Propagation of a Spatial Soliton in a System of Tunneling-Coupled Optical Waveguides with a Variable Coupling Coefﬁcient F

TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Propagation of a spatial soliton in a system of tunneling-coupled optical waveguides with a variable coupling coefﬁcient F. Kh. Abdullaev

S. S. Starodubtsev Physicotechnical Institute; Scientific-Production Unit ‘‘Fizika-Solntse,’’ Uzbekistan Academy of Sciences, 700084 Tashkent, Uzbekistan ͑Submitted October 24, 1996; resubmitted January 27, 1998͒ Zh. Tekh. Fiz. 68, 1–4 ͑June 1998͒ The problem of the propagation of a spatial soliton in a system of tunneling-coupled optical waveguides is studied. The tunneling coupling coefficient is assumed to be modulated in a transverse direction. The adiabatic dynamics is studied for the case of periodic modulation of the tunneling coupling coefficient. The stationary points for the beam parameters are found. The effective potential is calculated for the center of the beam in the equivalent-particle model. The condition for resonance emission of waves by a soliton is obtained. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00106-8͔

INTRODUCTION 2 Ϫiunzϭvn,nϩ1/2unϩ1ϩvnϪ1/2,nunϪ1ϩ͉un͉ un . ͑1͒ Systems of tunneling-coupled nonlinear optical Here ¯u(z,n) is the dimensionless intensity of the electric waveguides are attracting attention in connection with pos- field of the wave and vn,m is the coupling coefficient be- sible applications as optical decouplers, components in opti- 1,2 tween the waveguides. An approximate expression for vn,m cal logic circuits, and so on. Such configurations can be is8 obtained, for example, by preparing flat waveguides on a 3 v ϳexp͓Ϫ␥͑nϪm͔͒. GaxAl1ϪxAs substrate. This material possesses positive Kerr n,m nonlinearity when the wave frequency is less than half the We shall employ a continuum approximation to describe band gap. An important feature is that soliton propagation the evolution of a wave in the system of waveguides. This regimes of powerful optical beams—spatial optical soli- approximation can be used to study the variations of a wave 4,5 tons—are possible in such systems. Such systems also on scales much larger than the distance h between the wave- have the advantage that it is possible to obtain strong non- 9 guide centers. Then un and vn,m can be expanded in a Tay- linear effects in them by using weak nonlinear materials in lor series separate waveguides. For applications it is of interest to de- 2 velop methods for controlling the parameters of spatial soli- h u ϭu͑x͒Ϯhu ϩ u ϩ..., tons in these systems. In Ref. 6 it is suggested that variation nϮ1 x 2 xx of the tunneling coupling coefficient between the waveguides h h2 be used for such purposes. Using a variational approach and vn,nϮ1/2ϭv͑x͒Ϯ vxϩ vxxϩ... . the continuum approximation the authors were able to inves- 2 8 tigate the effect of linear and quadratic variation of the tun- Substituting these expressions in Eq. ͑1͒ we obtain for neling coupling coefficient. For their approach the choice of u(z,x) the equation trial function is critical. Since the continuum approximation 1 leads to a perturbed nonlinear Schro¨dinger equation, the 2 2 Ϫiuzϭ2vuϩh vuxxϩvxuxϩ vxxu ϩ͉u͉ u. ͑2͒ method of the inverse problem in soliton theory provides a ͩ 4 ͪ systematic approach. We shall study the case of periodic modulation of the In the present paper we shall study the effect of a peri- tunneling-coupling parameter along the direction x,vϭ1 odic variation of the tunneling-coupling coefficient on the ϩ␧ sin(ax). Setting hϭ1 and changing to the new variables propagation of a spatial soliton in a stack of nonlinear planar yϭx/&, zϭ2t, ␣ϭͱ2a, we obtain finally the equation waveguides. The investigation will employ perturbation 2 theory based on the inverse-problem method. The problem iutϩuyyϭ2͉u͉ uϭ␧R͑u,y͒ exhibits special features compared with that studied in Ref. ϭϪ␧ sin͑␣ ͒u Ϫ␧␣ sin͑␣ ͒u 6, specifically, in our case resonance emission of waves by a y yy y y spatial soliton appears. 1 ϩ ␧␣2 sin͑␣ ͒uϪ4␧ sin͑␣ ͒u. ͑3͒ 4 y y DESCRIPTION OF THE MODEL As a result, the problem reduces to investigation of the The system of equations describing the propagation of a nonlinear Schro¨dinger equation with a periodic perturbation. wave in an infinite system of tunneling-coupled nonlinear As follows from Eq. ͑2͒, the perturbation is conservative— planar waveguides is6,7 there exists a conserved energy integral

ϱ of the soliton lϭ1/2␩. The amplitude of the effective poten- 2 Nϭ ͉u͉ dxϭconst. tial depends strongly on the ratio l/␭. For l/␭Ӷ1 the effec- ͵Ϫϱ tive potential equals We note that the influence of a local nonuniformity of 8 the refractive index in a stack of waveguides on the dynam- u͑␨͒Ϸ ␧ sin͑␣␨͒␩2ϩ0͑l2/␭2͒. ics of a spatial soliton was recently analyzed on the basis of 3 a continuum approximation and the inverse-problem method In this limit the effective potential is virtually indepen- 10 for solitons. The corresponding perturbation on the right- dent of the ratio of the length scales and is proportional to hand side of Eq. ͑3͒ has the form R(u,x)ϭ␧␦(x)u. Numeri- the squared amplitude of the soliton. cal modeling confirmed that such an approach gives a good For l/␭ӷ1 we have description of beam dynamics. ␲␧␣3 u͑␨͒Ϸ exp͑Ϫ␲␣/4␩͒sin͑␣␨͒. 48␩ ADIABATIC DYNAMICS OF A SPATIAL SOLITON IN A One can see that the amplitude of the effective potential SYSTEM OF COUPLED WAVEGUIDES is exponentially small for solitons whose width is greater than the modulation period. There exists a corridor of param- We shall study the evolution of a single spatial soliton, eter values near ␣ϳ␧Ϫ1/3 where the amplitude of the poten- propagating in a system of tunneling-coupled waveguides. tial Շ1. When there is no modulation, i.e., vϭ0, the soliton solution It is difficult to find the general solutions of the system can be written in the form ͑4͒ and ͑5͒. Let us analyze the behavior of the system in the 2 2 usϭ2i␩ sech͓2␩͑yϪ␨͔͒exp͑Ϫ2i␰yϪ4i͑␩ Ϫ␰ ͒t͒. phase plane. For this, let us find the stationary points for the system ͑4͒ and ͑5͒. Let us rewrite the system of equations in Here ␩ is the amplitude of the soliton and ␨ϭϪ4␰t, where the form ␰ϭϪv/4 is the coordinate of the center and v is the velocity of the soliton. We note that in dimensional variables the d␰ ϭA cos͑␣␨͒͑␰2ϪB͒, ͑7͒ soliton velocity is the angle ␺ϭsinϪ1(␰/2) of propagation of dt a beam in the stack of waveguides. In this section we shall study the effect of large- and small-scale modulations of the d␨ ϭϪ4␰͓1ϩC sin͑␣␨͔͒, ͑8͒ tunneling-coupling coefficient on the dynamics of a spatial dt soliton. where a͒ Large-scale modulations of the tunneling-coupling coefficient. Using the perturbation theory,11,12 we find that the ␧␲␣2 Aϭ , amplitude of the soliton is conserved ␩ϭconst. This is a 2␩ sin h͑␲␣/4␩͒ reflection of the previously noted fact that for Eqs. ͑2͒ and 2 ͑3͒ there exists an integral of the motion, Nϭ͐ϱ u 2dx. 1 ␣ A Ϫϱ͉ ͉ BϭϪ ␩2ϩ Ϫ1 , Cϭ . We obtain the following system of equations for the velocity ͩ3 48 ͪ 2␣ and coordinate of the center of the soliton: We note that A, B, and C are decreasing functions of ␣. d␰ ␲␣2␧ ␩2 ␣2 Then the conditions for the stationary points can be written ϭ cos͑␣␨͒ ϩ␰2ϩ Ϫ1 , dt 2␩ sin h͑␲␣/4␩͒ ͩ 3 48 ͪ in the form ͑4͒ ␰ϭ0, ␣␨ϭ␲/2ϩn␲; ␰ϭϮͱB,

d␨ ␲␣␧ sin͑␣␨͒ Ϫ1 n ϭϪ4␰ 1ϩ . ͑5͒ ␣␨ϭϪsin ͑1/c͒͑Ϫ1͒ , nϭ 1, 2, ... . ͑9͒ dt ͫ 4␩ sin h͑␲␣/4␩͒ͬ It can be shown by analyzing the linearized system ͑4͒ 2 2 For low soliton velocities ␰Ӷ1, ␰ Ӷ␩ ͑‘‘heavy’’ soli- and ͑5͒ that the points ␣␨ϭ␲/2ϩ2n␲ are stable, while the ton͒ the system ͑4͒ and ͑5͒ is equivalent to the problem of points ␣␨ϭϪ␲/2ϩ2n␲ are unstable. The phase portrait of the motion of a particle of unit mass in a periodic potential the system is presented in Fig. 1. Since the phase trajectories u(␨) are periodic, we limited the figure to the values Ϫ␲Ͻ␣␨ 2␲␣␧ sin͑␣␨͒ ␩2 ␣2 Ͻ3␲/2. One can see regions of finite motion of an effective u͑␨͒ϭ ϩ Ϫ1 . ͑6͒ particle, which correspond to oscillations of a spatial soliton ␩ sin h͑␲␣/4␩͒ 3 48 ͩ ͪ as a whole in the system of tunneling-coupled waveguides, In the general case the effective potential depends on the and regions of unbounded motion, which correspond to a soliton velocity ␰. There are examples of velocity-dependent growing deflection of the beam. We note that two types of potentials in physics. For example, velocity-dependent po- effects materialize here. Effects of the first type are due to tentials have been used to describe the nucleon interactions the linear periodic potential, and effects of the second type in nuclei. are due to periodic modulation of the dispersion. The disper- Let us discuss the role of length scales in the problem. sion perturbation dominates for large ␰ in R(u) v(x)uxx There are two characteristic scales: The modulation period ϳ␰2v(x)u, which one can see in the phase portrait—region→ ␭ϭ2␲/␣ of the tunneling-coupling parameter and the size 1. For low velocities ␰ we have soliton oscillations around Tech. Phys. 43 (6), June 1998 F. Kh. Abdullaev 617

1ϩ␧2/8 dЈϭd . ͑12͒ 0ͱ1Ϫ3␧2/8 The adiabatic soliton dynamics studied in this section is valid when radiation effects can be neglected. We shall brieﬂy discuss the effect of the emission of waves by a spatial soliton. Analysis based on solving the equations for the Jost co- efﬁcient b(␭) of an associated linear spectral problem, where ␭ is the spectral parameter14 (␭ϭ2k and k is the wave number͒, shows that wave emission by a soliton arises. The radiation is concentrated around two spectral points

2 ␭1,2ϭϪ␰Ϯͱ␣␰Ϫ␩ . ͑13͒ The group velocity of the emitted waves is vϭϪ4␭1,2 . Maximum emission is observed when 2 FIG. 1. Phase portrait of the system ͑4͒ and ͑5͒ with ␧ϭ0.1, 2␩ϭ1, ␣ϭ1. ␣/2Ϫͱ␣␰Ϫ␩ Ӷ1. ͑14͒ For soliton amplitudes ␩ small compared with the velocity ␰ and modulation frequency ␣ we find that the emission is the bottom of each well of the periodic potential—region 2. maximum when The line 3, where the soliton velocity is constant ␰ϭϮͱB, ␣ϭv . ͑15͒ is of special interest. Here the effects of dispersion and po- s tential disturbances on the motion of the spatial soliton are One can see that resonance radiation occurs when the balanced. Since the soliton velocity ␰ is related with the modulation frequency equals the soliton velocity. On ac- beam propagation angle in the waveguide stack, we conclude count of the strong soliton-radiation coupling the soliton de- that a variation of the tunneling-coupling constant between cays as it propagates in the waveguide stack. In Ref. 15 a the waveguides leads to periodic variations of the location condition identical to Eq. ͑15͒ was obtained from qualitative where the soliton beam emerges from the waveguide stack. considerations in a description of numerical experiments on The latter effect can be used to develop optical devices based the propagation of a nonlinear Schro¨dinger soliton in a peri- on spatial optical solitons. odically nonuniform medium. Since the soliton velocity is b͒ Small-scale modulations of the tunneling-coupling co- actually the propagation angle of a beam in a system of efficient. A different approach must be used to describe the waveguides, we find that intense emission should occur for limit ␣ӷ␩, specifically, one or another variant of the propagation angles method of averaging over fast variations of ␯(y). Here we ␺ ϭsinϪ1͑␣/8͒. ͑16͒ shall use the method suggested in Ref. 13. The aim is to c obtain an equation for the slowly varying part of the wave Our results should be correct for these values of the pa- field, using the expansion rameters. Specifically, intense emission leads to the appearance of radiation damping and deceleration of the soliton. A uϭUϩA cos͑␣y͒ϩB sin͑␣y͒ complete analysis of the problem of wave emission by a spatial soliton in a system of waveguides and a calculation of ϩC cos͑2␣y͒ϩD sin͑2␣y͒ϩ..., ͑10͒ the radiation decay length of a soliton will be performed where the functions U, A, B, C, and D are assumed to be separately. slowly varying over distances ϳ1/␣. A closed equation for U(y,t) can be obtained by writing CONCLUSIONS out the equations for these functions and using asymptotic expansions in 1/␣. Assuming that ␧Ͻ1/␣ and ␣ӷ1, we ob- In this paper the propagation of a spatial soliton in a tain system of tunneling-coupled waveguides with a periodically varying tunneling-coupling coefficient was investigated. The 2 2 2 ␧ ␣ ␧ problem was studied in the long-wavelength approximation, iu ϩu ϩ2͉u͉2uϭϪ uϪ t yy 16 16 where it reduces to investigation of the propagation of a nonlinear Schro¨dinger soliton with periodic dispersion and po- ϫ 2iu Ϫ6u ϩ24 u 2u͒. ͑11͒ ͑ t yy ͉ ͉ tential perturbation. The adiabatic dynamics of a soliton was Therefore the beam dynamics in a system of tunneling- studied. It was shown that the motion of the soliton center is coupled waveguides is described by a renormalized nonlin- described by the motion of a particle of unit mass in a ear Schro¨dinger equation. The first term on the right-hand velocity-dependent effective potential which is a periodic side of the equation describes the variation of the soliton function of the coordinate of the soliton. The stationary phase and the remaining terms describe the variation of the points for the soliton were found. It was shown that the pre- soliton width. It follows from Eq. ͑11͒ that the soliton width dicted effects can be used to control the parameters of a increases beam in a system of waveguides. 618 Tech. Phys. 43 (6), June 1998 F. Kh. Abdullaev

The effect of small-scale modulations of the tunneling- 5 A. B. Aceves, C. de Angelis, S. Trillo, and S. Wabnitz, Opt. Lett. 19, 332 coupling coefficient on the propagation of a spatial soliton ͑1994͒. 6 was also investigated. The method of averaging for nonlinear R. Muschall, S. Schmidt-Hattenberger, and F. Lederer, Opt. Lett. 19, 323 partial differential equations was used to derive a renormal- ͑1994͒. 7 S. S. Abdullaev and F. Kh. Abdullaev, Izv. Vyssh. Uchebn. Zaved. Ra- ¨ ized nonlinear Schrodinger equation for a beam and to find diofiz. 25, 789 ͑1980͒. the change in the parameters of the soliton. 8 D. Marcuse, Integrated Optics, IEEE Press, New York, 1973; Mir, Mos- The effects due to wave emission by a spatial soliton cow, 1974. were analyzed. The condition of resonance emission of 9 S. Kawakami and H. A. Haus, IEEE J. Lightwave Technol. LT-4, 160 waves by a soliton was found, and the propagation angle for ͑1986͒. 10 W. Krolikowski and Yu. S. Kivshar, Phys. Rev. E 54, ͑1996͒. which the spatial soliton decays resonantly was calculated. 11 V. I. Karpman and E. P. Maslov, Zh. E´ ksp. Teor. Fiz. 73, 537 ͑1977͒ I thank B. A. Umarov and E. N. Tso for helpful discus- ͓Sov. Phys. JETP 46, 281 ͑1977͔͒. 12 D. J. Kaup and A. C. Newell, Proc. R. Soc. London, Ser. A 361, 413 sions. ͑1978͒. 13 Yu. S. Kivshar and S. K. Turitsyn, Phys. Rev. E 49, R2536 ͑1994͒. 1 ´ 14 A. Maer, Kvant. Elektron. ͑Moscow͒ 9, 2296 ͑1982͓͒Sov. J. Quantum F. Kh. Abdullaev, J. G. Caputo, and N. Flytzanis, Phys. Rev. E 52, 1552 Electron. 12, 1490 ͑1982͔͒. ͑1994͒. 2 S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 ͑1982͒. 15 R. Scharf and A. R. Bishop, Phys. Rev. A 43, 6535 ͑1994͒. 3 D. B. Mortimore and J. M. Arkwright, Appl. Opt. 29, 1814 ͑1990͒. 4 D. J. Christodoloudies and R. I. Joseph, Opt. Lett. 13, 794 ͑1989͒. Translated by M. E. Alferieff TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Critical fields and currents in a weakly nonlinear medium near the percolation threshold A. A. Snarski and S. I. Buda National Technical University, 252056 Kiev, Ukraine ͑Submitted June 30, 1997; resubmitted December 19, 1997͒ Zh. Tekh. Fiz. 68, 5–8 ͑June 1998͒ For fields above a critical value the expansion of the conductivity in powers of the field ceases to be valid and the weak-nonlinearity approximation no longer works. The density behavior of the critical fields in strongly inhomogeneous media near the percolation threshold is found on the basis of two criteria—an average criterion and a local criterion. The parameter values of the medium for which crossover—a change of the critical behavior—occurs are determined. Similar calculations are performed for the critical currents. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00206-2͔

Nonlinear phenomena play a special role in the study of the analogy between the critical behavior of ␹e and the rela- the transport properties of strongly inhomogeneous media tive spectral density Ce of 1/f noise, which were first estab- near the percolation threshold pc . This is because near pc lished in Refs. 8 and 9. there exist in a medium locations where the current density As a rule, two limiting cases are studied. Above the per- and voltage drop are substantial, making it necessary to take colation threshold, where the density of the high- into account the deviations from the linear Ohm’s law. To a conductivity phase pϾpc and the medium contains an infi- first approximation there appears in Ohm’s law, besides a nite cluster consisting of a phase with conductivity ␴1 (␴1 term linear in the field, a nonlinear cubic term ӷ␴2), the N/I ͑normal metal–insulator͒ case is studied. The low-conductivity phase is assumed to be an ideal insulator— j͑r͒ϭ␴͑r͒E͑r͒ϩ␹͑r͉͒E͑r͉͒2E͑r͒, ͑1͒ ␴2ϭ0. Below the percolation threshold, pϽpc , the S/N ͑superconductor–normal metal͒ case is studied. In this case where j͑r͒ is the current density, E͑r͒ is the electric field, the system does not contain an infinite cluster, and the cur- ␴͑r͒ the ordinary conductivity, and ␹͑r͒ is a constant char- rent necessarily flows through segments consisting of the acterizing the cubic nonlinearity. low-conductivity phase. It is assumed that the entire voltage Just as in the linear case (␹ϭ0), to describe the effec- drop occurs across these segments, i.e., the high-conductivity tive properties of a randomly inhomogeneous medium one phase is an ideal conductor ␳ ϭ1/␴ ϭ0. introduces effective transport coefficients, which by defini- 1 1 In Refs. 3–7 the following were obtained for these two tion relate the volume averages of the field and current den- cases: sity ͗E͘ ϳ␶Ϫtϩ␯͑dϪ1͒, ͗ j͘ ϳ␶␯͑dϪ1͒, pϾp , N/I, j ϭ␴ E ϩ␹ E 2 E , ͑2͒ c c c ͗ ͘ e͗ ͘ e͉͗ ͉͘ ͗ ͘ ͑5͒ Ϫ1 1/3 where ͗ ...͘ϭV ͐...dV and the size ϳV of the aver- ␯ Ϫqϩ␯ ͗E͘ ϳ͉␶͉ , ͗ j͘ ϳ͉␶͉ , pϽp , S/N, ͑6͒ aging region is assumed to be much larger than the charac- c c c teristic self-averaging length—the correlation length ␰. where ␶ϭ(pϪpc)/pc is the proximity to the percolation One of the main questions that must be answered in threshold, t and q are, respectively, the critical exponents of describing the effective properties of a medium taking into the effective linear conductivity above and below threshold, account the nonlinearity is the question of the values of the ␯ is the critical exponent of the correlation length ␰ Ϫ␯ electric field and current density for which the relation ͑2͒ is Ϸa0͉␶͉ , and a0 is the minimum length in the medium, 1,2 still valid. It is assumed that the relation ͑2͒ remains valid which for a network problem is the bond length in the net- if work. The analogy established in Refs. 8 and 9 between ␹e and the relative spectral density Ce of 1/f noise and the fact ͗ j͘Ͻ͗ j͘c , ͗E͘Ͻ͗E͘c , ͑3͒ that taking the finite ratio hϭ␴1 /␴2 into account may be essential for describing the critical behavior of Ce ͑in con- where ͗E͘c and ͗ j͘c are the so-called critical field and criti- 10–12 cal current density determined from the condition that the trast to ␴e͒ indicate that hϭ␴2 /␴1Þ0 must also be first term in Eq. ͑2͒ equals the first term: taken into account when calculating the critical fields and currents. 3 Models of a percolation structure above and below p ͗E͘cϭͱ␴e /␹e, ͗ j͘cϭͱ␴e /␹e. ͑4͒ c can be used to determine the fields and currents in a strongly Many papers have been devoted to the calculation of the inhomogeneous medium ͑Fig. 1͒.13–15 The principal ele- critical field and current density in percolation media ͑see, ments of this structure are a bridge consisting of a high- for example, Refs. 1–7͒. Some of these works are based on conductivity phase and a low-conductivity interlayer. Taking

␯͑dϪ1͒ tϩ␯ ͗ j͘c1ϭ␴1ͱ␴1 /␹1␶ , ͗ j͘c2ϭ␴2ͱ␴2 /␹2␶ . ͑10͒

For pϾpc the average ﬁelds and current densities for which relations ͑7͒ still hold and Eq. ͑2͒ can be used are the smaller of the two possible values:

͕E͖cϭmin͕͗E͘c1 ; ͗E͘c2͖, ͕j͖cϭmin͕͗ j͘c1 ; ͗ j͘c2͖. ͑11͒ Below the percolation threshold the entire current flows mainly through the bridge and interlayer ͑Fig. 1b͒.Asthe voltage increases, the local criteria for weak nonlinearity ͑7͒ are violated here first. Further calculations are most conveniently performed in terms of the resistivities. Then, to within a cubic term, the relation between the current density and electric field will have the form E͑r͒ϭ␳͑r͒j͑r͒ϩ␮͑r͉͒j͑r͉͒2j͑r͒, ͑12͒ 4 where to this accuracy ␳iϭ1/␴i and ␮iϭϪ␹i /␴i . The local critical current densities are determined by equating the terms in Eq. ͑2͒, i.e., jciϭͱ␳i /␮i. Knowing the dϪ1 cross-sectional area of the bridge a0 and the interlayer FIG. 1. Hierarchical model of a percolation structure with a finite ratio adϪ1 ϪqϪ␯(dϪ2), we find the average critical current den- hϭ␴ /␴ . 1—Bridge, N high-conductivity resistances connected in se- 0 ͉␶͉ 2 1 1 sities from the condition that the current j dϪ1 through the ries; 2—interlayer, N2 low-conductivity resistances connected in parallel. ͗ ͘␰ Ϫtϩ␯(dϪ2) ϪqϪ␯(dϪ2) According to Refs. 13 and 14, N1ϳ͉␶͉ , N2ϳ͉␶͉ . entire medium equals the critical currents for the bridge and dϪ1 dϪ1 ϪqϪ␯(dϪ2) interlayer ( jc1a0 , jc2a0 ͉␶͉ ): ␯͑dϪ1͒ into account the analogy between the weak nonlinearity and ͗j͘c1ϭ␴1ͱ␴1 /␹1͉␶͉ , 1/f noise (␹ and C ͒, we shall confine our attention to the e e j ϭ␴ ͱ␴ /␹ ␶ Ϫqϩ␯. ͑13͒ two structural elements indicated above, since these elements ͗ ͘c2 2 2 2͉ ͉ 14 give the main terms in Ce and ␹e for hÞ0. The N/I and From the condition that the total voltage drop across the S/N approximations correspond to taking into account only correlation length equals the sum of the voltages across the one element of the structure: above pc only the bridge, and bridge and interlayer we obtain the critical average fields below p only the interlayers. c ␴ ␴ ␴ We now require that besides the condition ͑3͒ a similar 1 1 qϩ␯͑dϪ1͒ 2 ␯ ͗E͘c1ϭ ͱ ͉␶͉ , ͗E͘c2ϭͱ ͉␶͉ . condition hold locally, i.e., that at each point of each phase ␴2 ␹1 ␹2 the local field and current density be less than the local criti- ͑14͒ cal values Both above and below the percolation threshold the critical current density and field are the smaller of the expres- Ei͑r͒ϽEci , ji͑r͒Ͻjci , iϭ1, 2, ͑7͒ sions in Eqs. ͑13͒ and ͑14͒͑see Eq. ͑11͒͒. where for each phase the critical local field and current den- Depending on the values of the parameters ␶, h sity are determined from a local law ͑1͒: ϭ␴2 /␴1 , and Hϭ␹2 /␹1 the minimum values in Eq. ͑11͒ E ϭͱ␴ /␹ , j ϭͱ␴3/␹ , iϭ1, 2. ͑8͒ can be the first or the second values of the field and current ci i i ci i i density. Equating the expressions for the fields in ͑9͒ and In contrast to the conditions ͑3͒, which can be called an current densities in Eq. ͑10͒, for the case pϾpc we obtain tϪ␯(dϪ2) average criterion, the conditions ͑8͒ can be called a local ͱ(␴2␹1)/(␴1␹2)␶ ϭ1, or, equivalently, criterion. Of course, there is no guarantee that the law ͑8͒ 1/2͑tϪ␯͑dϪ2͒͒ holds just because the condition ͑3͒ is satisfied. This means ␶0ϭ͑H/h͒ , hϭ␴2 /␴1 , Hϭ␹2 /␹1 . that in some cases the relations ͑5͒ and ͑6͒ should be re- ͑15͒ placed by different relations. Crossover—a change in the critical behavior of the criti- As one can see from Fig. 1a, the voltage across the cal field current density—occurs when ␶ passes through the tϩ␯ bridge equals the external voltage. Assuming that the electric value ␶0 . The critical current density depends on ␶ as ϳ␶ ␯(dϪ1) field on the bridge equals the critical field Ec1 and expressing for ␶Ͻ␶0 and ϳ␶ for ␶Ͼ␶0 . The critical field also ␯ the external voltage drop in terms of the average field, we undergoes crossover: For ␶Ͻ␶0 it is ϳ␶ and for ␶Ͼ␶0 it is find the average field corresponding to the local criterion ͑7͒ proportional to ␶Ϫtϩ␯(dϪ2). of weak nonlinearity to be ͗E͘c1ϭ(N1a0 /␰)Ec1 . Similar ar- The region 1 indicated in Fig. 2a corresponds to the case guments for the interlayer give ͗E͘c2ϭ(a0 /␰)Ec2 or when ͕E͖cϭ͗E͘c2 from Eq. ͑9͒ and ͕j͖cϭ͗ j͘c from Ϫtϩ␯͑dϪ1͒ ␯ Eq. ͑10͒. The region 2 corresponds to the situations when ͗E͘c1ϭͱ␴1 /␹1␶ , ͗E͘c2ϭͱ␴2 /␹2␶ . ͑9͒ ͕E͖cϭ͗E͘c1 , ͕j͖cϭ͗ j͘c1 , i.e., the critical exponents of the The critical current densities for the fields ͑9͒ are current density and electric field are identical to Eq. ͑5͒. Tech. Phys. 43 (6), June 1998 A. A. Snarski and S. I. Buda 621

FIG. 2. Surfaces bounding the regions of the constants ratios hϭ␴2 /␴1 and Hϭ␹2 /␹1 for a weakly nonlinear medium where the results of the standard approach3–7 are valid (N/I and S/N cases ͑5͒ and ͑6͒͒ and the critical current densities and ﬁelds obtained on the basis of a local criterion. Region 2—standard approach, 1—results of this work. The hatched region of values of ␶ corresponds to the region of broadening ⌬. The quantity ⌬ is determined on the basis of the analogy between the effective properties of the weakly nonlinear medium and the effective 1/f noise,10 ⌬ϭh1/(tϩq).

For pϽp the crossover condition for the critical current 1 R. Blumenfeld and D. J. Bergman, Phys. Rev. B 43, 13 682 ͑1991͒. c 2 density and electric ﬁeld can be found similarly to Eq. ͑15͒ P. M. Hui, Phys. Rev. B 49, 15 344 ͑1994͒. 3 K. W. Yu and P. M. Hui, Phys. Rev. B 50, 13 327 ͑1994͒. and has the form 4 P. M. Hui, Phys. Rev. B 49, 15 344 ͑1994͒. 5 K. H. Chung and P. M. Hui, J. Phys. Soc. Jpn. 63, 2002 ͑1994͒. 3 1/2͑qϩ␯͑dϪ2͒͒ 6 ͉␶0͉ϭ͑h /H͒ . ͑16͒ J. J. Lin, J. Phys. Soc. Jpn. 61, 4125 ͑1994͒. 7 G. M. Zhang, J. Phys.: Condens. Matter 8, 6933 ͑1996͒. 8 D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 ͑1988͒. The surface bounding the regions of ␴i and ␹i where the 9 A. Aharony, Phys. Rev. Lett. 58, 2726 ͑1987͒. results of the average criterion ͑6͒ are valid is shown in Fig. 10 A. A. Snarskii, A. E. Morozovsky, A. Kolek, and A. Kusy, Phys. Rev. E 2b. For the region 1 the results of the standard approach are 53, 5596 ͑1996͒. 11 ´ valid, i.e., Eq. ͑6͒. The region 2 corresponds to the case when A. E. Morozovski and A. A. Snarski, Zh. Eksp. Teor. Fiz. 102, 683 ͑1992͓͒Sov. Phys. JETP 75, 366 ͑1992͔͒. ͕j͖ ϭ͗ j͘ from Eq. ͑13͒ and ͕E͖ ϭ͗E͘ from Eq. ͑14͒. 12 ´ c c1 c c1 A. E. Morozovski and A. A. Snarski, Zh. Eksp. Teor. Fiz. 95, 1844 ´ ͑1992͓͒Sov. Phys. JETP 68, 1066 ͑1989͔͒. We thank E. M. Baskin for a discussion of the questions 13 A.-M. S. Tremblay, B. Fourcade, and P. Brenton, Physica A 157,89 touched upon here. ͑1989͒. 14 A. Kolek, Int. J. Elec. 73, 1095 ͑1992͒. This work was supported in part by the Russian Fund for 15 Fundamental Research under contracts Nos. 95-02-04432a A. Kolek, Phys. Rev. B 45, 205 ͑1992͒. and 97-02-16923a. Translated by M. E. Alferieff TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Multipole solutions of the wave equation V. V. Zashkvara† and N. N. Tyndyk

Physicotechnical Institute, Kazakhstan Academy of Sciences, 480082 Alma-Ata, Kazakhstan ͑Submitted March 22, 1996͒ Zh. Tekh. Fiz. 68, 9–14 ͑June 1998͒ The wave equation is solved by the operator separation method proposed in V. V. Zashkvara and N. N. Tyndyk, Zh. Tekh. Fiz. 61͑4͒, 148 ͑1991͓͒Sov. Phys. Tech. Phys. 36, 456 ͑1991͔͒. Solutions describing the evolution of circular-multipole ﬁelds are obtained in a cylindrical coordinate system. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00306-7͔

A new approach to solving Laplace’s equation by sepa- 1 ␸ ͑␰͒ϭ ␰2nϩ1, ration of variables has been proposed and substantiated in n ͑2nϩ1͒! Refs. 1–3. In this approach a solution is constructed in the form of a sum of paired products of functions of one variable nϭ0, 1, 2, ..., ͑NϪ1͒/2, if N is odd. ͑4͒ that satisfy chains of second-order ordinary differential equations and zero boundary conditions on a circle. The differen- The functions Fi(R) and ␸i(␰) and their derivatives at tial operators of these equations are coordinate-separated the point Rϭ1, ␰ϭ0 satisfy the boundary conditions parts of the Laplacian. A class of circular multipoles in cy- dFi lindrical and spherical coordinate systems has been obtained F ͑1͒ϭ ϭ0, iϭ0, 1, 2, ... i dR ͯ by the operator separation method.4 Rϭ1 In Ref. 5 a new approach is used to solve Poisson’s d␸i equation. It is shown that if the right-hand side of Poisson’s ␸ ͑0͒ϭ ϭ0, iϭ1, 2, ... . ͑5͒ i d␰ ͯ equation in a cylindrical system with dimensionless coordi- ␰ϭ0 nates R/r0 , ␰ϭz/r0 (r0 is the radius of an axial circle͒ is In Ref. 5 it was shown that for different values of n the represented by the function solutions Vn(R,␰) of Poisson’s equation possess the struc- ͑R,␰͒ϭ␰N␻͑R͒, ͑1͒ ture of modiﬁed circular multipoles, which were termed non- F Laplacian. In what follows we shall call f i(R) and Fi(R), where N is an integer and ␻(R) is a smooth function of R, respectively, the radial functions of Laplacian and non- then a solution of Poisson’s equation is given by the sum Laplacian circular multipoles.

n Our objective in the present paper is to use the operator separation method to solve the wave equation in a cylindrical Vn͑R,␰͒ϭ ␸m͑␰͒•FnϪm͑R͒. ͑2͒ m͚ϭ0 coordinate system, to obtain particular solutions on the basis of the multipole approach, and to determine the character of In this sum nϭN/2 if N is even and nϭ(NϪ1)/2 if N is the temporal processes described by these solutions. odd. The set of radial functions Fi(R) satisﬁes a chain of n The wave equation in dimensionless cylindrical coordi- differential equations nates R, ␰ and dimensionless time tϭct1 /r0 (t1 is the di- TF ϭN!␻͑R͒, mensional time and c is the propagation velocity of a distur- 0 bance in a uniform medium͒ is TF ϭϪF , 1 0 ͑⌬Ϫ␹͒U͑R, ␰, t͒ϭ0, ͑6͒ -t2 is the temporal opץ/2ץwhere ⌬ is the Laplacian and ␹ϭ ,...... erator. TF ϭϪF , ͑3͒ n nϪ1 According to the operator separation method,1,2 the so- where lution of Eq. ͑6͒ will be a sum of paired products of the coordinates and temporal functions ␯i(R, ␰) and ⌽i(t) 1 d d Tϭ R n R dR ͫ dRͬ Un͑R, ␰, t͒ϭ ␯s͑R, ␰͒•⌽nϪs͑t͒. ͑7͒ s͚ϭ0 is a second-order differential operator—the radial part of the Laplacian. If these functions satisfy the chains of equations The set of axial functions ␸i(␰)is ⌬␯0ϭ0, ⌬␯1ϭ␯0 , 1 ␸͑␰͒ϭ ␰2n, ⌬␯ ϭ␯ , ...... , n ͑2n͒! 2 1

nϭ0, 1, 2, ..., N/2, if N is even, ⌬␯nϭ␯nϪ1 , ͑8͒

␹⌽0ϭ0, Let us substitute expression ͑15͒ into Eq. ͑14͒. Perform- ing some manipulations, we arrive at a solution in a form ␹⌽1ϭ⌽0 , that facilitates further analysis of our problem: N n N N ␹⌽2ϭ⌽1 , ͑n͒ ͑n͒ V͑R,␰͒ϭ ␸mFnϪmϭ ␸m FnϪm ...... , n͚ϭ0 m͚ϭ0 m͚ϭ0 n͚ϭm

N NϪm ␹⌽nϭ⌽nϪ1 . ͑9͒ ͑mϩs͒ ϭ ͚ ␸m ͚ Fs . ͑16͒ Indeed, operating on the sum ͑7͒ with the operator mϭ0 sϭ0 ͓⌬Ϫ␹͔, we obtain Two successive operations lead to Eq. ͑16͒: switching n the order of m and n summations and then replacing the ͑⌬Ϫ␹͒ ͚ ␯s͑R, ␰͒⌽nϪs͑t͒ summation over n by summation over sϭnϪm. The radial sϭ0 (m) functions Fs form a triangular matrix n n s 012345.N ϭ ͑⌬␯s͒⌽nϪsϪ ␯s͑␹⌽nϪs͒ → s͚ϭ0 s͚ϭ0 m ↓ ͑0͒ n nϪ1 0F0, 1F͑1͒F͑1͒, ϭ ͚ ␯sϪ1⌽nϪsϪ ͚ ␯s⌽nϪsϪ1 . ͑10͒ 0 1 sϭ1 sϭ0 ͑2͒ ͑2͒ ͑2͒ 2F0 F1 F2, ͑3͒ ͑3͒ ͑3͒ ͑3͒ Let us now shift by 1 the summation index in the first 3F0 F1 F2 F3, sum sϪ1ϭk, ␯sϪ1ϭ␯k , ⌽nϪsϭ⌽nϪkϪ1 . Then ͑4͒ ͑4͒ ͑4͒ ͑4͒ ͑4͒ 4F0 F1 F2 F3 F4, n nϪ1 ͑5͒ ͑5͒ ͑5͒ ͑5͒ ͑5͒ ͑5͒ 5F0 F1 F2 F3 F4 F5, ␯sϪ1⌽nϪsϭ ␯k⌽nϪkϪ1 . s͚ϭ1 k͚ϭ0 ...... N F͑N͒ F͑N͒ F͑N͒ F͑N͒ .. .. F͑N͒ . Thus, the sums in Eq. ͑10͒ are equal and Eq. ͑6͒ is sat- 0 1 2 3 N (m) isfied. The solution of the system of equations ͑9͒ is given by The functions Fg , which together form the solution the functions ͑15͒ and correspond to individual terms of the polynomial ͑13͒, stand in the rows of the matrix (mϭconst). The solu- 1 2n tion of Poisson’s equation with the right-hand side in the ⌽n͑t͒ϭ t , nϭ0, 1, 2, ..., ͑11͒ ͑2n͒! form of the polynomial ͑13͒, according to Eq. ͑16͒, consists of paired products of ␸m times the sum of the functions t͉tϭ0ϭ0, or the mϩsץ/ ⌽nץ if ⌽0(0)ϭ1, ⌽n(0)ϭ0(nÞ0), and functions Fs . These functions lie along the diagonals of the matrix. The end point of the diagonal along which the index 1 sϭ0,1,2,..., NϪm varies is determined by specifying ⌽ ϭ t2nϩ1, nϭ0, 1, 2, ..., ͑12͒ n ͑2nϩ1͒! the value of m. The set of functions forming a diagonal is ͑m͒ ͑mϩ1͒ ͑mϩ2͒ ͑N͒ t͉tϭ0ϭ0(nÞ0). F0 , F1 , F2 , ...FNϪm . ͑17͒ץ/ ⌽nץ t͉tϭ0ϭ1, andץ/ ⌽0ץ ,if ⌽n(0)ϭ0 According to Eq. ͑8͒, to find the coordinate functions Remaining within the multipole approach, we shall solve ␯i(R, ␰) it is necessary to solve a chain of partial differential equations, consisting of Laplace’s equation for the functions the system of equations ͑8͒, choosing as ␯0(R, ␰) a circular multipole of order 2N, which is a harmonic polynomial in ␰ ␯0 and Poisson’s equations for the remaining functions ␯i . Before attempting to solve this problem we shall supplement ͑Refs. 1 and 2͒. Then the results of Ref. 5 and find the solution of Poisson’s equa- 1 ␻͑n͒͑R͒ϭ f ͑R͒, nϭ0, 1, ..., N, ͑18͒ tion in the case when the right-hand side of this equation is a ͑2n͒! NϪn polynomial in ␰ with arbitrary R-dependent coefficients. Let this be an even polynomial of degree 2N where f NϪn are radial functions of the first or second kind, studied in Ref. 4. N We shall show that in this case the functions F(mϩs) ͑R, ␰͒ϭ ␻͑n͒͑R͒␰2n. ͑13͒ s F n͚ϭ0 standing on the same diagonal of the matrix are equal, and the solution of the chain of equations ͑8͒ can be easily found. Then the solution of Poisson’s equation will be given by the Let us operate with the operator T on the function ͑17͒, tak- sum ing into account the differential equation ͑3͒ for Fi and the N system of differential equations satisfied by the radial func- ͑n͒ V͑R, ␰͒ϭ V ͑R, ␰͒. ͑14͒ tions f i ͑Refs. 1 and 2͒: n͚ϭ0 Tf ϭϪf , iϭ0, 1, 2, ..., n. ͑19͒ In accordance with Eq. ͑2͒ iϩ1 i n As an example, let us find ␯1(R, ␰). According to Eqs. (m) (m) ͑n͒ ͑n͒ ͑3͒ and ͑18͒, TF0 ϭ␻ (R)ϭfNϪm . Using Eq. ͑19͒,we V ͑R, ␰͒ϭ ͚ ␸mFnϪm . ͑15͒ mϭ0 obtain from this relation 624 Tech. Phys. 43 (6), June 1998 V. V. Zashkvara and N. N. Tyndyk

F͑m͒ϭϪf . ͑20͒ 1 N 0 NϪmϩ1 2s NϪm ␯s*͑R,␰͒ϳ ␳ ͑Ϫ1͒ (mϩ1) (mϩ1) s! m͚ϭ0 According to Eqs. ͑3͒ and ͑20͒, TF1 ϭϪF0 ϭf , and we have on the basis of ͑19͒ NϪm NϪmϩs ! ͑ ͒ m NϪm 2 ͑mϩ1͒ ϫ ͓␰ ␳ ͔ . ͑28͒ F1 ϭϪfNϪmϩ1 . ͑21͒ ͑NϪm͒!͑2m͒! (mϩ2) (mϩ2) According to Eqs. ͑3͒ and ͑21͒, TF2 ϭϪF0 A characteristic feature here is that the sum is a homo- ϭfNϪm . On the basis of Eq. ͑19͒ we have geneous polynomial of degree 2N in ␳ and ␰ containing sign- F͑mϩ2͒ϭϪf ͑22͒ alternating coefficients. This polynomial by itself describes a 2 NϪmϩ1 multipole structure with nodal point Rϭ1, ␰ϭ0, where the (mϩs) 2s and so on. Indeed, all NϪmϩ1 diagonal elements Fs 2N zero equipotential lines converge. The factor ␳ in front are equal and have the value Ϫ f NϪmϩ1 ; there are NϪm of the sum contributes to the structure of the field an addi- ϩ1 of them. Returning to formula ͑16͒, we conclude that if tional zero equipotential lying on the ␰ axis, which separates the right-hand side of Poisson’s equation is represented by a the single-potential region into two symmetric parts, thereby circular multipole ␯0(R, ␰) of order 2N, then the solution of increasing the multipole order by 2. In Ref. 5 such structures this equation ␯1(R, ␰) is a polynomial of the same order in ␰ were called incomplete non-Laplacian circular multipoles. and has the form We shall also determine the spatiotemporal structure of the N field ͑25͒ on a surface of section by the plane ␰ϭ0 near R ϭ1. In accordance with Eq. ͑4͒, we have for even functions ␯1͑R, ␰͒ϭϪ ͑NϪmϩ1͒␸m͑␰͒fNϪmϩ1͑R͒. ͑23͒ m͚ϭ0 ␸n(␰) Generalizing Eq. ͑23͒, one can show that the solution of 1, if mϭ0, an arbitrary link ⌬␯sϭ␯sϪ1 in the chain of differential equa- ␸m͑0͒ϭ ͑29͒ tions ͑8͒ is ͭ 0, if mϭ1, 2, . . . . N According to Eq. ͑25͒ we have ͑Ϫ1͒s ͑NϪmϩs͒! ␯s͑R,␰͒ϭ ␸m͑␰͒f NϪmϩs͑R͒. n s! m͚ϭ0 ͑NϪm͒! s ͑24͒ Un͑R,0,t͒ϭ ͑Ϫ1͒ fs͑R͒⌽nϪs͑t͒. ͑30͒ s͚ϭ0 Substituting expression ͑24͒ into Eq. ͑7͒, we write the solution of the wave equation ͑6͒ in the form As an example, let us consider even functions for the choice of ⌽n(t). Then n ͑Ϫ1͒s 1 Un͑R, ␰, t͒ϭ ⌽nϪs͑t͒ 2͑nϪs͒ s͚ϭ0 s! ⌽ ͑t͒ϭ t . ͑31͒ nϪs ͓2͑nϪs͔͒! N ͑NϪmϩs͒! Substituting expressions ͑27͒ and ͑31͒ into Eq. ͑30͒,we ϫ ␸m͑␰͒f NϪmϩs͑R͒. m͚ϭ0 ͑NϪm͒! arrive at the following expression for estimating the spatiotemporal structure of the field in the section ␰ϭ0 ͑25͒ n 1 The spatiotemporal structure of the solution ͑7͒ or ͑25͒ 2s 2͑nϪs͒ Un͑R,0,t͒ϭ ␳ t . ͑32͒ near the axial circle Rϭ1, ␰ϭ0 is determined by the leading ͚s ͓2͑nϪs͔͒! terms of the functions ͑24͒. To find them it is necessary to It follows from Eq. 32 that U (R,0,t) is a homoge- separate out the leading terms of the radial functions f i ap- ͑ ͒ n pearing in Eq. ͑24͒. It turns out that in consequence of the neous polynomial in R and t of degree 2n whose coefficients chain structure of the system of differential equations,1,2 all have the same sign. Near the point Rϭ1, tϭ0 the equipotentials U (R,0,t) of the wave field form a system of which f i satisfy with zero boundary conditions, we have n closed curves encompassing this point. The field Un(R,0,t) f / f ϳ␳2 as ␳ 0 ͑␳ϭRϪ1͒. ͑26͒ iϩ1 i → is not a multipole field. But after one makes the substitution It follows from this relation that the leading terms f i tϭi␶ and transforms to imaginary ␶ in Eq. ͑32͒, the field in equal the coordinates R,␶ becomes a circular multipole with a i 2iϩ1 node at Rϭ1, ␰ϭ0, ␶ϭ0. f iϳ͑Ϫ1͒ ␳ Thus the spatial–wave structure of the wave field ͑7͒ in for the radial functions of the first kind and all space is determined by a time-dependent polynomial i 2i whose coefficients ␯s(R, ␰) are circular multipoles. Initially f iϳ͑Ϫ1͒ ␳ (tϭ0) the structure of the wave field Un(R, ␰, t) will be for the radial functions for the second kind, represented by a non-Laplacian circular multipole ␯n(R, ␰) (nÞ0), and in the limit t ϱ it will be represented by a iϭ0, 1, ... . ͑27͒ → Laplacian circular multipole ␯0(R, ␰). We shall confine our attention to the radial functions of As an example, let us examine the structure of the mul- the second kind. Then the leading terms in ␯s(R, ␰) ͑24͒ can tipole solution of the wave equation ͑7͒ for the case Nϭ1. be represented by the formula The Laplacian circular multipole is Tech. Phys. 43 (6), June 1998 V. V. Zashkvara and N. N. Tyndyk 625

FIG. 1. FIG. 3.

␯0͑R, ␰͒ϭ f 1␸0ϩ f 0␸1 . ͑33͒ and so on. For even functions ⌽n(t) we have, according to Eq. ͑7͒, According to Eq. ͑24͒, the non-Laplacian circular multipoles can be calculated from the formula nϭ0, U0͑r, ␰, t͒ϭ␯0⌽0ϭ f 1␸0ϩ f 0␸1 , ͑36͒ 1 ͑Ϫ1͒s ͑sϩ1Ϫm͒! 1 ␯s͑R, ␰͒ϭ ͚ ␸m͑␰͒f sϩ1Ϫm͑R͒; nϭ1, U1͑R, ␰, t͒ϭ ␯s⌽1Ϫs s! mϭ0 ͑1Ϫm͒! s͚ϭ0 ͑34͒ 1 sϭ1, ␯ ϭϪ͓2␸ f ϩ␸ f ͔, ϭ ͑f ␸ ϩf ␸ ͒t2Ϫ͓2␸ f ϩ␸ f ͔, ͑37͒ 1 0 2 1 1 2! 1 0 0 1 0 2 1 1 sϭ2, ␯ ϭ3␸ f ϩ␸ f , 2 0 3 1 2 2 1 4 ϭ ϭϪ ϩ nϭ2, U2͑R, ␰, t͒ϭ ␯s⌽2Ϫsϭ ͑f1␸0ϩf0␸1͒t s 3, ␯3 ͓4␸0f4 ␸1f3͔ ͑35͒ s͚ϭ0 4!

FIG. 2. FIG. 4. 626 Tech. Phys. 43 (6), June 1998 V. V. Zashkvara and N. N. Tyndyk

FIG. 5. FIG. 6.

shots’’ of different times in the development of the spa- 1 2 tiotemporal process described by the wave multipole Ϫ ͓2␸0f2ϩ␸1f1͔t ϩ3␸0f3ϩ␸1f2 ͑38͒ 2! U1(R, ␰, t): tϭ0, 0.3, 0.5, 0.7, and 1.2, respectively. For t ϭ0 the wave multipole is an incomplete non-Laplacian cir- and so on. 4 cular sextupole ͑Fig. 1͒. Even for small t the contribution of To calculate U1 the radial functions of the second kind 1 2 2 4 the quadrupole (2 ln Rϩ1ϪR )ϩ␰ eliminates the zero were chosen for f (R) and the even functions of ␰ were 2 i equipotential Rϭ1 and destroys the non-Laplacian sextu- chosen for ␸ (␰): i pole. It is evident from these ﬁgures that in time the devel- 1 opment of this process leads to the formation of a structure f ͑R͒ϭ1, f ͑R͒ϭ ͓2lnRϩ1ϪR2͔, 0 1 4 close to a circular Laplacian quadrupole ͑Figs. 5, 6͒. The process unfolds similarly in the other cases which 1 2 2 4 we examined. It can be concluded on this basis that the op- f ͑R͒ϭ ͓Ϫ͑4ϩ8R ͒ln RϪ5ϩ4R ϩR ͔. ͑39͒ 2 64 erator separation method makes it possible to obtain a multipole solution of the wave equation that describes a ﬁeld 1 ␸ ͑␰͒ϭ1, ␸ ͑␰͒ϭ ␰2. ͑40͒ evolving from a more complicated to a simpler structure. 0 1 2 Then, according to Eq. ͑37͒, †Deceased. 1 1 2 2 2 U1͑R, ␰, t͒ϭ ͑2lnRϩ1ϪR ͒ϩ␰ t 4 ͫ2 ͬ 1 V. V. Zashkvara and N. N. Tyndyk, Zh. Tekh. Fiz. 61͑4͒, 148 ͑1991͒ ͓Sov. Phys. Tech. Phys. 36, 456 ͑1991͔͒. 1 2 V. V. Zashkvara and N. N. Tyndyk, Nucl. Instrum. Methods A 313, 315 Ϫ ͓Ϫ͑4ϩ8R2͒ln 5ϩ4R2ϩR4͔ 32 ͑1992͒. 3 V. V. Zashkvara and N. N. Tyndyk, Nucl. Instrum Methods A 321, 339 1 ͑1992͒. Ϫ ␰2͓2lnRϩ1ϪR2͔. ͑41͒ 4 V. V. Zashkvara, Nucl. Instrum. Methods 354, 171 ͑1995͒. 8 5 V. V. Zashkvara and N. N. Tyndyk, Zh. Tekh. Fiz. 65͑7͒, 154 ͑1995͒ ͓Tech. Phys. 40, 717 ͑1995͔͒. The computational results obtained for U1(R, ␰, t) using Eq. ͑41͒ are presented in Figs. 1–6, which show ‘‘snap- Translated by M. E. Alferieff TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Doppler spectroscopy of a beam in the channel of a H؊ ion source V. V. Antsiferov and G. I. Smirnov

Institute of Nuclear Physics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia ͑Submitted May 21, 1997͒ Zh. Tekh. Fiz. 68, 15–18 ͑June 1998͒ It is shown that high-accuracy contact-free measurements of the divergence and emittance of an accelerated HϪ ion beam at the exit from the source can in principle be performed by passive Doppler spectroscopy of a beam of excited hydrogen atoms produced by neutralization of the ions with excitation on the residual gas in the source channel. The intensity of the H␣-line radiation detected by the Doppler system is calculated, taking into account the principal processes leading to the excitation and deexcitation of the 3s,3p, and 3d levels of the hydrogen atoms in the beam, for residual gas densities of the order of 10Ϫ4 –10Ϫ5 Torr in the source channel. The computed H␣-line intensity was conﬁrmed experimentally, making it possible to perform photoelectronic detection of the spectral contour of the line in the current mode rather than the photon-counting mode. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00406-1͔

INTRODUCTION N͑H␣͒ϭN3sϩN3pϩN3d , ͑1͒ Monitoring of the parameters of a HϪ ion beam during where N is the number of photons emitted in the transitions continuous operation of the source is an extremely pressing m 3s 2p,3p 2s, and 3d 2p, respectively, by hydrogen problem. Contact sensors1–3 do not solve this problem be- atoms→ in the→ beam and is determined→ by the number of spon- cause they introduce perturbations into the beam and in- taneous decays occurring over the travel time t of excited crease the beam divergence, and it is virtually impossible to r hydrogen atoms in the detection section which are present in use contact sensors for continuous monitoring of the diver- the entire volume of the beam pulse and enter the detection gence of an ion beam during the operation of the source. The system of the Doppler recording system with angular aper- only method permitting such nonperturbative monitoring is ture ⌬␸ Doppler spectroscopy, which has been used in the past for A monitoring the divergence of high-intensity beams of atoms t D⌬␸AVAm r and negative ions of hydrogen in the channel of an Nmϭ nm͑x͒dt. ͑2͒ accelerator4,5 and for measuring the temperature of hydrogen 4␲f ͵0 atoms in the plasma discharge of surface-plasma sources of Here D and f are, respectively, the diameter and focal length HϪ ions.6,7 of the converging lens L ; A are the probabilities of In the present work we determined the possibilities of 1 m the corresponding spontaneous transitions—A ϭ0.63 performing high-accuracy contact-free monitoring of the di- 3s ϫ107 sϪ1, A ϭ2.2ϫ107 sϪ1, and A ϭ6.4ϫ107 sϪ1; V vergence and emittance of an accelerated beam of negative 3p 3d ϭ d2c /4 is the volume of a beam pulse of duration hydrogen ions in the source channel by passive Doppler ␲ ␶␤ ␶ with d being the beam diameter and ϭ /c, where is the spectroscopy without using a charge-transfer target. ␤ v v velocity of the ions in the beam; nm(x) is the density of excited hydrogen atoms at the detection point in the states H -LINE INTENSITY OF AN ACCELERATED PARTICLE ␣ 3s,3p, and 3d BEAM AT THE SOURCE EXIT 0 An organically integral component of the ion source of a nm͑x͒ϭnm͑x͒exp͑Amt͒. ͑3͒ modern accelerator is the turning magnet ͑see Fig. 1͒, which 0 separates particles according to the transverse velocities of The densities nm(x) of the excited hydrogen atoms after the beam extracted from the source and forms an ion beam the particles have passed through a section of length x in the with a ﬁxed emittance. Transport of the HϪ ion beam along source can be found from an equation that takes into account Ϫ 0 the channel of the source magnet is accompanied by neutral- the excitation of H ions and H atoms in collisions with the ization of the ions and excitation on the residual gas par- residual gas as well as radiative decay of the excited levels and collisional relaxation on the residual gas ticles. The number N(H␣) of photons spontaneously emitted in the H line by excited atoms in the hydrogen beam during ␣ 0 the time t that the beam atoms travel over the rectilinear dnm͑x͒ r ϭn ͓n0͑x͒␴ ϩnϪ␴ ͔ detection section at the exit from the magnet is dx g 0,m Ϫ1,m

dn0 ͑x͒ n0 x m ͑ ͒ Ϫ ϩ ϭng͓B␴0,mn ͑0͓͒exp͑Ϫx/␭2͒ dx ␭D Ϫ Ϫexp͑Ϫx/␭1͔͒ϩn ͑0͒␴Ϫ1,m

ϫexp͑Ϫx/␭1͔͒. ͑7͒ 0 Solving this equation with the initial conditions nm(0) ϭ0, we obtain

0 Ϫ Ϫ1 Ϫ1 Ϫ1 nm͑x͒ϭngn ͑0͓͒B␴0,m͑␭D Ϫ␭2 ͒ ͓exp͑Ϫx/␭2͒ Ϫ1 Ϫ1 Ϫ1 Ϫexp͑Ϫx/␭D͔͒ϩ͑␴Ϫ1,mϪB␴0,m͒͑␭D Ϫ␭1 ͒

ϫ͓exp͑Ϫx/␭1͒Ϫexp͑Ϫx/␭D͔͒. ͑8͒ 0 We shall calculate the density nm(x) of excited hydrogen atoms for two values of the residual gas density ng in the source for a HϪ ion source with the optimal parameters: ion-beam energy 40 keV, pulse current 0.2 A, pulse duration Ϫ4 0.1 ms, beam diameter 2 cm, beam emittance 10 rad•cm, and pulse repetition frequency 100 Hz. Then vϭ2.8 ϫ108 cm/s, nϪ(0)ϭ2ϫ109 cmϪ3, and the parameters ␤ are Ϫ Ϫ 3 FIG. 1. Doppler system for detecting a H ion beam in the source channel. ␤ʈϭv/cϭ7ϫ10 for the longitudinal component of the Ϫ Ϫ4 S—H ion source, M—turning magnet, O—optical window, beam velocity and ␤ЌϭvЌ /cϭ1ϫ10 for the transverse L1—converging lens, S p—detector slit, FOC—fiber-optic cable, component. L2—focusing lens of the spectral unit, MC—monochromator, SI—scanning 1. Let the residual gas pressure in the source be Pϭ5 interferometer, PM—photomultiplier. Ϫ5 12 3 ϫ10 Torr. Then ngϭ1.6ϫ10 molecules/cm . For 40 Ϫ 8 keV H ions the charge-transfer cross sections are ␴Ϫ1,0 Ϫ16 2 Ϫ17 2 ϭ8.7ϫ10 cm , ␴Ϫ1,1ϭ4ϫ10 cm , and ␴0,1ϭ1.43 0 Ϫ16 2 3s 3p 3d Ϫ15 2 n x A ϫ10 cm . Then ␴DCϭ␴DCϭ␴DCϭ1.6ϫ10 cm . m͑ ͒ m 0 Ϫ Ϫngnm͑x͒␴DC , ͑4͒ The cross sections for neutralization with excitation v 9 Ϫ18 2 are ␴Ϫ1,3sϭ2.1ϫ10 cm and ␴Ϫ1,3pϩ␴Ϫ1,3dϭ1.5 ϫ10Ϫ18 cm2. For a section of length xϭ20 cm in the source where ng is the density of residual gas particles in the source Ϫ 0 channel where charge transfer from H ions occurs the other channel, n (x) is the density of hydrogen atoms in the 1s 3 Ϫ Ϫ computational parameters are ␭1ϭ0.7ϫ10 cm, ␭2ϭ4.4 ground state in the beam, n (x) is the density of H ions in 3 3 3s 3p ϫ10 cm, ␭DCϭ0.4ϫ10 cm, ␭DRϭ44 cm, ␭DRϭ12 cm, the beam, ␴0,m is the cross section for the excitation of hy- 3d drogen atoms into the state m in collisions with the residual and ␭DRϭ4.3 cm. Then Eq. ͑8͒ simplifies substantially: gas, ␴ is the cross section for charge transfer from HϪ 0 Ϫ m m Ϫ1,m nm͑x͒ϭngn ͑0͓͒␴Ϫ1,m␭DR͓1Ϫexp͑Ϫx/␭DR͔͔͒. ͑9͒ ions accompanied by the excitation of hydrogen atoms into the state m in collisions with the residual gas, v is the lon- Substituting expression ͑9͒ into Eq. ͑4͒, integrating and Ϫ gitudinal velocity of the accelerated beam of H ions, ␴DC substituting into Eqs. ͑2͒ and ͑1͒, we find for the number ϭ␴m,0ϩ␴m,ϩ1ϩ␴m,Ϫ1 is the total cross section for the col- N(H␣) of photons emitted in the H␣ line lisional decay of the excited levels m. DV⌬␸ Since ␴ Ӷ␴ , the H0 atom and HϪ ion densities in A 0 Ϫ1,1 Ϫ1,0 N͑H␣͒ϭ ͚ nm͑x͓͒1Ϫexp͑ϪAmtr͔͒. ͑10͒ the beam are, from Eqs. ͑4͒, 4␲f m

Ϫ Ϫ For a diameter Dϭ6 cm of the lens L1 in the optical n ͑x͒ϭn ͑0͒exp͑Ϫxng␴0,1͒, ͑5͒ detection system the detection time for the H␣-line radiation Ϫ7 is trϭD/vϭ0.2ϫ10 s. Substituting into Eq. ͑10͒ the val- nϪ 0 0 ␴Ϫ1,0 ͑ ͒ ues of all parameters gives n ͑x͒ϭ ͓exp͑Ϫxn ␴ ͒Ϫexp͑Ϫxn ␴ ͔͒. Ϫ g 0,1 g Ϫ1,0 ␴Ϫ1,0 ␴0,1 5 ͑6͒ N͑H␣͒ϭ3ϫ10 photons/pulse. ͑11͒ 2. For a residual gas pressure Pϭ5ϫ10Ϫ4 Torr and a We introduce the following notation: ␭ ϭ1/n ␴ is DC g DC source-channel density n ϭ1.6ϫ1013 molecules/cm3,we the mean free path of excited hydrogen atoms in the state g have ␭ ϭ70 cm, ␭ ϭ440 cm, and ␭ ϭ40 cm, the other mϭ3 due to collisional decay of the excited levels; ␭ 1 2 DC DR parameters having similar values. In this case Eq. ͑8͒ be- ϭv/A is the mean free path of excited hydrogen atoms due m comes to the decay of the excited levels as a result of spontaneous 0 Ϫ m m emission; 1/␭Dϭ1/␭DCϩ1/␭DR ; and, ␭1ϭ1/ng␴Ϫ1,0 , ␭2 nm͑x͒ϭngn ͑0͓͒␴0,mϩB␭D͓1Ϫexp͑Ϫx/␭D͔͒ ϭ1/n ␴ , and Bϭ␴ /(␴ Ϫ␴ ). Substituting the g 0,1 Ϫ1,0 Ϫ1,0 0,1 Ϫm Ϫ1 Ϫ1 parameters into Eq. ͑2͒,weﬁnd ϩ͑␴Ϫ1,mϪB␴0,m͒͑␭D ϩ␭1 ͒ Tech. Phys. 43 (6), June 1998 V. V. Antsiferov and G. I. Smirnov 629

m spectrum will equal 2.3 nm, which makes it possible to sepa- ϫ͓exp͑Ϫx/␭1͒Ϫexp͑Ϫx/␭D͔͔͒. ͑12͒ rate quite simply the shifted H␣ line emitted by an acceler- Here ␴0,m is the cross section for the excitation of hydrogen ated beam of hydrogen atoms. The contribution to the width atoms into the 3s,3p, and 3d states in collisions with re- Ϫ17 2 ⌬␭D of the Doppler contour of the H␣ line with the detection sidual gas particles in the source: ␴0,3sϭ1ϫ10 cm , Ϫ17 2 Ϫ18 2 angle different from the magic angle will be of the order of ␴0,3pϭ1.5ϫ10 cm , and ␴0,3dϭ7ϫ10 cm ͑Ref. 10͒. Ϫ3 0 0 10 nm. The working values of the parameters nm(x) are: n3s(x) The contribution due to the angular aperture of the 5 Ϫ3 0 5 Ϫ3 0 ⌬␸A ϭ2.8ϫ10 cm , n3p(x)ϭ3.6ϫ10 cm , and n3d(x) receiving unit of the Doppler system to the width of the 5 Ϫ3 ϭ3.4ϫ10 cm . Substituting the values of all parameters Doppler contour is eliminated if4 into Eq. ͑10͒, we have in this case for the number N(H␣)of photons 0.1⌬␭D ⌬␸Aр р0.3°. ͑18͒ 6 ␭0␤ʈ N͑H␣͒ϭ4.2ϫ10 photons/pulse. ͑13͒ The intensity of the detected H␣ line can be increased by PARAMETERS OF THE DOPPLER DETECTION SYSTEM using in the detecting unit of the Doppler system ͑see Fig. 1͒ a lens L1 with diameter D and a slit of width The contribution from the spread ⌬␤ʈ of the longitudinal particle velocities in the accelerated beam to the Doppler ⌬lϭ f •⌬␸A ͑19͒ broadening of the H␣ line can be neglected for radiation at the focal point f . detection angles ␸r close to the ‘‘magic’’ angle ␸M , which For a lens with the focal length f ϭ20 cm the slit in the equals4 detecting unit should be 1 mm wide. For spectral analysis the ␸ ϭarccos ␤ . ͑14͒ H␣-line emission from the detecting unit of the Doppler sys- M ʈ tem is extracted with a ﬁber-optic cable into a spectral unit Since the average longitudinal velocity of beam particles consisting of a monochromator crossed with a scanning in the source is vϭ2.8ϫ108 cm/s, the magic angle equals Fabry–Perot interferometer. The monochromator separates ␸Mϭ89.6°. When the radiation detection angle equals the the shifted H␣ line emitted by the excited hydrogen atoms in magic angle ␸M , two H␣ lines will be observed in the Dop- the accelerated beam, while the width of the Doppler contour pler detection system: a line shifted by 0.03 nm, emitted by is detected with a scanning interferometer and photomulti- hydrogen atoms in the beam which are excited in the process plier, the signal from which is fed into a computer. of charge transfer from the ions on the residual gas, and an unshifted line, emitted by excited hydrogen atoms formed in the residual gas as a result of the dissociation of molecular CONCLUSIONS hydrogen in collisions with accelerated hydrogen ions and Ϫ Ϫ 0 The results of the present work show that passive Dop- atoms in the beam: H ͑H0͒ϩH2 H ͑H ͒ϩH0*(nϭ3)ϩH0. The density of the molecular hydrogen→ in the residual gas in pler spectroscopy of a beam of partially excited hydrogen atoms, obtained by neutralization of the ions in a beam and the source channel is quite high (ϳ104 Torr) and the inten- excitation of the hydrogen atoms in the beam on the residual sity of the unshifted H␣ line will be comparable to that of the shifted H line, while its spectral width will be gas in the source channel, can be used for nonperturbative ␣ monitoring of the divergence and emittance of a HϪ beam ⌬␭ ϭ2␭ v /c. ͑15͒ vT 0 T extracted from the source. There is no need to introduce a gas target into the source channel, and the Doppler contour For the average thermal velocity of the residual gas v T of the H line can be recorded using photoelectronic detec- ϭ1.38ϫ106 3E(eV) 1/2, the width of the unshifted H line ␣ ͓ ͔ ␣ tion of the H -line contour rather than the photon-counting equals 0.1 nm. ␣ mode. The method of passive Doppler spectroscopy devel- The width of the Doppler broadened H line emitted by ␣ oped was used to record the divergence of a beam of HϪ ions excited hydrogen atoms in the accelerated beam4 extracted from sources of negative hydrogen ions with plan- ⌬␭Dϭ2␭0␤Ќ cos ␪ ͑16͒ otronic and Penning electrode geometry. Its accuracy was conﬁrmed by comparing with contact slit measurements of will equal 0.13 nm in our case. To separate these two H ␣ the beam divergence. It was shown that for discharge current lines in the spectrum the detection angle ␸ must be chosen r densities р25 A/cm2 in the sources the divergence of the HϪ to be somewhat different from the magic angle ␸ . The M ion beam is smallest in the case of a planotronic source of deviation ⌬␸ of the detection angle from the magic angle r HϪ ions ͑Special Report No. 7504, Sukhumi Physicotechni- must satisfy the condition4 cal Institute, 1989͒.

