Perspectives of Polarons

consider two limiting cases. At high temperatures (e

(r) =

AKo(r/rD), K0 is the McDonald's function. In the region 0 < r < TO one can use an approximation

where a = e2//ccT is the ratio of the Coulomb interaction energy of the nearest electrons captured by a defect to their thermal energy, and In 7 = 0.577 is the

Euler constant. At low temperatures (e<(>/T) » 1 at 0 < r < R'. When R* > rD one can neglect the Debye screening and use the approximation ne <; nd. In this case, Eq. 27 takes the form

By using the boundary conditions (d/dr) \T=R*= 0 and
where •7rjZ*2n(j = //a. As is seen, R* is the distance where the electric field of charged defects is compensated for by the field of ionized impurities. Note that (R*/rD)* = 4e2f/aKT.

198

Let us write the whole effective potential for Eq. 6 at low temperatures

where Uc = etp(r) is the Coulomb potential, Uc — J 2 A 2 /2m*r 2 is the centrifugal energy, J = j for dislocations, and J = j — v for disclinations.

The problem of an electron localization near charged dislocations has been recently reviewed in 2 2 . For shallow electron levels Eq. 6 with Eq. 33 was solved

in 23 within the one-band approximation. When the effective potential well goes deeper, the region of an electron localization becomes smaller reaching a few lattice constants. In the case of deep electron levels it is possible to

use a rough approximation assuming that the Coulomb interaction results in a shift of discrete levels by a middle value £/c(0) ~ ef(c)- M Notice that the Coulomb potential depends on r logarithmically whereas the deformation potential behaves like 1/r.

Let us consider the wedge partial disclinations. It is interesting to note that the deformation potential due to wedge disclinations has a logarithmic

dependence on r, just as the Coulomb potential does. At low temperatures the effective potential is written as

Here D is determined in Eq. 8, and C = e2f/Ka. As is shown in Sec. 2, for

C = 0 and v > 0 the localized and/or resonance electron states can appear in the vicinity of the disclination line. It stands to reason that the Coulomb part

in Eq. 34 will modify this picture. In particular, the depth of the potential well decreases, so that the resonance electron states become preferable. This

fact is of importance for the scattering problem in defect semiconductors at low temperatures. Indeed, as was found in 1 3 , the relaxation time related with disclination vortices consists of two parts: the pure topological one and that

coming from the deformation-potential scattering

Here n&i, denotes the density of disclinations. As a result, the residual resistivity due to disclinations is

199

For deep electron levels the phase SQ is small, so that the deformation-potential scattering is ineffective. By contrast, for resonance electron states sin S0 ~ 1 thus giving the essential increase in resistivity. It is also interesting to note that the effective potential Eq. 34 depends now on the temperature as well as on the density of impurity via rD and R* ( R* depends on T through /). As a result, the discrete electron spectrum can be varied by changing T and rip. In particular, one can change the number of

discrete localized or resonance levels and even forbid their appearance. 5

Conclusion

In conclusion, we presented some results related to electronic states in materials with topological defects. As is shown, the presence of dislocations and disclinations can essentially modify the electron spectrum. This finding is of importance in the studies of different phenomena in defect crystals. It is expected that the presence of discrete electron levels due to defects will essentially affect the thermodynamic and kinetic properties of materials. In particular, the shallow electron levels in semiconductors influence the dislocation-induced resistivity at low temperatures. The deep electron levels serve as the centers of recombination for excess carriers. The polaron-type states would be of importance in dielectrics. Let us note that the existence of discrete electron levels

in dislocated semiconductors is well established whereas the results concerning the polaron-type states still remain to be confirmed experimentally. Disclinations are of importance in highly deformed materials that involve large mutual rotations of neighbouring elements. The experimental study of electrical properties of such materials has much potential for yielding information about disclinations. Notice also that in modern theoretical models disclinations are considered as important constituents of different disordered materials, like amorphous bodies, metal glasses, polymers, etc. However, up to now there is no direct evidence in favor of disclination structure of these materials. That is why any theoretical predictions concerning the electronic properties, especially specific transport properties, of disclinated materials are of considerable interest. The careful measurements of transport properties of these materials would be a good test to clarify both the availability of disclinations and their density.

Acknowledgements Part of this work has been financially supported by the grant from Russian Fund of Fundamental Research No 94-02-05867. The research described in this 200

publication was made possible in part by grant N RFR300 from the International Science Foundation.

References 1. R. Labusch and W. Schroter in Dislocations in Sohds, ed. F.R.N. Nabarro (North-Holland, Amsterdam, 5, 1980). 2. Yu.A. Osip'yan et al, Adv. in Phys. 35, 115 (1986). 3. R. Landauer, Phys. Rev. 94, 1386 (1954). 4. L.S. Kukushkin, Sov. Phys.- JETP Lett. 7, 251 (1968). 5. A.M. Kosevich, Sov. J. Low Temp. Phys. 4, 902 (1978). 6. V.P. Voronov and A.M. Kosevich, Sov. J. Low Temp. Phys. 6, 371 (1980). 7. V.A.Osipov, J. Phys. A 26, 1375 (1993). 8. A. Kadic and D.G.B. Edelen in Lecture Notes in Physics, ed. H. Araki, et al (Springer, Berlin, 174, 1983). 9. B.K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982). 10. V. Celli et al, Phys. Rev. Lett. 8, 96 (1962). 11. P.R. Emtage, Phys. Rev. 163, 865 (1967). 12. V.A.Osipov, Phys. Lett. A 164, 327 (1992). 13. V.A.Osipov, Phys. Lett. A 193, 97 (1994). 14. V.I. Vladimirov and A.E. Romanov, Disclinations in Crystals (Nauka, Leningrad, 1986) [in Russian]. 15. V.A.Osipov and S.E.Krasavin, /. Phys.: Cond. Matt. 7, L95 (1995). 16. V.A.Osipov, Physica A 175, 369 (1991). 17. V.A.Osipov, /. Phys. A 24, 3237 (1991). 18. V.M. Nabutovsky and B.Ya. Shapiro, Sov. Phys.- JETP 75, 948 (1978). 19. W. Shockley, Phys. Rev. 91, 228 (1953). 20. W.T. Read, Phil. Mag. 45, 775 (1954). 21. R.A. Vardanyan, Sov. Phys.- JETP 73, 2313 (1977). 22. V.B.Shikin and Yu. V.Shikina, Usp. Fiz. Nauk 165, 887 (1995). 23. V.L. Bonch-Bruevich and V.B. Glasko, Sov.Phys.-Solid State 3, 36 (1961).

201

BIPOLARONS IN ANISOTROPIC CRYSTALS N.I. KASHIRINA, E.V. MOZDOR, E.A. PASHITSKIJ, and V.I. SHEKA Institute of Physics of Semiconductors of National Ukraine Academy of Science, Kiev 25202, Ukraine We consider a simple bipolaron treatment for anisptropic crystals in the strongcoupling limit. We take into account anisotropy of effective band masses and dielectric constants of the crystals.

