Perspectives of Polarons

PERSPECTIVES OF

POLARONS

PERSPECTIVES OF

POLARONS

Editors

G. N. Chuev V. D. Lakhno Russian Academy of Sciences

World Scientific Singapore • New Jersey • London • Hong Kong

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Library of Congress Cataloging-in-Publication Data Perspectives of polarons / editors G. N. Chuev, V. D. Lakhno. p. cm. ISBN 9810227787 1. Polarons. I. Chuev, G. N. (Gennady N.), 1961II. Lakhno, V. D. (Viktor D.) QC176.8.P62P47 1996 530.4'16~dc20 96-35326 CBP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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The monograph "Perspectives of polarons" presents further development of the polaron theory and its applications in various fields of physics. Originally polaron arose in condensed media physics to explain peculiarities of electron states in semiconductors, ionic crystals and other polar media. Subsequently methods of the polaron theory were used to describe electron properties in other condensed media. The states of polaron type in nonpolar media were named as condenson, in magnetoordered media as magnet polaron, in the electrolytes as ionon, etc. Nowadays methods of the polaron theory are widely used in such sciences as biology, chemistry, nuclear physics, etc. Such wide applications result from the fact that polaron is one of the first substantial examples of a nonlinear quasiparticle. This monograph deals with only a small part of the problems indicated. It is based on the talks, given at the Workshops which were held in Pushchino in 1993 and 1994. Our book includes selected articles, which are representative of the exciting scientific presentations that marked the workshops. The first Chapter is devoted to theory and applications of large polaron. The combination of polaron theory with modern quantum chemical calculations is presented in Chapter 2. The polaron problem includes issues which are being studied intensively in the quantum field theory and nonperturbative models of elementary particles. We include three works (Chapters 3, 4, and 13) devoted to applications of the polaron in quantum field theory of particles. The polaron behavior in magnetic systems and magnetic fields are considered in Chapters 6, 10, and 12. Effects of disorder and order of the structure on self-trapped electron states are treated in Chapters 7 and 8. We include the review devoted to quasi-one-dimensional polaron (Chapter 5) in biopolymers, due to vast number of applications of polaron model in one-dimensional structures. The study of high-temperature superconductivity stimulates the bipolaron theory. A generalization of the bipolaron problem to the anisotropic case is considered in Chapter 9. We hope our book will be of interest to a broad range of readers, which can be able to sense the excitment of the articles. The monograph will hopely surve further development and contribute to the introduction of the polaron theory methods into various fields of physics.

Gennady N.Chuev Victor D.Lakhno

v

CONTENTS

PART 1. Papers based on the workshop "Perspectives in Polarons" (Pushchino, 1993). Theory and Applications of Strongly Coupled Large Polaron

1

G. N. Chuev and V. D. Lakhno Theoretical Investigation of Self-Trapped Hole in Alkali Halides.

Long-Range Effects Within the Model Hamiltonian Approach P. B. Zapol and L. N. Kantorovich

38

The Quantumfield Theory of Binucleon V. D. Lakhno

63

Generalized Functional Approach in the Theory Meson-Nucleon Interaction

85

V. D. Lakhno Nonlinear Mathematical Models in Biopolymer Science L. V. Yakushevich

The Ground State of Frohlich ID Fermi-Polaron E. A. Kochetov

94

144

PART 2. Papers based on the workshop "Autolocalized Electron States in Ordered and Disordered Systems" (Pushchino, 1994).

Mean-Field Theory of the Solvated Electron and Dielectron States

150

G. N. Chuev Resonance, Localized and Polaron-Type Electron States in Elastic Materials with Topological Defects V. A. Osipov

188

Bipolarons in Anisotropic Crystals N. I. Kashirina, E. V. Mozdor, E. A. Pashitskij, and V. I. Sheka

202

SU(2) Path Integral for the Heisenberg Ferromagnetic E. A. Kochetov

210

vii

Low-Temperature Electron Mobility of Acoustical Polaron B. A. Kotiya and V. F. Los

Polaron Effects on Magnitoresistance in Quasi-Two-Dimensional Conductors L. S. Kukushkin

216

223

Numerical Investigation of a Quantumfield Model for Strong-Coupled

Binucleon /. V. Amirkhanov, I. V. Puzynin, T. P. Puzynina, T. A. Strizh, E. V. Zemlyanaya and V. D. Lakhno

viii

229

AUTHOR'S ADDRESSES Dr. I.V. Amirkhanov

Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research, Dubna, 141980 Russia

Prof. G.N.Chuev Department of Quantum-Mechanical Systems, Institute of Mathematical Problem of Biology of Russian Academy of Sciences, Pushchino, 142292 Russia

Dr. L.N.Kantorovich Department of Physics, University of Keele, Keele,

Staffordshire, ST5 5BG.UK

Dr. N.I.Kashirina Institute of Physics of Semiconductors of National Ukraine Academy of Science, Kiev 25202, Ukraine

Dr.E.A.Kochetov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research Dubna 141980, Russia

Prof. B.A.Kotiya Georgian Technical University, Tbilisi, Georgia

Prof. L.S.Kukushkin Physical and Technical Institute

of Low Temperature, Kharkov, Ukraine

ix

Prof. V.D.Lakhno Department of Quantum-Mechanical Systems, Institute of Mathematical Problem of Biology of Russian Academy of Sciences, Pushchino, 142292 Russia

Prof. V.F.Los Institute of Metal Physics, Kiev, Ukraine

Dr. E.V. Mozdor Institute of Physics of Semiconductors

of National Ukraine Academy of Science, Kiev 25202, Ukraine

Dr. V.A.Osipov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research Dubna 141980, Russia

Prof. E.A.Pashitskij Institute of Physics of Semiconductors of National Ukraine Academy of Science, Kiev 25202, Ukraine Prof. I.V.Puzynin

Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research, Dubna, 141980 Russia

Dr. T.P. Puzynina, Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research, Dubna, 141980 Russia Prof. V.I. Sheka Institute of Physics of Semiconductors

of National Ukraine Academy of Science, Kiev 25202, Ukraine x

Dr. T.A.Strizh Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research,

Dubna, 141980 Russia Dr. L.V. Yakushevich Institute of Cell Biophysics, Russian Academy of Sciences,

142292 Pushchino, Moscow Region, Russia Dr. E.V. Zemlyanaya Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research,

Dubna, 141980 Russia Dr. P.Zapol Department of Physics,

Michigan Technological University, Houghton, MI, 49931, USA

xi

THEORY AND APPLICATIONS OF STRONGLY COUPLED LARGE POLARON G.N.CHUEV and V.D.LAKHNO Department of Quantum-Mechanical Systems, Institute of Mathematical Problems of Biology,

Russian Academy of Sciences, Pushchino, Moscow Region, 142292 Using the Bogolubov-Tyablikov method, we set forth a pol&ron theory. Taking into account the translational invariance of the free polaron, we consider the autolocaliaed electron states for different types of interaction. It is shown that the structure of these states is strongly related to the structure and peculiar features of the local phonon spectrum. We calculate the phonon spectrum in the strong-coupling limit for electron centers. We discuss the applications of the large pol&ron model and possibilities of its experimental verification. We generalize the BogolubovTyablikov treatment to the strongly coupled bipolaron and propose the criteria of stability and formation for the bipolaron states.