2 ⌬␭D Ϯ⌬␸rр10 р30°, ͑17͒ ␭0␤ʈ 1 G. E. Derevyankin and V. G. Dudnikov, IYaF SO AN SSSR Preprint No. for which the contribution from the longitudinal velocity 79-17 ͓in Russian͔, Institute of Nuclear Physics, Siberian Branch of the spread ⌬␤ of the beam to the width of the Doppler contour Academy of Sciences of the USSR, Novosibirsk ͑1979͒,15pp. ʈ 2 of the shifted H line can be neglected. Therefore the detec- G. E. Derevyankin, V. G. Dudnikov, and M. L. Troshkov, IYaF SO AN ␣ SSSR Preprint No. 82-110 ͓in Russian͔, Institute of Nuclear Physics, Si- tion angle of the Doppler system can be chosen as ␸r berian Branch of the Academy of Sciences of the USSR, Novosibirsk ϭ60°. In this case the separation of the two H␣ lines in the ͑1982͒,18pp. 630 Tech. Phys. 43 (6), June 1998 V. V. Antsiferov and G. I. Smirnov

3 7 G. E. Derevyankin and V. G. Dudnikov, AIP Conf. Proc. No. 11, pp. V. V. Antsiferov and V. V. Beskorovany, Zh. Tekh. Fiz. 63͑5͒,41 376–397 ͑1984͒. ͑1993͓͒Tech. Phys. 38, 390 ͑1993͔͒. 8 4 V. V. Antsiferov, Zh. Tekh. Fiz. 62͑5͒,71͑1992͓͒Tech. Phys. 37, 525 H. Tawara and A. Russec, Rev. Mod. Phys. 45, 178 ͑1973͒. 9 J. Geddes, J. Hill, and H. B. Gilbody, J. Phys. B 14, 4837 ͑1981͒. ͑1992͔͒. 10 R. H. Hughes, H. M. Peteﬁsch, and H. Kisner, Phys. Rev. 5, 2103 ͑1972͒. 5 V. V. Antsiferov, AIP Conf. Proc. No. 87, pp. 616–617 ͑1993͒. 6 V. V. Antsiferov and V. V. Beskorovany, Zh. Tekh. Fiz. 63͑4͒,50 ͑1993͓͒Tech. Phys. 38, 291 ͑1993͔͒. Translated by M. E. Alferieff TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Cooperative population dynamics of an ensemble of ⌳ atoms in a bichromatic ﬁeld B. G. Matisov, I. A. Grigorenko, N. Leinfell’ner, I. E. Mazets, and A. Yu. Snegirev

St. Petersburg State Technical University, 195251 St. Petersburg, Russia ͑Submitted June 3, 1997͒ Zh. Tekh. Fiz. 68, 19–24 ͑June 1998͒ Equations are derived which describe the dynamics of three-level atoms with a ⌳ level scheme, interacting with two coherent resonance ﬁelds under conditions such that cooperative relaxation predominates over incoherent spontaneous emission. A numerical calculation of the temporal dynamics of the values of the atomic populations is performed. It is shown that coherent population trapping in the presence of cooperative decay is possible. The quantities characterizing this phenomenon are calculated—the width of the black line and the transition time to coherent trapping in this scheme. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00506-6͔

INTRODUCTION EQUATIONS FOR THE DENSITY MATRIX

Research on the interaction of coherent electromagnetic Let us list the main assumptions employed in this work. radiation with multilevel quantum systems is today a rapidly It is assumed that an ensemble of N identical three-level developing field of nonlinear laser spectroscopy and quan- atoms interacts with two resonance electromagnetic waves. tum optics. In multilevel systems the presence of several Let L be the linear size of the medium. We denote the atomic ϭ channels of excitation and induction of coherences between states by ͉0͘l , ͉1͘l , and ͉2͘l , where l 1,...., N enumer- ates the coherent atoms. The corresponding wave function is long-lived quantum states by laser fields plays a fundamental role. This leads to the appearance of different quantum inter- ͉i͘lϭ͉i͑INT͉͒͘i͑MOT͒͘l , iϭ0, 1, 2, ͑1͒ ference effects in the internal dynamics of the atoms. The which consists of two parts describing the internal ͑INT͒ and interference of quantum states that arises as a result of co- translational ͑MOT͒ degrees of freedom. Let ͉0(INT) be the herent excitation and leads in turn to coherent population ͘ upper excited ⌳ state and ͉1(INT)͘ and ͉2(INT)͘ the low- trapping ͑CPT͒ is the basis of many directions of develop- energy states. Absorption of a photon corresponds to a shift ment in modern physics: ultradeep laser cooling of atoms, of the translational part of the wave function ͑1͒ in momen- production of lasers without an inversion, and others ͑see tum space by the amount of the photon momentum. For this Refs. 1 and 2 for a more detailed discussion͒. 3 reason, for further analysis it is convenient to perform the It is known that different relaxational processes, for ex- following phase transformation:4,5 ample collisional, strongly influence the evolution of the populations of atomic systems under CPT conditions. How- ͉j͑MOT͒͘lϭexp͑ikjzl͉͒0͑MOT͒͘l , jϭ1,2, ͑2͒ ever, in previous works concerning CPT in atomic systems, where z is the coordinate of the lth atom and k is the wave 3 l j the case n•␭ у1, where n is the atom density and ␭ is the number of the jth mode. wavelength of an optical transition, was not studied. Under This transformation makes it possible to eliminate in the these conditions cooperative effects have a determining in- equations below the explicit dependence on the coordinates. 4,5 fluence on the evolution of the system. As is well known, We define the collective atomic operator as the main such effect is the mutual coordination of the behav- N ior of the atoms. As a result of this, relaxation in such a ˆ Rijϭ ͉j͘II͗i͉. ͑3͒ system is determined not by ordinary spontaneous emission I͚ϭ1 but rather by a coherent process ͑superradiance͒. Effects ˆ similar to superradiance in a lumped Dicke model6 also oc- The commutator between the operators Rij is cur in elongated ͑needle-shaped͒ samples, where cooperative ͓Rˆ ,Rˆ ͔ϭ␦ Rˆ Ϫ␦ Rˆ . ͑4͒ emission has a narrow directional pattern. ␮␯ ␯Ј␮Ј ␯␯Ј ␮␮Ј ␮␮Ј ␯Ј␯ In the present paper we extend the well-known single- The commutation rules follow directly from the defini- mode Bonifacio model7 to the case of an ensemble of three- tion ͑3͒ and the orthonormality of the wave functions level ⌳ atoms interacting with a coherent bichromatic elec- ͗i͉iЈ͘ ϭ␦ ␦ . ͑5͒ tromagnetic field in the presence of cooperative relaxation. It l lЈ llЈ iiЈ is established on the basis of equations derived below and The electromagnetic field is described by bosonic cre- ˆ ϩ ˆ describing the evolution of the system that coherent popula- ation and annihilation operators a j and a j . The coupling tion trapping is possible and the main parameters character- constant characterizing the interaction of the atomic system izing this phenomenon are calculated. with the field is defined as

where 2␲ប␻0 j g ϭ d , jϭ1, 2, ͑6͒ j ͱ V 0 j i Mˆ ͑␴ˆ ͒ϭϪ ͓Hˆ ,␴ˆ ͔. ប where ␻0 j is the frequency, d0 j is the ͉0͘Ϫ͉j͘ transition dipole moment, and V is the quantization volume. Perturbation theory is applicable if As a simplification, we shall neglect incoherent decay ␶g jͱN from the upper excited level, which is a consequence of the Ӷ1, ͑12͒ interaction of the vacuum ͑zero͒ modes with the continuum. ប The reason for this simplification is that the cooperative pro- which should hold for both ( jϭ1, 2) modes. Therefore pho- cess is much faster than the incoherent process. Instead of tons escape in a time shorter than the duration of the super- slow atomic relaxation, fast relaxation of the electromagnetic radiance ͑SR͒ pulse ͑the duration of the SR pulse is ␶s 2 2 field into a laser-determined stationary state which produces Ϸ͉g j͉ /h N␶͒. For this reason, the variation of the field at a pure coherent state ͉␣1␣2͘ f , i.e., times tӷ␶ can be excluded adiabatically. Expanding expression ͑11͒ in a series in ␶, we obtain up ˆ 2 a j͉␣1␣2͘ f ϭ␣ j͉␣1␣2͘ f , ͑7͒ to terms of order ␶ ץ .where ␣ are complex numbers, is introduced j ␴ˆ ͑t͒ϭ␴ˆ Ј͑t͒Ϫ␶ ␴ˆ Ј͑t͒ϩ␶Mˆ ͑␴ˆ Ј͑t͒͒. ͑13͒ tץ The most important characteristic of the system is the residence time ␶ϭL/c of a photon in the medium. At times Substituting expression ͑13͒ into Eq. ͑9͒ and taking the shorter than ␶, the electromagnetic field is still in a state that trace of both parts of Eq. ͑9͒ over the field variables, we is coherently coupled with the state of the atomic system obtain an equation for the atomic density matrix which has emitted a photon. At times longer than ␶ the pho- i ␶ ץ i␶ ץ tons leave the region of interaction and coherence between ␳ˆ Ϫ Hˆ , ␳ˆ ϭϪ ͓Hˆ ,␳ˆ͔Ϫ Tr͕͓Hˆ ,͓Hˆ ,␳ˆ ͔͔͖ , t ͬ ប at ប2 fץ t ប ͫ atץ the matter and field states breaks down. The time ␶ is of the order of 1 ns for LϷ30 cm and can be chosen as the shortest ͑14͒ time interval in the system, since it is much shorter than where we have introduced the Hamiltonian 10Ϫ7 s—the characteristic spontaneous relaxation time. Thus ˆ ˆ the field density matrix arrives in its stationary state Hatϭf ͗␣a␣2͉H͉␣1␣2͘f ͉␣ ␣ ͘ ͗␣ ␣ ͉ with relaxation time ␶. 1 2 ff 1 2 which acts only on the atomic variables. Next, we apply to We shall describe the interaction of two waves with the both parts of Eq. 14 the operator entire collection of atoms as ͑ ͒ i␶ ˆ ˆ ˆ ˆ ˆ Qˆ ͑Xˆ ͒ϭXˆ ϩ ͓Hˆ , Xˆ ͔, ͑15͒ HϭϪប⍀1R11Ϫប⍀2R22ϩg1a1R01 ប at ˆ ϩ ˆ ˆ ϩ ˆ ˆ ˆ ϩg2*a2 R20ϩg1*a1 R10ϩg2a2R02 , ͑8͒ where Xˆ stands for Eq. ͑14͒. According to the accuracy adopted above, we neglect where ⍀ j is the detuning of the jth field from the frequency terms proportional to ␶2 of the corresponding atomic transition. i ␶ ץ The density matrix ␴ˆ (t) describing the atoms–field sys- ˆ ˆ ˆ ˆ 2 ˆ 2 ˆ ␳ϭϪ ͓Hat , ␳͔Ϫ 2 ͕͑͗H ͘ϪHat͒␳ ␶ ប បץ tem satisfies the Liouville–von Neumann equation with the relaxation term ˆ ˆ ˆ Ϫ2Tr͕H͑␳͉␣1␣2͘ff ͗␣1␣2͉͒H͖͖f i 1 ץ ˆ ˆ ˆ ˆ ˆ ␶ ␴ϭϪ ͓H,␴͔Ϫ ͑␴Ϫ␴Ј͒, ͑9͒ Ϫ ͑␳ˆ͑͗Hˆ 2͘ϪHˆ 2 ͒ϩ2Hˆ ␳ˆHˆ ͒, ͑16͒ t ប ␶ ប2 at at atץ ˆ ˆ 2 ˆ 2 where the equilibrium density matrix ␴Ј is where ͗H ͘ϭ f ͗␣1␣2͉H ͉␣1␣2͘ f . In the case of a ⌳ scheme the equation for the density ˆ ˆ ␴Јϭ␳ ͉␣1␣2͘ ff͗␣1␣2͉, ͑10͒ matrix is ץ and the reduced density matrix ␳ˆ describes only the atomic ˆ ˆ ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ ប ␳ϭϪi͓Hat ,␳͔ϩ␶͉g1͉ ͓͑R10 ,␳R01͔ϩ͓R10␳,R01͔͒ ␶ץ ˆ ˆ degrees of freedom ␳ϭTr͕␴͖ f ͑trace extends over the field variables͒. The equation ͑9͒ possesses a formal solution in 2 ϩ␶͉g ͉ ͓͑Rˆ ,␳ˆ Rˆ ͔ϩ͓Rˆ ␳ˆ ,Rˆ ͔͒. ͑17͒ the form of an infinite series 2 20 02 20 02 This equation is the extension of the Bonifacio model.7 ϱ tm 1 t ˆ ͑Ϫt/␶͒ ˆ m ˆ Ϫ͑tϪtЈ͒/␶ ␴͑t͒ϭe M ͑␴͑0͒͒ϩ e ATOMIC OPERATORS AND THEIR AVERAGE VALUES m͚ϭ0 m! ␶ ͵0

ϱ Exhaustive information about the dynamics of the ͑tϪtЈ͒m ϫ ͚ Mˆ m͑␴ˆ͑tЈ͒͒dtЈ, ͑11͒ atomic system is contained in the average value of the col- mϭ0 m! lective atomic operator Tech. Phys. 43 (6), June 1998 Matisov et al. 633

1 2 r ϭ Tr͕Rˆ ␳ˆ͖, i,jϭ0,1,2. ͑18͒ ij N ij Ϫ␦jЈ0 ͚ ⌫jЉNrjjЉrjЉ0ϩ͓⌫j͑1Ϫ␦j0͒ jЉϭ1 Switching to the new variables r , we write Eq. ͑17͒ as ij ϩ⌫jЈ͑1Ϫ␦jЈ0͔͒Nr0jЈrj0 , ͑23͒ i ␶ where we have introduced the notation ץ ˆ ˆ ˆ N r jjЈϭϪ Tr͕͓Hat ,␳͔RjjЈ͖ϩ 2 t ប ប ␶ץ ⌫ ϭ ͉g ͉2. 2 ␮ ប2 ␮ 2 ˆ ˆ ˆ ϫ ͚ ͉gjЉ͉ Tr͕͓͑RjЉ0,␳R0jЉ͔ jЉϭ1 It is important to note that ˆ ˆ ˆ ˆ r11ϩr22ϩr33ϭ1 ͑24͒ ϩ͓RjЉ0␳,R0jЉ͔͒RjjЈ͖. ͑19͒ and the approximation ͑22͒ has no effect on this condition. Permuting cyclically within the trace, we obtain After calculating the coefficients LmmЈ , which are either the jjЈ i ␶ ץ ˆ ˆ ˆ detunings ⍀ j or the atom–field coupling constants V j N r jj ϭϪ Tr͕␳͓Hat ,Rjj ͔͖ϩ 2 t Ј ប Ј ប ϭg j␣ j /ប, we can write Eq. ͑23͒ asץ 2 ˙ 2 r11ϭiV1r01ϪiV1*r10ϩ2⌫1N͉r01͉ , 2 ˆ ˆ ˆ ϫ ͚ ͉gjЉ͉ Tr͕␳͓R jЈjЈ ,R0 jЉ͔ j ϭ1 ˙ 2 Љ r22ϭiV2r02ϪiV2*r20ϩ2⌫2N͉r02͉ , ˆ ˆ ˆ ˆ ˆ ϫR j 0ϩR0 j ͓R j 0␳,R0 j ͔͖. ˙ Љ Љ Љ Љ r01ϭi⍀1r01ϩiV1*͑r11Ϫr00͒ϩiV2*r21 We now employ the commutation rules ͑4͒, which de- Ϫ⌫ Nr ͑r Ϫr ͒Ϫ⌫ Nr r , ˆ 1 01 11 00 2 02 21 crease on the right-hand side the order with respect to Rij .It ˙ is easy to see that the commutator with the Hamiltonian is r02ϭi⍀2r02ϩiV1*r12ϪiV2*͑r00Ϫr22͒ linear Ϫ⌫1Nr01r12Ϫ⌫2Nr02͑r22Ϫr00͒, 2 2 mmЈ ˙ Hˆ ,Rˆ ϭ L Rˆ , 20 r21ϭi͑⍀1Ϫ⍀2͒r21ϩiV2r01ϪiV*r20 ͓ at jjЈ͔ ͚ ͚ jjЈ mmЈ ͑ ͒ 1 mϭ0 mЈϭ0 ϩ͑⌫1ϩ⌫2͒Nr20r01 . ͑25͒ where the coefficients LmmЈ can be calculated easily. The jjЈ * The Hermiticity condition r jЈjϭr jj then holds. The sys- explicit form of LmmЈ in the Appendix gives Ј jjЈ tem of equations ͑25͒ for the average values of the collective 2 2 atomic operators is similar to the system of equations for the i ␶ ץ r ϭ LmmЈr Ϫ single-particle density matrix in the case of incoherent decay. t jjЈ ប ͚ ͚ jjЈ mmЈ ប2Nץ mϭ0 mЈϭ0 Indeed, the diagonal elements rii give us the populations of 2 the corresponding states ͑per atom͒. Nonetheless, there is an 2 ˆ ˆ ˆ important distinction: In the case of the cooperative effect ϫ ␦jЈ0 ͚ ͉gjЉ͉ Tr͕␳R jjЉRjЉ0͖ϩ␦j0 ͭ jЉϭ1 the relaxation terms are nonlinear and proportional to the total number N of atoms. As the initial condition in the sys- 2 tem ͑25͒, we assume that all atoms initially occupy the lower ϫ g 2Tr ␳ˆ Rˆ Rˆ Ϫ͑ g 2͑1Ϫ␦ ͒ ͚ ͉ jЉ͉ ͕ 0 jЉ jЉjЈ͖ ͉ j͉ j0 energy level ͉1(INT)͘, i.e., jЉϭ1

2 ˆ ˆ ˆ ϩ͉g jЈ͉ ͑1Ϫ␦ jЈ0͒͒Tr͕␳R0 jЈR j0͖ͮ . ͑21͒ As is customary in the theory of cooperative effects, we expand the correlator using the quasiclassical approximation 1 Tr͕␳ˆ Rˆ Rˆ ͖Ϸr r . ͑22͒ N2 ij iЈjЈ ij iЈjЈ The approximate nature of the simpliﬁcation ͑22͒ is expressed only in the fact that it is impossible to describe cor- rectly the early stage of the dynamics of the system in the case when there is no external laser ﬁeld. From Eq. ͑21͒ we obtain

2 2 2 FIG. 1. Temporal evolution of populations in the ⌳ system for ⍀ϭ0, i mmЈ ץ r ϭ L r Ϫ␦ ⌫ Nr r Vϭ0.3⌫N. The dimensionless time t/⌫N is plotted along the abscissa. Thin t jjЈ ͚ ͚ jjЈ mmЈ j0 ͚ jЉ 0jЉ jЉjЈ .ប mϭ0 mЈϭ0 jЉϭ1 curve—level 1, thick curve—level 2, dashed curve—level zero ץ 634 Tech. Phys. 43 (6), June 1998 Matisov et al.

FIG. 2. Same as Fig. 1 but for ⍀ϭ0, Vϭ3⌫N. FIG. 3. Same as Fig. 1 but for ⍀ϭ2⌫N, Vϭ3⌫N.

r jjЈ͉tϭ0ϭ␦j1␦jЈ1. ͑26͒ regime much earlier than the population of the lower levels ͑Figs. 1 and 2͒. In the case of incoherent relaxation, however, The off-diagonal elements of the matrix rij equal zero. the system of equations describing the population dynamics is linear,8 so that the evolution of all populations is deter- COMPUTATIONAL RESULTS mined by the roots of the same characteristic equation. In the We shall present the results obtained by integrating nu- nonlinear case of cooperative dynamics the oscillations of merically the system of equations ͑25͒ with the initial con- the populations predominate at the initial stage of evolution ditions ͑26͒. As a simpliﬁcation we set ⌫1ϭ⌫2ϵ⌫, V1 for VϽ⌫N ͑Fig. 1͒. In the case VϾ⌫N oscillations are sup- ϭV2ϵV, and ⍀1ϭ⍀2ϵ⍀. pressed ͑Fig. 2͒. Conversely, in the case when incoherent The temporal evolution of the atomic populations rii is relaxation predominates the oscillations are suppressed at displayed in Figs. 1–4. In the case ⍀ϭ0 the behavior of the Rabi frequencies less than the relaxation rate from the upper system corresponds to a transition to CPT,8 where the states level and are developed in the opposite case.8 We note also ͉1͘ and ͉2͘ form a noninteracting superposition, while the that the stationary state r11ϭ0.5, r22ϭ0.5, r12ϭϪ0.5, and population of the upper level ͉0͘ becomes zero ͑Figs. 1 and all other elements rijϭ0 is an exact solution of the system of 2͒. There is an important distinction from the ordinary tran- equation ͑25͒. sition to CPT: In the presence of cooperative relaxation in In the case ⍀Þ0 the population dynamics is more com- the system the population of the upper level reaches a steady plicated. The steady state of the system is an oscillatory re-

FIG. 4. Same as Fig. 1 but for ⍀ϭ1.5⌫N, Vϭ1.5⌫N. Tech. Phys. 43 (6), June 1998 Matisov et al. 635

Ϫ1 FIG. 6. ␶rel , ␦⍀BL , and F ͑in units of ⌫N͒ versus the dimensionless FIG. 5. Time-averaged population r00 of the upper level ͉0͘ versus the Ϫ1 coupling parameter k. Computed values: ᭺—␶rel , ᭝—␦⍀BL , ᭿—F. dimensionless detuning ⍀/⌫N. V/⌫N: a—0.5, b—1.0, c—2.0.

gime ͑Figs. 3 and 4͒, in contrast to CPT in the ordinary, CONCLUSIONS noncooperative case, where the steady state corresponds to The transition to CPT in an atomic ensemble interacting constant populations.1,8 These oscillations in the case of co- with two coherent resonance electromagnetic fields under cooperative dynamics are periodic ͑Fig. 3͒. When ⍀ and V operative relaxation conditions was investigated theoreti- become equal to one another, the period increases— cally. The results obtained make it possible to apply the phe- nonlinearity appears in the behavior of the quantum system nomenon of coherent population trapping, for example, for ͑Fig. 4͒. investigating electromagnetically induced transparency for Figure 5 shows the time-averaged value of the popula- purposes of coherent bleaching9,10 in dense optical media tion of the level ͉0͘ versus the dimensionless detuning ⍀/⌫N with average interatomic distance of the order of the wave- for different values of the atom-field coupling constant. One length of the atomic transition n␭3ϳ1. It was established can see that the width of the black line increases with the that cooperative relaxation introduces substantial changes in coupling constant V. the dynamics of the ensemble of atoms. The features that We now introduce the following three characteristics: distinguish the cooperative case from the case where inco- the characteristic time ␶ during which coherent population rel herent spontaneous relaxation predominates were clarified trapping is established in the system, the width ␦⍀ of the BL and parameters such as the width of the black line, the con- black line, and the width F of the overall contour. Analysis tour width, and the transition rate to CPT in the system stud- of the dependences of the introduced quantities on the value ied above were calculated. of the parameter kϭV/⌫N gives simple approximation formulas for all three quantities characterizing the quantum sys- This work was supported in part by the fund ‘‘Fonds zur tem: Fo¨rderung der wissenschaftlichen Forschung’’ under project No. S6508 and by the State Committee of the Russian Fed- Ϫ1 2 2 eration on Higher Education under Grant No. 5-5.5-139. ␶rel 0.03k ␦⍀BL 0.70k ϭ , ϭ0.005ϩ , ⌫N 0.17ϩk2 ⌫N 0.90ϩk

F APPENDIX ϭ2.3k0.92. ͑27͒ ⌫N We present here the explicit form of the nonzero coefﬁ- cients LmmЈ appearing in expression ͑20͒: L01ϭiV , L10 One can see that the transition rate ␶Ϫ1 to CPT under jjЈ 11 1 11 rel ϭϪiV*, L02ϭiV , L20ϭiV*, L01ϭi⍀ , L11ϭiV*, L00 two-photon resonance conditions ͑i.e., ⍀ ϭ⍀ ͒ behaves just 1 22 2 22 2 01 1 01 1 01 1 2 ϭϪiV*, L02ϭi⍀ , L12ϭiV*, L00ϭϪiV*, L22ϭiV*, as in the ordinary single-atom case: It increases quadratically 1 02 2 02 1 02 2 02 2 L21ϭi( Ϫ ), L01ϭiV , L20ϭiV . The coefﬁcients for kӶ1 and is constant in the opposite case. Further, the 21 ⍀1 ⍀2 21 2 21 1 LmmЈ satisfy the relation LmmЈϭ(LmЈm)*. dependence of the width of the black line is similar to the jjЈ jjЈ jЈj ordinary, noncooperative case.1 The width of the overall contour satisfying a power law with an exponent close to 1 in a wide range of values of the parameter k differs from the 1 noncooperative dynamics, where it is constant for kӶ1. The B. G. Agap’ev, M. B. Gorny, B. G. Matisov et al., Usp. Fiz. Nauk 163,1 computed points for the three characteristics ␶Ϫ1 , ␦⍀ , ͑1993͒. rel BL 2 E. Arimondo, in Progress in Optics, edited by E. Wolf, North-Holland, and F and the curves corresponding to the approximation Amsterdam, 1995, No. 35, pp. 257–269. formulas ͑27͒ are presented in Fig. 6. 3 E. Arimondo, Phys. Rev. A 4, 2216 ͑1996͒. 636 Tech. Phys. 43 (6), June 1998 Matisov et al.

4 8 ´ A. V. Andreev, V. I. Emel’yanov, and Yu. A. Il’inski, Cooperative Ef- E. A. Korsunski, B. G. Matisov, and Yu. V. Rozhdestvenski, Zh. Eksp. fects in Optics ͓in Russian͔, Nauka, Moscow, 1988, 288 pp. Teor. Fiz. 102, 1096 ͑1992͓͒Sov. Phys. JETP 75, 595 ͑1992͔͒. 5 9 ´ M. M. Al’perin, Ya. D. Klubis, and A. I. Khizhnyak, Introduction to the M. B. Gorny, B. G. Matisov, and Yu. V. Rozhdestvenski, Zh. Eksp. Physics of Two-Level Systems ͓in Russian͔, Naukova Dumka, Kiev, 1987, Teor. Fiz. 95, 1263 ͑1989͓͒Sov. Phys. JETP 68, 728 ͑1989͔͒. 256 pp. 10 K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 6 R. H. Dicke, Phys. Rev. 93,99͑1954͒. ͑1991͒. 7 R. Bonifacio, P. Schwendimann, and F. Haake, Phys. Rev. A 93, 302 ͑1971͒. Translated by M. E. Alferieff TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Thermophoresis of touching solid spheres in the direction along the line joining their centers S. N. D’yakonov and Yu. I. Yalamov

Moscow Pedagogical University, 107005 Moscow, Russia ͑Submitted February 3, 1997͒ Zh. Tekh. Fiz. 68, 25–31 ͑June 1998͒ A theory of the steady motion of an aggregate of two touching solid, nonvolatile, low-thermal- conductivity, spherical particles in the direction along the line joining their centers in a nonuniformly heated viscous gas is constructed in a hydrodynamic regime with slipping at low Reynolds and Peclet numbers. The thermophoretic transport velocity of an aggregate is determined in an approximation linear in the small parameters. The small parameters are the relative deviations of the thermal conductivity of the constituent particles of an aggregate from the thermal conductivity of the external medium. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00606-0͔

INTRODUCTION theories employing a bispherical coordinate system and the Stimson–Jeffery approach to the hydrodynamic problem.6 The study of particle dynamics in a viscous medium The endeavor to obtain an analytical solution of this problem with nonuniform temperature is of interest in connection for the linearized stationary equations of hydrodynamics and with investigations of thermophoretic motion in the physics heat transfer makes it possible to obtain more reliable results of aerodispersion systems for analysis of the interaction of for larger temperature differentials than those obtained ear- hot particles, in mechanics and rheology of suspensions, and lier in the limiting case of touching particles. in a number of other problems. Thus far the characteristics of thermophoretic motion in viscous media of single solid and FORMULATION OF THE PROBLEM liquid aerosol particles have been studied in greatest detail. A general bibliography concerning these questions is pre- We shall study the slow motion of an aggregate of sented in Ref. 1. two touching, solid, nonvolatile, low-thermal-conductivity It is more important to study the motion of a particle spherical particles in the direction along the line joining their ensemble, since particles which are sufficiently close to one centers in a temperature-nonuniform viscous gaseous me- another strongly influence the relative particle motions. In dium. aerosol systems, in practice, pairs of particles are most likely The problem of determining the thermophoretic velocity to approach one another. It is thus of interest to study the UT of an aggregate can be solved in tangential spherical co- dynamics of such pairs. ordinates ͑␨,␩,␸͒ related to the circular cylindrical coordi- The thermophoresis of two spherical aerosol particles in nates (␥,z,␸)as the direction along the line joining their centers was investi- 2␨ 2␩ gated in Refs. 2–5. Exact analytical solutions for particles ␥ϭ , zϭ , ␸ϭ␸. ͑1͒ ␨2ϩ␩2 ␨2ϩ␩2 located at an arbitrary distance but quite far from one another were obtained in a bispherical coordinate system on the basis The origin of the cylindrical coordinate system is rigidly of linearized stationary equations of hydrodynamics and heat fixed at the contact point of the particles. In this coordinate transfer. A numerical comparison of these solutions with ap- system the center of gravity of the exterior medium moves proximate solutions obtained by the method of reflections with the velocity UϭϪUT , which is sought, relative to the shows that for close-lying particles there is a degradation of stationary aggregate. (e) -the convergence of the approximate solutions. In the limiting A constant temperature gradient ATϭ(ٌT )ϱ is main case when the particles touch the exact analytical solutions tained in the gas infinitely far away from the particle aggre- are suitable only for estimating the instantaneous velocities gate. Here and below the superscripts e and i denote physical of the steady motion of the particles. These estimates are quantities in a region outside and inside an aggregate, re- strongly limited by the conditions of applicability of the lin- spectively, while a subscript ␣ (␣ϭ1,2) refers to a definite ear stationary equations of slow flow. particle. In the present paper we construct on the basis of a hy- Let the axis zϭ(r cos ␪) pass through the centers of the drodynamic analysis a theory of the motion of two solid touching spheres and be directed parallel to the vector AT .A nonvolatile contiguous spheres in the direction along the line nonuniform temperature distribution near the particles results joining their centers in a nonuniformly heated viscous gas. in the appearance of a directed ͑thermophoretic͒ motion of This work is necessitated by the fact that the limiting prob- the aggregate ͑on account of the thermal slipping of the gas lem of two touching spheres cannot be solved on the basis of along the surfaces of the solid spheres͒.

The external medium is assumed to be single- It is convenient to write the equations of hydrodynamics component, isotropic, incompressible, and continuous—the and heat transfer and the boundary conditions in a reduced Knudsen number Knϭ␭/R0Ӷ1 ͑where ␭, R0ϭR1ϩR2 , R1 form. The physical quantities entering in the equations are Ϫ1 Ϫ1 ϭ␩1 , and R2ϭ␩2 are, respectively, the average mean made dimensionless as follows: free path length of the gas molecules, the unit of length, the ␥ r r radii of curvature of the surfaces ␩ϭ␩1Ͼ0 and ␩ϭϪ␩2 ˜ ˜ ˜ ˜ ˜ ␥ϭ , zϭ , rϭ , ␨ϭ␨R0 , ␩ϭ␩R0 , Ͻ0 of the particles in an aggregate͒. R0 R0 R0 We assume that each particle consists of a material v͑e͒ v͑e͒ U ⌿͑e͒ which is uniform and has isotropic properties. ˜ ͑e͒ ␨ ˜ ͑e͒ ␩ ˜ ˜ ͑e͒ v␨ ϭ , v␩ ϭ , Uϭ , ⌿ ϭ 2 , Thermophoresis of an aggregate occurs at low Reynolds U͓0͔ U͓0͔ U͓0͔ R0U͓0͔ (e) (e) (e) (e) and Peclet numbers Re ϭUR0 /␯ Ӷ1 and ReT ϭUR0 /␹ Ӷ1. This makes it possible to drop the nonlinear ͑inertial and F T͑e͒ϪT͑e͒ ˜ z ˜ ͑e͒ 0 convection͒ terms in the equations of hydrodynamics and Fzϭ ͑e͒ , T ϭ , 6␲␩ R0U͓0͔ ATR0 heat transfer. The external mass forces are neglected. There 0 are no heat sources inside or outside the particles. ͑i͒ ͑e͒ T␣ ϪT0 The relative temperature differentials under the condi- ˜T͑i͒ϭ , ␣ A R tions of the problem are small and the temperature variation T 0 of the coefficients of molecular transport can be neglected. (e) ⌿ is the stream function and U͓0͔ is the velocity of the gas The density, kinematic viscosity, and thermal conductivity flow at infinity in the zeroth approximation in the small pa- (e,i) (e) (e,i) are assumed to be constants (␳0 ,␯0 ,୐0 ) at the unper- rameters ͑it is determined in the course of the solution͒. (e) turbed temperature T0 ͑the temperature of the external me- In what follows the tilde is dropped and the initial equa- dium at the location of the point of contact of the particles in tions and boundary conditions are written in the reduced an aggregate in the absence of the aggregate͒. However, the form as follows: existing temperature differentials are large enough so that in comparison the temperature variations due to heating as a E4⌿͑e͒ϭ0, ͑8͒ result of energy dissipation by internal friction can be ne- ͑e͒ ͑i͒ glected in the heat-transfer equation. ⌬T ϭ⌬T␣ ϭ0, ͑9͒ Since the thermal and hydrodynamic relaxation times of 1 the system are short, the motions of an aggregate can be r ϱ: ⌿͑e͒ϭϪ ␥2U, ͑10͒ described in a quasistationary approximation ͑a slow axisym- → 2 metric motion of the gas medium and a steady temperature ͑e͒ distribution inside and outside the particles͒. T ϭZ, ͑11͒ The following boundary conditions hold at infinity and S ␣ϭ1, 2͒: ⌿͑e͒ϭ0, ͑12͒ on the surface of the particles: ␣ ͑ ͑e͒ ͑e͒ ͑e͒ ͑e͒ ␯͑e͒ ͑e͒ Tץ ⌿ ͑e͒ 0ץ r ϱ: v ϭUiz , T ϭT0 ϩAzz, ͑2͒ → U͓0͔ ϭϪKTSL ͑e͒ AT␥ , ͑13͒ ␨ץ ␩ T0ץ e͒͑ S␣͑␣ϭ1,2͒: ͑i␩•v ͒ϭ0 ͑3͒ i ͑e͒ ͑e͒ ͑ ͒ ␯ T ϭT␣ , ͑14͒ ͑e͒ ͑e͒ 0 ͑e͒ i␨•v ͒ϭKTSL ͑e͒ ͑i␨•ٌT ͒, ͑4͒͑ To ͑e͒ ୐0 T͑e͒ϭ T͑i͒ , 15 ͒ ͑ ␣ e͒ ͑i͒ ͑i͒ ٌ␩ ٌ␩͑ T ϭT␣ , ͑5͒ ୐0␣

͑e͒ ͑e͒ ͑i͒ ͑i͒ ,୐0 ͑i␩•ٌT ͒ϭ୐0␣͑i␩•ٌT␣ ͒, ͑6͒ Fz͑aggregate͒ϭ0

2ץ ץ 1 2ץ 2ץ ץ 1 2ץ Fz͑aggregate͒ϭ0. ͑7͒ E2ϭ Ϫ ϩ , ⌬ϭ ϩ ϩ . z2ץ ␥ץ ␥ 2␥ץ z2ץ ␥ץ ␥ 2␥ץ .The conditions ͑2͒–͑7͒ physically signify the following At infinity the axisymmetric gas flow is uniform in space ͑16͒ and its velocity U is in the positive direction along the z axis, The reduced velocity U of the gas flow at infinity is while the external temperature field is unperturbed. sought in the form of a power series in the small parameters On the gas-impermeable surface S␣ of the solid nonvola- (e) ␧1 and ␧2 tile particles the normal velocity component v␩ of the external medium vanishes, while the tangential component v(e) ͑i͒ ͑e͒ ͑i͒ ␨ 0р␧1ϭ͑୐01 Ϫ୐0 ͒/୐01 Ӷ1, equals the velocity of thermal slipping ͑it is characterized by (e) ͑i͒ ͑e͒ ͑i͒ the coefficient KTSL , determined by methods of the kinetic 0р␧2ϭ͑୐02 Ϫ୐0 ͒/୐02 Ӷ1, theory of gases͒; the normal heat flux and temperature are ͑1͒ ͑2͒ 2 ͑1͒ continuous. U͑␧1 ,␧2͒ϭ1ϩ␧1U͓1͔ϩ␧2U͓1͔ϩ␧1U͓2͔ The resultant force F exerted on an aggregate by the ͑2͒ 2 ͑2͒ incident flow of the external medium equals zero. ϩ␧1␧2U͓2͔ϩ␧2U͓3͔ϩ... . ͑17͒ Tech. Phys. 43 (6), June 1998 S. N. D’yakonov and Yu. I. Yalamov 639

We shall conﬁne ourselves below to determining the T͑e͒͑␨,␩͉͒ ϭT͑i͒ ͑␨,␩͉͒ , ͑24͒ (1) (2) ͓0͔ S␣ ␣,͓0͔ S␣ quantities U͓0͔ , U͓1͔ , and U͓2͔ , which characterize the thermophoretic velocity of an aggregate in the zeroth and ﬁrst ͑e͒ ͑i͒ ␩T ͑␨,␩͉͒͒S ϭٌ͑␩T ͑␨,␩͉͒͒S , ͑25ٌ͒͑ approximations ͑the bracketed subscripts͒. ͓0͔ ␣ ␣,͓0͔ ␣ ͑e͒ ͑i͒ T ͑␨,␩͉͒r ϱϭz, T ͑␨,␩͉͒␨ ϱ Ͻϱ. ͑26͒ ͓0͔ ␣,͓0͔ ␩→ ϱ THERMAL PROBLEM → → An exact analytical solution of the thermal problem ͑9͒, The following algebraic equations follow from the ͑11͒, ͑14͒, and ͑15͒ cannot be obtained for arbitrary values of boundary conditions ͑24͒ and ͑25͒ with allowance for the (e) (i) integral transformations ͑A2͒ and ͑A3͒ in the Appendix: the reduced thermal conductivities ୐1ϭ୐0 /୐01 and ୐2 ϭ୐(e)/୐(i) . An approximate solution of the thermal ͑hydro- 0 02 2␭ exp͑Ϫ␭␩ ͒ϩA ͑␭͒cosh͑␭␩ ͒ϩB dynamic͒ problem can be constructed by the method of suc- 1 ͓0͔ 1 ͓0͔ cessive approximations in the small parameters ␧1 and ␧2 ϫ͑␭͒sinh͑␭␩1͒ϭC͓0͔͑␭͒exp͑Ϫ␭␩1͒, ͑the case of an aggregate of touching low-thermal- conductivity particles ୐1ϳ୐2ϳ1͒. The reduced axisymmet- Ϫ2␭ exp͑Ϫ␭␩1͒ϩA͓0͔͑␭͒sinh͑␭␩1͒ϩB͓0͔ ric temperature inside and outside the particles is sought in ϫ ϭϪ Ϫ the form ͑␭͒cosh͑␭␩1͒ C͓0͔͑␭͒exp͑ ␭␩1͒, ͑e͒ ͑e͒ ͑e͒ ͑e͒ Ϫ2 exp Ϫ ϩA cosh ϪB T ͑␨,␩͒ϭT͓0͔͑␨,␩͒ϩT͓1͔͑␨,␩͒ϩT͓2͔͑␨,␩͒ϩ..., ␭ ͑ ␭␩2͒ ͓0͔͑␭͒ ͑␭␩2͒ ͓0͔ ͑i͒ ͑i͒ ͑i͒ ͑i͒ ϫ͑␭͒sinh͑␭␩2͒ϭD͓0͔͑␭͒exp͑Ϫ␭␩2͒, T␣ ͑␨,␩͒ϭT␣,͓0͔͑␨,␩͒ϩT␣,͓1͔͑␨,␩͒ϩT␣,͓2͔͑␨,␩͒ϩ... . (e) (i) Here the temperature perturbations T͓1͔(␨,␩) and T␣,͓1͔ 2␭ exp͑Ϫ␭␩2͒ϩA͓0͔͑␭͒sinh͑␭␩2͒ϪB͓0͔ ϫ(␨,␩) in an approximation linear in the small parameters can be written as ϫ͑␭͒cosh͑␭␩2͒ϭϪD͓0͔͑␭͒exp͑Ϫ␭␩2͒, ͑e͒ ͑e͒ ͑e͒ whence we have T͓1͔͑␨,␩͒ϭ␧1t1 ͑␨,␩͒ϩ␧2t2 ͑␨,␩͒, ͑i͒ ͑i͒ ͑i͒ A ͑␭͒ϭB ͑␭͒ϭ0, C ͑␭͒ϭϪD ͑␭͒ϭ2␭, T␣,͓1͔͑␨,␩͒ϭ␧1t␣,1͑␨,␩͒ϩ␧2t␣,2͑␨,␩͒. ͓0͔ ͓0͔ ͓0͔ ͓0͔ ͑27͒ Since the Laplacian is a linear operator, the functions ͑e͒ ͑i͒ ͑e͒ ͑i͒ T͓0͔͑␨,␩͒ϭT␣,͓0͔͑␨,␩͒ϭz. ͑28͒ T͓0͔͑␨,␩͒, T␣,͓0͔͑␨,␩͒, t͑e͒͑␨,␩͒, t͑e͒͑␨,␩͒, t͑i͒ ͑␨,␩͒, t͑i͒ ͑␨,␩͒ Thus in the zeroth approximation ␧1ϭ␧2ϭ0 the tem- 1 2 ␣,1 ␣,2 perature gradient is constant everywhere in space. are solutions of Eqs. ͑9͒ in a tangential spherical system The functions ͑21͒–͑23͒ satisfy the conditions ͑␨,␩,␸͒ of coordinates of revolution7 t͑e͒͑␨,␩͒ ϭt͑i͒͑␨,␩͒ , ͑29͒ ϱ 1 ͉S1 1,1 ͉S1 T͑e͒͑␨,␩͒ϭzϩ͑␨2ϩ␩2͒1/2 ͑A ͑␭͒cosh͑␭␩͒ϩB ͓0͔ ͵ ͓0͔ ͓0͔ , t͑e͒͑␨,␩͒Ϫٌ T͑e͒͑␨,␩͒͒ ϭٌ͑ t͑i͒͑␨,␩͒͒ ٌ͑ 0 ␩ 1 ␩ ͓0͔ ͉S1 ␩ 1,1 ͉S1 ϫ͑␭͒sinh͑␭␩͒͒J0͑␭␨͒d␭, ͑18͒ ͑30͒ ϱ t͑e͒͑␨,␩͒ ϭt͑i͒͑␨,␩͒ , ͑31͒ ͑i͒ 2 2 1/2 Ϫ␭␩ 2 ͉S1 1,2 ͉S1 T1,͓0͔͑␨,␩͒ϭ͑␨ ϩ␩ ͒ C͓0͔͑␭͒e J0͑␭␨͒d␭, ͵0 t͑e͒͑␨,␩͒͒ ϭٌ͑ t͑i͒͑␨,␩͒͒ , ͑32͒ ٌ͑ 19͒͑ ␩ 2 ͉S1 ␩ 1,2 ͉S1 ϱ ͑i͒ 2 2 1/2 ϩ␭␩ ͑e͒ ͑i͒ T2,͓0͔͑␨,␩͒ϭ͑␨ ϩ␩ ͒ D͓0͔͑␭͒e J0͑␭␨͒d␭, t1 ͑␨,␩͉͒S ϭt2,1͑␨,␩͉͒S , ͑33͒ ͵0 2 2 ͑20͒ t͑e͒͑␨,␩͒͒ ϭٌ͑ t͑i͒͑␨,␩͒͒ , ͑34͒ ٌ͑ ␩ 1 ͉S2 ␩ 2,1 ͉S2 ϱ ͑e͒ 2 2 1/2 ͑ j͒ ͑ j͒ t j ͑␨,␩͒ϭ͑␨ ϩ␩ ͒ ͑A͓1͔͑␭͒cosh͑␭␩͒ϩB͓1͔ ͑e͒ ͑i͒ ͵0 t ͑␨,␩͒ ϭt ͑␨,␩͒ , ͑35͒ 2 ͉S2 2,2 ͉S2

ϫ͑␭͒sinh͑␭␩͒͒J ͑␭␨͒d␭, ͑21͒ e , t͑e͒͑␨,␩͒Ϫٌ T͑ ͒͑␨,␩͒͒ ϭٌ͑ t͑i͒͑␨,␩͒͒ ٌ͑ 0 ␩ 2 ␩ ͓0͔ ͉S2 ␩ 2,2 ͉S2 ϱ ͑i͒ 2 2 1/2 ͑ j͒ Ϫ␭␩ t1,j͑␨,␩͒ϭ͑␨ ϩ␩ ͒ C͓1͔͑␭͒e J0͑␭␨͒d␭, ͑22͒ ͑e͒ ͑i͒ ͵ t ͑␨,␩͉͒r ϱϭ0, t ͑␨,␩͉͒␨ ϱ Ͻϱ, ͑36͒ 0 j 1,j ␩→ ϱ → → ϱ ͑i͒ ͑i͒ 2 2 1/2 ͑ j͒ ϩ␭␩ t2,j͑␨,␩͉͒␨ ϱ Ͻϱ. ͑37͒ t2,j͑␨,␩͒ϭ͑␨ ϩ␩ ͒ D͓1͔͑␭͒e J0͑␭␨͒d␭ ␩→ ϱ ͵0 → From the boundary conditions ͑29͒–͑36͒ with relations ͑ jϭ1;2͒. ͑23͒ ͑A2͒ and ͑A3͒ taken into account it is easy to obtain the The functions ͑18͒–͑20͒ satisfy the conditions algebraic equations 640 Tech. Phys. 43 (6), June 1998 S. N. D’yakonov and Yu. I. Yalamov

͑1͒ ͑1͒ HYDRODYNAMIC PROBLEM A͓1͔͑␭͒cosh͑␭␩1͒ϩB͓1͔͑␭͒sinh͑␭␩1͒ ͑1͒ We seek the solution of the Stokes equation ͑8͒ in the ϪC 1 ͑␭͒exp͑Ϫ␭␩1͒ϭ0, ͓ ͔ form ͑1͒ ͑1͒ ͑1͒ A͓1͔͑␭͒sinh͑␭␩1͒ϩB͓1͔͑␭͒cosh͑␭␩1͒ϩC͓1͔͑␭͒ 1 ⌿͑e͒͑␨,␩͒ϭϪ ␥2Uϩ⌿˜ ͑e͒͑␨,␩͒. 2 1 2 ϫexp͑Ϫ␭␩1͒ϭϪ 2␭Ϫ exp͑Ϫ␭␩1͒, 3 ͩ ␩1ͪ An expression for the distortion ⌿˜ (e)(␨,␩) of the stationary velocity field near an aggregate, bounded on the sym- A͑2͒͑␭͒cosh͑␭␩ ͒ϩB͑2͒͑␭͒sinh͑␭␩ ͒ ͓1͔ 1 ͓1͔ 1 metry axis (␨ϭ0) of the flow and vanishing at infinity ͑2͒ (␨ϭ␩ϭ0), was first obtained in Ref. 8 ϪC͓1͔͑␭͒exp͑Ϫ␭␩1͒ϭ0, ␨ ϱ ͑2͒ ͑2͒ ˜ ͑e͒ A͓1͔͑␭͒sinh͑␭␩1͒ϩB͓1͔͑␭͒cosh͑␭␩1͒ ⌿ ͑␨,␩͒ϭ 2 2 3/2 W͑␭,␩͒J1͑␭␨͒d␭, ͑␨ ϩ␩ ͒ ͵0 ϩC͑2͒͑␭͒exp͑Ϫ␭␩ ͒ϭ0, ͓1͔ 1 W͑␭,␩͒ϭ͓a͑␭͒ϩ␩c͑␭͔͒sinh͑␭␩͒ ͑1͒ ͑1͒ A͓1͔͑␭͒cosh͑␭␩2͒ϪB͓1͔͑␭͒sinh͑␭␩2͒ ϩ͓b͑␭͒ϩ␩d͑␭͔͒cosh͑␭␩͒. ͑1͒ ϪD͓1͔͑␭͒exp͑Ϫ␭␩2͒ϭ0, We seek the functions X(␭)ϭa(␭), b(␭), c(␭), and d(␭) of the parameter ␭ in the form of a power series in ␧1 ͑1͒ ͑1͒ A͓1͔͑␭͒sinh͑␭␩2͒ϪB͓1͔͑␭͒cosh͑␭␩2͒ and ␧2 ͑1͒ ͑1͒ ͑2͒ ϩD͓1͔͑␭͒exp͑Ϫ␭␩2͒ϭ0, X͑␭,␧1 ,␧2͒ϭX͓0͔͑␭͒ϩ␧1X͓1͔͑␭͒ϩ␧2X͓1͔͑␭͒ϩ... . Each term in these expansions is determined from the A͑2͒͑␭͒cosh͑␭␩ ͒ϪB͑2͒͑␭͒sinh͑␭␩ ͒ ͓1͔ 2 ͓1͔ 2 boundary conditions on the surface of an aggregate of touch- ͑2͒ ing particles. Evidently, we can write ϪD͓1͔͑␭͒exp͑Ϫ␭␩2͒ϭ0, ͑1͒ ͑2͒ ͑2͒ ͑2͒ W͑␭,␩,␧1 ,␧2͒ϭW͓0͔͑␭,␩͒ϩ␧1W͓1͔͑␭,␩͒ A͓1͔͑␭͒sinh͑␭␩2͒ϪB͓1͔͑␭͒cosh͑␭␩2͒ϩD͓1͔͑␭͒ ϩ␧ W͑2͒ ␭,␩͒ϩ..., 2 1 2 ͓1͔͑ ϫexp͑Ϫ␭␩ ͒ϭ 2␭Ϫ exp͑Ϫ␭␩ ͒, 2 3 ͩ ␩ ͪ 2 W ͑␭,␩͒ϭ͑a ͑␭͒ϩ␩c ͑␭͒͒sinh͑␭␩͒ 2 ͓0͔ ͓0͔ ͓0͔

Hence we have ϩ͑b͓0͔͑␭͒ϩ␩d͓0͔͑␭͒͒cosh͑␭␩͒,

͑1͒ ͑1͒ ͑1͒ ͑ j͒ ͑ j͒ ͑ j͒ A͓1͔͑␭͒ϭB͓1͔͑␭͒ϭD͓1͔͑␭͒ W͓1͔͑␭,␩͒ϭ͑a͓1͔͑␭͒ϩ␩c͓1͔͑␭͒͒sinh͑␭␩͒ ͑ j͒ ͑ j͒ 1 1 ϩ͑b͓1͔͑␭͒ϩ␩d͓1͔͑␭͒͒cosh͑␭␩͒. ϭϪ 2␭Ϫ exp͑Ϫ2␭␩1͒, ͑38͒ 3 ͩ ␩1ͪ An expression for the resultant force exerted by the external medium on an aggregate of touching spheres moving 1 1 ͑1͒ in the direction along the line joining their centers is pre- C͓1͔͑␭͒ϭϪ 2␭Ϫ , ͑39͒ 3 ͩ ␩1ͪ sented in Ref. 8:

͑2͒ ͑2͒ ͑2͒ ϱ A͓1͔͑␭͒ϭϪB͓1͔͑␭͒ϭC͓1͔͑␭͒ Fz͑aggregate͒ϭ ␭b͑␭͒d␭. ͵0 1 1 ϭ 2␭Ϫ exp͑Ϫ2␭␩ ͒, ͑40͒ 3 ͩ ␩ ͪ 2 Then, since the small parameters ␧1Ӷ1 and ␧2Ӷ1 are 2 arbitrary, we have from the condition ͑16͒ of uniform motion of an aggregate ͑2͒ 1 1 D 1 ͑␭͒ϭ 2␭Ϫ . ͑41͒ ͓ ͔ 3 ͩ ␩ ͪ ϱ 2 ␭b͓0͔͑␭͒d␭ϭ0, ͑43͒ Using the results ͑28͒, ͑38͒, and ͑40͒, we have ͵0

ϱ ϱ 1 ϱ 1 ͑1͒ ͑2͒ ͑e͒ 2 2 1/2 ␭b͓1͔͑␭͒d␭ϭ ␭b͓1͔͑␭͒d␭ϭ0. ͑44͒ T ͑␨,␩͒ϭzϪ ␧1͑␨ ϩ␩ ͒ 2␭Ϫ ͵0 ͵0 3 ͵0 ͩ ␩ ͪ 1 In what follows we employ the temperature distribution ϫexp͑Ϫ2␭␩ ϩ␭␩͒J ͑␭␨͒d␭ 1 0 ͑42͒ near an aggregate of touching solid spheres. Using Eqs. 1 ϱ 1 A4 and A5 , we ﬁnd from the boundary conditions 12 2 2 1/2 ͑ ͒ ͑ ͒ ͑ ͒ ϩ ␧2͑␨ ϩ␩ ͒ 2␭Ϫ and 13 3 ͵0 ͩ ␩ ͪ ͑ ͒ 2 Ϫ1 W͓0͔͑␭,␩͉͒␩ϭ␩ ϭ2͑␩1ϩ␭ ͒exp͑Ϫ␭␩1͒, ͑45͒ ϫexp͑Ϫ2␭␩2Ϫ␭␩͒J0͑␭␨͒d␭. ͑42͒ 1 Tech. Phys. 43 (6), June 1998 S. N. D’yakonov and Yu. I. Yalamov 641

W͑1͒͑␭,␩͒ץ W ͑␭,␩͒ץ ͓0͔ ͓1͔ 1 ͑1͒ ϭ2␦␩1␭ exp͑Ϫ␭␩1͒, ͑46͒ ϭϪ2␩ ͑1ϩ␦͒ϩU ␭ exp͑Ϫ␩ ␭͒, ␩ ͯ 1ͭ3 ͓1͔ͮ 1ץ ͯ ␩ץ ␩ϭ␩ 1 ␩ϭ␩1 ͑52͒ W ͑␭,␩͒ ϭ2͑␩ ϩ␭Ϫ1͒exp͑Ϫ␭␩ ͒, ͑47͒ ͓0͔ ͉␩ϭϪ␩2 2 2 W͑1͒͑␭,␩͒ ϭ2U͑1͒͑␩ ϩ␭Ϫ1͒exp͑Ϫ␩ ␭͒, ͑53͒ ͓1͔ ͉␩ϭϪ␩2 ͓1͔ 2 2 W͓0͔͑␭,␩͒ץ ϭϪ2␦␩ ␭ exp͑Ϫ␭␩ ͒, ͑48͒ ␩ ͯ 2 2 ͑1͒ץ W͓1͔͑␭,␩͒ 2ץ ␩ϭϪ␩ 2 ϭ2U͑1͒␩ ␭ exp͑Ϫ␩ ␭͒Ϫ ␩ ͯ ͓1͔ 2 2 3ץ ͑e͒ ␯ ␩ϭϪ␩2 ͑e͒ 0 KTSL ͑e͒ AT T0 ␩1 ␦ϭ4 Ϫ1. ͑49͒ ϫ͑1ϩ␦͒͑␩1ϩ␩2͒ 2ϩ U 0 ͭ ␩1ϩ␩2 ͓ ͔ T The dimensional thermophoretic velocity U͓0͔ϭϪU͓0͔ is determined in the zeroth approximation by solving the Ϫ2␩1␭ͮ ␭ exp͑Ϫ␣1␭͒, ͑54͒ system ͑45͒–͑48͒ of linear inhomogeneous algebraic equations, taking Eq. ͑43͒ into account, W͑2͒͑␭,␩͒ ϭ2U͑2͒͑␩ ϩ␭Ϫ1͒exp͑Ϫ␩ ␭͒, ͑55͒ ͓1͔ ͉␩ϭ␩1 ͓1͔ 1 1 ϱ ␭ W͑2͒͑␭,␩͒ץ ⌽ ͑␭͒d␭ ͐ ͑e͒ D 1 ͓1͔ 4 ␯0 0 ͯ ␩ץ UT ϭϪ K͑e͒ A , ␦ϭ , ͑50͒ ͓0͔ 1ϩ␦ TSL T͑e͒ T ϱ ␭2 ␩ϭ␩1 0 ͐ ⌽͑␭͒d␭ D 2 0 ϭϪ2U͑2͒␩ ␭ exp͑Ϫ␩ ␭͒ϩ ͓1͔ 1 1 3 Ϫ1 ˜ ⌽1͑␭͒ϭ͑␩1ϩ␭ ͒exp͑Ϫ␭␩1͒D1 ␩2 ϩ ϩ Ϫ1 exp Ϫ D˜ , ϫ͑1ϩ␦͒͑␩1ϩ␩2͒ 2ϩ Ϫ2␩2␭ ␭ ͑␩2 ␭ ͒ ͑ ␭␩2͒ 3 ͭ ␩1ϩ␩2 ͮ ˜ ˜ ⌽͑␭͒ϭ␩1 exp͑Ϫ␭␩1͒D2Ϫ␩2 exp͑Ϫ␭␩2͒D4 , ϫexp͑Ϫ␣2␭͒, ͑56͒

2 2 2 ͑2͒ ͑2͒ Ϫ1 DϭϪsinh ͑␭͑␩1ϩ␩2͒͒ϩ͑␩1ϩ␩2͒ ␭ , W ͑␭,␩͒ ϭ2U ͑␩ ϩ␭ ͒exp͑Ϫ␩ ␭͒, ͑57͒ ͓1͔ ͉␩ϭϪ␩2 ͓1͔ 2 2 Ϫ1 D˜ϭϪ2 ϩ exp Ϫ D˜ ϩ2 ͑2͒ W ͑␭,␩͒ץ ␩1 ␭ ͒ ͑ ␭␩1͒ 1 ␦␩1␭͑ ͓1͔ 1 ͑2͒ ϭϪ2␩2 ͑1ϩ␦͒ϩU ␭ ␩ ͯ ͭ3 ͓1͔ͮץ ϫexp͑Ϫ␭␩ ͒D˜ Ϫ2͑␩ ϩ␭Ϫ1͒ 1 2 2 ␩ϭϪ␩2 ˜ ˜ ϫexp͑Ϫ␭␩2͒D3Ϫ2␦␩2␭ exp͑Ϫ␭␩2͒D4 , ϫexp͑Ϫ␩2␭͒, ˜ D1ϭϪ␩2␭͕͑␩1ϩ␩2͒␭ cosh͑␭␩1͒ϩsinh͑␭␩1͖͒ ␣1ϭ2␩1ϩ␩2 , ␣2ϭ␩1ϩ2␩2 . ͑58͒

ϩ͕␩1␭ cosh͑␭͑␩1ϩ␩2͒͒ The additional conditions ͑44͒ make it possible to write

2 ϩsinh͑␭͑␩1ϩ␩2͖͒͒sinh͑␭␩2͒, ϱ ␭ ͐ ⍀ ͑␭͒d␭ D j D˜ ϭϪ͑␩ ϩ␩ ͒␩ ␭ sinh͑␭␩ ͒ 1ϩ␦ 0 2 1 2 2 1 U͑ j͒ ϭ͑Ϫ1͒ j ␤ j, ␤ jϭ , ͑59͒ ͓1͔ 3 ϱ ␭ ϩ␩1 sinh͑␭͑␩1ϩ␩2͒sinh͑␭␩2͒, ͐ ⍀͑␭͒d␭ 0 D ˜ D3ϭϪ␩1␭͕͑␩1ϩ␩2͒␭ cosh͑␭␩2͒ϩsinh͑␭␩2͖͒ Ϫ1 ˜ ⍀͑␭͒ϭ͑␩1ϩ␭ ͒exp͑Ϫ␭␩1͒D1ϩ␩1␭ ϩ͕␩2␭ cosh͑␭͑␩1ϩ␩2͒͒ ˜ Ϫ1 ϫexp͑Ϫ␭␩1͒D2ϩ͑␩2ϩ␭ ͒ ϩsinh͑␭͑␩1ϩ␩2͖͒͒sinh͑␭␩1͒, ˜ ˜ ˜ ϫexp͑Ϫ␭␩2͒D3Ϫ␩2␭ exp͑Ϫ␭␩2͒D4 , D4ϭ͑␩1ϩ␩2͒␩1␭ sinh͑␭␩2͒ ˜ Ϫ␩2 sinh͑␭͑n1ϩ␩2͒͒sinh͑␭␩1͒. ⍀1͑␭͒ϭ␩1 exp͑Ϫ␭␩1͒D2ϩ͑␩1ϩ␩2͒ (1) (2) The corrections ␧1U͓1͔ and ␧2U͓1͔ ͑due to the difference ␩1 ϫ 2ϩ Ϫ2␩ ␭ exp͑Ϫ␣ ␭͒D˜ , of the thermal conductivities of the external medium and the ͩ ␩ ϩ␩ 1 ͪ 1 4 1 2 particles in the aggregate͒ are ﬁnally found with the aid of the transformations ͑A1͒, ͑A2͒, ͑A4͒–͑A6͒ from the system ␩2 ⍀ ͑␭͒ϭ͑␩ ϩ␩ ͒ 2ϩ Ϫ2␩ ␭ of algebraic equations ͑51͒–͑58͒: 2 1 2 ͩ ␩ ϩ␩ 2 ͪ 1 2 ͑1͒ ͑1͒ Ϫ1 W͓1͔͑␭,␩͉͒␩ϭ␩ ϭ2U͓1͔͑␩1ϩ␭ ͒exp͑Ϫ␩1␭͒, ͑51͒ ˜ ˜ 1 ϫexp͑Ϫ␣2␭͒D2ϩ␩2 exp͑Ϫ␭␩2͒D4 . 642 Tech. Phys. 43 (6), June 1998 S. N. D’yakonov and Yu. I. Yalamov

TABLE I. TABLE II.

(1) (2) (1) (2) R1 /R2 ␦ ␤ ␤ R2 /R1 ␦ ␤ ␤ 1 4.69309 7.61846ϫ10Ϫ2 Ϫ7.61846ϫ10Ϫ2 1 4.69309 7.61846ϫ10Ϫ2 Ϫ7.61846ϫ10Ϫ2 2 4.77724 1.31308ϫ10Ϫ1 Ϫ2.54592ϫ10Ϫ2 2 4.77724 2.54592ϫ10Ϫ2 Ϫ1.31308ϫ10Ϫ1 3 4.85906 1.49923ϫ10Ϫ1 Ϫ1.08006ϫ10Ϫ2 3 4.85906 1.08006ϫ10Ϫ2 Ϫ1.49923ϫ10Ϫ1 4 4.90815 1.57435ϫ10Ϫ1 Ϫ5.50003ϫ10Ϫ3 4 4.90815 5.50003ϫ10Ϫ3 Ϫ1.57435ϫ10Ϫ1 5 4.93750 1.61028ϫ10Ϫ1 Ϫ3.16567ϫ10Ϫ3 5 4.93750 3.16567ϫ10Ϫ3 Ϫ1.61028ϫ10Ϫ1 6 4.95576 1.62964ϫ10Ϫ1 Ϫ1.98498ϫ10Ϫ3 6 4.95576 1.98498ϫ10Ϫ3 Ϫ1.62964ϫ10Ϫ1 7 4.96762 1.64103ϫ10Ϫ1 Ϫ1.32536ϫ10Ϫ3 7 4.96762 1.32536ϫ10Ϫ3 Ϫ1.64103ϫ10Ϫ1 8 4.97563 1.64816ϫ10Ϫ1 Ϫ9.28434ϫ10Ϫ4 8 4.97563 9.28434ϫ10Ϫ4 Ϫ1.64816ϫ10Ϫ1 9 4.98121 1.65287ϫ10Ϫ1 Ϫ6.75435ϫ10Ϫ4 9 4.98121 6.75435ϫ10Ϫ4 Ϫ1.65287ϫ10Ϫ1 10 4.98522 1.65610ϫ10Ϫ1 Ϫ5.06618ϫ10Ϫ4 10 4.98522 5.06618ϫ10Ϫ4 Ϫ1.65610ϫ10Ϫ1 20 4.99735 1.66499ϫ10Ϫ1 Ϫ7.22619ϫ10Ϫ5 20 4.99735 7.22619ϫ10Ϫ5 Ϫ1.66499ϫ10Ϫ1 30 4.99911 1.66613ϫ10Ϫ1 Ϫ2.24074ϫ10Ϫ5 30 4.99911 2.24074ϫ10Ϫ5 Ϫ1.66613ϫ10Ϫ1 40 4.99960 1.66643ϫ10Ϫ1 Ϫ9.67350ϫ10Ϫ6 40 4.99960 9.67350ϫ10Ϫ6 Ϫ1.66643ϫ10Ϫ1

ANALYSIS OF RESULTS ϭ0 the velocity of an aggregate is higher than that of any The dimensional thermophoretic velocity of an aggre- single particle and the effect of the form of the aggregation is gate in an approximation linear in the small parameters can greatest when equal spheres touch ͑it equals at least 5%͒. be written as follows taking expressions ͑50͒ and ͑59͒ into Evidently, the theory of thermophoresis constructed on account: the basis of a hydrodynamic analysis is also valid for an ͑e͒ aggregate of touching solid hydrosol particles and high- ␯0 1 1 1 UTϭϪ4K͑e͒ A Ϫ ␧ ␤͑1͒ϩ ␧ ␤͑2͒ . viscosity pure drops. This case is most important for practi- TSL T͑e͒ Tͭ1ϩ␦ 3 1 3 2 ͮ 0 cal applications. ͑60͒ It is of interest to check in the course of the numerical analysis the agreement between Eq. ͑60͒ in the limiting cases APPENDIX of thermophoresis of an aggregate of touching solid nonvola- ϱ tile spheres (R1ӷR2 or R2ӷR1͒ with the result obtained 2 2 Ϫ1/2 1 exp͑Ϫ␭␩͒J0͑␭␨͒d␭ϭ͑␨ ϩ␩ ͒ , ͑A1͒ earlier for a single particle ͵0 2 ␯͑e͒ T ୐␣ ͑e͒ 0 ϱ U ϭϪ K AT . ͑61͒ 2 2 Ϫ3/2 1ϩ2୐ TSL T͑e͒ exp͑Ϫ␭␩͒J0͑␭␨͒␭d␭ϭ␩͑␨ ϩ␩ ͒ , ͑A2͒ ␣ 0 ͵0 Let us expand the expression 2୐ /(1ϩ2୐ ) in a power ␣ ␣ ϱ series in ␧ 2 2 2 2 Ϫ5/2 ␣ exp͑Ϫ␭␩͒J0͑␭␨͒␭ d␭ϭ3␩ ͑␨ ϩ␩ ͒ ͵0 2୐␣ 2 1 2 2 ϭ 1Ϫ ␧ Ϫ ␧ Ϫ... . 2 2 Ϫ3/2 1ϩ2୐ 3 ͩ 3 ␣ 9 ␣ ͪ Ϫ͑␨ ϩ␩ ͒ , ͑A3͒ ␣ Then, comparing the right-hand sides of Eqs. ͑60͒ and ϱ Ϫ1 ␨ 61 for the limiting cases R ӷR or R ӶR shows that the exp͑Ϫ␭␩͒͑␩ϩ␭ ͒J1͑␭␨͒d␭ϭ 2 2 1/2 , ͑ ͒ 1 2 1 2 ͵0 ͑␨ ϩ␩ ͒ following equalities are satisfied: ͑A4͒ 4 2 ϭ ͑whence ␦ϭ5͒ ͑62͒ ϱ ␨ 1ϩ␦ 3 exp͑Ϫ␭␩͒J1͑␭␨͒␭d␭ϭ 2 2 3/2 , ͑A5͒ ͵0 ͑␨ ϩ␩ ͒ in the zeroth approximation and ϱ 2 ␨␩ 1 ␧1 for R1ӷR2 , exp͑Ϫ␭␩͒J1͑␭␨͒␭ d␭ϭ3 2 2 5/2 . ͑A6͒ ͑1͒ ͑2͒ ͵0 ͑␨ ϩ␩ ͒ ␧1␤ Ϫ␧2␤ ϭ ͑63͒ 6 ͭ ␧2 for R2ӷR1 in the first approximation. 1 Yu. I. Yalamov and V. S. Galoyan, Dynamics of Droplets in Inhomoge- The limiting cases ͑62͒ and ͑63͒ are confirmed by the neous Viscous Media ͓in Russian͔,Lus, Erevan, 1985, 208 pp. 2 results of a numerical analysis ͑presented in Tables I and II͒. Yu. I. Yalamov, M. N. Gadukov, and A. P. Melekhov, Dokl. Akad. Nauk SSSR 287, 337 ͑1986͓͒Sov. Phys. Dokl. 31, 251 ͑1986͔͒. It should be noted that in the zeroth approximation in the 3 Yu. I. Yalamov, E. R. Shchukin, and S. I. Grashchenkov, Thermophoresis small parameters ͑corresponding to the thermal conductivity and Diffusiophoresis of Two Aerosol Particles with Allowance for Internal of the external medium being the same as that of the touch- Heat Sources ͓in Russian͔, MOPI, Moscow, 1989, 118 pp.; Deposited in VINITI, No. 7212-V89. ing nonvolatile solid spheres͒ the thermophoretic velocity of 4 Yu. I. Yalamov, M. N. Gadukov, and V. V. Levin, Thermophoresis of an aggregate remains unchanged when the particles are made Two Particles at Low Reynolds Numbers ͓in Russian͔, MPU, Moscow, to change places (R R , R R ͒. In the case ␧ ϭ␧ 1993, 9 pp.; Deposited in VINITI, No. 1029-V93. 1→ 2 2→ 1 1 2 Tech. Phys. 43 (6), June 1998 S. N. D’yakonov and Yu. I. Yalamov 643

5 L. D. Reed and F. A. Morrison, J. Aerosol Sci. 6, 349 ͑1975͒. 8 M. D. A. Cooley and M. E. O’Neill, Proc. Cambridge Philos. Soc. 66, 407 6 M. Stimson and G. B. Jeffrey, Proc. R. Soc. London, Ser. A 111, 110 ͑1969͒. ͑1926͒. 7 O. O. Olaru, Vestn. Mosk. Univ. Ser. 1, Matem. Mekh., No. 3, 73 ͑1992͒. Translated by M. E. Alferieff TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Determination of nonlinear aerodynamic characteristics from trajectory data of an object: modiﬁcation of the method for complicated cases A. B. Podlaskin

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia ͑Submitted March 26, 1997͒ Zh. Tekh. Fiz. 68, 32–36 ͑June 1998͒ The method of calculating the nonlinear aerodynamic characteristics of objects from trajectory data, based on the differential correction principle, is well known ͓N. P. Mende, FTI Preprint No. 1326, A. F. Ioffe Physicotechnical Institute, Leningrad ͑1989͒; G. T. Chapman and D. B. Kirk, AIAA J. 8, 753 ͑1970͔͒. A modiﬁcation of this method is proposed here, in which the solution is to be obtained in the form of a spline. This new approach, which has been tested on model problems, can provide a more reliable guarantee of adequacy of the solutions and an improvement in accuracy in cases where the functional relations sought have a complicated form. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00706-5͔

INTRODUCTION of the variance matrix of the regression coefficients of the unknown functions. The complexity of the question of ad- One problem of experiments on ballistic apparatus is to equacy of the solution found by the trial-and-error method is determine the aerodynamic characteristics, i.e., the forces well known, since the efficiency of the method in a given and torques acting on a body in free flight. In this context, case is determined by the errors of the experimental data. researchers have at their disposal trajectory data—an ordered set of readings of the coordinates of the object at fixed in- stants of time. These coordinates and the desired functional STATEMENT OF PROBLEM dependences of the aerodynamic forces and torques are in- Among the objects of ballistic studies, a few are encoun- 1 terrelated by a system of nonlinear differential equations, tered that possess a quite complicated form of dependence which are not analytically integrable. Thus the problem takes z(x) of the aerodynamic coefficients on the generalized co- the general form ordinate. Although the aerodynamic coefficients, as a rule, are described by smooth functions, they can, for example, be Az͑x͒ϭu͑x͒, ͑1͒ substantially nonmonotonic. Complexities of this kind usu- where u(x)ϭx is the vector of measured coordinates, z is the ally show up when the range of variation of the independent vector of aerodynamic coefficients ͑thus z(x) are the desired variable is increased. In such cases a model description of functional relations͒, and A is the integral operator of the the motion of the object using power series segments as the problem. This ill-posed problem is prescribed by an operator regression form can be assumed to be inefficient. The more equation of the first kind.2 complicated the form of the unknown function, the further Only the inverse operator AϪ1 can be written down ana- out must its approximating series be carried. As the number lytically. To solve this problem, the trial-and-error method of terms of the series increases, the conditionality of the has traditionally been used, i.e., for a series of approxima- least-squares matrices deteriorates. This is a well-known fact tions of the unknown functions the direct problem is solved in regression analysis.4 Computational algorithms exist ͑e.g., and the deviation of the calculated trajectory from the experi- Ref. 5͒ which allow one to solve the corresponding underde- mental trajectory is calculated. The first approximation is termined systems of equations. However, other shortcomings chosen on the basis of an a priori expert assessment. As the of higher-degree polynomials do not allow one to go this algorithm for constructing the minimizing sequence the dif- route in the case of a complicated form of z(x). Experience ferential correction method has proved to be successful3 ͑it is shows that the confidence interval for the sum of the series also known as the Gauss–Newton method͒. The class of segment increases severalfold at the end of the approxima- functions among which the stable solution is to be sought tion interval due to errors in the coefficients of the higher ͑the correctness class of the problem͒ can be restricted in powers of the independent variable. In addition, since a different ways. The only condition is that the functional sub- power-law basis is not orthogonal, introducing higher pow- space under consideration be compact.2 Convergence of the ers in the approximation of the unknown function on large method in the class of power-law polynomials containing a intervals alters the values of the regression coefficients of the finite number of terms of even or odd powers was confirmed lower powers. The presence of a large number of terms can in Ref. 1, where examples are given of successful determi- lead to the appearance of nonphysical oscillations. All this nation of aerodynamic coefficients on ballistic experiments motivates us to look for other forms of representation of the with sharp cones. The practical importance of a statistical unknown functions to use in the trial-and-error search for the assessment of the results was also demonstrated1 on the basis solution of the ill-posed problem under consideration here.