The bipolaron theory was first studied in1 by the adiabatic approximation for the case of strong coupling of electron-phonon interaction and isotropic continuum medium. Living aside the history of the bipolaron study presented in 2 ' 3 , we set forth the problem by the traditional Frohlich Hamiltonian which is generalized to the case of anisotropic two-atomic crystals. ' 5 For the bipolaron problem the corresponding Hamiltonian has the form

where

is the kinetic energy of the electron with coordinates r^ and T y ,

is the potential energy of electrons. The phonon frequencies with normal coordinates die and a£ polarization vectors BJ (k) satisfy the dispersion equation

202

where

while the polarization vectors follow the condition

It should be noted that Hamiltonian Eq. 1 describes the bipolaron states in crystals with the uniaxial anisotropy. In this case, if we reduce the tensors of effective mass and dielectric permittivity to the main axes, notations e0 and £00, mxx — myv = mi, mzz = m\\, txx = evy = £j_, f-zz — f\\ can be introduced. In contrast to works, 6 ' 7 where the limiting cases of low-dimensional systems

are considered, we study crystals with arbitrary anisotropy of effective band mass and dielectric permittivity (in the general case we also take into account their dispersion in time and space). In the weak-coupling limit there are no localized bipolaron states in the

continuum medium, for which is used Hamiltonian Eq. 1 holds. Therefore, the bipolaron formation is more probable in the strong-coupling limit. In the latter case, the phonon operators in Eq. 1 can be replaced by c-numbers /jy, so that the average two-participle energy takes on the form

The minimization of Eq. 6 over fu. and the summation over the polarization vectors (which is similar to minimization of the polaron energy, developed in 5 ) result in the adiabatic Pekar bipolaron functional

where

203

Good results were obtained in8 by the minimization of functional Eq. 7 for isotropic systems, where the trial wave functions took on the form

where r12 = |rj — r 2 |, and r t ,r2 are the coordinates of the first and second electron, respectively. The account of the term with multiplier /3r12 enables us to consider the electron correlation effects. But in the case of anisotropic crystals, the use of these trial wave functions leads to cumbersome mathematical problems in the evaluation of integrals for potential energy V\ 2 and for the term corresponding to the electron-phonon interaction. However, using Gaussian trial wave functions, we make the analytical calculations in the explicit form. In this paper, we use the trial wave function, which is axially symmetric and has the form

In this relation, the parameters are chosen to account the symmetry or antisymmetry with respect to exchange of the electron coordinates. The interference terms 2o2;.zi.z2 and 26 2 jp 1 /9 2 in the exponential index take into account the correlation effects. To calculate the minimum of functional Eq. 7, it is convenient to make the scale transformation r =>• Ar, where A is the scale multiplier, which can be found by the virial theorem, A = -(V"12 + Vef)/2T, and Ex = -("^12 + V'cf)/^TN, where N is the normalization multiplier. All averaged quantities in the functional Eq. 7 can be easily calculated by the wave functions Eq. 9. The minimization of E^ with respect to the parameters Cj,mj,bij (i = 1,2,3; j = 1,N) was made by the method of fast descent and it was applied to calculations of the energy of para- and ortohelium ground states. The calculations for N = 7 yield the following results

E0 = -2.8722374 for the parahelium, and El = -2.1183365 for the ortohelium, which coincide with the best calculations of corresponding quantities Eq. 9 in all significant digits. This application shows that our approximation of trial functions by Eq. 7 is rather convenient to minimize the bipolaron functional. We note that the choice of the wave functions in form Eq. 7 enables us to perform easily the symmetrization procedure (or antisymmetrization) over the electron coordinates. For instance, for the triplet states the coefficients Cj 204

must differ in sign for the terms different in the exchanging electron coordinates TJ and r 2 , while for the singlet state they coincide in sign. The following syrnmetrization (the choice of equivalent variational parameters corresponding to the terms, which are different in exchanging electron coordinates) can be made by the minimization of the functional. The possibility of calculation of singlet-triplet splitting for two-electron system by the wave functions chosen as a linear combination of gaussian orbitales enables us to estimate the dependence of the exchange energy on the inter-ion distance without ignoring the electron correlation effects which can play a key role. Figure 1. shows the dependence of the bipolaron ground state energy (line 1) and the polaron one (line 2) on the ratio of effective masses ( in logarithmic scale) for the cases of anisotropy "light axis" ("IX/TOII > 1) and "light plane" ( m _L/ m i| < 1> when EOO /CQ =>• 0- The corresponding bipolaron bound energy is plotted in Fig. 2. The bipolaron and polaron energies of the ground state are shown to decrease if the anisotropy rises, while the polaron bound energy increases (especially for crystals with quasi-one-dimensional anisotropy). In the isotropic case, we result in the lower energy of the bipolaron ground state and the larger bound energy AW (AW/2/ p = 0.253 as t^ /e 0 —» 0) than the best results calculated for the strong-coupling case, with the use of the conventional trial wave functions Eq. 8. The minimization of the polaron functional by trial function involving seven gaussian orbitales exactly leads to the energy evaluated in10 in the strong-coupling limit. We use this value to evaluate the bipolaron bound energy. The situation is similar to the case of isotropic crystals. In comparison with isotropic case, the regions of parameter 77 = £oo/ £ 0i where bipolaron exists, are slightly extended. For example, if m±/mii = 32 the bipolaron is stable as 0 < 7; < r)c = 0.150, and if rn±/m\\ = 1/64 it yields TJC = 0.156. In the isotropic case T)c = 0.140. We note that the calculation of the bipolaron energy in the ground state for anisotropic crystals is similar to the calculations for the crystals with anisotropic effective masses, which we study. Moreover, in the limit too/Co —* 0) using the scale transformation of electron coordinates (2^2 —h V e J_/£||Zi,2 we can reduce the problem to the case under consideration, where effective masses are anisotropic, while the dielectric constants are isotropic. The new effective masses are connected with the initial dielectric constants by the relation

To discuss the bipolaron formation in the crystals where high-temperature

conductivity is observed, we consider a concrete example La2CuOt, in which there is a great anisotropy of effective masses and dielectric permittivities. 205

This crystal is representative to illustrate the possibility for calculation of the bipolaron bound energy in the crystals where the anisotropy of effective masses and dielectric constants occur coincidentally. In our assumption, the data on static and high-frequency dielectric constants are only required, these constants are well-known from experiments 11 , eoo = 4, e0 = 50 in the plane of layers CuO?, and e0 = 23 in the perpendicular direction.

