On the Influence of Lattice Deformation by Electrons on the Optical and Electrical Properties of Crystals

S. I. Pekar

1. Introduction

The present article is devoted to the theory of nonmetallic crystals—dielectrics and semiconductors. The modern theory of semiconductors may be divided into two parts: a) the phenomenological theory and b) the microtheory of the states of electrons in a crystal. The first of these uses, for example, Ohm’s law or the more general equation of electrical conductivity and diffusion of “current carriers”

\[ J=-euN\frac{dV}{dx}-eD\frac{dN}{dx}, \tag{1} \]

where \(N\) is the concentration of current carriers, \(e\) their charge, \(u\) the mobility, \(D\) the diffusion coefficient, \(V\) the potential of the electric field, and \(J\) the current density. Further, characteristic of the phenomenological theory is Poisson’s equation

\[ \Delta V=-4\pi\rho(N), \tag{2} \]

where \(\rho(N)\) is the spatial charge density of the current carriers. To the phenomenological theory should also be assigned the dependence of the concentration of current carriers on temperature, of the form

\[ N=Ae^{-\frac{W}{2kT}}, \tag{3} \]

where \(W\) is the energy of thermal dissociation of current carriers (the case of thermal equilibrium), and the expression for the number of recombinations of current carriers per unit volume per second

\[ \beta NN_1, \tag{4} \]

where \(\beta\) is the recombination coefficient, and \(N_1\) is the concentration of recombination centers.

The equations given above may be called phenomenological, since in their justification the mechanism of microprocesses is ignored. The latter is evident already from the fact that these equations have the same form for the case of electronic and ionic conductivity, and also in the case of gases, liquids, and solids. Only the parameters of the theory \(u, D, A, W\), and \(\beta\) depend on the object.

Since the phenomenological theory ignores the mechanism of microprocesses, it remains almost unchanged even in those cases when our ideas about microprocesses undergo fundamental changes. Thus, for example, with the appearance of the quantum-mechanical electron theory of crystals, the form of equations (1)—(4) changed scarcely at all.

Over the last decade and a half the phenomenological theory of semiconductors has achieved considerable success. In 1938 D. I. Blokhintsev and B. I. Davydov, on the basis of equations (1) and (2), proposed a qualitative explanation of current rectification in semiconductors[^1]. Almost simultaneously S. I. Pekar gave a quantitative theory of rectifiers with blocking layers[^2]. This same theory, several months later, was independently proposed by N. F. Mott[^3]. The theory explained rather well the current-voltage characteristics of rectifiers with chemical blocking layers. Also, on the basis of equations (1), (2), and (3), a theory was constructed for the contact of a semiconductor with a metal both in the case of weak currents (B. I. Davydov, N. F. Mott[^4]) and in the case of strong currents (S. I. Pekar, Schottky and Schenke[^4]). The theory explained the pre-electrode resistances of semiconductors observed experimentally and the current-voltage characteristics of contacts. Another example of the use of phenomenological equations is the theory of the kinetics of photoconductivity of semiconductors, i.e. the consideration of photoconductivity and photo-e.m.f. under conditions of pulsed illumination (V. P. Zhuze and S. M. Ryvkin, V. E. Lashkarev, K. B. Tolpygo[^5]). The examples given above, of course, do not exhaust the questions successfully solved on the basis of phenomenological theory.

However, phenomenological theory is completely insufficient for a complete and comprehensive study of the properties of crystals, since it ignores the mechanism of microprocesses, on which many phenomena substantially depend. In addition, phenomenological theory is usually forced to introduce many parameters, for example mobility, the diffusion coefficient of current carriers, the energy of their thermal dissociation (and sometimes several such energies), the recombination coefficient, the coefficient of light absorption, the quantum yield of the internal photoeffect, etc. These parameters cannot be calculated within the framework of phenomenological theory; at best they are determined by comparing the theory with experiment. Sometimes the number of unknown parameters is so large that even the agreement of theory with experiment becomes little convincing. In these cases the role of phenomenological theory is reduced—

...are reduced only to the schematization and systematization of experimental data.

In order to calculate the above-mentioned parameters theoretically or to establish a relation between them, it is necessary to penetrate deeply into the mechanism of the microprocesses occurring in the crystal, which is precisely the task of microtheory. The present article is devoted precisely to questions of microtheory. In constructing microtheories of the states of electrons in a crystal on the basis of quantum mechanics, great mathematical difficulties are encountered. Therefore one has to resort to simplifying assumptions both with respect to the model of the crystal and in solving the wave equation for a system of many interacting particles. The errors arising from these simplifications are usually difficult to estimate in advance.

The most popular microtheory, which began to develop soon after the appearance of wave mechanics, is the theory based on the following simplifying assumptions:

1) The many-electron problem is replaced by a one-electron one. Instead of directly taking into account the Coulomb interaction between electrons, each electron is considered in some prescribed external potential field. In the case of an ideal crystal this potential field is assumed to be periodic, which leads to the generally known “band” theory of electrons in a crystal,^6 according to which individual electrons move independently of one another, the energy spectrum of each electron consists of alternating allowed and forbidden energy bands, and the wave function of each electron has the form

\[ \psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) e^{i\mathbf{k}\mathbf{r}}, \tag{5} \]

where \(\mathbf{k}\) is the wave vector of the electron, and \(u_{\mathbf{k}}(\mathbf{r})\) is a function possessing the periods of the crystal.

If, however, the matter at hand is not an ideal crystal but a crystal with a defect, an impurity, or some local “center” capable, for example, of absorbing and emitting light, capturing conduction electrons, etc., then, in addition to the periodic potential just mentioned, a potential well is introduced in which local electron states can exist.

2) The second of the above-mentioned simplifying assumptions is the assumption that the interaction of electrons with the vibrations of the atoms or ions of the crystal is small. In this case, in the zeroth approximation, the states of the electrons are calculated under the assumption that the atoms are fixed immovably in certain equilibrium positions, while the vibrations of the atoms are considered without taking into account the presence of electrons in the crystal. Thus, in the zeroth approximation, the crystal is subdivided into two noninteracting conservative subsystems: electrons and the normal vibrations of atoms elastically bound to certain equilibrium positions. Most phenomena considered...

continues in this zeroth approximation. Only in a few cases, as a small perturbation, is the interaction between electrons and atomic vibrations introduced. The latter is necessary, for example, in the problem of the mobility and mean free path of a conduction electron, since these quantities become infinite if one restricts oneself to the above-mentioned zeroth approximation.

Concerning the first of the simplifying assumptions indicated above, it should be emphasized that in metals the replacement of the many-electron problem by a one-electron problem has not received a satisfactory justification. The justifications for such a replacement usually given in textbooks cannot be regarded as correct\(^{7,8}\). Nor can this replacement be justified when considering the intrinsic (bound) electrons of a dielectric. However, when the question concerns the conduction electrons of a semiconductor or dielectric, and also the electrons of some local “center,” bound to the nuclei much more weakly than the intrinsic electrons of the dielectric or semiconductor, then in this case one may use the well-known adiabatic approximation: it is assumed that the weakly bound conduction electrons (or electrons of the local center) move sufficiently slowly and that the state of the strongly bound intrinsic electrons of the crystal follows adiabatically the motion of the conduction electrons. In this case, according to the adiabatic approximation, one may consider the motion only of the weakly bound slow electrons, while the presence of the intrinsic electrons will affect only the expression for the potential energy of the slow electrons\(^{7,8,9,10}\).

Thus, in nonmetallic crystals, as applied to weakly bound electrons, the first simplifying assumption can be partially justified: the Coulomb interaction of weakly bound electrons with the intrinsic electrons of the crystal may be replaced by the introduction of an appropriately chosen external potential field. In the case of a conduction electron in an ideal dielectric this field will be periodic\(^{9}\). Below we shall be concerned precisely with weakly bound electrons; therefore the first simplifying assumption is acceptable. Most of the results given below can be obtained by using, instead of the first simplifying assumption, the H.-L.-H. approximation\(^{1}\). The latter consists in the fact that the many-electron wave function of the crystal is constructed in the form of an antisymmetrized product of the wave functions of the individual atoms or molecules. In doing so, the molecular wave functions are assumed to be known, and the interaction between the molecules is assumed to be weak.

Turning to the discussion of the second simplifying assumption, it should be noted that the division of a crystal into two almost noninteracting subsystems—electrons and atomic vibrations—

\(^{1}\) The method of Heitler–London–Heisenberg; see \(^{6}\) Part III and \(^{7}\).