As a particular case of z(x), the lower graph plots the static aerodynamic moment coefficient as a function of the angle of attack, Cm(␣). This function 3 Cm͑␣͒ϭsign͑␣͓͒Cmax sin ͉͑␣͉ϩ␣0͒ϩC0͔, which I used as my model example, cannot in general be described by a power series with a finite number of terms. In regard to the problem under consideration this means that the Chapman–Kirk algorithm3 does not allow us to seek out a solution that would adequately describe the given moment function. Increasing the degree of the polynomial to 13 does not achieve the desired model adequacy even though the elements of the matrix of normal equations at this point have already reached the limit of machine accuracy. If we consider segments of its domain of definition, e.g., as shown by the dashed lines, then it is obvious that on these intervals the function can be well fitted by lower-degree polynomials. The upper curve is the ‘‘trajectory’’ of the oscillations of a body in flight, i.e., the dependence of the attack angle on the longitudinal coordinate along the trajectory. From the matching condition for the segments of the unknown function using a quadratic polynomial as the regression function to fit the functional dependence Cm(␣) on the domain ͉␣͉ ෈͓␣i ,␣iϩ1͔ we have

Cm͑␣͒ϭsign͑␣͒Cm͑␣i͒ϩ͓dCm͑␣i͒/d␣͔ 2 ϫ͑␣Ϫ␣i͒sign͑␣͒kiϩ1͑␣Ϫ␣i͒ ͑2͒ ͑the sign function was introduced in accordance with the physical requirement that the static aerodynamic moment be odd͒. Thus, the adopted regression form, possessing a piecewise-continuous second derivative, corresponds to a first-order spline. Many of the advantages of the previously developed technique for finding the aerodynamic characteristics deter- FIG. 1. Model functional dependence of the static aerodynamic moment mined the factors of continuity. Only, the use of the sign coefficient ͑lower graph͒ and example of the corresponding oscillations of function sign ͑␣͒ corresponds to the use of polynomials of the object as a function of the attack angle ͑upper graph͒. Dashed lines— even or odd degree in the Chapman–Kirk algorithm, which partition of the ͕␣͖ axis into intervals. is yet another example in the trial-and-error method of the necessity of invoking a priori information about the nature of the unknown function z(x). As in the case of an all-at- One variant that has been proposed is to partition the range once polynomial approximation of the aerodynamic charac- of variation of the independent variable x and construct a teristics, the unknown number of degrees of freedom can be regression of power series segments ‘‘matched’’ at the achieved by coprocessing the data of several experiments boundaries of the intervals in respect to the value of the with one model. In this case the number of new nodes of the function and the first derivative. In this case, within the lim- approximation increases more rapidly than the number of its of comparatively large intervals in x the unknown func- unknowns ͑the regression coefficients remain general, only tion will be described by substantially shorter polynomials. the initial conditions of the launch, which are treated as unknowns, are added on͒. Moreover, the coprocessing of the data of several launches of one object with different ampli- CONSTRUCTION OF THE METHOD tudes of oscillations in the attack angle allows one to make use of the nonisochronicity of these oscillations, which gives In light of the above considerations regarding the diffi- more information about the nonlinear characteristics of the culty of elucidating the adequacy of the desired model and moment than can be had from an analysis of the form of the the effect of hard-to-control experimental errors at the pro- oscillations. cessing stage of the algorithm, model problems were exam- As before, the main logical thrust of the algorithm con- ined, i.e., trajectory data calculated on the basis of prescribed sists in constructing a sequence of approximations of the aerodynamic characteristics of an ‘‘object.’’ Consider Fig. 1. desired solution which minimize the deviation of the calcu- 646 Tech. Phys. 43 (6), June 1998 A. B. Podlaskin lated trajectory from the experimental trajectory in a quadratic metric. To linearize the target function, i.e., the residual sum of squares of differences of the attack angle ␣,we expand it in a multidimensional Taylor series in the desired parameters in the vicinity of the solution, where this series is truncated at the level of the linear terms. In such a representation the deviation is written in terms of the derivatives of ␣ with respect to the regression coefficients of the unknown function and the initial conditions. The values of the derivatives are found from the equations of sensitivity, which are obtained by differentiating the equation of motion with respect to all the unknowns and are integrated together with the equation of motion. This approach allows one to com- pose the system of normal equations of the method of least squares—linear algebraic equations based on the equations of motion and sensitivity for all points of the scheme. How- ever, the solution of such a system of equations, which only approximate the given nonlinear problem, gives only a set of corrections to the coefficients. Therefore, the search for a solution reaches its goal after several iterations. Calculation FIG. 2. Spline-reconstructed dependence ͑crosses͒ of the static aerodynamic 6 moment coefficient from Fig. 1. Original function—squares. Solid line— of the variance matrix allows one in the course of the dif- estimate of the residual error of the ␦ approximation. ferential correction to estimate the error of the found approximations of the coefficients and assess its significance. The new form of representation of the unknown function an unpropitious choice of the partition of the domain ͕␣͖ it is necessitated certain modifications in the algorithm. In the possible to find a spurious solution. In practice it has been trial-and-error search for a solution of the operator equation found that good results in terms of efficiency of the algo- ͑1͒, when we are solving the direct problem, i.e., integrating rithm are had by assigning the boundaries at the extrema and the equations of motion, at each step the current value of ␣ is inflection points of the unknown function. For the indicated monitored with the aim of determining to which of the des- example ͑in Fig. 1 the moment is prescribed as the cube of ignated intervals it belongs. Depending on the interval to the sine of the attack angle͒, which has two inflection points which the argument belongs, the values of Cm(␣) are calcu- on the positive side and on the negative side in the range lated according to the formulas of the respective interval in ͉␣͉р40° ͑␣ХϮ13.25° and ␣ХϮ36.75°͒ and one extre- the procedures for calculating these values. Since on differ- mum (␣ϭϮ25°), acceptable results are achieved by starting ent intervals of ͕␣͖ the regression coefficients affect the tar- not with four intervals in the absolute value of ␣, but with get function in a different way: some of them directly, some six: it was necessary to divide up the regions of large curva- via their effect on the preceding segments ͓via formula ͑2͔͒, ture symmetrically near an extremum ͑increasing the number the calculation of the coefficients of the equations of sensi- of intervals above the minimum necessary serves no pur- tivity also depends on which interval ␣ belongs to. However, pose͒. Trajectory data of four ‘‘launches’’ were used, with from the derivatives calculated in this way a single matrix of amplitudes of oscillation in the attack angle equal to 7, 14, normal equations of the method of least squares is con- 27, and 38°. structed, which of course corresponds to a single equation The plan of the ‘‘experiment’’ called for 34 points. Ap- prescribing the motion of the object, and all coefficients used proximation with quadratic polynomials on six intervals to describe the unknown function turn out to be correlated. had 14 degrees of freedom. The accumulation of error of the found nonlinear regression coefficients ki went from 1 (͉␣ р␣ ) to 65% ( ␣ у␣ ). Here the achieved accuracy of ANALYSIS OF THE SOLUTION ͉ 1 ͉ ͉ 5 determination of the static moment was within 9% ͑Fig. 2͒. By varying the number and positions of the interval The error was calculated from the variance matrix of the boundaries ␣i , it is possible to achieve acceptable results in regression coefficients with their covariances taken into ac- determining Cm(␣). Acceptability is determined, on the one count. hand, by the errors of the found regression coefficients with Finding the solution of an ill-posed problem by the trial- allowance for their correlation ͑when the relative error of the and-error method2 is equivalent to minimizing the distance ␳ coefficient of the square of the independent variable exceeds ͑in the quadratic metric͒ from the intermediate approximate 100% for a prescribed confidence level, rejection of such an solution z␦ to the exact solution zT . The result of this pro- insignificant coefficient is indicated and the moment on this cess, as a rule, looks like this: the local deviations have op- interval is taken to be linear͒. On the other hand, the ad- posite signs at the boundaries of the approximation region equacy of the found solution for the model problem is easily and in its middle while its integral ␳(z␦ ,zT) over the entire monitored since the exact form of the desired function is region tends to zero. In the given case on each interval ␣i known. Experience shows that the problem does not have a Ͻ͉␣͉р␣iϩ1 the higher ͑one͒ coefficient of the approximat- unique solution in the considered class of functions and for ing parabolic segment is sought independently. This finds Tech. Phys. 43 (6), June 1998 A. B. Podlaskin 647

description of the moment function allows it to be found with less error. However, finding it is fraught with greater difficulties than finding the all-at-once polynomial approximation. There are additional problems associated with choice of the boundaries of the segments into which to divide the range of variation of ͕␣͖. Overcoming these difficulties will require time, but probably with accumulation of experience it will be possible to shorten the amount of time needed.

CONCLUSION

Analyzing the difficulties arising during processing of the algorithm for determining the aerodynamic characteristics from trajectory measurements using splines to describe the unknown functions, we can form the following picture regarding application of the method to actual experimental data. It is unrealistic to choose the initial approximation in FIG. 3. Comparison of the results of all-at-once ͑squares͒ and piecewise the form proposed using the given approach if all that is ͑crosses͒ approximation in the reconstruction of the nonlinear static aerody- available are raw data. Therefore it is advisable to begin namic moment of a sharp cone. Solid lines—values of the moment coef- processing the experimental results with the help of the ficient, dashed lines—estimate of the relative error ␦ of the corresponding Chapman–Kirk algorithm, for example. Having obtained a approximations. representation of the unknown functions, it is possible to shift over to the new algorithm and carry out a comparative analysis of the solutions found by the two approaches as to expression in the complex character of the dependence of the their adequacy ͑to the extent that it is possible to judge, error function of the found moment coefficient on the attack generally͒ and their errors. Features of the unknown func- angle: the error reflects the character of the local deviation of tions revealed in the course of fitting using the Chapman– z from z . Consideration of a model problem ͑where the ␦ T Kirk algorithm can help construct initial spline approxima- exact solution is known͒ makes it possible visually to trace tions. Here an additional optimization factor arises— out the variation of ␳ from the relative location of the curves deciding on the number of intervals in the independent of the prescribed and found functions C (␣). m variable and the positions of their boundaries. As has already The case of four launches of a sharp cone with a total been noted, for the observations just dealt with the most vertex angle of 30°, considered in Ref. 1, was also analyzed. successful locations, in the sense of convergence, for the In this case it is possible to compare the results for an all-at- boundaries are at the extrema and inflection points of the once approximation of the desired moment coefficient using unknown functions. When working with experimental data, the Chapman–Kirk algorithm ͑odd fifth-degree polynomial͒ the search for such points must be carried out one after the and for a piecewise-polynomial approximation. However, other. A future check of the fruitfulness of the proposed ap- use of the available experimental data had to be rejected by proach in this case will be possible by obtaining qualitative virtue of the sensitivity of the method to the systematic and experimental data. anomalous errors contained in them for a relatively small sample space and it was necessary to resort to modeled input I am grateful to N. P. Mende for a statement of the data. Model trajectories, calculated from the characteristics problem and for valuable discussions, and to S. V. Bobashev taken from Ref. 1, was ‘‘noised up’’ by adding normally for helpful remarks. distributed random error with a standard deviation of 0.5°, which corresponds to the error of the actual measurements. For this version of the function Cm(␣) it turned out to be sufficient to use four intervals ͑indicated by vertical line seg- 1 N. P. Mende, FTI Preprint No. 1326 ͓in Russian͔, A. F. Ioffe Physico- ments in Fig. 3͒. Comparison of the obtained curves reveals technical Institute, Leningrad ͑1989͒,44pp. 2 good agreement ͑the same solution is found͒. Estimates of A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Ill-Posed Prob- lems ͓in Russian͔͑Nauka, Moscow, 1979͒, 228 pp. the residual mean-square deviations of the attack angle ␣ of 3 G. T. Chapman and D. B. Kirk, AIAA Pap. 8, 753 ͑1970͒. 4 the trajectories calculated by the two different methods ͑␴␣ G. A. Seber, Linear Regression Analysis ͓Wiley, New York, 1977; Mir, 0.51° for the all-at-once approximation and 0.13° for Moscow, 1980, 456 pp.͔. Х ␴␣Х 5 the new method͒ favor the piecewise polynomial approxima- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Programming ͑Cam- tion. Construction of the variance matrix of regression coef- bridge University Press, New York, 1989͒, 759 pp. 6 ficients in both cases allows us to estimate the total error of S. M. Ermakov and A. A. Zhiglyavski, Mathematical Theory of the the moment within the given range ͕␣͖. Values of the confi- Optimal Experiment: A Textbook ͓in Russian͔͑Nauka, Moscow, 1987͒, dence interval of the static moment for a confidence level of 320 pp. 95% are shown in Fig. 3. It can be seen that the piecewise Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Steady-state leader breakdown. Nitrogen atmosphere A. V. Ivanovski ͑Submitted February 3, 1997͒ Zh. Tekh. Fiz. 68, 37–44 ͑June 1998͒ Physical arguments about the possible mechanism of formation of a leader channel are presented. A mathematical model describing steady-state leader breakdown is constructed. An algorithm for determining the propagation velocity, dimensions, and electric ﬁeld in the streamer zone is developed. A numerical simulation of the channel formation stage in the plasma of a streamer zone a nitrogen atmosphere is performed. The dependence of the leader velocity on the potential is obtained. The tentative model proposed here can be used to describe the leader breakdown of a long gap at high positive potentials. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00806-X͔

ϱ INTRODUCTION length, Gϭ2␲͐0 ␴rdr, is formed by a cylindrical ionization The problem of steady-state breakdown of an extended wave with nearly constant propagation velocity vr . The con- gas-filled gap with a prescribed potential U on one of the ductivity of the streamer ͑i.e., the electron concentration͒ de- 0 2 2 electrodes is considered. It is well known ͑see, e.g., Ref. 1͒ pends on the coordinate r as ␴Ϸvz /vrr (GϷvz ␶). It has that such a discharge has a complicated structure. The poten- been shown that such a regime can support the propagation tial U0 is transmitted along a highly conductive leader chan- of streamers many meters in length. Here the electric field nel at a temperature high enough for thermal ionization to varies weakly along the length of the streamer and is equal to take place. The channel is formed in the poorly conducting ϳ10 kV/m. This picture permits an explanation of the ex- streamer zone, whose dimensions are quite large Lу1m. perimentally observed3 linear dependence of the streamer ve- There are a number of conceptions about the mechanism of locity on the rate of growth of the potential for dU/dt 7 formation of the leader channel. According to one of them, it Ͼ400 kV/␮s(vzϾ4ϫ10 cm/s). We will proceed from the 2 is formed as a result of heating of a solitary streamer. Hy- concept of a streamer given in Ref. 5. potheses have been developed about the formation of the It is assumed that the leader channel is formed as a result channel against the background of a huge number of continu- of the development of an ionization–overheating instability1 3 ously forming streamers. We will proceed from the first. near the axis, where the ionization is maximum. For moder- If we neglect the potential drop in a highly conductive ate rates of current growth it is possible to restrict the dis- channel, then for gaps with large dimensions of the streamer cussion to processes at atmospheric pressure ͑without shock zone it is natural to consider the steady-state breakdown re- waves͒. The description of channel formation is complicated gime. In this case the problem of the formation of a leader in that the dimensions of the region in which the instability channel in a streamer discharge plasma is one-dimensional— develops are uncertain. The scale of this region can depend all quantities depend only on the transverse coordinate r. The z dependence is determined parametrically in terms of the on the initial conditions, e.g., the size of the streamer tip. We will neglect this scale р10Ϫ3 cm ͑Ref. 6͒, assuming that the retarded time ␶ϭtϪz/vx . For a given medium the described breakdown regime is completely determined by one channel parameters are determined by transport processes in the streamer plasma. parameter—the potential U0 , on which depend the propaga- In a quasineutral streamer plasma diffusion is ambipolar. tion velocity vz and the parameters of the streamer zone and channel. Determination of the dependence on U0 of the ve- However, heating of the electron gas is governed by the elec- locity vz and of the dimensions L and electric field E0 in the tron diffusion coefficient. On this basis, we neglect diffusion streamer zone is the aim of the present work. Here it is of charged particles in comparison with the electronic ther- assumed that vz is much greater than the electron drift ve- mal conductivity. Note that by virtue of the low electron locity, i.e., vz is determined by the charge transfer rate along concentration at the stage of channel formation the electronic the channel. An important aspect of these studies is the thermal conductivity does not contribute significantly to the model of the streamer zone of the leader. overall balance of energy. Its influence is manifested in an averaged way, through the removal of energetic electrons MODEL OF THE STREAMER ZONE capable of ionizing the gas, into the colder regions. The situation here is unclear.1 Reference 4 calls attention Below, on the basis of the concepts expounded above we to the fact that broadening of the current-carrying region of construct a calculational model of channel formation in a the cylindrical ionization wave plays a substantial role in the streamer discharge plasma. Here the main mechanism of propagation of a streamer. In Ref. 5 a regime of streamer cold gas entrainment into the channel is the electronic ther- propagation is constructed in which the conductance per unit mal conductivity.

BASIC EQUATIONS 5 ␧ϭ kTϩ␧V . ͑5͒ We assume that the plasma consists of molecular nitro- 2 gen, the corresponding ions and electrons with concentra- In writing Eq. ͑5͒ we have neglected the energy of the tions N, nϩ , ne . This assumption is justified up to tempera- electron-excited particles, assuming that it relaxes quite rap- tures of the medium Tр4000 K, where it is possible to idly into thermal energy. 1 neglect dissociation. The plasma is electrically neutral ne By virtue of retarded deactivation of the vibrational de- n and weakly ionized 10Ϫ3ϫN n , i.e., it is possible to Х ϩ у e grees of freedom, to determine ␧V it is necessary to solve the 1 neglect Coulomb collisions. Neglecting diffusive drift of the equation7 charged particles, the equation of continuity for the electron ␧Vץ ץ 1 ץ N␧V͒ 1͑ץ component takes the form ϩ ͑rv N␧ ͒Ϫ r␹N ͪ rץ ͩ rץ r 0 V rץ ␶ rץ ץ ne 1ץ ϩ ͑rv n ͒ϭk Nn Ϫk n2ϩS , ͑1͒ p r 0 e ion e r e str ␧VϪ␧Vץ ␶ rץ ϭneNSVϩN , ͑6͒ ␶V where v0 is the transverse hydrodynamic velocity. where S is the energy lost by an electron to excitation of The source term Sstr(vz ,vr ,E0) describes the creation of V electron–ion pairs at the front of the cylindrical ionization vibrations of a molecule, ␶V is the vibrational relaxation p wave forming the streamer discharge. To get the streamer time, ␧Vϭh␯/͓exp(h␯/kT)Ϫ1͔ is the vibrational energy un- discharge to propagate, it is necessary to set5 der conditions of thermodynamic equilibrium ͑h␯ ϭ0.291 eV is the energy of a vibrational quantum͒, ␹ C0 1vzvz ϭ␭/(Ncp) is the thermal diffusivity ͓c pϭ(7/2)k is the spe- Sstrϭ ␦͑␶Ϫr/vr͒, ͑2͒ 2␲ eke͑E0͒vr͑rϩr0͒ cific heat at constant pressure͔, and ␭ is the thermal conductivity. where ke(E0)ϭWe /E0 is the electron mobility, We(E0)is Transforming Eqs. ͑4͒–͑6͒, we obtain a system of hydro- the drift velocity, the capacitance of the discharge per unit dynamic equations in the temperature T, concentration N, length is C ϭ1/2Ln(2v /v ), and the quantity r has been 0 z r 0 and mass velocity v0 introduced to eliminate the singularity in the limit r 0. Tץ ץ T 1ץ Tץ →For a given electron energy distribution function Nc ϩv ϭ r␭ ϩn ͑S ϪNS ͒ r ͪ e e Vץ ͩ rץ r ͪ rץ ␶ 0ץ␧Ϫ f (␧) the ionization k and recombination k rate con- pͩ ion r stants are given by p ␧VϪ␧V ϱ ϪN , 2 ␶V kionϭͱ ͵ ␧qion͑␧͒f ͑␧͒d␧, me I Tץ ץ 1 ץ cpP0 1 ͑rv0͒ϭ r␭ ϩne͑SeϪNSV͒ ͪ rץ ͩ rץ r rץ ϱ ϱ k r 2 k ϭ ␧q ͑␧͒f ͑␧͒d␧, ͱ␧ f ͑␧͒d␧, ͑3͒ r ͱm ͵ r ͵ p e 0 0 ␧VϪ␧V ϪN , ␶V where qion and qr the ionization and recombination cross sections ͑I is the ionization potential͒; it is assumed that ion- p ␧V ␧VϪ␧Vץ ץ ␧V 1ץ ␧Vץ ization occurs from the ground state. ϩv ϭ r␹N ϩn S ϩ , r ͪ e V ␶ץ ͩ rץ r Nrץ ␶ 0ץ V To determine N and v0 it is necessary to solve the hydrodynamic equations which at constant atmospheric pres- P0 sure P take the form7 Nϭ . ͑7͒ 0 kT To determine the energy scattered by the electrons, it is ץ N 1ץ ϩ ͑rNv ͒ϭ0, P ϭNkT, r 0 0 necessary to solve the equation of energy balance, which canץ ␶ rץ be written in the form7 ץ 1 ץ 1 ץ Teץ 5 ץ 1 3 ץ N␧͒ϩ ͓rv0͑N␧ϩP0͔͒Ϫ ͑rJ͒ϭneSe , ͑4͒͑ r n kT ϩ r n kT v Ϫ␭ץ r rץ ␶ rץ ͬͪ rץ r ͫ ͩ2 e e 0 eץ ␶ ͩ 2 e eͪ rץ where we have neglected the kinetic energy in comparison e2E2 with the internal energy ␧, J is the energy flux, Se is the 0 2 ϭne e ϪneN͑SelϪSinϪSi͒Ϫne SrϩST , ͑8͒ energy transferred by an electron to the medium per second me␯m in elastic and inelastic collisions. By virtue of their small mobility the contribution of the where the electron temperature is understood as ions to heating of the medium can be neglected. The internal 3 ϱ 3/2 energy ␧ is equal to the sum of the kinetic energy ␧k kTeϭ ␧ f ͑␧͒d␧; 2 ͵0 ϭ(3/2)kT, the energy of rotational degrees of freedom, 2 which under conditions of thermodynamic equilibrium is the kinetic energy mev0/2 has been neglected in comparison equal to ␧RϭkT, and the vibrational energy ␧V with kTe ; ␭e is the electronic thermal conductivity; 650 Tech. Phys. 43 (6), June 1998 A. V. Ivanovski

2 2 e e E0/me␯m is the energy transferred to the electrons by the where INϭ14.5 eV is the ionization potential of atomic ni- field; S , S , and S are the rates of energy loss per mol- d ϩ el in i trogen and INϩϭ8.7 eV is the binding energy of the N2 ecule by an electron in elastic and inelastic collisions to ion- 2 molecule. ization of the gas; Sr is the rate of energy loss per ion by the e electrons during recombination; ␯m is the elastic scattering INTERACTION CROSS SECTIONS OF THE ELECTRON WITH frequency of the electrons; the source term ST 0 THE MEDIUM ϭ(3/2)kTeSstr describes the energy acquired by the electrons 0 at the front of the cylindrical ionization wave; and Te is the A detailed analysis of the interaction cross sections of equilibrium electron temperature in nitrogen at normal den- the electrons with molecular nitrogen was carried out in Ref. sity in the field E0 . 8. Let us describe briefly the set of constants which we will e The frequency ␯m and the electron energy loss rate are use here. To describe the elastic collisions, we employ the given by dependence of the transport cross section qm on the electron energy given in Ref. 9. We describe rotational excitation in 2 ϱ line with Refs. 10 and 11. Invoking the smallness of the ␯e ϭ N ␧q ␧ f ␧ d␧, m ͱ ͵ m͑ ͒ ͑ ͒ rotational constant B ϭ2.5ϫ10Ϫ4 eVӶkT, kT , we can me 0 0 e take the distribution over levels to be quasicontinuous. This 2 2m ϱ df allows us to replace the sum over levels j by an integral over e 2 Selϭͱ ␧ qm͑␧͒ f ͑␧͒ϩkT d␧, the corresponding continuum dj from jϭ0tojϭϱ. In par- m M ͵0 ͫ d␧ͬ e ticular, for the braking cross section we obtain

ϱ 2 R kT Sinϭͱ ͚ ␧i ␧Qi͑␧͒f ͑␧͒d␧ Sinϭ4B0␴0 1Ϫ , ͑13͒ m i ͵ ͩ ␧ ͪ e ͫ ␧i 2 ϱ where ␴0ϭ8␲/15(a0Q) , a0 is the Bohr radius, and Q Ϫ Ϫ Ϫ␧i ͵ ␧Qi ͑␧͒f ͑␧͒d␧ , ϭ1.05 is the electric quadrupole moment. 0 ͬ In the description of excitation of vibrations, we take account of only collisions of the first kind, i.e., we assume 2 ϱ S ϭ I ␧q ͑␧͒f ͑␧͒d␧, that the molecules are found in the ground state. The excita- i ͱ ion V m ͵I e tion cross sections qi of the first eight levels are given in Ref. 12. 2 ϱ 2 The electron excitation cross sections were taken from S ϭ ␧ q ͑␧͒f ͑␧͒d␧, ͑9͒ ϩ r ͱ ͵ r the following sources: A3⌺ and a1⌸ from Ref. 13; B3⌸ me 0 u g g 3 1 1 ϩ and C ⌸u from Ref. 14; b ⌸i , b ⌺u , and the sums of cross where qm is the transport cross section of electron scattering sections of the higher-lying states from Ref. 15. by a nitrogen molecule; QiϭPiqi(␧) is the excitation cross In describing electron impact ionization we restrict our- section of the ith level with energy ␧i ; Pi is the population selves to the process originating in the ground state. The Ϫ Ϫ of the ith level; Qi ϭPiqi (␧) is the deactivation cross sec- corresponding cross sections were taken from Ref. 16. Ϫ tion of the ith level with energy ␧i ; M and me are the mass In line with Ref. 17, we assign the dissociative recombi- of a nitrogen molecule and an electron, respectively. nation rate constants as To determine the rates of ionization, recombination, and 300 1/2 cm3 electron energy loss and the collision frequency, it is neces- Ϫ7 krϭ3ϫ10 , ͓͑Te͔ϭK͒. ͑14͒ sary to know the electron energy distribution function. It is ͩ Te ͪ s usually assumed to be Maxwellian For the vibrational relaxation time we use the expression18 2 1 ␧ 137 f ͑␧͒ϭ f M͑␧͒ϭ 3/2 exp Ϫ . ͑10͒ Ϫ9 ͱ␲ ͑kTe͒ ͩ kTeͪ ␶ ϭ6.5ϫ10 exp ,s͓͑T͔ϭK͒. ͑15͒ V ͩ T1/3ͪ Here the expression for the electronic thermal conduc- The dependence of the thermal conductivity of molecu- tivity takes the form lar nitrogen on the temperature of the medium is taken from Ref. 19. 5 5 k ␭eϭ kneDeϭ nekTe e , ͑11͒ 2 2 me␯m JUSTIFICATION OF THE HYDRODYNAMIC APPROACH where De is the electron diffusion coefficient. At high electron concentrations scattering takes place in The contribution of recombination to the energy trans- Coulomb collisions and use of the Maxwell distribution ferred to the medium per unit time Se can be calculated from function for the electrons is justified by the intense energy the formula exchange in electron–electron collisions. At low electron concentrations the distribution function is governed by the r 3 d interaction of the electrons with the neutral gas and use of Seϭkrne kTeϩINϪINϩ , ͑12͒ ͩ 2 2 ͪ the Maxwell energy distribution requires justification. Tech. Phys. 43 (6), June 1998 A. V. Ivanovski 651

FIG. 1. Dependence on the parameter E/N of the ratio eD/ke ͑1͒ and the FIG. 2. Dependence on the parameter E/N of the electron drift velocity ͑1͒ 19 3 electron temperature Te ͑2͒. ᭺—the hydrodynamic approximation (Te and the ionization rate ͑2͒. Nϭ2.5ϫ10 cm . Curves—the kinetic approxi- ϭeD/ke). mation, ᭺—the hydrodynamic approximation.

eE 2 ϱ ␧ df To describe the electron distribution function f under 0 WeϭϪ ͱ d␧, conditions of interaction with the neutral medium we use the 3 me ͵0 Nqm͑␧͒ d␧ following approach, which provides a good description of 1 2 ϱ ␧ the experimental data.20 Expanding f in a series of Legendre Deϭ ͱ f0͑␧͒d␧, polynomials and limiting it to the ﬁrst two terms 3 me ͵0 Nqm͑␧͒ 2 ϱ f ϭ f 0͑␧͒ϩcos ⌰ f 1͑␧͒, ͑16͒ 3/2 Teϭ ␧ f 0͑␧͒d␧. ͑18͒ 3k ͵0 where ⌰ is the angle between the ﬁeld vector E and the Under similar conditions, in the hydrodynamic approxi- electron velocity, we obtain the following equation for the mation the electron temperature T0 , see Eq. 8 can be steady-state value of f : ͓ e ͑ ͔͒ 0 found from the equation of energy balance 2 2 2 2 e E ␧ me e E ץ ϩkT␧2 q ͑␧͒ ␧ ͫ N2 3q ͑␧͒ M m ͬ ϭN͑SelϩSinϩSi͒, ͑19͒ץ ͭ m me␯e

-2me where the dependence of the collision frequency and the en ץ ϫ f ͑␧͒ϩ␧2 q ͑␧͒f ͑␧͒ ␧ 0 M m 0 ͮ ergy loss rates on the temperature are found from formulasץ ͑9͒ for f ϭ f M(Te ,␧) ͑we have neglected the energy losses to dissociation, assuming the electron concentration to be ϭ ͓␧Qi͑␧͒f0͑␧͒Ϫ͑␧ϩ␧i͒Qi͑␧ϩ␧i͒f0 ͚i small͒.

Ϫ Ϫ Ϫ Ϫ Figures 1, 2, and 3 compare the dependence on E/N of ϫ͑␧ϩ␧i͒ϩ␧Qi ͑␧͒f0͑␧͒Ϫ͑␧Ϫ␧i ͒Qi ͑␧Ϫ␧i ͒

ϱ Ϫ ϫf0͑␧Ϫ␧i ͔͒ϩ␧qion͑␧͒f 0͑␧͒Ϫ2 ␧Јqion͑␧Ј͒ ͵␧ϩI

ϫ f 0͑␧Ј͒␺͑␧Ј,␧͒d␧Ј. ͑17͒

The quantity ␺(␧Ј,␧)d␧ is the probability that one of the two electrons coming out of the collision will have energy in the interval from ␧ to ␧ϩd␧. It is normalized by the ␧ЈϪI j condition ͐0 ␺(␧Ј,␧)d␧ϭ1. For deﬁniteness, we assume the energies of the electrons to be identical ␺(␧Ј,␧)ϭ␦„␧ Ϫ(␧ЈϪI j)/2…. For purposes of comparison of the two approaches to description of the electron dynamics such an assumption is admissible. Using the distribution function f 0 the ionization and recombination rate constants are found from formulas ͑3͒, and the energy loss rate—from formula ͑9͒. The drift velocity, FIG. 3. Dependence of the relative energy losses ␩i on E/N. 1—excitation of rotations together with elastic collisions, 2—excitation of vibrations, 3— electron diffusion coefﬁcient, and electron temperature can excitation of electron levels, 4—ionization. Curves—the kinetic approxima- be calculated from the formulas tion, ᭺—the hydrodynamic approximation. 652 Tech. Phys. 43 (6), June 1998 A. V. Ivanovski

FIG. 4. Time dependence of the temperature of the medium along the dis- FIG. 5. Spatial distributions of the electron temperature and relative density Ϫ5 Ϫ4 Ϫ3 Ϫ3 charge axis. r0 , cm: 1—10 , 2—10 , 3—10 , 4—2.5ϫ10 ͑for E0 of the medium. 1,2—T ; 3,4—N/N ; E , kV/cm ␯, cm/s: 1,3—12.6 and 7 Ϫ5 Ϫ4 Ϫ3 e 0 0 ϭ12.6 kV/cm and ␯zϭ2ϫ10 cm/s͒, 5—10 , 6—10 , 7—10 , 8— 2ϫ107; 2,4—10.8 and 2ϫ108. Ϫ3 8 2.5ϫ10 ͑for E0ϭ10.8 kV/cm and ␯ϭ2ϫ10 cm/s͒.

to converge to the true solution obtained in the limit r0 0. the electron temperature, the drift velocities, the diffusion This testifies to the correctness of the mathematical state-→ coefficients, the ionization frequencies, and electron energy ment of the problem. loss rates for molecular nitrogen, obtained by numerical so- Figure 5 plots the electron temperature distribution and lution of the kinetic equation ͑17͒, ͑18͒ and in the hydrody- medium density distribution across the channel at the instant namic approximation ͑9͒, ͑11͒, ͑19͒. Equation ͑17͒ was of time when the value N/N0ϭ0.1 is reached. It can be seen solved by the method of Ref. 21. For E/Nр4 that at the earlier stage the scale of leader channel formation Ϫ16 2 Ϫ3 ϫ10 V•cm the difference in the quantities is large. In is ϳ10 cm and depends weakly on E0 and vz . To de- Ϫ16 2 the region E/Nу4ϫ10 V•cm the hydrodynamic ap- scribe the further evolution of the channel requires that we proach gives reasonable agreement with the results obtained introduce Coulomb collisions limiting growth of the electron from the kinetic treatment. temperature, take into account nitrogen dissociation limiting growth of the temperature of the medium, describe radiative transfer entraining new masses of gas into the channel due to ANALYSIS OF THE CALCULATED RESULTS heating of the cold periphery, etc. In addition, the growing conductivity of and consequently current through the channel The solution of the problem is defined by Eqs. ͑1͒–͑3͒ affect the magnitude of the electric field ͑feedback effect͒. and ͑7͒–͑12͒. At the initial time instant the nitrogen is found This requires a self-consistent approach to the description of 19 Ϫ3 under normal conditions: N͉␶ϭ0ϭN0ϭ2.5ϫ10 cm , the dynamics of channel formation and the magnitude of the T͉␶ϭ0ϭ293 K, ␧V͉␶ϭ0ϭ0, the electron concentration electric field. However, basing ourselves on a string of as- ne͉␶ϭ0ϭ0, and the electron temperature Te͉␶ϭ0ϭ293 K. The sumptions it is possible to obtain the dependence of the boundary conditions follow from the requirement of axial propagation velocity of the leader and the dimensions of the symmetry: at rϭ0 the particle and energy fluxes are equal to streamer zone on the potential without complicating the zero. The ionization and recombination rate constants, en- problem by bringing in a description of the enumerated ef- ergy loss rates, collision frequency, and electronic thermal fects. conductivity follow from formulas ͑3͒, ͑9͒, ͑11͒, and ͑12͒ The nature of the solution—an abrupt growth of the tem- assuming a Maxwellian electron energy distribution ͑10͒. perature of the medium ͑decrease in the density͒ at the stage The ionization source Sstr ͑2͒ and the field in the of channel formation—makes it possible to introduce the streamer zone are determined by the streamer parameters concept of a length of the streamer zone L and a potential of E0 , vz /vr , and vz , which are unknown. We take the value leader channel formation VϭE0L. For definiteness we take vz /vrϭ10 ͑Ref. 5͒.TofindE0 and vz , we proceed from the Lϭvz•t0.1 , where t0.1 is the time it takes the channel density fact that for a given potential U0 , of all possible values of E0 to reach the value N/N0ϭ0.1. Proceeding from the fact that that one will be realized for which vz is maximum. the leader channel possesses a high temperature ͑high con- Figure 4 plots the time dependence of the medium tem- ductivity͒, we estimate the value of the leader potential U0 perature along the discharge axis for different values of r0 ϷV, i.e., we neglect the potential drop in the channel. ͑2͒ in two series of calculations with E0ϭ12.6 kV/cm, vz Figure 6 plots the dependence of the potential V on the 7 8 ϭ2ϫ10 cm/s, and E0ϭ10.8 kV/cm, vzϭ2ϫ10 cm/s. As propagation velocity vz , calculated for E0 r0 is decreased, the dimensions of the streamer discharge ϭ11.4, 11.64, 12 kV/cm. The curves have a bell shape: Ϫ4 zone decrease and for r0р10 cm the solutions are indis- growth of the potential at small velocities gives way to a fall Ϫ4 tinguishable, i.e., for r0р10 cm the solution is observed at large velocities. Qualitatively, this may be understood Tech. Phys. 43 (6), June 1998 A. V. Ivanovski 653

FIG. 7. Diagram of the process: 1—leader channel, 2—streamer zone.

impossible since it is associated with a decrease in the ﬁeld strength in the streamer zone E0 . Thus, if we assume that for a given potential the leader

FIG. 6. Dependence of the potential V on the propagation velocity vz . E0 , propagates with the maximum possible velocity, then its ve- kV/cm: 1—11.4, 2—11.64, 3—12. locity is determined by the maxima in the V(vz) curves ͑the points c,e, f in Fig. 8͒. The dependence vz(U0), where U0 is v ϭ0 ͑Eץ/Vץ the potential determined by the condition z͉E0 0 is a parameter here͒, is plotted in Fig. 8. With growth of U0 from the following considerations. On the one hand, growth 0.75 the leader velocity grows roughly as vzϳU . This is clear of the velocity is accompanied by an increase of Lϭvz•t0.1 0 from comparison with the interpolating vz dependence also and consequently also of VϭE0L. On the other, in the shown there: adopted streamer zone model an increase in vz leads to growth of S ͑2͒, i.e., n in the streamer zone, and conse- 7 0.75 str e vzϭ1.47ϫ10 U0 cm/s͓͑U0͔ϭMV͒. ͑20͒ quently to increased energy release and a decrease in t0.1 and L, V. At small velocities the first of these prevails while at Thus, within the framework of the ideas expounded large velocities, the second, which leads to the appearance of above it is possible to construct the steady-state regime of leader propagation, describe the initial stage of channel for- a maximum in the dependence of V on vz . Thus, for a given potential V there is a family of curves mation, and obtain the dependence on the potential of the propagation velocity, the field, and the dimensions of the vz(E0) assigning the propagation velocity of the leader as a function of the electric field in the streamer zone. Thus, for streamer zone. These ideas depend in an essential way on the parameters of the plasma in the streamer zone. These param- VϭV* the field E0ϭ11.4 kV/cm corresponds to the velocity eters were chosen on the basis of a model of streamer at the point a, i.e., va , and at the corresponding point for breakdown.5 Presumably, in the case of a positive streamer large vz ; and the field E0ϭ11.64 kV/cm corresponds to the this model works at propagation velocities vzу3Ϫ4 velocities vb and vd ; the field E0ϭ12 kV/cm corresponds to ϫ107 cm/s. At least it makes it possible to explain the linear the velocity vc , etc. ͑Fig. 6͒, i.e., beginning its motion with dependence of the streamer velocity on the rate of growth of velocity va , the leader can accelerate to the velocities vb , vc , vd , and so on. Generally speaking, the propagation velocity and consequently the field in the streamer zone are determined by the propagation velocity of the channel. In turn, the channel parameters and consequently the channel propagation velocity are determined by processes in the streamer zone, i.e., by the field E0 and the velocity vz. Let us qualitatively consider the process of acceleration of a leader due to an increase in the channel propagation velocity ͑Fig. 7͒. Let at the initial time c str the channel velocity vz and streamer zone velocity vz be identical and equal to the leader propagation velocity vz . Increasing the field E0 will lead to an increase in the rate of c energy release, i.e., to a growth of vz , and, conversely, a c growth of vz leads to a shrinkage of the dimensions of the streamer zone L and consequently to a growth of E0 (E0 ϭV*/L). In other words, growth of the leader velocity is possible with simultaneous growth of E0 and vice versa. Having started its motion with velocity va , a streamer can accelerate to v since this acceleration is accompanied by a c FIG. 8. Dependence of the velocity vz on the potential U0 . Solid curve— growth of E0 . Further growth of the velocity, e.g., to vd ,is calculated; dashed curve—interpolation based on formula ͑20͒. 654 Tech. Phys. 43 (6), June 1998 A. V. Ivanovski

3,6 the potential observed in this region. Therefore we can depends weakly on the potential U0 and is determined by the assume that what has been said above is valid for positive value of E/N corresponding to the shift of the electron en- leaders for U0у3MV ͑Fig. 8͒, i.e., at high potentials. It ergy dissipation channel from excitation of vibrations to ex- cannot be ruled out that at small potentials formation of a citation of electron levels. Under normal conditions this cor- leader channel takes place via another mechanism, e.g., responds to the value E0Х12 kV/cm, i.e., in some sense it is against the background of a huge number of continuously possible to speak of the existence of a threshold field for forming streamers.3 The experimental data known to us do leader breakdown. not allow us to conjecture about the applicability of the 6. Presumably in the case of positive leaders the model model to negative leaders. works at potentials U0у3 MV. It is possible that at low It is interesting to note that the streamer zone fields E0 potentials channel formation takes place against a back- depend weakly on the potential U0 ͑on the velocity vz͒.As ground of a huge number of continuously forming 3 U0 varies from 6 to 26 MV, the field E0 varies within the streamers. The experimental data known to us does not al- range 11.4Ϫ12 kV/cm. This has to do with a peculiarity of low us to conjecture about the applicability of the model to energy dissipation in the given range of field strengths ͑Fig. negative leaders. 3͒, specifically with a shift of the main channel of energy 7. The model can be used to describe a leader break- scattering by the electrons from excitation of molecular vi- down of a long air gap. When additional processes are added brations to excitation of electron levels. By reason of re- to the model, it will be possible to obtain leader channel tarded deactivation of the vibrational levels of the molecules parameters matched to the potential U0 . heating of the medium and consequently channel formation are hindered in the first case. In this sense one may speak of 1 a threshold field for leader breakdown in nitrogen ͑air͒ equal Yu. P. Razer, Physics of Gas Discharge ͓in Russian͔͑Nauka, Moscow, to ϳ11Ϫ12 kV/cm. 1987͒, 591 pp. 2 M. V. Kostenko ͑Ed.͒, High-Voltage Technique ͓in Russian͔͑Vysshaya Shkola, Moscow, 1973͒, 351 pp. CONCLUSION 3 E´ . M. Bazelyan and A. Yu. Goryunov, Elektrichestvo, No. 11, pp. 27–33 1. A physical picture of the formation of a leader chan- ͑1986͒. 4 A. E´ . Bazelyan and E´ . M. Bazelyan, Teplofiz. Vys. Temp. 32, 354 ͑1994͒. nel has been presented. A leader channel is formed in the 5 A. V. Ivanovski, Zh. Tekh. Fiz. 65͑12͒,48͑1995͓͒Tech. Phys. 40, 1230 plasma of a streamer zone as a consequence of an ͑1995͔͒. 6 ´ ionization–heating instability. The streamer zone consists of E. M. Bazelyan and I. M. Razhanski, Spark Discharge in Air ͓in Russian͔ ͑Nauka, Novosibirsk, 1988͒, 164 pp. an isolated streamer formed by a cylindrical ionization 7 5 Ya. B. Zel’dovich and Yu. P. Razer, Physics of Shock Waves and High- wave. The main mechanism determining the channel param- Temperature Hydrodynamic Phenomena, 2nd ed. ͓in Russian͔͑Fizmatgiz, eters at the early stage is the electronic thermal conductivity. Moscow, 1966͒, 686 pp. 8 ´ 2. A mathematical model has been constructed, describ- N. L. Aleksandrov, A. M. Konchakov, and E. E. Son, Fiz. Plazmy 4, 169 ͑1978͓͒Sov. J. Plasma Phys. 4,98͑1978͔͒. ing the leader channel formation stage in a streamer zone 9 I. Shimamura, Sci. Pap. Inst. Phys. Chem. Res. ͑Jpn.͒ Vol. 82, pp. 1–51 plasma. By comparing with the solution of the kinetic equa- ͑1989͒. tions in the spatially homogeneous case it has been shown 10 E. Gerjuoy and S. Stein, Phys. Rev. 97, 1671 ͑1955͒. 11 E. Gerjuoy and S. Stein, Phys. Rev. 98, 1848 ͑1955͒. that use of the hydrodynamic approximation is justified for 12 Ϫ16 2 G. J. Schulz, Phys. Rev. 135, A988 ͑1964͒. E/Nу4ϫ10 V•cm . 13 W. L. Borst, Phys. Rev. A 5, 648 ͑1972͒. 3. Assuming smallness of the potential drop in a highly 14 P. N. Stanton and R. M. St. John, J. Opt. Soc. Am. 59, 252 ͑1969͒. conductive channel, an algorithm has been constructed for 15 A. E. S. Green and R. S. Stolarski, J. Atmos. Terr. Phys. 34, 1703 ͑1972͒. 16 D. Rapp and P. Englander-Golden, J. Chem. Phys. 42, 4081 ͑1965͒. determining the leader propagation velocity vz and the di- 17 A. V. Eletski and B. M. Smirnov, Usp. Fiz. Nauk 136, 254 ͑1982͓͒Sov. mensions L and electric field E0 in the streamer zone. The Phys. Usp. 25,13͑1982͔͒. 18 algorithm is based on the assumption that for fixed U0 the J. D. Lambert, Vibrational and Rotational Relaxation in Gases ͑Oxford, value of E is determined that provides the maximum pos- Clarendon Press, 1977͒. 0 19 ´ sible propagation velocity. V. S. Engel’sht and B. A. Uryukov ͑Eds.͒, Low-Temperature Plasma. Theory of the Column of an Electric Arc ͓in Russian͔͑Nauka, Novosi- 4. Steady-state leader breakdown in a nitrogen atmo- birsk, 1990͒, 215 pp. sphere has been modeled by a numerical simulation. It was 20 L. G. H. Huxley and R. W. Crompton, The Diffusion and Drift of Elec- trons in Gases ͑Wiley, New York, 1974͒; Mir, Moscow, 1977, 672 pp. found that the leader propagation velocity is proportional to 21 3/4 B. Sherman, J. Math. Anal. Appl. 1, 342 ͑1960͒. U0 . 5. It has been shown that the field in the streamer zone Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Lowering of ionization potentials in a nonideal plasma S. I. Anisimov and Yu. V. Petrov

L. D. Landau Institute of Theoretical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow District, Russia ͑Submitted February 10, 1997͒ Zh. Tekh. Fiz. 68, 45–50 ͑June 1998͒ Analytical expressions are obtained for the ionization potentials of neutral atoms and ions in the screened Coulomb potential of a nonideal plasma. Among all the chemical elements considered, cesium exhibits the greatest relative lowering of the ionization potentials in comparison with the case of an unscreened interaction. © 1998 American Institute of Physics. ͓S1063-7842͑98͒00906-4͔

INTRODUCTION variational method and by perturbation theory can be found in Ref. 4 . Analytical formulas convenient for practical cal- Many characteristics of plasmas, in particular the equa- ͒ culations can be obtained by using, instead of Eq. 2 , the tion of state and the kinetic coefﬁcients, depend substantially ͑ ͒ similar Hulthe´n potential13,14 on the electron density.1–11 In a not-too-dense plasma ionization is caused by thermal excitation of electrons to states of Z*/␭ U ͑r͒ϭϪ . ͑3͒ the continuum, and at high enough temperatures there exist H er/␭Ϫ1 in the plasma ions in different states of ionization. Such mul- tistep ionization in an ideal plasma is described the the Saha Like the Debye potential, potential ͑3͒ for rӶ␭ goes 12 equations, according to which the main parameters deter- over to the Coulomb potential Uc(r)ϭϪZ*/r and falls off mining ionization equilibrium at given plasma temperature exponentially at large r. Z* is the effective charge of the and density are the ionization potentials of the atoms and atomic radical for the valence electron with principal and ions. In an ideal plasma these are the ionization potentials of orbital quantum numbers n and l. We will determine Z* 0 the isolated neutral atoms and of ions in the various charge from the experimentally measured ionization potential In,l of states. With growth of the density of the plasma and its de- this valence electron in an isolated atom or ion viation from ideality a renormalization of the ionization po- Z*2 tentials takes place due to effects of interactions between the I0 ϭ . ͑4͒ n,l 2n2 particles of the plasma. For a weakly nonideal plasma effects of the Coulomb interaction can be taken into account within This gives the n- and l-dependent effective charge the framework of the Debye approximation. The correction 0 to the free energy per unit volume due to interaction in the Z*ϭnͱ2In,l. ͑5͒ 12 continuum can be expressed as ͑here and below we use Let us determine the energy levels of the bound states of atomic units͒ the electron in a centrally symmetric potential ͑3͒. Repre- 3/2 senting the electron wave function in the usual way in the 2 2 1 2 ⌬ f ϭϪ ͱ␲/T ͚ niZi ϭϪ ͚ niZi . ͑1͒ form of a product of a radial function R(r) and angular func- 3 ͩ i ͪ 3␭ i tions Y lm(␪,␸) and then transforming to the variable Here ni is the density of particles of the plasma with charge xϭr/␭, we obtain the radial Schro¨dinger equation for the Zi , T is the temperature, ␭ is the Debye screening length function ␹(x)ϭR(x)/x leading to an effective interaction of particles with charges 1 d2␹ Z* 1 l͑lϩ1͒ Z1 and Z2 at the distance r Ϫ ϩ Ϫ ϩ ␹ϭE␹. ͑6͒ 2␭2 dx2 ͩ ␭ exϪ1 2␭2x2 ͪ Z Z 1 2 Ϫr/␭ UD͑r͒ϭ e . ͑2͒ In this equation the states with lϭ0 are a special case. r 2 2 For them the centrifugal term Ul(x)ϭl(lϩ1)/(2␭ x )is Screening of the Coulomb interaction for bound electron generally absent, and Eq. ͑6͒ can be solved by transforming states in atoms and ions leads to a lowering of their ioniza- to the new argument uϭeϪx ͑Ref. 13͒. The energy levels of tion potentials. the discrete spectrum for lϭ0 are determined by the principal quantum number n and can be written as LOWERING OF THE IONIZATION POTENTIALS AS A 2 2 2 CONSEQUENCE OF SCREENING Z* n E ϭϪ 1Ϫ . ͑7͒ n 2n2 2Z*␭ The Schro¨dinger equation for the valence electron with ͩ ͪ potential ͑2͒ does not have an analytical solution in terms of Replacing Z* by the expression for it given in formula ͑5͒, known functions ͑approximate techniques of solution by the we obtain

2 Z*2 n2Ϫl͑lϩ1͒ 2 0 n ˜ E ϭϪI 1Ϫ . 8 En,lϭϪ 2 1Ϫ ͑18͒ n n,0 0 ͑ ͒ * ͩ 2␭ͱ2I ͪ 2n ͩ 2Z ␭ ͪ n,0 To obtain an analytical expression for the energy levels (nϭ1,2,3,..., 0рlрnϪ1͒. Expressing the effective charge Z* in terms of the ion- of the bound states with lÞ0, we replace the centrifugal 0 potential U (x)by ization potential In,l of an isolated atom or ion, we obtain the l ionization potential for an electron with quantum numbers n l͑lϩ1͒ 1 and l with screening taken into account U˜͑x͒ϭ . ͑9͒ l 2␭2 ͑exϪ1͒2 n2Ϫl͑lϩ1͒ 1 2 ˜ ˜ 0 ˜ I n,lϭϪEn,lϭIn,l 1Ϫ 0 . ͑19͒ The function Ul(x) approximates the function Ul(x) ͩ 2n␭ ͱ2I ͪ n,l very well. Indeed, for xӶ1(rӶ␭) the two functions coincide, but in the localization region of the bound electron As the particles of the plasma we may consider negative ions of the chemical elements. Loss of electrons by such ions rϳ1(xϳ1/␭). Therefore, in a plasma with large ␭ U˜ (x)is l can also be examined within the framework of the model a good approximation for U (x). With the true centrifugal l expounded above. In this case the ionization energy I0 is term U (x) replaced by the approximate centrifugal term ͑9͒ n,l l the electron affinity ⑀0 of isolated atoms of these chemical the radial equation for ␹(x) takes the following form: elements. As a rule, ⑀0Ӷ1 ͑e.g., for hydrogen ⑀0ϭ0.028͒; 1 d2␹ Z* l͑lϩ1͒ therefore, as follows from Eq. ͑19͒, the relative lowering of Ϫ ϩ Ϫ ϩ ␹ϭ˜E␹ 2␭2 dx2 ͩ ␭͑exϪ1͒ 2␭2͑exϪ1͒2ͪ the electron affinity as a result of screening turns out to be ͑10͒ still more significant than the lowering of the ionization potentials. Similarly, the critical value of the screening length, and can be solved exactly for any l. We make the transfor- ␭ , at which the renormalized electron affinity vanishes, mation of variable uϭeϪx in Eq. ͑10͒ and introduce the no- c tation n2Ϫl͑lϩ1͒ ␭ ϭ ͑20͒ c 0 ␣ϭ␭ͱϪ2˜E, ␤2ϭ2Z*␭. 2nͱ2␧ Taking the asymptotic behavior of ␹(x) turns out to be substantially greater than the critical screening length for vanishing of the renormalized first ionization ␹͑x͒ϳxlϩ1ϳ͑1Ϫu͒lϩ1 for x 0, → potential ͑the Mott screening length͒. This leads to the result ␹͑x͒ϳeϪ␣xϳu␣ for x ϱ, that with growth of the temperature the density of negative → ions in the plasma falls significantly faster in comparison into account, we seek the solution of Eq. ͑10͒ in the form with the situation where screening of the Coulomb interac- ␹͑u͒ϭu␣͑1Ϫu͒lϩ1w͑u͒. ͑13͒ tion is not taken into account. The above expressions for the ionization potentials and Here for the function w(u) we obtain the hypergeomet- the electron affinity were obtained by using the approximate ric equation centrifugal potential ͑9͒. The difference between Ul(x) and ˜ u͑1Ϫu͒wЉϩ͑2␣ϩ1Ϫ͑2␣ϩ2lϩ3͒u͒wЈϪ͑͑2␣ϩ1͒ Ul(x) can be taken into account in perturbation theory. In the first-order theory ϫ͑lϩ1͒Ϫ␤2͒wϭ0. ͑14͒ ␦U ͑x͒ϭU ͑x͒ϪU˜ ͑x͒ ͑21͒ Its finite solution in the limit u 1 has the form l l l → and the correction to the energy of an electron with quantum w͑u͒ϭF͑␣ϩ1ϩlϩ␥,Ϫn ,2lϩ1, u͒. ͑15͒ r numbers n and l is equal to Here ␥ϭ(␣2ϩ␤2ϩl(lϩ1))1/2, n ϭϪ(␣ϩ1ϩlϩ␥) is the r 1 1 radial quantum number taking nonnegative integer values, ϱ 2 ͐0 ␹n,l 2 Ϫ x 2 dx and F(␰,␩,␨,u) is the hypergeometric function l͑lϩ1͒ ͩ x ͑e Ϫ1͒ ͪ ␦En,lϭ 2 ϱ 2 . ͑22͒ ϱ m 2␭ ͐0 ␹n,ldx ͑␰͒m͑␩͒m u F͑␰,␩,␨,u͒ϭ . ͑16͒ ͚ Substituting the radial function ␹n,l ͑13͒ into expression mϭ0 ͑␨͒m m! ͑22͒, we write w2(u)as The symbol (␰)m is defined as 2nr 2 j ͑␰͒mϭ␰͑␰ϩ1͒...͑␰ϩmϪ1͒ for mϾ0, w͑u͒ϭ bju. ͑23͒ j͚ϭ0 ͑␰͒ ϭ1 for mϭ0. ͑17͒ m Here For nrϭ0,1,2,... the hypergeometric series ͑15͒ degen- j erates into a polynomial with degree nr . Introducing as in ama jϪm , if jрnr , the case of the Coulomb potential the principal quantum m͚ϭ0 b ϭ ͑24͒ number nϭnrϩlϩ1, which takes only positive integer val- j nr ues, we obtain the discrete spectrum for any orbital quantum if jϾn . ͚ ama jϪm , r number l Ά mϭ jϪnr Tech. Phys. 43 (6), June 1998 S. I. Anisimov and Yu. V. Petrov 657

The coefficients am here are equal to In an analogous way the coefficients dm can be expressed in terms of the coefficients bm : for lрnr ͑2␥Ϫnr͒m͑Ϫnr͒m amϭ . ͑25͒ ͑2␣ϩ1͒mm! m Calculating the integrals arising in expression ͑22͒,we 2l i if m 2n , obtain ͑Ϫ1͒ bmϪi , р r i͚ϭ0 ͩ i ͪ l͑lϩ1͒ dmϭ 2l ͑29͒ ␦E ϭ 2l n,l 2␭2 Ϫ1 i b , if mϾ2n , ͚ ͑ ͒ ͩ i ͪ mϪi r Ά iϭmϪnr dm ͚2n c ͑mϩ2␣͒ln͑mϩ2␣͒Ϫ͚2͑nϪ1͒ mϭ0 m mϭ0 mϩ2␣ϩ2 ϫ . cm for lϾnr ͚2n mϭ0 mϩ2␣

͑26͒ m 2l The coefficients c can be expressed in terms of the Ϫ1 mϪi b , if mр2l, m ͚ ͑ ͒ mϪi i coefficients b by way of the following formulas: iϭ0 ͩ ͪ m d ϭ ͑30͒ m 2nr lрnrϪ1 2l mϪi if mϾ2l. m ͚ ͑Ϫ1͒ bi , 2lϩ2 Ά iϭmϪ2l ͩ mϪiͪ i ͑Ϫ1͒ bmϪi , if mр2nr , i͚ϭo ͩ i ͪ cmϭ 2lϩ2 2lϩ2 In expressions ͑27͒–͑30͒ we have used the standard bi- Ϫ1 i b , if mϾ2n , ͚ ͑ ͒ ͩ i ͪ mϪi r nomial coefficient notation Ά iϭmϪ2nr ͑27͒ for lϾn rϪ1 i i! m ϭ . ͑31͒ 2lϩ2 ͩ j ͪ j!͑iϪ j͒! mϪi ͑Ϫ1͒ bi , if mр2lϩ2, i͚ϭo ͩ mϪi ͪ c ϭ m 2nr 2lϩ2 The expression for ␦E looks especially simple in the Ϫ1 mϪi b , if mϾ2lϩ2. n,l ͚ ͑ ͒ mϪi i Ά iϭmϪ2lϪ2 ͩ ͪ case lϭnϪ1(nrϭ0) corresponding to circular Coulomb or- ͑28͒ bitals. For electrons with such quantum numbers

2n ͑Ϫ1͒m 2nϪ2 ͚2n ͑Ϫ1͒m ͑2␣ϩm͒ln͑2␣ϩm͒Ϫ͚2nϪ2 l͑lϩ1͒ mϭ0 ͩ m ͪ mϭ0 2␣ϩ2ϩm ͩ m ͪ ␦E ϭ␦E ϭ . ͑32͒ n,l n,nϪ1 2␭2 ͑Ϫ1͒m 2n ͚2n mϭ0 2␣ϩm ͩ m ͪ

For large ␭ the expression for ␦En,l for all l is simply and inert gases ͑He, Ne, Ar, Kr, and Xe͒. The relative lowering of the first ionization potential of the alkali metals in ␣l ␦E ϭ ͑33͒ comparison with the case of an unscreened interaction in the n,l 2␭2 isolated atoms is plotted in Fig. 1 as a function of screening and the ionization potential with screening taken into ac- length, and of the second and third potentials, in Figs. 2 and count for large ␭ can be written in the form 3, respectively. For the first ionization potential the relative lowering turns out to be especially significant for cesium. n2Ϫl͑lϪ1͒ 1 2 l 1 0 This is due to the fact that the principal quantum number of In,lϭIn,l 1Ϫ ͱ 0 1Ϫ ͱ 0 . ͩ 2n␭ 2In,l ͪ ͩ ␭ 2In,l ͪ its valence electron (nϭ6) is the largest of all the above- ͑34͒ listed alkali metals for the orbital quantum number lϭ0 ͑which is the case for all of them͒ and the first ionization RESULTS 0 potential of an isolated atom is the lowest (I6.0ϭ3.893 V). The analytical expressions obtained above for the ioniza- The relative lowering of the ionization potential of cesium tion potentials were used to calculate the first three ionization turns out to be the largest for all of the chemical elements potentials of the alkali-metal atoms ͑Li, K, Na, Rb, and Cs͒ considered. 658 Tech. Phys. 43 (6), June 1998 S. I. Anisimov and Yu. V. Petrov

FIG. 3. The same as in Fig. 1 for the third ionization potential.

FIG. 1. Relative lowering of the first ionization potential of lithium, sodium, potassium, rubidium, and cesium as a function of the interaction screening the ionization potentials obtains for Xe. For all the atoms length. considered this lowering decreases as the charge state of the ion increases: the lowering of the first ionization potential is the most significant. The ionization potentials of isolated at- The relative lowering of the two ionization potentials of oms of all the elements are taken from Ref. 15. helium and the first three ionization potentials of the atoms Figure 7 plots the relative lowering of the electron affin- of the other inert gases ͑Ne, Ar, Kr, and Xe͒ are plotted in ity due to screening of the Coulomb interaction for the series Figs. 4, 5, and 6 as functions of the screening length. Among of elements ͑H, He, Cs͒. The relative lowering of the elec- the elements of this group the greatest relative lowering of tron affinity turns out to be substantially greater than the lowering of the first ionization potential of these elements.

FIG. 4. Relative lowering of the ﬁrst ionization potential of helium, neon, argon, krypton, and xenon as a function of the screening length in the FIG. 2. The same as in Fig. 1 for the second ionization potential. plasma. Tech. Phys. 43 (6), June 1998 S. I. Anisimov and Yu. V. Petrov 659

FIG. 7. Relative lowering of the electron afﬁnity of hydrogen, helium, and FIG. 5. The same as in Fig. 4 for the second ionization potential. cesium atoms as a function of the interaction screening length in the plasma.

This work was carried out with the ﬁnancial support of the Russian Fund for Fundamental Research, Grants No. 95- 02-04535a and No. INTAS-94-1105.

1 H. Renkert, F. Hensel, and E. U. Frank, Phys. Lett. A 30, 494 ͑1969͒. 2 W. Hefner and F. Hensel, Phys. Rev. Lett. 48, 1026 ͑1982͒. 3 A. Fo¨rster, T. Kahlbaum, and W. Ebeling, High Press. Res. 7, 375 ͑1991͒. 4 W. Ebeling, A. Fo¨rster, V. Fortov et al., Thermophysical Properties of Hot Dense Plasma, Teubner-Texte zur Physik, Vol. 25 ͑Teubner, Stuttgart–Leipzig, 1991͒, 315 pp. 5 A. A. Likal’ter, Usp. Fiz. Nauk 162, No. 7, 119 ͑1992͓͒Sov. Phys. Usp. 35, 591 ͑1992͔͒. 6 I. T. Yakubov, Usp. Fiz. Nauk 163,35͑1993͓͒Phys. Usp. 36, 365 ͑1993͔͒. 7 A. A. Likal’ter, Zh. E´ ksp. Teor. Fiz. 106, 163 ͑1994͓͒JETP 79, 899 ͑1994͔͒. 8 A. A. Likal’ter, Teploﬁz. Vys. Temp. 32, 803 ͑1994͒. 9 A. W. DeSilva and H.-J. Kunze, Phys. Rev. E 49, 4448 ͑1994͒. 10 A. Ng, P. Celliers, G. Xu et al., Phys. Rev. E 52, 4299 ͑1995͒. 11 A. A. Likal’ter, Zh. E´ ksp. Teor. Fiz. 107, 1996 ͑1995͓͒JETP 80, 1105 ͑1995͔͒. 12 L. D. Landau and E. M. Lifshitz, Statistical Physics, 2 vols., 3rd ed. ͑Pergamon Press, Oxford, 1980͒. 13 S. Flu¨gge, Practical Quantum Mechanics, 2 Vols. ͓Springer-Verlag, Ber- lin, 1971; Mir, Moscow, 1974, 341 pp.͔. 14 N. R. Arista, A. Gras-Marti, and R. A. Baragiola, Phys. Rev. 40, 6873 ͑1989͒. 15 I. K. Kikoin ͑Ed.͒, Tables of Physical Quantities. A Handbook ͓in Rus- sian͔͑Atomizdat, Moscow, 1976͒, 1008 pp.

FIG. 6. The same as in Fig. 4 for the third ionization potential. Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Probe diagnostics of strongly ionized inert-gas plasmas at atmospheric pressure F. G. Baksht, N. K. Mitrofanov, A. B. Rybakov, and S. M. Shkol’nik

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia ͑Submitted March 26, 1997͒ Zh. Tekh. Fiz. 68, 51–55 ͑June 1998͒ The technique of probe measurements ͑experiment and theory͒ is applied in a dense, strongly ionized inert-gas plasma at atmospheric pressure. The measurements are performed in a high-current ͑250–550 A͒ free-burning argon arc with a thermionic cathode. As a control technique we used spectroscopic measurements. Comparison of calculation with experiment reveals good agreement. The possibility of determining the plasma potential from the measured ﬂoating-probe potential is demonstrated. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01006-X͔

INTRODUCTION an atmospheric-pressure plasma. As the control technique we used spectroscopic measurements. Virtues of the probe method of plasma diagnostics in- The experiments were carried out in a freely burning clude locality, the possibility to carry out measurements in atmospheric-pressure arc in argon, the most frequently used devices from which extraction of light is hindered or com- plasma-forming material in plasma devices. Reference 4 pre- pletely impossible, etc. In practice, only the probe method sents results of calculations of the probe characteristics in a allows one to determine the plasma potential—the most im- xenon plasma. The present paper applies the theory devel- portant parameter in the study of processes occurring near oped in Ref. 4 to a calculation of the probe characteristics in the electrodes in various discharges. The latter circumstance an argon plasma. is the most important stimulus for ongoing studies directed at extending the range of plasma parameters accessible to probe EXPERIMENTAL SETUP AND MEASUREMENT TECHNIQUE diagnostics. In a strongly ionized plasma at atmospheric pressure The experiments were performed in a water-cooled probe measurements have been carried out for some time stainless-steel chamber with an inner diameter of 180 mm ͑see, e.g., Refs. 1 and 2͒. However, the technique of carrying and a height of 200 mm. The chamber was first pumped Ϫ3 out such measurements has still not been fully worked out. down to a pressure of pр10 Torr, flushed with argon and This is frequently explained by the technical difficulties in- then filled with argon to 3–5% above atmospheric pressure. volved in performing the measurements ͑high heat flux den- The arc was oriented vertically: a tungsten rod-shaped cathode with diameter Dϭ2 mm was located below the arc, and sities to the probe͒, but mainly by difficulties of interpreta- a flat copper water-cooled anode above it. Care was taken tion of the results by virtue of the absence of a consistent not to sharpen the end of the cathode, which had the shape of theory of current collection by the probe under these condi- a hemisphere. The interelectrode gap was 12 mm. Thermi- tions. The assertion made in Ref. 1 on the basis of only onic emission from the cathode was generated by self- qualitative arguments that the potential V of a floating probe f heating which was produced by a constant-current ͑50–70 relative to the plasma in a strongly ionized, high-pressure A͒ auxiliary ͑service͒ arc. The main pulsed discharge plasma has an absolute value of ͉V f͉ϭ(2Ϯ1) V has been source—a generator with low internal resistance, pumping widely accepted. However, this estimate, as results of recent 3,4 out single rectangular current pulses with Iр1000 A and du- calculations have shown, is probably incorrect even in or- ration up to 5 ms—was connected in parallel with the service der of magnitude, since a calculation gives ͉V f͉у10 V. arc. The leading edge of the pulse was formed by transitory Since measurements of the floating-probe potential rela- processes in the discharge and was ϳ1 ms in duration. Mea- tive to the electrode in atmospheric-pressure arc discharges surements were performed in the steady-state regime 3–4 ms usually give a value ͉␸ f͉р10 V, in the determination of the after onset of the pulsed discharge. The experimental setup is cathode potential drop Vk according to Ref. 1 it is possible to described in more detail in Ref. 5. obtain a value of Vk more than two times lower than the Spherical probes with diameter dϵ2aϭ0.45 value that follows from the calculation in Ref. 4. In the de- Ϫ0.50 mm were fabricated from a tungsten wire 0.35 mm in termination of the anode drop Va the choice of the correct diameter. The wire was insulated by Al2O3 ceramic 0.75 mm value of V f is no less important, since not only the magni- in diameter. The area of the cylindrical part of the probe tude but also the sign of Va depend on it. extending out from the insulator did not exceed 15% of the The aim of the present work is an experimental check of area of the sphere. With the help of an electromagnet the the theory developed in Ref. 4 and further development of probe was shot across the arc through the central zone of the the probe-measurement technique at high ionization levels in arc channel with a speed of у1 m/s. During the motion of

1063-7842/98/43(6)/4/$15.00 660 © 1998 American Institute of Physics Tech. Phys. 43 (6), June 1998 Baksht et al. 661 the probe a special device generated time markers which where collisions of electrons with ions and neutrals do not were formed with the help of a comb-type light chopper affect the electron temperature distribution Te(r) in the mounted to the shaft of the probe holder. probe sheath. In a large number of calculations the value of With the help of an S9-8 digital storage oscilloscope we Te in the probe sheath was in general not determined but was recorded the time dependence of the floating-probe potential assigned as a parameter ͑see, e.g., Refs. 6–9͒. In fact, here it ␸ f relative to the grounded cathode or the ion current Ii was assumed that Te coincides with the electron temperature when a negative bias of 5–15 V relative to ␸ f was applied to in the unperturbed plasma. In the present work we calculated the probe. The position of the probe was determined with the for a different situation, typical of rather high pressures, in help of the time markers. As a result of the unavoidable air which as a consequence of intense collisions between the gap of the probe holder inside the solenoid of the electro- electrons and heavy particles a single temperature T(r)is magnet, the accuracy of determination of the position of the established in the main part of the probe sheath for both the probe was Ϸ0.3 mm. Measurements in each of the regimes electrons and the heavy component. This temperature differs were carried out in the form of a series of 8–10 oscillograms, from the temperature of the unperturbed plasma Tϱ . First of after which the results were averaged, discarding outliers all, this difference is connected with a lowering of the tem- whose appearance was connected with the bias of the arc perature of the heavy particles in the probe sheath from Tϱ to channel. The position of the arc was monitored visually on a the temperature T0 of the probe surface. Second, it is con- screen onto which its magnified image was projected. nected with a lowering of the electron temperature as a con- Before commencement of the measurements a negative sequence of their motion toward the probe in a retarding bias of Ϸ20 V relative to ␸ f was applied to the probe and electric field in a quasineutral plasma, and also with energy the probe was shot through the plasma several times. As the losses by the electrons to ionization and overcoming the re- probe passed through the central part of the arc channel, a tarding potential barrier in the Langmuir sheath of the probe. cathode ‘‘spot’’ formed on the surface of the probe, cleaning In the theoretical treatment the probe sheath region was di- its surface without causing any noticeable erosion since the vided up into a series of layers in line with the role of the current was limited to 10 A. Such a preliminary cleaning predominant effects in them ͑in more detail see Refs. 4 and prevented the appearance of spots on the probe during ion 10͒. Under the conditions considered here the following hi- current measurements which were carried out at somewhat erarchy of characteristic lengths obtains: L0ӶliӶLMӶLi lower ͑in magnitude͒ biases. Such cleaning was repeated as ӶLTӶa. Here L0 is the thickness of the Langmuir sheath of needed during the measurements. the probe, li is the mean free path of the ions, LM is the To record the arc emission spectrum we used an MDR-3 Maxwellization length of the electrons, Li is the length of spectrometer joined to an OSA B&M Spektronik multichan- single ionization of the argon atoms by the Maxwellized nel optical analyzer. The half-width of the instrument func- electrons, LT is the relaxation length of the temperature of tion of the spectroscopic setup ͑for the width of the entrance the heavy component to the electron temperature. If the slit р0.08 mm͒ was 0.12 nm. During the measurements the above sequence of inequalities is fulfilled, it is possible to width of the entrance slit of the monochromator was set to analyze processes in the indicated regions separately, taking 0.15 mm, which ensured undistorted transmission of the ac- into account the presence of narrower layers in the effective tual intensity distribution in the emission spectrum. A 25-nm boundary conditions. The inner layers for rрLT can be segment of spectrum was recorded in each exposure. treated as planar. Numerical calculations using a technique With the help of a system of crossed mirrors and a two- we developed earlier4 were performed for a spherical probe lens quartz condenser a reduced, horizontally oriented image of radius aϭ0.25 mm in an argon plasma at atmospheric of the arc was formed in the plane of the entrance slit, which pressure. was scanned across the slit by rotating a plane-parallel quartz The condition for realization of the indicated hierarchy plate. We estimated the spatial resolution along the discharge of characteristic lengths, in addition to a high enough pres- axis to be 0.3 mm. The exposure time temporal resolu- р ͑ sure, includes a comparatively high temperature Tϱ of the tion͒ of the analyzer was varied within the limits 0.2–0.5 ms. plasma. In particular, if the condition LiӶLT is fulfilled, it is The delay of startup of the analyzer was chosen with the necessary that the electron temperature in the ionization re- intention that the start of recording of the spectrum corre- gion exceed 2 eV ͑Ref. 4͒. In this case the temperature of the spond to the moment the probe passed through the paraxial unperturbed plasma should exceed 2.5 eV. In this case the region of the discharge. The spectral sensitivity of the setup unperturbed plasma consists only of electrons and doubly was measured with the help of an SI8-200U reference band- charged argon ions Arϩϩ, while in the intermediate region lamp. LiӶrϽa a transition takes place from single to double ion- The probe and spectroscopic measurements were not ization of the plasma. The extent of the corresponding tran- performed simultaneously; however, processing of a large sition region under the conditions considered here exceed by quantity of measurements revealed good reproducibility of an order of magnitude the recombination length of the Arϩϩ the arc burning regimes. ions with the electrons. This allows us to use the condition of local thermodynamic equilibrium ͑LTE͒ to determine the composition of the argon plasma in the probe sheath every- THEORY where for rϾLi . The operation of the probe in dense plasma was previ- Calculated results are shown in Figs. 1 and 2. Figure 1 ously analyzed theoretically mainly in application to regimes plots the calculated dependence of the electron current den- 662 Tech. Phys. 43 (6), June 1998 Baksht et al.