Figure 1: Energy of the polaron ground state (line 1), and bipolaron ground sate (line 2). m* = min{m^y, mg }

Figure 2: Bipolaron bound energy for the aniaotropic cases: "light axis" (m* = "iz)i snd "light plane" (m* = m ^ y ) In the case of isotropical effective masses, the bipolaron bound energy 206

is 15.6% of the polaron energy, but in the limiting case of the maximum anisotropy of effective masses mxy "^ rnz > this quantity is 25.2% of the polaron energy. Therefore the bipolaron formation should be quite possible in crystals with high-temperature conductivity, especially, if we take into account that these crystals are the systems with "light plane" anisotropy. Figure 3 shows the bipolaron bound energies expressed in the units of the

double polaron energy at various values of the ratio mXY/mz. The lines 1, 2, 3, 4, 5 correspond to this ratio at 1, 2 ~ 2 , 2 ~ 4 , 2~ 1 5 , 2~ 2 0 , respectively.

Figure 3. Dependence of the bipolaron bound energy on parameter v = oo/eo- The lines 1, 2 , 3 , 4 , 5 , correspond to the parameter equal

f

to 1, 2 ~ 2 , 2 ~ 4 , 2~ 1 5 , 2~ 2 0 , respectively, mXY/Tnz = 1-

It is seen that the region of bipolaron existence extends, and the bound energy increases as the anisotropy of crystals grows. We point out that we can 207

consider the two-dimensional systems by this method using scale transformation Ip(2D)/Ip(3D) = 2/3(3ir/4) 2 (where Ip(3D) and / P (2Z>) are the polaron energies in the two- and three-dimensional systems, respectively). Similar relations can be derived for the bipolaron ground state energy and, consequently, for bipolaron bound energy. The possibility of this scale transformation to the two-dimensional systems is revealed in1 , and numerically tested by us for the case of the maximum anisotropy of effective masses. We note that we consider electron-optical-phonon interaction. Using the continuum approximation, we can easily express the bound energy for electronacoustic-phonon interactions. But the absence of reliable data on the tensor of the deformation potential makes it impossible to calculate the bound bipolaron energy with account of the condenson effect. We note that this effect can play a key role due to the increasing bound energy in this case. The bound energy of autolocalized state may change significantly as a result of the interaction with plasma vibrations of charge carriers in the conduction and valence bands, or in the systems of movable ions for crystals with ion conductivity. But in addition to the effect caused by screening Coulomb interactions of electrons and ion vacancies, for bounded systems, such as polarons, bipolarons, F, F', F2 -centers there is a screening of electron interactions with optical phonons. Therefore, without detailed calculations of the bound energy for the polaron and bipolaron surrounded by plasmons we cannot estimate the plasmon effect on these energies.

References 1. S.I. Pekar, Studies in the Electronic Theory of Crystls (Gostekhizdat, Moscow, 1951) [in Russian]. 2. G. Verbist et al, Phys. Scripta 39, 66 (1991). 3. V.I. Vinitskij et al, Ukr. Fiz. Zh. 37, 76 (1992) ( U k r . Phys. J.). 4. E. Bergman, J. Phys. and Chem. Sol. 35, 1433 (1974).

5. V.I. Orbukh V.I. and V.I. Sheka, Ukr. Fiz. Zh 30, 771 (1985) (Ukr. Phys. J.). 6. J.T. Devreese, Phys. Scripta 39, 309 (1989). 7. E.P. Pokatilov et al, Phys. Status Solidi (b) 171, 437 (1992). 8. S.G. Suprun and B.Ya. Moizhes, Fiz. Tverd. Tela 24 (5), 1571 (1982) (Sov. Phys. Solid State Phys.). 9. P. Gombash, Many Particle Problem in Quantum Mechanics (Inost. Lit., Moscow, 1953) [in Russian]. 208

10. 11. 12. 13.

S.J. Miyake, J. Phys. Soc. Jpn 38, 187 (1972). D. Reagor at al, Phys. Rev. Lett. 62, 2048 (1989). G. Verbist et at, Phys. Rev. B 43, 2712 (1991). V.D.Lakhno, Phys. Rev. B 46, 7519 (1992).

209

SU(2) PATH INTEGRAL FOR THE HEISENBERG FERROMAGNETIC E.A.KOCHETOV Joint Institute for Nuclear Research Bogoliubov Laboratory of Theoretical Physics 141980 Dubna, Moscow region, Russia We have derived the expression for the partition f u n c t i o n in terms of the SU(2) (spin) path integrals for a q u a n t u m Heisenberg ferromagnetic. We evaluate the excitation s p e c t r u m in the proximity of the mean field, and calculate corrections of effective interactions to the temperature Green functions, reproducing the results of the spin diagram technique at finite temperatures.

1

Introduction

One of the interesting problems of solid-state physics is the problem of the quantum Heisenberg ferromagnetic at finite temperatures. The partition function of the quantum Heisenberg ferromagnetic is expressed as

where Sj are the SU(2) generators of dimensionality 2S + 1, and Jij is the symmetrical matrix of exchange interaction (Ju = 0), which is positively determined and can be written as1

where T stands for a chronologically ordered product and the summation over all repeating indices is assumed. In Eq. 1 we imply the summation over all repeating indices. Deriving Eq. 1, we used the Hubbard-Stratonovich equality. The partition function of noninteracting spines in the external fluctuating field

210

V"(*) can be written as the SU(2) path integral

where we put •iji = tl>x + iipv, V1 — V"i ~ *V"y > and V1^ = *l>x • The functional measure in Eq. 2 is formally defined as a product of infinite number of SU(2) invariant point measures

According to 2 , integral Eq. 2 is evaluated by the substitution

where

As a result, we obtain

where the function z = u/v depends on if> via the Riccatti-type equation

Thus, we take the partition function of the form

211

Relations Eqs. 5-6 serve the starting point for expansion in the vicinity of the mean field point.

2

Mean-field theory

We choose a steady mean field in the z direction as «&j = (0, 0, *j). equation of the stationary phase

The

(subscript |0 denotes value at a stationary point) yields

where b(x) = SBs(Sx), and BS(X) is the Brillouin function. Deducing Eq. 7, we assume that JQ = S • J\j and $i = $ due to spatial homogeneity of the system. Expanding S(T/J) at the point $ up to the second order terms in the deviation 77 = 1/1 — ijj |Q, we get

where the inverse effective longitudinal and transverse interactions are

212

Taking variational derivatives of Eq. 4, we find

that results in the temperature Green function of noninteracting spines with the Hamiltonian H = H0 = $£s' Z ) ; n$ = (e?* - I)" 1 .