1

Introduction

A polaron concept being the simplest example of a quantum field theory has a huge number of interesting applications in condensed matter physics. We note that the polaron problem was originally formulated as a problem of an autolocalized electron state in an ionic crystal. l In this treatment, the polaron was assumed to correspond to the strong-coupling limit. Figure 1 plots schematically the Pekar polaron treatment. But for a long time the central problem in the polaron theory was to generalize the method to arbitrary coupling, since the strong-coupling criterion

is not satisfied for most of ionic crystals. Significant effort was made to develop various approaches to the calculation of the dependence of the ground electron state on the electron-phonon coupling constant a,3'4 to evaluate the effective mass at various a, to extend the treatment to finite temperatures, 6 to study the polaron transport problem. 4>5 ' 6 ' '8 On the other hand, there is a number of physical examples when the electron can be autolocalized and the strong coupling takes place. Among such examples are the magnetically ordered crystals, where magnetic polaron states are possible, 9'10'11 the polar liquids, where the autolocalized states are the solvated electrons12'13, and other systems. Moreover, even in ionic crystals where polaron states are treated by the weak or intermediate coupling description, the strong coupling is possible for the bipolaron states. We also note that in the important case of the bound 1

polaron arising in the .F-center formation, the strong-coupling criterion is much weaker and can be fulfilled for the ionic crystals. Thus, the study of the strongly coupled polaron is of interest to condensed matter physics, though being a limiting case. Besides, we show in the review that the concept of the strongly coupled polaron can be successfully used for a number of related problems, such as the meson theory of nuclei interaction, 14,15,16 tne evaluation of the mobility of ions implanted into liquid helium ir and so on.

Figure 1: The Pekar polaron treatment in NaCl crystal. Dashed circles denote displaced equilibrium positions of atoms, dotted area is the region of electron localization. In most works on the strongly coupled polaron, the main attention is attached to the study of the ground state. At the same time, the question of the possible existence of polaron states distinct from the ground state is basic to the study of processes related to electron excitation in polar media, for instance, to photoexcitation of ^-centers and other lattice defects. At present, the problem of the excited polaron states, besides being of

theoretical interest, is gaining attention due to the problem of the electron transfer in a wide variety of condensed media (solutions,18 biomacromolecules, 19,20,21,22

et 44

where g is the constant of the nucleon-meson coupling, JJ,Q is the meson mass, c is the light velocity.

In the case of the continuum exciton 29'45

where up is the plasma frequency, e(k) is the dielectric constant of a doped semiconductor, /CD is the inverse Debye radius. In a number of cases, Hamiltonian Eq. 1 describes the behavior of electrons in magnetically ordered media. 24 > 46 7

3 3.1

States of the electron subsystem The ground slate

Eq. 9 is the nonlinear Shrodinger equation. It has been repeatedly investigated for the ground electron state for various self-consistent potentials. For the polaron Eq. 12 and the .F-center Eq. 13 the ground spherically symmetrical state Vo( r ) was found in 1 by the variation method. The best numerical solution for the ground state was obtained in 4 7 . The results of these calculations were confirmed in 4 8 . The solution of the equation with the short-range potential Eq. 15, corresponding to the ground state was actively investigated in nonlinear optics, ' in physics of disordered systems, 5 1 > 5 2 > 5 3 in nonlinear field theory 54 either by different variation methods or by numerical iterative calculations. 5S In 42 it was investigated by variation methods, as applied to the problem of an autolocalized electron state. The results were revised in 56 by the direct numerical calculation of the problem Eq. 15. The numerical calculations of the ground state for the self-consistent potential Eq. 17 are given in 57 ' 58 .

3.2

Excited states

The detailed study of the nonlinear Shrodinger equation shows that in the general case its solution is not unique, it has a discrete set of solutions, the proper self-consistent potential corresponds to each of them. This is most clearly cut for a .F-center, where the solutions can be put into correspondence to the solutions of the linear problem with the hydrogen-like potential. Then, the ground state corresponds to the la-type solution, the first excited state with spherical symmetry corresponds to the 2s-type, and the state asymmetric about rotation corresponds to the 2p-type and so on. We note that the excited states are less investigated than the ground state. It was proved in 59 ' 60 ' 61 that there is a countable number of solutions (modes) for the self-consistent polaron potential Eq. 12 and for the potential Eq. 15. The spherically symmetrical polaron states were numerically found in 6 , and the ones for the .F-center in 6 3 . In 5O ' 58 , the excited states were calculated for a homopolar crystal with the short-range potential Eq. 15. Some of the first solutions and the corresponding self-consistent potentials are shown in Fig. 2. Generally, for the (n + l)-st spherically symmetrical mode the solution intersects n times the x axis. Table 1 lists the numerical values of the radii, effective masses and total energies for various spherically symmetrical polaron states. Note that in es the quasiclassical asymptotic behavior (when 8

n —» oo) was found for the electron energy of the polaron:

The data in Table 1 suggest that the radius of the state grows rapidly with the number of the solution, while the energy and the effective mass decrease. Thus, to observe self-consistent states with the energy higher than the ground state energy, one must use crystals with a coupling constant much greater than a > 10 (a = e2ch~ v/i/2ftw). Typical values of the coupling constants for the majority of crystals are in the range a ~ 2-3. So, a = 3.97 in KC1, a = 2.0 in AgCl, a = 1.69 in AgBr, a = 0.85 in ZnO.2 For this reason the calculation of self-consistent states of bound polarons is more interesting to be compared with the experimental data.

Table 1: Energies (in units e*fj,c2/h ), radii (in units h /^tce 2 ) and effective masses (in units /*a4) for different self-consistent polaron states.

The self-consistent states have the same origin for different types of interactions considered in Sec. 2, but their physical characteristics can be distinct for various types of interactions. For instance, the total and the electron energies are negative in the case of the polaron in an ionic crystal, so they are below the bottom of the conduction band. In the case of an electron in homopolar crystals Eq. 15, the total energy of each self-consistent state is positive, while the electron energy (equal in absolute value to the total energy) is negative.

9

Figure 2: Wave functions j/(z) and self-consistent potential z(x) for the ground, first, second, and third excited polaron states. 10

Hence, in homopolar crystals the electron energy of the excited state is lower than the energy of the ground state (nodeless state). There is also a great difference between effective masses of self-consistent states for these two cases. In the case of an ionic crystal, the effective mass rapidly decreases with the number n. On the contrary, for a homopolar crystal the effective mass fast increases with the number. In the case of the nonlocal screened potential (deuteron problem), the existence of solutions to the nonlinear Shrodinger Eq. 16 depends on the value of the coupling constant g. The nodeless solution exists if values of g are greater than the critical value 2\ > 103) indicates regions of the density condensation, while the sign " — " (IV" 2 1 < 103) corresponds to the regions with the low electron density. The hybrid states of 2s + 3d or 2s — 3d-type (depending on the sign

of the combination) result from the combination of these two states.

Figure 4: Electron density distribution for the self-consistent excited states with mixed symmetry 2s ± 3d.

These nonspherical states are similar in the shape to the hybrid 2s ± 3dstates of the linear Shrodinger equation with the Coulomb potential. The nonlinearity of Eq. 10 leads to the solutions, which are qualitatively different from solutions to a linear problem, especially for r —> 0. However, similarity to the linear problem remains in the shape and the asymptotic behavior for T —> oo, since asymptotics of linear and nonlinear problems only differ in

constant. 13

Nonlinear polaron-type equations have a remarkable feature: simple scaling transformations allow one to obtain an equivalent problem, but without the eigen value. When searching for a solution, we can introduce a parameter («r) into the latter problem. More precisely, in the right-hand side of our equations

we can write down f(r) • a (instead of 0), where f(r) is a certain prescribed function. In this problem, the procedure of continuation of the parameter lor . Table 3 lists dimensionless phonon frequencies for the ground and the first self-consistent polaron states. Figure 9 shows the phonon spectrum approximation.

Table 3. Squares of phonon frequencies ^(n, I) for the ground and the first

excited self-consistent polaron states.

