The Theory of Planar Bragg Resonators P

THEORETICAL AND MATHEMATICAL PHYSICS

The Theory of Planar Bragg Resonators P. V. Petrov All-Russia Research Institute of Technical Physics, Russian Federal Nuclear Center, Snezhinsk, Chelyabinsk oblast, 456770 Russia e-mail: [email protected] Received February 12, 2001; in ﬁnal form, June 8, 2001

Abstract—The reflecting properties of one-dimensional planar Bragg gratings are studied. A coupled resonator model for studying the diffraction of electromagnetic waves in an arbitrarily corrugated waveguide is suggested. It is based on exact relationships that follow from the two-dimensional boundary-value problem stated in terms of the Helmholtz equation. The specific relationships for the rectangular corrugation of the grating- forming plates are presented. The reflection coefficients of the Bragg gratings vs. corrugation length and incident radiation frequency are calculated. An analytical solution for the “narrow” corrugation is obtained. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION ematical model that adequately describes the diffrac- The use of Bragg resonators, which provide distrib- tion of electromagnetic waves by one- and two-dimen- uted feedback and the spatial coherency of radiation, sional Bragg gratings without invoking the perturbation seems to be a promising approach to implementing theory, has an accuracy close to that of direct numerical electrodynamic systems for free-electron masers [1Ð4]. calculations, and is free of awkward mathematics. Bragg resonators are waveguides with a single- or a In this work, we suggest a coupled resonator model double-periodic corrugation (Bragg gratings) [1, 5]. for the analysis of the electromagnetic fields in one- Mathematical models [1Ð5] used for describing the dimensional planar Bragg gratings and report their spectra of the modes and reflection coefficients in reflecting properties. Bragg gratings are based on the coupled mode theory, within which the mode coupling coefficients are found by the perturbation method [6, 7]. Namely, the EQUATION FOR MAGNETIC FIELD waveguide surface deformation (corrugation) defined by some function l(r) is replaced by the “equivalent” Consider a hollow planar waveguide with perfectly boundary condition on the undisturbed surface of a reg- conducting walls that is infinitely long in the 0Y direc- ular waveguide [7]: tion and symmetric about the plane 0X. Its undisturbed ± ω (regular) boundary is defined by the function x = Lx. Eτ = ∇()l()r En⋅ Ð i----l()r []nH, , (1) Let a finite number M of corrugations be patterned on c the side walls of the waveguide in the interval 0 < z < L. where n is the unit vector of the outer normal to the The corrugations are also symmetric about the plane waveguide surface; E, H ~ exp(Ðiωt) are the electric 0X. Let the contour of the qth corrugation be described and magnetic fields; ω is the radiation circular fre- by the function x = lq(z) (Fig. 1) quency; and ∇ is the Hamiltonian operator. The paren- () ∈ [], thesized and bracketed vectors mean the scalar product lq z , zaq, 1 aq, 2 and the vector product, respectively. x =  (2) 0, za< , , za> , , q = 1, M. Approximation (1) applies [7] if (i) the deformation  q 1 q 2 is small (the corrugation height is much smaller than the wavelength of the incident radiation and the corru- We assume that the Bragg grating is excited by a gation length) and (ii) the angle between the deformed natural mode of the regular waveguide from the side of and virgin surfaces is small. As a rule, waveguides used negative z (i.e., from left to right). The mode has a lon- in experiments with Bragg gratings have a rectangular gitudinal wave number hp and a transverse wave num- corrugation [1Ð5]. In this case, the deformation is usu- 2 2 ally small, while the angle is not. Even for a sinusoidal ber gp = k Ð h p . Let the TM modes of the waveguide corrugation on the surface, which is widely used in be excited (for them, only the components Ex, Ez, and numerical simulation, the angle between the surfaces is Hy are other than zero). Then, the initial problem of π/4 and can hardly be considered small. Because of finding the electromagnetic fields is reduced to the this, it would be appropriate to develop a physicomath- boundary-value problem that, in terms of the Helm-

142 PETROV Ω Ω X q (q = 1, M) (in 0, the magnetic field can be repre- x = lq(z); 1x = lq+1(z) sented as the sum of the incident and scattered fields); that is, Ω Ω 2 ∈ Ω a q a q+1 H0 + Hs, r 0 q1 q2 H =  (8) n ∈ Ω Hq, r q, Eτ where H meets boundary conditions (4)Ð(6). Ω 0 0 L x At the boundary x = Lx with the regular waveguide, the tangential field components are continuous, 0 () () () Hq z = H0 z + Hs z (9) Z Y and the scattered field Hs meets homogeneous boundary conditions (4) and the condition that the modes Fig. 1. Planar Bragg grating: 1, contour of the qth corrugation; 2, contour of the undisturbed (regular) waveguide. leave the system at z = 0, L. At the interface between the corrugation and the regular waveguide, the electric field component along ≡ holtz scalar equation for the magnetic field H Hy(x, z), the 0Z axis is assumed to be some function Ez(x = Lx, y, Ω is stated as z) = Eτ(z). Then, in the region 0, Hs is obtained by 2 2 solving the boundary-value problem ∂ ∂ 2 ------++------k H = 0 (3) 2 2 2 2 ∂ ∂ 2 ∂x ∂z ------++------k H = 0, (10) ∂ 2 ∂ 2 s in the domain x z M ∂H ∈ Ω Ω Ω ± ------s + ih H = 0 ()v = 012,,,… , (x, z) = ∪ q∪ 0 , v s ∂z , q = 1 z = 0 L (11) ∂ ∂ Ω ∞ ∞ Hs Hs where 0 = {0 < x < Lx, Ð < z < } is the region of the ==0, ÐikEτ()z . Ω ------∂ ------∂ regular waveguide and q = {Lx < x < lq(z), 0 < x < L} x x = 0 x xL= x is the region of the qth corrugation (q = 1, …, M). The boundary conditions are as follows: the Neumann Its solution can be represented through the Green homogeneous condition function [8]: ∂ L H = 0 ik ------H ()xz, = Ð------dz'G ()xzx,,' = L , z' Eτ()z' , (12) ∂n ∪{}()∪{} (4) s π∫ k x S = xl= q z x = 0 4 q 0 ∞ on the metallic surface of the grating and on the plane π ε (),, , 2 i v ()() of symmetry 0Z Gk xzx' z' = ------∑ ----- cos gv x cos gv x' Lx hv ∂ v H () = 0 ------∂ +2ihp H = ihp cos g p x (5) z × exp()ihv zzÐ ' , (13) at z = 0 if the grating is excited by the TM mode, and π  v ∂ v 2 2 2, = 0 H 2 2 gv ==------, hv k Ð,gv εv = ------Ð0,ihv H ==hv k Ð,gv L ≠ ∂z x 1, v 0. (6) π On the surface of the regular waveguide, expression g ==----- v ; v 01,,… v L s ≡ x (12) for Hs (z) Hs(x = Lx, z) can be written in the com- at z = L if the modes can freely leave the system. pact operator form The electric field components are given by the equa- s H = ÐikGˆ Eτ, (14) tions s R 1 ∂H 1 ∂H ∞ ε L i v Ez ==Ð------, Ex ------. (7) ˆ () ik ∂x ik ∂z GR = ------∑ ----- ∫dz'exp ihv zzÐ ' . (15) 2Lx hv The solutions to Eqs. (7) under boundary conditions v = 0 0 (4)Ð(6) will be sought separately for the regular Let us find a relation between the electric field Eτ(z) Ω waveguide region 0 and for each of the corrugations and the magnetic field Hq at the interface between the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE THEORY OF PLANAR BRAGG RESONATORS 143 Ω qth corrugation region q and the regular waveguide: XX' ≤ ≤ x = Lx, aq, 1 z aq, 2. Irrespective of the corrugation Ω shape, the magnetic field Hq inside the region q is the d Z' solution of Helmholtz equation (3) with Neumann a homogeneous conditions (4) on the metallic surface x = 0' Ez ≤ ≤ lq(z), aq, 1 z aq, 2 of the corrugation and must obey b Lx relationship (9) at the boundary with the regular waveguide. Using the Green function [8] for determin- 0 Z ing Hq(x, z) through the field value at the boundary Sq = ∪ ≤ {x = lq(z)} {x = Lx}, aq, 1 aq, 2 between the qth cor- Fig. 2. Periodic rectangular corrugation in a planar Ω waveguide. rugation region q and the regular waveguide, we find

Taking into account the corrugation geometry, one (), ()' (),,' ∈ Hq xz = °∫ dSq H rs Gq xzrs can express Eτ(z) on the regular waveguide surface z η Sq [0, L] through the Heaviside function (x) = {0, x < 0; (16) 1, x > 0} in the operator form aq, 2 = dz'()Hs ()z' + Hs()z' G ()xzz,,' , ∫ 0 s q () 1 ˆ ()s s Eτ z = Ð---- Gin H0 + Hs , aq, 1 ik M (20) q where Gq is the surface Green function for the Helm- Gˆ in = ∑ ()η()ηzaÐ , Ð ()zaÐ , Gˆ in. Ω q 1 q 2 holtz equation in the region q. Gq satisfies homoge- q = 1 neous Neumann conditions (4) on the metallic surface ≤ ≤ {x = lq(z), aq, 1 z aq, 2} and also the inhomogeneous Using relationships (14) and (20), we come to the Dirichlet conditions at the boundary x = Lx with the reg- s Fredholm integral equation of the first kind for Hs (z) ular waveguide. on the regular waveguide surface: In view of Eqs. (7), we obtain from (16) the expres- s ˆ ˆ s ˆ ˆ s q Hs Ð GRGinHs = GRGinH0. (21) sion for Ez (x, z) in the qth corrugation region: Its solution has the form ∂ (), q 1 H xz E ()xz, = Ð------q - z ik ∂x s ()ˆ ˆ ˆ Ð1 ˆ ˆ s Hs = I Ð GRGin GRGinH0 . (22) aq, 2 (17) 1 s s ∂ s = Ð---- dz'()H ()z' + H ()z' ------G ()xzz,,' . Then, knowing H (z), one can find Eτ(z) by formu- ik ∫ 0 s ∂x q s aq, 1 la (20) and determine the scattered magnetic field throughout the waveguide with (12). Accordingly, at the interface between the qth corru- q gation and the regular waveguide, we have Eτ (z) = CONJUGATION OPERATOR ∈ Eτ(z) = Ez(x = Lx, z) (z [aq, 1, aq, 2]) or, in the operator FOR RECTANGULAR CORRUGATION form, The operator Gˆ in specifies the dependence of the q 1 q s s scattered radiation parameters on the corrugation shape Eτ = Ð---- Gˆ ()H + H . (18) ik in 0 s and on the corrugation distribution over the Bragg lat-

tice length. The form of Gˆ in is the simplest for a rect- q Here, Gˆ in is the conjugation operator for the electric angular corrugation (Fig. 2). Let it have a period b. and magnetic ﬁelds in the qth corrugation: We introduce a local coordinate system X'0'Z' for the initial rectangular corrugation (Fig. 2). In the rect- aq, 2 q ∂ angular domain of width a and height d subject to the ˆ (),, Gin = ∫ dz'------Gq xzz' xL= , homogeneous Neumann conditions on the metallic ∂x x (19) a walls of the corrugation (z' = [0, a]; x' = d) and the inho- q, 1 mogeneous Dirichlet condition on the interface with , ∈ [], zz' aq, 1 aq, 2 . the regular waveguide (x' = 0), the Green function for

TECHNICAL PHYSICS Vol. 47 No. 2 2002 144 PETROV Helmholtz equation (3) has the form [8] In the infinitely narrow corrugation limit (a 0), (27) transforms into the boundary condition for the β () (),, 2 ()χ () χsinh n xdÐ impedances Gq xzz' = ---∑ cos nz cos nz' ------ε β -, a n cosh nd q ()()s () s() n (23) Eτ = ikqtan H0 bq + Hs bq , (28) πn 2 2 χ ==------, β χ Ð.k which was used in the analysis of surface waves over an n a n n interdigital structure [9], and the operator Gˆ in degener- According to (19), we obtain ates into the sum of delta functions: q Gˆ in()zz, ' M Ð 1 a Gˆ in = Ðkatan() kd ∑ d()zqbÐ . (29) β (24) 2 n ()β () χ ()χ q = 0 = Ð---∑----ε- tanh nd cos nz ∫dz'cos nz' , a n n 0 We will restrict our analysis by the mutual scattering s M Ð 1 of the TEM modes (one-mode scattering): ν = 0, H = 2 0 Gˆ in()zz, ' = Ð--- ∑ []η()ηz Ð ()z Ð a exp(ikz). Then, Eq. (21) takes the form a q q q = 0 M Ð 1 L s zq + a () β () Hs z = i ∑ ∫dz'exp ik zÐ z' (30) ×γ∑ cos()χ z dz'cos()χ ()z' Ð bd , (25) q = 0 0 n n q ∫ n n z × ()s() ()() q Hs z' + exp ikz' d z' Ð qb , β γ n ()β where the dimensionless parameter n ==----ε- tanh nd , zq zqbÐ . n a β = ------tan()kd (31) The explicit form of the operators Gˆ R and Gˆ in [see 2Lx (15) and (25)] makes it possible to numerically obtain s defines the dependence of the solution on the parame- Hs (z) from (21) and to determine the reflection coeffi- ters of the corrugation and the waveguide. For the spe- π cients of the grating: cific case hp = k = /b, which corresponds to the Bragg resonance condition [1] Lx 1 (), R = ----- ∫ dxSz xz0 , h p + h0 = h, (32) S0 0 (26) the solution of (30) can be written in the analytical form ∂ 1 Hy (), (), (), (), M Ð 1 Sz xz0 ==Ex xz0 H*y xz0 ---- H*y ------xz0 . ik ∂z s iβ q H ()z = ------∑ ()Ð1 exp()ik zÐ qb . (33) s 1 Ð iβM Here, Hy(x, z0) is determined from (12) for points with q = 0 the coordinate z outside the corrugation, Eτ(z) is calcu- 0 Accordingly, the reflection coefficients becomes lated by (20), and S0 is the electromagnetic wave energy flux incident on the Bragg grating. β2M2 R = ------. (34) 1 + β2M2 ANALYTICAL SOLUTION FOR ONE-MODE SCATTERING BY NARROW CORRUGATION Within such an approximation, we can find the resonance region. Here, it is appropriate to pass from inte- Generally, Eq. (21) is solved numerically; however, gral equation (30) to the second-order differential equa- if a corrugation is narrow (its width a is much smaller λ tion [8]. To determine the resonance region, it is suffi- than the incident wavelength ), this equation is analyt- cient to consider the homogeneous equation for a ically solvable. In this case, one can leave only the first grating with the infinite number of corrugations: term in expression (25) for Gˆ in . Then, relationship (18) 2 s() between the electric and magnetic fields on the surfaces d Hs z ()2 () s() ------+0,k + F z Hs z = of the corrugation and the regular waveguide is reduced to dz2 bq+ a ∞ (35) q() i () ()s () s() Ez z = --- tan kd dz' H0 z' + Hs z' . (27) F()z = 2kβ ()zqbÐ . a ∫ ∑ d bq q = Ð∞

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE THEORY OF PLANAR BRAGG RESONATORS 145

The function F(z) has a period b and can be R expanded into a Fourier series. Passing to the variable 1.0 x = k0z, we obtain 0.9 L = 10 cm 2 s 0.8 d H k2 βk ------s + ----+ 2 ------2  2 πk 0.7 dx k0 0 (36) 0.6 × ()() ()…  s 0.5 L = 5 cm - 12+++cos 2x 2cos 4x  Hs = 0. 0.4 It is known that Eq. (36) describes parametric reso- 0.3 nance [10]. If one considers only the first (the most Ӎ 0.2 intense) resonance, which takes place at k/k0 1 [10], and ignores higher order resonances, Eq. (36) is 0.1 reduced to the Mathieu equation [11] 0 0.94 0.96 0.98 1.00 1.02 1.04 1.06 k/k0 2 s d Hs ()() () () s ------2 +0,sk Ð 2 pkcos 2x Hs = Fig. 3. Reflection coefficient vs. wave vector for the single- dx mode scattering of electromagnetic wave from narrow-cor- (37) rugation (a = 0.01 cm, d = 0.09 cm) Bragg gratings with var- k2 βk βk sk()==----+2 2 ------; pk() Ð ------. ious lengths. Continuous curve refers to Eq. (20) with v = 0, 2 πk πk a = 0.02 cm, and d = 0.09 cm; ×, to Eq. (29) with β = k0 0 0 6.3 × 10Ð2. The conditions for parametric resonance correspond to the instability regions of the Mathieu functions and, R β = 6.3 × 10–2 ≤ 1.0 for p(k) 0, lie between the curves [11] 1 0.9 s ()p ≤≤sc ()p . (38) 2 2r + 1 2r + 1 0.8 3

Here, sr and cr are the eigenvalues of the Mathieu equa- 0.7 tion that correspond to the even and odd solutions at a ≈ 0.6 fixed p. For the first resonance r = 0 at k k0, the func- 4 0.5 tions s1(p) and c1(p) can be represented as the power series in p [11]: 0.4 0.3 p2 p2 (39) s (p ) = 1 +p Ð ----- +…, c ()p = 1 Ð p Ð ----- + … . 1 8 1 8 0.2 0.1 Substituting expressions (39) with the functions s(k), p(k) from (37) into (38), we find the range of k for 0 2 4 6 8 10 12 which the parametric resonance arises. Up to the terms L, cm of the order of β, we have Fig. 4. TEM mode (v = 75 GHz) reflection coefficient vs. 2β ≤≤ length of the one-dimensional Bragg grating with a period k01 Ð π--- kk0. (40) 0.2 cm for various corrugation depths and widths (β = 6.3 × 10Ð2). (1) a = 0.004 cm, d = 0.096 cm; (2) a = 0.01 cm, d = 0.09 cm; (3) a = 0.04 cm, d = 0.064 cm; and (4) a = 0.1 cm, Thus, Bragg resonance condition (32), used in [1Ð d = 0.036 cm. ×, analytical representation (32); ᭺, finite-ele- 5], sets the resonance position only approximately. This ment method (a = 0.1 cm, d = 0.036 cm). is corroborated by calculating the coefficient of electromagnetic wave reflection [see (22) for the one-mode REFLECTION COEFFICIENTS regime] as a function of the wave vector for narrow- OF ONE-DIMENSIONAL BRAGG GRATINGS corrugation Bragg gratings (Lx = 0.5 cm, a = 0.01 cm, d = 0.09 cm, and β = 6.3 × 10Ð2) of length L = 5 and The practical use of Bragg gratings is usually asso- 10 cm shown in Fig. 3. Here, the reflection coefficient ciated with their reflecting properties. Of most interest obtained from solution (30) with β ≅ 6.3 × 10Ð2 for the is the dependence of the reflection coefficient on the delta corrugation is also shown. corrugation length and incident radiation frequency for

TECHNICAL PHYSICS Vol. 47 No. 2 2002 146 PETROV

R R 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 66 68 70 72 74 76 78 80 82 68 70 72 74 76 78 80 v, GHz v, GHz

Fig. 5. Frequency dependence of the TEM mode reflection coefficient for the one-dimensional Bragg grating of length L = 5 cm with a period 0.2 cm and a rectangular corrugation Fig. 6. Frequency dependence of the TEM mode reflection (a = 0.01 cm, d = 0.09 cm). Continuous curve, Rtot; dashed coefficient for the one-dimensional Bragg grating of length curve, R ; ᭺, R obtained by finite-element method; L = 5 cm with a period 0.2 cm and a rectangular corrugation TEM02 tot (a = 0.1 cm, d = 0.03 cm). Designations are the same as in +, R ; and ×, R . Fig. 5. TEM TEM04 given corrugation period and depth. We determined the in form (25) was kept the same, β ≅ 6.3 × 10Ð2. For com- reflecting properties of one-dimensional Bragg gratings parison, the reflection coefficients for an infinitely nar- for the case of a planar waveguide with the plate spac- row corrugation [see (34)] and those calculated by the ing 2Lx = 1 cm and a rectangular corrugation with a finite-element method [12] are also shown. period of 0.2 cm. It is seen that formula (34) describes the dependence Figure 4 depicts the reflection coefficient vs. grating of the reflection coefficients R on the corrugation length length for the wave frequency v = 75 GHz and various β with a fairly good accuracy if the gratings are long. For corrugation width and depth. The parameter obtained the short gratings, the discrepancy between the values by numerically solving Eq. (21) with the operator Gˆ in calculated and those predicted by the analytical for-

Mode composition of the reﬂected radiation when the TEM mode of the planar waveguide with Lx = 0.5 cm strikes the Bragg grating of length L Mode composition, %, for narrow corrugation: Mode composition, %, for wide corrugation: L, cm a = 0.01 cm, d = 0.09 cm a = 0.1 cm, d = 0.36 cm

TEM TEM02 TEM04 TEM TEM02 TEM04 0.8 30.1 60.3 9.7 28.6 61.0 10.4 1 33.5 64.4 2.2 31.8 65.7 2.4 2 46.4 46.3 7.2 36.3 58.0 5.6 3 62.5 32.5 5.0 49.9 43.0 7.1 4 75.5 23.8 0.7 66.5 28.3 5.2 5 84.4 15.4 0.2 80.2 16.9 3.0 6 89.9 9.0 1.1 87.9 10.0 2.1 7 92.7 6.2 1.1 87.9 10.7 1.4 8 94.2 5.5 0.2 76.6 22.7 0.7 9 95.6 4.3 0.0 55.2 43.7 1.1 10 96.9 2.8 0.3 45.0 53.1 1.9 12 97.9 1.9 0.1 71.4 24.9 3.7

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE THEORY OF PLANAR BRAGG RESONATORS 147 mula is associated with the different mode composi- gratings with different lengths were numerically calcu- tions of the radiation. Formula (34) was derived under lated for different radiation frequencies. The data the assumption that both incident and reflected modes obtained are compared with the analytical results for a are the TEM modes. It turned out, however, that, when narrow corrugation and with the direct numerical calcu- the corrugation is short or “wide,” a considerable part lations by the finite-element method. of the energy is reflected into the TM02 and TM04 modes (see table). This part of the energy decreases with increasing grating length, and eventually the one-mode REFERENCES scattering regime TEM ⇒ TEM sets up. 1. G. G. Denisov and M. G. Reznikov, Izv. Vyssh. Uchebn. As follows from the calculations, for a narrow cor- Zaved., Radiofiz. 25, 562 (1982). rugation (a Ӷ λ), where boundary condition (28) for the 2. V. L. Bratman, N. S. Ginzburg, and G. G. Denisov, impedances is quite appropriate, the reflecting proper- Pis’ma Zh. Tekh. Fiz. 7, 1320 (1981) [Sov. Tech. Phys. ties of the grating are governed by the dimensionless Lett. 7, 565 (1981)]. parameter β [see (31)], which is consistent with the the- 3. N. S. Ginzburg, N. Yu. Peskov, and A. S. Sergeev, Pis’ma oretical prediction. As the corrugation widens, condi- Zh. Tekh. Fiz. 18 (9), 23 (1992) [Sov. Tech. Phys. Lett. 18, 285 (1992)]. tions (28) fail and higher order terms must be left to 4. N. S. Ginzburg, N. Yu. Peskov, A. S. Sergeev, et al., correctly include diffraction in the operator Gˆ in . This is Nucl. Instrum. Methods Phys. Res. A 358, 189 (1995). supported by the direct computation of electromagnetic 5. N. Yu. Peskov, N. S. Ginzburg, G. G. Denisov, et al., wave scattering with the finite-element method [12]. Pis’ma Zh. Tekh. Fiz. 26 (8), 72 (2000) [Tech. Phys. Figures 5 and 6 demonstrate the dependences of the Lett. 26, 348 (2000)]. total reflection coefficient and of the partial reflection 6. N. F. Kovalev, I. M. Orlova, and M. I. Petelin, Izv. Vyssh. coefficients (for the TM modes) on the frequency of the Uchebn. Zaved., Radiofiz. 11, 783 (1968). TEM mode incident on Bragg gratings with narrow (a = 7. B. Z. Katsenelenbaum, Theory of Irregular Waveguides 0.01 cm, d = 0.09 cm) and wide (a = 0.1 cm, d = with Slowly Varying Parameters (Akad. Nauk SSSR, 0.03 cm) corrugations for L = 5 cm. Moscow, 1961). 8. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953; Inostrannaya CONCLUSION Literatura, Moscow, 1958). We presented the theory of electromagnetic field 9. L. A. Vaœnshteœn, Electromagnetic Waves (Radio i Svyaz’, excitation by one-dimensional Bragg resonators in Moscow, 1988). terms of the coupled resonator method. It applies the 10. L. D. Landau and E. M. Lifshitz, Course of Theoretical Green function to the two-dimensional boundary-value Physics, Vol. 1: Mechanics (Nauka, Moscow, 1988; Per- problem for the Helmholtz equation, is rigorous (i.e., gamon, New York, 1988). does not use any approximation), and can thus be 11. Handbook of Mathematical Functions, Ed. by M. Ab- employed for analyzing corrugated structures of arbi- ramowitz and I. A. Stegun (National Bureau of Stan- trary shape and size. For a narrow corrugation, we dards, Washington, 1964; Nauka, Moscow, 1979). obtained an analytical solution, which is governed by a 12. Partial Differential Equation Toolbox: User’s Guide (The single dimensionless parameter depending on the cor- Mathworks Inc., 1997). rugation shape and sizes. The coefficients of the reflection of the TEM mode from one-dimensional Bragg Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

THEORETICAL AND MATHEMATICAL PHYSICS

The Cantor Distribution and Fractal Transition Scattering V. N. Bolotov Institute of Electromagnetic Research, Kharkov, 61022 Ukraine e-mail: renic@iemr. vl.net.ua Received January 9, 2001

Abstract—A new distribution is introduced for the elaboration of a theory of fractal electrodynamics. The construction of the new distribution is based on the Cantor set structure and Cantor measures. The basic properties of this Cantor distribution are discussed. The semigroup of operators deﬁned in terms of Cantor measures is examined. As an example, the theory of transition scattering of a dissipative permittivity burst by a stationary charge is considered. © 2002 MAIK “Nauka/Interperiodica”.

ξ INTRODUCTION interval 1 Ð 2 is rejected. In this way, the set E1 of two closed segments of length ξ is obtained. Then, the same The electrodynamics of fractal media examines ξ ξ electromagnetic processes in a space filled with a mate- is done with the segments [0, ] and [1 Ð , 1], which rial forming a fractal structure. Like any macroscopic enter into the set E1, etc. At the nth step, we come to the n ξn theory, the electrodynamics of fractal media operates set En consisting of 2 segments of length . The com- on physical quantities averaged over “physically infini- pact set En is called a pre-Cantor set (Fig. 1), and the tesimal” volume elements and does not touch upon the intersection of pre-Cantor sets produces a Cantor set ∞ atomic (molecular) structure of the material. Fractal E = ∩ E . structures are fractal clusters, fractal surfaces, percola- n = 0 n tion clusters, and other material formations observed in The HausdorffÐBezikovich dimension of a Cantor | ξ| ξ experiments. Of special importance in the development set E is dH = ln2/ ln , 0 < < 1/2 [1]. A Cantor set is of fractal electrodynamics is averaging, which can be characterized by points of the first and second kind. made with fractal and multifractal measures. Points of the set E that are the (left or right) ends of Thus, the electrodynamics of fractal media relies on intervals adjacent to the Cantor discontinuum are called the fundamental concept of fractal and multifractal or, points of the first kind of the Cantor set (left, EL, or more specifically, on the Cantor set structure and Haus- R R L right, E , respectively). Similar sets En and En can be dorffÐBezikovich fractional-dimension spaces. The constructed for any of the pre-Cantor sets E . In Fig. 1, mathematical apparatus of fractal electrodynamics is n R taking several paths. It can be based on fractional inte- the points belonging to the set En are marked by circles gration/differentiation, fractal wavelets, or iterative L functional systems. In this work, we will consider the and those belonging to the set En , by squares. These apparatus of distributions. It is the author’s opinion that points of the first order are ordered in a natural way and such an approach is the most general and includes frac- produce a countable set. All other points of the Cantor tional integration/differentiation and wavelets as spe- set are points of the second kind. The set of the points cific cases. of the second kind is similar to that of irrational numbers and has the power of continuum [1]. As an example, we will examine the formation of the fractal spectrum (the localization of the radiation Any point of the Cantor set E with a ratio ξ can be frequencies at points of the Cantor set) upon transition written as scattering. The permittivity wave is specified by a func- ∞ tion of dissipative burst γξ(x). The complex analytic n t = ω ξ , (1) form of the dielectric burst can be formed by shock ∑ n waves in the dielectric, as well as by turbulent flows in n = 1 plasmalike media and Kerr liquids. ω ξÐ1 where n = {0, Ð1} takes only two different values. ∈ ω Numbers t E for which all n equal each other, CANTOR SET starting with some number, belong to points of the first kind of Cantor set. Thus, the set of right points of the The structure of the conventional triadic Cantor set ξ R with a ratio is well known. The unit segment E0 = first kind En is defined by expansion (1) where all coef- [0, 1] is divided into three parts, and the middle open ficients, starting with the (n + 1)th term, are zero:

THE CANTOR DISTRIBUTION AND FRACTAL TRANSITION SCATTERING 149 ω ω n + 1 = n + 2 = … = 0. The set of left points of the first 0 1 E kind EL is deﬁned by expansion (1) where all coefﬁ- 0 n ξ ξ ξÐ1 011 – cients, starting with the (n + 1)th term, equal Ð 1: E ω ω ξÐ1 1 n + 1 = n + 2 = … = Ð 1.

R E2 To enumerate all points of a set, for example, En , it is necessary that each ω (k = 1, 2, …, n) be assigned k Fig. 1. Pre-Cantor sets. two values: 0 and ξÐ1 Ð 1. In doing so, we obtain 2n R R n points of the set En ; that is, N(En ) = 2 . Clearly, R L In the set Ω of finite-length words in the alphabet A, N(En ) = N().En one can separate the operation of occurrence or concatenation (*): … … … … FREE SEMIGROUP x1 x2 xm * y1 y2 yn = x1 x2 xm y1 y2 yn. (3) ξ α ∈ Ω A triadic Cantor set with a ratio is generated by an Thus, the concatenation of two words 1 n and α ∈ Ω alphabet A consisting of two letters: A = {a, a } (a can 2 k (of length n and k, respectively) gives the word α α α α ∈ Ω be viewed as the negation of a). This is due to the fact of length n + k: 1 * 2 = 1 2 n + k. that the one-to-one correspondence maps a 0, ξÐ1 The operation of concatenation is associative by a Ð 1 and only one point tα from the Cantor set Ω α Ω definition. Therefore, a set of finite words with con- corresponds to any word = a1a2 … ak … from a set catenation forms the free semigroup Ω* = (Ω, ) [4]. of (finite and infinite) words. The approach using a * word set is common in the theory of fractals and multi- Note that each word from Ω admits the single fractals [2]. In this case, any a entering into the word α expansion in the product of elements from A. Recall k Ω Ω equals either a or a . Thus, a ω and that by a semigroup * we mean a nonempty set k k with the binary operation * that meets the associative ∞ law x * (y * z) = (x * y) * z for any x, y, z ∈ Ω. k tα ==t()α ωξ . (2) ∑ k Let the set ER contain a binary operation (᭺) such k = 1 that For example, to the word α = aaaa …(aa ), where k tα ᭺ tβ = tα + ξ tβ (4) the combination (aa ) is repeated, there corresponds the ∞ R R ξÐ1 ξ2n ξ ξ for tα ∈ E and tβ ∈ E . Cantor number tα = ∑k = 1 ( Ð 1) = /(1 + ). For k p ξ = 1/3, tα = 1/4. In the other case, to the word β = It is easy to check by direct calculation [see (2)] that R aaaaaa…(aa ), there corresponds the Cantor number ᭺ ∈ ᭺ tα tβ = tαβ Ekp+ . On the other hand, tβ tα = tβ + tβ = 1/(1 + ξ). For ξ = 1/3, tβ = 3/4. ξp ∈ R ≠ tα = tβα Ekp+ . Thus, tαβ tβα by construction, so The numbers 1/3 and 3/4 are the points of the second that this binary operation is noncommutative but asso- kind of the Cantor set [3]. In fact, in the above examples α β ciative. The associativity property is checked by direct and in both the word and word the combination of calculation. two letters, rather than one letter, from the alphabet A is repeated. In the other case, we would deal with points Thus, the set ER with binary operation (4) forms the of the first kind. semigroup (ER, ᭺).

Ω Ω Ω R ᭺ The word set can be divided into subsets n (n = Theorem. If * is a free semigroup and (E , ) is 0, 1, 2, …) of words of ﬁnite length. The number of ele- the semigroup of right points of the ﬁrst kind in a Can- Ω k Ω ments N( k) in a set of words of length k is 2 . The ele- tor set, then there exists the isomorphism t : * Ω Λ R ᭺ ment 0 is an empty word . Two words x1x2…xm and (E , ). α β γ y1y2…yn are considered to be equal if they coincide as The proof is straightforward: t( * ) = t( ), since sequences: m = n, x = y , x = y , …, x = y . α β γ α ∈ Ω β ∈ Ω γ ∈ Ω 1 1 2 2 m m * = , where k, n, and n + k. On the R R Ω ᭺ ξk ∈ ∈ Note also that the bijective mapping t: n En other hand, tα tβ = tα + tβ = tγ, where tα Ek , tβ establishes a correspondence between a word of length R ∈ R En , and tγ Enk+ . However, since there exists a one- R α α ∈ Ω n and a point of the pre-Cantor set En : tα, to-one correspondence between the sets n + k and Ω ∈ R R γ n, tα Rn . Enk+ , then t( ) = tγ.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 150 BOLOTOV µ µ µ CANTOR DISTRIBUTION that is, lim (En) = (E). The measure (En) on the n → ∞ The introduction of distributions is associated with ∆ pre-Cantor set En defines the distribution n (n = 0, 1, the need for extending the representation of functions, 2, …) by the formula for example, in theoretical physics. A function can be considered as a mathematical object influencing a trial ()ϕα∆ , ϕ n = ∫ d n. (6) function rather than as a set of values at various points. Of most interest in this respect is the approach that As before, we will use the shorthand writing ∆ ξ ≡ ∆ deals with distributions generated by measures [5]. As n(x; ) n(x). Integral (6) should be read as the Leb- has been noted in the Introduction, a theory of electro- esgueÐStieltjes integral [3]. Note also that if n = 0, inte- dynamics of fractal media will be based on fractal or, gral (6) becomes the Riemann integral. Thus, the pre- more accurately, Cantor measures. In this work, such a α Cantor functions n(x) are generating functions for the measure is based on the Cantor function α, which is µ measures (En). called “devil’s ladder.” A Cantor function is a mono- ∆ The distribution n should be read as a functional tonic nondecreasing function bounded from above and ϕ ∈ from below that is the limit of the sequence of pre-Can- acting over the space of the basic functions D(Q), α α α D(Q). This space also covers all finite functions that are tor functions n; that is, (x) = lim n (x). ∞ n → ∞ indefinitely differentiable in Q: D(Q) = C0 . In our case, α ϕ ϕ The population of the pre-Cantor functions { n} is Q = R is the real axis. The convergence k in the constructed in the segment [0, 1] using the totality of space of the basic functions D(Q) depends on the con- α ϕ sets {En}. Figure 2 shows the function n(x) for n = 2. vergence of the functional sequence { k} and all the α () It is seen that the function 2(x) is constant in the open ϕ n ϕ(n) ∆ derivatives k . For the distributions n, the intervals rejected and linearly grows in the closed inter- ∆ ϕ functional ( n, ) has the form vals, which belong to E2 in our case. As n increases, so does the slope of the straight lines (2ξ)Ðn (or the deriva- ϕα ∆ ϕ ≡ ()∆ , ϕ α ∞ ∫ d n = ∫ n dx n . (7) tive of n(x) on En). In the limit n , the slope tends to infinity. Thus, the Cantor function α is a func- ∆ Thus, the pre-Cantor distributions n can be explic- tion that changes stepwise at points of the Cantor set. In itly expressed through the derivatives of α (x): this work, all constructions use a triadic Cantor set with n a ratio ξ. If this is quite clear, the parameter ξ is omitted  ∉ α ξ ≡ α 0, xEn for simplicity. For example, n(x; ) n(x). d  ------α ()x =  1 (8) On each of the pre-Cantor sets E , one can specify dx n ∈ n ------n, xEn. measures that are totally additive functions and, which ()2ξ is of special importance, are generated by the pre-Can- α tor functions n(x): From (7) and (8), it follows that ()∆ , ϕ 1 χ ()ϕ() µ()E = d α. (5) n = ∫ ------E x x dx, (9) n ∫ n ()ξ n n ⊂ 2 En R En µ where χ is the characteristic function of the set E : The set of the measures { (En)} form a sequence En n converging to a measure specified on the Cantor set; ∈ 1, xEn χ =  (10) α En ∉ n(x) 0, xEn. 1 Expression (9) can easily be transformed to the form 1 3/4 ()∆ , ϕ 1 ϕξ()n n = -----n∫ ∑ xx+ α dx, (11) 2 αΩ∈ 1/2 0 n R where xα ∈ E . Note that (∆ , 1) = 1 for any n. 1/4 n n Functional (11) shows the result of action of the dis- ∆ ϕ tributions n on the trial function (x) from the space of 0 ξ2 ξ 1–ξ 1–ξ2 1 x basic functions. This result is virtually the averaging of ξ ξ 2 ϕ ∆ (1– ) 1– ξ + ξ (x) on En. The supports of the distributions { n} are ∆ the pre-Cantor sets {En}; that is, supp n = En. Thus, the ∆ ∆ ∈ Fig. 2. Cantor function for n = 2. distributions { n} are functionals on D(Q) and n D'(R)

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE CANTOR DISTRIBUTION AND FRACTAL TRANSITION SCATTERING 151 for any n. Here, D'(R) is the linear space of all distribu- order of derivative. The operator of Fourier transformations that is conjugate to D(R). Convergence in D'(R) is tion (29) acts in a similar way. To express the operators introduced as the weak convergence of the functionals. {A } in the explicit form, we put n = 0 in expression (16). ∆ k Thus, the sequence n defines the Cantor distribution ∆ ϕ ∆ ϕ Then, (Ak 0, ) = ( 0, Ak ). In another form, ()∆ , ϕ ()∆ , ϕ lim n = ξ . (12) n → ∞ ϕαd = A ϕdx. (17) Using (11) and (12), we represent the Cantor distri- ∫ k ∫ k bution in the form R R Using expressions (7), (11), and (17), we come to ()∆ , ϕ 1 ϕ() ()∆ , ξ ==lim----- ∑ xα , ξ 1 1. (13) the explicit form of the operators {Ak} (k = 0, 1, 2, …): n → ∞ n 2 αΩ∈ n ∆ Clearly, supp ξ = E. ϕ() 1 ϕξ()k Ak t = ----k ∑ tt+ α . (18) Turning back to the introduction of the distributions 2 αΩ∈ k through measures (5), we can represent ∆ξ in the form that is the most convenient for calculations and shows At k = 0, A ϕ = ϕ. The operators A map the space the relation between the Cantor function α, generating 0 k of the basic functions into itself. They perform the the measure µ(E), and the Cantor distribution ∆ξ, which ξk ∈ ∈ ∈ R is a functional: transformation t t' = t + tα (t R, t' R, tα Ek ) and averaging over the right points of the first kind in ()ϕα∆ , ϕ ξ = ∫ d . (14) the Cantor set. R ∈ The operators {Ak} G retain the structure of a From (13), it readily follows that the distribution ∆ξ semigroup, since it is easy to check by direct calcula- is a singular distribution. Note that both the Dirac gen- tions using (18) that A A ϕ = A ϕ. Indeed, taking into eralized function δ and ∆ξ are singular distributions and k n k + n ξk α ∈ Ω β ∈ Ω γ ∈ their supports are sets of measure zero: suppδ = {0} and account that tβ + tα = tγ, where n, k, ∆ Ω supp ξ = E. n + k, we find The Cantor distribution ∆ξ defines the Cantor structure of a point. The author believes that it can be applied ϕ() 1 ϕξ()nk+ ξk for elaborating physical theories not on smooth mani- An Ak t = ------n -k ∑ ∑ t ++tα tβ 2 2 αΩ∈ βΩ∈ folds but on manifolds where each point is a Cantor set n k (for example, on fractal space-time). (19) 1 ϕξ()nk+ = ------nk+ ∑ tt+ γ . SEMIGROUP OF OPERATORS ON SET 2 γΩ∈ µ nk+ OF CANTOR MEASURES (En) From (15) and (16) at n ∞, one important prop- Let us introduce a set of operators {An} that act on the measure set {µ(E )} (n = 0, 1, 2, …) following the erty of invariance that is necessary for calculating the n ∆ law functionals ξ follows; namely, A µ()E = µ()E . (15) µ() µ() ()∆∆ , ϕ (), ϕ k n kn+ Ak E ==An E , ξ Ak ξ An . (20)

The operators {Ak} deﬁned by action (15) form a semigroup G, since it is easy to check that A A = A Equality (20) is valid for any n (k = 0, 1, 2, …). As k n k + n an example, let us calculate the speciﬁc integral and that the associative law Ak(AnAj) = (Ak An)Aj is ful-

ﬁlled. The operator A0, which leaves the measure 1 unchanged, is the unity of the semigroup. The action of v v I()v ==() ∆ξ, x x dα. (21) the operator {An} can be extended for functional ∫ spaces, for example, for the space of the basic functions 0 D(Q). To obtain the operators {Ak} in the explicit form, ∈ Using relationship (21) for A and expression (18), we assume that any Ak G has the property 1 v ()∆∆ , ϕ (), ϕ we easily obtain the functional I( ): Ak n = n Ak . (16) 1 1 This property of the operators {A } is identical to the ()ξ v ()ξ ξ v k () v α x + 1 Ð + x α action of the differentiation operator on the distribu- I v ==∫ A1 x d ∫------d . (22) n n n 2 tions from D'(Q): (D f, ϕ) = (Ð1) (f, D ϕ), where n is the 0 0

TECHNICAL PHYSICS Vol. 47 No. 2 2002 152 BOLOTOV

Thus, we can find the functional relationship for the basic functions ᏸ and the set of slowly increasing dis- functionals I(v): tributions ᏸ' [5]. The former space, ᏸ = ᏸ(Rn), con- ∞ n v sists of indefinitely differentiable (on R ) functions that ()1 Ð ξ v Ð1 Ðk ()v ()ξ () decrease, along with all derivatives, faster than any I = ------v ∑  Ð 1 Ik, 2 Ð ξ k power function of |x|Ð1 at |x| ∞. In this case, ᏸ ⊃ D. k = 0 (23) Any linear continuous functional on the set of basic v vv()…Ð 1 ()v Ð1k + ᏸ = ------. functions is called a slowly increasing distribution. k k! The space of the distributions ᏸ' = ᏸ'(Rn) consists of the set of all slowly increasing distributions. It is obvi- For v = m, m ∈ Z, and in view of the fact that ous that ᏸ' ⊂ D'. I(0) = 1, expression (23) takes the form Let f(x) ∈ ᏸ and ϕ(x) ∈ ᏸ; then, from the definition ()ξ m m Ð 1 of the Fourier transform for distributions, we have () 1 Ð m ()ξÐ1 Ðk () Im = ------∑  Ð 1 Ik. (24) ()ξm k ()Ff[]ϕ, = ()fF, []ϕ . (29) 21Ð k = 0 If m = 1, I(1) = 1/2. In this case, LebesgueÐStieltjes Since the operation ϕ F[ϕ] is linear and contin- integral (21) does not differ from the Riemann integral uous in going from ᏸ to ᏸ, the functional on the right- of the function xm. However, for m = 2, I(2) = 1/[2(1 + hand side of (29) represents a distribution from ᏸ'. The ξ)]. This result is markedly different from the value of operation f F[ f ] is linear and continuous in going the Riemann integral, 1/3. The results coincide for ξ = from ᏸ' to ᏸ' [5]. 1/2. Note also that 1/[2(1 + ξ)] > 1/3 if 0 < ξ < 1/2. If f is a distribution with a compact support, it is a Below are the expressions for several functionals slowly increasing distribution and its Fourier transform calculated with the above procedures: exists [5]. It is this case that is of interest, because the ∆ ∞ Cantor distribution has the compact support supp ξ = ()ξξ n ∆ ()∆ , ηt η/2 1 Ð η E. It should be noted that the distributions { n} also ξ e = e ∏ cosh ------, (25) ∆ 2 have the compact carriers supp n = En; hence, they also n = 0 have their associated Fourier transforms. Based on the iπat iπa/2 theorem proved in [5], the Fourier transform F[ f ] for ()∆ξ, e = e γ ξ()πa , f ∈ ᏸ' and, hence, for ∆ξ ∈ ᏸ' can be represented as ∞ k ()ξ1 Ð ξ (26) ikx γ ξ()πa = ∏ cos ------πa . []∆ () ∆, η()e 2 F n k = n x ------π , (30) n = 0 2 η ∈ ∆ From expressions (26) and (13) for the functionals, where D and equals unity in the vicinity of supp n. we arrive at the expression that will be used later: Using the algorithm for calculating functional (11), ∆ we easily find the Fourier transform for n: it ω ω 1 α i /2 n ----- γ ()ω n lim ∑ e = e ξ . (27) 1 1 ikxα ikξ /2 sin()kξ /2 n → ∞ n []∆ 2 αΩ∈ F n = -----n ------∑ e e ------n -. (31) n 2π ξ 2 αΩ∈ k /2 Thus, invariant (20) provides a simple way of calcu- n Thus, the Fourier transform of the Cantor distribu- lating the functionals ∆ξ. tion has the form []∆ []∆ 1 1 ikxα FOURIER TRANSFORM FOR CANTOR F ξ ==lim F n lim ------∑ e , (32) n → ∞ n → ∞ n 2π 2 αΩ∈ DISTRIBUTION n Let ϕ be a function locally integrable on Rn. The R where xα ∈ E . Fourier transform of this function is given by n With regard for previous result (27), F[∆ξ] can be []ξϕ () 1 Ði()ξ, x rearranged to F = ------n ∫ e dx. (28) ()2π ik/2γ () Rn e ξ k F[]∆ξ ()k = ------. (33) ξ ξ ξ 2π Let x = (x1, …, xn) and = ( 1, …, n) be pseudo- Euclidean coordinates in the general case. The scalar Let us define the inverse Fourier transform FÐ1 in ᏸ' product (ξ, x) of these vectors is introduced, as usual, as by means of the pseudo-Euclidean metrics. Integral (28) Ð1[] []()()π n is assumed to be convergent in principal value. F f = FfÐ x 2 , (34) To construct the Fourier transform of the distribu- where f(Ðx) is the image of f(x). The operation FÐ1 is tions, we introduce the space of rapidly decreasing linear and continuous from ᏸ' to ᏸ' and also is inverse

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE CANTOR DISTRIBUTION AND FRACTAL TRANSITION SCATTERING 153

γξ(x) γξ(x) (a) (b) 0.9 0.9 0.6 0.6 0.3 0.3 0 0 –0.3 –0.3 –90 –60 –30 0 30 60 90 x 0 200400 600 800 1000 x (c) (d) 0.9 0.9 0.6 0.6 0.3 0.3 0 0 –0.3 –0.3 –0.6 –0.6 –0.9 –0.9 100 300 500 700 x 0 2000 4000 6000 8000 10 000 x

Fig. 3. Plots of the function γξ(x) for ξ = (a, b) 1/2 and (c, d) 1/10. x varies from (a) Ð90 to +90, (b) 0 to 1000, (c) 100 to 800, and (d) 0 to 10 000. relative to F; that is, The electrostatic ﬁeld of a charge embedded in a medium with a permittivity ε is characterized by the Ð1[][] []Ð1[] ∈ ᏸ 0 F Ff ==f , FF f f , f '. (35) induction D: Thus, the Cantor distribution can be expressed qr through the inverse Fourier transform: Dr()== εE ()r , E ()r ------. (39) 0 0 0 ε 3 0r Ð1 ik/2 F []e γ ξ()k = 2π∆ξ. (36) When a permittivity wave appears, D(r) D(r, ε |ε | Ӷ |ε | On the other hand, it is easy to check that t) = (r, t)E(r, t). In the ﬁrst approximation 1 0 , the variable polarization appears around the charge [7]: 1 1 ∆ () ------γ () ε F ξx + --- k = ξ k . (37) δD 1E0 2 2π δP ==------cos()k r Ð ω t . (40) 4π 4π 0 0 Figure 3 plots the function γξ(x) for various ξ. This polarization generates a wave with a frequency ω 0 radiating from the charge. FRACTAL TRANSITION SCATTERING We will study the transition scattering due to a non- It is known that permittivity waves scattered by a periodic dissipative perturbation of the permittivity: charge generate electromagnetic (transition) radiation. ε()r, t = ε+ ε γ ξ()k r Ð ω t , (41) This process is called a transition process for two rea- 0 1 0 0 sons. First, the effect can also be observed if the charge where k0r is the scalar product. is immobile; second, radiation of this type takes place Dielectric burst (41) (the perturbation of the density over the entire space, not only in a region localized at of the medium) can be caused, for example, by shock the boundary. This causes the electromagnetic waves to waves in this medium or by concentration nonuniformi- interfere [6, 7]. The transition radiation that occurs ties in a plasma. We simulate the transition scattering ε when the permittivity varies by a periodic law has using the function γξ(x), whose Fourier image is related been well studied. The permittivity can depend only on to the Cantor set. The behavior of this function for var- the density N of the medium. If a longitudinal acoustic ious ξ is shown in Fig. 3. ∆ wave propagates in the medium, N = N0 + Ncos(k0r Ð ω ω The Fourier transform D(k, ) for the induction 0t). Thus, D(r, t) has the form of (28): εεε ()ω = 0 + 1 cos k0r Ð 0t , (38) 1 3 Ði()kr Ð ωt Dk(), ω = ------d rdte ε()r, t Er(), t . (42) ε ε ε ∆ ()π 4∫ where 1 is a change in due to a change in N ( 1 ~ N). 2

TECHNICAL PHYSICS Vol. 47 No. 2 2002 154 BOLOTOV Substituting ε(r, t) from (41) into (42) and using From (49) and (50), one can calculate the field of the (36) yields transverse waves scattered: 3 Dk()ε, ω = Ek(), ω τ iqω []kk, [], k 0 E (k, ω ) = Ð------0 1/2 πω()kc 2()k2 Ð ω2ε /c2 ()k Ð ωk /ω 2 (43) 0 0 0 0 πε v ()∆v , ω v ω v 1 +2 1 ∫ d EkÐ k0 Ð 0 ξ+ --- . ε ω (51) 2 × 1∆ 1 Ð1/2 ε---- ξ-----ω- + --- . 0 0 2 Consider the transition scattering if the charge is stationary. The Fourier transforms of the induction Expression (51) is identical to those for the Fourier D(k, ω) and the polarization vector δP(k, ω) are related components of the field that were obtained in [6] for by the well-known formula permittivity waves like (38). The basic difference is that formula (51) derived in our work involves the Cantor ()ε, ω (), ω πδ (), ω ∆ δ Dk = 0Ek + 4 Pk . (44) distribution ξ rather than the function. This means that, in our case, the electromagnetic radiation has Hence, some frequency range specified by the Cantor distribution. ε 1/2 1 1 It has been shown [8] that the transverse wave inten- δPk(, ω ) = ---- dx∆ξ x + --- Ek().Ð xk , ω Ð xω (45) 2 ∫ 2 0 0 sity is given by Ð1/2 4 k 6ω τ 2 W ω, --- dωdo = ()2π ------ε δP ()k, ω dωdo. If the charge is stationary, the equation for the field k 3 0 is written as follows: c (52) Here, δPτ(k, ω) is the transverse polarization, which is 1 ∂2 curl curlE +0.------D = (46) readily found from (49): 2 ∂ 2 c t τ []kk, [],,δPk(), ω δP ()k, ω =,------(53) From these equations, it is easy to derive the expres- k2 sion for the Fourier images in the form of (43): ε ω where k = k/k 0 /c. ω2 πω2 2δ ε δ (), ω 2 ε Thus, expressions (49), (51), and (52) imply that the k ij Ð kik j Ð ------0 ij E j k = ------1 c2 c2 electromagnetic spectrum of the fractal transition scat- ω 1/2 (47) tering is concentrated in the frequency range = ω ω ω 1 0( α Ð 1/2). The numbers α belong to points of the × dv E ()∆k Ð v k , ω Ð v ω ξ v + --- . ∫ i 0 0 2 Cantor set and are defined by expression (2). Ð1/2 As usual, the summation is over the repeating indi- CONCLUSION |ε | Ӷ |ε | ces. It is assumed that 1 0 . Then, the right-hand side of (47) can be considered as a perturbation due to This work elaborates on the theory of fractal transi- a change in the polarization near the charge. In the first tion scattering [9]. The results obtained in this article order of perturbation theory, the Fourier components of are more general and can be applied to developing the the stationary charge field are given by theory of fractal electrodynamics. Our considerations rely upon the Cantor distribution based on Cantor mea- 4πiqk sures. The algorithm for calculating the functionals Ek, ω = Ð------. (48) with the Cantor distributions was described, and the ()π 3ε 2 2 0k Fourier transform of the Cantor distribution was derived. Thus, the perturbation of the polarization near the charge can be obtained by substituting (48) into (45): By way of example, we considered the transition scattering of a dissipative burst of permittivity (41) ε ()ω ω incident on a charge. Being scattered by the charge, the iq 1 k Ð k0/ 0 ω 1 δPk(), ω = ------∆ξ ------+ --- . (49) space-time permittivity function of this type forms the π2ε ω ()ω ω 2 ω 2 4 0 0 k Ð k0/ 0 0 wide fractal radiation spectrum (the radiation frequencies are localized at points of the Cantor set). A dielec- If the waves scattered are assumed to be transverse tric burst of a complex shape may be generated by (relative to the vector k), expression (47) takes the form shock waves in an insulator, as well as by turbulent ω2 πω2 [], [], δ , (), ω flows in plasmalike media and Kerr liquids. 2 ε τ(), ω 4 kk Pk k Ð -----2- 0 E k = ------2 ------2 -. (50) Fractal transition scattering can serve as a source of c c k wide-band electromagnetic radiation, on the one hand,

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE CANTOR DISTRIBUTION AND FRACTAL TRANSITION SCATTERING 155 or be used for the diagnostics of chaotic dynamics in 5. V. S. Vladimirov, Generalized Functions in Mathemati- various media, on the other. cal Physics (Nauka, Moscow, 1979). 6. V. L. Ginzburg and V. N. Tsytovich, Zh. Éksp. Teor. Fiz. 65, 1818 (1973) [Sov. Phys. JETP 38, 909 (1973)]. REFERENCES 7. V. L. Ginzburg and V. N. Tsytovich, Transient Radiation 1. P. S. Alexandroff, Introduction to the Theory of Sets and and Transient Scattering (Nauka, Moscow, 1984). General Topology (Nauka, Moscow, 1997). 8. V. V. Kolesov, Zh. Éksp. Teor. Fiz. 106, 77 (1994) [JETP 2. E. Olivier, Nonlinearity 12, 1571 (1999). 79, 39 (1994)]. 3. A. N. Kolmogorov and S. V. Fomin, Elements of the The- 9. V. N. Bolotov, Zh. Tekh. Fiz. 70 (12), 98 (2000) [Tech. ory of Functions and Functional Analysis (Nauka, Mos- Phys. 45, 1604 (2000)]. cow, 1968; Graylock Press, Rochester, 1957). 4. G. Lallement, Semigroups and Combinatorial Applica- tions (Wiley, New York, 1983; Mir, Moscow, 1985). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

THEORETICAL AND MATHEMATICAL PHYSICS

A Neuron as a Quantum-Optical Device: A Model S. L. Grigor’ev Received April 4, 2001

Abstract—Inversion in the structure of polyaminoacids and proteins is discussed, and the electrodynamic quantum-optical model of a neuron is constructed. The feasibility of the quantum-logical summation of exciting pulses by a nanomembrane is considered. The model is consistent with the physiology foundations and assumes implementation by means of techniques used in microelectronics, integrated optics, and nanooptics. © 2002 MAIK “Nauka/Interperiodica”.

The physical properties of neurons depend on those from the relationship T = 2π I/D , where T is the pen- of proteins and amino acids that enter into the compo- dulum period, I is the moment of inertia of the pendu- sition of the membrane. These building blocks, in turn, lum, and D is the torsional stiffness of the bond. The may contain polar molecular groups (peptide, etc.) [1] moment of inertia is I = mr2, where m = 3 × 10Ð26 kg is and fit into the class of polar dielectrics, whose reso- Ð10 nance frequencies lie typically in the microwave and the mass of an oxygen atom and r = 10 m is the far-IR ranges [2, 3]. At microwaves, the classical length of a C1ÐO bond. The torsional stiffness depends approach [4, 5] to analyzing the atomic oscillation on the C1ÐC2 valence bond stiffness, OÐH hydrogen dynamics in these materials applies up to the resonance bond stiffness, and dipoleÐdipole Coulomb interaction frequencies of polar molecular groups (50Ð200 cmÐ1). [9]. Experimental data for molecular motion in crystals The peptide groups incorporated into protein chains of the simplest amino acid, glycine, is available from [4]. are shown in Fig. 1. Their dipole moment is about 0.3D Consider the behavior of a rotatable pendular dipole and arises because of the shift of the electron density subjected to an electromagnetic field (Fig. 5). Obvi- from a nitrogen atom to an oxygen one. The peptide ously, its response to the field will differ from that of a groups can readily rotate about the C ÐC and C ÐN 1 2 2 Hertzian classical dipole with longitudinally bound bonds. charges [10]. The maximum of the radiation scattered Proteins and amino acids have the property of self- (reflected) by a pendular dipole is observed when the assembling into secondary structures like crystals or incident wave is directed along the dipole vector and even higher order structures [6]. Among them, the the electric field vector of the incident wave is orthogo- “folded leaf” structure is one of the simplest (Fig. 2). The secondary structure consists of the primary poly- nal to the dipole vector. If the dipole moment of the mer molecules with hydrogen bonds. Such structures peptide molecule is reversed with the external field readily polarize and have a high permittivity in a wide direction remaining the same, the total reflection of the frequency range [1]. wave is changed to its total transmission (Fig. 6). Thus, The polarization mechanism is well described by the conformation of the structure causes the effect of the Pauling model [7] (Fig. 3). By analogy with this “optical switching” (but in the microwave range). Since model, the folded leaf structure can have two, base and the microwave component in the earth’s electromag- inverse, states in which the bonds lie in the plane of the leaf (Fig. 2a) and between the leaves (Fig. 2b). This is a specific case of conformation typical of many high- HHH O – molecular materials [8]. + Let us consider this structure from the viewpoint of N electrodynamics and microwave spectroscopy. In the ë2 ë secondary structure of the base type (Fig. 2a), the ele- ë1 N + ë mentary cell has four dipoles forming an electroneutral R system (Fig. 2, a1). The dipoles are arranged at differ- – O H H ent angles to each other, so that they variously respond to an external field. In addition, in the substances considered, the rocking (pendular) vibration of the peptide R = H, CH3, CH2–OH, ... group prevails (Fig. 4). The frequency of the vibration is the frequency of a torsion pendulum, which is found Fig. 1. HÐNÐCÐO peptide groups in the protein chain.

A NEURON AS A QUANTUM-OPTICAL DEVICE: A MODEL 157

--- (a) (a) H C N C= O --- C C 7 Å Oxygen H C= --- N O Hydrogen H C= NH N N O--- H C C= 3 Å N O--- C= C= (b) O--- O--- C H N E = 0 E Resultant field --- (b) a1 O = C N C H --- Fig. 3. Pauling model for polar molecules of water and ice:

O= --- C (a) disordered and (b) ordered structures. O C = C N NH C H N N θ

H --- C= O O --- = CH --- N HC E O C C = C

O N --- C H N H C E b1 CH

Fig. 2. (a) Base and (b) inverse secondary structures of Fig. 4. Rocking vibration (θ) of the peptide group. E is the polyalanine. a1 and b1, dipole configurations. electric vector of the dipole. netic field is absent, this switching effect seems to be (a) used in the biological memory of animals [3]. Z E0 Now let us turn to an elementary cell of four dipoles θ P0 and take advantage of electromechanical analogy to construct its model. We represent a peptide group as a + torsional pendulum having two charged poles that are D Pmax attracted to or repel from the poles of other pendulums _ according to the relation between the signs of the Y charges (Fig. 7a). For such a four-pendulum system, the Emax stable equilibrium state is lacking if the initial dipole X S directions are shifted by a quarter-period (π/2). As is known from quantum electrodynamics, this system (b) may exist only in motion [11]. The Coulomb forces provide the potential energy, while the kinetic energy is θ provided by the rotating masses. If the axes of the pendulums are parallel to each other, the system loses the anisotropy of interaction with the external electromag- D + Pmax netic field: its electric component accelerates two pen- _ dulums and slows down the two other. The anisotropy arises when the axes of one pair of pendulums are S rotated relative to the other, as in the case of the folded leaf structure (Fig. 7b). Here, two of four Coulomb bonds are replaced by torsional bonds for the analogy with the polyalanine structure to be complete. Fig. 5. Scattering of the incident wave P0, E0 by (a) pen- Let us trace the interaction of the dipoles with a dular and (b) Hertzian dipole (D). Pmax, Emax are the direction and the electric field vector of the wave scattered and weak external electric field (wave) for a short time S is the cross section of the toroid where the radiation scat- interval (much shorter than the vibration period). In this tered is concentrated.