ON THE INFLUENCE OF LATTICE DEFORMATIONS ON THE PROPERTIES OF CRYSTALS

immediately excludes the possibility of even a qualitative theoretical interpretation of many phenomena. It is known, for example, that photo-transitions of electrons in a crystal are, generally speaking, accompanied by the excitation of lattice vibrations, i.e. by heat release. The amount of heat released in this case can be calculated as the difference between the experimentally known energy of the photo-transition \(\hbar\omega\) and the energy of the corresponding thermal transition of the electrons \(W\). \(\hbar\omega\) is often one and a half to two times greater than \(W\) (see below). Thus, a considerable part of the energy of the absorbed light quantum, i.e. an energy of the order of one electron-volt, is converted into heat during the transition. This fact testifies to a strong interaction between electrons and atomic vibrations and cannot be explained by a theory in which this interaction is regarded as a weak perturbation. The latter also does not explain the frequently observed large magnitude of the Stokes shift in crystals (the distance between the absorption and luminescence bands corresponding to mutually inverse electronic transitions), the large half-width of the absorption and luminescence bands, sometimes reaching the order of an electron-volt when the electron passes from one discrete level to another discrete level, the shape and temperature dependence of these bands, and so on.

Furthermore, a theory that regards the interaction between electrons and atomic vibrations as a small perturbation is incapable of explaining nonradiative thermal transitions of electrons in a crystal, for in such a theory the larger the number of vibrational quanta created (absorbed) in a transition, the higher the order of smallness of the transition probability. Transitions in which several tens of vibrational quanta are created, according to such a theory, have a negligible probability, wholly insufficient to explain the facts. The examples given, of course, do not exhaust the difficulties of the theory based on the second simplifying assumption.

During the last decade in the USSR a new direction of theory has begun to develop and has led to significant successes; it does not assume the weakness of the interaction between electrons and atomic vibrations. This interaction, and sometimes the principal part of this interaction, is included in the Hamiltonian of the system already in the zeroth approximation. In this way not only quantitatively but also qualitatively new results are obtained. The shapes of the absorption and luminescence bands of various kinds of local “centers” in a crystal receive a natural explanation. The half-width and temperature dependence of these bands, the Stokes shift, the relation between the photo-transition energy \(\hbar\omega\) and the energy of the corresponding thermal transition \(W\), and so forth are calculated. In many cases the theory proves for the first time to be in quantitative agreement with experiment. Qualitatively new results are obtained in considering the quantum states of conduction electrons.

Below a brief survey of this new direction of theory is given.

2. STATIONARY STATES OF LOCALIZED WEAKLY BOUND ELECTRONS1

We shall be concerned with electrons localized near violations of the periodicity of the crystal lattice. It is assumed that these electrons are bound much more weakly and move much more slowly than the proper electrons of the dielectric. These may be, for example, conduction electrons localized at defects of the crystal lattice, valence electrons of impurity atoms, or proper electrons of the dielectric which, as a result of excitation, have passed into states with large effective radii and therefore have become weakly bound. The criterion for weak binding of an electron is the condition that the frequency of the light intensively absorbed by it be much smaller than the frequencies of the light absorbed by the proper (basic) electrons of the dielectric.

In this case one may use the well-known adiabatic approximation2, assuming that the state of the proper electrons adiabatically follows the motion of the atomic nuclei and of the weakly bound electrons. One may then consider the conservative motion only of the slow particles—the atomic nuclei and the weakly bound electrons—while the presence of the proper electrons of the crystal will enter only into the expression for the potential energy of the slow particles.

The Hamiltonian of the system of slow particles has the form:

\[ \hat H = -\frac{\hbar^2}{2m}\sum_{i=1}^{N}\Delta_i + V(r,q) + \frac{1}{2}\sum_{\chi}\hbar\omega_{\chi} \left( q_{\chi}^{2} - \frac{\partial^{2}}{\partial q_{\chi}^{2}} \right). \tag{6} \]

Here the first term is the kinetic-energy operator of the weakly bound electrons of the local “center” under consideration; the second term is the potential energy of interaction of these electrons with one another and with the atoms (ions) of the crystal; \(r\) is the set of coordinates of all the weakly bound electrons of the center; \(q\) is the set of all normal coordinates \(q_{\chi}\) describing vibrations of the atoms; and \(\omega_{\chi}\) are the frequencies of the proper vibrations of the atoms. The last term in (6) is the operator of the normal vibrations of the atoms of the crystal, from which the weakly bound electrons have been removed, but in which, in other respects, the local center is present.

In what follows we shall speak only of weakly bound electrons, which will be called simply electrons. The explicit expression \(V(r,q)\) is unknown; nevertheless, despite this, it is possible to obtain a number of concrete results. Assuming that the amplitudes of the atomic vibrations are small, the potential energy may be expanded in a series in powers of \(q_{\chi}\):

\[ V(r,q)=V_0(r)+\sum_{\chi}V_{\chi}(r)q_{\chi}+\cdots \tag{7} \]

To determine the stationary states and energy levels of the system it is necessary to find the eigenvalues and eigenfunctions of the operator (6). Since the atoms, having comparatively large masses, move much more slowly than even weakly bound electrons, the system under consideration can again be divided into two subsystems: a slow one—the vibrations of the atoms—and a fast one—the weakly bound electrons. Further, one may again assume that the state of the electrons adiabatically (inertialessly) follows the motion of the atoms. For each instantaneous configuration of the atoms \(q_x\), the corresponding stationary state of the electrons \(\psi_s(r,q)\) has time to become established. The latter is determined from the wave equation

\[ \left[-\frac{\hbar^2}{2m}\sum_{i=1}^{N}\Delta_i+V(r,q)\right]\psi_s(r,q)=E_s(q)\psi_s(r,q), \tag{8} \]

where \(s\) is the set of quantum numbers of the electrons, and \(q\) is a multidimensional parameter. In solving this equation we shall assume that the atoms vibrate mainly near configurations in which their potential energy is minimal; in what follows this assumption will be confirmed after the wave function of the system has been found. Let \(q_{xs}\) be the coordinates of such equilibrium configurations. For the atomic vibrations mentioned, \(V(r,q)\) varies only slightly near the function \(V(r,q_s)\). In the zeroth approximation one may replace \(V(r,q)\) in (8) by \(V(r,q_s)\). As a result one obtains

\[ \left[-\frac{\hbar^2}{2m}\sum_{i=1}^{N}\Delta_i+V(r,q_s)\right]\psi_s(r,q_s)=E_s(q_s)\psi_s(r,q_s). \tag{9} \]

Then the correction to the potential

\[ V(r,q)-V(r,q_s)=\sum_x V_x(r)(q_x-q_{xs})+\cdots \tag{10} \]

is introduced as a small perturbation, and in the first approximation one obtains

\[ E_s(q)=E_s(q_s)+\sum_x V_{xs}(q_x-q_{xs})+\cdots, \tag{11} \]

\[ \psi_s(r,q)=\psi_s(r,q_s)-\sum_{xs'}' \frac{V_{xs's}}{E_{s'}(q_s)-E_s(q_s)}\psi_{s'}(r,q_s)(q_x-q_{xs}), \tag{12} \]

where

\[ V_{xs}=\int V_x(r)\,|\psi_s(r,q_s)|^2\,dr, \tag{13} \]

\[ V_{xs's}=\int V_x(r)\psi_{s'}^{*}(r,q_s)\psi_s(r,q_s)\,dr. \tag{14} \]

Let us proceed to the consideration of the vibrations of the atoms. According to the adiabatic approximation\(^{11}\), the energy operator of the atoms must have the form

\[ \hat H_s=E_s(q)+\frac12\sum_x \hbar\omega_x\left(q_x^2-\frac{d^2}{dq_x^2}\right). \tag{15} \]

The role of the potential energy of the atoms is played by the function

\[ F_s(q)=E_s(q)+\frac12\sum_x \hbar\omega_x q_x^2. \tag{16} \]

Thus, with the introduction of the electron “center” into the potential energy of the atoms there appears the additional term (11), i.e. the total energy of the electrons.

If, on the right-hand side of (11), the expansion is restricted to the first term, independent of \(q_x\), then it turns out that the introduction of the electrons in no way affects the motion of the atoms. This is precisely equivalent to neglecting the interaction between the electrons and the vibrations of the atoms, i.e. to the second simplifying assumption of the old theory mentioned in the preceding paragraph.

If, on the right-hand side of (11), the expansion is restricted to the second term, linearly dependent on \(q_x\), then, according to formula (16), with the introduction of the electron “center” into the crystal, additional forces begin to act on the atoms; moreover, along the coordinates \(q_x\) constant generalized forces \(-V_{xs}\) appear, which lead to a displacement of the equilibrium positions of the atoms, but do not cause a change in the frequencies of their vibrations. Below we shall restrict ourselves precisely to this approximation, i.e. in the expansion (11) we shall take into account only the linear terms. It turns out that this approximation already makes it possible to explain the effects that interest us: heat liberation in phototransitions, the Stokes shift, the large difference between the energy of a phototransition and the energy of thermal activation of the corresponding transition, the large half-width of absorption and luminescence bands, etc. Taking account of the quadratic terms of the expansion introduces into these effects corrections of a higher order of smallness.