21

Figure 9: Approximation of the local phonon spectrum for the ground (a) and the first (b) excited states of the .F-center in KC1. 22

4.2

Phonon instability

If all the eigenvalues of the matrix MM are positive, then the amplitudes of phonon oscillations are small (0(C«) oc exp (—aj,C 2 /2)) and the system remains stable. If any of the eigenvalues is small or negative, the amplitude in this

approximation becomes of the order of the crystal's dimension, and the system looses its stability. Thus the problem of stability with respect to the motion

of phonons implies the study of the eigenvalues of the matrix Mfcj. In 1 0 6 > l o r ) we found numerically that for excited states of the polaron and the .F-center there are critical values of the effective F-center charge vcr = Z/(e0c) — 0.21, when the square of renormalized frequency becomes negative for one or several modes, and the phonon instability of the system takes place. As is shown in 102 , this phonon instability is due to the bifurcation of solutions to nonlinear problem Eq. 13. In r2 Gabdoulline found the critical values i/, at which the self-consistent

states of Is =p 3d type appear in addition to the excited self-consistent 2s state. Figure 10 shows the dependencies of the square frequency w^, for the phonon mode n = 2, / = 3 and the electron energy of self-consistent states E on the

effective .F-center charge. The arising phonon instability is similar to the Jahn-Teller effect. For the polaron in addition to self-consistent states, each of them corresponds to its proper potential, there is an infinite set of non-selfconsistent electron levels available for each potential. Each of these levels is characterized by its 0.14, where rj = e^ /eo was the ion coupling parameter. The current state of research is presented in reviews. 122 > 123 The treatment extended to the case of the presence of short-range interactions is given in 124 ' 125 .

Note that exact solutions have not been obtained in the bipolaron theory by now in contrast with the polaron theory, where asymptotically exact solutions are known in the limits of both the weak and the strong coupling. Moreover, in the bipolaron case there are no solutions at all for small and intermediate values of the coupling constant a. According to, 12a the bound polaron state is possible only at sufficiently large values of the coupling (a > 5, 2). In the adiabatic limit, both the electrons are believed to move in a single potential well, induced by their fast oscillations. For this reason the interaction of the electrons with the polarization (ri,r2) takes the form:

As a result, the problem is reduced to the calculation of a two-particle bipolaron wave function in a self-consistent potential. Since there are presently no methods to find asymptotically exact solutions of the problem, some additional assumptions are needed. As is mentioned above, the form of the approximation of the wave function can significantly affect the result. 26

5.2

Adiabatic theory of the bipolaron

An alternative approach can be exemplified by 127, where the BogolubovTyablikov method '31 was used to develop a consistent adiabatic translational invariant theory. Following this approximation the bipolaron motion is separated in the adiabatic limit, and the translational motion of the centre of the bipolaron mass is presented as a plane wave. Relative electron coordinates describe fast oscillations in a potential well, which has the form of the electron effective interaction:

The potential well is not fixed in space, but follows adiabatically the centre of the electron mass. The interaction Eq. 30 is clearly translational-invariant unlike the usual phenomenological approach Eq. 29, which does not possess translational invariance with a spatially fixed potential well. Within this approach we can treat the problem in a way similar to Sec.,2 of the paper. As a result, the problem becomes one-particle and is reduced to the study of the Shrodinger equation for the relative motion of the electron pair:

Then, we determine the bipolaron mass similarly to Eq. 10:

5.3

Results of calculations

In 127 calculations were performed for an ion crystal (uk = ui, cj. oc k~l), and the interaction U(r) between two electrons was the Coulomb repulsion screened by the high-frequency dielectric permittivity:

The solutions obtained depend significantly on the parameter ae = 0.125(1 — e

oo/ e o)- Figure 11 shows the particle-like solutions of the boundary problem

27

Eqs. 31 and 32 for some as values. It is evident from the figure, that the probability of the electrons occurrence at the same point decreases as ae grows, while the maximum of the electron density distribution displaces to the right and goes to infinity at the critical value aecr = 0-5. This results from the fact that at sufficiently large r the asymptotic of the potential Eqs. 31 and 32 has the form:

Figure 11: Quasiparticle solutions of Eq. 32 at various ae. The localized solution of Eq. 31 exists only on the condition that the righthand side of Eq. 35 is positive, i.e. for as < 0.5. Then, we have 7jCr = 0.75, for the ion coupling parameter r) = «oo/eo = (8ae — l)/(8ae), that greatly exceeds the value of this parameter obtained by the phenomenological theory. The critical value of the parameter rjs, at which the bipolaron is stable, is given by the condition of energetic advantage of the bipolaron state with respect to its decay into two independent polaron states

28

where Epo\ is the energy of a single polaron state. Calculations show that inequality Eq. 36 is valid if r\ < j/ s , where rj, = 0.56. Table 4 lists values of parameters for crystals satisfying the condition 77 < TJS and the calculated values of bipolaron energies, radii, effective masses. When the experimental data on the effective electron mass ra is not available, we present results depending only on the ratio m/mo, where mo is the mass of the free electron.

Table 4: Values * of energies W, total energies E, radii R and effective masses of bipolarons M.

The condition of the strong adiabatic coupling is that the frequency of the electron oscillations in the polaron well should be much greater than the

29

frequency of lattice oscillations. It follows from Table 4 that this condition is met for reasonable values of the electron effective mass. Thus, while a single polaron meets the condition of weak or intermediate coupling, a bipolaron follows the strong coupling condition in the case of the studied crystals. This enables us to evaluate critical values of electron-phonon coupling constants a,, when the bound bipolaron state is possible. It follows that the energetic requirement for the bipolaron formation is

Since E oc a2, the condition Eq. 37 allows us to estimate values of critical coupling constants. Table 5 lists these values.

Table 5: Critical values of electron-phonon coupling constants a8 for different values of r>.

Note that critical values of coupling constants, obtained from the numerical solution to the bipolaron equations, are much smaller than those evaluated by trial variational functions. Thus, according to l28 , the critical values are as w 5.4 for 77 = 0, as ss 7.2 for 77 = 0.1, they appear to be three times as large as those obtained from the exact solution to the bipolaron problem. Note also that at the polaron-bipolaron transition the solution symmetry changes, which can lead to significant rearrangement of the local phonon spectrum near the critical value of the parameter 77 due to the phonon instability

of the solution in the same way as it takes place in polaron excited states (see Sec. 4). Some of the results obtained can be easily extended to the two-dimensional case. The two-dimensional bipolaron problem has attracted much interest with the discovery of high-temperature superconductivity.

The simplest model used to describe polarons in three-dimensional space is obtained from Eq. 12 with cj, replaced by:

30

The physical parameters (frequencies, dielectric constants, effective masses) were assumed to be the same as in the three-dimensional case. The dependence of 0^,0 on the wave vector is chosen from the requirement that the electron-polarization interaction should be the Coulomb form (as 1/r) in the £>-dimensional case. The numerical factor in Eq. 38 is given by the condition that Ck,D corresponds to cj. at D = 3. To evaluate the energy and the critical constants in the two-dimensional case, the estimates obtained in by the Gauss approximation can be used. These estimates relate bipolaron energies in three- and two-dimensional cases. As a result in the two-dimensional case, we express the bipolaron energy as

where .EaD.bipol is tne bipolaron energy in the three-dimensional case. Hence, aejX) = 8630, 772.0 = TlaD • The critical value of the electron-phonon coupling constant is derived similarly and in the two-dimensional case takes the form:

where values of a, are listed in Table 5. The translational invariant bipolaron theory presented above yields results qualitatively different from those, obtained by the conventional adiabatic method. According to 127, the situation looks as follows. In the bipolaron formation, the electrons are localized in a deep potential well with the electron excitation energy W ~ 1 eV. This energy remains the same up to the critical value of the parameter r)s = 0.56, at which the bipolaron state decays into independent polaron ones. Up to the critical values 77 = 77, the frequency of the electron oscillations in the bipolaron potential well greatly exceeds the frequency of the lattice oscillations, and we can use the adiabatic approximation. The adiabatic criterion fails only for the crystals with very small electron-phonon coupling constants, such as PbSe (a = 0.215), and PbS (a = 0.317), where 77 < 77,, i.e. bipolaron states are conceptually possible. Bipolaron characteristics are best presented in crystals TIBr and T1C1, where continual approximations are fulfilled well; the radii of states are 20 Aand 16 Arespectively. The adiabatic condition is also met with a great safety, despite a relatively small constant a w 2.5. It is significant that in all the cases listed in Table 4, there is a great difference between the electron energy of bipolaron W and the total energy E. The absolute value of the electron energy of the bipolaron is approximately 5 31

times as great as that of the total energy, while for a single polaron in the strong coupling limit this ratio is equal to 3. This distinction can lead, particularly, to a great difference between the energies of photo- and thermodissociation. In some crystals the criterion of stable bipolaron formation is at bound of accuracy. Thus, in Rbl and KI the bipolaron is stable at room temperature and instable at helium temperature. Therefore, in Rbl and KI the cooling from room temperature to helium one leads to the bipolaron dissociation. This phenomenon could be observed on absorption spectra, changes in mobility and cyclotron frequency, etc.