FIG. 3. Saturation value of the ion current density to the probe as a function

of the arc current (z0Ϸ1 mm).

FIG. 1. Density of the electron and ion current to the probe as functions of the probe potential. T0ϭ0.15 eV; Tϱ , eV: 1—3.4, 2—3.2, 3—3.0, 4—2.8. The points show the values of the floating-probe potential V f . is on the order of the probe radius aϭ0.25 mm and is also quite small in comparison with z0 . In addition, it should be sity je and ion current density ji to the probe on the probe noted that near the cathode at distances zϽD there exists a 4 potential V p relative to the unperturbed plasma. Figure 2 stagnation zone; a cathode jet, whose velocity vϳ10 cm/s plots the dependence of the floating-probe potential V f ͑rela- ͑Ref. 1͒, is formed in the region zуD. Therefore probe mea- tive to the unperturbed plasma͒ on the temperature Tϱ . surements near the axis of the arc channel at distances aϽz0ϽD can be assumed to be correct and can be compared EXPERIMENTAL RESULTS with calculations for an immobile plasma. The probe measurements were carried out in the follow- As was shown above, the criteria of applicability of the ing way. For a given discharge current the floating-probe theory are met in a plasma with TϾ2.5 eV. Such a tempera- potential ␸ relative to the grounded cathode was deter- ture in a freely burning atmospheric-pressure arc is reached f mined. Then, different negative biases ͑5–15 V͒ relative to near the cathode. Earlier studies showed5 that the character- ␸ were applied to the probe and the ion current was mea- istic dimension of the region of plasma with TϾ2.5 eV for f sured. The dependence of the ion current I on time had a IϾ250 A is Ϸ2 mm and grows as the current is increased. i wide maximum which corresponded to passage of the probe By virtue of this fact, measurements were carried out for through the paraxial region of the arc. The maximum value Iу250 A at a distance z Ϸ1 mm from the cathode. As cal- 0 of I was reached near the arc axis. In the investigated re- culations have shown,4 the perturbations introduced by the i gimes we observed unmistakable saturation of the ion cur- cold probe in the plasma relax at the distance Ϫ1 rent with increase of the magnitude of the negative bias of LTϳ10 mmӶz0 . The size of the current collection region the probe relative to ␸ f . The dependence of the saturation ion current density jis measured in this way on the arc current I is plotted in Fig. 3. The plasma temperature at the distance z0 from the cathode on the discharge axis was measured as a function of the arc current by the method of relative line intensities without using the Abel inversion. As the results of Ref. 11 show, under the conditions of our measurements the associated error of determination of the temperature does not exceed a few percent. The high temperature of the near-cathode plasma was the reason for using the spectral lines of Ar III for diagnostic purposes. Isolated lines with known probabilities of radiative transitions were chosen12 which had been used earlier13,14 in arc plasma thermometry. In the working segment of the emission spectrum 315–340 nm against the background of the recombination–bremsstrahlung con- FIG. 2. Dependence of the floating-probe potential on the plasma tempera- tinuum we distinctly observed the two triplets of Ar III ͑six 3 0 3 5 0 5 ture Tϱ . T0ϭ0.15 eV. lines in all͒ 4sЈ D Ϫ4pЈ F and 4s S Ϫ4p P with ex- Tech. Phys. 43 (6), June 1998 Baksht et al. 663

plasma a transitional segment of the probe characteristic is distorted for IeϳIi . What is more, the significant decrease of the ion current to the probe in comparison with the saturation ion current at IeϳIi ͑Fig. 1͒ prevents us from using the measured probe current–voltage characteristic to directly determine Ie ͑by extrapolating Iis͒. All this makes the transitional segment of the characteristic unsuitable for diagnostic purposes. Nevertheless, the plasma temperature Tϱ can be determined by measuring the saturation ion current to the probe. Indeed, as calculation shows, jis is a sensitive function of the plasma temperature and depends only very weakly on the surface temperature of the probe ͑the dashed curves in Fig. 4͒. This is important since during the measurements the surface temperature of the probe T0 varies in an uncontrolled way. The maximum value of T0 depends on the FIG. 4. Dependence of the saturation value of the ion current density to the residence time of the probe in the plasma and the probe probe on the plasma temperature. Points—experiment, dashed curves— potential relative to the plasma. Estimates show that in our calculation. T0 , eV: 1—0.05, 2—0.25. experiments the surface temperature of the probe could reach 2500 K. The comparison of calculation with the results of experiment shown in Fig. 4 reveals good agreement. citation potentials of the upper levels Ϸ28.1 V and The results obtained confirm that the theory developed in Ϸ25.4 V, respectively. However, for diagnostic purposes we Ref. 4 adequately describes current collection by a floating were able to use only the Ar III line pairs with wavelengths probe in a strongly ionized atmospheric-pressure plasma. In ͑333.61 and 330.19 nm͒ and ͑333.61 and 328.58 nm͒. Mea- this work we have demonstrated the possibility of measuring surements on other line pairs gave significant spread and the plasma temperature indirectly from measurements of the unrealistically exaggerated values of the temperature. This is saturation ion current, and we have also demonstrated the a result of the impossibility of resolving the implemented possibility of determining the plasma temperature from mea- spectral lines of Ar III and the near-lying lines of atomic surements of the potential of the floating probe. tungsten and tungsten ion and also the lines of carbon ion. The studies described in this paper were carried out with The presence of carbon was due to its removal from the the support of the International Science Foundation ͑Grants cathode pin during arc burning, which was noted earlier.5 No. R5D000 and R5D300͒. The excitation temperature of the lines measured by this method under the conditions of our experiment coincides with the plasma temperature.15 The spectroscopic measurements were used to construct 1 W. Finkelnburg and G. Mecker, Electric Arcs and Thermal Plasma ͑IL, the dependence of the saturation ion current density jis to the Moscow, 1961͒, 370 pp. 2 probe on the plasma temperature ͑Fig. 4͒. B. S. Gavryushchenko, R. Ya. Kucherov, A. V. Postogarov et al., Zh. Tekh. Fiz. 45, 2119 ͑1975͓͒Sov. Phys. Tech. Phys. 20, 1333 ͑1975͔͒. 3 E. M. Sklyarova and I. B. Chekmarov, Zh. Tekh. Fiz. 64͑7͒,28͑1994͒ ͓Tech. Phys. 39, 649 ͑1994͔͒. DISCUSSION OF RESULTS 4 F. G. Bakit and A. B. Rybakov, Zh. Tekh. Fiz. 67͑12͒,16͑1997͓͒Tech. Phys. 42, 1385 ͑1997͔͒. The calculated results plotted in Fig. 1 show that the 5 G. A. Dyuzhev, N. K. Mitrofanov, and S. M. Shkol’nik, Zh. Tekh. Fiz. dependence of the logarithm of the electron current to the 67͑1͒ 35 ͑1997͓͒Tech. Phys. 42,30͑1997͔͒. 6 K. N. Ul’yanov, Zh. Tekh. Fiz. 40, 790 ͑1970͓͒Sov. Phys. Tech. Phys. 15, probe on the probe potential in a strongly ionized 613 ͑1970͔͒. atmospheric-pressure argon plasma is substantially nonlin- 7 F. G. Bakit, Zh. Tekh. Fiz. 43, 214 ͑1973͒. ͓Sov. Phys. Tech. Phys. 18, ear. An analogous result was also obtained in the case of 139 ͑1973͔͒. 8 xenon.4 This has to do with the fact that the electron tem- F. G. Bakit, G. A. Dyuzhev, N. K. Mitrofanov et al., Zh. Tekh. Fiz. 43, 2574 ͑1973͓͒Sov. Phys. Tech. Phys. 18, 1617 ͑1973͔͒. perature at the surface of the probe is lowered when the 9 K. N. Ul’yanov, Teplofiz. Vys. Temp. 16, 492 ͑1978͒. potential difference between the probe and the plasma is de- 10 F. G. Bakit and V. G. Yur’ev, Zh. Tekh. Fiz. 49, 905 ͑1979͓͒Sov. Phys. creased, i.e., when the electron current is increased. This Tech. Phys. 24, 535 ͑1979͔͒. 11 effect was discovered experimentally several years ago and A. A. Kurskov, E. A. Ershov-Pavlov, and L. V. Chvyaleva, Zh. Prikl. Spektrosk. 45, 753 ͑1986͒. investigated theoretically in the case of a strongly ionized 12 W. L. Wiese et al., Atomic Transition Probabilities, NSRDS-NBS 22, 16 low-pressure plasma. At a plasma density an order of mag- Vol. II, U.S. Dept. of Commerce, Washington, D.C. ͑1969͒. nitude lower than in the present work, the effect was signifi- 13 K. Behringer and P. Thoma, J. Quant. Spectrosc. Radiat. Transf. 16, 605 cant only when a large electron current was collected by the ͑1976͒. 14 S. Pellerin, B. Pokrzywka, K. Musiol, and J. Chapelle, J. Phys. III probe ͑in comparison with the ion current͒, but for IeϳIi the ͑France͒ 5, 2029 ͑1995͒. probe characteristics turned out to be undistorted and al- 15 H. W. Drawin, Z. Phys. 228͑2͒,99͑1969͒. lowed us to determine the electron temperature. 16 F. G. Bakit, G. A. Dyuzhev, S. M. Shkol’nik, and V. G. Yur’ev, Zh. Tekh. Under the conditions investigated in the present work, Fiz. 47, 2280 ͑1977͓͒Sov. Phys. Tech. Phys. 22, 1319 ͑1977͔͒. that is to say, in a strongly ionized atmospheric-pressure Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Numerical calculation of the heavy-ion energy spectrum in the cathode sheath of a glow discharge in a gas mixture V. I. Kristya

Scientiﬁc-Research Institute of Materials for Electronics Technology, 248650 Kaluga, Russia ͑Submitted April 23, 1997͒ Zh. Tekh. Fiz. 68, 56–59 ͑June 1998͒ A numerical method is developed for solving the equation for the heavy-ion total-energy distribution function in the cathode sheath of a glow discharge in an inert-gas mixture which requires much less computer time than the Monte Carlo method. It is shown that it allows one to calculate with satisfactory accuracy the energy spectrum of the heavy ions bombarding the cathode in glow-discharge devices. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01106-4͔

In many gas-discharge devices such as gas lasers and exceeds the elastic collision cross section and the electron displays, the working medium is a mixture of inert gases ionization cross section of the parent atom. In a mixture con- containing a light gas with a small admixture of a heavy one. taining only a moderate amount of heavy gas, the motion of Their service life in many cases is a function of the cathode the heavy ions can be strongly affected by their elastic col- sputtering time, where the main contribution to the sputtering lisions with atoms of the light gas between charge-transfer comes from ions of the heavy component.1,2 To model the events on gas atoms of their own species. Nonresonant evolution of the emission surface of the cathode in such a charge transfer on atoms of the light gas, on the other hand, discharge, it is necessary to know the energy distribution can be ignored, since its cross section for inert gases at ion 11 function of the ions of the heavy component of the medium. energies below 1 keV is small. The energy distribution function of the ions in the cath- If the mass ratio of the atoms of the two components is ode sheath of the discharge has been calculated by analytical large (M H /M Lӷ1), then the deviation of the trajectories of methods in a number of papers3–6 for a discharge in a pure the heavy ions from the normal to the cathode surface will be 10 gas. However, as was shown in Ref. 7, the use of such a ion small and their motion can be treated as one-dimensional. distribution function to calculate the sputtering rate of the Thus, if we let the z axis be directed along the normal to the cathode in a gas mixture can lead to qualitatively invalid cathode surface, the coordinate zϭ0 correspond to the results. On the other hand,8,9 Monte Carlo simulation of the boundary of the plasma with the cathode sheath, and the ion distribution function requires large amounts of computer coordinate zϭdc , to the surface of the cathode, then the heavy-ion distribution function f (z, ) in the cathode sheath time, which limits the applicability of this method for mod- ␧ will satisfy the equation10 eling cathode sputtering in a glow-discharge plasma in gas mixtures. In Ref. 10, I proposed an equation for the total- f 1 1ץ ␸ץ f ץ energy distribution function of the ions of the heavy compo- Ϫe ϭ ͓␦͑␧͒Ϫf͔ϩ ␧ ␭c ␭eץ zץ zץ nent of the mixture and found its analytical solution in the continuous-slowing-down approximation for a heavy ion in a ␧/͑1Ϫ␥͒ f͑z,␧Ј͒d␧Ј light gas. The expression I obtained there for the ion distri- ϫ Ϫf ͑1͒ ͫ͵␧ ␥␧Ј ͬ bution function allows one with satisfactory accuracy to calculate the cathode sputtering rate in a gas mixture. However, with the boundary condition f (0,␧)ϭ␦(␧Ϫ␧ ), where ␭ it does not describe a number of properties of the actual 0 c and ␭e are the heavy-atom resonant charge-transfer length distribution function of the ions ͑in particular, its high- and the elastic collision length of the heavy atoms with the energy tail, due to the stochastic nature of the ion–atom col- 2 light atoms, ␥ϭ4M HM L /(MHϩML) , ␸ is the electric field lisions, which is not taken into account in the continuous- potential, and ␧0 is the ion energy at the boundary of the slowing-down approximation for an ion in a gas͒. cathode sheath. In the present paper I develop a method for numerical Integration of Eq. ͑1͒ over ␧ gives the ion flux conser- solution of the equation proposed in Ref. 10 for the distribu- vation law tion function of the heavy ions which allows one to calculate the energy spectrum of the heavy ions without using the continuous-slowing-down approximation and requires much ͵ f ͑z,␧͒d␧ϭ1. ͑2͒ less computer time than the Monte Carlo method. When an ion moves in a gas of its own species in the Replacing ␧ by the new variable cathode sheath of a discharge, the main ion–atom interaction is resonant charge transfer, whose cross section significantly sϭ␧ϩe␸͑z͒, ͑3͒

1063-7842/98/43(6)/4/$15.00 664 © 1998 American Institute of Physics Tech. Phys. 43 (6), June 1998 V. I. Kristya 665 we obtain from Eq. ͑1͒ an equation for the function f (z,s), nonzero in the interval from e␸(z)to␧0, and the condition which contains a derivative in only one variable of ion flux conservation ͑2͒ after substitution of relation ͑6͒ takes the form f 1 1 1 1 sm f ͑z,sЈ͒dsЈ ץ ϩ ϩ f ϭ ␦͑sϪe␸͒ϩ , z ␭ ␭ ␭ ␭ ͵ ␥͑sЈϪe␸͒ ␧0 1 1ץ ͫ c eͬ c e s h͑z,s͒dsϭ1Ϫexp Ϫ ϩ z . ͑9͒ ͑4͒ ͵e␸ ͫ ͩ ␭ ␭ ͪ ͬ c e where To find a numerical solution to Eq. ͑8͒ we can use a method similar to that proposed in Ref. 12 to solve the ki- ͓sϪ␥e␸͔/͑1Ϫ␥͒, sϽ␧ ͑1Ϫ␥͒ϩ␥e␸, 0 netic equation for the electron distribution function in the smϭ ͭ ␧0 , sу␧0͑1Ϫ␥͒ϩ␥e␸. cathode sheath of a discharge. We divide the interval ͓0, d ͔ along the z axis into n Integrating Eq. ͑4͒,wefind c segments of length ⌬zϭdc /n, whose end-points are equal to z 1 1 ziϭi⌬z, iϭ0,1,...,n. We divide up the interval ͓e␸(zi),␧0͔ f͑z,s͒ϭ dzЈ exp ϩ ͑zЈϪz͒ of variation of s in each cross section zϭzi into m segments ͵0 ͫͩ ␭ ␭ ͪ ͬ c e i of length ⌬siϭ(␧0Ϫe␸(zi))/m, i.e., skϭe␸(zi)ϩk⌬si , 1 1 sm f ͑zЈ,sЈ͒dsЈ kϭ0,1,...,m. In the first cross section, as follows from the ϫ ␦͑sϪe␸͒ϩ 0 boundary condition for Eq. ͑8͒, we have h(z0 ,s )ϭ0. To ͫ ␭c ␭e ͵s ␥͑sЈϪe␸͒ ͬ k find the values of the function h(z,s) in other cross sections 1 1 from Eq. ͑8͒, we can use the Cauchy–Euler method,13 which ϩ␦͑sϪ␧ ͒exp Ϫ ϩ z . ͑5͒ 0 ͫ ͩ ␭ ␭ ͪ ͬ gives c e i i i We represent the function f (z,s) in the form h͑zi ,sk͒ϭh͑ziϪ1 ,sk͒ϩhz͑ziϪ1 ,sk͒⌬z. ͑10͒ i 1 1 The values h(ziϪ1 ,sk) are determined by interpolation i i iϪ1 f ͑z,s͒ϭ␦͑sϪ␧0͒exp Ϫ ϩ z ϩh͑z,s͒. ͑6͒ over the known values h(z ,s ) for s s and are set ͫ ͩ ␭ ␭ ͪ ͬ iϪ1 k kу 0 c e i iϪ1 i equal to zero for skϽs0 . The function hz(ziϪ1 ,sk)isa The first term in expression ͑6͒ is the ion distribution finite-difference approximation of the right-hand side of Eq. function of the primary ion beam entering the cathode sheath ͑8͒ from the discharge, and the second term is the distribution 1 1 function of the secondary ions formed in the cathode sheath i i i iϪ1 i hz͑ziϪ1 ,sk͒ϭϪ ϩ h͑ziϪ1 ,sk͒ϩ␪͑Ϫskϩs0 ͒␣k in charge-transfer and elastic collisions of the primary ions. ͩ␭c ␭eͪ Substituting expression ͑6͒ into formula ͑5͒, we obtain an i iϪ1 1 1 1 ␪͑skϩ␧m Ϫ␧0͒ equation for h(z,s) ϩ exp Ϫ ϩ z ␭ ͫ ͩ ␭ ␭ ͪ iϪ1ͬ ␧iϪ1 e c e m z 1 1 h z,s͒ϭ dz exp ϩ z Ϫz͒ 1 i h z ,s ͑ Ј ͑ Ј ␶m ͑ iϪ1 Ј͒ ͵0 ͫͩ ␭c ␭e ͪ ͬ ϩ dsЈ, ͑11͒ ␭ ͵ i ␥ s Ϫe␸ z ͒ e sk ͓ Ј ͑ iϪ1 ͔ 1 1 sm ϫ ␦͑sϪe␸͒ϩ where ͫ ␭c ␭e ͵s iϪ1 ␧m ϭ␥͓␧0Ϫe␸͑ziϪ1͔͒, ␦͑sЈϪ␧0͒exp͓Ϫ͑1/␭cϩ1/␭e͒zЈ͔ϩh͑zЈ,sЈ͒ ϫ dsЈ . ␥͓sЈϪe␸͑zЈ͔͒ ͬ ͓si Ϫ␥e␸͑z ͔͒/͑1Ϫ␥͒, k iϪ1 7 i i ͑ ͒ ␶mϭ skϽ␧0͑1Ϫ␥͒ϩ␥e␸͑ziϪ1͒, i After differentiating with respect to z it is possible to ͭ ␧0 , s у␧0͑1Ϫ␥͒ϩ␥e␸͑ziϪ1͒. k reduce this equation to the form i The function ␣k describes the distribution of the ions h 1 1 1 1 that have undergone charge transfer in the segment ͓ziϪ1 ,zi͔ץ ϭϪ ϩ h͑z,s͒ϩ ␦͑sϪe␸͒ϩ i iϪ1 z ͩ␭ ␭ ͪ ␭ ␭ in the interval ͓s0 ,s0 ͔. The correct choice of its form, asץ c e c e calculations show, has a substantial affect on the calculated 1 1 ␪͕sϪ͓␧0Ϫ␥͑␧0Ϫe␸͔͖͒ ion distribution function. Since ⌬zӶ␭ , this function can be ϫexp Ϫ ϩ z e ͫ ͩ ␭ ␭ ͪ ͬ ␥͑␧ Ϫe␸͒ found from Eq. ͑8͒ in neglect of the elastic scattering of ions c e 0 at a distance ⌬z after charge transfer, which gives 1 sm h͑z,sЈ͒ ϩ dsЈ, ͑8͒ i ͵ Ϫ exp͑zsk/␭c͒ ␭e s ␥͑sЈ e␸͒ i ␣kϭ i iϪ1 i , ͑12͒ ␭ce␸Ј͑zsk͓͒exp͑zs0 /␭c͒Ϫexp͑zs0/␭c͔͒ where ␪(x) is the Heaviside step function. i i iϪ1 The boundary condition for Eq. ͑8͒ follows from the where the quantities zsk , zs0 , and zs0 are found from the i i iϪ1 boundary condition for the function f (z,s) and has the form equation sϪe␸(z)ϭ0 for s equal to sk , s0 , and s0 . re- h(0, s)ϭ0. The function h(z,s), as follows from Eq. ͑3͒,is spectively. 666 Tech. Phys. 43 (6), June 1998 V. I. Kristya

FIG. 1. Ion distribution function at the cathode surface, as calculated by the proposed method ͑solid line͒ and the exact solution ͑13͒͑dashed line͒ in the absence of elastic collisions of the heavy ions with the light atoms (␭c /dc ϭ0.313, ␭e /dcӷ1͒.

To avoid nonconservation of the ion flux due to discreti- zation errors, at each step in z the function h(z,s) is renormalized on the basis of relation ͑9͒. Using relations ͑6͒, ͑9͒– ͑12͒ successively for iϭ1,2,...,n, it is possible to find the energy spectrum of the heavy ions over the entire extent of the cathode sheath from the boundary with the plasma to the cathode surface. To estimate the accuracy of the proposed method of so- FIG. 2. Distribution function of ions of the heavy component at the cathode lution of Eq. ͑1͒, calculations were carried out without taking surface in a helium–neon mixture, calculated by the proposed method into account the elastic collisions of the heavy ions with the ͑curves͒ and by the Monte Carlo method ͑histograms͒ for three discharge regimes. j, mA/cm2: a—1.0, b—0.6, c—0.2; U , V: a—291, b—272, c— light atoms (␭eӷdc). In this case the velocities of the ions c are directed along the z axis, and the ion distribution func- 250; dc , cm: a—0.230, b—0.269, c—0.409; ␭c /dc : a—0.313, b—0.226, c—0.124; ␭e /dc : a—0.125, b—0.090, c—0.050. tion, if the initial energy ␧0 of the ions at the boundary of the cathode sheath is neglected, is given by3,10 1 1 Figure 3 plots the dependence of the cathode sputtering f ͑z,␧͒ϭ ␦͓z ͑␧͔͒ϩ e͉␸Ј͓z ͑␧͔͉͒ ͫ 0 ␭ coefficient averaged over the ion energies 0 c

z0͑␧͒ z Rϭ Y f d , d 14 ϫexp exp Ϫ , ͑13͒ ͵ ͑␧͒ ͑ c ␧͒ ␧͑͒ ͩ ␭ ͪͬ ͩ ␭ ͪ c c on the discharge current density, calculated on the basis of where the dependence z (␧) is given by ␧ϭe͓␸(z ) 0 0 the ion distribution function found by different methods Ϫ␸(z)͔. ͓Y(␧ϭa(␧Ϫ␧ )2 is the ion cathode sputtering coefﬁcient,1 The results obtained for the case of a quadratic depen- t ␧ is the threshold sputtering energy, and a is a constant for dence of ␸(z) ͑Refs. 3 and 14͓͒␸(z)ϭϪU (z/d )2, where t c c the given kind of ions and cathode material͔. There is good U is the cathode potential drop͔ and mϭnϭ100 are plotted c agreement between the results obtained from the one- in Fig. 1, whence it follows that for the given number of dimensional and two-dimensional models. divisions of the spatial and energy intervals of variation of Consequently, the proposed method, based on a numeri- the function h(z,s) the numerical solution coincides quite cal solution of the one-dimensional equation for the ion dis- well with the exact solution. Therefore these values of m and tribution function, allows one to calculate the energy spec- n were used in further calculations of the ion distribution function in the presence of elastic collisions, where an exact solution of the problem is lacking. Figure 2 plots the ion distribution function for three values of the discharge current density j in a 15:1 helium–neon mixture at a pressure of 6 Torr and temperature of 300 K (␧0ϭ4 eV, with the values of Uc and dc determined from the Aston model1͒, as found by the given method and as obtained by a two-dimensional Monte Carlo simulation of the motion of the ions in the cathode sheath by the technique described in Ref. 10. It can be seen that the method of calculating the ion distribution function proposed here gives results which are very close to the two-dimensional Monte Carlo results. Computer time requirements using the one- FIG. 3. Dependence of the cathode sputtering coefﬁcient averaged over the dimensional model are roughly an order of magnitude less, energy spectrum of the ions, on the discharge current density in a helium– neon mixture for ␧tϭ30 eV, as found by the Monte Carlo method ͑points͒, and in contrast to the Monte Carlo method they do not grow by the proposed method ͑solid line͒, and on the basis of the ion distribution with the ratio dc /␭c . function ͑13͒ for a single-component gas ͑dashed line͒. Tech. Phys. 43 (6), June 1998 V. I. Kristya 667

6 trum of the heavy ions in the cathode sheath of a glow O. B. Firsov and V. V. Kuchinski, Fiz. Plazmy 18, 114 ͑1992͓͒Sov. J. discharge in an inert-gas mixture with satisfactory accuracy Plasma Phys. 18,61͑1992͔͒. 7 A. P. Korgiaviy and V. I. Kristya, J. Appl. Phys. 70, 5117 ͑1991͒. and without the use of large amounts of computer time. It 8 can be used to create a self-consistent model describing the B. E. Thompson and H. H. Sawin, J. Appl. Phys. 63, 2241 ͑1988͒. 9 R. T. Farouki, S. Hamaguchi, and M. Dalvie, Phys. Rev. A 44, 2664 interaction with the cathode surface for a glow-discharge ͑1991͒. plasma of complicated composition. 10 V. I. Kristya, Zh. Tekh. Fiz. 66͑6͒,8͑1996͓͒Tech. Phys. 41, 525 ͑1996͔͒. 11 J. B. Hasted, Physics of Atomic Collisions ͓Butterworths, London, 1964; 1 L. H. Hall, J. Appl. Phys. 64, 2630 ͑1988͒. Mir, Moscow, 1965, 710 pp.͔. 2 12 T. C. Paulick, J. Appl. Phys. 67, 2774 ͑1990͒. G. G. Bondarenko, A. P. Korzhavy, V. I. Kristya et al., Metally, No. 5, 54 ͑1996͒. 13 D. Potter, Computational Physics ͓Wiley, New York, 1973; Mir, Moscow, 3 I. Abril, A. Gras-Marti, and J. A. Valles-Abarca, Phys. Rev. A 28, 3677 1975, 392 pp.͔. 14 ͑1983͒. A. P. Korzhavy and V. I. Kristya, Zh. Tekh. Fiz. 63͑2͒, 200 ͑1993͓͒Tech. 4 V. V. Kuchinski, V. S. Sukhomlinov, and E. G. Shekin, Zh. Tekh. Fiz. Phys. 38, 156 ͑1993͔͒. 55,67͑1985͓͒Sov. Phys. Tech. Phys. 30,38͑1985͔͒. 5 V. P. Konovalov, J. Bretagne, and G. Gousset, J. Phys. D 25, 1073 ͑1992͒. Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Motion of the cathode spot of a vacuum arc in an external magnetic ﬁeld S. A. Barengol’ts, E. A. Litvinov, E. Yu. Sadovskaya, and D. L. Shmelev

Institute of Electrical Physics, Ural Branch of the Russian Academy of Sciences, 620049 Ekaterinburg, Russia ͑Submitted May 19, 1997͒ Zh. Tekh. Fiz. 68, 60–64 ͑June 1998͒ The problem of the motion of the cathode spot of a vacuum arc electrical discharge in a magnetic ﬁeld applied tangential to the cathode surface is considered. The treatment is based on concepts of the nonstationary, cyclical nature of processes occurring in the cathode spot and the key role of return electrons falling out of the near-cathode plasma back onto the cathode. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01206-9͔

INTRODUCTION greater, favorable conditions arise for the formation of a new center.5 The direction of motion of the spot is toward the In the present paper we consider the paradoxical phe- region where the current of return electrons concentrates. nomenon of the retrograde motion of the cathode spot of a The magnetic field created by the transport current from vacuum arc in a tangential magnetic field.1 This phenomenon the emission center i can be estimated from the formula6,7 is attracting research interest because no generally accepted 1 explanation has yet appeared. The existence and motion of ␮0 2i1 B ϭ ͑1͒ the cathode spot is treated as an nonstationary, cyclical pro- 1 4␲ r cess of appearance and dying off of emission centers or explosive centers.2,3 The direction of motion of the spot is the and is directed according to the ‘‘right-hand rule.’’ The mag- direction in which a new center will preferentially arise in netic field of the current i2 of the return electrons is given by place of the old one. On the basis of these ideas we will ␮0 2i2 attempt to explain the phenomenon of retrograde motion. B ϭ , ͑2͒ 2 4␲ r MODEL OF RETROGRADE MOTION OF THE CATHODE and this field is nonzero only inside the current torus, where SPOT it adds together with the field of the forward current. In the Numerical modeling of the expansion of a plasma jet presence of an external magnetic field tangential to the cath- from the emission center of a cathode spot has shown that ode surface, a torque equal to7 there exists in the vicinity of the center a ring current carried Mϭ͓p B ͔, ͑3͒ by return electrons moving from the plasma to the cathode m ⌺ and closed through the emission zone of the center ͑Fig. 1͒.4 will act on each current loop. Here the square brackets de- The motion of the return electrons forms current loops which note the vector ͑cross͒ product, B⌺ is the total magnetic field, together create a toroidal surface. The symmetry axis of the equal to the vector sum of all the fields, and the vector quan- torus is perpendicular to the cathode surface and passes tity pm is the magnetic moment of the current loop carrying through the center of the spot. The geometry of this arrange- a current i ment is shown in Fig. 2. The process of formation of a new emission center is pmϭiS, ͑4͒ linked with the return electron current. Where this current is where S is a vector equal in magnitude to the area encom- passed by the loop and having the direction prescribed by the right-hand rule. The torque will rotate the current loop in such a way that the vectors pm and B⌺ become parallel and the plane of the loop becomes perpendicular to B⌺ . The resulting action on the current torus will have the direction shown in Fig. 2 ͑F͒, i.e., the loop will tend to ‘‘swing around’’ to the ‘‘anti- Ampe`re’’ direction, or the direction of maximum magnetic field. In addition, in a nonuniform magnetic field the current loop is acted on by the force

Gϭgrad͑pmB⌺͒, ͑5͒

FIG. 1. Geometry of an emission center: 1—cathode, 2—plasma, 3— which pulls the loop into the region of higher magnetic ﬁeld. current lines. All this leads to a bunching up of the current lines and an

ing from the diagram shown in Fig. 4, we can write down the following relations: ␲ A A ϭR cos Ϫ␪ , ͑6͒ 1 2 ͩ 2 B ͪ and R ␲ ␸Ϸarctan cos Ϫ␪ . ͑7͒ ͩ Rϩr ͩ 2 B ͪͪ Here R is the radius of the current ring and r is the radius of the emission zone. Figure 5 provides a comparison of the picture developed above with experiment. Curves 1 and 2 correspond to experiments with a dirty cathode surface on which the craters are small and relatively far apart.8,1 In this case, relation ͑7͒ must be augmented by the condition Rӷr. The corresponding calculated curve ͑4͒ is also shown. In experiments with a good vacuum and a well-cleaned cathode surface the size of the craters is signiﬁcantly greater and they lie one on top of another ͑curve 3͒.9 In this case it may be assumed that RϷr ͑curve 5͒. In addition to this, one more circumstance should be noted. The cathode is separated from the near-cathode plasma by a space-charge sheath in which the charged particles, electrons and ions, move without collisions. To describe the particle motion in the sheath layer we will use the full system of Vlasov equations, with the goal of obtaining the momentum and energy conservation laws of the particles and the ﬁeld. We write the Vlasov equations together with Maxwell’s equations:

␣f ␣ e ץ ϩv•ٌ f ␣ϩ ͑Eϩ͓vHٌ͔͒v f ␣ϭ0, ͑8͒ ␣t mץ 1 1 Eϭ ͚ e␣ ͵ f ␣dvϭ ͚ e␣͗1͘␣ , ͑9͒•ٌ ␧0 ␣ ␧0 ␣

Eץ Eץ , ␣H͔ϭ␧0 ϩ e␣ vf ␣dvϭ␧0 ϩ e␣͗vٌ͓͘ ␣t ͚ץ ͵ ␣t ͚ץ ͑10͒ Bץ FIG. 2. a—schematic depiction of the arrangement of current contours of ٌ͓E͔ϭϪ , ͑11͒ tץ return electrons about an emission center, b—view from above and to the side. Dashed lines depict reaction of the contours to an external tangential magnetic ﬁeld. Arrows on rings show the direction of current. The direction ٌ•Bϭ0. ͑12͒ of motion of the electrons is opposite.

increase in the current density of the return electrons from the plasma to the cathode in precisely the retrograde direction. If the external magnetic ﬁeld is oriented at an angle with respect to the cathode surface, then the picture of motion of the spot changes. The so-called Robson angle effect arises. This is illustrated in Fig. 3. Figure 4 shows a diagram elucidating this phenomenon. In the given case the current loop not only swings around but also inclines relative to the cathode plane, tending to occupy the position in which its plane becomes perpendicular to the magnetic ﬁeld vector. Proceed- FIG. 3. Illustration of the Robson angle effect ͑1 is the track of a spot͒. 670 Tech. Phys. 43 (6), June 1998 Barengol’ts et al.

Here f ␣ denotes the distribution function of particles of sort ␣ ͑electrons and ions from the plasma, electrons from the cathode͒; e␣ and m␣ are the charge and mass of the particles; E and H are the electric and magnetic ﬁeld vectors; the magnetic ﬁeld induction Bϭ␮0H, ␧0 and ␮0 are the v; theץ/ץx; ٌvϭץ/ץelectric and magnetic constants, ٌϭ angle brackets denote averaging over the corresponding distribution function. We multiply Eq. ͑8͒ by m␣v, integrate over v, and carry out the sum over ␣. We then have

ץ ␣m␣v͘␣ϩٌ• ͗m␣vv͗͘ t ͚␣ ͚␣ FIG. 5. Dependence of the Robson angle on the inclination angle of theץ external magnetic ﬁeld relative to the cathode. 1—Robson’s experiment:8 Al, 0.15 T, 10Ϫ1 Torr, 28 A; 2—Kesaev’s experiment:1 Cu, 1 kOe, Ϫ3 9 Ϫ9 Ϫ͚ e␣͑E͗1͘␣ϩ͓͗v͘␣B͔͒ϭ0. ͑13͒ 10 Torr, 1–10 A; 3—Juttner’s experiment: Mo, C, 0.37 T, 10 Torr, ␣ 10–300 A; 4—formula ͑7͒ for Rӷr; 5—formula ͑7͒ for RϷr.

ץ Using Maxwell’s equation, we transform the two last ␣͚ ͗m␣v͘␣ϩ␧0␮0͓EH͔ ϩٌ•͚ ͗m␣vv͘ ␣ ͪ ␣ ͩ tץ terms in Eq. ͑13͒

␮0 2 Ϫ␧0Eٌ Eϩ ٌH Ϫ␮0H ٌH • E e␣͗1͘␣ϭ␧0Eٌ•E, • 2 ͚␣

␧0 ϩ ٌE2Ϫ␧ E ٌEϪ␮ Hٌ Hϭ0. ͑14͒ • E 2 0 • 0ץ , e v͘ B͔ϭٌ͓͓H͔B͔Ϫ␧ B͓͗ ͚ ␣ ␣ 0 t The last term in Eq. ͑14͒ has been added for symmetry ͬ ץ ͫ ␣ ͓to accommodate Eq. ͑12͔͒. According to the rules of vector 1 2 calculus, we can write ,Eٌ͓E͔͔ϭ ٌE ϪE•ٌE͓ 2 1 1 Aٌ AϪ ٌA2ϩA ٌAϭٌ AAϪ A2Uˆ , ͑15͒ • 2 • •ͩ 2 ͪ ␮0 -H͔B͔ϭϪ ٌH2ϩ␮H ٌH. where the expression AA is a tensor ͑direct product of vecٌ͓͓ 2 • tors͒ and Uˆ is the identity matrix. We now rewrite Eq. ͑13͒ in the following form: Taking the above vector identity ͑15͒ into account, we obtain from Eq. ͑14͒ the equation of momentum conservation of the particles and ﬁeld ץ ␣͚ ͗m␣v͘␣ϩ␧0␮0͓EH͔ ϩٌ• ͚ ͗m␣vv͘ ␣ ͭ ͮ ␣ ͭ tץ 2 2 ␧0E ϩ␮0H Ϫ ␧ EEϩ␮ HHϪ Uˆ ϭ0. ͑16͒ ͩ 0 0 2 ͪͮ In an analogous way we obtain the equation of energy conservation. Toward this end we multiply Eq. ͑1͒ by 2 (m␣v )/2, integrate it over v, and carry out the sum over ␣. We obtain 2 2 m␣v m␣v ץ .͚ ϩٌ•͚ v Ϫ͚ ͗e␣v•E͘␣ϭ0 ␣ ʹ t ␣ ͳ 2 ʹ ␣ ͳ 2ץ ␣ ␣ ͑17͒ From Eq. ͑10͒ Eץ e␣v•E͘␣ϭٌ͓H͔•EϪ␧0 E. ͑18͒͗ tץ ␣͚ Carrying out the vector product in Eq. ͑18͒, we have Hץ H͔ EϭϪٌ ͓EH͔Ϫ␮ H . ͑19ٌ͓͒ tץ FIG. 4. Diagram elucidating the origin of the Robson angle. • • 0 Tech. Phys. 43 (6), June 1998 Barengol’ts et al. 671

Employing Eqs. ͑18͒ and ͑19͒, we write the equation of energy conservation of the particles and ﬁeld in the form

2 2 2 m␣v ␧0E ϩ␮0H ץ ͚ ϩ ͪ t ͩ ␣ ͳ 2 ʹ 2ץ ␣ 2 m␣v ϩٌ• ͚ v ϩ͓EH͔ ϭ0. ͑20͒ ͩ ␣ ͳ 2 ʹ ͪ ␣ Note the following. The particles, electrons and ions, cross the space-charge sheath over times close to their respective inverse plasma frequencies. At the plasma densities near the center, nу1024 mϪ3, these times are very short in comparison with the characteristic lifetime of an explosive emission center ␶Ϸ10Ϫ8 s ͑Ref. 10͒. FIG. 6. Diagram of an emission center and the sheath layer separating the The dying out of a center is accompanied by a rapid drop cathode ͑1͒ from the near-cathode plasma ͑2͒. The hatched regions represent a cross section of the cylindrical cavity over which equations ͑16͒ and ͑20͒ of the current and the appearance of induced electric and are to be integrated. magnetic fields. The correction to the fields associated with this effect is given by r/(␶1c), where r is the radius of the crater, ␶1 is the characteristic time of current drop, and c is the speed of light.7 The corrections are substantial if r/(␶c) perscript P and subscript K together indicate the flux differ- у1. In our case rϷ10Ϫ6 m, ␶Ϸ10Ϫ9 s ͑Ref. 10͒, and ence at the boundaries of the sheath with the cathode and r/(␶c)Ͻ10Ϫ5Ӷ1, i.e., it is possible to ignore the variation of with the plasma. Since the quantity ⌬r is arbitrary, we finally the fields associated with variation of the current and use obtain only the time-independent forms of Eqs. ͑16͒ and ͑20͒ and of ␧ E2 P Maxwell’s equations. 2 0 ͚ ͗m␣v ͘␣Ϫ ϭ0. ͑22͒ ͩ ␣ 2 ͪ The particles in the cathode sheath are not magnetized; K however, in order to use the one-dimensional Vlasov equations it is still necessary to show that the tangential compo- Integration of Eq. ͑20͒ over the same region gives the nent of the electric field, associated with the Ohmic potential following expression: drop at the cathode and in the near-cathode plasma due to the 2 P m␣v arc current, is small. The potential drop at the cathode was r⌬r ͚ v ϩr2 ͩ ␣ ͳ 2 ʹ ͪ taken into account numerically within the framework of the ␣ K model described in Ref. 11. The calculations show that: a͒ L L the potential drop is almost purely Ohmic, and thermoelec- ϫ ͓EH͔2dzϪr1 ͓EH͔1dzϭ0, ͑23͒ tric effects are small, b͒ the potential drop can reach a value ͵0 ͵0 р10 V, i.e., it is comparable with the cathode fall, and c͒ the Ϫ6 where L is the thickness of the sheath, and the subscripts 1 potential drop is concentrated within р10 m of the emis- and 2 correspond to the value of the function at the inner and sion zone. outer radius of the hatched region in Fig. 6. The potential distribution in the near-cathode plasma Taking ⌬rϭr1Ϫr2 to be sufficiently small, we can write was considered in Ref. 4. It was shown there that it also has an Ohmic character and that the field is concentrated on a L L L Ϫ6 r2 ͓EH͔2dzϪr1 ͓EH͔1dzϷ⌬r ͓EH͔dz. scale of the order of 10 m. ͵0 ͵0 ͵0 Thus, we can neglect the tangential ͑to the cathode͒ component of the electric field in the cathode sheath and use Thus, Eq. ͑23͒ can be rewritten as follows: the one-dimensional time-independent Vlasov equations to 2 P L m␣v 1 calculate the particle characteristics. We integrate the time- ͚ v ϩ ͓EH͔dzϭ0. ͑24͒ ͩ ␣ ͳ 2 ʹ ͪ r ͵0 independent form of Eq. ͑16͒ over the region indicated in ␣ K Fig. 6. It can be assumed that the particle velocities and the electric field in the sheath have only one component, parallel Relation ͑21͒ is a consequence of balance of the momen- to the symmetry axis ͑Fig. 1͒. The volume integral over the tum flux through the end-faces of the cylindrical shell in Fig. hatched region in Fig. 6 is transformed into an integral over 6. It allows one to determine the field strength at the cathode its surface. By virtue of the cylindrical symmetry of the sys- without solving Poisson’s equation in the sheath if the par- tem, integration over the end-faces yields the relation ticle characteristics are known. Relation ͑24͒ shows that the energy flux transported by r E2 P 2 ␧0 the particles normal to the cathode surface increases with 2␲ rЈdrЈ ͚ ͗m␣v ͘␣Ϫ ϭ0, ͑21͒ increasing energy flux transported along the sheath by the ͵rϪ⌬r ͩ ␣ 2 ͪ K field. Hence the possibility opens up of a new interpretation since the magnetic field strength does not depend on the z of retrograde motion of the cathode spot. It is well known coordinate, which runs perpendicular to the cathode. The su- that the retrograde motion takes place in the direction of the 672 Tech. Phys. 43 (6), June 1998 Barengol’ts et al. maximum magnetic field. In this direction the vector ͓EH͔ is 1 I. G. Kesaev, Cathode Processes of an Electric Arc ͓in Russian͔͑Nauka, also maximum and, consequently, the normal particle flux Moscow, 1968͒. 2 toward the cathode has a large value, which in turn leads to G. A. Mesyats and D. I. Proskurovski, Pulsed Electrical Discharge ͓in Russian͔͑Nauka, Novosibirsk, 1984͒. a greater probability of appearance of new centers in the 3 G. A. Mesyats, Usp. Fiz. Nauk 165, No. 6, 601 ͑1995͒. direction of the retrograde motion. The additional accelera- 4 E. A. Litvinov, G. A. Mesyats, and A. G. Parfenov, Dokl. Akad. Nauk tion of the particles in the sheath has an electrodynamic char- SSSR 310, 344 ͑1990͓͒Sov. Phys. Dokl. 35,47͑1990͔͒. 5 acter. E. A. Litvinov, G. A. Mesyats, A. G. Parfenov, and A. I. Fedosov, Zh. Tekh. Fiz. 55, 2270 ͑1985͓͒Sov. Phys. Tech. Phys. 30, 1346 ͑1985͔͒. 6 R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 5, Electricity and Magnetism ͑Addison-Wesley, Reading, CONCLUSION Mass., 1965; Mir, Moscow, 1977͒. 7 R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Thus, we can claim reasonable agreement of the experi- Physics, Vol. 6, Electrodynamics ͑Addison-Wesley, Reading, Mass., mental data with calculations based on our quite simple 1965; Mir, Moscow, 1977͒. 8 model. A second mechanism, associated with the electrody- A. E. Robson, in Proceedings of the IVth International Conference on Phenomena in Ionized Gases, Uppsala, 1959, Vol. II b, pp. 346–349. namic character of the acceleration of the particles in the 9 B. Juttner, Thesis B. Academy of Sciences. Berlin, 1983. sheath layer separating the plasma from the cathode probably 10 E. A. Litvinov, G. A. Mesyats, and A. G. Parfenov, Dokl. Akad. Nauk plays a large role under dirty conditions, when the craters are SSSR 279, 864 ͑1984͓͒Sov. Phys. Dokl. 29, 1019 ͑1984͔͒. 11 widely separated from one another. S. A. Barengolts, E. A. Litvinov, A. G. Parfyonov, in Proceedings of the XIVth International Symposium on Discharge and Electrical Insulation in This work was supported by the Russian Fund for Fun- Vacuum, Santa Fe, New Mexico ͑1990͒, pp. 185–186. damental Research, Project No. 96-02-16194-a. Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Onset of rotating stall in induction magnetohydrodynamic ﬂows Yu. A. Polovko, E. P. Romanova, and E´ . A. Tropp

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia ͑Submitted June 3, 1997͒ Zh. Tekh. Fiz. 68, 65–70 ͑June 1998͒ The possible onset of unsteady induction magnetohydrodynamic ͑MHD͒ flows in wide channels or in channels of annular cross section is discovered on the basis of a proposed two- dimensional mathematical model. Such secondary flows have the character of rotating stall, which was previously known in high-pressure axial compressors. The existing experimental data confirm the possibility of observing this phenomenon, which can be interpreted as a new type of symmetry loss. It is shown for certain relations between the parameters that the system has a lower margin of stability against disturbances of the rotating-stall type than against steady-state disturbances. In particular, a loss of stability of a steady uniform flow can occur on the descending portion of the external characteristic of the machine. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01306-3͔

INTRODUCTION between the properties of the mathematical models of their different physical phenomena. At the same time, the second- The loss of stability of uniform induction flows in mag- ary flows in compressors are unsteady, have a helical char- netohydrodynamic ͑MHD͒ channels and the properties of the acter, and the velocity cell formed moves across the width of secondary flows that arise have hitherto been studied mainly the channel. The observed similarity between the phenomena on the basis of the so-called one-dimensional jet flow model, provides some basis to assume that flows of such structure which was first used for this purpose in Ref. 1. The main can also exist in induction MHD channels. In the present results were described in detail in Refs. 2–5. At the same work it is shown on the basis of a proposed two-dimensional time, transverse velocities, which lead to the development of model in the long-wavelength approximation that instability helical motion, were discovered in experiments devised to of the rotating-stall type can be observed in induction MHD study the structure of the flows in channels of induction flows. A linear analysis, which permits drawing several con- MHD machines.6,7 As was stressed in Ref. 8, the reason for clusions regarding the character of its appearance as a func- the observation of the transverse velocities could not be es- tion of the of the parameters of the problem, is performed. tablished unequivocally. This is because there was asymme- The previously studied, steady secondary flows are also de- try of the primary field with respect to the azimuth ͑the trans- scribed by the two-dimensional system used as special cases verse coordinate͒ in the experiments performed due to the of possible solutions for definite relations between the pa- engineering constraints. The question of whether the trans- rameters. verse motions are caused by this asymmetry or appear as a result of the instability of the symmetric problem remained MATHEMATICAL MODEL open, and an investigation of the stability of at least two- The flow of the conducting fluid in an induction MHD dimensional motion was needed to resolve it. Such an at- channel, whose scheme has been presented, for example, in tempt was apparently undertaken for the first time in Ref. 9, Ref. 4, is investigated. Here and below the notation and di- but the status of the results obtained is not entirely clear, rections of the coordinate axes correspond to those adopted since the stability of a uniform flow was studied in that work in Refs. 4 and 5. We shall use the so-called plane-parallel only with respect to a narrow class of disturbances, whose field model ͑Ref. 11, p. 155͒, assuming that there is only one wavelength was equal to or a rational fraction of the wave- magnetic field component normal to the channel wall, length of the external traveling field. The study of the properties of steady secondary flows in Bϭ͑0,0,˜b͑x,y,t͒exp͓i͑␣xϪ⍀t͔͒͒, Refs. 4 and 5 demonstrated their remarkable similarity to the ˜b x,y,t͒ϭb ϩib , ͑1͒ secondary flows in high-pressure axial compressors, which ͑ a r appear as a result of so-called ‘‘rotating stall.’’ 10 This simi- which corresponds to the fundamental mode of the external larity encompasses both external integral characteristics ͑the electromagnetic field traveling along the channel axis B0 presence of an extended horizontal segment on them and ϭBm sin(␣xϪ⍀t). This approximation is natural for chan- operation in an ‘‘ideal pressure source’’ regime5͒ and the nels of small height with walls having a high magnetic per- flow structure ͑the existence of internal boundary layers meability. Unlike the so-called jet model, despite the axial separating a ‘‘cell,’’ in which the stream has a velocity dif- symmetry of the machine, we allow the possibility of the fering strongly from the flow nucleus and is possibly even existence of an azimuthal velocity component, i.e., we set oppositely directed͒. Such similarity points out the similarity Vϭ(u,v,0). Substituting ͑1͒ into the induction equation

B v͉xϭ0ϭv͉xϭL , v͉yϭ0ϭv͉yϭS , ͑5͒ץ ⌬BϪ␮␴ ϩ␮␴ curl͑VϫB͒ϭϪcurl j0 , ͑2͒ dt which can be interpreted either as the conditions for a cylin- ͑␦ is the Laplacian operator͒, we obtain for the relative value drical induction MHD channel closed into a torus or as the of the complex amplitude of the magnetic induction b natural boundary conditions for wide and long channels. ˜ Here L and S are the relative length and width of the channel, ϭb/Bm respectively. The velocity of the external traveling field ͓v͔ ͓A1ϩA2͔bϭi, ϭ⍀/␣ was taken as the velocity scale in ͑3͒–͑5͒, and ͓t͔ ϭ1/⍀ and ͓b͔ϭB were taken for the time and the mag- m ץ 2ץ netic induction, respectively. We note that the applicability A1ϭ 2 Ϫ␧ Ϫ1ϩi␧͑1Ϫu͒, -t of the model ͑3͒–͑5͒ just formulated is restricted by the avץ yץ eraging procedure used, which requires weak variation of b ץ ץ ץ 2ץ A ϭ ϩ2i Ϫ␧u Ϫ␧v , ͑3͒ and v over lengths of the order of 2␶ and times of the order yץ xץ xץ x2ץ 2 of T, i.e., the problem is solved in the ‘‘long-wavelength’’ where ␧ϭ␮␴⍀ is the magnetic Reynolds number, A1 is the approximation. operator corresponding to the jet flow model, and A2 is an additional operator which is conditional on the presence of an azimuthal component of the velocity V and the possibility LINEAR ANALYSIS. SYSTEM OF EQUATIONS SPECIFYING of that the field amplitude can vary along the x axis. NEUTRAL SURFACES We note that in ͑2͒ j0 is the extrinsic current density 2B The problem ͑3͒–͑5͒ admits a solution corresponding to 2 0 ץ curl j0ϭϪ 2 ϭϪ␣ Bm i exp͓i͑␣xϪ⍀t͔͒. a steady flow that is uniform with respect to both coordi- :x natesץ The equations of motion are averaged with respect to the ␧͑1Ϫu0͒ period and wavelength of the external traveling electromag- uϭu0 , v0ϭ0, ba0ϭ 2 2 , netic field. After application of the averaging operator ␧ ͑1Ϫu0͒ ϩ1

T 2␶ 1 1 1 b ϭϪ , •dt •dx, r0 ␧2͑1Ϫu ͒2ϩ1 T ͵0 2␶ ͵0 0 2 p j ͑1Ϫu0͒ץ where Tϭ2␲/⍀ and ␶ϭ␲/␣, the equations of motion for ϭ Ϫu ͉u ͉. ͑6͒ xͪ ␧2͑1Ϫu ͒2ϩ1 0 0ץͩ the principal velocity component, which varies slowly in 0 0 comparison to the variation of the external ﬁeld, take the We seek solutions of the form form UϭU ϩ␦UϭU ϩC exp͓i͑2␲my/Sϩ2␲nx/LϪ␻t͔͒, u 0 0ץ uץ uץ z ϩu ϩv ͑7͒ ͪ yץ xץ ␶ץ ͩ where p j2 b b r 2 2 ץ a ץ ץ ϭϪ ϩ b Ϫb Ϫb Ϫuͱv ϩu , u u0 c1 ͪ xץ x rץ x ␧ ͩ a aץ v v0 c2

, v Uϭ ba , U0ϭ ba0 , Cϭ c3ץ vץ vץ z ϩu ϩv yͪ br br0 c4ץ xץ ␶ץͩ ͩ p ͪ ͩ p0 ͪ ͩ c5 ͪ 2 brץ baץ p jץ ϭϪ Ϫ b ϩb Ϫvͱv2ϩu2, which branches off from the uniform solution ͑6͒. After lin- yץ y rץ y ␧ aץ ͩ ͪ earizing the problem ͑3͒–͑5͒ in the vicinity of ͑6͒ and plug- v ging in ͑7͒, we obtain a system of linear equations, whoseץ uץ ϩ ϭ0, ͑4͒ y determinant equalsץ xץ where j is the relative current density due to the voltage ⌬ϭ͉Dik͉, supplied and z is the hydraulic inductance of the pump. where Here, as in the jet model, only the turbulent ﬂow at the D ϭi Ϫk2Ϫk2Ϫ1Ϫi k u , wall ͓the last terms of the ﬁrst two equations in ͑4͔͒ is taken 11 ␧␻ 1 2 ␧ 2 0 into account in view of the small height of the channel. D12ϭϪ␧͑1Ϫu0͒Ϫ2ik2 , Equations ͑3͒ and ͑4͒ are supplemented by the periodic- ity conditions D13ϭ␧br0 , D14ϭ0, D15ϭ0,

b͉xϭ0ϭb͉xϭL , b͉yϭ0ϭb͉yϭS , D21ϭ␧͑1Ϫu0͒ϩ2ik2 , 2 2 , b D22ϭi␧␻Ϫk1Ϫk2Ϫ1Ϫi␧k2u0ץ bץ bץ bץ ϭ , ϭ , yͯץ yͯץ xͯץ xͯץ xϭ0 xϭL yϭ0 yϭS D23ϭϪ␧ba0 , D24ϭ0, D25ϭ0, Tech. Phys. 43 (6), June 1998 Polovko et al. 675

2 2 2 2 2 2 2 2 j ij R3ϭ͑R1ϩ1͒ ϩ␧ ͑1Ϫu0͒ Ϫ␧ R Ϫ4k , D ϭϪ ͑1Ϫib k ͒, D ϭ b k , 2 2 31 ␧ a0 2 32 ␧ r0 2 R4ϭ2␧͓R2͑R1ϩ1͒ϩ2͑1Ϫu0͒k2͔. D33ϭzu0k2iϩ2͉u0͉Ϫi␻z, D34ϭ0, D35ϭik2 , ANALYSIS OF THE SYSTEM SPECIFYING THE BRANCH ij2 ij2 D ϭ k b , D ϭ b k , D ϭ0, POINT 41 ␧ 1 a0 42 ␧ r0 1 43 As we have already noted, the mathematical model D44ϭizu0k2ϩ͉u0͉Ϫi␻z, D45ϭik1 , adopted is applicable only in the case of k2Ӷ1 and ␻Ӷ⍀ because of the procedure used to average the equations of D51ϭ0, D52ϭ0, D53ϭik2 , D54ϭik1 , D55ϭ0. motion. In this context the form of the asymptotic formula Here we have introduced the following new notation: k 1 for ␻ and u0 at small values of k2 is of interest. After sub- ϭ2␲m/S and k2ϭ2␲n/L are wave numbers, which express stituting the series the relative density of the waves appearing along the width 1 ͑2͒ 2 2 and length of the channel, respectively. The condition for the ␻ϭ␻0ϩ␻1k2ϩO͑k2͒, u0ϭu0ϩu0 k2ϩO͑k2͒ existence of nontrivial solutions leads to a system of two into ͑8͒ and equating the terms with identical powers of k , nonlinear equations 2 in a ﬁrst approximation we obtain

Re͑⌬͒ϭ0, Im͑⌬͒ϭ0, 1 2 2 ␻0ϭ0, 2͉u0͉G͑GϩK Ϫ1͒Ϫ j ͑GϪKϪ1͒ϭ0, ͑9͒ from which the branch point u and the stall rotation fre- 0 2 2 2 1 quency ␻ are determined. The expanded form of this system where Kϭ1ϩk1, Gϭ1ϩ␧ s , and sϭ1Ϫu0. is Expression ͑9͒ is the familiar equation for determining the branch points of stationary solutions.1 Thus, the previ- 2 2 2 R3͉u0͉͑R1ϩk1͒ϪzR1R2R4Ϫk1j ͓ba0␧͑1Ϫu0͒ ously investigated steady ﬂows comprise a subset of the solutions of the two-dimensional model, whose branch points ϩbr0͑R1ϩ1͔͒ϭ0, are determined from ͑8͒ when k2ϭ0. Equating the terms of 2 2 2 2 R4͉u0͉͑R1ϩk1͒ϩzR1R2R3Ϫk1j ͓2ba0k2ϩ␧br0R2͔ϭ0, order k2 in the second equation and of order k2 in the ﬁrst ͑8͒ equation gives the following expressions: where 1 1 2 2 2 2 1 Gu0͓4␧͑Ku0ϩ2s͒ϩz͑K ϩ␧ s ͔͒Ϫ␧j ͑2sϪu0͒ 2 2 ␻1ϭ 1 2 2 2 2 , R1ϭk1ϩk2 , R2ϭk2u0Ϫ␻, G͓4␧u0Kϩz͑K ϩ␧ s ͔͒ϩ␧j

G͓͑K2ϩ␧2s2͒u1ϩ2k2u1͑2KϪ␧2W2Ϫ4͒Ϫ2z␧k2W͑KWϩ2s͔͒ϩk2j2 ͑2͒ 0 1 0 1 1 u0 ϭϪ 2 2 2 2 1 2 1 2 2 2 , 2k1͓͑K ϩ␧ s ͒͑GϪ2u0␧ s͒Ϫ2Gu0␧ sϩj ␧ s͔

where Wϭu1Ϫ␻ . It is seen that the phase velocity for the 2 0 1 on k2 for various values of z and the mode number n. The propagation of long waves along the channel cϭ␻/k2ϭ␻1 2 nonmonotonic character of the u0(k2) curves appearing for does not depend on the wavelength. fairly large values of z is of fundamental importance for the The case of large currents due to the voltage supplied analysis. It means that the region for the existence of flows of ( j ϱ) is also of practical interest. The asymptotic expres- the rotating-stall type can be wider than the region for the sion→ then has the simple form existence of steady secondary flows, for which k2ϭ0. Un- 2 2 steady flows should be observed experimentally for such pa- ͱk1ϩk2ϩ1 u ϭ1Ϫ ϩ␦͑ j͒, ␻ϭk ͑3u Ϫ2͒ϩ␦͑ j͒, 0 ␧ 2 0 where ␦( j) 0 and j ϱ. The system→ ͑8͒ was→ solved numerically by Newton’s method in the general case. The calculations were performed mainly for the pump operating regime of an MHD machine 0Ͻu0Ͻ1. The problem of flow in a channel closed into a torus is apparently artificial and is not of practical interest; therefore, we at once turn to an analysis of flow in an infinite tube of annular cross section. This corresponds to the continuous variation of k2 and the discrete set of values 2 FIG. 1. Dependence of u0 on the square of the wave number k2 for ␧ϭ4, k1ϭ2␲n/Lϭͱ୐n, nϭ1,2,3... . Figure 1 shows the neutral j2ϭ40, ͱ୐ϭ0.1, and zϭ5 ͑a͒,10͑b͒, and 20 ͑c͒ in the pump regime for curves, i.e., plots of the dependence of u0 at the branch point various azimuthal mode numbers nϭ1 ͑1͒,2͑2͒, and 3 ͑3͒. 676 Tech. Phys. 43 (6), June 1998 Polovko et al.

FIG. 2. Critical value of the angular frequency 2 ␻ ͑1a͒, the square of the wave number k2 ͑2a͒, and u0* ͑b͒ as functions of the hydraulic induc- 2 tance z for k1ϭ0.1, ␧ϭ4, and j ϭ40. The plot was constructed for the region of maximum pressure values in the pump regime.

rameters, since the stability reserve of the system toward EVOLUTION OF FINITE DISTURBANCES such disturbances is minimal. The results obtained demonstrate the possibility, in prin- Figure 2 shows the dependences of the critical values ciple, of the appearance of secondary unsteady induction u*ϭmax(u ), ␻, and k2 on z. The vertical line in Fig. 2a and 0 2 MHD flows of the rotating-stall type. However, at least two the horizontal line in Fig. 2b correspond to the highest value important questions can be solved only as a result of the of u possible for the steady uniform flow 6 in the pump 0 ͑ ͒ consideration of finite, but small disturbances. It is not clear, regime. The intersection of these lines by the u*(z), ␻(z), 2 first, whether the instability studied can lead to the twisting and k2(z) curves attests to the possibility of a loss of stability of an axisymmetric flow and, second whether the waves ͑7͒ on the descending branches of the external characteristic. 2–4 appearing can quench one another and lead to a wave which Such behavior is impossible for steady secondary flows. It is standing in the azimuthal direction, rather than to a ‘‘trav- is known that branching external characteristics which are eling cell.’’ In fact, as follows from a linear analysis, after smoother than those predicted by the one-dimensional theory the loss of stability at the critical point, the main part of a are observed experimentally. They do not exhibit a charac- disturbance of the uniform solution will have the form teristic peak in the vicinity of the branch point. The possibility of the branching of a static external characteristic on the ␦ Uϭ␩ j1U1 exp͓i͑k2xϩk1yϪ␻t͔͒ϩ␩ j2U1 descending portion can lead to smoothing of that peak. This ϫexp͓i͑k2xϪk1yϪ␻t͔͒ϩc.c.͑O͑␧͒͒, ͑10͒ becomes conclusively clear after a nonlinear analysis and complete numerical determination of the external character- where c.c. denotes the complex-conjugate terms. istics of the unsteady secondary flows. The amplitudes j1 and j2 are determined from the con- The limiting case of flow between parallel plates, where ditions of solvability of the equations for the next terms in the expansion of ␦ U into a series in the supercriticality ␩2 both wave numbers k1 and k2 vary continuously, is also 2 2 ϭu0*Ϫu0 . These conditions from a so-called system of interesting. Figure 3 shows the neutral surface u0(k1 ,k2) corresponding to the zeroth damping decrement of the dis- branching equations, which has the same form as in the case turbances for this case as an example. It is seen that it has a maximum corresponding to the critical point for the loss of stability for the parameters taken. The instability appears on the descending portion of the external characteristic, but the 2 fairly large values of k2 ͑0.4–0.7͒ at the critical point force us to question the applicability of the model used for studying the stability of flow between plates, at least for quantitative calculations. A rigorous analysis calls for studying the stability of a system which has not been averaged with respect to time and the wavelength. This leads to the need to solve the far more complex eigenvalue problem for equations with variable coefficients. We note that axial symmetry leads to equivalence of the stall rotation directions, i.e., small disturbances with either the azimuthal wave number k1 or Ϫk1 are always possible. However, as is seen directly from system ͑8͒, the sign of k2 does not vary, i.e., the direction of motion of a wave traveling along the x axis always coincides with the direction of 2 2 2 motion of the external field. FIG. 3. The neutral surface u0(k1 ,k2) for zϭ11.3, ␧ϭ4, and j ϭ40. Tech. Phys. 43 (6), June 1998 Polovko et al. 677 of the loss of stability of a hot jet.12 As follows from the several similarities to so-called rotating stall, which has pre- basic principles of symmetry and is indicated by the prelimi- viously been described for high-pressure compressors. nary analysis performed, the system of branching equations 3. At fairly high values of the hydrodynamic inductance, has solutions of the form the region for the existence of such twisting flows can be wider than that of steady secondary flows. j ϭ0, ͉j͉2ϭ j2, ͑11͒ 1 2 4. At fairly high values of the twisting hydrodynamic 2 2 inductance, flows can also appear on the descending j2ϭ0, ͉j͉1ϭ j , ͑12͒ branches of static external characteristics in the pump and which correspond to the generation of rotating stall with cells generator regimes, but this is ruled out for steady secondary moving in opposite directions, and the solution flows. 2 2 2 5. In all likelihood, the analysis performed accounts for j1ϭ j2ϭ␤ , ͑13͒ the previously observed twisting of the flows in the channels which corresponds to a wave which is standing in the azi- of cylindrical induction MHD machines and the smoothing muthal direction and traveling in the longitudinal direction. of the external characteristics of the secondary flows in com- The helical waves corresponding to solutions ͑11͒ and ͑12͒ parison to the flows predicted by the one-dimensional jet lead to deformation and twisting of the mean flow already in model. the next approximation with respect to ␧ using the Reynolds 6. The further study of rotating stall in induction MHD directions (u1•ٌ)u1 and nonlinear friction in ͑4͒. An analy- flows will involve a nonlinear analysis in the vicinity of the sis of the stability of these solutions near the neutral curve neutral curve and numerical construction of nonstationary reveals which of the two regimes, i.e., ͑11͒ and ͑12͒ with solutions and the corresponding external characteristics far twisting or ͑13͒ without twisting, is realized. The stability of from the branch point. In addition, a refinement of the pro- the solutions ͑11͒–͑13͒ will depend on the relation between j posed mathematical model, which would eliminate the con- and ␤, which, in turn, can be defined in terms of the param- straints on the density and velocity of the traveling waves eters of the original system ͑3͒ and ͑4͒. This analysis is our caused by the procedure used to average the equations of next subject of investigation. motion, is possible. We note that the steady flows generated by the evolution of the perturbations ͑7͒ for k2ϭ0 and their external charac- 1 teristics were constructed completely in Ref. 2 and that their A. K. Galitis and O. A. Lielausis, Magn. Gidrodin. No. 1, 106 ͑1975͒. stability was demonstrated in Ref. 3 within the one- 2 Yu. A. Polovko and E´ . A. Tropp, Magn. Gidrodin. No. 4, 106 ͑1986͒. 3 dimensional jet model. However, it is not known whether Yu. A. Polovko, Magn. Gidrodin. No. 3,81͑1989͒. 4 Yu. A. Polovko, E. P. Romanova, and E´ . A. Tropp, Zh. Tekh. Fiz. 66͑4͒, they are stable in the two-dimensional model ͑3͒–͑5͒. Dis- 36 ͑1996͓͒Tech. Phys. 41, 315 ͑1996͔͒. placement of the branch points of the nonstationary solutions 5 Yu. A. Polovko, E. P. Romanova, and E´ . A. Tropp, Zh. Tekh. Fiz. 67͑6͒, to the right, into the region of negative slope of the static 5 ͑1997͓͒Tech. Phys. 42, 591 ͑1997͔͒. 6 external characteristic for a uniform flow, leads to the fact A. O. Klavinya, O. A. Lielausis, and V. V. Riekstin’sh, ‘‘Nonuniform flow in the channel of a cylindrical pump,’’ in Abstracts of the 8th Riga that some stationary solutions of ͑4͒ can be totally absent Conference on Magnetohydrodynamics ͓in Russian͔, Riga ͑1975͒, Vol. 2, over a fairly broad range of variation of the external load pp. 79–81. after the loss of stability of the uniform flow. 7 I. R. Kirillov, A. P. Ogorodnikov, and V. P. Ostapenko, Magn. Gidrodin. No. 2, 107 ͑1980͒. 8 R. A. Valdmane, Ya. Ya. Valdmanis, and L. Ya. Ulmanis, Magn. Gidro- CONCLUSIONS din. No. 1, 103 ͑1986͒. 9 In summary, we formulate the main conclusions follow- S. Yu. Reutski, Magn. Gidrodin. No. 1, 121 ͑1987͒. 10 E. M. Greitzer, Trans. Am. Soc. Mech. Eng. 103, 193 ͑1980͒. ing from the analysis performed. 11 A. I. Vol’dek, Induction Magnetohydrodynamic Machines with a Liquid- 1. Twisting helical flows of the traveling-wave type can Metal Working Fluid ͓in Russian͔,Eńergiya, Leningrad ͑1970͒, 271 pp. 12 appear along with the previously studied static secondary E. M. Zhdanova, in Stability and Turbulence ͓in Russian͔,M.A. flows in induction MHD channels. Gol’dshtik and V. N. Shtern ͑Eds.͒, Novosibirsk ͑1985͒, pp. 71–81. 2. The appearance of secondary flows of such a kind has Translated by P. Shelnitz TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Stability of the normal phase in a bounded current-carrying superconducting film A. S. Rudy Yaroslavl State University, 150000 Yaroslavl, Russia ͑Submitted July 30, 1996; resubmitted June 27, 1997͒ Zh. Tekh. Fiz. 68, 71–77 ͑June 1998͒ The two-phase equilibrium states of a current-carrying thin superconducting film in the case of convective heat transfer on the free surface are considered, and their stability is investigated in a first approximation. It is shown that of the two equilibrium states, the state with the normal- phase region of larger size is stable. In the limiting case of an infinitely long film, the stable two-phase equilibrium state tends to a spatially uniform normal state, and the unstable state remains localized. In a definite range of values of the system parameters, the relaxation time of such a formation can be fairly long, and it should be regarded as a quasistable equilibrium state. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01406-8͔

INTRODUCTION dimensional. We introduce the following notation for the deviation of the temperature of the superconducting film from One of the problems associated with the development the thermostat temperature: T (x,t) is the deviation of the and use of cryoelectronic devices is thermal destruction of s temperature in the superconducting region; T (x,t) is the the superconducting state, which is accompanied by the for- n mation of a normal phase in the superconducting region. In deviation of the temperature in the region of the normal Ref. 1 such nonuniform equilibrium states were examined in phase. It is clear that the stationary distribution of the tem- reference to one-dimensional and planar structures as the perature is symmetric relative to the plane passing through most widely encountered elements in cryoelectronics. In par- the middle of the film xϭ␦. Let the system deviate from the ticular, the following was demonstrated in Ref. 1, where an equilibrium state in the initial moment so that the tempera- idealized model of a thin superconducting film carrying an ture distribution remains symmetric, as before. In this case alternating current and immersed in a cooling medium was the boundary-value problem for determining the temperature investigated. At values of the Stekly parameter ␴ exceeding field of the superconducting ͑S͒ and normal ͑N͒ phases has the critical value ␴c , along with the uniform superconduct- the form ing state there are nonuniform equilibrium states, in which the central part of the film is in the normal state. Because of the so-called external nonlinearity of the system ͑the discontinuity of the parameters and the source density on the phase boundary͒,at␴Ͼ␴c there are two such double-fronted nonuniform equilibrium states. In situations where the existence of some element of a cryoelectronic device in a nonuniform state is a necessary condition for its operation, as, for example, in Franzen’s bolometer,2 the problem of the stability of a localized normal state arises, which was not investigated in Ref. 1. The present work is devoted to an analysis of the stability of the stationary solutions obtained in Ref. 1 against symmetric perturbations of the temperature field and an investigation of the asymptotic behavior of the two equilibrium states.