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158 GRIGOR’EV

E0 E0 dipole is not absorbed; instead, it gains the energy from Pm = 0 Em the material. The material, a polar dielectric, has the Em = 0 P0 Pm P0 directional internal field and surface charge. Thus, the incident wave can be enhanced by induced radiation due to the conformation (inversion) of the structure. O C CO Consequently, a nanofilm consisting of several folded HH polypeptide leaves can act as a maser (the microwave E1 = 0 D N N D and IR analog of a laser) [12]. Similar properties may P1 = 1 E1 = E0 be found in proteins and polyaminoacids with the heli- P = P cal structure. This simple qualitative model can be 1 0 extended up to the mathematical one. Fig. 6. Reflection and transmission of the incident wave in Next, we will treat an invertible ferroelectric mem- the different states of the structure. 0, incident wave; brane in terms of the quantum-mechanical theory of m, reflected wave; and 1, transmitted wave. Other designations are the same as in Fig. 5. coherent radiation. The time of observation is assumed to be shorter than the time of excitation relaxation into thermal vibrations of the molecules. Then, the state of case, the following simple rule holds: if the direction of a charged particle on the membrane can be represented the wave (P) and the instantaneous direction of rotation as the superposition of the wave functions of its two of a pendular dipole (ω) coincide, the field transfers the possible states [13]: energy to the dipole (absorption takes place); if not, the ψ ψ ()() ψ ()() dipole gives the energy to the field (enhances it). = 1()r exp Ði/h E1t + 2()r exp Ði/h E2t , (1) Applying this rule and summing over all dipoles, we find that the maximal absorption is observed when the where r is the position of the particle, t is time, and E1 directions of the external field and one of the dipoles and E2 are the energies of the states. coincide; these maxima arise in 90° degree intervals. The polarization of the membrane by a pulse causes Such a phenomenon has been discovered in thin protein an electric field of potential energy U to appear. This films in the IR range [1]. field affects the energies of the states: Now consider the inverse structure, where some 0 0 dipoles are rotated through 90° and fixed by hydrogen E1 ==E1 Ð U, E2 E2 + U. (2) bonds between the leaves (Fig. 2b). The electromechanical model for this case is depicted in Fig. 8. Here, The polarized membrane may spontaneously pass a weak external wave directed along the vector of one into the initial (base) state by radiation. The probability

(a) ωω E0 – Y ϕ = π/2 + ϕ = π ω ω F T Es + + – X ω F ω F – + ϕ = 0; 2π – ϕ = 3/4π Z – + F F + – + X + – T (b) ω – Y Z + + ω ω F –– T T F – + ω – X X F + + – – Z + Z + F – – + F

Fig. 7. Electromechanical configuration of the vibratory system: (a) isotropic and (b) anisotropic. F, Coulomb Fig. 8. Electromechanical model of the inverse structure. ω forces; T, torsional bonds; and , instantaneous direction of E0, external field; Es, self-field of the system charge. Other rotation. designations are the same as in Fig. 7.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 A NEURON AS A QUANTUM-OPTICAL DEVICE: A MODEL 159

X1 CONCLUSIONS X1X2 Xn (1) The nervous system of higher animals and humans is a holographic neural network (see [16]) in 8 which signals are processed not only by the conven- 7 6 X2 tional electrochemical technique but also by the quan- Y(X) tum-optical way. Xn (2) A neuron as a quantum-optical device is akin to a parallel-pumped laser in which the ferroelectric mem- 1 5 brane is a lasing medium. 2 (3) The quantum-chemical properties of ferroelectric nanotubes and nanomembranes can lay the ground- work for creating artiﬁcial quantum-optical devices for energy and data conversion. 3 4 REFERENCES Y(X) 1. M. V. Vol’kenshteœn, Biophysics (Nauka, Moscow, 1981).

Fig. 9. Pyramidal neuron and its configuration (on the 2. J. S. Blakemore, Solid State Physics (Cambridge Univ. right): 1, neuron enclosure; 2, axon membrane; 3, axon Press, Cambridge, 1985; Mir, Moscow, 1988). (peripheral nerve fiber); 4, myelin enclosure (Schwann’s 3. O. V. Betskiœ and N. D. Devyatkov, Radiotekhnika, cell); 5, synapses (ends of other nerve fibers); 6, dendrite of No. 9, 4 (1996). neuron (fiber bandle); 7, neuron nucleus; and 8, cytoplasm and organelles. The nucleus is shown in the cross-sectional 4. K. N. Trueblood, in Accurate Molecular Structures, Ed. view. by A. Domenicano and I. Hargittai (Oxford Univ. Press, Oxford, 1992; Mir, Moscow, 1997). density of this radiation is 5. B. P. van Eijck, in Accurate Molecular Structures, Ed. by A. Domenicano and I. Hargittai (Oxford Univ. Press, S = 1/2m()ψ* pψψÐ pψ* , (3) Oxford, 1992; Mir, Moscow, 1997). where p is the operator of the particle momentum, 6. V. M. Stepanov, Molecular Biology (Vysshaya Shkola, ψ* is the complex conjugate wave function of the par- Moscow, 1996). ticle, and m is the mass of the particle. 7. J. M. Ziman, Models of Disorder: the Theoretical Phys- Substituting (1) and (2) into (3), leaving only the ics of Homogeneously Disordered Systems (Cambridge time-dependent terms, and separating the real part, we Univ. Press, Cambridge, 1979; Mir, Moscow, 1982). arrive at the expression for the probability density: 8. S. M. Shevchenko, Molecule in Space: Problems of ∼ []()()( () Modern Chemistry (Khimiya, Leningrad, 1986). SUEcos + 2 Ð E1 Ð cos UEÐ 2 Ð E1 . (4) 9. V. F. Antonov et al., Biophysics: Textbook (Arktos-Vika- It follows that there exists some polarization poten- Press, Moscow, 1996). tial of a ferroelectric membrane at which the probabil- 10. Ya. P. Terletskiœ and Yu. P. Rybakov, Electrodynamics: ity of coherent radiation is maximal. Textbook (Vysshaya Shkola, Moscow, 1990, 2nd ed.). Importantly, our model implies that a ferroelectric 11. E. Wichmann, Quantum Physics (McGraw-Hill, New membrane can serve as a logic adder of quantized York, 1967; Nauka, Moscow, 1986). polarizing pulses. Each pulse increases or decreases the membrane potential according to the polarization vec- 12. S. G. Ryabov, G. N. Toropkin, and I. F. Usol’tsev, tor. Once the radiation potential has been attained, the Devices of Quantum Electronics, Ed. by M. F. Stel’makh membrane emits a coherent pulse. (Radio i Svyaz’, Moscow, 1985, 2nd ed.). It is obvious that a pyramidal neuron [14] is the real 13. A. I. Akhiezer, Atomic Physics: Reference Book (Naukova device implementing our model (Fig. 9). Here, the axon Dumka, Kiev, 1988). membrane, containing high-molecular polar sub- 14. Human Physiology, Ed. by R. F. Schmidt and G. Thews stances, serves as a ferroelectric membrane. These sub- (Springer-Verlag, Berlin, 1989; Mir, Moscow, 1996), stances are all amino acids acting as neuromediators Vol. 1. [14]. Here, the membrane is pumped by pulses coming 15. S. L. Grigor’ev, Zh. Tekh. Fiz. 71 (11), 110 (2001) [Tech. from many synapses that are the ends of the nerve ﬁbers Phys. 46, 1457 (2001)]. of dendrites of the given and other neurons. Nerve 16. Problems of Optical Holography, Ed. by Yu. N. Deni- ﬁbers are waveguides, as was shown in the previous syuk (Nauka, Leningrad, 1981). article of the author [15]. In them, the receptors of sense organs and other neurons of various types serve as sources of quantized pulses (solitons) [14, 15]. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

THEORETICAL AND MATHEMATICAL PHYSICS

The Chaotic Layer of a Nonlinear Resonance under Low-Frequency Perturbation V. V. Vecheslavov Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia Received May 10, 2001

Abstract—Conditions whereby the chaotic layer of a nonlinear resonance is described in terms of low-frequency separatrix mapping are discussed. In this case, the accurate estimation of the size of the layer requires the arrangement of resonances at its edge to be known. The resonance picture is constructed using the separatrix mapping invariants of the ﬁrst three orders. The variation of the layer size with the mapping amplitude is traced with the criterion for resonance overlapping. Results obtained by direct calculation and by invariants analysis are compared. Issues that remain to be solved are noted. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION [7], which makes it possible to approximately describe the system dynamics near the separatrix at instants of The occurrence and the characterization of a chaotic stable equilibrium. Note that the second way is much layer appearing near the nonlinear resonance separatrix more feasible and is used much more frequently. in the presence of other resonances is a central and still poorly understood problem in modern nonlinear To date, the correct procedures for separatrix map- dynamics [1Ð3]. In many cases, the Hamiltonian of the ping and determining the sizes of the fundamental cha- problem can be represented as a pendulum (fundamen- otic layer have been worked out only for a symmetric ε ε ε Ω ω tal resonance) and two perturbing resonances: high-frequency perturbation 1 = 2 = , 1/ = −Ω ω λ ӷ 2/ = 1. Chirikov [1], using his resonance over- 2 p 2 lap criterion and the properties of standard mapping, Hypt(),, = ----- + ω cos()y + Vyt(), , 2 (1) has shown that separatrix mapping in the symmetric case has the form (), ε ()εΩ ()Ω Vyt = 1 cos y Ð 1t + 2 cos y Ð 2t . 32 ε ε Ӷ wwWx==+ sin(), x x + λln------mod 2π, (2) A perturbation is assumed to be weak ( 1, 2 1), w |Ω | |Ω | ω and the frequencies 1 , 2 > are not necessarily high. They may have both the same and opposite signs. where the amplitude is given by It is also assumed that the resonances of system (1) do πελ2 Ðπλ/2 not overlap. W = 8 e (3) Let the chaotic layer of the fundamental resonance and the three parts of the layer are of the same size: of system (1) be called the fundamental chaotic layer. It δ can be subdivided into three parts: upper (the phase y ≈ λ 4 ≤≤δ wu ==wm wl W1 + ------,0 1. (4) rotates at the top, p > 0), middle (the phase oscillates), λ and lower (the phase rotates at the bottom, p < 0). It is δ λ important that, in the general case of an asymmetric The correction 4 / appearing in (4) is a very complicated function of W and λ (see below), and its value perturbation, the sizes wu, wm, and wl of these parts may 2 ω2 depends on the resonance overlap at the edge of the be much different (hereafter, w = p /2 + cosy Ð 1 is layer. If λ ӷ 1, it is neglected; if, however, a perturba- the dimensionless energy deviation from the unper- tion is low-frequency or accurate estimates are needed, turbed separatrix) [4Ð6]. it should be taken into account and the resonance The width of any of the parts can be determined by arrangement becomes much more difﬁcult to visualize two methods. The former is ﬁnding the minimal period and analyze. It should be noted that the case when the Ω Ω of motion Tmin (T is the time interval between succes- frequencies 1 and 2 of system (1) are low is seldom sive instants the orbit crosses the stable phase y = π) and encountered and has not attracted much attention until the calculation of the maximal deviation from the unper- now. However, the situation noticeably changed after it ≈ ω turbed separatrix by the formula wmax 32exp(Ð Tmin) had been discovered that so-called secondary harmon- [1]. The second one is the so-called separatrix mapping ics have an anomalously strong effect on the formation

THE CHAOTIC LAYER OF A NONLINEAR RESONANCE 161 of the fundamental chaotic layer. Relevant data will be called resonant invariants are known. The construction discussed in the next section. of the invariants for system (2) will be discussed below.

SECONDARY HARMONICS RESONANCE INVARIANTS IN SEPARATRIX MAPPING Separatrix mapping (2), as well as the well-known Chirikov standard mapping, falls into the class of The early study of the general asymmetric perturba- explicit rotation mapping of type [2] tion of system (1) [4, 5] has shown that, along with the primary harmonics of frequencies Ω and Ω (entering dQ()Θ 1 2 JJ==Ð ξ------, Θ Θ + 2πν()J , (7) into Hamiltonian (1) in the explicit form), the separatrix dΘ mapping spectrum contains also secondary harmonics Θ ν with frequencies Ω = Ω + Ω and Ω = Ω Ð Ω . The where J and are action and angular variables, (J) is + 1 2 Ð 2 1 the frequency of unperturbed motion, and Q(Θ + 2π) = still more intriguing thing is that even very weak low- Θ frequency harmonics, which exponentially depend on Q( ) is a trigonometric polynomial in the general case. the frequency (3), may, under certain conditions, Difference equations (7) are known to be totally almost completely define the width of any of the parts equivalent to Hamilton continuous equations with a of the chaotic layer (for details, see [4Ð6]). Because of kick-like perturbation: this, the separatrix mapping of this part can be repre- dQ()Θ sented in one-frequency form (2) but the parameter λ Θú ==ν()J , Jú Ðξδ ()t ------, (8) can no longer be considered to be large. It appears as if * dΘ a one low-frequency harmonic, instead of two perturb- δ []() π ing high-frequency ones, exists (this assumption has where ∗ = 12+ ∑n ≥ 1 cos nt /2 is the periodic been confirmed numerically in [5]). Accordingly, it is delta function [1, 2]. necessary to construct and analyze the low-frequency separatrix mapping in order to determine the chaotic To separatrix mapping (2), there correspond the layer size. relationships λ 32 The first difficulty here is that the amplitudes of the Q()Θ ==cos() Θ, ν()J ------ln------, (9) secondary harmonics are unknown. A rigorous theory 2π J is still lacking, but, within the approximate approach whereas for the standard mapping, we have suggested in [4], the separatrix mapping amplitudes W at the sum and difference of the primary harmonic fre- Q()Θ ==cos() Θ, ν()J J. (10) quencies are given by the analytic expressions Recall that the arrangements of integer resonances πλ ν +/2 ( (Jn) = n, where n is an integer) on the phase plane 4π 1 1 e 2 2 W = ------a ε ε ----- + ------λ ()λ Ð 2 , (5) (J, Θ) for mappings (9) and (10) diverge. For the stan- + 3 + 1 2 λ2 λ2 sinh()πλ + + δ 1 2 + dard mapping, the resonance spacing Jn = Jn + 1 Ð Jn is Jn-independent (hence, standard mapping is also π ε ε referred to as a homogeneous model), while for the sep- aÐ 1 2 1 1 W = Ð------Ð,----- (6) Ð ()πλ 2 2 aratrix mapping, it depends on Jn. cosh Ð/2 λ λ 1 2 If λ Ω ω λ Ω ω λ Ω Ω ω λ where 1 = 1/ , 2 = 2/ , + = ( 1 + 2)/ , and Ð = δJ Ӷ J , λ ӷ 1, (11) Ω Ω ω n n ( 2 Ð 1)/ ; a+ and aÐ are adjusting coefficients. the separatrix mapping locally (near the resonance Jn) It has been shown [5] that formulas (5) and (6) fit transforms into the standard one (this fact was taken well the numerical data if we put a+ = 0.473 and aÐ = into account in deriving formulas (4) in [1]). 1.35. Yet, the nature of these corrections remains In [8], the function of dynamic variables and time of unclear. The parameter ranges where the secondary type harmonic completely specifies the size wu of the upper part of the chaotic layer were also reported [5]. Note m ξn that the outer parts of the layer are of the greatest prac- S ()Jt()Θ,,,()t t ξ = r ()J + ∑ -----G ()J,,Θ t , (12) m 0 n! n tical interest, since they define the resonance overlap n = 1 and contribute to the formation of global chaos. was called the invariant of order m. For this function, As was noted above, the accurate estimation of the the condition layer size requires the knowledge of the resonance D m + 1 arrangement at its edges. The arrangement can be found ------S ()t = ᏻ()ξ (13) without determining the trajectories of motion if so- Dt m

TECHNICAL PHYSICS Vol. 47 No. 2 2002 162 VECHESLAVOV is met along the true trajectories of system (8). Here, formulas in the initial variables w and x for this case are D/Dt is the generalized derivative that is the sum of the given in the Appendix. ordinary derivative and that due to component disconti- It is the author’s opinion that he was the first to intro- nuities [9]. duce the invariants S1 and S2 for the separatrix mapping It is general practice to express the periodic orbit of the first two orders [6]. In this work, we will also dis- and the associated resonance through the ratio N : M, cuss S3 and provide full information on all the three where M is the number of iterations of mapping onto N invariants. orbit periods [2]. In [8], the invariants of the first three orders for the standard mapping were constructed. They describe the integer, M = 1, and fractional, M = 2 DIRECT CALCULATION AND ANALYSIS and 3, resonances. It was also shown in that work that OF INVARIANTS the terms involved in (12) retain their values in the The multiple iteration of mapping (2) allows one to intervals between kicks and undergo jumps only at the find the sizes of all the parts of the fundamental chaotic instants of kicks. Consequently, the ordinary derivative layer and to trace their dependence on the amplitude W of Sm(t) in (12) is everywhere zero and condition (13) and frequency λ. However, it would be appropriate to should be realized as the restriction imposed on the dis- considerably simplify this process and, moreover, make continuity of the first kind in the step function. correct predictions. Formulas (1A)Ð(3A) for the separa- The potentials of systems (9) and (10) are the same, trix mapping invariants (see the Appendix) combined Q(Θ) = cos(Θ), so that the expressions obtained in [8] with the resonance overlap criterion seem to furnish can be rewritten in the form suitable for both mappings: such an opportunity. Below is an example of their use. () In [5], system (1) was investigated for the following r0' J ω ≤ ε ε ≤ ≤ G ()J,,Θ t = ------cosΩ, (14) parameter values: = 1.0, 0.01 1 = 2 0.5, 8.0 1 ()πν() Ω ≤ Ω 2sin J 1 18.0, and 2 = Ð10.0. The direct calculation and the spectral analysis of the separatrix mapping indi- ' () (),,Θ a1 J Ω cated that the secondary harmonic is the most intense at G2 J t = ------()πν()cos2 , (15) Ω Ω Ω ≈ 42sin J the sum of the frequencies + = 1 + 2 3.0. In this case, its contribution to the separatrix mapping of the r'''()J + 3a' ()J upper part of the layer is several hundred times larger G ()J,,Θ t = ------0 2 cos3Ω 3 16sin() 3πν()J than the net contribution from the primary frequencies. (16) For this reason, we will subsequently be interested in r'''()J + a' ()J λ δ λ Ð3------0 2 -cosΩ. mapping (2) with = 3.0. Clearly, the correction 4 / 16sin()πν()J in formula (4) cannot be neglected. The object of investigation will be the upper part of the layer, which is wu In these formulas, the primed quantities are deriva- thick. Based on formula (4), we introduce the quantity tives with respect to J and the following dependences () are used: () wu W Du W = ------λ , (18) Ω Θπ()ν() () ' ()() πν() W = +Ð t* J , a1 J = r0 J cot J , (17) which is the normalized size of this part [1]. In the high- () () ()πν() λ ∞ a2 J = a1' J cot 2 J , frequency limit ( ), Du(W) does not depend on W and tends to unity. where 0 < t∗ ≤ 2π is the local time between kicks. Figure 1 shows the dependence Du(W) obtained by Expressions (14)Ð(16) for the invariants of the first numerically iterating mapping (2) (the number of itera- three orders have small denominators. Following the tions for each of the points is 108). The dependence has DunnetÐLaingÐTaylor (DLT) method [10], they can be many discontinuities. Some of them are clearly seen suppressed by appropriately selecting the zero-order and others can be visualized on an enlarged scale. invariant r0(J) in formula (12) [more strictly, its first Within the modern dynamic theory, such a pattern is derivative r' (J)]. It turned out that the selection of explained by the fact that, as W grows, the invariant 0 curves with irrational numbers of rotation successively ' r0 (J) is not universal and depends on the number m. break down and are replaced by so-called Cantori [11]. The DLT invariant of order m adequately reflects the If an invariant curve is the boundary between the fun- topology of only “its own” resonances with M = m, damental chaotic layer and the N : M resonance nearest which appear for the first time just in this order and are to it, these objects merge together when the curve responsible for “their own” part of the phase space (for breaks down and the size of the layer increases by a details, see [8]). We have shown that the specific depen- finite quantity, i.e., by the phase volume of the resonance attached. The exact value of the mapping ampli- ' dences r0 (J) suggested in [8] for the standard mapping tude W at which the layer and the resonance merge is can also be used for separatrix mapping (2). Accurate called the threshold value and is designated as W(N : M).

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE CHAOTIC LAYER OF A NONLINEAR RESONANCE 163

≈ × Ð4 Du 6 : 1) and W 1.45 10 (resonance 5 : 1). Let us con- 3.0 sider the former in greater detail. On the left of Fig. 2, the fundamental chaotic layer is depicted immediately after the integer resonance 6 : 1 has been attached to it. 2.5 The islands of the 19 : 3, 13 : 2, and 7 : 1 stable resonances are seen in the chaotic sea. On the right of Fig. 2, the levels of the invariants for these four reso- 2.0 nances constructed by formulas (A1)Ð(A3) of the Appendix are demonstrated (the resonance 20 : 3 is omitted, since the lines of its invariant cross those of the 1.5 invariants 13 : 2 and 7 : 1 and the picture becomes very tangled).

1.0 From the picture of the invariants, one can make 024 6 8 10 12 14 16 18 20 some estimates. For example, the normalized size of W × 105 the upper part of the fundamental layer estimated from the maximal ordinate of the boundary invariant 6 : 1 is ≈ Fig. 1. Normalized width (18) of the upper part of the fun- Du 3.06 with the exact value being 3.10. The line of damental chaotic layer of system (1) as a function of the the invariant 19 : 3 allows one to estimate the size of the amplitude W of separatrix mapping (2) for λ = 3.0. layer prior to overlapping: 1.34 versus the exact value 1.53. The latter estimate is much cruder, which is explained as follows. Invariants, like any other analyti- It was noted [1] that the size increases to the greatest cal description, are basically unable to reﬂect the cha- extent when the fundamental layer overlaps with an otic component of motion (in particular, to take into integer resonance of the separatrix mapping. Figure 1 account the widths of the chaotic layers of resonances). shows two such cases near W ≈ 1.81 × 10Ð5 (resonance We performed our measurements inside, rather than at

w/W w/W 10 10

0 0 0 6.28 0 6.28 X X

Fig. 2. (Left) the upper part of the fundamental chaotic layer immediately after its overlapping with the integer resonance 6 : 1; (right) resonance invariants (from top to bottom) 6 : 1, 19 : 3, 13 : 2, and 7 : 1. W = 1.83 × 10Ð5, λ = 3.0.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 164 VECHESLAVOV

w/W w/W 6 6

0 0 0 6.28 0 6.28 X X

Fig. 3. (Left) three chaotic trajectories: the lower one is the fundamental chaotic layer; the other two are the chaotic layers of the resonances 17 : 3 and 11 : 2 of the separatrix mapping. The situation immediately before the fundamental layer overlaps with the resonance 17 : 3. (Right) the invariants of the resonances 6 : 1, 17 : 3, and 11 : 2 (from bottom to top). W = 6.7 × 10Ð5, λ = 3.0. the edge, of the fundamental layer, where the effect of values being in reasonable agreement with the data in this component is signiﬁcant, and obtained the inaccu- Fig. 1. Analyzing the arrangement of the invariants near rate value. Figure 2 shows two adjacent resonances (the W = 6.7 × 10Ð5 and applying the resonance overlap cri- lower 7 : 1 and the upper 6 : 1), so that we can say that terion, we can expect overlapping in the interval 6.0 × it depicts the instant of their overlapping. These reso- 10Ð5 < W < 7.5 × 10Ð5. The accurate value turns out to nances are seen to greatly differ, which is a direct con- be W(17 : 3) = 6.954 × 10Ð5. It is also seen that this value sequence of the separatrix mapping nonuniformity is far from that at which the overlapping with the reso- mentioned above (recall that for the standard mapping, nance 11 : 2 will occur (W(11:2) = 9.009 × 10Ð5, Fig. 1). all integer resonances are identical). Our nearest goal is to trace the overlapping evolution for the integer reso- Figure 4 illustrates the situation before the funda- nances 6 : 1 and 5 : 1 by comparing the direct calcula- mental layer merges with the next integer resonance 5 : 1. tion and the analysis of the invariant arrangement. The invariant picture depicts both integer resonances (5 : 1 at the top and 6 : 1 at the bottom). It follows then Far above the upper boundary in Fig. 2, the reso- that Fig. 4 describes the last stage in their overlapping. nances 17 : 3, 11 : 2, 16 : 3, and 5 : 1 of the separatrix Here, we also can indicate the interval (1.35 × 10Ð4 < mapping are arranged (from bottom to top). As W W < 1.5 × 10Ð4) in which the threshold value W(5 : 1) = grows, their phase regions will “sink” in the fundamen- 1.453 × 10Ð4 was found. tal chaotic layer and the attachment of the integer resonance 5 : 1 to this layer will be accompanied by a large The above analysis was restricted by the only fre- discontinuity (the right of Fig. 1). quency value λ = 3.0, and only one stage of overlapping the integer resonances of separatrix mapping (2) was Figure 3 shows the situation with W = 6.7 × 10Ð5 considered. It seems to be of interest to consider the immediately before the fundamental layer merges with transition from low frequencies to the asymptotic limit the resonance 17 : 3. Using the invariants, one can, as λ ∞. It would also be useful to gain some experi- before, estimate the size of the layer prior to overlap- ence in applying the effective resonance overlap crite- ping and the amount of the discontinuity expected, both rion to the problem discussed and to devise a procedure

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE CHAOTIC LAYER OF A NONLINEAR RESONANCE 165

w/W w/W 10 10

0 0 0 6.28 0 6.28 X X

Fig. 4. (Left) two chaotic trajectories: the lower one is the fundamental chaotic layer; the other is the chaotic layer of the integer resonance 5 : 1 of the separatrix mapping. The situation immediately before the layers overlap. (Right) the invariants of the resonances 5 : 1, 16 : 3, 11 : 2, and 6 : 1 (from top to bottom). W = 1.44 × 10Ð4, λ = 3.0.

for more accurately determining the threshold values of the Appendix, Sm(w, x) is written in the starting vari- the mapping parameter W(N : M). ables w and x. However, one can construct the generating functions Ᏺ(I, x) of the canonic transformation into ψ CONCLUSION other variables I and and find the invariant Sm(I) as a function of the new action I. First, this simplifies the From the previous section, we can conclude that study of the invariant itself, since the level line admits DLT invariants reflect the topology of “their” reso- the explicit representation in the form I(w, x) = const. nances fairly adequately and can be used for reliably Second, this allows one to construct the Hamiltonians estimating the sizes of the fundamental chaotic layer. for specific resonances and find, for example, the fre- Using the well-known Chirikov resonance overlap cri- quencies of motion. The associated procedure for the terion and knowing the arrangement of the invariants, standard mapping was described in [8]. one can find the interval of the parameter W where the chaotic layer may merge with the nearest resonance Unfortunately, the DLT theory of invariants has N : M of the separatrix mapping. To find the exact value never come to the attention of mathematicians. Because of the threshold W(N : M) corresponding to this event, of this, the construction of invariants is to a great extent it will suffice to perform numerical analysis only within a matter of guessing and intuition (the singularity-elim- ω this interval. inating dependences r0' ( ) in the Appendix have been Note that the construction of the complete invariant found just in this way). This circumstance makes the picture on the phase plane may be unnecessary. For advance to higher m very difficult. Another obstacle is rough estimates, it would suffice to find the length and the fact that the analytic expressions become much the arrangement of segments belonging to different res- more awkward as m grows. It has been reported [2, 8] onances on the dominant line x = 0, which is the locus that the DLT method requires the elimination of all the of the centers of all resonances. poles of all resonances M = m so that the invariant Sm It should be emphasized that the concept of invari- becomes an analytic function throughout the phase ants may lay the foundations of an advanced theory. In space. It seems plausible that the abandonment of the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 166 VECHESLAVOV global analyticity requirement might facilitate the solu- Below we give the complete expressions for the first tion to the problem [8]. Note that the construction of the three invariants of separatrix mapping (2): first invariant for the case when the singularities are λ λ eliminated only from some of neighboring resonances S ()zw(), x = eÐ2z/ sinz + --- cosz (of the entire countable set of integer resonances) is 1 2 exemplified in [12]. (A1) W λ2 It was the aim of this work to draw attention to the + ------1 + ----- cos()xzÐ , DLT method for constructing resonance invariants, 64 4 which, in our opinion, deserves further development ()(), Ð2z/λ and refinement. S2 zw x = 16e sin2z + λcos2z sin4z + 2λcos4z × ------Ð ------ACKNOWLEDGMENTS 1 + λ2 21()+ 4λ2 The author thanks B.V. Chirikov and É.A. Biberdorf (A2) W for the valuable discussions and advices. + ----- sinzsin 2zxzcos()Ð 2 This work was partially supported by the Russian 2 Foundation for Basic Research (grant no. 01-02-16836). 2z/λW Ð λe ------cos2zcos2()xzÐ , 16

APPENDIX λ S ()zw(), x = 4eÐ2z/ f ()z First Three Invariants of Mapping (2) 3 0 W 2 2z/λ In constructing the invariants of separatrix mapping + ----- sinzsin 2zsin3zxzcos()Ð Ð 3λe (2), it is convenient to return from the variables J, Θ to 2 the early variables w, x. Therefore, primed quantities in W 2 the Appendix refer to derivatives with respect to w. It × ------sinzsin2z()23++cos 2z 3cos 4z (A3) was found that the direct application of formulas (14)Ð 32 (17) to this mapping is difficult because of numerical λ instability involved in the construction of the contour λe2z/ 2 curves for the invariants. The situation is improved × cos2()xzÐ + ------6512 when the new variable × 3[]() () () () λ 32 W f 1 z cos3 xzÐ Ð 3 f 2 z cos xzÐ . zw()= --- ln------2 w In the last expression, we used the designations is introduced instead of w. () () () () () () f 0 z = gz ++3g 3z Ð2g 5z Ð g 7z g 9z , The resonance picture on the phase plane (w, x) is 2sinnz + nλcosnz correct if the instant of observation t∗ appearing in (17) gnz()= ------, is set equal to 2π. 1 + ()nλ/2 2 ' () λ() The specific dependences r0 (w) suggested in [8] f 1 z = 27+++ 35cos 2z 49cos 4z 81cos 6z with the aim of suppressing small denominators in the +14sin 2z ++ 14sin 4z 18sin 6z, case of the standard mapping proved to be applicable () λ() also for separatrix mapping (2). In terms of our desig- f 2 z = 23+++ 51cos 2z 69cos 4z 49cos 6z nations, we have +10sin 2z ++ 22sin 4z 14sin 6z. r' = sinz, 0 It is worth noting that if conditions (11) are met, the for the first-order invariant, invariants of separatrix mapping pass to those of standard mapping. 2 22sinz Ð sin 4z r' ==sin zsin2z ------0 4 REFERENCES for the second-order invariant, and 1. B. V. Chirikov, Phys. Rep. 52, 263 (1979). r' = sin2 zsin2 3zsin3z 2. A. Lichtenberg and M. Lieberman, Regular and Chaotic 0 Dynamics (Springer-Verlag, New York, 1992). 1 = ------()sinz ++ 3sin 3z Ð27sin 5z Ð9sin z sin z 3. R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Non- 16 linear Physics: from the Pendulum to Turbulence and Chaos (Harwood, Chur, 1988); G. M. Zaslavsky and for the third-order invariant. R. Z. Sagdeev, Introduction to Nonlinear Physics: from

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE CHAOTIC LAYER OF A NONLINEAR RESONANCE 167

the Pendulum to Turbulence and Chaos (Nauka, Mos- 9. I. M. Gel’fand and G. E. Shilov, Generalized Functions cow, 1988). (Fizmatgiz, Moscow, 1958; Academic, New York, 1964). 4. V. V. Vecheslavov, Zh. Éksp. Teor. Fiz. 109, 2208 (1996) 10. D. A. Dunnet, E. W. Laing, and J. B. Taylor, J. Math. [JETP 82, 1190 (1996)]. Phys. 9, 1819 (1968). 5. V. V. Vecheslavov, Pis’ma Zh. Éksp. Teor. Fiz. 63, 989 (1996) [JETP Lett. 63, 1047 (1996)]. 11. R. S. MacKay, J. D. Meiss, and I. C. Percival, Physica D (Amsterdam) 13, 55 (1984). 6. V. V. Vecheslavov, Physica D (Amsterdam) 131, 55 (1999). 12. V. V. Vecheslavov, Zh. Tekh. Fiz. 57, 896 (1987) [Sov. 7. G. M. Zaslavskiœ and N. N. Filonenko, Zh. Éksp. Teor. Phys. Tech. Phys. 32, 544 (1987)]. Fiz. 54, 1590 (1968) [Sov. Phys. JETP 27, 851 (1968)]. 8. V. V. Vecheslavov, Zh. Tekh. Fiz. 58, 20 (1988) [Sov. Phys. Tech. Phys. 33, 11 (1988)]. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 168Ð176. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 28Ð35. Original Russian Text Copyright © 2002 by Kazinets, Matisov, Mazets.

THEORETICAL AND MATHEMATICAL PHYSICS

Structural Factor for BoseÐEinstein Double-Component Condensate I. V. Kazinets*, B. G. Matisov*, and I. E. Mazets** * St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 194251 Russia e-mail: [email protected] ** Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia Received June 27, 2001

Abstract—The structural factors for the single- and double-component BoseÐEinstein condensates of neutral atoms in a spherically symmetric harmonic trap are derived. The approach adopted in this paper is applicable in the low excitation energy limit, where the well-known Stringari dispersion relation is valid. Our expression is compared with the quasi-classical one, which is shown to overestimate the structural factor. It is argued that the second component ﬁrst splits the frequency of collective condensate oscillations, which corresponds to the energy transferred by scattering, into two branches and then multiplies the single-component structural factor by some coefﬁcient that can be greater than unity. The analysis is used to estimate the properties of an experimentally implemented double-component condensate of 87Rb atoms. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION response of the system is sensitive to the discrete modes of its collective oscillations. For small transferred The dynamic structural factor is an important momenta, the dynamic structural factor of a single- parameter that specifies the behavior of many-particle component BEC has been theoretically analyzed in [8], systems. In particular, it provides information on the collective excitation spectrum upon small transferred where the formalism of quantizing the hydrodynamic momentum scattering and characterizes the momentum equations that describe collective oscillations in a BEC distribution in the case that a relatively large momen- and the ThomasÐFermi approximation were used. tum is transferred; i.e., that the system response is gov- In this work, we will derive the dynamic structural erned by single-particle effects. For rarefied gases, the factors for single- and double-component BECs in the structural factor can be measured experimentally, for ThomasÐFermi approximation. Our formalism is sim- example, by inelastic scattering of light by a boson gas ple and more illustrative than that used in [8]. Unlike confined by a magnetic trap [1]. [8], where only the general approach is stated, we As a many-particle system, we consider the BoseÐ derive formulas for the structural factors in a closed Einstein condensate (BEC) of a rarefied gas in a spher- analytical form. The expression for the single-compo- ically symmetric harmonic trap. Particle interaction is nent condensate is compared with the quasi-classical assumed to be weak. In this case, the well-known one obtained in [7]. Within the domain of its applicabil- Bogoliubov theory for a weakly interacting degenerate ity, the quasi-classical expression is an overestimation. boson gas [2] and its modifications taking into account However, both our expression and the quasi-classical the external trapping potential [1, 3] apply. one [7] equal zero at the same values of the transferred There are several works dealing with the calculation momentum បq. Therefore, within the domain of its of the structural factor for the BEC of a slightly imper- applicability, the latter expression can be used for cal- fect gas. Both spatially homogeneous [5, 6] and spa- culating those values of the transferred momentum បq tially inhomogeneous gases [7Ð9] have been consid- making the structural factor vanish. ered. The calculations for the former case are rather straightforward, while for a BEC in a harmonic trap the For a double-component condensate, the q depen- problem cannot be solved exactly and requires approx- dence of the partial structural factor (photon scattering imations to be used, such as the quasi-classical [7] or by atoms of one component) does not change, but the the local-density or momentum approximations [9]. energy transferred by scattering splits into two The above approximations apply if the momentum branches, for both of them the structural factor being transferred is relatively large. For example, in [9], multiplied by either the coefficient β+ or βÐ. It is note- q ӷ 1/R, where R is the BEC size and បq is the momen- worthy that β may exceed unity for a certain combina- tum transferred by scattering. For small momenta, the tion of the scattering amplitudes of the components,

STRUCTURAL FACTOR FOR BOSEÐEINSTEIN DOUBLE-COMPONENT CONDENSATE 169 suggesting that the presence of the second component The Bogoliubov coefﬁcients uf and vf are normal- may enhance the photon scattering by the other. ized as

dr[]u ()r u*()r Ð v ()r v*()r = δ . (4) SINGLE-COMPONENT CONDENSATE ∫ f f ' f f ' ff' As is known, the differential scattering cross section This condition yields the standard boson commuta- 2σ Ω ω d N/(d d ) for a particle being scattered by a sample tion relations for the quasi-particle operators. Substitut- of N atoms is expressed through the cross section ing expansion (2) into (1), we have σ Ω d 1/d of one-atom scattering as [7, 10] ˜ () ()[]ψ() () ()2 S f q = N0 ∫drexp iqr u*f r Ð v*f r 0 r .(5) d2σ dσ ------N = S()q, ω ------1, dΩdω dΩ Thus, to calculate the dynamic structural factor, one v ប must know analytical expressions for uf and f, which where q is the momentum imparted to the sample by are usually found by solving a set of differential equa- a particle scattered into a solid angle dΩ; បω is the ω tions. This set can be solved numerically [12]. How- associated transferred energy; and S(q, ) is the ever, the analytical solutions require additional approx- dynamic structural factor of the sample, which is given imations to be used. In particular, the ThomasÐFermi by [10] approximation [13, 14] is applicable if the number N0 ӷ (), ω ˜ ()δωω() of condensate atoms is large; that is, N0 aHO/a, where S q = ∑S f q Ð f , f (1) ប ()ω aHO = / M 0 (6) 2 S˜ ()q = 〈〉f | drψˆ †()r exp()ψiqr ˆ ()|r 0 . f ∫ is the length of a spherically symmetric harmonic trap ω | 〉 〈 | បω with a frequency 0, M is the atomic mass, and a is the Here, 0 is the ground state of the BEC; f and f are the excited state and the excitation energy of the BEC, amplitude of s-scattering of ultracold atoms by each other in the Born approximation. In this approximation, ψˆ respectively; and (r) is the field operator introduced the kinetic energy is assumed to be much smaller than in the secondary quantizing formalism for a many-par- the energy of particle interaction and the energy of ticle system (BEC). (The designation (qr) means the interaction with the external potential. scalar product.) By a particle being scattered, we mean a photon; however, it may also be another object (for At temperatures much lower than that of BoseÐEin- example, a neutron) in the general case. For simplicity, stein condensation, the mean value of the boson field we adopt the formalism T = 0, since modern experi- operator 〈ψˆ (r)〉 = Ψ(r, l)exp[iφ(r, l)] satisfies the mental equipment makes it possible to produce an GrossÐPitaevski equation [15]: ultracold atomic ensemble with a negligibly small ∂ above-condensate fraction [11]. iប-----Ψexp()iφ According to the Bogoliubov approximation, we ∂t (7) represent the field operator of an atom as the sum of the ប2∇2 µ () Ψ2 Ψ ()φ c-numerical part (corresponding to the BEC) and a = Ð ------Ð ++V ext r g exp i , small operator part corresponding to collective oscilla- 2M tions of the BEC. The latter, in turn, can be represented where g = 4πប2a/M and µ is the chemical potential of α† through the operators of production, ˆ f , and annihila- the BEC ground state and ∇ is the Hamiltonian opera- α tion, ˆ f , of quasi-particles responsible for the distur- tor. បω bance of a condensate with an energy f (see, e.g., Now we turn to the formalism [13], within which [4]): low-energy excitations are described in terms of the ψˆ ()r collisionless hydrodynamic approach that operates on the density perturbation δn(r, t) and hydrodynamic (2) velocity v(r, t). They are defined as ()ψφ () ()()α ()α† = exp i 0 0 r N0 +,∑ u f r ˆ f Ð v*f r ˆ f N ψ2()r + δn()r, t = Ψ2()r, t , f 0 0 (8) φ ψ Mvr(), t = ប∇φ()r, t . where 0 is the constant phase of the condensate; 0 is the real function satisfying the stationary GrossÐPitae- Considering δn and v as small parameters, we vski equation and normalized to unity, obtain, from (7), the linearized equation ψ2() ∫dr 0 r = 1, (3) ∂2 gN ------δn()r, t = ------0∇ψ[]2()r ⋅ ∇δn()r, t . (9) ∂ 2 M 0 and N0 is the number of atoms in the condensate. t

TECHNICAL PHYSICS Vol. 47 No. 2 2002 170 KAZINETS et al. When deriving (9), we assumed that the term ble confined by a trap and having a temperature much ω 2 less than that of BoseÐEinstein condensation: nl = ប 2 ------∇ Ψ()r, t ω K . Eventually, for an elementary perturbation of 2MΨ()r, t 0 nl a condensate with a frequency ω , we have of the kinetic energy is small compared with the energy nl of particle interaction gΨ2(r, t) and the external poten- n l + 2k ()nl r tial V (r). This assumption is consistent with the Tho- δn ()r = ∑ C --- Y ()Θφ, , (15) ext nlm k R lm masÐFermi approximation and is valid at large N0. k = 0 ψ The function 0(r) is determined from (7) and, in where the ThomasÐFermi approximation for a spherically k Ð 1 2 2 symmetric trap (V (r) = Mrω /2), has the form ()nl k ()nkÐ ' ()2n ++32l +2k' ext 0 C = C ()Ð1 ∏ ------k 0 ()k'1+ ()32++l 2k' ω2 2 k'0= 2 M 2 1 (16) ψ ()r = ------0 R 1 Ð --- (10) 0  k n! ()12+++n 2l 2k !! ()12+ l !! gN0 2 R = C ()Ð1 ------, 0 k!()nkÐ ! ()12++n 2l !! ()12++l 2k !! ≤ ψ2 µ ()ω2 if r R and 0 (r) = 0 if r > R. Here, R = 2 / M 0 R is defined by (11), and C is a normalizing constant µ 0 is the BEC radius defined by the condition = Vext(R). that is set equal to unity. Expression (15) applies to From normalization (3), we find lower order modes with energies much lower than the BEC chemical potential. a 1/5 Ra= HO15N0------, (11) To relate the Bogoliubov coefficients uf(r) and vf(r) aHO δ with nf(r), we assume that the ensemble of ultracold where aHO is given by (6). atoms is so excited that only the fth mode, which is the Let us expand the real function δn(r, t) in elemen- coherent state, is present; that is, only 〈〉αˆ = ω f tary perturbations with a frequency f: λ − ω 〈〉α† λ ω f exp( i f t) and ˆ f = *f exp(i f t) are other than δ δ ω δ ω n(r, t) = ∑[ nf (r)exp(Ði f t) + n*f (r)exp(i f t)]. zero. Thus, we have f 〈〉Ψψˆ ()r = ()r, l exp[]iφ()r, l Then, (9) in view of (10) yields ()ψφ [ () λ () ()ω = exp i 0 0 r N0 + f u f r exp Ði f t (17) ω 2 2 f δ () ∇ r ∇ δ () Ð2------n f r = r 1 Ð --- r n f r .(12) Ð λ*v*()r exp()]iω t . ω --- R --- f f f 0 R R On the other hand, using (8), we find For the trap considered, the appropriate quantum numbers for the elementary excitation of the fth mode Ψ()r, t exp[]iφ()r, t are radial quantum number n, as well as orbital moment ψ2() δ(), {}[]φ δφ(), l and its projection m onto the z axis: = N0 0 r + n r t exp i 0 + r t δn ()r == δn ()r F ()r/R Y ()Θφ, , δ (), f nlm nl lm Ӎ ψ n r t δφ()…, ()φ N0 01 +++------i r t exp i 0 n ψ2() (13) 2N0 0 r () ()nl l + 2k (18) Fnl r = ∑ Gk r , ()φ ψ () k = 0 = exp i 0 N0 0 r --- Θ φ where Ylm( , ) is the eigenfunction of the moment operator. δn ()r + ------f ++iNψ ()δφr ()r …exp()Ðiω t Substituting this expansion into (12), we come to ψ () 0 0 f the recurrence relation 2 N0 0 r δ () 2 n*f r () () K Ð ()ω /ω ψ ()δφ () … ()ω nl nl kl nlm 0 + ------++iN0 0 r *f r expi f t . Ck + 1 = Ck ------, ψ () ()k + 1 ()2l ++2k 3 (14) 2 N0 0 r 2 δ K = 2k +++2kl 3kl. Here, we used the Taylor expansion where n(r, t) and kl δφ(r, t) are assumed to be small in the first order. Dots δ | The boundedness condition for nf r = R requires that cover the terms of the second and higher orders of () the series be truncated at some finite n: C nl = 0. As a smallness, which will be rejected. Also, we used the n + 1 representation φ(r, t) = φ + δφ(r, t) = φ + result, we will come to the expression for the frequen- 0 0 δφ − ω δφ ω cies of collective oscillations in a large atomic ensem- f (r)exp( i f t) + *f (r)exp(i f t).

TECHNICAL PHYSICS Vol. 47 No. 2 2002 STRUCTURAL FACTOR FOR BOSEÐEINSTEIN DOUBLE-COMPONENT CONDENSATE 171

Using the real part of (7) in the form ~ S 1/4 2 2,2 ∂ ប 2 40 Ðប-----φ = ------()∇φ + gδn ∂t 2M and neglecting the terms with (∇φ)2 as the terms of the 30 second order of smallness, we arrive at δφ () g δ () 20 f r = ------បω- n f r . (19) i f Comparing expressions (17) and (18), we have 10 δ ()ψ () n f r 1 gN r u ()r = ------------+ ------0 0 , f λ ψ () បω 10 20 30 f 2 N0 0 r f qR (20) δ ()ψ () n f r 1 gN0 0 r Fig. 1. Structural factor as a function of transferred momen- v ()r = ------Ð ------+ ------, tum (continuous curve, formula (22); dashed curve, quasi- f λ ψ () បω f 2 N0 0 r f classical approximation [7]). The structural factor is plotted δ ≡ δ in the power 1/4 for better visualization. where nf (r) nnlm(r) is given by (15). Taking into account that the operator on the right- hand side of Eq. (12) is Hermitean and that its eigenval- The summation is over all nonnegative integers n ues (Ð2K ) depend both on n and l, we can write and l except for the pair (n, l) = (0, 0), which corre- nl sponds to the ground (unexcited) state of the conden- δ δ ≠ ≠ ()nl dr nnl(r) nn*'l' (r) = 0 if n n' or l l'. ∫ sate. The quantities Knl, Fnl(r ), and Ck are defined by Thus, from (4), we find for the normalizing constant (13), (14), and (16); r = r/R. |λ |2 បω f = 2gI/( f), Figure 1 shows the dependence of the structural fac- ˜ where tor Snl, (q) calculated by (22) and by the quasi-classical R formula [7] on the dimensionless parameter qR, where R is the condensate size defined by (11), for N = IF= []()r/R 2r2dr 0 ∫ nl a 107 atoms, ------HO- Ӎ 432, and n = l = 2. It is seen that the 0 (21) a n n () () C nl C nl ˜ 2g 3 k j quasi-classical expression estimates Snl, (q) to 2Ð200 = ប------ω-R ∑ ∑ ------(). f 2 kjl++ + 3 times higher than expression (22) for qR between 10 k = 0 j = 0 and 30. However, in both approximations, the function Using the well-known expansion of a plane wave ˜ [16, 17], Snl, (q) vanishes at the same values of q (this is also true ≡ ϑ for other parameters). Thus, in the domain of its appli- exp(iqr) exp(iqrexp ) = cability, the quasi-classical expression can be used for ∞ ˜ l π()ϑ ϕ finding the zeros of the function Snl, (q). ∑ i 4 2l + 1 Yl0( , )jl(qr), The evidence for the validity of expression (22) is l = 0 the fact that it obeys the rule of sum for the structural π () where jl(r) = /2r J 1 (r) is the spherical Bessel factor [10] with a great accuracy. According to this rule, l + --- 2 () ω()ω, ω function, we find the final expression for the structural m1 q = ∫ S q d factor of the single-component BEC in the ThomasÐ (23) Fermi approximation: ω ˜ () ប 2 () = ∑ nl, Snl, q = N0 q /2M . 3/5 TF a ˜ () 2l + 1 HOa nl, Snl, q = ------15N0------Knl 2 a aHO Figure 2 plots the common logarithm of 1 2 2nl+ = P ˜ TF 2 ω , S , ()q j ()qRr F ()r r dr (22) δ () nl nl ∫ l nl P q = ∑ ------() Ð 1 (24) m1 q × 0 nl, = ()00, ------n n . ()nl ()nl () ˜ TF ∑ ∑ Ck C j /2k +++2 j 2l 3 at P = 14, 18, and 22; the function Snl, (q) is calculated δ k = 0 j = 0 by (22). The dependence P(q) is the successive

TECHNICAL PHYSICS Vol. 47 No. 2 2002 172 KAZINETS et al.

δ log( P) orbital quantum numbers n and l is similar to that for a 0 single-component condensate:

± ± –3 ω = ω 2n2 +++.2nl 3nl (25) P = 14 nl 0 ± –6 However, the factor ω , in this case, takes two val- P = 18 0 ues corresponding to two branches of the velocity of –9 sound (two branches of the dispersion relation). This Ω P = 22 factor depends on the parameters a1, a2, a12, 1, and –12 Ω 2. Here, a1 and a2 are the s-scattering amplitudes for atoms of the ﬁrst and second components (atoms I and –15 II), respectively; a12 is the amplitude of scattering of an 06123915 Ω Ω atom I by an atom II; and 1 and 2 are the trap fre- qR quencies for the components I and II, respectively. δ Fig. 2. Semilogarithmic plot of the p(q) vs. qR dependence. Thus, a double-component condensate has two The higher qR, the greater the number of the series terms types of excitation, and, instead of (2), we can write necessary for estimating the series limit.

ψˆ = exp()ψiφ N ∑ approximation to the relative deviation of the quantity 1 01 01 1 TF P → ∞ ω ˜ , ប 2 δ (26a) ∑nl, nl, Snl(q) from m1(q) = N0 q /(2M): P(q) ∞ TF ()α α† α α† ω ˜ , + ∑ u11 ˆ 1 Ð v 11* ˆ 1 + u12 ˆ 2 Ð v 12* ˆ 2 , ∑()00, nl, Snl(q)/([m1(q)] Ð 1). Clearly, the larger q, f f f f f f f f the larger P (a larger number of the series terms) must f be taken to estimate the limit of the series. One may δ P → ∞ Ð14 conclude from Fig. 2 that P(q) 10 . Note that ψˆ = exp()ψiφ N ∑ the quasi-classical expression [7] does not obey the rule 2 02 02 2 of sum (there is no quadratic dependence on q). This (26b) suggests that the quasi-classical expression considerably overestimates the BEC structural factor, which + ∑()u αˆ Ð v* αˆ † + u αˆ Ð v* αˆ † . 21 f 1 f 21 f 1 f 22 f 2 f 22 f 2 f seems to be natural in view of difficulties associated with f the quasi-classical estimation of matrix elements [16]. ψ ψ Here, ˆ 1 (r) and ˆ 2 (r) are the field operators for the components I and II, respectively; αˆ † and αˆ are, DOUBLE-COMPONENT CONDENSATE 1 f 1 f The simultaneous condensation of two different respectively, the production and annihilation operators បω+ α† α atomic systems has been first demonstrated in [18]. The for quasi-particles with an energy f ; ˆ 2 and ˆ 2 87 f f two atomic systems were Rb atoms occupying the are the production and annihilation operators for quasi- | 〉 | 〉 87 hyperfine sublevel F, mF = 1, Ð1 and Rb atoms on បωÐ φ φ the |2, 2〉 sublevel. The systems formed slightly over- particles with an energy f ; 01 and 02 are the phase ψ lapping clouds with the separated centers of gravity. constants for the components I and II, respectively; 01 ψ Subsequently [19], a double-component BEC with the and 02 are real functions that satisfy the set of station- coinciding centers of gravity of the components has ary coupled GrossÐPitaevski equations and are normal- been implemented. In this case, the |1, Ð1〉 and |2, 1〉 ized to unity: hyperfine sublevels of 87Rb, which have the same mag- ψ2 () ψ2 () netic dipole moment, was used. We will consider the ∫dr 01 r ==∫dr 02 r 1; (27) situation where the centers of the condensates coincide and the trapping potential for either component has the and N1 and N2 are the number of atoms in the compo- Ω2 2 nents I and II of the condensate. form Vi = Mi i r /2, i = 1, 2. The ground state of a dou- The Bogoliubov coefficients u , v , u , v ble-component BEC has been theoretically considered 11 f 11 f 12 f 12 f in [20]; and collective excitations, in [21Ð23]. For a and u , v , u , v are normalized as purely double-phase condensate (when the clouds of 21 f 21 f 22 f 22 f both components completely overlap) in a spherically dr[u u* Ð v v* symmetric trap, it has been shown [21] that, in terms of ∫ 11 f 11 f ' 11 f 11 f ' the ThomasÐFermi approximation, the dependence of (28a) + u u* Ð v v* ] = δ , the collective excitation frequency on the radial and 12 f 12 f ' 12 f 12 f ' ff'

TECHNICAL PHYSICS Vol. 47 No. 2 2002 STRUCTURAL FACTOR FOR BOSEÐEINSTEIN DOUBLE-COMPONENT CONDENSATE 173

[ mation (neglecting the kinetic energy), we come to the ∫dr u21 u21* Ð v 21 v 21* f f ' f f ' (28b) linearized equations ] δ 2 + u22 u22* Ð v 22 v 22* = ff'. ∂ f f ' f f ' δ (), ------n1 r t ∂t2 These conditions yield the standard boson commuta- (31a) tion relations for the operators of quasi-particles of the N1 ∇ψ{}2 ()∇[]δ (), δ (), first and second type (the operators for quasi-particles = ------01 r g1 n1 r t + g12 n2 r t , M1 of the first type commute with those for quasi-particles ∂2 of the second type). δ (), ------n2 r t ∂t2 We will also assume that a photon interacting with a (31b) BEC resonates with and is scattered by atoms I and N2 ∇ψ{}2 ()∇[]δ (), δ (), ψ ψ = ------02 r g2 n2 r t + g12 n1 r t . does not interact with atoms II. Then, ˆ (r) = ˆ 1 (r) M2 should be substituted in expression (1) for the structural factor. In view of expansion (26), we have for the partial The ground state can be defined by substituting Ψ ψ φ φ structural factor: i(r, t) = Ni 0i(r) and i(r, t) = 0i(r) (i = 1, 2) into set (30) and neglecting the kinetic energy term. Eventu- (), ω { () ally, we find S1 q = N1∑ ∫drexp iqr 2 f ψ () 01 r

2 + µ µ ()Ω2 Ω2 2 (32a) × []ψu* ()r Ð v* ()r ()r δω()Ð ω g2 1 Ð g12 2 Ð g2M1 1/2 Ð g12M2 2/2 r 11 f 11 f 01 f = ------, (29) ()2 2 g1g2 Ð g12 N1 + durexp()iqr []ψ* ()r Ð v* ()r ()r ∫ 12 f 12 f 01 ψ2 () 02 r ×δ()} ω ωÐ 2 2 2 Ð f . g µ Ð g µ Ð ()g M Ω /2 Ð g M Ω /2 r (32b) = ------1 2 12 1 1 2 2 12 1 1 -. ()2 A set of GrossÐPitaevski coupled equations that g1g2 Ð g12 N2 describe the evolution of the mean ﬁeld operators The radii Ri of the condensates are deﬁned by the 〈ψˆ (r)〉 = Ψ (r, t)exp[iφ (r, t)] and 〈ψˆ (r)〉 = Ψ (r, ψ 1 1 1 2 2 condition 0i = 0 (i = 1, 2). Normalization (27) φ rR= i t)exp[i 2(r, t)] for the components of a double-compo- yields nent BEC has the form [21] ()2 1/5 15N1 a1a2 Ð a12 /aHO 2 2 1 ∂ ប ∇ R1 = aHO ------, (33a) ប-----Ψ ()φ µ () 1ΩÐ2 Ð3/2 i ∂ 1 exp i 1 = Ð ------Ð 1 + V 1 r a2 Ð a12 12 M12 t 2M1 (30a) 2 1/5 2 2 15N ()a a Ð a /a Ψ Ψ Ψ ()φ 2 1 2 12 HO2 ---+ g1 1 + g12 2 1 exp i 1 , R = a ------, (33b) 2 HO2 Ω2 3/2 a1 Ð a12 12M12 2 2 Ω Ω Ω ∂ ប ∇ where 12 = 1/ 2, M12 = M1/M2, and a = ប Ψ ()φ µ () HOi i ∂----- 2 exp i 2 = Ð ------Ð 2 + V 2 r t 2M2 ()Ω (30b) h/ Mi i (i = 1, 2). 2 2 Equating R and R from (33), we obtain the compo- ---+ g Ψ + g Ψ Ψ exp()iφ . 1 2 2 2 12 2 2 2 nent particle number ratio that provides a purely binary condensate: πប2 Here, gi = 4 ai/Mi (i = 1, 2) and g12 = Ω2 3/2 N2 ΩÐ2 Ð2 a1 Ð a12 12M12 πប2 ------= 12 M12 ------. (34) 4 a12/M1M2 are the interaction parameters, Vi is N ΩÐ2 Ð3/2 1 a2 Ð a12 12 M12 the trap potential for the ith component (for a spherical Ultimately, for the ground state of a double-compo- Ω2 2 µ trap, Vi = Mi i r /2), i is the chemical potential of the nent BEC in the ThomasÐFermi approximation, we ith component in the ground state of the BEC. have

δ 2 2 15 Ð3 2 Passing to the variables ni(r, t) and vi(r, t) (i = 1, 2) ψ ()r == ψ ()r ------R []1 Ð ()r/R , (35) deﬁned by (8) and using the ThomasÐFermi approxi- 01 02 8π 1 1

TECHNICAL PHYSICS Vol. 47 No. 2 2002 174 KAZINETS et al. where R1 is deﬁned by (33a) and the component ratio where satisﬁes (34). 2 2 3/2 Let us expand, as before, the real functions δn (r, t) N M a a Ð a Ω M a i ξ ==------2 1 -----2 ------1 12 12 12 -----2. (39) (i = 1, 2) in elementary perturbations with a frequen- 2 a Ω2 Ð3/2 a ω N1M2 1 a2 12 Ð a12M12 1 cy f: Substituting (38) into (37a) yields δn (r, t) = ∑[δn (r)exp(Ðiω t) i i f f f 1 --- 2 2 + δn* (r)exp(iω t)]. ± g1 2 a12 i f f Θ ξ ± ()ξ ξ nl = ------Ð11 Ð + 4 ------. (40) 2g12a1a2 Then, set (31) takes the form

Ðω2 δn ()r To find a relation of the Bogoliubov coefficients f 1 f δ (36a) u1i (r), v 1i (r), u2i (r), and v 2i (r) with n1 (r) and 2 f f f f f = X ∇ {}[]∇1 Ð r []g δn ()r + g δn ()r , ± 1 r r 1 1 f 12 2 f δn (r) (i = 1 and 2 refers to “+” and “–”, respectively), 2 f ω2 δ () Ð f n2 r we assume that the ensemble of ultracold atoms is so f (36b) excited that there exists only the fth mode with a fre- 2 = X ∇ {}[]∇1 Ð r []g δn ()r + g δn ()r , + + + † 2 r r 2 2 f 12 1 f quency ω (i.e., only 〈〉αˆ = λ exp(Ðitω ) and 〈〉αˆ = f 1 f f f 1 f π 5 λ+ ω+ where Xi = (15/4 )Ni/(Ri Mi) (i = 1, 2) and r = r/R1. (f )*exp(itf ) are other than zero) or the fth mode with ωÐ 〈〉α λÐ ωÐ Since functions (15) and (16) represent the complete a frequency f (i.e., only ˆ 2 = f exp(Ðitf ) and set of the eigenfunctions of operator (12), a solution of f 〈〉αˆ † = (λÐ )*exp(itωÐ ) are other than zero). Then, for set (36) for the same quantum numbers n, l, and m is a 2 f f f linear combination of functions (15). Then, up to the j = 1, 2, we have normalizing constant, we can write 〈〉Ψψ () ()φ δn ()r == δn ()r F ()r/R Y ()Θφ, , ˆ j r = j exp i j 1 f 1nlm nl 1 lm ()ψφ [ λ± ()ω± = exp i 0 j 0 j N j + f u j1 exp Ði f t (41) δn ()r == δn ()r ΘF ()r/R Y ()Θφ, , f 2 f 2nlm l nl 1 lm ± ± Ð ()λ *v* exp()]iω t . Θ Θ φ δ f j1 f f where nl is a constant and Fnl(r/R1)Ylm( , ) = nlmn(r) satisfies Eq. (12), where (ω /ω )2 = K [see (14)]; in f 0 nl On the other hand, for the mean field operator of the other words, F (r) is defined by (13) and (16). Then, nl jth component, we can use an expansion like (18). A from (36), we have relationship between the phase perturbation δφ (r) j f []ω2 Θ f Ð X1g1Knl Ð0,X1g12Knl nl = (37a) and the density perturbation δn (r) for the jth compo- j f ÐX g K +0.[]Θω2 Ð X g K = (37b) nent is found from the jth equation of set (30). By anal- 2 12 nl f 2 2 nl nl ogy with (19), we have From the solvability condition for (37), we determine the collective excitation frequencies of a double- ± g g ± δφ ()r = ------1 -δn ()r + ------12 -δn ()r , component BEC in the ThomasÐFermi approximation: 1 f បω± 1 f បω± 2 f i f i f ω2 ()ω± 2 f = nl δφ± () g2 δ ± () g12 δ () 1 2 r = ------±- n2 r + ------±- n1 r . --- f បω f បω f 2 2 i f i f X1g1 ξ ± ()ξ 2 ξ a12 = ------1 + 1 Ð + 4 ------Knl 2 a1a2 (38) Eventually, we come to Ð2 Ð3/2 2a ()a Ð a Ω M δ () Ω 1 2 12 12 12 n1 r = 1------() f ()2 u1i r = ------± 2 a1a2 Ð a12 f λ f 1 (42a) 2 --- + 2 1 N ψ ()r ()g + g Θ 2 a12 × 1 01 1 12 nl × ξ ± ()ξ ξ------+ ------± - , 1 + 1 Ð + 4 Knl, ψ () បω a1a2 2 N1 01 r f

TECHNICAL PHYSICS Vol. 47 No. 2 2002 STRUCTURAL FACTOR FOR BOSEÐEINSTEIN DOUBLE-COMPONENT CONDENSATE 175 δ () β± n1 r v ()r = ------f 1i f λ± 3 f (42b) + 1 N ψ ()r ()g + g Θ × Ð ------+ ------1 01 1 12 nl- , 2 ψ () បω± 2 N1 01 r f

δ () 1 n1 r u ()r = ------f 2i f λ± f (42c) + + Θ N ψ ()Θr ()g + g 4.8 5.0 5.2 a12 × nl 2 02 nl 2 12 ------+ ------± - , 2 N ψ ()r បω 2 02 f Fig. 3. β+ (dashed curve) and βÐ (continuous curve) vs. amplitude a of scattering of atoms I by atoms II. 87Rb δ () 12 n1 r atoms occupying the hyperfine sublevels |1, Ð1〉 and |2, 1〉. v ()r = ------f 2i f λ± f (42d) 2 2 + + a a 2 Θ N ψ ()Θr ()g + g ± 1 Ð2ξ Ð ------12 + 12Ð4ξ ++------12 ξξ × nl 2 02 nl 2 12  Ð ------+ ------± - . a1a2 a1a2 2 N ψ ()r បω × ------, 2 02 f 2 2 a12 ()ξ 2 ξ a12 21Ð ------1 Ð + 4 ------Here, δn (r) ≡ δn (r) are given by (15) and the con- a1a2 a1a2 1 f nlm λ± stants f are determined from normalization conditi- ˜ TF Snl, (q) is defined by (22) if a = a , a = a , N = N , δ 1 HO HO1 0 1 on (28) with regard for the orthogonality of nnlm(r): and ξ is given by (39). ± 2 2g I λ = ------1 - Comparing expression (44) for the partial structural f បω± f factor of a double-component BEC with expressions (1) and (22) for a single-component condensate allows the 2 2 following conclusions. The presence of the second a 2 a12 21Ð ------12 ()1 Ð ξ + 4ξ------(43) component, first, changes the frequencies (i.e., the a a a a × ------1 2 1 2 -, ω+ 2 2 nl ± ξ a12 ξ a12 ξξ2 ω 1 Ð2Ð ------+ 12Ð4++------energy transferred by scattering) so that nl a1a2 a1a2 ωÐ nl where I is given by (21) with R1 substituted for R. and, second, introduces the factors β+ and βÐ for the fre- ω+ ωÐ The final expression for the partial structural factor quencies nl and nl , respectively. of a double-component condensate is Ω Consider a double-component condensate at 1 = TF(), ω Ω S1 q 2 and M1 = M2. Such a situation is experimentally realized for sets of 87Rb atoms occupying the hyperfine (44) | 〉 | 〉 ˜ TF()β[]+δω()β ω+ Ðδω() ωÐ sublevels 1, Ð1 and 2, 1 [19]. According to [23], we = ∑Snl, q Ð nl + Ð nl , put a1 = 5.68 nm and a2 = 5.36 nm. The value of a12 will nl, be varied from 1.8 to 5.36 nm. For a2 < a12 < a1, the where purely binary condensate cannot be produced, as fol- β± lows from (34). Figure 3 plots against a12. It follows 1 ------from this figure that photon scattering may be enhanced 2 10 ± a Ð a /a in the presence of the other component. This is because β = ------2 12 1 - ΩÐ2 Ð3/2 the fluctuation amplitude of the BEC density grows as a2 Ð a12 12 M12 a12 approaches the critical value a1a2 above which 1 --- the components are spatially separated [24]. At anoma- 2 2 2 lously high density fluctuations, an associated ultracold ξ + 1 ()1 Ð ξ a12 × ------± ------+ ξ------(45) quantum system behaves physically similarly to ordi- 2 4 a1a2 nary liquids near the critical point.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 176 KAZINETS et al.

