ()p' p dp d used the following notation: () () (4.3) v σ i , ψ ωδ() ϕ i εε,,ψ ε σ σ()i = Nat ' ex ()p' d ex ()' d , ex, tr()p = ∑ ex, tr()p where i 1 (4.9) ()i ()i σ σ , ψ ω ()i d ex = ex ()p' d ' πξ()σξ , ξ = ∑2 ∫ d 1 Ð ex ().p is the differential cross section of excitation of state (i). i Ð1

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 712 BABICH Analogously to relation (3.23), taking into account the where relation σ()i ε ,,εψ ()σ ε ,,εψ ()i ε ω () d ion()' = ε', ω'()' iond 'd ' v∆p ==∆ε ε i , ex ()σ ε , ε ()i ()δϕ π ε ω = ε'()' ion ( )/2 d 'd ' we obtain for the second part of Stex is doubly differential (with respect to energy and angle) ()i 1 ∂ 2 ε St = ----- p ionization cross section of shell (i); ion is the ioniza- ex() 2 2∂ p p σ ε ε )()i (4.10) tion threshold of shell (i); and ( ε'( ', )ion is the differ- ∂f ential ionization cross section, which is symmetric rel- × fp(),,µ t F ()p Ð.F ()p µ------ex() 1 ex() 2 ∂µ ε ε ε()i ative to the secondary electron energy s = ( ' Ð ion )/2. The energy and momentum conservation law can be Here, we introduced the friction forces responsible for written in the form inelastic collisions (without ionization): ϕε,,εψ ψ µ ε , ε ()' ==cos Ð0.0()' (5.2) ε()i σ()i Fex() 1 = Nat∑ ex ex (),p (4.11) µ ε ε ε ε ӷ ε()i i For 0( ', ), in the approximation , ' ion , the following expression is valid [5, 14]: 1 ()i ()i σ πσ , ξ ξ 2 ex ()p = 2 ∫ ex ()p d , (4.12) εε() µ2 ε , ε '2+ mc 0()' = ------. (5.3) Ð1 ε'()ε + 2mc2 ε()i σ()i Fex() 2 = Nat∑ ex ex,tr().p (4.13) The total number of electron transitions in ionizing i collisions populating per unit time the phase volume element dudV in the vicinity of energy ε is given by Operators (4.8) and (4.10) satisfy the condition of conservation of the number of electrons. Since the scatter- ξ ,,µ σ ε , ε ()i ωδ ϕ ε ing through small angles dominates ( 1), we have NatdV∑∫ f() p' ' t v'()ε'()' iond ()dud ' () () σ i Ӷ σ i i ex, tr()p ex ()p and, hence, the second term in rela- ∞ 1 tion (4.10) can be neglected. The term proportional to 2 ()i γ' Ð 1 v ε ()σ ε , ε ------ξ Fex(1) is a part of the term describing small variations in = dVduNat ∑ ∫ d ' ∫ ε'()' ion 2 d (5.4) γ Ð 1 the electron momentum (see Section 7). In the RREA i εε()i Ð1 problem, in the atmosphere of the Earth (small values + ion 2π of Z), Stex is superﬂuous since atomic electrons can be δξ µ × ,,µ ()Ð 0 α regarded as free in view of the smallness of the binding ∫ fp()' ' t ------π -d . ()i 2 ε 0 energy (and, the more so, ex ) as compared to the ε energy ' of impinging electrons. BoundÐbound transi- Summation is carried out over all shells. Integrating tions slightly add to the contribution from the collisions with respect to ξ, we obtain the same operator as in transferring atomic electrons to the continuum. Opera- relation (1.4), but with a different lower limit of integra- tor Stex can be used in problems in which the binding tion with respect to ε': energy is comparable to ε'. ∞ ()i St = N v ∑ dε'()σε ()ε', ε 5. IONIZING COLLISIONS ion() 1 at ∫ ' ion i εε()i + ion (5.5) The probability that, as a result of a collision ioniz- 2π ing shell (i), the electron is moved per unit time from γ'2 Ð 1 fp()',,µ' t × ------α------element du'dV in the vicinity of kinetic energy ε' = 2 ∫ d , γ Ð 1 2π mc2(γ' Ð 1) to element dudV in the vicinity of ε = 0 mc2(γ Ð 1) is where, in accordance with relations (2.1), (5.2), and ()i ε ,,εψ ,ε ,ω (5.3), we have T ion()' d d (5.1) σ ε ,,εψ ()i ωδϕε()(),,εψ ε µ µµ µ2 µ2 α = Natv'()ε', ω'()' iond ' d , ' = 0 + 1 Ð10 Ð cos . (5.6)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 COLLISION OPERATOR FOR RELATIVISTIC ELECTRONS IN A COLD GAS 713

The operator describing the departure of electrons from du has the form

()εε()i Ð ion /2 ,,µ ()σ εε, ()i ε Stion() 2 = Natv fp()t ∑ ∫ ε()' iond ' i 0 (5.7) ε ε ε ε ε 0 ' Ð ion ' – ion ,,µ σ()i ε ------s = Natv fp()t ∑ tot(), 2 i Fig. 3. Dependence of the differential ionization cross sec- where tion on the secondary electron energy.

() ()εεÐ i /2 ion Repeating the procedure used in the derivation of elas- σ()i ε ()σ εε, ()i ε tot()= ∫ ε()' iond ' (5.8) tic collision operator (3.19) and taking into account the ξ 0 fact that, in accordance with relation (5.2), = ψ µ cos = 0, we single out from expression (5.5) the is the total ionization cross section of shell (i). In operator expressions (5.5) and (5.7), ε' and ε are transposed in accordance with the fact that St is responsible for () ion(1) εεi 2 + ion populating element du and St is responsible for the 2 ion(2) ()weak ()i p' departure of electrons from du. St = N v ∑ dε'()σε()', ε ---- ion() 1 at ∫ ionp i εε()i + ion 6. WEAK IONIZING COLLISIONS 1 (6.6) × fp()',,µ t + ---()1 Ð µ Lˆ µ fp(),,µ t In relation (5.5), we single out the “weak” interac- 2 0 tions in which the primary electron participating in an 2 2 2µ ∂ f ionization event preserves a large part of its energy; i.e., +1()Ð µ ------, 0 2 ∂µ2 ε' ≈ ε. (6.1)

We will take advantage of the symmetry of where the second and third terms in the brackets are analogous to the terms of operator (3.19), where inte- ()σε, ε ()i ()' s ion with respect to the secondary electron gration with respect to ξ is carried out in analogs of ε ε ε energy s = ( ' Ð ion)/2 (Fig. 3) [6]. Since electrons are relations (3.20) and (3.21). indistinguishable, we assume, for convenience, that the Let us calculate using relation (5.3) the value of secondary electron is the one possessing the lower − µ ∆ε kinetic energy, 1 0 in the approximation of small values of . Here, we take into account for the ﬁrst time the smallness of the ε ≤ ()ε ε s ' Ð ion /2. (6.2) binding energy of atomic electrons since relation (5.3) ε Ӷ ε ε was derived precisely in this approximation: Since the energy range s ( ' Ð ion)/2 dominates in () ()σε()', ε i , expressions (6.1) and (6.2) can be s ion ∂µ2 ∂ε regarded as compatible. Since µ ≈ µ2 ε , ε ≈ 0/ '∆ε 1 Ð 0 1 Ð 0()11' Ð + ------2 µ2 ε εε ε ε∆ε 0 ε' = ε (6.7) ' ==++s ion + , (6.3) 2mc2 ∆ε m∆ε = ------= ------. the following inequality holds for “weak” interactions: ()εε + 2mc2 2 p2 ε ε ε ε ' Ð ion ' + ion ∆ε ε ε ≤ ε ()i = s + ion ------+ ion = ------. (6.4) ε ε ()σε, ε 2 2 Replacing by s in ()' ion , we expand the integrand in expression (6.6) into a power series in ∆ε Substituting this relation into formula (6.3), we obtain and replace the integration with respect to ε' by inte- ε for weak interactions the formula (cf. (1.4)) gration with over s , substituting, by virtue of relation (6.1), ε for ε' in the upper limit of integration with ε ≤ εε ε '2+ ion. (6.5) respect to s, which is equal to (6.2). As a result, we

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 714 BABICH arrive at the differential representation of the operator nent in [3, 4] is due to the fact that the process was con- of weak ionizing collisions: sidered on a plane and the EDF f(p', µ', t) was ﬁxed for two values of µ'; as a result, an opportunity for expand-  ()εε()i Ð ion /2 ing into a series in µ' was missed. ()weak  ()i ,,µ ()σε, ε ε () Stion() 1 = Natv ∑ fp()t ∫ ()s iond s strong  Operator Stion() 1 , which is a part of ionization inte- i  0 gral (5.5) remaining after the replacement of the lower ()i ()εε()i ε ε Ð ion /2 limit of integration over energies by 2 + ion as in [3, 4] ∂ 1 2 ,,µ ∆ε()i ()σε, ε ()i ε (6.8) (see formula (1.4) in the Introduction), is responsible + ----- p fp()t ∫ ()s iond s p2∂ε for strong collisions. 0

() ()εεÐ i /2  ion  7. DESCRIPTION OF INTERACTIONS m ˆ ,,µ ∆ε()i ()σε, ε ()i ε OF ELECTRONS WITH ATOMIC PARTICLES + ------Lµ fp()t ∫ ()s iond s . 2 p2  0  The operators obtained in the previous sections have a large range of application. In this section, we give Using effective drag κ for “large” momentum trans- arguments enabling us to derive simpler and convenient fer [6], we introduce the friction force acting on elec- for numerical calculations representations of operators trons as a result of weak ionizing interactions, for electron energies considerably exceeding the ion- ε κ ization energies for the electron shells of atoms. Fion()= Nat () ()εεÐ i /2 To avoid errors in calculations, it is appropriate to ion (6.9) make the following remark. The literature data for cross ∆ε()i ()σε, ε ()i ε = Nat∑ ∫ ()s iond s, sections should be used bearing in mind that these data α i 0 are frequently given in a form integrated over angle (i.e., contain the factor 2π), while in the formulas which enables us to rewrite relation (6.8) in a more derived here (e.g., (3.14), (3.23), (4.7), and (5.4)), the compact form, integration with respect to α is carried out as well. ()weak σ ε ,,µ Stion() 1 = Natv tot()fp()t ε 1 ∂ 2 Fion() (6.10) 7.1. Elastic Interactions + ----- p F ()ε fp(),,µ t + ------Lˆ µ fp(),,,µ t 2∂ p ion 2γp p Using the Rutherford formula with the Mott factor σ ψ where the total ionization cross section is the sum for el(p, ) [6, 15], we obtain the following expression for transport cross section (3.20): () σ ()ε = ∑σ i ().ε (6.11) tot tot 2 4γ 2 Zate 2 2 i σ ()γ = 2π------2ln------Ð β . (7.1) tr 2 4 2 2ψ It can be seen that the ﬁrst term in relation (6.10) is ()mc ()γ Ð 1 min completely compensated by integral (5.7) Stion(2) responsible for the departure of electrons from the con- ψ 1/3 βγ sidered element du. Here, min = 0.0153Zat / in accordance with the ThomasÐFermi model [15, 16] and integration with Result (6.10) can be obtained by factorizing the ion- α ization cross section in relation (6.6), respect to is performed. In this approximation, we obtain the following dependence of the reciprocal ()σε, ε ()i σ()i ε χ()i ε , ε transport length on γ or ε = γmc2: ()' ion = tot()' (),' s (6.12) ε ε bearing in mind its symmetry relative to and s and π 2 4γ 2 2 taking into account the normalization 4 NatZate 131γβ β N σ ()γ = ------ln------Ð ----- . (7.2) at tr 2 2 2 2 1/3 εε()i ()()γ 2 Ð ion mc Ð 1 Zat χ()i ε , ε ε ∫ ()' s d s = 1. (6.13) 0 In accordance with formula (1.6), expression (7.2) for a gas of molecules consisting of two identical atoms A rigorous reduction procedure has made it possible (n = 2) can be written in terms of molecular quantities: to automatically single out from the ionization integral the differential component responsible for angular scat- 2 2 2 tering by atomic electrons. The absence of this compo- NatZat ==2Nmol()Zmol/2 NmolZmol/2. (7.3)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 COLLISION OPERATOR FOR RELATIVISTIC ELECTRONS IN A COLD GAS 715 ∆ε ≈ ε 7.2. Ionization (Bound-Free Transitions) ondary electron (see formula (6.4), where s) [6],

If electron energies ε' and ε before and after the ε/2 () interaction are considerably higher than the binding κ large = ε ()σε(), ε dε , (7.6) energy of atomic electrons, it is expedient, following ∫ s s ion s ε [3, 4], to use for the ionization cross section 1 ()σ ε , ε ()i ε'()' ion the Moeller formula for the cross section where of electron scattering by free electron, which was ini- ε 2 Ӷ 2 Ӷ 2 tially at rest [5, 6]. The total differential ionization cross 1 = q1/m, I q1/m mc ; section can be obtained by multiplying this formula by Zmol: i.e., I is in the overlap region. The total drag force appearing in the reduced oper- π 4 γ 2 σ ε , ε 2 Zmole ' 1 1 ator (9.1) (see, for example, the problem on p. 383 ε'()' ion = ------Ð ------mc2 γ'2 Ð 1 ε2 εε()' Ð ε in [6]), (7.4) () () ()2ε' + mc2 mc2 1 1 Fp()==()κ small + κ large N F ()p + F ()p (7.7) × ------mol ex ion 2 ++------2 ------2 . ()ε' + mc2 ()ε' Ð ε ()ε' + mc2 is the Bethe drag force F(γ) (see formula (1.2)) [5, 6]: In Section 4, we introduced the force F (p) origi- ex(1) π 4 γ 2 2 4()γγ 2 () nating from excitation processes (see formula (4.11)). γ 2 Zmole Nmol m c Ð 1 Ð 1 F()= ------ln------In reduced operator (6.10), we introduced the force mc2 γ 2 Ð 1 2I2 F (p) associated with weak ionizing interactions. We ion 2 (7.8) will express these forces via the well-known formulas 2 1 1 ()γ Ð 1 + --- Ð ----- ln2 ++------. of quantum electrodynamics. Total inelastic (“ioniza- γ γ 2 γ 2 8γ 2 tion” in the historically adopted terminology) specific losses of electron energy in the medium of particles Since formulas (7.4) and (7.8) were derived under with the mean “ionization” energy I can be calculated ε ε ε ӷ ε ε()i in terms of effective drag κ, including the range of the assumption that ', , s ion, quantity ion should small momentum transfer (in particular, including be omitted in the lower limit of the ionization integral transfer as a result of boundÐbound transitions) and the as well as the summation over index i. The inclusion of () range of large momentum transfer. In the range of ε i leads to an excessive accuracy, while the summa- small momentum transfer, which is defined by the ine- ion quality [6] tion was in fact carried out when the Moeller cross section was multiplied by Zmol. q2/mp= ()' Ð p 2/m Ӷ mc2, 7.3. Excitation Processes the value of κ is calculated as follows (see, for example, (BoundÐBound Transitions) formula (82.20) in monograph [6]): These processes are taken into account via force σ σ ψ 4 2 Fex(1)(p) and cross section ex, tr(p). Since el(p, ) and () 2πZ e γ κ small mol σ (p, ψ) are proportional to Z2 and Z, respectively (in = ------2 ------2 ex mc γ Ð 1 the range of small angles, both cross sections are (7.5) 2 4 2 described by the Rutherford formula [15]), quantity m c ()γ Ð 1 1 σ × ------ex, tr(p) can be neglected for large values of Z. The lnq 2 + -----2 . 2.73I γ quantity Fex(1)(p) appears in the first component of the operator with reduced ionization integral (9.1) and (9.2) This expression includes the processes of excitation via the total Bethe force F(p) (see Section 9). It follows κ(small) and ionization. The product Nmol gives the sum of from expressions (9.3)Ð(9.6) that the component the entire force Fex(p) and a part of force Fion(p). In the responsible for elastic energy losses is much smaller range of large momentum transfer, which is defined by than F(p): the inequality m p2 m p2Z -----N ----- σ ()/p F()γ = ------Γγ()F()/γ F()γ q2/mp= ()' Ð p 2/m ӷ I, M at m tr Mmγvp (7.9) Z m Ð4 the value of κ is calculated as the integral of cross sec- = ------mol------Γγ()≈ 10 . ε A m tion (7.4) multiplied by energy s transferred to the sec- mol nucl

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 716 BABICH ≈ ≈ Here, Zmol/Amol 1/2 is the ratio of the number of elec- m/mnucl 1/1840 is the ratio of the electron and nucleon trons to the number of nucleons in a molecule, masses, and the following notation is used [16]:

()2 2()γγ ()2 2 mc Ð 1 Ð 1  2 1 1 11 ------Ð2  --- Ð ---- - ln ++------1 Ð --- 2I2 γ γ 2 γ 2 8 γ Γγ()= ------. (7.10) ()Ð1/3 γ 2 ln 131Zat Ð 1

Γ ξ In accordance with calculations made in [16], for Stion{ f }, disregarding the terms quadratic in 1 Ð . weakly depends on γ, increasing for air from 0.262 to As a result, we obtain the following representation for 0.271 for electron energies in the range from 51 keV to the total collision operator: 1.53 MeV. In the factor preceding operator Lˆ µ in the St{}fp(),,µ t second component of Eq. (9.2), force Fion(p) can be replaced by the total Bethe force since the main contri- 2 1 ∂ 2m p bution to this force for large values of Z comes from = ----- p F ()p + -----N ----- σ ()p fp(),,µ t 2∂ ex() 1 at tr ionization processes (see the arguments on p. 732 in p p M m monograph [15]). In accordance with formula (9.5), in v the second component of operator (9.2), we have + N ----()σ ()p + σ , ()p Lˆ µ fp(),,µ t at 2 tr ex tr ∞ (8.1) v F 2 σ ӷ ()γ Nat---- tr()p ------, (7.11) ε ()σ ε , ε i ' Ð 1 2 2γp + Natv ∑ ∫ d ' ε'()' ion------γ 2 Ð 1 i εε()i + ion i.e., angular scattering by nuclei dominates over the 2π scattering from atomic electrons. By virtue of rela- dα × ------fp()',,µ' t Ð N v σ ()ε fp().,,µ t tion (7.11), the first component in the total operator ∫ 2π at tot with nonreduced ionization integral (8.2) cannot be dis- 0 regarded (see below). In the RREA kinetics, this com- σ ε ponent is most important, elevating the runaway thresh- Here and below, tot( ) is the total ionization cross sec- old for electrons and reducing the rate of avalanche tion (6.11); cosθ' = µ'(ε', ε, µ, α) as a function of ε', ε, enhancement [7, 8]. µ = cosθ, and α is defined by formulas (5.3) and (5.6); σ σ The operator in form (8.2) with the nonreduced ion- and cross sections tr(p), ex, tr(p) and forces Fex(1)(p), ization integral can be obtained from relation (8.1) due Fion(p) are described by formulas (3.20), (4.9), (4.11), to the fact that the first term, the only one containing and (6.9), respectively. F , can be ignored in the high-energy range since ex(1) The ionization integral in relation (8.1) was derived without any limitations except those imposed by the m p2 F ()p + -----N ----- σ ()p Ӷ Fp(). (7.12) conservation laws. The accuracy of a description is ex() 1 M at m tr determined by the differential cross section and depen- µ ε ε µ ε ε dence 0( ', ). If we use formula (5.3) for 0( ', ) and Indeed, it was mentioned above that excitation pro- Moeller formula (7.4) for the cross section, the ioniza- () cesses make a small contribution to the first component tion integral is limited by the condition ε', ε, ε ӷ ε i . of reduced operator (9.2), which includes the total s ion Bethe force. Since weak ionizing collisions for which To preserve the strictness in problems where the kinet- this component is responsible are contained in the non- ics should be taken into account in the range of energies ε()i reduced ionization integral of total operator (8.1), this on the order of ion , we must use the appropriate set of disregard, subject to relation (7.12), is justified with a ()σ ε , ε ()i µ ε ε high accuracy. (ε'()' ion and the exact formula for 0( ', ) taking into account the coupling of atomic electrons. 8. TOTAL COLLISION OPERATOR The first component of relation (8.1) is not signifi- Let us combine the parts of operator (3.25) for cant while describing the kinetics of high-energy elec- σ Stel{ f }, (4.8) and (4.10) for Stex{ f }, and (5.5) and (5.7) trons. We can also disregard ex, tr(p) in the second com-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 COLLISION OPERATOR FOR RELATIVISTIC ELECTRONS IN A COLD GAS 717 ponent according to the arguments given in Section 7. in accordance with the arguments given in Section 7, As a result, the total operator we obtain the operator

1 ∂ 2 St{}fp(),,µ t = ----- p Fp()fp(),,µ t v 2∂ p St{}fp(),,µ t = N ----σ ()p Lˆ µ fp(),,µ t p at 2 tr v F ()p ∞ σ ion ˆ ,,µ 2 + Nat---- tr()p + ------Lµ fp()t ()i γ' Ð 1 2 2γp (9.2) v ε ()σ ε , ε + Nat ∑ ∫ d ' ε'()' ion------2 - ∞ 2π γ Ð 1 (8.2) 2 i εε()i ()γ α + ion ε σ ε , ε i ' Ð 1 d ,,µ + Natv ∑ ∫ d '()ε'()' ion------∫ ------fp(),' ' t 2π γ 2 Ð 1 2π dα i εε()i 0 × ------fp()',,µ' t Ð N v σ ()ε fp(),,µ t 2 + ion ∫ 2π at tot 0 which differs from relations (1.2)Ð(1.4) only in the second component responsible for angular scattering. The same role is played by operator (1.3) combining the turns out to be simpler than operator (1.2)Ð(1.4). In description of angular scattering by the nucleus and by γ contrast to relations (1.2)Ð(1.4), where processes of atomic electrons in factor (Zmol/2 + 1)F(p)/4 p derived angular scattering by nuclei and atomic electrons are in [3, 4] using the results of analysis of Longmire and ε γ artiﬁcially combined in factor (Zmol/2 + 1)F( )/4 p, the Longley [16]). It should be recalled that angular scatter- scattering by nuclei is described in relation (8.2) by the ing is also contained in ionization integral (1.4) appear- transport cross section, while the scattering by atomic ing in (9.2) as well. electrons remains in the ionization integral. For molecules consisting of two identical atoms (see σ γ relation (7.3)), quantity (v/2)Nat tr( ) in relation (9.2) can be written, in accordance with the results obtained 9. ANALYSIS OF THE TOTAL REDUCED in [16], in terms of Bethe force F(γ): COLLISION OPERATOR () v σ γ v 2 Zmol/2 γ Γγ ----Nat tr()= ------F() () If we use the reduced form of the ionization operator 2 2 mc2()γ 2 Ð 1 with extracted weak interactions, the total operator (9.3) Z assumes the form = ------mol F()γ Γγ(). 2γp St[]fp(),,µ t In report [16], the contribution of the Mott factor is absent. Apparently, it was assumed in [3, 4] that Γ = ∂ 2 1 2m p σ ,,µ 0.25 since relation (9.3) in these publications has the = ----- p Fp()+ -----Nat----- tr()p fp()t p2∂ p M m form (see formula (1.3) in the Introduction) (9.1) v Zmol v Fion()p ----N σ ()γ = ------F().γ (9.4) + N ----()σ ()p + σ , ()p + ------Lˆ µ fp(),,µ t at tr γ at 2 tr ex tr 2γp 2 8 p ∞ 2π If we use this formula and assume that F(p) = Fion(p), 2 ()γ α γ 2 ӷ i ' Ð 1 d which is valid for small values of Zmol and ( Ð 1)mc + N v dε'()σε ()ε', ε ------fp(),',,µ' t at ∑ ∫ ' ion 2 ∫ π ε γ Ð 1 2 ion, max, the quantity i ()i 2εε+ 0 ion v F ()p N ----σ ()p + ------ion - at 2 tr 2γp in which after the separation of weak interactions (6.6) from relation (5.5); the lower limit of integration in the in relation (9.2) will be reduced to the expression remaining part of relation (5.5) describing strong inter- ()ε Zmol/2+ 2 F() ε ε()i ------, (9.5) actions is equal to 2 + ion (cf. relation (1.4)). In rela- 4γp tion (9.1), differing from relation (1.3) in the addend “2” in the numerator; however, this should not strongly affect the Fp()= Fion()p + Fex() 1 ()p results of the solution of the kinetic equation since Zmol/2 is usually much larger than unity. is the Bethe force. A consistent procedure of reducing the ionization integral permitted not only comparison with operator (1.2)Ð Omitting elastic energy losses in the ﬁrst component (1.4), but also made it possible to estimate the contribu- σ of operator (9.1) and ex, tr(p) in the second component tion from the components of the total collision operator

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 718 BABICH

(see Section 7) and, which is important, to considerably ACKNOWLEDGMENTS simplify the total operator by omitting some of these The author is deeply indebted to Acad. A.V. Gurev- components (cf. expressions (8.1) and (8.2)). The sepa- ich and Dr. R.A. Roussel-Dupré for valuable discussions ration of weak ionizing interactions into an individual and helpful recommendations, and to Acad. R.I. Il’kaev, differential operator is inexpedient since this lowers the director of the VNIIEF, and Dr. S.J. Gitomer for their accuracy of the description without providing any com- support in studies in the physics of atmospheric elec- putational advantages. tricity.

10. CONCLUSIONS REFERENCES We have derived the collision operator (8.1) for 1. A. V. Gurevich, G. M. Milikh, and R. Roussel-Dupré, electrons in a dense weakly ionized plasma with pre- Phys. Lett. A 165, 463 (1992). dominant interactions with atomic particles, which 2. A. V. Gurevich, G. M. Milikh, and R. Roussel-Dupré, takes into account the elastic scattering by nuclei, the Phys. Lett. A 187, 197 (1994). excitation of atomic particles, and their ionization. 3. R. A. Roussel-Dupré, A. V. Gurevich, T. Tunnell, and The operator in form (8.2) is intended for describing G. M. Milikh, Kinetic Theory of Runaway Air Break- the kinetics of high-energy electrons, for which the down and the Implications for Lightning Initiation (Los cross sections and arguments given in Section 7 are Alamos National Laboratory, 1993), Report LA-12601- valid. Predominant interactions in relation (8.2) are the MS, p. 51. elastic scattering by nuclei and the ionization of atoms. 4. R. A. Roussel-Dupré, A. V. Gurevich, T. Tunnell, and Operator (8.2) is simpler than operator (1.2)Ð(1.4) used G. M. Milikh, Phys. Rev. E 49, 2257 (1994). earlier [3, 4, 8Ð12] and requires less computer time in 5. H. Bethe and U. Ashkin, in Experimental Nuclear Phys- numerical calculations since it does not contain an ana- ics, Ed. by E. Segré (Wiley, New York, 1953; Inostran- log of component (1.2) separately describing small naya Literatura, Moscow, 1955), Vol. 1, Part 2. variations of the momentum modulus in ionizing colli- 6. V. B. Berestetskiœ, E. M. Lifshitz, and L. P. Pitaevskiœ, sions. Quantum Electrodynamics, 3rd ed. (Nauka, Moscow, 1989; Pergamon Press, Oxford, 1982), Vol. 4. A consistent procedure of separating “weak” interactions from the ionization integral made it possible to 7. L. P. Babich, I. M. Kutsyk, E. N. Donskoy, and obtain the operator component responsible for angular A. Yu. Kudryavtsev, Phys. Lett. A 245, 460 (1998). scattering, which differs from relation (1.3) in the factor 8. E. M. D. Symbalisty, R. A. Roussel-Dupré, L. P. Babich, et al., EOS Trans. Am. Geophys. Union 78, 4760 (1997). ˆ in front of Lµ and in a larger contribution of ionizing 9. E. M. D. Symbalisty, R. A. Roussel-Dupré, and V. Yukhi- collisions to the angular scattering of electrons. muk, IEEE Trans. Plasma Sci. 26, 1575 (1998). Operator (8.1) can be used for describing the kinet- 10. L. P. Babich, E. N. Donskoy, A. Yu. Kudryavtsev, et al., ics of electrons in a wide range of energies considerably Tr. Ross. Fed. Yad. Tsentra VNIIÉF, No. 1, 432 (2001). higher than the excitation energies. Limitations are 11. L. P. Babich, E. N. Donskoy, R. I. Il’kaev, et al., Dokl. imposed by the requirement of smallness of the scatter- Akad. Nauk 379, 606 (2001) [Dokl. Phys. 46, 536 ing angle. (2001)]. Operator (8.2) can be used not only in studies per- 12. L. P. Babich, E. N. Donskoy, I. M. Kutsyk, et al., IEEE taining to the problem of breakdown in planetary atmo- Trans. Plasma Sci. 29, 430 (2001). spheres in thunderstorm ﬁelds, which is only a particular 13. T. Holstein, Phys. Rev. 70, 367 (1946). physical problem in spite of its fundamental importance 14. L. D. Landau and E. M. Lifshitz, The Classical Theory for atmospheric electricity. A KE with operator (8.2) is of Fields, 7th ed. (Nauka, Moscow, 1988; Pergamon applicable in problems of high-energy electron trans- Press, Oxford, 1975), Vol. 2. port through dense gaseous media both in the presence 15. L. D. Landau and E. M. Lifshitz, Course of Theoretical and absence of an electric ﬁeld (the description of the Physics, Vol. 3: Quantum Mechanics: Non-Relativistic electronÐpositron component of cosmic-ray showers; Theory, 4th ed. (Nauka, Moscow, 1989; Pergamon Press, analysis of high-voltage discharges in dense gases with Oxford, 1977). runaway electrons and gas discharges sustained by an 16. C. L. Longmire and H. J. Longley, Improvements in the electron beam, including those intended for pumping Treatment of Compton Current and Air Conductivity in high-power gas lasers; analysis of propagation of rela- EMP Problems (Defense Nuclear Agency, 1973), Report tivistic electron beams in the atmosphere; and descrip- No. 3192T. tion of the kinetics of Compton electrons from a nuclear explosion in the atmosphere). Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 719Ð727. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 821Ð830. Original Russian Text Copyright © 2004 by Bonitz, B. B. Zelener, B. V. Zelener, Manykin, Filinov, Fortov. PLASMA, GASES

Thermodynamics and Correlation Functions of an Ultracold Nonideal Rydberg Plasma M. Bonitza, B. B. Zelenerb,*, B. V. Zelenerb, É. A. Manykinc, V. S. Filinovd, and V. E. Fortovd aLehrstuhl Statistische Physik, Institut für Teoretische Physik und Astrophysik, Christian-Albrechts-Universität Kiel, Kiel, D-24098 Germany bJoint Institute for High Temperatures, Russian Academy of Sciences, Moscow, 125412 Russia c Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia dInstitute of Thermal Physics of Extreme States, Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow, 125412 Russia *e-mail: [email protected] Received October 2, 2003

Abstract—A pseudopotential model is suggested to describe the thermodynamics and correlation functions of an ultracold, strongly nonideal Rydberg plasma. The Monte Carlo method is used to determine the energy, pressure, and correlation functions in the ranges of temperatures T = 0.1Ð10 K and densities n = 10Ð2Ð1016 cmÐ3. For a weakly nonideal plasma, the results closely agree with the Debye asymptotic behavior. For a strongly nonideal plasma, many-particle clusters and a spatial order in the arrangement of plasma electrons and ions have been found to be formed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION An anomalous slowdown of the recombination in the produced plasma was found in [1Ð3]. The recombina- The unique experimental works [1Ð3] have given us µ an insight into some of the physical properties of a tion time was on the order of 100 s. The recombina- dense ionized gas at cryogenic temperatures. Until now, tion time estimated by using a formula valid for a rar- a dense ionized gas has been traditionally produced at eﬁed plasma is several nanoseconds, which is many high temperatures and high pressures. orders of magnitude shorter than the observed value. In [1Ð3], in which a Xe plasma was studied, about Note that the produced plasma is strongly nonideal. × 6 5 10 metastable Xe atoms (the 6S[3/2]2 level, a life- In this plasma, the ratio of the mean potential energy of time of 43 s) were produced, decelerated by using the the particles to their kinetic energy (nonideality param- Zeeman technique, collected in a magneto-optical trap, eter), γ = βe2n1/3 (where β = 1/kT is the inverse temper- λ ≈ and radiatively cooled on the 6S[3/2]2 Ð6P[5/2]3 ( 1 ature, and e is the electron charge), is much larger than 882 nm) transition down to a temperature of 100 µK. unity. Thus, at T = 0.1 K and n = 2 × 109 cmÐ3, γ = 21. The maximum atomic density reached 5 × 1010 cmÐ3; At the same time, the electrons in this plasma are nonde- the density distribution was Gaussian with a root-mean- generate. The ratio of the thermal de Broglie wavelength σ ≈ µ ⑄ square radius of 180 m. e of an electron to the mean particle separation (degen- To produce a plasma, more than 20% of the atoms eracy parameter) at T = 0.1 K and n = 2 × 109 cmÐ3 is λ ≈ were photoionized over 10 ns. First, the 6P[5/2]3 ( 1 much smaller than unity: 882 nm) level was populated, and then the atom was λ ≈ ionized by photons with a wavelength 2 514 nm. The 1/3 1/3 n ប Ð2 difference between the photon energy and the ioniza- n ⑄ ==------0 . 2× 1 0 . (1) ∆ e tion potential, E, was distributed between electrons 2mekT and ions. Because of the small electron-to-ion mass × Ð6∆ ratio, only an energy of 4 10 E was acquired by ប ions, while the remaining energy was acquired by elec- Here, me is the electron mass, and is the Planck con- trons. In [1Ð3], ∆E/k was varied in a controllable way stant. between 0.1 and 1000 K. The maximum charged parti- In a strongly nonideal plasma, the estimates of any cle density was processes based on the formulas obtained in the approximation of γ Ӷ 1 are inapplicable for such non- × 9 Ð3 γ nn==e +210ni cm . ideality parameters .

720 BONITZ et al. 2. THERMAL RELAXATION where

Based on the properties of a nonideal plasma [4], the 3/2 authors of [1Ð3] concluded that thermal equilibrium ()τε T e M ei 0 = ------, sets in several tens of microseconds. It seems to us that 8n z2e4()2πm 1/2 this conclusion is unjustified, because a weakly non- i e ideal plasma with γ Ӷ 1 was considered in [4]. Since ε τ × Ð2 4 γ there is no quantitative kinetic theory for γ ≥ 1, only ()ei 0 = 3.4 10 s at Te = 0.1 K and ni = 10 , z = 1 ( = 10 qualitative estimates can be obtained. In this case, it fol- 0.67, M = 131.3 amu). At Te = 0.1 K and ni = 10 , lows from general physical considerations that the ther- τε mal relaxation time in a strongly coupled system (e.g., at ()ei 0 = 34 ns. fixed temperature T with variation in particle density n) We also assume that the differential scattering cross will be shorter than that in a weakly coupled system. section remains Coulomb, but either corrections to it To estimate the thermal relaxation time in a gas of arise or the integration limits change. However, the electrons and ions, we use a standard expression from [5] dependence of the relaxation time on n at T = const remains logarithmic. for γ Ӷ 1 and nλ3 Ӷ 1: τε The value of ei can change via the substitution of 3/2 ε T M * τ e 1 Le for Le by no more than a factor of 10 to 20. Of ei = ------2 4 1/2-----. (2) 8n z e ()2πm Le course, all of this reasoning must be confirmed by rig- i e orous estimations or numerical calculations. Thus, we may assume that thermal equilibrium at T = 0.1 K and Here, T is the electron temperature, M is the ion mass, e n = 109Ð1010 sets in less than 1 µs. z is the ion charge, ni is the ion density, and Le is the Coulomb logarithm: 3. THEORETICAL APPROACHES TO STUDYING 2 AN ULTRACOLD NONIDEAL RYDBERG aTe 1 ze ӷ Le ==ln ------ln------1 , (3) PLASMA 2 3/2 បv ze πγ Te The Xe plasma produced in the experiments [1Ð3] consists of singly charged Xe ions, electrons, and where vTe is the electron velocity that corresponds to Te, highly excited (n > 100) hydrogen-like Xe states. These and states are called Rydberg atoms. The possibility of the existence of condensed excited states of matter was first 1 considered in [6]. At present, these condensed states of a = ------(4) substance for Rydberg atoms (the so-called Rydberg π β 2 4 ne e substance) have been extensively studied theoretically [7, 8] and experimentally [9Ð11]. Subsequently, is the Debye screening length. this idea has been developed by several authors (see the review [12]). According to this approach, the interac- If the plasma is electrically neutral, then ne = ni. For- tion between Rydberg atoms as their density increases mula (2) contains a Coulomb logarithm Le that has no ultimately leads to a change in the phase state of the γ ≥ physical meaning for 1. The Coulomb logarithm Le system and to a qualitative change in all parameters. arises in this formula when the transport cross section Moreover, in contrast to free Rydberg atoms whose is calculated. The latter is generally determined in the lifetime in an excited state is about 10 ns, the lifetime rarefied case for a ≥ n (where n is the mean density). of a Rydberg substance is macroscopically long. However, it is clear that the transport cross section Despite the low density that is the gas density by its always has a physical meaning and is finite. parameters, the condensed excited state is a metastable ordered state of the substance. At present, there are By analogy, the estimate of the thermal relaxation experimental data [13, 14] for a cesium plasma at T = ε time τ for γ ≥ 1 can be represented by using the char- 500Ð1000 K that suggest the existence of a Rydberg ei substance. acteristic physical quantities as (2); in this case, how- * In [15, 16], the experimental data [13, 14] were ever, Le (the effective Coulomb logarithm that found to correlate with the ideas of an isolated region of includes the collective effects in a nonideal plasma) metastable states of an ultracold nonideal plasma. should be substituted for Le: In [6, 12], the methods of solid-state physics were used to produce a Rydberg substance by assuming the pres- ()τε ence of an electron Fermi liquid. In [17, 18], the recom- τε ei 0 ei = ------, (5) bination time of a dense plasma was calculated numer- Le* ically and was shown to be much longer than that for a

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 THERMODYNAMICS AND CORRELATION FUNCTIONS 721 weakly nonideal plasma. The authors of [19, 20] sug- tion function of a nonideal nondegenerate plasma. It gested a different approach to studying the thermody- involves the Slater partition function namics of a Rydberg substance in the case where the electron gas is nondegenerate. This approach uses pre- ∞ λ3N ψ 2 ()β viously developed methods for a strongly nonideal non- SN = N! ∑ n exp Ð En , (7) degenerate plasma [21]. n = 1 We consider the thermodynamic equilibrium of a where λ is the particle thermal wavelength. nonideal gas of electrons and ions (γ ≥ 1). This gas is peculiar in that there is absolutely no atomic discrete The essence of the pseudopotential model is that the spectrum or there are discrete states with n ≥ n*, where partition function (7) is represented as a product of the n* = 100 or more. Since n* ≥ 100, the states may be said pair electronÐelectron, ionÐion, and electronÐion Slater to be Rydberg ones. Strictly speaking, there is no full partition functions: thermodynamic equilibrium in our case. Therefore, when we talk about thermodynamic equilibrium 2N Ne degrees of freedom, we primarily have in mind the S ==∏ S ∏ S Ne + Ni ij ee translational degrees of freedom. The thermodynamic < < ij= 1 ij= 1 (8) equilibrium of all the remaining degrees of freedom N NN==N (rotational, vibrational, dissociation and chemical reac- i i e × tions, ionization and electron excitation) arises much ∏ Sii ∏ Sei. later. It may be assumed that in easily excited degrees ij< = 1 ij< = 1 of freedom, equilibrium exists at each instant of time, while the slow relaxation processes do not proceed at In the experimental conditions under consideration, all over the period under consideration. We will use this approximation is valid not only for γ < 1, but also results from [21Ð29] to describe this gas. for γ ≥ 1, because there are neither pair nor many-particle bound states in the Xe gas. By analogy with the classical case, product (8) may be substituted with 4. THE PSEUDOPOTENTIAL MODEL AND THE RANGE OF ITS APPLICABILITY  S = expÐβ∑U , (9) The thermodynamics of an equilibrium quantum- Ni + Ne ij mechanical system is completely determined if the par- ij tition function is known: where

lnSij ≈ ()()β ˆ Uij = Ð------(10) Z N Tr exp Ð H β ∞ (6) is the pseudopotential. = ψ 2 exp()ÐβE dq , ∫ ∑ n n N The pair Slater partition functions for electronÐion, N n = 1 V electronÐelectron, and ionÐion interactions can be calculated accurately. The expression for the long-range pseudopotential is identical to the Coulomb law, while the short-range pseudopotential is finite and tempera- where Tr()exp()ÐβHˆ is the trace of the density matrix ture-dependent. β ˆ ψ exp(Ð H ); n(qN) and En are the wave functions and energy levels of N particles, respectively; Hˆ is the 4.1. The ElectronÐIon Pseudopotential Hamiltonian of the N-particle system; qN are the coor- For the interaction of an electron with an ion, the Φ dinates of the N particles; and V is the volume of the pseudopotential ei is defined by the relation system. ()βΦ (), The authors of [25Ð29] developed an approach that exp Ð ei rT = Sei allows the thermodynamic properties of a dense plasma ∞ (11) to be described over a wide range of nonideality and π3/2λ3 Ψ ()2 ()β degeneracy parameters, including the region of strong = 8 ei ∑ α r exp Ð Eα , nonideality and degeneracy. However, a simpler Eα = E0 approach developed in [21Ð24] can be used for a nondegenerate plasma. The authors of these papers sug- where Sei is the two-particle Slater partition function, gested a pseudopotential model to calculate the parti- Eα(r) is the energy of the electron in the field of the ion

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 722 BONITZ et al. βΦ ei if 0 1 β β 2 2 E 'r < e ------λ ------λ -, 3 ei ei where –5 z yz()= 2πÐ1/2∫ exp()Ðt2 dt. 0 –9 The partition function S (r, T) was derived in the × 5 × 6 c 5 10 1 10 quasi-classical approximation for bound states at Eα > r, a0 E' and continuum states. |β 2 λ | Fig. 1. The ionÐelectron pseudopotentials at various tem- Expressions (13) and (14) for e / ei > 1 at r = 0 peratures, T = 1 (1), 0.5 (2), and 0.1 K (3), compared to the take the form Coulomb potentials (solid lines) 2 2 3 2 βe 1/2βe Ð3 βe S 0, ------≈ π ------n exp------, in state α, Ψα(r) is the corresponding wave function, dλ λ 0 3 ei ei n λ λ 0 (15) and ei = e/2 . The summation is over all possible β 2 β 2 states Eα. This pseudopotential is finite at r = 0, while , e ≈ π1/2 e the expression for the long-range pseudopotential is Sc0 ------λ - 2 ------λ -. ei ei identical to the Coulomb law. ≥ For a plasma without bound states up to n = n0, the At n0 100, it will suffice to use (14) and (15) and expression for Sei(r, T) can be written as [21] the hydrogen wave functions and energy levels to deter- Φ Φ mine the potential ei(r, T). When determining ei(r, T) (), Sei rT = Sd + Sc. (12) for the Xe ion, we must also take into account the fact that its size is finite (its crystallographic radius is about Here, 2 Å). Figure 1 shows the electronÐion pseudopotentials at T = 0.1, 0.5, and 1 K and, for comparison, the Cou- E' lomb potentials. π3/2λ3 ψ ()2 ()β Sd = 8 ei ∑ α r exp Ð Eα (13)

Eα = E0 4.2. The ElectronÐElectron and IonÐIon Pseudopotentials is the contribution of the part of the discrete spectrum from E to E', which can be calculated from the known The Slater partition function for the interaction 0 between two electrons is [21] wave functions Ψα(r) for the hydrogen atom, and the contribution of the remaining part of the spectrum is (), See rT β 2 (), e Sc zT = expÐ------, (14a) π3/2λ3 Ψ (),,σ σ 2 ()β r = 16 ee∑ α r 1 2 exp Ð Eα (16) α if ≡βΦexp()Ð ()rT, . β β 2 ee E 'r ≥ e ------λ ------λ -, ei ei The wave functions in expression (16) depend on the σ σ electron spins 1 and 2 and must be antisymmetric. and Expression (16) can be written as a sum of the contributions from the wave functions with symmetric and β 2 β 2 nonsymmetric parts: (), β e e Sc rT = 1 Ð yE ' + ------expÐ------r r 1 c 3 a (14b) S ()rT, = ---S ()rT, + ---S ()rt, 2 1/2 ee 4 ee 4 ee Ð1/2βe (17) π β + 2  E ' + ------, ≡βΦ()(), r exp Ð ee rT .

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 THERMODYNAMICS AND CORRELATION FUNCTIONS 723 Φ βΦ The potential ee(r, T) was numerically calculated ee by Barker [22] over a wide temperature range, T = 102−105 K. He suggested the following fitting formula Φ for ee(r, T): 5 × 107 Φ (), 2()()× Ð4 0.625 ee rT = --- 1Ð exp Ð10 8.35 rT , 3 r (18) 2 Φ (), × Ð4 0.625 ee 0 T = 16.7 10 T , where r is measured in a , T is in Kelvins, and Φ is in 1 0 ee 1 × 107 Rydbergs (Re = 0.5me4/ប2). At long range, formula (18) is identical to the Cou- 0.1 0.2 0.3 lomb law. For βe2/λ > 1, expression (17) at r = 0 can ee r, a0 be written [23] as Fig. 2. The electronÐelectron pseudopotentials at various β 2 π 1/2 β 2 4/3 temperatures, T = 1 (1), 0.5 (2), and 0.1 K (3), compared to , e ≈ 4 e the Coulomb potentials (solid lines). See0 ------λ - ------------λ - ee 3 ee (19) π 1/3 β 2 2/3 ×  π e The Monte Carlo method is a numerical method that --- expÐ3------λ . 2 2 ee uses Markov chains [24]. It allows us to select only the principal, most typical terms that define the integral β 2 λ sum. Therefore, it is also called a method of significant Thus, for e / ee > 1, fit (18) can be used down to T = 0.1 K. selection. Another peculiarity of the method is the use of periodic boundary conditions. The entire three- Figure 2 shows the electronÐelectron pseudopoten- dimensional space is broken down into equal cells of tials at T = 0.1, 0.5, and 1 K and, for comparison, the volume V with N particles in each cell. If one of the par- Coulomb potentials. ticles exits from a cell due to a change in its coordi- According to [21], the expressions for the ionÐion nates, then its image from a neighboring cell simulta- pseudopotentials are identical to the Coulomb law. We neously enters through the opposite cell face, and the must only take into account the fact that the ion size number of particles in the cell is conserved. (e.g., the crystallographic radius) is finite. The errors in the Monte Carlo results [24] are attributable to the choice of the number of particles in the cell 5. CALCULATING and to the finite length of the Markov chain. To estimate THE THERMODYNAMIC QUANTITIES the error in choosing the number of particles, we per- AND CORRELATION FUNCTIONS formed calculations for various N = 16, 32, 64, and 128 ∝ Ð1 5.1. The Method of Calculation and showed convergence (( N )). Our estimate of the statistical error due to the finite length of the Markov In the pseudopotential approach, the quantum parti- chain [24] allowed us to choose Markov chains of the tion function reduces to an expression that is classical required length. In addition, we discarded the nonequi- in form [21]. Therefore, all of the methods developed in librium part. We also calculated the electronÐelectron, the statistical thermodynamics of classical systems gee(r), ionÐion, gii(r), and electronÐion, gei(r), radial (both analytical and numerical) can be used to deter- correlation functions. mine the relevant thermodynamic quantities. We used the Monte Carlo method for a multicom- ponent plasma in a canonical ensemble developed 5.2. Results of the Calculations in [24, 25] to calculate the thermodynamic quantities Thus, we consider the pseudopotential model of an and correlation functions of an ultracold Rydberg ultracold Rydberg plasma. Experimental data suggest plasma. that this plasma consists of electrons, singly charged In this case, determining the various thermody- ions, and atoms in highly excited (n > 100) states. There namic quantities reduces to calculating the mean values are no low-excitation (n < 100) states in this plasma, of the known functions of coordinates q. For example, because it was produced through the laser excitation of for the energy, we obtain atoms at a certain wavelength, and because an anomalous increase in the recombination time was observed Ð1()…,, () (), N in the experiment. EQ= NVT∫ ∫EN q SN aTd q, (20) We performed calculations for the temperature V range T = 0.1Ð10 K and the density range n = 10Ð2Ð where Q(N, V, T) is the path integral. 1016 cmÐ3. The calculations in the range of low densities

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E/NkT P/nkT 700 4 300 2

E/NkT 0 0.2 0.6γ 1 1.4 150 300

0 20 40 60 γ 20 40 60 γ Fig. 3. Internal energy per particle, E/NkT, versus nonideality parameter γ. Fig. 4. Pressure P/nkT versus nonideality parameter γ.

g g 2.0 2.0 (a) (b) 1.5 1.5 1 1 1.0 1.0

2, 3 0.5 0.5 2, 3

0 0 5 × 107 1 × 108 5 × 105 1 × 106 r, a0 r, a0 g g 3.0 (c) 4 (d) 2.0 1 3 1 1, 2, 3 2 1.0 1, 2, 3 1

0 0 3 × 105 5 × 105 1 × 105 2 × 105 r, a r, a 0 g 0 g 12 4 (e) 2 (f) 2, 3

3 1 8 2 3 4 1 1

0 0 5 × 104 1 × 105 3 × 104 6 × 104 r, a0 r, a0 γ Fig. 5. The correlation functions at various densities and temperature T = 0.1 K: gei(r) (1), gee(r) (2), and gii(r) (3); (a) n = 10, = 0.036; (b) n = 107, γ = 3.6; (c) n = 108, γ = 7.7; (d) n = 109, γ = 16.7; (e) n = 1010, γ = 36; and (f) n = 1011, γ = 77.

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20.0 were needed to pass to the limit of the values that are consistent with the Debye–Hückel approximation for γ Ӷ 1 (see, e.g., [21]). 15.0 In Figs. 3 and 4, internal energy E/NkT per particle and pressure P/nkT are plotted against nonideality parameter γ. In the limit of small γ, there is agreement 10.0 1 with the Debye–Hückel approximation (see the inset in Fig. 3). At γ < 0.5, the dimensionless energy E/NkT reaches the Debye value 5.0 2 E 3/2 ------= Ð πγ . (21) 3 NkT 0 500 1000 1500 2000 2500 Figures 5 and 6 show the correlation functions r, a 0 gee(r), gii(r), and gei(r) for various densities and temper- Fig. 6. The correlation functions at density n = 1015 cmÐ3 and atures T = 0.1 and 10 K, respectively. For γ Ӷ 1, there temperature T = 10 K: gei(r) (1), gee(r) (2), and gii(r) (3). is good agreement with the Debye–Hückel approxima-

(a) (b)

(c)

Fig. 7. The graphical images of the particle coordinates that correspond to the correlation functions gee(r), gii(r), and gei(r). The open and ﬁlled circles represent the ions and electrons, respectively: (a) T = 0.1 K, n = 109 cmÐ3, the ions and electrons are at the lattice site; (b) T = 0.1 K, n = 1010 cmÐ3, the ions are at the sites of one lattice, while the electrons are at the sites of the other lattice; (c) T = 10 K, n = 1015 cmÐ3, the ions and electrons form droplets with lattice site nuclei, the transition state between the short-range order and the lattice.

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T, K from this function, we will obtain an expression similar to the standard Madelung law for an ionic crystal [30]:

E 2 1/3 ---- = Ae n , (22) N

10.0 where A = 8Ð9 is a constant (an analogue of the Made- 1.0 lung constant). 0.1 As follows from our calculations, the lattice constant is proportional to nÐ1/3. The form of Eq. (22) sug- 105 1010 1015 gests that an order similar to a crystal lattice is estab- n, cm–3 lished in the ionized gas produced at γ ≥ 1. In this case, the energy is a function of the mean particle separation, Fig. 8. The nÐT diagram. The crosses, squares, and circles represent the gaseous, liquid, and solid (lattice) states of the which is approximately equal to the lattice constant. plasma, respectively. The solid and dashed lines correspond 3 Figure 8 shows an nÐT diagram. The region of to γ = 1 and nλ = 1, respectively. parameters that corresponds to a Debye plasma is indicated by crosses; the regions where droplets and lattices γ ≥ appear are indicated by squares and circles, respec- tion (linearized or nonlinearized). For 1, the shape γ λ3 of the correlation functions suggests that a short-range tively. The = 1 and n = 1 lines are also shown in the figure. We see from this diagram that the formation of a order is formed among particles of both the same and γ ≈ opposite signs. This order is enhanced with increasing γ. short-range order begins only at 1; as was described The maxima of the correlation functions increase, above, the formation of an ordered structures shifts while their minima become zero, which is attributable toward higher particle densities as the temperature to the formation of a strict order in the spatial arrange- increases. In addition, under the given conditions, a ment of particles. There are almost no particles in the long-range order is formed long before the onset of region of zero correlation functions. degeneracy. We used a visualization program to better under- Our results also give us an insight into what the stand the situation related to ordering with increasing γ. authors of [1Ð3] call the anomalously slowed down This program visualizes the arrangement of particles in recombination. There are no Rydberg atoms that must recombine in an ionized gas at γ ≥ 1. However, there is various equilibrium configurations. Figures 7a–7c γ ӷ show some of the equilibrium configurations for vari- a short-range order (and a long-range order at 1) for ous T and n. charged particles of both the same and opposite signs, which reduces the probability of the approach and Let us discuss the results for the T = 0.1 K isotherm. recombination of oppositely charged particles. The order that corresponds to a lattice of size L = 2.2 × 5 9 Ð3 10 a0 at n = 10 cm arises as the density increases (Figs. 5d and 7a). The pairs of electrons and ions are 6. CONCLUSIONS × 5 located at the lattice sites at distance r = 2.2 10 a0. As We have considered a Rydberg ionized gas formed the density increases to n = 1010 cmÐ3 (Figs. 5e and 7b), from continuum electrons and ions. We investigated the the electrons and ions that form the pair move apart, temperature range T = 0.1Ð10 K and the density range and two (electron and ion) lattices are formed. This n = 10Ð2Ð1016 cmÐ3. As a result, we found the formation probably suggests that two nested lattices constituted of a structure at γ ≥ 1, which probably leads to the the initial lattice at the sites of which the pairs were experimentally observed slowdown of the recombina- located. tion. The structure is formed in the region where the As the temperature increases, the formation of an electron gas is far from being degenerate (nλ2 Ӷ 1) and ordered structure shifts toward higher particle densities. where the structure itself changes from a short-range Thus, for T = 10 K and at a much higher density, n = order (similar to the structure in a liquid) to a long- 1015 cmÐ3, only a short-range order in the form of elec- range order (similar to the lattice in solid bodies). Add- tronÐion clusters (as we see from the equilibrium con- ing states of the discrete spectrum to the gas under configuration) shown in Figs. 6 and 7c is established. These sideration will change the properties of this gas. When clusters are droplets of oppositely charged particles, these states are taken into account for specific densities with the electrons and ions in these droplets lining up in and temperatures, the energy decreases, which may minilattices. cause the phase diagram to change. As was noted above, the energy E/NkT at γ ≥ 0.1 in The suggested model contains no specific parame- the range T = 0.1Ð10 K is a linear function of γ (see ters of the elements. Therefore, it may be used for a gas Fig. 3). This implies that, eliminating the temperature of any element.