Note also that the considered method for separating out the translational invariant part of the motion and reducing a two-particle problem to the nonlinear Shrodinger equation is rather universal and can be applied to any two-particle problem in the strong-coupling limit, such as biexcitons, electronhole pair, and so on.

6

Conclusions

Despite the long history of the polaron study the interest to the problem of the strongly coupled polaron does not decrease. It seems to be due to the role, which the polaron plays in physics of particle-field interaction. Unlike many other quasiparticles (phonons, magnons, plasmons, etc.) given by the spectrum and type of linear excitations of the system, the strongly

coupled polaron is a 'nonlinear' quasiparticle formed as a result of a nonlinear self-consistent interaction. This, in turn, requires a nonstandard mathematics. 66,132

In our view, the potentialities of modern mathematical methods are little used in the polaron problem. The above example demonstrates that even the generalization of the well-known Bogolubov-Tyablikov method to the case of two-particle self-consistent states yields new results different significantly from the usual variational calculations. The above results show that strongly coupled large polaron possesses a complicated internal structure manifested in a wide variety of self-consistent states for the electron subsystem and in the local phonon spectrum. The peculiar behavior of the local phonon spectrum is closely related to the topology of self-consistent states of the electron subsystem and given by the symmetry of the self-consistent potential in the nonlinear Shrodinger equation. Note that the investigation of such equations has much to do with the development of numerical methods for the solution of nonlinear boundary problems. It seems likely that new interesting results in this field can be obtained with the active use of high-performance parallel computers. 32

The foregoing proves that the experimental test of the possible existence of the strongly coupled large polaron and bipolaron states is rather ambiguous. However, in our opinion, dramatic qualitative results of the theory (such as, the availability of excited self-consistent states, or phonon instability) can be revealed in specially designed experiments.

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(Nauka,

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24. V.D. Lakhno in Doctor's thesis (Pushchino, 1987); in Excited polaroni states in condensed media, ed. V.D. Lakhno (Manchester Univ. Press, Manchester, 1991).

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28. V.D. Lakhno in Excited polaron states in condensed media (Manchester Univ. Press, Manchester, 1991). 29. V.D. Lakhno in Polarons & Applications (Wiley, Chisheter, 1994).

30. N.N.Bogolubov, Ukr. Mat. Zh. 2, 3 (1950). 31. S.V.Tyablikov, Zh. Eksp. Tear. Fiz. 21, 377 (1951). 32. E.P. Solodovnikova et al, Tear. Mat. Fiz. 10, 162 (1972). 33. N.N. Bogoliubov and D.V. Shirkov in Introduction to the theory of quantized fields (Interscience, New-York, 1959). 34. H. Frohlich et al, Philos. Mag. 41, 221 (1950).

35. H. Frohlich, Adv. Phys. 3, 325 (1954). 36. E.I. Rashba in Excitons, eds. E.I. Rashba and M.V. Sturge (NorthHolland, Amsterdam, 1987). 37. D. Emin and T. Holstein, Phys. Rev. Lett. 36, 323 (1976).

38. E.P. Pokatilov, Fiz. Tverd. Tela 6, 2809 (1964). 39. A.A.Klukanov and E.P. Pokatilov, Fiz. Tverd. Tela 11, 766 (1969). 40. M. Engineer and G. Whitfield, Phys. Rev. 179, 869 (1969).

41. G. Whitfield and P.M. Platzman, Phys. Rev. B 6, 3987 (1972). 42. M.F.Deigen and S.I. Pekar, Zh. Eksp. Tear. Fiz. 21, 803 (1951). 43. Ch. Kittel in Introduction to solid state physiscs (Wiley, New York, 1976). 44. E.P. Gross, Ann. of Phys. 19, 219 (1962).

45. V.D. Lakhno, Phys. Rev. B 46, 7519 (1992). 46. V.D. Lakhno, Fiz. Tverd. Tela 26, 2547 (1984). 47. 48. 49. 50.

S.J.Miyake, /. Phys. Soc. Jpn. 38, 181 (1975). J. Adamowski et al, Phys. Lett. A 79, 249 (1980). G.B. Whitham in Linear and nonlinear waves (Wiley, New York, 1974). V.E.Zaharov et al, Prikl. Mekh. Tekh. Fiz. 1, 92 (1972).

51. B. Halperin and M. Lax, Phys. Rev. 148, 722 (1966). 52. J. Zittartz and J.S. Langer, Phys. Rev. 148, 747 (1966). 53. I.M. Livshitz et al, Introduction in theory of disordered system (Nauka,

Moscow, 1982) [in Russian]. 54. D.D.Ivanenko in Nonlinear quantum field theory (Inost. Lit., Moscow, 1959) [in Russian].

55. V.A.Mironov et al, Dokl. Akad. Nauk SSSR 260, (2) 325 (1981). 56. N.K.Balabaev et al, (Preprint Pushchino, ONTI NTsBI Akad. 34

Nauk

SSSR, 1983). 57. I.V. Amirkhanov et al in PTOC. of Int. Con. NEEDS, V.G. Makhankov and O.K. Pashaev (Springer, Berlin, 1991).

eds.

58. I.V. Amirkhanov et al, (Preprint Pushchino, ONTI NTsBI Akad. Nauk SSSR, 1990). 59. P.L. Lions, Nonlin. Anal, Theory Meth. Appl. 4, 1063 (1980). 60. E.I. Yankauskas, Izv. Vys. Uch. Zav., Radiofiz. 9, 430 (1968). 61. A.A. Kolokolov, Prikl. Mat. Mekhan. 38 (5), 922 (1974). 62. N.K.Balabaev and V.D. Lakhno, Tear. Mat. Fiz. 45, 139 (1980). 63. N.K.Balabaev and V.D. Lakhno, Opt. Spektrosk. 55 (2), 308 (1983). 64. G.L.Alfimov e< al, Phyaica D 44, 168 (1990). 65. J. Devreese et al, Solid State Commun. 44, 1435 (1982). 66. M.V. Karasev and V.P. Maslov in Sovr. Probl. Mat 13 (VINITI, Moscow, 1979) [in Russian].

67. 68. 69. 70.

G.E. Zilberman, Zh. Eksp. Tear. Fiz. 19 136 (1949) M.V. Karasev and Yu.V. Osipov, Tear. Mat. Fiz. 52, 263 (1982). D.E. Hagen et al, Phya. Rev. B 2, 553 (1970). R.R. Gabdoulline, (Preprint Pushchino, ONTI NTsBi Akad. Nauk SSSR, 1991). 71. R.R. Gabdoulline, Dokl. Akad. Nauk 333, 23 (1993). 72. R.R. Gabdoulline, Phys. Lett. A 185, 390 (1994). 73. W.B. Fowler, Treatise Solid State Chem. 2, 133 (1975). 74. L.F. Mollenauer and G. Baldacchini, Phya. Rev. Lett. 21, 465 (1971).