ACKNOWLEDGMENTS 12. K. G. Singh and D. S. Rokshar, Phys. Rev. Lett. 77, 1667 (1996); M. Edwards et al., Phys. Rev. Lett. 77, 1671 This work was supported by the program “Universi- (1996); F. Dalfovo et al., Phys. Rev. A 56, 3840 (1997). ties of Russia” (grant no. 015.01.01.04) and by the Min- 13. S. Stringari, Phys. Rev. Lett. 77, 2360 (1996). istry of Education of the Russian Federation (grant no. E00-3-12). 14. G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996). 15. D. Pines and P. Nozieres, The Theory of Quantum Liquids (Addison-Wesley, Redwood City, 1990), Vol. 2. REFERENCES 16. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic 1. J. Stenger, S. Inouye, A. P. Chikkatur, et al., Phys. Rev. Theory (Nauka, Moscow, 1989, 4th ed.; Pergamon, New Lett. 82, 4569 (1999); D. M. Stamper-Kurn, A. P. Chik- York, 1977, 3rd ed.). katur, A. Görlitz, et al., Phys. Rev. Lett. 83, 2876 (1999). 17. Handbook of Mathematical Functions, Ed. by 2. N. N. Bogoliubov, J. Phys. (Moscow) 11, 23 (1947); Izv. M. Abramowitz and I. A. Stegun (National Bureau of Akad. Nauk SSSR 11, 77 (1947). Standards, Washington, 1964; Nauka, Moscow, 1979). 3. E. P. Gross, Nuovo Cimento 20, 454 (1961). 18. C. J. Myatt, E. A. Burt, R. W. Ghrist, et al., Phys. Rev. 4. A. L. Fetter, Ann. Phys. (N. Y.) 70, 67 (1972). Lett. 78 (4), 586 (1997). 5. R. Graham and D. Walls, Phys. Rev. Lett. 76, 1774 19. M. R. Matthews, D. S. Holl, D. S. Jin, et al., cond- (1996). mat/9803310; D. S. Holl, M. R. Matthews, J. R. Ensher, 6. A. Gorlitz, A. P. Chikkatur, and W. Ketterle, cond- et al., cond-mat/9804138; D. S. Holl, M. R. Matthews, mat/0008067. C. E. Wieman, et al., cond-mat/9805327. 7. A. Csordas, R. Graham, and P. Szepfalusy, Phys. Rev. A 20. T.-L. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 3276 54, R2543 (1996); cond-mat/9711140. (1996). 8. Wen-Chin Wu and A. Grifﬁn, Phys. Rev. A 54, 4204 21. R. Graham and D. Walls, Phys. Rev. A 57, 484 (1998). (1996). 22. B. D. Esry and C. H. Greene, Phys. Rev. A 57, 1265 (1998). 9. F. Zambelli, L. Pitaevski, D. M. Stamper-Kurn, et al., Phys. Rev. A 61, 063608 (2000). 23. D. Gordon and C. M. Savage, Phys. Rev. A 58, 1440 (1998). 10. D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966; Mir, Moscow, 24. Yu. A. Nepomnyashchiœ, Zh. Éksp. Teor. Fiz. 70, 1070 1967), Vol. 1. (1976) [Sov. Phys. JETP 43, 559 (1976)]. 11. M. R. Andrews et al., Phys. Rev. Lett. 79, 553 (1997); M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998). Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

THEORETICAL AND MATHEMATICAL PHYSICS

Expansion of the Distance between Two Points in Spheroidal Functions as Applied to Problems of Mathematical Physics A. S. Baranov Pulkovo Astronomic Observatory, Russian Academy of Sciences, nab. Makarova 6, St. Petersburg, 196140 Russia Received August 21, 2000, in ﬁnal form, July 9, 2001

Abstract—A complete set of biharmonic functions is constructed in spheroidal coordinates. The distance between two points and its inverse value are expanded into a double series in terms of such functions. Possible applications in the theory of elasticity, in astrophysics, and other ﬁelds of mathematical physics are pointed out. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION k In formulas (1), Pn is the standard designation of It is known [1, 2] that in some problems of mathe- k k the associated Legendre functions, p (τ) = ik Ð n P (iτ), matical physics, functions constructed in the basis of n n k τ spheroidal coordinates are more convenient to use than and qn ( ) is the Legendre function of imaginary argu- spherical functions. Examples are often encountered in ment of the second kind: engineering mechanics, geophysics, and astrophysics. In a number of cases, not only harmonic but also bihar- 1 iÐn P ()x monic functions are of interest [3Ð5]. Under these con- q ()τ = ------n dxk()= 0 , ditions, the construction of elementary biharmonic n 2 ∫ τ Ð ix functions is rather simple, but the expansion of a given Ð1 biharmonic function in elementary biharmonic func- k tions poses problems. A key point here is the expansion k ()τ ()k()τ2 k/2 d ()τ ()> qn = Ð1 1 + ------qn k 0 . of the distance D between two arbitrary points and its dτk inverse value DÐ1 in these elementary functions. Second, the functions

BASIC FORMULAS k (),,τϕ []k ()τ k () k () k ()τ ikϕ Ln t = pn + 2 Pn t Ð Pn + 2 t pn e , First, we consider harmonic functions of type (3) ˜ k k k k k ikϕ Ln()t,,τϕ = []q ()τ P ()t Ð P ()t q ()τ e , k (),,τϕ k () k ()τ ikϕ n n + 2 n n + 2 V n t = Pn t pn e , (1) satisfy the biharmonic equation. k (),,τϕ k () k ()τ ikϕ Wn t = Pn t qn e , These functions were introduced and investigated in which are of prime concern in this paper. Here, t, τ, and [5]. In addition to them, the functions t and τ are also ϕ are spheroidal coordinates related to Cartesian coor- biharmonic, which can be verified directly. dinates x , x , and x as 1 2 3 We fix that of cofocal spheroids corresponding to r. Ð1 ϕ ϕ ϕ ()τ2 ()2 The quantity D is an analytical function of t and at x1 = Rcos, x2 = Rsin, R = c 1 + 1 Ð t , τ (2) fixed and is expanded in orthogonal functions Ynk = τ ϕ () k ϕ x3 ==c t, arccot x2/x1 ik Pn (t)e by the general rule [1]. We obtain the series

(the x3 axis is taken as the polar axis). The presence of Ð1 α some reference spheroid with the semiaxes a1(=a2) and D = ∑ nkYnk. a3 with a1 > a3 is assumed (i.e., the spheroid is oblate). nk, The parameter c introduced in (2) is deﬁned as the focal Hereafter, the summation over n and k is performed 2 2 ≥ ≥ ≥ α distance: c = a1 Ð a3 . in the ranges n 0, Ðn k n. The coefﬁcients nk are

2π 1 1 2 k k 1 Ð1 τ []β ()βτ ()τ π α ϕ --- 1 + eqn' 1 Ð i pn' 1 = Ð4 . nk = ------∫ ∫ D Ynk* dtd , c cnk 0 Ð1 Eventually, we obtain where ()nkÐ ! 2 k 2π 1 β = 4π------1 + τ q ()τ , π () i () n 2 ϕ 4 nk+ ! nk+ ! cnk ==∫ ∫ Ynk dtd ------() 2n + 1 nkÐ ! ()nkÐ ! 2 k 0 Ð1 β = 4π------1 + τ p ()τ e ()nk+ ! n (hereafter, asterisk means complex conjugation). with regard for formula (5). For clarity, we will use the expressions for an element of the spheroidal surface, dσ = Returning to formula (4), we arrive at 2 2 2 2 c ()1 + τ ()t + τ dtdϕ, and for an element of the 2 2n + 1 ()nkÐ ! k k normal in the neighborhood of the spheroid, dn = α = ------q ()τ p ()τ nk c ()nk+ ! n n 1 cd()t2 + τ2 /1()+ τ2 τ, which follow from relation (2). × k () ()ϕ ()τ < τ Then, Pn t1 exp Ðik 1 1 * or α 1 Ynk σ nk = ------∫∫------d , (4) 2 2 2 2 2 c c 1 + τ t + τ 2n + 1 ()nkÐ ! k k nk α = ------p ()τ q ()τ nk c ()nk+ ! n n 1 where integration is over the entire spheroid surface. (6) × k () ()ϕ ()τ > τ In this way, we come to the potential of an elemen- Pn t1 exp Ðik 1 1 . 2 τ2 tary layer with the density Ynk* /t + . This potential Now, straightforward calculations in view of defini- is locked for separately in the internal and external tions (1) yield β k τ k ϕ regions in the form vi = i pn ( 1)(Pn t1)exp(Ðik 1) and 2 Ð1 1 ()nkÐ ! k k v = β pk (τ )(Pk t )exp(Ðikϕ ), respectively, where β D = ---∑()2n + 1 ------V* ()r W ()r e e n 1 n 1 1 i c ()nk+ ! n n 1 β , and e are still unknown numerical coefficients. To find nk them, we use standard sewing condition on the surface: β β or the coincidence of i and e with the normal derivative having a jump that is equal to the density of the layer 2 π Ð1 1 ()nkÐ ! k k multiplied by the factor 4 . D = ---∑()2n + 1 ------W ()r V* ()r , (7) c ()nk+ ! n n 1 We take advantage of the relation known in the the- nk, ory of spherical functions (Wronski determinant) τ τ the first formula in (7) being used at < 1, and the k ()τ k ()τ Ð1 k dp k dq ()nk+ ! 2 Ð1 other, at τ > τ . The formulas for D are given in [1]. q ()τ ------n Ð p ()τ ------n - = ------()1 + τ , (5) 1 n dτ n dτ ()nkÐ ! They coincide with formulas (7) if we express the functions P and Q of imaginary arguments in terms of our p where the numerical coefficient is easily found from the and q. In the limit c 0, formulas (7) become well- τ ∞ k known formulas for spherical functions [6]. asymptotical expressions obtained at for pn and qk . Let us expand the quantity D in a way similar to that n used above for DÐ1; i.e., τ τ At 1 = , the sewing condition for the potential has β k τ β k τ D = ∑ξ ()r Y ()t, ϕ , the form i pn ( ) = e qn ( ). Then, by virtue of the rela- nk 1 nk tion nk, where ∂ 1 1 + τ2 ∂ ------= ------∂ 2 2 ∂τ * n c τ + t ξ 1 DYnk σ nk = ------∫∫------d . 2 τ2 2 τ2 which follows from formulas (2), the sewing together cnkc 1 + t +

TECHNICAL PHYSICS Vol. 47 No. 2 2002 EXPANSION OF THE DISTANCE BETWEEN TWO POINTS 179

Consider first the particular case n = k = 0, which, as The potential U(τ) is defined by the known will be demonstrated below, is associated with some expression computational difficulties. Then, ∞ R2 z2 U = πa'2c'1Ð ------1 Ð ------1 ξ 1 D σ ∫ 2 2 00 = ------∫∫------d . a' + s c' + s c2 1 + τ2 t2 + τ2 0 × ds η τ τ ------, We will analyze the auxiliary function ( ) = q0( ) Ð 2 2 τ τ τ ()a' + s c' + s q0( ∗), where ∗ denotes the quantity corresponding to the spheroid selected, because integration over the 2 2 spheroid surface τ = τ∗ should be replaced by integra- where the cylindrical coordinates R = x1 + x2 and z = tion over the volume at this point. The normal deriva- x3 refer to the point r1 and a' and c' are the semiaxes of τ tive is easily found: the spheroid corresponding to the parameter ; i.e., a' = c 1 + τ2 and c' = cτ. ∂η 1 ∂η 1 ------= ------= Ð------, ∂n 2 2 2 ∂τ 2 2 2 Taking the integrals over s, we obtain ct()+ τ /1()+ τ ct()+ τ ()1 + τ 2 τ π 2τ()τ2 1 R1 1 therefore, U = c 1 + 2arctan--- Ð ------arctan--- Ð ------τ c2 τ 1 + τ2 1 ∂η ξ ------σ 00 = Ð ∫∫D∂ d . 2 c n z1 1 1 Ð2------Ð arctan--- , 2τ τ τ τ c If 1 < ∗, we can apply the Green theorem [7] to the external space, more exactly, to the space between the 3 spheroid τ = τ∗ and greater spheroid with τ = τˆ , where U()τ π 2 2 2 1 τ ∫------dτ = ---c ()1 + τ arctan--- ++τ ---- τˆ will then tend to infinity. In our notation, 1 + τ2 2 τ 3

2 ∂η ∂D ∂η R1 2 2 1 3 D------Ð η------dσ Ð D------dσ Ð ----- ()1 + τ arctan--- + ττÐ ∫ ∫ ∂n ∂n ∫ ∫ ∂n 2 τ ττ ττ = ˆ = * η  ()∆η η∆ 2 ττ3 ()τ2 2 1 = ∫∫∫ D Ð D dr = Ð2∫---- dr Ð z1 3 + Ð 1 + arctan--- . D τ  (∆ is the Laplacian). Here, we at once have taken into η On the other hand, the surface integral is represented account the fact that (r) is a harmonic function. In the in the form preceding equalities, the volume integral is the potential at an internal point r1 that is induced by an inhomo- ∂η ∂D D geneous spheroid with cofocal layers of equal density. ∫ ∫ D------Ð η------dσ = ∫∫ Ð------Such potentials are well known [8]. Partition into layers ∂n ∂n 2 2 2 ct()+ τˆ ()1 + τˆ leads to the Stieltjes integral ττ= ˆ ∂D1 1 τˆ Ð ------arctan--- Ð arctan----- dσ. η ∂n τˆ τ ---- dr = ητ()dU()τ , * ∫ D ∫ τ * In the spheroidal coordinates, τ 2 2[ τ2 τ2 2 2 τ τ where U( ) is the potential at a ﬁxed internal point r1 D = c 2 ++1 Ð t Ð2t1 Ð t 1t1 that is induced by a homogeneous ellipsoid of unit den- ()ϕϕ ()τ2 ()τ2 ()2 ()]2 sity bounded by the surface τ = const. Integration by Ð2cos Ð 1 1 + 1 + 1 1 Ð t 1 Ð t1 , parts yields at large τ we have τ ˆ [τ τ ()ϕϕ η U()τ Dc= Ð t 1t1 Ð cos Ð 1 ητ()ˆ ()τˆ τ ∫---- dr = U + ∫ ------2d . D τ 2 2 2 Ð1 τ 1 + × ()τ ()()]()τ * 1 + 1 1 Ð t 1 Ð t1 + O .

TECHNICAL PHYSICS Vol. 47 No. 2 2002 180 BARANOV After the substitution of this asymptotic expression Note that we can use the formula (the proof is omit- into the surface integral, we use the limit τˆ tends to ∞; ted) then, τˆ as an auxiliary quantity disappears in the for- ()() ∆ nkÐ1+ nkÐ2+ k ------Ln mulas. The cylindrical coordinates R1 and z1 are  ()2n + 3 2 expressed in terms of the spheroidal ones according to (10) formulas (1). Asterisk at τ∗ can now be omitted, and we ()nk+ Ð 1 ()nk+ k  22()n + 1 k + ------L  = ------V eventually obtain 2 n Ð 2 2 n ()2n Ð 1  c  ()τ2 ()τ2 ()2 and construct exactly the same formula relating the ξ π 1 + 1 1 Ð t1 Ð 1 00 = Ð2 c ------˜ k  2 functions L and ÐW. In so doing, the functions Ln Ð 2 ˜ k and Ln Ð 2 should be defined carefully at n = k or n = 2 2 2 1 k + 1, because one has to check that the functions p and ---Ð1()+ t τ ()1 + τ arctan--- 1 1 τ k ˜ k q appearing in Ln Ð 2 and Ln Ð 2 obey standard recurrent formulas. It is easy to show that the functions pk τ  n τ()2τ2 ()τ2 ()τ2 should be considered to be identically zeros at n < k and + t1 1 Ð 1 Ð --- 1 + 1 1 Ð 1  2  k the definition of the functions qn should be completed as or, after the use of the particular expressions for the k ()τ ()()τ2 Ðk/2 Legendre functions and for their combinations V, W, L, qk Ð 1 = 2k Ð 2 !! 1 + , (11) and L˜ , k ()τ ()τ()τ2 Ðk/2 qk Ð 2 = 2k Ð 2 !! 1 + .  The case k = 0 is an exception. Formulas (11) ξ π τ 4[]0()[]˜ 0() () become invalid, we do not define the corresponding 00 = 2 c + --- 1 + V 2 r1 L0 r + P2 t  9 functions, and two of formulas (10) are replaced by (8) 2 1 0 τ 0 4 0 0  ∆ ˜ () () c Ð---L0 + --- = W0, + ---L0 r1 W0 r . 9 2 9   (12)  In this case, the difficulty arose since ξ (r ) is an 2∆ 1 ˜ 0 t 0 00 1 c Ð------L0 Ð --- = W1. ξ 25 6 unbounded function. This is not the case for other nk, because the principal term in the asymptotics for D dis- ξ Therefore, except for another singular case n = 1 and appears immediately on integration and nk(r1) proves k = 0, one can write immediately to be bounded. This function can be found if the value ()2 ()() of the Laplacian is known. Since ξ () nkÐ ! k ()τ nkÐ1+ nkÐ2+ nk r1 = c ------qn ------()nk+ ! ()2n + 3 2 ()1 2 ∆D ==∆ D ---- (9) ()() × k () nk+ Ð 1 nk+ k () ζ˜ () D Ln r1 ++------Ln Ð 2 r1 r1 ()2n Ð 1 2 (superscript (1) denotes the coordinate of the point r ), 1 at τ > τ and we merely have 1 2 ()nkÐ ! k ()nkÐ1+ ()nkÐ2+ () ξ ()r = Ðc ------p ()τ ------∆ 1 ξ α nk 1 ()n 2 nk = 2 nk, nk+ ! ()2n + 3 (13) ()() where α are defined by formulas (6). × ˜ k () nk+ Ð 1 nk+ ˜ k () ζ˜ () nk Ln r1 ++------Ln Ð 2 r1 r1 ()2n Ð 1 2 It is sufficient to construct a bounded and everywhere continuous solution of this Poisson equation; τ τ ζ ζ˜ at < 1. Here, the harmonic functions and are then, by virtue of the Liouville theorem [9], it will coin- defined from the sewing conditions for the functions ξ ξ nk cide with the function nk(r1) desired up to an additive and their normal derivatives on the spheroid τ = const constant (which proves to be zero while checking). (this is equivalent to comparing the derivatives with

TECHNICAL PHYSICS Vol. 47 No. 2 2002 EXPANSION OF THE DISTANCE BETWEEN TWO POINTS 181 τ τ τ respect to 1 at the point = 1). As a result of analysis, We treat formula (15) as a differential equation for which is clarified below, we arrive at the desired expres- Ωk . It is easy to integrate: sion, namely, n 2 2 k ()2n + 3 ()nk+ ! 2 ()Ω1 + τ = Ð------()τ + C .  ()nkÐ2+ !()nkÐ ! k k k n () ζ()  () ()τ ()τ nkÐ2+ ! r1 = ------2 2 Pn t1 pn 1 qn + 2 ()2n + 3 []()nk+ ! The constant of integration C can be determined, τ 2 e.g., by the substitution = i We obtain []()nkÐ ! k k k Ð ------P ()t p ()τ q ()τ 2 2 n 1 n 1 n Ð 2 k ()2n + 3 ()nk+ ! ()2n Ð 1 ()nk+ Ð 2 !()nk+ ! Ω = Ð------n ()nkÐ2+ ! 2 []()nkÐ2+ ! k k k ()nk+ ! ()nk++2 ! 1 + ------P ()t p ()τ q ()τ + ------+ ------. 2 n + 2 1 n + 2 1 n + 2 ()()2 ()2n + 3 ()nk+ !()nk++2 ! nkÐ ! nkÐ2+ ! 1 + τ ()() Ωk τ 'k τ nkÐ ! nkÐ2Ð ! k ()k ()τ k ()τ Along with n ( ) and Mn ( ), we also employ dif- + ------Pn Ð 2 t1 pn Ð 2 1 qn Ð 2 ()2n Ð 1 2[]()nk+ Ð 2 ! 2 ferential relation (5) while comparing the derivatives with respect to τ on both sides.  The above relations suffice to validate the choice of γ k ()k ()τ k ()τ ()ϕ ---+ nkPn t1 pn 1 qn cikexp Ð 1 , ˜  the functions ζ and ζ (it is not necessary to add a con- ξ stant, because the expression for nk(r1) shows the cor- ∞ where rect asymptotical behavior, vanishing at r1 at least on the average over the sphere). According to for- 44()k2 Ð 1 ()2n + 1 ()nkÐ ! 2 mulas (12), in the particular case n = 1 and k = 0, Ðt/6 γ = Ð------, (14) nk 2 2 () must appear in the second formula of (13) in place of ()2n Ð 1 ()2n + 3 nk+ ! the second term in brackets. It remains to collect all terms in the expansion for D. and ζ˜ is expressed in the same way but the symbols p In this case, for each k, the pairwise grouping of the and q change places everywhere. The verification is terms having the subscript n other than 2 seems to be based on several identities for the Legendre functions, the most appropriate. The singular cases n = 0 and n = 1 which are lacking in the reference books and call for at k = 0 give additional terms with separated τ and t. As special derivation. a result, we have ∞ ∞ First, we introduce the function Mk = qk pk Ð ()nkÐ ! 2 ()nkÐ1+ ()nkÐ2+ n n + 2 n Dc= ∑ ∑ ------Ð------k k ()nk+ ! ()2n + 3 2 qn pn + 2 . One can easily show (from the definition of the k = Ð∞ nk= k τ k k ()nk+ Ð 1 ()nk+ k k functions q) that Mn ( ) is a first-degree polynomial odd × ˜ () () ˜ () () Ln r V n* r1 Ð ------Ln Ð 2 r V n* r1 with respect to τ. Therefore, it is sufficient to determine ()2n Ð 1 2 one coefficient using the asymptotics for p and q at ()() large τ. The calculation yields nkÐ1+ nkÐ2+ k()k () + ------Ln* r1 Wn r (17) ()2n + 3 2 () k ()nk+ ! τ Mn = Ð 2n + 3 ------. (15) ()() ()nkÐ2+ ! nk+ Ð 1 nk+ k ()k () + ------Ln*Ð 2 r1 Wn r ()2n Ð 1 2 k Second, we use the combination Ωk = q' pk + n n + 2 n γ k()k () ()τ τ k k k k k k ---+ nkV n* r1 Wn r + c Ð tt1 1 qn' pn + 2 Ð qn + 2 pn' Ð qn pn' + 2 . Differentiating it and replacing the resulting second derivatives by the well- τ ≥ τ in the range 1; otherwise, r and r1 as well as the k k known differential equations for pn and qn [1], we respective coordinates, change roles. The terms with obtain k ˜ k | | ≥ Ln Ð 2 and Ln Ð 2 are omitted at k n Ð 1 [5]. γ k 2τ k 22()n + 3 k Thus, formula (17) with specific nk defined by rela- Ω' = Ð ------Ω + ------M , (16) n τ2 n τ2 n tions (14) is fully proved. For testing, one can, e.g., con- 1 + 1 + sider the case when both points are located on the polar τ τ ≥ k axis; i.e., t = t1 = 1, r = c , and r1 = c 1. If we put r r1 where the function Mn is defined by formula (15). for definiteness, the simple rearrangement of the terms

TECHNICAL PHYSICS Vol. 47 No. 2 2002 182 BARANOV on the right-hand side of Eq. (17) (only the terms with where dσ is an element of the surface S and µ is some |τ τ | 1 k = 0 remain, and all but the isolated term c Ð 1 can- surface density. cel out) immediately yields the correct expression for the distance: D = r Ð r . In another test, r and r tend to For an arbitrary sufficiently smooth biharmonic 1 1 (four-times differentiable) function ψ we identify v infinity simultaneously and proportionally with the ∆ψ angular coordinates retained. Then, we have the asymp- with inside the spheroid. Then, for the function totes r ≈ cτ and t ≈ cosθ (recall that θ is the polar angle); similar asymptotes hold for the other point. From for- ψ () 1 µ()σ 1 r = ---∫D r1 d 1 mula (17), we derive the relation 2 S ∞ ∞ n + 2 n ()nkÐ ! r r we obtain ∆ψ = v. Hence, the initial function ψ is rep- D = ∑ ∑ ------1 - Ð ------1 1 ()nk+ ! ()n + 1 ()n Ð 1 resented by the sum k = Ð∞ nk= 2n + 3 r 2n Ð 1 r × k ()θ k ()θ []()ϕϕ 1 Pn cos Pn cos 1 exp ik Ð 1 , ψ = --- Dµ()σr d + χ, (20) 2∫ 1 1 which also directly follows from the theory of spherical S functions. For this purpose, one multiplies the well- χ known series (see, for example, [1, p. 145]) where is some harmonic function. χ ∞ s However, (r) can always be expanded in the set of Ð1 1 r1 ()functions V [1]. According to our basic formula (17), D ==------∑ ------Ps cosv 2 2 rs + 1 the expansion of D represents the integral in formula (20) r + r1 Ð 2rr1 cosv s = 0 as the superposition of V, L, t, and τ, so that the completeness of this set in the above-mentioned class of 2 2 2 by D = r1 + r Ð 2rr1cosv and applies the recurrent for- sufficiently smooth biharmonic functions inside the k k spheroid is proved. The completeness of the set of the mula (n Ð k + 1)P (t) Ð (2n + 1)tP (t) + (n + n + 1 n ˜ τ k functions W, L , t, and outside the oblate spheroid in k)(Pn Ð 1 t) = 0. the class of sufficiently smooth biharmonic functions bounded at infinity is proved in a similar way. CONCLUSION Recall that sometimes the character of a problem The importance of expansions (7) and (17) is that requires just the spheroidal coordinates to be used. In they give a solution of the Poisson equation and the particular, such conditions appear in a natural way if the equation function f is most conveniently expressed in the spheroidal coordinates. Physically, such a situation can be ∆∆u()r = 4πf ()r , (18) associated with the specific role of some reference respectively, in a certain coordinate system. spheroid or, in the limiting case, a circle of finite radius in a particular problem. The example of astrophysical Indeed, as is known [10], applications is given in [11]. More complex examples may appear in the dynamics of self-gravitating fluids () 1 () u r = ---∫∫∫Df r1 dr1 (19) with regard for viscosity. The technique developed in 2 this study can find wide application, because Eq. (18) is a particular solution of Eq. (18). If the function f is and similar ones are often encountered in problems of different from zero in a finite region of the space, func- the theory of elasticity, the theory of plasticity, and tion (18) proves to be biharmonic in the entire remain- hydrodynamics [12, 13]. The specific role of some cir- ing region and integral (19) with D in the form of cle is also common in engineering problems of the equi- expansion (17) yields an expression for this biharmonic librium of a three-dimensional elastic medium [14]. function in the form of a series in some standard set of functions. At the same time, this validates the com- k ACKNOWLEDGMENTS pleteness of the set of the biharmonic functions V n and k The author thanks V.A. Antonov for the interest in Ln inside a specific compression of the spheroid S. this study. Indeed, the function v, which is harmonic inside the spheroid, can always be represented by the potential of an elementary layer REFERENCES µ()r v ()r = ------1-dσ , 1. E. W. Hobson, The Theory of Spherical and Ellipsoidal ∫ D 1 Harmonics (Cambridge Univ. Press, Cambridge, 1931; S Inostrannaya Literatura, Moscow, 1952).

TECHNICAL PHYSICS Vol. 47 No. 2 2002 EXPANSION OF THE DISTANCE BETWEEN TWO POINTS 183

2. V. A. Antonov, E. I. Timoshkova, and K. V. Kholshevni- 9. A. F. Timan and V. M. Troﬁmov, An Introduction to the kov, An Introduction to the Theory of Newtonian Poten- Theory of Harmonic Functions (Nauka, Moscow, 1968). tial (Nauka, Moscow, 1988). 10. J. Boussinesq, Application des Potentiels à l’Etude de 3. F. G. Tricomi, Lezioni sulle equazioni a devivate parziali l’Equilibre et du Mouvement des solides élastiques (Torino, 1954; Inostrannaya Literatura, Moscow, 1957). (A. Blanchard, Paris, 1969). 4. Yu. N. Rabotnov, Mechanics of Deformable Solid 11. S. Chandrasekhar and N. R. Lebovitz, Astrophys. J. 136, (Nauka, Moscow, 1988). 1037 (1962). 5. A. S. Baranov, Zh. Vychisl. Mat. Mat. Fiz. 37, 395 12. N. I. Bezukhov, Foundations of the Theory of Elasticity, (1997). Plasticity and Creep (Vysshaya Shkola, Moscow, 1968). 6. L. N. Sretenskiœ, The Theory of Newtonian Potential 13. G. K. Batchelor, An Introduction to Fluid Dynamics (Gostekhteorizdat, Moscow, 1946). (Cambridge Univ. Press, Cambridge, 1967; Mir, Mos- 7. H. Jeffreys and B. Swirles, Methods of Mathematical cow, 1973). Physics (Cambridge Univ. Press, Cambridge, 1966; Mir, 14. A. E. Andreœkiv, Three-Dimensional Problems of Theory Moscow, 1969), Vol. 1. of Cracks (Naukova Dumka, Kiev, 1982). 8. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, Conn., 1969; Mir, Moscow, 1973). Translated by D. Zhukhovitskiœ

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 184Ð189. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 42Ð47. Original Russian Text Copyright © 2002 by Ganzherli, Maurer, Chernykh.

GASES AND LIQUIDS

Investigation of a Free Convective Current Using Holographic Interferometry N. M. Ganzherli, I. A. Maurer, and D. F. Chernykh Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received April 9, 2001; in ﬁnal form, June 1, 2001

Abstract—By the method of holographic interferometry in real time, an investigation is carried out of the development of free convective currents in an enclosed cavity filled with different liquids exposed to heat fluxes from either linear or point sources. Characteristic times of the processes are measured. Temperature fields in the current are plotted. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION ing of the interferogram; and l is the light ray path through the medium. Nonstationary heat transfer processes in gases or liquids are complicated physical phenomena accompa- The light wave phase incursions change the interfer- nied by the development of currents with varying tem- ence conditions in the observation plane and cause variations in the field illumination. Using measured poral and spatial characteristics. Mathematical simula- ∆ϕ tion of such processes requires the solution of a set of (x, y) one can evaluate the distribution of the refractive nonlinear differential equations with boundary condi- index n(x, y, z) and such parameters of the medium related tions, also in the form of differential equations. The with the refractive index as the density, temperature, etc. objective of simulation is to obtain patterns of varying temperature and velocity fields in the flow. An experi- EXPERIMENTAL mental study allows the mathematical model of the phenomenon to be made more precise and adequate to As an example of a nonstationary process, we con- the physical process [1]. Probing nonstationary flows sider development of a quasi-free convective current in with the help of a dense grid of probes meets with dif- an enclosed cavity exposed to a localized heat source. ficulties because the probes interfere with the flow. Of importance to the development of the theory of cur- Besides, the finite response time of the probes restricts rents in liquids is the study of buoyancy, which, in a the possibilities of studying fast nonstationary pro- number of situations, can appreciably influence the heat cesses. Therefore, for studies of heat transfer processes, transfer intensity. Models of some convective processes noncontact optical methods [2, 3] are of special inter- that form currents due to buoyancy can be sorted, est—among them, interferometric and holographic according to [6], into two vast groups: buoyant currents techniques, which substantially expand capabilities of and thermal bubbles. In both cases, the motion in a optical experiments in visualization of phase inhomo- medium is caused by the force of gravity because of geneities [4, 5]. lower liquid density near the heat source. Fundamentals of the theory of buoyant currents are presented in [7, 8]. When light passes through an inhomogeneous Experimental investigations of heat transfer pro- medium, optical path differences arise caused by varia- cesses of a similar kind were carried out by a number of tions in the refractive index along the light path. These authors [9Ð11], for example, with the help of a grating variations cause a change in the light wave phase interferometer [10, 11]. ∆ϕ(x, y), which is defined by the expression We studied the development features of a quasi-free π convective flow in an enclosed cavity for a number of ∆ϕ(), 2 [](),, simple liquids with the Prandtl number Pr in a range xy = ------∫ nxyz Ð n0 dl, (1) λ from 4.6 to 34. The basic physical characteristics of the l liquids used were taken from [12, 13]. where x and y are the coordinates of a point in the plane As a heat source we used a standard MLT resistor λ of observation; is the light wavelength; n0 is the (with the resistance R = 430 Ω) 0.5 cm in length and refractive index of a medium in its undisturbed state at 0.18 cm in diameter. The amount of heat was controlled the initial time instant; n(x, y, z) is the distribution of the by the current passing through the resistor and varied refractive index in the medium at the moment of record- from 0.02 to 1.7 W. The heat source was placed inside

INVESTIGATION OF A FREE CONVECTIVE CURRENT 185 a metal cavity with two side walls of glass for interfer- M ometric-holographic diagnostics. The dimensions of P the cavity and the observation windows were 10 × 30 × H 50 mm and 30 × 50 mm, respectively. MO Two series of experiments were carried out with different positions of the resistor at the cavity bottom. In the first case, the resistor was located along the illumination direction, which made the heat flux more uniform across the liquid layer thickness and the resistor could be assumed close to linear. In the second series of BS MO L O D experiments, the heat source was housed inside a nozzle with an outlet diameter of about 1 mm. In this case, Fig. 1. Optical diagram of a holographic interferometer. the resistor could be considered a pointsource. BS beam splitter; MO microscope objective; L lens; M mirror; P plane-parallel plate; O object; D diffuser; H hologram. The development of a free convective current was studied with the use of a holographic interferometer. A feature of the optical scheme of the interferometer (a) (b) was a ground glass placed immediately behind the cavity (Fig. 1) [14]. In this case, the ground screen represented the localization plane of the resulting interfer- ence pattern, which simplified recording of the interferograms; however, it imposed a restriction on the character of variations of the refractive index in the medium under study acceptable for analysis. Rigorous analysis of the changes in the refractive index on the basis of interferograms could be made for plane objects and for objects in which the refractive index depended only on coordinates x and y and did not change along z axis coinciding with the direction of Fig. 2. Temperature fields for two development stages of a illumination of the object. In particular, the use of a free convective current in ethyl alcohol. (a) Formation and nearly linear heat source allowed us to evaluate changes detachment of a thermal bubble; (b) formation of a free con- ∆ vective current. in the refractive index n = n(x, y) Ð n0 using a simplified formula π Recording of the interferograms was implemented with ∆ϕ(), 2 l[](), xy = ------λ nxy Ð n0 , (2) the help of filming equipment, a TV camera and a photographic camera. The filming equipment could record or, changing from the phase incursion ∆ϕ(x, y) to the the process at speeds from 3 to 16 frames per second optical path difference expressed in terms of the num- and determine the temporal characteristics of the pro- ber of fringes k(x, y) [4, 5], by the formula cess with good accuracy, namely, the time of heating ()λ, (due to heat conduction) until detachment of a thermal (), kxy τ nxy = ------+ n0. (3) bubble 1, the time elapsed before the convective cur- l τ rent reached the top to the cavity, and the time 2 in A convective current caused by a point heat source which a quasi-stationary convection regime is established is an axisymmetric object, the axis of which is parallel (all times were measured from the start of heating). to the current direction. In this case, expression (1) The temperature was measured with three thermo- takes the form of Abel’s integral equation. To solve this couples: one directly above the heat source, another at equation we used a step approximation method pro- the top of the cavity where the current impinged, and posed by Pierce [15]. the third in a region undisturbed by the heated liquid Information on the development of a free convective flow. Data from the thermocouples were used to correct current obtained by the method of finite-fringe holo- the temperatures calculated from interferometric mea- graphic interferometry was supplemented with that surements with the use of the well-known dependence derived from infinite-fringe interferograms [4, 5]. of the refractive index of the liquids on temperature for Finite-fringe interferograms allowed us to determine a light wavelength of 0.63 µm. the sign of the refractive index change, and infinite- fringe interferograms gave directly isothermic lines in the flow field. RESULTS OF THE EXPERIMENT The method of holographic interferometry in real First, consider in detail the results of experiments on time gives a possibility to follow the entire process of a free convective current caused by a linear heat source. development and stabilization of a convective current. From an analysis of infinite-fringe interferograms, we

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186 GANZHERLI et al.

Θ Θ / max 1.0 (a) (b) (c)

x0 = 10.5 mm x0 = 10.5 mm x0 = 10.5 mm

0.5 5.3 5.3 1.3 1.3 5.3 1.3

0.5 0.5 0.5 y, mm

Fig. 3. Reduced temperature proﬁles in a free convective current at a heat source power of 0.23 W. Three cross sections at different heights x0 above the heat source.

Θ Θ / max 1.0 (a) (b) (c)

x0 = 10 mm x0 = 10 mm x0 = 10 mm

5 0.5 5 5 0.5 0.5 0.5

0.5 0.5 0.5 y, mm

Fig. 4. Reduced temperature profiles in a free convective current in acetone at different heat source powers. Three cross sections at different heights x0 above the heat source. obtained the temperature fields for different instants of formation of a thermal bubble (3 s after start of heat- time as the free convective current developed in an ing), and Fig. 2b corresponds to an encounter of the enclosed cavity filled with liquid. As an example, tem- quasi-free convective current with the top of the cavity perature fields obtained at different moments of time in (10 s after start of heating). The temperature difference a cavity filled with ethyl alcohol at a heat source power ∆Θ between adjacent isotherms is a characteristic of the of 0.23 W are presented in Fig. 2. Figure 2a depicts the particular liquid and depends on the temperature factor of the refractive index. The temperature increment ° Θ between isotherms in Fig. 2 is 0.395 C. log max 1.0 Analysis of the interferograms allowed us to obtain reduced temperature profiles in free convective currents that developed in the liquids under study at different heat fluxes. Figure 3 shows reduced temperature pro- 1 files in a free convective current along the coordinate y 2 in horizontal cross sections at various heights x0 above 3 the 0.23-W heat source in (a) acetone, (b) ethyl alcohol, 0.5 and (c) turpentine. 4 Figure 4 shows reduced temperature profiles in a free convective current in acetone at different heat source powers: (a) 0.02, (b) 0.08, and (c) 0.23 W. In Fig. 5, a profile is shown of the maximum excess (with respect to the liquid temperature in the thermally 1.5 1.0 logx0 Θ undisturbed area) temperature max along the symmetry axis of a free convective current with height x above Fig. 5. Lapse of the temperature Θ along the symmetry 0 max the heat source. In the area of the free convective cur- axis of a free convective current with x0 in (1) ethyl alcohol, (2) isopropyl alcohol, (3) acetone, and (4) distilled water. rent where the influence of the heat source, as well as

TECHNICAL PHYSICS Vol. 47 No. 2 2002 INVESTIGATION OF A FREE CONVECTIVE CURRENT 187

logFo tion of a temperature field at given heating conditions and has the form Fo = ατ/l2, where α is the temperature 0 conductivity, τ is the characteristic time of the process, and l is the characteristic size of the object. Fourier’s number expresses the ratio of the temperature distribution along the current axis to the temperature change with time at each point of the axis. Rayleigh’s number 3β Θ vα –0.5 1 Ra = gl 0 / characterizes the relationship between the heat flux in a medium due to the buoyancy and free convection, on the one hand, and two diffusion processes interfering with the motion and stabilizing it, namely, the internal friction due to viscosity and the heat conduction, on the other hand [8]. Here, g is the –1.0 gravitational acceleration, l is the characteristic size of β the cavity, 0 is the coefficient of volumetric thermal expansion, Θ is the temperature excess, v is the coeffi- 2 cient of kinematic viscosity, and α is the temperature conductivity. The dependence of Fourier’s number on Rayleigh’s number can be experimentally obtained by measuring 3 4 logRa the characteristic development times of convective current and the corresponding temperature increments. Fig. 6. Relationship between logFo and logRa for two Functions Fo = f(Ra) were found for the characteristic development regimes of a free convective stream. (1) Con- vection regime; (2) heat conduction regime. ᮏ Water, × iso- heating time by conduction and the corresponding tem- propyl alcohol, ᭹ ethyl alcohol, ᭺ carbon tetrachloride; perature increase in the thermal bubble, as well as for ᮀ turpentine; acetone. the characteristic time of formation of the free convection that developed and the corresponding temperature increase. The characteristic times for different liquids the effect of the current colliding with the cavity top, is and different heat fluxes were determined to an accu- Θ weak, variation of max in the current is roughly in pro- racy of 0.1 s. The temperature changes were obtained portion to xÐ0.5 . from the interferograms and compared with the ther- 0 τ mocouple measurement. The table contains the time 1 A relationship between Fourier and Rayleigh’s cri- of conduction heating until detachment of the thermal Θ teria of similarity, Fo and Ra, calculated for the cases bubble and the corresponding bubble temperature 1, τ considered, is of interest from the theoretical point of as well as the time 2 for formation of the quasi-station- view. Fourier’s number characterizes the rate of forma- ary convection regime and the corresponding tempera-

Θ Θ / max t, s 1.0 10

6 0.5 1 1 2 4 2 3 2

0.5 1.0 r, mm 0.2 0.6 1.0 P, W Fig. 7. Radial distributions of the dimensionless temperature changes in ethyl alcohol at the height x0 = 10 mm above Fig. 8. Time before detachment of the thermal bubble ver- the nozzle. The heat ﬂux is produced by a point source of sus heat source power in (1) isopropyl alcohol and (2) ethyl different powers: (1) 0.23; (2) 0.52; (3) 0.67 W. alcohol.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 188 GANZHERLI et al.

Times of formation of the thermal bubble, times of establishing the convection regime, and the corresponding temperature increments in the current at different heat ﬂuxes

0.02 W 0.08 W Liquid τ Θ ° τ Θ ° τ Θ ° τ Θ ° 1, s 1, C 2, s 2, C 1, s 1, C 2, s 2, C Water (7.5) Ethyl alcohol (16.8) 5 1.2 10 2.4 3 2.4 9 5.15 Isopropyl alcohol (34) 6 1.1 10 2.7 4.2 4.2 8 5.0 Acetone (4.2) 12 2.2 6 4.6 Carbon tetrachloride (4.6) 3.3 1.3 10 2.1 2 8 5.0 Turpentine (21.2) 9 3.15 7 6.0 0.23 W 0.48 W Water (7.5) 3.5 6.5 2.8 8.05 Ethyl alcohol (16.8) 2.7 8 9.1 2 7 11.8 Isopropyl alcohol (34) 3 8 2 6 Acetone (4.2) 5 6.6 3.5 6 Carbon tetrachloride (4.6) 1 6 1 4 Turpentine (21.2) 5 6.5 4 10

Note: The Prandtl number is given in parentheses.

Θ ture increase 2 in the current for various liquids with The reduced temperature profiles and the current devel- different Prandtl numbers. opment characteristics obtained are in agreement with The plot of logFo as a function of logRa is given in the theory. The developed interferometric-holographic Fig. 6. The points in the plane logFo, logRa were technique and the vast experimental data obtained with approximated by linear functions using the least- this technique can be of interest to specialists in the squares method. The slope of curve 1 corresponds to field of free convection. Fo ~ RaÐ0.21 which agrees with theoretical prediction for a free convective current in an enclosed cavity [8]. ACKNOWLEGMENTS In the case of a free convective current produced by The authors are grateful to P.V. Granskiœ for his kind a point heat source, using changes of the refractive ∆ assistance in computerized processing of the experi- index ni determined by Pierce’s method of step mental data. approximation, radial distributions of the refractive index in the convective current for a number of liquids The work was supported by the Russian Foundation were plotted. The calculations were performed for sev- for Basic Research (under the Program “Leading Sci- eral cross sections above the nozzle. Figure 7 demon- entific Schools”), project no. 00-15-96771). strates radial distributions of the temperature increases in ethyl alcohol at a height of x0 = 10 mm above the noz- REFERENCES zle at different heat source powers. 1. D. N. Popov, Nonstationary Hydromechanical Pro- In Fig. 8, variation of the time until detachment t of cesses (Mashinostroenie, Moscow, 1982). the thermal bubble with the increase in heat source 2. L. A. Vasil’ev, Schlieren Methods (Nauka, Moscow, power P are shown for isopropyl alcohol (curve 1) and 1968; Israel Program for Scientific Translations, Jerusa- ethyl alcohol (curve 2). lem, 1971). 3. W. Hauf and U. Grigull, Optical Methods in Heat Trans- fer (Academic, New York, 1970; Mir, Moscow, 1973). CONCLUSION 4. Yu. I. Ostrovskiœ, M. M. Butusov, and G. V. Ostrovskaya, The results of the investigation of development of a Holographic Interferometry (Nauka, Moscow, 1977). free convective current in an enclosed cavity provide 5. A. K. Beketova, A. F. Belozerov, A. N. Berezkin, et al., evidence of the efficiency of holographic interferome- Holographic Interferometry of Phase Objects (Nauka, try in problems of heat transfer in transparent media. Leningrad, 1979).

TECHNICAL PHYSICS Vol. 47 No. 2 2002 INVESTIGATION OF A FREE CONVECTIVE CURRENT 189

6. J. S. Turner, Buoyancy Effects in Fluids (Cambridge 12. N. B. Vargaftik, Tables of the Thermophysical Properties Univ. Press, Cambridge, 1973; Mir, Moscow, 1977). of Liquids and Gases (Nauka, Moscow, 1972; Halsted 7. Ya. B. Zel’dovich, Zh. Éksp. Teor. Fiz. 7, 1463 (1937). Press, New York, 1975). 8. O. G. Martynenko and Yu. A. Sokovishin, in Free-Con- 13. Tables of the Physical Quantities: Handbook, Ed. by vective Heat Exchange: Handbook (Minsk, 1982), I. K. Kikoin (Atomizdat, Moscow, 1976). p. 162. 14. A. L. Barannikov, N. M. Ganzherli, S. B. Gurevich, 9. K. S. Mustaﬁn, V. A. Seleznev, and E. I. Shtyrkov, Opt. et al., Izv. Akad. Nauk SSSR, Ser. Fiz. 49, 711 (1985). Spektrosk. 22, 319 (1967). 15. W. D. Pierce, in Receiving and Studying of High Temper- 10. Yu. I. Lyakhov, Prikl. Mekh. Tekh. Fiz., No. 2, 169 ature Plasma (Inostrannaya Literatura, Moscow, 1962). (1970). 11. V. L. Zimin and Yu. N. Lyakhov, Prikl. Mekh. Tekh. Fiz., No. 3, 159 (1970). Translated by N. Mende

TECHNICAL PHYSICS Vol. 47 No. 2 2002

GAS DISCHARGES, PLASMA

The Possibility of Using a Low-Tritium-Content DÐT Mixture Compressed with a High-Power Current Pulse M. V. Fedulov Received July 27, 2001

Abstract—The possibility of the propagation of a thermonuclear detonation wave in a deuterium pinch with a small (~0.5%) tritium admixture is considered. The required pinch parameters and the necessary value of the current produced by a high-power pulsed current generator are estimated. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION tion wave. For a low tritium content, it is the FBZ where Progress in the development of high-power pulsed the contribution from the reaction d + He3 p + He4 current generators with progressively increasing cur- with a released energy of 18.3 MeV can be neglected, τ rents gives hope that, in the foreseeable future, it will be because the He3 accumulation period g = 1/Sgnd (where possible to achieve the parameters required for the igni- Sg is the reaction rate) is long compared to the ratio of δ ≅ tion and maintenance of a thermonuclear detonation the FBZ length z to the detonation wave velocity vb wave with an acceptable energy gain ratio. In earlier 108T1/2 cm/s (here and below, the temperature is in keV, calculations of the propagation of a thermonuclear det- the particle energy is in MeV, the current J in MA, and onation wave [1Ð10], a 50/50 DÐT mixture was consid- other quantities are expressed in CGS units, unless oth- ered. At the same time, it is interesting to consider a erwise specified). The long duration of the He3 accumu- mixture with a tritium concentration corresponding to lation period means that the complete tritium reproduction. Since the ratio between the rates of the reactions d + d p + t λ ρδ Ӷ × 16 1/2 () 2 z = z 310T /Sg T g/cm . (1) (which will be designated by Sp) and d + t n + He4 (St) in the temperature range 20Ð40 keV varies approx- The left-hand side of inequality (1) is on the order of λ imately from 0.005 to 0.01 [11], it reasonable to con- r; i.e., as will be shown below, it is equal to several sider a DÐT mixture with a tritium concentration of 0.5 g/cm2. At the same time, the right-hand side decreases to 1%. To avoid misunderstanding, hereinafter, by the monotonically with temperature in the range 20Ð reaction rate is meant the averaged product of the cross 40 keV, reaching approximately 55 g/cm2 at T = 40 keV. section by the relative particle velocity 〈σv〉. Below, we τ ρ Note that, even for T = 40 keV, g is equal to ~100/ ns, will derive estimating relationships between the pinch ρ 3 τ parameters required for the detonation wave to propa- so that, for > 100 g/cm , g does not exceed the pinch gate in a low-tritium-content mixture (LTCM), pro- duration. Hence, the contribution from the reaction d + vided that the necessary temperature is attained in a cer- He3 p + He4 must be taken into account when cal- tain region of the pinch. culating the total energy yield. We assume that the temperature is uniform throughout the FBZ and the ratio of the average (more pre- EVALUATION OF THE PRODUCT ρ ρ OF THE DENSITY BY THE RADIUS FOR A FUEL cisely, effective) density to c in this region does not WITH A LOW TRITIUM CONTENT vary with time. We assume also that the energy loss per unit time through unit area of the FBZ because of the Calculations showed [5Ð10] that, for the detonation radial expansion and the fuel heating ahead of the wave wave to propagate through a compressed column of a front is equal to k (T)ρ, where the coefficient k 50/50 DÐT mixture, the sufficient value of the product 0 0 ρ describes the dependence of the energy loss in the FBZ of the mixture mass density c by the column radius rc on T. Assuming that the ratio of the pinch radius r to is about 0.25Ð0.5 g/cm2. Let us denote this product by p λ the effective radius of the expanding burning zone r and estimate its value for an LTCM. Since, for the (~2V/S, where V and S are the volume and the surface following analysis, it is necessary to know the qualita- area of the FBZ) is constant (more precisely, indepen- tive relation between λ = ρ r and the tritium concen- λ r c c dent of ct), we obtain that the value of r , accurate to a tration ct, we will reproduce a brief derivation of the constant factor, is determined by the expression λ dependence of r on the temperature T in the forepart of 2  the burning zone (FBZ), which determines the heating λ = constk ()ρT / ∑S n n ε Ð W , (2) of the cold plasma ahead of the thermonuclear detona- r 0 ij i j ij r

THE POSSIBILITY OF USING A LOW-TRITIUM-CONTENT 191

λ 2 where ni is the density of nuclei of species i; Sij are the r, g/cm ε 1 reaction rates; ij is the fraction of the reaction energy 10 left by fast particles in the FBZ; and Wr is the 0.005 bremsstrahlung power per unit volume, which is pro- 0.01 portional to ρ2. Expression (2) depends only on T, and a = const can be determined from the previous calculations for a 50/50 mixture. Since energy losses are determined pri- 100 marily by the pressure and the expansion velocity, we ct = 0.5 3/2 can assume that k0 is proportional to T . This allows us λ to estimate the dependence of r on T for different tritium concentrations. Below, we will assume that the minimum of the λ 10–1 curve r(T) (at T = Tm) for a 50/50 mixture corresponds 10 15 20 25 30 35 40 to 0.5 g/cm2. Calculations were performed under the T, keV assumption that all the energy of an α-particle pro- λ ρ duced in the dÐt reaction remains in the FBZ or is Fig. 1. Dependence of r = crc on the temperature for dif- expended on the fuel heating ahead of the wave front, ferent tritium concentrations ct. whereas a neutron escapes from the plasma without los- ing its energy. Figure 1 shows the dependence of λ on r of σ drops rapidly as E increases; consequently, the T calculated under the additional assumption that the n medium is transparent to bremsstrahlung. We note that probability of the first neutron collision can be rela- the left branch of the dependence of λ on T is unstable: tively low. At the same time, a neutron colliding with a r deuteron loses, on average, almost one-half of its as the temperature is slightly increased, it continues energy, so that the probability of subsequent collisions increasing up to Tm; in contrast, as the temperature T is increases sharply. For a rough upper estimate, we can decreased, the burning terminates. assume that the energy fraction left in the pinch plasma λ To calculate the dependence of r on T at low tritium by such a dÐt neutron is close to the probability of its ε concentrations, we should know the values of ij both first collision. for dÐt and dÐd reactions. We should also keep in mind In addition to the case of ct = 0.5, Fig. 1 also shows that (as will be seen from the results of calculations pre- λ λ the dependence of r on T for two other tritium concen- sented below) the minimum value of r should increase by at least one order of magnitude as compared to a trations. Calculations were performed under the 50/50 mixture. Let us show that, for any slowing-down assumption that the collisional cross section of a neu- length of charged particles due to collisions with field tron produced in the dÐt reaction is equal to 1 b. The results of calculations show that, as the tritium concen- particles, the full energy of charged particles remains in λ the FBZ due to magnetization. Indeed, for the maxi- tration decreases by two orders of magnitude, r increases by more than one order of magnitude, mum ratio of the Larmor radius rL of a fast charged particle to the pinch radius r , we have whereas the value of Tm increases insignificantly (from p 18 to 24 keV). In this case, the ratio between the dÐd ()1/2 and dÐt reaction rates increases only from 0.010 to rL/r p = 0.7 AE /J, (3) 0.012 and, although ct decreases, the dÐt reaction again where A and E are the atomic weight and the particle makes the main contribution to the heating of the FBZ, energy (in MeV) and J is the pinch current. largely due to the contribution from a neutron. For a For LTCM, we are dealing with currents of at least cross section equal to 1 b, the ratio of the mean free path λ several tens of MA; hence, the ratio rL/rp should be to the pinch radius is equal to ~3/ r; i.e., rp is nearly σ small. The total collision cross section n of a neutron twice as large as the mean free path. Perhaps more with E = 2.45 MeV produced in the dÐd reaction (in detailed calculations will show that the contribution deuterium) is about 6 b [12]; i.e., the ratio of its mean from a neutron produced in the dÐt reaction is some- σ free path to rp is equal to what overestimated. We note, however, that, for n σ λ ≅ λ equal to 0.5 b, the results of calculations change insig- In/r p =3.3/ n r 0.5/ r. (4) nificantly (the minimum value of λ increased by only λ 2 14%). Hence, at r values of several g/cm , we can assume that all the energy of a neutron produced in the reaction It should also be noted that, as the tritium concentra- d + d He3 + n is transferred to the plasma. More tion decreases, the role of bremsstrahlung losses complicated and important is the problem of the slow- increases. Thus, for ct = 0.5, bremsstrahlung losses ing down of neutrons produced in the dÐt reaction (E = amount to only 4% of the total energy lost by α-parti- 14.1 MeV). According to [12], at E > 4 MeV, the value cles in the FBZ, whereas, for ct = 0.005, this fraction

TECHNICAL PHYSICS Vol. 47 No. 2 2002 192 FEDULOV

J, MA At such strong plasma compression, the plasma 200 temperature Tc can be very high. Then, we can use the gas equation to determine the current J required for 180 maintaining the equilibrium. Assuming that the temper- 160 ature is spatially uniform, we can use the well-known M = 100 mg/cm formula 140 ()1/2 J = 440 MTc MA. (7) 120 For example, if, during compression, a fuel with a 100 M = 20 mg/cm mass per unit length of M = 0.1 g/cm is heated to only T = 0.5 keV, then a current as high as J ≅ 100 mA will * * c 80 be required for the pinch confinement. On the other hand, we cannot decrease the value of M with the pur- 60 ******************* 0.2 0.4 0.6 0.81.0 1.2 1.4 1.6 1.8 pose of decreasing the current and the total released energy in a shot, because it is necessary to compress the Tc, keV fuel to a very small radius rp. Fig. 2. Current required for confining the fuel heated up to Note that, when compressing solid deuterium (with λ 2 ρ 3 the temperature Tc for r = 6 g/cm : circles correspond to a mass density of 0 = 0.16 g/cm ), the initial radius is the ideal-gas model, asterisks correspond to the generate 1/2 at least larger than r = (M/πρ0) . This means that com- electron gas, and solid curves show the results obtained with 0 the TFP model. pression down to r = rp requires an increase in the inductance by a value more than increases to one-fourth. Although, in the latter case, the () ()()π ρ 1/2λ L* ==2lnr0/r p 2ln / 0M r FBZ is longer, it remains transparent to bremsstrahl- (8) ung, because the ratio of rp to the mean free path lr in ≅ 2ln() 4.4λ /M1/2 nH/cm deuterium is equal to r (for λ = 6 g/cm2 and M = 0.1 g/cm, we have L∗ ≅ ≅ λ2 7/2 r r p/lr 0.1 r /r pT , (5) 9 nH/cm). λ 2 In this case, the increase in the magnetic energy due so that, for r = 6 g/cm and T = 20 keV, we obtain 2 ≅ Ð4 to compression, EH = L∗J /2 kJ/cm, substantially rp/lr 10 /rp. 2 exceeds the pinch thermal energy Er = 0.75J kJ/cm. The possibility of decreasing J and E by decreasing CURRENT VALUE AND ENERGY YIELD T Tc is limited by the degeneration of the electron gas. Let Let us now determine the relation between the mass us consider a limiting case of Tc (expressed in energy ε of a DÐT mixture per unit length of the plasma column units) smaller than the Fermi energy F (for deuterium, and the current J required for creating an equilibrium ε ρ2/3 pinch with appropriate parameters. Keeping in view tri- we have F = 16c eV). If we take into account the tium reproduction or, in other words, assuming that the pressure of the degenerate electron gas, then, for deute- dÐt reaction is provided by the tritium produced in the rium, we obtain reaction d + d t + p, we should have a rather high ρ5/3 degree of burnout—no less than 10%. The utilization of P = 3.16 c Mbar. (9) six deuterons (with the contribution from the reaction For the current value required for maintaining the d + He3 p + He4 taken into account) results in an equilibrium, we obtain energy release of 43.25 MeV [13]. Hence, for 10% burnout, about 35M GJ per centimeter of plasma col- J ≅ 18M5/6/r2/3 ≅ 39λ2/3M1/6 ≅ 130M1/6 MA. (10) umn is released, where M is the mass per unit length in p r g/cm. In this case, it is necessary to compress the Hence, for an indefinitely low temperature, the plasma down to required value of the current exceeds the right-hand side of Eq. (10) (for M = 0.1 g/cm, we have J > 87 MA). r = M/πλ cm (6) λ 2 p r Note that, for a 50/50 DÐT mixture (for r = 0.5 g/cm ), 1/6 and, accordingly, increase the density up to ρ = the minimum current is equal to 25M MA (for M = c 0.1 g/cm, J > 17 MA). If, as a reference value at the πλ 2 3 λ 2 λ r /M g/cm . Taking the value r equal to 6 g/cm , we minimum of the curve r(T) for a 50/50 DÐT mixture, obtain that, even for M = 0.1 g/cm, the pinch diameter we take not 0.5 g/cm2, but one-half of this value [i.e., µ should be equal to ~100 m and the density should be we halve constk0(t) in formula (2)], then the calculated ρ ≅ 3 λ c 1 g/cm . value of r(T) for a fuel with ct = 0.005 will also

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE POSSIBILITY OF USING A LOW-TRITIUM-CONTENT 193 λ decrease by nearly one-half (the value of r has little 2. M. S. Chu, Phys. Fluids 15, 13 (1972). effect on the fraction of the energy lost by neutrons pro- 3. A. F. Nastoyashchiœ, At. Énerg. 32, 45 (1972). duced in the dÐt reaction). Hence, at c = 0.005, the min- t 4. A. F. Nastoyashchiœ and L. P. Shevchenko, At. Énerg. 32, λ 2 imum value of r will decrease to ~3 g/cm . Under this 451 (1972). assumption and with M = 0.1 g/cm, the minimum cur- 5. V. V. Cernukha and M. V. Fedulov, in Proceedings of the rent for a 50/50 mixture at a low temperature will 8th European Conference on Controlled Fusion and decrease to 10 MA; for a fuel with ct = 0.005, this value Plasma Physics, Prague, 1977, Vol. 11, p. 170. will decrease to 55 MA. 6. M. V. Fedulov and V. V. Chernukha, Preprint No. 2939 Asymptotic expressions (7) and (10) are quite sufﬁ- (Kurchatov Institute of Atomic Energy, Moscow, 1978). cient for the rough evaluation of the required current value. Computations of the dependence of the pressure 7. E. N. Avrorin et al., Fiz. Plazmy 10, 514 (1984) [Sov. J. Plasma Phys. 10, 298 (1984)]. on the temperature and density with the help of a TFP model [14] demonstrated that the required current value 8. V. V. Vikhrev and V. V. Ivanov, Preprint No. 4161/6 (Kur- did not increase substantially (Fig. 2). chatov Institute of Atomic Energy, Moscow, 1984). 9. V. V. Vikhrev and V. V. Ivanov, Dokl. Akad. Nauk SSSR 282, 1106 (1985) [Sov. Phys. Dokl. 30, 492 (1985)]. CONCLUSION 10. V. V. Ivanov, M. V. Fedulov, and V. V. Chernukha, Pre- Estimates show that, with a current of 100Ð200 MA, print No. 4572/8 (Kurchatov Institute of Atomic Energy, it is possible to produce a thermonuclear detonation Moscow, 1988). wave in a fuel consisting almost completely of deute- 11. B. N. Kozlov, At. Énerg. 12, 238 (1962). rium. The most complicated problem is, apparently, to provide the compression of the fuel with the smallest 12. Handbook of Physical Quantities, Ed. by I. S. Grigoriev possible mass to the required size. and E. Z. Meilikhov (Énergoatomizdat, Moscow, 1991; CRC Press, Boca Raton, 1997). 13. J. J. Duderstadt and G. A. Moses, Inertial Conﬁnement ACKNOWLEDGMENTS Fusion (Wiley, New York, 1982; Énergoatomizdat, Mos- I am grateful to S.L. Nedoseev for offering the sub- cow, 1984). ject of this study. 14. N. N. Kalitkin and L. V. Kuz’mina, Preprint (Institute of Applied Mathematics, USSR Acad. Sci., Moscow, 1975). REFERENCES 1. A. L. Fuller and R. A. Gross, Phys. Fluids 11, 534 (1968). Translated by N. Larionova

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 194Ð199. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 52Ð58. Original Russian Text Copyright © 2002 by Atroshenko, Krivosheev, Petrov.