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ACKNOWLEDGMENTS Matter, Ed. by V. E. Fortov (Inst. Probl. Khim. Fiz. Ross. This work was supported in part by the Russian Foun- Akad. Nauk, Chernogolovka, 2001), p. 110. dation for Basic Research (project nos. 02-02-16320 and 16. G. É. Norman, Pis’ma Zh. Éksp. Teor. Fiz. 73, 13 (2001) 04-02-17474). [JETP Lett. 73, 10 (2001)]. 17. A. N. Tkachev and S. I. Yakovlenko, Kvantovaya Élek- tron. (Moscow) 30, 1077 (2000). REFERENCES 18. A. N. Tkachev and S. I. Yakovlenko, Pis’ma Zh. Éksp. 1. T. C. Killian, S. Kulin, S. D. Bergeson, et al., Phys. Rev. Teor. Fiz. 73, 71 (2001) [JETP Lett. 73, 66 (2001)]. Lett. 83, 4776 (1999). 19. B. B. Zelener, B. V. Zelener, and É. A. Manykin, in Pro- 2. S. Kulin, T. C. Killian, S. D. Bergeson, and S. L. Rolston, ceedings of XVII International Conference on Equations Phys. Rev. Lett. 85, 318 (2000). of State of Substance, Ed. by V. E. Fortov (Inst. Probl. Khim. Fiz. Ross. Akad. Nauk, Chernogolovka, 2002). 3. T. C. Killian, M. J. Lim, S. Kulin, et al., Phys. Rev. Lett. 86, 3759 (2001). 20. V. S. Filinov, E. A. Manykin, B. B. Zelener, and B. V. Zelener, in Proceedings of 12th International 4. L. Spitzer, Jr., Physics of Fully Ionized Gases, 2nd ed. Laser Physics Workshop (Hamburg, 2003). (Wiley, New York, 1962; Mir, Moscow, 1957), p. 35. 21. B. V. Zelener, G. É. Norman, and V. S. Filinov, Perturba- 5. E. M. Lifshitz and L. P. Pitaevskiœ, Physical Kinetics tion Theory and Pseudopotential in Statistical Thermo- (Nauka, Moscow, 1979; Pergamon Press, Oxford, 1981). dynamics (Nauka, Moscow, 1981), p. 101. 6. É. A. Manykin, M. I. Ozhovan, and P. P. Poluéktov, Dokl. 22. A. A. Barker, J. Chem. Phys. 55, 1751 (1971). Akad. Nauk SSSR 260, 1096 (1981) [Sov. Phys. Dokl. 26, 974 (1981)]. 23. V. S. Vorob’ev, G. É. Norman, and V. S. Filinov, Zh. Éksp. Teor. Fiz. 57, 838 (1969) [Sov. Phys. JETP 30, 459 7. É. A. Manykin, M. I. Ozhovan, and P. P. Poluéktov, Zh. (1969)]. Éksp. Teor. Fiz. 84, 442 (1983) [Sov. Phys. JETP 57, 256 (1983)]. 24. V. M. Zamalin, G. É. Norman, and V. S. Filinov, The Monte Carlo Method in Statistical Thermodynamics 8. É. A. Manykin, M. I. Ozhovan, and P. P. Poluéktov, Zh. (Nauka, Moscow, 1977), p. 129. Éksp. Teor. Fiz. 102, 804 (1992) [Sov. Phys. JETP 75, 440 (1992)]. 25. V. S. Filinov, M. Bonitz, W. Ebeling, and V. E. Fortov, Plasma Phys. Controlled Fusion 43, 743 (2001). 9. C. Aman, J. B. C. Pettersson, and L. Holmlid, Chem. Phys. 147, 189 (1990). 26. V. S. Filinov, M. Bonitz, P. Levashov, et al., J. Phys. A: 10. R. S. Svensson, L. Holmlid, and L. Lundgren, J. Appl. Math. Gen. 36, 6069 (2003). Phys. 70, 1489 (1991). 27. V. S. Filinov, V. E. Fortov, M. Bonitz, and P. R. Levashov, 11. C. Aman, J. B. C. Pettersson, H. Lindroth, and L. Holm- Pis’ma Zh. Éksp. Teor. Fiz. 74, 422 (2001) [JETP Lett. lid, J. Mater. Res. 7, 100 (1992). 74, 384 (2001)]. 12. É. A. Manykin, M. I. Ozhovan, and P. P. Poluéktov, 28. V. S. Filinov, V. E. Fortov, and M. Bonitz, Pis’ma Zh. Khim. Fiz. 18, 88 (1999). Éksp. Teor. Fiz. 72, 361 (2000) [JETP Lett. 72, 245 (2000)]. 13. R. Svensson and L. Holmlid, Phys. Rev. Lett. 83, 1739 (1999). 29. V. S. Filinov, V. E. Fortov, M. Bonitz, and D. Kremp, Phys. Lett. 274, 228 (2000). 14. V. I. Yarygin, V. N. Sidel’nikov, I. I. Kasikov, et al., Pis’ma Zh. Éksp. Teor. Fiz. 77, 330 (2003) [JETP Lett. 30. C. Kittel, Introduction to Solid State Physics, 5th ed. 77, 280 (2003)]. (Wiley, New York, 1976; Nauka, Moscow, 1978). 15. G. É. Norman, in Proceedings of XVI International Con- ference on Impact of Intensive Energy Fluxes on the Translated by V. Astakhov

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 728Ð744. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 831Ð849. Original Russian Text Copyright © 2004 by Silant’ev. FLUIDS

A Nonlinear Theory of Turbulent Diffusion N. A. Silant’ev Instituto Nacional de Astrofísica, Óptica y Electrónica, Apartado Postal 51 y 216, CP 72000, Puebla, México Pulkovo Astronomical Observatory, Russian Academy of Sciences, Pulkovskoe shosse 65, St. Petersburg, 196140 Russia e-mail: [email protected] Received May 15, 2003

Abstract—It is shown that the well-known conservation laws for magnetic helicity and passive-scalar fluctuation intensity in the case of negligible molecular diffusion require that the hierarchy of nonlinear equations for the averaged Green function and the hierarchy of BetheÐSalpeter-type equations for fluctuation intensity be treated in a mutually consistent manner. These hierarchies are obtained to the sixth order in turbulent velocity correlators. Asymptotic formulas describing the evolution of scalar fluctuations and magnetic field are presented. A number of new effects are revealed that strongly affect diffusion, but are beyond the scope of the frequently used model of a delta-correlated turbulent field. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION and Diffusion of passive fields, such as particle density, ∂ (), Brt ∇2 (), ∇ × [](), × (), temperature, or magnetic field, is a problem of practical ------Ð Dm Brt = urt Brt . (2) importance in turbulence theory. Passivity means that ∂t the inverse effect of a passive field on turbulent flow can be neglected. This condition is most restrictive as Here, u(r, t) is the turbulent velocity field with pre- applied in the theory of magnetic field diffusion in tur- scribed statistical properties and Dm is molecular diffu- 4 πσ σ bulent plasmas, because turbulent motion may sivity (Dm = c /4 for magnetic field, where is strengthen both large- and small-scale components of a plasma conductivity and c is the speed of light). magnetic field. When the magnetic energy, B2/8π, is The problem of turbulent diffusion essentially comparable to the plasma kinetic energy, ζρu2/2, the reduces to finding evolution equations for the mean effect of magnetic field on turbulence must be taken fields 〈n〉 and 〈B〉 and for the second-order quantities into account. V()r, t; r', t' = 〈〉n()r, t n()r', t' It was argued in [1, 2] that magnetic field fluctuation (3) intensity rapidly grows to a level comparable to turbu- ≡ 〈〉n()1 〈〉n()2 +,〈〉n'1()n'2() lent energy, and therefore magnetic field can almost (), , 〈〉(), (), never be treated as a passive variable. The results of Hij r t; r' t' = Bi r t B j r' t' these studies are analogous to the well-known estimate (4) ≡ 〈〉()〈〉() 〈〉() () for the energy of magnetic field fluctuations in a two- Bi 1 B j 2 + Bi' 1 B'j 2 . dimensional fluid [3]. Computations of time-dependent magnetic fields in turbulent flows [4Ð6] do not support All variables are routinely represented as sums of mean these estimates. In [7, 8], the predictions made in [2, 3] and fluctuating components: n = 〈n〉 + n', B = 〈B〉 + B', was thoroughly analyzed and the estimates for mag- where 〈n'〉 = 0 and 〈B'〉 = 0. Suppose that turbulent flow netic energy fluctuations presented therein were shown can be characterized by an rms velocity fluctuation u0 to be highly overestimated. Thus, the evolution of mag- 2 〈 2 〉 〈 〉 (u0 = u (r, t) , u = 0) and the length and time scales netic field in a turbulent flow involves a period when R and τ associated with two-point correlations. The magnetic field can be treated as passive (particularly in 0 0 assumption that the mean fields are smooth over the the case of zero turbulent helicity). In numerous stud- τ ies, evolution of magnetic field is analyzed in this par- characteristic scales R0 and 0 (or t0 = R0/u0) leads to ticular approximation. diffusion equations for these fields, with diffusivity D + D , where D is a turbulent diffusivity and D ӷ D The starting stochastic equations for passive-scalar m t t t m (e.g., see [9]). Thus, the problem reduces to calculation density n(r, t) and magnetic induction B(r, t) are of Dt. ∂n()r, t 2 ------ÐD ∇ n()r, t = Ð div[]ur(), t n()r, t (1) In the model of turbulence in an unbounded flow ∂t m domain considered here, exact formulas can be

1063-7761/04/9804-0728 $26.00 © 2004 MAIK “Nauka/Interperiodica” A NONLINEAR THEORY OF TURBULENT DIFFUSION 729 obtained for Dt in both Lagrangian [10, 11] and Eule- any Green function by definition. In [20] and other rian [9, 12] representations. Since Eulerian calculations studies, a model was proposed for the evolution of mag- are more suitable from a practical perspective, the netic field fluctuations in a non-δ(τ)-correlated velocity present analysis relies on the Eulerian representation. field. However, the BetheÐSalpeter-type equation derived In this representation, the exact value of Dt is deter- therein is not consistent with conservation law (6). mined by finding the stochastic Green function G(r, t; Therefore, the results obtained in these studies must be r', t') ≡ G(1, 2) of Eq. (1) or (2) and averaging the result incorrect, at least, quantitatively. over the statistical ensemble of the u(r, t) components. Linear stochastic equations (1) and (2) imply that Hereinafter, we use the following convenient abbre- the averaged quantities 〈n〉 and 〈B〉 are related to the viated notation: fluctuations n' and B', and vice versa. Therefore, an attempt to write a separate equation for the averaged () (), ()(), f 1 ==f r1 t1 , f 12Ð f r1 Ð r2 t1 Ð t2 , Green function 〈G(1, 2)〉 of Eqs. (1) and (2) leads to a hierarchy of nonlinear equations for 〈G〉. This situation

dm=== drmdtm, dm drm, Rr1 Ð r2, is completely analogous to the classical closure problem in turbulence theory. τ , … = t1 Ð t2 . It is frequently assumed that the turbulent velocity ensemble is Gaussian; i.e., the averaged product of an The derivation and solution of evolution equations odd number of velocity components vanishes while the for correlators (3) and (4) is a much more complicated averaged product of an even number of components is task than the determination of the mean fields 〈n〉 and expressed as the sum of terms containing all possible 〈B〉. Indeed, it is shown below that correlators (3) two-point correlators. Under this assumption, a Dyson and (4) cannot be fully described in the diffusion equation can be written for 〈G(1, 2)〉 [13]. Formally, it approximation, and complex BetheÐSalpeter-type inte- is a linear integral equation with a kernel expressed as gral (or integrodifferential) equations must be solved the sum of an infinite number of terms describing ele- (e.g., see [13, 14]). Various approximate forms of this mentary interactions between the turbulent flow and the equation have been proposed and analyzed in [15Ð17]. passive field (so-called connected, or irreducible, inter- In this equation, fluctuation intensity is determined by actions). Similarly, one can easily derive a BetheÐSal- the difference of two terms of similar order of magni- peter-type linear integral equation for the averaged tude. On the other hand, Eqs. (1) and (2) written for quantity 〈G(1, 3)G(2, 4)〉, with a kernel also expressed homogeneous turbulence have the following important as the sum of an infinite number of terms corresponding corollaries: to irreducible interactions. Retaining a progressively increasing number of terms in the kernels of these equa- d 2 ----- 〈〉n ()r, t tions, one obtains a hierarchy of linear equations for dt (5) 〈G(1, 2)〉 and 〈G(1, 3)G(2, 4)〉. However, the linear 〈 〉 = Ð 2D 〈〉()∇n()r, t 2 Ð 〈〉n2()r, t divur(), t , equations for G(1, 2) are not very useful. In particular, m retaining the first term in the kernel (in the Bourret ξ d approximation [15]), one can calculate Dt only for 0 = ----- 〈〉AB⋅ = Ð2D 〈〉B ⋅ ()∇ × B . (6) τ Ӷ dt m u0 0/R0 1. This follows already from the fact that the Green function of the Bourret equation tends to that of According Eq. (5), decrease in the mean square of the standard diffusion equation in the diffusion (long passive-scalar density at a fixed point in an incompress- time, long distance) limit, with the diffusivity Dt = ible turbulent flow (divu = 0) can be caused only by 2 τ molecular diffusion. When molecular diffusion is u0 0/3 obtained as the first term of the expansion in ξ neglected, 〈n2(r, t)〉 is a conserved quantity. Accord- terms of 0 in the general theory [9, 12]. ingly, Eqs. (5) and (6) are referred to as conservation A more effective calculation of Dt relies on the hier- laws here. Equation (6) reduces to the well-known con- archy of nonlinear equations for 〈G(1, 2)〉. In particular, servation law for magnetic helicity A á B in a perfectly σ ∞ even the first equation of the hierarchy (direct interac- conducting, homogeneous plasma as and tion approximation equation [16]), which involves a Dm 0 (e.g., see [18, 19]). Here, A is the magnetic quadratic nonlinearity, can be used to calculate D ≡ vector potential: B = ∇ × A. t ()0 ξ 〈 〉 〈 〉 Dt for any value of 0. When the next term (contain- Evolution equations for n(1)n(2) and Bi(1)Bj(2) should be derived under the condition that conservation ing irreducible interaction of the fourth order in veloc- laws (5) and (6) hold; i.e., the aforementioned differ- ity) is retained in the hierarchy, the resulting correction ()1 ence of two terms of similar order of magnitude van- to Dt (denoted here by Dt ) is a negative quantity ishes in an integral sense. Note that the conservation ξ ()0 laws are satisfied automatically in the frequently increasing from zero at 0 = 0 to approximately 0.1Dt ξ ∞ employed approximation of δ(τ)-correlated velocity as 0 [21]. Thus, solving even the first two equa- 〈 〉 ∝ δ δ 〈 〉 field ( ui(1)uj(2) (t1 Ð t2)), since G(R, 0) = (R) for tions of the nonlinear hierarchy for G(1, 2) makes it

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 730 SILANT’EV possible to calculate turbulent diffusivity almost hold only if the averaged Green function 〈G(1, 2)〉 ≡ () () exactly as D = D 0 + D 1 . G0(1, 2) satisfies the hierarchy of nonlinear equations t t t derived in the next section. In particular, this means that The nonlinear equations take into account a much the widely used Bourret equation for the averaged greater number of elementary interactions between the Green function, being a linear one, cannot be applied to turbulent flow and the passive field, as compared to the the evolution of fluctuation intensity, because this leads corresponding ones in the hierarchy of linear Dyson to violation of conservation laws (5) and (6). equations. Mathematically, this implies that the asymp- Note that the simplest equations for V(1, 2) and totic behavior of the averaged Green function at ξ ӷ 1 0 H (1, 2) (where only the ladder diagrams describing is different from that of the Green function of the Bour- ij interactions between passive field and turbulence are ret equation (see details below). This makes it possible retained) require that the function 〈G(1, 2)〉 satisfy the to calculate D for large values of ξ . However, since D t 0 t first equation in the nonlinear hierarchy (the DIA equa- is primarily determined by large-scale turbulent tion). Allowance for interactions of the next order, motion, the accuracy of description of small-scale dif- which are described by a fourth-order irreducible corr- fusion by the first equations of the nonlinear hierarchy elator of u(r, t), requires that 〈G(1, 2)〉 satisfy the next ()0 remains an open question. The expansion Dt = Dt + equation in the hierarchy, which contains fourth-order ()1 correlators; and so on. Thus, conservation laws (5) and Dt + … is an asymptotic series even if it is based on (6) will hold if the hierarchical equations for V(1, 2) the nonlinear hierarchy for the Green function, because and Hij(1, 2) are consistent with the corresponding the number of different interactions allowed for in suc- equations in the nonlinear hierarchy for 〈G(1, 2)〉. It has cessive approximations rapidly increases. (Recall that been pointed out that the BetheÐSalpeter equations may the nth approximation for a Gaussian velocity field con- not be consistent with conservation laws (see [23] and tains (2n Ð 1)!! different terms!) Even though this num- references cited therein). However, integrodifferential ber is somewhat reduced in a nonlinear analysis (see equations for V(1, 2) and Hij(1, 2) consistent with (5) below), a mere improvement of the asymptotic conver- and (6) are derived for the first time in this paper. gence of the series is achieved as a result. The paper is organized as follows. First, a hierarchy In contrast to the hierarchy of linear Dyson equa- of nonlinear equations for the averaged Green function tions, a hierarchy of nonlinear equations can be derived 〈G(1, 2)〉 is derived from Eqs. (1) and (2) simulta- for a non-Gaussian velocity ensemble as well. This pos- neously with increasingly complex expressions for the sibility can be used to analyze the influence of non- fluctuating part G'(1, 2) of the Green function. These Gaussian velocity statistics on the behavior of the aver- results are somewhat similar to those obtained in [12], aged Green function, in particular, on the value of tur- but are obtained by a different method. It is important bulent diffusivity. The nonlinear hierarchy is derived in that the resulting hierarchy of equations is not restricted this paper. It is well known that real turbulence is non- to Gaussian turbulence. Next, correct equations are Gaussian [22]. Furthermore, the nonlinear hierarchy is derived for the correlators V(1, 2) and Hij(1, 2) up to characterized by a reduced number of irreducible inter- sixth-order irreducible velocity correlators. After that, actions in the kernels of the equations. For example, the equations for fluctuation spectra are written out and nonlinear hierarchy contains only one irreducible 〈 2 〉 fourth-order correlator instead of two in the corre- some asymptotic expressions are presented for n (r, t) 〈 2 〉 sponding Dyson equation, four sixth-order correlators and B (r, t) predicted by models of delta-correlated instead of nine, and so on. turbulent field and turbulence with a finite correlation time. (Owing to its relative mathematical simplicity, the 〈 〉 〈 〉 Evolution of n'(1)n'(2) and Bi' (1)B'j (2) is deter- latter model is still widely used to describe turbulent mined by the averaged quantity 〈G'(1, 3)G'(2, 4)〉 (i.e., advection of passive fields, as in [24].) By comparing by the fluctuating parts of the Green function). Since these expressions, it is shown that the delta-correlated ξ Ӷ conservation laws (5) and (6) involve 〈n2(1)〉 = approximation applies only to turbulence with 0 1 〈 〉〈 〉 〈 〉 〈 〉 n(1) n(1) + n'(1)n'(1) and Bi(1)Bj(1) , it is reason- and substantially overestimates the fluctuation intensity able to consider the correlation functions V(1, 2) and for both passive scalar and magnetic field. 〈 〉 Hij(1, 2) defined by (3) and (4). The mean quantities n 〈 〉 and B satisfy diffusion equations and decay over a 2. NONLINEAR EQUATIONS ≈ 2 ≥ short characteristic time t∗ R0 /12Dt. At t t∗, the cor- FOR THE GREEN FUNCTION relators V(1, 2) and Hij(1, 2) represent only the fluctua- The equations for the Green functions G(1, 2) of 〈 〉 〈 〉 tions n'(1)n'(2) and Bi' (1)B'j (2) . Eqs. (1) and (2) have the general form

The present derivation of integrodifferential equa- ∂ 2 ------Ð D ∇ G()12, ≡ L ()1 G()12, tions for V(1, 2) and Hij(1, 2) (resulting in a hierarchy ∂ m 1 0 t1 (7) of BetheÐSalpeter-type equations in the case of a Gaus- sian ensemble) shows that conservation laws (5) and (6) = L()1 G()12, + Iδ()δτR (),

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 731 where L(1)G(1, 2) = Ðdiv[u(1)G(1, 2)] for Eq. (1), in the second term of the first equation with while both Green function and operator L(1) of Eq. (2) are tensors: — =,Ð ᭺ () (), []∇()1 () δ∇()1 () (), which follows from (12); replace — in the inner part of Lik 1 Gkj 12 = k ui 1 Ð ik t ut 1 Gkj 12. the second term in the second equation with τ Recall that R = r1 Ð r2, = t1 Ð t2, summation over δ —= Ð ᭺ , repeated indices is assumed, and Iij = ij. The Green functions vanish at negative τ; i.e., they can be repre- and use (11) and (13) in the resulting expressions to sented as G(1, 2) ≡ θ(τ)g(1, 2), where θ(τ) is the Heavi- obtain the equivalent formulas side step function (θ(τ) = 0 at τ < 0 and θ(τ) = 1 at '= ᭺ Ð 〈〉᭺ , τ > 0). (15) The Green function of the left-hand side of Eq. (7), '= ᭺ Ð 〈〉᭺ . G ()θτ12Ð = ()()4πD τ Ð3/2 exp()ÐR2/4D τ , Adding the averaged Green function G0(1, 2) to m m m these formulas, one finally obtains equivalent equations can be used to write an integral equation for G(1, 2): for the total Green function that do not contain Gm: (), () ()() (), G 12 = Gm 12Ð + ∫d3Gm 13Ð L 3 G 32. = + ᭺ Ð 〈〉᭺ , (8) (16) = + ᭺ Ð 〈〉᭺ . In what follows, diagrammatic notation is used to abbreviate cumbersome equations and demonstrate Equations (15) and (16) play a key role in the deri- their symmetry: vation of a hierarchy of nonlinear equations for (), () 〈G(1, 2)〉 ≡ G (1, 2). Equations (16) written out in literal G 12 = , Gm 12Ð = , 0 form are () ᭺ 〈〉(), ≡ (), L 1 ==, G 12 G0 12 , (), (), G 12 = G0 12 = + . ' [](), () 〈〉(), () (), +3∫d G0 13L 3 Ð G 13L 3 G 32, In this notation, Eq. (8) has the form (17) (), (), G 12 = G0 12 = — + ᭺ . (9) (), []() (), 〈〉() (), A comparison of (8) with (9) shows that the integral +3∫d G 13 L 3 G0 32 Ð L 3 G 32 . is taken over the coordinates of the interaction operator 〈 〉 (circle) while the outer coordinates 1 and 2 are held To verify that G'(1, 2) = 0, by virtue of (15), one fixed. Iterating Eq. (9), one obtains the series must show that

= — ++᭺ ᭺ ᭺ +… . (10) 〈〉〈〉᭺ = ᭺ . (18) Averaging the series over the ensemble of u(r, t) leads to the expressions This is done by inserting (11) into both sides of Eq. (18) and using (13). 〈〉᭺ ≡ 〈〉᭺᭺ = — + — ——+ —. (11) Note that expressions (15)Ð(18) are valid when It is assumed here that the medium as a whole is at rest, 〈u(r, t)〉 ≠ 0 as well. Their derivation follows that pre- 〈 〉 ≡ i.e., 〈u〉 = 0. Applying the operator L to series (10), one sented above for u = 0, where one can set G0(1, 2) obtains the relation G0(1 Ð 2) for stationary homogeneous turbulence. Represent G(1, 2) as a series in powers of L: ᭺ ==᭺ Ð —. (12) Averaging this relation and using (11), one has (), ()()1 (), 2()…, G 12 = G0 12Ð +++G 12 G 12 . — 〈〉〈〉᭺ ==᭺ ——〈〉᭺᭺—. (13) (19) Combining (11) with (12) yields two expressions for Equations (17) entail the recursion relation the ﬂuctuating part of the Green function G'(1, 2) = = Ð : () () ' n (), n Ð 1 (), () () G 12 = ∫d3 G 13L 3 G0 32Ð ∫ ' = ᭺ Ð — 〈〉᭺᭺—, (14) (20) ᭺ 〈〉᭺᭺ ' = Ð — —. n ()nkÐ ()k Ð 1 To eliminate the molecular-diffusion Green function Ð ∑ G ()13, 〈〉L()3 G ()32, . Gm(1Ð2) from these formulas, replace the outer Gm = — k = 2

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 732 SILANT’EV

The expression for G(n)(1, 2) approximates the ﬂuctuat- equation is much simpler and the sixth-order correla- ing part of the Green function and contains n interaction tors can be included: operators L. Using (18), one can readily show that 〈G(n)(1, 2)〉 = 0. The analysis that follows relies on dia- ∂ 2 (n) ∇ (), τ δ()δτ() grammatic representations for the ﬁrst three G in the ∂τ----- Ð Dm G0 R = I R case of 〈u〉 = 0: ()1 + 〈〉᭺᭺ + 〈᭺᭺᭺ 〉 G ()12, = ᭺ , (21)

()2 (), ᭺ ᭺ 〈〉᭺ ᭺ G 12 = Ð , (22) + 〈〉〈〉᭺᭺᭺᭺᭺᭺ ()3 G ()12, = ᭺ ᭺ ᭺ (28) 〈〉᭺᭺ 〈〉〈〉᭺᭺ ᭺ ᭺ Ð 〈〉᭺ ᭺ ᭺ (23) +

Ð 〈〉᭺᭺ ᭺ Ð ᭺ 〈〉᭺᭺ . Substituting (21) into the ﬁrst expression in (11), + ᭺ 〈〉᭺ 〈〉᭺᭺᭺ ᭺ one obtains a nonlinear equation with a quadratic nonlinearity for the averaged Green function, + 〈〉᭺ 〈〉᭺᭺᭺ ᭺ ᭺ . = — + — 〈〉᭺᭺ , (24) known as the DIA equation [16, 25]. Similarly, the sub- Here, the use of nested angle brackets 〈〈…〉〉 and supe- stitution of the sum of (21) and (22) into (11) leads to rior braces is dictated by the complexity of combina- an equation with a cubic nonlinearity: tions of averaged pairs of operators L.

= — ++— 〈〉᭺᭺— 〈〉 ᭺᭺᭺ . (25) If series (10) were averaged directly without proceeding to the derivation of the nonlinear hierarchy, and For a Gaussian velocity ensemble, the three-point corr- the connected (irreducible) terms were singled out in elator in (25) vanishes and (25) reduces to (24). Thus, the averaged series, then the standard Dyson equation Eq. (25) is the lowest order equation that reflects the would be obtained [13]. This equation can be derived non-Gaussian nature of real turbulence in the nonlinear from (28) by replacing every inner G (1, 2) = with hierarchy. 0 the molecular-diffusion Green function Gm(1 – 2) = — The next equation of the hierarchy is obtained by and adding to the right-hand side the fourth-order irre- (1) (2) (3) substituting G'(1, 2) = G + G + G into (11): ducible term = — ++— 〈〉᭺᭺ — 〈〉᭺᭺᭺ 〈〉᭺ 〈〉᭺ ᭺ ᭺ , (29) + — 〈〉᭺᭺᭺ ᭺᭺᭺Ð — 〈〉〈〉᭺᭺ (26) Ð — 〈〉᭺ 〈〉᭺᭺ ᭺ . and five additional sixth-order connected terms, which The next-to-last term in this equation is disconnected. are not written out here. Terms analogous to (29) are The 17 sixth-order terms contained in the next equation already contained in (24), which can be demonstrated of the hierarchy include seven disconnected ones. Thus, by developing an iterative series for this nonlinear the resulting hierarchy of nonlinear equations for a non- equation. Similarly, additional sixth-order terms arise Gaussian ensemble of u(r, t) is not a Dyson equation, as the next equation of the hierarchy, Eq. (26), is iter- because it contains disconnected terms. In the Gaussian ated for a Gaussian velocity ensemble. Thus, the ker- limit, the disconnected (reducible) terms cancel out, nels in nonlinear equations contain fewer connected and a nonlinear analogue of the hierarchy of Dyson terms. equations is obtained. The connectedness of terms in (28) ensures that the Applying the operator L0(1) defined by (7) to kernels are narrowly supported. Therefore, the outer Eq. (26), one obtains a nonlinear integrodifferential Green function can be expanded into a Taylor series equation for G0(1 Ð 2): in coordinates and time, and the first terms of the resulting expansion can be used to obtain a diffusion approx- ∂ () ()≡ ∇2 (), τ imation: L0 1 G0 12Ð ∂τ----- Ð Dm G0 R ∂ 2 δ()δτ() 〈᭺᭺ 〉 (27) ∇ (), τ = I R + ∂τ----- Ð Dm G0 R (30) + 〈〉〈〉᭺᭺᭺ + ᭺᭺᭺ ᭺ = Iδ()δτR ()+ D ()∇τ 2G ()R, τ . Ð 〈〉〈〉᭺᭺ ᭺᭺ Ð 〈〉᭺ 〈〉᭺᭺ ᭺ . t 0 In the case of a Gaussian velocity ensemble, this Retaining only the first equation in the hierarchy (with

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 733 quadratic nonlinearity), one obtains density distributions. The ensemble is generally assumed to be homogeneous and isotropic, in which τ | | case V0(3, 4) = V0( 3 Ð 4 ). The absolute and variable ()0 ()τ 1 τ[ 〈〉() () 〈 〉〈 〉 Dt = ---∫dR∫d ui 1 ui 2 terms in (33) are n(1) n(2) and the fluctuation corre- 3 (31) lator 〈n'(1)n'(2)〉, respectively. Equation (33) can 0 readily be used to write an integral equation for ⋅]〈〉() () (), τ Ð Ru1 divu 2 G0 R . 〈n'(1)n'(2)〉, which is more complex than (33). This equation is not employed here. In the diffusion approximation, only the first term The analysis is simplified by applying the operator should be retained in the Taylor series expansion with L (1) or L (2) to Eq. (33) and using an equation for the respect to time, i.e., the value of the outer Green func- 0 0 tion at the point of the kernel’s maximum on the time averaged Green function (Eq. (28) in the present con- axis should be factored out (see details in [26]). An text) to obtain an integrodifferential equation for V(1, 2) ()1 independent of V0(3, 4). expression for Dt (contribution of the fourth-order The simplest form (ladder approximation) of nonlinear terms in the hierarchy) was given in [27]. The Eq. (33) is obtained when only the first connected term () () accuracy of the expressions for D 0 and G 1 is dis- is retained in the kernel. In the case of a δ(τ)-correlated t t field, the ladder approximation yields an exact solution, ()0 α cussed in Introduction. Expressions for Dt and the - because the contribution of remaining connected terms () δ τ effect factor α 0 for magnetic field diffusion can be to (33) vanishes (e.g., see [17]). In the non- ( )-corre- t lated case, this equation is written as found in [26]. (), (), (), (), V 12 = ∫d34∫d G0 1; 3 0 G0 2; 4 0 V 0 34 3. EQUATIONS (), (), (34) FOR THE CORRELATORS V(1, 2) AND Hij(1, 2) +34∫d ∫d G0 13G0 24 Suppose that the distribution of passive-scalar den- ×∇()3 ∇()4 〈〉() () (), sity in a turbulent medium at the initial moment t = 0 is i j ui 3 u j 4 V 34. n0(r). When the Green function G(r, t; r', t') of Eq. (1) is known, the statistical properties of n(r, t) are deter- To abbreviate diagrammatic representation of more mined by the expression complicated equations, an additional symbol is defined by writing Eq. (34) as n()r, t ≡ 〈〉n()r, t + n'()r, t (32) = + 〈〉᭺᭺ . (35) (),, () = ∫dr'G r t; r'0n0 r' . Henceforth, it should be borne in mind that coordi- An analogous expression relates the magnetic field nates 1 and 2 correspond to the extreme left and extreme right elements of a diagram, respectively. The B(r, t) to its initial distribution B0(r). Using (32), one can easily write expressions for operators on the right of a box (which represents V(1, 2) = 〈n(1)n(2)〉 and H (1, 2). The stochastic V(3, 4)) act on the terms on its left, and time decreases ij toward the box. The box crossed by a vertical line rep- Green function G(1, 2) is given by two expressions: resents V (3, 4). series (10) in molecular-diffusion Green function 0 When operator L (1) or L (2) is applied to (35), one Gm(1 Ð 2) and series (19) in the averaged Green func- 0 0 has to determine how many terms should be retained in tion G0(1, 2). For the commonly used Gaussian ensemble of u(r, t), the averaging of the expressions for Eq. (28). It was noted in Introduction that conservation n(1)n(2) and B (1)B (2) leads to a hierarchy of BetheÐ law (5) will hold only if the orders of the nonlinear i j ≡ Salpeter-type linear integral equations (see [13]) differ- equation for the averaged Green function G0 and ing only by the number of connected (irreducible) in the obtained integrodifferential equation for V(1, 2) terms retained in their kernels. These equations have are matched. This implies that only the first integral the following general form: term must be retained in (28), i.e., the analysis must be restricted to the DIA equation for G0(1, 2). The result- (), (), (), (), ing equations are V 12 = ∫d34∫d G0 1; 3 0 G0 2; 4 0 V 0 34

()〈〉᭺᭺ 〈〉᭺᭺ (), (), (33) L0 1 = + , (36) +∫d 3456∫d ∫d ∫d G0 13G0 24

() 〈〉᭺᭺ 〈〉᭺᭺ × K()34;,, 56V()56, , L0 2 = + . (37) 〈〈 〉〉 where V0(3, 4) = n0(r3)n0(r4) . Here, double brackets By virtue of the symmetry V(1, 2) = V(2, 1), Eq. (37) denote averaging over the ensemble of initial particle- is obtained from Eq. (36) by interchanging the points

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 734 SILANT’EV indexed by 1 and 2, which means inversion of dia- on the right-hand side of (36) is less than the second grams. This is true in any approximation. one, and vice versa. Equations (36), (38), and others The terms on the right-hand side of (36) can be obtained here ensure that conservation law (5) is con- obtained from the first diagram on the right-hand side sistent with each equation in the BetheÐSalpeter hierar- of (28) by successively replacing one of the averaged chy and the spurious effects are ruled out. However, it Green functions G0(1, 2) with V(1, 2). A direct verifica- remains unclear if these equations can adequately tion shows that this rule applies to higher order equa- describe the spectral distribution of scalar field fluc- tions in the hierarchy of BetheÐSalpeter equations as tuations. well. In particular, allowing for fourth-order irreducible To show that Eq. (36) (the simplest one in the hier- interactions (see (28)), one obtains archy) is consistent with conservation law (5), set 1 = 2 (i.e., r1 = r2 = r and t1 = t2 = t) and divu = 0. Then, the () 〈᭺᭺ 〉 〈〉᭺᭺ L0 1 = + first diagram on the right-hand side can be represented as

()1 ᭺᭺᭺ ᭺ ᭺᭺᭺ ᭺ []∇ ()〈〉() () + 〈〉+ 〈〉 (38) ∫d3 i G0 13Ð ui 1 u j 3 ×∇()3 (),, j V 13Ð t3 t1 + 〈〉᭺᭺᭺ ᭺ ᭺ + 〈〉᭺ ᭺ ᭺ . ∇()1 ()〈〉() () = i ∫d3G0 13Ð ui 1 u j 3 The equation containing the sixth-order correlators (39) has an analogous form. Since Eq. (28) has four sixth- ×∇()3 (),, j V 13Ð t3 t1 order terms with six Green functions G0 in each, the total number of sixth-order terms in (38) is 24. This ()〈〉() () Ð3∫d G0 13Ð ui 1 u j 3 equation is not written out here, because even its dia- () () grammatic representation is too cumbersome. Note ×∇1 ∇ 3 V()13Ð ,,t t . also that the nonlinear equations for G (1, 2) again con- i j 3 1 0 〈 〉 ≡ tain fewer connected terms, as compared to the BetheÐ Since the correlator ui(1)uj(3) Bij(1 Ð 3) depends on Salpeter equations derived directly from series (10) the difference of the arguments, the integral in the first (see [13]). However, this reduction starts only from the term on the right-hand side of (39) is independent of r1 sixth-order correlators: the 26 connected terms con- and therefore vanishes. The second term is obviously tained in the linear theory [13] reduce to 20. Each elim- equal to minus the second diagram in (36) (since the inated term has the form latter is invariant under inversion if 1 = 2). Thus, the right-hand side of Eq. (36) vanishes when 1 = 2. In 〈〉〈〉᭺ ᭺ ᭺ ᭺᭺ ᭺ ; other words, the extreme left operator L (circle) can be placed at the extreme right position in the diagram i.e., it contains a disconnected element inside a pair of (with opposite sign). averaging brackets. Note that the BetheÐSalpeter equa- Similarly, the first fourth-order term in (38) cancels tions are commonly represented as “two-level” dia- out with the last one, and the second and third terms grams (see [13, 17]). The diagrammatic expressions cancel out as well. When 1 = 2, the terms of the equa- presented here are more compact, and their symmetry tion containing 24 sixth-order correlators also cancel properties are demonstrated more explicitly. In particu- out pairwise; i.e., conservation law (5) holds for these lar, this representation is consistent with the rule of suc- equations as well. cessive replacement of terms in the BetheÐSalpeter In [20] and other studies, Eq. (35) was considered 〈 〉 equation with V(1, 2) (see Eq. (38)). Of course, these for the magnetic field correlator Hij(1, 2) = Bi(1)Bj(2) , equations can readily be recast in the standard two-level but the averaged Green function was assumed to satisfy form. the Bourret equation The evolution of 〈n2(r, t)〉 ≡ V(1, 1) is of primary = — + — 〈〉᭺ ᭺ , (40) interest. The derivative d〈n2(r, t)〉/dt is obtained by adding Eqs. (36) and (37) and setting t1 = t2 = t. This makes rather than nonlinear Eq. (24), which would be correct the right-hand sides of (36) and (37) equal, and twice in the ladder approximation. This led to the equation the right-hand side of (36) (or (38) in the next equation L ()〈〉1 = ᭺ ᭺ + 〈〉᭺᭺ , (41) of the hierarchy) can be taken. By virtue of conserva- 0 tion law (5), the right-hand side of the equation for which is inconsistent with conservation law (6), instead 〈 2 〉 d n (r, t) /dt must vanish if Dm = 0 for incompressible of (36) (here, boxes represent correlators Hij). Therefore, turbulence. It is important that the particle distribution the quantitative results obtained in [20] are not valid. must also be statistically homogeneous, i.e., V(1, 2) = The foregoing analysis shows that conservation V(R, t1, t2). Violation of (5) would lead to spurious laws (5) and (6) ensure that the hierarchy of BetheÐSal- effects: 〈n2(r, t)〉 would rapidly increase if the first term peter equations is correct if the orders of the nonlinear

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 735 equation for the averaged Green function G0(1 Ð 2) and only for Eq. (36) (with a quadratic nonlinearity), the corresponding BetheÐSalpeter equation are equal. because of the complexity of further calculations. One may use any plausible expression for the averaged Green function (e.g., a diffusion Green function) in properly structured Eqs. (36) and (38). The conserva- 4. EQUATIONS FOR PASSIVE-SCALAR SPECTRA tion laws will hold. It is important that the expressions Restricting analysis to the case of a Gaussian veloc- for the averaged Green functions in the kernels of the ity ensemble, consider first Eqs. (24) and (36) (with equations be identical, whereas different expressions quadratic nonlinearity) and then Eqs. (28) and (38) (of are contained in (41). Equation (41) is equivalent ∇()1 to (36) only in the model of delta-correlated turbulent fourth order in L(1) = Ð i ui(1)). The spectrum of the 〈 〉 ∝ δ field ( ui(1)uj(2) (t1 Ð t2)), when the Green functions correlator V(1, 2) = V(R; t1, t2) is related to the Fourier in the first terms on the right-hand sides of these equa- integral tions reduce to δ(R). 1 ipR⋅ VR(); t , t = ------dpe V˜ ()p; t , t (43) The analysis presented above made use of the com- 1 2 ()π 3∫ 1 2 mon assumption of Gaussian turbulent-velocity ensem- 2 ble. Real turbulent fields are not Gaussian [22]. The as follows: nonlinear equations for the averaged Green function ∞ G0(1, 2) (see Eqs. (26) and (27)) and the expressions for (), (), the fluctuating part G'(1, 2) of the Green function V 0; t1 t2 = ∫dpEV p; t1 t2 , (see (21)Ð(23)) are obtained here for an arbitrary veloc- 0 ity ensemble. Using (21)Ð(23), one can easily verify (44) 〈 〉 2 that G'(1, 3)G'(2, 4) cannot be decomposed into con- (), p ˜ (), EV p; t1 t2 = ------V p; t1 t2 . nected and disconnected parts. For example, the terms 2π2 of fourth order in L do not involve the corresponding fourth-order ladder diagram. Thus, no equation of Therefore, it is sufficient to analyze the equations BetheÐSalpeter type can be written. One can only for V˜ (p; t , t ). directly use the approximate expressions for G'(1, 2) 1 2 given by (21)Ð(23) to write down an approximate series A stationary, homogeneous, and isotropic turbulent expansion for 〈n(1)n(2)〉 = V(1, 2). Applying the opera- velocity ensemble is characterized by the correlator 〈 〉 τ tor L0(1) or L0(2) to the series and making use of (27), Bij(R, t) = ui(1)uj(2) (R = r1 Ð r2, = t1 Ð t2), and its one obtains Fourier transform has the form [28] 2 ᭺᭺ ᭺᭺ ˜ L ()〈〉1 = + 〈〉 Bkj()p, τ = () δp Ð p p fp(), τ 0 kj k j (45) 〈〉〈〉᭺᭺᭺ ᭺᭺᭺ (), τ (), τ + + (42) + pk p jWp + iekjt pt Dp , + 〈〉…᭺ ᭺᭺ + . where ekjt is the Levi-Civita permutation symbol (exyz = τ τ τ Here, the ten fourth-order terms are not written out. In Ðeyxz = 1 etc.). The functions f(p, ), W(p, ), and D(p, ) contrast to the Gaussian case, these terms do not satisfy characterize turbulence spectra: the rule of successive replacement of one of the aver- ∞ aged Green functions in (27) with the initial correlator 〈〉ur(), t ⋅ ur(), t + τ = ∫dpE() p, τ , V0(1 Ð 2) = . The method applied to Eq. (39) can readily be used here to show that expression (42) is con- 0 sistent with conservation law (5). Note that the third (), τ (), τ (), τ Ep = Einc p + Ecompr p , term on the right-hand side of (42) vanishes when 1 = (46) 2. As in the Gaussian case, the order of expression (42) (), τ 4 (), τ π2 Einc p = p fp / , equals that of the equation for G0(1 Ð 2). (), τ 4 (), τ π2 Once again, recall that the nonlinearity of (27) Ecompr p = p Wp /2 . ensures good asymptotic convergence of (42) even τ when the parameter ξ = u τ /R is large. Equation (42) For incompressible turbulent flow, W(p, ) = 0. The helic- 0 0 0 0 τ can be used as a basis for analyzing non-Gaussian ity spectrum is characterized by the function D(p, ): behavior of the correlation functions V(1, 2) and Hij if ∞ an expression for the three-point velocity correlator is H()τ ≡ 〈〉ur()∇, t ⋅ ()× ur(), t + τ = dpE ()p, τ , known or prescribed. ∫ h 0 All results obtained in this section can be applied to (47) 〈 〉 (), τ 4 (), τ π2 describe the correlator Hij(1, 2) = Bi(1)Bj(2) , in which Eh p = Ðp Dp / . case both operator L(1) and Green functions are ten- ˜ sors. Magnetic-helicity conservation (6) was verified The equation for V (p, t1, t2) in the ladder approxi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 736 SILANT’EV mation is the Fourier transform of Eq. (36): where µ is the cosine of the angle between the vectors p and q. For an incompressible flow (divu = 0), ∂ 2 1 τ ≡ ------+ D p V˜ ()pt,,t = ------Ecompr(p, ) 0 and Eq. (49) reduces to conservation ∂ m 1 2 3 t1 ()2π law (5). The second term on the right-hand side of (49) is an approximate expression for the term with divu t1 in (5) that corresponds to the ladder approximation of × ˜ (), () ∫dq Ð∫dt' pi Bij q t1 Ð t' pqÐ j the BetheÐSalpeter equation. The term containing divu (48) in (5) is responsible for particle clustering [14], i.e., for- 0 mation of regions of excessively high and low particle × (), ˜ (),, g˜ pqÐ t1 Ð t' V pt' t2 density in a compressible turbulent flow. In the frequently used model of delta-correlated tur- t2 ˜ τ τ δ τ ˜ ˜ (), (), ˜ (),, bulent field (Bij (p, ) = 0 ( )(Bij p)), Eq. (48) is exact + ∫dt' pi Bij q t1 Ð t' p jg˜ pt2 Ð t' V pqÐ t1 t' , since the remaining equations of the hierarchy vanish. 0 In this model, a closed time-dependent equation can be ˜ ˜ where the representation G0 (p, τ) ≡ θ(τ)(g˜ p, τ) is used written for V (p, t, t): (θ(τ) is the Heaviside step function, see the beginning ∂V˜ ()ptt,, ()0 2 of Section 2). It can readily be shown that (48) does ------2= Ð ()D + D p V˜ ()ptt,, ˜ ∂t m t not contain turbulent helicity; i.e., Bij can be treated as a symmetric tensor. Equation (48) holds for ∞ 1 ˜ ˜ ˜ ˜ 1 2 2 2 L0 (p, t )(V p, t , t ) ≡ L0 (p, t )(V p, t , t ) (see (37)), τ µ[]()µ () µ () 2 1 2 2 2 1 + --- p 0∫dq ∫ d 1 Ð Einc q + 2 Ecompr q (50) under the change t t and t t . 2 1 2 2 1 0 Ð1 By virtue of conservation law (5), the integral of the right-hand side of (48) over p must vanish if divu = 0. × V˜ ()pqÐ ,,tt. This can readily be verified. It may seem that Eq. (48) () can be analyzed in a diffusion approximation under the The diffusivity D 0 given by (31) can also be written as Ӷ ≈ ӷ τ ӷ τ t assumption that p p0 1/R0, t1 0, and t2 0 by ∞ analogy with Eq. (28) for the averaged Green function. t  ()0 () 1 τ (), τ However, the resulting equation is integrodifferential Dt t = ---∫d ∫dp[Einc p --- 3  because the second term contains V˜ (q, t , t'). More 0 0 1 (51) importantly, this equation is not consistent with conser- ∂ (), τ  vation law (5). ()], τ (), τ (), τ g˜ p + Ecompr p g˜ p + Ecompr p p------. The use of the Bourret equation (see (41)) is equiv- ∂p  | | alent to replacing g˜ ( p Ð q , t1 Ð t') in (48) with the molecular-diffusion Green function g˜ (|p Ð q|, t Ð t') = It should be noted that the last term in (51) vanishes m 1 ((g˜ p, 0) ≡ 1) in the case of a δ(τ)-correlated field and exp[ÐD (p Ð q)2(t Ð t')], which can be approximated by m 1 compressibility (divu ≠ 0) does not affect the diffusiv- unity if the diffusivity D is small. The first term in (48) m ()0 2 τ describes the decrease in particle density due to turbu- ity since Dt = u0 0/3 (diffusivity is determined by the lent diffusion. In the Bourret approximation defined total turbulent kinetic energy). However, this is not true by (41), its contribution is overestimated and conserva- for correlated scalar fluctuations, because the contribution law (5) is violated again. tions of Einc(q) and Ecompr(q) to the exact Eq. (50) have different angle-dependent weights. Setting t1 = t2 = t in the sum of (48) with its counter- ˜ ˜ In the δ(τ)-correlated model under consideration, (49) part for L0 (2)V (p, t1, t2) and integrating the result over p, one obtains is also an exact equation: d 2 2 d 2 2 〈〉(), 〈〉()∇ (), ----- 〈〉n ()r, t = Ð2D 〈〉()∇n()r, t ----- n r t = Ð2Dm n r t dt m dt (52) τ 〈〉2 〈〉2(), t ∞ +2 0 div u n r t . τ (),, τ + ∫d ∫dqEV qttÐ (49) According to (52), the fluctuating part of 〈n2〉 (i.e., 0 0 clustering) grows at the initial stage, when the density 〈 ∇ 2〉 ∞ 1 is nearly uniform. However, ( n) increases with the contribution of small-scale scalar fluctuations, and dif- × µ (), τ ˜ (), τ ()2 µ ∫dp ∫ d Ecompr p g pq+ p + pq , fusive dissipation of fluctuations tends to play a domi- 0 Ð1 nant role.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 737 ≈ ӷ τ ≈ ӷ At t1 t2 0 (or t1 t2 t0 = R0/u0), both (48) and where M = u0/c is the Mach number, x = p/p0, ˜ 2 η (49) can be approximated in terms of V (p, t, t). Let us Ecompr(p) = Ecompr(x)/u0 p0, and (x) = k(x)p0/c (c is the ≥ τ consider two cases. At moderate t (t 0), when the sca- sound speed). According to the model proposed in [30], η 2 lar turbulence has not yet concentrated in small-scale (x) = M Ecompr(x). motions, it can be assumed that V˜ (q, t, t Ð τ) ≈ V˜ (q, t, t) Ӷ ≈ It is obvious that the correlator (56) cannot be has a maximum at q q0 1/R0. Using the series approximated by δ(τ), and the growth of fluctuations expansion of g˜ (|p Ð q|, τ) in powers of the small param- can be evaluated only by using Eq. (53). For “acoustic” eter q, one obtains turbulence, Dt' > 0, i.e., the fluctuation intensity at ≥ τ 〈 2〉 d 2 t 0 (or t = t0) increases owing to both C n and ----- 〈〉n ()r, t 〈 ∇ 2〉 dt (53) Dt' ( n) . ()∇〈〉()(), 2 〈〉2(), = Ð22 Dm Ð Dt' n r t + Cn r t , Thus, a more realistic model of turbulence with a finite correlation time reveals new qualitative tends in where the initial evolution: fluctuations are damped by turbu- ∞ ∞ lent diffusion when acoustic effects are negligible, 〈 2 〉 τ 2 (), τ ˜ (), τ whereas additional growth of n (r, t) is predicted for Cp= ∫d ∫d p Ecompr p g p , (54) turbulence with a substantial acoustic component. 0 0 When t is much greater than the smaller of the times ∞ ∞ τ and t = R /u , only small-scale scalar fluctuations 1 ∂g˜ ()p, τ 0 0 0 0 D' = --- dp dτE ()p, τ p------. (55) may be taken into account, i.e., one may assume that t 3∫ ∫ compr ∂p ӷ Ӷ q p0 in (49). A series expansion in terms of p/q 1 0 0 yields

Here, Dt' is the direct contribution of compressibil- ≠ ()0 d 〈〉2(), 〈〉()∇ 2 ity (divu 0) to the turbulent diffusivity Dt . It van------n r t = Ð2Dm n ishes for a δ(τ)-correlated field. When acoustic effects dt ∞ in turbulence are negligible, ∂g˜ (p, τ)/∂p < 0 and, there- (),, (59) 4 〈〉2 EV ptt ' + ------div u ∫dp------. fore, Dt < 0. This implies that chaotic shock waves 3u p strongly damp fluctuations in a compressible gas flow. 0 0 Note that the coefficient C in (53) may be much smaller than the corresponding coefficient in (52), which leads This equation is derived by assuming that the function to additional damping of fluctuations. In the limit of g˜ (q, τ) has a narrower support in τ as compared to δ τ τ ( )-correlated turbulence, Eq. (53) reduces to (52). In Ecompr(p, ) and using the approximation other words, Eqs. (53)Ð(55) can be interpreted as an extension of Eq. (52) to the initial stage of scalar-field ∞ evolution in turbulent flows with finite correlation τ (), τ ≈ 3 ӷ ∫d g˜ q ------, q q0, (60) times. u0q 0 For “acoustic” turbulence, (), τ which follows from DIA equation (24) written for the Ecompr p ˜ τ (56) Laplace transform of g (p, ): () ()τ []()2τ = Ecompr p cos cp exp Ðkpp , ˜ (), ()0 g ps and the method for calculating Dt developed in [29] ∞ 1 ∞ can be used to obtain 2 p µτ[()µ2 (), τ = sD++m p ---∫dq ∫ d ∫d 1 Ð pEinc q ∞ 4 (61) π 3 0 Ð1 0 M u0 2 () Dt' = ------∫dxEcompr x , (57) Ð1 9 p0 Ðsτ 0 µ()µ (), τ (), τ -----+2 p Ð q Ecompr q ]e g˜ pqÐ .

Cu= 0 p0M ∞ 2 π 2 (58) Thus, Eq. (59) also substantially differs from its × dxx E ()ηx ()x + ---M E ()x , δ τ ∫ compr 6 compr counterpart, Eq. (52), written for a ( )-correlated 0 process.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 738 SILANT’EV In the δ(τ)-correlated case, the exact solution to Bourret-like linear theory, leading to an expression for ξ Eq. (61) is (see [17]) Dt that is valid for any 0.

() The equation for V˜ (p, t , t ) up to fourth-order irre- ˜ (), []()0 2 Ð1 1 2 g ps = sD+ m + Dt p , (62) ducible correlators is obtained by adding the four terms corresponding to the last four terms in (38) to the right- which entails hand side of Eq. (48). These additional terms are denoted by A, B, C, and D in accordance with their (), τ []()()0 2τ order in (38): g˜ p = exp Ð Dm + Dt p , τ τ τ t1 t1 Ð t1 Ð Ð ' i.e., the solution to the diffusion equation with the dif- ()ττ,, τ dq ds () () Apt1 t2 = ∫d ∫ d ' ∫ d ''∫------3∫------3 fusivity D + D 0 , where D 0 = u2 τ /3. This diffusive ()2π ()2π m t t 0 0 0 0 0 solution is identical to that predicted by the model of × (), τ (), τ (), τ Ӷ g˜ psÐ g˜ pqsÐ Ð ' g˜ pqÐ '' turbulence with a finite correlation time for p p0. (63) × ˜ (), ττ() However, the largest contributions to expressions (54) pi Bij s + ' pqsÐ Ð j and (55) are due to p ≥ p (rather than p Ӷ p ). Since dif- 0 0 × ()˜ (), τ τ () fusive solution (62) is not valid in this case, the psÐ n Bnm q ' + '' pqÐ m δ τ ( )-correlated approximation cannot be used either. × ˜ (),,τ τ τ Indeed, when the diffusion Green function is substi- V pt1 Ð Ð ' Ð '' t2 , tuted into (54), the resulting expression equals the dif- t t Ð τ t Ð τ Ð τ' 2ξ Ӷ 2 2 2 fusivity in (52) only for (p/p0) 0 1. In most turbu- dq ds Dpt(),,t = Ð d ττd ' dτ'' ------lence models, it is assumed that ξ = u τ /R ~ 1, i.e., the 1 2 ∫ ∫ ∫ ∫ 3∫ 3 0 0 0 0 ()2π ()2π δ(τ)-correlated approximation is not valid a priori (e.g. 0 0 0 δ τ × (), τ (), τ (), τ see [31]). In summary, the ( )-correlated approxima- g˜ p g˜ pqs+ Ð '' g˜ pq+ ' pi tion is applicable only to turbulence with ξ Ӷ 1, in (64) 0 × ˜ (), ττ() which case it is sufficient to use expansion (10) in terms Bij s t1 Ð t2 ++' pq+ j of molecular-diffusion Green functions. ˜ × p Bnm()q, τ' + τ'' ()pqs+ Ð The Green function of the Bourret equation (40) is n m × ˜ (),,τ τ τ given by expression (61), where g˜ (|p Ð q|, τ) on the V psÐ t2 Ð Ð ' Ð '' t1 , | | τ right-hand side should be replaced with g˜ m ( p Ð q , ) τ t1 t1 Ð t2 or with unity (if the small molecular diffusivity Dm is (),, τττ dq ds Ӷ Ӷ Bpt1 t2 = Ð∫d ∫ d '∫d ''∫------∫------neglected). In the diffusion approximation (p p0, s ()2π 3 ()2π 3 τ 0 0 0 1/ 0), the Green function of the Bourret equation is × (), τ (), τ (), τ given by (62). g˜ psÐ g˜ pqsÐ Ð ' g˜ p '' pi ξ ӷ (65) When 0 1, the Green functions of the DIA equa- × B˜ ()s, ττ+ ' ()pqsÐ Ð tion (61) and the Bourret equation (40) exhibit different ij j × ()˜ (), τ τ asymptotic behavior. While the exact asymptotic psÐ n Bnm q t1 Ð t2 Ð + '' pm expressions are not written out here (see [21]), it should × ˜ (),,τ τ τ ˜ ∝ ξ Ð1 V pqÐ t1 Ð Ð ' t2 Ð '' , be noted that, roughly, g˜ (p, s) ( 0p/p0) in the former τ ˜ ∝ ξ Ð2 t2 t2 Ð t1 case and g˜ (p, s) ( 0p/p0) in the latter. This difference explains why general expression (51) yields the ()ττ,, τ dq ds Cpt1 t2 = ∫d ∫ d '∫d ''∫------3∫------3 ()0 ξ ∞ ()2π ()2π physically reasonable Dt = constu0/p0 as 0 in 0 0 0 × (), τ (), τ (), τ () the former case (see also [12]), whereas the result g˜ p g˜ pqÐ ' g˜ ps+ '' pqÐ i obtained in the latter case tends to zero as 1/ξ . The fact (66) 0 × ˜ (), ττ that the DIA equation yields qualitatively correct val- Bij s t1 Ð t2 ++' p j ξ Ӷ ues of turbulent diffusivity both for 0 1 and for × ()˜ (), ττ ξ ӷ ps+ n Bnm q t1 Ð t2 + Ð '' pm 0 1 was first noted in [16], where the DIA equation for scalar diffusion was derived on the basis of an ear- × V˜ ()pqsÐ + ,,t Ð τ'' t Ð τ Ð τ' . lier paper by Kraichnan focused on a nonlinear theory 1 2 of turbulence per se [25]. Thus, the nonlinear theory of Consistency with conservation law (5) is verified turbulent diffusion does not rely on any expansion in by showing that the integrals of A + D and B + C over ξ terms of 0. The nonlinear analysis takes into account a p vanish if t1 = t2 = t and divu = 0. In contrast to much greater number of elementary interactions Eq. (48), additional terms (63)Ð(66) depend on turbu- between the scalar field and the flow, as compared to a lent helicity. When (63)Ð(66) are taken into account,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 739 〈 〉 the following expressions must be added to the right- homogeneous, and isotropic turbulence, Gij(1, 2) = θ τ τ 〈 〉 hand side of (49): ( )gij(R, ) and Hij(1, 2) = Bi(1)Bj(2) = Hij(R, t1, t2), τ where R = r1 Ð r2 and = t1 Ð t2. The Fourier transforms t t Ð τ t Ð τ Ð τ' dp dq ds of these quantities with respect to R are expressed as Et()= ∫d τ∫ d τ' ∫ dτ''∫------∫------∫------follows: ()2π 3 ()2π 3 ()2π 3 0 0 0 ˜ ()δ, τ ˜ ()δ, τ ()2 ˜ (), τ × (), τ (), τ (), τ g jk p = jkgm p + jk p Ð p j pk g2 p g˜ psÐ g˜ pqÐ Ð s ' g˜ pqÐ '' si (70) (67) + ie p g˜ ()p, τ , × ˜ (), ττ() jkt t 1 Bij s + ' pqÐ Ð s j ˜ ()δ,, ()2 ˜ (),, × ()˜ (), τ τ () H jk p t1 t2 = jk p Ð p j pk H0 pt1 t2 psÐ n Bnm q ' + '' pqÐ m (71) ˜ (),, × V˜ ()pt,,Ð τ Ð τ' Ð τ'' t , + iejkt pt H1 pt1 t2 . ˜ τ t t Ð τ t The term containing gm (p, ) is due to the absolute dp dq ds term in Eq. (9) for the Green function. The condition Ft()= Ð∫d τ∫ d τ'∫dτ''∫------∫------∫------()2π 3 ()2π 3 ()2π 3 divB = 0 (p á B˜ = 0) is satisfied, because the initial 0 0 0 magnetic field B (r) is solenoidal. Further analysis is × g˜ ()psÐ , τ g˜ ()pqÐ Ð s , τ' g˜ ()p, τ'' s 0 i (68) facilitated by introducing the function × ˜ (), ττ() Bij s + ' pqÐ Ð s j (), τ (), τ 2 (), τ g˜ 0 p = g˜ m p + p g˜ 2 p (72) × ()˜ (), τ τ psÐ n Bnm q ' Ð '' pm τ and deriving a system of equations for g˜ 0 (p, ) and × V˜ ()pqÐ , t Ð τ'', t Ð τ Ð τ' . τ τ g˜ 1 (p, ) from DIA equation (24). Defining g˜ ± (p, ) = ˜ τ ± ˜ τ The functions E(t) and F(t) are obtained by integrating g0 (p, ) pg1 (p, ), one obtains the sums A + D and B + C, respectively. In contrast to τ ττÐ ' ˜ 〈 2 〉 V (p, t1, t2), the quantity n (r, t) is independent of tur- (), τ (), τ τ τ (), τ g˜ ± p = g˜ m p Ð ∫d ' ∫ d ''g˜ m p ' bulent helicity even if the fourth-order correlators are (73) taken into account. It should be noted that both E(t) and 0 0 2 F(t) depend on the product of the energy spectrum × []p S ()p, τ'' ± pS ()p, τ'' g˜ ±()p, ττÐ ' Ð τ'' , τ 0 1 Ecompr(p, ) (associated with potential motions in compressible turbulence) with a correlator containing where τ Einc(p, ). Expressions (67) and (68) can be used to cal- ∞ 1 culate corrections to asymptotic equations (53) and (59). 2 1 2 2 p S ()p, τ = --- dq dµ{ p ()1 Ð µ The additional contribution to (59) has a relatively sim- 0 4∫ ∫ ple analytical form: 0 Ð1 ∞ × [ (), τ (), τ ˜ Einc q g˜ 0 pqÐ 2 2 2 EV ()qtt,, Ð------〈〉 div u 〈〉curl u dq------. (69) (74) 3 ∫ 3 (), τ ˜ ()], τ µ()µ 3u q Ð Eh q g1 pqÐ + 2 p p Ð q 0 0 × (), τ ()}, τ Expression (69) demonstrates that turbulent fluctua- Ecompr q g0 pqÐ , tions are additionally damped by vortex motion. When ∞ the molecular diffusivity is small, this effect plays a 1 1 2 2 2 dominant role in damping the fluctuations (dispersing S ()p, τ = --- dq dµ{()1 Ð µ [()p + q Ð pqµ 1 4∫ ∫ clusters of particles) at the initial stage. At a later stage, 0 Ð1 molecular diffusion takes over as a factor that flattens × [ (), τ (), τ out the particle density. This effect is beyond the scope Einc q g˜ 1 pqÐ of the δ(τ)-correlated model, because vortex motion (75) , τ , τ ] []2µ2 ()µ2 2 µ cannot be described by this model. Ð Eh()q g˜ 0()pqÐ + 2p + 1 + ()q Ð 2 pq × (), τ ()}, τ Ecompr q g˜ 1 pqÐ . 5. EQUATIONS FOR MAGNETIC ENERGY AND MAGNETIC HELICITY SPECTRA Expressions (73)Ð(75) have much simpler form for incompressible turbulent flow. The Laplace transform Magnetic field diffusion is described by a tensor of Eq. (73) is Green function Gij(1, 2) and a tensor interaction opera- ∇()1 δ ∇()1 ˜ (), []2 2 ˜ (), ± ˜ (), Ð1 tor Lij(1) = j ui(1) Ð ij t ut(1). For stationary, g˜ ± ps = sD++m p p S0 ps pS1 ps . (76)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 740 SILANT’EV Under the solenoidality condition p á B = 0, the term imation for the BetheÐSalpeter equation). First of all, p p g˜ (p, τ) vanishes and Green function (70) reduces ˜ i j 2 note that the function H0 (p, t1, t2) determines the spec- to trum of magnetic energy fluctuations: g˜ ()δp, τ = g˜ ()p, τ + ie p g˜ ()p, τ . (77) ∞ jk jk 0 jkt t 1 〈〉(),, ⋅ (),, (),, Brt1 t2 Brt1 t2 = ∫dpEB pt1 t2 , τ (82) In the case of zero turbulent helicity (D(p, ) = 0), both 0 g˜ (p, τ) = 0 and S (p, τ) = 0, and the equations for 1 1 E = p4 H˜ ()pt,,t /π2. τ B 0 1 2 g˜ 0 (p, ) are identical to Eqs. (61) and (62) for scalar diffusion. Use the relation B = — × A to express the spectrum of ˜ In the diffusion approximation, expression (76) has magnetic helicity fluctuations in terms of H1 (p, t, t): the form ∞ ˜ ()0 2 ()0 Ð1 H ()t ≡ 〈〉Ar(), t ⋅ Br(), t = dpE ()pt, , g˜ ±()ps, = []sD+ ()+ D p ± α p , (78) Mh ∫ Mh m t t (83) 0 where (), 2 ˜ (),, π2 EMh pt = Ðp H1 ptt/ . ∞ ∞ ()0 1 When D = 0 in a homogeneous turbulent flow, mag- D = --- dp dτ m t 3∫ ∫ netic helicity conservation law (6) implies that 0 0 d ˜ ----- dpH1()ptt,, = 0. (84)  dt∫ × [](), τ (), τ (), τ  Einc p + Ecompr p g˜ 0 p --- (79)  The validity of this relation is demonstrated below. Equation (38) entails the following system of equa- ∂ (), τ  g˜ 0 p tions for H˜ (p, t , t ) and H˜ (p, t , t ): + E ()p, τ p------Ð E ()p, τ g˜ ()p, τ . 0 1 2 1 1 2 compr ∂p h 1  ∂ 2 ˜ (),, τ ∂------+ Dm p H0 pt1 t2 In the case of zero turbulent helicity (Eh(p, ) = 0), this t1 expression is identical to turbulent diffusivity (51) for a t passive scalar (different coefficients D are obtained for 1 t τ[ 2 (), τ ˜ (),,τ magnetic and scalar fields only if irreducible inter- = Ð ∫d p S0 p H0 pt1 Ð t2 actions of fourth order in velocity are taken into 0 account [26]). The factor describing amplification of (), τ ˜ ()],,τ the mean magnetic field by turbulent helicity is + S1 p H1 pt1 Ð t2 ∞ ∞ t2 α()0 1 τ{ (), τ ˜ (), τ τ[ (),,,τ τ (), τ t = ---∫dp∫d ÐEh p g0 p + ∫d T 0 pt1 Ð t2 + t1 t2 Ð g˜ 0 p 3 (80) 0 0 0 + p2[]E ()p, τ + 2E ()p, τ g˜ ()}p, τ . (),,,τ τ ()], τ inc compr 1 + T 1 pt1 Ð t2 + t1 t2 Ð g˜ 1 p , δ τ For a ( )-correlated process, ∂ 2 (85) ------+ D p H˜ ()pt,,t ∂t m 1 1 2 ()0 2τ α()0 〈〉τ⋅ ()∇ × 1 Dt ==u0 0/3, t Ð u u 0/3.