Turning to Eq. 10, we obtain

which can be rewritten in the energy-momentum space as

Due to Eq. 12, the Larkin equation for the total propagator

yields

where (G0(un) = b/(iu>n + $),

un = 2irn//3), and

is the energy of elementary spin-wave excitations. We can easily calculate the path integral Eq. 8 contributing to the partition function due to Gaussian fluctuations in the vicinity of the mean field. The result coincides with estimations given in 1 . We note that our approach does not use the transformation from the spin representation to the Bose (Fermi) oscillatory representation of generators of SU(1) group, which essentially changes the initial Hamiltonian and complicates the calculations. Besides the proposed method enables us to find the expansions of the Green function in the vicinity of the mean field. 213

3

Green functions

For clarity we consider the transverse Green function

where G^. (T, c r \ i p ( t ) ) is the Green function of noninteracting spines in the external fluctuating field ip(t), which is calculated as a functional derivative of Eq. 3

where we introduce the notation

To calculate corrections to the Green function, caused by the Gaussian

fluctuations over the mean field, we should expand S(ifi) in Eq. 16 up to the

second order terms as in Eq. 8. Then, expanding the functional derivative Eq. 17 in powers of 77 and evaluating new integrals, we can easily solve the problem. As is seen from Eq. 8, the corrections increase as powers of the effective interactions Jefj-tin H Jejj.tr. At the stationary point, the required functional derivatives of zt(s) with respect to V'j(t) are calculated with the use of Eq. 4. The zero approximation yields

214

as it must. We extract the first order correction caused by the account of the forth term in Eq. 17,

Adding up these two terms, we find Eq. 14, and, consequently, we obtain the spectrum of spin-wave excitations. We should account Eq. 18 separately, since it has the zero order with respect to inverse effective volume v of the interaction, while the other first order terms with respect to Jcj j present one-loop corrections and have the first order with respect to v. Since the parameter v is small in the mean-field theory, the Green function Eq. 14 should be treated as the zero approximation of the total Green function calculated in the mean-field theory. It is clear, that relation Eq. 17 includes all diagrams of conventional spin diagram techniques at finite temperatures, where, however, interaction lines are replaced by effective lines.

4

Conclusion

Here we have developed a new method based on the spin path integration to calculate thermodynamic functions in the Heisenberg model. The method can be immediately generalized to more complicated groups describing dynamic

symmetries of complex systems. For instance, the supergroup J7(2|l) is a group of dynamic symmetry for the Hubbard model (t —- J model), 4 and the superpath integral constructed by this group can be used to study the model.

References 1. S. Leibler and H. Orland, Ann. Phys. 132, 277 (1981).

2. E.A. Kochetov, Phys. Lett. A 180, 383 (1993). 3. E. Manousakis, Rev. Mod. Phys. 63, 1 (1991). 4. P.B. Wiegmann, Phys. Rev. Lett. 60, 821 (1988).

215

LOW-TEMPERATURE ELECTRON MOBILITY OF ACOUSTICAL POLARON B.A.KOTIYA", and V.F.LOS b a

Georgia Technical University, Institute of Fundamental Research,

Tbilisi 380075, Republic Georgia Institute of Metal Physics of National Ukraine Academy of Science, Kiev S5214&, Ukraine Abstract

We derive a new exact evolution equations for correlation functions of a subsystem interacting with the Bose field. The electroconductivity for an electron in the acoustic polaron model is treated by the the linear Green-Kubo theory. For this model we evaluate the expressions for electroconductivity and low-temperature electron mobility, and calculate the correction to electron mobility caused by the account of electron-phonon correlations at the initial moment. 1

Introduction

In recent years, an interest to the problems of noneqiulibrium statistical mechanics has increased, for instance, one of the intensely studied fields

is kinetics of dynamic subsystem interacting with a thermostat (Bose field). 1 ' 2 We note that one of the important problems of condensed physics and quantum field theory is the problem of interaction between a particle and a quantum Bose field. The electron-phonon system is the simplest example of this electron-phonon interaction in a crystal. ~~ At present, a large number of problems concerning the behavior of slow carriers in polar semiconductors and ionic crystals are treated by the polaron model. The main attention has been focused on the theories of linear electron transfer processes in polar crystals. The investigation of polaron (electron) transfer and evaluation of polaron kinetic characteristics is one of pressing problems of the modern theory of large polarons. ~~ In this paper, we derive an exact quantum evolution equation (generalized quantum kinetic equation) for the two-time equilibrium correlation function of an (electron) subsystem interacting with the Bose (phonon) field. Our study is based on the Liouville superoperator formalism and the method of projective operators. Here we consider the acoustical polaron model in the continual limit. Using the second order perturbation theory for the case of weak electron-phonon interaction and low temperatures, we derive the equations with except phonon amplitudes and the expressions for the velocity-velocity correlation functions of the electron.

216

We also deduce the relations for the time and frequency of relaxation velocity and momentum of the electron. Using the above approach for the acoustic polaron at low temperatures, we evaluate the kinetic transfer coefficients (mobility, electroconductivity) for the electron. 2

Equation for correlation functions

We consider the dynamic subsystem S interacting with the Bose (phonon) field (thermostat) S. We write the Hamiltonian of the whole system (5 + E) in the form

where H,,Hz,Hi are the Hamiltonians of the subsystem, the (Bose) phonon field, and electron-phonon interaction, respectively.

where u(k) > 0 is the frequency of the quantum Bose (phonon) field, 6^, &J are the creation and annihilation Bose operators for the state, which is described by quantum numbers k, and Cfc(s), C^(s) are the operators related to subsystem 5. For the model of the continual acoustic large polaron, 10 we have

where p, k, and r are the electron momentum, wave vector, and coordinate, respectively, and m is the electron effective mass, V, is the sound velocity, a is the dimensionless electron-phonon coupling constant, V is the volume of a crystal, D is the constant of the deformation potential, and p is the density of the crystal. The main goal of our study is to calculate the low-temperature electron mobility (conductivity) for the acoustic polaron, which is described in the weak-coupling case (a < 1) by the Hamiltonian of Frohlich type. 1 " 3 Using the superLiuville formalism and the method of projective operators, we can derive an exact equation of quantum evolution for the two-time-dependent equilibrium correlation function (A,B,(t)), (here A,, and B, are the arbitrary operators related to the subsystem 5),

217

which describes the subsystem interacting with the Bose (phonon) field (see, for instance 11 ). This equation is written as

In this relation, as is follows from the definition, (A,B,(t)) = Z~1Sp(eKp[-/3H]A,ex.f[iLt]B,), L is the Liouville superoperator acting on the arbitrary operator C according to the rule

where L = L, + LS + Li, and L,, LE , Li correspond to the Hamiltonians of the subsystem and phonon field, determined by Eq. 1, /3 = 1/fcsT is the inverse temperature, Z = 5pexp[—flH] is the partition function of the whole system (S + E); P • • • = Sj>£(p% • • •) is the projection operator for averaging over the thermostat states.