SOLIDS

Crack Propagation upon Dynamic Failure of Polymethylmethacrylate S. A. Atroshenko, S. I. Krivosheev, and A. Yu. Petrov Research Institute of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 198904 Russia e-mail: [email protected] Received January 29, 2001; in ﬁnal form, July 9, 2001

Abstract—Crack propagation in polymethylmethacrylate shock-stressed by a pulsed magnetic ﬁeld is experimentally investigated. The failure type as a function of the distance from the crack tip is analyzed, and the failure diagram is constructed. The surface failure energy is estimated. The tilt angles of mesocracks are correlated to the failure type. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION was made. We studied the propagation of cracks from When subjected to high tensile stresses, glassy poly- the end of the sawcut when the specimen was subjected mers may become brittle and fail. Brittle failure is char- to a pulse of pressure uniformly distributed over the acterized by uniform strains up to the instant of failure, groove. The pressure pulse was generated by a 10-mm- which proceeds by fast crack propagation across the wide plane conducting line made of 250-µm-thick cop- specimen, i.e., in the plane normal to the tensile force per foil. A current pulser was discharged in the oscilla- applied. In this case, the cross-sectional area in the fail- tory mode with an oscillation period T = 5.5Ð6.0 µs and µ ure plane does not change significantly, and the residual the decay time constant T1 = 1Ð4 s. The current pulse longitudinal strains in both pieces of the specimen are amplitude was varied from 150 to 300 kA, which corre- absent. The failure type depends on experimental con- sponded to a stress amplitude in the specimen from 140 ditions. The brittle failure of viscoelastic materials to 320 MPa. The current was measured using a takes place at relatively low temperatures and small Rogowski loop having a passive RC integrator with an exposure times or at high strain rates. integration time constant of 68 µs and recorded with an S9-8 oscilloscope. EXPERIMENTAL MATERIALS If the capacitor bank is discharged in the oscillatory AND METHODS mode to the plane copper lines whose minimal cross An advantage of glassy polymers, such as polyme- section fits the maximum allowable action integral for thylmethacrylate (PMMA), is transparency, which copper and if the ratio of the line width to the spacing makes it possible to observe crack propagation and b/h ≥ 10, the pressure distribution can be considered control the process. uniform [2]. Then, under the assumption that the decay PMMA was used to determine the failure type upon dynamically stressing the specimen by a pulsed mag- L0 Rv netic field. As in [1], the stressing conditions provided the uniform distribution of the pressure along the crack Q A edge. The tests were performed at threshold stress P pulses.

We tested organic glass specimens with c1 = Cr 200 1/2 1970 m/s, c2 = 1130 m/s, and KIc = 1.47 MPa m , where c and c are the longitudinal and transverse 1 2 A velocities of acoustic waves, respectively, and KIc is the 100 limit stress intensity factor under static stressing. 0.3 3 3 The stressing conditions and the specimen geometry 200 are shown in Fig. 1. The specimens were prepared as follows. A 3-mm-wide groove was made in a PMMA Fig. 1. Stressing setup and test specimen. P, load; Cr and L0, slab 10 mm wide and 100 mm long. At the end of the capacitance and self-inductance of the current pulser, groove, a 3-mm-deep sawcut of width about 0.3 mm respectively; Q, switch. Sizes are given in millimeters.

CRACK PROPAGATION UPON DYNAMIC FAILURE 195 is not too high, the pressure can be determined by the 172 3 4 5 6 8 formula N

() 2t 2 π t Pt = P0 expÐ----- sin 2 --- . (1) T 1 T π π At the time instant tm = arctan( 2 T1/T)T/2/ , the 9 pressure reaches the maximal value 10 ()()2 2 π2 Pm = P0 exp Ð2tm/T 1 /1+ T /T 1/4/ . Figure 2 shows the optical scheme for the visualization of crack propagation under pulsed stressing. An Fig. 2. Optical detection scheme. 1, photographic film; 2, mirror; 3 and 6, objective lenses; 4, shutter; 5, adjustable SFR-2 streak camera was used as a photographic slit; 7, specimen; 8, running crack; 9, IFP-120 flash lamp; recorder. An IFP-120 flash lamp was employed as a and 10, pipeline. source of light. A part of specimen 7 is cut by adjustable slit 5 and is recorded by film 1. The image is transferred L , mm to the film path with rotating mirror 2. The film shows cr how the illuminance varies when the moving crack shuts off the optical path. The boundary between the 20 regions of different illuminance (different optical den- 15 sity of the image) is the interface between the cracked 10 and intact materials. To synchronize the crack propagation with the instant of stressing, the radiation from the 5 discharge gap is transferred to the field of the camera µ with a pipeline. 0 100 200 t, s Ten specimens were tested. The oscillation period µ µ Fig. 3. Typical Lcr vs. time dependence obtained by process- was T = 5.6 s, the decay time constant was T1 = 4.2 s, ing streak photographs. and the pressure pulse amplitude Pm was varied from 140 to 320 MPa. Typical results of processing the streak photographs (crack propagation dynamics) are given in prior to the arrival of the waves reflected from the Fig. 3. model boundaries. For the specimens used in our experiments, the time to arrival was about 100 µs. In our problem, the stress intensity factor has the form [3] RESULTS AND DISCUSSION t 1. Failure under Pulsed Stressing α () () Ps K I t = ∫------ds, (2) Note the stepwise character of the crack propaga- tsÐ tion. The crack travels a distance of 9Ð12 mm until it 0 stops for the first time. Initially, the crack moves with a α 2 2 π velocity of 100 to 500 m/s according to the stressing where = 2c2 c1 Ð c2 /(c1 c1 ). conditions. If the stress exceeds the threshold value, the Since in the experiments we varied only the stress initial portion of the distance to the first stop (several amplitude P(t), the value of the intensity factor millimeters) is traveled with a velocity of 420Ð450 m/s. remained the same up to multiplier and reached the The rest of this distance is traveled with a much lower maximum within t ≈ 1.7 µs after the start of stressing. velocity, about 100 m/s. The structural-temporal criterion [4] allows us to The time instant the crack stops correlates with the calculate the dependence of the failure initiation instant time it takes for the stress wave to doubly travel the dis- on the applied stress amplitude. In our case, this crite- tance from the crack edge to the boundary of the spec- rion has the form [5] imen. Tests with the smaller-size specimens confirm this observation. The stop can be explained by the t 1 () ≤ arrival of the waves reflected from the specimen bound- --- ∫ K I t' dt' K Ic, (3) ary at the tip crack. τ trÐ Processing the streak photographs, we constructed τ the dependence of the crack origination instant on the where is the incubation time (intrinsic material prop- applied stress amplitude. To compare the dependence erty). thus obtained with model data, we considered a semi- For PMMA, τ = 32 µs, as follows from experiments infinite crack with the normally stressed edges as a [1]. The calculation shows that the least failing ampli- mathematical model. The solution of this problem tude in our experiments is Pm = 94.7 MPa. Figure 4 yields the actual experimentally observed stress pattern depicts the calculated dependence of the applied stress

TECHNICAL PHYSICS Vol. 47 No. 2 2002 196 ATROSHENKO et al.

Pm, MPa scabbing [6] and spalling [7]. The failure delay is the 400 time between the instant the local force field reaches a maximum and the instant of failure initiation. This phenomenon cannot be explained in terms of the concept of 300 the critical stress intensity factor but finds an explanation within the structural-temporal approach. In Fig. 4, 200 the failure delay is the distance between the curve or a data point to the dotted line. 100 2. Surface Energy to Failure µ 0 5 1015202530t, s The surface energy to failure is a measure of the fracture strength of a material. It is a more general Fig. 4. Amplitude of the stress pulse applied vs. time to index of the material strength properties than the failure. strength itself, because it does not depend on local characteristics of the material or on the type of mechanical µm action. The discrepancy between the estimated and experimental values of the surface energy to failure and 9500 also the appearance of the fracture surface lead us to 9100 conclude that, when subjected to local stresses at the 8100 8800 crack tip, the material exhibits the plastic behavior 7700 7600 although its macroscopic properties are typical of a 7100 6800 brittle substance. It is of interest to compare the behav- 6000 ior of metals and polymers. At room temperature, the differences between the estimated and experimental Hackly values of the surface energy to failure for PMMA and 4200 Mist steel are close to each other [8], since both materials 3200 show similar local responds. In both cases, the energy dissipation mechanism involves a combination of shear Mirror stresses and diffusion; however, the type of diffusion and the parameters of the energy-dissipating field in these materials greatly differ. In steel, the energy is dis- 050100 150 µs sipated because of the dislocation motion in the metal lattice; in the polymer, the dissipation is due to the Fig. 5. Sequence of fracture types vs. stressing time and dis- motion of dislocation segments in the Van der Waals tance to notch. Figures indicate the distance traveled by the force field generated by surrounding molecules. Inelas- crack. tic processes taking place at the crack tip are a result of a high local stress concentration, to judge from the amplitude on the failure initiation instant. Circles value of the specific surface energy. In this respect, the denote data points. The vertical dotted line (t ≈ 1.7 µs) behavior of glassy polymers is akin to that of metals. indicates the time when the stress intensity factor Processes responsible for the failure of solids can be reaches the maximum. The time in Fig. 4 is counted judged from the fracture surface. Analyzing it, one can from the stress application time. The test data suggest clarify a failure mechanism and theoretically investi- that the crack starts to move within some time after the gate the strength properties of a material. local force field at the tip, the stress intensity factor, has If, under tension, the crack propagates from the frac- reached the maximum, that is, in the descending branch turing source slowly, the initial region of propagation is of the curve. Such a phenomenon is called failure delay. smooth and offers a high reflectivity. This region is called a smooth (mirror) zone or a zone with “silver” Failure delay has been observed in experiments on cracks—thin layers of a highly strained partially exfo- liated material that can specularly reflect light [9]. Next Table 1. Energy of cup fracture to the mirror zone, or the zone of slow crack propagation, is a transition zone, which is characterized by the Fracture Crack length Cup size Crack velocity increased fracture surface roughness. In notched speci- µ energy Lcr, mm d, m V, m/s mens, the transition zone is larger than in those without E, J/cup a notch and features irregular lines extended along the 5.0 32 92.4 17 × 10Ð18 crack propagation direction. The following is the zone × Ð18 containing a number of geometric figures, largely 17.5 18 50.696 5.3 10 parabolas and hyperbolas [10]. They arise when a sec- 25.5 13 194.25 2.8 × 10Ð18 ondary crack originates from a defect in front of the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 CRACK PROPAGATION UPON DYNAMIC FAILURE 197

20 µm 50 µm 1 2

17.5 mm 5 mm

6.5 mm

25 µm 3

Fig. 6. Typical fracture surfaces vs. distance traveled by the crack (1, mirror; 2, mist; and 3, hackly). propagating primary crack. The secondary crack prop- ure takes place. In this case, starlike structures arise on agates with a constant speed as a set of concentric cir- the fracture surface [12], and the cracks propagate more cles “radiating” from many local defects, which are or less uniformly. Under high dynamic overstressing, activated by the propagating stress wave. When cross- many of the defects act simultaneously and the material ing the primary crack, the secondary crack forms parab- undergoes brittle fracture irrespective of its behavior olas [11] if the rate of propagation of the primary and under static stressing. It has been found [13] that the secondary cracks are the same. The defect from which rate of PMMA destruction (crack propagation velocity) the secondary crack originates is the focus of the parab- grows with distance to the fracturing source and lies ola. A slight difference between the levels of the propagating stress waves renders these ﬁgures visible and between 400 and 700 m/s, depending on the static stress leads to the substantial absorption of the energy. The at the instant of failure (the crack velocity increases appearance of the fracture surface reﬂects basic defor- with the stress value). Note also that the destruction rate mation processes on which the fracture energy is spent. and the appearance of the fracture surface do not Outside this zone, the stress level and the crack propa- strictly correlate: the latter depends also on the stress- gation velocity increase, and high-rate (explosive) failing rate and history.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 198 ATROSHENKO et al.

rougher is the fracture surface, the higher energy is spent on crack propagation. The energy absorption mechanisms can be judged by the surface appearance. When a defect is initiated, the major part of the energy is absorbed because of plastic deformation. As has been mentioned above, the fracture surface is smooth in the mirror, or cleavage, zone, where the crack velocity is slow. In this zone, the energy needed for sustaining the crack propagation is small. The next region, the region of increased surface roughness (mist region), exhibits randomly arranged irregular triangles with a great spread in their sizes. Outside this region, the PMMA fracture surface becomes similar to the surface of metals at plastic (hackly or cup) fracture. A great number of parabolic cups requires a great energy for the crack propagation. A growth of the energy absorbed by the fracture surface increases the cup size, so that the degree of plastic fracture declines. The parabolic features in PMMA resemble the regions of cup fracture in high-viscosity steel. In the steel, material shear is seen near the cup crater. Upon cracking, parabolic surface features (cups), which look like those on the PMMA surface, appear. Parabolas reﬂect the conventional type of PMMA failure [13]. 300 µm The mechanism of parabola formation sheds light on the whole process of failure: crack initiation; intermit- tent evolution of cracking; further initiation of cracks at Fig. 7. Wedge crack in PMMA. defect sites; and the occurrence of secondary cracks and their relation to the position of the primary crack front (the secondary and primary cracks usually propagate at different levels).

3. Failure Type We analyzed the character of PMMA failure under dynamic stressing and constructed the diagram (Fig. 5) that shows how the failure type (vertical axis) varies with the crack length (counted from the notch top) and pulse duration (horizontal axis). As the crack propagates from the notch, the type of failure changes in the order: cleavage, quasi-cleavage, and cup fracture (in terms of metal science), which correspond to the smooth region, mist region, and parabolic (cup) region, respectively. As the crack propagates further, these basic types of failure alternate. The images of the fracture surface and the change-over from one fracture type to another are shown in Fig. 6. As the distance to the notch top increases, the cup size decreases from 32 to µ 80 µm 13 m, according to Fig. 6. Knowing the cup diameter and the PMMA speciﬁc surface fracture energy (γ = 2.1 × 102 J/mm2 [8]), one can calculate the fracture Fig. 8. Bifurcation of mesocracks in PMMA. energy per middle cup. This energy drops from 17 × 10Ð8 to 2.8 × 10Ð8 J as the crack propagates inward to the During the destruction, the major portion of the specimen (Table 1). energy is spent on plastic deformation around fractur- It should be noted that a small crack propagating ing sources. These are usually the foci of the parabolas into the specimen normally to the fracture surface was where substantial plastic strains are observed. The detected at the notch end on the fracture surface. More-

TECHNICAL PHYSICS Vol. 47 No. 2 2002 CRACK PROPAGATION UPON DYNAMIC FAILURE 199

Table 2. Tilts of mesocracks (iv) Mesocracks are initiated only in the smooth and rough (mist) regions. Their tilt to the direction of the Crack length Mesocrack tilt, Crack velocity main crack propagation was found to be independent of Lcr, mm deg V, m/s the crack velocity and distance to the notch. 1 32 271.6 The fracture type-pulse duration-notch distance dia- 3 33 119.8 gram was constructed. 6.5 29 70.58 The surface fracture energy per cup was shown to × Ð8 × Ð8 7.1 19 58.95 decrease from 17 10 to 2.8 10 J/cup, this energy being related to the characteristic crack size. 7.6 28 62.27 27 26 126.79 REFERENCES 1. S. I. Krivosheev, Preprint No. 142, IPMash RAN (Insti- over, a wedge crack originated at the cups was detected tute of Problems in Machine Science, Russian Academy at a distance of about 10 mm from the notch (Fig. 7). At of Sciences, St. Petersburg, 1997). a distance of about 4.2 mm, this crack bifurcates 2. G. A. Shneerson, Fields and Transient Processes in (Fig. 7): two branches normal to the main crack propa- Ultrahigh Current Equipment (Énergoatomizdat, Mos- gate in the direction coincident with that of the main cow, 1991). crack front. This crack causes cup (hackly) fracture. 3. G. P. Cherepanov, Mechanics of Brittle Fracture (Nauka, At the same time, mesocracks bifurcate from the Moscow, 1974; McGraw-Hill, New York, 1979). main crack (Fig. 8); however, they differ from those 4. N. F. Morozov, Yu. V. Petrov, and A. A. Utkin, Izv. Akad. described in [5Ð10]. Their tilt varies nonmonotonically Nauk SSSR, Mekh. Tverd. Tela, No. 5, 180 (1988). with distance to the notch end (Table 2). In addition, the 5. Yu. V. Petrov and A. A. Utkin, in Mechanics of Fracture: mesocracks are initiated in the smooth and rough Theory and Experiment (St. Petersb. Gos. Univ., St. regions. They are absent on the surfaces of cup (para- Petersburg, 1995), p. 94. bolic) fracture. 6. N. A. Zlatin, G. S. Pugachev, S. M. Mochalov, and As for the crack propagation velocity, it also varies A. M. Bragov, Fiz. Tverd. Tela (Leningrad) 17, 2599 nonmonotonically with distance to the notch, unlike (1975) [Sov. Phys. Solid State 17, 1730 (1975)]. data reported in [8]. 7. D. A. Shockey, D. C. Erlich, J. F. Kalthoff, and H. Homma, Eng. Fract. Mech. 23, 311 (1986). 8. J. P. Berry, J. Polym. Sci., No. 50, 107 (1961). CONCLUSIONS 9. V. I. Aleshin and E. V. Kuvshinskiœ, Mekh. Polim., No. 6, The method to visualize the brittle fracture of 989 (1978). cracked PMMA subjected to dynamic stressing (micro- 10. J. Fineberg and M. Marder, Phys. Rep. 313, 1 (1999). second pulses of threshold amplitude) is developed. 11. C. E. Felner, Theoretical and Applied Mechanics Report, The following features of the fracture were revealed. No. 224 (University Illiois, 1962). (i) Crack propagation delays relative to the time 12. J. A. Kies, A. M. Sallivan, and G. R. Irwin, J. Appl. Phys. instant the stress applied reaches the threshold value. 21, 716 (1950). (ii) Crack length is proportional to the amplitude of 13. Fracture Processes in Polymeric Solids (International above-threshold stress pulses. Publishers, New York, 1970). (iii) Fracture types alternate until the crack stops ﬁrst. Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

SOLIDS

Charged Defect Diagnostics from Luminescence Spectra L. P. Ginzburg and A. P. Zhilinskiœ Moscow Technical University of Communication and Informatics, Moscow, 111024 Russia Received November 25, 1999; in ﬁnal form, April 20, 2001

Abstract—A method allowing the detection of an absolutely rigid Coulomb gap formed by localized charged defects from the low-frequency part of the luminescence spectrum is proposed. Its efﬁciency is demonstrated with luminescence in amorphous semiconductors and optical ﬁbers, as well as with triboluminescence. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION by the methods usually applied to reveal a soft gap, in particular, by low-temperature experiments on hopping The subsystem of randomly distributed charged conduction or tunneling spectroscopy. defects (CDs) defines many of the physical properties of a material. One reason is that this subsystem gives On the other hand, according to [7], Eq. (1) involves rise to diagonal disorder, which is responsible for such two features, which can be detected by accurately mea- effects as the Anderson transition in the inverse layer of suring the low-frequency tail of the PL spectrum. The MIS structures [1], the position of the mobility gap in first one is the presence of the interval v-SiO [2, 3], the concentration dependence of the pho- 2 បω < បω ≤ បω toconductivity threshold in semiconductors [4], the 0 m, (3) scattering of carriers [5], etc. within which the value However, the presence of CDs is difficult to predict. The point is that, under the equilibrium condition, such ∆I()បω ξ()បω =,------(4) defects arise only if the occupied energy level of a neg- ()បω 2 ative center is below the empty level of a positive one. But even if there is an indirect evidence for the presence (∆I(បω) is the luminescence intensity) obeys the rule of CDs (for example, so-called UÐ centers in chalco- genide glassy semiconductors (CGSs)), mechanisms ξ()បω = ABÐ បω + C()បω 2, (5) behind their formation remain unclear and are discussed within different models (see, e. g., [6] and Ref. where A, B, and C are positive constants. therein). Therefore, the direct experimental detection of The second feature is that the coefficients A, B, and CDs seems to be of great importance. C relate as It has been demonstrated with CGSs [7] that CDs must affect the low-frequency region of the photolumi- AC Ӎ 0.25B2. (6) nescence (PL) spectrum. This effect can be understood from the fact that the spectral density of states in the CD As was mentioned above, CGSs provide an example subsystem is characterized by the Coulomb gap (CG) of materials where the presence of UÐ centers has been established [6, 9]. The analysis of all the PL spectra gE()∝ ()E Ð δ 2, (1) ever observed in CGSs has shown [7] that, in most cases, relationships (5) and (6) hold. The approach where developed in [7] was extended to v-SiO2 [3], in which δ Ӎ បω + Ð 0/2 (2) a significant amount of O3 and O centers is surmised បω ω by several authors (see, e. g., [10]). The result was in and 0 coincides with the PL threshold. Since 0 > 0, favor of this supposition. this CG can be called “absolutely rigid.” This means that the 2δ interval about the Fermi level remains “for- The aforesaid leads one to the question of to what bidden” [8].1 The expression for the electrical conduc- degree the technique of PL spectrum analysis devel- tivity near the gap edge acquires the factor oped in [7] can be applied for CD diagnostics. In other exp(−2δ/k T). That is why this gap cannot be detected words, whether one can conclusively establish the pres- 0 ence of CDs (or even estimate their concentration [3, 1 From this point on, the abbreviation CG implies an absolutely 11]) based only on a conventional “bell-like” curve rigid gap. ∆I(បω) or more precisely, its carefully measured low-

CHARGED DEFECT DIAGNOSTICS FROM LUMINESCENCE SPECTRA 201 frequency part where conditions (5) and (6) are fulfilled and taking into account (5), we obtain in interval (3). Ω m Our purpose is to show that, with the exception of ∝ ()βδ ()2 2 β 3 4 some specific situations, this sufficiency is justified. In AUFU∫ d + U ++2 U U , (13) Section 1, we demonstrate that, for the density of states 0 given by the general series Ω m ∞ BUFU∝ ∫ d ()β+ δ ()2U ++3βU2 2U3 , (14) gE()= a ()E Ð δ n, (7) ∑ n 0 n = 1 Ω m meeting conditions (5) and (6) within interval (3) leaves ∝ ()βδ ()2 (apart from some specific cases) only the term with CUFU∫ d + UU+ , (15) n = 2 in (7). Since no other model leads to a similar 0 dependence in the system of localized states (see, e. g., where [12, 13]), this proves the presence of CDs. Section 2 Ω δ considers examples illustrating how the application of m = Em Ð . (16) conditions (5) and (6) to a short low-frequency portion of seemingly identical PL spectra enables one to prove Following [7], we take advantage of the expansion or disprove the presence of the CD subsystem. ∞ ()δ ()δ n FU+ = ∑ an U , (17) 1. SUFFICIENCY OF CONDITIONS (5) AND (6) n = 1 FOR CD DIAGNOSTICS where the fact that F(δ) = 0 is taken into consideration. According to [7], the general formula for ∆I(បω) is Substituting (17) into (13)Ð(15) and integrating yields ∞ AA∝ ()β + a, BB∝ ()β + b, CC∝ ()β + c, (18) 2 ∆I()បω ∝ ()បω ∫dEF() E gE()gE()Ð បω , (8) where δ ()β β2γ ()Ω3 βγ()Ω4 γ ()Ω5 A ==2 m + 2 3 m, a 4 m, (19) where F(E) is the amount of localized radiative states with an energy E that may be occupied by an electron B()β = β2γ ()Ω1 2 +3βγ()Ω2 3 , b = 2γ ()Ω3 4 , (20) that absorbed a photon. m m m ξ បω ()β βγ()Ω2 γ ()Ω3 Let interval (3), where the value ( ) defined by C ==1 m, c 2 m (21) (4) follows law (5), be detected. The assumption that δ and the designation only the Ð Em interval contributes to (3) enables us to replace the upper limit of integration in (8) by Em. Then, ∞ Ωn before being substituted into (8), series (7) should be γ ()p = ∑ a ------m - (22) reduced to the sum nnp++1 n = 1 () α()αδ ()δ 2 gE = 1 E Ð + 2 E Ð . (9) is used. Combining Eqs. (18), we find Only under condition (9) does not the expansion of បω បω 2 g(E Ð ) in powers of contain terms with powers AC A()β C()β +++aC()β cA()β acb greater than two. Because of the proportional relation------= ------. (23) α 2 B2()β ++2B()β bb2 ac ship between the quantities in (8), we can put 2 = 1 B without loss in generality. Then, the density of states As was shown in [7], at p ≥ 4, the expression becomes γ ()γp ()p Ð 2 2 ------Ӎ 1 (24) gE()= β()E Ð δ + ()E Ð δ . (10) γ ()p Ð 1 Substituting (10) into the expression is fairly accurate.

Em Therefore, in view of (19)Ð(21), ξ()បω ()()()បω 2 = ∫ dEF E gEgEÐ (11) ac Ӎ 0.25b . (25) δ Let us assume that both conditions (5) and (6) are using the equality met. In this case, Eqs. (23) and (25) suggest that ()E Ð δ Ð បω 2 = ()E Ð δ 2 Ð 2()E Ð δ ␮ω + ()បω 2,(12) 4[]A()β C()β ++aC()β cA()β Ӎ B2()β + 2B()β b. (26)

TECHNICAL PHYSICS Vol. 47 No. 2 2002 202 GINZBURG, ZHILINSKIŒ

ξ(បω), 10–1 (eV)–2 One of the roots of (27) is evident. It corresponds to 3.5 the condition β Ӎ 0. (29) 3.0 From Eq. (28), it follows that other real positive root Ω (if any) must explicitly depend on m defined by (16). 2.5 According to (10), β is the rate of change of the density of states at the point E = δ. Therefore, the use of any root other than (29) entails the situation when β 2.0 depends not only on the behavior of g(E) in the vicinity δ of E = but also on the energy Em, which is separated 1.5 from δ by a finite interval. This seems to be a rather specific case. If it is out of consideration, only root (29) remains. As was mentioned above, it indicates the pres- 1.0 ence of the CG.

0.5 2. EXAMPLES Amorphous Semiconductors 5.5 6.0 6.5 បω × 10, eV We will compare two PL spectra taken from amor- Fig. 1. ξ(បω) function. Reference points are obtained from phous semiconductors [14, Fig. 3; 15, Fig. 2D]. These data reported in [14, Fig. 3]. spectra were chosen because of their apparent similarity. Both start from the zero value of ∆I(បω), peak at close energies (1.05 and 1.33 eV), and have comparable ξ(បω), 10–2 (eV)–2 half-widths (0.46 and 0.29 eV). However, as is seen 7 from Figs. 1 and 2, the low-frequency behavior of the ξ(បω) functions defined by (4) is entirely different. The 6 curve in Fig. 1 [14] is well fitted by (15). This can be checked by plotting the curve 5 dξ()បω η()បω = ------, (30) d()បω 4 which, under condition (5), must have the linear part 3 η()បω = 2Cបω Ð B. (31) 2 For testing, we used the least-squares method. Con- structing tangents to the curve in Fig. 1, we obtained a 1 set of closely spaced (0.005 eV apart) empirical points approximating the η(បω) function. The root-mean- 0.92 0.94 0.96 0.98 1.00 បω, eV square deviation for the first ten of them turned out to equal 6.7%. This value indicates that these points can Fig. 2. ξ(បω) function. Reference points are obtained from actually be approximated with a straight line. The coef- data reported in [15, Fig. 2D]. ficients of the line were determined by solving the corresponding normal equations: C = 19.11 (eV)Ð4 and B = 8.38 (eV)Ð3. The straight line plotted with these coeffi- The substitution of (19)Ð(21) into (26) and appropri- cients (Fig. 3) is in good agreement with dependen- ate rearrangements give ces (30) and (31). 3 2 Substituting C and B derived above into Eq. (5) and ββ()+++rβ sβ t Ӎ 0, (27) using the curve in Fig. 1, one finds that A Ӎ 1.78 (eV)Ð2. where Hence, it follows that AC Ӎ 0.24B2, which is consistent with condition (6). γ ()2 γ ()2 γ ()2 γ ()3 2 r ==2------Ω , s ------5------Ð 4------Ω , The behavior of the curve in Fig. 2 indicates that, γ ()1 m γ ()1 γ ()1 γ ()2 m conditions (5) and (6) fail in the case [15]. In [14], (28) a-As S , which belongs to the CGS class, was studied. γ ()γ () γ () 2 3 3 2 4 Ω3 As was mentioned in the Introduction, semiconductors t = 4γ------()-------Ð ------m. 1 γ ()1 γ ()3 of these type contain charged UÐ defects. The PL spec-

TECHNICAL PHYSICS Vol. 47 No. 2 2002 CHARGED DEFECT DIAGNOSTICS FROM LUMINESCENCE SPECTRA 203

η(បω), (eV)–3 Red Glow in Optical Fibers

2.75 Red glow (RG) observed in 3D samples of g-SiO2 near 1.9 eV is usually attributed to nonbridging oxygen ≡ ↑ 2.50 atoms, which form the SiÐO structure (the lines depict bonds; the arrow, an unpaired electron (spin)). However, the appearance of such defects during draw- 2.25 ing low-OH optical ﬁbers has not been completely understood. In [11], a process accounting for all atten- 2.00 dant factors (an excess of oxygen and the lack of E' centers) was proposed in the form 2()≡SiÐOÐOÐSi≡ 1.75 (32) ()≡≡SiÐO↑O+ÐSi≡ + ()SiÐOÐ↑OÐSi≡ .

5.3 5.4 5.5 5.6 បω, 10–1 eV A specific feature of Eq. (32) is that nonbrinding atoms are incorporated into oppositely charged com- Fig. 3. η(បω) function constructed with the tangents to the plexes; thus, it is impossible to exclude a priori the curve in Fig. 1. competition between the transitions within the CG. Analysis of the data reported in [17, 18] confirmed the fulfillment of conditions (5) and (6) [11]. Special atten- tra from a-Si, where the most typical defects are neutral tion should be paid to the spectra presented in [18, Fig. 4]. [16] dangling bonds, were considered in [15]. These spectra were obtained by two different ways. In the first case, measurements were performed at the end Note that the fundamental difference in the charac- of a 300-m-long optical fiber, so that transmission ter of the defects was revealed by studying the PL low- losses had to be taken into account (e measurement). frequency tails, comprising no more than 10% of the The other spectrum was detected in the immediate spectrum half-width. proximity to the side face of the fiber (s measurement). Being generally similar, the associated curves exhibit a The same result was obtained for other six PL spec- small yet visible difference in the low-frequency region tra from CGSs [7]. In [7], a-Si samples prepared under [18, Fig. 4]. The ξ(បω) and η(បω) functions plotted different conditions were also studied. Their PL spec- with the use of [18, Fig. 4] and data in [11] are shown trum was shifted towards lower frequencies. The corre- in Figs. 4 and 5, respectively. The difference in the low- sponding ξ(បω) function is presented in [7, Fig. 6] and frequency regions of the RG curves is in the different looks like that shown in Fig. 2. arrangements of the linear parts of the η(បω) functions

3.5 7 η(បω), 10–1 (eV)–3

3.0 6 4.0 2.5 5 –2 –2 3.5 s (eV) 2.0 4 (eV) –2 –2 3.0 10 10

, , s e ) ) 1.5 3 ω ω ប ប ( ( 2.5 ξ ξ 1.0 2 2.0 e 0.5 1 1.5

1.65 1.70 1.75 បω, eV 1.625 1.650 1.675 បω, eV

Fig. 4. ξ(បω) functions. Reference points were obtained from the data presented in [18, Fig. 4]. The curves are taken Fig. 5. η(បω) functions constructed with the tangents to the from [11]. curve in Fig. 4. The curves are taken from [11].

TECHNICAL PHYSICS Vol. 47 No. 2 2002 204 GINZBURG, ZHILINSKIŒ

ξ(បω), 10–2 (eV)–2 defined by (30) and (31). Nevertheless, as was mentioned above, Eqs. (5) and (6) hold in both cases. This 2.0 example illustrates the high reliability of the technique for CD diagnostics, since it proves to be insensitive to 1.5 small variations in the shape of the low-frequency SA4 region of the PL spectrum. Figure 6 shows the ξ(បω) functions constructed 1.0 OH1 from the RG spectra presented in [19, Fig. 12]. These curves markedly deviate from law (5). At first glance, 0.5 this looks surprising, because dehydrated optical fibers used in [19] were similar to those studied in [17, 18]. However, the former were subjected to γ irradiation, which is known to generate Compton electrons; 1.650 1.675 1.700 1.725 បω, eV thereby, the following processes seem to be plausible: Fig. 6. ξ(បω) functions. Reference points were obtained Ð បω ↑ បω Ð from the data presented in [19, Fig. 12]. SA4 and OH1 indi- O1 +++1 O1 2 e , (33) cate low-OH samples. ≡ Ð បω ≡ ↑ បω Ð SiÐO +++1 SiÐO 2 e . (34) ∆I(បω), arb. units 7 Both processes give rise to nonbrindging oxygen atoms, and process (34) destroys the CG formed by the charged complexes. 6

5 Triboluminescence Triboluminescence (TL) is luminescence induced 4 by mechanical action. This phenomenon still lacks adequate explanation. Our interest in TL in the context of this study is dictated by the following. In nonirradiated 3 optical fibers, red glow was attributed [11] to transitions between charged defects. But unlike other types 2 of glow, such as that from UÐ centers in CGSs, it displays no detectable Stokes shift. Hence, one could sug- 1 gest a correlation between process (32), which results from fiber drawing, and TL due to fiber breakage, which latter is centered near 1.97 eV [20]. Difficulties 1.4 1.5 1.6 1.7 បω, eV emerge when we analyze the TL spectrum presented in [20, Fig. 2b], since it is described by a polygonal line Fig. 7. Averaged low-frequency dependence ∆I(បω) with the spread of ∆I(បω) comparable to the value of according to [20]. ∆I(បω) in the low-frequency region. Visually, however,

ξ(បω), 10–1 (eV)–2 η(បω), 10–1 (eV)–3 2.5 5.5 2.0 5.0 1.5 4.5 1.0 4.0 0.5 3.5 1.4 1.5 1.6 1.7 បω, eV 1.4 1.5 1.6 1.7 បω, eV

Fig. 8. ξ(បω) function. Reference points were obtained Fig. 9. η(បω) function constructed with the tangents to the from Fig. 7. curve in Fig. 8.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 CHARGED DEFECT DIAGNOSTICS FROM LUMINESCENCE SPECTRA 205 the spread is highly symmetric about some center line. 6. K. D. Tséndin, in Electron Phenomena in Glassy Chal- Then, when applied to this line, or technique for CG cogenide Semiconductors, Ed. by K. D. Tséndin (Nauka, detection may give a valid result. The low-frequency St. Petersburg, 1996). region of this line is shown in Fig. 7. The dependence 7. L. P. Ginzburg, J. Non-Cryst. Solids 171, 172 (1994). ξ(បω) derived from the curve in Fig. 7 and the corre- 8. R. Chicon, M. Ortun˜ o, and M. Pollak, Phys. Rev. B 37, sponding η(បω) function are plotted in Figs. 8 and 9, 10520 (1988). respectively. At first glance, they show evidence of the 9. M. Kastner and H. Fritsche, Philos. Mag. B 37, 199 CG. Counting in favor of this are both the shape of the (1978). curve in Fig. 8 and the presence of the linear part in the 10. G. Lukovsky, Philos. Mag. B 41, 457 (1980). curve in Fig. 9 (the variation of the first-order differ- 11. L. P. Ginzburg, Fiz. Khim. Stekla 25 (2), 47 (1999). ences between the first five points obtained with the 12. J. M. Ziman, Models of Disorder: The Theoretical Phys- tangents to Fig. 8 amounted to 9.2%). However, the ics of Homogeneously Disordered Systems (Cambridge application of Eqs. (5) and (31) to Figs. 8 and 9 yields Univ. Press, Cambridge, 1979; Mir, Moscow, 1982). B = 0.21 (eV)Ð3, C = 0.21 (eV)Ð4, and A Ӎ Ð0.08 (eV)Ð2. 13. J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. B 29, These values are is consistent with Eq. (6); hence, the 4260 (1984). TL under study does not relate to the CG. Note that this 14. F. Mollot, J. Chernogora, and C. Benoit à la Gillaume, result reflects the well-know situation when the peaks Phys. Status Solidi A 21, 281 (1974). of TL and PL are coincident but the low-frequency 15. I. G. Austin, T. S. Nashashibi, T. M. Searle, and regions are considerably different [21]. W. E. Spear, J. Non-Cryst. Solids 32, 373 (1979). 16. B. L. Gel’fand and B. I. Shklovskiœ, in Physics of Com- pound Semiconductors. Proceedings of the IX Winter REFERENCES School on Physics of Semiconductors, Leningrad, 1979, p. 5. 1. B. I. Shklovskiœ and A. L. Efros, Electronic Properties of 17. P. Kaiser, J. Opt. Soc. Am. 64, 475 (1974). Doped Semiconductors (Nauka, Moscow, 1979; Springer-Verlag, New York, 1984). 18. Y. Hibino, H. Hanafusa, and S. Sakaguchi, Appl. Phys. Lett. 47, 1157 (1985). 2. A. Hartstein, Z. A. Weinberg, and D. Y. DiMaria, in 19. S. Munekuni, T. Yamanaka, Y. Shimogauchi, et al., Physics of SiO2 and Its Interfaces, Ed. by S. T. Pantelides J. Appl. Phys. 68, 1212 (1990). (Pergamon, New York, 1978), p. 51. 20. A. J. Smiel and T. A. Fisher, Appl. Phys. Lett. 41, 324 3. L. P. Ginzburg, J. Non-Cryst. Solids 171, 164 (1994). (1982). 4. L. P. Ginzburg, Fiz. Tekh. Poluprovodn. (Leningrad) 23, 21. J. I. Zink, Acc. Chem. Res. 11 (8), 289 (1978). 1629 (1989) [Sov. Phys. Semicond. 23, 1008 (1989)]. 5. G. Lucovsky, Philos. Mag. B 39, 531 (1979). Translated by A. Sidorova-Biryukova

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 206Ð208. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 65Ð67. Original Russian Text Copyright © 2002 by Reznichenko, Shilkina, Turik, Dudkina.

SOLIDS

Giant Piezoelectric Anisotropy in Sodium Niobate with a Composite-Like Structure L. A. Reznichenko, L. A. Shilkina, A. V. Turik, and S. I. Dudkina Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344090 Russia e-mail: [email protected] Received May 18, 2001

Abstract—On the basis of X-ray diffraction and electrophysical data for stoichiometric and nonstoichiometric sodium niobate, it is concluded that a mechanism responsible for the inﬁnite ratio of the electromechanical coupling factors for the thickness and plane modes may be related to the composite-like structure of the material. It is shown that such a structure may result from the presence of ordered extended voids (arising at the joints between perfect structure blocks), which behave as microcracks. Stresses generated at the joints may modify the domain structure and lead to the anisotropic distribution of polarized clusters in the ferroelectric phase. This also may enhance the piezoelectric anisotropy. © 2002 MAIK “Nauka/Interperiodica”.

In [1], we have discovered the piezoelectric effect in zero or even vanish and extremely small (down to sev- polarized stoichiometric and nonstoichiometric sodium eral unities) values of the mechanical quality factor niobate (NaNbO3) ceramics in the temperature range of QMm. Since such a combination of parameters is char- ° 20Ð300 C. This effect is accompanied by the piezores- acteristic of some NaNbO3-based ceramics, one can onant dispersion of the permittivity. It was noted that expect that their structure is also composite-like. this circumstance, together with the observation of dielectric hysteresis loops [2], indicates that polar It is likely that the mechanisms of forming the com- microdomains, or ferroelectric clusters, form in the posite-like structure in NaNbO3- and PbTiO3-based ceramics is the same, i.e., the development of the aniso- Pbma antiferroelectric rhombic phase of NaNbO3 subjected to a polarizing electric field. For the stoichiomet- tropic structure of extended defects (like microcracks ric composition, the values of the piezoelectric moduli [6Ð9]). However, since high-resolution optical and d measured at room temperature ranged from 25 to electron microscopy methods have failed to visualize 33 regularly arranged microcracks in NaNbO (unlike 50 pC/N. The values of the electromechanical coupling 3 coefficients for the thickness mode were found to be PbTiO3), one may assume that other extended defects K = 0.25Ð0.31; those of the mechanical quality factor, form voids similar to microcracks. The origination of t such voids can be related to the specific crystal-chemi- QMm = 2Ð7. All the compositions had very small planar ⇒ cal structure of the object, namely, its block structure electromechanical coupling coefficients (Kp 0) and ⇒ due to developing shear microstructures [10Ð12]. Their piezoelectric moduli (d31 0) and, hence, very large existence in NaNbO , as well as in more complex nio- anisotropies of the electromechanical coupling coeffi- 3 ∞ | | ∞ bium oxides and their solid solutions (as demonstrated cients, Kt/Kp , and piezomoduli, d33/ d31 . with the method of X-ray powder diffraction [13Ð15]), Earlier, a similar situation was investigated [3, 4] in is explained by the fact that Nb2O5, the thermally stablest compound of those participating in the synthesis several PbTiO3-based ferroelectric ceramics. There- fore, it was supposed [1] that the reason for the large of niobate materials, is prone to void formation [16, 17]. piezoelectric anisotropy in NaNbO3 and PbTiO3 is the same: small values and low anisotropy of the permittiv- A characteristic feature of the shear microstructures ity tensor components in single-domain metastable fer- is the elimination of anion vacancies, arising when the roelectric NaNbO3 crystals, as well as the specific degree of oxidation of Nb slightly changes Ð domain structure and distribution of clusters in the fer- ()Nb5+ Nb4+ O2Ð ᮀ (ᮀ is an anion vacancy), upon roelectric phase. 1 Ð x x 2 5 Ð x x shearing one part of the structure relative to the other In this study, we hypothesize that the very large along a certain crystallographic direction. In Nb2O5, piezoelectric anisotropy in NaNbO3 is related to its shear develops, as a rule, in two or three almost orthog- composite-like structure. The main distinctions onal directions, giving rise to rectangular infinitely long between composite-like materials and ordinary ceram- m × n columns of oxygen octahedrons or m × n × p (p = ics are [5] rather high Kt when plane modes are close to 1, 2) blocks [16, 17], respectively. At sites where the

GIANT PIEZOELECTRIC ANISOTROPY IN SODIUM NIOBATE 207 columns and blocks join together, voids (channels) (a) (b) 110 110 form, which may contain both octa- and tetrahedral 101 101 – – positions [16]. The existence of the latter was demon- 101 101 strated by X-ray emission spectroscopy [18]. Here, Nb atoms, retaining the octahedral configuration, occupy two levels (Z = 0 and Z = 1/2), being displaced by half- diagonal of the octahedron in neighboring blocks. Twelve of 14 known Nb2O5 modifications have the block structure. The stablest high-temperature poly- α morphic modification, -Nb2O5, is obtained from the

ReO3-type structure by alternating the (209 ) and (601) × planes. The resulting block structure has the form 3 C1 × × × ∞ × × ∞ C1 C3C2 4 1 + 3 5 , the columns 3 5 being parallel C2 to the c axis of the Nb2O5 unit cell. 2θ 39°2θ 39° In NaNbO3, we found three systems of crystallographic shear planes that are also parallel to the c axis ([010]) of the crystal. They form two types of the cross Fig. 1. Fragments of the X-ray diffraction patterns for the × NaNbO3 ceramics including the (100) and (101 ) reflec- sections of the m n octahedrons. One refers to the tions and (a) two (C1, C2) and (b) three (C1, C2, C3) satel- m × n × p blocks; the other, to the m × n × ∞ columns lites. of octahedrons. The direction of displacement of Nb atoms from their perfect sites, which was determined in the single crystals, suggests that the (102) and (201) NaNbO3 samples of stoichiometric and nonstoichi- planes are crystallographic shear planes and p in the ometric compositions were obtained by the conven- formula m × n × p equals two. tional ceramic technology (solid-phase synthesis at 800Ð850°C, followed by pressure-free sintering at The alternation of the crystallographic shear planes 1220Ð1240°C). We used extra-pure grade and commer- must cause satellite maxima to appear in the X-ray dif- cial-grade Nb2O5. Powders synthesized were (or were fraction pattern near the basic reflections. The spacing not) granulated by the two-stage grinding of pressed between the satellites and the basic reflections is blocks first through a sieve with a mesh of 0.7Ð0.9 mm inversely proportional to the wavelength (λ) of a distur- and after four-hour keeping in a closed vessel, through bance caused by crystallographic shear, deforming the a sieve with a mesh of 0.25Ð0.30 mm. Such a procedure ideal lattice, i.e., to the spacing between the crystallo- significantly increases the density of the granulated graphic shear planes [19, 20]. The X-ray diffraction material and (after its sintering) ceramic samples. patterns from the samples under investigation contain The table lists the density ρ; wavelength λ; the num- two (C1 and C2) (Fig. 1a) or three (C1, C2, and C3) (Fig. 1b) satellites corresponding to the spacing ber N of oxygen octahedrons between the crystallographic shear planes; factors Kt and Kp; and Irel, which between the crystallographic shear planes (70–80 Å (C1), is the ratio of the (430) reflection intensity for impurity 60–70 Å (C ), and 40 Å (C )) and indicating the forma- 2 3 compound NaNb3O8 to the (100) reflection intensity for tion of the columns and/or perfect blocks. The appear- NaNbO3 of stoichiometric and nonstoichiometric com- ance of the satellites only on the side of smaller angles position. θ seems to reflect the correlation of shift and substitution disorders. The infinitely long columns regularly It is seen that the less dense samples with a small arranged along the [010] direction join via ordered amount of the NaNb3O8 intermediate phase has the infi- channels with vacant or filled octahedral positions. nite anisotropy of properties. The finite value of the

Structure parameters and the electrophysical properties of stoichiometric and nonstoichiometric sodium niobate

ρ, C1 C2 Composition ρ 3 Kt Kp Irel /cm λ, Å N λ, Å N

NaNbO3 extra-pure grade granulated 4.31 0.31 0.10 69.4 12.6 42.12 7.6 0 NaNbO3 extra-pure grade nongranulated 3.1 0.31 0 79.3 14.4 40.9 7.4 8 NaNbO3 commercial-grade granulated 3.23 0.26 0 67.5 12.5 41.38 7.5 10 Na0.88NbO3 commercial-grade nongranulated 3.09 0.25 0 68.5 12.4 41.02 7.44 10 Na0.86NbO3 commercial-grade nongranulated 3.40 0.25 0 67.9 12.6 41.38 7.50 10

TECHNICAL PHYSICS Vol. 47 No. 2 2002 208 REZNICHENKO et al. ratio Kt/Kp is typical of high-density and pure NaNbO3 3. A. V. Turik and V. Yu. Topolov, J. Phys. D 30, 1541 obtained from the granulated powders. The reason may (1997). be as follows. 4. A. V. Turik, Ferroelectrics 222, 33 (1999). It is well known [21] that, in perovskites, shear 5. W. A. Smith and A. A. Shaulov, Ferroelectrics 87, 309 causes not only anion but also cubeÐoctahedral A sites (1988). 6. D. I. Makar’ev, A. N. Klevtsov, V. A. Servuli, and to disappear. As a result, NaNbO3 always contains an excess amount of Na atoms (4Ð6 at.%). If the sample L. A. Shilkina, in Proceedings of the International Con- ference “Fundamental Problems of Piezoelectric Instru- preparation conditions provide the maximum possible ment Making” (“Piezotechnology-99”), Rostov-on- solubility of the components (granulation at the stage of Don, 1999, Vol. 1, p. 44. synthesis, sintering by hot pressing, etc.), these free 7. E. A. Dul’kin, L. I. Grebenkina, D. I. Makar’ev, et al., atoms can occupy regular and irregular vacant sites in Pis’ma Zh. Tekh. Fiz. 25 (22), 21 (1999) [Tech. Phys. the NaNbO3 structure, including those inside the inter- Lett. 25, 894 (1999)]. columnar and interblock channels. Then, since evi- 8. D. I. Makar’ev, A. N. Klevtsov, and V. A. Servuli, Elec- dence for superlattice ordering of Na is lacking, it is tronic Journal Issledovano v Rossii, No. 57 (1999), reasonable to assume that its distribution in these chan- http://Zhurnal/mipt/rssi/ru/articles/1999/058.pdf. nels agrees with the model of random filling, as is the 9. D. I. Makar’ev, A. N. Klevtsov, and V. A. Servuli, in Pro- case for Ge atoms in the GeO2 : 9Nb2O5 compound ceedings of the XV All-Russia Conference on Physics of [22]. Sodium niobate crystallizing under these condi- Ferroelectrics (VKS-XV), Rostov-on-Don, 1999, p. 292. tions is pure and has a high density. When Na atoms 10. A. D. Wadsley, Rev. Pure Appl. Chem. 5, 165 (1955). form a side impurity compound rather than fill the 11. A. D. Wadsley, in Non-Stoichiometric Compounds (Aca- voids, these channels seem to behave as directional demic, London, 1964), p. 98. microcracks. High stresses arising at the sites of crack 12. J. S. Anderson, in Surface Defect Properties of Solids nucleation abruptly increase the internal friction and, as (Chemical Society, London, 1972), Vol. 1, p. 5. a consequence, decrease QMm. At the same time, the 13. L. A. Reznichenko, O. N. Razumovskaya, and L. A. Shil- voids suppress the plane mode, thereby increasing the kina, in Proceedings of the International Conference ratio Kt/Kp. The modification of the domain structure “Piezotechnology-99,” Obninsk, 1997, p. 191. and the anisotropic distribution of polarized ferroelec- 14. L. A. Shilkina, L. A. Reznichenko, and O. N. Razumov- tric clusters, which is induced by the stresses, may also skaya, in Proceedings of the 8th International Sympo- be factors contributing to the piezoelectric anisotropy. sium on Physics of Ferroelectric Semiconductors The validity of this assumption is supported by the infi- (JMES-8), Rostov-on-Don, 1998, Vol. 7, p. 192. nite piezoelectric anisotropy observed in high-porosity 15. L. A. Reznichenko, L. A. Shilkina, and O. N. Razumov- PbTiO3-based ceramics, having a large number of skaya, in Proceedings of the XV All-Russia Conference microcracks and, consequently, a low strength [6Ð9]. on Physics of Ferroelectrics (VKS-XV), Rostov-on-Don, 1999, p. 294. Thus, the additional reason for giant piezoelectric 16. C. N. R. Rao and J. Gopalakrishnan, New Directions in anisotropy in some types of stoichiometric and nonsto- Solid State Chemistry (Cambridge Univ. Press, Cam- ichiometric NaNbO3 is its composite-like structure bridge, 1986; Nauka, Novosibirsk, 1990). caused by the presence of ordered extended defects 17. F. Fairbrother, The Chemistry of Niobium and Tantalum (voids), which arise where the columns and perfect (Elsevier, Amsterdam, 1967; Khimiya, Moscow, 1972). blocks join and behave as microcracks. 18. L. A. Reznichenko, L. A. Shilkina, S. V. Titov, et al., in Note that since the shear microstructures “inherited” Proceedings of the International Symposium “Ordering from rutile TiO2 [16] also take place [23] in PbTiO3- in Minerals and Alloys” (OMA-2000), Rostov-on-Don, based ceramics and related Ti-containing materials, this 2000, p. 111. factor may contribute to the anisotropy of the piezo- 19. W. A. Wooster, Diffuse X-ray Reflections from Crystals electric coefficients in these materials. (Clarendon, Oxford, 1962; Inostrannaya Literatura, Moscow, 1963). 20. A. Guinier, Theórie et Technique de la Radiocristallog- ACKNOWLEDGMENTS raphie (Dunod, Paris, 1956; Fizmatgiz, Moscow, 1961). This work was financially supported in part by the 21. A. G. Petrenko and V. V. Prisedskiœ, Structure Defects in Russian Foundation for Basic Research (grant no. 99- Ferroelectrics (Uchebno-metodicheskiœ Kabinet po 02-17575). Vysshemu Obrazovaniyu pri Minvuze USSR, Kiev, 1989). 22. A. Ann, A. McConnell, J. S. Anderson, and C. N. R. Rao, Spectrochim. Acta A 32, 1067 (1976). REFERENCES 23. L. A. Reznichenko, L. A. Shilkina, S. V. Titov, and 1. L. A. Reznitchenko, A. V. Turik, E. M. Kuznetsova, and O. N. Razumovskaya, in Proceedings of the Interna- V. P. Sakhnenko, J. Phys.: Condens. Matter 13, 3875 tional Symposium “Ordering in Minerals and Alloys” (2001). (OMA-2000), Rostov-on-Don, 2000, p. 127. 2. R. H. Dungan and R. D. Golding, J. Am. Ceram. Soc. 47 (2), 73 (1964). Translated by Yu. Vishnyakov

TECHNICAL PHYSICS Vol. 47 No. 2 2002

SOLID-STATE ELECTRONICS

Multislot Microwave Transmission Lines Fabricated on an Insulating Substrate Covered with a Ferroelectric Film I. G. Mironenko and A. A. Ivanov St. Petersburg State Electrotechnical University, St. Petersburg, 197376 Russia e-mail: [email protected], [email protected], [email protected] Received January 25, 2001

Abstract—A rigorous electrodynamic model is used to calculate the dispersion characteristics of multislot microwave transmission lines fabricated on a layered structure containing a ferroelectric ﬁlm. The attenuation due to the ﬁnite conductivity of the metal electrodes is found. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION vectors aligned with one of the coordinate axes (z or y). A structure consisting of a ferroelectric film applied It is preferred to align the Hertz vectors with the y axis, on a dielectric substrate underlies slot and coplanar because they produce the LE and LM modes, which microwave transmission lines [1Ð3]. These lines can be turn out to be related through the electrode current at used to construct devices with electrically controlled the final stage of the analysis when the boundary con- characteristics basically because the permittivity ε of ditions are imposed in the plane y = 0. In the absence of the ferroelectric film can vary with an applied electric the electrodes, the structure illustrated in Fig. 1 is a lay- field. The permittivity of most ferroelectric materials ered inhomogeneous waveguide sandwiched in con- changes by a factor of 1.5–2 when the electric field ducting planes. A natural basis for this waveguide con- intensity is 2Ð3 kV/mm. Therefore, for the voltage consists of the LE and LM waves, which are related through trolling the line to be low, the slots must be sufficiently the electrode current in the hybrid mode. narrow. However, narrow slots introduce losses, because the conductivity of the electrodes is finite As is known, to analyze the dispersion and attenua- [1−3]. This contradiction is reconciled in a planar tion characteristics in such planar waveguides, one waveguide structure which we will further refer to as a should solve integral equations for the current distribu- multislot microwave transmission line. In this line, nar- tion over the electrodes or for the electric field distribu- row electrodes located inside of a wide slot on the sur- tion over the slots. An efficient procedure for solving face of the ferroelectric film serve simultaneously as a the equations is the Galerkin method, which provides as waveguide, transmitting the electromagnetic energy, high an accuracy as other ways of solving integral and as control electrodes, which produce the control equations. Therefore, we use the Galerkin method at electric field of a desired intensity in the narrow inter- the final stage of the analysis. electrode gaps over the entire wide slot. The purpose of this paper is to analyze the dispersion and attenuation characteristics of such multislot y transmission structures.

hi wi ε d STATEMENT OF THE PROBLEM 4 0 AND VALIDATION OF THE ELECTRODYNAMIC ε d ε2 2 MODEL d x ε Figure 1 illustrates the cross section of the transmis- 1 d1 sion line under study. We will deal with a rigorous electrodynamic model. Though relatively intricate, it accu- ε rately describes the real structure. The fundamental 0 d0 ≠ mode of this structure is a hybrid wave (Ez 0 and ≠ Hz 0). It can be described in terms of two scalar potentials, for example, the electric and magnetic Hertz Fig. 1. Cross section of the multislot planar structure.