t1 The diffusion Green function corresponding to (78) has τ 2[ (), τ ˜ (),,τ the form of (77) with = Ð ∫d p S1 p H0 pt1 Ð t2 0 ˜ (), τ () α()0 τ ()()0 2τ (), τ ˜ ()],,τ g0 p = cosh t p exp ÐDt p , + S0 p H1 pt1 Ð t2 (81) 1 ()0 ()0 2 (), τ ()α τ ()τ t2 g˜ 1 p = Ð--- sinh t p exp ÐDt p . p τ[ (),,,τ τ (), τ + ∫d T 1 pt1 Ð t2 + t1 t2 Ð g˜ 0 p Now, consider Eq. (38) for the tensor H˜ (p, t , t ), 0 ij 1 2 (),,,τ τ ()], τ retaining only second-order correlators (ladder approx- + T 0 pt1 Ð t2 + t1 t2 Ð g˜ 1 p ,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 741 where 2[]()0 ˜ ()α,, ()0 ˜ (),, = Ð2p Dt H1 ptt + t H0 ptt ∞ ∞ τ 1 (),,, τ 1 0 2 2 T 0 ptt1 t2 Ð = ---∫dq + ---- dq dµ{ p ()1 Ð µ [E ()q H˜ ()pqÐ ,,tt 4 2 ∫ ∫ inc 1 0 0 Ð1 1 ×µ{()µ2 [()2 2 µ × ()˜ (),, ()2µ2 µ ∫ d 1 Ð p + q Ð pq Eh q H0 pqÐ tt + 2 p Ð pq Ð1 × ()˜ ()},, Ecompr q H1 pqÐ tt , × (), ˜ (),, τ Einc qtH0 pqÐ t1 t2 Ð (86) where ˜ Ð E ()qt, H1()]pqÐ ,,t t Ð τ h 1 2 ()0 2τ α()0 〈〉τ⋅ ()∇ × Dt ==u0 0/3, t Ð u u 0/3. +2[]p2µ2 + ()1 + µ2 ()q2 Ð 2 pqµ For incompressible turbulent flow with zero helicity, × (), ˜ ()},, τ 2 Ecompr qtH0 pqÐ t1 t2 Ð , the first equation in (88) written for p H0(p, t, t) is iden- ∞ tical to Eq. (14) in [17]. (),,, τ 1 Now, magnetic helicity conservation law (84) for T 1 ptt1 t2 Ð = ---∫dq 4 flows with Dm = 0 can be obtained by performing some 0 simple changes of variables in the integral of the second

1 equation in (85). It should be noted that even calculation of magnetic energy (82) must be consistent with ×µ{ 2()µ2 [ (), ˜ (),, τ ∫ d p 1 Ð Einc qtH1 pqÐ t1 t2 Ð ˜ (87) conservation law (84), because the functions H0 (p, t1, t2) Ð1 ˜ and H1 (p, t1, t2) are interrelated. Ð E ()qt, H˜ ()pqÐ ,,t t Ð τ h 0 1 2 Equations (85)Ð(87) can be used to obtain an equa- 〈 2 〉 µ()µ (), ˜ ()},, τ tion for B (r, t) : +2p p Ð q Ecompr qtH1 pqÐ t1 t2 Ð . d 2 2 Note that S and S are associated with the diagram ----- 〈〉B ()r, t = Ð2D 〈〉()∇ × Br(), t 0 1 dt m 〈〉᭺᭺; T and T , with 〈〉᭺᭺ (see (38)). Therefore, 0 1 ∞ ∞ the corresponding expressions have similar structure. 1 t τ 1 µτ{ (),, τ In the case of zero turbulent helicity (D(p, ) = 0, + ---∫dpq∫d ∫ d ∫d EB pttÐ τ τ 2 g˜ 1 (p, ) = 0, S1(p, ) = 0), system (85) splits into sepa- 0 0 Ð1 0 ˜ ˜ rate equations for H0 (p, t1, t2) and H1 (p, t1, t2). × (), τ [()µ2 ()2 µ g˜ 0 pqÐ 1 Ð q Ð pq In the frequently used case of a δ(τ)-correlated veloc- ˜ τ τ δ τ ˜ × (), τ ()2()µ2 µ3 (), τ ] ity ensemble (Bij (p, ) = 0 ( )(Bij p)), system (85) sim- Einc q + q 1 + Ð 2 pq Ecompr q plifies to (89) (), τ (),, τ [ 3 ()µ2 + g˜ 1 pqÐ EMh pttÐ p q 1 Ð ∂ 2 ˜ (),, ∂-----2+ Dm p H0 ptt × , τ ()2 2 µ2 µ 3 3µ 2µ2 t Einc()q + p q ()21 Ð + p ()q ++2 p qp []()0 2 ˜ ()α,, ()0 ˜ (),, = Ð2 Dt p H0 ptt + t H1 ptt × ()], τ ()µ2 ()2 µ Ecompr q Ð 1 Ð q Ð 2 pq ∞ 1 × (), τ (),, τ ()}, τ τ Eh q EB pttÐ g˜ 1 pqÐ . + ----0∫dq ∫ dµ{()1 Ð µ2 [()p2 + q2 Ð pqµ 2 In the model of delta-correlated turbulent field, it 0 Ð1 simplifies to × ()˜ (),, ()˜ (),, ] Einc q H0 pqÐ tt Ð Eh q H1 pqÐ tt d 〈〉2 〈〉()∇ × 2 ----- B = Ð2Dm B +2[]p2µ2 + ()1 + µ2 ()q2 Ð 2 pqµ dt (90) 2τ 2 2 2 × E ()q H˜ ()}pqÐ ,,tt , (88) + ------0[]〈〉()∇ × u + 2〈〉 div u 〈〉B . compr 0 3

∂ 2 ˜ Note that Eq. (90) is independent of turbulent helic------2+ D p H1()ptt,, ∂t m ity, which determines the ampliﬁcation factor for the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 742 SILANT’EV mean magnetic field 〈B〉 (see (80)). Since 〈B2〉 = 〈B〉 á γ is entirely determined by compressible turbulent 〈B〉 + 〈B'2〉, this implies that the increase in the mean motion with nonzero helicity. In contrast to (90), the (large-scale) magnetic energy due to the α-effect is term proportional to 〈B2〉 in (91) contains a contribution compensated for by the damping of magnetic fluctua- due to turbulent helicity, because the function g1 is pro- tions due to the same mechanism. This observation was portional to the helicity spectrum. This implies that ≤ originally made in [7], where the diffusion approxima- C2 0; i.e., turbulent helicity inhibits the increase in tion was used without assuming short turbulent correla- magnetic energy at initial stages of evolution. Since tion time. Note that Eq. (90) was recently obtained ∂g˜ (p, τ)/∂p ≤ 0, it may be expected that D'' ≤ 0 (the in [24], where extension of Kazantsev’s equation [17] 0 t to two- and three-dimensional compressible turbulent term with Eh must be smaller than the first term in (92)). flows with zero helicity was analyzed in detail. Since Dt'' is much greater than Dm, the former coeffi- It should be noted that magnetic fields are generally cient controls the initial decay of magnetic energy. Note associated with rotating media (rotating stellar atmo- that Eq. (91) reduces to (90) in the limit of short corre- spheres, rotation of the Galaxy, etc.). Since rotation of lation time. a turbulent medium as a whole gives rise to helicity, ˜ ˜ In summary, the increase in magnetic energy in tur- system of equations (88) for H0 (p, t, t) and H1 (p, t, t) bulence with a finite correlation time is substantially must be solved even in δ(τ)-correlated models. However, slower than that predicted by the model of delta-corre- effects due to helicity are totally ignored in [20, 24] and lated turbulent field. other studies. As in the analysis of scalar transport [see (53)], When acoustic effects are important, one obtains ≡ Eq. (90) can be extended to describe an initial stage of Dt'' = 3Dt' /5, C2 0, and C1 = 4C/3, where the positive field evolution in the model of delta-correlated turbu- ' lent field: coefficients Dt and C are given by (57) and (58), respectively. Note that the magnetic field diffusivity in d 2 2 “acoustic” turbulence is equal to the scalar diffusivity ----- 〈〉B = Ð2()∇D Ð D'' 〈〉()× B dt m t obtained in [27]. Thus, additional growth of fluctua- (91) tions due to acoustic effects is predicted for both scalar γ 〈〉⋅ ()∇ × ()〈〉2 + B B + C1 + C2 B , and magnetic field diffusion. In the case of zero helicity, magnetic energy must where concentrate in small-scale fluctuations in the long-time ∞ ∞ limit. Equation (89) simplifies accordingly: 1  D'' = ------d p d τ  [] E () p , τ + 3 E ()p, τ --- t 15∫ ∫ inc compr  d 2 2 0 0 ----- 〈〉B = Ð2D 〈〉()∇ × B (92) dt m ∂g˜ ()p, τ ∂g˜ ()p, τ  ∞ × p------0 Ð 2E ()p, τ p------1 , (),, (96) ∂ h ∂ 8 2 2 E ptt p p  + ------[]〈〉()∇ × u + 2〈〉 div u dp------B -. ∫ p 53u0 ∞ ∞ 0 2 2 γ = --- dp dτ p E ()p, τ 3∫ ∫ compr In contrast to scalar fluctuations (see (55)), magnetic 0 0 (93) field fluctuations can be amplified not only by com- ∂ ˜ (), τ pressible turbulent motions (e.g., shock waves), but × (), τ g1 p g˜ 1 p Ð,p------also by rotational motions of a turbulent plasma. Since ∂p Ӷ EB(p, t, t) has a maximum at pmax p0, the value of 1/p Ӷ ≤ ∞ ∞ at some intermediate point p1 (p0 p1 pmax) can be 2 τ 2 factored out of the integral. As a result, the second term C1 = ---∫dp∫d p 3 (94) in (96) will become similar in structure to its counter- 0 0 part in (90), but its value will differ by a factor of ξ ξ ≈ × []E ()p, τ + 2E ()p, τ g˜ ()p, τ , (p0/p1)/ 0. When 0 1 (as usually assumed for well- inc compr 0 developed turbulence [31]), the resulting amplification ∞ ∞ factor will be much smaller than that predicted by 2 2 Eq. (90). A similar result is obtained for scalar fluc- C = Ð--- dp dτ p E ()p, τ g˜ ()p, τ . (95) 2 3∫ ∫ h 1 tuations. 0 0 Equations (91) and (96) differ substantially from (90), Note that two small terms on the order of 〈∇2B á (∇ × which corresponds to a δ(τ)-correlated process. All crit- B)〉 are ignored in Eq. (91). Furthermore, the term with ical remarks on the model of delta-correlated turbulent

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 A NONLINEAR THEORY OF TURBULENT DIFFUSION 743

field made in the preceding section apply to the model is controlled by molecular diffusion. Note also that the of magnetic field diffusion. amplification factor for well-developed turbulence with ξ ≈ 0 1 is much smaller than that predicted by the δ(τ)-correlated model. 6. CONCLUSIONS The correct hierarchies of equations for 〈G(1, 2)〉 The principal results of this study are summarized as and fluctuation intensities provide a solid basis for follows. Most importantly, it is shown for the first time more detailed analysis of time-dependent fluctuation that the BetheÐSalpeter-type time-dependent integrod- intensities. The method used to derive and match the ifferential equations for magnetic field and scalar fluc- hierarchies can also be used to analyze the background tuation intensities must be consistent with conservation turbulence. In particular, it would be interesting to gen- laws (5) and (6). The kernels of these equations are dif- eralize Kraichnan’s equation (which predicts pÐ3/2 ferences of two terms of similar order describing the instead of Kolmogorov’s pÐ5/3 for the energy spectrum balance of growth and decay of fluctuations at a fixed in the inertial subrange [25]) by retaining the fourth- point in a turbulent medium. The conservation laws order correlators. Such a generalization should be impose integral constraints on these processes. Consis- expected to yield a power exponent closer to Kolmog- tency with these laws rules out spurious growth or orov’s, which would mean a more accurate description damping of fluctuations. of real turbulence. It is found that a hierarchy of BetheÐSalpeter equations are consistent with the conservation laws only if REFERENCES the averaged Green function 〈G(1, 2)〉 satisfies a hierarchy of Dyson-type nonlinear equations. Moreover, the 1. S. I. Vainshtein and F. Cattaneo, Astrophys. J. 393, 165 highest orders of velocity correlators retained in both (1992). hierarchies must be equal. In particular, it is shown that 2. A. V. Gruzinov and P. H. Diamond, Phys. Rev. Lett. 72, the widely used approximate Bourret equation for 1651 (1994). 〈G(1, 2)〉 does not ensure consistency of BetheÐSal- peter equations with the conservation laws even in the 3. Ya. B. Zel’dovich, Zh. Éksp. Teor. Fiz. 31, 154 (1956) ξ τ Ӷ [Sov. Phys. JETP 4, 160 (1956)]. case of 0 = u0 0/R0 1, when this equation yields a correct value of turbulent diffusivity. 4. R. S. Peckover and N. O. Weiss, Mon. Not. R. Astron. Soc. 182, 189 (1978). The simple derivation of a hierarchy of nonlinear equations for the averaged Green function is not 5. L. L. Kichatinov, V. V. Pipin, and G. Rüdiger, Astron. restricted to the case of a Gaussian velocity ensemble. Nachr. 315, 157 (1994). Correct time-dependent equations are obtained for sca- 6. J. Cho and A. Lazarian, Astrophys. J. 589, L77 (2003). lar and magnetic fluctuations in compressible turbulent 7. N. A. Silant’ev, Pis’ma Zh. Éksp. Teor. Fiz. 72, 60 (2000) media with nonzero helicity. These equations are con- [JETP Lett. 72, 42 (2000)]. sistent with the conservation laws. They can also be used to analyze the influence of non-Gaussian velocity 8. N. A. Silant’ev, Astron. Astrophys. 370, 533 (2001). statistics on turbulent diffusivities and time evolution of 9. A. Z. Dolginov and N. A. Silant’ev, Geophys. Astrophys. scalar fluctuations. Fluid Dyn. 63, 139 (1992). These equations are used to derive asymptotic for- 10. G. I. Taylor, Proc. London Math. Soc. A 20, 196 (1921). mulas describing the evolution of scalar fluctuation 11. H. K. Moffatt, J. Fluid Mech. 65, 1 (1974). intensity at a fixed point of a turbulent medium. 12. N. A. Silant’ev, Zh. Éksp. Teor. Fiz. 101, 1216 (1992) It is shown that the model of delta-correlated turbu- [Sov. Phys. JETP 74, 650 (1992)]. 〈 〉 ∝ δ lent field ( ui(1)uj(2) (t1 Ð t2)), which is frequently 13. V. I. Tatarskiœ, Wave Propagation in a Turbulent Medium applied because of its mathematical simplicity, fails to (Nauka, Moscow, 1967; McGraw-Hill, New York, predict a number of important effects of turbulent diffu- 1961). sion on the time of energy transfer to small-scale fluc- 14. V. I. Klyatskin, Stochastic Equations by the Eyes of a tuations. The overall effect of a finite velocity correla- Physicist (Fizmatlit, Moscow, 2001). tion time is to inhibit the growth of fluctuation intensity, because fluctuations are damped by turbulent diffusion 15. R. Bourret, Can. J. Phys. 38, 665 (1960). (in the δ(τ)-correlated model, damping is due to molec- 16. P. H. Roberts, J. Fluid Mech. 11, 257 (1961). ular diffusion only) and the amplification factor is 17. A. P. Kazantsev, Zh. Éksp. Teor. Fiz. 53, 1806 (1967) reduced by taking into account nonlocal mechanisms of [Sov. Phys. JETP 26, 1031 (1967)]. turbulent transport. In contrast, turbulent diffusion in “acoustic” turbulence counteracts molecular dissipa- 18. H. K. Moffatt, Magnetic Field Generation in Electri- tion and simultaneously increases the amplification fac- cally Conducting Fluids (Cambridge Univ. Press, Cam- tor. However, when small-scale fluctuations play a bridge, 1978). dominant role at the final stage of evolution, their decay 19. N. Seehafer, Phys. Rev. E 53, 1283 (1996).

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20. A. P. Kazantsev, A. A. Ruzmaœkin, and D. V. Sokolov, 27. N. A. Silant’ev, Zh. Éksp. Teor. Fiz. 114, 930 (1998) Zh. Éksp. Teor. Fiz. 88, 487 (1985) [Sov. Phys. JETP 61, [JETP 87, 505 (1998)]. 285 (1985)]. 28. G. K. Batchelor, The Theory of Homogeneous Turbu- 21. N. A. Silant’ev, Zh. Éksp. Teor. Fiz. 111, 871 (1997) lence (Cambridge Univ. Press, Cambridge, 1953; Inos- [JETP 84, 479 (1997)]. trannaya Literatura, Moscow, 1955). 22. C. C. Lin, Turbulent Flows and Heat Transfer (Princeton 29. N. A. Silant’ev, Zh. Éksp. Teor. Fiz. 122, 1107 (2002) Univ. Press, Princeton, 1959). [JETP 95, 957 (2002)]. 30. V. E. Zakharov and R. Z. Sagdeev, Dokl. Akad. Nauk 23. G. Rickayzen, Green Functions and Condensed Matter SSSR 192, 296 (1970) [Sov. Phys. Dokl. 15, 439 (Academic, New York, 1980). (1970)]. 24. A. A. Schekochihin, S. A. Boldyrev, and R. M. Kulsrud, 31. S. A. Molchanov, A. A. Ruzmaœkin, and D. D. Sokolov, Astrophys. J. 567, 828 (2002). Usp. Fiz. Nauk 145, 593 (1985) [Sov. Phys. Usp. 28, 307 25. R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959). (1985)]. 26. N. A. Silant’ev, Zh. Éksp. Teor. Fiz. 112, 1312 (1997) [JETP 85, 712 (1997)]. Translated by A. Betev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 745Ð747. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 850Ð853. Original Russian Text Copyright © 2004 by Belim. SOLIDS Structure

Critical Dynamics of Three-Dimensional Spin Systems with Long-Range Interactions S. V. Belim Omsk State University, pr. Mira 55, Omsk, 644077 Russia e-mail: [email protected] Received October 21, 2003

Abstract—A ﬁeld-theoretic description of critical behavior of Ising systems with long-range interactions is obtained by using the Padé–Borel summation technique in the two-loop approximation directly in the three- dimensional space. It is shown that long-range interactions affect the relaxation time of the system. © 2004 MAIK “Nauka/Interperiodica”.

It was shown in [1] that effects due to long-range son, the analysis that follows is restricted to the case of interaction are essential for the critical behavior of 0 < a < 2. Ising systems. The renormalization-group approach to Relaxational dynamics of spin systems near the crit- spin systems with long-range interactions developed ical temperature can be described by a Langevin-type in [2] directly in the three-dimensional space made it equation for the order parameter: possible to calculate the static critical exponents in the two-loop approximation. However, analogous calcula- ∂ϕ δH tions of critical dynamics have never been performed ------= Ð λ ------++ηλh, (3) for these systems. ∂t 0 δϕ 0 In this paper, a field-theoretic description of critical λ η behavior of homogeneous systems with long-range where 0 is a kinetic coefficient, (x, t) is a gaussian interactions is developed directly for D = 3 in the two- random force (representing the effect of a heat reser- loop approximation. The model under analysis is the voir) defined by the probability distribution classical spin system with the exchange integral depending on the distance between spins. The corre- []()λ Ð1 d η2(), Pη = Aη exp Ð 4 0 ∫d xdt xt (4) sponding Hamiltonian is

with a normalization factor Aη, and h(t) is an external 1 H = ---∑ Jr()Ð r S S , (1) field thermodynamically conjugate to the order param- 2 i j i j eter. The temporal correlation function G(x, t) of the ij, order-parameter field can be found by solving Eq. (3) with H[ϕ] given by (2) for ϕ[η, h], averaging the result where S is a spin variable and J(|r Ð r |) is the exchange i i j over Pη, and retaining the component linear in h(0): integral. This model is thermodynamically equivalent to the O(n)-symmetric GinzburgÐLandauÐWilson δ model defined by the effective Hamiltonian Gxt(), = ------[]〈〉ϕ()xt, , (5) δh()0 h = 0  D 1()ϕτ a a ϕ4 where Hd= ∫ q--- 0 + q + u0 , (2) 2 []〈〉ϕ(), Ð1 {}ϕη (), xt = B ∫D xtPη, (6) where ϕ is a fluctuating order parameter, D is the space τ | | dimension, 0 ~ T Ð Tc (Tc is the critical temperature), {}η BD= ∫ Pη. (7) and u0 is a positive constant. Critical behavior strongly depends on the parameter a characterizing the interaction as a function of distance. It was shown in [3] that When applying the standard renormalization-group long-range interaction is essential when 0 < a < 2, procedure to this dynamical model, one must deal with whereas systems with a ≥ 2 exhibit critical behavior substantial difficulties. However, it was shown in [4] characteristic of short-range interactions. For this rea- that the model of critical dynamics in homogeneous

746 BELIM systems without long-range interaction based on a Lan- deﬁned for a homogeneous system as gevin-type equation is equivalent to that described by Gxt(), = 〈〉 ϕ()ϕ00, ()xt, the standard Lagrangian [5] = ΩÐ1∫D{}ϕ D{}ϕϕ* ()ϕ00, ()xt, exp()ÐL[]ϕϕ, * , where ∂ϕ δ d λÐ1ϕ2 ϕ λÐ1 H Ld= ∫ xtd 0 + i *0 ------+ ------, (8) Ω {}ϕ {}ϕ ()[]ϕϕ, ∂t δϕ = ∫D D * exp ÐL * . (9) Instead of dealing with the correlation function, it is reasonable to invoke the Feynman diagram technique where ϕ* denotes an auxiliary ﬁeld. The corresponding and represent the corresponding vertex in the two-loop correlation function G(x, t) of the order parameter is approximation as

ω Γ()2 ()τ, ω τ ,,λ a i 2 k ; 0 u0 0 = 0 + k Ð96-----λ- Ð u0 D0, (10) 0

3 dDqdD p D0 = ---∫------. 4 ()1 + q a ()1 + p a ()3 +++q a p a pq+ a Ð iω/λ

The next step in the ﬁeld-theoretic approach is the cal- An expression for β in the two-loop approximation culation of the scaling functions β, γτ, γϕ, and γλ in the was obtained in [2]: renormalization-group differential equation for vertexes: 2 2 2 β = Ð()4 Ð D 136ÐuJ +, 1728 2J Ð J Ð ---G u 0 1 0 9 ∂ ∂ ∂ ∂ m µ------++β------Ð γ ττ----- γ λλ------Ð ----γ ϕ ∂µ ∂u ∂τ ∂λ 2 (11) dDqdD p ×Γ()m (), ω τ,,,λµ k ; u = 0, J1 = ∫------2 a/2-, ()1 + q a ()1 + p a ()1 + q2 ++p2 2pq⋅ where the scaling parameter µ is introduced to change to dimensionless variables. dDq Further analysis requires the use of the function β J = ------, 0 ∫ a 2 and the dynamic scaling function γλ. ()1 + q

∂ dDqdD p G = Ð------a∫------a/2 ------a/2-. ∂ k ()1 + q2 ++k2 2kq⋅ ()1 + p a ()1 + q2 ++p2 2pq⋅

The function γλ calculated in the two-loop approxima- 2 2 β = Ð()4 Ð D 136Ðv +, 1728 2J˜ Ð 1 Ð ---G˜ v tion is 1 9 2 γ λ = ()4 Ð D 2()D' Ð G u , γ ()()˜ ˜ v 2 ∂ (12) λ = 4 Ð D 96 D Ð G , (14) D' =.D0 ∂------()ω λ- Ði / k ==0, ω 0 ˜ J1 ˜ G ˜ D' J1 ===-----2, G -----2, D -----2 . Defining the effective interaction vertex J0 J0 J0 u ≤ v = -----, (13) This redefinition is meaningful for a D/2. In this case, J0 J0, J1, G and D' are divergent functions. Introducing the cutoff parameter Λ, we obtain finite expressions for the one obtains the following expressions for β and γλ: ratios

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 CRITICAL DYNAMICS OF THREE-DIMENSIONAL SPIN SYSTEMS 747

Λ Λ 2 a ∫∫dDqdD p/1()()+ q a ()1 + p a ()1 + q2 ++p2 2pq⋅ J 1 ------0 0 ------2 = Λ 2 , J0 2 ∫dDq/1()+ q a 0 Λ Λ a 2 a Ð∂/()∂ k a ∫∫dDqdD p/1()()+ q2 ++k2 2kq⋅ ()1 + p a ()1 + q2 ++p2 2pq⋅ G ------0 0 ------2 = Λ 2 , (15) J0 2 ∫dDq/1()+ q a 0 D D ()()a ()a ()a a a 2 D' 3/4∫d qd p/1+ q 1 + p 3 +++q p pq+ ------2 = Λ 2 , J0 2 ∫dDq/1()+ q a 0 as Λ ∞. exponent for 1.5 ≤ a ≤ 1.9. When 0 < a < 1.5, the only The integrals are performed numerically. For a ≤ fixed point is the unstable gaussian one, v* = 0. 2 2 A comparison of the present results with the value D/2, a sequence of J1/J0 and G/J0 corresponding to Λ z = 2.017 of the dynamic critical exponent for three- various values of is calculated and extrapolated to dimensional systems with short-range interactions infinity. obtained in [6] demonstrates the essential role played Critical behavior is completely determined by the by long-range interactions in critical dynamics of spin stable fixed points of the renormalization group (RG) systems. In particular, the system’s relaxation time transformation. These points can be found from the | |νz ν increases according to the scaling t ~ T Ð Tc , where condition is the critical exponent characterizing the increase in β()v * = 0. (16) the correlation radius near a critical point. In three- dimensional systems with long-range interaction [2], The effective interaction vertexes were evaluated at both the critical dynamics and static behavior become the stable fixed points of the RG transformation in [2]. increasingly gaussian as the long-range interaction ≤ The dynamic critical exponent z characterizing crit- parameter a decreases. When a 1.8, the critical behav- ical slowing-down of relaxation is determined by sub- ior is virtually gaussian. stituting the effective charges at a fixed point into the scaling function γλ: REFERENCES γ z = 2 + λ. (17) 1. E. Luijten and H. Mebingfeld, Phys. Rev. Lett. 86, 5305 The table shows the stable fixed points of the RG (2001). transformation and the values of the dynamic critical 2. S. V. Belim, Pis’ma Zh. Éksp. Teor. Fiz. 77, 118 (2003) [JETP Lett. 77, 112 (2003)]. Fixed points and values of the dynamic critical exponent for 3. M. E. Fisher, S.-K. Ma, and B. G. Nickel, Phys. Rev. three-dimensional systems Lett. 29, 917 (1972). a v* z 4. C. De Dominicis, Lett. Nuovo Cimento 12, 567 (1975). 5. E. Brezin, J. C. LeGuillon, and J. Zinn-Justin, Phys. Rev. 1.5 0.015151 2.000072 D 8, 434, 2418 (1973). 1.6 0.015974 2.000180 6. V. V. Prudnikov, S. V. Belim, A. V. Ivanov, et al., Zh. 1.7 0.020485 2.000777 Éksp. Teor. Fiz. 114, 972 (1998) [JETP 87, 527 (1998)]. 1.8 0.023230 2.001529 1.9 0.042067 2.006628 Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 748Ð759. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 854Ð867. Original Russian Text Copyright © 2004 by Kuchinskiœ, Sadovskiœ, Strigina. SOLIDS Electronic Properties

Superconductivity in the Pseudogap State in the Hot-Spot Model: GinzburgÐLandau Expansion É. Z. Kuchinskiœ, M. V. Sadovskiœ, and N. A. Strigina Institute of Electrophysics, Ural Division, Russian Academy of Sciences, Yekaterinburg, 620016 Russia e-mail: [email protected]; [email protected]; [email protected] Received May 15, 2003

Abstract—Peculiarities of the superconducting state (s and d pairing) are considered in the model of the pseudogap state induced by short-range order fluctuations of the dielectric (AFM (SDW) or CDW) type, which is based on the model of the Fermi surface with “hot spots.” A microscopic derivation of the GinzburgÐLandau expansion is given with allowance for all Feynman diagrams in perturbation theory in the electron interaction with short-range order fluctuations responsible for strong scattering in the vicinity of hot spots. The superconducting transition temperature is determined as a function of the effective pseudogap width and the correlation length of short-range order fluctuations. Similar dependences are derived for the main parameters of a superconductor in the vicinity of the superconducting transition temperature. It is shown, in particular, that the specific heat jump at the transition point is considerably suppressed upon a transition to the pseudogap region on the phase diagram. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION hood of vector Q, which is determined by the corresponding correlation length ξ [6, 7]. The pseudogap state observed in a wide region on the phase diagram of HTSC cuprates leads to numerous Most of the previous theoretical publications were anomalies in the properties of these compounds both in devoted to analysis of the models of the pseudogap the normal and in the superconducting state [1, 2]. In state in the normal phase at T > Tc. In our earlier publi- our opinion, the most feasible scenario for the forma- cations [8Ð11], we considered superconductivity using tion of the pseudogap state in HTSC oxides is that [2] a simplified model of the pseudogap state, which is based on the existence of strong scattering of charge based on the assumption of the existence of hot (plane) carriers under short-range order fluctuation of the regions at the Fermi surface. In the framework of this “dielectric” type (antiferromagnetic AFM (SDW) or of model, we constructed the GinzburgÐLandau expan- the type of charge density waves (CDW)) in this region sion for various types of Cooper pairing [8, 10] and of the phase diagram. In the momentum space, this studied peculiarities of the superconducting state in the scattering occurs in the vicinity of the characteristic region of T < Tc on the basis of analysis of the solutions vector Q = (π/a, π/a) (a is the parameter of the 2D lat- to the Gor’kov equations [9Ð11]. It should be noted tice), corresponding to period doubling (the antiferro- above all that we considered an extremely simplified magnetism vector) and is a predecessor of the spectrum model of Gaussian short-range order fluctuations with rearrangement occurring during the establishment of an infinitely large correlation length, for which an exact the long-range AFM (SDW) order. Accordingly, an solution can be obtained for the pseudogap state [8, 9]. essentially non-Fermi-liquid rearrangement of the elec- A more realistic case of finite correlation lengths was tron spectrum takes place in definite regions of the analyzed both for model [10] (under the assumption of momentum space in the vicinity of the so-called hot self-averaging of the superconducting order parameter in spots at the Fermi surface [2]. In a number of recent short-range order fluctuations) and for an extremely sim- experiments [3Ð5], precisely this scenario of pseudogap plified, exactly solvable model [11], in which the role of formation was convincingly confirmed. In the frame- non-self-averaging effects could be analyzed [9, 11]. work of the picture described above, it is possible to The present study aims at analyzing the basic prop- construct a simplified model of the pseudogap state, erties of the superconducting state (for various types of which describes the main features of this state [2] and pairing) arising against the background of a “dielectric” takes into account the contribution from all Feynman pseudogap in a more realistic model of hot spots at the diagrams in perturbation theory relative to scattering Fermi surface. We will confine our analysis to a very from (Gaussian) short-range order fluctuations with a close neighborhood of the superconducting transition characteristic scattering momentum from a neighbor- temperature Tc based on the microscopic derivation of

SUPERCONDUCTIVITY IN THE PSEUDOGAP STATE IN THE HOT-SPOT MODEL 749 the GinzburgÐLandau expansion, assuming that the superconducting order parameter is self-averaging, thus generalizing the approach proposed for the model of a hot region developed in [10]. Q = (π, π) É 2. HOT-SPOT MODEL AND PAIRING INTERACTION

In the model of an “almost antiferromagnetic” (π, π) Fermi liquid, which is actively used for explaining the microscopic mechanism of HTSC [12, 13], the effec- Fig. 1. Fermi surface with hot spots connected by a scatter- tive interaction of electrons with spin fluctuations is ing momentum on the order of Q = (π/a, π/a). introduced. This interaction is described by the dynamic susceptibility characterized by the correlation length ξ of spin fluctuations (which must be determined least justified assumption from the standpoint of phys- from experiment), the vector Q = (π/a, π/a) of antifer- ics is the one that concerns the static (and Gaussian) romagnetic ordering in the dielectric phase, and the nature of fluctuations, which can be used only for quite ω characteristic frequency sf of spin fluctuations. This high temperatures [6, 7]. At low temperatures (includ- dynamic susceptibility and, hence, the effective interac- ing those corresponding to the superconducting phase), tion have peaks in the region of q ~ Q. Accordingly, two the spin dynamics and the non-Gaussian nature of fluc- types of quasiparticles appear in the system: hot quasi- tuations may also become significant for the micros- particles whose momenta lie in the vicinity of hot spots copy of Cooper pairing in the model of a nearly antifer- at the Fermi surface (Fig. 1) and cold quasiparticles romagnetic Fermi liquid [12, 13]. However, in our whose momenta are in the vicinity of regions at the opinion, the static Gaussian approximation considered Fermi surface, surrounding the diagonals of the Bril- louin zone [6]. As a matter of fact, quasiparticles from here might be sufficient for analyzing the qualitative the neighborhoods of hot spots are strongly scattered effect of the pseudogap formation on superconductivity over a vector on the order of Q due to their interaction (in particular, in the vicinity of the superconducting with spin fluctuations, while this interaction for parti- transition temperature), which will be henceforth cles with momenta far away from hot spots is quite described by using the simple approach of the BCS the- weak. ory and the GinzburgÐLandau phenomenology. Considering the range of high temperatures 2πT ӷ The spectrum of the initial (free) quasiparticles will ω sf, we can disregard the spin dynamics [6], confining be taken in the form [6] our analysis to the static approximation. Computations can be considerably simplified and the contributions ξ () p = Ð2tpcos xap+ cos ya from higher orders of perturbation theory can be ana- (2) µ lyzed if we pass to the model interaction of electrons Ð4t'cos pxapcos ya Ð , with spin (or charge) fluctuations of the form [7] where t is the integral of transfer between the nearest () V eff q neighbors, t' is the same for the next to nearest neigh- Ð1 Ð1 bors in a square lattice, a is the lattice parameter, and µ 2 2ξ 2ξ (1) = W ------, is the chemical potential. This expression provides a ξÐ2 ()2 ξÐ2 ()2 + qx Ð Qx + qy Ð Qy good approximation to the results of band calculations for real HTSC systems. For example, for where W is an effective parameter having the dimension YBa2Cu3O6 + δ, we have t = 0.25 eV and t' = Ð0.45t [6]. of energy. Here, as in [6, 7], W and ξ are treated as phe- Chemical potential µ is determined by the carrier con- nomenological parameters (which are determined from centration. experiment). Expression (1) is qualitatively similar to In the limit of an infinitely large correlation length the static limit of the interaction considered in [12, 13] ξ ∞ and quantitatively differs insignificantly from this limit ( ), the model of scattering from short-range in the most interesting region |q Ð Q| < ξÐ1, which deter- order fluctuations of the type considered here has an ξ mines scattering in the vicinity of hot spots, if the exact solution [14]. For finite values of , we can con- parameters appearing in this expression are appropri- struct an approximate solution [7] generalizing the 1D ately defined. In fact, we are talking about the replace- approach proposed in [15]. In this case, it is possible to ment of the actual interaction with dynamic short-range sum (approximately) the entire diagrammatic series for order fluctuations by the electron scattering from the the one-particle electron Green function. As a result, static random (Gaussian) field of such fluctuations. The the following recurrent procedure arises for the one-

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750 KUCHINSKIŒ et al.

ε p + q details corresponding to incommensurate fluctuations n of the CDW type can be found in [7, 14, 15]. The conditions for applicability of this approxima- χ q p p tion were discussed in detail in [6, 7]. In the limit ( ) = e( ) É e( ') ξ ∞, relation (3) is reduced to the exact solution [14], while in the limit ξ 0 for a fixed value of W, relation (3) gives a physically correct limit of free elec- ε – n Ðp trons. Passing to superconductivity in the system with Fig. 2. Diagrammatic representation for generalized sus- χ developed short-range order fluctuations considered ceptibility (q) in a Cooper channel. here, we assume that the superconducting pairing is due to the attractive potential acting between electrons with ε opposite spins of the simplest (BCS) form, particle Green function G( np) (representation in the (), ()() form of a chain fraction) [6, 7, 15]: V sc pp' = ÐVe p e p' , (7) ()ε Gk np where for e(p) we assume that 1 (3) = ------; 1 ()s pairing , ε ξ () v κ 2 ()()ε  i n Ð k p + ik k Ð W sk+ 1 Gk + 1 np cos()p a Ð cos()p a ()d 2 2 pairing , x y x Ð y κ ξÐ1 ε π  here, = , n = 2 T(n + 1/2) (for definiteness, we ()  ()()() ε e p = sin pxa sin pya dxy pairing , (8) assume that n > 0),  cos()p a + cos()p a ξ x y  pQ+ for odd k,  ξ ()p =  (4) ()anisotropic s pairing . k ξ  p for even k, As usual, the attraction constant V is assumed to be ω  v ()v () nonzero in a certain layer of width 2 c in the vicinity of x pQ+ + y pQ+ for odd k, ω v =  (5) the Fermi level ( c is the characteristic frequency of k () ()  v x p + v y p for even k, quanta, which ensures attraction between electrons). In the general case, the superconducting gap is anisotropic ∂ξ ∂ where v(p) = p/ p is the velocity of a free quasi- and has the form ∆(p) = ∆e(p). particle. The subsequent analysis will be carried out under The “physical” Green function is defined from rela- the assumption of self-averaging of the energy gap of ε ≡ ε tion (3) as G( np) G0( np). the superconductor over short-range order fluctuations, The combinatorial factor is given by which allows us to use the standard approach of the theory of disordered superconductors [16, 17]. Under the conditions when the short-range order correlation sk()= k (6) ξ Ӷ ξ ξ ∆ length is 0, where 0 ~ vF/ 0 is the coherence length of the BCS theory (i.e., when fluctuations corre- in the case of commensurate functions with Q = (π/a, π late over distances smaller than the characteristic size /a) [15], which will be considered below, if we disre- of Cooper pairs), the assumption concerning self-aver- gard their spin structure [6] (i.e., if we confine our anal- aging of ∆ must be preserved, being violated only in the ysis to fluctuations of the CDW type). If we take into region1 ξ > ξ [9Ð11]. account the spin structure of the interaction in the 0 model of a nearly antiferromagnetic Fermi liquid (the spin-fermion model [6]), the combinatorics of dia- 3. COOPER INSTABILITY. grams becomes more intricate. In particular, the spin RECURRENCE PROCEDURE and charge two-particle vertices differ considerably in FOR THE VERTEX PART this model. The spin interaction was described in [6] by using the isotropic Heisenberg model. If we adopt the It is well known that the superconducting transition Ising model for this interaction, we will be left only temperature can be determined from the equation for with scattering processes with electron spin conserva- Cooper instability of the normal phase, tion, for which the commensurate combinatorics of dia- 1 Ð0,Vχ()0; T = (9) grams (6) is valid both for the one-particle Green function and for spin and charge vertices. For this reason, 1 The absence of self-averaging of the superconducting gap even in we will confine our analysis only to the case of com- ξ ξ the region < 0 , which was obtained in our previous publication mensurate (6) “Ising” spin fluctuations (AFM, SDW) [11], is apparently due to the specific nature of the short-range and commensurate charge fluctuations (CDW). The order model used in that work.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 SUPERCONDUCTIVITY IN THE PSEUDOGAP STATE IN THE HOT-SPOT MODEL 751 where the generalized Cooper susceptibility is defined s(1) in Fig. 2 and is given by Γ G Γ G 1 1 χ()q; T =+ (10) G ()()Φ ()ε ,,ε G = ÐTe∑∑ p e p' pp, ' n Ð n q ; 1 ε , n pp' Φ ε ε here, p, p'( n, Ð n, q) is a two-particle Green function s(k) in the Cooper channel, which takes into account the Gk scattering from short-range order fluctuations. Gk – 1 We will first consider the case of charge fluctuations =+ Γ (CDW), where the interaction is independent of spin k Γ Gk – 1 variables. For the s and dxy pairing, the superconducting k – 1 Gk gap remains unchanged upon a transfer over Q (i.e., e(p + Q) = e(p)) and e(p') ≈ e(p). In the case of aniso- Fig. 3. Recurrence equations for the vertex part. tropic s and d 2 2 pairing, the superconducting gap x Ð y reverses its sign upon a transfer over Q (e(p + Q) = Thus, we must calculate the triangular vertices tak- −e(p)); consequently, e(p') ≈ e(p) for p' ≈ p and e(p') ≈ − ≈ ing into account all diagrams (including cross dia- e(p) for p' p + Q. Thus, for diagrams containing an grams) describing the interaction with dielectric fluctu- even number of interaction lines connecting the upper ε ε ≈ ations. The corresponding recurrence procedure for a ( n) and lower (Ð n) electron lines, we have p' p; Thus, 1D analog of our problem (and for real-valued frequen- we arrive at the same expression for the contribution to cies, T = 0) was formulated for the first time in [18]. For susceptibility as in the case of the s and dxy pairing. On the 2D model of the pseudogap with hot spots at the the other hand, for diagrams with an odd number of Fermi surface considered here, a generalization of this such interaction lines, we obtain an expression with the recurrence procedure is given in [19] in connection opposite sign for the contribution to susceptibility. This with optical conductivity calculations. The details of sign reversal can be attributed simply to the sign rever- the corresponding derivation can also be found in [19]. sal for the interaction connecting the upper and lower A generalization to the case of Matsubara frequencies electron lines of the loop in Fig. 2. In this case, we required for our problem can be carried out directly. For obtain for the generalized susceptibility the expression definiteness, we will henceforth assume, as before, that ε > 0. Ultimately, for a triangular vertex, we obtain the χ() ()ε ()ε , 2() n q; T = ÐTG∑∑ npq+ G Ð n Ðp e p recurrence relation represented by the graphs in Fig. 3 ε (11) n p (where the wavy line indicates the interaction with pseudogap fluctuations) and having the following ana- ×Γ±()ε ,,ε n Ð n q , lytic form: Γ± ε ε where ( n, Ð n, q) is the triangular vertex part taking Γ± ()ε ,,ε ± 2 () into account the interaction with short-range order fluc- k Ð 1 n Ð n q = 1 W skGkGk tuations, the superscript “±” allowing for the above- mentioned difference in the signs of interactions con- v κ × 2ik k necting the upper and lower electron lines. 1 + ------(12) 2iε Ð v ⋅ q Ð W2sk()+ 1 ()G Ð G Let us now consider the scattering from spin fluctu- n k k + 1 k + 1 ations (AFM (SDW)). In this case, the line of interac- z ×Γ±()ε ,,ε tion with the longitudinal spin component S , which k n Ð n q ; embraces the vertex and changes the direction of the spin, should be supplemented with an additional factor here, G = G (ε p + q) and G = G (Ðε , Ðp) are calcu- of (Ð1) [6]. From this point of view, in the case of inter- k k n k k n lated in accordance with expression (3), v is defined by action with spin fluctuations, the types of pairing con- k sidered above “change places”2 and the generalized formula (5), and vk have the form Cooper susceptibility is determined by triangular ver- ΓÐ Γ+  () tex for s and dxy pairing and by triangular vertex vp+ Q for odd k, vk =  (13) for anisotropic s and d 2 2 pairing. () x Ð y vp for even k .

2 Γ± ε ε ≡ This is due to the fact that the sign of the spin projection is A “physical” vertex is deﬁned as ( n, Ð n, q) reversed at the vertex of the interaction with the superconducting Γ± ε ε gap (we consider only the singlet pairing). 0 ( n, Ð n, q).

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Tc/Tc0 which characterizes the superconducting state in our model in the absence of pseudogap fluctuations (W = 0). 1.0 In this case, the equation for the corresponding superconducting transition temperature Tc0 has the standard form for the BCS theory (in the general case of anisotropic pairing) and can be written as 0.8 m π π 2VT e2()p 1 = ------∑ ∫dpx∫dpy------, (16) π2 ξ2 + ε2 n = 0 0 0 p n 0.6 ω π where m = [ c/2 Tc0] is the dimensionless cutoff parameter for the sum over Matsubara frequencies. All s2 calculations were made for a typical quasiparticle spec- µ 0.4 trum (2) in HTSC with = Ð1.3t and t'/t = Ð0.4. Choos- ing (quite arbitrarily) ω = 0.4t and T = 0.01t, we can s1 c c0 easily select the value of pairing parameter V in rela- d2 tion (16), which gives the same value of Tc0 for various types of pairing enumerated in (8). In particular, we 0.2 2 d1 obtain V/ta = 1 for the conventional isotropic s-type 2 pairing and V/ta = 0.55 for the d 2 2 -type pairing. For x Ð y the remaining types of pairing from relation (8), the values of the pairing constant for such a choice of param- 0 20 40 60 eters are found to be unrealistically high and we do not W/Tc0 give the results of the corresponding calculations.3 Fig. 4. Dependence of the superconducting transition tem- Figures 4 and 5 show typical results of numerical perature Tc/Tc0 on effective pseudogap width W/Tc0 for the calculations of the superconducting transition tempera- s-type pairing and scattering from charge (CDW) fluctua- ture Tc for a system with a pseudogap, which were tions (curves s1 and s2) and for the d -type pairing and x2 Ð y2 obtained directly from relation (9) using the recurrence scattering from spin (AFM (SDW)) fluctuations (curves d1 equations described above. It can be seen that and d2). The data are given for the following values of pseudogap (dielectric) fluctuations considerably reduce reciprocal correlation length: κa = 0.2 (s1 and d1) and κa = the superconducting transition temperature in all cases. 0.5 (s2 and d2). The d 2 2 pairing is suppressed much more rapidly x Ð y than the isotropic s pairing. At the same time, a To determine Tc, we must consider the vertex for decrease in correlation length ξ (an increase in param- + Ð eter κ) of pseudogap fluctuations facilitates an increase q = 0. In this case, Gk = G* and vertices Γ and Γ k k k in T . These results are quite analogous to those become real-valued, which considerably simplifies pro- c cedures (12) For ImG and ReG , we have the system obtained earlier in the model of hot regions [8, 10]. k k However, considerable differences also arise. It can be of recurrence equations seen from Fig. 4 that the curve describing the dependence of Tc on pseudogap width W has a characteristic ε + kv κ Ð W2sk()+ 1 ImG n k k + 1 plateau in the region of W < 10Tc0 for s pairing and scat- ImGk = Ð------, Dk tering from charge (CDW) fluctuations as well as for (14) 2 d 2 2 pairing and scattering from spin (AFM (SDW)) ξ () () x Ð y k p + W sk+ 1 ReGk + 1 4 ReGk = Ð------, fluctuations (i.e., in the cases when the upper sign in D k formulas (12) and (15) “operates”, leading to sign-con- ξ 2 2 ε κ stant recurrence procedure for a vertex), while a consid- where Dk = ( k(p) + W s(k + 1)ReGk + 1) + ( n + kvk Ð 2 2 W s(k + 1)ImGk + 1) and the vertex part for q = 0 can be 3 Of course, such a description on the basis of equations in the BCS determined from the equation theory with weak binding does not claim to be realistic in the cases of s and d pairing considered here as well. We must x2 Ð y2 ± − 2 ImGk ± just preset the characteristic scale of T to express all tempera- Γ = 1 + W sk()------Γ . (15) c0 k Ð 1 ε 2 () k tures in subsequent calculations in units of this temperature, n Ð W sk+ 1 ImGk + 1 assuming that a certain universality relative to this scale exists in the problem considered here. Passing to numerical calculations, it is convenient to 4 The latter case is realized, in all probability, in actual HTSC set the characteristic scale of energies (temperatures), materials based on copper oxides.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 SUPERCONDUCTIVITY IN THE PSEUDOGAP STATE IN THE HOT-SPOT MODEL 753 erable suppression of Tc takes place on a scale of W ~ Tc/Tc0 50Tc0. Qualitative differences appear in the case of s pairing and scattering from spin (AFM (SDW)) fluctu- 1.0 ations and in the case of d 2 2 pairing and scattering x Ð y from charge fluctuations. Figure 5 shows that, in the latter case (when the lower sign in formulas (12) and (15) operates; i.e., an alternating procedure arises for a ver- 0.8 tex), the rate of suppression of Tc is an order of magnitude higher. In the case of d 2 2 pairing, in the range x Ð y of W/Tc0 values corresponding to almost complete sup- 0.6 pression of superconductivity, the accuracy of our calculations becomes considerably worse in view of the alternating nature of the recurrence procedure for the vertex part. In particular, a typical ambiguity of Tc may 0.4 appear, which corresponds to possible existence of a narrow region of “recurrent” superconductivity on the 5 s2 phase diagram. Such a behavior of Tc slightly resem- bles similar peculiarities emerging in superconductors 0.2 d2 with Kondo impurities [20]. Our calculations show, s1 however, that the most probable scenario is the emer- d1 gence of the critical value of parameter W/Tc0, for which superconductivity is completely suppressed. In 0 2 4 6 8 10 12 this case, a region may appear, in which the transition to the superconducting state becomes a first-order W/Tc0 phase transition analogously to the known situation in superconductors with a strong paramagnetic effect in Fig. 5. Dependence of the superconducting transition temperature Tc/Tc0 on effective pseudogap width W/Tc0 for the an external magnetic field [21]. In any case, the effects s-type pairing and scattering from spin (AFM (SDW)) fluc- arising in this case deserve a separate analysis. All tuations (curves s1 and s2) and for the d -type pairing results considered below correspond to the region of x2 Ð y2 unambiguous behavior of T . and scattering from charge (CDW) fluctuations (curves d1 c and d2). The data are given for the following values of reciprocal correlation length: κa = 0.2 (s1 and d1) and κa = 1.0 (s2 and d2). 4. GINZBURGÐLANDAU EXPANSION In our earlier publication [8], the GinzburgÐLandau the order parameter, which can be written for various expansion was constructed in the exactly solvable ∆ ∆ model of a pseudogap with an infinitely large correla- types of pairing in the form (p, q) = qe(p). Expan- tion length for short-range order fluctuations. Subse- sion (17) is determined by the graphs of the loop expan- quently [10], these results were extended to the case of sion for free energy in the field of order parameter fluc- finite correlation lengths. In these publications, we con- tuations (shown by dashed lines) with a small wave sidered, in fact, only charge fluctuations and used a vector q [8], which are represented in Fig. 6. simple model of the pseudogap state, which was based It is convenient to write the GinzburgÐLandau coef- on the concept of hot (plane) regions existing at the ficients in the form Fermi surface. In this model, the sign of the supercon- AA===K , CCK , BBK , (18) ducting gap remained unchanged upon a transfer over 0 A 0 C 0 B vector Q both for s and d pairing [10]. Here, we carry where A0, C0, and B0 stand for the expressions for these out the generalization to a more realistic case of the coefficients in the absence of pseudogap fluctuations model of hot spots at the Fermi surface. (W = 0), which are derived in the Appendix for an arbi- ξ The GinzburgÐLandau expansion for the difference trary spectrum p and various types of pairing, in the free energies of the superconducting and normal states can be written in the standard form ()TTÐ c 〈〉2() A0 = N0 0 ------e p , T c ∆ 2 2 ∆ 2 B ∆ 4 Fs Ð Fn = A q ++q C q --- q , (17) ζ() 2 ()7 3 〈〉()2 2() C0 = N0 0 ------vp e p , (19) ∆ π2 2 where q is the amplitude of the Fourier component of 32 T c

7ζ()3 4 5 Such a peculiar behavior of T is manifested more strongly in the B = N ()0 ------〈〉e ()p , c 0 0 π2 2 case of scattering from incommensurate pseudogap ﬂuctuations. 8 T c

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ε n p + q The situation with coefficient B is more complicated in the general case. Calculations can be significantly ∆ ∆* simplified if we confine our analysis, as usual, to the qe(p) e(p') q |∆ |4 Fs – Fn = É case of q = 0 in the order of q and define coefficient B by the diagram show in Fig. 6b.Then we obtain the ε following expression for coefficient KB: – n –p (a) ε p T c 4()()()ε ()ε , 2 n K B = ----- ∑∑e p G np G Ð n Ðp B0 ε p (22) |∆ |2 n – q e(p) e(p') T = Tc É ×Γ()±()ε ,,ε 4 n Ð0n . ε – n –p It should be noted from the very outset that this expres- ∗ sion leads to a positive definite coefficient B. This fol- ∆ ∆ ε ε lows from the fact that G(Ð n, Ðp) = G*( np) so that ε ε (b) G( np)G(Ð n, Ðp) is real-valued; accordingly, vertex Γ± ε ε part ( n, Ð n, 0) defined by recurrence procedure (15) is also real. + 5. PHYSICAL CHARACTERISTICS OF SUPERCONDUCTORS WITH A PSEUDOGAP ∆∗ ∆ It is well known that the GinzburgÐLandau equations define two characteristic lengths of superconduc- Fig. 6. Graphical form of the GinzburgÐLandau expansion. tor, viz., the coherence length and the magnetic field penetration depth. For a given temperature, coherence length ξ(T) angle brackets denote conventional averaging over the gives the characteristic scale of inhomogeneities of Fermi surface, order parameter ∆: 1 〈〉… δξ()… 2 C = ------()∑ p , ξ () ---- N0 0 T = Ð . (23) p A and N0(0) is the density of states for free electrons at the In the absence of a pseudogap, we can write Fermi surface. ξ2 () C0 All peculiarities of the model in question, which are BCS T = Ð------. (24) associated with the emergence of a pseudogap, are con- A0 tained in dimensionless coefficients KA, KC, and KB. In the absence of pseudogap fluctuations, all these coeffi- In our model, we have cients are equal to unity. 2 ξ ()T KC ------It can be seen from Fig. 6a that coefficients KA and ------2 - = . (25) ξ ()T K A KC are completely determined by the generalized Coo- BCS per susceptibility [8, 10] χ(q; T) depicted in Fig. 2: For the magnetic field penetration depth, we have χ()χ0; T Ð ()0; T ------c 2 K A = , (20) 2 c B A0 λ ()T = Ð------. (26) 32πe2 AC χ()χ() q; T c Ð 0; T c KC = lim ------. (21) Analogously to relation (25), in the given model, we q → 0 2 q C0 can write

It was shown above that the generalized susceptibility 1/2 can be found from relation (11), where the triangular λ()T K B λ------()= ------. (27) vertices are determined by recurrence procedures (12); BCS T K AKC this allows us to directly calculate coefﬁcients KA and KC numerically. In the vicinity of Tc, the upper critical ﬁeld Hc2 is

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2 λ/λ ξ ξ BCS 2/ BCS 1.0 2.4 s

2.2 0.8 d 2.0

1.8 0.6

1.6

0.4 1.4 d s 1.2 0.2 1.0

0.8 0 20 40 60 0 20 40 60 W/Tc0 W/Tc0 Fig. 7. Dependence of the squared coherence length λ λ 2 Fig. 8. Dependence of the penetration depth / BCS on ξ2/ξ on effective pseudogap width W/T for the s-type BCS c0 effective pseudogap width W/Tc0 for the s-type pairing and pairing and scattering from charge (CDW) fluctuations scattering from charge (CDW) fluctuations (solid curve) (solid curve) and for the d -type pairing and scattering and for the d -type pairing and scattering from spin x2 Ð y2 x2 Ð y2 from spin (AFM (SDW)) fluctuations (dotted curve). The (AFM (SDW)) fluctuations (dotted curve). The data are data are given for the reciprocal correlation length κa = 0.2. given for the reciprocal correlation length κa = 0.2. defined in terms of the GinzburgÐLandau coefficients as model can be written as φ φ ()C Ð C 2 0 0 A s n Tc T c K A Hc2 ==------Ð------, (28) ∆C ≡ ------= ------. (32) πξ2() 2πC ()C Ð C T K 2 T s n Tc0 c0 B φ π | | where 0 = c / e is the magnetic flux quantum. Then Coefficients KA, KB, and KC were calculated numeri- the slope of the curve describing the upper critical field cally for the same typical parameters of the model as in near Tc is given by the calculations of Tc described above. The numerical values of these coefficients as such are not very interest- 16πφ 〈〉e2()p K 6 dHc2 0 A ing and are not given here. Figures 7Ð12 show the ------= ------T c------. (29) 2 2 K W/Tc0 dependences of the corresponding physical quan- dT Tc 7ζ()3 〈〉vp()e ()p C tities, defined by relations (23)–(32). In accordance The specific heat discontinuity at the transition point with the situation with Tc described above, two qualita- has the form tively different modes of the behavior are also observed in this case depending on whether the behavior of the T A 2 ()C Ð C = -----c ------, (30) vertex part in the recurrence equations is sign-constant s n Tc  B TTÐ c or alternating (the upper and lower signs in relation (12) and spin or charge fluctuations). The results of calcula- where Cs and Cn are the specific heats of the supercon- tions of physical quantities for the first case (the s-type ducting and normal states, respectively. At temperature pairing and scattering from charge (CDW) fluctuations

Tc0 (in the absence of a pseudogap, W = 0), we have as well as the d 2 2 -type pairing and scattering from x Ð y π2 〈〉2()2 spin (AFM (SDW)) fluctuations) are shown in Figs. 7Ð () ()8 T c0 e p Cs Ð Cn = N 0 ------. (31) Tc0 4 6 7ζ()3 〈〉e ()p The typical dependences of these coefficients on parameter W/Tc0 are functions rapidly decreasing from unity in the superconduc- Then the relative specific heat discontinuity in the given tivity range.