Substituting (11) into (16), it is easy to calculate the coordinates of the atoms for which the potential energy \(F_s(q)\) is minimal. One obtains

\[ q_{xs}=-\frac{V_{xs}}{\hbar\omega_x}. \tag{17} \]

The minimum value of the potential energy is equal to

\[ J_s=E_s(q_s)+\frac12\sum_x \hbar\omega_x q_{xs}^2. \tag{18} \]

The potential energy of the atoms may be written in the form

\[ F_s(q)=J_s+\frac12\sum_x \hbar\omega_x(q_x-q_{xs})^2, \tag{19} \]

and their energy operator (15) can be rewritten in the form

\[ \hat H_s=J_s+\frac{1}{2}\sum_x \hbar \omega_x \left[\left(q_x-q_{xs}\right)^2-\frac{\partial^2}{\partial q_x^2}\right]. \tag{20} \]

The eigenvalues of this operator are

\[ E_{s\ldots n_x\ldots}=J_s+\sum_x \hbar\omega_x\left(n_x+\frac{1}{2}\right), \tag{21} \]

where \(n_x\) are integers, and its eigenfunctions are

\[ \left. \begin{aligned} \Phi_{s\ldots n_x\ldots}(q)&=\prod_x \Phi_{n_x}(q_x-q_{xs}),\\ \Phi_{n_x}(q_x-q_{xs})&=A_{n_x}e^{-\frac{1}{2}(q_x-q_{xs})^2}H_{n_x}(q_x-q_{xs}). \end{aligned} \right\} \tag{22} \]

Here \(H_{n_x}\) are Chebyshev–Hermite polynomials of degree \(n_x\); \(A_{n_x}\) are normalization constants.

From the theory of the adiabatic approximation it follows that the eigenvalues of the operator (20), or (15), are simultaneously the energy levels of the whole system. The wave functions of the system have the form

\[ \Psi_{s\ldots n_x\ldots}(r,q)=\psi_s(r,q)\Phi_{s\ldots n_x\ldots}(q). \tag{23} \]

Along with the adiabatic approximation considered above, a direct variational method was also used to find the eigenvalues of the operator (6). The eigenvalues of the operator (6), as is known, coincide with the extremal values of the functional

\[ H[\Psi(r,q)]=\int \Psi^*\hat H\Psi\,dr\,dq,\quad dq=\prod_x dq_x, \]

\[ dr=\prod_{i=1}^{N} dx_i\,dy_i\,dz_i \tag{24} \]

under the additional normalization condition for \(\Psi\). The functions \(\Psi(r,q)\) that extremize the functional \(H[\Psi]\) are the eigenfunctions of the operator (6).

To find approximate extremal values of \(H[\Psi]\), \(\Psi\) was approximated by the multiplicative form

\[ \Psi(r,q)=\psi_s(r)\Phi_{s\ldots n_x\ldots}(q). \tag{25} \]

Extremization of the functional \(H[\psi_s\Phi_{s\ldots n_x\ldots}]\) with respect to \(\Phi_{s\ldots n_x\ldots}\), for fixed arbitrary \(\psi_s\), gives

\[ \Phi_{s\ldots n_x\ldots}(q)=\prod_x \Phi_{n_x}\left(q_x+\frac{V_{xs}[\psi_s]}{\hbar\omega_x}\right), \tag{26} \]

where \(\Phi_{n_\chi}\) is expressed by formula (22), and \(V_{\chi s}[\psi_s]\) is determined by relation (13). Substituting (26) into \(H[\psi_s\Phi_{s\ldots n_\chi\ldots}]\), we obtain a functional depending only on \(\psi_s\):

\[ H[\psi_s(r)] = J[\psi_s(r)] + \sum_\chi \hbar\omega_\chi\left(n_\chi+\frac{1}{2}\right), \tag{27} \]

\[ J[\psi_s(r)] = \int \psi_s^* \left[ -\frac{\hbar^2}{2m}\sum_{i=1}^{N}\Delta_i + V_0(r) \right]\psi_s\,dr -\frac{1}{2}\sum_\chi \frac{V_{\chi s}^2[\psi_s]}{\hbar\omega_\chi}. \tag{28} \]

This functional must be extremized by varying \(\psi_s\). In doing so, for \(\psi_s\) the following Euler equation is obtained:

\[ \left[ -\frac{\hbar^2}{2m}\sum_{i=1}^{N}\Delta_i +V_0(r) -\sum_\chi \frac{V_{\chi s}[\psi_s]}{\hbar\omega_\chi}\,V_\chi(r) \right]\psi_s = E\psi_s. \tag{29} \]

The direct variational method set forth above gives the best of the solutions within the class of functions restricted by the multiplicative form (25).

Equation (29) coincides with equation (9), as is easy to verify if one takes into account formula (7), and also takes into account that equation (9) is not linear, but represents a self-consistent problem: the parameters \(q_{\chi s}\) entering it, as is clear from formulas (17) and (13), are themselves functionals of \(\psi_s\). From the coincidence of equations (29) and (9) it follows that the function \(\psi_s(r)\) appearing in the direct variational method coincides with the function \(\psi_s(r,q_s)\) appearing in the adiabatic approximation presented earlier. The same coincidence follows also for the quantity \(V_{\chi s}\) and for the functions \(\Phi_{s\ldots n_\chi\ldots}(q)\), which appear in both methods. Further, substitution of \(\psi_s(r)=\psi_s(r,q_s)\) into the functional (28) gives

\[ J[\psi_s] = E_s(q_s) + \frac{1}{2}\sum_\chi \hbar\omega_\chi q_{\chi s}^2 = J_s. \tag{30} \]

The energy of the system is given by the extremal value of the functional (27), which just coincides with expression (21) of the adiabatic approximation.

Thus, the results of the direct variational method coincide with the results of the adiabatic method if, in the latter, applying the above-stated perturbation method, one restricts oneself to the first approximation in the energy and the zeroth approximation in the wave function of the system.

The exact solution of the nonlinear integro-differential equation (29) presents great mathematical difficulties. Its approximate solution and an explicit determination of \(\psi_s(r)\) will be given below for

of certain special cases, when the model of the local “center” is well known and simple. In other cases it is possible to obtain a number of interesting relations of a general character without using a model of the center and an explicit expression for the function \(\psi_s(r)\).

3. PHOTOTRANSITIONS OF ELECTRONS \(^{10}\)

In this section we shall consider phototransitions of weakly bound electrons of a local “center” from a discrete energy level to a discrete energy level. Let us consider the transition from the state \(s_1 \ldots n_x \ldots\) to the state \(s_2 \ldots n'_x \ldots\). In this transition not only the quantum numbers of the electrons \(s\) change, but also the quantum numbers of the vibrations \(n_x\), i.e. heat is released. The frequency of the light absorbed in such a transition is

\[ \omega=\frac{1}{\hbar}\left[E_{s_2\ldots n'_x\ldots}-E_{s_1\ldots n_x\ldots}\right] =\omega^0+\sum_x \omega_x\left(n'_x-n_x\right), \]

\[ \omega^0=\frac{1}{\hbar}\left[J_{s_2}-J_{s_1}\right], \tag{31} \]

where \(\omega^0\) is the frequency absorbed in a “purely electronic” transition, for which \(n'_x=n_x\).

The probability of a phototransition is calculated by means of the usual quantum-mechanical treatment \(^{10}\). The energy of the light absorbed by the above-mentioned electron-vibrational transition per second in a unit volume of the dielectric can be written in the form

\[ \tau S(\omega), \tag{32} \]

where \(S(\omega)\) is the spectral density of the intensity of the incident light, and the coefficient \(\tau\) is equal to

\[ \tau=\frac{4\pi^2 e^2 \omega N(s_1\ldots n_x\ldots)}{3\hbar c n(\omega)} \left|M_{s_1 n,\,s_2 n'}\right|^2 . \tag{33} \]

Here \(c\) is the velocity of light in vacuum, \(N(s_1\ldots n_x\ldots)\) is the concentration of centers in the initial state \(s_1\ldots n_x\ldots\), \(n(\omega)\) is the refractive index of light in the dielectric,

\[ M_{s_1 n,\,s_2 n'}=M_{s_1s_2}\,M_{n,n'}, \tag{34} \]

\[ M_{s_1s_2}= \int\sum_i \mathbf f(\mathbf r_i)\, \psi^*_{s_1}(\ldots \mathbf r_k \ldots)\, \psi_{s_2}(\ldots \mathbf r_k \ldots)\, d\tau_1\ldots d\tau_N, \tag{35} \]

\[ M_{nn'}= \prod_x\int \Phi_{n_x}(q_x-q_{xs_1})\, \Phi_{n'_x}(q_x-q_{xs_2})\,dq_x, \tag{36} \]

where \(\mathbf r_i\) are the coordinates of the “optical” electrons of the “center,” and the vectorial

coefficient \(\mathbf f(\mathbf r_i)\) is defined so that the product

\[ e\bigl(\mathbf f(\mathbf r_i),\mathbf E\bigr) \tag{37} \]

is the perturbation energy of the \(i\)-th optical electron by the field of the light wave \(\mathbf E\) and by the internal polarization field of the strongly bound electrons, which is also proportional to \(\mathbf E\). In formula (33) an averaging has been performed over all possible directions of the light incident on the center. This formula differs from the well-known formula for the absorption of light by atoms in vacuum by the more complicated expression for the matrix element \(M_{s_1 n,\,s_2 n'}\), and also by the factor \(n(\omega)\) in the denominator.