210 MIRONENKO, IVANOV BASIC STAGES OF THE SOLUTION Relationships (3) thus determine the Fourier trans- PROCEDURE forms of the scalar potentials up to arbitrary factors. (1) Formulas for the scalar potentials. Let us (2) Derivation of the integral equations. To derive specify the electric and magnetic potentials as the integral equations, we use the boundary conditions for the Fourier transforms of the tangential components ϕ(), ()()γ ω Ae= y xyexp Ð j z Ð t , of the electric and magnetic ﬁeld intensities in the plane y = 0. Let us write the relationships for the Fourier Fe= Ψ()xy, exp()Ð j()γz Ð ωt , y transforms of the ﬁeld components that directly follow where γ is the propagation constant. from the Maxwell equations:

Both potentials satisfy the wave equation in any Hx()ys, = jγϕ()ys, + ()∂Ψs/ωµ ()()ys, /∂y , domain of the cross section. Consider the Fourier trans- 0 forms of the potentials: (), ()∂ϕωε ε ()(), ∂ γΨ(), Ex ys = s/ 0 ys/ y Ð j ys, (4) +∞ (), ϕ()γ, ()∂Ψωµ ()(), ∂ ϕ()ys, ϕ()xy, Hz ys = js ys Ð / 0 ys/ y , = ∫ exp()Ð jsx dx. Ψ()ys, Ψ()xy, (), ()∂ϕγ ωε ε ()(), ∂ Ψ(), Ð∞ Ez ys = Ð / 0 ys/ y Ð js ys.

The Fourier transforms ϕ (y, s) and Ψ (y, s) satisfy The tangential components Ex and Ez, as well as the homogeneous equation their Fourier transforms, are continuous within the slots: ϕ ()ys, ()+ ()Ð 2 2 2 i (), (), ()∂ ∂ α  Ez 0 s = Ez 0 s , / y + i = 0 (1) (5) Ψi()ys, ()+ ()Ð Ex ()0, s = Ex ()0, s . α2 in any domain of the cross section. In Eq. (1), i = In other words, relationship (5) must be satisfied in γ 2 2 2ε 2 ω2ε µ the plane y = 0. Writing Eqs. (5) in view of (3) and (4) ( + s Ð k i), k = 0 0, and i is the index of the corresponding domain in Fig. 1. and eliminating two unknown functions, we obtain m() m()ε ε ()[]()+ (), []()Ð (), The solution to Eq. (1) in the ith domain has the B1 s = B / 2 A 0 s / A 0 s , form () () (6) Be ()s = Be()s ()[]F + ()0, s /[]F Ð ()0, s , ϕ (), ()() α ()() α 1 i ys = Ai s sinh i y + Bi s cosh i y , (2) where [A±(0, s)] = ∂A±(y, s)/∂y| . Ψ (), ()() α ()() α y = 0 i ys = Ci s sinh i y + Di s cosh i y . Using relationships (3), (4), and (6), Ex and Ez in Clearly, the boundary conditions for the Fourier the plane y = 0 can be written as transforms are the same as those for the electromag- (), ()ωε ε ()m()[]()+ (), netic potentials: (i) ∂ϕ (y, s)/∂y = 0 and Ψ (y, s) = 0 on Ex ys = s/ 0 2 B s / A 0 s (7) the perfectly conducting planes and (ii) the Fourier e ()+ Ð jγ B F ()ys, , transforms and their normal derivatives (∂ϕ (y, ∂ ε ∂ψ (), () γωε ε ()m()[]()+ (), s)/ y)(1/ ) and (y, s)/dy are continuous at the bound- Ez ys = Ð / 0 2 B s / A 0 s aries of the insulating layers. Using (2) and the bound- (8) e ()+ (), ary conditions, the scalar functions ϕ (y, s) and ψ (y, s) Ð jsB F ys. can be found up to constant factors. Let us represent These relationships for the Fourier transforms of the these functions in different domains of the cross section electric field components in the plane y = 0 are indepen- as dent of the electrode geometry. ≥ ϕ()+ (), m() ()+ (), As is known, the narrower the domain of definition y 0: ys = B s A ys; of unknown functions, the better the convergence of the Ψ()+ (), e() ()+ (), Galerkin method. In our structure, these domains are ys = B s F ys; slots between the electrodes. Therefore, we will derive () () (3) y ≤ 0: ϕ Ð ()ys, = Bm()s A Ð ()ys, ; integral equations for the fields in the slots. When the 1 conductivity (σ) of the electrode metal is finite, the Ψ()Ð (), e () ()Ð (), boundary conditions on the electrodes are written in ys = B1 s F ys, terms of the electrode currents. Assume that the electrodes are nontransparent for the field, though infinites- m e where B (s), …B1 (s) are arbitrary functions and imally thin. In this case, the tangential magnetic field A(+)(y, s), F(+)(y, s), A(Ð)(y, s), and F(Ð)(y, s) are the known experiences a discontinuity equal to the surface current functions presented in the Appendix. density js on the electrodes; i.e., the boundary condition

TECHNICAL PHYSICS Vol. 47 No. 2 2002 MULTISLOT MICROWAVE TRANSMISSION LINES 211 on the conducting electrodes (the plane y = 0) can be The left-hand side of formulas (12) and (13) is written as known from expressions (7) and (8). The Fourier trans- () () forms for the components of the magnetic field are []()ú + (), ú Ð (), found from (3) and (4). Therefore, in view of (12) and ey H x 0 Ð H x 0 = js, (9) (13), the remaining two unknown coefficients Bm and where the brackets mean the vector product. 1 e We introduce the surface impedance of the elec- B1 can be expressed through the field distribution in ωµ σ the slot as trodes in the form Zs = (1 + j)0/2 and write expression (9) in the equivalent form in terms of the  field on the electrodes m ωε ε γ () B = 0 2Ð ∑gi s () ()  ú (), []()ú + (), ú Ð (), i E x 0 = Zs ey H x 0 Ð H x 0 . (10) With (10), the Fourier transform of the component   2 2 ()+ m ú + sq∑ ()s  ()()γ + s []ξA ()0, s  , Ex of the field in the plane y = 0 can be represented as i   i +∞ (14) (), ú (), ()  Ex 0 s = ∫ Ex x 0 exp Ð jsx dx e γ () B = j ∑qi s Ð∞  i slot edge ()ú ()+ (), ú ()Ð (), () = Zs ∫ Hz x 0 Ð Hz x 0 exp Ð jsx dx   2 2 ()+ e Ð∞ + sg∑ ()s  ()()γ + s []ξF ()0, s  , i   i ú (), () + ∑ ∫Ex x 0 exp Ð jsx dx (11) over slots where

()ú ()+ (), ú ()Ð (), () ξe ()π () + ∑ ∫ Hz x 0 Ð Hz x 0 exp Ð jsx dx = 1 + Zs/120 k Gs, over electrodes ξm = 1 Ð ()Z 120π kε Ps(), +∞ s 2 ()+ ()Ð + Z ()Hú ()x, 0 Ð Hú ()x, 0 exp()Ð jsx dx. () () s ∫ z z Gs()= ()[]F + ()0, s /F + ()0, s slot edge () () Ð ()[]F Ð ()0, s /F Ð ()0, s , Since Hú z is continuous in the slots, formula (11) can () () be recast as Ps()= ()A + ()0, s /[]A + ()0, s ()Ð ()Ð Ex()0, s ()ε ε ()[](), (), Ð / 2 A 0 s /A 0 s . (12) ()ú ()+ (), ú ()Ð (), () = Zs Hz 0 s Ð Hz 0 s + ∑ qi s . The condition that the Fourier transforms of the tan- over slots gential magnetic ﬁeld components be continuous in the slot, A similar expression can be obtained for Ez (0, s): ()+ (), ()Ð (), Hx 0 s = Hx 0 s , E ()0, s z ()+ (), ()Ð (), Hz 0 s = Hz 0 s , ()+ ()Ð (13) = Z ()Hú ()0, s Ð Hú ()0, s + g ()s . s x x ∑ 1 yields the equations over slots () In these formulas, jγ []A + ()0, s Ps()Bm

() (), () ()+ e qi s = ∫ Ex x 0 exp Ð jsx dx, + ()sωµ F ()0, s Gs()B = 0, 0 (15) ith slot () js[] A + ()0, s Ps()Bm () (), () gi s = ∫ Ez x 0 exp Ð jsx dx. ()γ ωµ ()+ (), () e ith slot + / 0 F 0 s GsB = 0.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 212 MIRONENKO, IVANOV

y slot. Then, hi h0 hi w /2 h1 i () (), (), () x qi s = 2 ∫ Exi, x 0 cos x0, i s exp Ð jsx dx, w w1 wi i Ðw /2 i (17) y wi/2 w w w i 0 i () (), (), () gi s = 2 ∫ Ezi, x 0 cos x0, i s exp Ð jsx dx.

x Ðwi/2 hi h1 hi As is known, the integrals of the Chebyshev polyno- Fig. 2. Cross section of multislot planar structures with an mials are calculated exactly [5]. In view of this, (17) even and odd number of slots. can be written as

() ()n ()i ()(), With (14), Eq. (15) can be reduced, after some alge- qi s = 21∑ Ð an J2n sw1/2 cos x0, i s , bra manipulation, to n = 01,

() (), γ () (), γ () () ()m + 1 i f 11 s ú ∑qi s + f 12 s ú ∑gi s = 0, gi s = 2 j ∑ Ð1 bm 2m (18) m = 1 i i (16) × ()() (), γ (), γ () (), γ () J2m swi/2 /swi/2 cos x0, i s , ú sf21 s ú ∑qi s + f 22 s ú ∑gi s = 0, i i ()i ()i where Jv(z) is the Bessel function and an and bm are where the coefficients of Ex(x, 0) and Ez(x, 0) expansion in the (), γ Chebyshev polynomials for the ith slot. f 11 s ú Substituting (18) into Eq. (16) yields = ()k2s2Ps()ξe Ð γú 2Gs()ξm /()()ξγú 2 + s2 eξm , N ∞ (), γ (), γ ()n ()i ()(), γ (), f 12 s ú = f 21 s ú ∑ ∑ Ð1 an J2n swi/2 f 11 s ú cos x0, i s , ()2 ()ξe ()ξm ()()ξγú 2 2 eξm i = 1 n = 01 = Ð k Ps + Gs / + s , N ∞ () +2jγú ∑ ∑ ()Ð1 m + 12mb i ()J ()sw w f ()s, γú = ()k2γú 2Ps()ξe Ð s2Gs()ξe /()()ξγú 2 + s2 eξm , m 2m i/2 i 22 i = 1 m = 12, γú γ γ × (), γ (), = ' Ð j ''. f 12 s ú cosx0, i s = 0, ∞ (19) Equations (16) constitute a system of integral equa- N γ ()n ()i ()(), γ (), tions for the electric field distribution in the slots. ú s∑ ∑ Ð1 an J2n swi/2 f 21 s ú cos x0, i s i = 1 n = 01, (3) Solution of the integral equations. The technique for solving integral equations is well-known. The N ∞ ()m + 1 ()i ()()() electric field components Eúx and Eúz in the slots are + j∑ ∑ Ð1 2mbm J2m swi/2 / swi/2 expanded into Chebyshev polynomials of the first and i = 1 m = 12, × (), γ (), second kind with the weighting functions meeting the f 22 s ú cosx0, i s = 0, Meixner edge condition [4, 5]. where N is the number of slots on either side of the Let us calculate the integrals q (s) and g (s) in i i plane of symmetry. Eq. (16). We assume that the slots and the electrodes, () () which are arranged symmetrically about the axis of To find the unknown coefficients a i and b i and symmetry (magnetic axis for an even number of slots n m and electric for an odd number of slots) have equal the propagation constant γú , we multiply the first and p ω dimensions (Fig. 2). In this case, the components Ex and second equations in (19) by (Ð1) J2n(s i/2)cos(x0, t, s) q + 1 ω ω Ez of the fundamental mode in the symmetrically and j(Ð1) 2q(J2n(s t/2)/s t/2)cos(x0, t, s), respec- ∞ arranged slots are equal. Let x0i be the center of the ith tively, and integrate the results over the interval [0, ]

TECHNICAL PHYSICS Vol. 47 No. 2 2002 MULTISLOT MICROWAVE TRANSMISSION LINES 213

γ ', mm–1 γ '', dB/mm 3.0 0.06

0.05 2.5 0.04

2.0 0.03

0.02 1.5 0.01

1.0 0 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 Ne Ne

Fig. 3. Propagation constant versus number of electrodes Ne Fig. 4. Attenuation constant versus number of electrodes Ne at ε = (᭡) 500, (᭜) 1500, and (᭹) 2500. at the same ε as in Fig. 3. to obtain NUMERICAL RESULTS

N ∞ N ∞ Figures 3 and 4 show the real and imaginary parts of ()i ()11 ()γ ()i ()12 ()γ γú that were calculated for the structure with w = d = ∑ ∑ an K pnit,,, ú +0,∑ ∑ bm K pmit,,, ú = 0.05 mm, d = 0.34 mm, ε = 9.5, ε = 1, and σ = 5 × i = 1 n = 01, i = 1 m = 12, 1 1 2 (20) 7 Ω Ð1 N ∞ N ∞ 10 ( mm) at 30 GHz. The screening effect is ()i ()21 ()i ()22 neglected. A ferroelectric film can be characterized by a K ,,,()γú +0,b K ,,,()γú = ∑ ∑ n qnit ∑ ∑ m pmit the product εd. In the ranges ε ≤ 2500 and d ≤ 5 × i = 1 n = 01, i = 1 m = 12, 10−3 mm, the calculation error due to the use of this where combined parameter is negligibly small. The dependence of the propagation constant γ' and ∞ attenuation constant γ" of the multislot transmission ()11 ()γ ()np+ ()(), γ K pnit,,, ú = Ð1 ∫ J2 p swt/2 f 11 s ú line versus number of electrodes (Figs. 3, 4) corrobo- rates the starting idea of the paper: a multislot planar 0 transmission line fabricated on a dielectric substrate × ()(), (), J2 p swt/2 cos x0, i s cos x0, t s ds, covered with a ferroelectric film transmits the electromagnetic field like a slot line. The attenuations due to ()12 ()γ ()γ ()mp++1()12 ohmic losses in the electrodes (in our calculations, σ = K pmit,,, ú = j 4 m/wi Ð1 I , 5 × 107 (Ω mm)Ð1 of multislot and slot lines are close to ∞ each other when the slot sizes are comparable. In par- 12 ()(), γ () ticular, for a single-slot line based on our structure, γú = I = ∫ J2 p swt/2 f 12 s ú J2m swi/2 1.8 Ð j2 × 10Ð3 mmÐ1 at (εd) = 1.75 mm and w = 0.5 mm, 0 while for a multislot line with six d = 0.05-mm-wide × cos()x , , s cos()x , , s ds, 0 i 0 t electrodes spaced at 0.05 mm intervals, γú = 1.9 Ð j2 × Ð3 Ð1 () () 10 mm at the same parameters of the insulating 21 ()γ 12 ()γ Kqnit,,, ú = K pmit,,, ú , structure. Thus, the multislot planar transmission line fabri- ()22 ()γ ()mq++2 22 Kqmit,,, ú = Ð1 mqI , cated on a dielectric substrate covered by a ferroelectric film can be used for designing microwave devices with ∞ electrically tunable parameters. 22 [ ()(), γ () I = ∫ J2q swt/2 f 22 s ú J2m swt/2 0 APPENDIX × (), (), 2 ] cos x0, i s cos x0, t s /s wiwt ds. Relationships for calculating the scalar potentials are as follows. In the domain y > 0, The condition that the determinant of system of ()+ ()γ, ()+ ()α ()+ ()α equations (20) be zero yields an equation for γú . A s = B sinh 2 y + C cosh 2 y ,

TECHNICAL PHYSICS Vol. 47 No. 2 2002 214 MIRONENKO, IVANOV where ()Ð ()α () α ()α α B1 = tanh 1d1' tanh 0d0 + 0/ 1 , () B + = tanh()α d Ð ()εα /α tanh()α d , 2 2 0 2 2 0 0 () C Ð = ()αα /α tanh()d' + tanh()α d . ()+ ()εα α ()α () α 1 0 1 1 1 0 0 C = 0/ 2 2 tanh2d2 tanh 0d0 Ð 1, Here, ()+ ()γ, ()+ () ()+ ()α F s = B1 sinh a2 y + C1 cosh 2 y , 2 2 2 1/2 2 2 2 1/2 ()+ α ()γ α ()γ ε ()α α () α ()α 0 ==+ s Ð k , 1 + s Ð k 1 , B1 = 0/ 2 Ð tanh 2d2 tanh 0d0 , () C + = tanh()α d Ð ()αα /α tanh()d . α ()γ 2 2 2ε 1/2 α ()γ 2 2 2ε 1/2 1 0 0 0 2 2 2 2 ==+ s Ð k 2 , 1 + s Ð k , In the domain y < 0, d' = d + d. ()+ (), [ ()Ð ()()αα ε αε ()() α 1 1 A ys = B 1 / 1 Ð tanh 1d tanh d () + C Ð ()]αtanh()αd Ð ()αα ε/αε tanh()d sinh()y 1 1 1 REFERENCES () + [B Ð ()αα ε/αε tanh()()d Ð tanh() αd 1 1 1 1. O. G. Vendik, I. S. Danilov, and S. P. Zubko, Zh. Tekh. ()Ð ()]α()αα ε αε ()() α () Fiz. 67 (9), 94 (1997) [Tech. Phys. 42, 1068 (1997)]. + C 1 Ð 1 / 1 tanh 1d tanh d cosh y , where 2. I. G. Mironenko and A. A. Ivanov, Pis’ma Zh. Tekh. Fiz. 27 (13), 16 (2001) [Tech. Phys. Lett. 27, 536 (2001)]. () B Ð = tanh()α d' + ()()αα ε /α tanh()d , 1 1 0 1 1 0 0 3. I. G. Mironenko and A. A. Ivanov, Pis’ma Zh. Tekh. Fiz. ()Ð ()αα ε α ()() α (in press). C = 0 1/ 1 tanh0d0 tanh 1d1' + 1, 4. Integrated Ferroelectrics International Journal 22, 1420 ()Ð (), [ ()Ð ()()α α () α () α F ys = B1 1/ Ð tanh 1d tanh d (1998). () + C Ð ()]αtanh()αd Ð ()α /α tanh() αd sinh()y 5. K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, in Micros- 1 1 1 trip Lines and Slot Lines (Artech, Boston, 1996), p. 535. [ ()Ð ()()α α () α () α + B1 1/ tanh d Ð tanh 1d ()Ð ()]α()α α () α () α () + C1 1 Ð 1/ tanh 1d tanh d cosh y , Translated by A. Khzmalyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

SOLID-STATE ELECTRONICS

Electroluminescence Kinetics in Thin-Film ZnS-Based Devices at Ultralow Frequencies N. T. Gurin, A. V. Shlyapin, and O. Yu. Sabitov Ul’yanovsk State University, Ul’yanovsk, 432700 Russia e-mail: [email protected] Received March 13, 2001

Abstract—The variation of the instantaneous luminance from thin-film electroluminescent structures excited by alternating-sign sawtooth voltage pulses with a pulse rate of 0.1Ð2.0 Hz is studied. From the solution of the kinetic equation, the time dependences of the instantaneous luminance and the internal quantum yield are obtained. The time dependences of the luminance and the current exhibit two, “fast” and “slow,” rising portions, which correlate with various specific regions in the field and charge dependences of the luminance and other electrophysical parameters. The results are explained by the formation of space charges in the phosphor layer, causing a decrease in its effective thickness, and by a change in the luminescence mechanism. © 2002 MAIK “Nauka/Interperiodica”.

The voltage-luminance (VÐL) characteristic, or the Experimental TF ELDs were metalÐinsulatorÐsemi- dependence of the mean (apparent) luminance on the conductorÐinsulatorÐmetal (MISIM) structures. Here, µ alternating-sign exciting voltage amplitude, is a basic M refers to the lower transparent 0.2- m-thick SnO2- characteristic of thin-film electroluminescent devices based electrode applied on the glass substrate and to the (TF ELDs) [1]. The mean luminance, in this case, is the upper nontransparent thin-film (0.15 µm thick) Al elec- result of time averaging over instantaneous luminance trode of diameter 1.5 mm; S, to the 0.54-µm-thick pulses, so-called luminance waves. The shape of these ZnS : Mn (13 wt %) electroluminescent layer; and I, to µ waves depends on that of the exciting voltage. From the 0.15- m-thick ZrO2 : Y2O3 (13 wt %) insulating layers. decay time constant of the instantaneous luminance, the The phosphor layer was applied by thermal evaporation lifetime of excited luminescence centers, for example, in a quasi-closed chamber at a substrate temperature of Mn2+, was estimated at 1.3Ð1.5 ms in ZnS at low Mn 250°C, followed by annealing at 250°C for 1 h; the non- concentrations and was found to decrease with increas- transparent electrode, by thermal evaporation; and the ing Mn concentration [1, 2]. However, available data insulating layers, by electron-beam evaporation. for the VÐL curve shapes cannot shed light on physical processes responsible for the EL kinetics and do not We took the dependences of the instantaneous lumi- allow the detailed description of TF ELD properties. nance L and the current Ie through the TF structure on The point is that most studies have been performed for time t when the device was excited by alternating-sign the exciting voltage frequencies higher than 50 Hz, at sawtooth voltage pulses from a G6-34 generator which neighboring waves overlap. Another reason is equipped with a driver amplifier and an external G5-89 the use of a rectangular pulse or sinusoidal exciting trigger generator. The pulse amplitude was 160 V at a voltage, which makes the analysis of physical pro- nonlinearity coefficient of no more than 2%. In the “sin- cesses very difficult. However, the application of a saw- gle-shot” excitation mode, a train of two triangular tooth voltage with a frequency of lower than 50 Hz (in pulses with a repetition rate of 2 Hz is applied. The time this case, all characteristic luminescence times are Ts between single-shot excitations was varied from 1 to smaller than a one quarter of the exciting voltage cycle) 100 s. In the continuous excitation mode, the pulse rep- makes the in-depth analysis of EL in thin-film devices etition rate was 0.1, 0.2, 0.5, and 2 Hz. The current Ie possible [2]. was measured with a 100 Ω- to 10 kΩ-resistor series- In this work, we study the time variation of the TF connected to the EL structure. The voltage drop across ELD instantaneous luminance when the structure is the resistor was equal to or less than 1 V. The instanta- excited by sawtooth voltage pulses with a low repeti- neous luminance value was measured with an FÉU-84-3 tion rate. The time variation of the luminance is corre- photoelectric multiplier. The time dependences of the lated with those of the current and the field in the phos- exciting voltage, current through the structure, and phor layer, with the field and charge dependences of the instantaneous luminance were recorded with an S9-16 instantaneous luminance, and also with the current- two-channel storage oscilloscope connected to a PC voltage and voltage-charge characteristics of the phos- through the associated interface. This system can mea- phor. sure (and memorize) 2048 points within a discretiza-

216 GURIN et al. tion period with an accuracy of better than 2%. The which was subtracted from V(t). The values of Ci and mathematical and graphical data processing were car- Cp were determined starting from the TF ELD total ried out with the application programs Maple V capacitance Ce = 200 pF measured with an E7-14 Release4 Version 4.00b and GRAPHER Version 1.06 impedance meter and from TF ELD geometry. (from the 2-D Graphing System). The experimental curves L(t) and I (t) (Fig. 1), as For given variations of the exciting voltage V(t) and e well as the calculated curves Ip(t), Fp(t), L(Qp), Ip(Fp), current Ie(t) in the external circuit, the mean field Fp(t) in the phosphor layer is given by [3] and Qp(Fp) (Fig. 2), have the following distinctive features: (1) The weak asymmetry of the current pulses t Ip(t) in the continuous excitation mode when the posi- () 1 () 1 () Fp t = ----- Vt Ð -----∫Ie t dt + Fpol, (1) tive and negative voltage half-waves are applied to the dp Ci 0 upper electrode (+Al and ÐAl conditions, respectively), as was observed in [3, 4], the asymmetry diminishing where dp is the phosphor layer and Ci is the capacitance with increasing frequency f; and considerable asymme- of the capacitor made of two TF ELD insulating layers. try of the luminance waves L(t), which also diminishes The field Fpol includes the field Fpi due to the polar- with increasing frequency. At the frequencies f = 0.1 ization charge Qpi accumulated at the phosphor-insula- and 2 Hz, the amplitude of the luminance wave under tor interface and the field Fps due to the space charge in the +Al conditions is, respectively, 3.4 times and the phosphor layer. Generally, the latter may have sev- ~1.5 times greater than under the ÐAl conditions eral components associated with different space (Fig. 1). (2) The appearance of an extra peak of the current I , as was also observed in [3, 4], in the single-shot charges and differently oriented relative to Fpi. The field p F decays in the interval between two active states of excitation mode during the first half-cycle under the pol −Al conditions, the amplitude of this peak growing the device, in which the current Ip passes through the phosphor. Because of the space charges, this current with Ts, and the appearance of the associated peak of has the reactive component and can be assigned to a the luminance L (Figs. 2a, 2e), which is explained by conventional semiconductor device that has a phosphor the increased probability of emitting center excitation layer sandwiched in insulating layers. For the initial (the probability is proportional to the Ip current density [2, 4]. (3) The presence of two distinct, “fast” and geometric capacitance of this phosphor layer Cp, its “slow,” rising portions in the curves L(t) and Ip(t). Point thickness dp, and the voltage Vp(t) = Fp(t)dp across it [3, 4], the current through the phosphor, in view of (1), 1 is the separating point where the rate of rise of L and is given by Ip changes the sign (Figs. 2a, 2e). (4) The similar shapes of the L(t) and I (t) curves except for the case +Al, f = () p () ()Ci + Cp dV t 2 Hz (Figs. 2a, 2e). (5) The proximity of point 1', where Ip t = Ie t ------Ð Cp------C dt the rise in L and Ip markedly slows down (Figs. 2a, 2e), i (2) () to the point where the S-shaped portions appear in the dV() t dFp t = C ------Ð ()C + C d ------. current-voltage, or Ip(Fp), characteristic of the phos- i dt i p p dt phor layer (Fig. 2d) [3Ð5], in the curves L(Fp) (Fig. 2c), The charge transferred through the phosphor in the and in the curves Qp(Fp) (Fig. 2h). (6) The nearly linear absence of losses due to recombination and charge trap- early portion of the curve L(Ip), except for the very ping equals the charge accumulated at the interfaces beginning, where the curve is superlinear, and the during the active state and is related to the charge Qe decrease in the slope of this curve (Fig. 2b) when pass- passing in the external circuit as [2, 3] ing over point 1 and especially over point 1'. (7) The presence of two, fast and slow, rising portions in the Ci + Cp () []() () curve L(Qp), which are separated by point 1 (Fig. 2g), Qp t = ------Qe t Ð CeVt + Qpol, (3) Ci as was also observed in [2]. where To explain the above observations, we will consider t the EL kinetics. As is known [1, 6], the instantaneous luminance L(t) of TF ELDs depends on the excited () () Qe t = ∫Ie t dt (4) emitting centers N*(t) as 0 Aη d N*()t AN P d N*()t and Q is the residual polarization charge. Lt()==------int p ------1 r p -, (5) pol τ* τ* The values of Q and F , as well as the true posi- pol pol ν π tions of the Qp(t) and Fp(t) curves relative to the where A = K0 fλh / is the constant determined in the abscissa axis, were found as in [3] with an accuracy of approximation of emission monochromatism and dif- ±2%. The dependences calculated were obtained with fusely scattering surface, K0 is the coefficient of emis- formulas (1)Ð(4) at Ci = 986 pF and Cp = 250 pF with sion extraction from the device, fλ is the luminous effi- allowance for the voltage drop across the resistor, ciency, h is the Planck constant, ν is the emission fre-

TECHNICAL PHYSICS Vol. 47 No. 2 2002 ELECTROLUMINESCENCE KINETICS IN THIN-FILM ZnS-BASED DEVICES 217

L, arb. units V, V L, arb. units V, V (a) (c) 1.2 0.10 160 5.0 0.50 160 3 3 1 1 0.6 0.05 2 80 2.5 0.25 2 80

, A t, s , A t, s –7 0 0 –7 0 0 0 10 0 2 , 10 , 10 e e I I –0.6 –80 –2.5 –80

–1.2 –160 –5.0 –160

L, arb. units V, V L, arb. units V, V (b) (d) 2 0.2 160 1.4 1.2 160 3 3 1 1 1 0.1 2 80 0.7 0.6 2 80

, A , A t, s

–7 t, s –7 0 0 0 0 0 4 0 0.5 , 10 , 10 e e I I –1 –80 –0.7 –80

–2 –160 –1.4 –160

Fig. 1. (1) Instantaneous luminance, (2) current, and (3) exciting voltage vs. time for the continuous excitation. The exciting voltage frequency f = (a) 0.1, (b) 0.2, (c) 0.5, and (d) 2.0 Hz. The ﬁrst half-cycle corresponds to the conditions (ÐAl).

η σ α τ quency, int = N1Pr is the intrinsic quantum yield, N1 = If is constant, is independent of N*, and * is σ dp N [1] is the number of emitting centers excited by independent of t, the solution of Eq. (6) is [8] one electron passing through the phosphor, σ is the cross section of impact excitation of emitting centers, 1 Ð ∫α()t dt + ----- ∫dt N is the density of the emitting centers, P = τ*/τ is the τ* r r N*()t = e 1 (8) probability of radiative relaxation for an emitting cen- ∫α()t dt + ----- ∫dt ter, τ* is the lifetime of an excited emitting center, and ×α[]()t Neτ* dt+ C , τ ∫ r is the relaxation time constant of excited centers due to radiative transitions to the ground state. where C is a constant of integration. Under the assumption of the impact excitation of α emitting centers, N*(t) is found by solving the kinetic The maximal value of corresponds to the greatest ≈ × Ð6 equation [6, 7] current in our frequency range, Ip2 1.6 10 A at a frequency of 2 Hz at point 2 (Fig. 2e). At σ ≈ 2 × dN*()t N*()t −16 2 2 α ------= α()t []NNÐ *()t Ð ------, (6) 10 cm and Se = 2 mm [1, 6], the value of deter- dt τ* mined from (7) is α = 10Ð1 sÐ1. For τ* ≈ 1.5 × 10Ð3 s [2], 1/τ* ≈ 667 sÐ1. Hence, the condition α(t) Ӷ 1/τ* is valid where throughout the frequency range. Taking into account σI ()t that Ip(t) = 0 and N*(t) = 0 at t = 0, so that the constant α()t = ------p - (7) qS of integration C = 0, and that condition (7) is met, the e solution (8) of Eq. (6) takes the form is the probability of emitting center excitation per unit σ time, S is the TF ELD surface, and q is the electron () N Ðt/τ* () 1/τ* e N* t = ------e ∫Ip t e dt. (9) charge. qSe

TECHNICAL PHYSICS Vol. 47 No. 2 2002 218 GURIN et al.

–6 L, arb. units (a) V, V Ip, 10 A (Â) V, V 1.5 150 2 1' 150 1' 2 II 2

1.0 1 2 100 1' 100 1' 1 1 1 0.5 1 50 50 I, II I I

00.1 0.2 00.1 0.2 t, s t, s 8 L, arb. units Fp, 10 V/m 1.5 (b) (f) 2 1 1' 1' 2 2 1' I, II

1.0 1 II 2 I 1' 1 1 0.5 I 1

0210 0.1 0.2 –6 Ip, 10 A t, s L, arb. units L, arb. units (c) (g) 1.5 1.5 II 2 1' 2 II 1' 1 1.0 2 1 1.0 I 2 1' 1' 1 1 0.5 I 0.5 1 1

0 0 12–50 5 8 –8 Fp, 10 V/m Qp, 10 C –6 –8 Ip, 10 A (d) Qp, 10 C (h) 2 1' 5 2 2

1' 1 0 1 1' II I I 1 I, II 0 –5 120 1 2 8 8 Fp, 10 V/m Fp, 10 V/m Fig. 2. TF ELD characteristics for f = 2 Hz. Dashed lines, V(t); thick continuous lines, continuous excitation; thin continuous lines, single-shot excitation with Ts = 100 s. I, conditions (ÐAl); II, conditions (+Al).

TECHNICAL PHYSICS Vol. 47 No. 2 2002 ELECTROLUMINESCENCE KINETICS IN THIN-FILM ZnS-BASED DEVICES 219

In the fast-rising portion of Ip(t) (Fig. 2e), the curve In the descending portion, the curve Ip(t) is approx- Ip(t) under the continuous excitation mode is approxi- imated as mated as I Ðt/τ Ðt/τ () p2()4 5 ttÐ 1 Ip t = ------e + e , (19) ------τ 2 I ()t = I e 1 , (10) p p1 where τ and τ are the decay time constants of I (t). τ 4 5 p where 1 is the rise time constant of the current Ip(t) and In this case, the solution (8) of Eq. (6) is t1 is the time instant corresponding to point 1. Solution (9) is reduced to σ τ I τ Ðt/τ τ Ðt/τ () N * p24 4 5 5 ttÐ N* t = ------------------e + ------e 1 qS 2 τ Ð τ* τ Ð τ* σ τ τ* ------τ σ τ τ* e  4 5 () N 1 1 N 1 () N* t ==------τ------τ -Ip1e ------τ------τ -Ip1 t (11) qSe 1 + * qSe 1 + * τ τ τ τ  2 Ðt2/ 2 3 t2/ 3 and the dependence L(t) in view of (5) has the form + Ip1Ðτ------τ -e + τ------τ -e 2 Ð * 3 + * Ad2σ2 N2P τ (20) () p r 1 () τ τ τ  Lt = ------τ------τ -Ip t + ------1 - + ------2 - Ð ------3 - qSe 1 + * τ τ τ τ τ τ  (12) 1 + * 2 Ð * 3 + * η2 τ A int 1 = ------I ()t . I τ τ τ  qS P τ + τ* p p24 5 Ðt/ * e r 1 Ð ------τ------τ - + τ------τ - e , τ ӷ τ 2 4 Ð * 5 Ð *  For 1 *, η2 where t = 1/4f is the time instant corresponding to () A int () 2 Lt = ------Ip t . (13) point 2. The dependence L(t) is given by qSePr 2 In the slowly rising portion of the curve I (t) Aη I τ Ðt/τ τ Ðt/τ p () int p24 4 5 5 (Fig. 2e), it is approximated (under the continuous Lt = ------------τ------τ -e + τ------τ -e qSePr 2 4 Ð * 5 Ð * excitation mode) by Ðt/τ t/τ () ()2 3 τ Ðt /τ τ t /τ Ip t = Ip1 1 Ð e + e , (14)  2 2 2 3 2 3 + Ip1Ð τ------τ -e + τ------τ -e where τ and τ are the rise time constants, respectively, 2 Ð * 3 + * 2 3 (21) before and after the point where the rate of rise of the τ τ τ 1 2 3  current Ip(t) changes the sign in the slowly rising por- + τ------τ - + τ------τ - Ð τ------τ - tion (Fig. 2e). 1 + * 2 Ð * 3 + * Then, the solution (8) of Eq. (6) takes the form τ τ  Ip24 5 Ðt/τ* σ τ τ τ τ τ Ð ------+ ------e . () N * 2 Ðt/ 2 3 t/ 3 2 τ Ð τ* τ Ð τ* N* t = ------Ip1 1 Ð τ------τ e + τ------τ -e 4 5  qSe 2 Ð * 3 + * (15) τ ӷ τ τ τ For 2 * and 1 = 3, τ τ τ Ðt/τ* + ------1 -1Ð + ------2 - Ð ------3 - e τ τ τ τ τ τ 2 1 + * 2 Ð * 3 + * Aη I τ Ðt/τ τ Ðt/τ () int p24 4 5 5 Lt = ------------τ------τ -e + τ------τ -e and qSePr 2 4 Ð * 5 Ð * 2 η τ τ τ τ A int 2 Ðt/ 2 3 t/ 3 Lt()= ------I 1 Ð ------e + ------e τ τ τ p1 τ τ τ τ Ðt2/ 2 3 t2/ 3 qSePr 2 Ð * 3 + * + I 1 Ð e + ------e (22) (16) p1τ Ð τ* τ τ τ 3 1 2 3 Ðt/τ* + τ------τ -1Ð + τ------τ - Ð τ------τ - e . 1 + * 2 Ð * 3 + * I τ τ Ðt/τ*  Ð ------p2 ------4 - + ------5 - e . τ ӷ τ τ τ τ τ τ τ For 2 * and 1 = 3 (which holds in our fre- 2 4 Ð * 5 Ð *  quency range, as will be shown later), τ τ τ ӷ τ 2 If 3, 4, 5 *, then, in view of (14), η τ τ τ A int Ðt/ 2 3 t/ 3 Lt()= ------I 1 Ð e + ------e . (17) 2 2 p1τ τ Aη I Ðt/τ Ðt/τ Aη qSePr 3 + * ()≈ int p2()4 5 int () Lt ------e + e = ------Ip2 t . (23) τ τ τ ӷ τ qSePr 2 qSePr For 1, 2, 3 *, η2 The table lists the time constants determined from () A int () Lt = ------Ip t . (18) the experimental and calculated data and characterizing qSePr the EL kinetics. For the conditions (+Al, f = 2 Hz), the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 220 GURIN et al.

τ, s that the processes responsible for the corresponding portions of the curves Ip(t) and L(t) are similar in 0 τ τ τ 10 nature. The values of 1, 2, and 3 linearly decrease with rising frequency with nearly the same slopes τ τ (Fig. 3). The time constants 4 and 5 apparently char- 1 acterize the capture of free carriers by bulk and surface 10–1 traps. The detailed explanation of the curves Ip(t), as τ τ τ τ well as the of frequency dependences of 1, 2, 3, 4, τ 2 and 5, calls for special investigation into charge trans- 10–2 3 fer in the phosphor layer. 4 The results obtained can be interpreted as follows. 5 As follows from Figs. 2b and 4, for all the frequencies, 10–3 the dependences L(Ip) have the same slope of the initial 10–1 100 f, Hz portion within the measurement accuracy, in accordance with formulas (12) and (13). In the second por- Fig. 3. τ Ðτ vs. frequency. (1) τ , (2) τ and τ for the con- tion, for all the frequencies, the dependence of L on I 1 5 τ 2 τ1 3 τ p ditions (ÐAl) and (+Al), (3) 5, (4) 4 (ÐAl), and (5) 4 weakens, which means that the proportionality coeffi- (+Al). (1, 3) ᭿ (+Al) and ᭜ (ÐAl). cient in formulas (17) and (18) for L(Ip) decreases (compared to the initial portion). The luminances L1 L, arb. units and L and the currents I and I at points 1 and 2 101 2 p1 p2 (Figs. 2a, 2e) linearly depend on frequency f (Fig. 5), while the field Fp varies with frequency only slightly (Figs. 2b, 2d). This can be explained by the nearly linear dependence of I on the rate of change of the voltage IV 1 1' p 100 2 [see (2)] in this case. 1' 2 2 III 1' 1 According to (5), (13), and (18), the change in the V 1 slope of the L(Ip) curves in passing to the second por- II 1' 1' 2 tion (Figs. 2b, 4) can be explained by a decrease in the 10–1 1 2 cross section σ of impact excitation; in the probability 1' 1' VI Pr of radiative transitions; or in the phosphor effective 1 2 I 1 thickness dp, within which the impact excitation of 1' 2 emitting centers takes place. 1 σ 10–2 The reasons for such a variation of or Pr can be VII (i) a change in the Mn2+ ion excitation mechanism (from direct impact excitation to resonance excitation VIII involving neighboring atoms or ZnS lattice defects [9]; this seems to be hardly probable, because the excitation –3 10 –9 –8 –7 –6 –5 level at ultralow frequencies is low for all the condi- 10 10 10 10 10 tions except (+Al, 2 Hz); (ii) the excitation of emitting , A Ip complexes (along with ordinary centers) that combine Mn2+ ions and ZnS lattice defects, such as sulfur vacan- Fig. 4. L(Ip) for the continuous excitation. f = (I, VIII) 0.1, cies [10], and have other impact excitation cross sec- (II, VII) 0.2, (III, VI) 0.5, and (IV, V) 2. (IÐIV) +Al and (VÐ σ τ VIII) ÐAl; ᭿ (ÐAl) and ᭜ (+Al). tions , relaxation times *, and radiative relaxation probabilities Pr. However, these complexes form at ele- vated Mn concentrations (2.2 wt %), i.e., when the τ τ values of 2 and 3 are given for the curve Ip(t), while phosphor layer is deposited at a substrate temperature τ ≈ τ ≈ for the curve L(t), 2 2 ms and 3 560 ms. Here, the of 200°C without subsequent annealing, and their function decays starting from the additional portion room-temperature EL spectra do not depend on the with a time constant τ of 10Ð40 ms, which is 3 ms wide. exciting voltage [10]. τ τ ≈ The descending portion with a similar value of , The effective thickness d may decrease because of 14 ms, is also typical of higher excitation frequencies, p ≥ space charges left at the anode and the cathode after the f = 10 and 50 Hz [2]. It is observed at L 1.1 as well. preceding half-cycle. Also, d shrinks because of the τ p From the table, it follows that the time constants 1 occurrence and expansion of space charge regions τ and 3 are coincident with each other at all the frequen- when deep donors ionize at the anode. This decreases cies within the overall (measurement + calculation + the field at the anode and gives rise to the negative dif- approximation) accuracy (±10%). This may indicate ferential resistance region in the currentÐvoltage char-

TECHNICAL PHYSICS Vol. 47 No. 2 2002 ELECTROLUMINESCENCE KINETICS IN THIN-FILM ZnS-BASED DEVICES 221

L1, L2, arb. units L1, L2, arb. units (a) (c) 1.5 1.0 1.5 1.0 2

1 1.0 2 1.0 0.5 0.5 1 4 0.5 4 0.5 3 3

0 0 –6 –6 Ip1, Ip2, 10 A Ip1, Ip2, 10 A (b) (d) 3.0 1.0 3.0 1.0

2 2 1.5 0.5 1.5 0.5 4 1 4 1 3 3

0 0 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 f, Hz f, Hz

Fig. 5. (a, c) Luminances L1, L2 and (b, d) currents Ip1, Ip2 vs. frequency f. (1) L1, Ip1; (2) L2, Ip2; (3) L1, Ip1 and (4) L2, Ip2 normalized to the associated values at f = 2 Hz. acteristic (Fig. 2d). Finally, the effective thickness second portion. Subsequently, as the external field decreases because of the formation of space charges grows, this current begins to rise again (in the second τ τ when deep acceptors ionize at the cathode in the single- portion) with the same time constant ( 1 = 3). In this shot excitation mode. In this case, the cathode field situation, the effect of the residual space charges and decreases [3]. As a result, the free electron energy at the their fields is appreciably reduced in the single-shot 2+ anode may be insufficient for the ionization of Mn excitation mode with Ts = 100 s (Fig. 2b) [3] and the τ τ emitting centers. The fact that 1 and 3 equal each other effective thickness dp is the largest. supports our previous supposition. Namely, once the At the same time, under the conditions (+Al, f = deep donors have ionized, the field of the resulting 2 Hz), the slope of the curve L(Ip) in the second portion space charge at the anode has expanded, and the mean decreases much sharper (Fig. 2b), this decrease being field in the phosphor has decreased (Fig. 2f), the tunnel observed for higher levels of L (close to those at higher emission current from the cathode is first limited in the frequencies, 10 and 50 Hz). This can be explained by

Table τ τ τ τ τ f, Hz Conditions 1, ms 2, ms 3, ms 4, ms 5, ms 0.1 ÐAl 246 1440 230 44 220 +Al 240 1420 220 62 236 0.2 ÐAl 127 580 155 29 88 +Al 118 750 120 21 84 0.5 ÐAl 57 330 55 10 28 +Al 47.5 320 45 4 24 2 ÐAl 14 93 11 3.3 9.1 +Al 13.5 108 12 2.6 9.8

TECHNICAL PHYSICS Vol. 47 No. 2 2002 222 GURIN et al.

η η int, arb. units int, arb. units 1.5 1.5 (a) (c)

1 1' 1' 2 1 2 1.0 1.0 II II

11' 1' 2 1 2 0.5 0.5 I I

024 0 0.4 0.8 1.5 1.5 (b) (d)

1 2 1 1' 2 1' 1.0 1.0 II 1 ' II 1 2 1 1' 2 1' 1 0.5 0.5 I I

012 0 0.1 0.2 t, s t, s

η Fig. 6. int(t) for f = (a) 0.1, (b) 0.2, (c) 0.5, and (d) 2 Hz. Thick continuous lines, continuous excitation; thin continuous lines, single-shot excitation. I, ÐAl; II, +Al. the resonant interaction of excited Mn2+ ions with each remaining nearly the same (Fig. 2c). The last effect other and/or with surrounding atoms or ZnS lattice causes the energy of electrons to increase and the ion- defects [2]. ization efficiency N1 to rise. The superlinear initial region in the curve L(I ) (Fig. 2b) is explained by the The equal slopes of the normalized dependences of p same reasons. L1, L2, Ip1, and Ip2 on f in all the cases except for the conditions (+Al, f = 2 Hz) for the curve L(t) (Figs. 5aÐ5c) Using expressions (5), (12), (13), (16)Ð(18), and suggest that the same mechanism is responsible for the (21)Ð(23), we can estimate the instantaneous values of η rise in Ip in the initial (first) portion and at the end of the the internal quantum yield, int, and the external quan- η η second one (Fig. 2f), and also validate the use of sim- tum yield, ext = K0 int, both are most important param- plified expressions (13) and (18) for finding the depen- eters of TF ELDs. From formulas (5), (13), (18), and η dence L(t). (23), the instantaneous value int(t) for the entire curves L(t) and Ip(t) is The increased slope of the curves L(Ip) in the first portion for the single-shot excitation and also when the () frequency f declines in the continuous excitation mode η () qSe Lt int t = ------()-. (24) (Figs. 2b, 4) is explained by the large relaxation times A Ip t of the space charges at both electrodes, the decrease in η the fields of these charges [3], and the increase in the The curves int(t) constructed with the data tabulated field in the bulk with the mean field Fp in the phosphor and formulas (13), (16)Ð(18), and (21)Ð(24) (Fig. 6)

TECHNICAL PHYSICS Vol. 47 No. 2 2002 ELECTROLUMINESCENCE KINETICS IN THIN-FILM ZnS-BASED DEVICES 223 sharply rise in the first portion of L(t) and then ertheless provides the ionization of the Mn2+ emitting smoothly decline after point 1' in the second portion for centers. This means that kinetic equation (6), where the f = 0.1, 0.2, and 0.5 Hz. Under the conditions (ÐAl, f = generation term α(t)[N Ð N*(t)] is present, must be η 2 Hz), int slightly increases after point 1'. Note that the solved for the descending branch of L(t). decline in the second portion is much greater for the Thus, our study of the instantaneous luminance and single-shot excitation, although the absolute values of other physical characteristics of TF ELDs showed the η int at the beginning of the second portion exceed those presence of fast and slow portions in the time depen- for the continuous excitation (Fig. 6d). As the fre- dences of the instantaneous luminance and current η quency f increases, the maximal values of int slightly through the phosphor layer. The former is characterized increase (Fig. 6). Under the conditions (+Al, f = 2 Hz) by the nearly linear current dependence of the lumi- η (Fig. 6d), int(t) sharply drops in the second portion and nance and by a fast rise in the internal quantum yield the additional peak appears in the descending branches presumably because of an increase in the number of η of L(t) and Ip(t). Such behavior of int(t) can be emitting centers excited by an electron passing through η explained as follows. In the general form, int and the the phosphor layer when the mean field in this layer physical parameters of the phosphor layer are related as grows. In the slow portion, the current and the luminance grow with a lower rate and the current depen- η () () () σ() (), () () int t ==N1 t Pr t t NxtdP t Pr t , (25) dence of the luminance weakens, which is accompanied by a decrease in the internal quantum yield. It appears that, where N(x, t) is the distribution of the emitting centers at relatively low excitation levels (currents I ), the phos- across the phosphor layer. p phor effective thickness, within which the impact exci- η The fast rise in int in the first portion up to point 1 tation of Mn2+ emitting centers takes place, changes. (Fig. 6) is associated with the growth of the mean field The excitation is associated with the ionization of deep Fp(t) in the phosphor (Fig. 2f), the energy of acceler- donors and acceptors. The ionized donors and accep- ated electrons, and the number N1(t) of emitting centers tors, along with the field of residual space charges pro- excited by an electron. In the interval between points 1 duced during the preceding half-cycle of the exciting η and 1', int varies insignificantly, since the parameters voltage, diminish the local fields at the electrodes. 2+ of Mn ion excitation remain fairly constant and the When the current Ip exceeds some critical level, the 2+ mean field Fp(t) varies only slightly (Fig. 2f). From direct impact excitation of Mn centers is likely to η point 1 to point 2, the decline in int(t) can be explained, change to the resonant interaction of these centers with according to (25), by a decrease in the effective value of each other and/or with neighboring atoms or ZnS lattice dp(t) due to the field redistribution in the phosphor. In defects. the interval between the voltage pulses in the single- shot excitation, the space charges considerably relax; REFERENCES therefore, upon the subsequent ionization of deep cen- 1. Electroluminescent Sources of Light, Ed. by I. K. Ve- ters, the scattering of the carrier energy by these centers reshchagin (Énergoatomizdat, Moscow, 1990). η is more intense. As a result, int in the second portion 2. N. T. Gurin, O. Yu. Sabitov, A. V. Shlyapin, and sharply drops (Fig. 6d). The extra peak appearing in the A. V. Yudenkov, Pis’ma Zh. Tekh. Fiz. 27 (4), 12 (2001) descending portion under the conditions (+Al, f = 2 Hz) [Tech. Phys. Lett. 27, 138 (2001)]. (Fig. 6d) is due to the slower decay of the luminance 3. N. T. Gurin, O. Yu. Sabitov, and A. V. Shlyapin, Zh. Tekh. L(t) compared with the current Ip(t). This is because the Fiz. 71 (8), 48 (2001) [Tech. Phys. 46, 977 (2001)]. curve L(t) has the extra decay portion with τ of about 4. N. T. Gurin, A. V. Shlyapin, and O. Yu. Sabitov, Zh. Tekh. several milliseconds, which is presumably related to Fiz. 71 (3), 72 (2001) [Tech. Phys. 46, 342 (2001)]. another mechanism of Mn2+ center excitation and, 5. N. T. Gurin, A. V. Shlyapin, and O. Yu. Sabitov, in Pro- σ accordingly, with the changed parameters N1(t), (t), ceedings of the International Conference “Optics of Semiconductors,” Ul’yanovsk, 2000, p. 80. N(x, t), and Pr(t) in formula (25). For the same reason, η in this case, int in the second portion drops sharper 6. N. T. Gurin and O. Yu. Sabitov, Zh. Tekh. Fiz. 69 (5), 65 than under the conditions (+Al, f = 0.1, 0.2, 0.5 Hz). (1999) [Tech. Phys. 44, 537 (1999)]. η 7. N. A. Vlasenko, Yu. V. Kopytko, and V. S. Pekar, Phys. The asymmetry of L(t) and int(t) (Figs. 1, 2a, 6) at Status Solidi A 81, 661 (1984). various polarities of the excitation voltage [conditions 8. G. A. Korn and T. M. Korn, Mathematical Handbook for (ÐAl) and (+Al)] stems from the nonuniform distribu- Scientists and Engineers (McGraw-Hill, New York, tion of Mn2+ centers across the phosphor: it is higher at 1968; Nauka, Moscow, 1974). the upper electrode, hence, the higher values of L(t) and η 9. A. A. Zhigal’skiœ, E. V. Nefedtsev, and P. E. Troyan, Izv. int(t) under the conditions (+Al) [3, 4]. Vyssh. Uchebn. Zaved., Fiz., No. 2, 37 (1995). η It should be noted that the decay of int(t) down to 10. A. N. Georgobiani, A. N. Gruzintsev, Xu Xurong, and zero (Fig. 6) occurs in parallel with the same decrease Lou Zidong, Neorg. Mater. 35, 1429 (1999). in Ip(t) (Fig. 2e). The mean field Fp in the phosphor, while being reduced roughly to 108 V/m (Fig. 2f), nev- Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 224Ð226. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 84Ð87. Original Russian Text Copyright © 2002 by Kuz’menko, Ganzha, Domashevskaya, Ryasnoœ. OPTICS, QUANTUM ELECTRONICS

Combined Photoreflectance/Photoluminescence Studies of the Stability of Semiconductor Surface Passivation R. V. Kuz’menko*, A. V. Ganzha*, É. P. Domashevskaya*, and P. V. Ryasnoœ** * Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia e-mail: [email protected] (Kusmenko) ** Voronezh State Academy of Architecture and Civil Engineering, Voronezh, 394006 Russia Received March 30, 2001

Abstract—Combined photoreﬂectance/photoluminescence measurements are proposed for the study of the stability of semiconductor surface passivation. With such an approach, laser action on both charged and recombination-active electron states can be investigated. The efﬁciency of the method proposed is demonstrated with GaAs substrates passivated by selenium. © 2002 MAIK “Nauka/Interperiodica”.

The stationary laser irradiation of the semiconductor that the Ga2Se3/GaAs interface exhibits a lower density surface by photons with an energy exceeding the of surface states in comparison with the naturally oxi- energy E0 of the direct optical transition activates des- dized surface. However, the stability of this phenome- orption, chemisorption, and reoxidation processes, as non has not been studied. well as the generation of defects on the surface and in the near-surface area. The mechanisms of the corre- All the measurements were carried out in air at room sponding reactions for the naturally oxidized surfaces temperature with the setup described in [9]. A modulated laser beam (HeÐNe laser, λ = 632.8 nm) is focused of IIIÐV semiconductors have been discussed in details × in [1Ð6]. However, the associated models consider only on the surface area measuring 0.1 0.1 mm, and the changes in the integral intensity of luminescence. The E0 spectrum of GaAs photoreflectance is recorded (the works cited above studied only the effect of laser radi- pseudomorphic Ga2Se3 layer is transparent at 632.8 nm). ation on recombination-active electron states. The com- It is known that high-power laser irradiation may bination of luminescence excitation spectroscopy with induce photostimulated reactions near the GaAs sur- photoreflectance measurements based on the periodic face. Therefore, the laser power density was taken such electromodulation of reflected light by the near-surface that the integral intensity of luminescence remained area of a semiconductor (the irradiation of the surface nearly unchanged in the course of the measurements. by a laser light with a photon energy higher than the After recording the first photoreflectance spectrum, we band gap leads to the modulation of the surface electric increased the power density, stopped the modulation of self-field) makes it possible to gain additional informa- the laser beam, and recorded the time dependence of tion on the surface electric field [7, 8]. Thus, the com- the photoluminescence (PL) intensity IPL(t). After a bination of photoreflectance and photoluminescence time depending on the PL signal curve shape, the exper- measurements allows one to study the effect of laser iment was interrupted, the irradiation power density irradiation on both recombination-active and charged was decreased, and the next measurement of the photo- states at the surface. An additional advantage of such an reflectance spectrum was performed. Each time after approach is the possibility of making measurements recording the spectrum, the laser intensity was with the same setup described in [9]. increased to the previous level in such a way as to avoid discontinuities in the curve of the integral photolumi- In this work, our method is demonstrated by study- nescence intensity (IPLI), and the curve IPL(t) was ing the stability of the GaAs surface passivated by Se. taken again. Samples used were n-GaAs(100) substrates annealed in a Se or Se + As vapor at the temperature 660Ð680 K for We empirically found that the IPLI signal from our 5Ð45 min. The annealing was performed at a Se vapor samples varies insignificantly for several hours when pressure ranging from 0.1 to 1 Pa. The processing of a the power density is L ~ 1 mW/cm2. On the contrary, GaAs substrate in a chalcogen-containing medium at the signal markedly changes within several seconds at these temperatures results in the heterovalent substitu- L > 100 W/cm2. Therefore, we used the above values tion reaction and the formation of the Ga2Se3 surface for the measurements of the photoreflectance and IPLI, pseudomorphic layer. It has been demonstrated [10Ð12] respectively. The times of recording the photoreflec-

COMBINED PHOTOREFLECTANCE/PHOTOLUMINESCENCE STUDIES 225 tance spectrum and the curve IPL(t) were 600 and 1500 IPL s, respectively. Figure 1 shows a typical result of the combined 1 measurements for our samples. The decay (upper) 2 curve, obtained at L ~ 2500 W/cm , represents the first 2 type of the time dependence of the PL intensity. The ten-fold decrease in the power density affects the initial part of the dependence: the increase in the signal to a certain maximum level is followed by the decay of the 3 first type (the lower curve in Fig. 1). A further decrease in the laser intensity causes an extension of the rising portion, so that only the slow growth of the IPLI is 4 2 observed during the measurements at L ~ 1 W/cm . 3' 5 The simulation of the IPL(t) dependences showed that the curve of the first type is well described by a sin- 4' gle damped exponential. The curve of the second type can be approximated by the sum of rising and damped 2' exponentials: 5' ' PL() PL()∞ t t 1 I t = I t = CA+,expÐ--- Ð BexpÐ--- t1 t2 6' where t1 and t2 are the characteristic time constants of the laser-induced processes. 7' The photoreflectance spectra have a mid-field shape [13]. In the mid-field case, the spectrum consists of a central peak in the vicinity of the energy of the transition and high-energy FranzÐKeldysh oscillations (Fig. 2). The theory of photoreflection predicts that, when the photoreflectance spectra are measured near 0 500 1000 1500 t, s the transition E0 for a three-dimensional critical point, the FranzÐKeldysh oscillations can be approximated by the asymptotic formula [14, 15] Fig. 1. Results of the combined photoreflectance/photoluminescence measurements at L = 2500 (upper curve) and 2  250 W/cm (lower curve). The electric field intensities are ∆R 1 1 បω Ð E Γ 6 6 6 6 ------()បω ≈ Re------expÐ------0 (1) 3.01 × 10 , (2) 2.90 × 10 , (3) 2.89 × 10 , (4) 2.89 × 10 , បω 2 3/2 6 6 6 R  Ð E0 ()បω ()បΩ (5) 2.89 × 10 , (1') 3.12 × 10 , (2') 3.04 × 10 , (3') 2.99 × 106, (4') 2.97 × 106, (5') 2.95 × 106, (6') 2.95 × 106, and 2បω Ð E 3/2  (7') 2.95 × 106 V/m. × cos ------0 + Θ . 3បΩ 0  Here, បΩ is the electrooptical energy by the formula [7] 2 2ប2 1/3 εε 2 Q2 បΩ e F ϕ 0F ss = ------µ , ==------εε -. 8 || 2en 2 0en µ where || is the reduced effective electronÐhole mass in The quantitative analysis of the photoreflectance the direction of the electric field and e is the elementary spectra taken prior to the photostimulated reactions charge, which exceeds the phenomenological energy Γ yields the following mean values of the parameters: F ≈ of the spectral broadening of the transition. × 6 ϕ ≈ ≈ × 11 Ð2 3 10 V/m, e 0.3 eV, and Qss/e 2 10 cm . Fig- Thus, one can determine the value of the electroop- ure 1 shows the results of the photoreflectance mea- tical energy and, consequently, the surface electric field surements. The initial portion of the upper curve exhib- intensity (if the value of បΩ is known) from the period its the decrease in the surface electric field, which, how- of the FranzÐKeldysh oscillations. Knowing the surface ever, saturates in a short time. It is clear from the lower electric field intensity and the equilibrium concentra- curve that the decrease in the electric field intensity is tion n of charge carriers, one can obtain the surface related to the rising exponential. Thus, the photoreflec- ϕ potential and the density of charged surface states Qss tance measurements unambiguously confirm that the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 226 KUZ’MENKO et al.