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| | dHc2/dT 10. It can be seen that, with increasing pseudogap width 2.0 W, coherence length ξ(T) decreases, while penetration depth λ(T) increases as compared to the corresponding d2 s2 values in the BCS theory. Both these characteristic lengths exhibit a very weak dependence on parameter d1 κ s1 ; for this reason, the results in Figs. 7 and 8 are given 1.5 only for κa = 0.2. The slope (derivative) of the upper critical field at T = Tc first increases and then begins to decrease. The most typical is the decrease in the specific heat discontinuity as compared to the BCS value (see Fig. 10), which is in direct qualitative agreement 1.0 with experimental data [22]. It should be noted that the specific heat discontinuity in our model also has a characteristic plateau in the region of W/Tc0 < 10, which is similar to that noted above in the corresponding dependence of Tc. 0.5 0 20 40 60 The behavior of physical quantities in the case of the W/Tc0 s-type pairing and scattering from spin (AFM (SDW))

Fig. 9. Dependence of the derivative (slope) of the upper fluctuations and the d 2 2 -type pairing and scattering x Ð y critical field on effective pseudogap width W/Tc0 for the s-type pairing and scattering from charge (CDW) fluctua- from charge (CDW) fluctuations is illustrated in tions (curves s1 and s2) and for the d -type pairing and Figs. 11 and 12. Data on the characteristic lengths are x2 Ð y2 not shown since both coherence length ξ(T) and pene- scattering from spin (AFM (SDW)) fluctuations (curves d1 tration depth λ(T) are virtually the same as the corre- and d2). The data are given for the values of reciprocal correlation length κa = 0.2 (s1 and d1) and κa = 0.5 (s2 and sponding values in the BCS theory everywhere in the d2) and are normalized to the value of the derivative in the superconductivity range (except a small neighborhood absence of a pseudogap. of the region of ambiguity and vanishing of Tc, in which

|dH /dT| ∆C c2 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4 s2 s2 s1 s1 0.2 0.2 d2 d1 d2 d1 0 246810 0 20 40 60 W/Tc0 Fig. 11. Dependence of the derivative (slope) of the upper W/Tc0 critical field on effective pseudogap width W/Tc0 for the Fig. 10. Dependence of the specific heat discontinuity at the s-type pairing and scattering from spin (AFM (SDW)) fluc- transition point on effective pseudogap width W/T for the c0 tuations (curves s1 and s2) and for the d 2 2 -type pairing s-type pairing and scattering from charge (CDW) fluctua- x Ð y tions (curves s1 and s2) and for the d -type pairing and and scattering from charge (CDW) fluctuations (curves d1 x2 Ð y2 and d2). The data are given for the values of reciprocal cor- scattering from spin (AFM (SDW)) fluctuations (curves d1 relation length κa = 0.2 (s1 and d1) and κa = 1.0 (s2 and d2) and d2). The data are given for the values of reciprocal cor- and normalized to the value of the derivative in the absence relation length κa = 0.2 (s1 and d1) and κa = 0.5 (s2 and d2). of a pseudogap.

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∆C tions of the CDW or AFM (SDW) type and the separation of two classes of qualitatively different models of such suppression depending on the sign-constant or 1.0 alternating behavior of the vertex part in the recurrence equations (the upper or lower signs in expression (12) and spin or charge fluctuations). The version with scat- 0.8 tering from spin fluctuations and pairing with the d 2 2 -type symmetry is observed in high-T supercon- x Ð y c ductors based on copper oxides; however, we are not aware of systems in which the peculiar behavior 0.6 obtained above for the s-type pairing and scattering from spin (AFM (SDW)) fluctuations as well as for

d 2 2 -type pairing and scattering from charge (CDW) x Ð y 0.4 fluctuations is realized. The search for such systems is of considerable interest. s1 s2 The most important question in the description of 0.2 the pseudogap state of actual HTSC systems is the behavior of physical parameters upon a change in the d1 d2 carrier concentration. In our model, the concentration dependence must be expressed in terms of the corresponding dependence of effective width W of the 0 246810 ξ W/T pseudogap and correlation length . Unfortunately, such c0 dependences can be determined from experiment only 7 Fig. 12. Dependence of specific heat discontinuity at the indirectly and have been studied insufficiently [1, 2]. In transition point on the effective pseudogap width W/Tc0 for a very rough approximation, we can state that correla- s-type pairing and scattering from spin (AFM (CDW)) fluc- tion length ξ in a wide concentration range does not tuations (curves s1 and s2) and for d -type pairing and x2 Ð y2 vary very strongly, while pseudogap width W linearly scattering from charge (CDW) fluctuations (curves d1 and decreases with increasing charge carrier concentration d2). The data are given for the values of reciprocal correla- from values on the order of 103 K in the vicinity of the tion length κa = 0.2 (s1 and d1) and κa = 1.0 (s2 and d2). dielectric phase region to values on the order of the superconducting transition temperature as we approach the optimal doping level, vanishing at slightly higher these lengths sharply increase). As regards the deriva- carrier concentrations (see Fig. 6 in review [2], which is tive of the upper critical field and the specific heat dis- based on Fig. 4 in [3], where the corresponding set of continuity at the superconducting transition point, the data is given for the YBCO system). Using this regular- values of these quantities decrease quite rapidly with ity, one can easily recalculate the above dependences increasing parameter W/Tc0 apparently up to its critical on W to the corresponding dependences on the charge value at which Tc is completely suppressed (or to the carrier concentration. In the extremely simplified version value at which a narrow region of the first-order transi- of our model with an infinitely large correlation length tion is formed). and the Fermi surface with complete nesting, such an analysis was carried out in a recent publication [23] under the assumption that the value of Tc0 is also a lin- 6. CONCLUSIONS ear function of the concentration. The typical form of We have considered the peculiarities of the super- the phase diagram for HTSC cuprates was completely conducting state emerging in the pseudogap state due to reproduced qualitatively. At the same time, the obvious scattering of electrons from dielectric short-range order roughness of the model and the absence of reliable fluctuations in the model of hot spots at the Fermi sur- experimental data on the concentration dependences of ξ face. Our analysis was based on the microscopic deri- W, , and Tc0 do not make it possible to treat the vation of the GinzburgÐLandau expansion taking into attempts at “improving” these qualitative conclusions account all orders of perturbation theory in scattering very seriously. from pseudogap fluctuations. The condensed phase of In addition to the repeatedly mentioned disregard of such a superconductor can be described on the basis of the dynamics of short-range order fluctuations and the the corresponding analysis of the Gor’kov equations for confinement to Gaussian fluctuations alone, it should a superconductor with a pseudogap (see [10]) and is the be noted once again that the disadvantages of the model subject of special analysis. 7 The main result of this study is the demonstration of In addition, an analogous dependence of the value of Tc0, which superconductivity suppression by pseudogap fluctua- is completely unknown, may turn out to be significant.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 758 KUCHINSKIŒ et al. considered here also include the simplified analysis of As a result, coefficient A0 assumes the form the spin structure of interactions, which presumes that χ ()χ() 〈〉2() these interactions are of the Ising type. It would be A0 ==0 0; T Ð 0 0; T c ABCS e p , (A.3) interesting to also carry out a similar analysis for the general case of an interaction of the Heisenberg type. where χ ()χ () ABCS = BCS 0; T Ð BCS 0; T c ACKNOWLEDGMENTS (A.4) ()TTÐ c = N0 0 ------This study was partly supported by the Russian Foun- T c dation for Basic Research (project no. 02-02-16031); by the programs of fundamental studies in “Quantum is the standard expression for coefficient A in the case Macrophysics” (the Presidium of the Russian Academy of isotropic s pairing. of Sciences) and “Strongly Correlated Electrons in Coefficient C0 of the GinzburgÐLandau expansion is Semiconductors, Metals, Superconductors, and Mag- defined by generalized susceptibility (A.1) for small netic Materials” (Department of Physical Sciences, values of q: Russian Academy of Sciences); and by the project χ ()χ() “Investigation of Collective and Quantum Effects in 0 q; T c Ð 0 0; T c C0 = lim ------. (A.5) Condensed Media” (Ministry of Industry and Science q → 0 q2 of the Russian Federation). χ Expanding expression (A.1) for 0(q; Tc) into a series in small q, we obtain APPENDIX χ ()χ() 0 q; T c = 0 0; T c Ginzburg–Landau Coefficients for Anisotropic Pairing 3ε2 Ð ξ2 (A.6) in the Absence of a Pseudogap + T ∑∑------n p e2()p ()vp()q 2, c ()ε2 ξ2 3 ε 4 n + p In the absence of fluctuations (W = 0), the general- n p ized Cooper susceptibility, which is defined by the dia- so that we have for coefficient C the expression gram in Fig. 2, assumes the form 0 ∞ χ () 3ε2 Ð ξ2 0 q; T ξ n C0 = T c∑ ∫ d ------()ε2 ξ2 3 ε 4 n + (A.1) n Ð∞ 2() 1 1 = ÐTe∑∑ p ------ε ξ ------ε ξ-. i n Ð pq+ Ð i n Ð p 2 2 2 ε p ×δξξ()() () φ n ∑ Ð p e p vp cos p For the susceptibility at q = 0, which determines coef- ∞ ficient A , we obtain the expression 3ε2 Ð ξ2 0 ≈ T dξ------n - (A.7) c∑ ∫ 2 2 3 ε 4()ε + ξ 2 1 n Ð∞ n χ ()0; T = ÐTe∑∑ ()p ------0 ε2 ξ2 ε n + p 2 n p ×δξ()2() ()2 φ ∑ p e p vp cos ∞ p ξ 1 δξ() ξ 2() = ÐT∑ d ------∑ Ð p e p ζ() ∫ ε2 ξ2 ()7 3 〈〉2() ()2 2 φ ε ∞ n + = N0 0 ------e p vp cos , n Ð p 16π2T 2 ∞ (A.2) c 2 1 Σ δξ()e ()p ≈ ÐN ()0 T dξ------p p - where φ is the angle between vectors v(p) and q and 0 ∑ ∫ 2 2 () ε ξ N0 0 ε ∞ n + n Ð ∞ 1 ζ()3 = ----- ≈ 1.202. χ ()〈〉2() ∑ 3 = BCS 0; T e p , n = 1 n where the angle brackets denote averaging over the For a square lattice, the Fermi surface and, hence, χ Fermi surface and the standard susceptibility BCS(0; T) |v(p)| also possess a symmetry relative to rotation in the BCS model for isotropic s pairing is introduced. through angle π/2; the same symmetry is also inherent

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 SUPERCONDUCTIVITY IN THE PSEUDOGAP STATE IN THE HOT-SPOT MODEL 759 in e2(p) for the types of pairing considered here. Conse- REFERENCES quently, we can easily ﬁnd that 1. T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999).

2 2 2 2. M. V. Sadovskiœ, Usp. Fiz. Nauk 171, 539 (2001) [Phys. 〈〉e ()p vp()cos φ Usp. 44, 515 (2001)]. 3. J. L. Tallon and J. W. Loram, Physica C (Amsterdam) 1 2 2 = --- 〈〉e ()p vp()()12+ cos φ (A.8) 349, 53 (2000). 2 4. V. M. Krasnov, A. Yurgens, D. Winkler, et al., Phys. Rev. Lett. 84, 5860 (2000). 1 2 2 = --- 〈〉e ()p vp() , 5. N. P. Armitage, D. H. Lu, C. Kim, et al., Phys. Rev. Lett. 2 87, 147003 (2001). since quantity cos2φ reverses its sign when vector p 6. J. Schmalian, D. Pines, and B. Stojkovic, Phys. Rev. Lett. 80, 3839 (1998); Phys. Rev. B 60, 667 (1999). rotates through angle π/2. Indeed, the direction of velocity v(p) upon this rotation changes to the perpen- 7. É. Z. Kuchinskiœ and M. V. Sadovskiœ, Zh. Éksp. Teor. Fiz. 115, 1765 (1999) [JETP 88, 968 (1999)]. dicular direction; accordingly, cos2φ Ðcos2φ. As a result, we obtain the isotropic expression for coeffi- 8. A. I. Posazhennikova and M. V. Sadovskiœ, Zh. Éksp. Teor. Fiz. 115, 632 (1999) [JETP 88, 347 (1999)]. cient C , 0 9. É. Z. Kuchinskiœ and M. V. Sadovskiœ, Zh. Éksp. Teor. Fiz. 117, 613 (2000) [JETP 90, 535 (2000)]. 7ζ()3 2 2 C = N ()0 ------〈〉vp()e ()p ; (A.9) 0 0 π2 2 10. É. Z. Kuchinskiœ and M. V. Sadovskiœ, Zh. Éksp. Teor. 32 T c Fiz. 119, 553 (2001) [JETP 92, 480 (2001)]. 11. É. Z. Kuchinskiœ and M. V. Sadovskiœ, Zh. Éksp. Teor. in the case of isotropic s pairing and a spherical Fermi Fiz. 121, 758 (2002) [JETP 94, 654 (2002)]. surface, this expression acquires the standard form 12. P. Monthoux, A. V. Balatsky, and D. Pines, Phys. Rev. B 46, 14803 (1992). 7ζ()3 v 2 C = N ()0 ------F. (A.10) 13. P. Monthoux and D. Pines, Phys. Rev. B 47, 6069 (1993); BCS 0 π2 2 49, 4261 (1994). 32 T c 14. M. V. Sadovskiœ, Zh. Éksp. Teor. Fiz. 66, 1720 (1974) In the absence of pseudogap fluctuations (W = 0) [Sov. Phys. JETP 39, 845 (1974)]; Fiz. Tverd. Tela (Len- and for q = 0, coefficient B defined by the diagram in ingrad) 16, 2504 (1974) [Sov. Phys. Solid State 16, 1632 Fig. 6b has the form (1974)]. 15. M. V. Sadovskiœ, Zh. Éksp. Teor. Fiz. 77, 2070 (1979) [Sov. Phys. JETP 50, 989 (1979)]. 1 4 B = T ------e ()p 16. L. P. Gor’kov, Zh. Éksp. Teor. Fiz. 37, 1407 (1959) [Sov. 0 c∑∑ 2 2 2 ε ()ε + ξ Phys. JETP 10, 998 (1959)]. n p n p 17. P. G. de Gennes, Superconductivity of Metals and Alloys ∞ (Benjamin, New York, 1966; Mir, Moscow, 1968). ξ 1 δξ() ξ 4() 18. M. V. Sadovski and A. A. Timofeev, Sverkhprovodi- = T c∑ ∫ d ------∑ Ð p e p œ ()ε2 ξ2 2 most: Fiz. Khim. Tekh. 4, 11 (1991); J. Mosc. Phys. Soc. ε ∞ n + n Ð p 1, 391 (1991). ∞ (A.11) 4 1 Σ δξ()e ()p 19. M. V. Sadovskiœ and N. A. Strigina, Zh. Éksp. Teor. Fiz. ≈ N ()0 T ∑ dξ------p p - 122, 610 (2002) [JETP 95, 526 (2002)]. 0 c ∫ 2 2 2 N ()0 ε ()ε + ξ 0 20. E. Müller-Hartmann and J. Zittartz, Phys. Rev. Lett. 26, n Ð∞ n 428 (1971). = B 〈〉e4()p , 21. D. St. James, G. Sarma, and E. J. Thomas, Type II Super- BCS conductivity (Pergamon Press, Oxford, 1969; Mir, Mos- where cow, 1970). 22. J. W. Loram, K. A. Mirza, J. R. Cooper, et al., J. Super- ζ() ()7 3 cond. 7, 243 (1994). BBCS = N0 0 ------(A.12) 8π2T 2 23. A. Posazhennikova and P. Coleman, Phys. Rev. B 67, c 165109 (2003). is the standard expression for coefficient B in the case of isotropic s pairing. Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 760Ð769. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 868Ð878. Original Russian Text Copyright © 2004 by V. Gritsenko, D. Gritsenko, Novikov, Kwok, Bello. SOLIDS Electronic Properties

Short-Range Order, Large-Scale Potential Fluctuations, and Photoluminescence in Amorphous SiNx V. A. Gritsenkoa,*, D. V. Gritsenkoa, Yu. N. Novikova, R. W. M. Kwokb, and I. Belloc aInstitute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia bDepartment of Chemistry, Chinese University of Hong Kong, Shatin, Hong Kong, China cDepartment of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Hong Kong, China *e-mail: [email protected] Received September 12, 2003

Abstract—The short-range order and electron structure of amorphous silicon nitride SiNx (x < 4/3) have been studied by a combination of methods including high-resolution X-ray photoelectron spectroscopy. Neither random bonding nor random mixture models can adequately describe the structure of this compound. An intermediate model is proposed, which assumes giant potential ﬂuctuations for electrons and holes, caused by inhomogeneities in the local chemical composition. The characteristic scale of these ﬂuctuations for both electrons and holes is about 1.5 eV. The photoluminescence in SiNx is interpreted in terms of the optical transitions between quantum states of amorphous silicon clusters. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION SiH2Cl2; the nitrogen-containing gas is ammonium NH3. The process is carried out at a temperature of Amorphous silicon nitrides SiNx, together with sili- ° 700Ð800 C. The structure (short-range order) of SiNx con dioxide SiO2, are the main dielectrics used in mod- ern silicon-based electronic devices. Silicon nitride obtained by pyrolysis still remains unstudied. exhibits a unique memory effect, being capable of Although the memory effect in SiNx has been stud- localizing and capturing injected electrons and holes ied for more than a quarter of century, the nature of with a giant time of localized carrier trapping (about traps responsible for the localization of electrons and 10 years at 300 K) [1]. In recent years, the memory holes is still unknown [1, 3]. It was suggested [1] that effect in silicon nitride has been used for developing the role of traps for electrons and holes in SiNx can be electrically rewriteable ROM devices of Giga- and Ter- played by silicon clusters. Recently, Park et al. [17] abit capacity [2]. observed photoluminescence (PL) from quantum dots There are two alternative models describing the in plasma deposited SiNx. Therefore, we may suggest structure of amorphous layers of nonstoichiometric tet- that amorphous silicon quantum dots can exist in pyro- lytic SiN as well and can be detected by PL measure- rahedral silicon compounds (SiOx, SiOxNy, SiNx): the x random mixture (RM) model and the random bonding ments. (RB) model [3Ð17]. The RM model assumes that SiNx This paper reports on the results of investigations comprises a mixture of two phases, amorphous silicon into the short-range order, electron structure, and PL in (a-Si) and silicon nitride (Si3N4), and is composed of amorphous SiNx synthesized by pyrolysis. Based on the SiSi4 and SiN4 tetrahedra [8, 10]. According to the RB structural data, we propose a model assuming large- model, the structure of SiNx represents a network com- scale potential ﬂuctuations caused by inhomogeneities posed of SiNνSi ν tetrahedra of ﬁve types with ν = 0−4 of the local chemical composition of SiNx. The PL 4 Ð ≈ [3, 4, 11, 12]. Since amorphous SiNx is synthesized observed in SiNx (x 4/3) is interpreted in terms of the under thermodynamically nonequilibrium conditions, optical transitions between quantum states of amor- the product structure depends on the method of synthe- phous silicon clusters. sis. In particular, it was established that the structure of SiN obtained by plasma deposition is described by the x 2. SAMPLE PREPARATION RM model [8]. Silicon-based devices also widely AND EXPERIMENTAL TECHNIQUES employ SiNx synthesized by high-temperature pyrolysis of silicon- and nitrogen-containing gas mixtures. The experiments were performed with SiNx samples The silicon-containing component is typically silane synthesized in a low-pressure reactor by chemical ° SiH4, silicon tetrachloride SiCl4, or dichlorosilane vapor deposition (CVD) at 760 C from a SiH2Cl2ÐNH3

SHORT-RANGE ORDER, LARGE-SCALE POTENTIAL FLUCTUATIONS 761 gas mixture. The samples of SiNx with various compositions were obtained by changing the ratio of SiH2Cl2 and NH3 in the gas phase. The SiNx layers were deposited onto p-Si(100) substrates with a resistivity of ρ ≈ 10 Ω cm. The luminescence was studied in the samples ≈ of SiNx with x 4/3 synthesized by decomposition of a ° SiH4ÐNH3ÐH2 mixture at 890 C.

High-resolution X-ray photoelectron spectroscopy Intensity (XPS) measurements were performed in a Kratos AXIS-HS system using a source of monochromated AlKα X-ray radiation with បν = 1486.6 eV. The natural oxide film from the samples was removed by 2-min etching in an HFÐmethanol mixture (1 : 30), followed 106 104 102 100 98 by rinsing in methanol. Prior to XPS measurements, the samples were washed in cyclohexane and blown with Binding energy, eV flow of dry nitrogen. The binding energies were mea- Fig. 1. The XPS spectra of Si 2p levels in SiN0.51 measured sured relative to the 1s peak of carbon in cyclohexane for a photoelectron take-off angle of 20° (points) and 45° with a binding energy of 285.0 eV. In cases of signifi- and 90° (solid curves). cant positive charging of a sample, the charge was compensated using a beam of low-energy electrons. All the XPS measurements (except for the angle-resolved determined using a laser ellipsometer was about 800 Å. ones) were performed for the sample surface oriented The PL excitation spectra were measured using a deu- perpendicularly to the electron energy analyzer axis terium lamp of the DDS-400 type. (zero polar angle). In order to check that the XPS spectra reflect the 3. SHORT-RANGE ORDER bulk properties of SiNx, we performed angle-resolved IN SiNx BY XPS DATA measurements on a Phi Quantum 2000 spectrometer. Figure 1 shows the XPS spectra of Si 2p levels in Figure 2 shows the experimental XPS spectra (depicted by symbols) of Si 2p levels in SiN samples SiN0.51 measured for the photoelectron take-off angles x of 20°, 45°, and 90°. Weak angular dependence of the of various compositions. All these curves exhibit either signal shape shows evidence that the bulk of a sample two peaks or one peak with a shoulder and are analo- is probed, so that the XPS data obtained reflect the bulk gous to the spectra reported previously [7, 9, 13]. Applicability of the RB and RM models to description properties of SiNx. of the structure of SiNx with variable composition was In studying SiNx layers with different compositions checked by comparing the experimentally measured Si (i.e., with variable x), we used a nearly stoichiometric 2p spectra to the results of calculations based on these silicon nitride (Si3N4) as a reference sample for deter- models. mining the relative sensitivity factors with respect to Si Dashed curves in Fig. 2 show the XPS spectra cal- 2p and N 1s photoelectron lines. The reference sample ° culated in terms of the RB model. This model assumes synthesized at 800 C from a SiCl4ÐNH3 mixture with a that the structure consists of SiNνSi4 Ð ν tetrahedra of 1 : 10 ratio of components had a refractive index of 1.96 five types corresponding to ν = 0Ð4. The probability of and exhibited a characteristic IR absorption band at finding the tetrahedron with a given ν obeys a binomial Ð1 3300 cm due to the stretching vibrations of Si2NÐH distribution [6, 8] bonds. The calculated concentration of NÐH bonds in × 21 Ð3 this material was 2.1 10 cm . Thus, it was estab- ν 4 Ð ν lished that the reference sample had a composition of ()ν, 3x 3x 4! W x = ------1 Ð ------ν ()ν -. (1) SiN1.41H0.05. 4 4 !4Ð ! The Raman spectra were measured on a Renishaw Ramanscope spectrometer using HeÐNe laser radiation The idea of using five types of tetrahedra for model- (λ = 6328 Å). The IR absorption spectra were recorded ing the XPS spectrum of Si 2p levels is based on the on a Nicolet 550 spectrometer with a resolution of assumption that only nearest-neighbor silicon and/or 4 cmÐ1. nitrogen atoms contribute to the chemical shift of the Si 2p electron state. Equation (1) also assumes the ≈ The PL measurements for SiNx (x 4/3) samples absence of point defects such as =NÐN= bonds, dan- were performed at room temperature. The emission gling bonds (≡Si¥ and =N¥), and hydrogen bonds − spectra were normalized with respect to the sensitivity (Si H, NÐH) in SiNx (here, symbols “–” and “•” denote of the detection system. The sample film thicknesses a covalent bond and an unpaired electron, respectively).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 762 GRITSENKO et al. The results of electron paramagnetic resonance (EPR) measurements showed that the concentration of dan- gling bonds ≡Si¥ and =N¥ in our samples did not exceed 19 Ð3 SiN0.51 10 cm . Theoretical convolutions of the Si 2p spectra were obtained using the XPSPEAK 4.1 program package [14]. The theoretical spectra were calculated assuming the peaks corresponding to different SiNνSi ν tetrahedra SiN0.60 4 Ð to be equidistant (equally spaced on the energy scale). The peak halfwidth (deﬁned as the full width at half maximum, FWHM) was taken either the same for all peaks or linearly increasing as determined by extrapo- SiN0.72 lation between the peak halfwidth for Si and Si3N4. The results of calculations according to the RB model Intensity showed that the XPS spectrum of the Si 2p level in SiNx must contain a single maximum (Fig. 2, dashed SiN0.87 curves) with the peak position being shifted toward higher binding energies with increasing nitrogen content in SiNx. For the sake of generality, we also checked for the SiN1.04 applicability of the RB model to description of the structure of a disordered SiNx sample. The disorder was produced by irradiating a SiN sample with a beam of Si N Si 0.51 3 4 4-keV Ar+ ions (Fig. 3). As can be seen from Fig. 3a, the structure of the initial SiN0.51 sample is not adequately 105 103 101 99 97 95 described by the RB model: the experimental XPS spectrum exhibits two peaks corresponding (in the ﬁrst Binding energy, eV approximation) to the SiSi4 and SiN4 tetrahedra, Fig. 2. The XPS spectra of Si 2p levels in SiNx samples of whereas the theoretical model predicts a single peak various compositions: symbols represent the experimental with a maximum positioned at a Si 2p binding energy spectra; the results of theoretical calculations are depicted of the SiN2Si2 tetrahedron (Fig. 3b). After irradiation, by solid (RM model) and dashed (RB model) curves. nitrogen and silicon atoms are mixed and the sample exhibits a tendency to form a substitution solid solution (RB model). The XPS spectrum of the ion-bombarded (a) SiN0.51 sample displays a single peak with a binding energy of the maximum close to that calculated within the RB model (Fig. 3b).

The results of simulation of the XPS spectra of SiNx within the framework of the RM model are depicted by solid curves in Fig. 2. The spectra calculated using this model show, in agreement with experiment, a tendency to decrease in the fraction of a silicon phase in SiNx Intensity (b) with decreasing silicon content. However, the RM model somewhat overstates the silicon phase fraction and, in addition, predicts a dip approximately in the middle of the spectrum presented in Fig. 2. In experiments, however, such a dip is observed only for SiN0.51 and is absent in the XPS spectra of other samples. Thus, neither RB nor RM models can quantitatively 105 103 101 99 97 describe the structure of an SiNx compound. For this Binding energy, eV reason, we propose an intermediate model illustrated in Fig. 3. The XPS spectra (points) of Si 2p levels in SiN Fig. 4. This model suggests the presence of separate sil- 0.51 icon (SiSi tetrahedra) and Si N (SiN tetrahedra) measured (a) before and (b) after irradiation with 4-keV Ar+ 4 3 4 4 ions. Solid curves show the results of calculations using the phases, as well as subnitrides (composed of SiN3Si, RB model. SiN2Si2, and SiNSi3 tetrahedra) in the SiNx structure.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 SHORT-RANGE ORDER, LARGE-SCALE POTENTIAL FLUCTUATIONS 763

(a) E = 0 1234567 χ = 2.0 eV

Ec Φ e = 1.5 eV e

បω Eg(Si) = 1.6 eV Eg(Si3N4) = 4.6 eV

Φ h = 1.5 eV h Ev

A A

Si SiNx Si3N4 (b)

E Ec g e µ h

Fig. 4. Schematic diagrams illustrating the proposed intermediate model of SiNx: (a) a two-dimensional diagram of SiNx structure showing (bottom) the regions of a silicon phase, stoichiometric silicon nitride, and subnitrides and (top) the energy band profile of Φ Φ SiNx in the AÐA section (Ec is the conduction band bottom; Ev is the top of the valence band; e and h are the energy barriers for χ electrons and holes at the a-SiÐSi3N4 interfaces, respectively; Eg is the bandgap width; and is the electron affinity); (b) fluctuations of the ShklovskiiÐEfros potential in a strongly doped compensated semiconductor (µ is the Fermi level).

The proposed intermediate model assumes local spatial tribute to the IR absorption spectra either. At the same ﬂuctuations of the chemical composition of amorphous time, the method of Raman scattering allows excess sil- silicon nitride. This model will be considered in more icon in SiNx to be detected. detail in Section 6. Figure 5 shows the Raman spectra of SiNx samples of various compositions grown on silicon substrates. Here, the intense peak at 520 cmÐ1 is due to the longitu- 4. EXCESS SILICON IN SiNx ACCORDING TO RAMAN SCATTERING DATA dinal optical phonons in the silicon substrate. For SiNx samples with a low nitrogen content (x ≤ 0.72), there is The XPS measurements do not provide information an additional weak signal in the region of 460Ð480 cmÐ1, on the spatial distribution of silicon in the SiNx struc- that is, at a frequency coinciding with the position of the ture. The chemical shift of the Si 2p level is sensitive to peak of Raman scattering in amorphous silicon [18]. the local chemical environment, but not to the long- Previously, the Raman scattering from silicon in SiNx range order. Being nonpolar, SiÐSi bonds do not con- was studied in [15]. Thus, the Raman spectra provide

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 764 GRITSENKO et al. con and 2p orbitals of nitrogen [20]. In addition, Fig. 6b presents the high-resolution XPS spectra of Si for the valence bands of amorphous silicon nitrides Si3N4, SiN1.04, and Au measured relative to the top of the valence band of gold. SiN0.51 As can be seen, features of the valence band spectrum of SiN1.04 are generally analogous to those for Intensity SiN0.72 Si3N4, except for the region at the top of the valence band (Fig. 6b). The high-resolution XPS spectra show SiN that the top of the valence band of a-Si coincides with 1.04 that of gold. The electron work function of gold is 5.1 eV. Therefore, the top of the valence band of a-Si is 300 350 400 450 500 550 spaced 5.1 eV from the electron energy level in vacuum. The top of the valence band of Si N is spaced ∆k, cm–1 3 4 1.5 eV from the top of the valence band of a-Si Fig. 5. The Raman spectra of SiN1.04, SiN0.72, and SiN0.51 (Fig. 6b). Therefore, the energy barrier for holes at the Ð1 samples on silicon substrates. The peak at 520 cm corre- a-SiÐSi3N4 interface also amounts to 1.5 eV. sponds to scattering in silicon substrate; the arrow indicates the signal due to scattering from amorphous silicon clusters. 6. LARGE-SCALE POTENTIAL FLUCTUATIONS IN SiNx CAUSED unambiguous evidence for the presence of amorphous BY SPATIAL INHOMOGENEITIES silicon clusters in SiNx. IN THE CHEMICAL COMPOSITION

The bandgap width in Si3N4 is Eg = 4.6 eV. However, According to the XPS data, SiNx comprises a mix- experiments reveal the optical absorption in Si3Nx at ture of Si3N4, silicon subnitrides, and amorphous sili- photon energies below this value [1]. This signal can be con. The silicon nitride phase is composed of SiN4 tet- attributed to the absorption of light by silicon clusters rahedra; subnitrides are composed of SiN3Si, SiN2Si2, in a Si N matrix. The existence of silicon clusters in 3 4 and SiNSi3 tetrahedra; and the amorphous silicon clus- SiN is also confirmed by data on the fundamental x ters are composed of SiSi4 tetrahedra. The bandgap absorption edge, according to which long-wavelength width of compound Si3N4 is 4.6 eV, while that of amor- absorption takes place at 1Ð2 eV [16]. Silicon clusters phous silicon is 1.6 eV [19, 21] and that of silicon sub- with dimensions of 12–24 Å in hydrogenated silicon nitrides varies within 1.6Ð4.6 eV. Therefore, the band- nitride (SiNx:H) were observed in a high-resolution gap width in compound SiN also varies from 1.6 to electron microscope [17, 19]. The Raman spectra of x ≥ 4.6 eV. According to the data presented in Section 5, the SiNx with x 1.04 exhibit no signal related to the scat- maximum scale of potential fluctuations for holes is tering from silicon clusters. However, this result does 1.5 eV. Since the bandgap width of a-Si is 1.6 eV, the not exclude the existence of such clusters: the lack of energy barrier for electrons at the a-SiÐSi3N4 interface the signal can be explained by insufficient sensitivity amounts to 1.5 eV. Thus, the maximum scale of poten- of this method, related to small cluster size, low cluster tial fluctuations for electrons in SiN is also 1.5 eV. density, and small thicknesses of sample films (about x 1000 Å). Figure 4a presents the proposed model of large- scale potential fluctuations caused by variations in the local chemical composition of SiNx, as illustrated by a 5. DETERMINING ENERGY BARRIERS two-dimensional diagram showing all possible variants FOR HOLES AT THE SiÐSi3N4 INTERFACE of the local (spatial) structure of silicon nitride. The FROM XPS DATA energy band diagram refers to the AÐA section; the straight line indicates the level from which the electron For a comparative study of the valence bands of energies are measured (vacuum level). A decrease in Si3N4, SiNx, and a-Si, we have measured the corre- the bandgap width Eg is evidence of the presence of sponding XPS spectra. The samples of a-Si were pre- subnitrides in the silicon nitride matrix. The minimum + pared by irradiating crystalline silicon with 4-keV Ar bandgap width (Eg = 1.6 eV) corresponds to the silicon ions. The valence band of Si3N4 consists of two sub- phase. This model assumes smooth variation of the bands separated by an ion gap (Fig. 6a). The narrow chemical composition at the boundaries between sili- lower subband is formed by N 2s orbitals with an con clusters and the Si3N4 matrix. Our experimental admixture of Si 3s and 3p orbitals [3]. The broad upper data do not allow the size of this transition region to be subband is formed by the nonbonding 2pπ orbitals of estimated. We reckon that this size may be on the order nitrogen and the bonding 3s, 3p, and 3d orbitals of sili- of several dozens of ångströms.

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N(2s) (a)

Si(3s, 3p) (b)

Si3N4

SiN1.04 Ev

Si3N4 Intensity a-Si

2 0 –2 –4 Binding energy, eV Intensity

SiN1.04

a-Si

20 10 0 Binding energy, eV

Fig. 6. (a) XPS spectra of the valence band of Si3N4, SiN1.04, a-Si, and Au samples (arrows indicate the energy position of the top of the valence band); (b) the same spectra in the top of the valence band region recorded at a higher resolution.

Region 1 in Fig. 4a corresponds to a “quantum” that, here and below, we assume that the size of the cluster (with dimensions L on the order of the de Bro- transition region occupied by silicon subnitrides is sig- glie wavelength of quasi-free electrons in a silicon clus- nificantly greater than the length of Si–N and Si–Si ter) incorporated into the Si3N4 matrix. The ground bonds (amounting to 1.72 and 2.35 Å, respectively). state energy in this cluster is E = ប2/2mL2, where m is Region 4 corresponds to a silicon subnitride cluster in the effective electron mass. Region 2 represents large the silicon nitride matrix. Region 5 is a “quantum” sili- silicon clusters surrounded by Si N . In this case, there con cluster incorporated into the subnitride phase, and 3 4 regions 6 and 7 represent subnitride and nitride clusters, is no transition layer of silicon subnitrides and the respectively, surrounded by silicon. energy band diagram reveals a sharp SiÐSi3N4 interface. Large clusters do not feature quantization of the Thus, fluctuations of the local chemical composition energy levels of electrons and holes. Region 3 is a mac- of SiNx lead to large-scale spatial fluctuations of the roscopic silicon cluster surrounded by a silicon subni- potential for electrons and holes. Previously, similar tride phase. In this situation, the transition from silicon models of large-scale potential fluctuations have been to Si3N4 in the energy band diagram is smooth. Note developed for Si:H [22], SiC:H [23], SiCxOz:H [24],

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1.0 2.4 eV 5.2 eV 12

0.8

0.6

0.4 Relative intensity Relative

0.2

0 1 234567 Quantum energy, eV ≈ Fig. 7. The spectra of (1) PL and (2) PL excitation in SiNx (x 4.3) at room temperature. Points represent the experimental data, solid curves show the results of approximation by the Gauss function.

and SiOx [25]. When an electronÐhole pair is generated reported in [27, 30]. Park et al. [17, 19] studied the opti- in silicon subnitride, the electric field is directed in the cal absorption and PL spectra of amorphous silicon same direction for both electron and hole, thus favoring clusters in the Si3N4 matrix, observed quantum confine- their recombination (Fig. 4a). In the case of a radiative ment of electrons and holes at these quantum dots, and recombination mechanism, SiNx is an effective radia- determined [17] the PL peak energy as a function of the tive medium. Figure 4b illustrates the ShklovskiiÐEfros average size of amorphous clusters. As the cluster size model of large-scale potential fluctuations in a strongly decreased from 2.9 to 13 Å, the PL peak shifted from doped compensated semiconductor [26]. According to 1.8 to 2.7 eV. According to [17, Fig. 1], the PL quantum this model, the bandgap width is constant and the energy of 2.4 eV observed in our experiments corre- potential fluctuations are caused by the inhomogeneous sponds to a silicon cluster size of 17 Å. Figure 8 shows spatial distribution of charged (ionized) donors and acceptors. Here, the electronÐhole pair production is a configuration diagram of a defect for a strong elec- accompanied by spatial separation of electrons and tronÐphonon coupling constructed using the available holes, thus not favoring their recombination. PL and PL excitation data. The FranckÐCondon shift

7. PHOTOLUMINESCENCE OF SILICON NITRIDE SiNx Figure 7 shows the room-temperature PL spectrum ≈ of SiNx (x 4/3) with nearly stoichiometric composition excited by quanta with an energy of 5.2 eV (curve 1). The emission has a maximum intensity at an energy of 1.4 eV E2 Elum = 2.4 eV. Approximated by a Gauss function, the energy Total PL peak has an FWHM of 0.92 eV. The spectrum of 5.2 eV 2.4 eV excitation of the PL line at 2.4 eV has a maximum at 1.4 eV Eex = 5.2 eV. The PL excitation spectrum approximated E1 by a Gauss function has the same FWHM (0.92 eV) as Q Q that of the emission spectrum. Deviations of the shape 1 2 of the emission and excitation spectra from the Gauss Generalized coordinates function are probably caused by experimental errors. Fig. 8. A conﬁguration diagram of a defect according to PL In recent years, the PL spectra of silicon nitride data: the lower and upper terms refer to the energies of the (SiN ) and oxynitride (SiO N ) of variable composi- ground and excited state, respectively. The optical transi- x x y tions at 5.2 and 2.4 eV correspond to excitation and emis- tions have been observed in an energy interval of 2.2Ð sion; 1.4 eV (polaron energy) corresponds to the FranckÐ 2.8 eV [27Ð32]. The PL excitation peak at 5.2 eV was Condon shift.

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(polaron energy) Wt estimated from this diagram equals Basic differences between the proposed model of to half of the Stokes shift: Wt = (Eex Ð Elum)/2 = 1.4 eV. large-scale potential fluctuations in SiNx and the Shklovskii–Éfros model for compensated semiconduc- The width ∆ (FWHM) of the PL spectrum is related tors are as follows. to the phonon energy W in a single-mode approxima- ph (i) Large-scale potential fluctuations in compen- tion as sated semiconductors are of electrostatic nature, being related to the spatial fluctuations in the density of ∆ = 8WtWph ln2. charged donors and acceptors, while the bandgap width is constant (Fig. 4b). The electric field caused by spatial The phonon energy determined from this relation fluctuations of the potential favors separation of elec- amounts to W = 109 eV and the corresponding HungÐ trons and holes. In SiNx, the potential fluctuations are ph caused by inhomogeneities of the local chemical com- Rice ratio is Wt/Wph = 12.7. This Wph value is significantly greater than the phonon energy (60 meV) position. In the proposed intermediate model (Fig. 4a), reported for amorphous silicon [33]. The experimen- no space charge is formed (unlike the ShklovskiiÐEfros tally determined phonon energy coincides with the model) and the potential fluctuations favor the recombination of electrons and holes. energy of SiÐN bond oscillations in amorphous Si3N4 (900 cmÐ1 or 110 meV) [3]. The results can be (ii) The low-frequency dielectric permittivities of explained by a large surface to volume ratio for the sil- Si3N4 and Si are 7.0 and 11.8, respectively. Therefore, icon clusters studied. Thus, the interaction in the SiNx features spatial fluctuations of the permittivity. excited electronÐhole pair is related to local oscillations (iii) According to the proposed model assuming of the SiÐN bonds at the boundary between a silicon clus- potential fluctuations in SiNx, this material is capable of ter and the Si3N4 matrix. Previously, a strong interaction localizing electrons and holes in potential wells, as in the excited electronÐhole pairs in silicon nanoclusters experimentally observed in [1], with a giant time of occurring in a SiO2 matrix was studied by monitoring localized carrier trapping (about 10 years at 300 K). Si−O bond oscillations at low temperatures [34, 35]. The proposed intermediate model predicts the possibil- Thus, according to the proposed interpretation, the ity of electron and hole percolation in the large-scale emission at 2.4 eV is related to the optical transitions potential [25, 26]. between quantum states of amorphous silicon clusters The presence of silicon clusters (i.e., regions of sig- with an average size of about 1.7 Å. nificant excess of silicon) in SiNx is confirmed by Raman scattering data. We believe that the excess silicon is not detected by Raman spectroscopy in nearly ≈ 8. DISCUSSION OF RESULTS stoichiometric silicon nitride (SiNx with x 4/3) As was mentioned above, amorphous SiN can be because of insufficient sensitivity of this technique. The x existence of silicon clusters in SiN is confirmed by the synthesized only under thermodynamically nonequilib- x rium conditions. The structure and properties of this following experimental data: ≈ compound depend on the conditions of synthesis (the (i) The EPR spectrum of SiNx with x 4/3 displays temperature and gas pressure) and subsequent high- a signal with g = 2.0055 belonging to a Si atom with an temperature annealing [1]. We have studied the samples unpaired electron, bound to three other silicon atoms ≡ of SiNx prepared at relatively high temperatures. The ( Si3Si¥) [39]. structure of this material is adequately described by the (ii) The low-energy electron loss spectrum of SiNx proposed intermediate model. Ion irradiation of the with x ≈ 4/3 exhibits peaks at the energies of 3.2 and samples modifies the structure so that it approaches that 5.0 eV [40], which coincide with the energies of direct described by the RB model. When the deposition tem- electron transitions in silicon [41]. perature is decreased (plasma deposition), the structure of the synthesized SiN compound is described by the (iii) The fundamental absorption edge in nearly sto- x ichiometric silicon nitride SiN (x ≈ 4/3) is about 4.6 eV RM model [8]. x [1, 42]. However, experiments on the photothermal Previously, it was demonstrated that the intermedi- absorption [16] showed the presence of absorption in ate model describes the structure of an amorphous sili- the range from 1.7 to 3.9 eV. This result provides unam- con oxide SiOx [25], which is analogous to that of SiNx biguous evidence of the presence of excess silicon in ≈ and also depends on the conditions of synthesis. In par- SiNx with x 4/3. ticular, provided that the chemical composition is the (iv) The transport of electrons and holes in silicon same, the optical bandgap width in SiO1.94 can be varied nitride is conventionally interpreted within the frame- from 5.0 to 7.5 eV [36]. Note that the structure of silicon work of the Frenkel mechanism, according to which the oxynitrides with variable composition SiOxNy is quantita- Coulomb potential decreases in a strong electric field tively described (in contrast to the structures of SiNx and [43, 44]. However, our recent results showed that using SiOx) within the framework of the RB model [37, 38]. this model at low temperatures leads to unreasonably

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 768 GRITSENKO et al. small values of the frequency factor (ν ≈ 109 sÐ1) and an ACKNOWLEDGMENTS anomalously large tunneling mass of an electron (m* = The authors are grateful to V.V. Vasil’ev for kindly 5.0me) [45]. It was shown that the charge transfer in sil- providing experimental data on the photoluminescence icon nitride in a broad range of temperatures and fields and for fruitful discussions. can be quantitatively described using the theory of mul- This study was supported by the “Integration” Pro- tiphonon ionization [45]. gram of the Siberian Division of the Russian Academy The thermal energy of trap ionization amounts to of Sciences (project no. 116) and by the Program “Low- 1.4 eV [46], which coincides with the value of the Dimensional Quantum Structures” of the Presidium of FranckÐCondon shift estimated in this study. The the Russian Academy of Sciences. energy of local oscillations (60 meV) determined in [46] corresponds to the frequency of oscillations of sil- REFERENCES icon atoms in silicon clusters. These data provide independent evidence that amorphous silicon clusters act as 1. V. A. Gritsenko, in Silicon Nitride in Electronics, Ed. by traps for electrons and holes in silicon nitride. V. I. Belyi et al. (Elsevier, Amsterdam, 1988). 2. V. A. Gritsenko, K. A. Nasyrov, Yu. N. Novikov, et al., Nevertheless, this study does not provide straight- Solid-State Electron. 47, 1651 (2003). forward proof of the existence of amorphous silicon 3. V. A. Gritsenko, Atomic and Electronic Structure of clusters with dimensions below 17 Å in nearly stoichi- Amorphous Dielectrics in Silicon Based MIS Devices ≈ ometric SiNx with x 4/3. Previously [47, 48], we for- (Nauka, Novosibirsk, 1993). mulated a hypothesis that the role of traps in silicon 4. R. Karcher, L. Ley, and R. L. Johnson, Phys. Rev. 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25. V. A. Gritsenko, Yu. P. Kostikov, and N. A. Romanov, 38. V. A. Gritsenko, R. W. M. Kwok, H. Wong, and J. B. Xu, Pis’ma Zh. Éksp. Teor. Fiz. 34, 1 (1981) [JETP Lett. 34, J. Non-Cryst. Solids 297, 96 (2002). 3 (1981)]. 39. Y. Kamigaki, S. Minami, and H. Kato, J. Appl. Phys. 68, 26. B. I. Shklovskiœ and A. L. Éfros, Electronic Properties of 2211 (1990). Doped Semiconductors (Nauka, Moscow, 1979; 40. R. Hezel and N. Lieske, J. Appl. Phys. 53, 1671 (1982). Springer, New York, 1984). 27. V. V. Vasilev, I. P. Mikhailovskii, and K. K. Svitashev, 41. P. Y. Yu and M. Cardona, Fundamentals of Semiconduc- Phys. Status Solidi A 95, K37 (1986). tors (Springer, Berlin, 1996; Fizmatlit, Moscow, 2001). 28. K. J. Price, L. E. McNeil, A. Suvkanov, et al., J. Appl. 42. V. A. Gritsenko, E. E. Meerson, and Yu. N. Morokov, Phys. 86, 2628 (1999). Phys. Rev. B 57, R2081 (1998). 29. K. S. Seol, T. Futami, T. Watanabe, et al., J. Appl. Phys. 43. S. M. Sze, Physics of Semiconductor Devices, 2nd ed. 85, 6746 (1999). (Wiley, New York, 1981; Mir, Moscow, 1984). 30. K. S. Seol, Phys. Rev. B 62, 1532 (2000). 44. V. A. Gritsenko, E. E. Meerson, I. V. Travkov, and 31. T. Noma, K. S. Seol, M. Fujimaki, et al., J. Appl. Phys. Yu. V. Goltvyanskiœ, Mikroélektronika 16, 42 (1987). 39, 6587 (2000). 45. K. A. Nasyrov, V. A. Gritsenko, M. K. Kim, and 32. H. Kato, N. Kashio, Y. Ohki, et al., J. Appl. Phys. 93, 239 H. S. Chae, IEEE Electron. Device Lett. 23, 336 (2002). (2003). 46. K. A. Nasyrov, V. A. Gritsenko, and Yu. N. Novikov, 33. F. Giorgis, C. Vinegoni, and L. Pavesi, Phys. Rev. B 61, Phys. Rev. Lett. (in press). 4693 (2000). 47. P. A. Pundur, J. G. Shavalgin, and V. A. Gritsenko, Phys. 34. Y. Kanemitsu, N. Shimitzu, T. Komoda, et al., Phys. Rev. Status Solidi A 94, K107 (1986). B 54, 14329 (1996). 48. V. A. Gritsenko, H. Wong, I. P. Petrenko, et al., J. Appl. 35. Y. Kanemitsu and S. Okamoto, Phys. Rev. B 58, 9652 Phys. 86, 3234 (1999). (1998). 49. C. M. Gee and M. Kastner, Phys. Rev. Lett. 42, 1765 36. T. W. Hickmott and J. E. Baglin, J. Appl. Phys. 50, 317 (1979). (1979). 37. V. A. Gritsenko, J. B. Xu, R. W. M. Kwok, et al., Phys. Rev. Lett. 81, 1054 (1998). Translated by P. Pozdeev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 770Ð779. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 879Ð890. Original Russian Text Copyright © 2004 by Aleshchenko, Zhukov, Kapaev, Kopaev, Kop’ev, Ustinov. SOLIDS Electronic Properties

Peculiarities of Electron Spectrum Rearrangement for the Double-Well Heterostructure GaAs/AlGaAs with a Variable Dimensionality of Electronic States in an External Electric Field Yu. A. Aleshchenkoa,*, A. E. Zhukovb, V. V. Kapaeva, Yu. V. Kopaeva, P. S. Kop’evb, and V. M. Ustinovb aLebedev Physical Institute, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia bIoffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia *e-mail: [email protected] Received September 29, 2003

Abstract—Peculiarities of the electron spectrum rearrangement for the double-well heterostructure GaAs/AlGaAs with a variable dimensionality of electronic states in an external electric ﬁeld are investigated theoretically and experimentally. The structure is an important part of the active element of a quantum-well unipolar semiconductor laser proposed by the authors earlier. The possibility of controlling the dimensionality of the lower laser subband in such an active element by an external electric ﬁeld is demonstrated. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION inversion in such a system. Owing to the similarity between the initial and final electronic states in unipolar The possibility of obtaining stimulated radiation lasers, a single LO phonon with a nonzero momentum between subbands of semiconductor quantum wells is sufficient for electron relaxation between the parallel (QWs) was predicted by Kazarinov and Suris more subbands irrespective of their separation. At the same than 30 years ago [1]. The ideas expounded on in this time, it is impossible to increase the lifetime of elec- publication formed the basis of the injection semicon- trons by decreasing the overlap of the wave functions ductor quantum-well laser developed by specialists at since this reduces the optical efficiency of the laser. For Bell Laboratories in 1994 [2]. It was a unipolar laser this reason, the electron lifetimes for intersubband tran- based on intersubband transitions in the conduction sitions in the structures of unipolar lasers lie in the pico- band of tunnel-coupled QWs. Such a design of a quan- second range. Under these conditions, to attain consid- tum-cascade laser has a number of advantages over tra- erable gain, the structure of a unipolar laser must ditional diode lasers in which radiative recombination include up to 500 periods of the active element. As a occurs between the electron and hole states. In contrast result, the structure of a cascade laser remains to diode lasers, the wavelength emitted by a unipolar extremely complicated despite considerable advances laser is determined by the quantum constraint, i.e., by made in recent years [5, 6]. the thickness of the layers in the active region, rather than by the forbidden gap of the material. The advan- In an earlier publication [7], we proposed an origi- tages of unipolar lasers also include the high tempera- nal design of the active element for a unipolar semi- ture stability and the possibility of operating at room conductor laser. The structure is based on the physical temperature due to suppression of Auger relaxation idea of suppression of intersubband nonradiative relax- processes. Both factors are associated with the same ation by using the quasimomentum dependence of the sign of the effective mass in working subbands (parallel wave function of a QW with strongly asymmetric bar- subbands) of a unipolar laser. Contemporary lasers of rier heights [8, 9]. In such structures, a localized elec- this type can operate at room temperature in a wave- tronic state exists in a limited range (0, kc) of wave vec- length range of 3.57Ð16 µm [3, 4]. tors k in the direction along the layers of the structure. For k = kc , the 2DÐ3D transformation of the dimen- Unfortunately, parallel working subbands used in a sionality of states takes place [10]. This effect can be unipolar laser are responsible for a serious drawback used for a sharp increase in the nonradiative recombi- hampering the attainment of a noticeable population nation time in the active element of a unipolar laser, in

PECULIARITIES OF ELECTRON SPECTRUM REARRANGEMENT 771 which the lower laser subband corresponds to a QW U E = 0 with strongly asymmetric barriers. Such a subband 0 exists only for small momenta of longitudinal motion, which makes it possible to eliminate one-phonon inter- U Ᏹ Ᏹ L 1 subband transitions. Our calculations [7, 11] show that 2 1 optimized structures with a variable dimensionality of states, which are used as the active element of a unipo- 12 U0 lar laser, ensure a considerable increase in the nonradi- E ≠ 0 ative relaxation time between the laser subbands and a significant gain in the population inversion. Here, peculiarities of the electron spectrum Ᏹ 1 rearrangement for the double-well heterostructure U1 GaAs/AlxGa1 Ð xAs with a variable dimensionality of electronic states in an external electric field are studied Fig. 1. Simplified band diagram of the structure in zero field theoretically and experimentally. Such investigations of (E = 0) and in an external electric field. structures with asymmetric barriers are of considerable interest in view of the expected anticrossing of energy ers in the proposed design of the active element of a levels in complex systems and the passage of the elec- unipolar laser, but with a higher degree of localization tronic state in a QW with asymmetric barriers to the of the lower laser subband. continuum and also due to possible realization of the situation studied here in an injection unipolar laser. The The structure was designed so that an external elec- first step in this direction was made in [10] for a single- tric field could be applied to it. For this purpose, the well structure with asymmetric barriers. upper and the buffer n+-GaAs layers were used, in which control electrodes were formed. The doping level in the structure was selected in such a way that the 2. EXPERIMENT fulfillment of the condition of flat bands in the active quantum region was ensured in the absence of an exter- The structure, which is a fragment of the active ele- nal bias voltage. This condition is essential when the ment of a unipolar laser with a variable dimensionality structure is used in a unipolar laser with optical inter- of the electronic states, was grown by the method of subband pumping (fountain laser). The upper electrode molecular beam epitaxy on a semi-insulating GaAs in the form of a semitransparent Ni film was deposited substrate. The structure included the following layers × 2 (in the direction of growth): a 250-nm-thick buffer through a mask with a size of 3 7 mm . Subsequently, this electrode itself served as a mask for etching the layer of undoped GaAs; a 50-nm-thick silicon-doped + n+-GaAs layer (N = 1018 cmÐ3); a 25-nm-thick silicon- structure around this electrode down to the n lower D buffer layer used as the lower electrode. Leads were doped n-Al Ga As barrier layer (N = 5.3 × 0.09 0.91 D soldered to the upper and lower electrodes using 16 Ð3 10 cm ); an undoped 45-nm-thick i-Al0.09Ga0.91As indium solder. barrier; a 2.8-nm-wide GaAs QW; a 4-nm thick i-Al0.35Ga0.65As separating barrier; a 5.3-nm-wide The optical properties of the structure were studied by the photoluminescence (PL) method at a tempera- GaAs QW; a 10-nm-thick undoped i-Al0.35Ga0.65As barrier; a 30-nm-thick silicon-doped n+- Al Ga As ture of 80 K and excitation by radiation emitted by an 0.35 0.65 + × 17 Ð3 Ar laser Stabilite 2017 (Spectra-Physics) with a wave- barrier (ND = 6.5 10 cm ); and a 10-nm-thick silicon- + 18 Ð3 length of 5145 Å in the microprobe mode. The laser doped n -GaAs upper cap layer (ND = 10 cm ). A sim- emissive power density at the sample was ~100 W/cm2. plified band diagram of the structure in zero (E = 0) and Scattered radiation was analyzed by a monochromator a nonzero electric field is shown in Fig. 1. The structure Jobin Yvon T64000 and detected by a CCD matrix was designed so that it ensured the resonance of sub- Ᏹ Spectrum One (Spex) cooled with liquid nitrogen. Dur- band 1 (lower working subband) in well 2 with asym- ing measurements, a dc bias varying from +2 to −8 V Ᏹ metric barriers (QW2) and subband 2 in symmetric was applied to the upper semitransparent electrode. The QW1. In this case, the degree of localization of the elec- upper value of the positive bias was determined by the tron wave functions in QW2 increases and the effect of diode properties of the structure: for biases of +3 V and the existence of a subband in the QW with asymmetric higher, a considerable current passed through the struc- barriers only for small momenta of longitudinal motion ture, which partly compensated the applied field. The can be used most fully. The structure under study in fact minimum negative bias corresponded to the disappear- played the role of a single QW with asymmetric barri- ance of exciton peaks in the PL spectra due to the

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772 ALESHCHENKO et al.