Using Einstein’s known considerations on the relation between the probability of absorption and the probability of emission, and generalizing them to the case of a medium with optical density \(n(\omega)\), we obtain, for the energy emitted in the transition \(s_2\ldots n'_x\ldots \to s_1\ldots n_x\ldots\) per second per unit volume, the expression

\[ S_r= \frac{4\omega^4 n(\omega)\,N\bigl(s_2\ldots n'_x\ldots\bigr)} {3c^3} \left|M_{s_1 n,\,s_2 n'}\right|^2, \tag{38} \]

in which \(N\bigl(s_2\ldots n'_x\ldots\bigr)\) is the concentration of centers excited into the state \(s_2\ldots n'_x\ldots\). The approximate method used in § 2 for considering the states of the system, which leads to the multiplicative expression (25) for the wave function of the system and to the additive expression (21) for the energy, makes it possible, conventionally, to speak separately of electronic states characterized by the quantum number \(s\), and of the states of harmonic vibrations, each of which is characterized by a separate quantum number \(n_x\). It is by no means assumed here that the interaction between the electrons and the vibrations is small. Below we shall always be concerned only with two electronic states: the ground state \(s_1\) and the excited state \(s_2\). In each of these states the \(n_x\) may be arbitrary.

Consider the transition \(s_1\ldots n_x\ldots \to s_2\ldots n'_x\ldots\), absorbing the frequency

\[ \omega^0+\omega_l,\qquad \text{where}\quad \omega_l=\sum_x \omega_x\bigl(n'_x-n_x\bigr), \]

and the transition \(s_2\ldots n'_x\ldots \to s_1\ldots n_x\ldots\), emitting the frequency

\[ \omega^0-\omega_l; \]

the frequencies \(\omega^0+\omega_l\) and \(\omega^0-\omega_l\) are arranged mirror-symmetrically on both sides of the frequency of the “purely electronic” transition \(\omega^0\). On the basis of formulas (34), (35), (36), and (22) one can show\(^{10}\):

\[ \left|M_{s_1 n,\,s_2 n'}\right| = \left|M_{s_2 n,\,s_1 n'}\right|. \tag{39} \]

Thus, the probabilities of the above-mentioned absorbing and emit-

...of the transitions under consideration turn out to be related. If it is assumed that before each optical transition the vibrations are in thermal equilibrium, then one obtains\({}^{10}\):

\[ \left[\frac{\tau n}{\omega}\right]_{\omega^0+\omega_l} = \frac{\pi^2 c^2}{\hbar}\,\frac{N_1}{N_2} \left[\frac{S_r}{\omega^4 n}\right]_{\omega^0-\omega_l}. \tag{40} \]

Here \(N_1\) is the concentration of centers in which the electrons are in the state \(s_1\), irrespective of the vibrational state, and \(N_2\) is the concentration of centers whose electrons are in the state \(s_2\).

Formula (40) shows that if one plots graphically \(\frac{\tau n}{\omega}\) and \(\frac{S_r}{\omega^4 n}\) as functions of frequency, then the resulting graphs are mirror-symmetric with respect to the line \(\omega=\omega^0\). The mirror symmetry of absorption and luminescence spectra was first discovered experimentally by Levshin in 1931.\({}^{12}\) In his subsequent works he showed that this symmetry is characteristic of the spectra of a fairly large number of substances (dyes) in solutions. Mirror symmetry has also been found in the spectra of polyatomic molecules. The theoretical calculation given above was carried out in terms corresponding to the case of a crystal. However, nowhere above was a regular and periodic arrangement of atoms assumed. It was assumed only that the atoms perform harmonic vibrations about certain fixed equilibrium positions. But the latter also takes place in molecules and even in liquids, over sufficiently long intervals of time, until, as a result of an energy fluctuation of the vibrations, mixing of the liquid occurs and new equilibrium positions of the atoms are established. Usually the time of this mixing is greater than the time of a phototransition. Thus, the theoretical results obtained above (as well as part of the results below) are applicable not only to solids, but also to polyatomic molecules and, in part, to liquids.

It should be borne in mind that formulas (39) and (40) are approximate and could not have been obtained if, in the expansion (11), we had not restricted ourselves to the linear term but had also taken into account the quadratic term. The latter is equivalent to taking into account the dependence of the natural vibrational frequencies \(\omega_x\) on the electronic state \(s\).

4. SHAPE AND TEMPERATURE DEPENDENCE OF ABSORPTION AND LUMINESCENCE BANDS

Let us consider a light-absorption band formed by a series of lines corresponding to one definite electronic transition \(s_1 \to s_2\), but to all possible different vibrational transitions \(\ldots n_x \to \ldots n'_x \ldots\). The frequencies of the series mentioned are given by formula (31), if in it the quantum numbers \(s_1\) and \(s_2\) are fixed, and the numbers \(n_x\) and \(n'_x\) are assigned all possible positive integer values. The intensi-

the intensity of the line, according to (33)—(36), can be written in the form

\[ \tau=G_{s_1s_2}\left|M_{nn'}\right|^2, \tag{41} \]

where

\[ G_{s_1s_2}=\frac{4\pi^2 e^2\omega N(s_1..n_x..)}{3\hbar c\,n(\omega)} \left|M_{s_1s_2}\right|^2 . \tag{42} \]

If \(\dfrac{\omega}{n(\omega)}\) varies relatively little within the limits of the band, \(G_{s_1s_2}\) may be regarded as a constant coefficient. Thus, the shape and half-width of the band are determined by the factor \(\left|M_{nn'}\right|^2\). From formula (31) it is seen that the given frequency \(\omega\) is absorbed by an infinite number of elementary quantum transitions. We are speaking of the set of transitions for which the sum

\[ \sum_x \omega_x\left(n'_x-n_x\right) \tag{43} \]

has a prescribed value, denoted below by \(\omega_r\). In order to obtain the total intensity of absorption of light of frequency

\[ \omega=\omega^0+\omega_r, \tag{44} \]

it is necessary to sum the quantity (41) over all elementary transitions in which the given frequency \(\omega\) is absorbed. This summation leads to complicated calculations. For brevity, below we shall give only the final results.

It has been possible to carry out the summation exactly in the limiting case of small dispersion of the normal vibrations of the atoms (ions) in the crystal, i.e., in the case when all the proper frequencies \(\omega_x\) almost do not depend on \(x\) and are equal\(^3\) to the limiting frequency \(\omega_0\). In this case formula (31) may be rewritten in the form

\[ \omega=\omega^0+p\omega_0, \tag{45} \]

where

\[ p=\sum_x\left(n'_x-n_x\right); \tag{46} \]

\(p\) is an integer, taking both positive and negative values. Thus, the absorption spectrum consists of a series of discrete lines separated from one another by the same distance \(\omega_0\). The total amount of light energy absorbed by the line \(\omega^0+p\omega_0\) per unit time in \(1\ \mathrm{cm}^3\) of the crystal is equal to \(^{13}\)

\[ \tau_p S(\omega^0+p\omega_0), \]

where

\[ \tau_p=G_{s_1s_2}\cdot e^{-a\left(\bar n_0+\frac12\right)} \left(1+\frac1{\bar n_0}\right)^{p/2} I_p\left(a\sqrt{\bar n_0(\bar n_0+1)}\right), \tag{47} \]

\(\bar n_0\) is the mean Planck value of the oscillatory quantum number \(n_x\):

\[ \bar n_0=\frac{1}{e^{\frac{\hbar\omega_0}{kT}}-1}, \tag{48} \]

\(I_p(z)\) is the Bessel function of imaginary argument:

\[ I_p(z)=\frac{1}{i^p}J_p(iz), \tag{49} \]

\(a\) is a constant equal to

\[ a=\sum_x (q_{xs_1}-q_{xs_2})^2 . \tag{50} \]