75. C.S. Ewig, Chem. Phys. Lett. 188, 501 (1992). 76. J. Adachi and N. Kosugi, Bull. Chem. Soc. Jpn 66, 3314 (1993). 77. N.N. Kristofel" in Theory of impurity centers of small raddii in ionic crystals (Nauka, Moscow, 1974) [in Russian].

78. L. Bosi et al, Phys. Status Solidi (b) 68, 603 (1975). 79. L. Bosi and M. Nimis, ffuovo Cimento D 13, 1483 (1991). 80. S.I. Pekar, Usp. Fiz. Nauk 50, 197 (1953). 81. S.I. Pekar and Yu.E. Perlin, Zh. Eksp. Tear. Fiz. 43, 1108 (1962).

82. V.A. Moskalenko, Zh. Eksp. Tear. Fiz. 34, 46 (1958). 83. V.D. Lakhno, Dokl. Akad. Nauk SSSR 272 (1), 85 (1983). 84. H. Run and A. Rhys, PTOC. Roy. Soc. 204, 406 (1950). 85. M.D.Deigen, Zh. Eksp. Tear. Fiz. 26, 300 (1954).

86. 87. 88. 89. 90.

L. Kevan and D.F. Feng, Chem. Rev. 80, 1 (1980). V.D. Lakhno and O.V. Vasil'ev, Chem.Phys. 153 (1,2), 147 (1991). G.N.Chuev, Zh. Eksp. Tear. Fiz. 105 (3), 626 (1994). R. Kubo and Y. Toyozawa, Prog. Theor. Phys. 13, 160 (1955). R.A.Marcus, Annu. Rev. Phys. Chem. 15, 155 (1964).

35

91. R.R. Dogonadze and A.M. Kuznetsov in Physiscai Chemistry.

92. 93. 94. 95.

Kinetics

(VINITI, Moscow, 1973) [in Russian]. R.A.Marcus and N. Sutin, Biochim. Biophys. Ada. 811, 265 (1985). J.R. Miller in Adv. in Chem. Ser., eds. J.R. Bolton et al, (ACS, Washington, 1991) p. 228. N.K.Balabaev and V.D. Lakhno, Chem. Phys. Lett. 36, 585 (1995). N.K.Balabaev and V.D. Lakhno, Izv. Akad. Russ. Nauk. Ser. Fiz. (Bull Russ. Acad. Sci. Phys. Ser.) 36, 585 (1995).

96. N.K.Balabaev and V.D. Lakhno, Izv. Akad. Russ. Nauk. (Bull. Russ. Acad. Sci. Phys. Ser.) 36, 585 (1995).

Ser. Fiz.

97. H. Haberland in Clusters of Atoms and Molecules II (Springer, Berlin, 1995).

98. O.B. Evinson and E.I. Rashba, Usp. Fiz. Nauk 111, 683 (1973). 99. A.A. Maradudin, Solid State Physics 13, 273 (1966).

100. Sh.M. Kogan and R.A. Suris, Zh. Eksp. Tear. Fiz. 50, 1279 (1966). 101. G.N.Chuev, PhD Thesis (Pushchino, 1990); V.D. Lakhno and G.N. Chuev

102. 103. 104. 105. 106. 107. 108. 109. 110. 111.

in

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polaron

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in

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Tear.

Fiz. 21 (11), 1219

(1951). 120. V.L. Vinetskii and V.L. Gitterman, Zh. Eksp.

(1957). 36

Tear. Fiz. 33 (3), 730

121. S.G.Suprun and B.Ya. Moizhes, Fiz. Tverd. Tela 24 (5), 1571 (1982). 122. V.I. Vinitskij et al, Ukr. Fiz. Zh. 37 (1), 76 (1992). 123. M.A. Smondyrev and V.M. Fomin in Polarons & Applications, ed. V.D. Lakhno (Wiley, Chisheter, Wiley, 1994). 124. D. Emin and M.S. Hilery, Phys. Rev. B 39, 6575 (1989). 125. H. Hiramoto and Y. Toyozawa, J. Phys. Soc. Jpn 54, 245 (1985). 126. J. Adamowski and S. Bednarek, J. Phys. Condens. Matter. 4, 2845 (1992). 127. V.D. Lakhno, (Preprint Pushchino, Pushchino Research Center, 1993); Phys. Rev. B 51, 3512 (1995). 128. G. ladonisi et al, Phys. Status Solidi (t) 153, 611 (1989). 129. G. Verbist et al, Solid State Commun. 76, 1005 (1990); Phys. Rev. B 43, 2712 (1991). 130. J. Devreese in Polarons & Applications, ed. V.D. Lakhno (Wiley, Chisheter, 1994). 131. P.M. Peeters in Polarons & Applications, ed. V.D. Lakhno (Wiley, Chisheter, 1994). 132. V.E. Zaharov et al in Soliton theory: method of inverse problem (Nauka, Moscow, 1980) [in Russian].

37

THEORETICAL INVESTIGATION OF THE SELF-TRAPPED HOLE IN ALKALI HALIDES. LONG-RANGE EFFECTS WITHIN THE MODEL HAMILTONIAN APPROACH L.N. KANTOROVICH Laboratory of Med.-Phys. Problems, Latvian Medical Academy,

Dzirciema str.16, Riga LV1007, Latvia

P.B. ZAPOL Surface Physics Laboratory, Institute of Physics of Latvian Academy of Sciences,

Salaspils, LVZ169, Latvia A simple small-radiua polaron model of the self-trapped hole (Yk~ center) in alkali halide crystals is presented. Along with the usual contributions like the internal hole part and the long-range hole lattice interaction, used in the Hamiltonian

adopted in this study, the electronic polarization is also carefully included. The latter is treated within the framework of Toyozawa's electronic-polaron model. It is shown that the exact solution of the problem w i t h i n the adopted here LandauPekar approximation really leads to multi-hole q u a n t u m states accompanied by the relevant electronic and lattice polarizations. While solving our equations, the hole wave f u n c t i o n was assumed to be spread over a comparatively large local region of the crystal. The local (^3/1) symmetry of the defect is taken into account to simplify the solution of the corresponding non-linear equations. As an example the KCl crystal is considered, for which the Vfc-center electronic and spatial s t r u c t u r e as well as the self-trapping energy are computed. The results obtained demonstrate quite evidently the importance of taking into account a cAemtcafinteraction within the Vfc-center.

1

Introduction

It was found both experimentally and theoretically 1 that the hole prefers to be self-trapped in alkali halides. The self-trapped hole (STH) represents a point defect in a crystal and is mostly shared between two adjacent anions along the (110) direction forming a quasimolecule X^ , (X stands for the halogen ion). Questions arise why the hole prefers to be self-trapped in the two-site form

in the alkali halides and why the STH is energetically more favorable than a free hole having almost negligible coupling with lattice distortions. To answer these fundamental questions it is necessary to calculate the hole self-trapping (ST) energy. But in order to define the notion of the ST energy more carefully,

the consideration of the STH structure is required. Due to the periodic symmetry of the crystal lattice, all the anions of the 38

perfect crystal are equivalent as traps for a free hole, and the wave functions of both the free (F) hole and STH have the usual Bloch-like form. 2 ' 3 As a result, in both the states we deal with bands of energetic levels having widths Bp and BSTi respectively. As was explained in 4 , at least in the case of alkali halides the

STH can be simulated by a localized state, using a localized variational ansatz for the hole wave function, and, in addition, the STH band can be substituted by a single level. Thus, the Vjt-center can be treated by any method appropriate for point defect calculations. Therefore, the hole ST energy is defined as the energy difference between the middle of the STH band chosen as a single level (narrow STH bands are assumed) and the bottom of the free hole band coinciding in the case of the

alkali halides with the point k = 0 in the Brillouin zone (BZ). 4 ' 5 A number of various theoretical approaches have been developed to study the electronic and spatial structure of the V^-center in alkali halides: atom-atom potentials method, 6 semiclassical static-lattice method, 7 quantum-chemical (QC) methods, based on the molecular cluster approach, both semiempirical4'5'8'9 and ab