In the last equation Z is the partition function of thermostat S, it can be presented as

In Eq. 4 we introduce the mass and integral superoperators, which are determined by

where H0 = H, + H-E • We note that Eq. 4 for the correlation function (A,B,(t)) includes the operators (amplitudes) of thermostat (Bose field). We suppose that the characteristic times are rather different in the system (S + E) due to weak interaction of Hi(Li), i.e. Trci ^
from Eq. 4, we obtain the Markovian kinetic equation for the correlation function (A,B,(t)). This equation with account of the second order perturbations for electron-phonon interactions (with respect to V^ for the considered acoustic polaron) can be written at rather long time (t ^> t 0 )

where a < 1, J\f^(fl) = (exp[/3fi.w(k)] — 1)-1 is the average number of

completing phonons in the state k, r(z) = exp [i/hT(P)Z] r = exp[-i/h]T(P)Z,

and the relation [E, C"] ± ^ (k) =

EC - exp[±/3hu>(k)]CE holds for operators E and C'. In accordance with Eq. 5, the equation for two-time dependent equilibrium function, which is the z component of the velocity operator of the electron (v^v^^)), takes the form

where a < 1, T >• ta , and VM = dl(p)/dpf

— p^/m.

We can derive the evolution equation for correlation function (v,(i)v,) in a similar way. Actually, using the relation

we can easily obtain the equation for correlation function (7) by the substitution t —» — t into Eq. 6. We note that the evolution of the initial

electron-phonon correlations is described by the last terms of Eqs. 5 and 6.

219

3

Calculation of low-temperature electron mobility

To estimate the low-temperature electron mobility (conductivity) for the case of the acoustic polaron, we use the Green-Kubo theory of linear response. 1 -1 According to the Green-Kubo theory, we express the dissipative part of the electroconductivity tensor as

where w is the frequency of the external field, while *n(l) is the symmetrized equilibrium time-correlation function of z, which is the component of the current operator

Let us consider the one-band approximation for the electron in the isotropic case when a^v = crxf = 1, where 7 is the dimensionless parameter of the theory, To = mV s 2 /2ifcfl is the characteristic temperature of the model. Then, calculating the trace in Eqs. 6 and 7 over electron states, and solving the equations for correlation functions, we arrive at expressions for the velocityvelocity correlation functions of the electron at long time (t > rT=i ^> to)

where a < 1,7 3> 1. In this relation P is the dimensionless electron momentum, P = p/V,m. For the relaxation frequency (characteristic time) of the electron velocity we have

where dQ^ = sin &d®dip, P cos * = sin © cos ipP^ + sin & sin tpPy + cos ©P., and 0, ip are the spherical angles of the wave vector k, while $ is the angle between the vectors k and P.

220

We consider a slowly moving electron at P ^ 1 (p <; V,m, v ^ •»,). According to Eq. 11 we evaluate the relaxation frequency of the electron velocity

Here we consider the weak coupling case a < 1. Therefore, for a slowly moving electron (when the electron velocity is much less than the sound velocity in the crystal v
where a < l ; 7 ^> 1, t > Trel . Substituting these asymptotic expressions Eq. 13 into Eqs. 8 and 9, OW»(:tO) = ne3(v*vm(±t)), we arrive at the expression for low-temperature electric conductivity

where a < 1; 7 ^ 1, w <; mV^/hf, and n is the concentration of electrons. For low-temperature dc electron mobility of the acoustic polaron we have

where

A/i =

32a(^v

exp[47] sin 2 (32c*7 exp[-47]) ,

a < 1,

•y > 1 .

This correction A/i to the low-temperature electron mobility ;*o is caused by the fact that the model takes account of the electron-phonon correlations at the initial moment. References 1. N.N. Bogolubov, Preprint JINR E 17-1822 (JINR, Dubna, 19T8). 2. N . N . Bogolubov and N . N . J r . Bogolubov, Tear. Mat. Fiz. 43 (1), 3 (1980). 3. N.N. Jr. Bogolubov et al, Tear. Mat. Fiz. 67 (1), 115 (1985).

221

4. V.F.Los and A.G. Maitinenko, Physisca A 138, 518 (1986). 5. K.Rodiiges and V.K. Fedyanin, Element. Chast. i Yadra 15 (40), 871 (1984) (Sov. J. Part, and Nucl.). 6. J. Appel in Polarons, ed. Ya.A. Firsov (Nauka, Moscow, 1975) [in

Russian]. 7. F.M.Peetres and J.T. Devieese, Solid State Physics 38, 81 (1984).

8. R. Zwanzig, Phyaica 30 (6), 1109 (1964). 9. V.F.Los, Doklady AN SSSR 240 (5), 1078 (1978) (Sov. Phys. Dokl.). 10. F.M.Peettes and J.T. Devieese, Phys. Rev. B 32 (6), 3515 (1985). 11. B.A. Kotiya and V.F.Los in Polarons and Applications, ed.

V.D. Lakhno. (Wiley, Chishestei, 1994). 12. R. Kubo, /. Phys. Soc. Jpn. 19 (6), 570 (1957). 13. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics

(Wiley, New-York, 1975). 14. A. Isihara, Statistical Physics (Academic Press, New-York, 1971).

222

POLARON EFFECTS ON MAGNITORESISTANCE IN QUASI-TWO-DIMENSIONAL CONDUCTORS L.S.

KUKUSHKIN

Institute for Low Temerature Physics and Engineering of National Ukraine Academy of Science, Kharkov 310164, Ukraine We evaluate a low-temperature magnitoresistance of a quasi-two-dimensional conductor in the strong transverse magnetic field, where the scattering of carriers is mainly caused by phonons. The m u l t i p h o n o n effects play a key role, since the electron-phonon interaction is supposed to be strong. We estimate the t e m p e r a t u r e and field dependence of the conductivity.

The discovery of the quantum Hall effect and high-temperature conductivity have aroused interest in the study of properties of quasi-two-dimensional conductors. As a result, a large number of works devoted to the study of two-dimensional conductors have come out. In the present communication, we calculate the conductivity of a two-dimensional electron gas in strong magnetic field H, which is perpendicular to the plane of the electron motion. Our calculations are based on the method developed in . We consider the case when the scattering of carriers is primarily caused by electron-phonon interaction. We note that the works concerned with the problem (see Refs. 2 and 3) are usually based on the second order perturbation theory and assume the diffusion mechanism for the centers of the cyclotron motion of orbits. The direct consideration of multiphonon effects resulting from strong electron-phonon interactions is a complicated problem, because there is no gradient invariance in this approximation, since the centers of the cyclotron orbits are numbered by a quantum number depending on calibration of the vector potential. But at low temperatures T
223