If there is some dispersion of the normal vibrations, but the \(\omega_x\) are close to \(\omega_0\), then in this case, instead of the above-mentioned discrete spectral lines, narrow bands of finite width are obtained, and formula (47) gives the integral absorption intensity of each such band. With further increase of the dispersion the width of these bands grows, they overlap, and the absorption spectrum becomes continuous, but has a “structure”: it is represented by an oscillating curve with alternating maxima and minima, the distance between the maxima being approximately equal to \(\omega_0\). This curve may be represented as a broad bell-shaped curve on which a sinusoid with period \(\omega_0\) is superimposed. The exact equation of this curve the reader will find in \(^{14}\). With further increase of the dispersion the amplitude of the sinusoid tends to zero and the spectrum assumes the form of a smooth bell-shaped curve. The form of the latter can be obtained if one takes into account that in the spectral interval \(\Delta\omega\) there are \(\dfrac{\Delta\omega}{\omega_0}\) bands, each of which absorbs energy \(\tau_p S\) per second. Thus, in the spectral interval \(\Delta\omega\) an energy

\[ \frac{\tau_p S(\omega)\Delta\omega}{\omega_0} \]

is absorbed. This energy is customarily written in the form \(\tau(\omega)S(\omega)\Delta\omega\), where \(\tau(\omega)\) is the usual absorption coefficient of a continuous spectrum. Equating these two quantities, we obtain

\[ \tau(\omega)=\frac{\tau_p}{\omega_0}. \tag{51} \]

In this formula \(p\) should no longer be regarded as an integer index, but as a continuously varying parameter related to the frequency \(\omega\) by relation (45).

Formulas (47) and (51) determine the shape of the absorption spectrum and its temperature dependence, since \(n_0\) depends on the temperature. In deriving these formulas it was assumed that there is only one dispersion branch with limiting frequency \(\omega_0\) and that the \(\omega_x\) differ little ...

of \(\omega_0\). The above-mentioned overlap of bands often occurs already when the dispersion \(\omega_x\) amounts to only \(10\%\) of \(\omega_0\).

It has been possible to consider also the more general case of a crystal with an arbitrary structure of the elementary cell, any number of dispersion branches, and an arbitrarily large dispersion ensuring the overlap of bands\(^{14}\). However, unlike the preceding case, this treatment is approximate and is valid for two limiting cases: for the case of large and small heat release in the optical transition, more precisely, when the average number of vibrational quanta \(\hbar\omega_x\) produced in the optical transition is, respectively, considerably greater than and considerably less than unity.

In the case of large heat release the form of the absorption spectrum is determined by the formula

\[ \tau=\tau_m \exp\left\{-\frac{(\omega-\omega_m)^2}{2g''} +\frac{1}{6}\frac{g'''(\omega-\omega_m)^3}{g''^3}+\cdots\right\}, \tag{52} \]

where \(\omega_m\) is the frequency at which the absorption coefficient \(\tau\) is maximal; it is equal to

\[ \omega_m=\omega^0+\frac{1}{2}\sum_x(q_{xs_1}-q_{xs_2})^2\omega_x, \tag{53} \]

\[ g''=\sum_x(q_{xs_1}-q_{xs_2})^2\omega_x^2\left(\bar n_x+\frac{1}{2}\right), \tag{54} \]

\[ g'''=\frac{1}{2}\sum_x(q_{xs_1}-q_{xs_2})^2\omega_x^3, \tag{55} \]

\[ \bar n_x=\frac{1}{e^{\frac{\hbar\omega_x}{kT}}-1}, \tag{56} \]

\[ \tau_m=\frac{G_{s_1s_2}}{\sqrt{2\pi g''}}. \tag{57} \]

As is seen from formula (52), in the region of the absorption maximum, since \(\omega-\omega_m\) is small, the power series in the exponent of the exponential converges rapidly and one may confine oneself to the first quadratic term. As one moves away from the maximum, the cubic term gradually begins to play some role as well, as a consequence of which the absorption on the red side falls off somewhat more rapidly than on the violet side. Investigation shows that, in the case of large heat release, the cubic term is much less than unity and may be discarded right up to the points at which \(\tau(\omega)\) is equal to one half of its maximum value. In this region the absorption spectrum is represented by a Gaussian curve, whose half-width is equal to

\[ \delta\omega=2\sqrt{2\ln 2}\,\sqrt{g''}. \tag{58} \]

This result, as well as formulas (53), (54), and (57), were obtained by the author first for the case of small dispersion\(^{13}\), and then generalized to the case of arbitrary dispersion by the author jointly with Krivoglaz\(^{14}\).

From formulas (57) and (58) it follows that the product

\[ \tau_m \cdot \delta\omega = G_{s_1s_2}\,2\sqrt{\frac{\ln 2}{\pi}} \tag{59} \]

is constant. This means that the area of the bell-shaped curve depicting \(\tau(\omega)\) does not depend on temperature.

There exist two limiting cases in which the inconvenient, difficult-to-compute infinite sums can be eliminated from formulas (53) and (54).

A) The case of absence of dispersion and arbitrary temperature. In this case \(\omega_x\) may be replaced by \(\omega_0\), and \(\omega_0\) and \(\bar n_0\) taken outside the summation sign. Then from (53) and (54) one obtains

\[ g'' = 2\omega_0(\omega_m-\omega^0)\left(\bar n_0+\frac{1}{2}\right), \tag{60} \]

\[ \delta\omega = 2\sqrt{2\ln 2}\, \sqrt{2\omega_0(\omega_m-\omega^0)\left(\bar n_0+\frac{1}{2}\right)}. \tag{61} \]

B) The case of arbitrary dispersion and high temperature, for which

\[ \bar n_x+\frac{1}{2}\simeq \frac{kT}{\hbar\omega_x}\gg 1. \tag{62} \]

In this case from (53) and (54) one obtains

\[ g''=\frac{2kT}{\hbar}(\omega_m-\omega_0), \tag{63} \]

\[ \delta\omega=2\sqrt{2\ln 2}\, \sqrt{\frac{2}{\hbar}(\omega_m-\omega_0)kT}. \tag{64} \]

Everything set forth in this paragraph referred to the absorption band. The law of mirror symmetry (40) makes it possible immediately to write down all the results obtained above also for the luminescence band associated with the reverse electronic transition \(s_2\to s_1\). According to formula (40), the quantities \(\dfrac{\tau_n}{\omega}\) and \(\dfrac{S_r}{\omega+n}\) as functions of \(\omega\) are represented by mirror-symmetric graphs. The center of symmetry is the point \(\omega^0\). When the width of the bands is not too large, the variation of these quantities, their half-width, the position of the maximum, etc., approximately coincide with those for \(\tau(\omega)\) and \(S_r(\omega)\). It was shown above that the quantity \(\dfrac{\tau_n}{\omega}\) has a maximum at the frequency \(\omega_m\), determined by the formu-

… by (53). Consequently, the quantity \(\dfrac{S_r}{\omega^4 n}\) has a maximum at the mirror-symmetric point

\[ \omega_m=\omega^0-\frac{1}{2}\sum_x (q_{xs_1}-q_{xs_2})^2\omega_x . \tag{65} \]

If the first of these quantities decreases more rapidly toward the red side than toward the violet side (owing to the cubic term in (52)), then the second, conversely, decreases more rapidly toward the violet side. The graphs of the quantities \(\dfrac{\tau n}{\omega}\) and \(\dfrac{S_r}{\omega^4 n}\) have the same half-width. Thus, the half-width of the absorption and luminescence bands is approximately the same.

The Stokes shift, i.e. the distance between the maximum of \(\tau(\omega)\) and the maximum of \(S_r(\ )\), is approximately equal to

\[ \Delta \omega=\omega_m-\omega'_m=2(\omega_m-\omega^0) =\sum_x (q_{xs_1}-q_{xs_2})^2\omega_x . \tag{66} \]

Therefore, in case A) we have:

\[ \delta\omega=2\sqrt{2\ln2}\, \sqrt{\Delta\omega\cdot \omega_0\left(\bar n_0+\frac{1}{2}\right)} \tag{67} \]

and in case B)

\[ \delta\omega=2\sqrt{2\ln2}\, \sqrt{\frac{\Delta\omega}{\hbar}\,kT}. \tag{68} \]

Formulas (67) and (68) are convenient because they contain no unknown parameters, and therefore they are easy to compare with experiment.

For mutually mirror-symmetric absorbing and emitting phototransitions, the same heat release occurs, equal to

\[ \sum_x \hbar\omega_x(n'_x-n_x). \tag{69} \]

In the region of the absorption maximum this heat release, according to formulas (31) and (53), is equal to

\[ \hbar(\omega_m-\omega^0)=\frac{\hbar\Delta\omega}{2} =\frac{1}{2}\sum_x \hbar\omega_x(q_{xs_1}-q_{xs_2})^2 . \tag{70} \]

This result admits a simple classical interpretation: if one takes into account that in our notation the potential energy of the vibrations of atom \(B\) has the form

\[ \frac{1}{2}\sum_x \hbar\omega_x(q_x-q_{xs})^2, \tag{71} \]

then the right-hand side of (70) is the vibrational energy acquired by the atoms as a result of the instantaneous change of the forces and the displacement of the equilibrium positions of the atoms from \(q_{xs_1}\) to \(q_{xs_2}\).