«7it 2 ) in an isotropic and homogeneous space to depend only on the difference of their coordinates

This is just the form of the Yukawa interaction between two nucleons in the weak coupling limit . The situation with the strong coupling limit is rather different. In this case, the meson field is often considered as classical. But in a classical field two nucleons move in a classical potential well. Hence, the nucleon-field interaction can be presented as

This interaction is not translational invariant and differs in form from the Yukawa interaction. It is clear, however, that this interaction should be presented in the form of Eq. 1, if the interaction between nucleons is caused by the exchange the meson field quanta, In Sec. 2 we present the results of the translation ally invariant theory for the nucleon in a meson field. The general expressions for energy, total energy and effective mass of a nucleon are derived in the strong coupling limit. Numerical solutions of nonlinear differential equations are given in Sec. 3 for the nucleon in a scalar meson field. The energy, total energy, and effective radius of the nucleon are calculated. A consequent translational invariant

64

theory is developed in Sec. 4 for two particles in a quantum field. It is based on the method of collective coordinates, which is used in the nonperturbative quantum field theory to quantize near classic solutions. This method was first used in the polaron treatment to develop translational invariant theory of a nonrelativistic particle in a quantum field. The canonical BogolubovTyablikov transformations are used to introduce collective coordinates in Sec. 4. According to general treatment, the interaction between nucleons in a meson field was shown to depend only on the difference in the nucleon coordinates. The obtained result are independent of the nucleon-meson coupling constant. The problem of two particles in the field is reduced in Sec. 5 to the oneparticle problem. Its solution yields the bounding energy and effective radius of the binucleon close to the experimental data for deuteron. We discuss the binucleon effective mass in Sec. 6, and the divergence problem of the theory of meson-nucleon interaction in Sec. 7. As was pointed out above, these divergencies result from the fact that the Hamiltonian of nucleonmeson interaction is roughly determined. The functional of the total energy can have the local minimum in the strong coupling limit. But finite values of calculated parameters can be obtained by the perturbation theory for any power of the inverse coupling constant. In this Section we also present the equation determining the renormalized frequencies of the meson field. In Sec. 8 we use variational approach to calculate the energy of the binucleon ground state. In limit fj,/m —> 0, where p, is the binucleon mass, these variational estimations coincide with those obtained in the strong coupling limit. Sees. 9 and 10 are devoted to the case of pseudoscalar and vector mesonnucleon interactions. These interactions are significant to treat the value of quadrupole moment of deuteron and deuteron stability. The obtained results are discussed in Sec. 11. 2

Interaction of a nucleon with a meson field

In this Section we use the strong coupling theory to consider a nucleon interacting with a meson field. The total Hamiltonian H includes the complex scalar nonrelativistic field 9 and the real scalar meson field 0 is the vacuum wave function, being applied to Hamiltonian (7) yields infinitely large negative value of the ground state. This has led to the conclusion that divergencies in the strong coupling theory cannot be eliminated in principle. The fact that the contribution causing the divergencies can be isolated does not solve the problem. The decisive point here is that actually the variational estimate is given not to Hamiltonian (7) but to Hamiltonian :H:. Thus, the variational calculation refers only to a part of the total Hamiltonian, i.e. to a part isolated from the infinite positive value of the zero field energy. These procedures cannot be considered as consequent: they may lead to any value of the full energy, including the senseless one. The correct approach consists in a consistent application of the strong coupling theory to Hamiltonian (7). It implies the second order of the perturbation theory with respect to the inverse coupling constant g and yields the following equation for the renormalized frequencies of the meson field u>, in

77

where the functions Xt(T) satisfy the orthogonality condition

Vp(°) corresponds to the solution of nonlinear problem (12) associated with the ground state. Similar equations are derived for the binucleon, but with m^

replaced by ro^/2, g2 replaced 8g2 and KA replaced by Kj^jl. The solution of boundary problem (62) and (63) (boundary conditions result from /X 2 ( r )^ 3 r to be finite) is a nontrivial problem. It allows is to calculate the total energy of the system nucleon plus meson field by the formula

where u>j are given by Eq. 6, and the renormalized meson frequencies by Eq. 62. We cannot guarantee here the finite value of f^E^. But we can expect that &E^ will be finite by analogy with the corresponding quantity calculated in the strong coupling theory for the case of optical phonons. In our case this would correspond to the limit p, —> 0.

Thus, the exact solution of Hamiltonian (7) in the strong coupling limit may give finite values of the nucleon energy and does not require separation of isolated contributions. If we turn back to the perturbation theory where the logarithmic divergence of energy at large momenta is treated as the contribution made by virtual processes of absorption and emission of mesons, we recognize that the idea of invariance of mesons is incorrect at large momenta, i.e. they are not described by Eqs. 6 and 61 in this case.

8

Variational estimates for binucleon

Previously we derived exact equations for the binucleon and found some of their numerical solutions. It seems interesting to compare these results with the variational approach. Unitary transformation of Hamiltonian (21) with TO = 0 by operator

78

eliminates the center mass coordinates from the Hamiltonian. As a result, the Hamiltonian takes the form

To study the ground state of Hamiltonian (67) we choose the Lee, Low, Pines (LLP) probe wave functions in the form

As a result, the mean value of (H1) is

where

Variation of functional (70) under condition of zero total momentum leads to the expression for the coefficients £f

79

Substituting £j from Eq. 72 to Eq. 70 and varying the resultant functional over ^>o( r ii r 2)> we obtain the following Schrodinger equation for the binucleon

given that the probe wave function is normalized. Here

We choose the probe wave function V > o ( r i , r 2 ) in the form

Substituting Eq. 75 into Eq. 74 and taking into account that the meson mass is much less, than the nucleon mass, i.e. n/m^
where el

are unit orthogonal vectors, i.e. (ej ,e.

parallel to momentum /, i.e. (e™ f) =\ f \, (e™f) = (e

) = Srs, and ek

81

is

f) = 0, [e f] =

( a) f

( l) f

0, [e^f] = - I / I 43), t6/3^] =l / Ie / 2); *i is ihe

vecior

°f spine operator

of j'-th nucleon. Repeating the derivation of previous section, we obtain the functional of total energy JF in the strong-coupling limit

ta

where F is determined by Eq. 78, while FT is expressed as

where V" is the deuteron wave function. The functional FT determines the influence of tensor forces on the interactions between nucleons. Varying f over i/> results in the set of nonlinear differential equations which have spherically symmetrical solutions in contrast to the scalar field. In this case, f can be minimized by the probe deuteron wave function 8

where U(r),u(r) are the radial parts of the wave function, Yfim are spherical harmonics, Xi m are 'ne spin deuteron wave functions.

11

Further discussions

The Bogolubov-Tyablikov theory has been widely used to describe the mesonnucleon interaction. 9 ' 10 In the case of two nucleons interacting with the meson field, this theory was considered in11. There the nucleon mass is supposed to be infinitely large, i.e. the nucleons are assumed to be static sources. The Bogolubov-Tyablikov theory was one of the first translational invariant theories which took into account the symmetry of the Hamiltonian in separating 82

the classic component of the boson field. At present, there are various methods where the collective coordinates are introduced directly in the initial Hamiltonian similarly to r0 in Eqs. 20 and 21. Different ways are possible to impose the

additional conditions retaining the total number of independent variables in the system. For instance, the number of the Fourier components are decreased in 5 ' 6 by the value equal to the number of the introduced collective coordinates

instead of the Bogolubov-Tyablikov condition. In nonrelativistic theory the Bogolubov-Tyablikov method was used in 12 ' 13 . In these works the BogolubovTyablikov transformation was applied to Hamiltonian (2), but the collective variables were not introduced. Zero approximation in these methods yields the same results as in the theories with broken symmetry. As in the polaron case,

these methods use only splitting coordinates, which have no physical meaning. Therefore, these works have not developed the theory which would use the only variables having physical meaning such as relative coordinates r = r\ — r 2 and the potential energy U(TI — r?).