"ω, eV 1.64 1.66 1.64 1.62 1.60 (a) 2400 3 1.63

( ) 0 V 2000 1 1.62 (2) Ð1.5 V (3) Ð3.0 V 2 1.61 1600 (4) Ð6.0 V

Position of PL peaks, eV 1.60 1 600 1200 3000 2500 500 (b) 800 2000 400 PL intensity, pulse/s 1500 300 4 400 1000 200 × 10 500 100 PL intensity, pulse/s 0 PL intensity, pulse/s 7500 7600 7700 7800 0 0 λ, Å 20 180000 Fig. 2. PL spectra of the structure for various biases applied 18 (c) to the upper semitransparent electrode. 16 140000

14 100000 rupture of corresponding excitons by the applied elec- 12 tric ﬁeld. 60000 10 3. PHOTOLUMINESCENCE SPECTRA 8 20000

0 Area of PL peaks, rel. units Half-width of PL peaks, meV Figure 2 shows the PL spectra of the structure for –8 –6 –4 –2 0 2 biases at the upper semitransparent electrode V = 0, V, V −1.5, Ð3.0, and Ð6.0 V. The spectra are given on both the Fig. 3. Bias dependences of (a) the positions of peaks at wavelength scale and on the energy (upper) scale. The energies 1.604, 1.637, and 1.619 eV, (b) their intensities, steps on the curves in the vicinity of 7640 and 7775 Å and (c) the half-width of the main PL peak and the total area are due to joining of different spectral regions encom- of all peaks. The results for the main PL peak and for the passed by the multichannel matrix during one mea- high-energy peak with a lower intensity are shown by trian- surement. The PL spectra recorded at zero bias across gles and squares, respectively. the structure display four peaks at energies of 1.604, 1.619, 1.637, and 2.031 eV. The peak in the region of erably higher than the intensity of the PL spectrum of 1.604 eV has the highest intensity in the spectrum, while the peak at 2.031 eV (not shown in Fig. 2) has a single QW with asymmetric barriers [10] in view of the lowest height. The intensity of the latter peak is two a considerably stronger localization of the electron orders of magnitude lower than the intensity of the wave function in this subband in the case of its reso- main peak corresponding to 1.604 eV. The peak at nance with the subband in a QW with symmetric bar- 2.031 eV is due to the contribution of the higher barrier riers. It can be seen from Fig. 2 that the position of the to the PL spectrum. Its position corresponds to a peaks in the energy range of 1.604 and 1.619 eV varies insigniﬁcantly upon a change in the bias across the composition of the solid solution AlxGa1 Ð xAs with x = 36%. Accordingly, the low-intensity peak in the structure from 0 to Ð6 V; however, the considerable energy range of 1.637 eV (marked by the arrow in change in the intensity and half-width of the peaks Fig. 2) is due to the contribution of the lower barrier indicates that the structure is controlled by the external and corresponds to x = 9.3%. electric ﬁeld. In accordance with our calculations, the peaks at Figure 3 showing the bias dependences of the posi- energies of 1.604 and 1.619 eV are associated with the tions of the peaks at energies of 1.604 and 1.619 eV (a) quantum region of the structure with layer parameters and their intensity (b) as well as the half-width of the close to nominal values. It should be noted that the main PL peak and the total area of the peaks at energies intensity of the peak associated with the lower electron of 1.604 and 1.619 eV (c) demonstrates these peculiar- subband in the QW with asymmetric barriers is consid- ities most visually. Dark circles in Fig. 3a also show the

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PECULIARITIES OF ELECTRON SPECTRUM REARRANGEMENT 773 bias dependence of the position of the PL peak corre- W sponding to the lower barrier. The behavior of the main PL peak (1.604 eV) can be traced down to V = −8 V. At 0.14 (a) lower biases, the peak cannot be detected since the corresponding exciton is torn by the strong field. The peak 0.12 L at an energy of 1.619 eV can be traced approximately h to Ð6 V. At lower biases, it becomes very weak and 0.10 merges with the main PL peak. Approximately at the same biases, the electric field tears the exciton of the 0.08 lower barrier also. The weak dependence of the positions of the PL peaks on the applied bias in the quantum 0.06 region (Fig. 3a) is worth noting. The main feature of the spectrum is the nonmonotonic behavior of all dependences depicted in Fig. 3 in the field. Indeed, as the 0.04 value of V changes from 0 to Ð3 V, the intensity of the main PL peak in the energy range of 1.604 eV attains 0.02 its maximal value (the peak height is almost doubled and its width is reduced almost by half). As bias V 0 decreases further, the intensity of the main peak decreases and the peak becomes broader. The same ten- 0.14 dency can also be traced for the PL peak with the max- (b) 250 500 imum at 1.619 eV. In order to explain the observed 0.12 peculiarities, we analyzed theoretically the spectrum of 750 electronic states of the given structure and its modifica- 1000 tion in an external electric field. 0.10 1250 1500 0.08 4. THEORY 0.06

In studies of the PL spectra of the GaAs/AlxGa1 Ð xAs structures with a single QW and asymmetric barriers in 0.04 an external electric field [10], we demonstrated the possibility of exciton transitions involving the electron 0.02 states lying above the lower barrier. Consequently, the continuous spectrum must be taken into account in the- 0 oretical analysis of relaxation processes in such struc- 50 100 150 200 250 300 350 tures. We developed an original method for calculating the electron spectrum of structures with a variable Ᏹ, meV dimensionality of electronic states, which takes into Fig. 4. (a) Dependence of probability W of finding an elec- account the contribution from the continuum, including tron in the QW region of width h = 24 Å on energy Ᏹ mea- the situation in an external electric field. sured from the QW bottom; (b) the case when the potential in the QW region is equal to the height of the low barrier. Strictly speaking, the bound states vanishes due to The results are given for various values of L (in ångströms). an indefinite decrease of the potential at infinity when an external electric field is applied to quantum wells. Nevertheless, to describe the behavior of such systems tra), such an approach ensures a quite admissible in external fields, the eigenvalue spectrum can be cal- accuracy. culated in most cases by erecting an artificial boundary with a finite (or infinite) potential at a certain dis- The application of this approach for QWs with tance L to the right of the structure (see Fig. 1). As a asymmetric barriers under transformation of the dimen- result, instead of the actual continuous spectrum, we sionality of states requires additional investigations. In obtain a set of discrete states (the model of a quasi-con- this connection, we consider the behavior of a single tinuous spectrum). The energies of the states and the GaAs QW with asymmetric AlxGa1 Ð xAs barriers for a probability of the electron location in the well proper finite width L of the lower barrier in zero electric field weakly depend on L in a wide range of the values of L (the level diagram of the structure is shown in the inset up to several thousands angstroms. As long as the time to Fig. 4a). In systems with asymmetric barriers and of tunneling through the triangular barrier remains with different effective masses of carriers in the layers, longer than the characteristic times of the problem the bound state is absent starting from a certain well (e.g., the recombination time in analysis of PL spec- width hc. For h > hc, this state exists for zero wave vec-

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W 0.14

0.12 0

U0 –1 –2 0.10 –3 U1 –4 0.08 h L –5 –6 –7 0.06 –8 –9 0.04

0.02

0 –50 0 50 100 150 200 250 300 350 Ᏹ, meV Fig. 5. Dependences of probability W of finding an electron in the QW on energy Ᏹ for L = 1000 Å for various values of electric field E (in 104 V/cm). tor k of motion along the layers of the structure and dis- recombination of this state with hole states localized in appears for a certain finite wave vector k = kc since the the well even in the absence of a bound electron sate. “transformation of the dimensionality” of electronic Thus, a PL signal can be observed for structures in states takes place. This conclusion follows from analy- which the bound state of electrons formally does not sis of the solution of the problem with bounding barri- exist. ers of an infinite width. We have theoretically studied the behavior of states Figure 4a shows the dependence of probability W of in a QW with asymmetric barrier heights in an external finding an electron in the vicinity of QW of width 24 Å electric field. Proceeding from the results obtained for on the energy (measured from the bottom of the well), a QW in zero field, we can conclude that, in simulating when fraction x of aluminum in the left (high) barrier is the continuous spectrum, it is sufficient to extend the 0.36, while x = 0.093 in the right (low) barrier. The region of the low barrier to a distance of L = 1000 Å. parameters of the barriers are taken close to the charac- A schematic diagram of the structure for which the cal- teristics of the sample. The critical width for such a sys- culations were made is shown in the inset to Fig. 5. The tem is 28 Å; i.e., the bound state is absent in a well with heights of the right and left barriers are chosen the same h < 28 Å. For comparison, Fig. 4b shows the situation as in Fig. 4 and the QW width is h = 28 Å. For such a when the well is absent altogether (the potential in the width, the bound state exists in zero external field. region of the quantum well is equal to the height of the low barrier). The results of calculations are represented Figure 5 shows the energy dependences of probabil- for various values of L (the corresponding notation is ity W of finding an electron in a QW for L = 1000 Å for given in Fig. 4b). It can be seen from Fig. 4a that, due various values of electric field E. For E = 0, a preferred to the presence of a QW, probability W for energies state exists, for which W = 0.16 and the energy is lower close to the edge U1 (about 82 meV) of the low barrier than U1 (bound state). For a finite value of the field, has a maximum; the dependence of the probability on L there exists a set of states with energies close to the low at the maximum becomes weak starting from L = potential barrier height and with a high probability of 1000 Å. Since the effective mass of a hole exceeds the electron location in the QW; i.e., in this case we can corresponding parameter for an electron, the hole state speak of the existence of quasi-bound states that can be in the QW is strongly localized. As a result, the exist- manifested in PL spectra. The existence of relaxation in Ᏹ ≈ ence of a W peak for electrons at U1 may lead to the system and tunneling in the presence of an electric

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field leads to a certain ambiguity in determining the Ᏹ, meV specific state from the set of quasi-bound states, which 90 will be manifested in the PL spectrum. Depending on the times of intersubband relaxation, 80 0 tunneling, and radiative recombination, PL can be asso- 1500 70 Ᏹ 1500 ciated with the states at the peaks of W(Ᏹ) as well as max with the states separated from the maximum by a cer- 750 60 tain distance. The positions of energy levels in the quasi-continuous spectrum slightly change with the 50 value of L. To determine the characteristic energies, we Ᏹ Ᏹ approximate the discrete function W( n) by a continu- 40 1/2 ous function (spline) and calculate the value of energy Ᏹ Ᏹ max at the peak as well as the value 1/2 for which the 30 Ᏹ probability is equal to one-half the probability W( max) at the maximum. 20 As usual in such problems, we investigate the con- 10 vergence of the results upon an increase in model Ᏹ parameter L. Figure 6 shows the dependences of max 0 Ᏹ and 1/2 on electric field E for values of L equal to 750, 1000, and 1500 Å. A characteristic feature of these –10 0 –2.5 –5.0 –7.5 curves is the weak dependence on model parameter L; 4 in other words, the use of the model of a quasi-contin- E, 10 V/cm uous spectrum indeed makes it possible to describe the Ᏹ Ᏹ Fig. 6. Dependences of max and 1/2 on electric field behavior of such systems in an electric field. The strength E for various values of L (in ångströms). dashed line in Fig. 6 corresponds to L = 0, when the electric field exists only in the QW (the right barrier of an infinitely large width is shown by the dashed line in Figure 7 shows examples of the field dependences ≈ × 4 the inset to Fig. 5). In the field E = Ec Ð7.6 10 V/cm, of the interband transition energy for double-well struc- the bound state in such a system disappears. In the tures with an aluminum fraction of x1 = 0.36 in the left model of a quasi-continuous spectrum, the state in the barrier, x2 = 0.09 in the right barrier, and with a separat- QW is localized quite strongly in fields significantly ing barrier width of 40 Å. The widths of the wells are exceeding Ec as well. As a consequence, the PL line 50 and 30 Å in Fig. 7a and 45 and 35 Å in Fig. 7b. The associated with recombination of such states is parameters of the model structures are chosen in such × observed in fields considerably stronger than Ec. a way that, in fields with a strength on the order of 3 Another typical feature of the dependences shown in 104 V/cm, transitions occur with energies (including the figure is a comparatively large value of the differ- the exciton binding energy of about 10 meV) close to Ᏹ Ᏹ ence max Ð 1/2 and its increase with electric field. This the experimentally observed energies in the given sam- must lead to an increase in the PL linewidth with elec- ple. For the first structure, the ground state for an elec- Ᏹ tron is localized in the QW with symmetric barriers tric field. The noticeable difference between max and Ᏹ (50 Å), while for the second structure, it is localized in 1/2 leads to ambiguity in the interpretation of experimental results on PL in an electric field. the well with asymmetric barriers (35 Å). The figure shows only the transitions for which the overlap Let us now describe the PL spectra for a system of integral for the electron and hole wave functions tunnel-coupled QWs, one of which has barriers with exceeds 0.01. asymmetric heights. The general methods of computa- tion in this case are quite similar to the case of a single We will describe the types of transitions in the fields QW with asymmetric barriers. An additional difficulty corresponding to characteristic points in Fig. 7. We in the interpretation of the PL spectra in this case introduce the following notation for the electron and emerges due to the presence of several electron and hole states. We denote by na the state with level number hole energy levels in the system, their formation and n, which is localized predominantly in the well with disappearance upon a change in the field and, finally, symmetric barriers (QW1), and by nb, the state local- due to anticrossing of the levels belonging to different ized in QW2 (with asymmetric barriers). In Fig. 7a, for QWs. For convenience of interpretation, we will first fields E > E1, two electron subband and one hole sub- calculate the transition energies, disregarding the field band are localized and transition 2bÐ1b takes place. in the low barrier, and then solve the problem for a finite This transition takes place from the second electron (sufficiently large) L. As in the case of a single QW, it subband to the first hole subband; the corresponding is sufficient to take L = 1000 Å. wave functions are predominantly localized in QW2

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បω, eV បω, eV 1.63 1.63 (b) (a) 4 1.62 1.62 3 4 2

1.61 1 1.61 2

3 1 1.60 1.60

1.59 1.59 E4 E3 E2 E1 E4 E3 E2 E1 –4 –2 0 2 4 –4 –2 0 2 4 E, 104 V/cm E, 104 V/cm Fig. 7. Dependences of energy បω of interband transitions on electric field E for double-well structures with aluminum fraction x1 = 0.36 in the left barrier and x2 = 0.09 in the right barrier and with a separating barrier width of 40 Å. The widths of QWs are 50 and 30 Å (a) and 45 and 35 Å (b). with asymmetric barriers. For the 1aÐ1b transition, the scale) are observed. The anticrossings of electron and × 4 overlap integral is smaller than 0.01. In fields E ~ E1, hole levels take place for Ee = 2.1 10 V/cm and Eh = × 3 the second hole level localized in QW1 is formed; as a 3.2 10 V/cm, respectively. For E > E1, only one tran- result, transition 1aÐ2a becomes possible. In the range sition 2bÐ1b can be observed. For E < E1, a weak non- E2 < E < E3, anticrossings of both electron (for a field diagonal transition 1aÐ1b is manifested. For E < E , the × 3 2 strength of E = Ee = 5.0 10 V/cm) and hole (for E = second hole subband is formed, and the electron sub- × 4 Eh = 1.3 10 V/cm) subbands take place. In the anti- bands converge. As a result, the (high-intensity) 1aÐ2a crossing region, the probability of finding an electron and the (weak) 2bÐ2a transitions take place, and the (hole) in both wells becomes appreciable for each state; intensity of the 1Ð1 transition increases. In the region of as a result, in addition to the above-mentioned transi- E = E3, the second electron subband passes to the con- tions, transitions 1Ð1 and 2Ð2 begin to be manifested. In tinuum (as in the previous case, this field is quite close the vicinity of Ee, the upper electron subband reaches to Ee) and we are left only with transitions 1bÐ1b (with the continuum (transformation of dimensionality); as a high intensity) and 1bÐ2a. After attainment of anti- result, only one localized electron subband remains for crossing of the hole subbands (E = Eh), transition 1bÐ2b E < E3 . In this case, the PL spectrum acquires the main plays the major role. In field E = E4, the second hole transition 1bÐ2b and the weaker 1bÐ1b transition. The subband disappears and only one transition 1bÐ2b intensity of the latter decreases smoothly with the × 4 remains possible. Finally, in field E5 = Ð5 10 V/cm, field. Finally, for E < E4 , only one transition 1bÐ2b is transformation of the dimensionality for the electron × 4 left. In the field E5 = Ð5 10 V/cm, the first electron subband takes place and PL disappears. level reaches the continuum and the PL line disappears. On the whole, for the 50/40/30 structure, in Figure 8 shows the field dependences of transition the entire range of applied fields, we can single out energies for the same structures, but taking into account curves 1 and 2 (outside the anticrossing region), which the field in the region of the low barrier. The size of cir- are associated with transitions in QW2, and curves 3 cles in Fig. 8 is proportional to the intensity of the cor- and 4, for which transitions for states in QW1 play the responding transition. A comparison of Figs. 7 and 8 major role. shows that the inclusion of the contribution of the continuum leads to an increase in the range of fields in For the 45/40/35 structure (Fig. 7b), the reverse (rel- which the states localized in the symmetric QW are ative to the previous case) arrangement of curves 1, 2 manifested, to a decrease in the field range for transi- and 3, 4 on the energy scale and reverse positions of tions involving the states in the asymmetric QW, and, anticrossings of electron and hole states (on the field finally, to blurring of singularities in the regions of anti-

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ω ω " , eV (a) " , eV (b) 1.63 1.63

1.62 1.62 1.61

1.61 1.60

1.59 1.60

Ec Eb Ea 1.58 Eb Ea 1.59 1.57

1.58 1.56 –10 –5 0 5 –10 –5 0 5 E, 104 V/cm E, 104 V/cm

Fig. 8. Dependences of interband transition energies បω on electric field E for structures with the same geometrical parameters as in Fig. 7, but taking into account the field in the region of the low barrier. The size of the circles in the figure is proportional to the intensity of corresponding transitions. crossing of electron and hole states. The low-energy ity of the sample geometry did not allow us to carry out curve exists in Fig. 8a in the entire field range under direct recalculation of the bias applied to the structure investigation, while the high-energy peak disappears to the field in the region of quantum wells. Moreover, for a field strength slightly exceeding Eb. For the struc- dependence E(V) may be essentially nonlinear due to ture corresponding to Fig. 8b, the two curves can coex- currents present in the structure. For this reason, a com- ist in the entire range of fields in view of what was said parison with experiment can be carried out only on above about the possibility of the emergence of PL qualitative level. states from the continuum, which are localized in the asymmetric QW (dashed lines in the figure). A comparison of Figs. 3 and 8 shows that Fig. 8b is in better agreement with the experimental data. Indeed, the theory predicts for the 45/40/35 structure a consid- 5. DISCUSSION erable intensity of two transitions with energy values close to experiment both below and above the anti- Taking into account the theoretical dependences of crossing region. If we assume that, for a positive bias energies and intensities of transitions on the external across the sample, the field in the quantum region cor- electric field presented in Fig. 8, we return to the dis- responds to Ea, the changes in the spectra in accordance cussion of the experimental results. It follows from with Fig. 8b must be as follows. When the external field Fig. 3 that, in a wide range of biases applied to the varies in interval EaÐEb, the system gets in the region of structure, two peaks associated with the quantum anticrossing of states. This field interval corresponds to region are present in the PL spectra at energies of 1.604 the range of biases from +2 to Ð1 V in Fig. 3. Delocal- and 1.619 eV. For V2 = Ð2 V, a low-intensity maximum ization of electronic states in this region leads to split- is observed in the position of the low-energy PL peak as ting of each line into two lines, which is manifested in well as a minimum in the half-width of this peak. The experiment as a slight broadening of PL peaks with a intensity of the same peak and the total area of both simultaneous decrease both in the line intensities and in peaks associated with the quantum region attain their the total area of the peaks since delocalization of the − minimum values for V1 = 1 V, while the maximal val- electron wave functions facilitates the departure of ≈ ues of these quantities are observed for V3 Ð3 V. electrons from QW1 with symmetric barriers to the Finally, the intensity minimum of the high-energy PL states of the continuum (i.e., the departure of electrons peak corresponds to lower biases as compared to V1. from the quantum region). For E < Eb, the system The lack of exact information on the actual distribution leaves the state of anticrossing of electron levels and the of the doping impurity concentration and the complex- contribution from the continuum is still virtually

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 778 ALESHCHENKO et al. absent; this can be attributed to the strong localization the departure of electrons to the states of the continuum of the lower electronic state in QW2 with asymmetric is hampered by a broad (40 Å) barrier. barriers and to the low intensity of electron tunneling It should be noted that the bias dependences of the from the state in QW1 in the absence of resonance energies of PL peaks in the quantum region (Fig. 3a) between subbands due to the large width of the separat- are more gently sloping than those predicted by the the- ing barrier. As a result, the PL peaks associated with the ory (Fig. 8b). Indeed, the range of transition energy quantum region become narrower, while their intensity variation with the field in Fig. 8b is 15 meV for the low- increases (bias range from Ð1 to Ð3 V). At the same energy PL peak and about 5 meV for the high-energy time, the total area of the peaks in this bias range peak, while this range in Fig. 3a does not exceed 5 meV increases. for both peaks. This is due to the fact that the decrease A peculiar feature of the structure considered here is in the transition energy with increasing field is partly that the bias at which the transformation of the dimen- compensated by the decrease in the binding energy of sionality of the lower electronic state takes place is the corresponding exciton. close to the bias corresponding to anticrossing of states. Thus, a comparison of experimental and theoretical The effect of the 2DÐ3D transformation of the dimen- results for a double-well structure enclosed between sionality of the subband belonging to QW2 plays a strongly asymmetric (in height) barriers shows that the decisive role in the suppression of nonradiative relax- observed singularities in the variation of the character- ation to this subband. Due to the asymmetry in the bar- istics of PL spectra in an external electric field are asso- rier heights in the given structure, the localization of the ciated with the transformation of the dimensionality of wave functions of the subband corresponding to QW2 the electronic state in the QW with asymmetric barriers is determined by wave vector k (the state can be local- and with anticrossing of this state with a state in the ized in the well for k = 0 or in the region of the low bar- QW with symmetric barriers. rier for k > kc). Let us define the “width” of the subband ∆ Ᏹ Ᏹ as = (kc) Ð (0). In Fig. 8b, the effect of the 2DÐ3D 6. CONCLUSIONS transformation of the dimensionality of the electron subband in QW2 takes place in region EbÐEc. A Peculiarities of rearrangement of the electron spec- decrease in ∆ results in an additional decrease in the trum for the double-well GaAs/AlGaAs heterostructure width (proportional to ∆) and in the area of the PL peak with variable dimensionality of electronic states in an associated with the lower electronic state due to the external electric field are investigated theoretically and suppression of relaxation from the second subband to experimentally. The structure is an important compo- the first. As a result of the competition with the effects nent of the active element of the quantum-well unipolar associated with anticrossing of energy levels, the posi- semiconductor laser proposed by us earlier. An original tions of the minima of the width of the PL peak, its method developed for calculating the electron spectrum intensity, and area for different lines do not coincide. of structures with variable dimensionality takes into The transformation of the dimensionality leads to a account the contribution from the continuum, including decrease in the exciton binding energy, which might be the situation in an external electric field. It is shown that responsible for the formation of a low-intensity maxi- the dimensionality transformation effect in the lower mum in the position of the low-energy PL peak at Ð2 V subband associated with the QW with asymmetric bar- in Fig. 3a. An additional argument in favor of this riers plays a decisive role in the variation of the PL explanation is the coincidence of the positions of this spectra in an external electric field. The possibility of maximum and the minimum of the linewidth on the controlling the dimensionality of the lower laser sub- bias scale. band in such an active element by an external electric field is demonstrated. This makes it possible to con- A further increase in the field enhances the pro- struct the active element of a quantum-well unipolar cesses of tunneling and relaxation involving the states laser with record-high characteristics on the basis of localized in the region of the low barrier; this first slows QWs with asymmetric barriers. the variation of line widths and intensities and leads to ≈ − the attainment of intensity maxima at biases V 3 V. ACKNOWLEDGMENTS In the region of fields E < Ec, the contribution of the continuous spectrum becomes predominant, which The authors thank S.S. Shmelev for preparing elec- leads to a decrease in the transition energy and to a trical contacts for the structure and to V.G. Plot- sharp decrease in the PL peak intensities down to their nichenko and V.V. Koltashev for their assistance in opti- complete disappearance as a result of the rupture of an cal measurements. exciton. The shift in the position of extrema in the This study was partly financed in the framework of intensity for the high-energy PL peak towards lower the program “Low-Dimensional Quantum Structures” biases as compared to the low-energy peak (see Fig. 3b) of the Presidium of the Russian Academy of Sciences, can be explained by the facts that the first extremum is the program “Physics of Solid-State Nanostructures” of associated with transitions from the states of QW1 and the Ministry of Industry, Science, and Technology of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 PECULIARITIES OF ELECTRON SPECTRUM REARRANGEMENT 779 the Russian Federation, and the Federal Targeted Pro- 7. Yu. A. Aleshchenko, V. V. Kapaev, Yu. V. Kopaev, and gram “Integration” (grant no. B0049). N. V. Kornyakov, Nanotechnology 11, 206 (2000). 8. V. V. Kapaev and Yu. V. Kopaev, Pis’ma Zh. Éksp. Teor. REFERENCES Fiz. 65, 188 (1997) [JETP Lett. 65, 202 (1997)]. 1. R. F. Kazarinov and R. A. Suris, Fiz. Tekh. Poluprovodn. 9. V. F. Elesin, V. V. Kapaev, Yu. V. Kopaev, and A. V. Tsu- (Leningrad) 5, 797 (1971) [Sov. Phys. Semicond. 5, 707 kanov, Pis’ma Zh. Éksp. Teor. Fiz. 66, 709 (1997) [JETP (1971)]. Lett. 66, 742 (1997)]. 2. J. Faist, F. Capasso, D. L. Sivco, et al., Science 264, 553 10. Yu. A. Aleshchenko, I. P. Kazakov, V. V. Kapaev, and (1994). Yu. V. Kopaev, Pis’ma Zh. Éksp. Teor. Fiz. 67, 207 3. F. Liu, Y. Zhang, Q. Zhang, et al., Semicond. Sci. Tech- (1998) [JETP Lett. 67, 222 (1998)]. nol. 15, L44 (2000). 11. V. V. Kapaev, Yu. V. Kopaev, and N. V. Kornyakov, in 4. M. Rochat, D. Hofstetter, M. Beck, and J. Faist, Appl. Proceedings of 9th International Symposium on Nano- Phys. Lett. 79, 4271 (2001). structures: Physics and Technology (St. Petersburg, Rus- 5. C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, Rep. sia, 2001), p. 522. Prog. Phys. 64, 1533 (2001). 6. F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Phys. Today 55, 34 (2002). Translated by N. Wadhwa

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 780Ð792. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 891Ð905. Original Russian Text Copyright © 2004 by Zaœtsev. SOLIDS Electronic Properties

Peculiarities of the Electron Mechanism of Superconductivity R. O. Zaœtsev Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia e-mail: [email protected] Received October 2, 2003

Abstract—A two-particle scattering amplitude calculated within the framework of the Hubbard model has been used for establishing the region of development of the Cooper instability. Dependence of the superconducting transition temperature Tc on the electron density has been determined. The temperature dependence of the time of relaxation with electron spin ﬂipping has been determined taking into account spin ﬂuctuations. ∆ Factors responsible for an increase in the 2 0/Tc ratio as compared to the classical value have been established. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION transition temperature is observed for δ = 0, while the compound with δ = 1/2 exhibits no superconductivity. There are only two independent undetermined quan- This paper is devoted to the so-called kinematic tities in the theory of ideal superconductors: the mechanism of superconductivity, which is capable of BardeenÐCooperÐSchrieffer (BCS) constant and the explaining the aforementioned peculiarities in the ⑀ preexponential factor . In the superconductivity the- behavior of high-Tc superconductors. ory based on the Hamiltonian of the electronÐphonon The notions that superconductivity is possible in the interaction, the ⑀ value is on the order of the Debye fre- Hubbard model with strong repulsion were criticized, quency, whereas the BCS constant can be expressed via first because the anomalous Gor’kov mean values [1] the slope of the temperature dependence of resistivity appearing below the point of development of the Coo- in the region above the Debye temperature. In these per instability have to obey an additional sum rule, the terms, the superconducting transition temperature is existence of which implies infinite Hubbard energy. inversely proportional to the square root of the total However, researchers formulating these objections mass of atoms in a unit cell and hence decreases with (N.M. Plakida, Yu.A. Izyumov, V.Yu. Yushankhai, increasing mass of the introduced isotope. There is a et al.) had no doubts about the existence of the kine- large group of simple metals (Hg, Pb, Sn, Tl, Zn) pos- matic interaction discovered by Dyson [2] and about sessing a correct sign and order of the isotope effect. the correctness of the equation derived by the author [3] However, the power of the isotope effect in complex for determining the superconducting transition temper- compounds exhibiting a rather high transition tempera- ature. The objection concerning the anomalous ture (T > 20 K) is always below 0.4 and keeps decreas- Gor’kov means was based on intuitive (so-called c “physical”) considerations and, until now, has not been ing with increasing Tc. Therefore, we may suggest that confirmed by rigorous mathematical calculations. there exists a nonphonon mechanism of the Cooper A solution to this problem was suggested by Val’kov electron pairing in high-Tc superconductors. et al. [4]. It was demonstrated that “the inclusion of a singular contribution to the spectral intensity of the The experiments with tunneling in high-Tc superconductors showed that the 2∆ /T ratio in these mate- anomalous correlation function is shown to regain the 0 c sum rule and remove the unjustified forbidding of the rials is almost two times that according to the BCS the- s-symmetry order parameter in superconductors with ory. However, the most intriguing peculiarity is an strong correlations.” extremely sharp dependence of the transition tempera- Another critical remark is related to possible ferro- ture on the dopant concentration. For example, the magnetic ordering in the region where the supercon- superconductivity in La2 Ð xSrxCuO4 exists for 0.05 < ductivity appears. As will be shown below, a tendency x < 0.34 and the maximum Tc corresponds to x = 0.15Ð toward ferromagnetism is manifested only in the region 0.16. Compounds of the Nd2 Ð xCexCuO4 system can be of small concentrations, where superconductivity is superconducting within a rather narrow interval of impossible. It will be demonstrated that the presence of 0.14 < x < 0.18, where the maximum Tc corresponds to paramagnetic fluctuations in the superconducting x = 0.15. In the YBa2Cu3O7 Ð δ system, the maximum region leads to the appearance of a temperature-depen-

PECULIARITIES OF THE ELECTRON MECHANISM OF SUPERCONDUCTIVITY 781 dent finite time of relaxation with electron spin flip- ing only the interaction between electrons having oppo- ping. In the region of ultimately low temperatures, the site spins, we obtain the following ladder equation, effect of paramagnetic fluctuations is insignificant, Γ ,, Γ()0 ,, whereas a significant decrease in the effective BCS s()p1 p2 p3 p4 = s ()p1 p2 p3 p4 constant at a finite temperature leads to a decrease in the superconducting transition temperature T and to a () c Γ 0 ,, (4) ∆ Ð T∑ s ()p1 p2 p s Ð p Gω()p corresponding increase in the 2 0/Tc value. ω, p The third problem under dispute is the behavior of × GΩω()spÐ Γ (),ps,,Ð pp p the concentration dependence of the superconducting Ð s 3 4 transition temperature, T (n), in the limit as n 1. It c where s = p1 + p2 = p3 + p4, p are the operators of will be shown below that the transition temperature in () Γ 0 | this limit tends to zero for any finite Hubbard energy. momenta, and s (p1, p2 p3, p4) is the vertex part irre- With allowance for a finite value V of the Coulomb ducible for cutoffs with respect to parallel momenta. interaction between adjacent cells, the superconductiv- This vertex part will be calculated using the method ity disappears within a finite interval of concentrations developed by Dyson [2] for calculating the spin wave in the vicinity of n = 1, beginning with a certain critical scattering amplitude. value V = Vc. In the limiting case of infinite Hubbard energy, it is Although the integral equations obtained admit sufficient to use a Hamiltonian corresponding to the solutions of different symmetry, we will consider solu- lower Hubbard subband, tions of the s type symmetry, which are free of nodes σ, 0 0, σ and lead to a phase diagram with a maximum tempera- Hˆ = ∑ t()rrÐ ' Xˆ r Xˆ r' , (5) ture of the superconducting transition. rr, '()σrr≠ ' , where X operators obey the commutation relations 2. CALCULATION {}+0, , 0Ð, +Ð, δ OF THE SCATTERING AMPLITUDE Xr Xr' = Xr rr, ', (6) FOR INFINITE HUBBARD ENERGY {}σ, 0, 0, σ ()δ00, σσ, Xr Xr' = Xr + X rr, ' , Our task here is to calculate the BCS constant and the other anticommutators are zero. directly as a function of the dopant concentration nd. For definiteness, let us consider the lower Hubbard sub- Now let us determine the ground (|0〉), one-particle σ, , , band for nd < 1 and, for simplicity, assume that the Hub- (X 0|〉0 ), and two-particle (X+0XÐ0|〉0 ) states and bard energy is infinite (U ∞). The spectrum of r r r' excitations is expressed via the product of the Fourier calculate their energies. Denoting by E0 the energy of ˆ |〉 | 〉 component of the hopping integral tp and the so-called the ground state, H 0 = E0 0 , we define the one-parti- end factor f equal to the sum of the occupation numbers cle excitation energy (measured from the ground state of the initial and final states, ˆ σ, 0 energy level) via the commutator [H , Xr ]:

ξ ===ft Ð µ, f n +1nσ Ð n σ. (1) σ, σ, σ, p p 0 Ð ˆ 0|〉 0 ˆ |〉 ()0|〉 HXr 0 Ð Xr H 0 = E1 Ð E0 Xr 0 (7) The equation of state in zero external field is as follows: ⑀()σ σ, 0|〉 = p Xr 0 . n By the same token, the two-particle excitation energy is n ==2 fn∑ (),ξ f 1 Ð -----d . (2) d F p 2 ˆ +0, Ð0, p defined via the commutator [H , Xr Xr' ]: ˆ +0, Ð0, |〉 +0, Ð0, ˆ |〉 The one-particle Green function is conventionally HXr Xr' 0 Ð Xr Xr' H 0 defined via the excitation spectrum as (8) ()+0, Ð0, |〉 = E2 Ð E0 Xr Xr' 0 . 1 Gω ()p ==------, ω πT()2n + 1 . (3) , , n ωξ n ˆ +0 Ð0 i Ð p The two-particle commutator [H , Xr Xr' ] is calculated using the operator identity relation The Cooper instability develops when the two-particle vertex part acquires a singularity for zero total momen- []ˆ , +0, Ð0, +0, []ˆ , Ð0, H Xr Xr' = Xr H Xr' tum, spin, and energy. The condition for the appear- (9) Ð0, []ˆ , +0, {}[]ˆ , +0, Ð0, ance of this singularity can be formulated as the condi- Ð Xr' H Xr + H Xr Xr' , tion of solvability of the corresponding homogeneous system [1]. For the model under consideration featur- where the first two terms can be transformed using

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1 1 mutator (11). Passing to the momentum representa- (0, +) 4 (0, –) (0, +) 4 (0, –) tion in Eqs. (11) and (12),

+0, Ð0, |〉 ψ , ()⋅ ⋅ ––(–, 0) (+, 0) Xr Xr' 0 = ∑ ()pqexp ipr+ iqr' , pq, (0, +) (0, –) (0, +) (0, –) 23 23 we obtain an explicit expression for the two-particle interaction energy Fig. 1. The Born amplitudes of kinematic interaction (14). ()ψ, []ψ⑀ ()+ ⑀ ()Ð , E2 Ð E0 ()pq = ˆ p + ˆ p ()pq T/|t| (13) Ð ∑()ψt()Ðk + t()kpÐ Ð q ().kp, + qÐ k 0.08 k

0.06 Thus, the scattering amplitude in the Born approximation depends only on the momenta of scattered particles [2, 3]: 0.04 Γ()0 ,, ()p1 p2 p3 p4 = Ðt()p3 Ð t().p4 (14) 0.02 The Born amplitudes corresponding to this interaction are depicted in Fig. 1. 0.7 0.8 0.9 1.0 1.1 1.2 1.3 n 3. SUPERCONDUCTING TRANSITION Fig. 2. The curve of transition temperature Tc versus electron density n calculated using Eq. (16). The arrows indicate TEMPERATURE critical electron densities. Substituting expression (14) into the homogeneous part of Eq. (4), we obtain an equation for determining the point of appearance of the Cooper instability definition (7) as

() Γ [] Γ +0, []ˆ , Ð0, ⑀ Ð +0, Ð0, s = Tt∑ p + tspÐ Gω()p GÐω()spÐ s. (15) Xr H Xr' = ˆ p Xr Xr' , (10) ω, p Ð0, []ˆ , +0, ⑀ ()+ Ð0, +0, Xr' H Xr = ˆ p Xr' Xr Using Green functions (3) and summing over frequencies ω = πT(2n + 1) for s = 0, we arrive at the equation (here and above, p and p' are the operators of momenta for determining the temperature Tc of the superconduct- acting upon r and r', respectively), and the third term is ing transition, directly calculated as ξ t()p p ∑------ξ - tanh------= 1, (16) {}[]ˆ , +0, Ð0, p 2T c H Xr Xr' p (11) ξ +0, Ð0, +0, Ð0, where p is the excitation energy deﬁned by Eq. (1). = Ð∑t()r Ð r ()δX X + X X , . 1 r1 r' r r1 rr' Equation (16) together with the equation of state (2) r1 and the condition of electroneutrality determine the dependence of the superconducting transition tempera- The action of the two-particle commutator on the ture Tc on the dopant concentration. Figure 2 shows a ground state gives the following equation: plot of the transition temperature versus the electron density n, which was calculated using Eq. (16) (see [3Ð5]). +0, Ð0, ()+ ()Ð +0, Ð0, ()E Ð E X X |〉0 = []⑀ˆ p + ⑀ˆ p X X |〉0 It should be noted that the integration in Eq. (16) is 2 0 r r' r r' (12) performed predominantly in the vicinity of the Fermi {}[]ˆ , +0, Ð0, |〉 ξ ≈ µ + H Xr Xr' 0 . surface, where p = 0 or tp / f. From this it follows that the Cooper instability is not developed for negative An approximate expression for the one-particle values of the chemical potential. This result appears as quite natural because small values of the chemical energy operator ⑀ˆ()p is provided by Eq. (1). The potential correspond to small occupation numbers. The effect of scattering is determined by the double com- scattering amplitude will have the same sign as that for

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 PECULIARITIES OF THE ELECTRON MECHANISM OF SUPERCONDUCTIVITY 783 two particles possessing a small relative velocity and exhibiting infinitely strong repulsion at small distances. (0, +) (0, +) (–, 2) (–, 2) (0, +) (0, +) (0, +) (0, +) As the energy of the relative motion increases, the + + + scattering amplitude decreases and changes sign for a wavevector on the order of the inverse radius of the (0, –) (0, –) (+, 2) (+, 2) (+, 2) (+, 2) (+, 2) (+, 2) interaction potential. These properties are characteristic Fig. 3. The Born amplitudes of scattering on the Coulomb of the scattering amplitude (14) for zero total momen- potential in terms of X operators. tum s = p1 + p2 = p3 + p4 = 0. According to Eq. (16), the scattering amplitude calculated on the Fermi surface changes sign at zero chem- In order to determine the superconducting transition ical potential. Using the equation of state for T = 0, we temperature, let us consider the simplest model in obtain a critical value of nd = 2/3. Beginning with this which the influence of the direct Coulomb potential is dopant concentration, the scattering amplitude is nega- bounded by the given potential of interaction V between tive and the transition temperature is finite within the nearest neighbors, entire interval of 2/3 < nd < 1. The analysis of filling of the upper Hubbard subband reveals the symmetry with ϕ()q = Vi∑ exp()Ð qr⋅ . respect to the particleÐhole transformation: n 2 Ð d n.n. nd. For this reason, the Cooper pairing according to this simplest model also takes place in the region of 1 < Using the Coulomb potential in this form, it is possible n < 4/3. d to solve the homogeneous equation for Tc by means of Thus, even the Born approximation stipulates the separating variables. possibility of a change in the sign of the scattering Assuming that the total momentum is zero, we amplitude over the entire Fermi surface. This provides obtain a solution ψ(p) of the homogeneous equation explanation of a rapid change in the superconducting depending only on the relative momentum p = p = Ðp transition temperature and the existence of narrow 1 2 regions of high-temperature superconductivity with of colliding particles, respect to the dopant concentration. As the electron density increases so that n 1 and ψ Γ()0 () ψ ()p = ÐT∑ pqGω()q GÐω()Ðq (),q (19) the lower Hubbard subband is completely filled, the ω, system passes to a semiconductor state. The radius of q screening of the Coulomb potential rapidly grows to where the scattering amplitude for a simple cubic or become infinite for n = 1 and zero temperature. From square lattice is this we infer that, in the region of n ≈ 1, it is necessary to take into account the direct intercell Coulomb interaction, Γ()0 () ()pq = Ð 2tq + ∑2Vpcos k Ð qk k 1 Vˆ = --- ∑ nˆ nˆ ϕ()rrÐ ' , 2 r r' = Ð22t + ∑ Vpcos cosq (20) rr, '()rr≠ ' (17) q k k k ˆ ++, ˆ ÐÐ, ˆ 22, nˆ r = Xr ++Xr 2Xr ,

+2∑ Vpsin k sinqk. where nˆ is the electron density operator expressed in r k terms of the Hubbard operators X. Using the commutation relations, Under the assumption that the unknown function ψ(p) is even with respect to a change in the sign of momen- []ˆ 0, σ, ˆ 0, σ ˆ ++, ˆ ÐÐ, ˆ 22, tum, integration of the sum of sines yields zero. By vir- X nˆ ==X , nˆ X ++X 2X , tue of the cubic symmetry, integration of the sum of cosines yields equal values for all summands. There- we obtain an expression for the Born scattering fore, with respect to the integration over cosq , the ver- amplitude, k tex part in Eq. (20) is equivalent to ()0 Γ ()Ðp ,,p p p = t()p Ð t()p + ϕ(),p Ð p (18) () 1 2 3 4 3 4 3 1 Γ˜ 0 β ()pq = Ð 2tq + tptq, (21) which is a generalization of relation (14). Figure 3 shows the Born amplitudes of the Coulomb scattering, where β = V/Dt2. For a bcc lattice, we obtain an analo- which represent only the ﬁrst term corresponding to gous expression for β without the factor 1/D, that is, not scattering in the lower Hubbard subband. divided by the number of measurements D.

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T/|t| potential increases further, the second critical concentration determined from the system of equations 0.06 β Tc = 2.36 appears, 0.05 µ/ f 0.04 β 2 f ρ ⑀ ⑀ ==------, n 2 f ∫ 0()d . (25) 0.03 µ Ð1 0.02 n' nc1 nc2 c2 nc1' Since f = 1 Ð n/2, the function n (β) can be set in the 0.01 c2 parametric form, 0.7 0.8 0.9 1.0 1.1 1.2 1.3 n ζ β 2 ζ 2K() ==--ζ-, nc2() ------ζ , Fig. 4. The curve of transition temperature Tc versus elec- 1 + K() tron density n calculated using Eq. (23). The arrows indicate ζ (26) µ critical electron densities nc1 = 2/3, nc2 = 0.96, nc1' = 4/3, ζ ρ ⑀ ⑀ ζ <<ζ K()= ∫ 0()d , = ---,0 1, and n' = 1.04. f c2 Ð1 ρ ⑀ where 0( ) is the dimensionless seeding density of Thus, we arrive at the following equation for deter- states. mining Tc: In the case of a constant density of states, we obtain n = 2(2 + β)/(2 + 3β). When the parameter β increases, ψ()p = 2Tt∑ Gω()q G ω()Ðq ψ()q c2 q Ð n decreases and, in the limit of β ∞, n 2/3 ω, c2 c2 q (22) so that the region of existence of the superconducting state disappears. As can be checked, this behavior is β ψ Ð tpTt∑ qGω()q GÐω()Ðq ().q retained in the general case of an arbitrary seeding den- ω, p sity of states. This equation can be readily solved by separating vari- Figure 4 shows the plot of the transition temperature ψ ables as (p) = A + Btp. As a result, we obtain the con- versus the electron density n, which was calculated for dition of solvability that generalizes relation (16), a constant density of states. The value of β = 2.36 corresponds to the experimentally observed critical con- ξ 2 ξ t()p p β t ()p p centration nc2' = 1.04. A comparison of the Tc(n) curves ∑------ξ - tanh------= 1 + ---∑------ξ - tanh------, (23) presented in Figs. 2 and 4 shows that the inclusion of p 2T c 2 p 2T c p p the direct Coulomb interaction leads to a significant ξ µ shift of the position of maximum toward lower (and where p = ftp Ð . In the limit of Tc = 0, this integral is higher) electron densities: from n = 0.93 to n = 0.8 finite provided that either of the two conditions is satis- m m ' ' fied, (and accordingly, from nm = 1.07 to nm = 1.16). Thus, by selecting the magnitude of the direct Cou- β µ = 0or ft Ð µ ≡ C 1 Ð ---t . (24) lomb repulsion, it is possible to fit the critical electron p 2 p density corresponding to the disappearance of superconductivity. However, the value of the transition tem- The first condition is independent of the Coulomb perature turns out to be overstated because the calcula- potential and determines a lower critical concentration tion was performed using the Born approximation for the scattering amplitude. As will be shown below, the (nd = 2/3) above which the superconductivity appears. The second condition reduces to the relation β = (2 Ð inclusion of scattering on the spin fluctuations provides n)/µ determining the upper critical dopant concentra- for a decrease in Tc. tion which depends on the ratio βw ~ V/w of the Cou- lomb potential V and the seeding energy width w of the electron subband. 4. ALLOWANCE FOR RELAXATION PROCESSES For T = 0 and n = 1, the maximum value of the chemical potential is µ = 1/2 and, hence, the upper crit- As is known from the original Hubbard papers, elec- ical electron density for β < 2 remains equal to unity. In tron excitations in the normal phase are characterized other words, the superconductivity under these condi- by a finite free path length determined by scattering on tions exists inside the fixed interval 2/3 < n < 1. The the charge and spin fluctuations. In the limit of infinite transition temperature significantly decreases when β Hubbard energy, the fluctuations of one-particle charge increases from zero to two. However, as the Coulomb and spin states are determined by the longitudinal and

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(a) (0, +) (b) (0, +) (0, –) (0, –) ω (0, +) ω (–, 0) ω Γ Γ ω Γ ω = ω' + – ω' (+, 0) –ω (0, –) –ω (0, +) (0, +) (0, –) (0, –) (c)(d) (e) (0, +) (0, +) (0, +) ω (+, 0) ω (–, 0) ω (0, +) (0, +) (0, –) – Γ – Γ – (– +) Γ ω' ω (+ –) ω –ω ω ω (0, –) – (–, 0) (0, –) – (+, 0) (0, +) (0, –) (0, –) (0, –)

Fig. 5. A homogeneous equation for the vertex part with allowance for (a, b) kinematic and (c) Coulomb interaction and the scattering on (d) longitudinal, and (e) transverse spin ﬂuctuations. transverse parts of spin correlators calculated in the tions have a graphical representation as in Fig. 5 and same unit cell, can be analytically represented as

〈〉∆{}ˆ +0, , ˆ 0+, ∆{}ˆ +0, , ˆ 0+, K|| = X X r X X r , Γ 2 Γ ω()p + K2∑Gω()p' GÐω()Ðp' tp' ω()p' , , (27) 1 +Ð Ð+ p' K⊥ = --- 〈〉{}Xˆ , Xˆ , 2 r r (31) = 2T ∑ Γω ()p' Gω ()p' G ω ()Ðp' t , where ' ' Ð ' p' ω', p' +0, 0+, ++, 00, ++, 00, ∆{}Xˆ , Xˆ r = Xˆ r + Xˆ r Ð 〈〉()Xˆ r + Xˆ r . where 〈〉∆{}ˆ +0, , ˆ 0+, ∆{}ˆ Ð0, , ˆ 0Ð, In the one-loop approximation, K2 = K⊥ Ð X X r X X r .

()σ Ð1 ()σ , ()0 Ð1 ()σ After the separation of variables, the conditions of solv- []Gω ()p = []Gω ()p Ð Σω (),p ability of Eqs. (31) can be represented in the following form: ()σ ()σ Σω ()p = K||t()p ∑Gω ()p' t()p' (28) ()1 Sω p' 2T∑------()2- = 1, (32) ω 1 + K2Sω

()Ðσ + K⊥t()p ∑Gω ()p' t().p' where the sums over momenta are expressed via the σ p' function ω determined from the self-consistency condition (30), For the normal paramagnetic phase, Eqs. (28) are rewritten as ()1 Sω = ∑t()p Gω()p GÐω()Ðp ()σ ()Ð1 []ω ()σ µ p Gω ()p = i Ð fK+ 1 ω tp + , (29) σ []σ r K1 s + f where = µ------σ ω ω σ , K1 r Ð 2i f Ð i K1 s (33) 1 + x ()2 2 f ==------, σω ∑Gω()p t(),p Sω = ∑t ()p Gω()p GÐω()Ðp 2 (30) p p µσ ωσ x = 1 Ð n, K1 = K|| + K⊥. r + i s = µ------σ ω ω σ , K1 r Ð 2i f Ð i K1 s Equations for the transition temperature with allow- σ σ ± σ ance for the scattering on the charge and spin ﬂuctua- and s, r = ω Ðω. Substituting these expressions

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 786 ZAŒTSEV into (32), we obtain an equation Introducing the electron spin operators σ ()σ z 1 ++, ÐÐ, r fK+ s s Sˆ = ---()Xˆ Ð Xˆ , 2T∑------µ σ ω ω σ = 1, (34) 2 Ks r Ð 2i f Ð i Kr s ω + x y +Ð, (38) Sˆ ==Sˆ + iSˆ Xˆ , ± where Ks, r = K1 K2. In the low-temperature limit ˆ Ð ˆ x ˆ y ˆ Ð+, where K1, 2 0, we obtain Eq. (16) for the supercon- S ==S Ð iS X , ducting transition temperature in the ideal model. we obtain an expression for K via a sum of the mean- σ s The quantities r, s can be expressed via the density square spin fluctuations: of states ρ = δξ() on the Fermi surface: 2 ∑p p 〈〉ˆ + ˆ Ð ()∆ ˆ z Ks = 2 Sr Sr + 2 Sr σ 1 1 (39) s = ∑tp------ω ξ + ∑tp------ω ξ ˆ k 2 i n Ð p Ð i n Ð p = 2 ∑ ()∆Sr . ∞ (35a) kxyz= ,, 2ξ 2ξ = Ð2t ------p - ==Ð0,t∗ρξd ------Based on the isotropy considerations, this sum is ∑ p 2 2 ∫ 2 2 ω + ξ ω + ξ z 2 n p Ð∞ n ()∆ expressed via the mean-square fluctuation Sr ∞ which, in turn, can be written in terms of the static spin iω 1 σ ==Ð22 t ------Ð iωt∗ρξd ------susceptibility as r ∑ p 2 2 ∫ 2 2 ω + ξ ω + ξ z n p Ð∞ n (35b) z 2 ∂ 〈〉S ()∆ ˆ ------χ π Ks ===6 Sr 6T ∂ 6T (),T (40) ω ∗ρ ∗πρ() ω H = Ð22i t ------ω - = Ð it sgn n , n where χ(T) is the static susceptibility of the normal where the t* value is determined from the condition phase. ξ(t*) = 0. The density of states on the Fermi surface is Recently, the author demonstrated [6] that the spin ρ ⑀ expressed via the seeding density of states 0( ) = magnetic susceptibility in the Hubbard model with infi- δ()⑀ nite repulsion is given by the formula ∑ Ð tp as ∂ σ χ , n ξ()t∗ ==ft∗ Ð0,µ ()2nT = ------(36) ∂h (41) 1 µ Ð2 fD ρδ==∑ ()ft Ð µ ρ ()⑀ δ()f⑀ Ð µ d⑀ =---ρ --- . = ------0 -. p ∫ 0 f 0f () ()2 p 1 Ð K Ð1fD1 Ð Ð K D1 Ð fD0 D2 Ð D1 As can be seen, Eq. (34) in fact contains only the com- By virtue of the equation of state, the K value can be µ bination Ks, which has the meaning of the inverse time expressed via the electron density and the other coeffi- of relaxation with spin flipping. cients, via the integrals of a derivative of the Fermi function: In order to determine the temperature dependence of the time of relaxation with spin flipping, let us express 21()Ð n n 1 Ð K ==------, f 1 Ð ---, Ks via mean-square spin fluctuations. To this end, the ()2 Ð n 2 initial expression for Ks can be transformed as (42) k , , , , ' ξ ∆{}ˆ +0, ˆ 0+ ∆{}ˆ +0, ˆ 0+ Dk = ∑tpnF().p Ks = 2K⊥ + X X r X X r p +0, 0+, Ð0, 0Ð, Ð ∆{}Xˆ , Xˆ r∆{}Xˆ , Xˆ r For T 0, these coefficients can be expressed via the seeding density of states ρ (⑀) and, hence, via the den- 1 ++, 00, ++, ÐÐ, 0 = 2K⊥ + --- ()∆∆Xˆ r + ∆Xˆ r ()Xˆ r Ð ∆Xˆ r sity of states on the Fermi level: 2

ÐÐ, 00, ÐÐ, ++, ρ ⑀ δ()⑀ ρ ⑀ θµ ⑀ ⑀ 1 (37) 0()==∑ Ð tp , K 0() ()Ð f d , + --- ()∆ ∆ X ˆ r + ∆Xˆ r ()Xˆ r Ð ∆ Xˆ r ∫ 2 p (43) 1 ++, ÐÐ, ++, ÐÐ, ⑀sρ ⑀ δ()⑀ µ ⑀ = 2K⊥ + --- ()∆∆Xˆ Ð ∆ Xˆ () Xˆ Ð ∆ X ˆ Ds = Ð∫ 0() f Ð d . 2 r r r r In the limit of T = 0, the coefﬁcients obey the relation 1 ++, ÐÐ, 2 = 2K⊥ + --- ()∆Xˆ r Ð ∆Xˆ r . 2 2 D0D2 = D1 and, hence, the susceptibility can be

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 PECULIARITIES OF THE ELECTRON MECHANISM OF SUPERCONDUCTIVITY 787 expressed as a function of the electron density with wχ (n, T = 0) only two coefﬁcients (D0 and D1):

χ , Ð2 fD0 8 ()n 0 = ------() 1 Ð1K Ð Ð K + f D1 II (44) 6 2ρ ()µ/ f = ------0 . ()µ()ρ2 µ 4 1 Ð1K + Ð K + f / f 0()/ f I I ρ ⑀ 2 Since both the density of states 0( ) and the factor 1 Ð II K + f are always positive, we may ascertain that the sys- µ 0 tem with small electron densities ( < 0) always exhib- 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 n its a tendency toward ferromagnetism. In the case of a constant seeding density of states Fig. 6. Plots of the magnetic susceptibility versus electron ρ ⑀ θ ⑀2 density at T = 0 for (I) a semielliptic density of states and 0( ) = (1 Ð )/2w at T = 0, the magnetic susceptibility (II) a ﬂat band model. goes to inﬁnity only for n = 0 (see Fig. 6):

1 22()Ð n 3 limit of sgn()σω ()0+ instead of the function σ (ω). χ()nT, = 0 = ------, (45) r r wn()86Ð n Ð n2 Then, the right-hand part is calculated with a logarithmic accuracy, while the differential sum in the left-hand where n is the electron density and w is the energy half- part can be expressed in terms of the digamma function ψ Γ Γ width of the lower Hubbard subband. (x) = '(x)/ (x) as For a square or bcc lattices, the density of states log- µσ∗ 1 ψ 1 Ks ψ1 arithmically tends to infinity on the zero energy level. --- + --- + ------π Ð --- However, the influence of the singularity in the denom- g 2 4 fTc 2 inator is compensated by the energy factor µ/f so that ω (46) ()ξ γω the magnetic susceptibility remains constant for all tanh /2Tc ξ 2 = ∫------ξ -d = ln------π . electron densities. T c At a finite temperature and a constant density of 0 states, all integrals are explicitly calculated and the Here, all values are determined on the Fermi level ⑀* = chemical potential and, hence, the magnetic suscepti- µ/f and expressed via the seeding density of states bility can be determined (Fig. 6). At a small electron ρ (⑀) = ∑ δ()⑀ Ð t : density, we obtain the CurieÐWeiss law corresponding 0 p p to the gas phase. In this limit, the mean-square correla- σ∗ π⑀∗ρ ()⑀∗ ⑀∗ρ ⑀∗ tor is temperature-independent and acquires the classi- ==2 0 / f , g 2 0()/f . (47) cal value S(S + 1) = 3/4. For intermediate densities, the The correlation function K goes to zero in the limit as CurieÐWeiss law is valid only at a sufficiently high tem- s Ӷ T 0 and is a linear function of the temperature at perature. At low temperatures (T t), the system ӷ exhibits the Pauli spin susceptibility with a Stoner fac- T w. For this reason, the influence of spin fluctua- tor significantly above unity (Fig. 6). Here, the suscep- tions reduces to renormalization of the BCS constant: tibility never goes to infinity and the system remains 1 1 1 1 µσ∗K 1 paramagnetic at all electron densities (see [7, 8]). ψ s ψ ------∗- = --- + --- + ------π Ð --- , g g g 2 4 fTc 2 Thus, it can be ascertained that, for T Ӷ t and n ӷ (48) T/t, the value of K linearly tends to zero with decreas- s π ≈ γω 1 T c expÐ------. ing temperature, Ks 0, with a proportionality factor g∗ equal to the limiting value of the magnetic susceptibility at T = 0. From this we infer that the spin correlators This relation is supplemented by an equation for deter- are linear functions of the temperature and strongly mining the energy gap at T = 0: depend on the degree of doping x = 1 Ð n. ω Performing calculations for zero magnitude of spin 1 2 --- = ln------∆ -. (49) fluctuations and subtracting the corresponding sum g 0 over frequencies from the left-hand part of Eq. (34), we ∆ can obtain a simpler form of the equation for the super- Thus, the ratio 2 0/Tc is independent of the parameter ζ = µσ*K /4π fT calculated at T = T : conducting transition temperature Tc (for analogous s c calculations, see [9]). It can noticed that, after such sub- 2∆ 1 traction, the sums over frequencies are taken in the ------80 = πψexp --- + ζ . (50) |ω | ≈  region of n Tc and, hence, it is sufficient to take the T c 2

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∆ ∆ 2 0/Tc Accordingly, the ratio 2 0/Tc is overstated as compared to the well-known value approximately equal to 3.53 25 according to the classical BCS theory. This can be 20 observed for most of high-Tc superconductors. 15 ∆ The measured values of 2 0/3.53Tc for various 10 high-Tc superconductors fall within the interval from 5 1.27 to 3.12. As can be seen from Fig. 7, these values correspond to the parameters ζ ≈ 0.05Ð0.55. For exam- 0 0.2 0.4 0.6 0.8 1.0 ζ ple, the experimental values for YBa2Cu3O7 are ∆ ≈ ζ ≈ 2 0/3.53Tc 1.98Ð2.27 and 0.2. ∆ Fig. 7. A plot of the dimensionless 2 0/Tc ratio versus parameter ζ constructed using relation (50). 5. FINITE HUBBARD ENERGIES

Since the value of Ks is proportional to the ratio of the In order to write the equation of state, let us obtain temperature to the absolute value of the hopping inte- the Green functions in the simplest zero-loop approxi- gral, the parameter ζ is eventually on the order of unity. mation. For zero external ﬁeld and a given spin projec-

(0, –) (0, +) (0, +) (–, 0) (+, 2) β (0, –) β Γ Γ 11 11 α α (0, +) (+, 0) (0, +) (0, –) (0, –) (–, 2)

(–, 2) (0, –) (+, 2) β (–, 2) (2, +) (+, 2) β Γ 22 Γ α 22 (2, –) (0, +) α (+, 2) (+, 2) (–, 2) (–, 2)

(0, +) (0, –) (2, +) β (0, +) (+, 2) β Γ (2, +) 12 Γ α 12 (+, 0) (0, +) (+, 2) α (–, 2) (–, 2) (+, 2)