∆R/R, arb. units rather than on the surface of the semiconductor, follows 4 from the constant value of the surface electric field intensity in the decay portion of the curve, whereas the 3 incomplete-to-stable passivation transition is character- 2 2 ized by the variable surface field. Based on this fact, one can assume that the states due to this transition 1 1 must either appear inside the band gap near the band 0 edges or eliminate the states at the midgap. Our study of the stability of the selenium-passivated –1 GaAs surface showed that combined photoreflec- –2 tance/photoluminescence measurements are an effective way for investigating the electronic properties of –3 the semiconductor surface. The generation of non-radi- 1.38 1.40 1.42 1.44 1.46 1.48 eV ation-induced defects in the near-surface area of the semiconductor is responsible for the decay of the pho- Fig. 2. Experimental E0 spectra of the photoreflectance for toluminescence intensity. The photostimulated reac- GaAs samples passivated by Se. The difference in the tions on the passivated surface enhance the photolumi- FranzÐKeldysh oscillation period implies the difference in nescence signal. × 6 the surface electric field intensities: (1) F1 = 3.19 10 V/m × 6 and (2) F2 = 2.91 10 V/m. REFERENCES 1. M. Y. A. Raja, S. R. J. Brueck, M. Osinski, and J. McIn- first type of the time dependence of the IPLI also has erney, Appl. Phys. Lett. 52, 625 (1988). the growing component with the small time constant. 2. T. Suzuki and M. Ogawa, Appl. Phys. Lett. 31, 473 (1977). Our dependences of the IPLI can be interpreted 3. N. M. Haegel and A. Winnacker, Appl. Phys. A A42, 233 within the model assuming that photoinduced chemical (1987). reactions on the GaAs surface and the generation of 4. D. Guidotti, E. Hasan, H.-J. Hovel, and M. Albert, Appl. “non-radiation-induced” defects in the near-surface Phys. Lett. 50, 912 (1987). area proceed simultaneously. The laser-induced gener- 5. J. Ogawa, K. Tamamura, K. Akimoto, and Y. Mory, Appl. ation of non-radiation-induced defects in the near-sur- Phys. Lett. 51, 1949 (1987). face area of GaAs has been reported in [1Ð6]. 6. D. Guidotti, E. Hasan, H.-J. Hovel, and M. Albert, We assume that the surface of the samples studied is Nuovo Cimento D 11, 583 (1989). inhomogeneous and has the areas of both stable and 7. R. B. Darling, Phys. Rev. B 43, 4071 (1991). incomplete passivation. In the former areas, the het- 8. N. Bottka, D. K. Gaskill, R. S. Sillmon, et al., J. Elec- erovalent substitution reaction is completed, and they tron. Mater. 17, 161 (1988). can be assigned the minimum density of states. In the 9. J. Schreiber, S. Hildebrandt, W. Kircher, and T. Richter, latter, the density of states is increased, since the intrin- Mater. Sci. Eng. B 9, 31 (1991). sic states of the semiconductor surface add up with Se- 10. B. I. Sysoev, V. F. Antyukhin, V. D. Strygin, and induced states. The existence of the areas with different V. N. Morgunov, Zh. Tekh. Fiz. 56, 913 (1986) [Sov. surface state densities was confirmed by taking the Phys. Tech. Phys. 31, 554 (1986)]. high-resolution (10 × 10 µm) photoreflectance spectra 11. B. I. Sysoev, N. N. Bezryadin, G. I. Kotov, and from various sites of the passivated substrates. It was V. D. Strygin, Fiz. Tekh. Poluprovodn. (St. Petersburg) demonstrated that the surface electric field intensity 27, 131 (1993) [Semiconductors 27, 69 (1993)]. varies over the passivated surface, unlike the unproc- 12. N. N. Bezryadin, É. P. Domashevskaya, I. N. Arsent’ev, essed substrate. et al., Fiz. Tekh. Poluprovodn. (St. Petersburg) 33, 719 (1999) [Semiconductors 33, 665 (1999)]. The laser-induced transformation of the areas of incomplete passivation into the areas of stable passiva- 13. D. E. Aspnes, Surf. Sci. 37, 418 (1973). tion decreases the rate of surface recombination and 14. D. E. Aspnes, Phys. Rev. B 10, 4228 (1974). enhances the luminescence. On the contrary, the gener- 15. D. E. Aspnes and A. A. Studna, Phys. Rev. B 7, 4605 ation of the non-radiation-induced defects in the near- (1973). surface area causes the degradation of the signal. The fact that these defects appear in the near-surface area, Translated by A. Chikishev

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 227Ð234. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 88Ð95. Original Russian Text Copyright © 2002 by Balakirev, Onishchenko, Sidorenko, Sotnikov.

RADIOPHYSICS

Broadband Emission from a Relativistic Electron Bunch in a Semi-Infinite Waveguide V. A. Balakirev, I. N. Onishchenko, D. Yu. Sidorenko, and G. V. Sotnikov Kharkov Institute for Physics and Technology, National Science Center, Akademicheskaya ul. 1, Kharkov, 61108 Ukraine e-mail: [email protected] Received March 27, 2001; in ﬁnal form, July 6, 2001

Abstract—A study is made of the excitation of a transition radiation pulse during the injection of a charged particle bunch through the end metal wall into a semi-infinite cylindrical waveguide. Exact analytic expressions for the fields of a thin ring-shaped bunch are obtained in terms of the Lommel functions of two variables. The energy efficiency, power, and spectrum of radiation emitted from a finite-size charged bunch in a vacuum waveguide are calculated numerically with allowance for the multimode nature of the excited field. It is shown that, under certain conditions, the bunch can generate a short, high-intensity electromagnetic pulse with a broad frequency spectrum. The effect of various parameters of the charged bunchÐwaveguide system (such as the bunch current, bunch duration, and waveguide radius) on the generation efficiency of a transition radiation pulse is investigated. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION Gaponenko et al. [8]. Balakirev et al. [9] reported the results of experimental investigations on the emission At present, much attention is focused on the prob- of UWBPs by a TEM-horn antenna during its impact lem of the generation of short intense electromagnetic excitation by an IREB. High-power UWBPs can be pulses such that the width of their frequency spectra is generated not only in direct processes of the excitation comparable to their mean frequency. Such pulses are of ultrawide-band antennas by IREBs but also due to called “ultrawide-band pulses” (UWBPs) [1, 2]. The the effect of coherent transition radiation [10]. current interest in the problem of pulsed emission of high-power electromagnetic signals stems primarily The transition radiation from an individual charged from their application in ultrawide-band radio location. particle is one of the fundamental elementary radiation UWBPs have also found use in, e.g., solving ecological processes [11]. This effect occurs when a charged par- problems, detection of objects under the Earth’s sur- ticle moves through an electrically nonuniform face, geologic exploration of the marine shelf, and the medium. The emission of radiation by a charge propa- goal of ensuring flight and road safety [1–6]. gating in such a medium can be interpreted as follows. The spatial electromagnetic field distribution of a mov- Traditional methods for generating short UWBPs ing charged particle is determined by both its velocity are based on the use of ultrawide-band antennas (TEM- and the dielectric permittivity (magnetic permeability) horns, spiral and biconical antennas, and others [4]). of the medium through which it propagates. Even in the However, UWBPs can also be generated by intense case of uniform straight-line motion in an electrically pulsed relativistic electron beams (IREBs). Short (and nonuniform medium, the particle emits radiation, ultrashort) IREBs with durations of 1Ð100 ps, energies resulting in the spatial redistribution of its electromag- of 0.5Ð1 MeV, and peak currents of 1Ð100 kA can be netic field. The situation with a particle undergoing either produced by converting a continuous IREB into accelerated motion in a uniform medium is similar; the a sequence of electron pulses (modulated beams) [7] or only difference is that the electromagnetic field of the generated directly by high-current devices (explosive- particle is redistributed because of the change in the emission diodes). particle velocity. The transition radiation effect can be The most efficient way of generating electromag- greatly enhanced by using charged particle bunches netic UWBPs is to exploit nonresonant (impact) mech- [12]. The radiation fields of the particles in a bunch anisms for the production of IREBs. In this case, the whose dimensions are substantially smaller than the duration of the excited UWBP is determined by the wavelength of the emitted radiation are coherent; as a duration of a pulsed IREB and the power of the emitted result, the radiation from a bunch of N particles is N2 UWBP is determined by the beam current and energy. times more intense than that from one particle. Because The mechanism for the impact excitation of a UWBP of this effect, electron bunches provide an efficient by a relativistic electron beam during the charging of a means of generating intense electromagnetic transition whip antenna was studied experimentally by radiation.

228 BALAKIREV et al. τ ξ ε ω λ ε 2 2 Here, we theoretically investigate the excitation of where = t Ð t0, = z /c, 0n = nv0/(b 1 Ð v 0/c ), ultrawide-band transition radiation during the injection α λ ε λ of a short-pulse IREB into a semi-infinite circular- n = nc/(b ), and n is the nth root of the Bessel cross-section waveguide whose end wall (through function J0. which the beam is injected) is short-circuited by a con- Integral (2) describes the Coulomb field of a charge ducting diaphragm. In such a waveguide, a distinguish- moving in an infinite waveguide and is equal to ing feature of the transition radiation is a large dispersion of electromagnetic waves, so that the shape of a  π z z ------exp Ðω ttÐ Ð ------for ttÐ Ð ------≥ 0 UWBP of transition radiation propagating in a ω 0n0 v 0 v waveguide will deform permanently.  0n 0 0 I1n =   π z z ------exp ω ttÐ Ð ------for ttÐ Ð ------< 0.  ω 0n0 v 0 v THE FIELD OF A THIN-WALLED  0n 0 0 RING-SHAPED BUNCH (4) We consider a semi-infinite (0 ≤ z < ∞) cylindrical Integral (3) corresponds to free oscillations of a metal waveguide of radius b that is filled with a homo- cylindrical waveguide and describes the transition radi- geneous dielectric with permittivity ε. The waveguide ation. The analytic solution with integral (3) was first end (z = 0) is short-circuited by a metal wall transparent obtained by Denisov [13] in studying the propagation to relativistic electrons. An axisymmetric monoener- of an electromagnetic signal in an ionized gas. In what getic electron bunch is injected through the metal wall; follows, we will use the method proposed in [13]. the bunch then moves with a constant velocity v0 < ω2 α2 ω ±α The function Ð n has branch points = n. ε α α c/ along the symmetry axis of the waveguide (the z We draw a cut along the closed interval (Ð n, n) in the axis). We neglect the influence of a narrow vacuum drift complex plane ω. Integral (3) should be taken along a channel on the electrodynamics of the waveguide sys- contour lying above the cut because, in this case, the tem and, in order to simplify the calculations, assume total field (1) vanishes for t < t0. that the dielectric fills the entire waveguide. Under ε these assumptions, we will obtain an exact solution in For t Ð t0 Ð z /c < 0, integral (3) is given by the res- ε ω ω the important case of a vacuum waveguide ( = 1). idue of the integrand at the pole = i 0n: We start by determining the field of an infinitely π ω z short and infinitely thin charged ring with the charge I2n = ------ω - exp 0nttÐ 0 Ð ------. (5) density 0n v 0 eN ε ρ δ()δ()v For t Ð t0 Ð z /c > 0, we can apply the residue the- = Ð------π - rrÐ 0 ttÐ 0 Ð z/ 0 , 2 r0v 0 orem to obtain where e is the charge of an electron, N is the number of π z ------ω electrons in the ring, v is the ring velocity, r is the ring I2n = Iel + ω exp 0nttÐ 0 Ð ------. (6) 0 0 0n v 0 radius, and t0 is the time at which the ring enters the waveguide. Here, the integral Iel of the integrand in (3) is taken along a closed contour (in the negative direction) that Solving Fourier-transformed Maxwell’s equations ω ± ω α envelopes both the poles, = i 0n, and the cut (Ð n, with allowance for the boundary condition Er = 0 at the α end metal wall, we obtain the following expression for n). As the integration contour, we choose an ellipse Ð ω ±α the radial electric field of an axisymmetric E-wave: Cω whose foci are the branch points = n. Let us appropriately transform the integral Iel. First, we make (),,, , 2Ne ω Er trzt0 r0 = ------π ε - the replacement p = Ði . Second, we switch to the new b v 0 (1) variable α ζ = p2 + α2 Ð p. Third, we make another ω2 ()λ ()λ n n × 0n J0 nr0/b J1 nr/b {} ∑------I2n Ð I1n , replacement ζ = Ðβω, where β = ()τξÐ /()τβ+ . As λ 2()λ n J1 n n a result, the integral Iel becomes +∞ []ω ω() α exp Ði ti+ t0 + z/v 0 i 2 2 I = dω------, (2) ω n τ ξ ω 1 1n ∫ ()ωω ω ()ω Iel = ------ω ∫ d exp ----- Ð Ð ω---- Ð i 0n + i 0n 2 0n 2 Ð∞ + Cω (7) +∞ exp()Ðiωτ + iξω2 Ð α2 1 1 1 1 ω n × ------Ð ------Ð ------+ ------, I2n = ∫ d ------()ωω ω ()ω , (3) ωω ωω ωω ωω Ð i 0n + i 0n Ð 1 Ð 2 Ð 3 Ð 4 Ð∞

TECHNICAL PHYSICS Vol. 47 No. 2 2002 BROADBAND EMISSION FROM A RELATIVISTIC ELECTRON BUNCH 229

Finally, using expressions (4), (5), and (12) and tak- ω βÐ1 ()− ε ()± ε Ð1 where 1, 2 = c + v 0 c v 0 and ing into account relationships (13), we arrive at the fol- ω ω 3, 4 = Ð 1, 2. lowing expression for the total electric field (1), which These transformations convert the elliptic integra- is generated by a thin ring-shaped electron bunch prop- Ð + agating with constant velocity in a semi-infinite tion contour Cω into a circular contour Cω , which is waveguide and is represented as a linear superposition centered at the point ω = 0 in the complex plane ω and coul encloses no singular points except ω = 0. Along the of the Coulomb field of a moving charge, Er , and the + trans contour Cω , the integration is carried out in the positive transition radiation field Er : direction and we can use the expansions (),,, , Er trzt0 r0 ∞ ω k 1 1  = Ecoul()trzt,,, ,r + Etrans()trzt,,, ,r , Ð------ωω= ω----- ∑ ω----- (8) r 0 0 r 0 0 Ð j j j k = 0 coul(),,, , 2Ne because the contour does not enclose the poles ω = ω . Er trzt0 r0 = Ð------j 2ε ε 2 2 We can also use the relationship [14] b 1 Ð v 0/c (14) ()λ ()λ 1 ωωk xω 1 ()k + 1 () J0 nr0/b J1 nr/b ------π ∫ d exp ---Ð ---- = Ð1 Jk + 1 x . (9) × ∑------2 i 2 ω 2()λ + J1 n Cω n We substitute the series expansions (8) into integ-  × ⑀ ω z ral (7), change the order of integration and summation,  1 exp 0nttÐ 0 Ð ------and take into account relationship (9). As a result, we  v 0 obtain ∞ z  2πi m ⑀ ω  I = ------()Ð1 + 2 exp Ð 0nttÐ 0 Ð ------, el ω ∑ v 0  0n m = 0 (10) 12+ m 12+ m × []() () () trans 4Ne ir2 Ð ir1 J12+ m yn , (),,, , Er trzt0 r0 = ------b2ε 1 Ð εv 2/c2 α τ2 ξ2 ωÐ1 0 where yn = n Ð and r1, 2 = 21, . ()λ ()λ According to [15], the nth-order Lommel function × J0 nr0/b J1 nr/b ∑------2 - Un(q, x) of two variables is defined as ()λ n J1 n ∞ n + 2m mq ∞ (), ()--- ()  Un qx = ∑ Ð1  Jn + 2m x . (11) × ⑀ ()2m + 1 2m + 1 () x  1 ∑ r1 Ð r2 J2m + 1 yn m = 0  m = 0 ε (15) For t Ð t0 Ð z /c > 0, expressions (10) and (11) put integral (6) into the form 1 z + --- exp ω ttÐ Ð ------2 0n0 v 2π  0  (), (), I2n = ω------iU1 ir2 yn yn Ð iU1 ir1 yn yn --- 0n ∞ (12) ⑀ 2m + 1 1 () + 2 ∑ r1 Ð ------J2m + 1 yn 1  2m + 1 ω z m = 0 r2 + --- exp 0nttÐ 0 Ð ------. 2 v 0  1 z  The properties of the Lommel functions [15] imply ∑ + --- exp Ðω ttÐ Ð ------. 0n0 v that 2 0  (), iU1 iry y ε Here, we use the notation vpr = c/ and introduce the ∞  functions 2m + 1 () ≤ Ð ∑ r J2m + 1 y for r 1 z z  (13) 1 for ttÐ Ð ------≤ 0 and ttÐ Ð ------≥ 0 =  m = 0  0 v 0 v ∞ ⑀ 0 pr  () 1 =  ry y J2m + 1 y Ð sinh ----- Ð ----- Ð ∑ ------for r > 1.  z > z <  2 2r 2m + 1 0 for ttÐ 0 Ð ------0orttÐ 0 Ð ------0, m = 0 r v 0 v pr

TECHNICAL PHYSICS Vol. 47 No. 2 2002 230 BALAKIREV et al.

∞  z >  1 for ttÐ 0 Ð ------0 × ⑀ ()2m + 1 2m + 1 ()  v  1 Ð ∑ r1 + r2 J2m + 1 yn ⑀  0  2 = m = 0  z ≤ (17) 0 for ttÐ 0 Ð ------0. v 0 1 ω z ∑ + --- exp0nttÐ 0 Ð ------. The expression for the magnetic field generated by 2 v 0 a thin ring-shaped bunch can be obtained in an analo- ∞ gous way (in which it is also represented as a linear ⑀ 2m + 1 1 () superposition of the magnetic field associated with the + 2 Ð ∑ r1 + ------2m + -1 J2m + 1 yn curnt m = 0 r2 bunch current, Hϕ , and the transition radiation field trans Hϕ ): 1 z  ∑ + --- exp Ðω ttÐ Ð ------. 2 0n0 v (),,, , 0  Hϕ trzt0 r0 The quasistatic field components (14) and (16) and curnt(),,, , trans(),,, , = Hϕ trzt0 r0 + Hϕ trzt0 r0 , the transition radiation field components (15) and (17) ε are defined (and are nonzero) in the region z ≤ (t Ð t )v . curnt v 0 coul 0 pr Hϕ ()trzt,,, ,r = ------E ()trzt,,, ,r , (16) The quantity v introduced above is merely the highest 0 0 c r 0 0 pr velocity of the propagation of electromagnetic pertur- v bations in a waveguide. This indicates that, for t > t0, trans(),,, , 4Ne 0 Hϕ trzt0 r0 = ------neither Coulomb nor transition radiation fields enter the 2 ε 2 2 v b c 1 Ð v 0/c region to the right of the point zpr = (t Ð t0) pr . The fastest and highest-frequency component of the electro- J ()λ r /b J ()λ r/b magnetic signal is the precursor, which propagates with × ∑------0 n 0 1 n - 2()λ the velocity vpr [16]. Since the bunch propagation n J1 n velocity is v0 < vpr, the field overtakes the bunch. A qualitative pattern of the propagation of a transition radiation pulse is illustrated in Fig. 1, which shows the 0.01 longitudinal profiles of the first harmonics of the total 0 radial electric field and its Coulomb and transition radi- –0.01 ation components. The bunch generates an ultrawide- band transition radiation pulse, whose shortest-wave-

, rel. units –0.02 r length and highest-frequency components are the com- E –0.03 ponents of the precursor. The oscillation amplitude in the pulse decreases toward the precursor, so that the total ﬁeld vanishes in the cross section z = z . 0.01 pr 0 –0.01 NUMERICAL RESULTS For simulations, we chose an electron bunch with rel. units , –0.02 coul r the current density distribution

E –0.03 2 (), λ r0 2t0 jz r0 t0 = j0 J01---- exp Ð,4------1Ð (18) 0.01 a T b ≤ ≤ where j0 is the peak current density, 0 t0 Tb, Tb is the bunch duration, 0 ≤ r ≤ a, and a is the bunch radius. 0 0 , rel. units Analogous current density distributions are typical

r of many experiments. Such a bunch can be modeled by trans E a number of macroparticles in the form of thin rings –0.01 024681012with corresponding radii, charges, and longitudinal z/b coordinates. The field generated by the bunch can be represented as a sum of the fields of all constituent mac- Fig. 1. First harmonic of the field of a bunch in the form of roparticles. a thin disc of radius a/b = 0.125. The observation time is tc/b = 10, and the radius of the observation region is r /b = The qualitative pattern of the propagation of a tran- ε obs 0.25. The remaining parameter values are t0 = 0, = 1.0, and sition radiation pulse (see Fig. 1) was obtained by tak- v0/c = 0.5. ing into account only the first harmonic of the total

TECHNICAL PHYSICS Vol. 47 No. 2 2002 BROADBAND EMISSION FROM A RELATIVISTIC ELECTRON BUNCH 231

20 8 10 4

, kV/cm 0 0 , kV/cm trans r trans –10 r –4 E E –8 0 1 2 t, ns 0 1 2 3 4 5 6 7 8 9 t, ns

30 3 20

, MW 2 P , MW

10 P 1

0 1 2 t, ns 0 1 2 3 4 5 6 7 8 9 t, ns 0.3

0.2 0.3 ) f ( S 0.2 )

0.1 f ( S 0.1 0510 15 20 25 30 35 40 f, GHz

Fig. 2. Parameters of the transition radiation signal gener- 0510 15 20 25 30 35 40 f, GHz ated by a Gaussian bunch. The vertical lines are the critical frequencies fn. The signal is calculated at the observation point with the coordinates z /b = 1 and r /b = 0.25 for γ obs obs b = 4.0 cm, = 2.29, a = 0.5 cm, Lb = 2.0 cm, and Imax = 1.9455 kA. Fig. 3. The same as in Fig. 2 but at zobs/b = 20. radial field. However, in reality, a charged bunch in a the energy efficiency η, defined as the ratio of the radi- cylindrical waveguide generates the full set of radial ation energy to the kinetic energy of the bunch elec- field harmonics. This results in the narrowing of the trons, and the power-conversion efficiency χ, defined as field maxima and the appearance of characteristic the ratio of the maximum radiation power to the maxi- peaks whose amplitudes are much larger than the mum power of the bunch current. The efficiencies η and amplitude of the first radial field harmonic [17]. The χ so defined are directly proportional to the bunch numerical results presented in this section were charge. The parameters of the bunchÐwaveguide sys- obtained for the first ten field harmonics. For the above tem were chosen so that the maximum bunch current parameters of the waveguide and of the bunch, compu- was lower than the vacuum limiting current [18]. tations with a larger number of harmonics revealed no significant changes in the shape of the field profile. We Let us consider transition radiation signals near the simulated a vacuum waveguide (ε = 1) in which case waveguide end and far from it. The first case is illus- the radial inhomogeneity of the dielectric filling and trated in Fig. 2. We can see that an electromagnetic radiation losses in the dielectric are absent. pulse near the waveguide end is characterized by a large field amplitude (15 kV/cm), high maximum power In what follows, we restrict ourselves to analyzing (about 33 MW), and short duration (less than 1 ns). The the transition radiation component of the field. We cal- spectrum of this signal is broadband and has sharp nar- culated only the measurable radiation parameters. For row peaks at frequencies close to the critical frequen- this reason, we first investigated the time evolutions of λ π ε the field at certain spatial points. Second, we computed cies fn = nc/(2 b ) of the waveguide. The presence the time evolutions of the transition radiation power. of the low-frequency (f < f1) part of the spectrum can be Finally, we determined the frequency spectra S( f ) of explained as being due to the fact that the transition the generated electromagnetic pulses (here, f is the lin- radiation signal has propagated only a short distance ear frequency). The efficiency with which the oscilla- and, therefore, did not have enough time to form com- tions are generated was characterized by the following pletely. Far from the waveguide end (see Fig. 3), the quantities (in percentages of their maximum values): field amplitude is smaller than 8 kV/cm, the maximum

TECHNICAL PHYSICS Vol. 47 No. 2 2002 232 BALAKIREV et al.

S(f) than the corresponding critical frequencies fn and con- 0.4 tains no low-frequency part (below the frequency f1). We can see that the gaps between the neighboring crit- 1 ical frequencies disappeared and that the entire spec- 0.3 trum is somewhat broader. Such changes in the shape of the transition radiation signal may be explained as fol- 2 lows. When a charged bunch enters the waveguide, it 0.2 excites all possible natural waveguide modes simulta- 3 neously with different amplitudes. The dispersion properties of the waveguide are such that the natural modes 0.1 4 propagate at different velocities. As a result, the initially short wave packet spreads out continuously as it moves away from the waveguide end. Figure 3 shows 0 4 5 6 that the frequency of the recorded oscillations 10 8 15 10 decreases with time because higher frequency oscilla- 20 12 tions propagate at higher velocities and are the ﬁrst to f, GHz 25 14 Lb, cm 30 reach the observation point. The radiation energy remains constant; consequently, as the pulse duration Fig. 4. Dependence of the shape of the transition radiation increases, both the radiation power and the ﬁeld spectrum on the bunch length at the point with the coordi- γ strength decrease. nates zobs/b = 20 and robs/b = 0.25 for b = 4.0 cm, = 2.29, a = 0.5 cm, and Imax = 1.9455 kA. Figure 4 shows how the shape of the transition radiation spectrum changes as the length Lb of a charged 8 bunch increases. The total bunch charge was adjusted to be proportional to the bunch length in such a way that 6 the maximum current in bunches of different lengths was the same. The spectrum width is seen to decrease

, % 4 η signiﬁcantly as the bunch length Lb increases. In a 2 waveguide with the radius b = 4 cm, a bunch with the length Lb = 4 cm efﬁciently excites at least three 042 6 8 10 12 14 16 18 waveguide modes (curve 1) and a bunch with the length L , cm b Lb = 8 cm excites two modes (curve 2). In contrast, a bunch with the length Lb = 12 cm (curve 3), as well as 0.3 a bunch with the length Lb = 16 cm (curve 4), excites only one mode; moreover, the amplitude of the mode 0.2

, % excited by the longer of these two bunches is smaller. χ 0.1 This dependence of the shape of the radiation spectrum on the bunch length can be explained as a consequence of the current density distribution (18), which was cho- 042 6 8 10 12 14 16 18 sen above for an electron bunch. Figure 5 displays the Lb, cm dependence of the efficiencies η and χ on the bunch length Lb and the corresponding radial structure of the 8 η 1 2 transition radiation field. The energy efficiency is 6 seen to decrease as Lb increases. This is explained by 3

, kV/cm the smaller number of the excited waveguide modes | 4 4 and higher bunch kinetic energy, which is proportional trans r

E 2 to the number of bunch electrons. The power-conver- | χ sion efficiency has a maximum at Lb = 4 cm. In order 012 3 4 to explain this maximum, we turn to the field structure. r, cm As the bunch length Lb increases from 2 cm (curve 1) to 4 cm (curve 2), the spectral content of the generated Fig. 5. Dependence of η and χ on the bunch length and the radial structure of the field strength at the time at which the signal changes and the field maximum is displaced radiation power is maximum. The parameters of the bunchÐ toward the side wall of the waveguide. As a result, the waveguide system are the same as in Fig. 4. transition radiation power increases. With a further increase in Lb under such conditions that the first field radiation power is about 3 MW (which is one order of harmonic is dominant (curves 3 and 4 for Lb = 6 and magnitude lower than that in the previous case), and the 8 cm, respectively), the radial profile of the field does pulse duration is significantly longer (more than 5 ns). not change qualitatively but the maximum field ampli- The spectrum peaks at frequencies somewhat higher tude and, accordingly, the radiation power decrease.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 BROADBAND EMISSION FROM A RELATIVISTIC ELECTRON BUNCH 233

It is evident from Figs. 4 and 5 that the generation of 5 transition radiation in the waveguide is inefficient when 4 the waveguide diameter is smaller than or comparable to the length of the charged bunch. Let us consider how 3 , % the waveguide radius influences the generation effi- η 2 ciency of the transition radiation. The dependence of η 1 and χ on the waveguide radius b and the radial structure 0 of the transition radiation field that are shown in Fig. 6 4 8 12 b, cm were calculated for a bunch with fixed parameters. The maximum bunch current was lower than the vacuum 1.0 limiting current in a waveguide with the largest radius, and the dimensionless longitudinal coordinate of the observation point was zobs/b = const. With increasing , % 0.5 waveguide radius, the energy efficiency η increases χ monotonically and tends to saturate at large radii. On 0 the other hand, as the waveguide radius increases, the 4 8 12 , cm current of the bunch that can be injected into the b waveguide decreases. As b increases, the power-conversion efficiency χ, first increases to its maximum 4 1 2 value then gradually decreases. The radial structure of 3 the transition radiation field at the time at which the , kV/cm radiation power is maximum changes as follows. In | 2 3 trans waveguides with radii 4 and 8 cm (curves 1, 2), the field r 1 4 E is maximum near the side wall; the radiation power is | higher in a waveguide with the larger radius. In 024 6 8 10 12 14 r, cm waveguides with radii 12 and 16 cm (curves 3, 4), a 10-cm-long bunch excites a multimode signal and the Fig. 6. Dependence of η and χ on the waveguide radius and field maximum is displaced toward the waveguide axis. the radial structure of the field strength at the time at which the radiation power is maximum. The curves were calcu- The maximum field amplitude and, accordingly, the γ radiation power are both lower. Note that curves 3 and lated at the waveguide cross section zobs/b = 3 for = 2.29, 4 are qualitatively similar in shape to the radial profile a = 0.5 cm, Lb = 10.0 cm, and Imax = 0.973 kA. of the transition radiation field generated by a 2-cm- long bunch in a waveguide with a radius of 4 cm (curve 1 S(f) in Fig. 5). Figure 7 illustrates how the shape of the radi- 0.5 ation spectrum changes as the waveguide radius increases. We can see that, in a narrow waveguide (with 0.4 1 radius b = 4 cm), the bunch excites an essentially single-mode signal (curve 1). In a waveguide with radius 0.3 b = 8 cm, the bunch excites two waveguide modes 2 (curve 2), and, in a waveguide with radius b = 12 cm, it excites four modes (curve 3). In a waveguide with 0.2 3 radius b = 16 cm, at least five modes are excited (curve 4). The left boundary of the radiation spectrum 0.1 4 is displaced toward the lower-frequency range, the spectrum width remains essentially unchanged, and the 0 critical frequencies of the waveguide with the largest 2 4 4 radius are spaced most closely. 6 8 f, GHz 8 12 10 16 b, cm CONCLUSION Fig. 7. Dependence of the shape of the transition radiation When a charged bunch enters a semi-infinite cylin- spectrum on the waveguide radius. The parameters of the drical waveguide, it generates transition radiation. If bunchÐwaveguide system are the same as in Fig. 6. the Cherenkov resonance condition is not satisfied, the excited electromagnetic field is a superposition of the The spectrum of the transition radiation signal con- quasistatic field of a moving charge and the transition radiation field. The fastest component of the field (the tains waveguide’s eigen oscillations with all possible frequencies and wavenumbers. Because of the disper- precursor) propagates with the velocity c/ε , which is sion properties of the waveguide under consideration, higher than the bunch velocity. these oscillations propagate at vastly different veloci-

TECHNICAL PHYSICS Vol. 47 No. 2 2002 234 BALAKIREV et al. ties. The longitudinal profile of the transition radiation REFERENCES field calculated at a certain time shows that the wave- 1. H. F. Harmuth, Nonsinusoidal Waves for Radar and lengths of oscillations are the shortest in the “head” of Radio Communication (Academic, New York, 1981; the wave packet and become progressively longer Radio i Svyaz’, Moscow, 1985). toward the waveguide end (toward the “tail” of the 2. L. Yu. Astanin and A. A. Kostylev, Foundations of Super- packet). From the head to the tail of the packet, the spa- broadband Measurements (Radio i Svyaz’, Moscow, tial envelope of the signal first increases from zero to 1989). maximum then decreases gradually. The time evolution 3. L. Yu. Astanin and A. A. Kostylev, Zarubezhn. Radioéle- of the transition radiation field calculated at a suffi- ktron., No. 9, 3 (1981). ciently large distance from the waveguide end shows 4. N. V. Zernov and G. V. Merkulov, Zarubezhn. Radioélek- that the frequency of the recorded oscillations tron., No. 9, 85 (1981). decreases with time. By analogy with the spatial enve- 5. V. G. Stroitelev, Zarubezhn. Radioélektron., No. 9, 95 lope, the temporal envelope of the signal first reaches a (1981). maximum then decreases. 6. S. P. Pan’ko, Zarubezhn. Radioélektron., No. 9, 106 (1981). The spectrum of the transition radiation signal is 7. M. Friedman, V. Serlin, Y. Y. Lay, and J. Krall, in Pro- broadband and contains peaks corresponding to several ceedings of the 8th International Conference on High- harmonics of the radial electric field. The wide and Power Particle Beams, Novosibirsk, 1990, Vol. 1, p. 53. asymmetric peaks that are clearly distinguished in the 8. M. I. Gaponenko, V. I. Kurilko, S. M. Latinskiœ, et al., spectrum occur at frequencies somewhat higher than Vopr. At. Nauki Tekh., Ser. Yad.-Fiz. Issled. 11 (4, 5), the corresponding critical frequencies of the 151 (1997). waveguide. 9. V. A. Balakirev, M. I. Gaponenko, A. M. Gorban’, et al., Vopr. At. Nauki Tekh., Ser. Fiz. Plazmy, No. 3, 118 The transition radiation signal calculated near the (2000). waveguide entrance is characterized by high maximum 10. V. A. Balakirev and G. L. Sidel’nikov, Zh. Tekh. Fiz. 69 power and short duration. At a sufficiently large dis- (10), 90 (1999) [Tech. Phys. 44, 1209 (1999)]. tance from the waveguide entrance, the maximum 11. V. L. Ginzburg and I. M. Frank, Zh. Éksp. Teor. Fiz. 16, power of the signal is lower and the duration of the sig- 15 (1946). nal is significantly longer. 12. V. L. Ginzburg and V. N. Tsytovich, Transient Radiation The longer the bunch, the smaller the number of the and Transient Scattering (Nauka, Moscow, 1984). excited field harmonics. A sufficiently long bunch 13. N. G. Denisov, Zh. Éksp. Teor. Fiz. 21, 1354 (1951). excites an essentially single-mode signal, in which case 14. M. A. Lavrent’ev and B. V. Shabat, Methods of the The- the generation efficiency of a transition radiation pulse ory of Functions of a Complex Variable (Nauka, Mos- sharply decreases. The efficiency with which a UWBP cow, 1973). is generated by a long-duration bunch can be raised by 15. G. N. Watson, Treatise on the Theory of Bessel Func- tions (Cambridge Univ. Press, Cambridge, 1945; Inos- using a waveguide whose radius is larger than the trannaya Literatura, Moscow, 1949). bunch length. In this way, however, the bunch current will be low because it will be restricted by the vacuum 16. L. Brillouin, Ann. Phys. (Leipzig) 44 (10), 203 (1914). limiting current in the waveguide. 17. T.-B. Zhang, J. L. Hirshfield, T. C. Marshall, and B. Hafizi, Phys. Rev. E 56, 4647 (1997). In order to achieve the greatest efficiency of the gen- 18. A. A. Rukhadze, L. S. Bogdankevich, S. E. Rosinskiœ, eration of UWBPs, it is necessary to use short electron and V. G. Rukhlin, Physics of High-Current Relativistic bunches with currents close to the limiting current at Electron Beams (Atomizdat, Moscow, 1980). which the bunches can be transported through a vacuum waveguide. Translated by O. Khadin

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 235Ð237. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 96Ð98. Original Russian Text Copyright © 2002 by Vishnevskiœ, Mikherskiœ, Dubinko. SURFACES, ELECTRON AND ION EMISSION

Visualization of Nonuniform Magnetic Fields by Garnet Ferrite Films V. G. Vishnevskiœ, R. M. Mikherskiœ, and S. V. Dubinko Domen Design Ofﬁce, Vernadsky National University, Studencheskaya ul. 12, Simferopol, 95001 Ukraine e-mail: [email protected] Received March 5, 2001; in ﬁnal form, July 20, 2001

Abstract—The magnetization distribution in epitaxial garnet ferrite films subjected to the harmonic field of a signal from a magnetic tape is studied. By means of image processing techniques and diffraction spectrum analysis, it is shown that the distortion of the signal shape grows with increasing amplitude and period of the field. © 2002 MAIK “Nauka/Interperiodica”.

Epitaxial garnet ferrite films (EGFFs) are used as effective magnetooptic transducers, which are needed, in particular, for phonoscopic examination in criminal- istics [1]. They help to visualize magnetically recorded data. In some cases, EGFFs with easy-plane anisotropy are promising, since they, unlike the films with perpendicular anisotropy, are locally magnetized by rotating the moment rather than by domain wall displacement. Ä Ç In this respect, the question of to what extent the external field and the magnetization distribution induced in an EGFF coincide becomes of prime interest. Fig. 1. Magnetization structure in the EGFF with easy- plane anisotropy. The structure is induced by the sinusoidal In [2], it has been shown theoretically that the dis- µ tortions of the magnetization distribution that is field of the magnetic tape with the period d = 105 m. induced by a harmonic field depend on the amplitude and period of this field. The easy-magnetization axis lay in the film plane and was parallel to the shear com- I(x), M(x) 0.8 ponent of the external field. The aim of this work is to experimentally check the theoretical findings obtained 0.7 in [2]. 0.6 4 In the experiments, we used the diffraction of light by a phase grating induced in the EGFF by the sinuso- 0.5 1 3 idal field of a magnetic tape. It is assumed that the spec- 0.4 trum of a perfect sinusoidal grating contains mostly the first orders of diffraction. In addition, we studied the 0.3 distribution of the intensity of the grating image 0.2 observed in a polarization microscope. In the latter case, digital image processing techniques were 0.1 2 employed. 0 (Bi,Lu,Ca)3(Fe,Ga)5O12 and (Bi,Tm)3(Fe,Ga)5O12 EGFFs with easy-plane anisotropy were grown on 0 50 100 150 200 250 300 x, µm Gd3Ga5O12(111) substrates. The films were covered by a TiN reflective layer and brought into contact with the Fig. 2. Results of the digital image processing of image 1 magnetic tape. The tape contained sinusoidal signals taken from the segment AB. 1, intensity profile; 2, after fil- with a discrete set of frequencies and an amplitude gra- tering higher frequencies in 1; 3, magnetization distribution dation within 20 dB. The linear distortion of the records reconstructed from 2; and 4, cosinusoid

236 VISHNEVSKIŒ et al. reconstructed from curve 2 according to the expression ϕ + arcsin()Ix()/I Mx()∼ ------0-. (1) kθ 5 0 Here, I(x) is the intensity distribution along the coordi- 4 nate x, I0 is the intensity of light incident on the polar- 3 izer-EGFF-analyzer system, ϕ is the angle of deflection θ θ 1 2 of the polarizer axes from the crossed state, 0 = h is π Ð1 the total Faraday rotation, k = H0(HA + 4 Ms) < 1, H0 is the external field amplitude, HA is the axis of Fig. 3. Geometry of the magnetooptic diffraction experiments. anisotropy, and M(x) is the magnetization normal component distribution along the x axis. Comparing curve 3 and cosinusoid 4, we can judge the presence of distortions when the film images the signal shape. Diffraction spectrum analysis provides more accurate estimates of these distortions. A beam from a helium-neon laser passes through the side face of a polarizing prism-cube and strikes the EGFF; beams (except for the zero-order ones) dif- fracted by the induced magnetization distribution in the film leave the cube through its top side. The intensities of the diffraction orders are measured with a photoelectric multiplier. The experimental geometry is shown in Fig. 3, Fig. 4. Diffraction of light by the phase grating induced in where the polarization state of each of the beams is also the EGFF by the sinusoidal field of the magnetic tape. indicated. Such a geometry excludes both the effect of the magnetization tangential component on the Faraday rotation and polarization effects due to inclined inci- was no greater than 1.5%. The thicknesses of the EGFF dence of light [3]. The scattered fields of the active and the TiN coating were 3.5 and 0.2 µm, respectively. layer of magnetic tape 1 act on EGFF 3 through trans- π ≤ The saturation magnetization of the films was 4 Ms parent layer 2, producing a magnetic diffraction grat- 23 900 A/m and the specific Faraday rotation, θ = 0.9°Ð ing. Prism-cube 5 is arranged on substrate 4 via the 1.1° µmÐ1. The EGFF either had a spontaneous single- immersion contact so as to provide the film rotation in domain structure or was brought to the saturation state the plane. During the experiments, the in-plane EMA of by a uniform field applied along the easy magnetization the film was oriented either parallel (||) or perpendicular axis (EMA). The magnetization structure induced in (⊥) to the tangential component of the tape field. Pre- the EGFF by the tape field is shown in Fig. 1 (as it is liminary control of the EMA position was accom- seen in the polarization microscope). plished by the inductive method. The actual diffraction spectrum for the orientation (⊥) is depicted in Fig. 4. Figure 2 demonstrates the digital processing of this The spectrum has the well-defined symmetry (rela- image. Curve 1 is the intensity profile along segment AB, tive to the zero order), the energy maxima are at the curve 2 is the result of filtering high frequencies of orders ±1, and the intensity of even (second) orders is curve 1, and curve 3 is the magnetization distribution negligible.

Table

I , I3 I5 I7 I ++I I EMA Sample no. d, µm H, A/m 1 ------3 5 7 arb. units ------orientation I1 I1 I1 I1 1 100 11937 4.0 0.081 0.015 0.0025 0.3142 || 1 50 11937 2.25 0.044 0.009 0.0022 0.2357 || 2 100 11937 2.50 0.080 0.012 0.0028 0.3079 || 2 100 3979 1.50 0.047 0.011 0.0040 0.2620 || 2 100 11937 3.50 0.066 0.014 0.0035 0.2891 ⊥ 2 100 3979 1.00 0.030 0.005 0.0010 0.1897 ⊥

TECHNICAL PHYSICS Vol. 47 No. 2 2002 VISUALIZATION OF NONUNIFORM MAGNETIC FIELDS 237

The distortion of the actual grating relative to the tortions can be suppressed if the EMA is oriented nor- sinusoidal one, like the nonlinear distortion level, was mally to the tangential component of the field applied. estimated by the test Such a geometry is harder to treat theoretically, since the moment has two rotations different in phase: in the I ++I I L = ------3 5 7 , (2) EGFF plane and normally to it. I1 where I1, I3, I5, and I7 are the intensities of the first and REFERENCES higher orders, respectively. 1. S. V. Levy, Yu. S. Agalidi, and V. G. Vishnevskiœ, Izv. The experimental data are summarized in the table. Vyssh. Uchebn. Zaved., Radioélektron. 41 (7Ð8), 74 It is easy to see that the experimental data agree well (1998). with the theoretical results; namely, the maximal angle 2. R. M. Mikherskiœ, S. V. Dubinko, V. I. Butrim, and between the magnetization and the film plane is the V. G. Vishnevskiœ, Uch. Zap. Tavrichesk. Nats. Univ. 12 larger and the distortions of the induced grating are the (2), 156 (1999). higher, the larger the period d of the inducing field and 3. Yu. F. Vilesov, V. G. Vishnevskiœ, and N. A. Groshenko, the higher its amplitude. The level of the distortions Pis’ma Zh. Tekh. Fiz. 15 (13), 14 (1989) [Sov. Tech. depends on the applied field period stronger than on the Phys. Lett. 15, 500 (1989)]. field amplitude. As follows from the table, when the films with easy- plane anisotropy are used for the visualization, the dis- Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 238Ð243. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 99Ð104. Original Russian Text Copyright © 2002 by Shiryaev, Atamanov, Guseva, Martynenko, A. Mitin, V. Mitin, Moskovkin.

SURFACES, ELECTRON AND ION EMISSION

Production and Properties of MetalÐCarbon Composite Coatings with a Nanocrystalline Structure S. A. Shiryaev, M. V. Atamanov, M. I. Guseva, Yu. V. Martynenko, A. V. Mitin, V. S. Mitin, and P. G. Moskovkin Russian Research Center Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia e-mail: martyn@nﬁ.kiae.ru Received May 4, 2001

Abstract—It is suggested to produce metal–carbon composite coatings by magnetron sputtering of mosaic cathodes, which are Group IV, V, and VI metals. The mosaic structure of the cathode elements are computer- optimized for each of the metals. Reﬂection electron diffraction studies show that the coatings have the amorphous or nanocrystalline structures, which are thermally stable. The coatings offer speciﬁc physical properties, in particular, low friction factor and high hardness. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION ode is equal to the number of ions incident on a unit area per unit time: Modern methods for solid surface modification, such as ion implantation and magnetron sputtering, as jn= v T /4, (1) well as their combination, allow one to alter the surface properties of structural materials in a wide range. Coat- where vT is the ion thermal velocity and n is the plasma ings made of transition metal (Groups IVAÐVIA) car- density. bides with the face-centered cubic (fcc) lattice are of The ion current decreases at the recessed cathode particular interest, because this lattice is typical of met- parts, i.e., at the materials with the higher sputtering als characterized by high hardness, excellent high-tem- coefficient. Indeed, the plasma boundary is located at perature strength, and good corrosion resistance. the level of the highest cathode point along the mag- In this work, we synthesize the coatings of transition netic field line. Downward, i.e., toward the cathode, the metal carbides (TiC, Ta2C, Ta3C2, CrC, NbC, Mo2C, plasma density n rapidly decreases, because plasma and (TiC)N(Mo2C)N) by co-sputtering of the mosaic electrons are magnetized and be deflected from the cathodes. The metallic cathodes with graphite inserts plasma boundary by no more than the Larmor radius were sputtered by Ar+ ions or by an Ar+ + N+ mixture in r = mv /eB, the magnetron plasma. L e Magnetron sputtering is widely used for the deposi- where m is the electron mass, ve is the electron velocity, tion of various coatings [1], in studying composite coat- e is the electron charge, and B is the magnetic field. ings. In the latter case, the cathode is usually made from The ions and the electrons cannot be spaced more an alloy of necessary composition, which is sputtered in than the Debye length apart. The Debye length is given the magnetron [2]. by We suggest to apply composite coatings by using a ()π 2 1/2 mosaic cathode. The mosaic structure of the cathode rD = T/4 ne , (target) is tailored in such a way as to provide coatings where T is the plasma temperature. of uniform composition. Therefore, the characteristic distance of plasma density decrease can be estimated as THEORETICAL GROUNDS OF FILM DEPOSITION USING MAGNETRON r0 = rL + rD. SPUTTERING OF A MOSAIC CATHODE As a result, the erosion rates of the different cathode If the cathode is inhomogeneous, the erosion of the materials become equal. In this case, the material sur- different materials proceeds at a various rate due to a face moves due to erosion with a velocity difference in the sputtering coefficients. As a result, the uYj= /N, (2) materials with a higher sputtering coefficient turn out to be farther from the plasma than those with a lesser sput- where Y is the sputtering coefficient and N is the vol- tering coefficient. The ion current density on the cath- ume atom concentration.

PRODUCTION AND PROPERTIES OF METALÐCARBON COMPOSITE COATINGS 239

Thus, assuming that the ion velocities do not depend 1 2 3 on the distance from the cathode, we obtain the following condition for the erosion of two materials: () () S1nx1 /N1 = S2nx2 /N2, (3) where x1 and x2 are the surface coordinates of the first and second materials, respectively, along the axis normal to the surface. Fig. 1. Geometry of the composite target used in simulation. Knowing the dependence n(x), one can calculate ∆ | | from Eq. (3) the displacement x = x1 Ð x2 of the surfaces for the same erosion rates. If the plasma-facing surfaces of different materials constituting the target used for the deposition of multi- component coatings are coplanar, the ratio of the components being deposited will first be proportional to the 0.2 14 product of the area of sputtering by the sputtering coef- 12 ficient. As time passes, the erosion process becomes 0.1 10 8 steady-state, the erosion rates of all the components 0 6 become equal, and the ratio of the components is pro- 4 4 portional to the area of sputtering on the cathode. The 2 2 transient period of the erosion process is 0 τ∆∆ = x/ u, (4) Fig. 2. Concentration of one component in the coating for ∆ | | the coatingÐcathode distance h = 2 cm. where u = u1 Ð u2 is the difference in the rates of surface sputtering. Therefore, in practice, the cathode composition should be selected, starting from the equal erosion rates for all components. However, the coating deposited for the early period of time τ should be removed. Subse- quently, when the values of the surface displacement ∆x 0.08 for the components with higher sputtering coefficients 0.06 are accurately determined experimentally, these com- 14 ponents can be recessed at once by ∆x when manufac- 0.04 12 ≈ Ð4 0.02 10 turing the cathode. In our conditions, r0 10 m and 8 τ = ∆x/∆u ≈ 103 s. 0 6 4 4 A computational code has been developed to simu- 2 2 late the percentage of the components in the film depos- 0 ited and the film thickness. The code makes it possible to analyze the film deposition process when the targets Fig. 3. The same for h = 6 cm. of different geometry and composition are employed. A mosaic target used in the simulation is composed of base material 1 and overlying inserts made of mate- component in the film is shown. It is seen that, at a rials 2 and 3 (Fig. 1). However, our model applies to tar- small distance from the target, the components are gets consisting of an arbitrary number of components. deposited nonuniformly. The point is that, at a small To determine the percentage of the components in distance from the substrate, the material of each of the the coating, the contributions from each of the target islands is deposited directly underneath the insert. As inserts were calculated and then summed. The calcula- the distance increases, the distribution becomes more tions, using several angular distribution functions for uniform. At the same time, the amount of the material particles sputtered, showed that the distribution of par- reaching the substrate decreases. Therefore, the expo- ticles deposited slightly depends on the distribution sure time should be increased to obtain a coating of a function; therefore, the distribution function in the form desired thickness. From geometric considerations, one f(Θ) = cosΘ was used. can conclude that the material distribution in the film The distribution of the components in the film becomes uniform when the target–film distance R depends on the distance h between the target and the exceeds the insert spacing. At the same time, the den- plane of deposition. Therefore, the calculations were sity of the deposit is proportional to the ratio of the film performed for different values of h. The results are area S to R2. Hence, to minimize the deposition time, given in Figs. 2 and 3, where the distribution of one the substrate should be placed nearer to the target.

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240 SHIRYAEV et al.

Concentration, at. % density 1.81Ð1.86 × 103 kg/m3, which can be mechani- Cr cally treated and polished. 50 To perform complex investigations, the coating was deposited onto different metals; glass; ceramics; and hard-alloy foils, plates, and beads with a polished sur- C face. The TiC and TiCN, NbC, WC, CrC, MoC, and MoCN coatings were synthesized by sputtering the 40 cathodes made of corresponding metals with graphite inclusions by Ar+ ions or an Ar+ + N+ mixture. Depending on the material to be sputtered, the cathode voltage was varied from 400 to 600 V; the current, 30 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 in the range of 5Ð10 mA. Depth, µm Before the deposition, the substrates were cleaned by a high-energy (300 eV) Ar+ ion beam extracted from Fig. 4. Cr and C distributions in the coating. The chromium the plasma accelerator. After the deposition, the com- and graphite areas on the cathode relate as 50 : 50%. position and the thickness of the coatings were studied with Rutherford backscattering. Note also that the film thickness near the edges is The phase composition and the structure of the coat- smaller, which should be taken into account when ings were examined by high-energy electron diffraction arranging the substrates. The simulation showed the with a transmission electron microscope and an elec- following: (1) the magnetron sputtering of the mosaic tron diffractometer, X-ray diffraction, X-ray photoelec- target provides the deposition of the homogeneous tron spectroscopy, Auger electron spectroscopy, scan- films; (2) to produce a film of uniform component dis- ning electron microscopy, X-ray microspectral analysis tribution over the surface, the substrate surface should with an SEM, and optical microscopy. be placed at a distance far exceeding (about 3 times) the The coating thickness was estimated on the cleavage distance between the islands; (3) as the distance surface using a scanning electron microscope and a increases, the number of target atoms reaching the film profilometer. decreases, so that the exposure time must be increased The coating adhesion to the substrate was tested by to obtain the film of a desired thickness. a REVETEST standard automatic tester available from the LSRH using a Rockwell C indenter. The loading rate was 100 N/min; the indenter velocity, 10 mm/min. EXPERIMENTAL The coating microhardness was determined using a The experiments were carried out using a Vita ion PMT-3 hardness meter and an MI-7 microscope under accelerator equipped with a high-energy ion source, a a load of 0.2Ð2.0 N. plasma accelerator, and a magnetron. MetalÐgraphite The wear resistance of the coatings was determined compositions were used as the cathode. The metals by rubbing against a loosely fixed abrasive (diamond were tantalum, molybdenum, niobium, vanadium, paste on a thick paper) of granulometric composition chromium, zirconium, titanium, and oxygen-free cop- (10Ð15) × 10Ð6 m on a friction machine at a constant per. The refractory metals were prepared by vacuum load of 26 Pa. The coating thickness was found to be melting. We also used high-purity fine-grain graphite of (3Ð25) × 10Ð6 m. The wear resistance was estimated

(‡) (b) (c)

Fig. 5. Diffraction patterns for the (a) Ta2C, (b) TaC, and (c) Ta3C2 in the TaÐC coating.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 PRODUCTION AND PROPERTIES OF METALÐCARBON COMPOSITE COATINGS 241

Composition and properties of coatings Cathode com- Microhar- Phase composition Lattice types Crystallite position, ratio Chemical dness H Compound of coating (X-ray data), and parameters, size, V Me : C over composition, wt % (0.2Ð1 H), wt % ×10Ð1 nm ×10Ð1 nm surface in % GPa ± (WG) 50W + 50C WC0.97 = 96.4 0.4 hcp 30Ð40 16.00Ð27.00 C = 3.6 ± 0.4 a = 2.907 (~24)* b = 2.839 ± (7GTa) 50Ta + 50C TaC0.97Ð0.1 = 97.6 0.4 fcc 60Ð80 15.30Ð27.00 (13TaG) C = 3.4 ± 0.4 a = 4.44 (15.7) ± (6MoG) 70Mo + 30C Mo2C = 67.6 0.4 hcp Texture Mo(tot) = 77.6 + 2.3 13.00Ð30.00 Mo = 32.4 ± 0.4 a = 3.01 22Ð50 C(tot) = 2.3 ± 0.3 β ± Closer to -Mo2C, c = 4.74 Mo(fr) = 38.6 2.3 high-temperature ± α ± (18MoG) 50Mo + 50C MoC0.95Ð1.0 = 95.6 0.4 fcc -MoC 60Ð80 Mo(tot) = 82.6 2.6 18.00Ð31.00 (19MoG) C = 4.2 a = 4.27 84.5 ± 0.6 C(tot) = 15.5 ± 0.6 16.2 ± 0.1 C(fr) = 1.4 ± 1.0 5.6 ± 1.5 ± (11NbC) 70Nb + 30C Nb2C Hexagonal Texture Nb(tot) = 2.6 0.7 23.00Ð29.00 a = 3.115, 10Ð30 C(tot) = 6.07 ± 1.5 (24) c = 4.948 ± (NbG) 50Nb + 50C NbC0.95Ð1.0 fcc Texture Nb(tot) = 84.3 3.0 18.90Ð23.00 a = 4.436 20Ð50 C(tot) = 11.45 ± 0.8 (24) ± (10GV) 50V + 50C VC0.85Ð0.88 fcc 15Ð50 V(tot) = 84.3 3.0 13.00Ð26.00 a = 4.188 C(tot) = 9.5 ± 0.5 (28) ± (8GCr) 50Cr + 50C Cr3C + Cr Rhombic 80Ð120 Cr(tot) = 84.3 2.8 17.00Ð27.00 ± Closer to Cr6C phase a = 2.86, C(tot) = 13.4 0.2 b = 5.58, C(fr) = 0.59 ± 0.05 c = 11.48 ± (2TiG) 50Ti + 50C TiC0.95Ð0.98 = 98.5 0.4 Cubic, type V-1 Texture 111 21.50Ð37.40 4.330 30Ð50 (29.3) ± (3TiG) 80Ti : 20C TiC0.5Ð0.6 = 68.6 0.4 Cubic, type V-1 80Ð120 11.00Ð22.00 Ti = 31.4 ± 0.4 4.334 ± (4ZrG) 50Zr : 50C ZrC0.9Ð0.95 = 96.5 0.4 Cubic, type V-1 50Ð60 18.50Ð30.00 4.677 (28.6) ± (5ZrG) 80Zr : 20C ZrC0.5Ð0.6 = 76.5 0.4 Cubic, type V-1 Texture 10.00Ð21.00 Zr = 23.5 ± 0.4 4.683 120Ð180 Note: tot, total content; fr, in free state; * refers to tabulated value. from the rate of linear wear and friction track colora- states, was determined by specially developed tech- tion. niques with a Leco C-212 AH-7529 analyzer. The friction coefficient was determined using a slid- The thermal oxidation resistance in air and in a pure ing friction machine as follows: a coated rotating disk oxygen flow was tested by the method of thermogravi- rubbed against an uncoated fixed ball under dry friction metric oxidation using a Setaram instrument (France). conditions in air. The corrosion resistance of the coatings was esti- The chemical composition of the coatings, i.e., the mated electrochemically using a PS4 potentiostat by content of the metal and carbon in bound and free plotting anode potentiodynamic polarization curves at a

TECHNICAL PHYSICS Vol. 47 No. 2 2002 242 SHIRYAEV et al.

30 45 60 75 90 105 120 135 150 2Θ

Fig. 6. X-ray reﬂections for the Mo–C coating. The phases are Mo2C with a crystallite size of 4.8 nm, Mo4C (5.1 nm), and a small amount of MoC (2.29 nm).

rate of 7.8 V/h in H2SO4 and HCl solutions at room The measurement of residual stress in the 6MoG, temperature. 19MoG, and 8GCr samples (see table) by the X-ray diffractometer equipped with a special ψ attachment shows that in the Mo phase, compressive stresses in the EXPERIMENTAL RESULTS coating plane are ≈0.3 GN/m2 (6MoG), while in MoC For discharge power densities from 5 × 105 to 11 × phases, they are 0.19 GN/m2 (19MoG). In both cases, 105 W/m2, we obtained smooth homogeneous conden- the coatings were deposited on carbon ball bearing sates with metallic luster on different substrates. On the steel. polished and glass substrates, the condensate had the specular surface, which is typical of refractory metal The estimation of the adhesion for metalÐcarbon layers. coatings as a whole (both qualitative and quantitative) shows that the highest adhesion (about the mean car- The investigation of the phase composition and the bide tensile strength, 0.18Ð0.25 GPa; incomplete structure of the coating shows that when the mosaic detachment: film islands remain on the substrate) is cathode is sputtered, the metal and carbon atoms homo- observed for the ductile substrates (such as copper, geneously mix to form carbides, if the metals interact nickel, low-carbon and stainless steel). The adhesion with carbon, or pseudosolid carbon solutions in metals was 1.5Ð2 times lower for the substrates made of the otherwise. All the Group IVAÐVIA refractory metals hard allow, high-speed and ball bearing steel, as well as studied in this work interact with carbon at tempera- of glass and ceramics. This is especially true for the tures no more than 473 K to form carbide phases, coatings of a thickness >5 × 10Ð6 m. including high-temperature phases (see table). The formation of these phases is also confirmed by photoelec- The reflection and transmission electron micros- tron energy spectrum analysis. The variation in the ele- copy studies of the coating surface confirm the pres- ment composition of the coating across the depth is typ- ence of the ultrafine-grain nearly amorphous structure, ical of almost all of the compositions. The Auger as evidenced by the diffraction patterns (Fig. 5). spectroscopy data for CrÐC, TaÐC, and MoÐC demonstrate that ion etching causes the C concentration to X-ray reflections for the Mo–C coating are shown in decrease and the metal concentration to increase Fig. 6. Their identification shows that the coating is (Fig. 4). Apparently, this results from graphite diffusion multiphase, containing Mo2C with a crystallite diame- into the coating and its segregation at the surface. ter of 4.8 nm, Mo4C (5.1 nm), and a small amount of

TECHNICAL PHYSICS Vol. 47 No. 2 2002 PRODUCTION AND PROPERTIES OF METALÐCARBON COMPOSITE COATINGS 243

HV, GPa microhardness as a function of load is shown in Fig. 7. The surface layer is superhard at a load of 0.2 N, HV = 50 50 GPa. The wear tests using a diamond paste showed the coatings with the composition Me C + Me and 2 CB 40 MeC + C offer the highest wear resistance. CB Thus, it is shown that the co-sputtering of a metal 30 and carbon provides their homogeneous mixing. The resulting deposits may contain both bound and free metal and carbon atoms. The coatings have the ultraﬁne-grain near-amorphous nanocrystalline struc- 20 ture, and this state is retained up to an annealing temperature of ≈1000 K. In comparison with crystalline materials of this 10 class, the metalÐcarbon coatings with the nanocrystalline structure offer higher corrosion resistance in a 0 0.5 1.0 1.5 2.0 number of acids, higher oxidation resistance in air, Load, N lower friction coefﬁcient, and higher wear resistance. The MoC coating is superhard (HV = 50 GPa at 0.2 N). Fig. 7. Microhardness vs. load for the MoÐC coating.