There are serious difficulties in the application of the strong coupling theory to a nucleon moving in the meson field. In this case the criterion of adiabatic approximation consists in the requirement that the characteristic frequency of

nucleon oscillations in the induced potential well should exceed greatly the frequency of the meson field. This requirement can hardly be satisfied at reasonable coupling constants for the nonrenormalized meson frequencies. But it can be satisfied for the "smooth" meson frequencies witch are determined by Eq. 69. In the case of binucleon (or nuclei with greater number of nucleons) the adiabatic condition changes and requires that the nucleon should interact with the meson field induced by the averaged distribution of the second nucleon (other nucleons if n > 2). Since on the average nucleons are spaced from each other in the deuteron (being spot in our model), the time necessary to exchange the meson distanced from the nucleon significantly exceeds the time taken to exchange the meson close to the nucleon. Thus, the potential at a distance from the nucleon is determined by mean distribution of the density of nucleon probability, rather than its instant position, as in the case of nucleon interacting with the induced meson field.

The main reasons of errors arising in the use of the simplest variant of the quantumfield theory are: neglect of relativistic effects (which gives the error of about 10%), nondistinguishing between protons and neutrons, neglecting of nonspot character of the interaction (formfactor), neglecting of noncentral nature of nuclear forces (in particular, the forces depending on the orientation of nucleon spins). Neglecting of the latter fact in our model leads, in particular, to zero quadrupole binucleon moment which is inconsistent with the 83

experiment. In conclusion the author express his gratitude to R.R.Gabdoulline for numerical calculation of problem Eq. 14.

references

1. V.D. Lakhno in Excited Polaron States in Condensed V.D. Lakhno (Manchester Univ. Press, Manchester, 1991).

Media,

ed.

2. E.P. Gross, Annals of Phys. 19, 219 (1962).

3. 4. 5. 6.

N.N.Bogolubov, Ukr. Math. Journ. 2, 3 (1950). S.V.Tyablikov, Zh. Eksp. Tear. Fiz. 21, 377 (1951). N.H.Christ and T.D. Lee, Phys. Rev. D 12, 1606 (1975). R. Rajaraman in Solitons and Instantons (Noth-Holland, Amsterdam, 1982). 7. Ch. Kittel in Introduktion to Solid State Physics (Wiley, New York, 1976); G. Wentzel, Quanttim Theory of Fields (N.Y., 1949). 8. J.M.Blatt and V.E. Weisskopf in Theoretical Nuclear Physics (Wiley,

N.Y., 1952). 9. A.V.Rasumov and O.A. Khrustulev, Tear. Mat. Fiz. 29, 300 (1976). 10. A.N.Tolstenkov et al, Preprint (Ins. Phys. High. Energy 73 - 82, Serpukhov, 1973).

11. V.G. Bornuakov, Tear. Mat. Fiz. 51, 247 (1982). 12. V.A. Moskalenko, Uchenye Zapiski (Kishinev State Univ.) XVll, 103 (1955). 13. E.P. Solodovnikova and A.N. Tavkhelidze, Tear. Mat. Fiz. 21,13(1974). 14. M.A.Smodyrev, J.T.Devreese Phys. Rev. B 1996 (in press).

84

GENERALIZED FUNCTIONAL APPROACH IN THE THEORY OF MESON-NUCLEON INTERACTION V.D.LAKHNO Department of Quantum-Chemical Systems, Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 142292 A generalized Feynman variationat principle for path integrals is applied to the problem of a nucleon in a scalar meson field. It is shown that for the formfactor chosen in the self-consistent form the results of the study correspond to the strong coupling limit. In this case, the nonlinear differential equations for a nucleon in a meson field are obtained. The nucleon ground state energy and effective nucleon radius are found by the variational method for various nucleon—meson coupling constants.

1

Introduction

Variational methods are widely used in physics to obtain approximate non— perturbative solutions. In contrast, the applications of variational principles in the quantum field theory are rather limited (for a review see l > a ) .

One of the central problems in the quantum field theory of meson-nucleon interaction is the ultraviolet divergence of the perturbation theory, which arises even in the simplest case of point nucleon-meson scalar interaction. The use of formfactor leads to finite results of the perturbation theory in the weak coupling limit, but the convergence radius of the perturbation theory remains finite. In the strong coupling limit, the Bogolubov—Tyablikov adiabatic theory results in asymptotically exact nonlinear equations for the particle in a quantum field. ' It still remains unclear how the results obtained in this limit can be applied to real values of nucleon-meson coupling constants.

In this communication

we develop a generalized functional approach based on the Feynman path integrals to describe nucleon—meson interaction. This approach, where the trial potential plays the role of a variation parameter allows us to obtain a functional and, hence, differential equations for strong-to-weak coupling transition. It is shown, that for the form-factor chosen in the self-consistent form, i.e. the nucleon density distribution determined by the square of its wave function, the best results correspond to the strong coupling limit for any coupling constant. In this limit the nonlinear differential equations for the nucleon in a meson

field are obtained. The nucleon ground state energy is found by the variational method for various nucleon—meson coupling constants.

85

2

Application of the generalized functional approach to the problem of a nucleon in a meson field

The main problem in determining the properties of quantum mechanical systems by the method of path integrals is the calculation of a propagator in evolution of the system studied from the state r ' , t ' to the state r",t". The propagator can be presented in the form of path integrals as

In this paper, we apply the generalized functional approach 5 to a nucleon interacting with a scalar meson field. In this case, the action S can be expressed as

where m^ is the nucleon mass, g is the constant of nucleon- meson field interaction. Action S is obtained directly from Lagrangian L for the nucleon interacting with a scalar meson field

The form of S is analogous to S for the electron in a scalar phonon field. 6 Turning in Eq. 1 to complex time t" — t' = —i/SK, we obtain the density matrix p(r" , r ' ; / 3 ) , which yields the expression for the ground state energy of the studied system

where

86

Following,

we choose the trial action So(r,r')

in the form

Below, we use this expression to calculate the ground state energy Eq. 4.

3

Energy inequality

The upper bound of the ground state energy Eq. 4 is given by the Jensen's inequality which can be written by Eqs. 8 and 9 as

where

We are interested in calculating the values

and

4

Calculation of e0 — lim^,*, j(Vo)s

for the nucleon

To calculate Eq. 13, we introduce the coordinate of center mass as

87

Taking into account that Jacobian of transformation Eq. 15 is equal to unity in view of relations Eqs. 8 and 12, eo can be expressed as

where p. = m'/(m1 + m^). It follows from Eq. 16, that «o presents the ground state energy described by the Hamiltonian

where «o( z ) is tne wave function of the ground state of Hamiltonian H0. Using the equality

we get

Finally we have

5

Calculations of lim^oo j(v)s

for the nucleon

Using Eq. 6, we can present Eq. 14 in the form

88

In terms of relations Eq. 15 and definition Eq. 12, the mean value exp(ik[r(T) — r( oo the trial particle's mass rn' also tends to infinity, i.e. H —» 1. Finally, the total energy functional is

In deriving Eq. 35, the scalar meson field frequency is u/i, = VA^ + * 2 . Varying functional Eq. 35 over UQ under conditions that UQ is normalized, we obtain the 91

nonlinear Schrodinger equation

In the weak coupling limit as g —* 0 the trial particle's mass m' also tends to zero. If the delocalized states: u o (x) —> -4= are used, where V is the system's volume then as V —» oo, the total energy functional $1 tends to zero.

Table 1: Nucleon total energies (in nucleon mass units) and effective radius (in Fermi units).