We suppose that a carrier moves in the plane (z, y), and a magnetic field is directed along the axis Z, then the one-electron Hamiltonian of the system can be presented in terms of the Landau calibration as

where po

=

(e^0~

*8 the magnetic length (A = c = 1); n^ = <** a^ is

the number operator of phonons with frequency u>(k). Here, to simplify the consideration, we assume that the effective mass is isotropic, and phonons are three-dimensional (as it takes place in heterostructures and bicrystals) or twodimensional (as in thin films and for the electrons above the helium surface). The method developed in is used in an unitary transformation of Hamiltonian Eq. 1. After this transformation Hamiltonian Eq. 1 takes the form

where c* and c are the creation and annihilation operators of the electron oscillator c* = 2~l/2(popy + ip^y). We note that the unitary operators

may be reduced to unit ones by some general assumptions. However, in the calculation of low-temperature conductivity with due regard for the multiphonon effect the difference between these operators is essential. The behavior of the system is determined by the operators

where a^ = 2~1'2 po(kx+iky ). The first multiplier in Eq. 4 restricts the phonon contribution for wave vectors k > kjj = Ip^ 1, and corresponds to an increased electron-phonon coupling as the magnetic field increases, especially for the interaction of deformation type. Therefore, there are no operators px and z in the transformed Hamiltonian. Moreover, they do not arise in the expression for conductivity tensor derived by 224

the Kubo relation, if using the unitary operator U, we transform the expression for the spur in the Kubo relation. Then, the diagonal components {o\ f c } of nondegenerate electron gas are presented as

where i/e = v — it(e —> +0); p@ = Z~l exp(—/3H), Z is the partition function, /0 = T~l, C±(T) = C* (T)±C(T) are the operators in the Heisenberg presentation, and n is the number of electrons. In Eq. 5, index (2) denotes that pp and c-k(-r) are determined by Hamiltonian Eq. 2. For degenerate gas, the conductivity (2^

is described by expression Eq. 5, with np\

(2^

replaced by Fermi function pp

depending on H^3'. To derive the Hall conductivity we should change one of

the multipliers c^ 2 ) (<) into »c(_2)(<). The direct calculation of Eq. 5 is still a complicated problem, but the evaluation of conductivity by Eq. 5 is possible, if we use the adiabatic approximation to separate the electron and phonon subsystems. The criterion of this approximation is Qc ^> uj, where LJ is the characteristic phonon frequency, for example,

for acoustic phonons it is skjj 105 Oe. If the wave number kr of thermal phonons for which uj(kf) = T is less than magnetic number kjj = 2pg

, then the adiabatic approximation

results in the inequality T
The method proposed in5 allows to calculate (ryy(Q,T) taking into account the two-phonon effects in the low-temperature limit, when ky < kjf • In this case, operators F^ can be expanded in power series of y and py, including quadratic 225

terms. As a result, the adiabatic functions V'JvCj/i Q) (where N is the number of Landau level), calculated at fixed medium coordinates Q = {Q-^}, are the wave functions of oscillator with frequency and equilibrium position depending on Q. The power dependence of cryv on temperature is related to the intraband processes, and leads only to the dependence of matrix elements on Q

In optics this dependence results from deviations on the Frank-Condon principle. If the number of carriers is less than the multiplicity of degenerated

Landau levels, (n
It should be mentioned that the type of electron-phonon interaction and the form of phonon dispersion affect strongly the temperature and field dependence of tryy- For acoustic phonons with ai(ifc) = sk and deformation type of their interaction with electrons, when Vk = iD(hk/psl0SY , (where /0 is the localization radius along the magnetic field, p is the medium density, and D is the constant of the deformation potential), the conductivity is expressed as

Here fl lo = 5/66 is the Bernoulli number. Expression Eq. 9 differs from the corresponding expression in6 by multiplier ~ (T/fi,.) 2 , which is related to the adiabatic parameter. It seems to result from the fact that the processes of phonon entertainment are not adequately taken into account in . (The total entertainment is easily seen to 226

lead to (7j fy (0,T) = 0, and crxy = ne/H). For three-dimensional phonons at deformation interaction crvv ~ D*T12H~8. The latter also differs from the corresponding expression in6 by the same multiplier. At piezoelectric interaction,

when Vk ~ k~3'2 and phonons are acoustic, (k) ~ k ' (neglecting the narrow region near k = 0, which is linear), while matrix elements of electron-ripplon interaction are derived in . At T < 0.1 K our calculation yields
itable, moreover, the state extends as the magnetic field rises. Account of this phenomenon should have effect on the behavior of
IB n

°t fulfilled, then we cannot expand operators

{Fjj.} in power series of y and py, because of the effect of the cut-off multiplier exp( — fc 2 />o/4) in Ffc. Hence, the phonons with k w ku play a key role in the conductivity instead of thermal phonons, that results in the distinct (more weak) dependence of conductivity on temperature and magnetic field. At kf > KH, to find the adiabatic wave function, it is necessary to solve the differencedifferential equation, since {F^} are the translation operators

We point out that, for the second order of {Vj.} this effect results in the expres-

sion of KU lies beyond the scope of this work. In conclusion we note that the carriers have an additive effect on
227

case

Expression Eq. 10 can be used as a notation of the effective time of carrier free path. This fact justifies our calculations of the effects of other contributions in the longitudinal conductivity, which are not the main ones. The similar situation takes place for the diagonal components of the electroresistance tensor, since in the twc-dime,nsional case

and Hall conductivity at Slcr* ^ 1 is not dissipative and greatly exceeds the diagonal conductivity.

References 1. L.S. Kukushkin, Zh.

Eks. Tear.

Fiz.

78, 1020 (1980) (Sov.

Phys.

JETP). 2. 3. 4. 5. 6.

R. Kubo et al, Solid State Phys. 17, 269 (1965). F. Stern et al, Rev. Mod. Phys. 2, 54 (1982). R. Kubo and Y. Toyozawa, Progr. Theor. Phys. 13, 160 (1955). L.S. Kukushkin et al, Fiz. Niz. Tern. 20 (5), (1994) (Low. Tern. Phys.). Yu.A. Bychkov et al, Pis. Zh. Eks. Tear. Fiz. 34, 496 (1981) (Sov.

Phys. JETP Lett.). 7. V.B.Shikin and Yu.P.Monarkha, Two-dimensional Charged Systems in Helium (Nauka, Moscow, 1989) [in Russian].

8. Yu.P.Monarkha, Fiz. Niz. Tern. 19, 235 (1993) (Low. Tern. Phys.). 9. E.D.Vol and L.S. Kukushkin, Fiz. Niz. Tern. 9, 97 (1983) (Low. Tern. Phys.).

228

NUMERICAL INVESTIGATION OF A QUANTUMFIELD MODEL FOR STRONG-COUPLED BINUCLEON I.V. AMIRKHANOV, I.V.PUZYNIN, T.P. PUZYNINA, T.A. STRIZH, E.V. ZEMLYANAYA Joint Institute for Nuclear Research, Laboratory of Computing Techniques and Automation 141980 Dubna, Moscow Region, Russia V.D.LAKHNO Department of Quantum-mechanical Systems, Institute of Mathematical Problems of Biology, RAS Pushchino, Moscow Region, 14SS9S, Russia We consider a quantumfield model for binucleon in the case of nucleon spot interaction with scalar and pseudoscalar meson fields. The nonrelativistic problem of two nucleons is shown to reduce to the one-particle problem. We derive in the strong coupling limit the nonlinear equations describing two nucleons in the meson field. We also discuss the applicability of the model to obtain the deuteron and bineuteron characteristics.