ACKNOWLEDGMENTS MoC (2.29 nm). One can conclude that the molybde- Yu.V. Martynenko and P.G. Moskovkin thank the num carbides have the nanocrystalline structure. Grant Council at the President of the Russian Federa- As the carbon content decreases and phases, such as tion and the Leading Scientific School Foundation Ta2C, Nb2C, and Mo2C appear, i.e., when hcp MeC sub- (grant no. 00-15-96526) for the support. carbides form instead of the fcc MeC carbides (NaCl- like structure), the structure of coatings sharply REFERENCES changes. The lines 200, 220, and 111, which are typical of the fcc lattice with a parameter of 0.444 nm, are suc- 1. P. J. Kelly and R. D. Arnell, Vacuum 56, 159 (2000). cessively smeared. The ultrafine-grain nanocrystalline 2. J. Musil and J. Vicek, in Proceedings of the 5th Confer- structure of the coatings provides high resistance to ence on Modification of Materials with Particle Beams oxidation and corrosion. They also offer improved and Plasma Flows, Tomsk, 2000, p. 393. mechanical properties: hardness and wear resistance. The MoÐC coating is the hardest. For this coating, the Translated by M. Astrov

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 244Ð249. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 105Ð110. Original Russian Text Copyright © 2002 by Tumareva, Sominskiœ, Efremov, Polyakov. SURFACES, ELECTRON AND ION EMISSION

Tip Field Emitters Coated with Fullerenes T. A. Tumareva, G. G. Sominskiœ, A. A. Efremov, and A. S. Polyakov St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia e-mail: [email protected], [email protected] Received July 6, 2001

Abstract—Formation and characteristics of fullerene coatings on the surface of tungsten tip field emitters and emitters with ribbed crystals formed on their surface are studied. It is found that electric fields and heat treatments affect the structure and emission characteristics of the fullerene coatings. Methods of creating microprotusions on the surface of the coatings that considerably enhance the electric field have been developed and tested, and emitters with a single microprotrusion yielding emission current densities up to 106Ð107 A/cm2 are demonstrated. Emitters with fullerene coatings in the form of a distributed microcluster structure can yield currents in excess of 100 µA from a single micron-sized tip in a stationary regime. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION rizes the results of our research on fullerene coatings on the surface of tip field emitters of tungsten. One of the important unresolved tasks of vacuum electronics is the development of long-service-life field emitters for producing intense electron beams under EXPERIMENTAL TECHNIQUE technical vacuum conditions. It is known (see, for example, [1Ð3]) that coatings containing carbon are, as The experiments were performed in a field emission a rule, resistant to ion bombardments and the harmful projector. The fullerene coatings were deposited onto effects of gaseous ambients and can be used as protec- tip field emitters from a Knudsen cell [7]. The coating tive coatings for increasing the service life of field emit- structure was determined from emission images of the tip apex observed on the projector screen at magnifica- ters. Carbon, as a readily available and inexpensive 5 6 material, appears to be an obvious choice for such coat- tions of 10 Ð10 . To reveal the effect of the coating on ings. However, a protective coating should at the same the performance of the field emitters, the current–volt- time ensure high efficiency of the field emitter. Because age characteristics of the entire tip and of separate areas of the large work function value of carbon materials on its surface were measured. In addition, values of (for example, for graphite eϕ ~ 4.7 eV) there appears to voltage UI corresponding to certain fixed field emission be only practical means for obtaining high currents at currents I were monitored. Characteristic features of low voltages with tip emitters coated with carbon: the UI dependence of deposition time (“deposition microprotrusions should be formed at the emitter sur- curves”) were used to determine the approximate coat- face to enhance the electric field. Standing in the way of ing thickness or identify the moments of time at which the wide use of carbon are problems relating to prepa- structural transformations occurred corresponding to ration of this sort of structure and implementation of the the emergence (or disappearance) on the emitter sur- process of depositing this high-temperature material. It face of microprotrusions enhancing the electric field. might seem that good results for coatings with micro- Images on the projector screen were registered with structures enhancing the electric field could be a sensitive TV camera, then digitized in a special con- expected from the application of nanotubes [3, 4], but, troller and transferred to a PC for storage and subse- so far, acceptable current characteristics for tip cath- quent processing. This system could reliably register odes with nanotubes could not be obtained because a images obtained with extremely low currents from the simple enough technology for forming ordered struc- tip, down to I ~ 0.01Ð0.02 µA, at a rate of 24 images per tures of nanotubes is not yet available. Our purpose is second. The high sensitivity and speed of this image to study the possibility of using C60 fullerene molecules registration system afforded the possibility, where nec- as a field emitter coating. This material has not been essary, of minimizing the effect of the electric field on used earlier for such applications. Our presumption is the coating structure during registration, even in a sta- that the large spherical C60 molecules are better suited tionary regime. The effect of the electric fields on the for creating field-enhancing microstructures on the sur- emitter characteristics being registered could be further face of refractory materials than nanotubes are. The very reduced using a regime of short (~4 µs) pulses at a low first data obtained [5, 6] mostly supported these expecta- controllable repetition rate, f ≤ 400 Hz. With reduced tions, and research was continued. This study summa- effect from electric fields during the measurements, the

TIP FIELD EMITTERS COATED WITH FULLERENES 245 effect of the field specially applied for coating treatment could be better understood. The effect on the field emitter parameters of interest of the quantity of deposited substance (an approximate coating thickness Θ of up to 50Ð100 monolayers), substrate temperature T (from room temperature up to 2800 K), and the electric fields created by application of voltage between the cathode and the screen (up to approximately 108 V/cm) was studied. Most of the experiments were carried out at a residual gas pressure in the experimental setup of ~ 10Ð10Ð10Ð9 torr. Before deposition of the fullerene coating, a single crystal was formed on the apex of a tungsten tip. Heat (‡) treatment at ≥950Ð1000 K of the deposited fullerene layers caused decomposition of the fullerene molecules and formation on the tip apex of a so-called ribbed crystal having a layer of tungsten carbide on its surface [8]. Transition from the ribbed crystal back to pure tungsten single crystal (“purification” of the emitter) was performed by heating the tip to temperatures just below the limiting temperature or by heating to lower temperatures after a prior considerable (by a few orders of mag- nitudes) increase of the pressure in the setup. Represen- tative images of the tungsten tip and the ribbed crystal are shown in Fig. 1. The tungsten tip emitters studied had a tip apex diameter of approximately 0.3 to 2 µm. Formation of the fullerene coatings deposited both on the tungsten single crystal and the ribbed crystal was (b) studied. Although the measurement technique used introduced a slight perturbation, the main results pre- Fig. 1. Images of emission from (a) tungsten crystal and sented below were obtained in the stationary regime. (b) ribbed crystal. On the left side in frame (a) (and some Therefore, below unless otherwise indicated, the subsequent emission images as well), a shadow of the field regime is assumed stationary. emitter holder can be seen.

EXPERIMENTAL RESULTS are compared for different fullerene deposition rates AND DISCUSSION (different heating of the Knudsen cell). 1. Characteristics of the fullerene coatings Setting aside for the time being the reasons for formed without applying a strong electric field. As changes of the deposition curves with emission current, found in the experiments, if a coating is deposited with let us make some inferences concerning characteristics no electric field about the tip, the layer of fullerene mol- of the fullerene coatings from the data obtained at I = ecules formed both on pure tungsten crystals and ribbed 0.02 µA, i.e., under conditions of minimum influence crystals in early deposition stages is fairly uniform over of the electric fields involved in the measurement pro- the surface, without pronounced protrusions. As its cess. Under these conditions, uI, after attaining a certain thickness grows, the current to the screen at fixed volt- peak value uIm, slowly diminishes with increasing coat- age U diminishes because of the increasing work func- ing thickness in the case of a ribbed crystal and stays tion of the surface and, for the same reason, the voltage practically unchanged for a long time for deposition on UI corresponding to the fixed current I increases. Fur- pure tungsten. uI(t) characteristics measured at differ- ther variations of UI depend essentially on the regis- ent deposition rates are qualitatively similar and differ tered magnitude of the field emission current I. This is mainly in the time required for reaching uIm. The time illustrated in Fig. 2, where “deposition curves” uI(t) tm diminishes with deposition rate. Usually the deposi- (dependences on deposition time t of the quantity ut, tion rate is determined from deposition curves. The which is a quotient of the voltage UI measured at the shape of deposition characteristics in Figs. 2 and 3 is time moment t and its initial value UI(0) at time t = 0) typical of field emission systems where the work func- are compared. Curves 1 and 2 were obtained at currents tion increases with the coating thickness. The approach 0.02 and 0.25 µA, respectively. Deposition characteris- of the voltage uI towards the maximum uIm of the depo- tics for a ribbed crystal are shown also in Fig. 3, where sition curves might be related to formation of a mono- µ u0.02(t) curves measured at the same current I = 0.02 A layer coating. If this is the case, then from the charac-

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246 TUMAREVA et al. Θ uI tungsten with a fullerene coating of a thickness of ~ 1.5 (‡) 1.5Ð2 monolayers (mL) yields the value of the work 1 function eϕ of the fullerene coating. From the data obtained it follows that the work function equals approximately 5.5 eV. In our opinion, estimates of the deposition rate and 1.0 work function values are possible only under condi- 2 tions where the coating is fairly uniform and there are no highly localized regions of enhanced electric field as a result of formation of 3D microstructures. Such conditions, even with the weakly perturbing measurement 0.5 technique employed, could be obtained only for coatings of a thickness of Θ < 2Ð4 mL. Figure 4 shows rep- 1.5 resentative emission patterns for different stages of the (b) fullerene coating deposition onto a tungsten tip. The experimental observations indicate that at coating 1 thicknesses Θ ~ 0.2–1.1 mL, only insignificant emission nonuniformities in the form of separate spots exist. 1.0 Nonuniformities become more pronounced as the coating thickness increases; however, up to Θ ~ 2Ð4 mL, 2 their contribution to the total emission current from the tip is relatively small. For this reason, the initial distribution of emission from the tungsten tip is well discern- 0.5 ible. Only on thick coatings bright emission spots are 0 30 60 90 observed from emerging microprotrusions, which sig- t, min nificantly enhance the electric field and obscure the image of the substrate. Fig. 2. Normalized deposition curves uI for (a) pure tungsten and (b) ribbed crystal. Curves 1 and 2 were obtained at 2. Effect of the electric field on characteristics of I = 0.02 and 0.25 µA, respectively. fullerene coatings. The above-mentioned formation of 3D microprotrusions on the fullerene coating can be caused by even those relatively weak perturbations u0.02 from the electric field applied briefly for obtaining an 1.3 1 3 emission image on the projector screen. This supposition is supported by considerable changes of the form 2 of deposition curves with increasing registered current 1.2 (compare characteristics 1 and 2 in Fig. 2). Probably, the image control at high emission currents (and voltages) causes reconstruction of the coating already in the course of the measurement process. In Fig. 5, a series 1.1 of emission images is shown that were obtained at an emission current of 0.25 µA. During observations in this regime, microprotrusions on the tungsten tip sur- 1.0 face contributing considerably to the emission arose at 0 50 100 150 t, min coating thicknesses as small as Θ ≤ 0.5 mL. At large thicknesses, the nonuniformities grow to such an extent Fig. 3. Deposition curves u0.02(t) for ribbed crystal at I = that already at Θ = 1.7 mL the substrate cannot be dis- 0.02 µA and different deposition rates of fullerene coatings (different temperatures of the Knudsen cell). (1) T = 600; tinguished. The field enhancement at nonuniformities (2) 580; and (3) 550 K. is what causes the abrupt decline of the deposition curves measured at the current of 0.25 µA compared with curves measured at 0.02 µA. teristics given in Fig. 3, it can be inferred that a mono- These data are not the only evidence of the effect of layer coating is formed on a ribbed crystal in times of electric field. Special experiments were carried out in ~15 (curve 1), ~30 (curve 2), and ~175 min (curve 3). which the obtained coatings were treated in an electric For the same reasons, at the given deposition rate, a field. The dynamics of the emission characteristics of a monolayer coating on pure tungsten is formed in a time fullerene coating under applied electric field can be of ~20 min (Fig. 2a) seen in Fig. 6. An emission system was formed in the Comparison of the U0.02 values necessary for obtain- measurement process of curve 1 in Fig. 2b after 90 min ing a specified fixed current from pure tungsten and of the coating deposition on a ribbed crystal. At this

TECHNICAL PHYSICS Vol. 47 No. 2 2002 TIP FIELD EMITTERS COATED WITH FULLERENES 247

(‡)

(b)

(c)

Fig. 4. Emission images of a tungsten tip registered at I = 0.02 µA; Θ: (a) 0.2; (b) 1.1; and (c) 5 mL.

(c) moment, the voltage was increased to U0.25, a value corresponding to the current I = 0.25 µA. Figure 6 shows Fig. 5. Emission images of the tungsten tip registered at I = the dependence of the voltage quotient u0.25(t) = 0.25 µA; Θ: (a) 0.5; (b) 1.7; and (c) 4.5 mL. U0.25(t)/U0.25(0) on the time in the electric field starting from the moment (t = 0) when the voltage was applied. Rapid drop of this specific quantity is due to formation field treatment of fullerene coatings deposited on pure after voltage increase of the number of centers of tungsten. intense emission on the cathode surface. In time, some Summarizing the data obtained in studies of the of these centers break away from the tip (or disinte- effect of electric field on the structure of a fullerene grate), which makes the u0.25(t) curve nonmonotonic. coating, the mechanism of this effect appears to be as The microprotrusions on a ribbed crystal coating sur- follows. In highly nonuniform electric fields, C60 mole- face tend to accumulate near the ribs, i.e., in areas of cules can be polarized and the molecular dipoles initially enhanced field (Fig. 7a). Similar transforma- attracted to the regions of enhanced electric field tions of the deposition curves have been observed after emerging as a result of the roughness or microprotru-

TECHNICAL PHYSICS Vol. 47 No. 2 2002 248 TUMAREVA et al.

u0.25

1.0

(‡) 0.5 01020 t, min

Fig. 6. Time variation of the quantity u0.25 under applied electric field (“field treatment”). sions on the initial surface. Completion of small nonuniformities enhances the electric field still further, and the emission current from them increases. Efficient growth of microprotrusions on the emitting surface starts, according to our estimates, at electric field strengths of E ~ (1Ð5) × 107 V/cm. (b) It should be noted that the formation of microprotrusions on the fullerene coating surface has been observed not only in the stationary regime but in pulsed measurements as well. Formation of these nonuniformities only occurred at much higher collected currents (and electric field strengths) than in the stationary regime, ≥10 µA. So far, not enough data has been accumulated on processes in the pulsed regime. Therefore, the temporal characteristics of the growth of microprotrusions cannot be clearly recognized as yet. 3. Formation of stable emitting structures on the surface of fullerene coatings. As mentioned above, field treatment of the fullerene coatings on tungsten can (c) be used to create microformations enhancing the electric field at the surface and considerably reduce the Fig. 7. Images of emission from the ribbed crystal with voltages required for reaching a given current. How- fullerene coatings of various types: (a) several microprotru- ever, emitters with nonuniformities of such type do not sions on ribs; (b) a separate efficiently emitting center; and show stable emission even at currents I ≤ 1 µA. (c) a distributed microcluster structure providing emission currents of >100 µA. These studies demonstrated that fullerene microstructures displaying stable emission are possible to create only using ribbed crystals having a layer of tung- center. Formation of such a center results in a consider- sten carbide on the surface. It has been found that on able (up to 50%) reduction of U from its value for the Θ I thick enough coatings ( > 5Ð10 mL), separate micro- initial ribbed crystal. Initially, the unstable spot stabi- protrusions initially accumulating on the ribs (along lized after brief (1Ð5 min) heating at a temperature of [111] zonal lines) in an applied electric field combine about 950 K, which is close to the fullerene molecule and form one large intensively emitting center in the decomposition temperature. Prolonged (≥20Ð30 min) region of {023} planes close to the central (110) plane. Sometimes this effect takes place precisely during heating at this temperature caused its destruction; the observations or while a FowlerÐNordheim characteris- image returned to that of the original ribbed crystal. An tic is being measured. In Fig. 7, screen emission pat- emitting center treated at ~950 K yielded a current of terns are shown before (Fig. 7a) and after (Fig. 7b) the ~10 µA. The estimated current density reached ~106Ð merging of separate microprotrusions into one emitting 107 A/cm2.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 TIP FIELD EMITTERS COATED WITH FULLERENES 249

Substantially different coatings in regards to their (ii) It has been found that thermal and field treat- structure and emission properties could be obtained on ments produce isolated microprotrusions on the a ribbed crystal after a special treatment of its surface fullerene coatings, which considerably enhance the consisting in repeated cycles of deposition/thermal field and ensure stable field emission with high current decomposition of fairly thick deposited layers (Θ > 5Ð densities of up to 106Ð107 A/cm2 in the stationary 10 mL). After two to four cycles of the surface treat- regime. ment under an electric field of a strength necessary for (iii) The possibility has been demonstrated of fabri- producing a current of a few microamperes, numerous cating emitters with fullerene coatings in the form of a (a few tens) 3D clusters formed. The cluster structure distributed multicluster system yielding stable currents (Fig. 7c) produced in this way was stable under applied exceeding 100 µA from a single micron tip in the sta- electric field, and in the stationary regime a micron-size tionary regime. tip could yield currents of more than 100 µA. Such structures are resistant to gaseous media. Exposure to vacuum for several days produced practically no REFERENCES changes in their properties. Measurements have shown 1. B. V. Bondarenko, V. A. Seliverstov, A. G. Shakhovskoœ, that in pulsed regime currents from a field emitter with and E. P. Sheshin, Radiotekh. Élektron. (Moscow) 32, the fullerene coating could be increased severalfold. 199 (1987). Physical phenomena that take place during the pro- 2. N. Chubun, N. Lazarev, E. Sheshin, and A. Suvorov, in cedure required for creating the above-described effi- Abstracts of the 42th IFES, Madison, 1995. ciently emitting distributed structures on the surface of 3. Yu. V. Gulyaev, Yu. A. Grigor’ev, N. I. Sinitsyn, et al., in a ribbed crystal is as of yet not quite clear. At this stage Proceedings of All-Russia University Conference “Mod- it can only be supposed that the multiple deposi- ern Problems of Electronics and Radiophysics of Ultra- tion/decomposition acts to increase the thickness of the High Frequensies” (Kolledzh, Saratov, 1997), p. 90. transitional layer of tungsten carbide (or tungsten car- 4. N. I. Sinitsyn, Yu. V. Gulyaev, G. Torgashov, et al., Appl. bide with inclusions of carbon) at the base of the multi- Surf. Sci. 111, 145 (1997). cluster system being formed. 5. G. G. Sominski, T. A. Tumareva, A. S. Polyakov, and K. K. Zabello, in Proceedings of the International Uni- CONCLUSIONS versity Conference “Electronics and Radiophysics of Ultra-High Frequensies” (Nestor, St. Petersburg, 1999), In summary, the main results of this work are as fol- p. 327. lows. 6. T. A. Tumareva and G. G. Sominskii, J. Commun. Tech- (i) The mechanism has been established by which nol. Electron., Suppl. 1 45, S110 (2000). fullerene coatings form on the surface of tungsten tip 7. V. I. Karataev, Pis’ma Zh. Tekh. Fiz. 24 (5), 1 (1998) field emitters, and their characteristics have been deter- [Tech. Phys. Lett. 24, 167 (1998)]. mined. It was found that electric field affects the struc- 8. M. V. Loginov and V. N. Shrednik, Pis’ma Zh. Tekh. Fiz. ture and the emission characteristics of the fullerene 24 (11), 45 (1998) [Tech. Phys. Lett. 24, 432 (1998)]. coating on the surface of tungsten tips and ribbed crystals formed by thermal decomposition of fullerene molecules on tungsten. Translated by B. Kalinin

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Technical Physics, Vol. 47, No. 2, 2002, pp. 250Ð254. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 111Ð115. Original Russian Text Copyright © 2002 by Tumareva, Sominskiœ, Polyakov. SURFACES, ELECTRON AND ION EMISSION

Formation on Field Emitters Coated with Fullerenes of Microformations Producing Ordered Emission Images T. A. Tumareva, G. G. Sominskiœ, and A. S. Polyakov St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia e-mail: [email protected], [email protected] Received July 6, 2001

Abstract—Field emission projector studies of fullerene coatings deposited on tungsten tip field emitters reveal specific ordered patterns in the form of doublets, quadruplets, rings, disks, and other forms in the emitter images. The ways in which these types of ordered emission patterns arise and their relation to C60 microformations have been established. Possible causes of the emergence of the ordered emission images are analyzed on the basis of published data and experimental results obtained. A modification of the model of field emission from the surface of microformations taking into account internal reflection of the electronic waves from the formation boundaries has been proposed. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION taking into account propagation of electronic waves (de Broglie waves) along a sort of waveguide formed by the In the 1950s and 1960s, in studies of tip field emit- molecular complex. ters using a field emission projector, a number of researchers discovered an unusual phenomenon. After Authors of this work, in studies by the method of deposition onto an emitter of large organic molecules field emission projector coatings of tungsten tip emit- (such as copper phthalocyanine and anthraquinone), ters consisting of C60 fullerene molecules [7, 8], found carbon clusters, and some other materials, ordered that in this case emission images with ordered structure structures in the form of two- or four-petal figures similar to those described in [1Ð6] can be observed on (“doublets” and “quadruplets”), rings, disks and other the projector screen. The results obtained made it pos- forms appeared in the emission images (see, for exam- sible to explain the process of emergence and character- ple, [1Ð6]). The most detailed studies of the formation istics of the complexes of C60 molecules producing the of such emission patterns, termed by E.W. Müller ordered emission images. On this basis, the formation “molecular patterns,” were performed on emitters with model of the ordered field emission patterns from the coatings of organic molecules. In particular, character- emitter surface was elaborated. Below, the results of istic dimensions of the complexes of copper phthalocy- this study are presented. anine molecules responsible for the molecular patterns were determined [2, 5, 6]. According to [2], 13 molecules are sufficient for the emergence of such pictures. MEASUREMENT TECHNIQUE Savchenko [6] estimated that for a cross-sectional AND EQUIPMENT dimension of ~16–22 Å, the copper phthalocyanine Detailed description of the design of a field emission complexes have a height of ~70 Å; their long side is ori- projector and methods of depositing and studying the ented along the electric field lines; and they include a fullerene coatings on the field emitter surface have been considerably larger number of molecules, about 50. For given elsewhere [7, 8]. Therefore, we consider here other materials, including carbon, little reliable data only the details of the experimental technique and have been accumulated. There are indications that car- equipment to be taken into account for better under- bon clusters responsible for producing such images are standing of the results of this work. formed not on the field emitter surface but in the evap- Emission images of the surface of tip field emitter orator [6]. This is plausible but gives no information on magnified by 106 was obtained on the projector screen the carbon cluster structure. Regrettably, information of by applying voltage U between the screen and the cath- their dimensions is also lacking. ode. Tungsten tip emitters had a tip apex diameter of Two theories have been developed to explain the about 0.3 to 2 µm. On the tip apex, a tungsten single nature of molecular pictures. In one of these (monomo- crystal or a ribbed crystal with a coating of tungsten lecular) [2], their shape is related to the spatial and elec- carbide on its surface was formed [9]. The coatings tronic structure of the large adsorbed molecules. In a were deposited onto the tip by evaporation of C60 mol- waveguide model [3, 6] the same pictures are explained ecules from the Knudsen cell. Preparation of a ribbed

FORMATION ON FIELD EMITTERS COATED 251

(‡) (b)

Fig. 1. Typical ordered emission images of microprotrusions on the surface of fullerene coatings. (a) Dou- blet; (b) quadruplet; (c) semicircles separated with a central bright line; (e) (d) cross; (e) hollow ring. crystal on the tungsten tip apex was achieved by ther- observed at fullerene coating thicknesses of Θ > 0.1Ð mal decomposition at temperatures T ≥ 950Ð1000 K of 0.2 monolayers (mL). Magnified images of typical fullerene coatings deposited earlier. The deposition rate ordered patterns are shown in Fig. 1. Figure 2 shows the of the fullerene coating and its thickness Θ were deter- general appearance of the emission images observed at mined by a technique described in [8], from measured different ribbed crystal coating thicknesses. In the gen- “deposition curves,” which represented variations with eral view of the field emitter emission image, the fea- time t of the voltage UI necessary for obtaining a fixed tures of the formations of this sort are hard to distin- current I from the tip. As a rule, the deposition of coat- guish; however, at moderate coating thicknesses (less ings was done in the absence of an electric field about than ~2Ð4 mL), these images carry information on their the cathode. In order to determine the effect of electric locations relative to the characteristic details of the sub- fields, variations of the structure and emission charac- strate. teristics of the fullerene coatings were then examined for different fixed currents from the cathode. It was noted that the cross-sectional dimensions and the brightness of a spot of ordered structure on the pro- Most of the experiments were carried out at a resid- jector screen sometimes varied during observation. Ð10 Ð9 ual gas pressure of ~ 10 Ð10 torr in the experimental A decrease in their brightness and dimensions can be setup. related, for example, to reduction, as a result of ionic bombardment, of the number of fullerene molecules in EXPERIMENTAL RESULTS a microformation on the tip surface, which produces a given image. The reverse process of increasing bright- Ordered patterns in the emission images from the ness and dimensions is probably caused by additional tips similar to those described in [1Ð6] could be C60 molecules joining the structure. An increase in the

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252 TUMAREVA et al.

(‡) (b)

Fig. 2. Emission images of (a) an uncoated ribbed crystal and ribbed crystals with fullerene coatings of different thicknesses Θ: (b) 0.3; (c) 0.5; and (d) 3 mL. dimensions of microformations on the emitter surface In order to determine the formation mechanism of can be caused by the electric field. The mechanism of ordered spots on the projector screen, we thought it this process is supposedly as follows. Nonuniform elec- necessary to estimate the dimensions of the corre- tric fields existing near the emitter surface polarize the sponding microformations on the tip surface. For this C60 molecules and displace them toward nearby surface purpose, it was necessary to measure the currentÐvolt- nonuniformities, which enhance the field, that is, along age characteristics of the emitter as a whole and sepa- the field gradient. This results in additional enhance- rate well-defined microformations. Using these data, ment of the field about the microformation, increasing currentÐvoltage characteristics in the FowlerÐNord- the brightness of its image on the screen. The supposed heim coordinates were plotted. The data obtained were strong influence of nonuniform fields on the emergence used for determining the apex radius of the tip emitter, of ordered microformations is supported by data the cross section of the microformations D, and their obtained in studies of the fullerene coatings on ribbed height h. Figure 3 shows a typical FowlerÐNordheim crystals. In this case it is hard to provide a different plot of a microformation in the form of quadruplet. It explanation of the fact that at coating thicknesses up to was obtained by measuring currents directly from this about two monolayers, all major microformations on microformation under conditions where the currents the emitter surface concentrate along the ribs, where the from the rest of the emitter surface were negligible. electric field is stronger (Fig. 2). The cross-sectional dimension of the microforma- Strong electric field can also explain experimental tion was calculated by Oostrom’s method [10] and their observations of the sudden emergence and disappear- height by the technique suggested by Gomer [11]. The ance of the ordered structures. The former can be characteristic cross-sectional dimension of the micro- related to the grouping of fullerene molecules near a formations producing doublets and quadruplets was minor surface nonuniformity, and the latter with field found to be variable from 15 to 30 Å. The maximum evaporation of molecules or destruction of the micro- height of this sort of microformations did not exceed formations as a result of, for example, explosive emis- their cross-sectional dimension even in those instances sion. In some cases, it was possible to observe fast where these formations were seen as bright spots on the reconstruction of the existing pattern of spots on the projector screen and the current from their surface projector screen when voltage U was varied. For exam- exceeded by far the currents from other areas of the tip. ple, repeated transitions could be observed from two- The reason was probably that the height of consider- petal to four-petal structure and back. ably less bright ordered microformations discernible

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FORMATION ON FIELD EMITTERS COATED 253 against the background of the substrate image (for ln(I/U2) example, against the image of the tungsten single crys- –37 tal) can only be less. The data obtained on the characteristics of the fullerene coatings studied seem to provide grounds for making corrections to the existing models of formation –36 of ordered emission patterns on the projector screen. Estimates that we made taking into account data on the dimensions of microformations giving ordered images and the information available in the literature on solid –35 formations of fullerene molecules (see, for example, [12]) indicate that the largest microformations of this type include not more than a few tens of C60 molecules. Calculations have shown that the above-noted change –34 of the emission spot structure taking place in electric field is accompanied by marked variations of the dimensions of initial microformations on the coating –33 surface. So, a microformation producing an image in 0.11 0.12 0.13 0.14 103/U the form of doublet had a cross-sectional dimension of D ≈ 30 Å and a height of h ≈ 15 Å. Transformation into Fig. 3. FowlerÐNordheim plot of a microformation on the a quadruplet practically did not change its height but fullerene coating producing an emission image in the form reduced the cross-sectional dimension down to 15 Å. of quadruplet (current I and voltage U are in A and V, respectively). An abrupt change of the image structure of microformations with variation of their dimensions probably indicates that the image depends not so much on the emission come mainly from the upper filled levels in type of molecule as on the dimensions of microforma- the energy structure. It is generally assumed (see, for tion; i.e., a sort of dimensional effect takes place. It example, [3]) that the field emission from microforma- would seem that this is an argument in favor of the tions made up of large organic molecules is determined waveguide model of formation of ordered emission pat- by collective π-electrons. For explaining our data, it can terns. However, our data also provide arguments be supposed that in the purely carbon fullerene micro- against the existing waveguide model [3, 6]. Anyway, formations there exist such collective electrons. These the microformations in our experiments, of a height not electrons moving inside the microformation will be exceeding (or even smaller) than its cross-sectional “bombarding” the boundary with a lower potential bar- dimension, can hardly be considered waveguides. Even rier as well. A fraction of these will tunnel into the vac- in [3, 6], where the height of copper phthalocyanine uum. Electrons withdrawn from the microformation microformations, from which field emission was regis- should be replenished by an influx of electrons from the tered, was well in excess of the cross-sectional dimen- substrate through the opposite boundary. sion, the emitting microformation could hardly be iden- tified with a waveguide channeling de Broglie elec- Spatial distribution of electrons exiting into a vac- tronic waves without posing any obstacles (not uum is obviously determined by the distribution of the reflecting). Indeed, the probability of electrons exiting amplitude of standing electronic waves in the potential into a vacuum under conditions typical for field emit- well (resonator) considered, in particular, by character- ters is considerably less than unity, which means that in istics of this distribution near the boundary with a constructing a model of the phenomenon, reflection of reduced potential barrier. The shape of the distribution the electronic waves inside the solid-state system should vary in a discrete manner with the geometrical should be taken into account. Therefore, we consider dimensions of the microformation because of changes that the resonator model is more adequate for describ- in the oscillation modes in the resonator, which ing the field emission from the surface of ordered explains the dimensional effect noted above. This microformations. In its essence, it is very close to the essentially qualitative model of the origin of ordered standard quantum-mechanical model describing emission images must obviously be analyzed theoreti- allowed energy states of electrons in solid-state micro- cally, but it appears attractive to us because it eliminates structures. some drawbacks in the waveguide theory [3, 5] and A microformation producing an ordered emission allows us to understand the specifics of field emission image can possibly be treated as a sort of 3D potential from microformations on the surface of a field emitter. well (resonator) with walls that reflect electrons. In field emission studies, it is necessary to take into CONSLUSION account decreasing by electric field the potential barrier at the outer microformation boundary relative the sub- In summary, note the most important results of this strate. It is evident that electrons taking part in field work. For the first time, microformations of C60

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254 TUMAREVA et al. fullerene molecules on the surface of a tip field emitter, 4. V. A. Shishkin, Candidate’s Dissertation (Inst. Élek- which yield emission images with ordered structure, trokhimii Akad. Nauk SSSR, Moscow, 1962). have been observed and studied. Major characteristics 5. A. P. Komar and V. P. Savchenko, Dokl. Akad. Nauk of these microformations were determined, including SSSR 158, 1310 (1964) [Sov. Phys. Dokl. 9, 917 their dimensions and the transformations taking place (1965)]. under applied electric field and ion bombardment. 6. V. P. Savchenko, Candidate’s Dissertation (Fiz.-Tekh. Inst. im. A. F. Ioffe Akad. Nauk SSSR, Leningrad, 1966). A “resonator model” has been proposed that quali- 7. T. A. Tumareva and G. G. Sominskii, J. Commun. Tech- tatively explains the mechanism of formation and nol. Electron., Suppl. 1 45, S110 (2000). reconstruction of the ordered patterns in the emission 8. T. A. Tumareva, G. G. Sominskiœ, A. A. Efremov, and images from microformations on the surface of a field A. S. Polyakov, Zh. Tekh. Fiz. 72 (2), 105 (2002) [Tech. emitter. Phys. 47, 244 (2002)]. 9. M. V. Loginov and V. N. Shrednik, Pis’ma Zh. Tekh. Fiz. 24 (11), 45 (1998) [Tech. Phys. Lett. 24, 432 (1998)]. REFERENCES 10. A. G. J. Oostrom, Philips Res. Rep., Suppl., No. 1, 1 1. E. W. Müller, Z. Naturforsch. 27, 290 (1955). (1966). 2. A. J. Melmed and E. W. Müller, J. Chem. Phys. 29, 1937 11. R. Gomer, J. Chem. Phys. 28, 457 (1957). (1958). 12. A. V. Eletskiœ, Usp. Fiz. Nauk 170, 113 (2000). 3. A. P. Komar and A. A. Komar, Zh. Tekh. Fiz. 31, 231 (1961) [Sov. Phys. Tech. Phys. 6, 166 (1961)]. Translated by B. Kalinin

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Technical Physics, Vol. 47, No. 2, 2002, pp. 255Ð259. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 116Ð120. Original Russian Text Copyright © 2002 by Novoselov, Denisov, Tkachenko.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

Nitrogen Oxide Conversion in NitrogenÐOxygen Mixtures Excited by a Microsecond Electron Beam Yu. N. Novoselov, G. V. Denisov, and R. M. Tkachenko Institute of Electrophysics, Ural Division, Russian Academy of Sciences, Yekaterinburg, 620016 Russia e-mail: [email protected] Received June 14, 2001

Abstract—The removal of nitrogen oxides from gas mixtures under the action of an ionizing microsecond pulsed electron beam is studied experimentally. Experiments were carried out with gas mixtures simulating ﬂue gases of thermal power plants. The current density and pulse duration of the electron beam, as well as the concentrations of oxygen and the impurity to be removed, are shown to affect the cleaning characteristics. It is found that the nitrogen oxide conversion initiated by a pulsed electron beam cannot be described by models developed for steady-state electron beams. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION of SO2 oxidation occurs in the gas to be cleaned [8]. The main reactions responsible for this mechanism are NOx nitrogen oxides, which are in abundance in flue reactions involving charged and excited particles. With gases of thermal power plants, metallurgy, and some this method, an energy cost as low as ~1 eV/mol [9] and other industries, are very dangerous from the environ- a decontamination factor of more than 90% were mental standpoint. Removing them from industrial gas- attained. Such a high removal efficiency is related to the eous wastes is one of the most important environmental repeated involvement of thermalized beam electrons in problems. One of the methods for removing NOx is attachment reactions resulting in the production of neg- based on pretreating the flue gas by electron beams ative molecular oxygen ions. By varying the beam cur- and/or electric discharges (see, e.g., [1]). According to rent density, the optimum concentration of active parti- theoretical concepts [1Ð3], the thermalized electrons of cles can be obtained, whereas varying the beam pulse an electron beam or electric discharge in a wet flue gas duration allows one to rule out competitive processes stimulate the formation of the O, OH, O2H, and other [7]. In [10, 11], it was experimentally shown that the free radicals, which leads to the oxidation of NO and use of pulsed electron beams to remove nitrogen oxides NO2 to HNO3 nitric acid. Adding ammonia results in also reduces the energy cost. In this study, the conver- reactions with nitric acid leading to the production of sion of NOx nitrogen oxides in flue gas mixtures irradi- powdery NH4NO3 ammonium salt, which can be ated by a microsecond electron beam is studied experi- caught with different types of filters. mentally. A cleaning technique based on steady-state electron accelerators was approved under industrial conditions EXPERIMENTAL SETUP [1, 4, 5]. It was shown that, with this method, the NOx concentration can be reduced by more than 80%. How- In experiments, a facility based on a pulsed electron ever, the high energy cost of the removal of an impurity accelerator with a plasma cathode was used. The accel- by means of a steady-state electron beam did not allow erator design was similar to that described in [12]. The widespread use of this method. Depending on specific accelerator produced a radially divergent electron beam conditions, the energy cost ranges from 20 to 50 eV per with a cross-section area of 1.44 m2 behind the output molecule (eV/mol), which results in a high power con- foil. The beam electron energy was 280Ð300 keV. The sumption and makes the method noncompetitive. Sim- beam current density j and the FWHM pulse duration τ ilar problems occur when removing SO2 sulfur oxide: could be varied within (0.2Ð1.2) × 10Ð3 A/cm2 and 32Ð the use of steady-state electron beams enables a high 90 µs, respectively. The electron beam was injected into decontamination factor, but the energy cost is unaccept- a 0.17-m3 gas chamber, bounded by a grounded output ably high. foil of the accelerator and a metal cylinder, to which a In [6, 7], it was shown that microsecond pulsed elec- 9-µF capacitor charged to a maximum voltage of 20 kV tron beams can be efficiently used to remove sulfur could be connected. The distance between the chamber oxides. In this case, at the optimum current density and walls was 10 cm. The energy dissipated in the gas gap pulse duration of the electron beam, a chain mechanism was determined as W = jτD, where D is the beam

256 NOVOSELOV et al.

[NOi], ppm chamber volume and on the chamber wall during the measurements. 2100 Different methods were used to monitor the gas mixture composition. A Tsvet-500 gas chromatograph 1700 1 was used to determine the nitrogen and oxygen concentrations. The concentrations of oxygen and NOx nitro- 1300 3 2 gen oxide were measured with the help of a TESTO- 350 gas analyzer. In addition, a conductometric method 900 [14] with cryogenic separation of NO from NO was 4 2 used to monitor the NO and NOx contents in the mix- 500 0 100 200 300 N ture. The [NO] concentration was determined after the cryogenic separation of NO from NO2, whereas the [NO ] = [NO] + [NO ] concentration was measured Fig. 1. Concentrations of (1) NOx, (2) NO2 [calculated by x 2 reaction (1)], (3) NO2, and (4) NO nitrogen oxides vs. num- without separation. The concentration of nitrogen ber of irradiation pulses N for j = 0.3 × 10Ð3 A/cm2 and τ = oxide was found from the calibration graph of the [NO] 42 µs. concentration versus the conductance of the absorbing solution, which was measured in a separate experiment ε, eV/mol η, % for reference NO/N2 gas mixtures with the NO concentration ranging from 200 to 5500 ppm. The statistical error in measuring the concentrations of the mixture 200 30 components, including nitrogen oxides, was no higher than 0.03. The gas mixture was irradiated by single pulses. 2 20 100 Each experiment was performed with a series of 300Ð 500 pulses. The nitrogen oxide content was checked 10 after every 50 pulses. In experiments, the number of the 1 removed impurity molecules, the decontamination fac- η ∆ 0 tor = [NOx]/[NOx]0 (in %), and the energy cost per 010002000 3000 4000 5000 ε ∆ [NO ], ppm molecule removed = WN/e [NOx] (in eV/mol) were x ∆ Ð3 determined. Here, [NOx]0 and [NOx] (in cm ) are the Fig. 2. Dependences of (1) the energy cost ε and (2) the initial impurity density and its change after an exposure decontamination factor η on the initial NO concentration to a series of pulses, respectively; W (in J/cm3) is the ≈ x in the N2 : O2 : H2O 87 : 10 : 3 mixture. specific energy deposited in the gas in one electron beam pulse; N is the number of pulses in the series; and e (in C) is the electron charge. The total error in mea- energy absorbed by the gas per unit length. The product ε jτ was determined by integrating the current and volt- suring (with allowance for the error in determining age waveforms, and D was measured with the help of the deposited energy W) was no higher than 0.3. TsDP-F-2 film detectors by a routine technique [13]. ≈ Experiments were carried out with N2 : O2 90 : 10 EXPERIMENTAL RESULTS mixtures with an admixture of NOx. A N2/NO model Preliminary experiments were carried out with mixture with a controlled component content was pre- N2/O2 mixtures in the absence of NOx. In this case, pared in a separate mixer. Just before the beginning of whatever the electron beam parameters, no nitrogen the experiment, this mixture was puffed into the gas oxides were formed. In the presence of NOx, we always chamber, which had already been filled with oxygen. observed an almost linear decrease in the NO and NO The mixture stayed in the chamber for several hours; x during this period, the density of a nitrogen monoxide concentration with increasing the number of the elec- impurity decreased by 15Ð20% due to the oxidation tron beam irradiation pulses N. The higher the initial reaction impurity concentration [NOx]0, the higher the slope of the [NOx] = f(N) linear dependence [10]. These results 2NO + O2 2NO2. (1) indicate that (i) the conversion of nitrogen oxides While the mixture stayed in the chamber, the NO depends on their initial concentration and (ii) the higher concentration was measured at regular intervals. After the initial impurity concentration [NOx]0, the larger the reaching the dynamic equilibrium among the compo- number of molecules removed per irradiation pulse. nents of the mixture in the chamber (this mixture com- In accordance with the model of nitrogen oxide position was assumed to be “initial”), the mixture was removal based on the assumption that the oxidation exposed to the electron beam. Thus, we excluded the processes of free radicals are prevalent, the main final influence of natural oxidation of NO to NO2 both in the product of the NOx conversion is nitric acid. In the

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NITROGEN OXIDE CONVERSION IN NITROGENÐOXYGEN MIXTURES 257

ε, eV/mol η, % ε, eV/mol η, % 60 (a) (b) 1 50 40 30 1 40 40 30 20 30 20 20 2 20 10 2 10 10 0 0 0 0 2040 60 80 0.2 0.4 0.6 0.8 1.0 τ , µs j, 10–3 A/cm2

Fig. 3. (1) Decontamination factor η and (2) energy cost ε as functions of (a) the pulse duration τ and (b) current density j of the electron beam for an initial nitrogen oxide concentration of 1000 ppm. experiments, when water vapor was added to the mix- 42 µs. The behavior of the energy cost ε complies with ture, droplets of highly-diluted nitric acid arose on the the behavior of the dependences in Fig. 1. At low NOx chamber wall after irradiation. However, the technique concentrations, ε is fairly large and amounts to tens of and apparatus used did not allow us to carry out quan- electronvolts per molecule. Thus, for [NOx]0 ~ 1000 ppm, titative measurements. Therefore, we failed to correlate the energy cost is ~50 eV/mol. The increase in [NO ] the number of nitric acid molecules with the concentra- x 0 ∆ leads to a decrease in the energy cost. Thus, at an initial tion [NOx] of the impurity removed. No other prod- nitrogen monoxide concentration of 5500 ppm, ε ucts in either liquid or solid state were observed. decreases to ~2 eV/mol. For comparison, the dissocia- Figure 1 presents the characteristic dependences of tion energies of an NO molecule and molecular oxygen the nitrogen oxide concentrations on the number of are 6.5 and 5.12 eV, respectively [15]. Hence, it is electron beam pulses. The initial impurity concentra- unlikely that the direct electron-impact dissociation of tions plotted on the ordinate axis correspond to N = 0. the impurity is the main process of NOx conversion. Curves 3 and 4 are the dependences of the NO2 nitrogen dioxide and NO nitrogen monoxide concentrations, These results conﬁrm the above remark that the pro- respectively. In the case of nitrogen monoxide, the cess of removing nitrogen oxides from a gas mixture is dependence is linear, whereas, in the case of NO , it greatly affected by the initial impurity concentration. It 2 was also shown that the electron beam parameters (the deviates from being linear. When the mixture is τ exposed to an electron beam, NO oxidizes to NO , current density j and the pulse duration ) affect the 2 impurity conversion, namely, the decontamination fac- whose concentration ﬁrst slightly increases with the η ε number of irradiation pulses and then decreases along tor and the energy cost [16]. The dependences of the measured values of η and ε on the electron beam dura- with the NO concentration. Curve 1 corresponds to τ NO concentration, which is the sum of NO and NO tion are shown in Fig. 3a for a current density of x 2 × Ð3 2 concentrations. Such behavior of the impurity concen- 0.7 10 A/cm . It is seen that both the decontamination factor and the energy cost per toxic molecule trations allows us to suggest that the removal of impu- τ rities from the mixture is not related to reaction (1). removed decrease with increasing pulse duration . The dependences of these quantities on the electron beam Indeed, the decrease in the NO concentration in reac- µ tion (1) should be accompanied by an increase in the current density at a constant pulse duration (32 s) demonstrate a similar behavior (Fig. 3b). The minimum NO2 concentration in accordance with the stoichiometric ratio of the oxides for this reaction (Fig. 1, curve 2). energy cost is as low as 3Ð4 eV/mol, which corresponds to the minimum possible electron beam duration and current In this case, the NOx concentration, as well as the sum of all the oxide concentrations, would have been con- density allowed by the experimental facility used. stant. However, this is not the case in our experiments. The above results cannot be explained (even quali- Thus, the irradiation of the mixture by an electron beam tatively) within theoretical concepts based on the free- reduces the impurity concentrations and this reduction radical mechanism for NOx oxidation. Indeed, accord- is not described by the simple reaction (1). ing to this mechanism [1Ð3], the active particles, such The nitrogen oxide removal is characterized by the as O, O3, and OH and O2H radicals, should play an energy ε spent on the oxidation of one impurity mole- important role in the conversion of nitrogen oxides. The cule. Figure 2 presents the dependences of ε (curve 1) concentrations of these agents increase with increasing and the decontamination factor η (curve 2) on the initial energy deposition in a gas (i.e., with increasing beam NOx concentration in the mixture. In Fig. 2, the number current density at a constant pulse duration). In this of irradiation pulses is 300, the electron beam current case, both the number of the molecules removed and density is 0.3 × 10Ð3 A/cm2, and the pulse duration is the decontamination factor should increase with the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 258 NOVOSELOV et al.

ε, eV/mol η, % tion among electrons, negative oxygen ions, and impurity ions on the one hand and O+ , N+ , and other posi- 60 60 2 2 tive ions on the other hand. In ionized air at atmo- 50 1 50 spheric pressure, the characteristic time of these reactions is about several microseconds [18, 19]. 40 40 In addition to the electron beam parameters and the 30 2 30 initial nitrogen oxide concentration, the characteristics of NOx conversion also depend on the oxygen content 20 20 in the mixture. The oxygen percentage in real flue gases 10 varies in the range from several to ten percent. The 10 dependences of the energy cost ε per impurity molecule 0 removed and the decontamination factor η on the oxy- 0 246810 ≈ gen concentration are shown in Fig. 4. We used N2 : O2 [O2], % (88Ð98) : (10Ð2) mixtures with a 100- to 1000-ppm nitrogen oxide admixture. When preparing the mix- Fig. 4. (1) Energy cost ε and (2) decontamination factor η tures, the oxygen concentration was increased at the vs. oxygen concentration for an initial nitrogen oxide concentration of 600 ppm. cost of increasing nitrogen concentration. It is seen in the figure that the increase in the oxygen concentration in the mixture reduces the number of the NOx impurity [NOi], ppm molecules removed; i.e., it decreases the decontamination factor and increases the energy cost, thus worsen- ing the removal characteristics. 500 Typical dependences of [NO], [NO2], and [NOx] concentrations (where [NOx] = [NO] + [NO2]) on the 400 1 number of irradiation pulses are shown in Fig. 5. Curves 1, 2, and 4 correspond to the 10% oxygen con- 300 centration, and curve 3 was obtained in the absence of oxygen. In all cases, the oxide concentrations decrease 2 with the increasing number of electron beam pulses. 200 3 According to the existing free-radical models of 4 NOx conversion in an ionized gas, the increase in the 100 oxygen concentration must accelerate the reactions of nitrogen oxide oxidation and, consequently, decrease 0100 200 300 the concentration of nitrogen oxides. This is not the N case in our experiments. The results obtained indicate that the oxidation processes are not the only mechanism for nitrogen oxide conversion. Moreover, the highest Fig. 5. Concentrations of (1) NOx, (2) NO, (3) NO, and (4) NO2 nitrogen oxides as functions of the number of irra- reduction in the nitrogen oxide concentration (corre- diation pulses N for different oxygen contents in the gas sponding to the maximum decontamination factor) mixture: [O2] = (1, 2, 4) 10 and (3) 0%. The solid lines show occurs in the oxygen-free mixtures (Fig. 5, curve 3). An the experimental data, and the dashed lines show the computed results. alternative to the free-radical mechanism for nitrogen oxide removal with nitric acid as a final product can be the dissociation of NOx resulting in the creation of current density. However, the experimental results atoms and, subsequently, molecules of nitrogen and (Fig. 3b, curve 1) do not show such a behavior. oxygen [20]. Note that the number of the nitrogen oxide mole- When fast electrons are injected into a nitrogenÐ cules removed and, consequently, the decontamination oxygen mixture, their energy is spent not only on the factor depend on the duration of ionizing exposure in ionization and excitation of the mixture components, the microsecond range (Fig. 3a, curve 1). The typical but also on the dissociation of nitrogen. Knowing the characteristic times of chemical reactions with the par- formation energies of the mixture components, we can ticipation of free radicals taken into account in the mod- estimate their concentrations. Under our experimental els of [1Ð3] are on the order of 10Ð3Ð10Ð4 s [17, 18]. The conditions, the estimated concentration of atomic nitro- experiments demonstrate that, in the case of pulsed ion- gen produced due to the direct-impact dissociation of ization, the processes with characteristic times of N2 molecules by the fast beam electrons does not ~10−6 s are of importance. Such processes can be, e.g., exceed 1012 cmÐ3. The concentration of atomic nitrogen the reactions of electronÐion and ionÐion recombina- significantly increases (to ~8 × 1013 cmÐ3) due to the

TECHNICAL PHYSICS Vol. 47 No. 2 2002 NITROGEN OXIDE CONVERSION IN NITROGENÐOXYGEN MIXTURES 259

+ REFERENCES dissociative recombination of N2 ions. Then, the reactions with the participation of atomic nitrogen and 1. Non-Thermal Plasma Techniques for Pollution Control, impurity molecules lead to a decrease in the nitrogen Ed. by B. M. Penetrante and S. E. Schultheis (Springer- Verlag, Berlin, 1993), NATO ASI Ser., Ser. G 34. oxide concentration and the production of N2 and O2 molecules. The behavior of the nitrogen oxide concen- 2. J. C. Person and D. O. Ham, Radiat. Phys. Chem. 31, 1 tration computed in accordance with a mechanism pro- (1988). posed in [20] is shown in Fig. 5 by the dashed lines. In 3. O. Tokunaga and N. Suzuki, Radiat. Phys. Chem. 24, calculations, we used the reaction rate constants from 145 (1984). [8]. 4. N. Frank, S. Hirano, and K. Kawamura, Radiat. Phys. When studying the removal of sulfur oxides under Chem. 31, 57 (1988). the action of a microsecond electron beam, it was 5. A. G. Chmielewsky et al., Radiat. Phys. Chem. 46, 1063 shown that applying a weak electric field (E ~ 200Ð (1995). 500 V/cm) on an ionized model mixture intensifies the 6. D. L. Kuznetsov, G. A. Mesyats, and Yu. N. Novoselov, process of plasmochemical oxidation of SO2 [7]. This is related to the appearance of an additional channel for in Novel Application of Lasers and Pulsed Power, Ed. by R. D. Curry, Proc. SPIE 2374, 142 (1995). the production of O* vibrationally excited oxygen 2 7. D. L. Kuznetsov, G. A. Mesyats, and Yu. N. Novoselov, molecules, which participate in the chain mechanism Teplofiz. Vys. Temp. 34, 845 (1996). [21]. It was interesting to find out whether the external electric field influences the conversion of nitrogen 8. E. I. Baranchikov, G. S. Belen’kiœ, V. P. Denisenko, et al., oxides. Such experiments were carried out at an electric Dokl. Akad. Nauk SSSR 315, 120 (1990). field of E ~ 0Ð2000 V/cm in mixtures with the NOx con- 9. A. V. Ignat’ev, D. L. Kuznetsov, G. A. Mesyats, and tent ranging from 100 to 3300 ppm and with an addition Yu. N. Novoselov, Pis’ma Zh. Tekh. Fiz. 18 (22), 53 of 0.1Ð1.0% water vapor. It turned out that, within the (1992) [Sov. Tech. Phys. Lett. 18, 745 (1992)]. measurement accuracy, the experimental results 10. G. V. Denisov, Yu. N. Novoselov, and R. M. Tkachenko, obtained with and without an external field were Pis’ma Zh. Tekh. Fiz. 24 (4), 52 (1998) [Tech. Phys. approximately the same. This fact indicates the insig- Lett. 24, 146 (1998)]. * 11. Y. Nakagawa and H. Kawauchi, Jpn. J. Appl. Phys. 37, nificance of O2 molecules in nitrogen oxide conversion. 91 (1998). 12. A. M. Efremov, B. M. Koval’chuk, V. S. Tolkachev, et al., Prib. Tekh. Éksp., No. 1, 99 (1987). CONCLUSION 13. V. V. Generalova and M. N. Gurskiœ, Dosimetry in Radi- The results obtained allow us to draw the following ation Technology (Izd. Standartov, Moscow, 1981). conclusions. The initial impurity concentration signifi- 14. E. A. Khudyakova and A. P. Kreshkov, Theory and Prac- cantly affects nitrogen oxide conversion under the tice of Conductimetric Analysis (Khimiya, Moscow, action of a pulsed electron beam. The minimum energy 1976). cost per one molecule removed (ε ~ 2 eV) is achieved min 15. A. A. Radtsig and B. M. Smirnov, Reference Data on at a high impurity concentration. The electron beam Atoms, Molecules, and Ions (Atomizdat, Moscow, 1980; parameters and the oxygen content in the irradiated gas Springer-Verlag, Berlin, 1985). significantly affect the characteristics of the NO x 16. G. V. Denisov, D. L. Kuznetsov, Yu. N. Novoselov, and removal process. Oxidation reactions are not the only R. M. Tkachenko, Pis’ma Zh. Tekh. Fiz. 24 (15), 47 process of NOx conversion in an ionized nitrogenÐoxy- (1998) [Tech. Phys. Lett. 24, 601 (1998)]. gen mixture. At low oxygen concentrations, the disso- 17. V. N. Kondrat’ev, Gas-Phase Reaction Rate Constants ciation of nitrogen oxides is the most probable process. (Nauka, Moscow, 1970). The mechanism governing NOx conversion under the conditions of steady-state gas ionization and based on 18. L. I. Virin, R. V. Dzhagatspanyan, G. V. Karachevtsev, the reactions involving free radicals fails to account for et al., Ion-Molecule Reactions in Gases (Nauka, Mos- the experimental results obtained with pulsed electron cow, 1979). beams. To explain these results, it is necessary to 19. B. M. Smirnov, Ions and Excited Atoms on Plasma develop an adequate non-steady-state model with (Atomizdat, Moscow, 1974). allowance for plasmochemical reactions among neu- 20. G. V. Denisov, Yu. N. Novoselov, and R. M. Tkachenko, tral, charged, and excited particles, as well as reactions Pis’ma Zh. Tekh. Fiz. 26 (16), 30 (2000) [Tech. Phys. involving free radicals. Lett. 26, 710 (2000)]. 21. Yu. N. Novoselov, Pis’ma Zh. Tekh. Fiz. 19 (23), 58 ACKNOWLEDGMENTS (1993) [Tech. Phys. Lett. 19, 760 (1993)]. This study was supported by the International Sci- ence and Technology Center, project no. 271. Translated by N. Ustinovskiœ

TECHNICAL PHYSICS Vol. 47 No. 2 2002

Technical Physics, Vol. 47, No. 2, 2002, pp. 260Ð267. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 121Ð128. Original Russian Text Copyright © 2002 by Noskov, Malinovski, Sack, Schwab.

EXPERIMENTAL INSTRUMENTS AND TECHNIQUES

The Simulation of Dendrite Growth and Partial Discharges in Epoxy Resin M. D. Noskov*, A. S. Malinovski*, M. Sack**, and A. J. Schwab** * Tomsk Polytechnical University, pr. Lenina 30, Tomsk, 634050 Russia e-mail: [email protected] ** University of Karlsruhe, D-76128 Karlsruhe, Germany e-mail: [email protected] Received October 26, 2000; in ﬁnal form, March 13, 2001

Abstract—A self-consistent model of the development of dendrites and partial discharges in a dielectric solid under a variable voltage is suggested. Dendrites originate at sites with the enhanced local ﬁeld strength and also where the dielectric is broken down under the action of partial discharges. The numerical simulation is used to quantitatively describe the spaceÐtime dynamics of the dendrites and partial discharges in epoxy resin for the tip–plane electrode conﬁguration. The simulated data are compared with electrical measurements of partial discharges and with optical images of dendrite growth under the same conditions. © 2002 MAIK “Nauka/Inter- periodica”.

INTRODUCTION breakdown model [7Ð11] and the model of avalanche discharge [12, 13]. In the former, the evolution of the The breakdown of polymeric insulation during channels is assumed to be dependent on the electric long-term exposure to a variable voltage is associated field strength; namely, the growth probability is propor- with the evolution of gas-filled channels, so-called den- tional to the local electric field. The field distribution is drites [1, 2]. Dendrites originate at sites with the calculated with the Laplace equation. The dendrite enhanced local field strength (electrode surface irregu- potential is taken to equal the electrode potential or, larities, conducting inclusions, microcracks, pores, otherwise, the voltage drop along the channel is speci- etc.). The breakdown takes place soon after a dendrite fied. With various modification of the dielectric break- has covered the electrode spacing. Dendrites grow, down model, tree- and bushlike dendrites have been because partial discharges in their channels break down simulated [8, 10]. Also, the effect of various barriers the dielectric. Thus, the evolution of dendrites and par- and inclusions with different permittivities [10], con- tial discharges are interrelated processes, causing the ductivities [11], and dielectric strengths [9] have been breakdown of insulation. The dynamics of dendrite analyzed. In the model of avalanche discharge [12, 13], growth and the parameters of partial discharges depend microbreakdowns in the dielectric material due to elec- on the physical and chemical properties of the dielec- tron avalanches is considered. The channel evolution is tric, the frequency and amplitude of the applied voltage, assumed to be related to random fluctuations of the electrode configuration, etc. local electric field. Within this model, various shapes of The study of dendrite growth has become of partic- the dendrites have been predicted for different fluctua- ular interest because of the recent advent of cables with tion ranges. the polymeric insulation that can withstand voltages as The disadvantage of both models is that they ignore high as 400Ð500 kV. Using electronic and electron- partial discharges in the dendrite channels and, hence, optical tools, the researchers have determined the do not describe the actual dynamics of the charge and parameters of the partial discharges and the time-space field distributions in the dielectric. On the other hand, characteristics of the dendrite growth [3Ð6]. Yet, in the available models of partial discharges (see, e.g., spite of the extensive studies, a quantitative theory of [14, 15]) do not touch upon the growth of dendrites and the dendrite growth due to the partial discharges has not cannot be used to quantitatively characterize insulation been elaborated to this point. This is related to a variety breakdown. Therefore, the development of a unified of interrelated physical and chemical phenomena model that covers both the mechanisms of partial dis- responsible for the dielectric breakdown and also to the charges in the available channels and the dendrite fact that the growth of the dendrites is a complex sto- growth due to the partial discharges seems to be quite chastic process. In recent years, a number of mathemat- topical. Such a model has been suggested in [16]. With ical models for the stochastic description of branching it, we succeeded in discovering the basic mechanisms dendrite growth has been suggested [7Ð13]. The most and laws behind the dendrite growth and in determining interesting results have been obtained with dielectric the parameters of the partial discharges [17]. In this

THE SIMULATION OF DENDRITE GROWTH 261 work, this model is extended for the quantitative characterization of the dendrite growth and the partial discharges in epoxy resin for the tipÐplane electrode con- 1 ﬁguration subjected to a varying voltage. The results of simulation are compared with experimental data. 2

A SELF-CONSISTENT MODEL OF DENDRITE GROWTH AND PARTIAL DISCHARGES The Basic Postulates of the Model 3 Our self-consistent model considers the growth of dendrites as the spatial formation of branching channels. A new channel appears as a result of the breakdown of the dielectric surrounding the existing chan- Fig. 1. Domain of simulation: 1, tip; 2, dendrite; and 3, pla- nels or at sites with the enhanced local field strength. nar electrode. The dielectric breakdown is associated with the partial discharges in the channels. It is assumed that the material damage is proportional to the discharge energy lomb law and the principle of superposition as applied being released in the channel. The channel grows if the to electric fields produced by point charges. The field energy per unit channel length reaches the critical value distribution for such an electrode configuration is simulated by placing a sufficiently large number of point Wc and the field strength near the channel damaged exceeds the critical strength E . The growth direction of charges on the surface of a cone representing the tip. c The effect of the grounded planar electrode on the field a new channel stochastically depends on the local elec- distribution is taken in consideration by using the mir- tric field strength: the probability that the channel will ror image method. The value of each of the charges on grow in the direction n varies as the square of the pro- the tip is calculated by iteration so as to obtain a given jection En of the local strength onto this direction if En > time variation of the applied voltage ϕ(t). The dendrite Ec and equals zero otherwise. The random selection of channels are simulated by the branches of point charges the growth direction in the model reflects the inhomo- separated by a fixed distance d from one another. Thus, geneity of the dielectric material and also the stochastic a dendrite can be represented as a set of contacting nature of the formation of a new channel. The introduc- spheres (Fig. 1). The sphere radius r = d/2 defines the tion of the critical strength Ec means that there is some minimal spatial scale of simulation and is an important threshold strength for the formation of a new channel parameter of the numerical model. even if the material is damaged. The quadratic dependence of the growth probability on the field strength To describe the evolution of the dendrite growth and relates the channel growth to the electric field energy the partial discharges, the model uses discrete time density. steps ∆t of equal width. The step ∆t should be suffi- A partial discharge in the dendrite channel is initi- ciently small so that individual partial discharges can be distinguished. At each nth step corresponding to the ated when the local field along a channel segment ∆ time instant tn = n t, the state of each of the spheres is exceeds the threshold initiating field Ein. The discharge n is quenched when the local field drops below the characterized by the specific damage energy Wi and quenching field Eq. The electric field is defined by both n the electrode potential and the charge distribution along the electrical charge qi (i is the sphere number). The the channel. In the absence of the partial discharges, the ϕn potential i of the ith sphere is calculated as the sum channels are nonconducting. Charge transfer is accom- n plished only during the partial discharges. Since the of the potential due to the charge qi and the potentials partial discharge duration (10Ð100 ns) is much smaller due to all other charges (including the imaginary ones): than the characteristic time of dendrite growth (>103 s), it is assumed that the partial discharges proceed instan- qn qn taneously [1]. ϕn = ------i - + ∑ ------j - i 4πεε r 4πεε r Ð r 0 ≠ 0 i j j 1 (1) Numerical Implementation of the Model qn Ð ------j . ∑ πεε The three-dimensional numerical implementation 4 0 ri Ð r*j of the model allows the computer-aided simulation of j dendrite growth and partial discharges in a dielectric. Here, ε, ε are the relative and absolute permittivities The electrodes are a tip to which a variable voltage is 0 applied and a grounded plane. The electric field is cal- and rj, r*j are the coordinates of the real charge qj and culated with the charge method [18] based on the Cou- the associated specularly reflected charge, respectively.