(0, –) (–, 2) (–, 0) (–, 2) (+, 2) β (0, –) β Γ 21 Γ 21 α α (0, –) (–, 2) (2, –) (0, +) (0, –) (–, 2)

Γ Fig. 8. The right-hand part of a homogeneous system of equation for four vertex parts of i, k. Cross-hatched squares indicate the Γ α β vertex parts of α, β. Indices “ ” and “ ” denote the transitions (0, +), (–, 2) and (0, –, (+, 2), respectively.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 PECULIARITIES OF THE ELECTRON MECHANISM OF SUPERCONDUCTIVITY 789 tion, the inverse Green function is represented by the Taking into account the commutation relations for matrix the electron density operators, 0, σ 0, σ Ðσ, 2 Ð2σ, []Xˆ , nˆ ==Xˆ , []Xˆ , nˆ Xˆ , ω ⑀ σ Ð1 i Ð 1 Ð f 1tp, Ð f 1tp (55) Gω p = , (51) ++, ÐÐ, 22, σ ω ⑀ nˆ = Xˆ ++Xˆ 2Xˆ , Ð f 2tp, i Ð 2 Ð f 2tp we readily obtain the Coulomb corrections to four scat- ⑀ µ ⑀ µ where 1 = Ð and 2 = U Ð are the energies of transi- tering amplitudes (see Fig. 3). tions between the empty and one-particle and the one- Using the double commutation relations between 0, σ Ðσ, 2 σ, 0 2, Ðσ particle and two-particle levels, respectively (differing four operators X, (Xˆ , Xˆ ), and (Xˆ , Xˆ ), we by the Hubbard energy U); f = 1 Ð n/2 and f = n/2 are 1 2 obtain a system of homogeneous equations for four ver- the end factors expressed via the electron density n. tex parts: Within the framework of the zero-loop approxima- Γ ==Γ()0+ 0Ð , Γ Γ()Ð2 +2 , tion, the equation of state is written using the sum of all 11 22 Γ Γ()Γ Γ() possible products of the components of one-particle 12 ==0+ +2 , 21 Ð2 0Ð , Green function and the end factors: Γ Γ 11 = Aα', β' α', β' σ σ σ, 0 2, Ðσ 0, σ Ðσ, 2 〈〉aˆ + aˆ = ()Xˆ r + σXˆ r ()Xˆ r + σXˆ r r r β 0+ α' 0Ð β' Γ Ð t()p TG∑ ω ()p' GÐω ()Ðp' t()p' α', β', = Ti∑ exp()ωδ (52) ω, p' Γ Γ ω, p 22 = Bα', β' α', β' × {}()11 σ 21 ()22 σ 12 Gω ()p + Gω ()p f 1 + Gω ()p + Gω ()p f 2 . β Ð2 α' +2 β' Γ Ð t()p TG∑ ω ()p' GÐω ()Ðp' t()p' α', β', Upon calculation and substitution of the matrix ele- ω, p' (56) ments and summation over the spin projections, we 1 arrive at the relation Γ = ---()ΓAα , β Ð Bα , β α , β 12 2 ' ' ' ' ' ' ()iω Ð ⑀ f + ()iω Ð ⑀ f n = 2Ti∑ exp()ωδ ------2 1 1 -2 , (53) β 0+ α' +2 β' Γ ()ωξ+ ()ωξÐ Ð t()p TG∑ ω ()p' GÐω ()Ðp' t()p' α', β', ω, i Ð p i Ð p p ω, p' 1 where Γ = ---()ΓBα , β Ð Aα , β α , β 21 2 ' ' ' ' ' ' ξ± U tp ± 1 2 2 ()µ p = ---- + ------U + tp Ð 2U 1 Ð n tp Ð β Ð2 α' 0+ β' Γ 2 2 2 Ð t()p TG∑ ω ()p' GÐω ()Ðp' t()p' α', β' , ω, p' are the two branches of the energy spectrum of one-par- α β ticle excitations. Expanding the summand into simple where indices “ ” and “ ” independently run over two multipliers, we obtain couples (0, +), (Ð, 2) and (0, Ð), (+, 2), respectively. Direct calculation of the matrix elements Aα, β and Bα, β leads to the following system of relations: λ ξλ n = 2TA∑ ()p nF(),p A = T Φt {()iω Ð ⑀ ()iω Ð ⑀ Ð f t p,±λ = 11 ∑ p 1 2 2 2 2 p (54) ω, () p ± 1 tp Ð U 1 Ð n ()ω ⑀ ()}ω ⑀ A = --- 1 ± ------. + i 1 Ð 2 i Ð 2 Ð f 2tp , 2 2 2 () U + tp Ð 2U 1 Ð n tp Φ 2 {}()ω ⑀ ()ω ⑀ A22 = ÐT∑ tp f 1 i 1 Ð 1 + i 2 Ð 1 , The conditions of appearance of the Cooper instability ω, p at a ﬁnite Hubbard energy consist of two pairs of equa- σ σ tions for the two types of excitations, (0, ) and (Ð , 2), A = ÐT Φt {()iω Ð ⑀ f t differing by the direction of spin projection and the sign 12 ∑ p 1 2 1 p ω, p of the transition energy ⑀ = ⑀σ Ð ⑀ and ⑀ = ⑀ Ð ⑀ σ (see 1 0 2 2 Ð ()ω ⑀ ()}ω ⑀ Fig. 8). + i 1 Ð 2 Ð f 2tp i 2 Ð 1 ,

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Φ {()ω ⑀ ()ω ⑀ The coefficients Aα, β and Bα, β are symmetric: the sub- A21 = T∑ tp i 1 Ð 1 i 2 Ð 2 Ð f 2tp ⑀ ⑀ stitution 1 2, f1 f2 corresponds to A11 ω, p − B22, B11 ÐA22, A12 ÐB21, and A21 ÐB12. + ()iω Ð ⑀ f t }, 2 2 1 p Separating the variables, we obtain the following (57) equations: Φ 2 {}()ω ⑀ ()ω ⑀ B11 = T∑ tp f 2 i 1 Ð 2 + i 2 Ð 2 , ω, p ()1 ()2 ()α ()α xi ==Aik xk + Aik yk, Aik ÐAki , (59) Φ {()ω ⑀ ()ω ⑀ ()1 ()2 , ,, B22 = ÐT∑ tp i 1 Ð 1 i 2 Ð 1 Ð f 1tp yi ==Bik xk +Bik yk, ik 123. ω, p ()ω ⑀ ()}ω ⑀ + i 1 Ð 1 i 2 Ð 1 Ð f 1 t p , Here, the first index corresponds to the scattering of (0, +) on the (0, Ð) excitation; the second index corre- Φ {()ω ⑀ ()ω ⑀ sponds to the scattering of (Ð, 2) on the (+, 2) excitation; B12 = ÐT∑ tp i 1 Ð 2 i 2 Ð 1 Ð f 1tp and the third index corresponds to a mixed scattering of ω, p (0, +) on the (Ð, 2) excitation (the fourth index is miss- ()ω ⑀ } + i 2 Ð 1 f 2tp , ing because x4 = Ðx3 and y4 = Ðy3). It can be also noted that x3 = (x1 Ð x2)/2; this relation follows from the (α) B = T Φt {()iω Ð ⑀ ()iω Ð ⑀ Ð f t explicit form of A matrices. Upon summation over 21 ∑ p 2 2 1 1 1 p the frequencies, it turns out that summands depend on ω, p the momentum only via the function t , which makes ()ω ⑀ } p + i 2 Ð 1 f 2tp , expedient the introduction of the seeding density of ρ ⑀ δ()⑀ states 0( ) = ∑ Ð tp . In addition, it can be noted Φω[]()ω2 ξ2 ()2 ξ2 Ð1 p = n + ()+ n + ()Ð , that integration for determining the pole part of the sin- (58) gular integrals is performed near the Fermi surface ξ U tp µ ± 2 2 () ± = ---- + ---- Ð U +,tp Ð 2U 1 Ð n tp ξÐ 2 2 determined from the relation p = 0. Using this condition, we can fix the hopping integral tp = t* (instead of where the chemical potential µ) determined from the condi- ω ω ω π () tion i 1 ===Ði 2 i n i T 2n + 1 , ⑀ µ ⑀ µ ∗ 1 ==Ð , 2 U Ð , U t 1 2 2 µ = ---- + ---- Ð --- U +.()t∗ Ð 2U()1 Ð n t∗ (60) ()σσ, (), 2 2 2 σ, ()ˆ ˆ 00 f 1 ==f ()0 X + X , ⑀ ()σ, σ (), As a result, all integrals can be expressed via t*, 1 = , σ ()ˆ Ð Ð ˆ 22 −µ ⑀ µ f 2 ==f ()2 Ð X + X . , and 2 = Ð + U:

 2 ⑀2⑀Ð1 2 ⑀ 2 ⑀ Ð22t f 1 2 1 t 1 f 1 2t 2 f 1 ()1 * * * Aˆ = L 2 ⑀ 2 ⑀2 ⑀Ð1 2 ⑀ , Ð22t f 2 2 t 1 f 2 2 2t 1 f 2 * * * 2 Ð1 2 Ð1 2 Ðt ⑀ ⑀ ()f ⑀ Ð f ⑀ t ⑀ ⑀ () f ⑀ Ð f ⑀ t () f ⑀ Ð f ⑀ (61) * 2 1 1 2 2 1 * 1 2 1 2 2 1 * 1 2 2 1

()2 ()1 ⑀ ⑀ 1 Aˆ ==t Aˆ , t Ð------1 2 , L =T ------. ⑀ ⑀ ∑ 2 2 2 2 * * f 1 2 + f 2 1 ()ωω ξ ()ξ ω, p n + ()+ n + ()Ð (2) (1) Matrices B = t*B also exhibit a strong degeneracy,

 t4 f 2⑀2⑀Ð2 Ðt4 f 2 Ðt4 ⑀ ⑀Ð1 f 2 * 1 2 1 * 1 * 2 1 1 ()2  Bˆ = ÐβL 4 2 4 ⑀2 2⑀Ð2 4 ⑀ 2⑀Ð1 . (62) Ðt f 2 t 1 f 2 2 t 1 f 2 2 * * * 4 Ð1 4 Ð1 4 Ðt f f ⑀ ⑀ t f f ⑀ ⑀ t f f * 1 2 2 1 * 1 2 1 2 * 1 2

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t/U

1.0 β = 2.5 β = 2 β = 1.75 0.8

0.6

0.4

0.2

0 0.97 0.98 0.99 1.00 0.98 0.99 1.00 0.98 0.99 1.00 n n n Fig. 9. Phase diagrams showing the regions of existence of the superconducting phase at T = 0 for the various values of the Coulomb potential β and the rectangular density of states.

The variables x1, 2 and y1, 2, 3, 4 in Eq. (59) are expressed Substitution of these expressions into Eqs. (64) shows via each other using the relations following from the that, for a given finite value of the interstitial Coulomb form of A and B matrices: repulsion between electrons (parameter β), the phase f ⑀ ()x Ð x diagram is symmetric relative to the simultaneous sub- 1 2 1 2 µ µ x1 ===x2------⑀ , x3 Ðx4 ------, stitution Ð and x Ðx. It can be also shown f 2 1 2 µ (63) that (x) is antisymmetric if the seeding density of 2⑀2 ⑀ f 1 2 f 2 1 states ρ (⑀) = δ()⑀ Ð t() is an even function. For y ===y ------, y Ðy Ðy ------. 0 ∑p p 1 2 2⑀2 3 4 1 f ⑀ f 2 1 1 2 this reason, the phase diagram describing the existence Substituting these relations into Eqs. (59), we obtain of superconductivity in terms of the variables (x, t/U) is the following conditions of solvability: symmetric with respect to the particleÐhole transformation x Ðx. ω ()ξ Λ tanh /2T ξ For the sake of illustration, let us perform further ∫------ξ -d = 1, calculations for a constant seeding density of states ρ ⑀ 0 (64) 0( ) with a unity halfwidth. In this case, the equation 2U⑀ ⑀ ∂ξ Ð1 of state (53) can be represented in the form of a full Λ 1 2 β 2 ρ ()Ð = Ð ------+ t 0()t ------. derivative, ⑀2 ⑀2 * * ∂ f 2 1 + f 1 2 tp Here, the calculations were performed for the lower δξÐ ()tp θξ()Ð Hubbard subband, so that the partial derivative is calcu- n = 2∑------δ - Ð ()tp (Ð) tp (66) lated for ξ (p) = 0 or t(p) = t*. p The superconductivity takes place for the positive = ξÐ()t∗ Ð ξÐ().Ð1 values of Λ, which has the meaning of the effective BCS constant. The coefficients in Eqs. (64) can be expressed For T = 0, this yields the following explicit function via the degree of underfilling x = 1 Ð n and the chemical t*(n) and the energies: potential µ = µ Ð U/2. Eventually, we obtain t∗ = [2n2 ++u()1 Ð n Ð12n ⑀ µ U ⑀ µ U 1 ==Ð Ð ----, 2 Ð + ----, 2 2 2 +1()Ð 2n ()1 + u Ð 2nu] (67) 2 µ2 1 + x 1 Ð x U Ð 4 2 Ð1 U f 1 ===------, f 2 ------, t ------, × []2n Ð u () 1 Ð n Ð1 1 Ð () + u Ð 2 nu , u = ----, 2 2 * 2()Ux Ð 2µ t (65) 2U⑀ ⑀ 1 2 β 2 ()2 µ2 −u 1 1 2 Ð ------Ð t = U Ð 4 ⑀ , = +--- Ð n ++------()1 + u Ð.2nu ⑀2 ⑀2 * 12 f 2 1 + f 1 2 2 2 2 2U ()U2 Ð 4µ2 These relations, together with the effective BCS con- × ------Ð β------. stant determined above, allow the phase diagram 2 2 2 U ++4µ 4xUµ 4()Ux Ð 2µ describing the existence of superconductivity can be

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t/U

1.0 β = 3 β = 5 0.8 0.6 0.4

0.2 0 0.7 0.8 0.9 1.0 0.7 0.8 0.9 1.0 n n

Fig. 10. Phase diagrams showing the regions of existence of the superconducting phase at T = 0 for the various values of the Cou- lomb potential β and the rectangular density of states. constructed for all values of the dimensionless Hub- energy. In the limit of infinite Hubbard energy, only the bard energy U/t, Coulomb parameter β, and electron region of s-pairing is retained. density n. In the nonsuperconducting part of the phase dia- For any strength of the Coulomb interaction, there gram, the effect of magnetic fluctuations reduces to the exists a critical electron density, corresponding to zero appearance of relaxation with spin flipping, the rate of chemical potential, above which the Cooper instability which is proportional to the first power of the tempera- can arise. In the flat band model, the corresponding ture. This leads to a decrease in the superconducting ⑀ value can be determined from the condition 1(n, u) = transition temperature. In the superconducting part of 0, which yields the phase diagram, the relaxation rate exhibits an additional decrease caused by a rapid drop of the supercon- 1 3u 1 2 ducting gap and the corresponding decrease in the den- nc1 = --- Ð ------+ --- 9u ++.4u 4 (68) sity of state on the Fermi level. This corresponds to an 2 4 4 ∆ increase in the 2 0/Tc ratio. As the intercell Coulomb repulsion increases, the BCS constant decreases and the region of existence of ACKNOWLEDGMENTS the superconducting state diminishes. Beginning with a β certain critical value c = 2 in a system with arbitrarily This study was supported by the Scientific Council large Hubbard repulsion, the superconductivity appears on Superconductivity within the framework of the Fed- eral Scientific-Technological Program “Basic Direc- when there is a finite underfilling nc2 of the Hubbard subband (cf. Figs. 9 and 10). The temperature of the tions in the Physics of Condensed Media.” superconducting transition exhibits two zeros, corresponding to the electron densities nc1 and nc2. This very REFERENCES form of the phase diagram is observed for high-T c 1. L. P. Gor’kov, Zh. Éksp. Teor. Fiz. 34, 735 (1958) [Sov. superconductors. Phys. JETP 7, 505 (1958)]. 2. F. Dyson, Phys. Rev. 102, 1217, 1230 (1956). 6. CONCLUSIONS 3. R. O. Zaitsev, Phys. Lett. A 134, 199 (1988). It was demonstrated that the phase space of the Hub- 4. V. V. Val’kov, D. M. Dzebisashvili, and A. S. Kravtsov, bard model with strong repulsion contains a finite Pis’ma Zh. Éksp. Teor. Fiz. 77, 604 (2003) [JETP Lett. 77, 505 (2003)]. region in which the scattering amplitude is negative. It was also found that the Fermi level can occur entirely 5. J. Zelinski, M. Matlak, and P. Entel, Phys. Lett. A 136, in the region where the scattering amplitude corre- 441 (1989). sponds to the Cooper pairing. In this system, the super- 6. R. O. Zaœtsev, Zh. Éksp. Teor. Fiz. 123, 325 (2003) [JETP 96, 286 (2003)]. conductivity can exist only in a limited interval of electron densities. 7. J. Hubbard and K. R. Jain, J. Phys. C 1, 1650 (1968). In this study, various phase diagrams were obtained 8. J. W. Schweitzer, Phys. Rev. B 3, 2357 (1971). for the most symmetric case of superconductivity of the 9. L. P. Gor’kov and A. I. Rusinov, Zh. Éksp. Teor. Fiz. 46, Cooper type with s-pairing. As for the phase diagram in 1361 (1964) [Sov. Phys. JETP 19, 920 (1964)]. a system with d-pairing, it can be obtained proceeding from the same Eqs. (56–59) written for a finite Hubbard Translated by P. Pozdeev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 793Ð810. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 906Ð926. Original Russian Text Copyright © 2004 by Sluchanko, Bogach, Glushkov, Demishev, Ignatov, Samarin, Burkhanov, Chistyakov. SOLIDS Electronic Properties

Genesis of the Anomalous Hall Effect in CeAl2 N. E. Sluchankoa,b,*, A. V. Bogacha,b, V. V. Glushkova,b, S. V. Demisheva,b, M. I. Ignatova,b, N. A. Samarina, G. S. Burkhanovc, and O. D. Chistyakovc aInstitute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia bMoscow Physicotechnical Institute, Institutskiœ proezd 9, Dolgoprudnyœ, Moscow oblast, 141700 Russia cBaœkov Institute of Metallurgy and Materials Science, Russian Academy of Sciences, Leninskiœ pr. 49, Moscow, 119991 Russia *e-mail: [email protected] Received October 6, 2003

ρ Abstract—The Hall coefficient RH, resistivity , and Seebeck coefficient S of the CeAl2 compound with fast electron density fluctuations were studied in a wide temperature range (from 1.8 to 300 K). Detailed measure- ϕ ≤ ments of the angular dependences RH( , T, H 70 kOe) were performed to determine contributions to the anom- am a alous Hall effect and study the behavior of the anomalous magnetic RH and main RH components of the Hall signal of this compound with strong electron correlation. The special features of the behavior of the anomalous am − magnetic component RH were used to analyze the complex magnetic phase diagram H T determined by magnetic ordering in the presence of strong spin fluctuations. An analysis of changes in the main contribution a RH (H, T) to the Hall effect made it possible to determine the complex activation behavior of this anomalous component in the CeAl2 intermetallic compound. The results led us to conclude that taking into account spinÐ polaron effects was necessary and that the Kondo lattice and skew-scattering models were of very limited applicability as methods for describing the low-temperature transport of charge carriers in cerium-based intermetallic compounds. The effective masses and localization radii of manybody states in the CeAl2 matrix were esti- ρ mated to be (55Ð90)m0 and 6–10 Å, respectively. The behaviors of the parameters RH, S, and were jointly analyzed. The results allowed us to consistently describe the transport coefficients of CeAl2. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION localized magnetic moments of rare-earth metal ions. However, our preliminary study of the Hall coefficient One of the most interesting and complicated proper- performed comparatively recently [12] for a typical ties of rare-earth metal-based compounds with interme- representative of this class of compounds, the so-called diate valence and heavy fermions is the Hall coefficient magnetic Kondo lattice CeAl , revealed a complex acti- R [1Ð4]. In particular, in the overwhelming majority 2 H vation behavior of the R (T) parameter, which did not of cases, the Hall effect in cerium-based intermetallic H compounds is anomalous both in the magnitude and fit in with the concept [1, 2] of the determining role played by the scattering effect in the formation of sign of RH. Indeed, the RH value for conducting cerium compounds is dozens of times larger than the Hall coef- anomalies of the RH coefficient in this intermetallic ficient of their nonmagnetic analogues (La, Y, Lu, etc., compound. compounds) and positive at temperatures comparable to the characteristic temperature of spin fluctuations In this work, we thoroughly studied the Hall effect T [1, 2]. Studies performed by various authors for var- in CeAl2 in a wide temperature range from 1.8 to 300 K sf and in magnetic fields of up to 70 kOe. Our goal was to ious Ce-based intermetallic compounds (CeAl3 [5], CeCu Si [6], CeCu [7], CePd [8], CeNiSn [9], CeOs experimentally examine if the existing theoretical 2 2 6 3 2 approaches could be used to describe the anomalous [10], CePb3 [11], etc.) showed that their RH(T) temperature dependences contained large-amplitude maxima Hall effect in rare-earth metal compounds with long- range magnetic order and heavy fermions. To elucidate RH at T max in the neighborhood of the Tsf temperature. the special features of the low-temperature transport of According to [1, 2], the most correct explanation of this charge carriers in CeAl2, we also measured the temper- behavior of the RH(T) parameter can be obtained using ature dependences of the Seebeck coefficient S(T) and ρ the model of skew-scattering of charge carriers by resistivity (H0, T) at fixed magnetic field values.

794 SLUCHANKO et al. 2. EXPERIMENTAL ∆T/T between the “cold” and “warm” sample ends was no more than 5%. The transport characteristics were measured in this work for polycrystalline CeAl samples with a cubic An experimental setup of an original design was 2 used to measure the Hall coefficient; its block diagram Laves phase structure. The samples were synthesized is shown in Fig. 1. Measurements were taken in cryostat from stoichiometric amounts of high-purity compo- 1 with superconducting magnet 2 while the sample was nents in an electric arc furnace with a nonconsumable step-by-step rotated in a magnetic field. Sample 3 pre- tungsten electrode on a water-cooled copper hearth in pared for direct-current measurements in the four-con- an atmosphere of purified helium. Composition homo- tact configuration (Fig. 1, inset a) was placed on copper geneity in the bulk was attained by repeatedly remelt- plate 4 (Fig. 1, inset b) of a rotary device together with ing the samples with subsequent homogenizing anneal- Hall probe 5 and CERNOX 1050 standard resistance ing in evacuated quartz ampules. X-ray (DRON-3) and thermometer 6 (Lake Shore Cryotronics, USA). Hall microstructure (optical microscopy) analyses showed probe 5 measured the magnetic field vector component the products to be single-phase. normal to the surface of the sample. The assemblage on copper plate 4 was rotated in the magnetic field of the The Seebeck coefficient was measured by the four- superconducting magnet step-by-step (steps of ∆ϕ = 1.8°) probe method using a setup of an original design simi- by step motor 7. After each rotation through 2Ð5 steps, lar to that described in [13]. The temperature gradients the position of the assemblage was fixed (see insets in ∆T on the samples were varied in the region of a linear Fig. 1), and signals from the sample Hall contacts, Hall response of the thermal electromotive force voltage probe, and resistance thermometer were measured. ∝ ∆ U1, 2 T measured for various pairs of sample con- A model 2182 (Keithley, USA) nanovoltmeter was tacts [13]. The relative maximum superheating value used to directly perform precision measurements of the

(a) UH Superconducting solenoid current source ϕ H n I 7 Up Microprocessor device SM Temperature controller ∝ ϕ UH RH (T, H) H cos

8 Digital voltmeters

∆ϕ = 1.8° (b) Nanovoltmeter He4 1 3 2 6 PC

4 5

Fig. 1. Block diagram of setup for measuring transport characteristics: (1) cryostat, (2) superconducting magnet, (3) sample, (4) copper plate, (5) Hall probe, (6) resistance thermometer, (7) step motor, and (8) double-walled ampule. The geometry of contacts to the sample and the position of the sample on the plate are shown in insets a and b, respectively.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 GENESIS OF THE ANOMALOUS HALL EFFECT IN CeAl2 795

80 (a) 1 20 80 60 6 10 25 V/K cm µ

70 40 0 , S µΩ

, 20 ρ –10 20 60 –20 15 ∆ 0 1 = 100 K 200 T, K 50 10 cm V/K µ , µΩ 1 S , 0 ρ 40 S 2 Tinv –10 30 3 5 –15 (b) 20 ∆ 2 = 170 K –20 6 4 2F 10 5/2 ∆ –25 1 = 100 K

010TN 100 T, K

Fig. 2. Temperature dependences of the resistivity of CeAl2 in various magnetic fields: H0 = 0 (1), 32 (2), 50.5 (3), 60 (4), and 70.7 kOe (5). Curve 6 is the temperature dependence of the Seebeck coefficient. Curves 1 and 6 are shown in inset a on a linear 2 temperature scale; a scheme of crystal-field splitting of the F5/2 Ce ground state in CeAl2 is shown in inset b.

Hall voltage from the sample. The temperature of the dependence ρ(T) in the absence of an external magnetic measuring cell with the sample, which was placed in field (Fig. 2, curve 1) is close to linear in the temperature double-walled ampule 8, was stabilized and controlled range 100Ð300 K (Fig. 2, inset a). At T < 100 K, the by a temperature controller of an original design. The resistance decreases more sharply as temperature low- controller provided a 0.01 K accuracy of temperature ers. After the ρ(T) curve passes through a minimum stabilization. The system for recording and controlling near 13 K (Fig. 2, inset a), ρ increases almost logarith- low-temperature experiments was connected to a PC mically as temperature decreases and passes through a (Fig. 1) through a microprocessor device. The PC was ρ ≈ used to accumulate and process experimental informa- maximum at T max 5.5 K. Below this maximum, the ρ ≈ tion and set the required parameters and operating con- (T) curve has a kink at T = TN 3.8 K (marked by a ditions for the electronic components of the setup. dashed line in Fig. 2), which corresponds to long-range The resistance of CeAl was measured by the four- magnetic ordering (antiferromagnetic modulated struc- 2 ture [14]) of the system of magnetic moments localized probe direct-current method. In magnetic fields, the on cerium centers in the CeAl matrix. behavior of transverse (I ⊥ H, Fig. 1, inset a) magne- 2 toresistance was studied. An external magnetic field of up to 70 kOe noticeably changes the form of the ρ(T) curve. The appear- 3. RESULTS ance of substantial (up to 50%) negative magnetoresistance in the specified range of field values at T < 20 K The results of measurements of the resistivity ρ(T, (Fig. 2, curves 1Ð5) is accompanied by a broadening of ≤ H0) in a fixed magnetic field H0 70 kOe and of the the low-temperature ρ(T) maximum and its shift Seebeck coefficient S(T) of CeAl2 are shown in Fig. 2 upward along the temperature axis. At the same time, (curves 1Ð5 and 6, respectively). The temperature the suppression of the low-temperature magnetic con-

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ρ Ω ρ Ω H, m H, m 0.08 70.7 kOe (a) T = 4.17 K (b) 0.09 0 60.6 kOe 0.07 50.3 kOe 0.08 40.2 kOe 20.0 kOe 0.06 0.07 9.9 kOe

0.06 0.05

0.05 0.04 0.04

0.03 0.03

0.02 9.9 kOe 0.02 30.2 kOe 35.2 kOe 0.01 0.01 60.6 kOe 70.7 kOe T0 = 3.4 K 0 0 90° 180° 270° 360° 0 90° 180° 270° 360° ϕ ϕ

Fig. 3. Angular dependences of the Hall resistance of CeAl2 in magnetic ﬁelds up to 70 kOe at temperatures T0 = 4.17 (a) and 3.4 K (b).

ρ tribution to (T) in magnetic fields H0 > 30 kOe is much Note that, as far as we know, this result is the first exper- more effective at helium temperatures T < 5 K. This in imental observation of the S(T) curve singularity that ρ turn transforms the low-temperature (T, H0) maximum corresponds to the transition of CeAl2 to the magneti- into a step (Fig. 2, curves 1Ð5). Note that the experi- cally ordered phase. The small amplitude of S(T) ≈ ∆ ≤ µ | | µ mental data presented in Fig. 2 (curves 1Ð5) are in close changes at T TN ( S 1 V/K at S > 20 V/K) agreement with the results of measurements performed allows us to exclude the formation of a magnetically in [15] for polycrystalline CeAl2 samples at low and ordered state of CeAl2 as a factor responsible for the superlow temperatures in magnetic fields up to deep negative Seebeck coefficient minimum. 200 kOe. As mentioned above, we measured the Hall resis- ρ The temperature dependence of the Seebeck coeffi- tance component H(H, T) on the setup shown in Fig. 1 cient S(T) of CeAl2 measured in this work is shown in by rotating the sample through a certain angle and then Fig. 2 (curve 6). In the temperature range 100Ð300 K, by fixing its position in a magnetic field. Typical fami- ρ ϕ the Seebeck coefficient is a slowly varying function of lies of the angular dependences H( , H0, T0) obtained ≤ temperature, which takes on positive values; S(T), how- for CeAl2 in various magnetic fields H 70 kOe both at ever, sharply decreases as temperature lowers (T < temperatures above the Néel temperature (for instance, ≈ ≈ 100 K) and changes sign at T = Tinv 45 K (Fig. 2, see at T0 = 4.17 K > TN 3.8 K) and in the magnetically also inset a). In addition to a negative S(T) minimum of ordered state (T = 3.4 K < T ) are shown in Figs. 3a ≈ 0 N a considerable amplitude at Tmin 10 K, our precision and 3b, respectively. Measurements with sample rota- Seebeck coefficient measurements revealed the pres- tions in a magnetic field, when the amplitude of the nor- ence of a singularity in the vicinity of the Néel temper- mal component of the external magnetic field vector ≈ || ϕ ature TN 3.85 K (the dashed line in Fig. 2); this singu- H⊥ n changes by the harmonic law H⊥ = H0cos , larity corresponded to the transition to the magnetically should usually be expected to give a sine Hall voltage ∝ ϕ ordered state of localized cerium magnetic moments. dependence of the form UH RH(T, H0)H0cos (Fig. 1,

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(a) 4.14 K (b) 0.07 0.118 3.70 K H0 = 32 kOe 3.50 K 3.00 K 0.06 0.116 2.50 K 2.06 K 0.05 0.114 0.04

0.112 0.03

0.110 0.02

10.00 K 0.108 5.00 K 3.2 K 0.01 4.14 K 2.6 K H = 3.4 kOe 0 3.80 K 2.1 K 0.106 0 0 90° 180° 270° 360° 0 90° 180° 270° 360° ϕ ϕ

Fig. 4. Angular dependences of the Hall resistance of CeAl2 at various temperatures in magnetic ﬁelds H0 = 3.4 (a) and 32 kOe (b).

≈ inset a). However, with CeAl2 samples, such a form of field H0 60 kOe [Fig. 5a, all experimental curves were ρ ϕ H( , H0, T0) curves is only observed in limited temper- obtained at temperatures T > TN(H)]. Another visual ature and magnetic field ranges. In particular, at helium example of the appearance of the specified additional temperatures, an angular dependence of the Hall signal contribution of even harmonics is provided by the fam- ρ ≈ close to sinusoidal is only observed in fields up to ily of H(T) curves recorded at temperature T0 5.5 K > 35 kOe (Fig. 3a). At temperatures below TN, the shape TN (Fig. 6a, curves in the field range 40–80 kOe). At the ρ ϕ of the H( ) curves becomes much more complex and same time, the angular dependences of the Hall resis- ≥ a contribution of even harmonics is added to the main tance again become sinusoidal already at T0 8 K (e.g., ρ ϕ ∝ ϕ signal constituent H( ) cos over the whole range see the family of curves shown in Fig. 6b) over the of magnetic fields used in this work (Fig. 3b). Changes whole filed range H ≤ 70 kOe studied in this work. ρ ϕ in the form of the H( ) curves as temperature lowers To conclude this section, we stress that the anoma- from T > TN to T < TN are most visually shown in lous component observed in this work and caused by Figs. 4a and 4b, where the angular dependences of the the appearance of even harmonics in the Hall signal Hall resistance measured at various temperatures in cannot be explained by asymmetry in the arrangement ≈ magnetic fields H0 3.4 and 32 kOe, respectively, are of Hall contacts on the sample of the intermetallic com- plotted. pound and, therefore, cannot be related to the addition of a contribution of CeAl2 magnetoresistance, which is ρ ϕ ϕ Another special feature of the H( ) dependences in an even magnetic field function, to UH(T, H, ). The CeAl2 is the appearance of a contribution of even har- angular dependences of magnetoresistance ρ(ϕ, H, T) monics to the Hall effect in high magnetic fields H ≥ measured simultaneously with Hall effect dependences 40 kOe. This contribution is observed both in the tem- allow us to exclude such an influence of the ordinary ≤ ≤ perature region 4 T 7.5 K (above TN) and at T < TN resistive component, which arises because of “non- for T and H values outside the region of the low-tem- equivalence” in the arrangement of Hall contacts to the perature antiferromagnetic modulated phase in the HÐT sample, as a factor determining the shape and character ϕ magnetic phase diagram of CeAl2. This anomalous of variations of UH(T, H, ) for all CeAl2 samples stud- Hall resistance behavior most clearly manifests itself in ied in this work.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 798 SLUCHANKO et al. 4. DISCUSSION imental data by (1) can be estimated from the differences 4.1. Separation of Contributions to the Hall Effect ρexpt()ρϕ,, ρ ()ϕϕ H T 0 H0 Ð H0 Ð H1 sin Ð 01 We analyzed the results obtained for the angular Ð ρ sin[]2()ϕϕÐ , dependence of the Hall resistance (Figs. 3Ð6) of the H2 02 CeAl2 intermetallic compound using the representation which are also plotted in Fig. 5b. Note that (1) is a fairly good approximation everywhere except the region of high magnetic fields H > 50 kOe, where a noticeable ρ ()ϕ,, H T 0 H0 additional contribution of higher even harmonics (1) ρ ρ ()ϕϕ ρ []()ϕϕ ρexpt ϕ = H0 ++H1 sin Ð 01 H2 sin 2 Ð 02 , appears in the H ( ) curves (Fig. 5b, curves D). We applied the approach described above to deter- which includes not only the main component ρ (odd mine the temperature and magnetic field dependences H1 of the amplitudes of both the main contribution to the with respect to the magnetic field) and the constant shift ρ Hall effect ρ [mΩ]d[cm] = ρa [mΩ cm], where d is H0, but also the second harmonic contribution. The H1 H ρa procedure for separating the contributions to (1) is most the sample thickness and H is the anomalous Hall clearly shown in Fig. 5b, where the contributions with resistance component (according to the classification ρ ρ the amplitudes H1 and H2 found for the family of suggested in [1Ð4], this is the anomalous contribution curves recorded at T ≈ 3.8 K in various magnetic fields of skew scattering) and the anomalous magnetic com- H ≤ 70 kOe are plotted along with the experimental ρ ρam ponent H2d = H . The results are plotted in Figs. 7 ρexpt H curves. The accuracy of approximating the exper- and 8, respectively. Figure 7 shows that the field depen-

ρ Ω ρ Ω H, m (a) (b) H, m ρ 2.00 K H1 0.07 ρexpt ρ 2.80 K H H2 0.03 3.60 K 0.06 4.14 K A 0.02 5.50 K 9.9 kOe 10.00 K expt ρ 0.04 ρ 1 0.05 H H ρ H2

0.02 0.04 B 20 kOe ρexpt 0.03 H T0 = 3.8 K 0

0.02 35.2 kOe C –0.02

ρexpt ρ 0.02 0.01 H H1 ρ H2 0

0 H0 = 60 kOe D 60.6 kOe –0.02 0 90° 180° 270° 360° 0 90° 180° 270° 360° ϕ ϕ

Fig. 5. (a) Angular dependences of the Hall resistance of CeAl2 at various temperatures in magnetic ﬁeld H0 = 60 kOe and (b) sep- ρexpt ρ aration of contributions to (1) (see text) at various H values and T0 = 3.8 K; H are experimental data, H1 is the main component ρ ρ ρ ρ contribution, H2 is the second harmonic contribution, and H Ð H1 Ð H2 is the difference signal (triangles).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 GENESIS OF THE ANOMALOUS HALL EFFECT IN CeAl2 799 ρ Ω ρ Ω H, m (a) (b) H, m

0.18

0.17 0.17

0.16 0.16

0.15

0.15 0.14 9.9 kOe 15.0 kOe 0.13 9.9 kOe 20.0 kOe 20.0 kOe 25.0 kOe 0.14 30.2 kOe 30.2 kOe 0.12 40.2 kOe 40.3 kOe 50.3 kOe 50.3 kOe 60.6 kOe 60.6 kOe 0.11 T = 10 K 0.13 T0 = 5.5 K 70.7 kOe 0 70.7 kOe 0 90° 180° 270° 360° 0 90° 180° 270° 360° ϕ ϕ

Fig. 6. Angular dependences of the Hall resistance of CeAl2 in magnetic ﬁelds up to 70 kOe at temperatures T = 5.5 (a) and 10 K (b).

ρa ρam dences of the Hall resistivity H (H, T0) are essentially (Fig. 8). Such a behavior of H (T, H) is, on the whole, nonlinear not only for the antiferromagnetic modulated in agreement with the conclusions drawn in [16Ð18], phase of CeAl2 but also in the immediate vicinity of the where quasi-elastic neutron scattering by CeAl2 was Néel temperature (Fig. 7a). In magnetic fields H ≤ studied, and with the HÐT magnetic phase diagram of 70 kOe, they remain nonlinear up to temperatures this compound [19] (see also inset in Fig. 8). In partic- above 10 K, and, only at T ≥ 30 K do the field depen- ular, our data lend support to the conclusion [16] of the a dences ρ (H) become linear (Fig. 7b). existence in CeAl2 of a region of strong magnetic fluc- H tuations with the coherence length ξ ≥ 20 Å at temperatures up to 12 K. Note that, in the series of Laves phases LnAl (Ln = Ce, Nd, Tb, Dy, etc.), all trivalent 4.2. The Anomalous Magnetic Hall Effect Component 2 rare-earth metal dialuminides except CeAl2 are ferro- The temperature dependences of the anomalous magnets. This leads us to suggest that the main factor ρam responsible for the complex long-range antiferromag- magnetic contribution to the Hall resistivity H (H, T0) netic ordering in the magnetic Kondo lattice of CeAl2 is were studied at various fixed external magnetic field precisely the competition between magnetic RKKY values; the results are shown in Fig. 8. At comparatively ≤ exchange interaction and the mechanism of spin-flip low fields H 30 kOe, the appearance of the anomalous scattering. The latter mechanism is responsible for the magnetic contribution can be unambiguously related to compensation (commonly attributed to the Kondo the transition to the antiferromagnetic modulated phase ≈ effect) of the localized rare-earth metal magnetic at temperatures below TN 3.85 K (Fig. 8). The appear- moment. This reduction of the magnetic moments of ρam ance of the anomalous magnetic contribution H rare-earth metal ions determines the trend toward the becomes noticeable also at T > T as field H increases, formation of a nonmagnetic ground state and results in N a substantial instability of the ferromagnetic structure. ≈ ρam and, in field H0 60 kOe, the presence of a small H It appears that, in this situation, the presence of strong ≈ component is observed at temperatures up to T 8Ð10 K ferromagnetic fluctuations in the CeAl2 matrix in high

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ρa µΩ ρa µΩ H , cm H , cm

0.6 (a) (b) 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 5.5 K 2.20 K 7.5 K 0.1 3.00 K 10.0 K 0.1 3.40 K 30.0 K 4.14 K 40.0 K

0 20 40 60 0 20 40 60 H, kOe H, kOe

ρa Fig. 7. Field dependences of the main anomalous component H of the Hall resistivity of CeAl2 at various temperatures. magnetic fields H ≥ 60 kOe at both helium and interme- have a singularity (a maximum) in the vicinity of ≈ diate (5Ð10 K) temperatures can result from the sup- Hmax 15 kOe (Fig. 9a), which increases in amplitude pression of Kondo magnetic moment fluctuations on as temperature decreases below liquid helium tempera- ≈ cerium centers by an external magnetic field H ~ HK µ ≈ CeAl2 ≈ kBTK/ B 70 kOe (T K 5 K [16]). Another conse- ρa m µΩ H , cm quence of magnetic ordering in CeAl2 at low temperatures under the conditions of the competition between M || H various mechanisms (see above) appears to be the for- 0.3 60 mation of new phases in the HÐT magnetic diagram of 40 CeAl2. In particular, the phase transition from the anti- , kOe H P ferromagnetic modulated phase to a noncollinear mag- 0.2 AFM HAF netic structure was observed in [19] (Fig. 8, inset). 20 c 0 Yet another interesting feature of the anomalous 12345 magnetic component of the Hall resistivity is the non- 0.1 3.4 kOe T, K 32.0 kOe ρam 60.0 kOe monotonic magnetic field dependence of the H (H) amplitude at temperatures T < TN (Fig. 8). These results a 0 can more conveniently be discussed using the RH and 2510TN am T, K RH Hall coefficients, which can be directly obtained a am Fig. 8. Temperature dependences of the anomalous mag- from ρ and ρ and the magnetic field value. The am H H netic component ρ (see text) of the Hall resistivity in a am H field dependences of RH and RH found from the data magnetic fields H0 = 3.4, 32.0, and 60.0 kOe; the HÐT phase shown in Figs. 3Ð6 are plotted in Fig. 9. According to diagram of CeAl2 is shown in the inset: AFM is the antifer- a am romagnetic modulated phase, P is the paramagnetic phase, Fig. 9a, the RH and RH Hall coefficients are compara- AF and H is the phase boundary of the AFM phase; the ble in order of magnitude at helium temperatures, and c am lower hatched region corresponds to the mode of the reori- the anomalous magnetic contribution RH (H, T0) is entation of antiferromagnetic domains by an external field am (see text), and the upper hatched region corresponds to a indeed essentially nonmonotonic. The RH (H) curves noncollinear magnetic structure (according to [19]).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 GENESIS OF THE ANOMALOUS HALL EFFECT IN CeAl2 801

–3 3 –3 3 RH, 10 cm /C RH, 10 cm /C 40° (a) (b) 4.14 K 2.0 20° 2.0 5.50 K a RH ∆ϕ 0 7.50 K

–20° a 10.0 K RH –40° 16.0 K 1.5 1.5 0 20 40 60 30.0 K H, kOe 40.0 K

1.0 1.0

am RH 0.5 2.70 K 0.5 2.20 K 3.40 K 3.80 K 5.50 K 4.14 K 7.50 K 0 0

0204060H, kOe 0204060H, kOe

a am Fig. 9. Field dependences of the anomalous RH and anomalous magnetic RH components of the Hall coefﬁcient of CeAl2 at various temperatures. Field dependences of the phase shift between harmonics (see text) are shown in the inset.

a ≤ ture and becomes equal to RH at T 3.4 K. An analysis lends support to the conclusion [19] on the existence of based on (1) also makes it possible to determine the a “horizontal” (Hc = const) phase boundary in the HÐT ∆ϕ ϕ ϕ phase shift = 01 Ð 02 between the main and even magnetic phase diagram of this compound (see inset in Hall signal harmonics, which offers an additional pos- Fig. 8). sibility of quantitative characterization of the change in am the anomalous magnetic contribution and related scat- While discussing the nature of the RH (H, T0) maxima in the vicinity of H ≈ 15 kOe (Fig. 9a), we must tering of charge carriers by CeAl2 magnetic structure max features as the temperature and magnetic field vary. The mention the results obtained in [20Ð22] for the thermal inset in Fig. 9a shows that, to within the accuracy of expansion and magnetostriction of CeAl2. According measurements, the phase shift ∆ϕ takes on fixed values to [20, 21], the reorientation of antiferromagnetic at H ≤ 30 kOe (∆ϕ ≈ 25°) and H ≥ 40 kOe (∆ϕ ≈ Ð35°). domains occurs in the antiferromagnetic modulated A sharp change in the sign and value of ∆ϕ occurs in a phase of CeAl2 as the magnetic field value increases to fairly narrow neighborhood of H* ≈ 35 kOe, and this 15 kOe. The antiferromagnetic domains, initially ori- ≈ H* value remains virtually unchanged both at T < TN ented chaotically (uniformly), assume the orientation in 3.85 K (this corresponds to the antiferromagnetic mod- which the antiferromagnetic polarization direction is ulated phase in the CeAl matrix) and in the region from transverse with respect to the external magnetic field. 2 ≤ 4 to 8 K characterized by strong magnetic fluctuations As CeAl2 magnetization reversal at H 20 kOe virtually does not change the T temperature of the mag- in the CeAl2 matrix (Fig. 9a). For a more visual and N convenient representation, the change by 60° in the netic transition in this compound (the lower hatched ∆ϕ ϕ ϕ phase diagram region in the inset in Fig. 8), the authors phase shift = 01 Ð 02 of the second harmonic with respect to the main signal in the vicinity of H* ≈ 35 kOe of [20, 21] arrived at the conclusion of the reorientation am of antiferromagnetic domains by an external magnetic is shown in Fig. 9a as a reversal of sign of the RH com- field without noticeable changes in their size and anti- ponent. In our view, the observed behavior of the ∆ϕ ferromagnetic polarization. These findings lead us to parameter is an additional argument in favor of the rear- conclude that the appearance of the anomalous mag- rangement of the magnetic structure of CeAl2. It also netic scattering of charge carriers and, accordingly,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 802 SLUCHANKO et al.

0.2 2.0 III II I

3.4 kOe /C –1 3 1.5 32.0 kOe )

60.0 kOe Ce cm eN –3 a 0.4 H 1.0 R , 10 a H = ( R

0.6 ν 0.5 0.8 1.0

0 TN 10 100 T, K

Fig. 10. Temperature dependences of the main anomalous component of the Hall coefficient of CeAl2 in magnetic fields H0 = 3.4, 32.0, and 60.0 kOe. essentially nonmonotonic behavior of the anomalous magnetic fields occurs almost equally effectively at ≥ ≈ magnetic component of the Hall coefficient Ram at H < both T0 TN (e.g., see Fig. 9a, curves for T0 4.14 and H 3.8 K) and T < T (e.g., T ≈ 3.4 K in Fig. 9a). It follows 30 kOe are likely caused by magnetic domain magneti- 0 N 0 zation reversal in the magnetically ordered state of that the nature of the appearance of the anomalous positive Hall effect in the vicinity of Tsf may be common to CeAl2. CeAl2, which is a magnetic Kondo lattice according to the classification suggested in [23] (Tsf = TK), and other 4.3. The Main Hall Effect Component in CeAl cerium-based intermetallic compounds with strong 2 spin (nonmagnetic Kondo lattices [23]) and charge The temperature dependences of the main (odd in (intermediate-valence compounds) fluctuations, in magnetic field) anomalous component of the Hall coef- which nonmagnetic ground states are formed as tem- a perature decreases because of manybody effects. Simi- ficient R (T, H ) of CeAl measured in this work in a H 0 2 ρ ≈ wide temperature range from 1.8 to 300 K at several larly, the resistivity maximum at T max 5.5 K (Fig. 2, fixed magnetic field values are shown in Fig. 10. In curve 1) and the negative Seebeck coefficient minimum agreement with the results obtained in [5Ð10] for other (Fig. 2, curve 6) should in our view be treated as special cerium-based intermetallic compounds with charge and features of low-temperature transport in compounds a with electron density fluctuations. In the CeAl matrix, spin fluctuations, a positive large-amplitude R (T) 2 H these fluctuations are caused by the formation of many- maximum was observed in the vicinity of the character- body states. The problem of a consistent interpretation istic temperature of spin fluctuations Tsf equal to Tsf = ≈ of ρ(T), Ra (T), and S(T) singularities will be consid- TK 5 K for the compound under study [16]. As the H µ ≈ ≤ condition BH kBTsf is satisfied in fields H 70 kOe ered in the next section in more detail. used in this work (see above), the amplitude of the a a The suppression of the R (T) maximum in mag- R (T) maximum essentially depends on the external H H netic fields is accompanied by its noticeable shift magnetic field value. The magnetic field-induced suppression of spin fluctuations that are caused by spin-flip upward along the temperature axis (see Fig. 10). For a RH scattering of charge carriers results in nonlinear field instance, the RH (T max ) Hall coefficient decreases more ρa ≈ dependences of the Hall resistivity H (T) (see Fig. 7) than twofold in magnitude in magnetic field H 60 kOe and, as a consequence, a sharp decrease in the ampli- RH (Fig. 10), whereas the T max value increases to a tude of the RH (T) maximum (Fig. 10). Note that the RH ≈ T max (60 kOe) 6.5 K. Such a noticeable shift of the a anomalous positive component R (T) cannot be a H low-temperature singularity of R (T) in high magnetic related to the magnetic (antiferromagnetic) ordering of H the CeAl matrix at temperatures T < T ≈ 3.85 K. fields upward along the temperature axis correlates well 2 N with the behavior of the low-temperature resistivity a Indeed, the width of the Hall coefficient RH (T) maxi- maximum in magnetic fields (see Fig. 2 and [15]). As a mum is fairly large compared with T (∆T ~ 10 K, a ρ N result, while the absolute RH (4.2 K) and (4.2 K) val- a Fig. 10), and the suppression of the RH amplitude in ues decrease substantially and in parallel to each other

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(both parameters decrease approximately threefold in 80 a Θ = –350 ± 20 K magnetic field H ≈ 70 kOe), their ratio µ = R /ρ is a 0.2 2 H H 2 slowly varying magnetic field function (Fig. 11). 70 3.4 kOe 32.0 kOe We must stress that this noticeable shift of the low- V s/cm 0.1 , a 1 60.0 kOe H temperature singularities of R (T) and ρ(T) upward Ð H µ Θ 1 = –3.6 ± 0.5 K along the temperature axis, which depends on the exter- 60 nal magnetic field strength, cannot be given a simple 0 100 200 T, K explanation within the traditional Kondo lattice model. Indeed, in this approach, the specified anomalies of the 50 galvanomagnetic properties of the system with strong /(V s) electron correlation appear to be related to the forma- 2 tion of a manybody resonance in the density of elec- 40 , cm tronic states in the neighborhood of Fermi energy E H F µ having a width about kBTK. It is expected in this situation that the resonance spin-flip scattering of conduc- 30 tion electrons by localized cerium magnetic moments, 7.5 K which determines both the renormalization of the den- 10.0 K 3.00 K sity of states mentioned above and the appearance of 20 30.0 K 4.17 K a ρ 40.0 K 5.50 K the anomalies of RH (T) and (T), should be almost µ fully suppressed by magnetic field H ~ kBTK/ B, and 10 R ρ that the Ra (T H ) and ρ(T ) parameters should H max max 0 10203040506070 decrease without noticeable changes in the positions of the low-temperature galvanomagnetic singularities. H, kOe µ a ρ Another special feature of the behavior of the anom- Fig. 11. Field dependences of the H = RH / parameter for alous Hall coefficient in CeAl2, which also does not fit CeAl2 at various temperatures. Temperature dependences in with the traditionally used approach, is the complex µÐ1() H T in magnetic fields H0 = 3.4, 32.0, and 60.0 kOe are a activation dependence of RH (T) in this intermetallic shown in the inset. compound; this dependence was for the first time studied in [12]. The temperature dependence of the anoma- 3 lous Hall coefficient is plotted in the coordinates I II III Ea3/kB = 2.53 ± 0.02 K ()a LaAl2 log RH Ð RH = f(1/T) in Fig. 12. In these coordi- E /k = 2.20 ± 0.02 K nates, we easily see three characteristic temperature /C a3 B a 3 2

intervals of the R (T) dependence and, accordingly, cm H , three asymptotic behaviors. In the temperature intervals 2 Ea3/kB = 1.66 ± 0.02 K H 50Ð300 K (I) and 10Ð40 K (II), an anomalous activation LaAl growth is observed as the temperature decreases, R Ð E /k = 7.6 ± 0.2 K H a a2 B a R ()T ∝ exp()E , /k T (2) R H a12 B 3.4 kOe 32.0 kOe (see Fig. 12); the activation energies in these intervals 1 Ea1/kB = 12.0 ± 0.5 K ≈ ± ≈ ± 60.0 kOe are Ea1/kB 12.0 0.5 K and Ea2/kB 7.6 0.2 K, respectively. Note that, as distinct from the results 0 0.1 0.2 0.3 0.4 0.5 –1 –1 reported in [12], these Ea1 and Ea2 values were obtained T , K with the Hall coefﬁcient of the LaAl2 nonmagnetic ana- Fig. 12. Temperature dependences of the Hall coefﬁcient

a a LaAl2 log of cerium dialuminate used as the RH anomalous RH Ð RH (see text) of CeAl2 in reciprocal logarithmical component of the CeAl2 compound; for LaAl2, coordinates at various magnetic field values. LaAl2 ≈ × Ð4 3 RH Ð6 10 cm /C [24]. In temperature interval ≤ a The Ea3/kB value, which lies in the interval 1.5Ð2.6 K, III (T 5 K), the behavior of RH (T) below its maxi- depends on the external magnetic field (see Fig. 12). mum is fairly well described by the dependence As has been mentioned above, such a behavior of a ()∝ () a RH T exp ÐEa3/kBT . (3) the Hall coefficient RH (T), very unusual for metallic

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 804 SLUCHANKO et al.