DYNAMICAL MODEL OF THE SYSTEM

We consider the thin superconducting film carrying an alternating current in Fig. 1, whose central portion is in the normal state. Let the film be immersed in a thermostat filled with a liquid or a gas, and let the entire system, with the FIG. 1. Superconducting film in the circuit of an ac source with convective exception of the film, have a temperature below the critical heat transfer on the free surface. The deviation of the film temperature in the two-phase state from the thermostat temperature is shown. Inset— value and be in thermal equilibrium. We examine the case in displacement of the phase boundary upon perturbation of the temperature which the nonuniform temperature field of the film is one- field.

␣ ⌰¯ ͓¯␰ ϩ␰ ͑t͔͒ϩu ͓¯␰ ϩ␰ ͑t͒,t͔ c T˙ ϭ␭ TЉϪ2 T , s b b s b b Vs s s s h s ¯ ¯ ¯ ϭ⌰n͓␰bϩ␰b͑t͔͒ϩus͓␰bϩ␰b͑t͒,t͔ϭ1, ␣ c T˙ ϭ␭ TЉϩ␳ ͑1ϩ␤T ͒j2Ϫ2 T , ␭ ͕⌰¯ Ј͓¯␰ ϩ␰͑t͔͒ϩuЈ͓¯␰ ϩ␰͑t͒,t͔͖ Vn n n n 0 n h n s s b s b ¯ ¯ ¯ ϭ␭n͕⌰nЈ͓␰bϩ␰͑t͔͒ϩusЈ͓␰bϩ␰͑t͒,t͔͖, ͑5͒ Ts͑0,t͒ϭ0, TnЈ͑x,t͉͒xϭ␦ϭ0, can be expanded in a Taylor series in the vicinity of the point ¯ 3 Ts͓xb͑t͒,t͔ϭTn͓xb͑t͒,t͔ϭTc , ␰b with respect to the variable ␰b(t). After discarding the terms which are nonlinear with respect to the perturbations, ␭ TЈ͑x,t͒ ϭ␭ TЈ͑x,t͒ . ͑1͒ s s ͉xϭxb͑t͒ n n ͉xϭxb͑t͒ we obtain a linear approximation of the continuity conditions: Here cVs , cVn , ␭s , and ␭n are the heat capacities per unit ¯ ¯ ¯ ¯ ¯ volume and the thermal conductivities of the superconduct- ⌰s͑␰b͒ϩ⌰sЈ͑␰b͒␰b͑t͒ϩus͑␰b ,t͒ ing and normal phases, respectively; ␳ is the resistance of 0 ¯ ¯ ¯ ¯ ¯ the normal phase; ␤ is the temperature coefficient of resis- ϭ⌰n͑␰b͒ϩ⌰nЈ͑␰b͒␰b͑t͒ϩun͑␰b ,t͒ϭ1, tance; j2 is the period-averaged value of the square of the ␭ ͓⌰¯ Ј͑¯␰ ͒ϩ⌰¯ Љ͑¯␰ ͒␰ ͑t͒ϩuЈ͑¯␰ ,t͔͒ current density; ␣ is the heat transfer coefficient, h is the s s b s b b s b thickness of the film, xb is the coordinate of the phase bound- ¯ ¯ ¯ ¯ ¯ ϭ␭n͓⌰nЈ͑␰b͒ϩ⌰nЉ͑␰b͒␰b͑t͒unЈ͑␰b ,t͔͒. ͑6͒ ary, and Tc is the critical temperature. The last terms on the right-hand sides of Eqs. ͑1͒ take into account the heat loss Separating the stationary and nonstationary components in due to convective heat transfer on the surface. In addition, in the problem ͑2͒, with consideration of ͑6͒ we have view of the small value of h, we assume that there is no ⌰¯ Љ͑␰͒ϭϪ2Bi ⌰¯ ͑␰͒ϭ0, ⌰¯ Љ͑␰͒Ϫ␬⌰¯ ͑␰͒ϩKϭ0, thermal resistance or temperature gradient along a normal to s s s n n the surface. ¯ ¯ ⌰s͑0͒ϭ0, ⌰nЈ͑␰͉͒␰ϭ1ϭ0, Going over to the normalized variables ⌰ϭT/Tc and ␰ϭx/␦, we bring the problem ͑1͒ into the dimensionless ⌰¯ ͑¯␰ ͒ϭ⌰¯ ͑¯␰ ͒ϭ1, ␭ ⌰¯ Ј͑␰͒ ¯ ϭ␭ ⌰¯ Ј͑␰͒ ¯ ; ͑7͒ s b n b s s ͉␰ϭ␰b n n ͉␰ϭ␰b form ␦2 2 ˙ ␦ us͑␰,t͒ϭusЉ͑␰,t͒Ϫ2Bisus͑␰,t͒, ˙ as ⌰s͑␰,t͒ϭ⌰sЉ͑␰,t͒Ϫ2Bis⌰s͑␰,t͒, as ␦2 u˙ ␰,t͒ϭuЉ ␰,t͒Ϫ␬u ␰,t͒, ␦2 n͑ n͑ n͑ ˙ an ⌰n͑␰,t͒ϭ⌰nЉ͑␰,t͒Ϫ␬⌰n͑␰,t͒ϩK, an us͑0, t͒ϭ0, unЈ͑␰,t͉͒␰ϭ1ϭ0,

⌰s͑0,t͒ϭ0, ⌰nЈ͑␰,t͉͒␰ϭ1ϭ0, ¯ ¯ ¯ ¯ ¯ ¯ us͑␰b ,t͒ϩ⌰sЈ͑␰b͒␰b͑t͒ϭun͑␰b ,t͒ϩ⌰nЈ͑␰b͒␰b͑t͒ϭ0,

⌰s͓␰b͑t͒,t͔ϭ⌰n͓␰b͑t͒,t͔ϭ1, ¯ ¯ ¯ ¯ ¯ ¯ ␭s͓usЈ͑␰b ,t͒ϩ⌰sЉ͑␰b͒␰b͑t͔͒ϭ␭n͓unЈ͑␰b ,t͒ϩ⌰nЉ͑␰b͒␰b͑t͔͒. ͑8͒ ␭s⌰sЈ͓␰,t͔͉␰ϭ␰ ͑t͒ϭ␭n⌰nЈ͓␰,t͔͉␰ϭ␰ ͑t͒ . ͑2͒ b b The stationary problem ͑7͒ was considered in Ref. 1, Here Biϭ␣␦2/h␭ is the product of the intrinsic Biot number where it was shown that the system has only the uniform ␣␦/␭ and the similarity criterion of parametric form ␦/h, solution ⌰¯ (␰)ϭ0, which corresponds to the superconducting 2 2 2 2 and ␬ϭ2BinϪ␤␳0j ␦ /␭n and Kϭ␳0 j ␦ /␭nTc are auxil- state of the film, for values of ␴ϭK/␬ that are smaller than iary parameters. We seek the solution of the problem ͑2͒ in the critical value ␴c . When ␴ӷ␴c , the problem ͑7͒ has the form of a sum of the stationary and nonstationary solu- nonuniform solutions of the form tions sinh ͱ2Bi ␰ ¯ s ¯͑m͒ ¯ ¯ ⌰s͑␰͒ϭ ; ␰Ͻ␰b , ⌰͑␰,t͒ϭ⌰͑␰͒ϩu͑␰,t͒, ␰b͑t͒ϭ␰bϩ␰b͑t͒, ͑3͒ ¯͑m͒ sinh ͱ2Bis␰b where u(␰,t) is a small symmetric perturbation of the equi- cosh ͱ␬͑1Ϫ␰͒ librium state: ¯ ¯ ͑m͒ ⌰n͑␰͒ϭ␴ϩ͑1Ϫ␴͒ ; ␰Ͼ␰b , cosh ͱ␬͑1Ϫ¯␰͑m͒͒ ¯ ¯ ¯ b u͑␰b ,t͒/⌰͑␰b͒Ӷ1. ͑4͒ ͑9͒ It is not difficult to show ͑Fig. 1͒ that the displacement of the where ␴ϭK/␬ has the same meaning as the Stekly 1͒ 2 coordinate of the phase boundary ␰b(t) for such a perturba- parameter ␴0ϭ j ␳oh/2␣Tc for a classical superconductor, tion of the temperature field will also be small. Then the to which it is related by the equality ␴ϭ␴0 /(1Ϫ␤Tc␴0). ¯ (m) continuity conditions of the problem ͑2͒, which are written The coordinate of the phase boundary ␰0 is defined as preliminarily as the root of the equation 680 Tech. Phys. 43 (6), June 1998 A. S. Rudy

␴ ␭ ␴ ␦2 ␦2 0 ¯ ¯ n 0 ␮2ϭ ␯ϩ2Bi, ␮2ϭ ␯ϩ␬. coth ͱ2Bin ͑1Ϫ␰b͒coth ͱ2Bis␰bϭ͑␴Ϫ1͒ͱ . s n ␴ ␭s ␴ as an ͑10͒ The solutions which satisfy the first three boundary con- For planar structures based on classical superconductors ditions have the form the number of equilibrium states m in the supercritical region (␴Ͼ␴ ) is equal to two. According to Ref. 1, more than two c V ␰͒ϭC sinh ␮ ␰, ͑16͒ stationary states are permissible for high-temperature super- s͑ s conductors. It should be stipulated that we are dealing with ¯ multifronted two-phase states, whose analysis requires ap- ␭s sinh ␮s␰b Vn͑␰͒ϭC cosh ␮n͑1Ϫ␰͒, ͑17͒ propriate formulation of the boundary-value problem. ¯ ␭n cosh ␮n͑1Ϫ␰b͒ INVESTIGATION OF THE STABILITY OF NONUNIFORM where C is a constant. STATIONARY STATES Substituting ͑9͒, ͑16͒, and ͑17͒ into the last boundary To determine the stability of the two-phase states in a condition, we obtain an equation for the eigenvalues of the first approximation we seek solutions of the problem ͑8͒. The operators ͑15͒ last two boundary conditions of this problem assign the con- ¯ ¯ tinuity of the nonstationary components of the temperature ␮s coth ␮s␰bϩ␮n tanh ␮n͑1Ϫ␰b͒ and the heat flux on the phase boundary. Let us consider ¯ ¯ these conditions in greater detail. ϭͱ2Bis tanh ͱ2Bis␰bϩͱ␬ coth ͱ␬͑1Ϫ␰b͒. ͑18͒ Substitution of the solution sought in the form u(␰,t) ϭV(␰)exp(␯t) into the continuity condition for the tempera- Thus, to determine the stability of the stationary solu- ture leads to the following law for the motion of the phase tions of ͑9͒, we must find the roots of the characteristic equa- ¯ boundary: tion ͑18͒ for the corresponding values of ␰b . ¯ Vs͑␰b͒ ␰b͑t͒ϭϪ exp͑␯t͒, ¯ ¯ ⌰sЈ͑␰b͒ FURTHER IDEALIZATION OF THE MATHEMATICAL MODEL V ͑¯␰ ͒ n b The analysis of Eq. ͑18͒ in its general form is a fairly ␰b͑t͒ϭϪ exp͑␯t͒. ͑11͒ ¯ ¯ formidable problem; therefore, we shall confine ourselves to ⌰nЈ͑␰b͒ an investigation of the two-phase state of a superconductor Eliminating ␰ (t) from ͑11͒ and taking into account that b with a low temperature coefficient of resistance in a small ¯ ¯ ⌰sЈ͑␰b͒ ␭n vicinity of the transition point. In this case the thermal con- ϭ , ͑12͒ ductivities of the superconducting and normal phases, as well ¯ ¯ ␭ ⌰nЈ͑␰b͒ s as the Biot numbers, can clearly be assumed to be equal: we transform the first of the continuity conditions into the ␭sϭ␭n ,BisϭBinϵBi. In view of the small value of ␳0␤,it simpler condition is permissible to take ␬ϭ2Bi and set ␮nϭ␮sϵ␮ and to ¯ thereby significantly simplify Eq. ͑18͒: Vs͑␰b͒ ␭n ϭ . ͑13͒ ¯ ␭s ¯ ¯ Vn͑␰b͒ ␮͓coth ␮␰bϩtanh ␮͑1Ϫ␰b͔͒ϭA, ͑19͒ We write the continuity conditions for the heat flux with consideration of ͑11͒ as where ¯ ¯ VsЈ͑␰͒ ⌰sЉ͑␰͒ VnЈ͑␰͒ ⌰nЉ͑␰͒ ¯ ¯ Ϫ ϭ Ϫ . ͑14͒ Aϭͱ2Bi ͓tanh ͱ2Bi␰bϩcoth ͱ2Bi͑1Ϫ␰b͔͒. ͑20͒ ͫ Vs͑␰͒ ⌰¯ Ј͑␰͒ͬ ͫ Vn͑␰͒ ⌰¯ Ј͑␰͒ͬ s ¯ n ¯ ␰ϭ␰b ␰ϭ␰b Now the proof of the stability of the stationary solutions Separating the variables in ͑8͒ and replacing the conti- ¯ reduces to calculating ␮ for assigned values of ␰b and Bi and nuity conditions by ͑13͒ and ͑14͒, we arrive at the Sturm– determining the sign of the parameter Liouville problem VЉ͑␰͒ϭ␮2V ͑␰͒, VЉ͑␰͒ϭ␮2V ͑␰͒, a s s s n n n 2 ␯ϭ 2 ͑␮ Ϫ2Bi͒. ͑21͒ ¯ ␦ Vs͑␰b͒ ␭n Vs͑0͒ϭ0; VЈ͑␰͉͒␰ϭ1ϭ0, ϭ , ¯ Vn͑␰b͒ ␭s The left-hand side of Eq. ͑20͒ is a function of the complex variable ␮ϭ␮ ϩi␮ , while the right-hand side of the ¯ ¯ Ј Љ VsЈ͑␰͒ ⌰sЉ͑␰͒ VnЈ͑␰͒ ⌰nЉ͑␰͒ Ϫ ϭ Ϫ , ͑15͒ equation, which is an implicit function of the control param- ¯ ͫ Vs͑␰͒ ⌰¯ Ј͑␰͒ͬ ͫ Vn͑␰͒ ⌰¯ Ј͑␰͒ͬ eter AϭA͓␰b(␴)͔, does not depend on ␮. Separating the real s ␰ϭ¯␰ n ␰ϭ¯␰ b b and imaginary parts in ͑20͒ and eliminating A, we arrive at where the equation Tech. Phys. 43 (6), June 1998 A. S. Rudy 681

FIG. 2. Dependence of the roots of the characteristic equation on the coordinate of the phase boundary: a—the solid line in the upper half plane corresponds to the position of the ﬁrst root of Eq. ͑23͒ on the imaginary axis, and the dotted line corresponds to the position on the real axis. Biot number: 1—2, 2—1, 3—0.1; b—second root of Eq. ͑23͒.

␮Љ cosh ␮Ј cos ␮Љϩ␮Ј sinh ␮Ј sin ␮Љ The stability of the equilibrium states ͑9͒ is determined ¯ by the further course of the ␮Ј(␰b) curve. As is shown in ␮Ј cosh ␮Ј cos ␮ЉϪ␮Љ sinh ␮Ј sin ␮Љ ¯ Fig. 3, the plot of ␰b(␴0) is symmetric relative to ␰ϭ0.5; ¯ ¯ therefore, in one of the equilibrium states, for example, cosh ␮Ј sin ␮Љϩcosh ␮Ј͑2␰bϪ1͒sin ␮Љ͑2␰bϪ1͒ ϭ . mϭ1, we always have ¯␰ Ͻ0.5, while in the other equilib- ¯ ¯ b sinh ␮Ј cos ␮Љϩsinh ␮Ј͑2␰bϪ1͒cos ␮Љ͑2␰bϪ1͒ ¯ rium state ␰bϾ0.5. It is easy to see that upon passage of a ͑22͒ real root through the point ␰ϭ0.5, i.e., upon passage from one equilibrium state to the other, ␯ changes sign for any Bi. The results of a numerical search for roots of Eq. ͑22͒ ¯ indicate that the equation does not have any complex roots For this reason, in the second of Eqs. ͑23͒ we must set ␰b 2 and that all the points of the spectrum lie either on the real ϭ0.5 and solve it numerically with respect to ␮ . The plot 2 2 axis or on the imaginary axis. Alternately setting ␮ϭi␮Љ of ␮ ϭ␮ (2Bi) in Fig. 4 shows that, regardless of the Biot ¯ and ␮ϭ␮Ј in ͑21͒, we arrive at the following equations: number, reversal of the sign of ␯1 always occurs at ␰bϭ0.5 and, therefore, the second equilibrium state is always un- 2␮Љ cos ␮Љ ¯ stable. ϭͱ2Bi͑tanh ͱ2Bi␰b ¯ sin ␮Љϩsin͑2␰bϪ1͒␮Љ ¯ ϩcoth ͱ2Bi͑1Ϫ␰b͒͒, 2␮Ј cosh ␮Љ ¯ ϭͱ2Bi͓tanh ͱ2Bi␰b ¯ sinh ␮Јϩsinh͑2␰bϪ1͒␮Ј ¯ ϩcoth ͱ2Bi͑1Ϫ␰b͔͒. ͑23͒

The ﬁrst of Eqs. ͑23͒ has an inﬁnite number of roots ␮kЉ , two of which are shown in Figs. 2a and 2b. It is seen from

Fig. 2a that the ﬁrst pair of complex-conjugate roots Ϯi␮1Љ ¯ tends to zero as ␰b increases. At the value of the Stekly ¯ parameter for which ␰bϭ␰c , where ␰c is the root of the equation ¯ ¯ ¯ ͱ2Bi␰b͓tanh ͱ2Bi␰bϩcoth ͱ2Bi͑1Ϫ␰b͔͒ϭ1, ͑24͒ the ﬁrst pair of roots moves over to the real-number axis. The negative root moves over to the positive semiaxis and vice versa. Thus, the second of Eqs. ͑23͒ has a single pair of FIG. 3. Dependence of the coordinate of the phase boundary on the Stekly roots. parameter. Bi: 1—0.1, 2—0.5, 3—1, 4—3, 5—7. 682 Tech. Phys. 43 (6), June 1998 A. S. Rudy

cosh ͱ2Bi ͑1Ϫ␰͒ ¯ s ⌰s͑␰͒ϭ , ¯͑m͒ sinh ͱ2Bis͑1Ϫ␰b ͒

cosh ͱ␬␰ ¯ ⌰n͑␰͒ϭ␴ϩ͑1Ϫ␴͒ , ͑27͒ ¯ ͑m͒ cosh ͱ␬␰0 and the condition ͑10͒ takes the form ␴ ␭ ␴ 0 ¯ ¯ n 0 coth ͱ2Bin ␰b coth ͱ2Bis͑1Ϫ␰b͒ϭ͑␴Ϫ1͒ͱ . ␴ ␭s ␴ ͑28͒ ¯ (1) ¯(2) Assuming that ␰b Ͻ␰b , we consider the mϭ2 state. FIG. 4. Plot of ␮2(2Bi) at the branch point of the stationary solutions for As the length of the film tends to infinity, i.e., when the ␴ϭ␴c . system becomes degenerate with respect to ␦, the coordinate ¯ (2) of the free boundary ␰b also tends to infinity, and the temperature of the normal phase ͑27͒ tends to a spatially uniform ¯ ¯ In the range ␰bϽ␰c , where all the eigenvalues of the distribution: operator ͑16͒ are imaginary ͓␮ ϭi␮Љ(¯␰ )͔, the solutions k k b ⌰¯ ͑␰͒ϭ␴. ͑29͒ ͑17͒ and ͑18͒ of the boundary-value problem ͑8͒ have the n form It follows from ͑29͒ that the solution ͑27͒ corresponds to ϱ a state whose inhomogeneity is caused only by the proximity of the boundary. When the latter is moved away to infinity, ⌰s͑␰,t͒ϭ Ck sin͑␮kЉ␰͒exp͑␯kt͒, k͚ϭ1 the system tends to the normal state with a uniform temperature distribution. ϱ ¯ ¯ ␭s sin͑␮kЉ␰b͒ In the case of mϭ1, the limiting transition xb ϱ leads ⌰n͑␰,t͒ϭ ͚ Ck cos ␮kЉ͑1Ϫ␰͒exp͑␯kt͒, → kϭ1 ␭ ¯ to the expressions n cos ␮kЉ͑1Ϫ␰b͒ ͑25͒ 2␣ ¯ ¯ ͑1͒ 2 2 ⌰s͑x͒ϭexp ͱ ͑xb Ϫx͒, where ␯kϭϪ as /␦ (␮Љkϩ2Bi) is negative and the equilib- h␭s rium states are stable. cosh ͱ2␣␴ /h␭ ␴x When ¯␰ Ͼ¯␰ , the first eigenvalue of the operator ͑16͒ ¯ 0 n b c ⌰n͑x͒ϭ␴ϩ͑1Ϫ␴͒ , ͑30͒ ¯ cosh ͱ2␣␴ /h␭ ␴¯x͑1͒ becomes real ͓␮1ϭ␮1Ј(␰b)͔, and the solutions ͑25͒ trans- 0 n b form into where

⌰s͑␰,t͒ϭC sinh͑␮1Ј␰͒exp͑␯1t͒ h␭n ␴ ␭n ␴ ¯x͑1͒ϭ arctanh Ϫ1 . 31 ϱ b ͱ ͑␴ ͒ͱ ͑ ͒ 2␣ ␴0 ␭s ␴0 ϩ Ck sin͑␮kЉ␰͒exp͑␯kt͒, k͚ϭ2

¯ ␭s sinh͑␮1Ј␰b͒ ⌰n͑␰,t͒ϭC cosh ␮1Ј͑1Ϫ␰͒ ␭ ¯ n cosh ␮1Ј͑1Ϫ␰b͒ ϱ ¯ ␭s sin͑␮kЉ␰b͒ ϫexp͑␯1t͒ϩ ͚ Ck kϭ2 ␭ ¯ n cos ␮kЉ͑1Ϫ␰b͒

ϫcos ␮kЉ͑1Ϫ␰͒exp͑␯kt͒. ͑26͒ ¯ When ␰bϾ0.5, the sign of ␯ changes, and the solution ͑26͒ becomes exponentially unstable.

INVESTIGATION OF THE ASYMPTOTIC BEHAVIOR OF THE STATIONARY SOLUTIONS Let us conclude with a discussion of the physical nature of an unstable equilibrium state. For reasons of convenience FIG. 5. Locally nonuniform equilibrium state of an infinite thin superconducting film. The temperature field corresponds to the system parameters we move the origin of coordinates to the center of the film. 6 ␤ϭ0, ␴ϭ3, and 2␣/h␭nϭ10 . The additional plots illustrate the law of Then the stationary solutions ͑9͒ transform into equal areas. Tech. Phys. 43 (6), June 1998 A. S. Rudy 683

The same expressions were obtained in Ref. 4 as self- where the plus sign corresponds to the right-hand phase similar solutions describing a localized, spatially nonuniform boundary. equilibrium state of an infinitely long superconducting film Equation ͑34͒, like ͑22͒, does not have complex roots, ͑Fig. 5͒. and only one root exists for real z, it being such that zϾy. It is noteworthy that relation ͑31͒, like ͑10͒, follows di- Hence it follows that the parameter rectly from the energy conservation law. It is not difficult to obtain these relations by equating the total quantity of heat ␭ ␯ϭ ͑z2Ϫy 2͒ evolved and the heat loss in the system. For a film of infinite ¯ 2 cvnxb length, relation ͑31͒ allows the following simple geometric interpretation. We find the areas S1 and S2 of the hatched is always positive and that the solution defined by ͑30͒ and objects in Fig. 5: ͑31͒ is unstable. This result means that the localized nonuniform equilib- ¯x h␭ ␴ 2␣ ␴ b ¯ n 0 ¯ rium state considered in Ref. 4, is not a stationary autosoli- S1ϭ ͵ ⌰n͑x͒dxϭ͑␴Ϫ1͒ͱ tanh ͱ xb , 0 2␣ ␴0 h␭n ␴ ton. It cannot appear spontaneously, and the artificially created temperature distribution defined by ͑30͒ and ͑31͒ is ϱ h␭ ¯ s unstable. On the other hand, it is clear that at large values of S2ϭ ⌰s͑x͒dxϭ . ͑32͒ ͵¯ ͱ xb 2␣ the variables the implicitly assigned function ͑34͒ tends to the explicit form yϭz. Setting zϪyϭ␧, where ␧Ӷ1, it is Equating S and S , we obtain ͑31͒ again, whence it 1 2 not difficult to obtain the following estimate: follows that S1ϭS2 . The latter relation is a specific addition 5 to the law of equal areas. 4 Self-sustained localized nonuniform formations in dissi- ␧Ϸ z exp͑Ϫ2z͒, 2ϩ⌬c/c pative systems are customarily called autosolitons. In Ref. 4 nv the equilibrium state defined by ͑30͒ and ͑31͒ was interpreted ␭ 8 as an autosoliton normal phase, since the question of its sta- ␯ϭ z2 exp͑Ϫ2z͒, ͑35͒ ¯ 2 2c ϩ⌬c bility had not been investigated at that time. The same state xb n was obtained above as an asymptotic form of an unstable according to which the characteristic exponent of the solu- solution when the system became degenerate; therefore, it tion ͑26͒ is practically equal to zero when yӷ1. Such a qua- can be expected that it also remains unstable in the case of a sistable nonuniform formation can remain long-lived even film of infinite length. ¯ for comparatively small sizes of the normal region (xb Let us investigate the asymptotic behavior of the charac- Ϫ3 teristic equation defined by ͑19͒ and ͑20͒. When the origin of ϳ10 m). coordinates is displaced to ␰ϭ1, Eqs. ͑19͒ and ͑20͒, which were written for the right-hand half plane (␰Ͼ0), transform into CONCLUSIONS c 2␣ c 2␣ ¯ vs vs ¯ The results of the analysis performed allow us to state xbͱ ␯ϩ coth ͱ ␯ϩ ͑␦Ϫxb͒ ␭s ␭sh ␭s ␭sh that of the nonuniform equilibrium states found in Ref. 2 for a thin current-carrying superconducting film, the state which 2␣ 2␣ ¯ ¯ corresponds to a normal-phase region of larger size is stable. Ϫxbͱ tanh ͱ ͑␦Ϫxb͒ ␭sh ␭sh The fact that the stability was investigated only with respect to symmetric perturbations is not significant in the present c 2␣ c 2␣ ¯ vn vn ¯ case, since of the two equilibrium states only one is generally ϭϪxbͱ ␯ϩ tanh ͱ ␯ϩ xb ␭n ␭nh ␭n ␭nh unstable. In the equilibrium states with a normal phase of smaller 2␣ 2␣ ¯ ¯ dimensions the conditions on the film boundary weakly in- ϩxbͱ coth ͱ xb . ͑33͒ ␭nh ␭nh fluence the heat balance, and the nonuniform solution re- We assume that the thermal conductivities of the two mains localized upon passage to a film of infinite length. phases are equal (␭sϭ␭nϭ␭) and that the specific heat un- This solution is also unstable in an infinitely long film, but dergoes a jump upon passage into the superconducting state the relaxation time to the locally uniform equilibrium state (cvsϭcvnϩ⌬c). Introducing the notation can be fairly long when the parameters of the system have certain values. The instability of a given equilibrium state c 2␣ 2␣ ¯ vn ¯ means that spontaneous localization of the normal phase ͑the zϭxb ␯ϩ , yϭxb ͱ ␭ ␭h ͱ␭h formation of a stationary autosoliton͒ is impossible in the and allowing the film length to tend to infinity (␦ ϱ), we system under consideration. Thin films based on new super- obtain → conductors may be more promising in this sense. As was shown in Ref. 2, more than two equilibrium states are pos- ⌬c y 2 sible in films with a large temperature coefficient of resis- z tanh zϮ 1ϩ 1Ϫ Ϫy͑coth yϮ1͒ϭ0, ͫ ͱ c ͩ z2 ͪͬ tance; therefore, it would be interesting to investigate the vn ͑34͒ asymptotic behavior of the corresponding stable solutions. 684 Tech. Phys. 43 (6), June 1998 A. S. Rudy

1͒ 2 The dimensionless quantity ␴0 , which serves as a measure of the ratio of W. Franzen, J. Opt. Soc. Am. 53, 596 ͑1963͒. 2 3 the characteristic flux of heat evolved j ␳0dV to the heat removal rate A. S. Rudy and A. Yu. Kolesov, Nonlinear Analysis 1 ͑1997͒. 4 ␣(TcϪT0)dS in superconductors with a transport current, is called the A. S. Rudy, Pis’ma Zh. Tekh. Fiz. 22͑20͒,62͑1996͓͒Tech. Phys. Lett. Stekly parameter. In particular, for thin films based on classical supercon- 22, 848 ͑1996͔͒. 2 5 ductors ␴0ϭ j ␳0h/2␣(TcϪT0). As was shown in Ref. 1, the ratio ␴ A. V. Gurevich, R. G. Mints, and A. L. Rakhmanov, The Physics of ϭK/␬ can serve as the Stekly parameter for materials with a large tem- Composite Superconductors, CRC Press, Boca Raton, Florida ͑1995͒ perature coefficient of resistance. ͓Russian original, Nauka, Moscow ͑1987͒, 240 pp.͔.

1 A. S. Rudy, Pis’ma Zh. Tekh. Fiz. 22͑9͒,85͑1996͓͒Tech. Phys. Lett. 22, 382 ͑1996͔͒. Translated by P. Shelnitz TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Collinear light scattering on dipole-exchange spin waves in inhomogeneous ferromagnetic ﬁlms L. V. Lutsev

Domen Scientific-Research Institute, 196084 St. Petersburg, Russia ͑Submitted June 19, 1995; resubmitted February 9, 1998͒ Zh. Tekh. Fiz. 68, 78–84 ͑June 1998͒ A theory is developed for the collinear TE–TM scattering of optical waveguide modes on dipole- exchange spin waves in perpendicularly magnetized ferromagnetic films that are inhomogeneous across their thickness. It is found in homogeneous ferromagnetic films and in films with small deviations from homogeneity that the TE–TM scattering on higher spin- wave modes is strongest when the synchronism conditions for the transverse phases and for the longitudinal and transverse wave vectors are satisfied. When the thickness of the planar optical waveguide does not match the thickness of the ferromagnetic film, the phase synchronism condition is violated with the resultant appearance of an oscillating type of dependence of the TE–TM scattering on the spin-wave mode number. The scattering of light on spin-wave modes in films with a magnetization gradient is investigated in the presence of turning points for the magnetostatic potential. It is found that the existence of a turning point in the region of the antinode for the optical modes leads to an increase in the scattering amplitude. The formation of inhomogeneous magnetooptical structures and superlattices based on ͑Lu,Y,Bi͒3͑Fe,Ga͒5O12 is discussed. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01506-2͔

INTRODUCTION netic layers. The formation of inhomogeneous magnetooptical structures and superlattices based on The interaction of light with spin waves in iron garnet ͑Lu,Y,Bi͒3͑Fe,Ga͒5O12 is discussed. film has been studied intently in recent years. This interaction can be utilized for both practical and research purposes: for ultrahigh-frequency optical modulators and for studying 1. DIPOLE-EXCHANGE SPIN WAVES the spin-wave processes occurring in iron garnet thin films. Let us consider a perpendicularly magnetized ferromag- The results of investigations of the noncollinear interaction netic planar structure of thickness d with magnetic and di- of optical waveguide modes with spin waves and TE–TM electric parameters that are inhomogeneous across its thick- mode conversion were presented in Refs. 1 and 2. The fea- ness ͑Fig. 1͒. The 0z axis is perpendicular, and the 0x and tures of the TE TM conversion of optical modes upon col- 0y axes are parallel to the film surface. We assume that the ↔ linear scattering were studied in Refs. 3–7 by both theoreti- spin-wave and optical modes propagate along the 0x axis. cal and experimental methods. A theoretical analysis of the diffraction of optical modes on surface and bulk spin waves for an arbitrary angle of incidence of the optical mode was conducted in Ref. 8. The purpose of the present work is to take into account the exchange interaction accompanying the scattering of optical waveguide modes on spin waves in inhomogeneous ferromagnetic films. The exchange interaction must be taken into account, if the inhomogeneous ferromagnetic film has a layer with a turning point for the magnetostatic potential of the spin wave. The magnitude of the variable magnetic moment is greater in this layer than in other layers, leading, in turn, to an increase in the scattering of the optical waveguide modes. This paper is divided into three parts. The first two parts describe the properties of dipole-exchange spin waves and present the dispersion relations and eigenfunctions of optical waveguide modes. In the third part the optical-mode coupling equations are derived, and the conditions for achieving FIG. 1. Geometry of a planar structure for collinear light scattering on spin waves: f—ferromagnetic layer; 1, 2—cladding and transitional nonferro- maximum TE–TM scattering are analyzed for various film magnetic layer with the thicknesses dc and ds ; A—antenna used to excite structures with homogeneous and inhomogeneous ferromag- the spin waves.

Ϫ1 Spin waves are described in the magnetostatic approximation ␸͑x,z,t͒ϭ͑2␲͒ ␸n͑z͒exp͑ikxϩi␻nt͒, ͑3͒ by the magnetostatic potential ␸(x,z,t), which is found from the equations9 where cos͓k͑n͒͑zϪd/2͒ M ␦ zץ ϭϪ␥ M H , ,t ͫ • ␦Mͬ ϩ͑nϪ1͒␲/2͔͑0рzрd͒ץ k͑n͒ nϪ1z div͑hϩ4␲m͒ϭ0, ͑1͒ ␸n͑z͒ϭPn ͑Ϫ1͒͑n͒exp͓͉k͉͑dϪz͔͒ ͑zϾd͒, k0 where ͑n͒ kz exp͉͑k͉z͒ ͑zϽ0͒, 1 k͑n͒ 2 2 Ά 0 ϭ ϪM͑Hϩh͒ϩ2␲͑M,n͒ Ϫ ␤ ͑M,n͒ H ͵ ͫ 2 a ͑4͒ nϭ1,2,3,... is the mode number, P is the normalization M nץ Mץ 1 Ϫ ␣ dV ͑2͒ parameter, k(n)2ϭk2ϩk(n)2 , k is the wave vector, and k(n) is 2 r r 0 z z i ͬ defined by the relation ץ i ץ is the effective classical dipole-exchange Hamiltonian, k͑n͒ ͑n͒ z ͉k͉ MϭM0ϩm(x,z,t) is the magnetic moment density ͑M0 2 cot kz dϭ Ϫ ͑n͒ . ͑5͒ ͉k͉ kz does not depend on x and t, and ͉m(x,z,t)͉ӶM 0͒, ␥ is the gyromagnetic ratio, Hʈ0z is the external constant magnetic The dispersion relation for the nth mode has the form field, h(x,z,t)ϭϪٌ␸(x,z,t) and ␸(x,z,t) are the variable 2 ͑n͒2 ͑n͒2 magnetic field and the magnetostatic potential of the spin ␻nϭ͑⍀ϩ␥␣M0k0 ͒͑⍀ϩ␥␣M0k0 wave, ␣ is the exchange coupling constant, ␤a specifies the 2 ͑n͒2 ϩ␥4␲M0k /k0 ͒, ͑6͒ uniaxial anisotropy field Haϭ␤a(M,n)n with the n axis, which is perpendicular to the film surface, the term where ⍀ϭ␥(HϪ4␲M 0ϩHa). 2 (1) 2␲(M,n) in ͑2͒ describes the energy of the demagnetizing Taking into account the values of the group velocity vg (n) ri is an abbreviated form of the first mode and kz for ͉k͉Ӷ␲/d ͑Ref. 10͒, it followsץ/ץ magnetic field of the film, and z. from the dispersion relation ͑6͒ that the dispersion curves ofץ/ץ y, andץ/ץ ,xץ/ץ for writing the derivatives The system of equations ͑1͒ was investigated in Ref. 10 the first and higher modes cross when in a linear approximation with respect to m(x,z,t) for the k ϭ␣␥M k͑n͒2/v͑1͒ϭ␣␲͑nϪ1͒2/d3. ͑7͒ case of films that are inhomogeneous across their thickness n 0 z g with the magnetic parameters ␥(z), ␣(z), M0(z), and The second equality in ͑1͒ gives the relation between the Ha(z). The calculation of TE–TM scattering on a spin wave variation of the magnetic moment density mx(z) and the requires finding the distribution of m across the thickness of magnetostatic potential ␸n(z) of the nth mode: the ferromagnetic structure. We shall consider ferromagnetic ik͑n͒2 films that are homogeneous and weakly inhomogeneous 0 mx͑z͒ϭ ␸n͑z͒. ͑8͒ across their thickness, as well as ferromagnetic films with a 4␲͉k͉ magnetization gradient across their thickness. In the latter For a weakly inhomogeneous ferromagnetic film, in a case it is assumed that the magnetostatic potential ␸ of the first approximation with respect to the degree of deviation of spin wave has a turning point within the film. the magnetic parameters from the parameters of the homo- A. Ferromagnetic films that are homogeneous and geneous film structure the dispersion relation of a spin wave weakly inhomogeneous across their thickness. We assume is specified by the expression that the spin-wave frequency ␻ is far from the ferromagnetic ͑n͒2 2 ͑n͒2 resonance frequencies of any layer of the ferromagnetic film. ␻nϭ͗n͉⍀͑z͒ϩ␥͑z͒M 0͑z͒͑␣͑z͒k0 ϩ2␲k /k0 ͉͒n͘, In this case the potential ␸ does not have turning points ͑9͒ d within the film, and the magnetic susceptibility tensor ␹ik(␻) where ͗n͉f (z)͉l͘ϵ͐0␸n*(z) f (z)␸l(z)dz with the normaliza- ( j)2 Ϫ1/2 does not have singular points. Ferromagnetic structures that tion factors P jϭ(d/2ϩ͉k͉/k0 ) ( jϭn,l). are weakly inhomogeneous across their thickness are under- To determine the coupling coefficient of TE and TM stood to be structures with weak deviations ⌬␹ik from the modes for scattering on a spin wave it is convenient to relate ¯ ¯ mean values ␹ik ͑Ref. 10͒. The small value of ⌬␹ik /␹ik the normalization of the eigenfunctions ␸n(z) ͑4͒ to the en- makes it possible to use perturbation theory, where the first ergy U(n) of the spin wave per unit area of the film. Accord- approximations for calculating the dispersion relations and ing to Ref. 11, the energy of a spin wave U(n) is related to the magnetostatic potential are the dispersion relations and the magnon number density Nn : the potential ␸ of a homogeneous film with the mean param- d d m¯ 2␻ eters ¯␹ , ¯␥, ¯␣, M¯ , and H¯ . The expansion parameters are ͑n͒ x n ik 0 a U ϭ Nnប␻ndzϭ dz ͑10͒ ¯ ͵0 ͵0 2␥M0 the ⌬␹ik /␹ik , which, in turn, are determined by the relative ¯ ¯ ¯ ¯ deviations ⌬␥/␥, ⌬␣/␣, ⌬M 0 /M 0, and ⌬Ha /Ha . ͑the bar denotes a time average͒. The spin-wave eigenfunctions of a homogeneous film With consideration of ͑3͒, ͑4͒, and ͑8͒, from ͑10͒ we find are the normalization parameter Tech. Phys. 43 (6), June 1998 L. V. Lutsev 687

2 ͑n͒ The equations for the TE and TM modes are derived in ͑4␲͒ ͉k͉ 2␥M 0U P ϭ . ͑11͒ analogy to the equations for the TE and TM modes in a n k͑n͒2 ͱ␻ ͑dϩ2 k /k͑n͒2͒ 0 n ͉ ͉ 0 planar structure in Refs. 14 and 15 by taking the Fourier B. Ferromagnetic films with a magnetization gradient transform with respect to t and retaining the first terms of the across their thickness. Let us consider a ferromagnetic film approximation with respect to ␧lj /␧0 (lÞj). A TE mode is ¯ with linear variation of the magnetization 4␲M 0ϭ4␲M 0 completely characterized by the field component Ey : Ϫ␮(zϪd/2) and the frequency range in which there is a ␧ ͑z͒␻2 i␻2 2ץ 2ץ turning point for the potential ␸ within the ferromagnetic 0 TE TE n 2 ϩ 2 ϩ 2 Eyϩ x c ͪ c␧0͑z͒␻TMץ zץ ͩ film structure. According to Ref. 10, the distribution of mϮϭmxϮimy across the thickness is described by the ex- Hyץ Hyץ pression ϫ ␧yx* Ϫ␧yz* ϭ0. ͑16͒ ͪ xץ zץ ͩ z i͉k͉ Ϯ ␸͑␰͒ Ϯ mϮ͑z͒ϭ v2 ͑z͒ v1 ͑␰͒d␰ A TM mode is completely characterized by Hy : w͑0͒ ͫ ͵0 ␣͑␰͒ 2H ␧ ͑z͒␻2ץ Hץ 1 ץ d y y 0 TM Ϯ ␸͑␰͒ Ϯ ␧0͑z͒ * ϩ 2 ϩ 2 Hyϩ␧0͑z͒ x cץ ͪ zץ z ͩ ␧ ͑z͒ץ ϩv ͑z͒ v ͑␰͒d␰ , ͑12͒ 1 2 0 ͵z ␣͑␰͒ ͬ Hץ ␧ ץ Hץ ␧ ץ Ϯ ¯ 1/3 xz y zx y where w(0) is a Wronskian, v (z)ϭAi͓(␮/␣M 0) (z ϫ 2 * ϩ 2 * zץ x ␧ ͑z͒ץ xץ z ␧ ͑z͒ץ 1 Ϯ Ϯ ¯ 1/3 Ϯ ͫ ͩ 0 ͪ ͩ 0 ͪͬ Ϫz0 )͔ and v2 (z)ϭBi͓(␮/␣M 0) (zϪz0 )͔ are Airy func- ␧ ץ ␧ ץ Ϯ 2 ¯ ¯ i␻ ␧ ͑z͒ tions, and z0 ϭ͓ϯ␻/␥Ϫ␣k M 0ϪHϩ4␲M 0ϪHa(0)͔/␮. TM 0 xy zy ϩ * Ey Ϫ *Ey ϭ0. ͬͪ x ͩ␧ ͑z͒ץ ͪ z ͩ ␧ ͑z͒ץͫ The substitution of mϮ(mx ,my) into the second equality c 0 0 in ͑1͒ gives an integrodifferential equation in ␸(z). This ͑17͒ equation was solved numerically. The dispersion relation ␻(k) was found from the requirement that ␸(z) and In ͑16͒ and ͑17͒ the symbol * denotes the convolution -z be continuous on the boundary of the ferromagץ/(␸(zץ netic film. Outside the film ␸(z)ϳexp(Ϫ kz ). Ϫ1/2 ͉ ͉ ͑u*w͒͑␻͒ϭ͑2␲͒ u͑␻Ϫ␻1͒w͑␻1͒d␻1 . When k 0, we can employ a simplified formula speci- ͵ → fying ␻n ͑Refs. 12 and 13͒: The terms containing the convolution describe effects 2 2 ¯ 2 1/3 with a change in frequency upon scattering. If ␻TE and ␻TM ␻nϭmin ⍀͑z͒ϩ␥͓2␲ ␮ ␣M 0͑nϪ1/2͒ ͔ , ͑13͒ are much greater than the spin-wave frequency ␻n in ͑6͒, ͑9͒, where or ͑13͒, the convolution is replaced by the product. The terms with Hy in ͑16͒ and with Ey in ͑17͒ lead to TE TM ⍀͑z͒ϭ␥͑z͓͒HϪ4␲M 0͑z͒ϩHa͑z͔͒. ↔ mode conversion. The fourth term with Hy in ͑17͒ describes In this approximation the turning point zr is given by the the modulation of the TM mode by the spin wave and will relation not be taken into account below. Because of the condition ͑15͒, the terms with H in ͑16͒ and with E in ͑17͒ can be z ϭ͓2␲2␣M¯ ͑nϪ1/2͒2/␮͔1/3. ͑14͒ y y r 0 regarded as perturbations. Normalization of the eigenfunctions ␸n(z) per unit of For films with constant values of ␧0(z) ͑␧c is the value (n) the energy density of the spin wave U , which is needed to in the cladding, ␧ f is the value in the film, and ␧s is the value compare the TE–TM scattering amplitudes in different fer- in the substrate͒ the eigenfunctions of the unperturbed equa- romagnetic structures, was found numerically using Eqs. tions ͑16͒ and ͑17͒ are orthogonal to one another and have ͑10͒ and ͑12͒. the form14

⌿͑p͒͑x,y͒ϭ͑2␲͒Ϫ1/2␺͑p͒͑z͒exp͑i␤͑p͒x͒, 2. OPTICAL WAVEGUIDE MODES ␺͑p͒͑z͒ϭP͑p͒ The equations describing TE and TM optical waveguide ͑p͒2 Ϫ1/2 ͑p͒ modes and the effects of TE TM conversion are obtained ͑1ϩac ͒ exp͓Ϫ␥c ͑zϪd͔͒ ͑zϾd͒, 14 ↔ from Maxwell’s equations. We consider the case in which ͑p͒ ͑p͒ ϫ cos͑k f zϪ␪s ͒ ͑0ϽzϽd͒, the diagonal component of the dielectric constant ␧0(z)isa ͭ ͑1ϩa͑p͒2͒Ϫ1/2 exp͑␥͑p͒z͒ ͑zϽ0͒, function only of z and is much greater than the off-diagonal s s components. The off-diagonal components take into account ͑18͒ the gyrotropic effects, i.e., the dependence on m(x,z,t):15 (p)2 (p)2 where pϭ0,1,2,... is the mode number, k f ϭϪ␤ ␧0͑z͒ӷ␧ljϭigeljkmk͑t͒͑lÞj͒, ͑15͒ 2 2 (p)2 (p)2 2 2 (p) (p) ϩ␧f␻ /c , ␥s,c ϭ␤ Ϫ␧s,c␻ /c , and ␪s,c ϭarctan as,c . where gϭF f ␭ͱ␧ f /␲M 0 , F f is the Faraday coefﬁcient, ␭ is For a TE mode the wavelength of the light in a vacuum, ␧ is the mean value f 2 1/2 of ␧ (z) in the ferromagnetic ﬁlm, e is a totally antisym- ͑p͒ ͑p͒ ͑p͒ ͑p͒ 0 ljk as,c ϭ␥s,c /k f , P ϭ ͑p͒Ϫ1 ͑p͒Ϫ1 . metric tensor, and l, j,kϭ͕x,y,z͖. ͩ dϩ␥c ϩ␥s ͪ 688 Tech. Phys. 43 (6), June 1998 L. V. Lutsev

For a TM mode 2 ͑p͒ ͑p͒ ͑p͒ 2 ␻TE͑␤TE Ϫk͉͒͗Ey ͉␧yz͉Hy ͉͘ ␯͑p͒2ϭ . ͑22͒ ͑p͒ 2 ͑p͒ ␥ ␧ 4c ␧f␤TE ͑p͒ s,c f as,c ϭ ͑p͒ , (p) k f ␧s,c The coupling coefficient ␯ determines the period of the spatial oscillations along the 0x axis for TE TM con- ͑p͒ ͑p͒ ͑p͒ ͑p͒ ͑p͒ ͑p͒ Ϫ1/2 ↔ k f ϩ␥s as k f ϩ␥c ac version: P͑p͒ϭ& dϩ ϩ . ͫ k͑p͒␥͑p͒͑1ϩa͑p͒2͒ k͑p͒␥͑p͒͑1ϩa͑p͒2͒ͬ f s s f c c 2␲ ͑p͒ L ϭ ͑p͒ . ͑23͒ The subscripts f, s, and c indicate that the quantity refers ␯ to the film, the substrate, and the cladding. In the phase vari- We shall now examine some special cases of TE–TM (p) 14 ables ␪s,c the dispersion relations take the simple form scattering on spin waves in ferromagnetic films that are homogeneous and weakly inhomogeneous across their thick- k͑p͒dϪ␪͑p͒Ϫ␪͑p͒ϭ␲p pϭ0,1,2,... . ͑19͒ f s c ͑ ͒ ness and in films with a magnetization gradient. The functions ͑18͒ together with the radiative modes A. Ferromagnetic films that are homogeneous and form a complete orthonormal system. They will be used to weakly inhomogeneous across their thickness. Substituting calculate the coupling coefficient of TE and TM modes and the eigenfunctions ͑4͒ and ͑18͒ with consideration of ͑8͒, the conditions for synchronism in an inhomogeneous film in ͑11͒, and ͑15͒ into ͑22͒, we obtain the coupling coefficient of first-order perturbation theory. a pair of TE and TM modes upon scattering on the nth spin- wave mode

͑p͒ ͑n͒ 1/2 ͑␤TE Ϫk͒␥U ␯͑p͒ϭP͑p͒P͑p͒ F 3. TE–TM SCATTERING OF OPTICAL MODES ON SPIN n TE TM f ͑p͒ ͑n͒2 ͫ 2␤ ␻n͑dϩ2͉k͉/k ͒M 0ͬ TE 0 WAVES 2 ͑n,p͒ ͑n,p͒ ͑n,p͒ sin͑K jl dϪ⌶j,l ͒ϩsin ⌶ j,l Variation of the magnetic moment density m(x,z,t) ϫ ͚ ͑n,p͒ , ͑24͒ ͯ j,lϭ1 K ͯ jl ϭ͕mx ,my,0͖ of a spin wave leads to the interconversion of (n,p) (n) (j) (p) (l) (p) TE and TM modes. We represent the electromagnetic field in where K jl ϭkz ϩ␩ kf,TEϩ␩ Kf,TM , a planar waveguide in the form of the superposition of a pair ͑n,p͒ ͑n͒ ͑ j͒ ͑p͒ ͑l͒ ͑p͒ of nearby TE and TM modes: ⌶jl ϭ͑kz dϪ␲͑nϪ1͒͒/2ϩ␩ ␪s,TEϩ␩ ␪s,TM , ͑1͒ ͑2͒ ͑p͒ ͑p͒ ͑p͒ ␩ ϭ1, ␩ ϭϪ1. ⌿B ͑x,z͒ϭF͑x͒⌿TE ͑x,z͒ϩG͑x͒⌿TM͑x,z͒, ͑20͒ (p) (p) An analysis of the relation obtained ͑24͒ shows that the where ⌿TE and ⌿TM are the functions ͑18͒. scattering will be strongest when the conditions We take into account that a͒ the derivatives of the am- ͑n,p͒ x are terms that are small in K je ϭ0 ͑j,lϭ1,2͒, ͑25͒ץ/(G(xץ x andץ/(F(xץ plitudes (p)Ϫ1 Ϫ1 (p)Ϫ1 Ϫ1 x ͑n,p͒ץ/Gץ xӶ1 and ␤TM Gץ/Fץ first order, i.e., ␤TE F ⌶ jl ϭ2␲r ͑j,lϭ1,2, rϭ0,Ϯ1,Ϯ2,...͒, ͑26͒ Ӷ1, that b͒ under the condition ␻TE ,␻TMӷ␻n the convolution in ͑16͒ and ͑17͒ is replaced by the product, and that c͒ which can be called the conditions for synchronism of the the dependence of ␧lj (lÞj)onmhas the form ͑15͒ and is transverse wave vectors and phase synchronism, are satis- proportional to exp(ikxϩi␻nt). In these approximations we fied. The physical meaning of these conditions can be ex- obtain the coupling equations plained in the following manner. Let us consider two adjacent layers with antinodes for the spin-wave mode. Since m 2 ͑p͒ ͑p͒ ͑p͒ -F͑x͒ i␻TE␤TM͗Ey ͉␧yz͉Hy ͘ has opposite values in these layers, the rotation of the polarץ ϭ ͑p͒ exp͑Ϫi⌬x͒G͑x͒, x 2c␧f␻TM␤TE ization plane is determined by the difference between theץ Faraday effects in these layers. The maximum total rotation ͑p͒ ͑p͒ ͑p͒ G͑x͒ i␻TM͑␤TE Ϫk͒͗Hy ͉␧zy͉Ey ͘ of the polarization plane occurs in the case in which there isץ ϭ ͑p͒ exp͑i⌬x͒F͑x͒, an antinode of the optical mode in one of the layers and there x 2c␤TMץ ͑21͒ is a node in the other layer, as is reflected in the conditions ͑25͒ and ͑26͒. where To illustrate the importance of the fulfillment of relations ͑25͒ and ͑26͒ for obtaining maximum TE TM conversion, E͑p͒ ␧ H͑p͒ ϭ H͑p͒ ␧ E͑p͒* ͗ y ͉ yz͉ y ͘ ͗ y ͉ zy͉ y ͘ we performed numerical calculations for a→ YIG/GGG struc- (p) (p) d ture and kϭk . Fulfillment of the condition ⌬ϭ␤ Ϫ␤ ͑p͒ ͑p͒ n TE TM ϭ ␺TE *͑z͒␧yz͑z͒␺TM͑z͒dz, Ϫkϭ0, which can be called the synchronism condition for ͵0 longitudinal wave vectors, was achieved by introducing layer ͑p͒ ͑p͒ 2 ͑Fig. 1͒ with linear variation of the dielectric constant ac- ⌬ϭ␤TE Ϫ␤TMϪk, ␧yzϭigmx . cording to the law ␧(z)ϭ␧ f ϩ(␧ f Ϫ␧s)z/ds (z෈͓0,Ϫds͔) When the synchronism condition ⌬ϭ0 is satisfied, the within the layer ͑other ways of achieving the condition coupling coefficient of the TE and TM modes reaches a ⌬ϭ0 were described in Ref. 16͒. The eigenfunctions of the maximum and is given by the expression optical modes of such a film structure were found from ͑16͒ Tech. Phys. 43 (6), June 1998 L. V. Lutsev 689

FIG. 2. Ratio between the coupling co- (4) (4) efﬁcients ␯n /␯1 for TE4 TM4 scattering in a homogeneous ferromagnetic→ layer ͑a YIG/GGG structure͒ as a function of the spin-wave mode number for

kϭkn ͓Eq. ͑7͔͒. dc , ␮m: a—0, b—0.5, c—1.0.

¯ and ͑17͒ using perturbation theory in the form of a series in tion theory. The expansion parameter here is ⌬␹ik /␹ik , powers of the deviation from a homogeneous film. In a first which, in turn, is determined by the relative deviations approximation the eigenfunctions have the form ͑18͒ with ¯ ¯ ¯ ¯ (p) ⌬␥/␥, ⌬␣/␣, ⌬M 0 /M 0 , and ⌬Ha /Ha . altered values of ␤ B. Ferromagnetic films with a magnetization gradient across their thickness. The coupling coefficient of TE and ͑p͒2 ͑p͒2 2 ͑p͒ ͑p͒ TE:␤ ϭ␤0 ϩ͑2␲/␭͒ ͗Ey ͉͑␧͑z͒Ϫ␧0͉͒Ey ͘, TM modes upon scattering on the nth spin-wave mode in a ¯ film with the magnetization gradient 4␲M 0ϭ4␲M 0Ϫ␮(z ͑p͒2 ͑p͒2 2 ͑p͒ ͑p͒ TM:␤ ϭ␤0 ϩ͑2␲/␭͒ ͗Hy ͉͑␧͑z͒Ϫ␧0͉͒Hy ͘ Ϫd/2) was found from ͑22͒ after plugging in the eigenfunctions ͑18͒ with consideration of the distribution of mϮ ͓Eqs. ץ ␧͑z͒ץ Ϫ H͑p͒ H͑p͒ . ͑27͒ ͑12͒ and ͑15͔͒. A numerical calculation was performed for a z ͯ y ʹ YIG/GGG structure. The external magnetic field ͑forץ zץͳ y ͯ ␧͑z͒ ␻n/2␲ϭ9 GHz͒ was selected so that there would be a turn- Figure 2 presents the ratios between coupling coeffi- ing point for the magnetostatic potential within the ferromag- (4) (4) cients ␯n /␯1 for TE4 TM4 scattering as a function of the netic film. Figure 3 presents the ratios of the coupling coef- → (4) number n of the spin-wave mode for various values of the ficient ␯n for TE4 TM4 scattering in a film with a gradient → (4) cladding thickness dc . The mean values for YIG were used: to the coupling coefficient ␯1hom for TE4 TM4 scattering in Ϫ12 2 → 4␲M 0ϭ1750 Oe, Haϭ0, ␣ϭ4␲•3.2ϫ10 cm , ␥ϭ2␲ a homogeneous ferromagnetic film as a function of the spin- ϫ2.83 MHz/Oe for dϭ10 ␮m, ␻n/2␲ϭ9 GHz, wave mode number n of the for k 0. The values 4␲M¯ → 0 ␭ϭ1.15 ␮m, ncϭͱ␧cϭ1.0, n f ϭͱ␧ f ϭ2.220, and nsϭͱ␧s ϭ1750 Oe and ␮ϭ10 Oe/␮m were used. All the remaining ϭ1.945. The transition layer 2 ͑Fig. 1͒, which was employed parameters were the same as for the homogeneous ferromag- to achieve the condition ⌬ϭ0, had a thickness netic structure. The condition ⌬ϭ0 was achieved just as in dsХ0.6 ␮m. The presence of this layer required recalcula- the case of the homogeneous ferromagnetic film by introduc- (4) (n,4) (n,4) tion of the values of ␯n , Kjl , and ⌶ jl with altered ing intermediate layer 2 ͑Fig. 1͒. An analysis of the distribu- (4) values of ␤ according to Eqs. ͑18͒, ͑24͒, and ͑27͒. The tion of mϮ ͑12͒ reveals that mϮ has its greatest amplitude in cladding 1 had a dielectric constant identical to the dielectric the vicinity of the turning point zr ͑14͒. If the antinodes of constant of the YIG layer (ͱ␧ f ϭ2.220) and was nonmag- the TE and TM optical modes are located in the layer with netic. Thus, the thickness of the spin-wave waveguide did the turning point zr , the TE–TM scattering is strongest. In not coincide with the thickness of the optical waveguide. Fig. 3 this is observed for the spin-wave modes with nϭ2, (p) (n,p) This led to the additions k f dc to the ⌶ jl , to violation of 7–8, and 14–16. Variation of the cladding thickness dc leads the phase synchronism conditions ͑26͒, and, thus, to the os- to displacement of the antinodes relative to zr . It is seen cillating character of the dependence of the TE–TM scatter- from a comparison of the plots in Figs. 2 and 3 that if there ing on the spin-wave mode number. The synchronism con- is a turning point for the magnetostatic potential ␸(z), the (n) (4) (4) dition ͑25͒ kz Ϫk f ,TEϪkf,TMϭ0 was satisfied for nϭ9 amplitude of the TE–TM scattering in inhomogeneous fer- Ϫ10. romagnetic films can take a larger value than in homoge- (p) Formula ͑24͒ for the coupling coefficient ␯n was ob- neous films. It can be concluded on this basis that the most tained for a homogeneous ferromagnetic layer. The employ- promising magnetooptical materials are those in which ment of this formula for weakly inhomogeneous ferromag- TE–TM scattering is possible on a periodic distribution of netic structures in the absence of turning points for the several layers with turning points for ␸(z). They can be magnetostatic potential is permissible in first-order perturba- magnetic superlattices, i.e., film structures with spatially pe- 690 Tech. Phys. 43 (6), June 1998 L. V. Lutsev

FIG. 3. Ratio between the coupling co- (4) (4) efﬁcients ␯n /␯1hom for TE4 TM4 scattering in an inhomogeneous→ ferromagnetic layer as a function of the spin-wave

mode number for k 0. The values of dc are the same as in→ Fig. 2.

riodic deviations of the magnetic parameters from constant of the optical modes increases the amplitude of the TE–TM values across the thickness, with dӷͱL and a fairly large scattering in comparison to the scattering in homogeneous number of periods. It was noted in Ref. 10 that superlattice films. structures can be obtained on the basis of multicomponent iron garnets. Iron garnets with ͑Lu3ϩ,Y3ϩ͒,͑Lu3ϩ,Y3ϩ,Bi3ϩ͒, or (Lu3ϩ,Y3ϩ,La3ϩ) in dodecahedral lattice sites were proposed. Periodic variation of the growth conditions during epitaxial growth leads to variations in the entry of these ions 1 D. Young and C. S. Tsai, Appl. Phys. Lett. 53, 1696 ͑1988͒. into the dodecahedral sites and to variations in the entry of 2 C. S. Tsai and D. Young, Appl. Phys. Lett. 54, 196 ͑1989͒. Pb2ϩ and Pb4ϩ. This, in turn, leads to variations in the 3 A. D. Fisher, J. N. Lee, E. S. Gaynor, and A. B. Tveten, Appl. Phys. Lett. uniaxial growth anisotropy.12 More detailed investigations 41, 779 ͑1982͒. 4 Yu. V. Gulyaev, I. A. Ignat’ev, V. G. Plekhanov, and A. F. Popkov, were performed for the iron garnet ͑Lu,Y,Bi͒3͑Fe,Ga͒5O12 in Radiotekh. Elektron. 30, 1522 1985 . Ref. 17. Trial runs of the epitaxial growth of films showed ͑ ͒ 5 O. G. Rutkin, N. G. Kovshikov, A. A. Stashkevich et al., Pis’ma Zh. that the formation of these iron garnet structures during a Tekh. Fiz. 11, 933 ͑1985͓͒Sov. Tech. Phys. Lett. 11, 386 ͑1985͔͒. single production cycle can be achieved by varying the pro- 6 N. Bilaniuk and D. D. Stancil, J. Appl. Phys. 67, 508 ͑1990͒. duction parameters ͑the growth temperature, the rotation rate 7 V. V. Matyushev and A. A. Stashkevich, J. Appl. Phys. 69, 5972 ͑1991͒. 8 of the substrate, etc.͒. A. A. Solomko, Yu. A. Gada, A. V. Dovzhenko et al., Opt. Spektrosk. 66, 190 ͑1989͓͒Opt. Spectrosc. ͑USSR͒ 66, 110 ͑1989͔͒. 9 A. I. Akhiezer, V. G. Bar’yakhtar, and S. V. Peletminskii, Spin Waves, CONCLUSIONS North-Holland, Amsterdam ͑1968͒. The following conclusions can be drawn as a result of 10 L. V. Lutsev, Zh. Tekh. Fiz. 65͑2͒,41͑1995͓͒Tech. Phys. 40, 139 ͑1995͔͒. the theoretical analysis performed. 11 a͒ In homogeneous ferromagnetic films and in films with A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves, CRC Press, Boca Raton ͑1996͓͒Russian original, Nauka, Moscow ͑1994͒, slight deviations from homogeneity TE–TM scattering on 464 pp.͔. higher spin-wave modes is strongest when the synchronism 12 L. V. Lutsev, V. O. Shcherbakova, and G. Ya. Fedorova, Fiz. Tverd. Tela. conditions for the transverse phases and the longitudinal and ͑St. Petersburg͒, 35, 2208 ͑1993͓͒Phys. Solid State 35, 1098 ͑1993͔͒. transverse wave vectors are satisfied. When the thicknesses 13 L. V. Lutsev, I. L. Berezin, and Yu. M. Yakovlev, Elektron. Tekh. Ser. of the planar optical waveguide and the ferromagnetic film Elektron. Sverkhvys. Chastot., No. 5͑419͒, pp. 5–8 ͑1989͒. 14 do not match, the phase synchronism condition is violated H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir ͑Ed.͒, Springer- Verlag, Berlin–New York ͑1988͒; Mir, Moscow ͑1991͒, pp. 18–131. with the resultant appearance of an oscillating type of depen- 15 A. K. Zvezdin and V. A. Kotov, Magnetooptics of Thin Films ͓in Rus- dence of the TE–TM scattering on the spin-wave mode num- sian͔, Nauka, Moscow ͑1988͒, 192 pp. 16 ber. A. M. Prokhorov, G. A. Smolenski, and A. N. Ageev, Usp. Fiz. Nauk. b͒ A layer with a turning point for the magnetooptical 143,33͑1984͓͒Usp. Fiz. Nauk 27, 339 ͑1984͔͒. potential in a film with a magnetization gradient across its 17 L. V. Lutsev, V. L. Ivashintsova, Yu. M. Yakovlev et al.,inFirst Joint thickness makes the largest contribution to TE–TM scatter- Conference on Magnetoelectronics ͓in Russian͔, Moscow ͑1995͒, pp. 105–106. ing. The juxtaposition of this layer, whose position depends on the spin-wave mode number, to the region of the antinode Translated by P. Shelnitz TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Postgrowth residual stresses in polycrystalline zinc selenide L. K. Andrianova, I. I. Afanas’ev, and A. A. Dunaev

S. I. Vavilov State Optical Institute All-Russia Science Center, 199034 St. Petersburg, Russia ͑Submitted August 30, 1996; resubmitted December 8, 1997͒ Zh. Tekh. Fiz. 68, 85–90 ͑June 1998͒ The residual stresses in samples of polycrystalline ZnSe are studied by measuring the photoelasticity in the visible part of the spectrum with transillumination parallel and perpendicular to the growth axis. The thermal and growth components of the birefringence, which exhibit different types of distributions among samples, are investigated. It is established that the thermal component has a nearly equilibrium distribution, while the growth component has an asymmetric distribution, which reﬂects individual features in the growth of each speciﬁc sample. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01606-7͔

The residual stresses in industrial metals, which com- treated in both planes and investigated in polariscope- prise a widely encountered class of polycrystalline metals, polarimeters with a field of vision from 150 to 300 mm with- are customarily divided into stresses of the first and second out magnification using a diffuse source of white light. A kinds.1 The former include macrostresses, which affect the KSP-7 polariscope with a magnification of 8 – 10ϫ and a volume of a body as a whole, and the latter include micros- Seńarmont compensator for an optical wavelength of 546.1 tresses, which act within the grains of a material and on their nm, which was isolated by a filter, was employed for more boundaries. Unlike metals, polycrystalline ZnSe is transpar- exact measurements. Templates cut from disks so that in a ent in the visible region of the spectrum. Under the influence section one pair of their sides would be equal to the thickness of mechanical stresses and strains, the spherical optical indi- of the disk and the length would be equal to the radius of the 2 catrix (Xi /n )ϭ1 is slightly distorted and takes the form of disk or a chord, depending on the cutting geometry, were a uniaxial or biaxial ellipsoid, depending on the character of investigated in a similar manner. The disks and templates the stressed ͑strained͒ state. This permits the application of were marked to show which planes belonged to the substrate the ordinary tools and methods of photomechanical analysis and the growth surface. to it. Optical effects of the action of stresses of both kinds A right-hand system of axis, in which the growth axis can be observed in the field of vision of a polariscope. In the coincides with the direction of the Z coordinate, was used to present work only stresses of the first kind are investigated. describe the geometry of the piezobirefringence and the The application of the methods of photomechanics to stresses ͑Fig. 1͒. In such a system the field of values of the zinc selenide is significantly simpler than in the case of other birefringence Nʈϭ⌬nrϪ⌬n␣ , where ⌬nr and ⌬n␣ are photoelastic crystalline materials. First, like AgCl or KRS, it small increments of the refractive index along a radius of the has a fairly high optical sensitivity,2,3 but, unlike those ma- disk and in the direction perpendicular to it, can be measured terials, it is less plastic and, therefore, behaves essentially when the disks are transilluminated along Z. In an axisym- like an elastic medium, and the application of the law of metric distribution of the birefringence, the central part of photoelasticity, which does not take into account the time the circular disks is free of birefringence, since ⌬nrϭ⌬n␣ . factor when a force is applied, is more correct for just such The birefringence in directions perpendicular to Z cannot be media. Second, it can be regarded as a proven fact that in any measured in an intact disk, since there are no immersion case polycrystalline ZnSe is an isotropic photoelastic mate- liquids with a refractive index close to 2.60; therefore, such rial within the sensitivity range of polariscope-polarimeters.3 measurements were performed on templates. When the tem- This facilitates application of the photoelasticity law to it plates are transilluminated in directions perpendicular to Z, without consideration of the crystallographic coordinates of the field of values of the birefringence NЌϭ⌬nzϪ⌬n␣ , the medium, which must be taken into account in the case of where ⌬nz is the increment of the refractive index along the 4 single crystals. Thus, it can be regarded as a simple and Z axis and ⌬n␣ is the increment of the refractive index along effective sensor of the piezobirefringence and stresses that a direction perpendicular to Z measured in the perpendicular appear or are manifested in various stages of a production direction, can be determined. In the central parts of the axi- process. symmetric field Nʈ and NЌ depend on the influence of the The original disks of polycrystalline ZnSe, with a diam- thermal stresses caused by the axial temperature gradient eter of 150–400 mm and a thickness of 15–40 mm, were controlling the vapor condensation process. grown by vacuum desublimation.5 An analysis of the micro- It is known from the theory of photoelasticity of opti- structure and x-ray diffraction investigations provide evi- cally and mechanically isotropic bodies that differences be- dence of the anisotropy of the external form of the grains and tween normal stresses can be calculated from measured val- their textured character, which become stronger as the con- ues of the birefringence, if the photoelastic constant of the 6,7 densate thickness increases. The disks were optically material is known. Using the notation tʈ(r,␣)ϭ␴rϪ␴␣ ,we

FIG. 1. Asymmetric arrangement of ellipses in sections of the optical indicatrix in planes of the samples investigated far from the edge part when disks are mea-

sured along the Z axis and ⌬nrϪ⌬n␣ Ͼ0 ͑a͒, when a template is measured along the Z axis and ⌬nrϪ⌬n␣ϭ0 ͑b͒, and when a template is measured perpendicularly to the Z axis ͑c͒: 1—neutral lines with zero birefringence.

can write the relation between the stresses and the birefrin- birefringence of the second kind͒. A similar phenomenon is gence in the form of the formula observed when templates are transilluminated in the same direction ͑Fig. 1b͒. t ϭBN cos 2␸ , ͑1͒ ʈ ʈ ʈ Regardless of whether a distribution of birefringence is 3 where Bϭ2/n (␲11Ϫ␲12) is the photoelastic constant of the observed in the disks when they are observed along the Z material and ␸ʈ is the azimuth of the principal directions of axis or not, the piezobirefringence NЌ is always observed the indicatrix and of the stresses ␴r and ␴␣ . when they are observed in the directions perpendicular to Z, When the distribution of the optical anisotropy and the i.e., in templates of the radial and chord type ͑Fig. 1c͒. The stresses in the original disk is axisymmetric, the principal pattern of this birefringence in the form of indicatrices and mechanical and optical directions are oriented radially; there- their orientation contains an odd number of zones with bire- fore, ␸ʈϭ0. Otherwise, the principal stresses are not radial, fringence of opposite sign. If Nʈϭ0 over the entire plane of and this situation leads to the appearance of tangential a disk, and NЌÞ0, as is always observed, the observed bire- stresses, ␶ʈ , which can be defined by the formula fringence is associated with the formation of uniaxial optical ␶ ϭBN sin 2␸ /2. ͑2͒ indicatrices of opposite sign ͑Fig. 1c͒. Such indicatrices ʈ ʈ ʈ clearly form under the action of an axial temperature gradi- Knowing t and ␶ , we can also define the difference ʈ ʈ ent ⌬Tʈ and in the absence of a radial gradient ⌬TЌ .Ifa 2 2 between the principal stresses tϭͱtʈ ϩ4␶ʈ , the azimuth of radial gradient still appears for a number of technological the principal optical and mechanical directions being defined reasons, it leads to the distortion of circular sections of the by an angle ␸ʈÞ0 directly during the polarization measure- uniaxial indicatrices ͑Fig. 1c͒. Biaxial indicatrices, whose ments. In addition, circular sections can be at different angles to the observation tan 2␸ ϭ2␶ /t . ͑3͒ directions taken in the present work, form. The sections of ʈ ʈ ʈ such indicatrices in the plane of the sample display ellipses in peripheral zones ͑Fig. 1a͒. A comparison of this figure and RESULTS Fig. 1c reveals the similarity between them, which is con- 1. Residual birefringence and stresses. The magnitude fined to the fact that both patterns are caused by the removal and distribution of the optical anisotropy clearly depend on of thermal energy from the surface of the growing preform. the magnitude of the axial temperature gradient, which as- With respect to the temperature gradients ⌬Tʈ and ⌬TЌ , the signs the growth rate. If the thickness of the layer grown is indicatrices have identical signs. However, the conditions for small, the amount of heat removed from the cylindrical sur- mechanical equilibrium of a disk and a template differ sig- face of the disk is also small. However, above a certain nificantly. In a disk the thermal stresses created on the edge thickness, at which the cylindrical surface acquires a suffi- by the condition ⌬TЌÞ0 are balanced by the stresses of the cient area, the latter becomes a source of heat losses, which central circular zone, in which the thermal strains are equal cause the appearance of a radial temperature gradient. Tran- in all directions lying in the plane of the disk. This leads to silluminating the disk along the Z axis, we discover axisym- the formation of a uniaxial indicatrix at the center of the metric fields of Nʈ ͑Fig. 1a͒. When there is no radial tem- sample. When ⌬TʈÞ0 and the sample is observed in a di- perature gradient, there are likewise no peripheral zones with rection perpendicular to the Z axis, the central zone of a birefringence. In this case the disk does not contain piezobi- template contains birefringence of opposite sign ͑Fig. 1c͒, refringence of the first kind, but it has birefringence, which is and its thickness is divided into three zones with birefrin- noticeable within the grain structure of the material ͑piezo- gence of opposite sign. Thus, the distribution of Nʈ measured Tech. Phys. 43 (6), June 1998 Andrianova et al. 693

FIG. 3. Inﬂuence of mechanical treatment on residual thermal stresses of the symmetric type ͑stress diagrams͒: a—original state, b—after reduction of the template thickness.