1

Introduction

The calculation of binucleon states has been in focus of many investigations.1'2 This problem is of interest, since its solution can yield direct data on nuclear forces. In this paper we try to calculate a bound state of two nucleons in a meson field by using the consistent translation-invariant theory. It is well known that in the strong coupling limit the field can be considered as classical in zero approximation.3 Therefore the interaction $(PI, r 2 ) of two particles with the classical component of the field should be written as

On the other hand, since the medium is supposed to be isotropic and homogeneous, two quantum particles should interact as

This contradiction is absent in the quantumfield theory of weak and intermediate coupling which describes the particles interacting by exchange of field quanta. The use of the perturbation theory in this limit leads to a translation - invariant expression Eq. 2 for the interaction between two particles. 229

The aim of this paper is to consider the translation-invariant quantumfield model for the interaction between nucleons and pion field in the strong coupling limit and to study nonlinear differential equations for nucleons, arising in this limit.

2

Translation-invariant theory in the strong coupling limit

According to current concepts, the main contribution to the long - range nucleon-nucleon interaction is caused by ir-mesons. In the nonrelativistic approximation the pion-nucleon interaction is determined by the pseudoscalar coupling Hamiltonian:

where fj, is the pion mass, / is the interaction constant, a and r are spin and isospin operators, respectively. Expanding the pion field into Fourier series

where ro is an arbitrary reference point, we can present Eq. 3 as

K

where

The total Hamiltonian for two nucleons in a meson field is expressed as

where

Jip corresponds to two free nucleons with coordinates r t and r 2 , m is the nucleon mass,

230

where i = 1,2 corresponds to the first and the second nucleons, respectively;

is a Hamiltonian of a free meson field. We can use the results of strong coupling theory ' for Hamiltonian Eqs. 9 and 10. According to this theory, coordinates TO and r divide, and the Schrodinger

equation in zero approximation is written for the binucleon wave function i/>o as

where r = ri — r 2 is relative coordinate, and

where

where a is the unit vector. The Bogolubov - Tyablikov method was used in5 for the case of scalar interaction. According to 5 , the potential of scalar interaction He is written as

where g is the constant of the interaction with the scalar field. We take into account both contributions into the interaction, putting in Eq. 13

231

3

Differential equations for binucleon.

The Schrodinger equation for binucleon with potential Eqs. 16—18 can be presented in the variational form as the functional:

where

Taking into account the functional Eq. 20 to be minimum for functions V"( r ) with the normalizing conditions

we get the set of equations

Using the notation

and assuming A = 1, we can present Eq. 23 as

Energy levels e and wave functions i/> of the binucleon can be determined by solving the system Eq. 24 for functions V>(r), which satisfy the normalizing 232

condition Eq. 22 and are limited within 0 < r < oo, with boundary conditions

(Here r = V z2 + J/2 + z2.) Then for the binucleon radius R and the quadrupole moment Q the following formulas can be used

Using in Eqs. 24 the dimensionless variables

we get the following system of equations (bar mark for vector f was omitted)

where

Using the last equation we can determine -y, for example, supposing ro = 10- 13 m. Taking the change of variables Eq. 28 into account, we rewrite

Eqs. 22, 26, 27 and 19 as

233

Then we use the relations between the variables:

Here the expressions in the brackets are dimensionless variables. The set of differential equations for bipolaron Eq. 29 is the nonlinear threeparameter eigenvalue problem. Numerical integration of Eq. 29 was performed in5 in the limit kfic —* <x>. Thus, the solution of Eq. 29 for finite hue is an actual problem.

4

Statement of the boundary value problem for the set of equations.

We discuss some alternatives for the statement of the boundary problem for the system Eq. 29 in the special case of axial symmetry, i.e. $(p,z,) —» 3t(p, z) (for the cylindrical coordinates ) or $(r, 6,<j>) —* $(r, 6) (for the spherical coordinates ). In addition, instead of the semi-infinite integration interval

we use the finite interval: —ZM <

z

<

z

u,

^ < p < PM for cylindrical

coordinates and 0 < r < rM for the spherical coordinates.

4-1

Alternative I.

We are supposed to solve the system Eq. 29 in cylindrical coordinates using the boundary conditions determined on the rectangle —ZM^KfGD (see Fig. 1), 234

that is

on the interval — ZM < z < ZM and

for 0 < p < PM • Here

where r = v p2 + z2 , and
235

Thus, the determination of the physical parameters of the deuteron is to solve the spectral problem for the set of partial differential Eq. 29 with the boundary conditions (Eqs. 39 and 40) and the normalizing conditions Eq. 31. Taking into account the symmetry of the problem, the boundary conditions can be determined on rectangle QzMCpM (the hatched region in Fig. 1).

Figure 1.

Then on the rectangle border OzM, zMC and ptfC *ne boundary conditions are as in Eqs. 39, 40 and for the border OpM they take the form

4-S.

Alternative II.

We can formulate the problem in terms of the ordinary differential equations.

236

The solution of the system Eq. 29 in cylindrical coordinates we present as

where function {nn>$n} is determined as the solution of a spectral problem

Here AfM

is determined from the following relations:

The function (p^ ig replaced by V\A or VZA , an i n } or {MS™, *3n}- In special case we can use only one set of functions {$„}. Substituting the expansion Eq. 42 into the system Eq. 29, we have:

where

237

Functions yn, Vj,-, V3j satisfy the following boundary conditions

where

The coefficients A( — ZM ), -<4.i(—ZM) and A$(—ZM ) are determined in the same way.

Using the problem symmetry, as in the first alternative, the set Eq. 44 can be solved on the interval 0 < 0 < ZM • The boundary conditions are as in

Eqs.

49 and 50 is replaced by:

238

The normalizing conditions for Eq. 44 take the form:

if we use boundary conditions Eqs. 49 and 50 and the form

for the boundary conditions Eq. 49 and 51. Expressions Eqs. 19, 26 and 27 for the expansion Eq. 42 can be written as:

where

4-3

Alternative HI.

Let us present the solution of the system Eq. 29 as an expansion in spherical coordinates:

239

where

pi are the Legendre polynoms. Substituting the expansion Eq. 55 into the system Eq. 29, we obtain

where

240

Taking into account the asymptotic behavior of solutions

the system of Eq. 57 will be solved with the following boundary conditions:

Thus, physical characteristics of binucleon can be found by the solution to the spectral problem Eq. 57 with boundary conditions Eq. 58 and normalizing conditions

Taking into account Eq. 55, the expressions Eqs. 19, 26 and 27 used for calculating the physical parameters take the form

where

241

5

Numerical methods and numerical results.