Assuming that the crystal is in thermal equilibrium, applying the usual canonical distribution and using formula (21), one can calculate the probability \(w(s)\) that the electrons of the “center” are in state \(s\), while the \(n_k\) have arbitrary values. As a result of such a calculation one obtains

\[ \frac{w(s_2)}{w(s_1)}=e^{-\frac{W_{12}}{kT}}, \tag{72} \]

\[ W_{12}=J_{s_2}-J_{s_1}=\hbar\omega^{0}=\hbar\left(\omega_m-\frac{\Delta\omega}{2}\right) =\hbar\left(\omega_m+\frac{\Delta\omega}{2}\right). \tag{73} \]

\(W_{12}\) is called the energy of thermal excitation of the electrons from the ground state \(s_1\) to the state \(s_2\). Formula (73) shows that \(W_{12}\) does not coincide with the energies of the corresponding optical transitions \(\hbar\omega_m\) and \(\hbar\omega'_m\).

It should be emphasized that the effects considered above—heat release in phototransitions, the Stokes shift, the difference between the energy of thermal excitation of electrons and the energy of the corresponding optical transition, the large broadening of the absorption and luminescence bands corresponding to an electron transition from a discrete to a discrete energy level, and the temperature dependence of the bands—all these effects have been obtained as a result of the fact that \(q_{xs_1}\ne q_{xs_2}\). These effects are caused by the change in the equilibrium positions of the atoms as a result of the phototransition of the electrons; they are associated with the deformation of the lattice by optical electrons.

It is very important that the results presented above have been obtained without specifying the model of the light-absorbing “center,” without specifying the type of crystal (or amorphous dielectric), and without using the explicit expression for the electronic part of the wave function, \(\psi_s(r)\). Therefore these results possess great generality and indicate the existence of similarity in the absorption spectra of various kinds of local “centers” in different crystals. Thus, for example, in the case of large heat release in a phototransition, not too small a dispersion \(\omega_k\), and provided the other criteria of applicability of the present theory are satisfied (the adiabatic condition, harmonicity of the normal vibrations of the atoms, and the possibility of neglecting the quadratic terms in (11)), the shape of the absorption and luminescence bands must always approach a “Gaussian curve,” characterized by a single parameter \(g''\) or by the half-width (58). The latter is related to the Stokes shift by the general formulas (67) and (68).

Above, the case of large heat release in a phototransition was considered. It has also been possible to consider the opposite limiting case—

of small heat release, i.e., the case when the mean number of vibrational quanta \(\hbar \omega_x\) arising in a phototransition is much less than unity. In this case it turned out that the form of the absorption spectrum depends essentially on the details of the spectrum of eigenfrequencies \(\omega_x\). Therefore there is no similarity between the absorption spectra in the case of different crystals and different types of “centers”; \(\tau(\omega)\) could not be expressed by such simple and general formulas as above, in the case of large heat release. Nevertheless, general formulas for \(\tau(\omega)\) have been obtained. To avoid overloading the present article, we shall not give them here, but shall give only a simpler expression for the integral intensity of the band, i.e., for

\[ T=\int \tau(\omega)\,d\omega, \]

where the integral is taken over the region of the band.

Let the index \(\nu\) enumerate those dispersion branches of the normal vibrations in which the number of vibrational quanta as a result of the phototransition has increased by \(q_\nu>0\):

\[ q_\nu=\sum_x (n'_{x\nu}-n_{x\nu}). \]

\[ \sum_x \]

denotes summation within a single dispersion branch. Let, analogously, the index \(\lambda\) enumerate those branches of the normal vibrations of the atoms in which the number of vibrational quanta as a result of the phototransition has decreased by \(r_\lambda>0\):

\[ r_\lambda=\sum_x (n_{x\lambda}-n'_{x\lambda}). \]

Consider an absorption band due to phototransitions with specified numbers \(q_\nu\) and \(r_\lambda\). This band extends from the frequency

\[ \omega_{\min}=\omega^0+\sum_\nu q_\nu \omega^{(1)}_{x\nu}-\sum_\lambda r_\lambda \omega^{(2)}_{x\lambda}, \tag{73a} \]

to the frequency

\[ \omega_{\max}=\omega^0+\sum_\nu q_\nu \omega^{(2)}_{x\nu}-\sum_\lambda r_\lambda \omega^{(1)}_{x\lambda}, \tag{73б} \]

where \(\omega^{(1)}_{x\nu}, \omega^{(1)}_{x\lambda}\) and \(\omega^{(2)}_{x\nu}, \omega^{(2)}_{x\lambda}\) are the smallest and largest vibration frequencies in the corresponding dispersion branches. The integral intensity

of this band turns out to be equal to

\[ T(\ldots q_\nu \ldots, \ldots r_\lambda \ldots) = G_1 \prod_\nu \frac{\sigma_\nu^{q_\nu}}{q_\nu!} \prod_\lambda \frac{{\sigma'_\lambda}^{r_\lambda}}{r_\lambda!}, \tag{73в} \]

where

\[ \sigma_\mu=\sum_x (q_{\mu x s_1}-q_{\mu x s_2})^2(\bar n_{\mu x}+1); \]

\[ \sigma'_\mu=\sum_x (q_{\mu x s_1}-q_{\mu x s_2})^2\bar n_{\mu x}; \]

\[ G_1=G\exp \sum_\mu \frac{\sigma_\mu+\sigma'_\mu}{2}; \]

the index \(\mu\) numbers all branches of dispersion without exception.

In the particular case when only one branch of normal vibrations interacts more intensely with the optical electrons, \(\mu=1\), there is obtained, on the violet side of the line of the purely electronic transition, a progression of bands almost equidistant from one another, with integral intensities

\[ T(q,0,0\ldots,00\ldots)=G_1\frac{\sigma_1^{q_1}}{q_1!}, \tag{73г} \]

and on the red side—an analogous progression of bands with integral intensities

\[ T(00\ldots,r_1 00\ldots)=G_1\frac{{\sigma'_1}^{r_1}}{r_1!}. \tag{73д} \]

The integers \(q_1\) and \(r_1\) are the ordinal numbers of the bands in the progressions, and formulas (73г) and (73д) give the dependence of the intensity of a band on its ordinal number, and also on the temperature.

At high temperatures \(\sigma'_1\simeq\sigma_1\) and, consequently, the integral intensities of the absorption bands on the violet and on the red sides of the line of the purely electronic transition \(\omega^0\) are approximately mirror-symmetric.

At low temperatures the intensity of the bands on the red side decreases exponentially to zero as the crystal is cooled \((\sigma'_1\to 0)\); on the violet side, however, the intensity of the bands tends to a finite limit.

Formulas (73в), (73г), and (73д) were derived in a work by the author together with M. A. Krivoglazov\(^{14}\).

It should be emphasized that, in the case of small heat release, from the absorption spectrum one can make much more detailed judgments about the spectrum \(\omega_x\) than in the case of large heat release.

5. COMPARISON OF THE CALCULATED SHAPE AND TEMPERATURE DEPENDENCE OF BANDS WITH EXPERIMENT

The theoretical results presented above, concerning the shape and mirror symmetry of bands of light absorption and luminescence, apply only to simple bands, i.e., to bands corresponding to a single electronic transition, \(s_1 \rightleftarrows s_2\), combined with all possible vibrational transitions \(n_x \to n'_x\). If the experimentally observed spectrum is a superposition of several simple bands, then its shape depends substantially on the distances between the simple bands, their half-widths, and the ratio of the intensities of the simple bands. In order to compare such a complex

Fig. 1. Luminescence spectrum of \(CaWO_4\) with \(PbWO_4\) impurity at different temperatures. Wavelength of the exciting light \(\lambda = 2537\) Å. \(\circ\) — measurement results; solid line — Gaussian curve.

spectrum with the theory, it must first be resolved into simple bands, which in most cases can be done only approximately. Therefore, when comparing theory with experiment, preference should be given to cases in which simple bands were observed, or, at worst, well-resolved double bands.

The resemblance of a simple absorption or luminescence band to a “Gaussian curve” was noted by many experimentalists long before the appearance of the theory. Refraining from presenting numerous experimental data, we shall give only a few typical examples.

In Fig. 1 is shown^15 the luminescence spectrum of CaWO₄ with an admixture of PbWO₄. As is seen from the figure, the experimental points lie well on the “Gaussian curve.”