2 ∆ systems, not only does not fit in with the Kondo lattice field splitting parameters of the F5/2 cerium state, 1 = ∆ model, but also cannot be given a simple explanation 100 K and 2 = 170 K [16Ð18, 28Ð30], see Fig. 2, in terms of the model of skew-scattering of charge car- inset b). As a consequence, the parameters of spin- riers [1Ð4]. Indeed, both approaches under consideration polaron states in CeAl2 change. The localization radii are based on the assumption of a prevailing role played * * by resonance spin-flip scattering of conduction electrons a p12, and effective masses m12, of manybody states ≈ by localized magnetic moments of rare-earth metal ions. with the binding energies Ea1/kB 12 K (Fig. 12, inter- ≈ From the point of view of the authors of [1Ð4], both the val I) and Ea2/kB 7.6 K (Fig. 12, interval II) can be anomalous positive Hall effect in compounds with estimated by the equations heavy fermions (including cerium-based compounds) τ µ and resistivity anomalies are consequences exclusively m12*, = e eff/ H, (5) of the special features of scattering effects. In particular, the contribution of skew-scattering at temperatures ប a a*p12, = /2Ea12, m12*, . (6) above the RH (T) maximum (that is, at T > TK) is esti- τ mated in [1Ð4] by the approximate equation The relaxation time eff at various temperatures between 4 and 300 K can be determined from the half- a ()∝ ρ()χ() width Γ/2 of the quasi-elastic peak in the neutron scat- RH T T T , (4) tering spectra of CeAl2 (e.g., see [16]). The equation χ where (T) is the reduced magnetic susceptibility of the Γ/2 = ប/τ , (7) system. On the whole, applying (4) to analyze the data eff a µ ρ ∝ χ × Ð12 × Ð13 τ on the H = RH (T)/ (T) (T) parameter (Figs. 2, 10), gives 1.3 10 and 4.1 10 s for eff at 5 and 60 K, µ which characterizes the scattering of charge carriers, respectively. Using the experimental data on H(T) pre- leads to qualitative agreement between the experimen- sented in Figs. 2, 10, and 11 and Eqs. (5) and (6), we tal results obtained in this work (see inset in Fig. 11) obtain the following effective masses of heavy charge and the conclusions drawn in [1Ð4]. The CurieÐWeiss carriers and the corresponding localization radii of µÐ1()∝ Θ ∝ χÐ1 spin-polaron states: behavior of H T (T Ð 1, 2) (T) is character- Θ ized by the paramagnetic Curie temperatures 1 = ()≈ − ± Θ ± m1* 60 K 90m0, a*p1 = 6.4 Å, 350 20 K and 2 = Ð3.6 0.5 K. However, as has been mentioned above, the whole set of manybody ()≈ effects in the low-temperature transport of charge carri- m2* 5 K 57m0, a*p2 = 10 Å ers, namely, the activation behavior of the Hall coefficient, shifts of Ra (T) and ρ(T) singularities in magnetic (m0 is the mass of the electron). Note that these esti- H * * fields, etc., fall outside the scope of this approach. In mates of m12, and a p12, are comparable in order of ≈ ≈ addition, the influence of crystal-field splitting of the magnitude with the me* 30m0 and aep* 6 Å parame- 2 F5/2 cerium state (see Fig. 2, inset b) on the behavior of ters of manybody exciton-polaron states found in [31] µ a ρ for the classic compound with fast charge and spin fluc- the Hall mobility H = RH (T)/ (T) was not consistently taken into account in [1Ð4], which impedes a quantita- tuations, samarium hexaboride SmB6. tive analysis of these results. The magnetic field-induced shift of the anomalies of the galvanomagnetic characteristics upward along the The use of the approach based on the formation of temperature axis observed in this work for CeAl2 (see spin-polaron states in Hubbard bands to interpret the Figs. 2, 10) also finds a natural explanation within the anomalies of the low-temperature transport in the com- framework of the suggested approach. The formation of pound with heavy fermions that we are studying is in our spin polarons as a result of fast spin fluctuations on Ce view much more preferable (e.g., see monograph [25] sites is accompanied by the appearance of exchange and [26, 27]). Such states arise as a result of fast spin field H , which is to a great extent responsible for the fluctuations in the immediate vicinity of localized ex cerium magnetic moments in the CeAl matrix. This anomalous increase in the Hall coefficient and the 2 a approach provides a natural explanation of the activa- appearance of RH (T) singularities in the vicinity of ≈ µ tion behavior of the Hall coefficient in CeAl2 (see kBTsf BHex. In addition to the suppression of fast spin Fig. 12). In our view, the activation energies Ea1, 2 fluctuations on Ce sites, the summation of the compo- should be put in correspondence to the characteristics nents H + Hex substantially shifts the anomalies of the of spinÐpolaron complexes formed in the vicinity of Ce transport characteristics upward along the temperature ≥ ∆ sites. The transition from T 1, 2 (interval I in Fig. 12) axis as the external magnetic field increases. Note that ∆ ≈ ≈ to T < 1 100 K determines the change in the mode of the Hex 75 kOe value in CeAl2 at low temperatures ∆ fast spin 4fÐ5d fluctuations (here, 1, 2 are the crystal- was estimated in [14, 32, 33] by analyzing the polarized

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 GENESIS OF THE ANOMALOUS HALL EFFECT IN CeAl2 805 neutron diffraction spectra. Close Hex values were also magnetic modulated phase of CeAl2. In these works, an obtained in [22] from CeAl2 magnetostriction measure- “energy gap in the excitation spectrum of magnons” ≈ ± ± ± ments at helium temperatures, Hex 79 2 kOe. An with Eg = 0.87 0.08 meV [39] (Eg = 11 3 K [40]) additional “estimate from below” of the exchange field was observed. In addition, the inelastic neutron scatter- µ Θ ≈ ± BHex/kB is provided by the 2 Ð3.6 0.5 K value ing spectra of the magnetically ordered CeAl2 phase found in this work (Fig. 11, inset), which reproduces contained two absorption singularities, or “magnon Θ ≈ ± ± well the paramagnetic Curie temperature Ð3.9 K peaks,” at Ea1 = 1.2 0.8 meV and Ea2 = 0.7 0.4 meV obtained in [34] from CeAl2 magnetic susceptibility [17]. To within the error of measurements performed measurements. in [17], these values correspond to the binding energies of spin polarons E found in this work (see Fig. 12). Another no less important argument in favor of the a1, 2 spin-polaron approach that we propose for interpreting At the same time, it is necessary to stress important differences between the formation of manybody states in the low-temperature properties of CeAl2 is, in our view, the Schottky anomaly of the low-temperature heat the narrow-band FeSi semiconductor with a fairly low ≈ concentration of spin polarons at low temperatures T < capacity at about T 6 K observed in [35], where mea- 17 18 Ð3 surements were performed in magnetic field H ≈ 40 K (10 Ð10 cm [36, 37]) and in the CeAl2 inter- 50 kOe. Recall that, according to the result obtained in metallic compound with a fairly broad conduction this work (see Fig. 10), the Zeeman splitting of the band. In addition to substantial screening effects, the ground state of the system in the effective field Heff = strong magnetic interaction of ferromagnetic microre- ≈ gions through RKKY electron density oscillations H + Hex (external magnetic field H 60 kOe) causes a (indirect exchange) arises in CeAl2. The related special the appearance of an RH (T) maximum at about features of magnetic ordering in CeAl2 were discussed R T H (60 kOe) ≈ 6.5 K. The calculations of two-level in [19] comparatively recently. It appears that so com- max plex a magnetic structure in CeAl at medium- and system parameters in the resonance level model per- 2 formed in [35] using the results of low-temperature long-range magnetic order distances (antiferromagnetic ordering in the system of ferrons of submicron CeAl2 heat capacity measurements then acquire special significance. In [35], the activation energy E /k ≈ dimensions) is the main reason why the identification a B of magnetically ordered phases in this compound 9.6 K was found from the Γ/2 ≈ ប/τ = k T ≈ 0.5 meV eff B K encounters serious difficulties (e.g., see [41–47]). value as an estimate of the width of two-level system levels. To within calculation errors [35] and taking into Let us return to the experimental results shown in account the contribution of the external magnetic field a ∝ Γ Figs. 10 and 12. Note that the RH (T) exp(ÐEa3/kBT) H = 50 kOe to the Zeeman splitting of the 7 doublet, this value fairly closely agrees with that found in the dependence observed in this work for the anomalous ≈ ± contribution to the Hall coefficient of CeAl at T < 10 K present work, Ea/kB 7.6 0.2 K. 2 is similar to that predicted in [48] on the basis of calcu- The suggested approach predicts that temperature lations of the behavior of the Hall coefficient in a lowering in the region T < 50 K (Fig. 10, region II) will system with topologically nontrivial spin configura- not cause only an increase in the manybody resonance tions (Berry phases). According to [48, 49], the Hall amplitude in the vicinity of the Fermi energy EF but effect is modified in this situation because of the 〈 〉 ∝ also an essential rearrangement of the magnetic struc- appearance of the internal magnetic field Hint = hz ture of spin polarons. By analogy with the result (1/kBT)exp(ÐEa/kBT), which adds to the external field H. obtained for the FeSi compound with strong electron correlation [36, 37], the transition to coherent spin fluc- To conclude this section, let us fairly roughly esti- tuations in the vicinity of Ce sites should be accompa- mate the localization radius of manybody states from the nied by the formation of ferromagnetic microregions a results presented in Fig. 10. The RH parameter will be (ferrons) from spin polarons in the CeAl2 matrix. According to the study performed in [36, 37] for iron put in correspondence to the effective reduced concentra- ν Ð1 monosilicide, an additional special feature of such a tion of carriers per cerium atom = (RHeNCe) (the “phase transition” in a system of nanosized magnetic right-hand axis in Fig. 10). In this situation, an increase regions is the retention of virtually unchanged activa- in ν in the interval 0 < ν ≤ 1 can be treated as an increase tion characteristics (band structure parameters) both for in the effective volume per carrier; we believe this spin polarons and for ferromagnetic nanoclusters increase to be caused by manybody effects in CeAl2. formed from them. It appears that the situation with The characteristic CeÐCe distance in the crystal struc- ≈ CeAl2 is similar. At T < 20 K, a strong dispersion of ture of the CeAl2 Laves phase is aCe−Ce 3.5 Å [50]. elastic constants and the related substantial anomaly in Using this value, we obtain the crude estimate a*p = 6Ð the absorption of ultrasound was observed in this com- ν a pound [38]. Also note the result obtained in [39, 40] in 16 Å for = N/NCe = 0.2Ð0.3 in the vicinity of the RH studying NMR spin-lattice relaxation in the antiferro- maximum.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 806 SLUCHANKO et al. 4.4. The Separation of Contributions account a qualitative rearrangement of the density of ρ a electronic states in the neighborhood of EF. to the (T), S(T), and RH (T) Transport Coefficients For this reason, it is more correct to use the standard As has been mentioned in Section 3, the resistivity σ a ρ(T) (Fig. 2, curve 1), Seebeck coefficient S(T) (Fig. 2, equations for , S, and RH in the form [53] a curve 6), and Hall coefficient RH (T) (Figs. 10, 12, H = 3 σσ 3.4 kOe) dependences contain several anomalies in the = ∑ i, (8) temperature range 1.8Ð300 K. These anomalies are evi- i = 1 dence of concerted changes in the specified CeAl2 parameters in the temperature ranges 50Ð300 K (I), 5Ð 3 σσ 50 K (II), and T < 5 K (III). The distinguishing feature S = ∑ iSi, (9) of the transport of charge carriers in region I is a deter- i = 1 mining role played by the contribution of inelastic scat- Γ 3 3 tering related to the transitions between the ground ( 7) a σ 2 σ 2 a 2 RH ∑ i = ∑ i RHi. (10) and two excited doublets (Fig. 2, inset b) of the F5/2 cerium state. The transport of carriers in mode I at i = 1 i = 1 100−300 K is characterized by a linear ρ(T) depen- We could not analyze the experimental data of this dence combined with slow variations in the positive work using the approach suggested in [52] and based on (12Ð15 µV/K) Seebeck coefficient and an almost acti- ()1 () ≈ summing the nonmagnetic S0, positive Kondo Sd T , vation behavior of the Hall coefficient with Ea1/kB ()2 () 12 K. The transition from mode I to II (see Figs. 2, 10, and negative resonance Sd T terms (in [52], the 12, 13) is accompanied by sharp changes in the mea- inversion temperature of the Seebeck coefficient was ρ a S sured , S, and RH values. In addition to substantial found to be T < 0.6T ≈ 3 K, which is obviously at ρ inv K deviations of the (T) dependence from linearity, S(T) S ≈ a variance with the value for CeAl2, T inv 46 K). For this sign reversal (Fig. 2, curve 6) and a kink in the RH (T) reason, we applied a phenomenological approach to sep- ≈ dependence (Fig. 12) are observed at T 50 K. The σ a low-temperature features of the behavior of ρ, S, and arate the contributions to , S, and RH in this compound. a Also note that, as far as we know, no self-consistent RH in the transition region 4Ð12 K between intervals II σ a and III (Figs. 2, 10, 12, 13) also provide clear evidence analysis of contributions to the , S, and RH transport of a change in the asymptotic behaviors of the specified coefficients for rare-earth metal-based compounds with CeAl2 characteristics. heavy fermions has been performed as yet. Currently, neither reliable and consistent explana- In region I, of the greatest importance is inelastic tion can be suggested for the transport characteristics of scattering of carriers accompanied by transitions cerium-based compounds with heavy fermions, includ- 2 ∆ ≈ between cerium F5/2 state doublets spaced 1 100 K ing the nature of various contributions to the conductiv- and ∆ ≈ 170 K apart from the ground-state doublet Γ . ity, Seebeck coefficient, and Hall coefficient nor can 2 7 σ a these contributions be identified. It is therefore of inter- For this reason, the contributions 1, S1, and RH1 (see est to perform a comparative analysis based on the Fig. 13) were approximated by the analytic equations results of measurements performed in this work for σ = σ ()T exp()Ð ∆/k T , high-quality polycrystalline CeAl2 samples. Among the 1 0 1 B α few investigations in which resistance and thermoelec- σ ()T ==1.03/T , α 0.73, tric data on cerium intermetallic compounds were con- 0 sidered jointly, we must mention work [51], where the ()I ()I µ Nordheim equation S1 ==S0 +11BT, S0 V/K, 2 ρS = ρ S + ρ S B = 0.0085 µV/K , 0 0 mag mag (11) was used to analyze the impurity and magnetic contri- RH () E LaAl ρ a a 0 a1 2 butions to (T) and S(T) of CeNi2Sn2. Such a simplified RH1 = RH1 exp------Ð RH , representation of the sum of the low-temperature trans- kBT port components for a tetragonal compound with ∆ ≈ ≈ ()0 ≈ × Ð3 3 20 K and Tsf = TK 1.6 K is at least, insufficiently accu- RH1 0.89 10 cm /C, rate. In addition, according to [52], both the GorterÐ RH ≈ ρ ρ Ea1 /kB 12 K. Nordheim equation and the Matissen rule = ∑i i σ are not good approximations for compounds with The preexponential factor 0(T) was found by fitting heavy fermions, which should be treated taking into the experimental dependence with the use of the opti-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 GENESIS OF THE ANOMALOUS HALL EFFECT IN CeAl2 807

0.06 III II I (a) –1 0.04 cm) µΩ

, ( 0.02 σ σ σ 2 3 σ 1 0 20 S1 10 0 V/K

(b) µ –10 , S S 2 –20 S3

3 2

cm a

–3 RH3 a RH1 , 10 1 a

a R H H2 R

0 10 100 T, K σ a Fig. 13. Decomposition of the temperature dependences of the transport characteristics (a) (T), (b) S(T), and (c) RH (T) of CeAl2 into contributions in temperature intervals I, II, and III.

≈ × 22 Ð3 mization procedures implemented in the ORIGIN 6.1 and NCe 1.5 10 cm in CeAl2. It follows from (12) program. that the concentration of carriers in the temperature ν ≈ The “inelastic” contribution σ is described in (11) range 100Ð300 K can be estimated at I 0.53 and, 1 ≈ × 21 Ð3 by a very simplified equation, which gives an activation therefore, NI 8 10 cm . Note that (12) describes dependence of conductivity in the transition region at the asymptotic behavior of S, which, in our case, corre- T < 100 K. The main requirement imposed on (11) is sponds to the formation of a band of spin-polaron states σ Ӷ Β ≈ the vanishing of the 1 component at T 100 K. of width Ea2/k 12 K in the vicinity of EF. Clearly, the high-temperature contribution also con- According to [36], an analogous activation depen- tains the σΓ component determined by the scattering dence of the Hall coefficient of spin-polaron transport 7 Γ in (11) can also be used to approximately estimate the of carriers by the ground-state doublet 7. Unlike the inelastic contribution, this component is also retained at concentration of carriers for the spin-polaron states of low temperatures. Comparative estimates of its relative the FeSi matrix, value in region I, however, give σΓ ≤ 0.1σ , which in 7 1 CeAl2 1 ≈ × 21 Ð3 our view justifies the approximation that we use. N I = ------()- 710 cm . eRa 0 In addition, (11) contains a large constant contribu- H1 tion to the Seebeck coefficient (S1) alongside a small linear term. We can therefore use the Heikes’ formula This estimate agrees satisfactorily with the results to describe the Seebeck coefficient under strong Hub- obtained above. bard correlation conditions, The asymptotic behaviors of all charge carrier transport parameters of CeAl2 change as temperature decreases in the interval 50Ð100 K (see Figs. 13, 14). I kB 1 Ð ν S0 = ----- ln------, (12) σ a e ν The corresponding contributions to , S, and RH are comparable in magnitude. The situation in the transi- ν where, as previously, we use the notation = N/NCe, tion temperature region 5Ð10 K between intervals II

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 808 SLUCHANKO et al.

(a) 0.2 σ S1 1 0

–0.2 σ cm K) σ S2 2 Ω S3 3 –0.4 , V/( σ

–0.6 S

–0.8

(b) ) 2 10 Ω a σ2 RH3 3 cm/(C 5 , 10 2

σ III II I H a

R 1 a σ2 RH2 2 a σ2 RH1 1

10 100 T, K σ a σ2 Fig. 14. Decomposition of (a) S and (b) RH into partial contributions in temperature intervals I, II, and III for CeAl2.

σ σ and III is similar. In this region, the 2 and 3, S2 and described the activation behavior of the thermoelectric a a and Hall coefficients in interval II, S3, and RH2 and RH3 components have comparable values (see Figs. 13, 14). The procedure used within the S phenomenological approach to separating the contribu- ∝ kB Ea2 S2 ------, tions to low-temperature transport was as follows: In e kBT interval III, the data given in Figs. 2 and 10 were (14) approximated by the analytic dependences RH a a()0 Ea2 LaAl2 RH2 = RH2 exp------Ð RH . kBT σ Ðβ β 3 ==AT , 1.44, σ Figures 13, where the components of , S, and RH are plotted, and 14, where their additivity is verified III III µ S3 ==S0 +18CT, S0 Ð V/K, [Eqs. (9) and (10)], show that, on the whole, the sug- (13) µ 2 gested procedure for separating contributions, in spite C = Ð0.8 V/K , of its approximate character, gives a quantitative description of the behavior of the transport coefficients Ra ==DT0.7, D 0.76386. of CeAl2. It can therefore be used to estimate certain H3 microscopic parameters that characterize the electronic structure of this compound. Within this procedure, the Equations (13) fairly accurately describe the σexpt, Sexpt, activation energy of the Seebeck coefficient in interval II expt S ≈ and RH curves at T < 4 K. is estimated at Ea2 /kB 3.6 K, which is noticeably H ≈ Next, additivity condition (8) for σ was used to smaller than the value for the Hall coefficient Ea2 /kB σ σ σ () obtain the 2 component by subtracting the sum 1 + 3 a 0 ≈ × Ð3 3 7 K. The use of RH2 1.03 10 cm /C in (14) for σexpt a from the experimental curve. The S2 and RH2 con- estimating the concentration of carriers in interval II × 21 Ð3 ν ≈ tributions were analyzed using the equations that gives NII = 6 10 cm or = NII/NCe 0.4. The

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 GENESIS OF THE ANOMALOUS HALL EFFECT IN CeAl2 809 III µ S0 = Ð18 V/K value can also be formally used in (13) 10 K is caused by the special features of medium- and within the framework of the spin-polaron approach to long-range magnetic ordering and the complex HÐT estimate the reduced concentration of carriers in the magnetic phase diagram of CeAl2 at low temperatures. Θ ≈ ≈ transition temperature region 5Ð10 K. The use of the This result and the estimates of the p 3.6 K and Hex high-temperature asymptotic behavior of the Seebeck 75 kOe magnetic exchange parameters appear to pro- coefficient [Eq. (12)] is justified in this temperature vide evidence in favor of the formation of nanosized region, because the formation of a narrow band of ferromagnetic regions in the CeAl2 matrix at tempera- manybody states in the vicinity of EF with the activa- tures substantially higher than the Néel temperature ≈ ≈ tion energy Ea2/kB 7.6 K results in the appearance of TN 3.85 K of this compound. It was shown that the ferromagnetic nanoclusters based on spin-polaron temperature dependence of the main anomalous com- states at T < 20 K (see the argumentation given above). ponent Ra in this compound with heavy fermions has As a result, we have H a complex activation character. The behavior of Ra (T) ν ≈ ≈ × 21 Ð3 H III 0.45, NIII 6.8 10 cm . observed in CeAl2 does not fit in with the interpretation It should also be stressed that a comparison of the within the framework of the skew-scattering model, S ≈ according to which scattering effects play a determin- activation energies of the thermoelectric (Ea2 /kB ing role in the formation of Hall coefficient anomalies RH ≈ in concentrated Kondo systems. 3.6 K) and Hall (Ea2 /kB 7 K) coefficients and the paramagnetic Curie temperatures found in this work The special features of the suppression of the anom- Θ ≈ Θ ≈ ( p 3.6 K) and measured in [34] ( p 3.9 K) lead us alous Hall effect in high magnetic fields studied in this to suggest that there are two approximately equal com- work are likely to be evidence that spin-polaron effects ponents that determine the formation of manybody should be taken into account to interpret the behavior of states in CeAl at low temperatures. It can be expected the transport characteristics of cerium-based interme- 2 tallic compounds. We estimated the parameters charac- that the spin-polaron (magnetic) contribution to Ea2, although it does not manifest itself in the temperature teristic of manybody states that arise in the CeAl2 dependence of the Seebeck coefficient [53], at the same matrix at low and intermediate temperatures (their Θ ≈ effective masses and localization radii). The nontrivial time plays a key role in determining the p and Hex 75 kOe magnetic exchange parameters. This sugges- analysis of contributions to the transport characteristics tion allows us to expect that the sum of the contribu- of CeAl2 performed in this work based on the results of tions of the exciton (4f +Ð5dÐ) and spin-polaron compo- Hall effect measurements combined with resistivity and Seebeck coefficient data led us to conclude that the RH nents should determine the Ea2 value, which character- approach that uses the Kondo lattice model has serious izes the low-temperature behavior of the Hall limitations as applied to the totality of properties of coefficient. At the same time, the experimental results cerium-based concentrated Kondo systems. presented in this work are obviously insufficient as reliable evidence of excitonÐpolaron nature of manybody ACKNOWLEDGMENTS states in CeAl2 with fast electron density fluctuations. Note in conclusion that, within the framework of the This work was financially supported by the Russian approach that we use to separate the contributions to the Foundation for Basic research (project nos. 01-02-16601 low-temperature charge transport, the most complex and 03-02-06531); the “New Materials” project of the problem is, in our view, a quantitative analysis of the Ministry of Education of Russian Federation (no. 202.07.01.023); the program “Strongly Correlated σ a 3, S3, and RH3 components at T < 5 K. In this region, Electrons in Semiconductors, Metals, Superconduc- we must take into account not only the establishment of tors, and Magnetic Materials” of the Russian Academy coherence (the formation of heavy carrier bands), but of Sciences (Division of Physical Sciences); the pro- also effects related to complex magnetic ordering of the gram for the development of the instrumental base of cerium-based intermetallic compounds under conside- scientific organizations of RF Ministry of Industry and ration. Science; INTAS project no. 00-807; and the program of the Russian Academy of Sciences for support of young scientists. Special thanks for individual support are due 5. CONCLUSIONS to the Foundation for Promoting Science in this country The detailed measurements of the Hall effect in (V.V.G. and S.V.D.) and Government of Moscow and CeAl2 with fast electron density fluctuations performed Soros Foundation (A.V.B. and M.I.I.). in this work allowed us to separate and classify the contributions to the anomalous Hall effect in this compound with heavy fermions. The appearance of an REFERENCES anomalous magnetic contribution “even in magnetic 1. P. Coleman, P. W. Anderson, and T. V. Ramakrishnan, field” to the Hall resistance observed in this work at T < Phys. Rev. Lett. 55, 414 (1985).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 810 SLUCHANKO et al.

2. A. Fert and P. M. Levy, Phys. Rev. B 36, 1907 (1987). 29. P. Thalmeier and P. Fulde, Phys. Rev. Lett. 49, 1588 3. P. M. Levy and A. Fert, Phys. Rev. B 39, 12224 (1989). (1982). 4. P. M. Levy, Phys. Rev. B 38, 6779 (1988). 30. G. Guntherodt, A. Jayaraman, G. Batlogg, et al., Phys. Rev. Lett. 51, 2330 (1983). 5. N. B. Brandt, V. V. Moshchalkov, N. E. Sluchanko, et al., Solid State Commun. 53, 645 (1985). 31. N. E. Sluchanko, V. V. Glushkov, B. P. Gorshunov, et al., Phys. Rev. B 61, 9906 (2000). 6. V. V. Moshchalkov, F. G. Aliev, N. E. Sluchanko, et al., J. Less-Common Met. 127, 321 (1986). 32. B. Barbara, M. F. Rossignol, J. X. Boucherle, et al., Phys. Rev. Lett. 45, 938 (1980). 7. T. Penney, F. P. Milliken, S. von Molnar, et al., Phys. Rev. B 34, 5959 (1986). 33. A. Benoit, J. X. Boucherle, J. Flouquet, et al., in Valence Fluctuations in Solids, Ed. by L. M. Falicov, W. Hanke, 8. A. Fert, P. Pureur, A. Hamzic, and J. P. Kappler, Phys. and M. B. Maple (North-Holland, Amsterdam, 1981), Rev. B 32, 7003 (1985). p. 197. 9. T. Hiraoka, E. Kinoshita, T. Takabatake, et al., Physica B 34. M. C. Croft, R. P. Guertin, L. C. Kupferberg, and (Amsterdam) 199Ð200, 440 (1994). R. D. Parks, Phys. Rev. B 20, 2073 (1979). 10. H. Sugawara, H. R. Sato, Y. Aoki, and H. Sato, J. Phys. 35. C. D. Bredl, F. Steglich, and K. D. Schotte, Z. Phys. B Soc. Jpn. 66, 174 (1997). 29, 327 (1978). 11. U. Welp, P. Haen, G. Bruls, et al., J. Magn. Magn. Mater. 36. N. E. Sluchanko, V. V. Glushkov, S. V. Demishev, et al., 63Ð64, 28 (1987). Zh. Éksp. Teor. Fiz. 119, 359 (2001) [JETP 92, 312 12. N. E. Sluchanko, A. V. Bogach, V. V. Glushkov, et al., (2001)]. Pis’ma Zh. Éksp. Teor. Fiz. 76, 31 (2002) [JETP Lett. 76, 37. N. E. Sluchanko, V. V. Glushkov, S. V. Demishev, et al., 26 (2002)]. Phys. Rev. B 65, 064404 (2002). 13. N. E. Sluchanko, V. V. Glushkov, S. V. Demishev, et al., 38. G. Hampel and R. H. Blick, J. Low Temp. Phys. 99, 71 Zh. Éksp. Teor. Fiz. 113, 339 (1998) [JETP 86, 190 (1995). (1998)]. 39. D. E. MacLaughlin, O. Peca, and M. Lysak, Phys. Rev. 14. B. Barbara, J. X. Boucherle, J. L. Buevoz, et al., Solid B 23, 1039 (1981). State Commun. 24, 481 (1977). 40. J. L. Gavilano, J. Hunziker, O. Hudak, et al., Phys. Rev. 15. F. Lapierre, P. Haen, A. Briggs, and M. Sera, J. Magn. B 47, 3438 (1993). Magn. Mater. 63Ð64, 76 (1987). 41. S. M. Schapiro, E. Gurewitz, R. D. Parks, and L. C. Kup- 16. F. Steglich, C. D. Bredl, M. Loewenhaupt, and ferberg, Phys. Rev. Lett. 43, 1748 (1979). K. D. Schotte, J. Phys. Colloq. 40 (C5), 301 (1979). 42. A. Schenk, D. Andreica, M. Pinkpank, et al., Physica B 17. S. Osborn, M. Loewenhaupt, B. D. Rainford, and (Amsterdam) 259Ð261, 14 (1999). W. G. Stirling, J. Magn. Magn. Mater. 63Ð64, 70 (1987). 43. A. Schenk, D. Andreica, F. N. Gygax, and H. R. Ott, 18. M. Loewenhaupt, W. Reichardt, R. Pynn, and E. Lindley, Phys. Rev. B 65, 024444 (2002). J. Magn. Magn. Mater. 63Ð64, 73 (1987). 44. A. Amato, Rev. Mod. Phys. 69, 1119 (1997). 19. N. E. Sluchanko, A. V. Bogach, I. B. Voskoboœnikov, 45. E. M. Forgan, B. D. Rainford, S. L. Lee, et al., J. Phys.: et al., Fiz. Tverd. Tela (St. Petersburg) 45, 1046 (2003) Condens. Matter 2, 10211 (1990). [Phys. Solid State 45, 1096 (2003)]. 46. F. Giford, J. Schweizer, and F. Tasset, Physica B 20. M. Croft, I. Zoric, and R. D. Parks, Phys. Rev. B 18, 345 (Amsterdam) 234Ð236, 685 (1997). (1978). 47. T. Chattopadhyay and G. J. McIntyre, Physica B 21. M. Croft, I. Zoric, and R. D. Parks, Phys. Rev. B 18, (Amsterdam) 234Ð236, 682 (1997). 5065 (1978). 48. J. Ye, Y. B. Kim, A. J. Millis, et al., Phys. Rev. Lett. 83, 22. E. Fawcett, V. Pluzhnikov, and H. Klimker, Phys. Rev. B 3737 (1999). 43, 8531 (1991). 49. Y. B. Kim, P. Majumdar, A. J. Millis, and B. I. Shraiman, 23. N. B. Brandt and V. V. Moshchalkov, Adv. Phys. 33, 373 cond-mat/9803350 (1998). (1984). 50. E. Walker, H. G. Purwins, M. Landolt, and F. Hulliger, 24. M. Christen and M. Godet, Phys. Lett. A 63A, 125 J. Less-Common Met. 33, 203 (1973). (1977). 51. J. Sakurai, H. Takagi, T. Kuwai, and Y. Isikawa, J. Magn. 25. N. F. Mott, MetalÐInsulator Transitions (Taylor and Magn. Mater. 177Ð181, 407 (1998). Francis, London, 1974; Nauka, Moscow, 1979). 52. K. H. Fisher, Z. Phys. B 76, 315 (1989). 26. S. H. Liu, Phys. Rev. B 37, 3542 (1988). 53. P. M. Chaikin, in Organic Superconductivity, Ed. by 27. T. Portengen, Th. Ostreich, and L. J. Sham, Phys. Rev. B V. Z. Kresin and W. A. Little (Plenum, New York, 1990), 54, 17452 (1996). p. 101. 28. M. Loewenhaupt, B. D. Rainford, and F. Steglich, Phys. Rev. Lett. 42, 1709 (1979); M. Loewenhaupt and U. Witte, J. Phys.: Condens. Matter 15, S519 (2003). Translated by V. Sipachev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 811Ð819. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 927Ð937. Original Russian Text Copyright © 2004 by Konyukhov, Likhachev, Oparin, Anisimov, Fortov. NONLINEAR PHYSICS

Numerical Modeling of Shock-Wave Instability in Thermodynamically Nonideal Media A. V. Konyukhova,b, A. P. Likhacheva, A. M. Oparinb, S. I. Anisimovc, and V. E. Fortova aInstitute of Thermophysics of Extreme States, Joint Institute of High Temperatures, Russian Academy of Sciences, Moscow, 125412 Russia bInstitute for Computer-Aided Design, Russian Academy of Sciences, Vtoraya Brestskaya ul. 19/18, Moscow, 123056 Russia cLandau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected] Received August 27, 2003

Abstract—A numerical analysis of the nonlinear instability of shock waves is presented for solid deuterium and for a model medium described by a properly constructed equation of state. The splitting of an unstable shock wave into an absolutely stable shock and a shock that emits acoustic waves is simulated for the ﬁrst time. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION shock wave splits into multiple stable elementary For a variety of media (in particular, for media with waves (shock waves, isentropic rarefaction/compres- phase transitions), the shock Hugoniot curves include sion waves, and/or contact discontinuities) [3]. These segments corresponding to shock waves that are unsta- findings led to the hypothesis that exponential growth ble with respect to small disturbances of the wave front. cannot be observed because an unstable initial shock The linear analysis developed in [1, 2] predicts two wave must evolve into an admissible split-wave config- types of shock-wave evolution: small periodic shock- uration [3]. front disturbances may either grow exponentially with To facilitate further analysis, we recall here some time or persist for an indefinitely long time without facts concerning the problem of shock-wave stability. It being either damped or amplified. In the latter case, the was shown in [4] that the point in the pÐV diagram shock front emits acoustic and entropy waves propagat- where the shock Hugoniot curve is tangent to a line ing downstream. The D’yakovÐKontorovich linear the- passing through the initial state is a sonic point; i.e., the ory was the first to provide criteria for shock-wave Mach number behind the shock front is unity in a refer- instability. In terms of the stability parameter ence frame tied to the front: uDÐ DuÐ 2∂V M ==------=1.------Lj= ------∂ - , c c p H The slope of the tangent line to the corresponding isen- 2 where j = (p2 Ð p1)/(V1 Ð V2) is the mass flux across the trope satisfies the relation ρ shock front, V = 1/ is specific volume, p is pressure, 2 ∂p j 2 and the derivative is taken along the Hugoniot curve, ------==Ð------Ð j . ∂ 2 the following regions of qualitatively different behavior V S M are identified: absolute stability (–1 < L < L0), acoustic By definition, emission (L0 < L < 1 + 2M), and exponential growth (L < Ð1, L > 1 + 2M). We use the following notation ∂p 2 here: ∂------= Ð j V H θ 2 2 1 Ð M Ð M θ V 0 u at the tangency point. Therefore, a point where the Ray- L0 ===------, ------, M ---. 1 + θM2 Ð M2 V c leigh line is tangent to the Hugoniot curve is a point of tangency of the Hugoniot with the isentrope: Subsequently, it was shown that the Hugoniot seg- ∂ ∂ ment corresponding to exponential growth lies within a p 2 p ∂------==Ð j ∂------. region of shock-wave nonuniqueness, where a single V S V H

812 KONYUKHOV et al.

P P i.e., acoustic disturbances (with velocity u + c) lag 1.0 behind the shock, transforming into an isentropic com- (a) (b) pression wave [3]. 0.9 4 400 The lower point of tangency of the Rayleigh line 4 with the shock Hugoniot is the lower boundary of 0.8 3 acoustic emission:

0.7 2 2 2 1 1 Ð θM Ð M L0 ===------1Ð L. 0.6 300 1 + θM2 Ð M2

0.5 If the Hugoniot curve is smooth (see Fig. 1a), then the L(P) and L (P) curves are either mutually tangent or 0.4 1 0 200 they intersect at this point. If the Hugoniot curve has a kink (Fig. 1b), then the values of L and L are not 0.3 0 deﬁned at this point, whereas L Ð L either reverses or 0 0 0.2 retains its sign across the kink (as in the case of inter- section or tangency, respectively). However, the L(P) 0.1 100 and L0(P) curves must intersect at the upper point of 12345 0.40.50.60.7Rayleigh-line tangency with a Hugoniot of any shape. V V This point separates Hugoniot segments associated Fig. 1. Typical Hugoniots (a) without and (b) with a kink with exponential growth and acoustic emission. Shock corresponding to the model equation of state and the waves with parameters corresponding to a portion of SESAME tabular equation of state for deuterium, respec- the latter segment adjoining the former one emit acous- tively. The intervals of stability (0Ð1), exponential growth tic waves at angles close to the shock propagation (1Ð2), acoustic emission (2Ð3), and shock-wave splitting (1Ð4) are indicated. The dotÐdash curve is an isotherm. direction. As L Ð1 and M 1, the equation for the cosine of this angle tends to a limit form [4] that has a unique solution: cosα = Ð1. When cosα < 0, the At such points, the D’yakovÐKontorovich stability acoustic wave vector points in the direction of shock parameter is propagation. The wave is “outgoing” because of advection by the ﬂow.

2∂V The upper boundary of shock-wave splitting is Lj==------Ð1. ∂p determined by equating the velocities of the leading H wave and the downstream shock. The pressure at this Thus, the points where the Rayleigh line is tangent boundary can be found from the equation to the Hugoniot curve correspond to the boundaries of exponential growth. Between these boundaries, L < Ð1. p Ð p c 2 ------4 1- = Ð-----1- V Ð V 2 If the Hugoniot is smooth, then the convexity condi- 4 1 V 1 tion for equation of state (EOS) is violated below the lower tangency point: as a tangency condition for a Rayleigh line and an isentrope. (Here, subscripts correspond to points in Figs. 1a ∂2 p and 1b.) This relation also means that the downstream------< 0 . ∂ 2 shock velocity relative to the medium equals the speed V S of sound:

Above this point, a single initial shock wave evolves 1/2 into the following split wave conﬁguration: p4 Ð p1 DII Ð u1 ==V 1------c1.

V 1 Ð V 4 → S WTCS, A detailed analysis of the splitting wave conﬁgura- where S is a shock wave, W is a shock or rarefaction tions corresponding to smooth and kinked Hugoniot wave, T is a tangential discontinuity, and C is an isen- curves was presented in [3]. tropic compression wave [3]. This is explained by the Even though shock-wave stability has been ana- fact that u + c < D below the point where lyzed in numerous theoretical studies for almost half a century, experimental observations supporting the theoretical predictions mentioned above remain scarce to DuÐ M ==------1, this day. On the one hand, the corresponding thermody- c namic conditions are difﬁcult to implement experimen-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 NUMERICAL MODELING OF SHOCK-WAVE INSTABILITY 813 tally. Another reason is the lack of data required to gral conservation laws written as expressions for the specify the conditions of an appropriate physical exper- grid pressure, velocity, and ρ1/2: iment and the expected manifestations of shock-wave instability. To obtain data of this kind, a parametric 1/2 n + 1 ()()n 2 ξ numerical analysis of the behavior of a shock wave z = z + 1 , must be performed by varying its intensity under suitable choice of a properly constructed or realistic equa- ()n 2 n ξ tion of state. Only a few numerical simulations of n + 1 z u + 2 u = ------2 -, shock-wave splitting [5, 6] and acoustic emission [7] ()zn + 1 have been performed to date, and their results are far from complete. Thus, systematic numerical analysis of nonlinear shock-wave instability remains a challenging ε()εpn + 1, ρn + 1 = ()pn, ρn task. This problem is addressed in the present study. 1 n n 2 n + 1 n + 1 2 + ---()ξ()z u Ð ()z u + , In this paper, we analyze the stability of a shock 2 3 wave in solid deuterium (using an EOS borrowed from the SESAME library [8]) and in a model medium where z = ρ1/2, ε is the internal energy per unit volume, described by the equation [5]

2 2 τ (), ρ ()Ðp ()Ð()41/Ð ρ x = Ð---()F Ð F ep = 1 Ð e 4 Ð e . h i + 1/2 i Ð 1/2 This equation of state is thermodynamically consistent, is the net flux of the conserved quantities across a cell, and the corresponding shock Hugoniot curves include τ segments associated with instabilities of all known is the time step, and h is the mesh size. The numerical types. The realistic three-phase SESAME equation of flux across a cell face can be written in a more compact state (given in tabular form) describes the first-order form in terms of pressure, density, and velocity (as molecular-to-metallic phase transition in solid deute- compared to the conserved quantities): rium [8]. 1 F = ---[]F ++F R F , The shock Hugoniot curve based on the model equa- i + 1/2 2 i i + 1 i + 1/2 i + 1/2 tion of state characterizes shock compression processes involving phase transitions or endothermic chemical ρ reactions. This curve allows for both shock splitting u 2 and acoustic emission. Figure 1a shows a low-pressure F = p + ρu , portion of this curve.  ρuH Figure 1b shows the shock Hugoniot curve for low- temperature deuterium compressed from an initial state  × 3 3 ρ ρ characterized by p0 = 1.6 10 GPa and V0 = 0.7 cm /g. 1 ------Unlike the curve shown in Fig. 1a, it has two kinks 2c 2c  where adiabatic compressibility decreases and ρ()ucÐ ρ()uc+ increases stepwise (points 1 and 3, respectively). Since R = u ------, 2c 2c a kinked curve is a limit case of a smooth curve, one  may say that a smooth region of EOS convexity viola- 1 2 ρ()HucÐ ρ()Huc+ ερ + ---u ------tion reduces to a point in this limit and tangency at a 2 2c 2c kink is interpreted accordingly.  ∆ρÐ ∆p/c2 2. NUMERICAL METHOD a = ∆ ()∆ρ , p/ c Ð u To solve the governing equations, we use Roe’s ∆p/()∆ρc + u method [9] extended to an arbitrary equation of state of the form ε = (p, ρ). In this conservative method based T ± ll± on exact characteristic flux splitting, a stationary shock l ==()uu, Ð cu, + c , l ------. satisfying the RankineÐHugoniot jump conditions is an 2 exact solution to the finite-difference problem. We use φl a computationally efficient formulation of the method The components of the vector Fj + 1/2 are expressed as in terms of the vector Y = (p, ρ1/2, u)T [10]. 1 1 l 1 1 1 g Ð g i Following the approach developed in [10], we find φ = g + g Ð ψλ+ ------i + 1 i α . i + 1/2 i i + 1 i + 1/2 α1 i + 1/2 the solution vector on the (n + 1)th time layer from inte- i + 1/2

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 814 KONYUKHOV et al.

In TVD1 (ﬁrst-order accurate scheme), 1 1 Ð σ 1 1 1 g = ------ψλ()m[]α , α . i 2 i + 1/2 j + 1/2 j Ð 1/2 φl ψλ()αl 1 i + 1/2 = Ð i + 1/2 i + 1/2. In the UNO3 scheme relying on the third-order ENO In TVD2 (Harten’s second-order accurate scheme [11]), interpolation [12],

l 1 Ð σ 1 1 1 g = ------ψλ()m[]α , α i 2 i + 1/2 j + 1/2 j Ð 1/2

 2 2 1+ 23Ð σ + σ 1Ð σ Ð 1 1 1 1 1  λ ------+ λ ------m[]∆ α , ∆ α , if α ≤ α , i + 1/2 6 i + 1/2 6 Ð j Ð 1/2 + j Ð 1/2 i Ð 1/2 i + 1/2 +   σ σ2 σ2 λ1Ð 23Ð + λ1+ Ð 1 []∆ α1 , ∆ α1 α 1 > α1 i + 1/2------+ i + 1/2------m Ð j + 1/2 + j + 1/2 , if i Ð 1/2 i + 1/2 ,  6 6

with The EOS derivatives are calculated numerically by using states “1” and “2:”1 τ σλ1 = j + 1/2---, 0.5 h ε ------(ε()ε˜ , ρ˜ (), ρ˜ ρ = ρ ρ p2 2 + p1 2 ˜ 2 Ð 1 ε()ε˜ , ρ ()), ρ 0, xy ≤ 0, Ð p2 1 Ð p1 1 , mxy[], =  min()xy, sgn()x , xy > 0, 0.5 (1) ε = ------(ε()εp˜ , ρ˜ + ()p˜ , ρ p ρ˜ Ð ρ 2 2 2 1 x, xy≤ , 2 1 mxy[], =  ε()ε, ρ˜ ()), ρ y, xy> , Ð p1 2 Ð p1 1 ,

≥ δ  p2, p2 Ð p1 p1 ,  ≥ ⑀ p˜ =  x , x , 2 δ < δ   p1 + p1 , p 2 Ð p 1 p1 , ψ() 2 2 x = x + ⑀ ------, x < ⑀,  2ε ρ , ρ Ð ρ ≥ ρ δ, ρ 2 2 1 1 ˜ 2 =  ρ + ρ δ, ρ Ð ρ < ρ δ, where ⑀ is the entropy correction parameter [11]. 1 1 2 1 1 An approximate solution to the Riemann problem δ for two states corresponding to adjacent cells labeled where is a small positive number. “1” and “2” is given by the relations This differentiation procedure ensures that the jump condition ρ = z z , 1 2 ε ε ε ()ερ ρ () 2 Ð 1 = ρ 2 Ð 1 + p p2 Ð p1 z u + z u u = ------1 1 2 -2, and hence the relation z1 + z2 ∆FA= ∆U, z H + z H H = ------1 1 2 -2, hold across a shock. (Here, A is the Jacobian matrix of z1 + z2 the ﬂux vector for an averaged state.) Then, an arbitrary

1 2 h Ð ερ This approach is analogous to Glaister’s solution [13], where dif- c = ------ε . ferent states were combined for the ﬁrst time in calculating pres- p sure derivatives to extend Roe’s method to an arbitrary EOS.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 NUMERICAL MODELING OF SHOCK-WAVE INSTABILITY 815 steady shock satisfying the Hugoniot relations, i.e., deﬁned by a grid function of the form 10–3 ≤ n U1, ii0, U =  (2) –5 i > 10 U2, ii0,

10–7 such that F(U2) Ð F(U1) = 0, is an exact solution to the

ﬁnite-difference equations (i is a particular value of Norm of residual 0 10–9 grid index).2

In the region of shock-wave nonuniqueness, single 10–11 shock waves are exact solutions to the finite-difference UNO 3 TVD 2 problem in a reference frame tied to its front and small –13 disturbances introduced into initial conditions either 10 TVD 1 decay (when the shock is stable) or grow exponentially. In the course of computations, single waves split and 0 100 200 300 400 500 600 700 evolve into alternative scale-invariant configurations, because solutions to the finite-difference problem are n not unique either. Figure 2 demonstrates that the norm Fig. 2. Norm of residual for split-wave solutions versus time of residual computed by the schemes employed in this step number. study levels off as a split-wave solution is approached under nonuniqueness conditions. Between points 2 and 3, the split-wave configuration involves two shocks separated by an isentropic com- 3. ANALYSIS OF SHOCK-WAVE SPLITTING pression wave:

The Cauchy problem for the Euler equations supple- →→→→ → mented with initial conditions (2) was solved in a coor- S WTSCS. dinate system moving with the front; i.e., the initial shock wave was stationary on the numerical grid. Fig- At point 3, M = 1. The difference in propagation veloc- ure 3 shows the resulting scale-invariant solutions as ity between shock waves can be determined as the dis- functions of x/t for various Hugoniot points character- tance between the corresponding fronts on the x/t axis. ized by post-shock pressures P. In the right panel, 0 is As P increases along the Hugoniot, the leading-wave the initial state, 1 is the lower boundary of exponential intensity remains invariant, the downstream-shock growth and shock-wave splitting, 2 is the upper bound- intensity and velocity increase, and the pressure rise in ary of EOS convexity violation, 3 is the upper boundary the isentropic compression wave decreases. At point 4, of exponential growth and the lower boundary of the leading-wave and downstream-shock velocities are acoustic emission, and 4 is the upper boundary of equal, and the isentropic-wave pressure rise vanishes. shock-wave splitting. This means that point 4 is the upper boundary of shock- wave splitting. The solution is depicted by constant-pressure con- In contrast to the case of smooth violation of EOS tours in the (P, x/t) plane. The pressure values on the convexity, the split-wave conﬁguration corresponding contours correspond to those on the P axis. Hereinafter, to a kinked Hugoniot curve (Fig. 1b) does not involve this representation of results obtained for various any isentropic compression wave:

Hugoniot curves is called a splitting diagram. →→→ → Below point 1, shock waves are absolutely stable. S WTS S. Between points 1 and 2, a single shock wave splits into Figure 4 compares the scale-invariant pressure proﬁles a conﬁguration involving a shock and an isentropic obtained for the model equation of state at P = 0.54 (left

compression wave: panel) and for deuterium at P = 3.6 × 103 GPa (right →→→→ panel). The pressure interval where the isentropic com- S WTCS . pression wave (ICW) exists is indicated here. The velocity difference between the leading wave and 2 This is true only when the Harten entropy correction parameter is downstream shock resulting from shock-wave splitting zero. Generally, entropy correction is applied to eliminate the rar- in deuterium is about 2.8 × 103 m/s. efaction shock arising in Roe’s method. In the present context, rarefaction shocks do not arise when EOS convexity holds. How- These results were veriﬁed by using various numer- ever, we have to deal with the “symmetric” problem of compres- ical techniques (other than those described above), sion shocks when the EOS convexity condition is violated. including Eulerian and Lagrangian schemes. All of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 816 KONYUKHOV et al.

P P 1.0 1.0

0.9 0.9 4

0.8 0.8

0.7 0.7

0.6 0.6 3

0.5 0.5 2

0.4 0.4 1

0.3 0.3

0.2 0.2

0 0.1 0.1 0.2 0.1 0 –0.1 –0.2 1 2 3 4 5 x/t V Fig. 3. Pressure versus scaling variable x/t (left panel) for one-dimensional shock waves corresponding to Hugoniot curve for model equation of state (right panel). these results are mutually consistent and agree with PP previous computations [6]. 0.6

350 4. COMPARISON OF RESULTS 0.5 FOR SEVERAL FINITE-DIFFERENCE SCHEMES AND THE REQUIREMENT ICW OF ENTROPY CORRECTION 0.4 300 Test computations were performed with the first- and second-order accurate TVD schemes and with fluxes calculated by using the third-order one-dimen- 0.3 250 sional ENO interpolation [12]. It was shown that physical solutions satisfying the law of entropy increase under conditions of EOS convexity violation can be 0.2 200 computed by the TVD schemes only with entropy corrections. Among the tested schemes, only UNO3 pro- 0.1 vides a correct solution without entropy correction. For 150 example, the first-order accurate scheme without 0.2 0.1 0 –0.1 –0.2 1 0.5 0 –0.5 –1

x/tx/t entropy correction yields the incorrect two-shock solu- →

→

→→→ → →→→→ Fig. 4. Shock-wave splitting conﬁgurations: S tion S WTS S instead of S WTSCS. Fur- → → → →

thermore, the upper boundary of shock-wave splitting W TS C S for Hugoniot without kinks (model equation of → → → → predicted by using higher order accurate schemes is state, left panel); S WT S S for kinked Hugoniot lower than its experimental value. This result was (SESAME equation for deuterium, right panel).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 NUMERICAL MODELING OF SHOCK-WAVE INSTABILITY 817

P P P 1.0 1.0 1.0 UNO 3 TVD 2 TVD 1 0.9 0.9 0.9

0.8 0.8 0.8

0.7 0.7 0.7

0.6 0.6 0.6

0.5 0.5 0.5

0.4 0.4 0.4

0.3 0.3 0.3

0.2 0.2 0.2

0.1 0.1 0.1 0.2 0.1 0 –0.1 –0.2 0.2 0.1 0 –0.1 –0.2 0.2 0.1 0 –0.1 –0.2 x/t x/t x/t

Fig. 5. Splitting diagrams obtained by using different ﬁnite-difference schemes.

L, L0 L, L0 2.0 (a) (b) 0.5 1.5 P P P 1 1.0 1 0 L L 0.5 0 L0 L0 0

–0.5 –0.5

–0.85 –1.0 –0.90

–1.5 –0.95 –1.0

–1.00 –2.0 0.36 0.38 123 123 P P

Fig. 6. D’yakovÐKontorovich stability parameter L and its critical value L0 versus post-shock pressure P. Pressures behind (a) initial, leading, and (b) downstream shocks are denoted by P0, P1, and P2, respectively.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 818 KONYUKHOV et al. improved by introducing additional diffusion, as in 5. SIMULATION OF ACOUSTIC EMISSION Harten entropy correction (1). A satisfactory solution UNDER SHOCK-SPLITTING CONDITIONS was obtained near the upper shock-splitting boundary with an entropy correction parameter of 0.125ω, where In computations, acoustic emission can be observed ω is the spectral radius of the Jacobian matrix of the if the post-shock state lies above point 4 on the Hugo- ﬂux vector. niot curve in Fig. 1, during a limited time interval under shock-splitting conditions before the distance between Figure 5 compares splitting diagrams computed for split shocks exceeds the acoustic wavelength (if acous- the model equation of state (see Fig. 3) by using differ- tic emission develops faster than does shock splitting), ent schemes with this value of the entropy correction or when acoustic waves are emitted by the downstream parameter. shock in a split-wave conﬁguration.

Y Y

Z X Z X

t = 139.8 t = 1342.3 Y Y

Z X Z X

t = 461.3 t = 1370.2 Y Y

Z X Z X

t = 908.8 t = 1398.2

Fig. 7. Magnitude of pressure gradient. Darker areas correspond to steeper gradients.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 NUMERICAL MODELING OF SHOCK-WAVE INSTABILITY 819

Numerical results illustrating the last manifestation 6. CONCLUSIONS are presented here for an initial shock wave correspond- The evolution of a shock wave into a split-wave con- ing to the Hugoniot point at P = 0.82 in Fig. 3. At this figuration involving a stable shock and a shock emitting point, the original wave simultaneously satisfies the acoustic waves has been simulated for the first time Kontorovich criterion for acoustic emission and the under conditions of shock-wave nonuniqueness. The shock-splitting conditions. Figure 6 shows the direction of the simulated acoustic wave propagation is D’yakovÐKontorovich stability parameter L and its consistent with predictions of the linear theory. critical value L for two Hugoniot curves: the original 0 An analysis of scale-invariant solutions represented Hugoniot emanating from point 0 and the Hugoniot by splitting diagrams in the (P, x/t) plane supports the corresponding to the downstream shock. The pressures theoretical results of [3] in terms of structure of solutions indicated by vertical lines in Fig. 6a correspond to the →

initial and leading shocks. The vertical line in Fig. 6b and boundaries for the split wave conﬁgurations S

→ → represents the pressure corresponding to the down- → → → → → → → → → → stream shock. The figure demonstrates that both initial W T,C S S W TS C S , and S W T.S S and downstream shocks satisfy the acoustic emission A numerical simulation of shock splitting performed condition L0 < L < 1 + 2M. The inset in the left panel with the use of generalized Roe’s method (approximate shows that the equation L(P) = L0(P) has two nearly solution of the Riemann problem for an arbitrary equa- equal roots, which implies that there is a small region tion of state) has demonstrated that entropy correction is of acoustic emission below point 1 in Fig. 3a. required to obtain physical solutions near the upper shock-splitting boundary and solutions satisfying the The splitting into stable and unstable shocks was condition of entropy increase for split-wave configura- simulated numerically by solving a two-dimensional tions involving isentropic compression waves. Riemann problem on a 400 × 300 grid in a coordinate system tied to the initial shock front. Initial conditions were set to approximate an unstable shock-front frag- ACKNOWLEDGMENTS ment and adjoining uniform flow regions. The initial This work was supported by Presidium of the Rus- disturbance was a 1% drop in the velocity profile at the sian Academy of Sciences under the program “Mathe- center of a cell contiguous to the shock front. No qual- matical Modeling, Intelligent Systems, and Control of itative change in the ensuing flow pattern was induced Nonlinear Mechanical Systems.” by varying the perturbed velocity component or perturbation amplitude. The gray-scale plots of pressure-gradient magnitude shown in Fig. 7 illustrate the numerical REFERENCES solution obtained for P = 0.82 with the time step corre- 1. S. P. D’yakov, Zh. Éksp. Teor. Fiz. 27, 288 (1954). sponding to a Courant number of 0.3 (darker areas cor- 2. V. M. Kontorovich, Zh. Éksp. Teor. Fiz. 33, 1525 (1957) respond to steeper gradients). Time is measured here in [Sov. Phys. JETP 6, 1179 (1957)]. arbitrary units since the system does not have any 3. N. M. Kuznetsov, Zh. Éksp. Teor. Fiz. 88, 470 (1985) intrinsic time scale. In the three-wave configuration [Sov. Phys. JETP 61, 275 (1985)]. shown here, the cusps in the downstream shock front 4. L. D. Landau and E. M. Lifshitz, Course of Theoretical emit weak downstream-propagating acoustic waves Physics, Vol. 6: Fluid Mechanics, 3rd ed. (Nauka, Mos- and move along the front. The acoustic-emission cow, 1986; Pergamon, New York, 1987). parameters have the following values: L = Ð0.59, M = 5. A. L. Ni, S. G. Sugak, and V. E. Fortov, Teplofiz. Vys. 0.83, and θ = 2.8. The cosine of the angle between the Temp. 24, 564 (1986). acoustic wave vector and the positive x axis is deter- 6. V. A. Gushchin, A. P. Likhachev, N. G. Nechiporenko, mined by the equation [4] and E. R. Pavlyukova, News in Numerical Simulation: Algorithms, Computer Simulation, and Results (Nauka, 2 Moscow, 2000), p. 165. 24 θ 2 α 3 + M α M ------+ Ð 1 cos + 2M------Ð 1 cos 7. J. W. Bates and D. C. Montgomery, Phys. Rev. Lett. 84, 1 + L 1 + L 1180 (2000). 2 8. SESAME Report on the Los Alamos Equation-of-State 21()+ M 2 + ------Ð0,() 1 + θM = Library (1983), Report LANL-83-4. 1 + L 9. P. L. Roe, J. Comput. Phys. 43, 357 (1981). whose roots are Ð1 and Ð0.66 in this particular case. The 10. O. M. Belotserkovskiœ and A. V. Konyukhov, Zh. 2α Vychisl. Mat. Mat. Fiz. 42, 235 (2002). range of the latter root is ÐM2 < cos < 1, which corresponds to an outgoing acoustic wave with wave vec- 11. A. Harten, J. Comput. Phys. 49, 357 (1983). tor making an angle of ±131° with the positive x axis. 12. A. Harten and S. Osher, SIAM J. Numer. Anal. 24, 279 The snapshots shown for three successive instants in (1987). the right-hand part of Fig. 7 illustrate the propagation of 13. P. Glaister, J. Comput. Phys. 74, 382 (1988). acoustic wave fronts and demonstrate good agreement with the theoretical value of the angle. Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 4, 2004, pp. 820Ð836. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 4, 2004, pp. 938Ð956. Original English Text Copyright © 2004 by Bezrukov, Levkov. MISCELLANEOUS

Dynamical Tunneling of Bound Systems through a Potential Barrier: Complex Way to the Top¦ F. Bezrukova and D. Levkova,b aInstitute for Nuclear Research, Russian Academy of Sciences, Moscow, 117312 Russia bDepartment of Physics, Moscow State University, Moscow, 119899 Russia e-mail: [email protected], [email protected] Received January 15, 2003

Abstract—A semiclassical method of complex trajectories for the calculation of the tunneling exponent in systems with many degrees of freedom is further developed. It is supplemented with an easily implemented technique that enables one to single out the physically relevant trajectory from the whole set of complex classical trajectories. The method is applied to semiclassical transitions of a bound system through a potential barrier. We ﬁnd that the properties of physically relevant complex trajectories are qualitatively different in the cases of potential tunneling at low energy and dynamical tunneling at energies exceeding the barrier height. Namely, in the case of high energies, the physically relevant complex trajectories describe tunneling via creation of a state close to the top of the barrier. The method is checked against exact solutions of the Schrödinger equation in a quantum-mechanical system of two degrees of freedom. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION in which case the function S(q) (and, therefore, the exponential part of the wave function) is calculated as Semiclassical methods provide a useful tool for the the action functional on this complex trajectory. study of nonperturbative processes. Tunneling phenomena represent one of the most notable cases where A particularly difficult situation arises when one semiclassical techniques are used to obtain otherwise considers transitions of a nonseparable system with a unattainable information on the dynamics of the transi- strong interaction between its degrees of freedom such tion. One standard semiclassical technique is the WKB that the quantum numbers of the system change consid- approximation to tunneling in quantum mechanics of erably during the transition. Methods based on the adi- one degree of freedom. In this case, solutions S(q) of abatic expansion are not applicable in this situation, the HamiltonÐJacobi equation are purely imaginary in while the method of complex trajectories proves to be the classically forbidden region. Therefore, the func- extremely useful. tion S(q) can be obtained as the action functional on a real trajectory q(τ), which is a solution to the equa- The method of complex trajectories in the form suit- tions to motion in the Euclidean time domain, able for the calculation of S-matrix elements was formulated and checked by direct numerical calculations ti= Ð τ, in [3Ð5] (see [6] for review). Further studies [7Ð12] showed that this method can be generalized to the cal- with the real Euclidean action culation of the tunneling wave functions and tunneling probabilities, energy splitting in double-well potentials, and decay rates from metastable states. Similar meth- S = ÐiS. E ods were successful in the study of tunneling in high- energy collisions in field theory [13–16], where one con- This simple picture of tunneling is no longer valid siders systems with a definite particle number (ᏺ = 2) in for systems with many degrees of freedom, where solu- the initial state, and in the study of chemical reactions tions S(q) of the HamiltonÐJacobi equation are known and atom ionization processes, where the initial bound to be generically complex in the classically forbidden systems are in definite quantum states [6, 17, 18], etc. region (see [1, 2] for recent discussion). This leads to The main advantage of the method of complex trajecto- the concept of “mixed” tunneling, as opposed to “pure” tunneling, where S(q) is purely imaginary. Mixed tun- ries is that it can be easily generalized and numerically neling cannot be described by any real tunneling trajec- implemented in the cases of a large and even infinite tory. However, it can be related to a complex trajectory, (field theory) number of degrees of freedom, in contrast to other methods, such as the Huygens-type construction in [1, 2] and the initial value representation ¦This article was submitted by the authors in English. (IVR) in [3, 19Ð23].

DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 821

In this paper, we develop the method of complex tra- gies exceeding the barrier height is called “dynamical jectories further. Namely, we concentrate on the follow- tunneling.”1 ing problem. It is known [3] that a physically relevant Examples of dynamical tunneling are well known in complex trajectory satisfies the classical equations of scattering theory [4]. This type of tunneling between motion with certain boundary conditions. However, this bound states was discovered in [25], and the generality boundary value problem generically has also an infi- of dynamical tunneling in large molecules was stressed nite, although discrete, set of unphysical solutions. In in [26, 27]. Dynamical tunneling is of primary interest one-dimensional quantum mechanics, all solutions can in our study. easily be classified. In systems with many degrees of We observe a novel phenomenon that dynamical freedom, such a classification is extremely difficult, if տ at all possible. In the case of a small number of degrees tunneling at E V0 (more precisely, at E > E1, where of freedom (realistically, N = 2), one can scan over all E1 is somewhat larger than V0) occurs in the following solutions and find the solution giving the largest tunnel- way: the system jumps on top of the barrier and restarts ing probability [3, 9, 10], but in systems with a large or its classical evolution from the region near the top. infinite number of degrees of freedom, the problem of From the physical standpoint, this is not quite what is choosing the physically relevant solution becomes a normally meant by “tunneling through a barrier.” Yet, formidable task. the transitions remain exponentially suppressed, but the reason is different: to jump above the barrier, the sys- The problem of choosing the appropriate solution tem has to undergo considerable rearrangement, unless becomes even more pronounced when the qualitative the incoming state is chosen in a special way (see foot- properties of the relevant complex trajectory are differ- note 1). This rearrangement costs an exponentially ent in different energy regions. This may happen when small probability factor. We note that a similar expo- the physically relevant classical solution “meets” an nential factor was argued to appear in various field the- unphysical one at some energy value E = E1, or in other ory processes with multiparticle final states [28–31]. words, when solutions of the boundary value problem, We find that the new physical behavior of the system viewed as functions of the energy, bifurcate at E = E1. is related to a bifurcation of the family of complex-time In this paper, we give an example of this kind, which classical solutions, viewed as functions of energy. This appears to be fairly generic (see also [11, 12, 15, 16, 24]). is precisely the bifurcation mentioned above. Our We then develop a method that allows one to choose the method for dealing with this bifurcation is to regularize physically relevant solution automatically, implement it the boundary value problem so that the bifurcations dis- numerically, and check this method against the numer- appear altogether (at real energies), and the only solu- ical solution to the full Schrödinger equation. tions recovered after removing the regularization are physical ones. We study inelastic transitions of a bound system This paper is organized as follows. The system to be through a potential barrier. To be specific, we consider discussed in what follows is introduced in Section 2.1. a model with one internal degree of freedom in addition In Section 2.2, we formulate the boundary value prob- to the center-of-mass coordinate. We consider a situa- lem for the calculation of the tunneling exponent. In tion in which the spacing between the levels of the Section 2.3, we then examine the classical overbarrier bound system is small compared to the height of the solutions and find all initial states that lead to classi- barrier and assume a sufficiently strong coupling cally allowed transitions. In Section 2.4, we present a between the degrees of freedom to make sure that the straightforward application of the semiclassical tech- quantum numbers of the bound system change consid- nique outlined in Section 2.2 and find that it ceases to erably during the transition process. This is precisely produce relevant complex trajectories in a certain the situation in which the method of complex trajecto- region of initial data, namely, at E > E1. In Section 3, ries shows its full strength. we introduce our regularization technique and show Transitions of bound systems involve a particular that it indeed enables us to find all the relevant complex trajectories, including those with E > E1 (Section 3.1). energy scale, the barrier height V0. At energies below V , classical overbarrier transitions are forbidden ener- We check our method against the numerical solution of 0 the full Schrödinger equation in Section 3.2. In Sec- getically; the corresponding regime is called “potential tion 3.3 and Appendix C, we show how our regulariza- tunneling.” For E > V0, it is energetically allowed for the system to evolve classically to the other side of the 1 It is clear that the properties of transitions of a bound system at barrier. However, overbarrier transitions may be forbid- E > V0 depend on the choice of the initial state. Namely, there den dynamically even at E > V0. Indeed, inelastic inter- always exists a certain class of states the transitions from which actions of a bound system with a potential barrier gen- are not exponentially suppressed. To construct an example, one erally lead to the excitation of the internal degrees of places the bound system on top of the barrier and evolves it clas- freedom with the simultaneous decrease of the center- sically backwards in time to the region where the interaction with the barrier is negligibly small. On the other hand, even at E > V0, of-mass energy, and this may prevent the system from there are states the transitions from which are exponentially sup- the overbarrier transition. The tunneling regime at ener- pressed (dynamical tunneling).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004

822 BEZRUKOV, LEVKOV

V parameter. At the classical level, this parameter is irrel- evant: after rescaling the variables2 as λ λ x1 x1/ , x2 x2/ , E m m the small parameter enters only through the overall multiplicative factor 1/λ in the Hamiltonian. Therefore, ω the semiclassical technique can be developed as an 0 asymptotic expansion in λ. The properties of the system are made clearer by X replacing the variables x1 and x2 with the center-of- mass coordinate Fig. 1. An oscillator hitting a potential barrier, with only the x + x “dark” particle interacting with the barrier. X ≡ ------1 -2 2 tion technique is used to smoothly join the “classically and the relative oscillator coordinate allowed” and “classically forbidden” families of solu- x1 Ð x2 tions in the respective cases of two- and one-dimen- y ≡ ------. sional quantum mechanics. 2 In terms of these variables, the Hamiltonian becomes 2. SEMICLASSICAL TRANSITIONS 2 2 2 2 p p ω 2 1 λ()Xy+ THROUGH A POTENTIAL BARRIER H = -----X- ++-----y ------y +--- exp Ð------. (3) 2 2 2 λ 2 2.1. Model The interaction potential The situation discussed in this paper is a transition 1 λ()Xy+ 2 through a potential barrier of the bound system consid- U ≡ --- exp Ð------ered in [11, 12], namely, the system made of two parti- int λ 2 cles of mass m, moving in one dimension and bound by ±∞ a harmonic oscillator potential of frequency ω (Fig. 1). vanishes in the asymptotic regions X and One of the particles interacts with a repulsive potential describes a potential barrier between these regions. At X ±∞, Hamiltonian (3) corresponds to an oscillator barrier. The potential barrier is assumed to be high and ω wide, while the spacing between the oscillator levels is of the frequency moving along the center-of-mass much smaller than the barrier height V . The Hamilto- coordinate X. The oscillator asymptotic state is charac- 0 terized by its excitation number N and total energy nian of the model is p 2 2 2 2 X ω1 p p mω 2 E = ------+ N + --- . H = ------1 ++------2 ------()x Ð x 2 2 2m 2m 4 1 2 (1) We are interested in the transmissions through the x 2 potential barrier of the oscillator with given initial val- 1 + V 0 exp Ð------, ues of E and N. 2σ2 where the conditions on the oscillator frequency and 2.2. T/θ Boundary Value Problem potential barrier are The probability of tunneling from a state with a បω Ӷ fixed initial energy E and oscillator excitation number V 0, (2) N from the asymptotic region X Ð∞ to any state in σ ӷ ប ∞ / mV0. the other asymptotic region X + is Because the variables do not separate, this is certainly a ᐀()EN, nontrivial system. (4) 〈| ()ˆ ()|〉, 2 We choose units with ប = 1, m = 1. It is also conve- = lim ∑ f exp ÐiH t f Ð ti EN , → ∞ t f Ð ti nient to treat the frequency ω as a dimensionless param- f eter, so that all physical quantities are dimensionless. In ω where it is implicit that the initial and final states have our subsequent numerical study, we use the value = support only well outside the range of the potential, 0.5 but keep the notation “ω” in formulas. The system is semiclassical, i.e., conditions (2) are satisfied, if we 2 To keep the notation simple, we use the same symbols x , x for σ λ λ λ 1 2 choose = 12 and V0 = 1/ , where is a small the rescaled variables.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 823 with X < 0 and X > 0, respectively. Semiclassical meth- Im t A ods are applicable if the initial energy and excitation T/2 number are parametrically large, B D ˜ λ ˜ λ EE==/ , NN/ , A' C Re t where E˜ and N˜ are kept constant as λ 0. The tran- Fig. 2. Contour in the complex time plane. sition probability has the exponential form real, while the complex amplitudes of the decoupled ᐀ 1 ()˜ , ˜ oscillator must be linearly related, = DexpÐλ---FEN , (5) θ ImX 0, v e u*, as t Ð∞. (6c) where D is a preexponential factor, which is not consid- Boundary conditions (6b) and (6c) in fact make eight ered in this paper. Our purpose is to calculate the lead- real conditions (because, e.g., ImX(t') 0 implies ()˜ , ˜ ing semiclassical exponent FEN . The exponent for that both ImX and ImXú tend to zero) and completely tunneling from the oscillator ground state is obtained determine a solution, up to the time translation invari- in [11Ð13, 32] by taking the limit N˜ 0 in FE()˜ , N˜ . ance (see the discussion in Appendix A). In what follows, we rescale the variables as It is shown in Appendix A that a solution of this boundary value problem is an extremum of the functional XX/ λ, yy/ λ FXy[],,; X* y*; T, θ = Ð iS[] X, y + iS[] X*, y* (7) θ and omit the tilde over the rescaled quantities E˜ and N˜ . Ð ETÐ N + Boundary Terms. The exponent F(E, N) is related to a complex trajec- The value of this functional at the extremum gives the tory that satisfies a certain complexified classical exponent for the transition probability (up to the large boundary value problem. We present the derivation of overall factor 1/λ, see Eq. (5)), this problem in Appendix A. The outcome is as follows. FEN(), = 2ImS ()T, θ Ð ET Ð Nθ, (8) There are two Lagrange multipliers T and θ, which are 0 related to the parameters E and N characterizing the where S0 is the action of the solution, integrated by incoming state. The boundary value problem is conve- parts, niently formulated on the contour ABCD in the com- 2 plex time plane (see Fig. 2), with the imaginary part of  1 d X S0 = ∫dtÐ---X------the initial time equal to T/2. The coordinates X(t) and 2 dt2 y(t) must satisfy the complexified equations of motion (9) 2 in the interior points of the contour and must be real in 1 d y 1 2 2  Ð ---y------Ð ---ω y Ð U ()Xy, . the asymptotic future (region D): 2 int  2 dt 2 δ δ Here, the integration runs along the contour ABCD. The S S θ ------δ ()- ==δ------() 0 , (6a) values of the Lagrange multipliers T and are related Xt yt to the energy and excitation number as ∂ Imyt() 0, Im Xt () 0, as t + ∞ . (6b) ET(), θ = ------2ImS ()T, θ , (10) ∂T 0 ∂ In the asymptotic past (region A of the contour, where NT(), θ = ------2ImS ()T, θ . (11) t = t' + iT/2, t' is real negative), the interaction potential ∂θ 0 U can be neglected and the oscillator decouples, int Using Eq. (8), it is also straightforward to verify the inverse Legendre transformation formulas 1 y = ------()uiexp()Ð ωt' + v exp()iωt' . ∂ ω TEN(), = Ð------FEN(), , (12) 2 ∂E ∂ The boundary conditions in the asymptotic past, t' θ()EN, = Ð------FEN(), . (13) Ð∞, are that the center-of-mass coordinate X must be ∂N

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 824 BEZRUKOV, LEVKOV It can also be verified that the right-hand side of solution to the real time axis. For solutions satisfying Eq. (10) coincides with the energy of the classical solu- (6c), this gives tion and the right-hand side of Eq. (11) is equal to the classical counterpart of the occupation number, ω () 1  T ()ω yt = ------uexpÐ------exp Ði t 2 2ω 2 Xú E ==------+ ωN, N uv . (14) ωT  2 + u*exp θ + ------exp()iωt , 2  Therefore, we can either seek the values of T and θ that correspond to given E and N or, following a computa- T tionally simpler procedure, solve boundary value prob- ImXt()= Ð--- p lem (6) for given T and θ and then find the correspond- 2 X ing values of E and N from Eq. (14). We note that initial conditions (6c) complemented by Eqs. (14) are equiva- at large negative time. We see that the dynamical coor- lent to the initial conditions in [3Ð5], the latter being dinates on the negative side of the real time axis are expressed in terms of actionÐangle variables. The generally complex. For solutions approaching the boundary conditions in the asymptotic future (6b) are asymptotic region X +∞ as t +∞ (such that X different from those in [3Ð5], because we consider an and y are exactly real at finite t > 0), this means that inclusive, rather than fixed, final state. there should exist a branch point in the complex time plane: the contour A'ABC in Fig. 2 winds around this We now discuss some subtle points of boundary point and cannot be deformed to the real time axis. This value problem (6). First, we note that the asymptotic argument does not work for solutions ending in the reality condition in (6b) does not always coincide with interaction region as t +∞, and, hence, branch the reality condition at finite time. Of course, if the points between the AB part of the contour and the real solution approaches the asymptotic region X +∞ time axis may be absent. We see in Section 3.1 that this on the part CD of the contour, asymptotic reality condi- is indeed the case in our model in a certain range of E tion (6b) implies that the solution is real at any finite and N. positive t. Indeed, the oscillator decouples as X +∞, and, therefore, condition (6b) means that its phase and amplitude, as well as X(t), are real as t +∞. Due 2.3. Overbarrier Transitions: the Region to the equations of motion, X(t) and y(t) are real on the of Classically Allowed Transitions entire CD part of the contour. This situation corre- and Its Boundary E0(N) sponds to the transition directly to the asymptotic region X +∞. However, the situation can be drasti- Before studying the exponentially suppressed tran- cally different if the solution on the final part of the time sitions, we consider the classically allowed ones. For contour remains in the interaction region. For example, this, we study the classical evolution (real time, real- we can imagine that the solution approaches the saddle valued coordinates) in which the system is initially point of the potential X = 0, y = 0 as t +∞. Because located at large negative X and moves with a positive center-of-mass velocity towards the asymptotic region one of the perturbations around this point is unstable, X +∞. The classical dynamics of the system is there may exist solutions that approach this point expo- specified by four initial parameters. One of them (e.g., nentially along the unstable direction, i.e., the initial center-of-mass coordinate) fixes the invariance under time translation, while the other three are Xt(), yt()∝ exp()Ðconst ⋅ t the total energy E; the initial excitation number of the y ≡ ω oscillator, defined in classical theory as N Eosc/ ; and ϕ with possibly complex prefactors. In this case, the solu- the initial oscillator phase i. tion may be complex at any finite time and become real Any initial quantum state of our system can be fully only asymptotically as t +∞. Such a solution cor- determined by the energy E and the initial oscillator responds to tunneling to the saddle point of the barrier, excitation number N; we can represent each state by a after which the system rolls down classically towards point in the EN plane. There is, however, one additional X +∞ (with probability of order 1, inessential for classically relevant initial parameter, the oscillator ϕ the tunneling exponent F). We see in Section 3.1 that a phase i. An initial state (E, N) leads to unsuppressed situation of this sort indeed takes place for some values transmission if the corresponding classical overbarrier 3 ϕ of the energy and excitation number. transitions are possible for some value(s) of i. These Second, because the interaction potential disappears 3 We note that the corresponding classical solutions obey boundary at large negative time (in the asymptotic region X conditions (6b) and (6c) with T = θ = 0; i.e., they are solutions to Ð∞), it is straightforward to continue the asymptotic the boundary value problem (6).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 825 states form some region in the EN plane, which is to be NS • found in this section. For given N, at sufﬁciently large E, the system can 0.6

certainly evolve to the other side of the barrier. On the N other hand, if E is smaller than the barrier height, the EPI(N) system deﬁnitely undergoes reﬂection. Thus, there 0.2 exists some boundary energy E0(N) such that classical E1(N) E0(N) transitions are possible for E > E0(N), while, for E < ϕ 00.5 ES = 1 1.5 E E0(N), they do not occur for any initial phase i. The line E0(N) represents the boundary of the region of clas- Fig. 3. The boundary E0(N) of the region of classically sically allowed transitions. We have calculated E0(N) allowed transitions, the bifurcation line E1(N), and the line 4 numerically: the result is shown in Fig. 3. of the periodic instantons EPI(N).

An important point of the boundary E0(N) corresponds to the static unstable classical solution X(t) = y(t) = 0. In the ﬁeld theory context, such a solution is called “sphaleron” [33], and we keep this terminology in what follows. This solution is the saddle point of the U(X, y) potential

()ω, ≡ 2 2 (), UXy y /2 + Uint Xy 0 1 and has exactly one unstable direction, the negative 0 y – X mode (see Fig. 4). The sphaleron energy ES = U(0, 0) = 0 1 determines the minimum value of the function E0(N). y + X Indeed, classical overbarrier transitions with E < E are S Fig. 4. The potential (dotted lines) in the vicinity of the impossible, but the overbarrier solution with a slightly sphaleron (X = 0, y = 0) (marked by the point), the excited higher energy can be obtained as follows: a momentum sphaleron (thick line) corresponding to the point (E, N) = along the negative mode is added at the point X = y = 0, (1.985, 3.72) at the boundary of the region of classically “pushing” the system towards X +∞. Continuing allowed transitions, and the trajectory of a solution that is this solution backwards in time shows that the system close to this excited sphaleron (thin line). The asymptotic regions X ±∞ are along the diagonal. tends to X Ð∞ for large negative time and has a certain oscillator excitation number. Solutions with the energy closer to the sphaleron energy correspond to a and final times, t and t . If, within this time interval, a smaller “push” and, thus, spend a longer time near the i f solution with the energy E1 evolves to the other side of sphaleron. In the limiting case where the energy is the barrier and a solution with the energy E and the equal to E , the solution spends an infinite time in the 2 S same oscillator excitation number is reflected, there vicinity of the sphaleron. This limiting case has a defi- exists an intermediate energy at which the solution nite initial excitation number NS, so that E0(NS) = ES ends up at t = tf in the interaction region. Taking the (see Fig. 3). The value of NS is unique because there is ∞ limit as tf + and E1 Ð E2 0, we obtain a point exactly one negative direction of the potential in the at the boundary E (N) and a solution tending asymptot- vicinity of the sphaleron. 0 ically to some unstable time-dependent solution that In complete analogy to the features of the overbar- spends an infinite time in the interaction region. We call rier classical solutions near the sphaleron point (ES, NS), the latter solution the excited sphaleron; it describes we expect that, as the values of E and N approach any some (in general, nonlinear) oscillations above the other boundary point (E0(N), N), the corresponding sphaleron along the stable direction in the coordinate overbarrier solutions spend more and more time in the space. Therefore, every point of the boundary (E (N), N) ≠ 0 interaction region, where Uint 0. This follows from a corresponds to some excited sphaleron. In the phase continuity argument. Namely, we first fix the initial space, solutions tending asymptotically to the excited sphalerons form a surface (separatrix) that separates 4 We note that the boundary E0(N) of the region of classically regions of qualitatively different classical motions of allowed transitions can be extended to N > NS. Because E = ES is the system. the absolute minimum of the energy of classically allowed transi- In Fig. 4, we display a solution found numerically in tions, the function E0(N) grows with N at N > NS. In fact, it tends our model that tends to an excited sphaleron. We see as ω ∞ to the asymptotics E0 = N as N + . In what follows, we that the trajectory of the excited sphaleron is, roughly are not interested in transitions with N > NS, and, therefore, this speaking, orthogonal to the unstable direction at the part of the boundary E0(N) is not shown in Fig. 3. saddle point (X = 0, y = 0).

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2.4. Suppressed Transitions: technique, with the known solution at (T, θ) serving as 5 Bifurcation Line E1(N) the initial approximation. If the solutions end up in the correct asymptotic region at each step, i.e., X +∞ We now turn to classically forbidden transitions and on part D of the contour, the solutions obtained by this consider the boundary value problem in Eq. (6). It is procedure of small deformations are physically rele- relatively straightforward to obtain solutions for θ = 0 vant. But the method of small deformations fails to pro- numerically. In this case, boundary conditions (6b) and duce relevant solution if there are bifurcation points in (6c) take the form of reality conditions in the asymp- the EN plane, where the physical branch of solutions totic future and past. It can be shown [34] that the phys- merges to an unphysical branch. Because there are ically relevant solutions with θ = 0 are real on the entire unphysical solutions close to physical ones in the vicin- contour ABCD in Fig. 2 and describe nonlinear oscilla- ity of bifurcation points, the procedure of small defor- tions in the upside-down potential on the Euclidean part mations cannot be used near these points. BC of the contour. The period of the oscillations is equal to T, and, hence, points B and C are two different We have found numerically that, in our model, the method of small deformations produces correct solu- turning points where Xú = yú = 0. These real Euclidean tions of the T/θ boundary value problem in a large solutions are called periodic instantons. A practical region of the EN plane where E < E1(N). However, at տ technique for obtaining these solutions numerically on sufficiently high energy E > E1(N), where E1(N) ES, the Euclidean part BC consists in minimizing the the deformation procedure generates solutions that Euclidean action (for example, with the method of con- bounce back from the barrier (see Fig. 5), i.e., have a jugate gradients, see [11, 12] for details). The solutions wrong “topology.” This occurs deep inside the region of on the entire contour are then obtained by solving the classically forbidden transitions, where the suppression Cauchy problem numerically, forward in time along the is large, and one naively expects the semiclassical tech- line CD and backward in time along the line BA. From nique to work well. Clearly, solutions with a wrong the solution in the asymptotic past (region A), we then topology do not describe the tunneling transitions of calculate its energy and excitation number (14). The interest. Therefore, if the semiclassical method is appli- solutions of this Cauchy problem are obviously real, cable in the region E1(N) < E < E0(N) at all, there exists and, hence, boundary conditions (6b) and (6c) are another, physical, branch of solutions. In that case, the indeed satisfied for θ = 0. It is worth noting that solu- line E (N) is the bifurcation line where the physical θ 1 tions with = 0 are similar to the ones in quantum solutions meet the ones with a wrong topology. Walk- mechanics of one degree of freedom. The line of peri- ing in small steps in θ and T is useless in the vicinity of odic instantons in the EN plane in our model is shown this bifurcation line, and a special trick is required to in Fig. 3. find the relevant solutions beyond that line. The bifur- Once the solutions with θ = 0 are found, it is natural cation line E1(N) for our quantum-mechanical problem to try to cover the entire region of classically forbidden of two degrees of freedom is shown in Fig. 3. transitions in the EN plane with a deformation proce- The loss of topology beyond a certain bifurcation dure, by moving in small steps in θ and T. The solution line in the EN plane is by no means a property of our of the boundary value problem with (T + ∆T, θ + ∆θ) model only. This phenomenon has been observed in may be obtained numerically by applying an iteration field theory models in the context of both induced false vacuum decay [14] and baryon-number violating transitions in gauge theory [15] (in field theory models, the Re X parameter N is the number of incoming particles). In all cases, the loss of topology prevented one from comput- 20 Physical ing the semiclassical exponent for the transition probability in the interesting region of relatively high 0 energies. Reflected Returning to quantum mechanics of two degrees of –20 freedom, we point out that the properties of tunneling solutions with different energies approaching the bifur- –40 cation line E (N) from the left of the EN plane are in –20 –10 0 10 20 1 Re t some sense similar to the properties of tunneling solutions in one-dimensional quantum mechanics whose Fig. 5. The dependence of the tunneling coordinate X on energy is close to the barrier height (see Appendix C). time for two solutions with nearly the same energy and ini- Again by continuity, these solutions of our two-dimen- tial excitation number. The physical solution tunnels to the sional model spend a long time in the interaction asymptotic region X +∞, while the unphysical one is region; this time tends to infinity on the line E (N). reflected to X Ð∞. The physical solution has E = 1.028, 1 N = 0.44, while the unphysical one has E = 1.034, N = 0.44. These two solutions are close to the point on the bifurcation 5 In practice, the NewtonÐRaphson method is particularly conve- line E1(N = 0.44) = 1.031. nient (see [11, 12, 14, 15]).

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Hence, at any point of this line, there is a solution that ization. At the same time, the excited sphaleron conﬁg- ∞ starts in the asymptotic region left of the barrier and uration has Tint = , which leads to the inﬁnite value of ends up on an excited sphaleron. Such behavior is the regularized functional indeed possible because of the existence of an unstable direction near the (excited) sphaleron, even for com- F⑀ ≡ F + 2⑀T . plex initial data. In the next section, we suggest a trick int to deal with this situation—this is our regularization technique. Hence, the excited sphalerons are not stationary points of the regularized functional.

3. REGULARIZATION TECHNIQUE For the problem under consideration, Uint ~ 1 in the interaction region and Tint can be defined as In this section, we develop our regularization technique and find the physically relevant solutions 1 T = ---[]dtU ()Xy, + dtU ()X*, y* . (15) between the lines E1(N) and E0(N). We see that all solu- int 2 ∫ int ∫ int tions from the new branch (and not only on the lines E0(N) and E1(N)) correspond to tunneling onto the excited sphaleron (“tunneling on top of the barrier”). We note that Tint is real and that the regularization is These solutions would be very difficult, if at all possi- equivalent to the multiplication of the interaction ble, to obtain directly by numerically solving the non- potential with a complex factor, regularized classical boundary value problem (6): they ⑀ are complex at finite times and become real only U ()1 Ð i⑀ U = eÐi U + O()⑀2 . (16) asymptotically as t +∞, whereas numerical meth- int int int ods require working with finite time intervals. This results in the corresponding change of the classical As an additional advantage, our regularization tech- equations of motion, while boundary conditions (6b) and nique allows one to obtain a family of overbarrier solu- (6c) remain unaltered. tions that covers all the region of the initial data corresponding to classically allowed transitions, including We still have to understand whether solutions with its boundary. This is of interest in models with a large ⑀ ≠ 0 exist at all. The reason for the existence of such number of degrees of freedom and in field theory, solutions is as follows. We consider a well-defined (for ⑀ where finding the boundary E0(N) by direct methods is > 0) matrix element difficult (see, e.g., [35] for a discussion of this point). ᐀⑀ 3.1. Regularized Problem: 〈| []()ˆ ⑀ ()|〉, 2 = lim∑ f exp Ð iH Ð Uint t f Ð ti EN , Classically Forbidden Transitions → ∞ t f Ð ti f The main idea of our method is to regularize the equations of motion by adding a term proportional to a | 〉 ⑀ where, as before, E, N denotes the incoming state with small parameter such that configurations staying near given energy and oscillator excitation number. The the sphaleron for an infinite time no longer exist among quantity ᐀ has a well-defined limit as ⑀ 0, equal the solutions of the T/θ boundary value problem. After ⑀ performing the regularization, we explore all the region to tunneling probability (4). Because the saddle point of of classically forbidden transitions without crossing the the regularized functional F⑀ gives the semiclassical bifurcation line. Taking the limit ⑀ 0, we then exponent for the quantity ᐀⑀, we expect that such a reconstruct the correct values of F, E, and N. saddle point indeed exists. In formulating the regularization technique, it is Therefore, the regularized T/θ boundary value prob- more convenient to work with the functional F[X, y; X*, lem is expected to have solutions necessarily spending y*; T, θ], Eq. (7), itself rather than with the equations of a finite time in the interaction region. By continuity, motion. We prevent F from being extremized by config- these solutions do not experience reflection from the urations approaching the excited sphalerons asymptot- barrier if the procedure of small deformations starting ically. To achieve this, we add a new term of the form from solutions with the correct topology is used. The ⑀ 2 Tint to the original functional (7), where Tint estimates line E1(N) is no longer a bifurcation line of the regular- the time the solution “spends” in the interaction region. ized system, and the procedure of small deformations The regularization parameter ⑀ is the smallest one in the therefore enables us to cover the entire region of classi- problem, and, hence, any regular extremum of the func- cally forbidden transitions. The semiclassical suppres- tional F (the solution that spends a finite time in the sion factor of the original problem is recovered in the ≠ ⑀ region Uint 0) changes only slightly after the regular- limit 0.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 828 BEZRUKOV, LEVKOV ≡ c c– this, we have plotted in Fig. 6 the solution x(t) (X(t), + 0.02 y(t)) at large times after taking the limit ⑀ 0 numer- 0.9 Re c– ically. To understand this figure, we recall that the 0 potential near the sphaleron point X = y = 0 has one pos- 0.8 –0.02 itive mode and one negative mode. Namely, introduc- Im c– 0.7 ing new coordinates c+, cÐ as –0.04 0.6 α α X = cos c+ + sin cÐ, Re c+ –0.06 0.5 α α –0.08 y = Ð sin c+ + cos cÐ, 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ω2 t cot2α = Ð------, 2 Fig. 6. Large-time behavior of a solution with ⑀ = 0 at (E = 1.05, N = 0.43). The coordinates X and y are decomposed on the basis of the eigenmodes near the sphaleron. We note that we write, in the vicinity of the sphaleron, Imc+ = 0. p2 p2 ω 2 ω2 H = 1 +++-----+ -----Ð ------+ c 2 Ð ------Ðc 2, 2 2 2 + 2 Ð It is worth noting that the interaction time is Leg- endre conjugate to ⑀, where 1 ∂ ------(),,⑀ 2 4 T int = ∂⑀F⑀ EN . (17) 2 ω ω 2 ω± = ± Ð 1 + ------+ 1 +0.------> 2 4 This equation can be used as a check of numerical cal- θ culations. Because the solutions of the T/ boundary value problem are complex, the coordinates c+ and cÐ are also We implemented the regularization procedure complex. In Fig. 6, we show real and imaginary parts of numerically. To solve the boundary value problem, we c+ and cÐ at a large real time t (part CD of the contour). use the computational methods described in [11, 12]. We see that, while Rec oscillates, the unstable coordi- To obtain the semiclassical tunneling exponent in the + nate cÐ asymptotically approaches the sphaleron value: region between the bifurcation line E1(N) and the ∞ boundary of the region of classically allowed transi- cÐ 0 as t +. The imaginary part of cÐ is nonzero at any finite time. This is the reason for the failure tions E0(N), we began with a solution to the nonregular- ized problem deep in the “forbidden” region of the ini- of straightforward numerical methods in the region E > E1(N): the solutions from the physical branch do not tial data (i.e., at E < E1(N)). We then increased the value of ⑀ from zero to a certain small positive number, keep- satisfy the reality conditions at any large but finite final ing T and θ fixed. We next changed T and θ in small time. We have pointed out in Section 2.2 that this can steps, keeping ⑀ finite, and found solutions to the regu- happen only if the solution ends up near the sphaleron, larized problem in the region E (N) < E < E (N). These which has a negative mode. This is precisely what hap- 1 0 pens: for ⑀ = 0 at asymptotically large t, our solutions solutions had the correct topology; i.e., they indeed ∞ are real and oscillate near the sphaleron, remaining in ended up in the asymptotic region X +. Finally, the interaction region. we lowered ⑀ and extrapolated F, E, and N to the limit ⑀ 0. 3.2. Regularization Technique We now consider the solutions in the region E1(N) < ⑀ versus Exact Quantum-Mechanical Solution E < E0(N), which we obtain in the limit 0 more carefully. They belong to a new branch and may, there- Quantum mechanics of two degrees of freedom is a fore, exhibit new physical properties. Indeed, we found convenient testing ground for checking the semiclassi- that, as the value of ⑀ decreases to zero, the solution at cal methods and, in particular, our regularization tech- any point (E, N) with E1(N) < E < E0(N) spends more nique. We have found solutions to the full stationary and more time in the interaction region. The limiting Schrödinger equation and exact tunneling probability solution corresponding to ⑀ = 0 has infinite interaction ᐀ by applying the numerical technique in [11, 12]. Our time: in other words, as t +∞, it tends to one of the numerical calculations were performed for several excited sphalerons. The resulting physical picture is small values of the semiclassical parameter λ, namely, λ that, at a sufficiently large energy (i.e., at E > E1(N)), for = 0.01Ð0.1. Transitions through the barrier for the system prefers to tunnel exactly onto an unstable these values of the semiclassical parameter are well classical solution, excited sphaleron, that oscillates suppressed. In particular, for λ = 0.02, the tunneling about the top of the potential barrier. To demonstrate probability ᐀ is of the order eÐ14. To check the semi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 829 classical result with better precision, we have calcu- F lated the exact suppression exponent ()λ ≡ λ ᐀ FQM Ð log 0.8 (cf. (5)) for λ = 0.09, 0.05, 0.03, and 0.02 and extrapo- λ lated FQM to = 0 by polynomials of the third and fourth degree. The extrapolation results are indepen- 0.4 dent of the degree (3 or 4) of polynomials with a precision of 1%. The extrapolated suppression exponent FQM(0) corresponds to infinite suppression and must 1.0 0 coincide exactly (up to numerical errors) with the cor- 1.2 1.4 rect semiclassical result. 1.6 We performed this check in the region E > E = 1, S 0.1 0.2 0.3 0.4 0.5 0.6 0.7 which is most interesting for our purposes. The results 0.8 N of the full quantum-mechanical calculation of the sup- E λ pression exponent FQM in the limit 0 are repre- Fig. 7. The tunneling exponent F(E, N) in the region E > sented by points in Fig. 7. The lines in that figure repre- ES = 1. The lines show the semiclassical results, and the sent the values of the semiclassical exponent F(E, N) dots represent exact ones, obtained by solving the for constant N, which we obtain in the limit ⑀ 0 of Schrödinger equation. The lines across the plot are the boundary of the region of classically allowed transitions the regularization procedure. In practice, instead of tak- E (N) and the bifurcation line E (N). ing the limit ⑀ 0, we calculate the regularized func- 0 1 tional (), (), ()⑀ responding to the excited sphaleron configurations in F⑀ EN = FEN + O the final state. But the excited sphalerons no longer exist among the solutions of the regularized boundary for sufficiently small ⑀. We used the value ⑀ = 10Ð6, and value problem at any finite value of ⑀. This suggests that the value of the suppression exponent was then found the regularization makes it possible to enter the region with a precision of the order 10Ð5. We see that, in the of classically allowed transitions and, after taking an entire region of classically forbidden transitions appropriate limit, obtain classical solutions with finite (including the region E > E1(N)), the semiclassical values of E and N. result for F coincides with the exact one. By definition, the classically allowed transitions have F = 0. We therefore expect that, in the “allowed” 3.3. Classically Allowed Transitions region of initial data, the regularized problem has the property that We now show that our regularization procedure allows one to obtain a subset of classical overbarrier 2 F⑀()EN, = ⑀fEN(), + O()⑀ . solutions existing at sufficiently high energies. This subset is interesting because it extends to the boundary of the region of classically allowed transitions, E = In view of inverse Legendre formulas (12) and (13), the values of T and θ must be of order ⑀, E0(N). In principle, finding this boundary is purely a problem of classical mechanics; indeed, in mechanics T ==⑀τ()EN, , θ ⑀ϑ()EN, , of two degrees of freedom, this boundary can be found numerically by solving the Cauchy problem for where the quantities τ and ϑ are related to the initial given E and N and all possible oscillator phases (see energy and excitation number (see Eqs. (12), (13)) as Section 2.3). However, if the number of degrees of freedom is much larger, this classical problem becomes ∂ ∂ τ F⑀ 1 (), quite complicated, because a high-dimensional space ==Ðlim ------Ð------T int EN, (18) of Cauchy data has to be spanned. As an example, a sto- ⑀ → 0 ∂E ⑀ 2∂E chastic Monte Carlo technique was developed in [35] to ∂ ∂ deal with this problem in the field theory context. ϑ F⑀ 1 (), ==Ðlim ------Ð------T int EN, (19) The approach below is an alternative to the Cauchy ⑀ → 0 ∂N ⑀ 2∂N methods. We first recall that all classical overbarrier solutions where we have used Eq. (17). We thus expect that the with given energy and excitation number satisfy the T/θ region of classically allowed transitions can be invaded boundary value problem with T = 0, θ = 0. We cannot by taking a fairly sophisticated limit ⑀ 0 with τ ≡ reach the “allowed” region of the EN plane without reg- T/⑀ = const, ϑ ≡ θ/⑀ = const. For the allowed transitions, τ ϑ θ ularization, because we have to cross the line E0(N) cor- the parameters and are analogous to T and .

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 830 BEZRUKOV, LEVKOV

–10 –6 –2 2 6 3 2 1 0 –1 –2 –3 –4 (a) –10 –6 –2 2 6 3 2 1 0 –1 –2 –3 –4 (b) –10 –6 –2 2 6 3 2 1 0 –1 –2 –3 –4 (c) π 0 –π

Fig. 8. The phase of the tunneling coordinate in the complex time plane at three points of the curve τ = 380, ϑ = 130. Figures 8a, ⑀ ⑀ ⑀ ⑀ ⑀ ⑀ ∞ 8b, and 8c correspond to = a = 0.01, = b = 0.0048, and = c = 0, respectively. The asymptotics for X Ð and X +∞ correspond to argX = π and 0. The contour in the time plane is plotted with the white line.

θ Solving the regularized T/ boundary value problem the interaction time Tint is, therefore, infinite. This is allows one to construct a single solution for given E and consistent with (18) and (19) only if τ and ϑ become N. On the other hand, for ⑀ = 0, there are more classical infinite at the boundary. Hence, to obtain a point of the overbarrier solutions: they form a continuous family boundary, we take the further limit, labeled by the initial oscillator phase. Thus, taking the ⑀ limit 0 gives a subset of overbarrier solutions, ()(), ()()τϑ, , ()τϑ, E0 N N = lim E N . which should therefore obey some additional con- τ/ϑ = const straint. It is almost obvious that this constraint is that τ → +∞ the interaction time Tint (Eq. (15)) is minimal. This is shown in Appendix B. Different values of τ/ϑ correspond to different points of The subset of classical overbarrier solutions the line E0(N). We thus find the boundary of the region obtained in the ⑀ 0 limit of the regularized T/θ proof classically allowed transitions without initial-state cedure extends to the boundary of the region of classi- simulation. cally allowed transitions. We now consider what hap- We have checked this procedure numerically. The pens as this boundary is approached from the “classi- limit ⑀ 0 indeed exists—the values of E and N tend cally allowed” side. At the boundary E0(N), the to the point of the EN plane that corresponds to the clas- unregularized solutions tend to excited sphalerons, and sically allowed transition. The phase of the tunneling

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 831

T to the real time axis. These solutions still satisfy the reality conditions asymptotically (see Fig. 6) but show 4 tunneling (a) nontrivial complex behavior at any finite time. T1 reflection The regularized T/θ procedure makes it possible to 3 approach the boundary of the region of classically allowed transitions from both sides. The points at this boundary are obtained by taking the limits T 0, 2 tunneling T/θ = const of the tunneling solutions and τ +∞, onto τ/ϑ = const of the classically allowed ones. Because sphaleron τ ≡ τ ϑ θ τ 1 * / = T/ by construction, the lines * = const are continuous at the boundary E0(N), although they may τ reflection over-barrier have discontinuity of the derivatives. The variable * 0 can be used to parameterize the curve E0(N). 1.0 1.2 1.3 1.5 1.6 E1(N = 0.1) E0 (N = 0.1) E 4. CONCLUSIONS T 6 We conclude that classical solutions describing (b) transmissions of a bound system through a potential tunneling barrier with different values of energy and the initial reflection oscillator excitation number form three branches. TS ⑀ These branches merge at bifurcation lines E (N) and = 0.01 tunneling 0 onto E1(N). Solutions from different branches describe phys- sphaleron ically different transition processes. Namely, solutions at low energies E < E1(N) describe conventional poten- 2 tial-like tunneling. At E > E0(N), they correspond to unsuppressed overbarrier transitions. At intermediate energies, E1(N) < E < E0(N), physically relevant solu- reflection over-barrier 0 tions describe transitions on top of the barrier. This branch structure is shown in Fig. 9a, where the period T = Ð∂F/∂E obtained numerically for solutions from 0.8 ES = 1.0 1.2 the different branches is plotted as a function of energy E for N = 0.1. Fig. 9. Dependence of the parameter T = Ð∂F/∂E on the We note that the qualitative structure of branches in energy for (a) the two-dimensional model with fixed N = 0.1 a model with internal degrees of freedom is similar to and (b) the one-dimensional model (see Appendix C). Dif- the structure of branches in one-dimensional quantum ferent lines correspond to different branches of classical solutions of the T/θ boundary value problem. The branches mechanics (see Appendix C). The latter is shown in labeled “reflection” end up on the wrong side of the barrier. Fig. 9b. The features of solutions in both cases are sim- Figure 9b also contains a line with nonzero ⑀. ilar, although the solutions ending up on top of the barrier are degenerate in energy in the one-dimensional case and, hence, are not physically interesting. coordinate X(t) in the complex time plane is shown in Fig. 8 for the three points (Figs. 8aÐ8c) of the curve τ ≡ In this paper, we introduced a regularization tech- T/⑀ = 380, ϑ ≡ θ/⑀ = 130. Point (a) lies deep inside the nique that allows smoothly connecting solutions from different branches. Its advantage is that it automatically tunneling region, Ea < E1(Na); point (c) corresponds to θ ⑀ chooses the physically relevant branch. This technique the overbarrier solution with T = 0, = 0, = 0; and is particularly convenient in numerical studies: we have point (b) is in the middle of the curve. The branch seen that it makes it possible to cover the whole inter- points of the solution, the cuts, and the contour are 6 esting region of the parameter space. We applied this clearly seen on these graphs. technique to baryon-number violating processes in It is worth noting that the left branch points move electroweak theory [16]. down as T and θ approach zero. Solutions close enough to the boundary E0(N) have the left branch point in the ACKNOWLEDGMENTS lower complex half-plane (see Fig. 8). Therefore, the corresponding contour can be continuously deformed The authors are indebted to V. Rubakov and C. Rebbi for numerous valuable discussions and criticism; A. Kuznetsov, W. Miller, and S. Sibiryakov for helpful 6 The phase of the tunneling coordinate changes by π around the branch point. The points where the phase of the tunneling coordi- discussions; and S. Dubovsky, D. Gorbunov, A. Penin, nate changes by 2π correspond to zeros of X(t). and P. Tinyakov for stimulating interest. We wish to

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 832 BEZRUKOV, LEVKOV thank the Boston University Center for Computational have the familiar path integral representation Science and Ofﬁce of Information Technology for allo- Ꮽ [] ()[] cation of supercomputer time. This research was sup- fi = dx exp iS x , ∫ () ported by the Russian Foundation for Basic Research x ti = xi x()t = x (grant no. 02-02-17398), grant of the President of the f f (A.5) Russian Federation no. NS-2184.2003.2, US Civilian Ꮽ [] ()[] i*' f = dx' exp ÐiS x' , ∫ () Research and Development Foundation for Independent x' ti = xi' States of FSU (CRDF) award no. RP1-2364-MO-02, and x'()t = x under the DOE (grant no. US DE-FG02-91ER40676). f f F.B. is supported by the Swiss Science Foundation where x = (X, y) and S is the action of the model. To obtain a similar representation for the initial-state (grant no. 7SUPJ062239). Ꮾ matrix elements, we rewrite ii' as Ꮾ 〈|, ˆ ˆ |〉, APPENDIX A ii' = Xi yi PEPN Xi' yi' , (A.6)

T/θ Boundary Value Problem where Pˆ N and Pˆ E denote the projectors onto the respective states with the oscillator excitation number The semiclassical method for calculating the proba- N and the total energy E. It is convenient to use the bility of tunneling from a state with a few parameters coherent state formalism for the y oscillator and choose fixed was developed in [13Ð15, 32] in the context of the momentum basis for the X coordinate. In this repre- field theory models and in [3–5, 11, 12] in quantum mechanics. Here, we outline the method adapted to our sentation, the kernel of the projector operator Pˆ EPˆ N model of two degrees of freedom. becomes 1. Path integral representation of the transition 1 〈|qb, Pˆ EPˆ E|〉pa, = ------dξηd probability. We begin with the path integral represen- ()π 2∫ tation for the probability of tunneling from the asymp- 2 ∞ totic region X Ð through a potential barrier. Let i 2 | 〉 × exp Ð iEξ Ð iNη ++--- p ξ exp()iωξ + iη ba δ()qpÐ , the incoming state E, N have fixed energy and oscilla- 2 tor excitation number, and have support only for X Ӷ 0, well outside the range of the potential barrier. The where |p, a〉 is the eigenstate with the respective eigen- inclusive tunneling probability for states of this type is values p and a of the center-of-mass momentum pˆ given by X and the y-oscillator annihilation operator aˆ . It is ∞ ∞ + + straightforward to express this matrix element in the ᐀(), EN = lim  dX f dy f coordinate representation using the formulas → ∞ ∫ ∫ t f Ð ti  0 Ð∞ ω 1 2 1 2 (A.1) 〈|〉ya = 4 ---- exp Ð2---a + ωay Ð ---ωy ,  π 2 2 × 〈|, ()ˆ ()|〉, 2 X f y f exp ÐiH t f Ð ti EN ,  1 〈|〉Xp = ------exp()ipX . 2π where Hˆ is the Hamiltonian operator. This probability Evaluating the Gaussian integrals over a, b, p, and q, we can be reexpressed in terms of the transition amplitudes obtain Ꮽ = 〈|X , y exp()ÐiHˆ ()t Ð t |〉X , y (A.2) fi f f f i i i  ()' 2 Ꮾ ξη ξ η i Xi Ð Xi and the initial-state matrix elements ii' = ∫d d expÐ iE Ð iN Ð ------ 2 ξ Ꮾ 〈|〉, , 〈|, ', ' 〉 ii' = Xi yi EN ENXi yi (A.3) ω + ------as 12Ð exp()Ð2iωξ Ð iη 2 (A.7) +∞ 0 2  yi + yi' ᐀(), × ------()12+ exp()Ð2iωξ Ð iη EN = lim  dX f dXidXi' → ∞ ∫ ∫ 2 t f Ð ti  ∞ 0 Ð (A.4) +∞   -Ð2y y'exp()Ð iωξ Ð iη , × Ꮽ Ꮽ Ꮾ i i ∫ dyidyi'dy f fi i*' f ii' .   ∞ Ð where we omit the preexponential factor depending on The transition amplitude and its complex conjugate η and ξ. For the subsequent formulation of the bound-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 833 ary value problem, it is convenient to introduce the It is convenient to write the conditions at the initial time notation (obtained by varying Xi, yi, Xi' , and yi' ) in terms of the Ti==Ð ξ, θ Ðiη. asymptotic quantities. At the initial time instant ti Ð∞, the system moves in the region X Ð∞, well out- Then, combining integral representations (A.7) and side the range of the potential barrier. Equations (A.11a) (A.5) and rescaling the coordinates, energy, and excita- in this region describe free motion of decoupled oscil- tion number as x x/,λ E E/λ, N N/λ, we lators, and the general solution takes the form finally obtain () () Xt = Xi + pi ttÐ i , +i∞  1 ᐀()EN, = lim  dTdθ []dxxd ' yt()= ------[]aiexp()Ð ω()ttÐ + aiexp()ω()ttÐ , → ∞ ∫ ∫ i i t f Ð ti  ω ∞ 2 Ði (A.8) and similarly for X'(t) and y'(t). For the moment, a and 1  × expÐ---F[]xx, '; T, θ , a are independent variables. In terms of the asymptotic λ  variables Xi, pi, a, a , the initial boundary conditions become where X Ð X' F[]xx, '; T, θ = Ð iS[] X, y + iS[] X', y' p ==p' Ð------i i, (A.9) i i iT Ð ETÐ Nθ + B ()x , x'; T, θ . (A.11c) i i i a' + a' = aexp()ωT + θ + aexp()Ð ωT Ð θ , Here, the nontrivial initial term B is i aa+ = a'exp()Ð ωT Ð θ + a'exp()ωT + θ . ()2 Xi Ð Xi' ω Variation with respect to the Lagrange multipliers T and Bi = ------Ð ------θ 2T 12Ð exp()ωT + 2θ gives a relation between the values of E, N, and the initial asymptotic variables (where we use boundary 1 2 2 conditions (A.11c)): × ---()y + y' ()12+ exp()ωT + 2θ (A.10) 2 i i p 2 E = -----i + ωN, ()ω θ 2 (A.11d) ---Ð2yi yi'expT + . Naa= . In (A.8), x and x' are independent integration variables, Equations (A.11a)Ð(A.11d) constitute the complete set ≡ while x'f xf (see Eq. (A.5)). of saddle-point equations for the functional F. 2. The boundary value problem. For small λ, path The variables X' and y' originate from the conjugate Ꮽ integral (A.8) is dominated by a stationary point of the amplitude i*' f (see Eq. (A.5)), which suggests that functional F. Therefore, to calculate the tunneling prob- they are complex conjugate to X and y. Indeed, the ability exponent, we extremize this functional with ansatz X'(t) = X*(t), y'(t) = y*(t) is compatible with respect to all the integration variables X(t), y(t), X'(t), boundary value problem (A.11). The Lagrange multi- θ y'(t), T, and . We note that, because of the limit tf Ð pliers T and θ are then real, and problem (A.11) may be ∞ ti +, variation with respect to the initial and final conveniently formulated on the contour ABCD in the values of the coordinates leads to boundary conditions complex time plane (see Fig. 2). ±∞ imposed at asymptotic t rather than at finite We now have only two independent complex vari- times ti, tf. We also note that the stationary points may ables X(t) and y(t), which have to satisfy the classical be complex. equations of motion in the interior of the contour, Variation of functional (A.9) with respect to the δ δ coordinates at intermediate times gives second-order S S δ------()- ==------δ () 0 . (A.12a) equations of motion, in general complexified, Xt yt δS δS δS' δS ' The final boundary conditions (see Eq. (A.11b)) ------δ ()- ==δ------() ------δ ()- =δ------()-0. =(A.11a) become the reality conditions for the variables X(t) and Xt yt X' t y' t y(t) at the asymptotic part D of the contour, ∞ The boundary conditions at the final time tf + are ImX f ==0, Imy f 0, obtained by extremization of F with respect to X ≡ X' (A.12b) f f ú ∞ ≡ ImX f ==0, Imyú f 0, t + . and yf y'f . These are Seemingly complicated initial conditions (A.11c) sim- ú ú' X f ==X f , yú f yú'f . (A.11b) plify when written in terms of the time coordinate t' =

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 834 BEZRUKOV, LEVKOV t + iT/2 running along part AB of the contour. We again APPENDIX B write the asymptotic from of a solution, but now along the initial part AB of the contour, A Property of Solutions to the T/ θ Problem in the Case of Overbarrier Transitions XX= + p ()t' Ð t , 0 0 i For given E and N, there is only one overbarrier classical solution, which is obtained in the limit ⑀ 0 of 1 []()ω() ()ω() θ y = ------uiexp Ð t' Ð ti + v exp i t' Ð ti . the regularized T/ procedure. To see what singles out 2ω this solution, we analyze the regularized functional [] [] ⑀ [] In terms of X0, y0, u, and v, boundary conditions (A.11c) F⑀ q = Fq + 2 T int q , (B.1) become where q denotes the variables x(t), x'(t) and T, θ ImX0 ==0, Imp0 0, together. The unregularized functional F has a valley of e θ (A.12c) extrema q (ϕ) corresponding to different values of the v = u*e . initial oscillator phase ϕ. Clearly, at small ⑀, the extremum of F⑀ is close to a point in this valley with the Finally, we write Eqs. (A.11d) in terms of the asymp- e ϕ totic variables along the initial part of the contour, phase extremizing Tint[q ( )],

2 d e p ------T []q ()ϕ = 0. (B.2) E = -----0 + ωN, dϕ int 2 (A.13) e N = ωuv . Hence, the solution q⑀ of the regularized T/θ boundary value problem tends to the overbarrier classical solu- These equations determine the Lagrange multipliers T θ tion, with Tint extremized with respect to the initial and in terms of E and N. Alternatively, we can solve oscillator phase. problem (A.12) for given values of T and θ and find the values of E and N from Eqs. (A.13), which is more con- Because Uint(x) > 0, Tint is a positive quantity with at venient computationally. least one minimum. In a normal situation, there is only one saddle point of F⑀, and, hence, solving the T/θ Given a solution to problem (A.12), the exponent F boundary value problem gives the classical solution is the value of functional (A.9) at this saddle point. We with the time of interaction minimized. thus obtain expression (8) for the tunneling exponent. The exponent F is now expressed in terms of S0 in Eq. (9), the action of the system integrated by parts. APPENDIX C The nontrivial boundary term Bi (Eq. (A.10)) is can- celed by the boundary term coming from integration by Classically Allowed Transitions: parts. We note that we did not use constraints (A.13) to A One-Dimensional Example obtain formula (8), and we therefore still have to extremize (8) with respect to T and θ (see discussion in The difficulties with bifurcations of classical solu- Section 2.2). tions emerge in quite a general class of quantum- mechanical models. To illustrate this statement, we Classical problem (A.12) is conveniently called the consider one-dimensional quantum mechanics, where T/θ boundary value problem. Equations (A.12b) and the result is given by the well-known WKB formula. (A.12c) imply eight real boundary conditions for two We show that the origin of the above difficulties can complex second-order differential equations (A.12a). also be seen in one-dimensional model. Implementa- However, one of these real conditions is redundant: tion of the regularization technique is explicit in the Eq. (A.12b) implies that the (conserved) energy is real, one-dimensional case. This makes it easy to see how and, therefore, the condition Imp0 0 is auto- our technique allows us to smoothly join the classical matically satisfied (we note that the oscillator energy solutions relevant to the tunneling and allowed transi- ω ω θ Eosc = uv = e uu* is real). On the other hand, sys- tions. tem (A.12) is invariant under time translations along In quantum mechanics of one degree of freedom, the real axis. This invariance is fixed, e.g., by requiring only one variable X(t) is present, which describes the that ReX take a prescribed value at a prescribed large motion of a particle of mass m = 1 through a potential negative time t' (we note that other ways may be used barrier U(X). The motion is free in the asymptotic 0 ±∞ instead; in particular, for E < E (N), it is convenient to regions X . The semiclassical calculation of the 1 tunneling exponent is performed by solving the classi- impose the constraint ReXú (t = 0) = 0. Together with the cal equation of motion latter requirement, we have exactly eight real boundary δS conditions for the system of two complexified (i.e., four ------0= real) second-order equations. δXt()

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 DYNAMICAL TUNNELING OF BOUND SYSTEMS THROUGH A POTENTIAL BARRIER 835 on contour ABCD in the complex time plane, with the cally allowed transitions but also well inside the region condition that the solution is real in the asymptotic past of classically forbidden transitions (but still at E > ES; (region A) and asymptotic future (region D). The rele- see the Introduction and Section 2.3). vant solutions tend to X Ð∞ and X +∞ in To illustrate the situation, we consider an exactly regions A and D, respectively. The auxiliary parameter solvable model with T is related to the energy of the incoming state by the requirement that the energy of the classical solution 1 is E. The exponent for the transition probability is UX()= ------. cosh2 X F = 2ImS Ð ET. (C.1) We implement our regularization technique by formally changing the potential We note that these boundary conditions resemble the ones on the tunneling coordinate X in the two-dimen- ⑀ sional system. UX() eÐi UX(), (C.3)

In quantum mechanics of one degree of freedom, which leads to the corresponding change of the classi- contour ABCD may be chosen such that points B and C cal equations of motion. Here, ⑀ is a real regularization are the turning points of the solution. Then, the solution parameter, the smallest parameter in the model. At the is also real at part BC of the contour. Indeed, a real solu- end of the calculations, we take the limit ⑀ 0. tion at part BC of the contour oscillates in the upside- down potential, T/2 is equal to the half-period of oscil- We do not change the boundary conditions in our lations, and points B and C are the two different turning regularized classical problem, i.e., we still require X(t) to be real in the asymptotic future on the real time axis points, Xú = 0. Continuation of this solution from and X(t') to be real as t' Ð∞ on part A of contour point C to the positive real times in accordance with the ABCD. Then, the conserved energy is real. The sphale- equation of motion corresponds to real-time motion, ∞ ron solution X(t) = 0 now has a complex energy with zero initial velocity, towards X +; the coor- (because the potential is complex). Hence, the solutions dinate X(t) stays real on part CD of the contour. Like- of our classical boundary value problem necessarily wise, the continuation back in time from point B leads avoid the sphaleron, and we can expect that the solu- to a real solution in part AB of the contour. The reality tions behave smoothly in energy. conditions are, thus, satisﬁed at A and D. The only contribution to F comes from the Euclidean part of the con- The general solution to the regularized problem is tour, and it can be checked that expression (C.1) reduces to E ()() ------⑀ sinhX = Ðcosh 2EtÐ t0 , eÐi Ð E XC () ()() FE = 22∫ UX Ð E dX, (C.2) where t0 is the integration constant. The value of Imt0 is

XB ﬁxed by the requirement that ImX = 0 at positive time t +∞, which is the standard WKB result. T 1 []Ði⑀ The solutions appropriate for the classically forbid- Imt0 = --- Ð ------arg e Ð E . 2 22E den and classically allowed transitions apparently belong to different branches. As the energy approaches The residual parameter Ret0 represents the real-time the height of the barrier U0 from below, the amplitude of the oscillations in the upside-down potential translational invariance present in the problem. The decreases, while the period T tends to a ﬁnite value condition that the coordinate X is real on the initial part determined by the curvature of the potential at its max- AB of the contour gives the relation between T and E, imum. On the other hand, the solutions for E > U0 T 1 Ði⑀ always run along the real time axis, and, hence, the --- = ------{}π + arg()e Ð E . (C.4) parameter T is always zero. Therefore, the relevant 2 2E solutions do not merge at E = U0, and T(E) has a discontinuity at E = U0. The regularization technique of For ⑀ = 0 and E < 1, the original unregularized result Section 3.1 removes this discontinuity and allows T/2 = π/2E is reproduced. smooth transitions through the point E = U0. The only difference with quantum mechanics of multiple degrees We now analyze what happens in the regularized of freedom is that, in the latter case, bifurcation points case in the vicinity of the would-be special value of ≡ exist not only at the boundary of the region of classi- energy, E = ES 1. It is clear from Eq. (C.4) that T is

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004 836 BEZRUKOV, LEVKOV now a smooth function of E. Away from E = 1, Eq. (C.4) 7. M. Wilkinson, Physica D (Amsterdam) 21, 341 (1986). can be written as 8. M. Wilkinson and J. Hannay, Physica D (Amsterdam) 27, 201 (1987). T --- 9. S. Takada, P. Walker, and W. Wilkinson, Phys. Rev. A 52, 2 3546 (1995). 10. S. Takada, J. Chem. Phys. 104, 3742 (1996).  π ------, forbidden region, 1 Ð E ӷ ⑀, (C.5) 11. G. F. Bonini, A. G. Cohen, C. Rebbi, and V. A. Rubakov,  2E quant-ph/9901062. =  ⑀ 12. G. F. Bonini, A. G. Cohen, C. Rebbi, and V. A. Rubakov,  ӷ ⑀ ------, allowed region, E Ð 1 . Phys. Rev. D 60, 076004 (1999).  2EE()Ð 1 13. V. A. Rubakov, D. T. Son, and P. G. Tinyakov, Phys. Lett. B 287, 342 (1992). Deep enough in the region of forbidden transitions, 14. A. N. Kuznetsov and P. G. Tinyakov, Phys. Rev. D 56, where 1 Ð E ӷ ⑀, the argument in Eq. (C.4) is nearly 1156 (1997). zero and we return to the original tunneling solution. 15. F. Bezrukov, C. Rebbi, V. Rubakov, and P. G. Tinyakov, When E crosses the region of size of order ⑀ around hep-ph/0110109. E = 1, the argument rapidly changes from O(⑀) to Ðπ, 16. F. Bezrukov, D. Levkov, C. Rebbi, et al., Phys. Rev. D and, hence, T/2 changes from π/2 to nearly zero. 68, 036005 (2003). Thus, at E > 1, we obtain a solution that is very close to 17. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. Éksp. Teor. Fiz. 51, 309 (1966) [Sov. Phys. JETP 24, 207 the classical overbarrier transition, and the contour is (1966)]. also very close to the real axis. This is shown in Fig. 9. We conclude that, at small but finite ⑀, the classically 18. V. S. Popov, V. Kuznetsov, and A. M. Perelomov, Zh. Éksp. Teor. Fiz. 53, 331 (1967) [Sov. Phys. JETP 26, 222 allowed and classically forbidden transitions merge (1967)]. smoothly. 19. N. Makri and W. Miller, J. Chem. Phys. 89, 2170 (1988). For E < 1, the limit ⑀ 0 is straightforward. For 20. K. Thompson and N. Makri, J. Chem. Phys. 110, 1343 E > 1, a somewhat more careful analysis of the limit (1999). ⑀ 0 is needed. It follows from Eq. (C.5) that the 21. K. Kay, J. Chem. Phys. 107, 2313 (1997). limit ⑀ 0 with a constant finite T < π /2 leads to 22. N. Maitra and E. Heller, Phys. Rev. Lett. 78, 3035 solutions with E = 1. Classical overbarrier solutions of (1997). ≡ the original problem with E > ES 1 are obtained in the 23. W. Miller, J. Phys. Chem. A 105, 2942 (2001). limit ⑀ 0 if T also tends to zero while τ = T/⑀ is kept 24. A. N. Kuznetsov and P. G. Tinyakov, Mod. Phys. Lett. A finite. Different energies correspond to different values 11, 479 (1996). τ of . This is what one expects: classical overbarrier 25. M. Davis and E. Heller, J. Chem. Phys. 75, 246 (1981). transitions are described by the solutions on the contour 26. E. Heller and M. Davis, J. Chem. Phys. 85, 309 (1981). with T ≡ 0. 27. E. Heller, J. Phys. Chem. 99, 2625 (1994). 28. T. Banks, G. Farrar, M. Dine, et al., Nucl. Phys. B 347, REFERENCES 581 (1990). 1. Z. Huang, T. Feuchtwang, P. Cutler, and E. Kazes, Phys. 29. V. I. Zakharov, Phys. Rev. Lett. 67, 3650 (1991). Rev. A 41, 32 (1990). 30. G. Veneziano, Mod. Phys. Lett. A 7, 1661 (1992). 2. S. Takada and H. Nakamura, J. Chem. Phys. 100, 98 31. V. A. Rubakov, hep-ph/9511236. (1994). 32. V. A. Rubakov and P. G. Tinyakov, Phys. Lett. B 279, 3. W. Miller, J. Chem. Phys. 53, 3578 (1970). 165 (2002). 4. W. Miller and T. George, J. Chem. Phys. 56, 5668 33. F. R. Klinkhamer and N. S. Manton, Phys. Rev. D 30, (1972). 2212 (1984). 5. T. George and W. Miller, J. Chem. Phys. 57, 2458 34. S. Y. Khlebnikov, V. A. Rubakov, and P. G. Tinyakov, (1972). Nucl. Phys. B 367, 334 (1991). 6. W. H. Miller, Adv. Chem. Phys. 25, 69 (1974). 35. C. Rebbi and R. Singleton, hep-ph/9502370.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 4 2004