FIG. 2. Principal types of postgrowth optical anisotropy observed in the metric, and the diagram in Fig. 2b asymmetric. According to direction perpendicular to the growth axis: a—equilibrium diagram of re- Ref. 8, the diagram in Fig. 2a should correspond to heating. sidual thermal stresses of the quenching type ͑the symmetric type͒,b— During cooling, the signs of the stresses are reversed due to equilibrium diagram of residual thermal stresses with growth stresses due to an axial temperature gradient ͑the asymmetric type͒. the plasticity under the high-temperature conditions. There- fore, this diagram could have formed only after plastic deformation, apparently of the grain-boundary type,9 which along the Z axis can be represented with motion along the would lead, precisely as in the case in Fig. 2, to reversal of diameter by an NʈϪ0ϪNʈ pattern, and NЌ can be repre- the signs of the residual stresses, whose structure reflects the sented with motion along the Z axis by an NЌϪ(ϪNЌ) action of the axial temperature gradient ⌬Tʈ as a whole to ϪNЌ pattern. Knowing the photoelastic properties of the within the signs. The diagram in Fig. 2b is also frequently material,2,3 its behavior upon heating in the field of vision of encountered and has a more complicated type of distribution the crossed polariscope in the temperature range 20–200 °C of NЌ and tЌ than in the preceding diagram. They are, as it ͑Ref. 8͒, and relations ͑1͒–͑3͒, we can make the transition were, two diagrams of opposite sign, which are combined from indicatrices to stresses using the classical model of a with one another so that the opposite sides of a template have simple birefringent plate for this transition. The diagram of residual stresses of opposite sign. On the substrate side the stresses caused by the radial temperature gradient during the upper part of the diagram has a form similar to a diagram of growth and cooling of the sample consists of peripheral com- the symmetric type. This diagram, however, smoothly joins pressive stresses, which are balanced at the center by stresses the analogous diagram of opposite sign in the lower part of of opposite sign that do not produce birefringence. Cooling the template. The tensile stresses on the lower edge of the from the growth temperatures 800–1000 °C was followed by template provide evidence that it was, as it were, additionally plastic deformation,9 as a result of which the sign of the heated at a low temperature, which is such that even upon residual stresses was the reverse of the sign observed during further cooling the diagram of opposite sign does not manage 1 growth and cooling under the action of ⌬TЌ . Residual pe- to form because of the low plasticity of the material at those ripheral compressive stresses, which are balanced by uniform temperatures. The material behaved like a hard spring, but in-plane stresses that are equal to one another, but cannot be this was not previously detected in Ref. 9, since the thermal determined by the polarization method, appear. The magni- strains were not frozen. tude of these stresses is small and amounts to about 2. Influence of mechanical treatment on NЌ and tЌ . The 2 Ϫ30 kgf/cm . These stresses vanish after the templates are birefringence Nʈ in disks has been investigated fairly thor- cut in accordance with the scheme in Fig. 1b. oughly, since it is often monitored during the fabrication of Figures 2a and 2b present two types of diagrams of the optical elements from glass and crystals. The behavior of residual stresses tЌϭ␴zϪ␴␣ , which were observed on sev- NЌ , which is observed in templates, is not so well known. eral tens of templates. We call the diagram in Fig. 2a sym- The influence of the mechanical treatment of templates 694 Tech. Phys. 43 (6), June 1998 Andrianova et al.

FIG. 4. Influence of mechanical treatment on residual thermal stresses of the asymmetric type ͑stress diagrams͒: a—original state, b, c—after reduction of the template thickness; 1—substrate, 2—growth surface. across their thickness is shown in Fig. 3a. After a total reduction of the thickness of a template on both sides by 16%, the original symmetric diagram ͑Fig. 3a͒ exhibited slight FIG. 5. Influence of isothermal annealing on residual thermal stresses. changes in its form in the part adjacent to the substrate, pro- viding some basis to regard the contact with the substrate as an additional factor which increases the stresses in this part. 1000 °C with holding for 5 h. Each template was annealed The opposite side underwent significantly smaller changes. once. The annealing temperature variation step was 100 °C. For this reason, it would be useful to investigate the influ- All the templates had the same original birefringence pattern ence of mechanical treatment of the plane of a template ad- down to the grain structure. The band patterns were photo- jacent to the substrate in greater detail, rather than the varia- graphed before and after annealing, and stress diagrams were tion of NЌ and tЌ in templates with diagrams of the constructed from them according to the method adopted. The asymmetric type. Figure 4 shows the variation of the stress templates were held in a gradient-free portion of the furnace, diagram when 16, 33, 50, and 66% of the thickness is uni- and the cooling to room temperature after each anneal was laterally ground off. After the thickness is reduced by 33%, not forced. It was found that annealing at temperatures from the diagram remains asymmetric. After 50% is ground off, 400 to 700 °C did not cause changes in the diagrams of the the diagram becomes symmetric, but has a sign which is residual birefringence NЌ and the stresses tЌ . Annealing at opposite to that of Fig. 3a, and after further treatment ͑66%͒, temperatures from 800 to 1000 °C with the same cooling it transforms into a diagram which is similar to a diagram for procedure lowers NЌ and tЌ . The decrease in the residual four-point bending, where the tensile stresses from the stresses at 800 °C amounts to 25%, while annealing at growth surface increase significantly and compressive 1000 °C lowers them by 80–90%, i.e., increases the optical stresses appear on the surface on the substrate side. homogeneity of the material. Figure 5 shows the character of 3. Influence of isothermal postgrowth annealing. A se- the decrease in the level of the residual stresses as a function ries of templates from a single original disk was subjected to of the annealing temperature for a diagram of the asymmetric isothermal annealing at constant temperatures from 400 to type. Tech. Phys. 43 (6), June 1998 Andrianova et al. 695

4. Influence of mechanical treatments. The four-point 2. The thermal component of the birefringence and bending of templates was employed to study the influence of stresses is observed when disks or templates are transillumi- the grain inhomogeneity across the thickness and the residual nated parallel to the axis of the temperature gradient provid- stresses on the mechanical behavior of the material in the ing for the growth process. This component has a nearly region corresponding to its elasticity at room temperature. equilibrium distribution. The bending was performed so that the stresses appearing as 3. The growth component of the birefringence and the a result would stress the surface facing the growth front, the stresses is most clearly expressed in templates transillumi- surface facing the substrate, and the surface parallel to the Z nated perpendicularly to the growth axis. It very often has an axis upon transillumination of the sample in the same direc- asymmetric distribution, which is far from equilibrium, and a tion. The loading was carried out in the field of vision of the tendency toward stratification, which reflects individual fea- polariscope by a lever press with a graduated load. The me- tures in the growth of each concrete sample. chanical moment step during the loading was 0.25 kgf. An analysis of the band pattern for the loading cases indicated demonstrates their completely identical nature, with the exception of the local edge deviations caused by individual features of the grain structure. The higher was the mechani- 1 Ya. B. Fridman, Mechanical Properties of Metals ͓in Russian͔, Oboron- cal moment, the more similar the band patterns became. We giz, Moscow ͑1952͒, 554 pp. 2 observed a similar response of the material upon the diamet- K. K. Dubenski, A. A. Kaplyanski, and N. G. Lozovskaya, Fiz. Tverd. Tela ͑Leningrad͒ 8, 2068 ͑1966͓͒Sov. Phys. Solid State 8, 1644 ͑1967͔͒. ric compression of disks with a diameter of 30 mm and a 3 thickness of 4 mm, which were cut parallel, perpendicularly, L. K. Andrianova, I. I. Afanas’ev, A. A. Demidenko et al., Opt. Mekh. 9 Promst. 57͑10͒,36͑1990͓͒Sov. J. Opt. Technol. 57, 615 ͑1990͔͒. and at a 45° angle to the growth axis. This is evidence that, 4 I. I. Afanas’ev and N. V. Ionina, Opt. Spektrosk. 67, 319 ͑1989͓͒Opt. as a whole, the material is homogeneous and isotropic ac- Spectrosc. ͑USSR͒ 67, 184 ͑1989͔͒. cording to its photomechanical behavior and that its grain 5 I. A. Maksimova, I. A. Mironov, and V. N. Pavlova, ‘‘Method for obtain- inhomogeneity, texture, growth features, and residual ing polycrystalline blocks of zinc and cadmium chalcogenides for optical ceramics’’ ͓in Russian͔, Inventor’s Certificate No. 844609. stresses do not create any macroscopic mechanical features 6 A. A. Demidenko, A. A. Dunaev, S. N. Kolesnikova, and I. A. Mironov, such as reinforcement, anisotropy, and the like in the mate- Vysokochist. Veshchestva, No. 1, pp. 103–109 ͑1991͒. rial. 7 G. V. Anan’eva, A. A. Dunaev, and T. I. Merkulyaeva, Vysokochist. Veshchestva, No. 4, pp. 114–119 ͑1995͒. 8 L. K. Andrianova, I. I. Afanas’ev, A. A. Dunaev et al., Zh. Tekh. Fiz. CONCLUSIONS 62͑8͒, 102 ͑1992͓͒Sov. Phys. Tech. Phys. 37, 708 ͑1992͔͒. 9 1. The induced optical anisotropy in polycrystalline zinc I. I. Afanas’ev, L. K. Andrianova, and A. A. Demidenko, Opt. Mekh. Promst. 57͑8͒,42͑1990͓͒Sov. J. Opt. Technol. 57, 490 ͑1990͔͒. selenide grown by a desublimation technology has two components: a growth component and a thermal component. Translated by P. Shelnitz TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Analysis of structural defects in boron-implanted silicon single crystals on the basis of the results of double- and triple-crystal x-ray diffractometry A. P. Petrakov, N. A. Tikhonov, and S. V. Shilov

Syktyvkar State University, 167001 Syktyvkar, Russia ͑Submitted May 6, 1996; resubmitted December 19, 1997͒ Zh. Tekh. Fiz. 68, 91–96 ͑June 1998͒ The structural defects in Si single crystals are analyzed on the basis of diffraction reﬂection curves and triple-crystal spectra. The relative variation of the lattice period and its distribution as a function of depth are calculated, and the type of defects appearing and the behavior of the implanted impurity in response to high-temperature annealing are determined. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01706-1͔

INTRODUCTION between them, which was measured by an x-ray diffractometric method, was about 10Љ. The samples were implanted Ion implantation is an effective tool for altering the elec- ϩ trical properties of semiconductor materials. Boron is em- with B having an energy of 25 keV in doses from Dϭ6.2 14 15 Ϫ2 ployed quite often as the implantant. Ion implantation causes ϫ10 to 6.25ϫ10 cm . The implantation was carried out damage to the subsurface structure of crystals subjected to at room temperature under conditions which rule out chan- irradiation. This structural damage is investigated by differ- neling. The use of a fairly weak ion current with a density of ent methods. Among the nondestructive methods, double- 0.2 ␮A/cm2 also ruled out the phenomenon of self-annealing and triple-x-ray diffractometry are very informative. These during implantation. After implantation, some of the samples methods have been used in numerous studies ͑see, for ex- were annealed in a nitrogen atmosphere. The annealing tem- ample, Refs. 1–5͒ of the kinds of defects formed in silicon perature was varied from 300 to 1000 °C. The annealing single crystals as a result of implantation. At the same time, times were 10, 60, and 120 min. the development of a method for treating experimental spec- The structure of the subsurface layers of the silicon tra, particularly triple-crystal spectra, would make it possible single crystals was diagnosed using an automatic double- to obtain new data on defects. The present paper analyzes the and triple-crystal x-ray diffractometer, which was assembled results of systematic triple-crystal x-ray diffractometric in- on the basis of a DRON-UM1 x-ray diffractometer. vestigations of silicon single crystals that were implanted Dispersion-free double-crystal (n,Ϫn) and triple-crystal with boron in various doses and subjected to isothermal an- (n,Ϫn,n) geometries6,7 and Cu K radiation were em- nealing at various temperatures and for various annealing ␣1 times. ployed. The double-crystal diffraction reflection curves and the triple-crystal x-ray diffraction spectra were measured in the ␧ scan mode ͑rotation of the analyzer͒. The sample azi- EXPERIMENTAL METHOD muthal angle ͑␻͒ was varied from Ϫ500Љ to ϩ500Љ. High- Nearly perfect single-crystal wafers of KDB-10 silicon perfection silicon single crystals with a symmetric ͑111͒ re- with a thickness of 500 ␮m were investigated. The surfaces flection served as the monochromator and analyzer. The half- of the samples coincided with the ͑111͒ plane. The angle widths of the diffraction reflection curves of the

FIG. 1. Diffraction reﬂection curves of crystals. Implantation dose, cmϪ2: 1—6.25 ϫ1014, 2—1.875ϫ1015, 3—3.125ϫ1015, 4—6.25ϫ1015; P—curve for a perfect crys-

tal, I/I0—ratio of the reﬂected to the incident intensity.

FIG. 2. Plots of the reduced intensity function P(␻). The notation corresponds to Fig. 1.

monochromator and the analyzer amounted to 10Љ, which is from a layer with an increased lattice constant. As is seen close to the theoretical value. from the plot, the maximum value of ⌬d/d, which is approximately equal to 0.003, corresponds to a depth of 0.05 EXPERIMENTAL RESULTS ␮m. The effective thicknesses9 of the damaged layers were A. Dependence of the structural damage on the dose. also calculated for the irradiated samples: L from the half- Samples with dose loads from 6.25ϫ1014 to 6.25 i widths of the maxima on the reduced intensity function, and ϫ1015 cmϪ2 were investigated to find the dose dependence L by the integral-characteristic method, without consider- of the structural damage in ion-implanted silicon. Figure 1 p ation of a region of 20 or 30 around 0. These data are presents the ‘‘tails’’ of the corresponding diffraction reflec- Љ Љ presented in Table I. tion curves. The zero corresponds to the exact Bragg position B. Influence of annealing on the structure of ion- of the reflection from the ͑111͒ plane of an undistorted crys- implanted silicon. To investigate the influence of annealing tal. There is an appreciable increase in the intensity of the on the structure of boron-implanted silicon, we measured the diffraction reflection curve at small angles. Here the intensity diffraction reflection curves and triple-crystal x-ray diffrac- increases with the dose, and a well resolved additional peak tion spectra of samples which were irradiated with a dose of is observed for the largest dose. The triple-crystal x-ray dif- 1.875ϫ1015 cmϪ2 and subjected to annealing in a nitrogen fraction spectra did not display diffuse peaks. Plots of the atmosphere at 300, 600, 800, 900, and 1000 °C with anneal- reduced intensity function P(␻)ϭI ␻2/P ͑I is the intensity • id ing times equal to 10, 60, and 120 min. Figure 5 presents the of the principal peak, ␻ is the sample azimuthal angle, and ‘‘tails’’ of the diffraction reflection curves for the samples P ϭIid ␻2 is the reduced intensity function for an ideal id • subjected to annealing at 600 and 800 °C for 10 min. An- crystal and is approximately constant for all values of the nealing at the lower temperatures did not alter the form of sample azimuthal angle͒ constructed from the triple-crystal the diffraction reflection curve shown in Fig. 1 for a dose x-ray diffraction spectra are shown in Fig. 2. All the samples equal to 1.875ϫ1015 cmϪ2. Increasing the temperature to display a distinct maximum on the negative-angle side, 600 °C led to an increase in intensity on both the negative- whose intensity increases with the dose. In addition, the shift and positive-angle sides. Annealing at 800 °C led to an ap- of the maxima toward negative angles increases. The posi- preciable drop on the negative-angle side. tion of the peaks on the plot coincides with the region of The plots of the reduced intensity function P(␻) con- elevated intensity on the diffraction reflection curve at nega- structed from the triple-crystal x-ray diffraction spectra re- tive angles. The presence of a maximum on the plot of P(␻) veal the presence of a maximum on the negative-angle side attests to the occurrence of coherent scattering from a layer ͑Fig. 6͒, which coincides with the region of elevated inten- with an altered lattice constant. The mean relative strains were calculated from the positions of the maxima in Fig. 2 using the formula ⌬d/dϭcot ⌰b•⌬⌰ and are presented in Table I. TABLE I. Values of the strain and thickness of the damaged layer as a A plot of the dependence of the mean strain on the dose function of the implantation dose. is presented in Fig. 3. Figure 4 presents the strain profile ⌬d L ͓from P(␻)͔, L (20), L (30), ϫ103 p i i calculated using a program developed according to the Dose (cmϪ2) d ␮m ␮m ␮m method described in Ref. 8 for a sample irradiated with a 14 dose equal to 6.25ϫ1015 cmϪ2. The program calculates the 6.25ϫ10 0.8 0.16 0.043 0.031 1.875ϫ1015 1.2 0.14 0.067 0.054 variation of the strain as a function of depth from the sample 3.125ϫ1015 1.9 0.14 0.11 0.09 surface on the basis of an analysis of the additional peak on 6.25ϫ1015 2.9 0.13 0.12 0.11 the diffraction reflection curve caused by coherent scattering 698 Tech. Phys. 43 (6), June 1998 Petrakov et al.

FIG. 3. Dependence of the mean relative strain ⌬d/d on the implantation dose D. FIG. 4. Strain profile as a function of depth Z calculated from diffraction reflection curves. sity on the corresponding diffraction reflection curve. At positive angles the plots of the reduced intensity function the use of the integral method to estimate the thickness of the ͑for annealing temperatures equal to 300 and 600 °C͒ show a damaged layers is incorrect because of the presence of dif- tendency to rise, which reflects the increase in intensity on fuse scattering, which makes a contribution to the diffraction the diffraction reflection curve ͑Fig. 5͒. However, annealing reflection curve. at 800 °C leads to the appearance of a maximum in this Plots of the dependence of the intensity of the diffuse range of angles. The values of the mean strain ⌬d/d calcu- peak on the sample azimuthal angle were constructed in log- lated from the positions of the maxima on the plot of P(␻) log coordinates from the diffuse peak on the triple-crystal are presented in Table II. x-ray diffraction spectra for the samples annealed at 1000 °C Thus, annealing at 800 °C led to the appearance of two for 120 min ͑Fig. 7͒. Each plot has two linear segments with regions with strains of different sign in the subsurface layer. slopes close to 1 and 3. It should also be noted that there was no diffuse peak on the The most significant structural changes are observed at triple-crystal x-ray diffraction spectra obtained in the ana- doses close to the doses (Ϸ1016 cmϪ2) which cause amor- lyzer scan mode for any of the samples subjected to heat phization of a subsurface layer under the implantation con- treatment at temperatures from 300 to 900 °C. ditions employed. This can be observed for the samples im- Annealing at higher temperatures leads to further planted with boron in a dose equal to 6.25ϫ1015 cmϪ2. changes in the structure of the subsurface layer. At 1000 °C Figures 1 and 2 show the diffraction reflection curves and the triple-crystal x-ray diffraction spectra display a diffuse plots of P(␻) for this implantation dose. Annealing at 400 peak, which intensifies as the annealing time is increased. –700 °C for 10 min led to the appearance of one or two The plot of P(␻) for 1000 °C displays only one maximum, additional peaks on the diffraction reflection curve at angles which attests to the presence of a layer with a negative mean smaller than the Bragg angles. Two maxima associated with strain ͑Table II͒, whose magnitude decreases. coherent scattering are observed on the plot of P(␻) at nega- Table II presents the values of Li and L p calculated for tive sample azimuthal angles. After annealing at 800 °C for all temperatures. The values of Li for the heavily annealed 10 min the diffraction reflection curve of the sample practi- samples ͑1000 °C, 60 and 120 min͒ are not presented, since cally coincided with the analogous curve for an unimplanted

FIG. 5. Diffraction reﬂection curves of crystals implanted with a dose equal to 1.875ϫ1015 cmϪ2. Annealing temperature, °C: 1—600, 2—800 ͑the annealing time was 10 min͒; P—curve for a perfect crystal. Tech. Phys. 43 (6), June 1998 Petrakov et al. 699

FIG. 6. Plot of the reduced intensity function P(␻). The implantation dose FIG. 7. Dependence of the logarithm of the intensity of the diffuse peak was 1.875ϫ1015 cmϪ2, the annealing temperature was 800 °C, and the an- ln I on the logarithm of the sample azimuthal angle ln ␻. The annealing nealing time was 10 min. d temperature was 1000 °C, and the annealing time was 120 min; 1—positive sample azimuthal angles, 2—negative angles. sample. In this case the plot of P(␻) did not display any peaks. A further increase in the temperature to 900 °C again led to an increase in the intensity on the diffraction reflection curve at large and small angles, and the plot of P(␻) shows method toward highly strained layers. The value of Li is four maxima ͑Fig. 8͒. There were no diffuse peaks on the calculated from the difference between the areas under the triple-crystal x-ray diffraction spectra in the range of anneal- diffraction reflection curves of the damaged and perfect crys- ing temperatures 400–900 °C. Annealing at 1000 °C for 60 tals. Because the reflection coefficient of the latter crystal is min led to the appearance of an intense diffuse peak. higher than that of the ion-implanted crystal, the part of the diffraction reflection curve in the region of the Bragg angle must be excluded. The data in Table I demonstrate the criti- DISCUSSION OF RESULTS cal nature of this exclusion. It should also be noted on the The presence of maxima on the plots of the reduced basis of the data in Table I that the total thickness of the intensity function ͑Fig. 2͒ and the absence of a diffuse peak damaged layer with ‘‘large’’ and ‘‘small’’ degrees of strain on the triple-crystal x-ray diffraction spectra are caused by varies only slightly ͑0.20–0.25 ␮m͒ for all the samples ex- coherent scattering in a subsurface layer with an increase in cept the sample irradiated with a dose equal to 6.25 the lattice constant. This quite trivial result is associated with ϫ1015 cmϪ2. This can be attributed to amorphization of a the generation of a large number of point defects upon the certain part of the layer, which takes place in the region of implantation of boron. Silicon ions displaced from regular the maximum on the defect distribution profile. positions into interstitial sites cause expansion of the lattice. When the samples are annealed, restructuring of the de- As would be expected, the lattice strain increases with in- fects in the subsurface structure should be expected. Anneal- creasing dose ͑Fig. 3͒. The absence of a diffuse peak on the ing at 300 and 600 °C does not lead to significant changes in triple-crystal x-ray diffraction spectra indicates that no ex- the x-ray diffraction pattern. However, annealing at 800 °C tended defects form in silicon following implantation by bo- leads to the appearance of a layer with a negative strain in ron ions with doses up to 6.25ϫ1015 cmϪ2. the subsurface region ͑Fig. 6͒ along with the layer with a The thickness of the subsurface layer damaged by im- positive strain formed as a result of the implantation of ions. plantation, which was estimated from the half-width of the Its formation can be attributed to a significant increase in the reduced intensity function, varies only slightly. concentration of substituent boron ions in lattice sites. The The increase in the value of Li calculated by the integral layer with an increased lattice constant is probably located method can be attributed to the higher sensitivity of this somewhat closer to the surface because of the stronger dis-

TABLE II. Variation of the strain and the thickness of the damaged layer as a result of annealing.

⌬d ϫ103 d L p , ␮m Tempe- Annealing rature, time, positive negative positive negative

°C min strain strain strain strain Li(20), ␮m Li(30), ␮m 300 10 1.2 ••• 0.13 ••• 0.079 0.059 600 10 1.4 ••• 0.14 ••• 0.070 0.050 800 10 1.4 Ϫ1.8 0.13 0.09 0.057 0.043 900 60 1.6 Ϫ0.97 0.12 0.14 0.045 0.009 1000 60 ••• Ϫ0.38 ••• 0.18 ••• ••• 1000 120 ••• Ϫ0.29 ••• 0.26 ••• ••• 700 Tech. Phys. 43 (6), June 1998 Petrakov et al.

The data presented in Fig. 7 show that, for the most part, defects of the dislocation-loop type make contributions to the diffuse peak for the crystal annealed at 1000 °C for 120 min. Dislocation loops are associated with stacking faults formed by interstitial silicon atoms.12 It should be stressed that defects of these types are not the only defects which make a contribution to the x-ray diffraction. Other crystal lattice imperfections of silicon in the form of rod-shaped defects, vacancy clusters, interstitial atoms, pores, cracks, tetrahedra of stacking faults, etc. are known.11 We express our thanks to V. A. Bushuev for his support FIG. 8. Plot of the reduced intensity function P(␻). The implantation dose and advice in all aspects of this work. was 6.25ϫ1015 cmϪ2, and the annealing temperature was 900 °C.

1 A. Yu. Kazimirov, M. V. Koval’chuk, and V. G. Konn, Metalloﬁzika, placement of the defect distribution toward it in comparison 9͑4͒,54͑1987͒. 2 to the distribution of the implanted impurity. V. Holy and J. Kubena, Czech. J. Phys. 32, 750 ͑1982͒. 3 M. Servidori and F. Cembaly, J. Appl. Crystallogr. 21͑5͒, 176 ͑1988͒. The gradual decrease in the thickness of the layer with a 4 P. Zaumseil and U. Winter, Phys. Status Solidi A 120,67͑1990͒. positive strain revealed by the integral-characteristic method 5 V. A. Bushuev and A. P. Petryakov, Kristallograﬁya 40, 1043 ͑1995͒ as the annealing temperature is increased from 300 to 900 °C ͓Crystallogr. Rep. 40, 968 ͑1995͔͒. 6 is most likely attributable to the diffusion of interstitial sili- A. Iida and K. Kohra, Phys. Status Solidi A 51, 533 ͑1979͒. 7 A. M. Afanas’ev, P. A. Aleksandrov, and R. M. Imamov, X-Ray Diffrac- con ions, which leads to spreading of the distribution of in- tion Diagnostics of Submicron Layers ͓in Russian͔, Nauka, Moscow terstitial silicon, the passage of silicon ions into regular po- ͑1989͒, 152 pp. sitions, and the occupation of lattice positions by boron ions, 8 V. G. Kohn, M. V. Kovalchuk, R. M. Imamov, and E. F. Labonovich, Phys. Status Solidi A 64, 435 ͑1981͒. which actively displace silicon ions at high annealing 9 10,11 V. A. Bushuev and A. P. Petrakov, Fiz. Tverd. Tela ͑St. Petersburg͒ 35, temperatures. 355 ͑1993͓͒Phys. Solid State 35, 181 ͑1993͔͒. Heat treatment at 1000 °C leads to the disappearance of 10 G. A. Kachurin, I. E. Tyschenko, and M. Voelskow, Fiz. Tekh. Polupro- vodn. 21, 1193 ͑1987͓͒Sov. Phys. Semicond. 21, 725 ͑1987͔͒. the layer with a positive strain due to the implantation of an 11 even larger amount of boron in lattice sites. In addition, the J. W. Mayer, L. Eriksson, and J. A. Davies, Ion Implantation in Semicon- ductors: Silicon and Germanium, Academic Press, New York ͑1970͒; Mir, diffusion of boron ions into the bulk leads to a decrease in Moscow ͑1973͒, 296 pp. the negative strain in the subsurface layer ͑Table II͒. 12 F. F. Komarov, A. P. Novikov, V. S. Solov’ev, and S. Yu. Shiryaev, The presence of a diffuse peak on the triple-crystal x-ray Structural Defects in Ion-Implanted Silicon ͓in Russian͔, Universitetskoe diffraction spectra following high-temperature annealing at- Izd., Minsk ͑1990͒, 319 pp. tests to the association of point defects in extended defects. Translated by P. Shelnitz TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Conversion of optical modes in an absorbing magnetogyrotropic waveguide A. M. Shuty and D. I. Sementsov Ulyanovsk Branch of the M. V. Lomonosov Moscow State University, 432700 Ulyanovsk, Russia ͑Submitted March 28, 1996͒ Zh. Tekh. Fiz. 68, 97–104 ͑June 1998͒ The features of mode conversion in an absorbing magnetogyrotropic waveguide are investigated by a coupled-wave approach. It is shown that absorption leads to an additional contribution to the coupling of both identically and orthogonally polarized modes. A new waveguide regime for mode conversion, in which there is no oscillatory energy exchange between modes, is revealed. The possibility of controlling the damping of the total ﬁeld in a waveguide by varying the orientation of the magnetization in the waveguide layer is demonstrated. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01806-6͔

INTRODUCTION faces between the layers. The xϭ0 and xϭϪL planes separate the waveguide layers from the cladding and the sub- In most planar waveguides in use the losses associated strate. The dielectric constant of the cladding (␧ ) and the with optical absorption in the film material are insignificant 1 substrate (␧ ) are assumed to be real, and the dielectric con- and are generally disregarded in theoretical analyses.1–3 2 stant of the waveguide layer is assumed to be the complex However, in waveguides based on epitaxial iron garnet films, quantity ␧ϭ␧ Ϫi␧ . The magnetic permeabilities of all the the absorption in the near-IR range has values in the range Ј Љ layers for the optical range used are virtually equal to unity. ␣ϳ1–10cmϪ1, which cannot always be considered The field of a mode propagating along the z axis with the small.2,4 The research reported in the literature deals mainly complex propagation constant ␤ϭ␤ Ϫi␤ is proportional to with the influence of absorption on waveguide eigenmodes, Ј Љ the factor exp(Ϫi␤z). It is easy to show ͑Ref. 3͒ that two and the features of mode conversion in an absorbing wave- different waveguide modes with the fields E and H obey guide has scarcely been analyzed. For example, mode damp- i i the relation ing coefficients were obtained on the basis of the ray approach in Ref. 5, and waveguide light propagation was bϵٌ͑E H*ϩE*H ͒ϭ2k ␧*E E*, ͑1ٌ͒ investigated in Ref. 6 with consideration of the absorption in 1 2 2 1 0 1 2 the metallic coating of the waveguide. The contributions where k ϭ␻/c, ␻ is the frequency of the radiation, and c is from various absorption mechanisms to the damping coeffi- 0 the speed of light in free space. cients of modes of different orders were considered in Refs. -Separating the vector b and the operator ٌ into trans 2 and 7. Mode damping associated with the conversion of verse and longitudinal components, we integrate ͑1͒ over a part of the energy of a waveguide mode into radiant energy transverse cross section of the waveguide was described in Ref. 8. Experimental results on mode conversion in a magnetogyrotropic waveguide with consider- bץ ation of the complex character of the propagation constants z ϩٌtbt dsϭ2k0␧Љ E1E2*ds, ͑2͒ ͵ ͪ zץ ͩ ͵ of the modes were discussed in Ref. 9. Finally, an exact solution and numerical analysis of the problem of waveguide y,0), and the integration isץ/ץ,xץ/ץ)light propagation in an absorbing, transversely magnetized where dsϭdxdy, ٌtϭ waveguide, for which the TE and TM modes are eigen- carried out over the cross section of the waveguide. modes, were presented in Ref. 10. In the present work the Replacing the integral over the area from the second features of mode conversion in an absorbing planar magne- term on the left-hand side of ͑2͒ by an integral over a contour togyrotropic waveguide are investigated on the basis of the enclosing the waveguide, and taking into account the expo- coupled-mode theory for an arbitrary orientation of the mag- nential damping of the field of waveguide modes with in- netization in the waveguide layer. creasing distance from the boundaries of the waveguide layer, we obtain

ORTHOGONALITY RELATION FOR AN ABSORBING WAVEGUIDE tbt͒dsϭ Ͷ ͑btet͒dlϭ0, ͑3ٌ͒͑ ͵ The orthogonality relations of modes play an important role in constructing waveguide solutions on the basis of the where et is a unit vector that is perpendicular to the integra- coupled-wave theory. Let us ﬁnd the form that they take in tion contour. the presence of absorption. We consider a planar waveguide With consideration of ͑3͒ we arrive at the following sys- structure consisting of a substrate, a waveguide layer, and a tem of equations for the real and imaginary parts of the com- cladding. We direct the x axis perpendicularly to the inter- ponent bz :

ЈϪ Ј bЈϪ Љϩ Љ bЉϪ2k Im E E* dsϭ0, AЈ ϭϪik exp i z ˆE dx, 7 ͵ ͓͑␤2 ␤1͒ z ͑␤2 ␤1͒ z 0␧Љ ͑ 1 2 ͔͒ ␮ 0 ͑ ␤␮ ͒ ͵ ͑⌬␧ ͒E␮ ͑ ͒ where the prime denotes the derivative with respect to the ЈϪ Ј bЉϩ Љϩ Љ bЈϩ2k Re E E* dsϭ0. ͵ ͓͑␤2 ␤1͒ z ͑␤2 ␤1͒ z 0␧Љ ͑ 1 2 ͔͒ coordinate z and the integration is carried out over the a ͑4͒ section of the waveguide layer of thickness L. Of the complete set of modes, the strongest coupling is It follows from the relations obtained that only modes observed between the modes having the greatest phase syn- with orthogonal polarization (E ЌE ) are orthogonal, i.e., 1 2 chronism. There are generally two such modes, and energy is do not interact with one another in an absorbing scalar wave- transferred between them as they propagate in the waveguide. The orthogonality relation ͐b dsϭ0 holds for these z guide. We write the coupling equations for these modes modes. The orthogonality relation does not hold between dif-

ferent modes of the same polarization; therefore, a relation A␮Ј ϭϪi⌬␤␮A␮Ϫi␥␮␯A␯ exp͓i͑␤␮Ϫ␤␯͒z͔, stipulated by the absorption of the waveguide should exist * for them. A␯ЈϭϪi⌬␤␯A␯Ϫi␥␮␯A␮ exp͓Ϫi͑␤␮Ϫ␤␯͒z͔, ͑8͒ ˆ where the coupling coefficient ␥␮␯ϭk0͐E␮*⌬␧E␯dx. The corrections to the propagation constants of the TE EQUATIONS OF COUPLED MODES and TM modes are defined in the following manner For a further analysis of the influence of absorption on Eϭk E dx, mode coupling, a waveguide without absorption having a ⌬␤␯ 0 ͵ ⌬␧yy͉ ␯y͉ dielectric constant ␧ϭ␧Ј should be taken as the unperturbed structure, and the imaginary part of the dielectric constant of M ⌬␤ ϭk0 ͓⌬␧xx͉E␯x͉ϩ⌬␧zz͉E␯z͉ the waveguide layer should be taken as the perturbation ␯ ͵ ⌬␧(␣)ϭϪi␧Љ, where ␧ЉϷ␣ͱ␧Ј/k0 and ␣ is the absorption ϩE* ⌬␧ E ϩE*⌬␧ E ͔dx. ͑9͒ of the material. If the dichroism of TE and TM waves is ␯x xz ␯z ␯z zx ␯x taken into account, the perturbed part of the dielectric con- The solution of Eqs. ͑8͒ with the boundary conditions stant should be regarded as a diagonal tensor with noniden- A␮ϭA␮(0) and A␯ϭ0, which were taken at zϭ0, have the tical diagonal components, and the absorption will be differ- following form: ent for modes with different polarization. In a i⌬␮␯ magnetogyrotropic waveguide the perturbation of the dielec- A ͑z͒ϭA ͑0͒ cos ␹ zϪ sin ␹ z ␮ ␮ ␮␯ ␹ ␮␯ tric constant is also determined by the orientation of the ͩ ␮␯ ͪ 11 magnetization. The total perturbation of the dielectric con- ϫexp͓i͑⌬␮␯Ϫ⌬␤␮͒z͔, stant in this case can be represented in the following manner: ␥␮*␯ ⌬␧ˆ͑␣,M͒ϭ⌬␧ˆ͑␣͒ϩ⌬␧ˆ͑M͒. ͑5͒ A␯͑z͒ϭϪiA␮͑0͒ sin ␹␮␯z exp͓Ϫi͑⌬␮␯ϩ⌬␤␯͒z͔, ␹␮␯ Assuming that there are no radiative modes, we expand 2⌬ ϭ␤ ϩ⌬␤ Ϫ␤ Ϫ⌬␤ , ␹2 ϭ͉␥ ͉2ϩ⌬2 . the components of the electric field of the perturbed wave- ␮␯ ␮ ␮ ␯ ␯ ␮␯ ␮␯ ␮␯ ͑10͒ guide in the complete set of modes of the unperturbed waveguide: With consideration of ͑10͒ the mode conversion effi- 2 ciency ␩␮␯ϭ͉A␯(z)/A␮(0)͉ takes the form E ϭ A z͒exp Ϫi␤ z͒ϮB z͒exp i␤ z͒ x͒. 2 j ͚ ͓ ␯͑ ͑ ␯ ␯͑ ͑ ␯ ͔E j͑ ␥␮␯ ␯ ␩ ϭ sin ␹ z exp͓Ϫ͑␤Љ ϩ␤Љ͒z͔. ͑11͒ ␮␯ ͯ␹ ␮␯ ͯ ␮ ␯ ͑6͒ ␮␯ Here the upper signs in the square brackets refer to the transverse field components ( jϭx,y); the lower signs refer to the DAMPING CONSTANT AND COUPLING OF IDENTICALLY POLARIZED MODES longitudinal components ( jϭz); A␯(z) and B␯(z) are the amplitudes of the forward and backward eigenmodes, which The damping constant specifying the damping of a mode propagate in the ϩz and Ϫz directions and vary along the in a waveguide is found as the imaginary part of the pertur- waveguide as a result of mode coupling and perturbation of bation of the propagation constant, i.e., as ␤␯ЉϭϪIm ⌬␤␯ . their propagation constants; and the E j(x) are profile func- Figure 1 presents the dependence of the damping constant ␤␯Љ tions, which define the distribution of the field across the for the TE ͑solid curve͒ and TM ͑dashed curve͒ modes of the thickness of the waveguide. The coupling of counterpropa- first three orders ␯ϭ0,1,2 on the waveguide thickness L. The gating modes is significant in waveguides with a periodically following parameters are used here and below in the calcu- 12 varying dielectric constant. In the case considered here we lations: ␧Јϭ4.5371, ␧2ϭ3.8, ␧1ϭ1; wavelength of the ra- shall confine ourselves to an analysis of the interaction of diation in free space, ␭ϭ1.15 ␮m; damping parameter, ␣ only the forward modes. ϭ1cmϪ1. The symbols for the zeroth- and second-order The representation of a field in a waveguide in the form modes in the figure show the solutions obtained numerically ͑6͒ permits derivation of the following equation for the mode from the dispersion expressions found by matching the fields amplitudes: on the interfaces of the media. Both methods give coinciding Tech. Phys. 43 (6), June 1998 A. M. Shuty and D. I. Sementsov 703

thickness L␯ϭL␯0 and that this dependence asymptotically approaches the straight line ␤␯Љ(␣)ϭ␣/2 as the deviation from that thickness increases. The coupling coefﬁcients of the identically polarized modes are deﬁned by the expressions

EEϭk * dx, ␥␮␯ 0 ͵ ⌬␧yyE␮yE␯y

MMϭk * ϩ ␥␮␯ 0͵͓E␮x͑⌬␧xxE␯x ⌬␧xzE␯z͒

ϩE␮*z͑⌬␧zzE␯zϩ⌬␧zxE␯x͔͒dx. ͑13͒ It follows from ͑5͒ that the perturbation of the dielectric constant contains a component which depends on the absorp-

FIG. 1. Dependence of the damping constant ␤␯Љ on the waveguide thickness tion ␣ and leads to the coupling of identically polarized L. modes even in a scalar waveguide, i.e., in the absence of magnetization. This is confirmed by the orthogonality relations obtained above ͑4͒. When ␣ϳ1–10cmϪ1, ⌬␧͑␣͒ is of values for ␤Љ(L), attesting to the good approximation ␯ the same order as the perturbation which is quadratic with achieved using the coupled-mode formalism. It is seen from respect to the magnetization in the diagonal terms of the the figure that in the case of equality between the imaginary dielectric constant in the iron garnet film. Therefore, the con- parts of the diagonal components of the dielectric tensor, the tribution of the absorption introduced to the coupling of damping of the TM modes differs significantly from the identically polarized modes is comparable to the coupling damping of the TE modes only near the cutoff thickness, caused by the magnetization,11 but the large phase mismatch where the following relation holds: of modes of different orders greatly weakens the mode inter- E E E 2 E ͑␤␯ ͒Љ C␯ ␧Јk0q␯ v␯ action. The mode coupling coefficient appearing as a result M Ϸ M M E M , ͑12͒ Ϫ2 Ϫ1 ͑␤ ͒Љ ͩ C ␤ h ͪ v of absorption is of the order of ͉␥␮␯͉ϳ10 cm for ␣ ␯ ␮ ␯ ␯ ␯ ϭ1cmϪ1, while the phase mismatch ⌬Ј ϳ102 cmϪ1. For a E,M E 2 ␮␯ where the C␯ are normalized mode coefficients, q␯ ϭ(␤␯ mismatch ⌬Ј ӷ͉␥ ͉ the mode conversion efficiency ␩ 2 1/2 M E E,M 2 2 1/2 ␮␯ ␮␯ ␮␯ Ϫk0␧1) , q␯ ϭq␯ ␧Ј/␧1 , h␯ ϭ(d0␧ЈϪ␤␯) , and v␯ р ␥ /⌬Ј Ӷ1. Therefore, in an absorbing waveguide, as in 2 ͉ ␮␯͉ ␮␯ ϵLϩ2 sin (h␯L)/q␯Ϫsin(2h␯L)/2h␯ . a transparent waveguide, identically polarized modes with Figure 2 presents the dependence of the damping con- different propagation constants (␤␮Þ␤␯) can be considered stant ␤␯Љ of the zeroth ͑␯ϭ0, curves 1 and 2͒ and first ͑␯ noninteracting modes, i.e., eigenmodes, of the waveguide in ϭ1, curves 3 and 4͒ TE modes on the absorption ␣ of the the absence of other ͑for example, periodic͒ perturbations of waveguide-layer material for two waveguide thicknesses: L the dielectric constant. ϭ1.5 ␮m ͑curves 1 and 3͒ and Lϭ1 ␮m ͑curves 2 and 4͒. An analysis of these curves reveals that the linear depen- COUPLING OF DIFFERENTLY POLARIZED MODES dence of ␤␯Љ(␣) has its smallest slope at the mode cutoff In an absorbing waveguide, as in a transparent waveguide, orthogonally polarized modes are coupled only when the dielectric tensor has off-diagonal components. In a magnetogyrotropic waveguide the TE␮ and TM␯ modes are coupled with the coupling coefficient

ϭk * ϩ dx. 14 ␥␮␯ 0 ͵ E␮y͑⌬␧yzE␯z ⌬␧yxE␯x͒ ͑ ͒ To find the profile functions appearing in ͑14͒,we should take into account the perturbation of the dielectric constant of the waveguide layer and the propagation constants of the modes, which lead to a dependence of the coupling coefficient ␥␮␯ on the absorption. Figure 3 presents the dependence of the coupling coefficient which provides for TE␮ TM␯ conversion in the first three orders (␮ϭ␯ ϭ0,1,2)→ on the thickness of the waveguide layer for orientation of the magnetization along the x axis and ␣ ϭ20 cmϪ1 ͑the dashed curves correspond to a transparent waveguide, ␣ϭ0͒. It is seen that the difference between the values of ͉␥ ͉ for absorbing and transparent films increases FIG. 2. Dependence of the damping constant ␤␯Љ on the absorption ␣ for two ␮␯ different waveguide thicknesses. as the deviation from the cutoff thickness increases, i.e., as 704 Tech. Phys. 43 (6), June 1998 A. M. Shuty and D. I. Sementsov

coupled waveguides with strongly different absorption properties.13 An analysis of ͑15͒ shows that the conversion efﬁciency reaches its maximal values at the waveguide lengths zϷ␲(1/2ϩn)/(␶ cos ␺)(nϭ0,1,2, ...) and its minimal values at the lengths zϷ␲n/(␶ cos ␺), where, in contrast to the situation in a transparent waveguide, for un- synchronized modes the intensity of the ␯th mode I␯ 2 ϭ͉A␯(z)͉ is nonzero: 2 ␥␮␯ I͑min͒ϭ A ͑0͒ sinh2͑␲n tan ␺͒ ␯ ͯ ␮ ␹ ͯ ␮␯

ϫexp͓Ϫ͑␤␮Љ ϩ␤␯Љ͒␲n/␶ cos ␺͔. ͑16͒ Hence it follows that in an absorbing waveguide the polarization of the total ﬁeld of two coupled modes for ␺Þ0 and any waveguide length differs from the polarization of the ﬁeld of the input mode. In the case of complete phase syn-

chronism (⌬␮Ј ␯ϭ0) the expressions for the intensities of the 2 FIG. 3. Dependence of the coupling coefficient for TE TM conversion input mode I␮ϭ͉A␮(z)͉ and the intensity of the excited ␮→ ␯ on the thickness of the waveguide layer. mode I␯ take the form ⌬Љ 2 2 ␮␯ I␮ϭ͉A␮͑0͉͒ cos ␹␮␯zϩ sin ␹␮␯z the film thickness increases. This is due to the fact that, as ͩ ␹␮␯ ͪ the deviation from the cutoff thickness increases, the local- ϫexp͓Ϫ͑␤Љ ϩ␤Љ͒z͔, ization of each mode in the film increases with resultant en- ␮ ␯ 2 hancement of the influence of absorption on the coupling ͉␥␮␯͉ I ϭ͉A ͑0͉͒2 sin ␹ z exp͓Ϫ͑␤Љ ϩ␤Љ͒z͔, coefficient, which is determined by the overlap of the profile ␯ ␮ ͩ ␹ ␮␯ ͪ ␮ ␯ ␮␯ functions in the layer with the perturbed dielectric constant. ͑17͒ Deflection of the magnetic moment away from a normal to- 2 1/2 ward the Faraday orientation leads to a significant ͑by more where ␹␮␯ϭ(͉␥␮␯͉ Ϫ⌬␮Љ ␯) . The energy of the ␯th mode in a waveguide of length z than two orders of magnitude͒ increase in ͉␥␮␯͉, while the ϭ␲n/␹␮␯ is equal to zero, while the energy of the ␮th mode difference ͉␥␮␯(␣)͉Ϫ͉␥␮␯(0)͉ varies only slightly; therefore, it can be stated that the influence of absorption on the remains nonzero over the entire course of the waveguide. If the condition ͉⌬␮Љ ␯͉Ӷ͉␥␮␯͉ holds at the minima at the lengths coupling of modes, i.e., on (͉␥␮␯(␣)͉Ϫ͉␥␮␯(0)͉)/͉␥␮␯(0)͉, decreases. Here and in the following we shall use a magne- zϭ␲(1/2ϩn)/␹␮␯ (nϭ0,1,2, ...),theintensity of the input togyrotropic waveguide with the ͓111͔ crystallographic axis mode takes the values normal to the surface of the film for the calculations; its ⌬Љ 1 ͑min͒ 2 ␮␯ linear and quadratic magnetooptical parameters have the fol- I␮ ϭ͉A␮͑0͉͒ exp Ϫ␲ ϩn ͑␤␮Љ ϩ␤␯Љ͒/␹␮␯ . Ϫ4 Ϫ4 ͩ ␹ ͪ ͫ ͩ 2 ͪ ͬ ␮␯ lowing values: f ϭ3.07ϫ10 , ⌬gϭϪ0.73ϫ10 , g11 Ϫ4 Ϫ4 ͑18͒ ϭ5.07ϫ10 , and g44ϭ2.4ϫ10 . We bring the expression for the mode conversion effi- To analyze mode coupling it is useful to introduce the 2 ciency ͑11͒ into the form quantity r␮␯(z)ϭ͉A␯(z)/A␮(z)͉ , which characterizes the 2 contribution of each mode to the intensity of the total field in ␥␮␯ ␩ ϭ ͓sin2͑␶z cos ␺͒ϩsinh2͑␶z sin ␺͔͒ the waveguide. The form of the function r␮␯(z) is largely ␮␯ ͯ ␹ ͯ ␮␯ determined by the difference ⌬␮Љ ␯ between the damping parameters of the modes. In real waveguide structures the ab- ϫexp͓Ϫ͑␤Љ ϩ␤Љ͒z͔, ␮ ␯ sorption of modes of different polarization can differ by sev- 1/2 9 ␶ϭ͑␹␮␯␹␮*␯͒ ,2⌬␮Ј␯ϭ␤␮Ϫ␤␯ϩRe͑⌬␤␮Ϫ⌬␤␯͒, eral fold even far from the cutoff thickness. The parameter ͉⌬␮Љ ␯͉ increases significantly when waveguides with a clad- 2⌬␮Љ ␯ϭ␤␯ЉϪ␤␮Љ , ding of a conducting material or waveguides obtained by 3,14 1 proton exchange are used. ␺ϭ arctan͓2⌬Ј ⌬Љ ͑⌬Ј2 ϩ͉␥ ͉2Ϫ⌬Љ2 ͒Ϫ1͔. ͑15͒ Figure 4 presents plots of r (z) for TE TM conver- 2 ␮␯ ␮␯ ␮␯ ␮␯ ␮␯ ␮␯ 0→ 0 sion and various values of ⌬␮Љ ␯ and the waveguide thick- While the conversion efficiency for a transparent wave- nesses Lϭ3.6 ͑a͒, 3.7 ͑b͒, and 2.0 ␮m ͑c͒. The difference guide is a periodic function, the conversion efficiency for a between the mode damping parameters takes the values Ϫ1 waveguide with absorption is a decreasing oscillating func- ⌬␮Љ ␯ϭϪ0.25, Ϫ0.125, 0, 0.125, and 0.25 cm ͑Fig. 4a, Ϫ1 tion. When the condition ͉⌬␮Љ ␯͉у͉␥␮␯͉ holds for ⌬␮Ј ␯Ϸ0, the curves 1–5͒, ⌬␮Љ ␯ϭϪ0.25 cm ͑Fig. 4b͒, and ⌬␮Љ ␯ conversion efficiency ceases to be an oscillating function. ϭϪ0.25, Ϫ0.5, Ϫ0.75, and Ϫ1cmϪ1͑Fig. 4c, curves 1–4͒. This effect is similar to the effect considered in a system of The growth anisotropy of the iron garnet film was taken into Tech. Phys. 43 (6), June 1998 A. M. Shuty and D. I. Sementsov 705

r␮␯(z) curve with increasing maxima was obtained as a result of an investigation of TE0 TM0 conversion for ⌬␮Љ ␯ Ϸ0.29 cmϪ1. However, our analysis→ ͑Fig. 4a͒ shows that the maxima of r␮␯(z) should obey a dependence which decreases with the waveguide length in this case. When there is complete phase synchronism between the modes ͑Fig. 4b͒ and the condition ͉⌬␮Љ ␯͉Ͻ͉␥␮␯͉ holds, the intensity of the ␮th input mode reaches a minimum, and the function r␮␯(z) takes very large values at certain waveguide lengths. The polarization of the total ﬁeld approximates the polarization of the ␯th mode in this case. The strong mode mismatch at the waveguide thickness Lϭ2 ␮m ͑Fig. 4c͒ makes the maxima more frequent and dramatically lowers their height.

Also, if ␤␮Љ Ͼ␤␯Љ , as the difference between the damping parameters of the input and excited modes ͉⌬␮Љ ␯͉ increases, the contribution of the excited mode to the total intensity increases, rising with the waveguide length z.

NONOSCILLATING INTERMODE INTERACTION REGIME

In the case of phase synchronism (⌬␮Ј ␯ϭ0) and a fairly large difference between the damping parameters of the

coupled modes (͉␤␮Љ Ϫ␤␯Љ͉у2͉␥␮␯͉), the trigonometric functions in ͑16͒ transform into hyperbolic functions. As a result, the periodic energy exchange between modes propagating in a waveguide that takes place in a transparent waveguide and in the case of a small difference between the mode damping parameters is not observed. In this case the intensity of the 2 ␯th excited mode I␯ϭ͉A␯(z)͉ has one maximum at the waveguide length

1 ␤␮Љ ϩ␤␯Љϩ2␴␮␯ z1ϭ ln , ͑19͒ 2␴␮␯ ͩ ␤Љ ϩ␤ЉϪ2␴␮␯ ͪ ␮ ␯

and the intensity of the ␮th input mode I␮ has one minimum and one maximum at the waveguide lengths

Ϫ 1 ␦␮␯ z2ϭ ln ϩ , 2␴␮␯ ͩ ␦ ͪ ␮␯

Ϫ 1 ͑␤␮Љ ϩ␤␯Љϩ2␴␮␯͒␦␮␯ FIG. 4. Plot of r␮␯(z) for TE0 TM0 conversion. → z3ϭ ln ϩ , ͑20͒ 2␴␮␯ ͩ ͑␤Љ ϩ␤ЉϪ2␴␮␯͒␦ ͪ ␮ ␯ ␮␯

account in constructing the curves in Fig. 4b by adding 1.2 respectively, where Ϫ3 11 ϫ10 to the diagonal component ⌬␧xx . As a result of 2 2 1/2 Ϯ ␴␮␯ϭ͑⌬Љ␮␯Ϫ͉␥␮␯͉ ͒ and ␦␮␯ϭ⌬␮Љ ␯Ϯ␴␮␯ . this anisotropy, the TE0 and TM0 modes are completely synchronized, i.e., ⌬00Ј ϭ0, when LϷ3.7 ␮m. It follows from It is significant that the interaction of modes results in the plots presented that if the absorption of the ␮th input changes in their damping, which is determined for the por- mode is greater than the absorption of the ␯th excited mode, tion of the waveguide at zϾz3 by the quantity the maxima and minima of r␮␯ increase with the waveguide ¯ length. This means that the contribution of the ␯th mode to ␣␮,␯ϭ2͑␤␮Љ ϩ␤␯ЉϮ␴␮␯͒. ͑21͒ the field propagating in the waveguide increases with the ¯ distance z. When there is no dichroism, the height of the This permits regulation of the mode damping ␣␮,␯ in a maxima of r␮␯ does not vary, but the minima take a zero certain range by varying the coupling coefficient. In addition, value, at which the field takes on the original polarization. If the total intensity I␮ϩI␯ at the waveguide exit can be regu- the absorption of the input mode is weaker than the absorp- lated by varying the coupling coefficient, although this could tion of the excited mode, the height of the maxima decreases not be done when identically absorbed modes are coupled. with only a slight increase in the minima of r␮␯ .InRef.9an The results presented are also valid for a small phase mis- 706 Tech. Phys. 43 (6), June 1998 A. M. Shuty and D. I. Sementsov match 2⌬␮Ј ␯Ͻ͉␥␮␯͉. Under the conditions ⌬␮Ј ␯ϭ0 and ⌬␮Љ ␯ϭ͉␥␮␯͉ the expressions for the intensities of the modes take the form

2 I␮ϭI0͑z⌬␮Љ ␯ϩ1͒ exp͓Ϫ͑␤␮Љ ϩ␤␯Љ͒z͔,

2 I␯ϭI0͑z⌬␮Љ ␯͒ exp͓Ϫ͑␤␮Љ ϩ␤␯Љ͒z͔, ͑22͒ 2 where I0ϭ͉A␮(0)͉ , and the intensity maxima of the ␯th and ␮th modes and the minimum of the ␮th mode, respectively, are realized at the lengths

2 2 z1ϭ2/͑␤␮Љ ϩ␤␯Љ͒, z3ϭ2␤␮Љ /͑␤␮Љ Ϫ␤␯Љ ͒,

z2ϭ1/͑␤␮Љ Ϫ␤␯Љ͒. If the ␮th input mode damps more slowly than the excited mode, its intensity does not have extremum values and decays monotonically. As follows from an analysis of the relations presented, in the absence of intermode coupling the possibilities of regu- lating the energy damping in a waveguide are increasingly better the more strongly the absorption coefficients of the modes differ. To obtain a quantitative estimate of the effect described, let us consider conversion of the TM1 TE1 type in a magnetogyrotropic waveguide consisting of an↔ iron garnet film on a gadolinium–gallium garnet substrate with a cladding of a conducting material. The conducting layer mainly influences the absorption of the modes by strongly Ϫ1 enhancing the dichroism ͑␤ЉMϭϪ4.5 cm and ␤EЉ ϭϪ0.5 cmϪ1͒ and has practically no influence on other parameters of the modes.14 The film thickness Lϭ6.8 ␮m, at which the TE1 and TM1 modes are synchronized (⌬␮Ј ␯Ϸ0) in the waveguide considered, was used for the calculations. Figure 5 presents the dependence of the relative ͑normalized to I0 , i.e., the intensity of the input mode at zϭ0͒ intensity of the input mode J␮(z)ϭI␮(z)/I0 , ͑curves 1͒, the relative intensity of the excited mode J␯(z) ͑curves 2͒, the total rela- FIG. 5. Dependence on z of the relative intensities J␮ ͑1͒, J␯ ͑2͒, J␮ϩJ␯ ͑3͒, tive intensity J␮(z)ϩJ␯(z) ͑curves 3͒, and the relative inten- and J␮ϭexp(2␤␮Љz) for ␥␮␯ϭ0 ͑4͒. sity of the input mode in the absence of coupling J␮(z) ϭexp(Ϫ2␤␮Љz) ͑␥␮␯ϭ0, curves 4͒ on the waveguide length z for TM1 TE1 ͑a͒ and TE1 TM1 ͑b͒ conversion. The mag- tively varied by altering the coupling coefficient as a result netization→ lies in the plane of→ the film at the angle ␸ϭ59° to of rotation of the magnetic moment of the film. Figure 6 the z axis, at which the mode-synchronism condition (⌬␮Ј ␯ shows the dependence of the total relative intensity of the Ϸ0) and the equality ⌬␮Љ ␯ϭ͉␥␮␯͉ hold. It follows from the modes on the waveguide length for TM1 TE1 conversion dependences presented in Fig. 5a that the intensity of a and various orientations of the magnetic moment→ in the plane strongly absorbed input mode interacting with a less strongly of the film: ␸ϭ0, 30, 50, 59, 80, 85, and 90° for curves 1–7, absorbed mode passes through the minimum value J␮(z2) respectively. Under conditions close to phase synchronism of and then damps more slowly than the intensity of the ␮th the coupled modes, the most effective control of the energy eigenmode. If the input mode is absorbed more slowly than transferred in the waveguide is possible, if the coupling co- the excited mode ͑Fig. 5b͒, mode coupling accelerates its efficient ͉␥␮␯͉ is smaller than ͉⌬␮Љ ␯͉ ͑curves 4–7͒ or exceeds damping. The total intensity is greater in the former case and it only slightly ͑curve 3͒. When the coupling coefficient is less in the latter case than the intensity of the noninteracting increased further ͑curves 1 and 2͒, the possibilities of con- mode. trolling the waveguide energy by varying ͉␥␮␯͉ are greatly When the magnetization deviates from the angle ␸ reduced, and in the limiting case of ͉␥␮␯͉ӷ͉⌬␮Љ ␯͉ they vanish ϭ59°, the phase synchronism is violated to a slight extent completely, since the energy of two synchronized modes Ϫ1 Ϫ1 (⌬␮Ј ␯ϭϪ2.13 cm for ␸ϭ0 and ⌬␮Ј ␯ϭ0.8 cm for ␸ damps according to an exp͓Ϫ(␤␮Љϩ␤␯Љ)z͔ law, being deter- ϭ90°͒, and the character of the mode interaction in this case mined only by the imaginary parts of the propagation param- is determined mainly by the relation of the coupling coeffi- eters of the modes. For large coupling coefficients (͉␥␮␯͉ cient to the difference between the damping parameters ⌬␮Љ ␯ . ӷ͉⌬␮Љ ␯͉) variation of the damping of the total energy is pos- The total energy propagated in a waveguide can be effec- sible only by significantly increasing the phase mismatch Tech. Phys. 43 (6), June 1998 A. M. Shuty and D. I. Sementsov 707

strongly different damping parameters, the oscillatory energy exchange between the modes vanishes and the damping of the modes themselves becomes dependent on the coupling coefﬁcient under conditions close to phase synchronism. Ef- fective control of the damping of the ﬁeld in the waveguide becomes possible under such conditions. The effect described can be utilized to create radiation amplitude modulators based on absorbing planar waveguides.

1 Integrated Optics, T. Tamir ͑Ed.͓͒Springer-Verlag, Berlin ͑1975͒; Mir, Moscow ͑1978͒, 575 pp.͔. 2 A. M. Prokhorov, G. A. Smolenski, and A. N. Ageev, Usp. Fiz. Nauk 143,33͑1984͓͒Sov. Phys. Usp. 27, 339 ͑1984͔͒. 3 D. I. Sementsov, Zh. Tekh. Fiz. 56, 2157 ͑1986͓͒Sov. Phys. Tech. Phys. 31, 1294 ͑1986͔͒. 4 A. K. Zvezdin and V. A. Kotov, Magnetooptics of Thin Films ͓in Rus- sian͔, Nauka, Moscow ͑1988͒, 192 pp. 5 P. V. Adamson, Opt. Spektrosk. 66, 1172 ͑1989͓͒Opt. Spectrosc. ͑USSR͒ 66, 684 ͑1989͔͒. 6 A. Reisinger, Appl. Opt. 12, 1015 ͑1973͒. FIG. 6. Dependence of the total relative intensity of the coupled modes for 7 E´ . P. Stinser, A. N. Ageev, and S. A. Mironov, Pis’ma Zh. Tekh. Fiz. 3, TM1ϪTE1 conversion on the waveguide length. 913 ͑1977͓͒Sov. Tech. Phys. Lett. 3, 373 ͑1977͔͒. 8 H.-G. Unger, Planar Optical Waveguides and Fibres, Clarendon Press, Oxford–New York ͑1978͒; Mir, Moscow ͑1980͒, 656 pp. 9 ⌬Ј , the mode coupling is then destroyed, and the energy G. Hepner, J. P. Castera, and B. Desormiere, Physica D ͑Amsterdam͒ 89, ␮␯ 264 ͑1977͒. transferred by the waveguide, consisting practically com- 10 D. I. Sementsov and A. M. Shuty, Opt. Spektrosk. 84͑2͒, 280 ͑1998͒ pletely of the energy of the ␮th input mode, damps according ͓Opt. Spectrosc. 84, 238 ͑1998͔͒. 11 D. I. Sementsov, A. M. Shuty , and O. V. Ivanov, Radiotekh. Elektron. to an exp͓Ϫ2␤␮Љz͔ law. The analysis performed here allows us to conclude that 41͑3͒,1͑1996͒. 12 D. I. Semenotsov, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 2,94͑1993͒. absorption in a magnetogyrotropic waveguide inﬂuences the 13 P. V. Adamson, Opt. Spektrosk. 69, 453 ͑1990͓͒Opt. Spectrosc. ͑USSR͒ interaction of both identically and orthogonally polarized 69, 269 ͑1990͔͒. 14 modes, making an additional contribution to the intermode M. Adams, An Introduction to Optical Waveguides ͓Wiley, New York 1981 ; Mir, Moscow 1984 , 512 pp. . coupling and altering its character when certain conditions ͑ ͒ ͑ ͒ ͔ are satisﬁed. For example, for interacting modes with Translated by P. Shelnitz TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Temperature stability and radiation resistance of holographic gratings on photopolymer materials T. N. Smirnova, O. V. Sakhno, I. A. Strelets, and E. A. Tikhonov

Institute of Physics, Academy of Sciences of Ukraine, 252650 Kiev, Ukraine ͑Submitted January 8, 1997͒ Zh. Tekh. Fiz. 68, 105–110 ͑June 1998͒ Data are presented from studies of the temperature dependence of the diffraction efficiency and radiation resistance of volume phase hologram/transmission gratings. The reversible diffraction efficiencies are described by the phase equilibrium diagram for the polymer–diffusate system. The radiation resistance of these hologram/gratings is determined by the thresholds for photothermolytic decay of the diffusate, bromonaphthalene, that was used. Composites containing diffusates with high thresholds for photolysis and thermolysis are studied. As a result, modified versions of the photopolymer recording medium with radiation resistances exceeding 200 MW/cm2 are proposed. © 1998 American Institute of Physics. ͓S1063-7842͑98͒01906-0͔

INTRODUCTION ture, they may cause irreversible changes and destroy a hologram. Researchers encounter a similar situation when Photosensitive polymer composites ͑PPCs͒ for holo- gratings interact with high-power laser beams. During inter- graphic recording are constantly being developed and actions with high-power laser beams, a third stage may de- improved.1–6 The advances of recent years have led to new velop between these two, owing to the nonlinear optical re- possibilities for photopolymers, such as the recording of sponse of the grating material ͑a cubic nonlinearity with negative holograms in real time4,5 and a thermal technique respect to the field in an isotropic medium͒. This stage will for developing latent holographic images.6 The known ad- be the subject of later studies; the results of a study of the vantages of PPCs compared to bichromated gelatin include first and second stages are presented in this paper. the ability to obtain holograms without having to go through One of the important practical achievements of this work the stages of chemical development of a latent image and has been the creation of a modification of the basic compos- extensive possibilities for optimizing the composites and ite which has a higher radiation resistance than bichromated achieving better reproducibility of the basic parameters. The gelatin. phase character of the recorded information over a wide range of thickness and area of the recording medium makes STRUCTURE AND PROPERTIES OF PHOTOSENSITIVE these materials attractive for the fabrication and development POLYMER COMPOSITE HOLOGRAPHIC MEDIA of various kinds of holographic optical elements. We have for sometime been using the basic photopoly- Holographic media based on FPK-488 are made in the mer holographic recording composite FPK-488 and its modi- form of a triplex: substrate–PPC–substrate, with a gap rang- fications for fabricating various types of holographic optical ing from a few microns to 1 mm, depending on the angular elements. These include transmission and autocollimating selectivity specifications for the fabricated holographic opti- diffraction gratings,7 diffraction optical filters,8 and selec- cal element. The gap is typically formed by spacers which do tively reflecting diffraction gratings.9 not inhibit shrinkage of the polymer layer. This kind of me- Applications of optical devices based on polymer holo- dium is prepared before recording and, for sizes up to 10 cm, graphic optical components in instruments or systems inevi- presents no technical difficulty. PPCs of this class incorpo- tably rely on the stability of the main operating parameters. rate three interacting subsystems: a monomer–oligomer mix- The primary factor which affects photopolymer holographic ture which is capable of polymerizing; a mixture of two or optical elements is the temperature. Humidity or mechanical more components which form an effective initiator for a effects are not as important, since this composite does not radical polymerization process; and, a chemical component swell in water and the optical element is a rigid structure which plays a fundamental role in the diffusion-induced ir- consisting of a pair of substrates with the holograms between reversible spatial modulation of the refractive index—a them. The temperature, on the other hand, is an unavoidable chemically neutral diffusate ͑CND͒. parameter, since local temperature rises may occur because Some characteristics of the photosensitive polymer com- of dissipative losses during channeling of high power laser posites are listed in Table I. FPK-488M differs from the beams. basic FPK-488 composite5 in the monomer–oligomer com- In the first stage, the effect of general and local heating ponent, and FPK-488N differs in the diffusate. of holographic optical components on their spectral and an- The basis of the mechanism for holographic recording in gular characteristics is reversible. With increasing tempera- these composites is the radical photopolymerization of the

TABLE I.