7

Numerical results

To study the minimum of functional Eq. 34, we choose the trial wave function in the Gaussian form for the nucleon

Substitution of Eq. 37 into Eq. 34 leads to the problem of finding the minimum of *!(a,/i)

with respect to parameters o C (0,oo) and /i C (0, 1). Numerical investigation of Eq. 38 demonstrates that min $j(a, /J,) is realized at fj, = 1 for any values of

the coupling constant g. Table 1 lists the values of the nucleon total energies $! (in the units of nuclear mass ma) and effective radius < z >= v2/7r amfn,

92

where o m j n is the parameter at which $j (a, 1) is minimum for various values of the nucleon-meson coupling constant g. Table 1 suggests that there is a critical value gcr ~ 22, at which the bound nucleon state is absent in a meson field. For g < gcr but near to this value, the bound state is metastable.

8

Discussions

The approach developed yields the finite nucleon energy in a meson field for any values of the nucleon-meson coupling constant. The fact that at g < gcr the total nucleon energy tends to zero is due to the self-consistent expression for the nucleon formfactor Eq. 33. But the idea of the nucleon in a meson field, whose state can be described by a wave function il>(r), and, hence, by formfactor Eq. 33 is valid only in the strong coupling limit. The appearance of the total negative nucleon energy at g > gCT gives a qualitative estimate of the nucleon-meson coupling constant, at which the coupling may be considered to be strong. In this paper we have studied only the nucleon ground states. The excited states may be found from Eq. 36. The states like these should contribute to various processes of the nucleon scattering. In particular, the scattering resonances, associated with excited states, may be used in choosing the value of the nucleon-meson interaction constant. The approach may be also useful in calculation of binucleon states, particularly, in generalizing the results of the strong coupling theory, obtained for the binucleon quantumfield model 4 to the case of intermediate values of g and to relativistic field theory of scalar meson—nucleon interaction.

References

1. R.W. Haymaker,.R™. Nuovo dm. 14 (8), 1 (1991). 2. D.I. Diakonov and V.Yu. Petrov, Nucl. Phys. B 245, 259 (1984). 3. V.D. Lakhno in Excited Polaron States in Condensed Media, V.D.Lakhno (Manchester University, N.Y., 1991). 4. V.D.Lakhno, Tear, i Mat. Fiz. 100, 219 (1994) [in Russian].

5. 6. 7. 8.

J.M. Luttinger R.P. Feynman, R.P. Feynman, R. Rosenfelder

and C.Y. Lu, Phys. Rev. B 21 (10), 4251 (1980). Phys. Rev. 97, 660 (1955). Statistical Mechanics (Massachusetts, 1972). et al, Zeit. Phys. A 350, 131 (1994).

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ed.

NONLINEAR MATHEMATICAL MODELS IN BIOPOLYMER SCIENCE L.V. YAKUSHEVICH Institute of Cell Biophysics, RUssian Academy of Sciences, 14S292 Pushchino, Moscow region, Russia Applications of the nonlinear mathematical models, some of them being known in the polaron theory, to biopolymer science are described. The main nonlinear equations and their soliton-like solutions imitating nonlinear excitations in biopolymers

are presented. Experimental data on the nonlinear properties of biopolymers and examples of nonlinear mechanisms of biopolymer functioning are discussed.

1

Introduction

Nonlinear mathematical models are widely used now in many fields of science.

In this review we describe the main nonlinear mathematical models applied to bipolymer science and especially to the study of nonlinear dynamical properties of DNA and alpha-helical proteins. Nonlinear dynamics of biopolymers is a new and rapidly developing field of biophysical science. It can be considered as a part of general dynamics

which deals with the internal mobility of biopolymers (Fig. 1). Theoreticians define it also as the next (anharmonic or nonlinear) approximation after the first (harmonic or linear) one. The nonlinear approximation is used when the amplitudes of internal motions in biopolymers are large, and the linear approximation which is valid only for small amplitudes, becomes incorrect. Conformational transitions, denaturation processes, formation of opening states in the DNA-protein recognition processes are only some of the well known examples of the large amplitude motions. Mathematical modeling of the nonlinear dynamics of biopolymers made its appearance in 1973 when Davydov and Kislukha published their pioneer work on solitary excitations in one-dimensional chain which was considered as a simple model of alpha-helical fragment of proteins.1 The first work on the DNA nonlinear dynamics was published only in 1980, and the first one on the RNA nonlinear dynamics in 1990. Now the number of articles devoted to the nonlinear dynamics of biopolymers rapidly increases, and specialists in different fields of science make contribution to the problem: mathematicians work out mathematical basis of the nonlinear theory, physicists suggest nonlinear models imitating internal dynamics of biopolymers and propose experimental methods

of detecting the nonlinear excitations, biologists develop various applications. However, the articles devoted to the theme are scattered among many special

94

journals and very often specialists in one field, for example in mathematics, can not understand their colleague-biologist who publishes the articles on the same problem in biochemical journals. Therefore, we decided to try to gather

in this review the results on nonlinear dynamics of biopolymers obtained in different fields: mathematics, physics and biology, and to show the relation

between them.

Figure 1: Scheme illustrating two approximations in the dynamical theory of biopolymers.

2

Internal mobility of biopolymers

As we mentioned above, the nonlinear dynamics is a part of general dynamics. Therefore we begin our review with a brief description of the main features of the general dynamics of biopolymers.

Because the structure of biomolecules is rather complex, it can be expected that the general picture of their internal mobility is also very complex. To describe the picture, different types of classifications of internal motions according to their energies, amplitudes and characteristic times were proposed. Authors of 4 ' 5 ' 6 ' 7 gave examples of the classifications of internal motions in proteins. Analogous examples for DNA presented i n 8,».io,n,l2 gave the detailed

classification for both DNA and proteins. Most of the classifications used are based on the time characteristics of internal motions. The time scale is usually divided into several diapasons: picosecond, nanosecond, microsecond and so on, and internal motions which

belong to these diapasons are described. To illustrate the general picture of 95

the internal mobility, we present here one of the classifications (Table 1). It consists of several diapasons, each being described by the main internal motions, the main structural elements involved in the motions, the energies of activation of the motions, the amplitudes of the motions and the main experimental and theoretical methods of the motion study. To understand when biopolymers can exhibit the nonlinear behavior, let us consider one of the diapasons in detail. Let us choose the nanosecond diapason. This diapason is of special interest because corresponding internal motions are very important in many biophysical phenomena: conformational transitions, gene regulation, DNA-protein recognition, energy transmission, ion conductivity through biomembranes and others which deal with the energies E = several kcal/mol, frequencies v = several cm"1 and r = several nanoseconds. According to Table 1, the nanosecond diapason includes among others the so-called solid-like motions of sugars, phosphates and nitrous bases which are known as the main structural units of DNA. Let us assume that these motions are very important in a biological phenomenon which we study. Let us consider them in detail. In the general case, these motions can be described mathematically by a set of coupled dynamical equations in unknowns which are the amplitudes of the motions and their derivatives. When the amplitudes of the motions are small, the linear approximation can be used, and the nonlinear terms in the corresponding dynamical equations can be neglected. Solutions of the resulting equations will have the form of usual linear waves. However, when the amplitudes of the internal motions are large, the linear approximation is not valid, and we must take into account anharmonic terms and consider the nonlinear mathematical problem. In this case the solutions having the form of solitary waves are possible. This point of view is illustrated by the scheme shown in Fig. 2. Thus, we can conclude that biopolymers could exhibit the nonlinear behavior in the cases when their internal motions have large amplitudes. These large amplitude motions can be generated in different ways. It is supposed, for example, that they can be generated by thermal fluctuations or that they can result from the collisions of "not" molecules of the solutions with biopolymers.13 Some authors suppose that large amplitude motions can be activated due to the chemical energy released in the ATP hydrolysis 14 or due to the local interactions of biopolymers with ligands. In applications we shall consider many examples of large amplitude motions in biopolymers and different sources of their activation.

96

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