TECHNICAL PHYSICS Vol. 47 No. 2 2002

262 NOSKOV et al. At each of the steps, the following procedures are bility of the iterative procedure. In this work, λ = 0.3 × πεε performed. (i) The instantaneous value of the voltage 4 0r. The charge transfer changes the charges on the applied to the tip is changed, and the tip charge is recal- spheres belonging to at least one conducting segment. culated; (ii) the partial discharges, as well as changes in New charges are calculated by the formula the distributions of the specific damage energy and the n() n() ∆n () charges on the spheres participating in the partial dis- qi k = qi k Ð 1 Ð ∑ qij, k , (5) charges, are simulated; and (iii) dendrite growth is sim- j ulated by adding new spheres to the existing structure or to the tip. where the summation is over all points j connected to a point i by the conducting segments. Once the charge The charge on the tip is calculated as follows. The transfer has been completed, the charges on the tip are potential of each of the tip points is set equal to the recalculated so that the tip potential remains equal to ϕn instantaneous value of the applied voltage: i = the potential ϕ(n∆t) corresponding to a given time step. ϕ ∆ n (n t). The dendrite charges are taken to be fixed. New potentials of the dendrite points, ϕ (k), are calcu- Equation (1) is written for each of the tip points, and the i lated by formula (1) according to the new charge distri- resulting set of linear equations is solved by the Gauss butions over the tip and the dendrite. At each of the iter- method. ation steps, the damage energies of the spheres belong- No more than one partial discharge is simulated at ing to the conducting segments are increased. Only a each of the steps. The dendrite channels are divided into part of the energy being released in the channels during segments formed by a pair of neighboring spheres. the spatial discharges is spent on damage to the dielec- Each of the segments may be either conducting or non- tric material. For simplicity, this fact is taken into conducting. The conducting (nonconducting) state cor- account by appropriately raising the critical damage responds to the presence (absence) of a discharge energy Wc; that is, Wc is now the energy per unit chan- between the spheres. The simulation of a partial dis- nel length that must be released during the partial discharge includes the self-consistent calculation of the charges for a new channel to form. The increment of the segment-state dynamics, charges, and sphere poten- specific damage energy equals the energy being tials. At the initial step of the iterative procedure (the released when the charge is transferred from the jth iteration number k = 0), all the segments are noncon- sphere to the ith sphere along the dendrite divided by ducting and the charges and specific damage energies the segment length d. Thus, after the kth iteration step, of all the spheres equal those obtained at the preceding n n n Ð 1 n n Ð 1 the damage energy Wi (k) will be time step n Ð 1: qi (0) = qi , Wi (0) = Wi . The n n n n transition from the (k Ð 1)th step to the kth step is as fol- W ()k = W ()∆k Ð 1 + q , ()∆ϕk , ()k Ð 1 /d. (6) lows. First, the states of the segments are changed, if i i ij ij necessary. A segment passes to the conducting state or The iterative procedure is repeated until none of the remains conducting if the absolute value of the poten- segments is in the conducting state. After the nth itera- tial difference between adjacent spheres i and j, tion step, the charges qn and the specific damage ener- ∆ϕn ϕn ϕn i ij, (k Ð 1) = i (k Ð 1) Ð j (k Ð 1), exceeds the partial n ϕ gies Wi of the dendrite points depend on the last itera- discharge initiation voltage in = Eind: tion step kl: n ∆ϕ , ()ϕk Ð 1 > . (2) ij in n n() n n() qi ==qi kl , Wi Wi kl . (7) A segment passes to the nonconducting state or remains nonconducting if the absolute value of the The partial discharge propagates through all the seg- potential difference is lower than the partial discharge ments that were in the conductive state at least once ϕ n quenching voltage q = Eqd: during the iterative procedure. The amount Q of the partial discharge is calculated as the total change in all ∆ϕn ()ϕ< ij, k Ð 1 q. (3) the charges on the tip and on the dendrite: Then, the charge is transferred to the conducting Qn = ∑()qn Ð qn Ð 1 . (8) ∆ n i i segments. The value of the charge transferred qij, (k) i ∆ϕn is proportional to the potential difference ij, (k Ð 1) The dendrite growth is simulated by adding new between the spheres: spheres to the existing dendritic structure or to the tip. A new sphere can be attached to any of the dendrite ∆ n ( ) λ∆ϕn () qij, k = ij, k Ð 1 , (4) n spheres for which the specific damage energy Wi has λ where is a numerical iteration parameter. reached the critical value Wc: The parameter λ is selected so as to provide the n ≥ maximal computation speed without affecting the sta- Wi Wc. (9)

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE SIMULATION OF DENDRITE GROWTH 263

With condition (9) met, an ensemble of possible least spatial scale of simulation but also relates the positions of the new spheres attached to a given sphere charge and the potential of a sphere [the first term on is constructed. The number of the positions must be the right of Eq. (1)]. Thus, the parameter r defines the sufficiently large to realize the stochastic growth law size of the space charge region around the dendrite adopted in the model. We constructed ensembles of ten channels. The size of the region of the charge injected positions. The possible positions are regularly arranged into the dielectric depends on the field strength and may around a given sphere so that the new spheres do not exceed the channel diameter determined by optical cross the existing ones. The position of a new sphere is means. randomly selected with a probability P that is propor- The model parameters can be determined by analyz- tional to the ∆ϕ squared (where ∆ϕ is the potential dif- ing physical processes attendant on the dendrite growth ference between a given point and a position being con- or by comparing the results of simulation with experi- sidered) if the absolute value of ∆ϕ exceeds the critical mental data. In our case, the parameters were deter- |∆ϕ| value Ecd, > Ecd (otherwise, the probability is mined by comparing test calculations with experimen- zero); that is, tal data for the dendrite growth in epoxy resin. The

2 parameters used in the simulations were Wc = 0.07 J/m, ∆ϕ /Z, ∆ϕ > E d × 8 × 7 × P =  c (10) Ec = 1.0 10 V/m, Ein = 8 10 V/m, Eq = 1.5 ∆ϕ ≤ 7 µ 0, Ecd, 10 V/m, and r = 10 m. The voltage parameters and the electrode configurations were the same as those where Z is the normalizing factor defined as Z = Σ(∆ϕ)2 used in the experiments. (the summation is over all possible positions for which |∆ϕ| > E d). c MATERIAL AND EXPERIMENTAL TECHNIQUE If the growth has occurred, the value Wc is subtracted from the specific damage energy of that sphere The experiments were performed with Araldit to which a new one has been attached. The specific CY221 epoxy resin (HY2966 hardener) from Ciba Spe- n n cialty Chemicals Inc. Prior to the preparation of the damage energy W j and the charge q j of a new sphere samples, a steel tip of radius 10 µm was inserted in a j are set equal to zero. Once the simulation has been rectangular mold. For the experiments, the sample was completed, we pass to the next time step. The simula- placed on a planar grounded electrode inside a transpar- tion lasts until the dendrite reaches the planar electrode. ent vessel filled with transformer oil for insulation. The Initially, while the dendrite does not exist, a new sphere tip-plane spacing was 1.5 mm. A sinusoidal voltage can be attached only to the tip. Since we do not consider with an effective value of 8 kV and a frequency of the initiation of the dendrite, the tip is assigned the 50 Hz was applied to the tip. damage energy higher than Wc to make the growth pos- The basic components of the computerized setup are sible. a partial discharge measuring system, a video camera for shadow filming, and a photoelectric multiplier for Parameters of the Model detecting the optical radiation from the partial discharges (Fig. 2). The basic model parameters are Wc, Ec, Ein, Eq, and r. They, along with the electrode configuration and the applied voltage parameters, are responsible for the fea- RESULTS AND DISCUSSION tures of the dendrite growth and the partial discharges. At the initial stage of dendrite formation, one or sev- The parameters have the simple physical interpretation. eral channels appear at the tip. The channels develop by The critical damage energy Wc is the energy needed for stochastic branching and bending. Typical dendrite the formation of the channel of unit length. The critical structures obtained by the simulation and in the experi- field Ec is related to the channel formation due to the ments are shown in Fig. 3 for four successive instants partial discharges. It may be much lower than the of the growth. Structures of this type are called tree-like breakdown field of the undamaged dielectric. Both Wc [1]. An increase in the voltage applied causes intense and Ec depend on the physical and chemical properties branching and, accordingly, the growth of bush-like of the dielectric and on the channel sizes. The threshold structures. The dendrite shape can quantitatively be described in terms of fractal dimension [19]. In this fields Ein and Eq specify the conditions for gas discharge initiation and quenching in the dendrite chan- work, the fractal dimension is estimated by the cover- nels. It has been shown that the threshold fields in the age method. The dendrite projection is covered by thin channels increase as the channel diameter squares of side l. Then, the number N(l) of squares that decreases and may considerably exceed the associated contain dendrite fragments is calculated. For a fractal values for the discharge in free space. In addition, the structure of dimension D, the relationship threshold fields E and E vary with the gas composi- ÐD in q Nl()∝ l (11) tion and pressure in the channel. The radius r of the spheres used in the simulation not only specifies the holds.

TECHNICAL PHYSICS Vol. 47 No. 2 2002 264 NOSKOV et al.

234 (a)

6 0.2 mm 0.2 mm 12

1 7 8 8

Fig. 2. Experimental setup: 1, variable voltage source; 2, step-up transformer; 3, illuminating lamp; 4, working chamber; 5, photoelectric multiplier; 6, video camera for shadow ﬁlming; 7, partial discharge measuring system; and 0.2 mm 8, controlling computers. 0.2 mm 34

Thus, the fractal dimension can be found from the (b) slope of a straight line approximating the lnN(l) vs. lnl dependence. Dendrite fractal dimensions thus obtained vary between 1.45 and 1.55. Similar values for treelike structures were found in [20, 21]. The dendrite growth rate drops with time both upon the simulation and in the experiments. Figure 4 shows the time dependences of the dendrite length R (the distance from the tip to the farthest point of a dendrite). The slowing-down of dendrite growth at the initial stages has been noticed in 0.2 mm 0.2 mm many works [1, 3Ð5]. 1 2 The growth of a dendrite is accompanied by partial discharges in its channels. The simulation indicates that the partial discharges are initiated at the tip and propagate towards the channel end. Each subsequent partial discharge propagates through a new channel. This can be explained by charge transfer and the reduction of the ﬁeld along the channels having taken part in the preceding partial discharge. Typical discharge paths simulated are shown in Fig. 5a. Actual partial discharges radiate 0.2 mm 0.2 mm light. As follows from the radiation patterns obtained 3 4 with an image-converter tube, different channels emit at different time instants (Fig. 5b). This observation can Fig. 3. Dendrite growth: (a) simulation and (b) experiment. be viewed as a demonstration of the fact that different t = (1) 2, (2) 4, (3) 6, and (4) 9 min. partial discharges occupy different channels. The simulation shows that the amount Q of a partial and the total length of radiating channels has been discharge is proportional to the total length S of all found [6]. During the dendrite growth, the maximal channels covered by this discharge. For the same path path length of the partial discharges increases and, length L (the distance between the tip and the farthest accordingly, so does their maximal amount. Figure 6 point of the path), the values of S may be different demonstrates the time dependences of the maximal because of different numbers of the channels taking amount Qmax of the partial discharges that were part in the partial discharge. Therefore, the path length observed upon the simulation and in the experiments. deﬁnes only the maximal amount of the discharge. The For the same dendrite length, the amount of a partial relationship between the total path length S and the discharge depends on the phase of the voltage applied. amount of partial discharges that was established dur- The maximal amount of the discharge rises with instan- ing the simulation is corroborated by the experiments taneous value of the voltage. The polarity of the dis- where the correlation between the discharge amount charge coincides with the sign of the time derivative of

TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE SIMULATION OF DENDRITE GROWTH 265

R, mm (a) 1.0

0.8

0.6

0.4

0.2 12

024 6 8 10 12 (b) t, min

Fig. 4. Dendrite length vs. growth time. Continuous curve, experiment; dotted line, simulation. the voltage. The correlation between the amount of the partial discharge and the phase (ϕÐqÐn diagram) is shown in Fig. 7. ϕÐqÐn diagrams of this type, which are often called wing-shaped, have been also observed in other works (see, e.g., [5]) when dendrites grew at a variable voltage. The discharge repetition rate increases Fig. 5. (a) Partial discharge path and (b) dendrite channel with the absolute value of the time derivative of the emission pattern. The discharge paths are shown black and voltage and reaches a maximum at phase angles of 0° the dendrite channels, gray. The partial discharge amount is and 180°. The experimental and simulated histograms (1) 44 and (2) 181 pC. The arrow points to the tip position. showing the distribution of the partial discharges over the phase are depicted in Fig. 8. Qmax, pC The behavior of the partial discharges that was 250 observed upon the simulation and in the experiments can be explained by the existence of the threshold fields 200 responsible for the initiation and quenching of the discharges. The discharge starts if the field strength along 150 the channels is higher than the critical value Ein and 100 ceases when the field drops to Eq because of charge transfer. Thus, after the discharge, the voltage drop per 50 unit channel length is roughly Eq. Therefore, the maximal discharge path length and, hence, the maximum possible amount of the partial discharge grow with volt- 024 6 8 10 12 age applied. This explanation has been corroborated t, min experimentally. In [15], where the emission from the dendrite channels was recorded for various phases of Fig. 6. Maximal amount of the partial discharge vs. time. the voltage, it was established that the emitting part of Continuous curve, experiment; dotted line, simulation. the dendrite expands with increasing absolute value of the voltage. other polymers as well, such as polyethylene, polyme- These results are in qualitative agreement with thylmethacrylate, etc. Thus, the model can serve as the experimental data for the dendrite growth and partial basis for a technique that allows us to measure the discharges under similar conditions in epoxy resin and dielectric strength as a function of the electrode config- other polymers (see, e.g., [3Ð5]). This means that the uration, voltage parameters, and material properties. model suggested adequately predicts the general laws Another important application of our work is insulation of dendrite growth due to partial discharges and that the diagnostics based on partial discharge analysis. basic ideas of the model reflect real physical processes. The model can be extended further by considering The quantitative agreement of the simulated results and the wall conductivities of the channel and the dielectric, the experimental data for epoxy resin also counts in the dependences of the initiation and quenching fields favor of the model. It should be noted that the model is on the pressure and channel radius, etc. This will make to a great extent general, so that it can be used for the model more accurate and enable us to study such describing the discharge-related dendrite growth in effects as the transition from the treelike growth to the

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Q, pC bushlike one as the voltage rises, the temporary ceasing and renewal of the discharges and dendrite growth, and 150 (a) others. 100

50 CONCLUSION 0 The self-consistent model of dendrite growth and partial discharges in polymeric insulation is developed. –50 It is based on the energy approach, assumes the thresh- –100 old character of channel expansion, and introduces the local initiation and quenching fields of the discharge. –150 The nonlattice three-dimensional implementation of 150 (b) the model makes it possible to perform numerical experiments on the dendrite growth and partial dis- 100 charge initiation for a tip–plane electrode configuration 50 at a variable voltage. We studied a possible correlation between the par- 0 tial discharge parameters and dendrite growth in epoxy –50 resin. The simulation shows that the amount of a partial discharge is proportional to the total length of the chan- –100 nels involved in this discharge. The maximal amount of –150 the partial discharges grows with the dendrite length. 090180 270 360 As follows from the phase diagrams, within a period, ϕ, deg the maximal amount of the discharge increases with the instantaneous value of the voltage, while the discharge repetition rate rises with the voltage frequency. The Fig. 7. Correlation between the partial discharge amount and the applied voltage phase (t = 9 min). (a) Simulation quantitative agreement between the simulated results and (b) experiment. and the experimental data confirm the adequacy of the model and its reliability.

n, 1/degree cycle ACKNOWLEDGMENTS 0.12 This work was supported by the German Research (a) 0.10 Society (DFG) and the Russian Foundation for Basic Research. 0.08 0.06 REFERENCES 0.04 1. L. A. Dissado and J. C. Fothergill, Electrical Degrada- tion and Breakdown in Polymers (Peregrinus, London, 0.02 1992). 0 2. G. S. Kuchinskiœ, Partial Discharges in High-Voltage Structures (Énergiya, Leningrad, 1979). 0.12 3. J. V. Champion and S. J. Dodd, J. Phys. D 29, 862 (b) (1996). 0.10 4. J. V. Champion, S. J. Dodd, and J. M. Alison, J. Phys. D 0.08 29, 2689 (1996). 5. Suwarno, Y. Suzuoki, F. Komori, et al., J. Phys. D 29, 0.06 2922 (1996). 0.04 6. Y. Ehara, M. Naoe, K. Urano, et al., IEEE Trans. Dielectr. Electr. Insul. 5, 728 (1998). 0.02 7. L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Phys. 0 Rev. Lett. 52, 1033 (1984). 0 90 180 270 360 8. H. J. Wiesmann and H. R. Zeller, J. Appl. Phys. 60, 1770 ϕ, deg (1986). 9. P. J. Sweeney, L. A. Dissado, and J. M. Cooper, J. Phys. D 25, 113 (1992). Fig. 8. Density function of the partial discharges n vs. applied voltage phase (t = 9 min). (a) Simulation and 10. V. R. Kukhta, V. V. Lopatin, and M. D. Noskov, Zh. Tekh. (b) experiment. Fiz. 65 (2), 63 (1995) [Tech. Phys. 40, 150 (1995)].

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11. D. I. Karpov, V. V. Lopatin, and M. D. Noskov, Élek- 17. M. D. Noskov, A. S. Malinovski, M. Sack, et al., IEEE trichestvo, No. 7, 59 (1995). Trans. Dielectr. Electr. Insul. 7, 725 (2000). 12. L. A. Dissado and P. J. Sweeney, Phys. Rev. B 48, 16261 18. H. Singer, H. Steinbigler, and P. Weiss, IEEE Trans. (1993). Power Appar. Syst. 93, 1660 (1974). 13. J. C. Fothergill, L. A. Dissado, and P. J. Sweeney, IEEE 19. K. Kudo, IEEE Trans. Dielectr. Electr. Insul. 5, 713 Trans. Dielectr. Electr. Insul. 1, 474 (1994). (1998). 14. J. V. Champion and S. J. Dodd, J. Phys. D 31, 2305 20. S. Kobayashi, S. Maruyama, H. Kawai, et al., in Pro- (1998). ceedings of 4th International Conference on Properties and Applications of Dielectric Materials, 1994, p. 359. 15. K. Wu, Y. Suzuoki, T. Muzutani, et al., J. Phys. D 33, 1197 (2000). 21. H. Uehara and K. Kudo, in Proceedings of 6th Interna- tional Conference on Conduction and Breakdown in 16. A. S. Malinovski, M. D. Noskov, M. Sack, et al., in Pro- Solid Dielectrics, 1998, p. 309. ceedings of IEEE 6th International Conference on Con- duction and Breakdown in Solid Dielectrics, 1998, p. 305. Translated by V. Isaakyan

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Technical Physics, Vol. 47, No. 2, 2002, pp. 268Ð271. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 72, No. 2, 2002, pp. 129Ð133. Original Russian Text Copyright © 2002 by Abramova, Vettegren’, Shcherbakov, Svetlov.

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The Emission of Photons and the Dynamics of Submicrodefects on the Surface of Noble Metals K. B. Abramova, V. I. Vettegren’, I. P. Shcherbakov, and V. N. Svetlov Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received February 15, 2001

Abstract—The shape and concentration of defects, as well as the intensity of electroluminescence from the back side of Cu, Ag, and Au samples with their front side irradiated by a laser shot, are studied experimentally. It is shown that there exists a correlation between the electroluminescence intensity and the concentration of radiation-induced defects, which is consistent with the dislocation model of luminescence. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION samples were 0.1Ð0.5 mm and 30 × 30 mm2, respec- It has been discovered [1] that the mechanical fail- tively. ure of metals is accompanied by the emission of pho- The emission was detected by an FÉU-136 photo- tons. Later, the same emission was found to attend electric multiplier sensitive to the 300Ð800 nm range. metal stressing, including stressing by a laser shot The multiplier operated in the analog mode and was [2, 3]. These findings were explained [4Ð6] by the integrated with a digital storage oscilloscope. release of the dislocation core energy when the disloca- The front surface of the plates was stressed by tions emerge on the surface. The radiation spectra con- applying 1.06-µm laser pulses of width 1.5 ms. Their tain lines that are assigned to electron transitions in dis- energy in the free-running mode was 24 J. The diameter location cores. The luminescence intensity has been of the optical beam was 2 mm. The incident beam found to be proportional to the dislocation density [2, 7, 8]. intensity was varied with neutral optical filters so as not It has been demonstrated [9, 11] that upon stressing to damage the material within the irradiated zone on the Cu, Mo, Pd, and Au samples, “nanodefects” (pyramidal front surface and still provide reliable luminescence indentations) with the sides parallel to the easy slip from the back side of the plate. In our experiments, the ≈ planes are generated. The generation of these defects energy delivered to the sample per pulse was P 0.2Pthr ≈ was explained by the emergence of dislocations on the ( 0.7 J), where Pthr is the threshold energy, i.e., the surface due to stressing [11]. The density of disloca- energy at which a plasma torch appears on the back tions crossing the surface can be judged from the shape side. and concentration of the nanodefects. The surface pattern was examined with an RTP-1 Upon studying mechanoluminescence from the scanning tunnel profilometer designed in the Research back side of the metal samples with their front side irra- Institute of Physics, St. Petersburg State University. To diated by laser pulses of equal power [7, 8, 12, 13], it provide the desired resolution and the operating stabil- was found that the luminescence intensity decreases ity of the instrument, we used a diffraction grating onto with increasing number of irradiating pulses. It was which a gold layer was evaporated. The measuring tips speculated that the luminescence decays because the were made of a tungsten wire by electrochemical etch- number of dislocations emerging on the surface ing. The adequacy of their shapes was estimated from decreases during the irradiation. The aim of this work is the images of the finest defects on the topograms. The to check this supposition. tip of the profilometer can travel normally to the surface by a distance of no more than 1 µm. To remove surface microirregularities higher than 1 µm, the samples were EXPERIMENTAL polished by diamond pastes and then rinsed by acetone We measured the luminescence from the surface of and alcohol. metal plates under stress cycling and examined their The condensation of water vapor from air on the sur- submicropattern after each of the cycles. The plates face produces an electrolyte. The ionic current flowing were made of 99.96% pure copper, 4N pure silver, and in the electrolyte excludes the possibility of imaging 4N pure gold. The thickness and the surface area of the the surface. To suppress this effect, the samples were

THE EMISSION OF PHOTONS 269

mV covered by polyurethane and the resulting space was 12 purged by dry nitrogen. 11 10 RESULTS OF MEASUREMENTS 9 Figure 1 shows the waveforms of luminescence sig- 8 1 nals taken from the polished silver sample under cyclic 7 stressing, i.e., when the sample was successively irradi- 6 2 ated by three pulse of the same power. It is seen that the 5 radiation intensity decreases from pulse to pulse. 4 Within the dislocation model of luminescence, this is 3 explained by a decrease in the number of dislocations 3 emerging on the surface. Fragments of the topograms 2 from the back side of the sample (whose luminescence 1 signal is shown in Fig. 1) before and after irradiating the front side by the three laser pulses. In all of the topo- 0 0.5 1.0 1.5 2.0 grams, similarly shaped defects marked by 1 in Fig. 2 ms are the basic surface features. The enlargement of such Fig. 1. Waveforms of photon emission signals from the pho- a defect is shown in Fig. 3. Their depth (height) vary toelectric multiplier: 1, ﬁrst cycle; 2, second cycle; and from 15 to 30 nm, and their cross sizes lie between 50 3, third cycle. and 200 nm. The 3D image of the defect looks like a

(a) (b)

1 1 200 nm (z)

600 nm (y)

Fig. 2. Fragments of the topograms taken from the silver sample. (a) Nonirradiated polished surface; (b)Ð(d) surface after the ﬁrst, second, and third cycle, respectively.

TECHNICAL PHYSICS Vol. 47 No. 2 2002

270 ABRAMOVA et al. pyramidal indentation, three side walls of which are nearly perpendicular to the surface and the fourth one makes an angle of ≈30° with it. The vertex angle of the defect is ≈70°. The orientation of the defect walls implies that the defects form (Figs. 2bÐ2d) when dislocations emerge on the surface under irradiation- induced stressing [9, 12, 13]. The same is also true for the defects appearing on the nonirradiated surface: it is 15 nm known that during the mechanical polishing of the surface, dislocations move along the planes of easy slip, 50 nm emerge on the surface, and produce the specific pattern shown in Fig. 2a [14]. 50 nm It follows from the topograms that each stressing cycle changes the surface submicropattern, the changes Fig. 3. Enlarged view of the resulting defect. being the same as upon long-term (static) stressing [10]. After the first cycle, the polished surface becomes even smoother. The coarse defects flatten, and the con- ∆ × 10 2 centration of fine defects is low (Fig. 2b). One can Ilum, arb. units N 10 , 1/cm 1.0 0.16 assume that polishing generates loosely pinned dislocations at the surface. During stressing (during the first 0.9 0.14 cycle in our work and within the initial half an hour of 0.8 stressing in experiments described in [10]), they 0.12 emerge on the surface. After the second laser pulse of 0.7 the same power, the number of defects grows and they 0.10 0.6 2 1 become closer spaced (Fig. 2c). Once the third pulse 0.5 0.08 has been applied (Fig. 2d), the number of defects continues to grow and the defects coarsen [9Ð11]. To trace 0.4 0.06 the defect evolution on the surface under stress cycling, 0.3 we measured the total length of the nanodefect walls on 0.04 the topograms [12] and calculated the difference ∆N 0.2 between the values obtained after two successive 0.1 0.02 cycles. The difference decreases with increasing num- 0 ber of the irradiating pulse 012 3 4 n , number las DISCUSSION

Fig. 4. (1) Concentration of the surface defects and According to the dislocation theory of mechanolu- (2) intensity of photon emission vs. cycle number. minescence [4Ð6], its intensity is given by

IC= 1 Nd, (1) × –7 2 Ilum 10 , W/cm where 0.5 P v C = η 〈〉hv ------h -d exp()Ðαt ; 1 a 0.4 η is the probability of radiative recombination, 〈hv〉 is the mean energy of photons emitted, α is a constant 0.3 depending on the metal properties, t is time, Ph is the probability of hole generation, a is the lattice parame- 0.2 ter, vd is the dislocation velocity, and Nd is the mobile dislocation density. 0.1 Since it is assumed that luminescence occurs when the mobile dislocations emerge on the surface, we looked for a correlation between its intensity and the 0 0.04 0.08 0.12 0.16 ∆ ∆N × 1010, 1/cm2 value of N after each of the cycles. We measured the total length of the nanodefects and estimated the number of dislocations that had emerged on the surface Fig. 5. Luminescence intensity vs. ∆N. from the value of ∆N [11].

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THE EMISSION OF PHOTONS 271

Figure 4 plots ∆N (curve 1) and the luminescence ble to study the dynamics of mobile dislocations in the intensity (curve 2) against the number of laser pulses. surface layer of metals. The correlation expected is obvious: both ∆N and the intensity linearly decrease with increasing number of pulses. Such behavior can be explained by a decrease in ACKNOWLEDGMENTS the number of dislocation sources. This work was supported by the Russian Foundation Figure 5 demonstrates the dependence of the lumi- for Basic Research (grant no. 97-032-18097) and the nescence intensity (in absolute units obtained from Federal Program “Integration” (project no. Fig. 4 and the geometry of the experiments [11]) on A0142/KO854). ∆N. The intensity varies with ∆N as ∆ ∆ IC==2 NC1 Nn. (2) REFERENCES From Fig. 5, it follows that C = 3.42 × 10Ð17 W. 1. K. B. Abramova, V. P. Valitskiœ, N. A. Zlatin, et al., Zh. 2 Éksp. Teor. Fiz. 71, 1873 (1976) [Sov. Phys. JETP 44, Using the above sizes of the defect walls and the value ≈ 983 (1976)]. of the Burgers vector for silver, b 0.3 nm, we ﬁnd the 2. K. B. Abramova and I. P. Shcherbakov, Zh. Tekh. Fiz. 64 average number of dislocations per surface defect for (9), 76 (1994) [Tech. Phys. 39, 901 (1994)]. ≈ × Ð19 n 600. Then, C1 = C2/n = 0.57 10 W for one dis- × 3. A. M. Kondyrev, I. P. Shcherbakov, K. B. Abramova, and location. This value agrees with the value C1 = 2.6 A. E. Chmel’, Zh. Tekh. Fiz. 62 (1), 206 (1992) [Sov. 10−19 W calculated from (1) by substituting the tabu- Phys. Tech. Phys. 37, 110 (1992)]. lated parameters for silver. The direct proportionality 4. M. I. Molotskiœ, Fiz. Tverd. Tela (Leningrad) 20, 1651 between the number of defects and the luminescence (1978) [Sov. Phys. Solid State 20, 956 (1978)]. intensity was also observed for Cu and Au. For these 5. M. I. Molotskii, Sov. Sci. Rev. 13 (3), 1 (1989). metals, the calculated and experimental values of C1 6. B. R. Chandra, M. S. Ryan, Seema R. Simon, and were also found to be close to each other. M. Y. Ansari, Cryst. Res. Technol. 4, 495 (1996). 7. K. B. Abramova, I. P. Shcherbakov, I. Ya. Pukhonto, and A. M. Kondyrev, Zh. Tekh. Fiz. 66 (5), 190 (1996) [Tech. CONCLUSION Phys. 41, 511 (1996)]. Our experiments have shown that stress cycling of 8. K. B. Abramova, I. P. Shcherbakov, and A. I. Rusakov, metal samples by laser pulses of equal power that leave Zh. Tekh. Fiz. 69 (2), 137 (1999) [Tech. Phys. 44, 259 the surface intact changes their surface microrelief and (1999)]. initiates luminescence from the back side. The lumines- 9. V. I. Vettegren’, V. L. Gilyarov, S. N. Rakhimov, and cence intensity decreases from cycle to cycle. As the V. N. Svetlov, Fiz. Tverd. Tela (St. Petersburg) 40, 668 number of the irradiation cycles grows, the concentra- (1998) [Phys. Solid State 40, 614 (1998)]. tion of surface nanodefects changes but the rate of their 10. V. I. Vettegren’, S. N. Rakhimov, and V. N. Svetlov, Fiz. accumulation drops. Tverd. Tela (St. Petersburg) 37, 913 (1995) [Phys. Solid The shape and the orientation of the nanodefects State 37, 495 (1995)]. agree with the supposition that they form when mobile 11. V. I. Vettegren’, S. N. Rakhimov, and V. N. Svetlov, Fiz. dislocations emerge on the surface. Starting from this Tverd. Tela (St. Petersburg) 40, 2180 (1998) [Phys. Solid supposition, we estimated the density of dislocations State 40, 1977 (1998)]. emerging under irradiation. It has been found that the 12. K. B. Abramova, V. I. Vettegren’, I. P. Shcherbakov, rate of accumulation of dislocations decreases with et al., Zh. Tekh. Fiz. 69 (12), 102 (1999) [Tech. Phys. 44, time upon stress cycling. 1491 (1999)]. It has been demonstrated that the mechanolumines- 13. K. B. Abramova, A. M. Kondyrev, I. Ya. Puchonto, et al., cence intensity is proportional to the number of dislo- Proc. SPIE 3093, 22 (1997). cations emerging on the metal surface. The proportion- 14. I. I. Novikov, Defects of Crystal Structure of Metals ality coefﬁcient is close to that calculated from the dis- (Metallurgiya, Moscow, 1975). location model of mechanoluminescence. Investigation into the mechanoluminescence process makes it possi- Translated by V. Isaakyan

TECHNICAL PHYSICS Vol. 47 No. 2 2002

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The Structure of a Shock Electromagnetic Wave Synchronous with Several Waves Propagating in Coupled Transmission Lines with Different Types of Dispersion A. B. Kozyrev Institute of Physics of Microstructures, Russian Academy of Sciences, Nizhni Novgorod, 603600 Russia e-mail: [email protected] Received February 20, 2001

Abstract—A shock electromagnetic wave is studied numerically in the case when its structure is equally deﬁned by two or more synchronous waves. The analysis relies on the previously suggested electrodynamic system of two coupled transmission lines one of which is normally dispersive (coaxial line) and the other is anomalously dispersive (ladder-type bar slow-wave structure). © 2002 MAIK “Nauka/Interperiodica”.

The instability of the front of a shock electromag- mechanisms of front stabilization. In particular, such a netic wave (SEMW) propagating in nonlinear disper- situation is observed in two coupled transmission lines sive transmission lines has been extensively studied with normal and anomalous dispersions [4]. The equiv- from the standpoint of its application for efficient direct alent circuit is shown in Fig. 1. In such a system, within video-to-radio pulse conversion [1Ð3]. However, these a certain range of its velocities, the SEMW is synchro- works restrict their analysis to the case in which the nous with two normal waves simultaneously. Numeri- SEMW parameters (such as the duration of the leading cal simulations have shown that, as a rule, the energy edge) and the behavior of the field components as they efficiency of the low-frequency wave excitation is low approach constant values behind the SEMW front are (because the SEMW weakly penetrates into the anom- governed by one synchronous wave. This work is the alously dispersive transmission line) and does not sta- first to numerically detail the SEMW structure when it bilize the SEMW front [4]. The authors of [4] consid- ω ≈ is equally defined by two or more synchronous waves. ered a high-frequency system with c 24.4 GHz The analysis is exemplified by the electrodynamic sys- (because their purpose was to increase the operating tem of two coupled transmission lines that has been frequency of a pulsed RF oscillator built around a fer- proposed earlier (Fig. 1) [4]. One of these transmission rite-filled transmission line). The final stabilization of lines (coaxial line) exhibits the normal dispersion; the the SEMW front occurred because of the incoherent other (ladder-type bar slow-wave structure), the anom- magnetization reversal loss in the ferrite and the alous dispersion. SEMW structure was defined by the low-frequency As is known, when a step in the electromagnetic synchronous wave alone. The second (high-frequency) field propagates through a nonlinear transmission line synchronous wave was not excited, because, in this filled with ferrite or with a medium exhibiting a nonlin- case, the excitation would be a higher frequency pro- ear CÐV characteristic, the slope of the wave front cess compared with the incoherent magnetization increases, producing an SEMW. In spectral terms, a reversal of the ferrite. nonlinear increase in the wave front slope means that This paper numerically simulates nonlinear SEMW higher frequency components are generated and the propagation processes in a low-frequency system (in energy is transferred from lower to higher spectral com- coupled transmission lines with the critical frequency ponents. Conversely, the stabilization of the SEMW ω ≈ c 6.24 GHz), where the excitation of the second syn- front duration means the development of physical pro- chronous wave is a lower frequency process compared cesses that absorb the energy and compensate for with the magnetization reversal of the ferrite and the energy transfer to the high-frequency spectral compo- SEMW front stabilizes by the excitation of two syn- nents, thereby preventing a further increase in the chronous waves. SEMW front slope. As a rule, the SEMW structure is governed largely by the lowest-frequency mechanism. This situation is illustrated in Fig. 2, which plots the However, if the characteristic frequencies of different oscillograms of the voltage across the 600th cell of the mechanisms are closely spaced or if the lowest-fre- normally and anomalously dispersive coupled trans- quency mechanism is inefficient and cannot stabilize mission lines fed by a semiinfinite voltage pulse. In this the SEMW front completely, one should expect that the case, the group velocities of the waves synchronous SEMW structure will also depend on higher frequency with the SEMW front are close to each other and the

THE STRUCTURE OF A SHOCK ELECTROMAGNETIC WAVE 273

(a) (b) x Φ Φ Φ (1) L0/2 R1 R1 R1 n–1 R1 n R1 n+1 R1 R1 R1 Vnsec L0/2 z y (1) (1) (1) Vn (1) Rin1 I1 In Insec+1 C C C C C0 Rout1 0 0 0 0 C AAC0 C C C C C 0 1 coupl coupl coupl coupl coupl

2 (2) In L01 C01 L01 L01 L01 L01 L01 L01 Rin2 I0 Rout2 R2 R2 R2 R2 R2 R2 – R C01 A A 2 Insec+1

(2) (2) 2C01 C01 C01 C01 C01 Vn C01 C01 C01 C01 Vnsec 2C01

Fig. 1. (a) Design of the coupled quasi-coaxial transmission line and ladder-type bar slow-wave structure (T waveguide with an bar array: (1) metal and (2) ferrite) and (b) the equivalent circuit.

2.0 0.2 (a) (c)

1.5 s V

/ 1.0 0.1 (1) 600 V 0.5

0 0 1.0 0.2 (b) (d)

0.5 s V / 0 0.1 (2) 600 V

–1.5

–1.0 600 800 1000 1200 0 0.1 0.2 0.3 τ τ t/ 0 f 0

()1 ()2 Fig. 2. Oscillograms of the voltage across the 600th cell of the line with the (a) normal (V600 /Vs) and (b) anomalous (V600 /Vs) dispersion normalized by the SEMW amplitude Vs = 50 kV and (c, d) their spectra for the relative SEMW velocity vs/v0 = 0.85 [4]. ω τ τ 1/2 Ω Parameters of the equivalent circuit are c = 2/ 0 = 6.24 GHz ( 0 = (L0C0) = 0.32 ns), Z0 = L0/C0 = 25 , C01/C0 = 0.1, L01/L0 = 22.5, and Ccoupl/C0 = 0.16. energy of the oscillations generated is transferred with Thus, the oscillogram behind the SEMW front is the a constant rate equal to the difference between the superposition of two radio pulses of equal length, as SEMW velocity and the group velocities of the syn- indirectly indicated by the same spectral widths of the chronous waves. peaks in the oscillogram’s spectra. The shape of the

TECHNICAL PHYSICS Vol. 47 No. 2 2002

274 KOZYREV trailing edge of the radio pulse is defined by the disper- neously. One of them travels to the input of the anoma- sion of the group velocities of the synchronous waves lously dispersive line and the other, to its output, so that [3]. The front has a sufficiently complex structure. they can be released on the associated matched loads. Although its duration depends on the period of the To conclude, we note that, by analogy with coupled lower frequency synchronous wave, the field (voltage) normally and anomalously dispersive lines, one can varies with a frequency on the order of that of the higher consider a multiconductor electrodynamic system con- frequency synchronous wave in some regions of the sisting of several coupled transmission lines. Here, the front. SEMW structure will be defined by a specified number In the case we study, the energy efficiency of the of synchronous waves, their excitation efficiency being synchronous wave excitation (the ratio of the power controlled by the coupling between the lines. The exci- spent on the excitation of the synchronous wave to the tation efficiency of the higher frequency waves will total power applied to the SEMW front) is 0.46 for both depend on the coefficients of line coupling and on the waves. The total portion of the energy spent on the gen- excitation efficiencies of the lower frequency waves. eration of the RF waves is high (92%). The remaining This would make it possible to generate multifrequency portion of the energy applied to the SEMW front goes pulses with the energy uniformly distributed over the into the incoherent magnetization reversal of the ferrite. frequencies. The spectrum of such a radio pulse can be Note that, in the electrodynamic system in the form of controlled electronically (by varying the bias current or a transmission line with the next nearest links coupled the SEMW amplitude). [1, 2], the simultaneous synchronism with two waves (for example, with the lower frequency forward normal wave and the higher frequency backward spatial har- ACKNOWLEDGMENTS monic) can also be achieved at certain parameters of the I am grateful to A.M. Belyantsev for his interest in transmission line and a certain SEMW velocity. How- the work and valuable remarks. ever, in this case, the lower frequency synchronous This work was supported by the Russian Foundation wave is excited efficiently and completely stabilizes the for Basic Research (grant no. 99-02-18046). SEMW front. The equally efficient excitation of two synchronous waves becomes impossible and only one frequency dominates in the spectrum of the oscillo- REFERENCES gram. 1. A. M. Belyantsev, A. I. Dubnev, S. L. Klimin, et al., Zh. In Fig. 2, the group velocities of the synchronous Tekh. Fiz. 65 (8), 132 (1995) [Tech. Phys. 40, 820 waves have the same direction as the SEMW. As fol- (1995)]. lows from the dispersion characteristics of the system 2. A. M. Belyantsev and A. B. Kozyrev, Zh. Tekh. Fiz. 70 of coupled transmission lines [4], the situation where (6), 78 (2000) [Tech. Phys. 45, 747 (2000)]. the group velocities are directed oppositely may arise. 3. A. M. Belyantsev and A. B. Kozyrev, Zh. Tekh. Fiz. 68 The wavetrain produced in this case also has an irregu- (1), 89 (1998) [Tech. Phys. 43, 80 (1998)]. lar two-frequency structure depending on the group velocities of the synchronous waves and on their local 4. A. M. Belyantsev and A. B. Kozyrev, Zh. Tekh. Fiz. 71 dispersion. However, unlike the previous case, the ener- (7), 79 (2001) [Tech. Phys. 46, 864 (2001)]. gies of the waves are transferred in opposite directions, so that actually two radio pulses are generated simulta- Translated by A. Khzmalyan

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Canonical Averaging of the Schrödinger Equation A. G. Chirkov St. Petersburg State Technical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251 Russia Received June 13, 2001

Abstract—The representation of the Schrödinger equation in the form of a classical Hamiltonian system makes it possible to construct a uniﬁed perturbation theory that is based on the theory of canonical transformations and covers both classical and quantum mechanics. Also, the closeness of the exact and approximate solutions of the Schrödinger equation can be approximately estimated with such a representation. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION tonian system (well-known in nonlinear mechanics) The main objective of investigation in the nonrela- and accordingly to use rigorous and simpler methods of tivistic quantum theory is the Schrödinger equation [1] classical mechanics for its investigation. Consider Eq. (1) with Schrödinger operators (2), i.e., the prob- ∂Ψ(), ប qt ˆ Ψ(), lem i ------∂ = H qt, (1) t ∂Ψ()qt, iប------= ()ΨHˆ 0 + εVˆ ()qt, , where i2 = Ð1; ប = 1.054 × 10Ð27 erg s is the Planck con- ∂t (3) stant; q = (q1, q2, …, qn) is, generally speaking, a point Ψ()Ψq, 0 = ()q , in the conﬁgurational space of the respective classical 0 system; t is time; Ψ(q, t) is a complex function with the where the operator Hˆ 0 is time-independent and ε is a integrable squared modulus; and Hˆ is the self-(adjoint formal small parameter. (symmetric) operator acting in the Hilbertian) space. Along with problem (3), we consider the generating Schrödinger equation (1) should be solved with the approximation, which is obtained from (3) at ε = 0: Ψ Ψ initial condition (q, 0) = 0(q) and some boundary 0 ∂Ψ ()qt, 0 0 conditions. iប------==Hˆ Ψ ()qt, , Ψ ()Ψq, 0 ()q . (4) ∂t 0 0 The case treated in the perturbation theory arises if the operator Hˆ can be represented as the sum The general solution of problem (4) found by the Fou- rier method can be represented in the form (for the sake of simplicity, the spectrum is assumed to be discrete) Hˆ = Hˆ 0 + εVˆ , 0 < ε Ӷ 1 (2) ∞ of two self-adjoint operators. Here, problem (1) for the Ψ0(), 0ψ0() ()ω0 qt = ∑ cn n q exp Ði nt , operator Hˆ is assumed to be exactly solvable (gener- 0 n = 0 (5) ating approximation), whereas the other operator is assumed to be small in a sense (perturbation) [2]. ()0 Ψ ()ψ0*() ω0 0 ប cn ==∫ 0 q n q dq, n En/ , In such a representation, most physically interesting ψ0 0 problems prove to be mathematically ill-posed, because where n (q) and En are the eigenfunctions and the perturbation operators are usually unbounded and even eigenvalues of the problem not self-adjoint. The latter fact is because physicists ˆ ψ0() 0ψ0() almost never distinguish the concepts of self-adjoint H0 n q = En n q (6) and symmetric operators. Therefore, the domain of def- (asterisk means complex conjugation). inition for an operator in the Hilbertian space remains obscure, which prevents the use of methods of the spec- It follows from the self-conjugacy of the operator Hˆ 0 tral theory [2, 3]. Because of this, in practical calcula- that the expansion tions, it is hard to set the applicability condition of the ∞ perturbation theory and estimate the difference Ψ(), ()ψ0() ()ω0 between the exact and approximate solutions. qt = ∑ cn t n q exp Ði nt (7) However, it becomes possible to reduce the n = 0 Schrödinger equation to the form of a classical Hamil- is valid for Ψ(q, t) ∈ L2.

1063-7842/02/4702-0275 $22.00 © 2002 MAIK “Nauka/Interperiodica” 276 CHIRKOV Substituting expansion (7) into (3), we find the Hamiltonian function (12), in the form equations for the expansion coefficients cn(t) and ∞ () cn* t : ε ()ε,,ψ H1 I t = ∑ v mn In Im ∞ mn, = 0 (14) () ε () ()ω0 × []()ψ ψ ω0 cún t = Ði ∑ v nmcm t exp Ði mnt , exp Ði m Ð n + mnt . m = 0 ∞ (8) In (13), prime means that the fundamental resonance (i.e., the terms with n = m) is separated. cú*()t = iε v c*()t exp()iω0 t , n ∑ mn m mn The Planck constant does not appear in sets (8) and m = 0 (13) explicitly; hence, these are classical Hamiltonian systems with the infinite number of internal resonances. 1 0 0 v () ψ *()ˆ ()ψ, () mn t = ប---∫ m q V qt n q dq, (9) The methods of their investigation were developed by Bogolyubov, Mitropol’skiœ, Grebenikov, Arnold, and their disciples [4Ð8]. In particular, the estimation of the ω0 ω0 where dot denotes the time derivative and mn = m Ð difference between the exact and approximate solutions ω0 . by norm and finding the applicability condition for the n averaging method for these systems are given by the Set of equations (8) proves to be Hamiltonian (in the Los’ theorem [4, 9], which is a generalization of the classical meaning) with the Hamiltonian Bogolyubov theorem for the case of an infinitely dimensional coordinate Hilbertian space. ∞ Using the method of averaging, one can construct a ε (),, ε * ()ω0 H1 cc* t = Ði ∑ v nmcn cm exp i nmt , (10) solution of sets (8) and (13), which approximates the nm, = 0 exact one with any given accuracy. which describes the classical distributed system with The canonical form of sets (8) and (13) makes it the infinite number of internal resonances. The Hamil- possible to pass to the evolutionary equations by using tonian property is provided by the fact that the matrix Hamiltonian functions (10), (11), or (14) alone, i.e., by vnm is Hermitian (the perturbation operator is self- calculating the average Hamiltonian function. For ω0 example, for (10) and (11), the second-order approxi- adjoint). We separate the fundamental resonance n = () mation H 2 for the averaged Hamiltonian is con- ω0 m to represent Hamiltonian function (10) in the form structed using the formulas

()2 2 ε (),, ε H ()εcc, * = H1()εcc, * + H2()cc, * , H1 cc* t = Ði v nncncn* ∞ ∂ ˜ ∂{} (15) (11) 〈〉 H1 H1 ε ()ω0 H1 ==H1 , H2 Ð ------∂ ------∂ , Ð i ∑ v' nmcn*cm exp i nmt . c* c nm, = 0 where cn and cn* are the evolutionary components of * Now we pass from the variables cn and cn to the real the variables c and c* , respectively, and ψ n n variables action and angle, In and n, by the formulas t0 + T ()ψ * 〈〉 1 (),, cn ==In exp Ði n , In cncn , f = lim --- ∫ fcc* t dt, T → ∞ T ()c Ð c* (12) t0 c* ==I exp()iψ , ψ Ðarctan------n n . n n n n ()* icn + cn ˜f ()cc,,* t = fcc(),,* t Ð 〈〉f , (16) Then, set (8) is represented in the form {}f = ∫ ˜f ()cc,,* t dt. ∞ ú ε ' {}[]()ψ ψ ω0 Here, an arbitrary function of the slow variables c and In = 2 ∑ In ImIm v nmexp Ði m Ð n + mnt , m = 0 c* is put equal to zero. ∞ Note that H is the integral of motion for the aver- ψ ε ε ' In ú n = v mn + ∑ ----- (13) aged equations, i.e., the adiabatic invariant [7, 8]. Im m = 0 ()1 The ﬁrst approximation cn for the expansion coef- × {}[]()ψ ψ ω0 1 Re v nmexp Ði m Ð n + mnt ; ﬁcients cn is found using the formula cn = cn , where

TECHNICAL PHYSICS Vol. 47 No. 2 2002 CANONICAL AVERAGING OF THE SCHRÖDINGER EQUATION 277 cn satisfies the equations REFERENCES 1. L. D. Landau and E. M. Lifshitz, Course of Theoretical ∂H Physics, Vol. 3: Quantum Mechanics: Non-Relativistic cú = ε------1. (17) Theory (Fizmatgiz, Moscow, 1963; Pergamon, New n ∂ cn* York, 1977, 3rd ed.). 2. T. Kato, Perturbation Theory for Linear Operators () The second-order approximation c 2 for the expan- (Springer-Verlag, Berlin, 1966; Mir, Moscow, 1972). n 3. M. C. Reed and B. Simon, Methods of Modern Mathe- sion coefficients cn is calculated using the formulas matical Physics (Academic, New York, 1978; Mir, Mos- cow, 1982), Vol. 4. ()2 ∂{}H 4. Yu. A. Mitropol’skiœ, Averaging Method in Nonlinear c = c + ε------1 , (18) n n ∂c* Mechanics (Naukova Dumka, Kiev, 1971). n 5. Yu. A. Mitropol’skiœ and B. I. Moseenkov, Asymptotic Solutions of Partial Differential Equations (Vishcha where the second-order approximation for the evolu- Shkola, Kiev, 1976). ()2 tionary components cn is derived from the equations 6. E. A. Grebenikov, Averaging Method in Applied Prob- lems (Nauka, Moscow, 1986). ∂H ∂H 7. V. I. Arnold, Mathematical Methods of Classical cú = ε------1 + ε2------2. (19) Mechanics (Nauka, Moscow, 1989; Springer-Verlag, n ∂ ∂ cn* cn* New York, 1989). 8. V. I. Arnold, Supplementary Chapters to the Theory of Thus, one can identify the variables cn and their evo- Ordinary Differential Equations (Nauka, Moscow, lutionary components only in the first approximation. 1978). The relations obtained make it possible to perform cal- 9. F. S. Los’, Ukr. Mat. Zh. 2 (3), 87 (1950). culations in classical and quantum mechanics by unified formulas. Translated by D. Zhukhovitskiœ

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A Dislocation Mechanism of Friction between a Nanoprobe and Solid Surface S. Sh. Rekhviashvili Institute of Applied Mathematics and Automation, Design Ofﬁce of the Research Center, Russian Academy of Sciences, Nal’chik, 360000 Russia Received June 18, 2001

Abstract—A dislocation mechanism of friction between an atomic-force microscope (AFM) probe and an atomically smooth solid surface is put forward. In this mechanism, the contact region is represented by an edge dislocation. The triboacoustic emission measured with an AFM shows the dislocation nature of friction. The friction force is calculated for a parabolic tip. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION (ii) From the dislocation theory, it is known that the tangential stress is minimal upon gliding close-packed According to modern concepts, direct proportional- planes or planes with incommensurate structures. This ity between the friction force and the load for macro- seems to be the reason for low values of friction factors scopic rubbing bodies (the Amonton law) stems from and wear rates under solid friction at the nanostructure the superposition of many micro- and nanocontacts level. between friction surfaces. As a result, the actual total contact area is several orders of magnitude smaller than (iii) The simulation of contact interaction between the apparent area [1]. At present, friction in nanocon- an AFM probe and a solid surface by molecular dynam- tacts is investigated by an AFM, which is among the ics methods and the Monte Carlo procedure confirms most powerful instruments used for studying solid sur- the formation of the “bad” crystal region [3Ð6]. faces at the atomic or molecular level and measuring (iv) Edge dislocation glide generates, as a rule, low- ultralow forces (of the order of 1 pN), as well as in nan- frequency acoustic waves. Within the wave zone, the otechnology. An AFM mechanically scans the surface velocity vector of elements of the medium is propor- (in the contact or contactless mode) with a special sen- tional to the integral [7] sor, a cantilever, involving a support and an elastic ∂2 tipped microbeam. ∼ V ∫------2 jksdr, The atomism of friction is still in its infancy. To ∂t date, most theoretical works either proceed from poorly where jks is the dislocation flux density tensor (single- substantiated model representations, which do not component in our case) and r is the radius vector of a allow us to explicitly express the friction force [2] or dislocation. use numerical simulation by the molecular dynamics method [3, 4] and the Monte Carlo technique [5, 6]. The As follows from this expression, acoustic emission results of both dramatically depend on the number of must be observed under the nonstationary motion of the atoms in a system and on the approximation of the probe, i.e., when the second time derivative of the dis- interaction potential. However, in the latter case, one location flux density is nonzero. As will be shown later, can simulate the images and observe the processes this fact is confirmed in experiments. occurring immediately in the contact area. In this paper, a dislocation mechanism of friction EXPERIMENTAL between the nanoprobe and an atomically smooth sur- To verify the dislocation mechanism of friction, we face is put forward for the first time. Here, the contact used a specially designed AFM whose block diagram is area is represented by an edge dislocation enclosing an shown in Fig. 1a. A rectangular pulse from driving gen- area Ω. This assumption is based on the following. erator 1 is applied to piezoelectric element 2. As a (i) As follows from X-ray diffraction data, the dislo- result, thin tungsten 3, fixed on the piezoelectric ele- cation displacement is a multiple of the Burgers vector ment, executes mechanical vibration, slipping over the or, virtually, of the lattice spacing. A dislocation moves surface of specimen 4. The vibration amplitude and the jumpwise in the field of the Peierls–Nabarro forces, slip velocity of the probe depend on the voltage applied which causes the stickingÐglide effect, often observed to the piezoelectric element and on the triggering signal in experiments with AFMs. frequency. During scanning, pickup 5 measures the

A DISLOCATION MECHANISM OF FRICTION 279 intensity of sound waves generated by the friction of (a) the tip with the surface. The signal from the pickup is applied to low-noise ampliﬁer 6, tunable band-pass ﬁl- 1 2 ter 7, and then to oscilloscope 8. The waveform of the triboacoustic signal from the 3 AFM is shown in Fig. 1b, where 1 is the signal applied 5 to the piezoelectric element and 2 is the output signal. 4 The leading and trailing edges of triggering pulse 1 correspond to two end positions of the probe. In either position, the piezoelectric element abruptly stops, while the probe executes damped vibrations near the 876 surface. As follows from experiments, the acoustic emission intensity nearly linearly grows with the load (b) applied to the probe. The peaks in curve 2 correspond 1 to the turning points of the probe. In other words, the triboacoustic emission peaks when the probe acceleration is maximum, which is typical of the dislocation 2 mechanism of acoustic wave generation [7].

ANALYSIS 0.5 ms The friction force is calculated starting from the classical concept of edge dislocation. The work needed Fig. 1. (a) AFM block diagram and (b) waveform of the tri- to displace a dislocation is A = τbΩ, where τ is the shear boacoustic signal upon interaction between the tungsten stress, b is the Burgers vector, and Ω is the area probe and the single-crystal Si (111) surface. The radius of enclosed by the dislocation [8]. Let an elementary slip curvature of the probe is about 100 nm; load, no more than µ (microslip) of the AFM probe be equal to b; then, the 0.1 N. tangential force is given by F = τΩ. (1) G, GPa 200 The force needed to break interatomic bonds under isobaric isothermal conditions is W 150 ∆G ∆F = ------, Mo dNA 100 where ∆G is the Gibbs energy, d is the interatomic spac- Co Fe Ni ing, and N is the Avogadro number. A Pa V Zn Cu Ω 50 Ag In the contact area, the number of atoms is N = ns , Au Ti Cd Mg Dy Gd where ns is the sheet atom concentration; therefore, in Bi Sn La order to break the whole contact, one should apply the 0 Nd force 100 200 300 400 500 600 700 800 900 ∆H , kJ/mol ∆Gn Ω s F ==∆FN ------s -. (2) dN A Fig. 2. Correlation between the shear modulus and heat of Comparing (1) and (2) gives sublimation. n τ = ------b-∆G, (3) N A 0.186x Ð 19.132. The correlation factor was found to be 0.842. where nb = ns/d is the volume concentration of the atoms. The experiments with AFMs indicate that the fric- The shear modulus vs. heat of sublimation for vari- tion is nonzero even in the absence of an external load ous solids is shown in Fig. 2, which approves the valid- [11, 12]. This is due to the fact that after unloading, the ity of formula (3) (i.e., the direct proportionality adhesive force and the capillary force, which appears between τ and ∆G). The shear stress was assumed to be during the AFM operation in a humid atmosphere, still proportional to the shear modulus; the Gibbs energy, to act. In calculating the contact area Ω, these forces can the heat of sublimation. The data points are taken from be taken into account in the DeryaginÐMullerÐToporov [9, 10], and the regression line is given by y(x) = approximation [13]. As a result, we obtain the friction

TECHNICAL PHYSICS Vol. 47 No. 2 2002

280 REKHVIASHVILI force in the form of REFERENCES π ∆ 2/3 1. Physics Encyclopedia, Ed. by A. M. Prokhorov (Sov. nb G R()π γ F = ------F⊥ ++2 R Fc , Éntsiklopediya, Moscow, 1983), pp. 765Ð766. N A E (4) 2. G. V. Dedkov, Pis’ma Zh. Tekh. Fiz. 24 (19), 44 (1998) 2 2 1 3 1 Ð µ 1 Ð µ [Tech. Phys. Lett. 24, 766 (1998)]; G. V. Dedkov, Zh. --- = ------1 +,------2 Tekh. Fiz. 70 (7), 96 (2000) [Tech. Phys. 45, 909 E 4 E1 E2 (2000)]. where R is the radius of curvature of the probe tip; F⊥ is 3. U. Landman et al., Science 248, 454 (1990). the load; γ is the specific energy of adhesion for flat sur- 4. A. V. Pokropivnyœ et al., Pis’ma Zh. Tekh. Fiz. 22 (2), 1 µ (1996) [Tech. Phys. Lett. 22, 46 (1996)]; Zh. Tekh. Fiz. faces; Fc is the capillary force; and E1, 2 and 1, 2 are the moduli of elasticity and the Poisson ratios for the tip 67 (12), 70 (1997) [Tech. Phys. 42, 1435 (1997)]. and the specimen, respectively. 5. B. S. Good and A. Banerjea, J. Phys.: Condens. Matter 8, Note that the friction force depends on the load as 1325 (1996). 2/3 6. S. Sh. Rekhviashvili, in Proceedings of IV All-Russia F ~ F⊥ , which is typical of experiments with AFMs Symposium “Mathematical Simulation and Computer [11, 12]. Technologies,” Kislovodsk, 2000, Ed. by A. A. Samar- skiœ, Vol. 2, Part 1, p. 25. 7. V. D. Natsik and K. A. Chishko, Fiz. Tverd. Tela (Lenin- CONCLUSION grad) 14, 3126 (1972) [Sov. Phys. Solid State 14, 2678 In this paper, various physical processes in the AFM (1972)]. probeÐspecimen system were analyzed. The experi- 8. J. Friedel, Dislocations (Pergamon, Oxford, 1964; Mir, mental and theoretical investigations performed com- Moscow, 1967). pletely confirmed the dislocation nature of the interac- 9. Handbook of Physical Quantities, Ed. by I. S. Grigoriev tion between an atomically smooth surface and the nan- and E. Z. Meilikhov (Énergoatomizdat, Moscow, 1991; oprobe. CRC Press, Boca Raton, 1997). One more point is noteworthy. When the glide of a 10. A. R. Regel’ and V. M. Glazov, The Periodic Law and little cluster of atoms over the crystal surface was Physical Properties of Electronic Melts (Nauka, Mos- numerically simulated within the framework of the dis- cow, 1978). location model, it was found that the AFM images 11. H. Tanaka et al., Thin Solid Films 342, 4 (1999). should possess the typical fractal properties: (i) self- 12. R. E. Carpick and M. Salmeron, Chem. Rev. 97, 1163 similarity when the scale (in our case, the size of the (1997). contact area) changes and (ii) the fractional Hausdorff dimension in the range between 2 and 3. The images 13. N. A. Burnham and A. J. Kulik, in Surface Forces and Adhesion: Handbook of Micro and Nanotribology (CRC may be heavily distorted, up to the total disappearance Press, Boca Raton, 1997), p. 21. or inversion of the contrast, when the maxima and min- ima on the scan change places. These phenomena call for special consideration. Translated by B. Malyukov

TECHNICAL PHYSICS Vol. 47 No. 2 2002