Sensitivity,* Resolution, Amplitude of the modulaion Type of PPC mJ/cm2 mmϪ1 of the refractive index, ␦n**

FPK-488 300 Ͼ6000 0.012–0.015 FPK-488M 60 Ͼ6000 0.025 FPK-488N 150 Ͼ6000 0.012–0.014

*The photosensitivity of the recording medium is defined as the exposure required to attain a maximum diffraction efficiency for write beam intensities of 0.2 mJ/cm2 at a writing wavelength of ␭ϭ0.488 ␮m. **The amplitude of the modulation of the refractive index was calculated for grating readout by a ␭ϭ0.6328 ␮m beam with the electric field vector E parallel to the ‘‘rulings’’ ͑s polarization͒. monomer–oligomer mixture and interdiffusion of the monomer and the CND between the phase planes of the incipient FIG. 1. Temperature-induced variations in the amplitude of the modulation hologram which correspond to the maximum and minimum of the refractive index of a grating based on FPK-488 ͑1͒, and in the dif- light intensities in the interference light field.10,11 The result- fraction efficiency of gratings based on FPK-488M ͑2͒ and FPK-488 ͑3͒. ing spatial modulation in the concentration of the CND en- hances, by many times, and stabilizes the modulation of the refractive index of the polymer layer. is destroyed. For gratings based on FPK-488, destruction sets in at Tϭ218 °C; for the gratings recorded on FPK-488M, at TEMPERATURE STABILITY OF THE PARAMETERS OF FPK- 244 °C. 488 HOLOGRAPHIC GRATINGS A polymer layer without the neutral component re- We have studied the temperature dependences of the dif- mained undamaged up to the highest observation tempera- fraction efficiency ␩ and transmission T of holographic ture, Tϭ250 °C. The damage for all the gratings ͑with ␯ 0 Ϫ1 gratings recorded on layers of FPK-488 and FPK-488M. For ϭ2000 mm ͒ had a specific and fixed character: cracks this purpose we measured the variation in the diffraction ef- formed parallel to the direction of the phase planes of the ficiency over temperatures of Ϫ50 to ϩ250 °C. A transmis- grating, with a division ratio of roughly 20:1. The cracks sion diffraction grating was placed in a temperature con- occur periodically and thus convert the thick transmission trolled cabinet with a temperature T that was controlled over grating into a thin one ͑according to the Klein–Cook crite- the range ϩ20–ϩ250 °C. The diffraction efficiency was rion͒ and lead to the appearance of a multiwave diffraction measured over the interval TϭϪ100 to ϩ20 °C using a liq- pattern of the probe beam. Polymer layers of the same pho- uid nitrogen cryostat with a variable temperature. The varia- topolymer recording material with no hologram recorded on tion in the diffraction efficiency of the grating was tested them crack in a disordered fashion at the same temperatures. using a He–Ne laser beam (␭ϭ0.6328 ␮m) incident on the As a grating is cooled to around Ϫ5 °C, its diffraction sample grating at the Bragg angle. The powers of the inci- efficiency remains practically constant. At low temperatures (TϽ0 °C), a reversible reduction in the diffraction effi- dent (Pin), transmitted (Pout), and diffracted beams (Pdif) were monitored using FD-26 photodiodes whose outputs ciency is observed ͑Fig. 2͒. For gratings with FPK-488 with were fed to an automatic data recording and processing sys- an initial ␩ϭ98%, the minimum diffraction efficiency stabi- tem. lizes at ϳ0.81. When a grating is heated at an average rate of The diffraction efficiency was defined as the ratio ϳ0.5 deg/min, such that a quasiequilibrium temperature distribution can develop over the layer, the diffraction effi- Pdif /(PdifϩPout). Losses due to light scattering (␳ϭ1 ciency increases. The initial value is recovered at Tϭϩ8 °C. ϪT0) were taken into account by measuring the grating Hysteresis appears in the diffraction efficiency for gratings of transmission T0ϭ(PinϪPfrϪPout)/Pin , where Pfr represents the Fresnel losses measured when the probe beam was inci- FPK-488M on cooling and heating. Thus, when a grating ͑with an initial ␩ϭ0.95͒ is initially cooled to Ϫ60 °C and dent on the grating at the angle for which Pdifϭ0. These studies yielded the following data. Heating the then heated, its efficiency at first increases but then, without gratings to 100 °C causes a negligible change in the diffrac- reaching its original value, stabilizes at a level of ϳ0.8. tion efficiency. On going to TϾ100 °C, a reduction in the However, the initial diffraction efficiency is recovered when diffraction efficiency is observed, with an approach to a the grating is kept Tϭ20 °C for a long time ͑over an hour͒ steady value which remains invariant for fixed T throughout or heated to ϩ30– 40 °C. In addition, for gratings based on the observation time ͑up to 8 h͒. Cooling the grating to the the two modified PPCs, the light scattering increases as the initial ͑room͒ temperature is accompanied by a return of the temperature is reduced. diffraction efficiency to its original value. The diffraction efficiency of a transmission volume Figure 1 shows the diffraction efficiency ϭ f (T)of phase grating at the Bragg angle ␤ is given by the Kogelnik ␩ 12 transmission gratings for the FPK-488 and FPK-488M com- formula for a given temperature posites. A temperature-reversible recovery of the diffraction efficiency occurs up to temperatures T such that the grating ␩͑T͒ϭsin͓␲␦n͑T͒d͑T͒/␭ cos ␤͑T͔͒, ͑1͒ 710 Tech. Phys. 43 (6), June 1998 Smirnova et al.

FIG. 3. The refractive indices of FPK-488 ͑1͒ and of a polymer based on it ͑2͒ as functions of the concentration of the chemically neutral diffusate ͑CND͒. FIG. 2. Temperature-induced variations in the diffraction efﬁciency ͑1,2͒ and light scattering losses ͑3͒ for gratings based on FPK-488 ͑1,3͒ and FPK-488M ͑2͒. Let us see which properties of the recording material determine the magnitude and temperature behavior of ␦n(T). By deﬁnition, where ␦n is the amplitude of modulation of the refractive index and d is the grating thickness. ␦nϭ͉͑nmaxϪnmin͉͒/2, ͑2͒

It might be expected that ␩(T) is determined by tem- where nmax,min are the refractive indices of the polymer in the perature variations in the grating parameters which enter in phase planes formed at the antinodes and nodes of the inter- Eq. ͑1͒. ference field, where the concentration of the CND is differ- As our measurements showed, the diffraction efficiency ent. of the grating is essentially unchanged over temperatures of Our measurements ͑Fig. 3͒ show that the dependence of 1–100 °C. This appears to be related to the specific structural the refractive index of the polymer on the amount of CND features of the holographic optical element and its thermal within this range of concentrations c is quite well described mechanics: because of the adhesion of the polymer to the by a linear dependence of the form14 substrate, the change in the size of the polymer layer along the grating vector is determined principally by the thermal nϭn0ϩc͑dn/dc͒, ͑3͒ expansion ͑contraction͒ of the glass substrate. For T where dn/dc is the rate of increase of the refractive index ϭ200 °C and a coefficient of linear thermal expansion of the and n0 is the refractive index of the polymer without the Ϫ6 Ϫ1 glass of ␣cϭ8ϫ10 deg , the change in the grating pe- neutral component. riod is roughly 0.07% and the change in the diffraction effi- Substituting Eq. ͑3͒ in Eq. ͑2͒ yields ciency is 0.02%. For TϽ0, ␣ is smaller and the effect of c ␦n T͒ϭ dn T͒/dc c T͒, ͑4͒ this factor on the grating period is even more negligible. ͑ ͉ ͑ ͉ 1N͑ The changes in the grating thickness is determined by where c1N is the amplitude of the modulation of the concen- the coefficient of linear thermal expansion of the polymer, tration of the neutral component. ␣ p , and not of the substrate. The PPCs used here are new The absolute value of the rate of increase of the refrac- systems of polymer ͑more often, copolymer͒ and solvent tive index is known14 to increase in proportion to the differ- ͑CND͒. There are no published data on their temperature ence between the refractive indices of the polymer and sol- coefficients of expansion. The estimates given here are based vent. The magnitude of ⌬c depends on the thermodynamic on values of ␣ p for polymers with similar properties. Ac- affinity of the neutral component of the polymer. ͑We shall cording to data13 for polymers similar to those studied here, discuss this in more detail below.͒ Thus, ␦n(T) is deter- Ϫ4 Ϫ1 the maximum value of ␣ pϭ5ϫ10 deg . For this value mined by the thermodynamic properties of the polymer– of ␣ p , the grating thickness d increases by ϳ10% when T is CND system and by the difference in their refractive indices. raised to 200 °C. The rate of increase of the refractive index is constant over a The observed decrease of the diffraction efficiency wide range of temperatures for various polymer–solvent ͑rather than an increase with grating thickness͒ is evidence systems.13 The slight change in ␦n over 20–100 °C indicates that its temperature behavior is determined primarily by the that the change in ͉dn(T)/dc͉ is small for the PPCs studied dependence ␦n(T), which decreases by more than the thick- here. Based on this, we may assume that the character of the ness d increases. temperature dependence of ␦n(T) is determined mainly by The temperature variation ␦n(T) calculated from Eq. ͑1͒ that of ⌬c(T). on the basis of the measured ␩(T), with allowance for the The temperature dependence of ⌬cϭ f (T) is explained change in d for ␤ϭconst, is shown in Fig. 1 ͑curve 1͒. by a change in the phase equilibrium of the polymer–CND Tech. Phys. 43 (6), June 1998 Smirnova et al. 711

The situation is different when the gratings are cooled below 20 °C. The monotonic drop in the diffraction efficiency can be explained by a decrease in the thickness of the grating with temperature, since c1N changes little according to the phase diagram. In addition, the drop in the diffraction efficiency may be related to increased light scattering in the grating ͑Fig. 2, curve 3͒. This last effect may be caused by crystallization of the CND, whose freezing point is Ϫ6°C. The further drop in the diffraction efficiency as the temperature is brought below the freezing point of the CND and the FIG. 4. Phase diagrams of the states of a polymer-CND system for two above-described hysteresis may be related to a change in the models described in the text ͑c is the concentration.͒ diffusion times of the system owing to the high viscosity of the polymer phase and the peculiarities of the crystallization system. The phase equilibrium of such a system is described kinetics of the neutral component in the polymer matrix. by a state diagram. In the following, we shall consider two models which are capable of describing ⌬cϭ f (T) qualita- RADIATION RESISTANCE OF POLYMER HOLOGRAPHIC tively ͑Fig. 4a and 4b͒. We consider phase equilibrium GRATINGS curves with an upper critical temperature of displacement T* as the most common type of diagram for systems similar to The radiation resistance of the photosensitive polymer those studied here.13,15 composite gratings was studied by irradiating them with Our measurements imply that for FPK-488 and FPK- pulsed Nd3ϩ:YAG ͑␭ϭ1.064, 0.532 ␮m͒ and dye (␭ 488M, T* lies above the temperature at which the holograms ϭ0.630 ␮m) lasers. The transmission and diffraction effi- are destroyed. Prior to recording, these composites are mul- ciency of the gratings and their thresholds for N-pulse and ticomponent single phase solutions. Polymerization in the many-pulse damage were measured. The threshold for N- gradient interference field destroys the equilibrium state of pulse damage is defined as the intensity (IN) at which N the system and shifts it to point A ͑Fig. 4a͒. Here system 1 pulses on a given point of the sample will produce micro- breaks up into two phases. When equilibrium is reached, the scopic damage on its surface and in its volume. The multi- compositions of the phases correspond to the points C1 and pulse damage threshold is defined as the intensity I* which C2 . The phase planes of the gratings in this case differ in will cause damage for NϾ1000 pulses. having different amounts of these phases. The way the diffraction efficiency and transmission T0 Diagrams 2 and 3 of Fig. 4b allow for the difference in varied was similar for laser irradiation at different wave- solubility of the polymer formed at the antinodes and nodes lengths, intensities, and pulse repetition rates. As a rule, the of the interference field. Such a difference may be caused by diffraction efficiency and transmission of a grating are ob- a dependence of the degree of polymerization ͑density factor served to decrease in an irradiated region ͑the latter is mea- of the polymer network͒ and, therefore, of the solubility of sured at orientations off the Bragg angle͒ all the way up until the polymer, on the spatial distribution of the intensity of the destruction of the grating. Visually, a blackening of the grat- light field.16 ing is observed where later a microscopic hole is formed During the recording process, at the antinodes of the owing to evaporation of the polymer. Here the transmission field the system becomes two-phased, and microsyneresis of a polymer layer without a neutral component does not ͑squeezing out͒ of the excess neutral component from the change until it is destroyed at I*ϳ1 GW/cm2. polymer mass takes place. For the polymer formed at the Our results indicate that the damage threshold for holo- nodes of the field, the concentration of the neutral compo- graphic optical components made of these photosensitive nent remains in equilibrium. As a result, the phase planes of polymer compositions is determined primarily by the photo- the grating consist of polymer phases with different equilib- physical and thermal properties of the CND component of rium concentrations of the CND ͑C3 and C4 in Fig. 4b͒. the polymer. Here the processes leading to destruction of the For the system described by curve 1, an elevated tem- grating have a cumulative character. A reduction in the in- perature causes the thermodynamic affinity of the CND to tensity is observed as the frequency of the laser light ap- the polymer to increase and, accordingly, the compositions proaches the edge of the fundamental absorption band of the of the phases move closer together ͑points C1Ј and C2Ј͒. Then diffusate. ⌬c(T) and ␦n(T) decrease, which leads to a reduction in the Without pretending to an exhaustive description of the diffraction efficiency. In the second case ͑Fig. 4b͒, the equi- damage mechanism, we may assume that under intense irra- librium concentration of the neutral component in both poly- diation, photochemical processes develop in the polymer ow- mer phases increases, with an accompanying drop in the con- ing to multiphoton absorption. The consequence of these centration gradient of the CND and, therefore, a reduction in processes is the formation of photodissociation products of ␦n. In both cases, the change in ⌬c(T) is reversible, so that the component with the lowest photodissociation threshold. the diffraction efficiency recovers when the grating is cooled. The products of photodissociation that exhibit fundamental The choice of a specific model for the holographic re- absorption at the wavelength of the high-power laser are ex- cording process in these photosensitive polymer composite tremely important, since in this case there is positive feed- requires additional study of their thermodynamic properties. back in the chain of events formed by absorption–photoly- 712 Tech. Phys. 43 (6), June 1998 Smirnova et al.

TABLE II.

Diffraction efﬁciency, 2 No CND nd n pϪnd Copt , vol.% l*, MW/cm ␭ϭ0.6328 ␮m 1 Bromonaphthalene 1.66 Ϫ0.14 30 16.0 0.99 2 Quinolene 1.627 Ϫ0.107 35 12 0.85 3 Petachlor Diphenyl 1.636 Ϫ0.116 30 15 0.9 4 Triﬂuorethanol 1.29 0.23 35 Ͼ200 0.94 5 Acetonitryl 1.344 0.176 40 Ͼ200 0.04 6 Methanol 1.328 0.192 40 Ͼ200 0.03 7 Heptane 1.387 0.133 12 Ͼ200 0.98 8 Hexane 1.375 0.145 20 Ͼ200 0.98

Note: The multi-pulse damage threshold I* was measured using light from a Nd laser with the following parameters: ␭ϭ1.06 ␮m, pulse length 10–15 ns, pulse repetition rate 12.5 Hz; ‘‘Ͼ’’ means that no damage to the sample was observed during continuous operation for 6 h.

sis–absorption–heating–thermolysis, etc. An increase in the and 7, 8 have similar affinities to the polymer, and the affin- concentration of these products leads to local heating of the ity decreases in the series ͑5,6͒–͑1,4͒–͑7,8͒. polymer and its evaporation in the region of the interaction Diffusates 2 and 3 ensure highly efficient recording at with the light beam. Apparently in the case of bromonaph- roughly the same optimum concentrations as for the bro- thalene as the CND, bromine is photodissociated from the monaphthalene in the basic FPK-488. The multi-pulse dam- main naphthalene molecule and this then opens up the pos- age thresholds for them lie between 12 and 16 MW/cm2 and sibility of direct absorption of visible light by molecular bro- are roughly the same as for the basic FPK-488. In both cases, mine. The process proceeds with steady autoacceleration and damage to the grating was accompanied by the formation of thus is cumulative. Soot formation and the evaporation of the absorbing products, in agreement with earlier observations.17 polymer, as a final stage of damage, take place owing to On the other hand, the CNDs 4–8 greatly increased the ra- thermolysis during the rapid local heating of the polymer in diation resistance of the gratings, although the photo- the irradiated regions. sensitive polymer composites based on them differed in their In order to enhance the radiation resistance of holo- recording efficiencies. Thus, the maximum diffraction effi- graphic optical components based on these photopolymers, ciency for additives 5 and 6 was less than 30–40%. CND 4 we have studied replacing the CND contained in the basic yielded a diffraction efficiency of up to 96%, but noticeably composite FPK-488. A list of the diffusates that were studied reduced the photosensitivity of the PPC. This was a result of and some characteristics of the new FPK-488N are shown in a chemical interaction between this diffusate and the initia- Table II. tor. The preliminary choice of the compounds for use as the The best results for the magnitude of the diffraction ef- CND was based on the following considerations. Aromatic ficiency and the radiation resistance, simultaneously, were 1–3 and aliphatic 4–8 compounds were investigated. Since obtained using the normal paraffins 7 and 8 as CNDs. Be- the aromatic compounds undergo thermolysis more easily sides increasing the radiation resistance, going to the normal than the aliphatic,17,18 using them as a diffusate in a photo- paraffins made it possible to increase the photosensitivity of polymer recording medium lowers the radiation resistance of the material by a factor of 1.5–2 as a result of increasing the holographic optical components. In order to ensure highly rate of polymerization of the composite while the concentra- efficient recording, compounds were chosen for which the tion of the CND was lowered. This last result suggests that absolute value of the difference between the refractive indi- for materials such as FPK-488, reducing the thermodynamic ces of the polymer-forming part (n p) and the CND (nd) was affinity of the neutral component leads to an improvement in at least 0.1. such parameters of the materials as its photosensitivity. In addition, materials with a different solubility relative to the polymer were examined, and this made it possible to establish a relation between the holographic parameters and CONCLUSION the thermodynamic properties of the photosensitive polymer composite. The Huggens parameters, which characterize the Based on the results obtained here, we reach the follow- thermodynamic affinity of a CND to a polymer, were not ing conclusions: determined. However, the change in the affinity in a series of 1. The temperature variation in the diffraction efficiency additives can be estimated qualitatively from the value of the of gratings based on photopolymer materials such as FPK- optimum concentration copt of the CND which ensures maxi- 488 is characterized by the existence of two temperature mum recording efficiency. A reduction in this efficiency is ranges. Within the range ϩ100ϾTϾϪ10 °C, the grating evidence of a drop in the equilibrium concentration of the efficiency is essentially constant. The decrease in the diffrac- neutral component in the polymer and, accordingly, of a re- tion efficiency for TϽϪ10 °C and for TϾ100 °C up until duction in the thermodynamic affinity in the system. As the grating is destroyed is reversible. The change in the dif- Table II implies, the groups of compounds Nos. 1–4, 5, 6, fraction efficiency with temperature in the latter case is de- Tech. Phys. 43 (6), June 1998 Smirnova et al. 713 termined by temperature-induced changes in the phase equi- 5 E. A. Tikhonov, T. N. Smirnova, and E´ . S. Gyul’nazarov, Kvant. E´ lektron. librium conditions for the polymer–diffusate system. ͑Moscow͒ 40,1͑1991͓͒sic͔. 2. The radiation resistance of gratings using FPK-488 6 A. M. Weber, W. K. Smouthers, T. J. Truot, and D. J. Mickish, Proc. Soc. polymers is determined by the properties of the chemically Photo-Opt. Instrum. Eng. ͑SPIE͒ 1212, 30–39 ͑1990͒. 7 T. N. Smirnova, E. A. Tikhonov, and E´ . S. Gyul’nazarov, Abstacts of the neutral diffusate. By choosing an optimum diffusate from the VI Conference on Holography ͓in Russian͔, Vitebsk ͑1990͒,p.20. class of aliphatic compounds, we were able to raise the laser 8O. V. Sakhno, T. N. Smirnova, and E. A. Tikhonova, Zh. Tekh. Fiz. 2 damage threshold of the gratings to տ200 MW/cm , which 63͑12͒,70͑1993͓͒Tech. Phys. 38, 1071 ͑1993͔͒. makes it possible to use gratings based on FPK-488N for 9 T. N. Smirnova, T. A. Sarbaev, and E. A. Tikhonov, Kvant. Elektron. controlling the output from high-power lasers. ͑Moscow͒ 21, 373 ͑1994͒. 10 ´ 3. By choosing constituents of the photosensitive poly- E. S. Gyul’nazarov, T. N. Smirnova, and E. A. Tikhonova, Opt. Spek- trosk. 67, 175 ͑1989͓͒Opt. Spectrosc. ͑USSR͒ 67,99͑1989͔͒. mer composite with an optimum combination of physical 11 ´ E. S. Gyul’nazarov, V. V. Obukhovski, and T. N. Smirnova, Opt. Spek- and thermodynamic parameters, it is possible to minimize trosk. 69, 178 ͑1990͓͒sic͔. the concentration of the diffusate in the medium without 12 R. Kohler, K. Berhart, and L. Lin, Optical Holography, Moscow ͑1973͒, lowering the recording efficiency and, thereby, to raise the 686 pp. photosensitivity of these materials. 13 A. E. Nesterov, Handbook of the Physical Chemistry of Polymers. Prop- 4. The recording of holographic gratings can serve as a erties of Solutions and Mixtures of Polymers ͓in Russian͔, Kiev ͑1984͒, method for determining the relative change in the affinity of 374 pp. 14 B. V. Ioffe, Refractometric Techniques in Chemistry ͓in Russian͔, Lenin- the solvents to the polymer, based on the change in the op- grad ͑1974͒, 400 pp. timum concentration of solvent which ensures a maximum 15 A. E. Nesterov and Yu. S. Lipatov, Phase States of Solutions and Mixtures recording efficiency for a given pair. of Polymers ͓in Russian͔, Kiev ͑1987͒, 168 pp. 16 J. A. Jenney, J. Opt. Soc. Am. 6, 1155 ͑1970͒. 17 ´ 1 H. J. Caulfield, Handbook of Optical Holography, New York ͑1979͒, 375 A. V. Butenin and B. Ya. Kogan, Kvant. Elektron. ͑Moscow͒ 3, 1136 pp. ͑1976͓͒Sov. J. Quantum Electron. 6, 611 ͑1976͔͒. 2 W. J. Tomlinson and E. A. Chandross, Adv. Photochem. 12, 202 ͑1980͒. 18 R. Z. Magaril, Mechanisms and Kinetics of Homogeneous Thermal Con- 3 V. A. Barachevski, Zh. Nauch. Prikl. Fotogr. Kinematogr. 38,75͑1993͒. version of Hydrocarbons ͓in Russian͔, Moscow ͑1970͒, 175 pp. 4 T. N. Smirnova, E. A. Tikhonov, and E. S. Gulnazarov, Proc. Soc. Photo- Opt. Instrum. Eng. ͑SPIE͒ 1017, 190 ͑1988͒. Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Experimental study of the focusing of submicrosecond pressure pulses in liquids Yu. V. Sud’enkov

St. Petersburg State University, 198904 St. Petersburg, Russia E´ . V. Ivanov

St. Petersburg State Institute of Precision Mechanics and Optics (Technical University, 197101 St. Petersburg, Russia ͑Submitted March 28, 1997͒ Zh. Tekh. Fiz. 68, 111–117 ͑June 1998͒ Experimental data are presented from a study of the focusing of single, submicrosecond pressure pulses in water. The effects of the initial amplitude distribution, the initial pressure level, and the geometric parameters of the opto-acoustic concentrator are studied. It is found that the focusing efﬁciency can be substantially enhanced by going from a bell-shaped distribution of the initial amplitude to an annular distribution. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02006-6͔

INTRODUCTION 2 ͑4LD /LN͒Ӷ1, LDϭ␲͑F␤͒ /␭, The focusing acoustic pulses in liquids currently has a 2 wide range of applications in different areas of science, tech- LNϭ␳0c0␭/͑2␲␧p0͒, ͑3͒ nology, and medicine.1–3 Thus the problem of concentrating pulsed pressures in the smallest possible spatial region is where LD and LN are, respectively, the diffraction length and extremely timely. the distance for formation of a discontinuity in a plane wave, The Kirchhoff–Helmholtz integral equation4 is used to ␧ϭ(1ϩ␥)/2 is the acoustic nonlinearity parameter, and ␳0 describe the focusing of monochromatic waves in the linear and c0 are the density of the liquid and the speed of sound approximation. If the local distance F is much greater than in it. the wavelength ␭, while the geometric angle of convergence, Therefore, for sufficiently short waves and large effec- ␣, is not too large (␣р1 rad), then this approach yields the tive angles of convergence, nonlinear effects, whose com- following result:5 bined effect on the focusing efficiency for monochromatic waves is complicated, must be taken into account.6–8 2 2 ͑p f /p0͒ϳF␤ /␭, L f ϳ␭/␤ , Dfϳ␭/␤, ͑1͒ It has been found that the processes involved in the propagation of pulsed and monochromatic waves in both the where p f and p0 are, respectively, the maximum pressure amplitudes in the focal plane and at the surface of the focus- linear and nonlinear regimes have many features in common.4,9,10 In particular, to examine the focusing of an ing source, L f and D f are the length and diameter of the initially monopolar pressure pulse of duration , one can use focal region at a level of 0.5p f , and ␤ is the effective con- ␶ vergence angle. Eqs. ͑1͒–͑3͒ with the wavelength ␭ replaced by the ‘‘pulse 5 11,12 The parameter ␤ is defined by the formulas length’’ c0␶. Thus, according to the linear theory, the efficiency with which a single pulse is focused will increase 2␲ ␣ 2 Ϫ1 as its duration is reduced. Then, however, the influence of ␤ ϭ␲ d␸ f s͑␪,␸͒sin ␪d␪, ͵0 ͵0 nonlinear effects should become greater, and this may be qualitatively different, depending on the ratio L /L . f ␪,␸͒ϭp ␪,␸͒/p , ͑2͒ D N s͑ s͑ 0 Hence, studies of the focusing of short, single pressure Ϫ7 where ps(␪,␸) and f s(␪,␸) are, respectively, the pressure pulses (␶р10 s), which are most conveniently generated amplitude at the surface of the focusing source and its dis- using an opto-acoustic approach,13 are of great interest from tribution function, while ␪ and ␸ are the latitude and longi- the standpoint of both fundamental and applied problems. tude angles in a spherical coordinate system. ͑The origin of In both theoretical and experimental studies of the focus- the coordinate system lies at the geometric focus, with the ing of pressure pulses it is customary to consider bell-shaped angle ␪ reckoned from the acoustic axis and ␸ in the focal profiles of their initial amplitude. An analysis of the avail- plane.͒ able experimental data14–18 indicates that for enhancing the Thus in order to enhance the focusing efficiency for lin- focusing efficiency for submicrosecond pulses it is necessary ear monochromatic waves, it is necessary to reduce to increase the geometric angle of convergence. This conclu- the wavelength ␭ and increase the effective convergence sion is consistent with Eqs. ͑1͒ and ͑2͒:as␣is increased, so angle ␤. does ␤, and therefore the focusing conditions are improved. The criterion for applicability of the linear approach, There is, however, another approach to the problem which takes only diffraction into account, is the following:4 of increasing the effective convergence angle. According to

FIG. 1. The experimental apparatus ͑a͒ and temporal proﬁles of the pressure pulses ͑b͒: 1—near the surface of the absorbing layer, 2—at the focus of the opto-acoustic concentrators.

Eq. ͑2͒, the largest contribution to ␤ is from the periphery of Figure 1a shows a sketch of the experimental apparatus. the beam, rather than from its axial region. Thus, the effec- The laser beam 1 was expanded by a negative lens 2.In tive angle ␤ can be increased without changing ␣, by going order to smooth out its transverse multimode structure, a from a bell-shaped distribution of the initial amplitude to an diffuser 3 was placed at the inlet of the opto-acoustic con- annular distribution. centrator 4. Under nonlinear conditions a bell-shaped initial distribu- The laser pulse was converted into a single pressure tion is clearly even more inferior from the standpoint of fo- pulse of duration 0.2 ␮s at half maximum in a solid absorb- cusing because of the negative effect of nonlinear refraction. ing layer of thickness Х0.4 mm deposited on the concave In the case of a bell-shaped distribution of the initial ampli- spherical surface of the glass substrate. The layer material tude, nonlinear refraction straightens the wave front and was well matched with water in terms of its acoustic imped- thereby makes focusing more difficult. At the same time, the ance ͑␳Ϸ103 kd/m3, cϷ1.8ϫ103 m/s͒ and this essentially nonlinear distortion of the pulse profile, which enriches the eliminated energy losses owing to reflection as the pressure initial spectrum with higher-frequency harmonics, promotes pulse passed from the opto-acoustic layer into the water focusing until a shock front develops. Then the deleterious tank 5. effect of nonlinear refraction is augmented by nonlinear ab- The thermoelastic mechanism for opto-acoustic genera- sorption. Thus, different situations can occur, depending on tion ensured that the absorbing layer could be used many the intensity of these nonlinear processes. times and provided good reproducibility of the parameters of It appears that using beams with an annular distribution the exciting pressure pulses with a high light-to-sound con- of the initial pressure can greatly attenuate the deleterious version efficiency. The parameter ␩ϭp0 /J0 ͑where J0 is the influence of nonlinear effects. In this case, in the initial stage maximum energy density of the laser pulse at the aperture of of propagation of the wave, nonlinear refraction will not in- 2 the concentrator͒ was ϳ20 MPa/͑J/cm ͒ for 0ϽJ0 hibit the focusing process but, on the contrary, will facilitate Ͻ0.5 J/cm2. it. In addition, for a fixed acoustic energy, p0 and, therefore, The change in the distribution of the initial amplitude of the influence of nonlinear effects will be smaller for larger the pressure pulse was measured by transforming the inten- values of ␤. sity distribution of the laser light at the inlet of the opto- In order to verify these assumptions, we have made an acoustic concentrator. experimental study of the dependence of the focusing of sub- The pressure in the water was recorded using a piezoce- microsecond pressure pulses in water on the level of p and 0 ramic probe 6 with a sensitive area of diameter 0.5 mm. The on the geometric parameters of the opto-acoustic concentra- signal from the probe was fed to the input of an S8-14 re- tor for the cases of bell-shaped and annular initial distribu- cording oscilloscope ͑bandwidth ϳ50 MHz͒. A differential tions. interferometer with stabilized sensitivity and photoelectron pulse counting, operating in a linear mode, was used to cali- EXPERIMENTAL TECHNIQUE brate the probe. A multimode Q-switched neodymium glass laser ͑wave- Studies of focusing were done on two opto-acoustic con- length 1.06 ␮m͒ was used to excite the pressure pulses. The centrators: 1͒ Fϭ68 mm and Dϭ54 mm, and 2͒ F duration of the laser pulse at half maximum was 20 ns and its ϭ91 mm and Dϭ110 mm, where D is the diameter of the energy was varied over the interval 0.1–0.3 J. concentrator. 716 Tech. Phys. 43 (6), June 1998 Yu. V. Sud’enkov and E´ . V. Ivanov

Figure 1b shows typical profiles of the pressure pulses recorded near the surface of the absorbing layer and at the focus of the opto-acoustic concentrators for different levels of the initial amplitude. At low pressures, one can observe a characteristic diffractional transformation of the temporal profile, i.e., the pulse is differentiated. As the initial amplitude is increased, the influence of nonlinearity shows up distinctly:4 the leading edge of the pulse becomes steeper, the duration of the compression phase is shorter, and its relative amplitude increases, while the rarefaction phase is extended and smoothed out. It should be noted that in our experiments, as opposed those of Musatov and Sapozhnikov,16 no shock front formation was observed, so only two nonlinear effects operated: nonlinear distortion of the temporal profile, and nonlinear refraction.

RESULTS AND DISCUSSION

The axially symmetric functions f s(␪,␸)ϭ f s(␪) constructed on the basis of the experimental data for the case of a bell-shaped distribution of the initial amplitude are shown in Fig. 2a. The changes in the amplitude of the compression phase of the pressure pulse along the acoustic axis and in the focal plane are shown in Figs. 2b and 2c for two laser pulse energies E. The absolute magnitudes of the initial and focal pressures, the gain coefficient Gϭp1 /p0 , the length and diameter of the focal region, and the effective convergence angle are listed in Table I. These data show that as the effective convergence angle and the focal length are increased ͑i.e., on going from the first concentrator to the second͒, the focal pressure and the gain increase substantially. At the same time, there is a noticeable reduction in the size of the focal region, both the dimension along the acoustic axis ͑2͒ and the diameter. For low initial pressures (p0ϳ0.2– 0.3 MPa), these data are in fairly good agreement with the predictions of the linear theory. According to Eq. ͑1͒, going from the first to the second concentrator should yield an increase in G by a factor of 2.4 as L f and D f are reduced by factors of 1.8 and 1.3, respectively. In the experiment the gain increased by a factor of 2.4 and the size of the focal region was reduced to roughly half. FIG. 2. Bell-shaped distribution of the initial pressure in the opto-acoustic At higher initial pressures (p0ϳ2 – 3 MPa), the agree- concentrator. The dashed curve is the first concentrator and the smooth ment between the linear theory and the experimental data is curve, the second. E ͑J͒: 0.3 and 3.0 ͑1͒, 1.85 ͑2͒, 0.15 ͑3͒. poorer owing to the greater role of nonlinear phenomena. Here the changes indicating an enhancement in the focusing efficiency on going from the first concentrator to the second other at low initial pressures. As p0 is increased, nonlinear are less marked than at low p0 : the gain increases by a factor refraction begins to predominate and the focusing deterio- of 1.7, while the size of the focal region is reduced by a rates. factor of 1.3. Let us examine the experimental data for the case of an The influence of nonlinear effects on the focusing pro- annular distribution of the initial amplitude. The functions cess is clearly demonstrated by the plots of p f (p0) and f s(␪,␸)ϭ f s(␪) for this sort of distribution are shown in Fig. G(p0) for the second concentrator shown in Fig. 3. For p0 4a. Figure 4b shows the variations in the amplitude of the р1 MPa, these curves are still linear, while for higher initial compression phase of the pressure pulse along the acoustic pressures the focusing efficiency falls off. That the range of axis for the first and second concentrators for two laser pulse linearity was much wider than predicted by Eq. ͑3͒ is appar- energies. The corresponding radial distributions in the focal ently explained by the fact that nonlinear refraction and non- planes of the opto-acoustic concentrators are given in Fig. linear distortion of the temporal profile counterbalance each 4c. The normalization parameters, gain coefficient, length Tech. Phys. 43 (6), June 1998 Yu. V. Sud’enkov and E´ . V. Ivanov 717

TABLE I.

Opto-acoustic p0 , p f , L f , D f , concentrator E, J MPa MPa G mm mm ␤0

1 0.30 0.30 2.0 6.7 22 2.8 13.5 1 3.00 3.00 20.0 6.7 22 2.8 13.5 2 0.15 0.17 2.7 15.9 10 1.6 18.0 2 1.85 2.00 22.9 11.5 18 2.2 18.0

and diameter of the focal region, and the effective convergence angle are listed in Table II. The data for an annular distribution of the initial amplitude also show that as the effective divergence angle and focal length are increased ͑on going from the first to the second concentrator͒, there is a significant rise in the gain and focal pressure with a simultaneous decrease in the size of the focal region. As opposed to the case of a bell-shaped initial distribution, these changes are larger than predicted by the linear theory. In the experiment, even for p0 ϳ1 – 2 MPa, an increase in the focusing efficiency by a factor of 2.7 was observed with respect to all three parameters ͑G, L f , and D f ͒, while for p0ϳ0.1– 0.2 MPa these indicators were still higher. Here it should be noted that the measured value of the diameter of the focal region for the second concentrator was comparable to the size of the sensitive area of the pressure probe. Thus, the dimensions of the focal volume were over- stated, while the pressures measured in the neighborhood of the focus were understated. The effect of the finite probe size on the measurement results can be estimated by writing the amplitude distribution of the compression phase of the pressure pulse in the focal plane in the form

p͑r͒ϭp*f F͑r/D*f ͒, ͑4͒ where F(r/D*f ) the dimensionless proﬁle function, p*f is the true value of the pressure at the focus, and D*f is the actual diameter of the focal region at the 0.5p*f level.

FIG. 3. The focal pressure and gain coefﬁcient as functions of the initial FIG. 4. Annular distribution of the initial pressure in the opto-acoustic con- pressure for the second opto-acoustic concentrator in the case of a bell- centrator. The dashed curve is the ﬁrst concentrator and the smooth curve, shaped distribution of the initial amplitude. the second. E ͑J͒: 0.3 and 3.0 ͑1͒, 1.85 ͑2͒, 0.15 ͑3͒. 718 Tech. Phys. 43 (6), June 1998 Yu. V. Sud’enkov and E´ . V. Ivanov

TABLE II.

Opto-acoustic p0 , p f , L f , D f , 0 concentrator E, J MPa MPa G mm mm ␤ M LG 1 0.30 0.18 2.2 12.2 22 2.8 17.0 1 3.00 1.80 22.0 12.2 22 2.8 17.0 2 0.15 0.08 2.7 33.8 5 1.0 22.5 1.42 2 1.85 1.00 29.2 29.2 9 1.3 22.5 1.16

Introducing the dimensionless averaging coefﬁcient M to be related to a lessening of the negative contribution of

ϭp*f /p f , we find nonlinear phenomena in the case of an annular distribution of d/2 Ϫ1 the initial pressures. An analysis of the geometry of the an- 2 Mϭd 8 F͑r/D*f ͒rdr , ͑5͒ nular distributions of the initial amplitude ͑Fig. 4a͒ shows ͫ ͵0 ͬ that for the second concentrator, the ring of the initial distri- where d is the probe diameter. bution stands substantially farther away from the center than for the first concentrator, a circumstance which also showed Since D f Ͼd, for the purpose of estimates it is sufficient up in the character of the nonlinear refraction. This assump- to approximate F(r/D*f ) by some bell-shaped function, as- tion is confirmed by the plots of p/p f (z) in Fig. 4b. For the suming that D*f ϷD f Ϫd. Let us compare the values of M calculated for Lorentzian ͑L͒ and Gaussian ͑G͒ focal distri- first concentrator the pressure equals zero only in the neigh- butions borhood of the point zϭrϭ0, while for the second concentrator the region in which there is no acoustic perturbation on 2 Ϫ1 2 M Lϭx ln ͑1ϩx ͒, the beam axis is much wider (zр20 mm). 2 2 Ϫ1 In order to determine the influence of the geometry of an M Gϭx ln 2͓1Ϫexp͑Ϫx ln 2͔͒ , ͑6͒ annular initial distribution on the focusing efficiency, we per- where xϭd/D* . f formed the following experiment. At p0ϳ0.1 MPa, an annu- M L and M G begin to differ noticeably only for xϾ1.5 lar distribution of the initial amplitude for the second con- and, therefore, the parameter M LGϭ(MLϩMG)/2 intro- centrator was shifted from the center to the edge in a way duced in Table II can be used for our estimates. For D1 such that the effective width of the annulus remained con- у1.6 mm, the calculation yields values of M LG which differ stant. The diameter of the focal region was found to decrease little from unity (M LGр1.09), so they have not been indi- to 0.5 mm, i.e., to the size of the sensitive area of the pres- cated in Table II and were not taken into account in analyz- sure probe, with the measured focal pressure p f essentially ing the experimental data. constant owing to the strong spatial averaging. These results Thus, in the case of the second type of opto-acoustic confirm the important influence of the geometric parameters concentrator, the growth in the focusing efficiency was un- of an annular distribution on the focusing efficiency and the derestimated by roughly factors of 1.5 and 1.2, respectively, possibility of further increasing the efficiency by optimizing for low and high initial pressures. In order to determine the f s(␪,␸). focal parameters more accurately, the spatial resolution of The suppression of the deleterious contribution from the measurements must be increased severalfold; that is an nonlinear refraction compared to the positive influence of the independent and by no means simple problem. nonlinear distortion of the temporal profile shows up most Let us analyze the enhancement in the focusing effi- clearly in the behavior of p f (p0) and G(p0) for the case of ciency on going from a bell-shaped distribution of the initial an annular distribution of the initial amplitude ͑Fig. 5͒.At amplitude to an annular one. For the first concentrator, the initial pressures below 1 MPa, G(p0) has a characteristic gain increases by a factor of 1.8, while the changes in the maximum. This confirms the earlier conclusion6,8,10,11,14 that dimensions of the focal region are negligible. At the same at low initial pressures, when a shock front is unable to de- time, for the second concentrator, the gain increases by a velop, nonlinear effects can improve the focusing conditions factor of 3.0, while the length and diameter of the focal re- compared to the linear case, if the nonlinear distortion of the gion are reduced to half. temporal profile predominates over nonlinear refraction. At Going from a bell-shaped distribution of the initial am- higher p0 , as in the case of a bell-shaped initial distribution, plitude to an annular distribution, therefore, makes it pos- the focusing efficiency falls off owing to the dominant con- sible to greatly enhance the focusing efficiency, and these tribution of nonlinear refraction. changes are most noticeable for the second concentrator. In the meantime, according to Eq. ͑1͒, the increase in the focus- CONCLUSIONS ing efficiency should be the same for the first and second concentrators, since on replacing a bell-shaped initial distri- These studies have shown that the distribution of the bution by an annular one, ␤ increases by the same factor for initial amplitude of submicrosecond pressure pulses in a liq- both of the concentrators ͑Tables I and II͒. uid has a significant effect on the process by which they are The observed additional improvement in the focusing focused. In particular, we have demonstrated the possibility conditions for the second opto-acoustic concentrator appears of greatly increasing the focusing efficiency by going from a Tech. Phys. 43 (6), June 1998 Yu. V. Sud’enkov and E´ . V. Ivanov 719

cantly on the geometric parameters of the annular distribution of the initial amplitude.

1 L. D. Rozenberg ͑Ed.͒, High-Power Ultrasonic Waves ͓in Russian͔, Nauka, Moscow ͑1968͒. 2 H. Reichenberger, Proc. IEEE 76, 1236 ͑1988͒. 3 C. R. Hill ͑Ed.͒, Physical Principles of Medical Ultrasonics ͓Halstead Press, New York, 1986; Mir, Moscow, 1989͔. 4 M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, The Theory of Waves ͓in Russian͔, Nauka, Moscow ͑1990͒, 432 pp. 5 I. N. Kanevski, The Focusing of Acoustic and Ultrasonic Waves ͓in Rus- sian͔, Nauka, Moscow ͑1977͒, 336 pp. 6 L. A. Ostrovski and A. M. Sutin, Dokl. Akad. Nauk SSSR 221, 1300 ͑1975͒. 7 N. S. Bakhvalov, Ya. M. Zhilekin, and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams ͓in Russian͔, Nauka, Moscow ͑1982͒. 8 S. N. Makarov, ‘‘Finite amplitude near-field modelling of ultrasonic fields using a transfer matrix formulation,’’ PTB-Bericht MA-42, Physikalische- Technische Bundesanstalt, Braunschweig; Berlin ͑1995͒,57pp. 9O. V. Vasil’eva, A. A. Karabutov et al., Interaction of One-Dimensional FIG. 5. The focal pressure and gain coefficient as functions of the initial Waves in Dispersionless Media ͓in Russian͔, Izd. MGU, Moscow ͑1983͒, pressure for the second opto-acoustic concentrator in the case of an annular 152 pp. 10 distribution of the initial amplitude. Al. M. Kolomenski, A. A. Maznev, and V. G. Mikhalevich, Izv. Akad. Nauk SSSR, Ser. Fiz. 54, 2451 ͑1990͒. 11 O. A. Sapozhnikov, Akust. Zh. 37, 760 ͑1991͒. 12 A. G. Musatov, O. V. Rudenko, and O. A. Sapozhnikov, Akust. Zh. 38, 502 ͑1992͒. bell-shaped initial distribution to an annular one. This phe- 13 V. E´ . Gusev and A. A. Karabutov, Laser Optoacoustics ͓in Russian͔, nomenon is observed over a wide range of initial pressures. Nauka, Moscow ͑1991͒, 304 pp. 14 A. I. Bozhkov, F. V. Bunkin et al., Izv. Akad. Nauk SSSR, Ser. Fiz. 46, At low levels of p0 , the positive effect originates primarily in an enhancement in the effective convergence angle ␤ 1624 ͑1982͒. 15 I. I. Komissarova, G. V. Ostrovskaya et al., Zh. Tekh. Fiz. 62͑2͒,34 when the distribution function is changed appropriately. At ͑1992͓͒Tech. Phys. 37, 130 ͑1992͔͒. higher initial pressures there is an additional factor: the sup- 16 A. G. Musatov and O. A. Sapozhnikov, Akust. Zh. 39, 315 ͑1993͒. 17 A. G. Musatov and O. A. Sapozhnikov, Akust. Zh. 39, 510 ͑1993͒. pression of the negative influence of nonlinear refraction 18 compared to the nonlinear distortion of the temporal profile A. G. Musatov, Akust. Zh. 41, 117 ͑1995͒. of the pulse. Then the degree of suppression depends signifi- Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Magnetostatic surface waves produced by an inhomogeneity of the anisotropy with a turning point of the spectral function on a ferromagnet surface I. A. Ka bichev and V. G. Shavrov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, 103907 Moscow, Russia ͑Submitted February 3, 1997͒ Zh. Tekh. Fiz. 68, 118–123 ͑June 1998͒ Magnetostatic surface waves with fixed frequency and wave vector are predicted to exist in a ferromagnet with an inhomogeneity of the magnetic anisotropy such that the spectral function has a turning point on the surface. This result is most important for the case when an external magnetic field magnetizes the ferromagnet perpendicular to its surface. The frequency of the surface wave is determined by the frequency of the magnetostatic volume wave at the surface of the ferromagnet, and the wave vector is determined by the surface values of the local magnetic anisotropy field and its derivative. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02106-0͔

INTRODUCTION EXPERIMENTAL AND THEORETICAL STUDIES OF MAGNETOSTATIC SURFACE WAVES IN INHOMOGENEOUS Ferrite epitaxial films, mainly of yttrium iron garnet, FILMS are widely used in microwave electronics. They are Most studies have been made of the case where the ex- grown on nonmagnetic substrates, such as gallium gado- ternal magnetic field and the magnetization are in the same linium garnet. The existing film growth technologies mainly direction and lie in the plane of the film, while the wave, yield films that are inhomogeneous over the sample itself, propagates perpendicular to them.17–24 The inhomoge- thickness.1,2 Inhomogeneities can also be created artificially neity in the distribution of the magnetization is modeled by a by ion implantation.3–8 In this regard, a need has arisen for multilayer film consisting of two or three homogeneous lay- 19–23 studying the properties of magnetostatic waves in films that ers with different magnetic parameters. Numerical calculations have been done for a specified inhomogeneity pro- are inhomogeneous over their thickness. The distribution of 17,18,24 the magnetization in the ground state of a ferromagnet is file by several authors. It is important to obtain analytical results for a rather general form of the inhomoge- determined to a great extent by the profile of the magnetic neities. anisotropy constant; here the inhomogeneous exchange con- The case in which an external magnetic field magnetizes stant and the saturation magnetization can be regarded as an inhomogeneous ferromagnet in a direction perpendicular 6 constant. We will therefore consider a ferromagnet with to its surface is of greatest interest. An inhomogeneity in a only an inhomogeneity in the magnetic anisotropy. We limit film of yttrium iron garnet with this geometry has only been ourselves to inhomogeneities with a single turning point of considered once,25 in an examination of the direct and in- the spectral function on the surface of the ferromagnet. A verse spectral problems for magnetostatic forward volume magnetostatic volume wave is thereby allowed to propagate waves, while the experimental spectra were interpreted in on the surface of the ferromagnet. We shall choose the pro- terms of a spatial inhomogeneity in the uniaxial anisotropy. file of the inhomogeneity in the magnetic anisotropy constant to be such that magnetostatic volume waves cannot propa- STATEMENT OF THE PROBLEM AND BASIC EQUATIONS gate inside the ferromagnet. In that way, we obtain exponential damping of the magnetic potential of the wave in the Let us consider a uniaxial semi-infinite ferromagnet interior of the ferromagnet, and this leads to localization of which occupies the region zϾ0 ͑see Fig. 1͒ and has an ar- the wave near the surface. A wave of this sort can be classed bitrary profile of the inhomogeneity of the local magnetic as a magnetostatic surface wave. This result is of greatest anisotropy field HA(z)ϭ2K(z)/M0 , where K(z) is the mag- interest for a ferromagnet magnetized by an external magnetic anisotropy constant and M 0 is the saturation magnetization. We assume that the magnetic anisotropy field H (z) netic field perpendicular to its surface, since then only mag- A is less than the absolute value of the demagnetizing field netostatic volume waves will exist in the homogeneous ͉H ͉ϭ4␲M . Then in weak magnetic fields H Ͻ4␲M case,9–11 while surface excitations of the spin system show M 0 0 0 ϪHA(z) the ground state is inhomogeneous: up only when exchange is included.12–16 Thus, in a normally magnetized ferromagnet, magnetostatic surface waves can ␺0͑z͒ϭϮarccos͑H0 /͓4␲M 0ϪHa͑z͔͒͒, ͑1a͒ exist as a result of the inhomogeneity in the magnetic anisot- while in strong fields H0Ͼ4␲M 0ϪHA(z) a homogeneous ropy. ground state is realized,

where ␥ is the gyromagnetic ratio, Heff(z)ϭH0ϩHMϩHA(z) ϫ(Mn)n/M0 is the effective magnetic field, and n is the unit vector characterizing the direction of the anisotropy axis of the ferromagnetic crystal ͑directed along the easy axis z of the ferromagnet͒. The magnetization vector in the ground state has components (M 0 sin ␺0(z), 0, M 0 cos ␺0(z)), and the demagnetization field is (0,0,Ϫ4␲M 0 cos ␺0(z)). We assume that the deviations of the magnetization vector m and the demagnetizing field h from these equilibrium values are small. We linearize the Landau–Lifshitz equation ͑3͒ to obtain the coupling between the components of the vectors m and h, which we write in the form

miϭ␹ijhj , i,jϭx,y,z, ͑4͒

where ␹ij is the high-frequency magnetic susceptibility tensor of the ferromagnet. FIG. 1. The geometry of the problem. H0 is the external magnetic ﬁeld; n is a unit vector characterizing the direction of the easy axis z of the ferromag- Its components are given by net, which is perpendicular to its surface; M is the magnetization ͑in general 2 it deviates from the easy axis by an angle ␺0(z)͒; and, k is the wave vector ␹XXϭ⌫͑z͒⍀1͑z͒cos ␺0͑z͒, of the magnetostatic surface wave. ␹XYϭϪ␹YXϭ⌫͑z͒i␻ cos ␺0͑z͒,

␹YYϭ⌫͑z͒⍀2͑z͒, ␹YZϭϪ␹ZYϭ⌫͑z͒i␻ sin ␺0͑z͒, ␺0͑z͒ϭ0. ͑1b͒ ␹ZXϭ␹XZϭϪ⌫͑z͒⍀1͑z͒cos ␺0͑z͒sin ␺0͑z͒, This latter condition also occurs in the case of HA(z) 2 Ͼ4␲M 0 and H0Ͼ0. In a situation with HA(z)Ͼ4␲M 0 and ␹ZZϭ⌫͑z͒⍀1͑z͒sin ␺0͑z͒. 4 M ϪH (z)ϽH Ͻ0, the homogeneous phase 1b is ␲ 0 A 0 ͑ ͒ Here we have used the notation metastable. Thus, the results for the magnetostatic surface 2 wave spectrum in the latter case will be valid only when the ⌫͑z͒ϭ␥M 0 /͓⍀1͑z͒⍀2͑z͒Ϫ␻ ͔, energy of the magnetic excitations of the ferromagnet is ͑i͒ small compared to the potential barrier preventing a transi- ⍀1͑z͒ϭ␥͓H0 ϩHA͑z͒cos ␺0͑z͔͒cos ␺0͑z͒, tion of the ferromagnet into the homogeneous stable state ͑i͒ ⍀2͑z͒ϭ␥͓H0 cos ␺0͑z͒ϩHA͑z͒cos 2␺0͑z͔͒, ␺0(z)ϭ␲. The exchange interaction was not taken into account in determining the ground state. This is valid if the i while H0ϭH0Ϫ4␲M 0 cos ␺0(z) is the internal magnetic dimensions of the ferromagnet film and the characteristic field. Note that on going to a system of coordinates with its Z scale length of the inhomogeneity of the magnetic anisotropy axis coincident with the magnetization of the ground state of field ͑the length over which HA(z) changes from its surface the ferromagnet, we essentially obtain a form of the high- to its bulk value͒, L, exceed the exchange length frequency magnetic susceptibility tensor which was known 9 ␦ϭͱ2A/(HA0M0). ͑A is the inhomogeneous exchange con- previously. The differences are related to the dependence of stant and HA0 is the anisotropy field in the interior of the the magnetic anisotropy field on the vertical coordinate z. film.͒ The components of the magnetic susceptibility tensor also Surface magnetostatic waves propagate along the Y axis, become functions of z. Substituting Eq. ͑4͒ into the equation so we assume that all the variables in the problem are proof magnetostatics ͑2͒, and then introducing the magnetic sca- -portional to exp(i␻tϪiky), where ␻ is the frequency and k is lar potential ⌽(hϭϪٌ⌽), we obtain a second-order differ the wave vector. We consider frequencies up to several GHz, ential equation with variable coefficients: since they are usually employed in practice.19–23 At these 2 2 frequencies, the wave vector kϽ105 cmϪ1. In this region, D ⌽͑z͒Ϫk Q͑␻,z͒⌽͑z͒ϭ0, ͑5͒ the contribution of the exchange interaction is small com- 2 where Q(␻,z)ϭ1ϩ4␲␥M 0⍀2(z)/͓⍀1(z)⍀2(z)Ϫ␻ ͔ pared to the other terms in the magnetic energy: dipole– 2 ϩ␻D͕4␲␥M0 sin ␺0(z)/͓⍀1(z)⍀2(z)Ϫ␻ ͔͖/k is a function dipole and Zeeman. In an examination of magnetostatic which determines the character of the solutions, which we z2. Theץ/2ץwaves, it can be neglected. We shall begin with the system of shall refer to as the spectral function, and D2ϭ equations of magnetostatics: form of the spectral function Q(␻,z) depends on the choice

curl HMϭ0, div͑HMϩ4␲M ͒ϭ0, ͑2͒ of ground state: 2 where HM is the demagnetization field. Q͑␻,z͒ϭϪ͓␻V1͑z͒Ϫ␻͔͓␻Ϫ␻V2͑z͔͒/␻ ͑6a͒ The magnetization M satisfies the Landau–Lifshitz equation for the inhomogeneous ground state ͑1a͒ and Q͑␻,z͒ϭ͓␻2 ͑z͒Ϫ␻2͔/͓␻2 ͑z͒Ϫ␻2͔ ͑6b͒ tϭϪ␥͓M•Heff͑z͔͒, ͑3͒ V2 V1ץ/Mץ 722 Tech. Phys. 43 (6), June 1998 I. A. Ka bichev and V. G. Shavrov for the homogeneous phase ͑1b͒. The quantities ␻V1(z) and Thus, the second-order differential equation ͑5͒ with the ␻V2(z) have the same physical significance in both cases: spectral function ͑6͒ and the boundary conditions ͑7͒ can be they are the lower and upper local boundaries of the magne- used to describe the propagation of a surface magnetostatic tostatic volume wave spectrum in a homogeneous slab with wave in an inhomogeneous ferromagnet. parameters equal to their values at the point z. Their forms, however, depend on the choice of ground state. For the inhomogeneous state ͑1a͒, MAGNETOSTATIC SURFACE WAVE SPECTRA ␻ ϭA͑z͒ϩ͑Ϫ1͒jͱA2͑z͒ϪB͑z͒, Vj Of all the possible profiles of the local magnetic anisot- A͑z͒ϭ2␲␥M0 cos ␺0͑z͒D␺0͑z͒/k, ropy field HA(z), we shall consider only those which have a 2 2 single turning point in the spectral function Q(␻,z), at the -z, surface of the ferromagnet, i.e., Q(␻,zϭ0)ϭ0. This condiץ/ץB͑z͒ϭ4␲␥ M 0HA͑z͒sin ␺0͑z͒, Dϭ while for the homogeneous phase ͑1b͒, tion uniquely determines the frequency of the surface magnetostatic wave: ␻V1͑z͒ϭ⍀0͑z͒, ␻V2͑z͒ϭͱ⍀0͑z͓͒⍀0͑z͒ϩ4␲␥M 0͔, ␻Sϭ␻V2͑0͒. ͑8͒ where ⍀0(z)ϭ␥͓H0Ϫ4␲M 0ϩHA(z)͔ is the local ferromagnetic resonance frequency. The frequency ␻ cannot take other values. The upper In the beginning we examine a uniform material, where local limit on the magnetostatic volume wave spectrum in a there is no dependence on z and Q(␻,z)ϭQ(␻). In an in- slab, ␻V2(0), is given by different expressions ͑6a͒ or ͑6b͒, finite material there is a single magnetostatic volume wave, depending on whether an inhomogeneous ͑1a͒ or homoge- whose frequency is determined from the condition Q(␻) neous ͑1b͒ state is realized. In the interior of the ferromagnet ϭ0. For a slab with Q(␻)Ͻ0, the solutions of Eq. ͑5͒ are for zϾ0 we assume that the spectral function Q(␻S ,z)is expressed in terms of a linear combination of the sine and positive. Then the upper local limit ␻V2(z) for any zу0 cosine and describe an infinite set of magnetostatic volume must satisfy the inequality wave spectral modes.10,11 It should be noted that these 26,27 waves, by analogy with elastic waves, occupy an inter- ␻V2͑z͒р␻V2͑0͒. ͑9͒ mediate position between surface and volume waves, but the The equal sign holds only for zϭ0. Thus, the situation term ‘‘magnetostatic volume waves in a slab’’ has become we are examining is realized only for profiles of the local established in the literature. In a half space with Q(␻)Ͼ0, magnetic anisotropy field which ensure that the upper local the solution of Eq. 5 is expressed in terms of a linear com- ͑ ͒ limit ␻ of the magnetostatic volume wave spectrum in the bination of exponents with different signs and describes a V2 slab satisfies inequality ͑9͒. magnetostatic surface wave ͑if it satisfies the boundary con- In the short-wavelength approximation, ͉k͉Lӷ1 ͑wave- ditions . ͒ length much shorter than the inhomogeneity scale length L͒, Everything is a bit more complicated in an inhomoge- 28,29 Eq. ͑5͒ has the solution neous medium. For example, if there is point z0 , referred to 2/3 as the turning point, such that Q(␻,z0)ϭ0, then only one ⌽͑z͒ϭ⌽0 Ai͓͉k͉ ␰͑z͔͒, magnetostatic volume wave can propagate at the point z0 .If 2/3 Q(␻,z)Ͻ0 in some layer, then an infinite set of magneto- 3 z ␰͑z͒ϭ dtͱQ͑␻S ,t͒ , zу0, ͑10͒ static volume wave modes exist in it, while for Q(␻,z)Ͼ0a ͫ 2 ͵0 ͬ magnetostatic volume wave cannot exist, but a magnetostatic surface wave can propagate. Let us choose a situation such where Ai(␣) is the Airy function of the first kind. that Q(␻,zϭ0)ϭ0, while for any other zϾ0 the spectral Near the surface, for z 0, the function ␰(z) 3 → function Q(␻,z)Ͼ0. The conditions for propagation of a ϭͱDQ(␻S,0)z, while the Airy function is written in the magnetostatic volume wave are thereby created on the sur- form face of the ferromagnet, while inside it the magnetic poten- 1 tial will fall off exponentially. Ultimately, this should result ϭ0␣ϭϪ 2/3␣͔␣ץ/͒␣Ai͑ ץAi͑␣͒ϭAi͑0͒ϩ͓ in localization of the wave near the surface, so we shall refer 3 ⌫͑2/3͒ to this as a surface wave. 2/3 3 z͉k͉ ͱDQ͑␻S,0͒ We now formulate the boundary conditions for the prob- Ϫ , 34/3 4/3 lem. They involve the continuity at the ferromagnet surface ⌫͑ ͒ (zϭ0) of the normal component of the magnetic induction where ⌫͑␧͒ is the gamma function. and the tangential component of the magnetic field strength, Then the distribution of the magnetic potential of the which reduces to the following conditions for the magnetic surface magnetostatic wave with frequency ͑8͒ has the potential: asymptotic form

⌽͑0͒ϭ⌽B͑0͒, ϪD⌽͑0͒ϩ4␲mz͑0͒ϭϪD⌽B͑0͒, 2/3 3 1 z͉k͉ ͱDQ͑␻S,0͒ ͑7͒ ⌽͑z͒ϭϪ⌽ ϩ ͑11a͒ 0ͭ32/3⌫͑2/3͒ 34/3⌫͑4/3͒ ͮ where ⌽B(z) is the magnetic potential in the vacuum region (zу0). near the surface (z 0) and → Tech. Phys. 43 (6), June 1998 I. A. Ka bichev and V. G. Shavrov 723

z In the case where there is no external magnetic ﬁeld ⌽0 4 Q͑␻S͒ ⌽͑z͒ϭ exp Ϫ͉k͉ dtͱQ͑␻ ,t͒ (H ϭ0) one has ␻ ϭ␻ (0). To determine the absolute 2␲ ͱQ͑␻ ,z͒ ͵ S 0 S V S ͭ 0 ͮ value of the wave vector according to Eq. ͑12͒, we calculate ͑11b͒ the derivative of the spectral function on the surface; the in the interior of the ferromagnet (zӷ0). Outside the ferro- dispersion of the surface magnetostatic wave at short wave- magnet, for the vacuum region (zр0), we have lengths can be neglected. As a result, we obtain

⌽͑z͒ϭ⌿0 exp͕͉k͉z͖. ͑11c͒ 2 DHA͑0͒ 2H0HA͑0͒ ͉k͉ϭϪ␤ 1Ϫ 3 . ͑18͒ Substituting the expressions for the magnetic potential HA͑0͒ ͭ ͓4␲M0ϪHA͑0͔͒ ͮ ͑11͒ into the boundary conditions ͑7͒ yields an expression for the wave vector of the magnetostatic surface wave, All these results have been obtained for short wavelengths, such that ͉k͉ӷLϪ1. This leads to a condition on the ͉k͉ϭ␤DQ͑␻S,0͒, ͑12͒ surface value of the logarithmic derivative of the local mag- where ␤ϭ͕⌫(2/3)/⌫(4/3)͖3/9Ϸ0.3874. netic anisotropy field, In calculating ␤ we have used tabulated values of the 2 30 1 2H0HA͑0͒ gamma function. A result for the wave vector ͑12͒ was D ln͓ϪHA͑0͔͒ӷ 1ϩ 3 , ͑19͒ obtained in the short-wavelength range, where the condition ␤L ͭ ͓4␲M 0ϪHA͑0͔͒ ͮ Ϫ1 ͉k͉ӷL must be satisfied. This leads to a requirement for i.e., the local magnetic anisotropy field must increase near the surface value of the derivative of the spectral function, the surface. For other values of the logarithmic derivative of Ϫ1 the local magnetic anisotropy field at the surface, the short- DQ͑␻S,0͒ӷ͓␤L͔ . ͑13͒ wavelength approximation does not hold and our equations are not applicable. If condition ͑19͒ is satisfied, then the final MAGNETOSTATIC SURFACE WAVE SPECTRUM IN A FERROMAGNET WITH AN INHOMOGENEOUS GROUND expression for the magnetostatic surface wave frequency is STATE obtained from Eq. ͑16͒ by substituting the wave vector, i.e., 2 We now make the results for the frequency of the sur- H0ͱϪ␲M 0HA͑0͒ ␻Sϭ␻V͑0͒ 1Ϫ␴ sign ␺0͑0͒ 3 , face magnetostatic wave ͑8͒ and the absolute value of the ͭ ͓4␲M 0ϪHA͑0͔͒ ͮ wave vector ͑12͒, as well as the conditions ͑9͒ and ͑13͒ for the inhomogeneous state ͑1a͒, more specific. A magneto- ␴ϭsign k, ͑20͒ static volume wave spectrum with lower ␻ (z) and upper V1 where ␻V2(z) local boundaries ͑7͒ exists in an inhomogeneous medium if A2(z)ϾB(z)or 2 H0 2 ␻V͑0͒ϭ2␥ͱϪ␲M 0HA͑0͒ 1Ϫ 2 DHA͑z͒ ͭ 2͓4␲M 0ϪHA͑0͔͒ ͮ H ͑z͒Ͻ␲M cot2 ␺ ͑z͒ . ͑14͒ A 0 ͓4␲M ϪH ͑z͔͒k 0 ͭ 0 A ͮ is the magnetostatic volume wave frequency in a homoge- Restricting ourselves to low fields H0Ӷ4␲M 0 neous ferromagnet with parameters equal to their values on ϪHA(z), we have the surface of the inhomogeneous medium. 4 2 If ␺0(0)Ͼ0, then for propagation in the positive Y di- ␲M 0H0͓DHA͑z͔͒ rection (␴ϭϩ1), the magnetostatic surface wave frequency HA͑z͒ϽHkpϭ 6 2 ͓4␲M0ϪHA͑z͔͒ k ␻S ͑20͒ will be somewhat lower than the magnetostatic vol- 4 ume wave frequency ␻V(0), and if the wave propagates in ␲M0H0 Ϸ . ͑15͒ the negative Y direction (␴ϭϪ1), then ␻S is higher than ͓4␲M ϪH ͑z͔͒4͓kL͔2 0 A ␻V(0). In the case ␺0(0)Ͻ0, for the positive Y direction This condition is clearly satisfied in an ‘‘easy plane’’ (␴ϭϩ1) the magnetostatic surface wave frequency ␻S ͑20͒ will be somewhat higher than the magnetostatic volume ferromagnet with HA(z)Ͻ0. For short wavelengths, with kLӷ1, the term A(z) in the expression for the upper limit wave frequency ␻V(0), and for the negative direction (␴ϭϪ1), lower. Thus, a nonreciprocity effect shows up. It ␻V2(z) can be regarded as small. Then, for the magnetostatic surface wave frequency ͑8͒, we obtain involves a difference in the frequencies for identical magnitudes of the wave vector but different propagation directions ␻Sϭ␻V͑0͓͒1Ϫ␣͔, ͑16͒ of the wave. Equation ͑20͒ implies that it is observed only in an external magnetic field H0 . If there is no field (H0Þ0), where ␻V(0)ϭ2␥͉sin ␺0(0)͉ͱϪ␲M 0HA(0) is the frequency of a magnetostatic volume wave in an infinite homogeneous then ␻Sϭ␻V(0), and the nonreciprocity effect does not oc- ferromagnet with parameters equal to their surface values in cur. the inhomogeneous medium, while Condition ͑11͒, which ensures that the spectral function is positive in the absence of a magnetic field (H0ϭ0), is 2 sign ␺0͑0͒ ␲M 0 satisfied for any profile of the local magnetic anisotropy field ␣ϭH0DHA͑0͒ 3 ͱϪ ͑17͒ with ͓4␲M 0ϪHA͑0͔͒ k HA͑0͒ Ϫ1 is a small correction leading to a dispersion ϳk . HA͑z͒уHKϭHA͑0͒. ͑21͒ 724 Tech. Phys. 43 (6), June 1998 I. A. Ka bichev and V. G. Shavrov

At low fields H0Ӷ4␲M 0ϪHA(z), the expression for frequency and wave vector in an inhomogeneous ferromag- the critical field is somewhat different, net for which the spectral function has a turning point on the surface. This kind of magnetostatic surface wave is obtained H2H ͑0͒ͱϪ␲M H ͑0͒ DH ͑z͒ 0 A 0 A A from a magnetostatic volume wave propagating at the sur- HkϭHA͑0͒Ϫ 3 1Ϫ . ͓4␲M 0ϪHA͑0͔͒ ͭ DHA͑0͒ͮ face of the ferromagnet by choosing a profile for the local ͑22͒ magnetic anisotropy field which precludes the existence of a If DHA(z)/DHA(0) is not a constant equal to 1, then HK magnetostatic volume wave in the interior of the ferromagnet should depend weakly on the propagation direction of the and ensures exponential damping of the magnetic potential. wave. Thus we have determined the fixed values of the fre- This significantly new result is of greatest interest for a fer- quency ͑20͒ and wave vector ͑18͒ of the magnetostatic sur- romagnet that has been magnetized perpendicular to its surface wave in the case where the ferromagnet has an inhomo- face, where only a magnetostatic volume wave can exist in geneous ground state. We have shown that these values are the homogeneous case. The fact that the frequency and wave different in an external magnetic field for different directions vector are fixed is nontraditional. They are uniquely deter- of propagation of the wave ͑i.e., a nonreciprocity effect oc- mined by the values of the local magnetic anisotropy field curs͒. The local magnetic anisotropy field inside the ferro- and its derivative at the surface. The conditions for the exis- magnet must be greater than the critical field and increase tence of the predicted magnetostatic surface wave depend on near the surface. The critical field is determined mainly by the choice of the ground state of the ferromagnet. the local magnetic anisotropy field at the surface.