In Fig. 2 analogous data are given^15, ^16 for an MgWO₄ crystal. In Fig. 3 is shown^15 the luminescence spectrum of a ZnS crystal. It consists of two simple bands: the green (main) band is usually associated with a copper impurity, and the blue one with an excess of zinc.

Fig. 2. Luminescence spectrum of MgWO₄ at different temperatures. Wavelength of the exciting light \(\lambda = 2537\ \text{Å}\). ○ — results of measurement; solid line — Gaussian curve.

The figures presented show that, in the corresponding four cases, the luminescence spectrum is well described by formula (52) without the cubic term in the exponent. The latter means that the criterion of large heat release is strongly satisfied; this will be confirmed below independently.

Each of the luminescence bands mentioned must correspond to a band of impurity absorption of the same shape and half-width. However, these absorption bands already fall in the ultraviolet region and are masked by absorption of the host substance. Up to the present time experimentalists have not succeeded, with sufficient accuracy, in separating from the total absorption spectrum the contribution of the above-mentioned impurity absorption; the latter therefore cannot yet be compared with theory.

For the experimental determination of the shape of the impurity absorption band one should prefer those cases in which this band falls within the transparency region of the host substance. As a classical example we shall cite the well-known \(F\)-absorption bands in ionic crystals. A typical \(F\)-absorption band is shown in Fig. 4[^17]. The shape of the spectrum is well described by the theoretical formula (47), (51). In the limiting case of large thermal broadening this formula passes into formula (52). A slight asymmetry of the curve (a slower decrease of the absorption coefficient toward the violet side) is in complete agreement with the theory.

Fig. 3. Luminescence spectrum of \(\mathrm{ZnS:S:Cu}\). Wavelength of the exciting light \(\lambda = 3650\,\text{\AA}\). \(\circ\)—measurement results; dashed lines—Gaussian curves; solid line—sum of the Gaussian curves.

According to the theoretical formula (59), the area bounded by the curve \(\tau(\omega)\) must not depend on temperature. This fact does indeed hold and has been repeatedly noted by experimenters. The independence of the area from temperature is seen in Fig. 4.

Passing to consideration of the dependence of the half-width of the band on temperature, let us rewrite formula (67) in the following form:

\[ \frac{\delta \omega}{2{,}354 \sqrt{\omega_0 \Delta \omega}} = \sqrt{ \frac{1}{e^{\hbar \omega_0/kT}-1} + \frac{1}{2} }. \tag{67a} \]

This formula is applicable in those cases where the dispersion of the intrinsic

frequencies \(\omega_\chi\) is small and when all of them can be replaced by some average effective frequency \(\omega_0\). More precisely, it is sufficient that the dispersion be small only for that part of the normal vibrations which interacts most strongly with the optical electrons. These are the normal vibrations for which, as a result of the photo-transition, the greatest displacement of the equilibrium positions \(q_{\chi s_1} - q_{\chi s_2}\) occurs.

In Fig. 5 the solid curve represents the quantity (67a). From this figure it is seen that at the very lowest temperatures \(\delta\omega\), and consequently the whole shape of the band, does not depend on temperature,

Fig. 4. \(F\)-band of light absorption in KBr at different temperatures.

which agrees with numerous experimental data. In the region of intermediate temperatures, when \(0.3 \leq \dfrac{kT}{\hbar\omega_0} \leq 1.8\), \(\delta\omega\) depends almost linearly on temperature. With increasing temperature the solid curve asymptotically approaches the dotted curve, representing \(\sqrt{\dfrac{kT}{\hbar\omega_0}}\). The latter depicts formula (68), applicable in the case of arbitrary dispersion and high temperature.

In the case of ionic crystals with a rock-salt-type lattice, the optical electrons interact most strongly with longitudinal polarization vibrations. The corresponding frequencies \(\omega_0\) for alkali-halide crystals were determined by K. V. Tolpygo\(^{18}\) from data on the absorption and dispersion of infrared light in these crystals. For KBr it was found that \(\hbar\omega_0 = 0.020\ \text{eV}\).

Application of formula (67a) to the \(F\)-absorption band in KBr and a comparison of the results with experiment are given in Table I.

Table I

\[ 2.354\,\hbar\sqrt{\omega_{0}\Delta\omega}=0.323\ \text{eV} \]

\(T^\circ\mathrm{K}\)	\(\hbar\delta\omega\) in eV, calculated	\(\hbar\delta\omega\) in eV, measured
28	0.23	0.22
293	0.37	0.37
473	0.48	0.49
873	0.63	0.66

Fig. 5. The solid curve corresponds to

\[ \sqrt{\frac{1}{e^{\frac{\hbar\omega_{0}}{kT}}-1}+\frac{1}{2}}. \]

The dashed curve is

\[ \sqrt{\frac{kT}{\hbar\omega_{0}}}. \]

The dashed straight line is

\[ 0.6+0.436\,\frac{kT}{\hbar\omega_{0}}. \]

Here the Stokes shift \(\Delta\omega\) was determined by exact matching of the calculated and observed \(\delta\omega\) at room temperature.

(On the Influence of Lattice Deformation on the Properties of Crystals)

(Below, in § 7, \(\Delta\omega\) will be calculated theoretically.) The agreement of the calculated and observed \(\delta\omega\) at three other temperatures serves as confirmation of the theory. Similar results are also obtained for \(F\)-bands in other alkali-halide crystals.

Figure 6 gives the observed\(^{16}\) temperature dependence of the half-width of the luminescence bands in the above-mentioned crystals \(\mathrm{CaWO_4}\) and \(\mathrm{MgWO_4}\). The course of the experimental curves agrees qualitatively with the theoretical curve presented in Fig. 5. For

Fig. 6. Dependence of the luminescence half-width on temperature in \(\mathrm{CaWO_4}\) and \(\mathrm{MgWO_4}\).

a more accurate quantitative comparison of the curves it is necessary to know the natural frequencies of the longitudinal polarization oscillations of the ions in these crystals. Comparison of Fig. 6 with Fig. 5 shows that, for the two above-mentioned tungstates, temperatures above \(400^\circ\mathrm{K}\) already approach the high-temperature region, in which, instead of formula (67), one may use formula (68). The latter, although it gives a somewhat underestimated value of \(\delta\omega\) (as is seen from the comparison of the dashed and solid curves in Fig. 5), nevertheless has the advantage that the atomic vibration frequencies do not enter into it at all; it is sufficient to know only the Stokes shift \(\Delta\omega\).

To determine \(\Delta\omega\) in \(\mathrm{MgWO_4}\), we use experiments in which both the luminescence spectrum and the corresponding excitation spectrum were measured on one and the same crystal sample\(^{19}\). According to these measurements, the excitation maximum lies near \(4.8\ \mathrm{eV}\), and the luminescence maximum at \(2.3\ \mathrm{eV}\). Thus, the Stokes shift is \(\hbar\Delta\omega = 2.5\ \mathrm{eV}\). Calculating from formula (68) the absolute value of \(\delta\omega\) at a temperature of \(451^\circ\mathrm{K}\), we obtain \(\hbar\delta\omega_{\mathrm{calc}} = 0.74\ \mathrm{eV}\). The corresponding measured value is

\[ \hbar\delta\omega_{\mathrm{meas}} = 0.84\ \mathrm{eV}^{16}. \]

Agreement of the theory with experiment should be regarded as good, especially if one takes into account that formula (68) gives an underestimated value of \(\delta\omega\).

The mean heat evolution in a phototransition, according to formula (70), is equal to

\[ \frac{\hbar\Delta\omega}{2}. \]

For the \(F\)-absorption band in a KBr crystal it is equal to \(0.5\) ev, which amounts to 25 quanta of the polarization vibrations of the ions. Thus, we are indeed dealing with a case of large heat evolution. Heat evolution of the same order also occurs for \(F\)-absorption of light in other alkali-halide crystals. In \(\mathrm{MgWO}_4\), for the phototransitions considered above, the mean heat evolution is equal to \(1.2\) ev, which also means large heat evolution (under any plausible assumptions about the magnitude of \(\hbar\omega_\varkappa\)). In the latter case the heat evolution exceeds half the energy of the emitted quantum.

Figure 7 gives the absorption and emission spectra of rhodamine 6G extra molecules dissolved in ethyl alcohol. This figure clearly demonstrates the mirror symmetry according to Levshin\({}^{12}\) (p. 99).

Fig. 7. Mirror symmetry of the absorption and luminescence spectra of a solution of rhodamine 6G extra in ethyl alcohol at different temperatures (after Levshin).