We investigated numerically the set of Eq. 44. Calculations of the basic functions {$„} = {*i n } = {*3n}, n = l,L, L = 1,2,3,4,5, for a given pM were performed. The code SLIPl 6 was used to solve numerically the problem

Eq. 43 for the base functions {$ n } with normalizing conditions Eq. 43. The algorithm suggested in was used to solve the problem Eq. 44. This algorithm is based on the combination of continuous analog of the Newton method (CANM) and the continuation method and allows us to reduce the solution to simple linear problems. The following iterative scheme was used: • The problem Eq. 43 was solved with the boundary conditions

for the set of basic functions.

• Using the initial approximation of yn(z) and a set of basic functions calculated at the first step, we evaluate the right side of the second and the third equations of system Eq. 44 J\j and J3j, (j = n) using Eqs. 4648. Then we determine functions V\j and V3j (j = n) as the solution of a linear boundary problem by the sweep method. • Substituting the approximations of Vjy and V$j into the first equations of system Eq. 62, we have an eigenvalue problem for the set of linear differential equations, which can be solved by CANM in the modified code based on the program START.9 The function exp(— \z\) was used as the initial approximation for j/ n (*) together with the solutions obtained for the other set of parameters. The number of Newtonian iterations do not usually exceed 10-13. • Solutions yn from the previous step are used to calculate Jij and J3j in the next iteration and so on. The process of iterations stops when the

solutions obtained after two sequential iterations would coincide with each other with a given accuracy e. Actually 5-8 iterations are enough to reach the accuracy e = 10~3. 242

• Parameter Ap is calculated with the help of the solutions yn. Then the problem Eq. 43 is solved again with the corresponding boundary conditions, and a new set of basic functions is determined. The solution to the problem Eq. 44 are calculated again for this set by the above algorithm.

The process will continue until stabilization, i.e. until the newly-obtained solutions and the value APM coincide with the preceding ones with the accuracy given a priori. Actually 3-4 iteration steps are required to reach the stabilization. The discrete approximation of the problem was made on a h-step uniform

grid {zi} with the help of the difference schemes of the second order convergence O(A 2 ). The calculations on a sequence of the twice compressible grids

were shown in Table 1 and confirm the 2nd order of the convergence( see Table 1). Table 1:

Table 2 lists the results obtained on a sequence of expanding intervals [—ZM,ZM] showing convergence on parameter ZM• Table 2:

It should be noted that in practice we use a finite number L of functions

n (n = 1, L) in expansion and, therefore, the accuracy of the obtained results essentially depends on PM • H this parameter is small we cannot approximate the solution with good accuracy.

On the other hand, the big value of PM 243

requires a large number of basic functions. For the optimal solving, the number of basic functions L and pM can be determined by the numerical experiments. These results are given in Tables 3 and 4. In the top raw of each column the value A, normalization Nn and amplitude <j>n(z) for n = 1, L, obtained for the boundary conditions Eq. 63 are given. The same values after the solution stabilization for parameter AfM are given in the bottom row. Table 3:

Table 4:

It can be seen from Tables 3 and 4 that pM = 5 — 7 can be considered as optimal for this set of parameters, since in this case the solution does not depend on PM • In addition, the initial norms and the expansion amplitudes do not differ much from the limiting values. With increasing or decreasing PM the dependence of the results on this parameter makes stronger. Let us note that the contribution on the norm of each next expansion component decreases. This allows us to assume that when increasing a number L of functions $ n , one can obtain a solution with a required accuracy.

Functions yn(n = 1, ..., 5), V\ n (n = 1, ..., 5), V-^n(n = 1, ..., 5) for this set of parameters are presented in Fig. 2. The function f>(p, z) (see Eq. 42) is 245

presented in Fig.3.

Figure 2.

246

Figure 3.

247

Figure 4.

248

Now let us turn our attention to experimental data. According to10 — radius R = 1.963 • 10~13 m, — quadrupole momentum Q = 2.86 • 10~

cm ,

— binding energy 2.246eV. It follows from Eqs. 52-54 that at r0 = 10~13 the calculated dimensionless values correspond to the experimental ones if the following relations are fulfilled:

According to Eq. 30, the parameters of the model must be as follows

and the B and Nc may be arbitrary. However, we failed to get satisfactory results for these parameters. Considering ki as arbitrary parameter and fitting its value by the mentioned relations Eq. 64, we obtained the best results at

Hence, we have

Turning to dimension values with the help of relations Eqs. 30-36 we find

The same calculations were done for the problem statement by alternative

III. For the parameters presented in Tables 1-4 at / = 0, • • • , 5 the achieved eigenvalue equals A = 0.67 and is in satisfactory agreement with the results presented in Tables 3 and 4 for alternative II. For more precise calculations the increasing number of basic functions are needed in both statements.

6

Conclusion

Calculating numerically the nonlinear Eq. 41 for two nucleons in a scalar and pseudoscalar field, we evaluate binding energy, effective radius, and deuteron quadrupole momentum, which fit the experimental data Eq. 82, if we assume the effective mass of scalar meson to be near the mass of pseudoscalar meson

fj,a w 140 Mev and k* = k% = 1. At the same time, this fitting yields the coupling constant g and / one order less than the experimental data determined 249

by the local linear model of nucleon-meson interactions.

But our model is

nonlocal and nonlinear.

Note that we investigated only the simplest case of spot interaction. The nucleon formfactor included in the equations can change significantly the final results. The other important trend is to generalize the theory to the case of intermediate coupling constants and take into account relativistic effects.

Authors are grateful to the Russian Foundation for Basic Research (Grant N 94-01-01119) for financial support.

References 1. T. Ericson and W. Weise in Fiona and Nuclei (Claredon Press, Oxford, 1988). 2. G.E. Brown and A.D. Jackson in The Nucleon - Nucleon Interaction (North - Holland, Amsterdam, 1976).

3. 4. 5. 6.

E.P. Gross, Ann. Phys. 19, 219 (1962). N . N . Bogolubov, Ukr. Mat. Zh. 2, 3(1950). V.D.Lakhno, Tear. Mat. Fiz. 100, 219 (1994). I.V. Puzynin and T.P. Puzynina in Algorithms and programs for solving

some problems of physics (KFKI-74-34, Budapest (93-111), 1974). 7. I.V. Amirkhanov et al, in Polaron and Applications, ed. V.D. Lakhno,

(Willey, Chishester, 1994). 8. A.P. Zhidkov et al, PEPAN 4 127, (1973); I.V. Puzynin et al, Zh. Vych. Math. Fiz. 32, 846 (1992).

9. Yu.S. Smirnov, JINR Russian]. 10. K.N. Mukhin,

Communications (Ell-88-912, Dubna, 1988) [in

Experimental

Nuclear Physics (Energoizdat,

1983) [in Riussian]. 11. V.D. Lakhno Chapter 3 in this book.

250

Moscow,