MAGNETOSTATIC SURFACE WAVES IN A FERROMAGNET WITH A HOMOGENEOUS GROUND STATE We now obtain these results for the homogeneous state 1 C. Vittoria and J. H. Schelleng, Phys. Rev. B 18, 4020 ͑1977͒. 2 ͑1b͒. The surface magnetostatic wave frequency ͑8͒ in this C. Borghese, P. De Gasperies, and R. Tappa, Solid State Commun. 25,21 ͑1978͒. case is given by 3 A. H. Eschenfel’der, Magnetic Bubble Technology, Springer-Verlag, Ber- lin ͑1980͒; Mir, Moscow ͑1983͒, 496 pp. ␻Sϭ␻V͑0͒, ͑23͒ 4 V. S. Speriosu and C. H. Wilts, J. Appl. Phys. 54, 3325 ͑1983͒. 5 where ␻V(0)ϭͱ⍀0(0)͓⍀0ϩ4␲␥M0͔ is the magnetostatic C. H. Wilts, H. Awano, and V. S. Speriosu, J. Appl. Phys. 57, 2161 volume wave frequency for a homogeneous ferromagnet ͑1985͒. 6 C. H. Wilts and S. Prased, IEEE Trans. Magn. MAG-17, 2045 ͑1981͒. with parameters equal to their values on the surface of a 7 ´ V. E. Osukhovski, D. E. Linkova, Z. Z. Ditina et al., Fiz. Tverd. Tela inhomogeneous medium. For the wave vector k, Eq. ͑12͒ ͑Leningrad͒ 26, 1533 ͑1984͓͒Sov. Phys. Solid State 26, 933 ͑1984͔͒. implies that 8 G. A. Shmatov, V. N. Filippov, V. B. Sadkov, and I. I. Kryukov, Pis’ma Zh. Tekh. Fiz. 15͑17͒,86͑1989͓͒Sov. Tech. Phys. Lett. 15, 699 ͑1989͔͒. 9 DHA͑0͒ ⍀0͑0͒ϩ2␲␥M 0 A. I. Akhiezer, V. G. Bar’yakhtar, and S. V. Peletminski, Spin Waves, ͉k͉ϭϪ␤ . ͑24͒ North-Holland, Amsterdam ͑1968͓͒Russian original, Nauka, Moscow 18␲M 0 ⍀0͑0͒ ͑1967͒, 368 pp.͔. Thus, this sort of magnetostatic surface wave can exist 10 V. G. Bar’yakhtar and M. I. Kaganov, ‘‘Inhomogeneous resonance and only for inhomogeneities in the local magnetic anisotropy spin waves,’’ in Ferromagnetic Resonance ͓in Russian͔, Fizmatgiz, Mos- cow ͑1961͒, pp. 266–284. field which fall off near the surface, i.e., DHA(0)Ͻ0. Since 11 G. A. Vugal’ter and I. A. Gilinski, Izv. Vyssh. Uchebn. Zaved. Radiofiz. our results have been obtained for the short-wavelength case, 32, 1187 ͑1989͒. Eq. ͑13͒ can be satisfied and is equivalent to the requirement 12 B. N. Filippov, Fiz. Met. Metalloved. 32, 911 ͑1971͒. that 13 B. N. Filippov and I. G. Tityakov, Fiz. Met. Metalloved. 35,28͑1973͒. 14 R. E. De Wames and T. Wolfram, J. Appl. Phys. 41, 987 ͑1970͒. 15 18␲M 0 ⍀0͑0͒ B. R. Tittman, R. E. De Wames, R. D. Henry, and P. J. Besser, Trudy DH ͑0͒ӶϪ . ͑25͒ MKM-73, Vol. 2, pp. 19–27 1974 . A ␤L ⍀ 0͒ϩ2␲␥M ͑ ͒ 0͑ 0 16 J. T. Yu, R. A. Turk, and P. E. Wigen, Phys. Rev. B 5, 420 ͑1972͒. 17 F. R. Morgenthaler, IEEE Trans. Magn. MAG-13, 1252 ͑1977͒. In the interior of the ferromagnet for zϾ0, the spectral 18 function Q( ,z) was assumed positive. Then the upper lo- F. R. Morgenthaler, J. Appl. Phys. 52, 2267 ͑1981͒. ␻S 19 P. Hartemann and D. Fontaine, IEEE Trans. Magn. MAG-18, 1595 cal limit ⍀V2(z) for arbitrary zу0 should satisfy the in- ͑1982͒. 20 equality ͑9͒, which leads to an inequality for the local mag- N. I. Lyashenko and V. M. Talalaevski, Ukr. Fiz. Zh. 31, 1716 ͑1986͒. netic anisotropy field, 21 Yu. V. Gulyaev, I. A. Ignat’ev, A. F. Popkov, and V. M. Shabunin, in Abstracts of the 10th All-Union School–Seminar on New Magnetic Mate- HA͑z͒рHA͑0͒. ͑26͒ rials for Microelectronics ͓in Russian͔, Riga ͑1986͒, Part 1, pp. 176–177. 22 I. G. Kudryashkin, D. G. Krutogin, E. A. Ladygin et al., Zh. Tekh. Fiz. Thus, a magnetostatic surface wave with the frequency 59͑3͒,70͑1989͓͒Sov. Phys. Tech. Phys. 34, 294 ͑1989͔͒. ͑23͒ and wave vector ͑24͒ exists for a local magnetic anisot- 23 Yu. M. Yakovlev, E. G. Rzhakhina, T. A. Kyrlova et al., Fiz. Tverd. Tela ropy field which is smaller than that on the surface and falls ͑Leningrad͒ 30, 622 ͑1988͓͒Sov. Phys. Solid State 30, 360 ͑1988͔͒. 24 off near the surface. I. V. Zavislyan and A. F. Kalada, Vestn. Kievsk. Univ. Ser. Fiz., No. 23, pp. 75–79 ͑1982͒. 25 I. Yu. Gaovich, G. P. Golovach, I. V. Zavislyan, and V. F. Romanyuk, CONCLUSIONS Fiz. Tverd. Tela ͑St. Petersburg͒ 34, 1680 ͑1992͓͒Sov. Phys. Solid State 34, 893 ͑1992͔͒. In summary, we have demonstrated the possible exis- 26 I. A. Viktorov, Acoustic Surface Waves in Solids ͓in Russian͔, Nauka, tence of a single magnetostatic surface wave with a fixed Moscow ͑1981͒, 288 pp. Tech. Phys. 43 (6), June 1998 I. A. Ka bichev and V. G. Shavrov 725

27 E. Dieulesant and D. Royer, Elastic Waves in Solids, Wiley, New York Equations ͓in Russian͔, Nauka, Moscow ͑1983͒, 352 pp. ͑1980͒; Nauka, Moscow ͑1982͒, 434 pp. 30 E. Jahnke, F. Emde, and F. Loesch, Tables of Higher Functions, 6th ed., 28 A. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York McGraw-Hill, New York ͑1960͒; Nauka, Moscow ͑1977͒, 344 pp. ͑1981͒; Mir, Moscow ͑1984͒, 534 pp. 29 M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Inﬂuence of the edge ﬁeld on the focusing properties of a coaxial cylindrical lens L. P. Ovsyannikova and T. Ya. Fishkova

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia ͑Submitted May 22, 1997͒ Zh. Tekh. Fiz. 68, 124–127 ͑June 1998͒ The effect of the distance between the grounded ﬂat entrance diaphragm and the outer cylindrical electrode ͑which determines the edge ﬁeld of the lens͒ is investigated over a wide range of variation of the geometry of a coaxial cylindrical lens. It is found that focusing of a charged particle beam on the lens axis is achieved over a wide range only in the case of small clearances between the diaphragm and the outer electrode. It is shown that for an alternative power feed arrangement, in which the outer cylindrical electrode is grounded and the voltage is applied to the inner electrode, beam focusing is in general degraded, in particular on the lens axis. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02206-5͔

In Refs. 1 and 2 we identified the working regimes of a dependence for the entrance edge field. It should be stated coaxial cylindrical lens focusing an annular charged-particle that near the diaphragm but far from the field-assigning elec- beam on its axis and investigated it over a wide range of trode, the field distribution is similar to the field distribution variation of its geometrical and electrical parameters. We away from the diaphragm. The values of the voltages deter- also obtained simple empirical formulas for the cardinal el- mining the form of the equipotentials and the rate of falloff ements of this lens. The clearance between the grounded of the edge field away from the field-assigning electrode de- end-face diaphragm at the lens entrance and the external cy- pend on the geometry of the lens. lindrical electrode was significantly less than the transverse The charged particle beam trajectories were calculated and longitudinal dimensions of the lens. Since this clearance numerically using the TEO computer code for two- determines the configuration of the edge field, it should have dimensional electrostatic fields. We considered beams enter- a substantial influence on the focusing properties of the lens. ing the coaxial cylindrical lens parallel to its longitudinal The influence of the position of the entrance diaphragm is in fact the subject of the present paper. Figure 1a presents a diagram of a coaxial cylindrical lens consisting of two cylindrical electrodes, a flat end-face diaphragm at the entrance with an open back face. The inner electrode and the diaphragm were constructed as a single unit with a grounded housing, onto which the outer electrode was mounted through an insulator. Application of a voltage V to the outer electrode while the diaphragm and inner electrode remain grounded leads to the appearance of a field on the lens. Therefore, in what follows we will call the outer cylindrical electrode the field-assigning electrode. In such a lens, the equipotentials of small absolute values of the potential pass near the inner electrode, then run nearly parallel to the end-face diaphragm, and finally close in the space between the housing and the inner electrode. Here, as a consequence of the open back face they penetrate into the space beyond the electrodes. The equipotentials of large absolute values of the potential have a simpler configuration, smaller extent, and surround the field-assigning electrode from the inside and out. It can be seen from Fig. 1b that in the region of the diaphragm located near the field-assigning electrode ͑the edge-field region at the lens entrance͒, the values of the potential at fixed radius grow with increasing value of the longitudinal coordinate whereas away from the diaphragm ͑the edge-field region at the lens exit͒, near the inner electrode, the potential at first grows and then falls. In FIG. 1. a:—Coaxial cylindrical lens: 1,2—cylindrical electrodes, 3—flat the rest of the exit edge-field region the dependence of the diaphragm, 4—housing, 5—charged particle trajectories; b—equipotential potential on the longitudinal coordinate is analogous to this plot.

1063-7842/98/43(6)/4/$15.00 726 © 1998 American Institute of Physics Tech. Phys. 43 (6), June 1998 L. P. Ovsyannikova and T. Ya. Fishkova 727 axis and determined the working regimes focusing the beam increased according to a linear law, whereas with increase of on the lens axis. For the characteristic geometry of the lens the clearance it at first grows ͑curves 1–3͒ and then falls with ratio of the radii of the large and small cylindrical elec- ͑curves 4 and 5͒. trodes R/␳ϭ10 the distance from the diaphragm to the field- From the aforesaid it can be concluded that for focusing assigning electrode was varied over wide limits 0Ͻs/R of a parallel beam onto the lens axis, coaxial cylindrical р1.5. The length of the latter was varied within the limits lenses should be designed with a small clearance between the 0.5рL/Rр1.9 so that the length of the lens from the dia- entrance diaphragm and the field-assigning electrode. phragm to the back face did not change (lϭLϩsϭ2R It is of interest to investigate the influence of the edge ϭconst). The minimum clearances s/R were governed by field on focusing of a coaxial cylindrical lens for which the the magnitude of the breakdown voltage, the maximum length of the field-assigning electrode was not varied. The clearances lead to vigorous growth of the lens excitation due latter was chosen from the condition that the potential differ- to decrease of the length of the field-assigning electrode. ence between the electrodes not exceed the accelerating po- Figure 2 displays the calculated results. It plots the focal tential, i.e., a lens excitation Fр1. We calculated a lens with length, measured from the back of the lens, and the entrance R/␳ϭ10 and LϭRϭconst for which we varied the clearance radius of the paraxial beam trajectory—the trajectory around within the limits 0Ͻs/Rр2. In this case the overall length of which the other beam trajectories are focused, as functions of the lens from the diaphragm to the back face varied within the lens excitation F ͑FϭeV/␧, where e is the charge of the the range 1ϽlRр3. Results of the numerical calculation are particle and ␧ is its energy͒. The small brackets mark off the plotted in Fig. 3. As can be seen, focusing takes place over minimum and maximum excitations bounding the existence the entire range of variation of the clearance which deter- region of the focus. For small clearances (sр0.3R) focusing mines the edge field. However, starting at sϭ0.4R ͑curves is realized over a wide range of variation of the distance 3–6͒ the focal region is found near the lens and does not from lens exit to focus ͑focal length͒ (100р␭/Rр0.2); here extend beyond ␭Х2R, and as the clearance is increased this the value of the minimum achievable focal length decreases region decreases in extent. Note that the focal length and the as the clearance is decreased. At large clearances sХR the initial radius of the paraxial beam trajectory for the clear- focal region substantially narrows, and for sϭ0.5R focusing ances sϭR and sϭ2R practically coincide. Consequently, is generally absent. The entrance radius of the paraxial beam from the point of view of focusing it is without sense to trajectory in the focusing regime falls as the excitation is make the clearance greater than the radius of the outer cylindrical electrode. For small clearances between the entrance diaphragm and the field-assigning electrode (sр0.3R) focusing exists within wide limits, and while the focal length varies only slightly with the clearance ͑curves 1 and 2 in Fig. 3a͒, the radius of the paraxial trajectory at the entrance to the coaxial cylindrical lens varies considerably ͑curves 1 and 2 in Fig. 3b͒. Note that with increase of the ratio of the electrode radii the region in s in which focusing is realized increases somewhat and for R/␳ϭ100 is 0Ͻsр0.4R. As this ratio is decreased, the indicated region narrows substantially, and for R/␳ϭ2 the maximum clearance is sϭ0.1R. In the choice of the optimal geometry of a coaxial cylindrical lens, the quality of focusing is important, where the latter is defined by the radius of the spot formed by the lens when focusing a ring beam on the lens axis. Figure 4 plots the radius of the spot in the plane passing through the intersection point of the longitudinal axis of the lens and the paraxial beam trajectory ͑the trajectory around which the other trajectories are focused͒ as a function of the clearance between the diaphragm and the field-assigning electrode. The solid curves correspond to ring thickness at lens entrance ⌬r0ϭ0.05R, and the dashed ones, to ⌬r0ϭ0.1R.In the small clearance region the radius of the focal spot is two orders of magnitude smaller than the thickness of the entrance ring while for sуR this radius grows substantially. In addition, for sу0.4R, as can be seen from Fig. 3, the focal region is small. Therefore the geometry of a coaxial cylin- FIG. 2. Dependence of the focal length ͑solid curves͒ and radius of the drical lens with a large clearance between the entrance dia- paraxial beam trajectory ͑dashed curves͒ on the excitation for a lens with phragm and the field-assigning electrode is not of interest R/␳ϭ10 and l/Rϭ2ϭconst for various clearances between the diaphragm and the field-assigning electrode. s/R: 1—0.1, 2—0.25, 3—0.75, 4—1.0, from the point of view of beam focusing on the lens axis. 5—1.5. Figure 5 plots the radius of the axially focused beam 728 Tech. Phys. 43 (6), June 1998 L. P. Ovsyannikova and T. Ya. Fishkova

FIG. 4. Radius of the focused spot as a function of the clearance between the diaphragm and the ﬁeld-assigning electrode for a coaxial cylindrical lens

with R/␳ϭ10 and L/Rϭ1. Solid curves—⌬r0ϭ0.05R, dashed curves— ⌬r0ϭ0.1R; 1,2—␭ϭ0.5R; 3,4—␭ϭR.

Thus, for a coaxial cylindrical lens, over a wide range of variation of the ratio of electrode radii for various lengths of the field-assigning electrode, the optimum clearance between the entrance diaphragm and the field-assigning electrode from the point of view of focusing the beam on the axis is sр0.1R. In this case, to determine the focal length and the radius of the paraxial beam trajectory ͑the trajectory about which focusing takes place͒ it is possible to use the corresponding formulas from Ref. 2. In conclusion, note that we have also investigated a coaxial cylindrical lens with an alternative power feed arrangement, where the voltage is fed to the inner electrode while the outer electrode and diaphragm remain grounded. In this case the design of the lens simplifies since the need for a housing falls away and the inner electrode is mounted to the diaphragm through an insulator. However, such a power feed arrangement leads to a substantial change in the edge field at the entrance to the lens and, as a consequence, causes the beam to behave in a different way, leading to degradation of

FIG. 3. Same as in Fig. 2 for a lens with R/␳ϭ10 and L/Rϭ1 ͑a,b͒. s/R: 1—0.1, 2—0.25, 3—0.4, 4—0.5, 5—1.0, 6—2.0.

over a wide range of variation of the focal length for small clearances between the diaphragm and the field-assigning electrode for a coaxial cylindrical lens with ratio of electrode radii R/␳ϭ10. It can be seen from the figure that only in the region 2р␭/Rр10 is the spot radius for a lens with clearance sϭ0.25R smaller than for sϭ0.1R. Outside this region of focal length values the clearance value sϭ0.1R provides FIG. 5. Dependence of the radius of the focal spot on position measured the smallest spot size, whose magnitude varies only slightly from the back of the lens for small clearances between the diaphragm and for r ϭ0.005R the spot radius r 0.003R, while for ͑ ⌬ 0 iХ the field-assigning electrode. s: 1—0.1R, 2—0.25R; solid curves—⌬r0 ⌬r0ϭ0.1R it is equal to riХ0.009R͒. ϭ0.05R, dashed curves—⌬r0ϭ0.1R. Tech. Phys. 43 (6), June 1998 L. P. Ovsyannikova and T. Ya. Fishkova 729 focusing in general and on the axis of the lens in particular. grow with the radius. This has to do with the fact that the particles in the edge field 1 L. P. Ovsyannikova and T. Ya. Fishkova, Pis’ma Zh. Tekh. Fiz. 22͑16͒, are accelerated, and this acceleration and also the radial force 39 ͑1996͓͒Tech. Phys. Lett. 22, 660 ͑1996͔͒. of the lens decrease with growth of the radius. For the main 2 L. P. Ovsyannikova and T. Ya. Fishkova, Zh. Tekh. Fiz. 67͑8͒,89͑1997͒ voltage feed arrangement ͑Fig. 1͒ the particles are slowed ͓Tech. Phys. 42, 935 ͑1997͔͒. down, and this slowing down and the radial force of the lens Translated by Paul F. Schippnick TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Focusing of electrons reﬂected from a crystal with loss of energy M. V. Gomoyunova, I. I. Pronin, and N. S. Faradzhev

A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia ͑Submitted November 3, 1997͒ Zh. Tekh. Fiz. 68, 128–133 ͑June 1998͒ A study is made of the features arising in the spatial distributions of reflected electrons as a result of a focusing effect. Experiments are conducted on single-crystal Mo ͑100͒ with primary electron energies of 0.5–2 keV and detection of electrons which lose fixed amounts of energy up to 300 eV. An analysis of the data establishes the dependence of the electron focusing efficiency on the amount of energy loss. It is shown that when electrons are reflected with single losses through plasmon excitation, the magnitude of the effect is determined mainly by the average number of scattering atoms encountered by an electron along its path to the surface. When the energy losses are high, defocusing owing to multiple elastic and inelastic scattering of the electrons is found to predominate. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02306-X͔

INTRODUCTION polar angle ␪ of the escaping electrons. The azimuthal exit angle ␸ could be varied by rotating the sample about an axis In recent years, two types of diffraction effects observed perpendicular to its surface. This instrument design made it during bombardment of test objects by medium energy elec- possible to measure the distributions I͑␸͒ of the reflected trons have come into active use for the structural analysis of electrons with respect to the azimuthal exit angle ␸ for dif- solid surfaces. The first is the diffraction of reflected elec- ferent polar angles ␪ and thereby to obtain almost complete trons, which shows up in the form of peaks in their spatial diffraction patterns. The energy ⌬E lost by the electrons distributions oriented along close-packed atomic rows and upon reflection was varied as a parameter. The intensity of planes of the crystal,1–3 and the second is the dependence of the electron flux was measured by modulating the current of the intensity of electron reflection and Auger electron emis- the primary beam. Measurements were made for primary sion on the angle of incidence of the primary electrons rela- electron energies E of 0.5–2 keV. Principal attention was tive to the crystal axes.4,5 Here the emission peaks appear for r devoted to an energy of 1.25 keV, which is high enough for the same orientations of the incident electrons at which the the focusing effect to show up but not so high that the energy peaks in the reflected electron diffraction patterns are ob- resolution of the analyzer no longer allows the main peaks in served. Thus far, it has been established that electron focus- the characteristic electron energy loss spectrum to be re- ing effects in the crystal play a dominant role in the forma- solved. tion of these diffraction features. However, not all aspects of The method of preparing the test sample has been de- this phenomenon have been studied adequately. For ex- scribed elsewhere.11 The cleanliness of the surface of the ample, there have been almost no studies of the focusing molybdenum single crystal was monitored by electron Auger behavior of electrons reflected with different energy losses. spectroscopy and the structure of its surface region, by low- Only a few papers touch on this problem.6–9 In the mean- energy electron diffraction ͑LEED͒. The measurements were time, it is known that the probe depth, the contrast of the made at room temperature in a vacuum of 5ϫ10Ϫ10 Torr. diffraction patterns, etc., depend on the magnitude of the energy loss. Thus, it seemed appropriate to us to make a systematic study of the spatial distributions of electrons re- RESULTS OF THE MEASUREMENTS AND DISCUSSION flected with different energy losses over a wide range of exit angles. Single crystal Mo ͑100͒, which has been used in ear- A general idea of the appearance of the electron focusing lier work, was chosen as the object of study. effect in the simplest case of quasielastic reflection (⌬E Ͻ1 eV) is provided in Fig. 1a, which shows a two dimensional map of the distribution of the intensity I(␪,␸) of the EXPERIMENTAL TECHNIQUE electrons over the polar and azimuthal exit angles, obtained The measurements were made in a special ultrahigh- by synthesis of a family of azimuthal scans measured at an vacuum angle-resolved secondary electron emission spec- energy Erϭ2 keV. The data are shown in a stereographic trometer that has been described elsewhere.10 The energy projection. The center of the circle corresponds to the normal resolution of the modified-plane-mirror analyzer was 0.4%, to the surface of the sample, and the outer circle to emission and the angular resolution was about 1°. The electron beam of electrons along the surface. The distribution is shown in a of the spectrometer bombarded the sample with electrons linear scale of gray shadings in which the maximum reflec- along the normal to its surface. The energy analyzer could be tion corresponds to white and the minimum, to black. Quasi- rotated around the sample, making it possible to vary the elastic electron reflection from single-crystal Mo ͑100͒ is

FIG. 1. A two dimensional map of the intensity distribution I(␪,␸)of quasielastic electron reflection over the polar and azimuthal exit angles obtained at an energy Erϭ2 keV for single crystal Mo ͑100͒͑a͒and a stereographic projection of this face indicating the most close-packed planes and directions of the crystal ͑b͒. evidently highly anisotropic and has a distinct diffraction pattern. Its symmetry reflects the symmetry of the ͑100͒ face of the body-centered cubic crystal, as illustrated in Fig. 1b, which shows the corresponding stereographic projection. A FIG. 2. Azimuthal angular distributions of electrons inelastically reflected comparison of the data of Fig. 1a with this projection makes from single crystal molybdenum, measured for a primary electron energy it possible to identify the main peaks in this pattern at once. Erϭ1.25 keV and a polar angle of emission of 55° ͑from the normal to the Clearly, they are caused by the escape of electrons along sample surface͒. The energy losses ⌬E for different groups of electrons: close-packed directions of the crystal, such as ͗111͘ and 1—less than 1 eV ͑quasielastically reflected electrons͒, 2—24 eV ͑electrons which have lost energy exciting a bulk plasmon͒, 3—50, 4—100, and 5— ͗110͘. These peaks were observed over the entire range of 250 eV. energies that was studied and are caused by focusing of electrons moving along strings of atoms with small interatomic distances. These results are all in good agreement with data where I and I are the intensities of the electron flux in on quasielastic scattering of electrons which we have ob- max min the analyzed peak of the I(␸) distribution and in its deepest tained previously11 for another face of molybdenum, minimum, respectively. Mo ͑110͒. By analyzing a family of azimuthal distributions mea- The focusing effect also shows up in the spatial distribu- sured for a number of values of ⌬E and ␪, it is possible to tions of the electrons reflected from the crystal with a loss of determine the electron focusing efficiency ␹(⌬E) as a func- energy. As an illustration, Fig. 2 shows several typical azi- tion of the magnitude of the energy losses for the major muthal distributions of the electrons reflected with losses close-packed directions of the crystal. In order to obtain this ⌬EϽ300 eV. ͑In accordance with the symmetry of the information, the distributions I(␸) were measured for 6 dif- Mo ͑100͒ face, the range of variation of the azimuth here has ferent angles ␪ with a step in ⌬E amounting to 2 eV. These been limited to half a quadrant͒. These data were obtained data are represented in Fig. 3. It is clear that for small energy for E ϭ1.25 keV and a polar emission angle of ␪ϭ55°. A r losses ͑in the region ⌬EϽ50 eV͒, the focusing is quite distinct diffraction structure is visible in all the curves. The strong and ␹ can reach 75%. It is noteworthy that ␹ depends strongest feature is the peak observed at ␸ϭ0, which is monotonically on ⌬E. For most of the curves, nonmonoto- caused by focusing of electrons along the most close-packed nicities are observed at the same values of ⌬E. The fine crystal direction, ͗111͘. Weaker peaks are observed at azi- structure of the ␹(⌬E) curves for small angles ␪ looks like it muths of ␸ϭ18 and 45°, corresponding to orientations of the is superimposed on a horizontal straight-line background, escaping electrons along the ͑130͒ and ͑100͒ planes. The and as the polar angle is increased, the background falls off shape of the I(␸) curves depends on the magnitude of the with increasing ⌬E. This is typical of the ␹(⌬E) curves for energy losses experienced by the electrons. As ⌬E increases, large energy losses (⌬EϾ50 eV), as well. However, even there is a noticeable drop n the focusing peak for electrons here the rate of decrease in the degree of focusing with rising along the ͗111͘ direction, which essentially vanishes by ⌬ ⌬E depends on the polar angle, increasing with rising ␪. For Ϸ200 eV, and then is inverted to form a minimum. Such sufficiently large angles, the ␹(⌬E) curves are observed to behavior is typical of the focusing peaks observed for elec- settle into a region of negative values corresponding to the trons moving along other close-packed directions. above-noted inversion in the diffraction structure of the an- For a quantitative estimate of the focusing effect for gular distributions I(␸) for electrons reflected with large en- electrons reflected along the low-index directions hkl with ͗ ͘ ergy losses. different energy losses we can use In order to clarify the nature of the observed nonmono- ␹ϭ͓͑ImaxϪImin͒/Imax͔•100%, tonicities in ␹(⌬E), let us compare them with the character- 732 Tech. Phys. 43 (6), June 1998 Gomoyunova et al.

ϭ11 eV, corresponds to overlapping excitation peaks for a surface plasmon (h␻sϭ9.5 eV) and a low-energy bulk plas- 12 mon (h␻vϭ10.4 eV). The second, larger loss at ⌬E ϭ23.5 eV is caused by the generation of the fundamental bulk plasmon of molybdenum. The third peak at ⌬E ϭ48 eV is usually attributed to dynamic polarization of electrons in the shallow core 4p level,13 but some part of it may come from double excitation of bulk plasmons. Besides these peaks, the spectrum includes a noticeable background of multiple losses, which increases with rising ⌬E. Comparing the data of Figs. 3 and 4 shows that electron focusing increases if the electrons experience single energy loss through excitation of bulk plasmons as they are reflected. For small polar escape angles, the anisotropy of the distributions for these electrons is greater than in the case of quasielastic scattering. Let us analyze these data in terms of a simple model that we have used before to study the focusing of quasielastically scattered14 and primary electrons.15 This model is based on the assumption that the reflection of electrons from a solid is the result of single, large-angle electron–phonon scattering events. Although a rigorous solution of the problem will include multiple quasielastic scattering processes,16 this model is entirely appropriate for describing the motion of electrons in a thin subsurface layer of the crystal. In this case the trajectory of the electrons inside the solid can be approxi- mated by a broken line consisting of two straight segments. FIG. 3. Plots of ␹(⌬E), the electron focusing efficiency as a function of the The first of these (l ) corresponds to motion of the electron energy lost by the electrons during reflection. These data were obtained for 1 into the interior of the crystal ͑to a point where a quasielastic Erϭ1.25 keV and apply to the following close-packed directions of the crystal: 1—͗012͘, 2—͗112͘, 3—͗110͘, 4—͗111͘, and 5—͗122͘. All the scattering event occurs͒ and the second segment (l2), to the curves are plotted on the same scale as that indicated for curve 5. For the motion of the scattered electron toward the surface. Note other curves, only the lines corresponding to the shifted zero ordinate are that, since the focusing of the reflected electrons takes place indicated. along path l2 , this is the parameter in the problem which should have a controlling influence on the magnitude of the istic electron energy loss spectrum in molybdenum. One observed effect. Estimating l2 is simplest in the case of quasielastic scat- such spectrum is shown in Fig. 4. It was taken for Er ϭ0.5 keV and a polar angle of 45°. There are three distinct tering of the electrons, with l2ϭ␭/(1ϩcos ␪), where ␭ is the maxima in this spectrum. The first, observed at ⌬E mean free path of the electron with respect to an inelastic interaction. It is clear that the length of the focusing string in this case becomes greater as ␪ increases. For inelastic reflection of electrons involving single excitation of plasmons, yet another elementary act is involved which may take place before or after the quasielastic scattering into the backward hemisphere. Since plasmon excitation takes place through a long-range Coulomb interaction with the electronic sub- system of the crystal, the probability of generating plasmons is essentially independent of electron focusing.15 In addition, since the electron generates long-wavelength, small- momentum plasmons with a higher probability,17 there is no significant change in the direction of motion ofa1keV electron. Thus, the events in which plasmons are generated should have little effect on the focusing process. Thus, we assume that the observed differences in ␹ for electrons which excite plasmons and are scattered only quasielastically origi- nate primarily in a difference in the average exit depths of these groups of electrons and, therefore, in the different lengths of the focusing strings. Since the exit depth is FIG. 4. Characteristic electron energy loss spectrum in molybdenum mea- roughly twice as large for electrons reflected with plasmon sured for Erϭ500 eV. excitation as for quasielastically scattered electrons, we may Tech. Phys. 43 (6), June 1998 Gomoyunova et al. 733

both dependences are nonlinear and reveal a distinct slowing down in the growth of ␹ with increasing n. This is evidence of the onset of an electron defocusing process,11 which arises because of multiple elastic scattering of electrons in sufficiently long atomic strings. The effect of defocusing becomes especially noticeable when points on curves 1 and 2 corresponding to the same direction are compared pair by pair. In fact, while a doubling of the length of strings having a small number of scatterers ͑going from nϭ1–2 to nϷ3͒ slightly increases the degree of focusing of electrons that have excited plasmons, in the case of the ͗111͘ and ͗110͘ directions, where the initial n are already quite large, a further increase in n has the contrary effect of making the points on curve 2 appear to lie below those on curve 1. The weakening of the diffraction structure of the distributions for high energy losses ͑the drop in the curves in Fig. FIG. 5. Plots of the electron focusing efficiency ␹(n) as a function of the 3 for ⌬EϾ50 eV͒ is evidently also related to enhanced de- number of scatterers encountered on the path of the electrons toward the focusing of the electrons. Here inelastic interactions of the surface for emission along different close-packed directions of the crystal. 1—Quasielastic reflection, 2—reflection with single excitation of a bulk electrons with the crystal play an important role, along with plasmon; Erϭ1.25 keV. multiple elastic interactions. The point is that the focusing of reflected electrons, which causes them to escape manly along the atomic rows, leads to an increase in the electron density assume that the length l2 is also twice as large. Thus, by near the ion core of the crystal, and this should be accompa- comparing the ␹(⌬E) curves obtained in the single energy nied by increased elastic, as well as inelastic, scattering of loss region for different angles ␪, we can understand how the the electrons at large angles. As a result, the intensity of the focusing efficiency for electrons moving along different flux of electrons moving toward the surface along the atomic crystallographic directions depends on their path lengths. rows falls off more rapidly than in other orientations and, in The focusing properties of the atomic strings depend on particular, in those for which minima of the I(␸) curves are the characteristic distance between the atoms, as well as on observed for quasielastic reflection. The influence of these their length. Thus, it is appropriate to present the data of Fig. processes increases as the path traversed by the electrons in 3 as plots of ␹(n), where n is the average number of scat- the direction toward the surface becomes longer; this hap- terers encountered along an electron’s path as it moves to- pens when the energy loss ⌬E and angle ␪ increase. This ward the surface along a given string. In terms of our model, factor also evidently explains the faster drop in the ␹(⌬E) this number can be estimated as follows: nϭl2 /d, where d is curves seen upon an increase in the polar angle of escape of the interatomic distance along the given direction. Plots of the electrons. The observed inversion in the structure of the ␹(n) obtained on the basis of data for quasielastic scattering spatial distributions of the electrons which have experienced and scattering with single excitation of a bulk plasmon are large energy losses during reflection is of a similar nature. shown in Fig. 5 ͑curves 1 and 2, respectively͒. They charac- Thus, an extension of the range of energy losses by the terize the focusing of electrons by different strings of atoms reflected electrons detected in medium energy electron dif- oriented along 6 low-index directions of the crystal ͑͗111͘, fraction greatly complicates the picture of the phenomena, ͗110͘, ͗133͘, ͗012͘, ͗112͘, ͗122͒͘ with interatomic distances since multiple elastic and inelastic scattering processes come varying between 2.72 and 9.45 Å. Pairs of points referring to into play which defocus the electrons. This, on one hand, the same direction are denoted by the same symbol. It is makes it more difficult to analyze the patterns and model clear from the figure that all the data fit fairly well on two them numerically, and, on the other, should reduce the con- monotonically increasing curves. This means that for each trast in the measured distributions. group of electrons, the number of scatterers in the string plays a controlling role in focusing them, while the inter- CONCLUSION atomic distance is less important. The data for the ͗122͘ direction are especially characteristic in this regard. For the An experimental study of the focusing effect for elec- other atomic strings, an increase in the polar angle ␪, leading trons with energies of the order of 1 keV using an analysis of to lengthening of l2 and an increase in n, is accompanied by the spatial distributions of the electrons reflected from a monotonic rise in ␹, but in this case, although the path single-crystal Mo ͑100͒ with different energy losses ⌬E has length l2 is quite large (␪ϭ70°), the values of n are quite revealed the following behavior: small ͑1.6 and 3.2͒ because of the large interatomic distance 1. The effect shows up over a fairly wide range of elec- and, accordingly, the values of ␹ are also lower. tron energy losses up to roughly 200 eV. The size of this A monotonic growth in the focusing efficiency of range depends on the polar angle of escape ␪ of the electrons. quasielastically reflected electrons is observed at least up to 2. For electrons reflected with small ͑single event͒ en- four scatterers, and for electrons reflected in processes in- ergy losses, the focusing efficiency is determined principally volving plasmon excitation, up to eight. At the same time, by the number of atoms n encountered by an electron along 734 Tech. Phys. 43 (6), June 1998 Gomoyunova et al. its path as it moves toward the surface along a close-packed 5 S. Valeri and A. Di Boni, Surf. Rev. Lett. 4, 141 ͑1997͒. direction, and it increases with n. The spatial orientation of 6 A. Mosser, Ch. Burggraf, S. Goldsztaub, and Y. H. Ohtsuki, Surf. Sci. 54, the exit directions and the packing density of the atoms in 580 ͑1976͒. 7 M. V. Gomoyunova, I. I. Pronin, and S. L. Zaslavski , Fiz. Tverd. Tela them are of less importance. ͑Leningrad͒ 24, 2006 ͑1982͓͒Sov. Phys. Solid State 24, 1145 ͑1982͔͒. 3. The focusing efficiency for electrons reflected with 8 S. Hufner, J. Osterwalder, T. Greber, and L. Schlapbach, Phys. Rev. B 42, large energy losses declines, as a rule, with increasing ⌬E, 7350 ͑1990͒. although the ␹(⌬E) curves are nonmonotonic and correlate 9 M. Erbudak, M. Hochstrasser, and E. Wetli, Phys. Rev. B 50, 12973 ͑1994͒. with the energy loss spectrum. The weakening of the focus- 10 ing is caused by multiple elastic and inelastic scattering of I. I. Proninn, M. V. Gomoyunova, D. P. Bernatski, and S. L. Zaslavski, Prib. Tekh. E´ ksp., 1, 175–179 ͑1982͒. the electrons by the crystal ͑defocusing processes͒. The 11 M. V. Gomoyunova, I. I. Pronin, and N. S. Faradzhev, Zh. E´ ksp. Teor. peaks of the distributions are damped most rapidly along the Fiz. 110, 311 ͑1996͓͒JETP 83, 168 ͑1996͔͒. most close-packed directions of the crystal, for which an 12 J. H. Weaver, D. W. Lynch, and C. Golson, Phys. Rev. B 10, 501 ͑1974͒. inversion of the diffraction patterns is observed at large ⌬E 13 M. V. Zharnikov, V. D. Gorobchenko, and I. L. Serpuchenko, Zh. E´ ksp. and ␪. Teor. Fiz. 92, 228 ͑1987͓͒Sov. Phys. JETP 65, 128 ͑1987͔͒. 14 I. I. Pronin, M. V. Gomoyunova, and N. S. Faradzhev, Fiz. Tverd. Tela This work was performed as part of the Program on ͑St. Petersburg͒ 39, 752 ͑1997͓͒Phys. Solid State 39, 752 ͑1997͔͒. 15 M. V. Gomoyunova and I. I. Pronin, Zh. Tekh. Fiz. 67͑8͒, 117 ͑1997͒ Surface Atomic Structures, Project No. 95-1.21. ͓Tech. Phys. 42, 961 ͑1997͔͒. 16 M. A. Vicente Alvares, H. Ascolani, and G. Zampieri, Phys. Rev. 53, 1 S. A. Chambers, Surf. Sci. Rep. 16, 261 ͑1992͒. 7524 ͑1996͒. 2 M. Erbudak, M. Hochstrasser, and E. Wetli, Mod. Phys. Lett. B 8, 1759 17 H. Raether, Excitation of Plasmons and Interband Transitions by Elec- ͑1994͒. trons. Springer Tracts in Modern Physics, Vol. 88. Springer-Verlag, Ber- 3 N. S. Faradzhev, M. V. Gomoyunova, and I. I. Pronin, Phys. Low-Dim. lin, Heidelberg, New York ͑1980͒. Struct. 314,93͑1997͒. 4 S. Mroz, Prog. Surf. Sci. 48, 157 ͑1995͒. Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

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Mechanisms for formation of a population inversion in the levels of metal atoms and ions in a plasma jet V. P. Starodub

Institute of Electron Physics, National Academy of Sciences of Ukraine, 294016 Uzhgorod, Ukraine ͑Submitted May 5, 1997͒ Zh. Tekh. Fiz. 68, 134–135 ͑June 1998͒ The mechanisms for formation of population inversions in plasma jets of lithium, sodium, cadmium, and strontium vapor are studied. The primary contribution to forming a population inversion over the transitions of the alkali atoms is found to be three-body electron–ion recombination, and for transitions between the ion levels of strontium and cadmium, inelastic collisions with the buffer gas play an important role. Using helium instead of argon as a buffer leads to a substantial increase in the magnitude of the inversion. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02406-4͔

It is important to examine the conditions and analyze the strontium plasmas, reaction ͑1͒ does not make a signiﬁcant mechanisms for population inversions in moving plasmas contribution to the formation of a population inversion. More 2 with rapidly varying density and temperature in order to as- likely, a population inversion is created over the 6 S1/2 2 certain the feasibility of plasmadynamic lasers. In this paper, 5 P3/2 transition of strontium through processes involving a study is made of plasma jets formed from the vapors of →the buffer gas He, as is indicated by the dependence of the lithium, sodium, cadmium, and strontium. The experiments magnitude of the inversion on the helium pressure ͑Fig. 2͒. were conducted on a device described elsewhere.1 The Note that all the curves shown in Fig. 2 were obtained with a source of the jet was a dc plasmatron with a 3-mm-diam constant feed rate for the metals. An analysis of published acoustic nozzle. Argon and helium were used as buffer data4–7 and of the level diagrams for the strontium and he- gases. The excited level populations were determined opti- lium atoms and the strontium ion shows that the following cally from the intensities of spectral lines, which in turn were elementary processes may play an important role in populat- determined by comparison with the intensity of a standard ing these levels: source which has a known spectral energy distribution. These studies showed that population inversions develop between certain levels of lithium and sodium atoms and cadmium and strontium ions. These atoms and ions each have one electron in their outer shell, so they are similar. The data show, however, that the behaviors of the population inversions over these atoms and ions are not at all identical, either in terms of the initial conditions or along the jet. Figure 1 shows plots of the magnitudes of the population inversions as functions of distance along the jet for the transitions of these elements which are the most promising from the standpoint of lasing. For lithium and sodium the maximum population inversion is observed near the exit section of the nozzle and falls off rapidly along the jet, following the variation in the electron density.2,3 This indicates that processes involving electrons, in particular, three-body electron–ion recombination

AϩϩeϩeϭA*ϩe, ͑1͒ play an important role in creating the population inversion. As opposed to the alkali metals, the peak inversion in the ϩ ϩ FIG. 1. The variations in the magnitudes of population inversions over populations of the Sr and Cd ions is observed a signiﬁcant 2 2 transitions of Li I, Na I, Cd II, and Sr II along the jet: 1—3 S1/2 2 P3/2 distance away from the nozzle exit Fig. 1, curves 3 and 4 . 2 2 2 2 → 2 ͑ ͒ Li I, 2—5 S1/2 3 P3/2 Na I, 3—6 S1/2 5 P3/2 Sr II, and 4—5s D5/2 This is unambiguous evidence that in the cadmium and 5 2P Cd II.→ → → 3/2

In order to clarify the role of processes ͑1͒–͑4͒ we have performed an additional experiment. Some cesium, which is easily ionized, was added to the He–Sr plasma jet. The cesium density was 1013–1014 cmϪ3 at most. In this experiment, the cesium impurity enhanced the electron density by roughly a factor of two, without affecting the electron tem- 2 perature significantly. If the 6 S1/2 level is populated by recombination ͑reaction ͑1͒͒, then the intensity of the corresponding line should have increased. If processes ͑2͒–͑4͒ predominate in filling the level, then the intensities of the 430.5 and 416.2 nm lines should have decreased, since in the first case ⌬NϰNe and in the second case ⌬Nϰ1/Ne . Intro- ducing a cesium impurity into the He–Sr plasma jet did not increase the intensities of these lines but decreased them by ϳ20%. It can thus be inferred that reactions ͑2͒–͑4͒ make a 2 significant contribution to filling the 6 S1/2 state of Sr II. To clarify the role of the buffer gas in creating a population inversion over the cadmium ion lines, helium was replaced FIG. 2. The magnitudes of population inversions over transitions of Li I, by argon. It was found that in an Ar–Cd plasma the inver- 2 Na I, Cd II, and Sr II as functions of the helium pressure: 1—3 S1/2 2 22 2 2 2 sion is 6–8 times smaller than in the He–Cd plasma jet. This 2 P3/2 Li I, 2—5s D5/2 5 P3/2 Cd II, 3—5 S1/2 3 P3/2 Na I, and →4—6 2S 5 2P Sr II. → → is explained by the fact that the cross section for the endot- 1/2→ 3/2 hermic charge exchange on the Cdϩ level is roughly an order of magnitude smaller than the corresponding cross section for the endothermic charge exchange in He–Cd. In addition, SrϩϩϩSr*͑41D ,53P ͒ Srϩ*͑62S ͒ϩSrϩ, ͑2͒ a Penning reaction takes place in the He–Cd plasma, but not 2 0,2 → 1/2 in argon. SrϩϩϩHe*͑21S ,23S ͒ Srϩ*͑62S ͒ϩHeϩ, ͑3͒ 0 1 → 1/2 Based on data obtained from a study of metal vapor Sr͑0͒ϩHe*͑nl1,3L͒ Sr*͑4p55s26s,4d͒ plasma jets, therefore, it has been found that three-body → electron–ion recombination predominates in the formation of ϩHe Srϩ*͑62S ͒ϩHeϩe. ͑4͒ → 1/2 population inversions over transitions of the alkali atoms, Two of these reactions involve atoms of the buffer gas. while inelastic collisions with the buffer gas play a signifi- The rate constants for reactions ͑2͒ and ͑3͒ are large and, cant role for transitions between levels of the strontium and according to the data of Ref. 5, are equal, respectively, to cadmium ions, and if helium is used instead of argon as a 1ϫ10Ϫ8 cmϪ3 sϪ1 and 2ϫ10Ϫ8 cmϪ3 sϪ1. It was not pos- buffer gas, there is a substantial increase in the magnitude of sible to find the rate for reaction ͑4͒, but there is some the inversion. indication4 that above 19–25 eV there are some autoionizing levels of the strontium atom which can decay to form a 1 S. P. Bogacheva, M. F. Veresh, I. P. Zapesochnyy et al., Ukr. Fiz. Zh. 30, 2 strontium ion in the 6 S1/2 state. The cross section for this 186 ͑1995͒. 2 kind of reaction for different levels of Sr* I is estimated to be S. P. Bogacheva, L. V. Voronyuk, I. P. Zapesochnyy et al., Zh. Prikl. Ϫ14 Ϫ17 2 Mekh. Tekh. Fiz., No. 6, 10 ͑1984͒. in the range 10 –10 cm . 3 M. F. Veresh, I. P. Zapesochny, and V. P. Starodub, Zh. Tekh. Fiz. 57, The inversion does not exist for low concentrations of 572 ͑1987͓͒Sov. Phys. Tech. Phys. 32, 347 ͑1987͔͒. helium ͑Fig. 2͒, since the helium concentration is too low for 4 A. A. Borovik, I. S. Aleksakhin, V. F. Bratsov, and A. B. Kuplyauskene, 2 Opt. Spektrosk. 53, 976 ͑1982͓͒Opt. Spectrosc. ͑USSR͒ 53, 583 ͑1982͔͒. pumping the 6 S1/2 level ͑reactions ͑3͒ and ͑4͒͒. As the he- 5 B. M. Smirnov, Excited Atoms ͓in Russian͔,Eńergoizdat, Moscow ͑1982͒, lium concentration is raised, a population inversion appears, 232 pp. reaches a maximum, and then falls off. The decrease in the 6 V. N. Kondrat’ev and E. E. Nikitin, Kinetics and Mechanism of Gas- magnitude of the inversion is explained by the fact that, as Phase Reactions ͓in Russian͔, Nauka, Moscow ͑1975͒, 397 pp. 7 the helium concentration is raised, relaxation processes in the B. M. Smirnov, Asymptotic Methods in Collision Theory ͓in Russian͔, Atomizdat, Moscow 1973 , 294 pp. plasma are faster and the metastable states of strontium and ͑ ͒ helium are destroyed more rapidly. Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Mathematical modeling of the nonstationary operation of thermoelectric current sources Yu. I. Dudarev and M. Z. Maksimov

Sukhumi Physicotechnical Institute, Academy of Sciences of the Abkhaz Republic, Sukhumi, Abkhazia ͑Submitted April 14, 1997͒ Zh. Tekh. Fiz. 68, 136–137 ͑June 1998͒ ͓S1063-7842͑98͒02506-9͔

Thermoelectric energy conversion based on the Seebeck, where T0(x,t) is the temperature of the system in neglect of Thomson, and Peltier effects has been used extensively for the dependence of the thermal properties and thermal sources several decades in various areas of science, technology, and on the temperature and current effects. low-level power generation. In refrigeration technology, as Going from a multilayer system to an effective homoge- well as in thermoelectric generators, stationary operating neous system using the WKB method,4 i.e., making the sub- regimes have traditionally been used. Studies of nonstation- stitution ary cooling and current generation began only much 1 x later ͑Stil’bans and Fedorovich, E. K. Iordanishvili and dx ୐0 x ⌸͑x͒ϭ , a ϭ , ͑4͒ 2 ͵ 0 C co-workers, and others͒ and certain advantages of these op- → 0 ͱa0͑x͒ 0 erating regimes were identified at once, especially in the ini- 4 tial stage of operation, when the difference in the time con- and using a Green function formalism, we obtain stants of the thermal and electrical processes are important. ⌸͑L͒ Thermoelectric power supplies are multilayer structures T0͑x,t͒ϭ ͵ TH͑⌸Ј͒G͑⌸,t;⌸Ј,0͒d⌸Ј in which the temperature distributions are determined by 0 solving a system of nonlinear differential heat conduction ϩ t Q1ͱa0͑0͒ equations with the corresponding boundary conditions for Ϫ ͵ ୐ ͑0͒ the heat sources and sinks. In their general form in the one 0 ͭͩ 0 dimensional approximation, these equations are Q2ͱa0͑L͒ ϩb1ͱa0͑0͒T1͑␶͒ G͑⌸,t;0,␶͒Ϫ TiTi ͪ ͩ ୐ ͑L͒ץ i͑T͒␣ץ Tiץ ץ 0 ୐ ͑T͒ ϩ j ͑t͒ ϩ j2͑t͒␳ ͑T͒ x i iץ Tץ x ͬ iץ x ͫ iץ i ϩb2ͱa0͑L͒T2͑␶͒ G͑⌸,t;⌸͑L͒,␶͒ d␶. ͑5͒ ͮ ͪ Tiץ ϭC ͑T͒ , i t To determine U(x,t) we have the integral equation ץ T Q ͑t͒ץ 1 1 tϩ ⌸͑L͒ Ϫ ϩb1͑TϪT1͑t͒͒ϭf1͑t,T͒, xϭ0, U x,t ϭϪ , G ,t; , d x ୐0͑0͒ ͑ ͒ ⌿͑⌸Ј ␶͒ ͑⌸ ⌸Ј ␶͒ ⌸Јץ ͵0 ͭ͵0 T Q ͑t͒ץ L 2 ϩ a 0 f G ,t;0, Ϫ ϩb2͑TϪT2͑t͒͒ϭf2͑t,T͒, xϭL, ͱ 0͑ ͒ 1 ͑⌸ ␶͒ x ୐0͑L͒ץ Ϫͱa ͑L͒f G͑⌸,t;⌸͑L͒,␶͒ d␶, ͑6͒ T͑x,0͒ϭTH͑x͒, ͑1͒ 0 2 ͮ where j is the current density, ୐ and ␣ are the thermal conductivity and thermopower ͑Seebeck͒ coefficient, ␳ is the where Tץ Tץ ץ electrical resistivity, C is the specific heat, and i is the layer number. ⌿ϭϪ ⌬୐͑x,T͒ ϩ⌬C͑x,T͒ tץ xͬץ ͫ xץ Usually the thermal characteristics of the material in Tץ ␣ץ each layer can be represented in the form Ϫj͑t͒ TϪj2͑t͒␳͑T͒, xץ Tץ ␴⌬ ␴͑x,T͒ϭ␴ ͑x͒ϩ⌬␴͑x,T͒, Ͻ1. ͑2͒ 0 ␴ Tץ ୐k⌬ 0 fkϭϪ ϩ␻k͑t,T͒, kϭ1,2, xץ The system of Eqs. ͑1͒ must be supplemented by equa- ୐0k tions for determining the current density j, which depend on Q and ␻ are known functions; the upper limit tϩ means the electrical circuit joining the thermal elements. In accor- k k that the integration with respect to ␶ is taken to tϩ␧ fol- dance with the method outlined previously,3 the solution of lowed by a transition to the limit ␧ 0; G(⌸(x),t;⌸(xЈ),␶) this problem is conveniently represented in the form is the Green function for the system→ of equations ͑1͒, which T͑x,t͒ϭT0͑x,t͒ϩU͑x,t͒, ͑3͒ is found approximately using the WKB method as

1 ␦ϩiϱ G͑⌸,t;⌸Ј,␶͒ϭ N͑⌸,⌸Ј,P͒ 2␲i ͵␦Ϫiϱ

ϫexp͓p͑tϪ␶͔͒dp, ͑7͒ where

b1ͱa0͑0͒ cosh ͱp⌸ЈϪ sinh ͱp⌸Ј ͫ ͱp ͬ NӍ R

b2ͱa0͑L͒ ϫ sinh ͱp͓⌸͑L͒Ϫ⌸͔ ͭ ͱp ϩcosh ͱp͓⌸͑L͒Ϫ⌸͔ͮ , max FIG. 1. W/Wt as a function of time: 1—calculated by the method pro- b2ͱa0͑L͒ b1ͱa0͑0͒ posed here, 2—experiment, 3—exact solution. Rϭͱp Ϫ cosh ͱp⌸͑L͒ ͫͩ ͱp ͱp ͪ of these results shows that their are in satisfactory agreement, b1b2ͱa0͑0͒a0͑L͒ ϩ 1Ϫ sinh ͱp⌸͑L͒ , ͑8͒ so the method developed here can be recommended for cal- ͩ ͱp ͪ ͬ culating and analyzing the characteristics of specific thermo- and ⌸Ͼ⌸Ј. electric devices. For ⌸Ͻ⌸Ј, ⌸ and ⌸Ј must be switched. Equation ͑6͒ is Mathematical modeling of the nonstationary operation of easily solved by successive approximations, with T0(x,t) thermoelectric generators shows that, in order to accelerate taken as a zeroth approximation. the approach of a thermoelectric generator to a given power It should be noted that using asymptotic methods,5 in production regime, it is necessary to reduce the thickness of particular the reduced matching technique,6 makes it possible the transition layers on the hot-junction side and to use ma- to simplify the computational formulas greatly. For short terials with a high thermal diffusivity in them. In order to times sustain prolonged operation of the system, its heat capacity must be increased and the temperature or calorific capacity t 1 of the heat source must be raised. The amplitude of the out- 2 Ͻ ⌸0͑L͒ 4 put electrical signal also depends significantly on these fac- and low current densities tors. ୐ jϽ0.1 , 1 ␣L L. S. Stil’bans and N. A. Fedorovich, Zh. Tekh. Fiz. 28, 489 ͑1958͓͒sic͔. 2 E. K. Iordanishvili and V. P. Babin, Nonstationary Processes in Thermo- the influence of current effects on the temperature can be electric and Thermomagnetic Energy Conversion Systems ͓in Russian͔, neglected, and this substantially shortens the computational Nauka, Moscow ͑1983͒, 216 pp. 3 Yu. I. Dudarev and M. Z. Maksimov, Teplofiz. Vys. Temp. 26, 824 procedure. The effective characteristics for multilayer sys- ͑1988͒. tems can be obtained using the sum rule for the eigenvalues 4 F. M. Morse and H. Feshbach, Methods of Mathematical Physics, of the original problems.7 McGraw-Hill, New York ͑1953͒; IL, Moscow ͑1958͒, Vols. 1 and 2. 5 Figure 1 shows the time variation in the relative electri- Yu. I. Dudarev, A. P. Kashin, V. I. Lozbin, and O. V. Marchenko, Inzh. max Fiz. Zh. 42, 492 ͑1982͒. cal power of a thermoelectric generator, W/WT , obtained 6 A. P. Kashin, T. M. Kvaratskheliya, M. Z. Maksimov, and Z. E. Chiko- through calculations employing the method proposed here vani, Teor. Mat. Fiz. 78, 392 ͑1989͒. 7 ͑curve 1͒, an exact solution ͑curve 3͒, and experimentally Yu. I. Dudarev, A. P. Kashin, and M. Z. Maksimov, Inzh. Fiz. Zh. 48, 333 ͑1985͒. ͑curve 2͒. The experimental data were obtained by A. A. Sokolov ͑Sukhumi Physicotechnical Institute͒. A comparison Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

On the problem of estimating the relaxation time of local deformations on metal surfaces Yu. I. Dudarev, A. V. Kazakov, and M. Z. Maksimov

Sukhumi Physicotechnical Institute, Academy of Sciences of the Abkhaz Republic, Sukhumi, Abkhazia ͑Submitted April 28, 1997͒ Zh. Tekh. Fiz. 68, 138–139 ͑June 1998͒ Asymptotic methods for evaluating the change in the relief of local surface formations are used to obtain simple and fairly exact equations for the proﬁle of the relief and characteristic relaxation times of local deformations on metal surfaces in kinetic and diffusion models. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02606-3͔

1. In creating ultrahigh-capacity information storage sys- which develop on metallic surfaces owing to the interaction tems based on solids, information is coded through the action with the tip of a tunneling microscope are much larger1,3 than of the tip of a scanning tunneling microscope made of a their vertical dimensions, similarly to natural surface refractory metal. The resulting changes in the relief ͑promi- asperities.2–4 In this case, nences and indentations͒ can be associated with bits of r b information.1 Self-diffusion of the material of the storage 0 ͉f r͉ϳ Ͻ1, medium can, however, cause a gradual smoothing of the re- R R lief and reduce the bit lifetime. Studies of the relaxation ki- so that Eq. ͑1͒ can be linearized to take the compact form netics and dynamics of surface structures are therefore of f ץ some interest for predicting the operating lifetime of this ϭK ͑Ϫ1͒nLn f . ͑4͒ t nץ type of device. This question is the subject of a detailed 1 paper. We believe, however, that the solutions of the equa- 2. Here it should be pointed out that describing a rough- tions obtained there for the function f (r,t) which describes ness profile by the system of equations ͑2͒ and ͑4͒ is fully the surface relief were not examined analytically to a suffi- equivalent to the problem of the time evolution of the distri- cient extent. We shall discuss this in more detail. First of all, bution function f 0(r) in studies of the kinetics of fine pul- these equations can be written in a unified form verization and other technological processes for working 5,6 f r,t 1 rf ores and raw materials. This makes the analysis of the r ץ n nϪ1 ͒ ͑ ץ ϭϪKn͑Ϫ1͒ L , corresponding solutions much easier. In fact, according to r ͩͱ1ϩf2ͪץ t rץ r Refs. 5 and 6, the solution of the system of Eqs. ͑2͒ and ͑3͒ :f can be written in the following integral formץ ץ ץ 1 Lϭ r , f ϭ , ͑1͒ r ϱץ rͪ rץ ͩ rץ r f n͑r,t͒ϭϪb0 f0͑r0͒Gn͑r,t͉r0,0͒dr0 , ͑5͒ where L is the radial Laplacian operator on the plane, nϭ1 ͵0 corresponds to the kinetic model (K ϵK) and nϭ2, to the 1 where Gn(r,t͉r0,0) is the Green function of the operator ͑4͒, diffusion model (K2ϵ␭). whose Fourier representation has the form Here it is assumed that the initial profile of the depres- sions on the surface of the metal are described by a Gaussian 1 G ͑r,t͉r ,t͒ϭ dq exp͓iq͑rϪr ͒ϪK tq2n͔. distribution n 0 ͑2␲͒2 ͵ 0 n 2 ͑6͒ ␲b0r f ͑r,0͒ϵϪb f ͑r͒ϭϪb exp Ϫ , ͑2͒ It is easy to confirm that, as t 0, 0 0 0 ͩ V ͪ 0 → Gn͑r,t͉r0,0͒ϭ␦͑rϪr0͒, ͑7͒ where b0 is the initial depth of the depression and V0 is its volume with a root-mean-square radius of the equivalent cyl- and on the basis of Eq. ͑5͒ we immediately obtain the result inder of ͑2͒ for any f 0(r). Furthermore, since the main contribution to the integral ͑6͒ is from finite qϳq , by making the sub- V V 0 2 2 0 2 2 0 stitution K q2ntϭ␳ it is easy to obtain a second leading ͗r ͘ϭR ϭ ϭb0␮ , ␮ ϭ 3 . ͑3͒ n ␲b0 ␲b asymptotic term of G as t ϱ n → This representation of the initial conditions for the pro- 1 1 file is entirely natural, since it conforms with the overall G ͑r,t͉r,0͒ϭ ⌫ 1ϩ ͑K t͒Ϫ1/n, ͑8͒ n 2 n n distribution of roughness on the surface.2–4 In the general ͩ ͪ case, the solution of the system of Eqs. ͑1͒ and ͑2͒ is diffi- where ⌫(z) is the gamma function. cult; however, the horizontal dimensions 2R of the pits Substituting Eq. ͑8͒ in Eq. ͑5͒ yields

V0 1 from Eq. ͑1͒ to Eq. ͑4͒ becomes better justified. This is con- Ϫ1/n 1 f ͑r,t͒t ϱϭϪ ⌫ 1ϩ ͑Knt͒ , ͑9͒ firmed by numerical calculations of two approximations for → 4␲ ͩ nͪ t1 in the kinetic model. In these cases, we can also assume since f 0Ӎ1; then Eq. ͑17͒ for the kinetic (nϭ1) and diffusion ϱ (nϭ2) models becomes 2␲b0 f ͑r0͒r0dr0ϭV0 . ͑10͒ ͵0 t ␶ ͑1/␰Ϫ1͒, ␶ ϭR2/4K; ͑18͒ In the case of the kinetic model ͑nϭ1, K1ϭK͒ and the kinӍ 1 1 Gaussian initial condition ͑2͒, all the calculations become simpler,1 since the corresponding integrals of the Bessel 7 2 4 functions in Eqs. ͑5͒ and ͑6͒ are complete Weber integrals. tdifӍ␶2͑1/␰ Ϫ1͒, ␶2ϭ␲R /64␭. ͑19͒ We have 2 f 1͑r,t͒ϭϪb1͑t͒exp͑Ϫ␲b1͑t͒r /V0͒, ͑11͒ Their ratio is so that f 0,t ϭϪb t ϭϪb 1ϩ4␲Kb t/V Ϫ1. ͑12͒ 1͑ ͒ 1͑ ͒ 0͑ 0 0͒ tdif /tkinϭ͑1/␰ϩ1͒␶2 /␶1 , ͑20͒ For arbitrary n and, in particular, for the diffusion model ͑nϭ2, K ϭ␭͒, it is not possible to obtain such simple equa- 2 i.e., for other conditions the same, it depends on the degree tions for f n(r,t). However, for approximate estimates we can use the reduced matching method8 in the parameter t. of filling. For example, for ␰ϭ0.5 we have According to the procedure of this method, for the leading asymptotic terms of the function ͓Ϫ f (r,t)͔n we have from ϭ Eqs. ͑2͒ and ͑9͒ tdif /tkin 3␶2 /␶1 . ͑21͒

n n f ͑r,t͒ ͓ f 0͑r͔͒ t 0 Ϫ ϭ → ͑13͒ ͩ b ͪ ␶ /ttϱ, Next, for ␰ 0, Eq. ͑19͒ for the diffusion model gives 0 ͭ n → 4 4 2 → the correct result ␭tdif /␲b0␮ ϭ1/64␰ , as opposed to the which, when matched, give estimates of Ref. 1, which gave 1/96␰. n Ϫ1/n 3. By using a rigorous analysis of the initial kinetic f ͑r,t͒ϭϪb ͑t/␶ ϩ1/f ͑r͒͒ , ͑14͒ n 0 n 0 equations and asymptotic methods for the change in the re- and lief of local surface formations, we have obtained simple and n fairly accurate equations for the relief profile and for the V0⌫͑1ϩ1/n͒ ␶nKnϭ . ͑15͒ characteristic relaxation times of local deformations on metal ͩ 4␲b0 ͪ surfaces in the kinetic and diffusion models. All of this, to- From this, first of all for the kinetic model (nϭ1) and gether with methods for calculating the energy parameters 1 the initial condition ͑2͒ we obtain and transport coefficients, makes it much easier to predict the operating lifetime of memory structures, to determine the 2 2 2 f 1͑r,t͒ϷϪb0͑exp͑r /R ͒ϩt/␶1͒, ␶1ϭR /4K, ͑16͒ conditions for their reliable operation, to choose the storage- where Eq. ͑3͒ has been used. medium material, and other tasks which are indispensable Here, on the one hand, the function ͑16͒ satisfies the steps in nanotechnologies. initial conditions, and its series expansion and that of expression ͑11͒, in powers of r2/R2 give the same values for the first few coefficients, and, on the other hand, for b1(t) ϭ f 1(0,t) we obtain the correct result, Eqs. ͑11͒ and ͑12͒. 1 Thus Eqs. ͑14͒ and ͑15͒ can be used with sufficient accuracy A. M. Dobrotvorski and V. K. Adamchuk, Zh. Tekh. Fiz. 64͑8͒, 132 for other r and n. If here we introduce the degree of preser- ͑1994͓͒Tech. Phys. 39, 816 ͑1994͔͒. 2 V. I. Trofimov and V. A. Osadchenko, Poverkhnost’, No. 9, 5 ͑1987͒. vation of a dip in the surface, ␰ϭϪf(r,t)/b0Ͻf0(r), then 3 V. I. Vettegren’, S. Sh. Rakhimov, and V. N. Svetlov, Fiz. Tverd. Tela ͑St. for the time t for filling it from the initial f (r) to a given Petersburg͒ 37, 913 ͑1995͓͒Phys. Solid State 37, 495 ͑1995͔͒. 4 level ␰,wefind V. V. Boko, A. P. Kashin, M. Z. Maksimov, and Z. E. Chikovani, Pov- erkhnost’, No. 11, 40 ͑1993͒. Ϫn Ϫn 5 E. A. Nepomnyashchi ,inTheoretical Foundations of Chemical Technol- tnϭ␶n͑␰ Ϫf ͑r͒͒, ␰рf0͑r͒. ͑17͒ 0 ogy ͓in Russian͔, Vol. 7, No. 5, p. 754 ͑1973͒; ibid., Vol. 11, No. 3, p. 477 Here it should be noted that none of the above estimates are ͑1977͒; ibid., Vol. 12, No. 4, p. 576. 6 very sensitive to the shape of the initial distribution f (r), A. P. Kashin, M. Z. Maksimov, and Z. E. Chikovani, Ukr. Fiz. Zh. 36, 973 0 ͑1991͒. because, as in Ref. 1, it is assumed that condition ͑10͒, nor- 7 M. M. Agrest and M. Z. Maksimov, Theory of Incomplete Bessel Func- malization to volume V0 for a given depth f 0ϭb0 , and con- tions and their Applications, Springer-Verlag, Berlin ͑1971͒. 8 dition ͑3͒, for the average radius, are satisfied. All together, A. P. Kashin, T. M. Kvaratskheliya, M. Z. Maksimov, and Z. E. Chiko- this makes it possible to choose an initial distribution of the vani, Teor. Mat. Fiz. 78, 392 ͑1989͒. simplest form, close to rectangular, for which the transition Translated by D. H. McNeill TECHNICAL PHYSICS VOLUME 43, NUMBER 6 JUNE 1998

Possible design for a cryogenic ferromagnetic gyroscope L. A. Levin and S. L. Levin

͑Submitted May 27, 1997͒ Zh. Tekh. Fiz. 68, 140–142 ͑June 1998͒ Three of the simplest schemes for the design of a new physical apparatus based on the Barnett effect are examined. It is shown that the most realistic is a design with a special type of electrically open-circuited, superconducting magnetic shield. The design of this device is described. © 1998 American Institute of Physics. ͓S1063-7842͑98͒02706-8͔

The possibility, in principle, of creating a new type of arrangement4 of a ‘‘snake swallowing its tail’’͒. If the ratio cryogenic ferromagnetic gyroscope, an angular velocity sen- of the height of the gap to the length of the gap overlap zone sor, has been discussed.1 The cryogenic ferromagnetic gyro- is kept at 1:10, then the shielding coefficient will be of the scope is based on the Barnett effect,2 in which a magnetic order of 105 –106. A further reduction in this ratio will in- field is generated when a ferromagnetic body is rotated. It crease the shielding coefficient. This scheme has been used has been shown1 that there are three physical phenomena for by us in the device to be discussed below.5 exciting a magnetic field when different bodies are rotated. An example of such a shield is a cylinder with a very These are the magnetoresonance effect in 3He, the London narrow slit, of the order of 0.01–0.1 mm, along a generator moment in superconductors, and the Barnett effect. In order and with a large wall thickness of 2–3 mm. Besides increas- to obtain a magnetic induction in the cryogenic ferromag- ing the shielding coefficient, a shield of this sort makes it netic gyroscope equal to the induction produced in the possible to shield the sensor winding placed on it from the nuclear gyroscope with 3He developed so long ago,3 it would ferromagnetic material and to reduce its inductance by be necessary for the relative dimensionless magnetic perme- roughly a factor of ␮. ability of the ferromagnetic body to be у800. It was shown The ferromagnetic rod is mounted inside the shield. The that the cryogenic ferromagnetic gyroscope should be much diameter of the rod and the inner diameter of the shield must simpler than the nuclear gyroscope. be close in value (DcϷDe), so that a field close to the Bar- In this paper we examine some design versions of a nett induction BB acts on the inner surface of the shield. The cryogenic ferromagnetic gyroscope. It was pointed out current created on the inner side of the shield to compensate previously1 that the simplest scheme for a cryogenic ferro- the field BB flows over the outer surface of the shield, trans- ferring the field B to the surface of the shield. This field magnetic gyroscope is a ferromagnetic rod located inside a B excites a current in the superconducting sensor winding, cylindrical superconducting, electrically open-circuited which is wound on the surface of the shield and connected to shield. A superconducting sensor winding is placed on the a short-circuiting superconducting loop incorporating the in- rod and connected to a superconducting short-circuiting loop put winding of a SQUID. In order to prevent current leakage, incorporating the input winding of a SQUID. This scheme, the width of the shield must be somewhat less than the length however, has two disadvantages. First, a simple cylindrical of the ferromagnetic rod. open-circuited magnetic shield gives a low shielding coeffi- The current excited in the sensor winding, which is pro- cient and, second, a winding placed directly on the ferromag- portional to the rotational speed of the apparatus, flows along netic rod has a high inductance, which reduces the current in the input winding of the SQUID, creating a flux which is the input winding of the SQUID and makes it difficult to measured by the SQUID. An electrically open-circuited match the winding to the input of the SQUID. In principle, it shield can obviously be used in place of a sensor winding. is possible to mount the ferromagnetic rod in a cylindrical, The magnetic flux created in the ferromagnetic rod by closed-circuited superconducting shield, but in this case the the Barnett effect is overall size of the superconducting shield becomes much larger owing to the compensating magnetic field which de- ␮⍀ velops inside the shield, or the dimensions of the rod must be ⌽BϭBBSϭ S, ͑1͒ ␥B reduced, which results in a lower sensitivity for the device. In sum, the two versions with simple, open- or closed- where ␮ is the relative dimensionless magnetic permeability circuited shields are poorly suited to making a realistic cryo- of the rod material, ⍀ is the angular velocity of rotation, S is genic ferromagnetic gyroscope. the transverse cross section of the rod, and ␥B is the gyro- 11 Ϫ1 In order to create a functional cryogenic ferromagnetic magnetic ratio (␥Bϭ1.7ϫ10 A•s•kg ). gyroscope we propose using a special electrically open- When DcϭDe , the flux at the shield is circuited superconducting shield. It is an essentially cylindrical shield, slit along a generator of the cylinder, with a large ␮⍀ ⌽eϭleLeϭ⌽Bϭ S, ͑2͒ overlapping zone at the site of the slit ͑for example, in the ␥B

1063-7842/98/43(6)/2/$15.00 741 © 1998 American Institute of Physics 742 Tech. Phys. 43 (6), June 1998 L. A. Levin and S. L. Levin where Ie and Le are the current and inductance of the shield. shield 3, To eliminate the influence of the London moment The magnetic flux at the input winding of the SQUID is on the measurement of the Barnett field, it is necessary that the end-cap shields 7 be slit appropriately. Here the ratio of ␮⍀S Lc ⌽cϭI0Lcϭ , ͑3͒ the gap height to the overlap zone must be р1:10, i.e., the ␥B L0ϩLc cap must be comparatively thick and the slit very narrow. where I0 is the current in the short-circuited superconducting There are still some details involved in making and us- circuit flowing through the input winding of a SQUID with ing a cryogenic ferromagnetic gyroscope. Cooling the appa- inductance Lc , and L0 is the inductance of the sensor wind- ratus to the working temperature to eliminate trapping of the ing. magnetic flux by superconducting components should be We refer to Lc /(L0ϩLc)ϭK as the flux transfer coeffi- done under conditions of essentially zero magnetic field, es- cient. In complicated circuits this coefficient is more cum- pecially near the superconducting transition temperature of bersome than in Eq. ͑3͒, but, as a rule, it consists of relation- the material. ships among the inductances and mutual inductances of the It is desirable to use a single crystal for the ferromag- components in the device. In practice, Kϳ10Ϫ1 –10Ϫ3. The netic rod material and have the easy axis of magnetization Barnett magnetic flux can be measured if ⌽cу⌽nc , where coincide with the axis of the rod. ⌽nc is the SQUID noise. Then Ferromagnetic materials are noisy even at extremely low temperatures. In order to reduce the level of magnetic noise ␮•⍀•S•K ⌽cϭ⌽ncϭ , ͑4͒ it is appropriate to coat the surface of the ferromagnetic rod ␥B with a nonmagnetic material such as pure copper that has a which implies that the sensitivity of the cryogenic ferromag- low Ohmic resistance at the operating temperature. netic gyroscope to angular velocity will be The noise level of SQUIDs increases sharply below a certain frequency f nϷ0.1– 1.0 Hz. To eliminate this effect it ⌽nc␥B f ϭ . ͑5͒ is appropriate to modulate a higher frequency signal and de- 2␲•␮•S•K modulate at the output of the SQUID. Equation ͑5͒ implies that in order to raise the sensitivity If an additional winding ͑or windings͒ is placed on the of the cryogenic ferromagnetic gyroscope, it is necessary to open shield, then the prospects for the cryogenic ferromag- Ϫ5 netic gyroscope are greatly expanded; with a suitable control increase ␮, S, and K. For example, for ⌽ncϭ10 ⌽0 ͑⌽0 ϭ2ϫ10Ϫ15 Wb is the quantum of magnetic flux͒, ␮ϭ800, system it can measure angular acceleration and rotation Sϭ10Ϫ3 m2, and Kϭ10Ϫ2, we obtain for f a value of the angle. order of 10Ϫ7 sϪ1. We have proposed a design for a new physical appara- A simplified sketch of the cryogenic ferromagnetic gy- tus, the cryogenic ferromagnetic gyroscope. The sensitive roscope is shown in Fig. 1. Inside the case 1 are a ferromag- element of the cryogenic ferromagnetic gyroscope ͑a rod netic rod 2 and a superconducting magnetic shield 3, both with a shield͒ has no moving parts and does not require any rigidly connected to the case; possible shapes of the shield additional energy to operate; it does not release heat. Thus, it are illustrated to the right and left in the figure. A supercon- is very convenient for cryogenic devices and its operating ducting sensor winding 4 is wound on the shield 2 and con- lifetime is unlimited. nected to the input winding 5 of a squid 6. Superconducting A comparison with the known designs for cryogenic end-cap shields 7 are mounted on the ends to improve the gyroscopes—angular velocity sensors ͑the cryogenic nuclear screening coefficient. The ratio of the height of the gaps to gyroscope͒3 and the cryogenic ferromagnetic gyroscope— the overlap lengths of the shields 3 and 7 is 1:10. shows that the cryogenic ferromagnetic gyroscope design is It is known that during rotation of a superconducting simpler and more reliable. In terms of sensitivity, the cryo- body, such as a solid or hollow closed cylinder, a magnetic genic nuclear gyroscope and the cryogenic ferromagnetic gy- field ͑the London moment͒ develops in it.1 A London mo- roscope are similar. There is a possibility, in principle, ment does not develop in the electrically open-circuited of increasing the sensitivity of the cryogenic ferromagnetic gyroscope by employing a ferromagnetic material with a higher ␮.

1 L. A. Levin, Zh. Tekh. Fiz. 66͑4͒, 192 ͑1996͓͒Tech. Phys. 41, 399 ͑1996͔͒. 2 S. J. Barnett, Rev. Mod. Phys. 7, 129 ͑1935͒. 3 K. F. Woodman, P. W. Franks, and M. D. Richards, Rev. J. Navigation 40, 336 ͑1987͒. 4 D. B. Sullivan and R. F. Dziuba, Rev. Sci. Instrum. 45, 517 ͑1974͒. 5 A. P. Buravlev, B. E. Landau, L. A. Levin, and S. L. Levin, ‘‘Cryogenic ferromagnetic gyroscope’’ ͓in Russian͔, Inventor’s Certiﬁcate No. 2084825; Publ. Byull. Izobret., No. 20 ͑1997͒.

FIG. 1. A design sketch of the cryogenic ferromagnetic gyroscope. Translated by D. H. McNeill