It follows from the mirror symmetry that the absorption and luminescence bands are most likely simple. At room temperature the half-width of both bands is \(0.22\) ev. The Stokes shift is equal to \(0.135\) ev. Calculating by formula (67) \(\omega_0\)—the mean effective frequency of the vibrations of the atoms that interact most strongly with the optical electron—we obtain

\[ \hbar\omega_0 = 0.13\ \text{ev}. \]

The mean heat evolution is approximately \(0.07\) ev, i.e. less than \(\hbar\omega_0\). Consequently, in this example the criterion of large heat evolution is violated. It is precisely this that explains the large deviation of the band shape from the “Gaussian curve.”

6. THE CASE OF IONIC CRYSTALS AND LARGE RADII OF ELECTRONIC STATES

The results presented above were obtained without specifying the model of the light-absorbing “center,” and without an explicit expression for the wave functions of the electrons \(\psi_s(\mathbf r)\). However, for the theoretical solution of many important questions it is necessary to specify the model of the center, the structure of the crystal, and to find an explicit expression for \(\psi_s(\mathbf r)\). Such questions include: the calculation of the frequency of a purely electronic transition \(\omega^0\), the theoretical calculation of heat release and the Stokes shift, the calculation of the energies of thermal excitation and thermal dissociation of the electrons of the center, etc.

Below we shall consider ionic crystals with a cubic lattice, in which the unit cell consists of two ions. Many results will also be applicable to any isotropically polarizable ionic crystals and polar liquids.

Of the various types of normal vibrations of the ions, the longitudinal polarization waves interact most strongly with the electrons. As for acoustic and transverse polarization vibrations, we shall neglect their interaction with the electrons. As a result these vibrations are separated off as a conservative closed subsystem, and in what follows they may be ignored.

Assuming the radius \(|\psi_s|^2\) of the electron cloud to be sufficiently large, the crystal may be regarded as a dielectric inertially polarizable continuum. Let \(\mathbf P(\mathbf r,t)\) be that part of the vector of the specific polarization of the dielectric which is associated only with longitudinal polarization vibrations. Then the potential energy of interaction of the electrons of the center with one another and with the ions can be written in the following form:

\[ V(\mathbf r, q) = \sum_i W(\mathbf r_i) + \sum_i W'(\mathbf r_i) + \frac{e^2}{n^2} \sum_{i>j} \frac{1}{r_{ij}} - e\sum_i \int \frac{\mathbf P(\mathbf r')(\mathbf r' - \mathbf r_i)} {|\mathbf r' - \mathbf r_i|^3} \,d\tau' . \tag{74} \]

Here \(W(\mathbf r)\) is the ordinary periodic potential of an electron in a crystal with immovably fixed ions, \(W'(\mathbf r)\) is the local change of the periodic potential associated with the defect (the center), \(i\) and \(j\) are the numbers of the optical electrons of the center, and \(r_{ij}\) is the distance between them. \(n^2\) is the square of the refractive index of light, playing the role of the dielectric constant of polarization of the strongly bound electrons of the dielectric. The last term in (74) is the energy of interaction of the electron with the inertially polarizable dielectric continuum.

The polarization of a dielectric can be subdivided into a non-inertial and an inertial part. The first is the polarization of a crystal in which the ions are fixed immovably at the lattice sites. This polarization is associated with the deformation of the electron orbits of the dielectric; it can be caused only by an external electric field whose frequency, in order of magnitude, lies between the infrared frequencies of the natural vibrations of the ions and the characteristic frequencies of the bound electrons of the dielectric. The weakly bound optical electrons of the “center” mentioned above, as well as conduction electrons, moving past the ions of the crystal, create fields precisely of such a frequency and, consequently, cause non-inertial polarization of the ions. The polarization of the ions non-inertially follows the motion of the electron producing it, as a result of which an additional force acts on the electron, possessing the period of the lattice. We shall assume that this force is already included in the periodic potential \(W'(\mathbf r)\) appearing in formula (74).

By the inertial part of the polarization we shall mean the polarization associated with displacements of the ions from the lattice sites. It is composed of dipole moments caused by displacements of the ions, and of that part of the polarization of the bound electrons which is caused by the displacement of the ions. It is precisely this inertial polarization that is meant when one speaks of the natural (thermal) polarization oscillations of the ions. Inertial polarization can be created in a crystal if one first slowly switches on an external electric field and then switches it off so rapidly that the ions do not have time to move from their positions, but the strongly bound electrons do have time to change their polarization. The polarization that remains in the crystal immediately after the field is switched off will be the inertial part of the polarization. If the electrostatic induction of the applied field \(\mathbf D(\mathbf r)\) varies sufficiently smoothly in space (changes little over a distance of the order of the lattice constant), then macroscopic electrodynamics may be used. Then, upon slow switching-on of the field, the total polarization appears

\[ \mathbf P_{\mathrm{sum}}=\frac{\varepsilon-1}{4\pi\varepsilon}\,\mathbf D \tag{75} \]

(\(\varepsilon\) is the dielectric constant of the crystal).

Upon rapid switching-off of the field, the change in polarization is equal to

\[ \delta \mathbf P=-\frac{n^{2}-1}{4\pi n^{2}}\,\mathbf D. \tag{76} \]

Thus the remaining inertial polarization is equal to

\[ \mathbf P=\mathbf P_{\mathrm{sum}}+\delta\mathbf P=\frac{c_{0}}{4\pi}\,\mathbf D, \tag{77} \]

where

\[ c_{0}=\frac{1}{n^{2}}-\frac{1}{\varepsilon}. \tag{78} \]

It is not difficult to compute the potential energy of an inertially polarized crystal as the work performed by the field \(\mathbf D\) under slow switching on and rapid switching off. This potential energy turns out to be equal to

\[ U_p=\frac{2\pi}{c_0}\int \mathbf P^2\,d\tau . \tag{79} \]

The integral is taken over the entire volume of the crystal.

It should be emphasized that in formula (74) the last term is the potential energy of the interaction of the electrons only with the inertial part of the crystal polarization \(\mathbf P(\mathbf r,t)\). The interaction with the noninertial polarization, as was already explained, is included in the first term of formula (74).

For further calculations it is necessary explicitly to introduce the normal coordinates of the dielectric vibrations. To this end let us expand the inertial polarization in a trigonometric series of the form

\[ \mathbf P(\mathbf r,t)=\sum_{\chi}\frac{\chi}{|\chi|}P_{\chi}(t)\chi_{\chi}(\mathbf r),\qquad \chi_{\chi}(\mathbf r)=\sqrt{\frac{2}{L^3}} \begin{cases} \cos \chi r, & \text{for } \chi_x<0,\\ \sin \chi r, & \text{for } \chi_x>0 \end{cases} \tag{80} \]

(\(L\) is the size of the basic cubic region in which the \(\chi_{\chi}\) are orthonormal). Here and below \(\chi\) is a three-dimensional vector. As the normal coordinates one may, as is known, choose the expansion coefficients \(P_{\chi}\). Then the energy of the natural vibrations of the dielectric is expressed as follows:

\[ \sum_{\chi}\frac{2\pi}{c_{\chi}}\left(P_{\chi}^{2}+\frac{1}{\omega_{\chi}^{2}}\dot P_{\chi}^{2}\right). \tag{81} \]

Here \(\omega_{\chi}\) are the natural frequencies of the vibrations, and \(c_{\chi}\) are certain positive constants. For long-wavelength vibrations, in which the polarization varies smoothly in space, it is necessary that the first term in formula (81)—the potential energy of the ions—coincide with the quantity (79). Hence it follows that, as \(\chi\to 0\), \(c_{\chi}\) tends to the constant \(c_0\) appearing in formula (78).

In order that the mathematical apparatus developed in § 2 can be used directly, the normal coordinates must be normalized in such a way that the energy operator (81) has the form

\[ \frac{1}{2}\sum_{\chi}\hbar\omega_{\chi} \left(q_{\chi}^{2}-\frac{\partial^{2}}{\partial q_{\chi}^{2}}\right). \tag{82} \]

For this it is necessary to put

\[ q_{\chi}=\sqrt{\frac{4\pi}{\hbar\omega_{\chi}c_{\chi}}}\,P_{\chi}. \tag{83} \]

Formula (74) also goes over into formula (7), if one sets

\[ V_0(r)=\sum_i |W(r_i)+W'(r_i)|+\frac{e^2}{n^2}\sum_{i>j}\frac{1}{r_{ij}}, \tag{84} \]

\[ V_\chi(\mathbf r)=-e\sqrt{\frac{\hbar\omega_\chi c_\chi}{4\pi}}\sum_i \int \frac{\dfrac{\chi}{|\chi|}\chi_\chi(\mathbf r')(\mathbf r'-\mathbf r_i)} {|\mathbf r'-\mathbf r_i|^3}\,d\tau'. \tag{85} \]

Further calculations should be carried out directly by the formulas of § 2. Namely, from formula (13) we obtain:

\[ V_{\chi s}=-\sqrt{\frac{\hbar\omega_\chi c_\chi}{4\pi}}\,D_{\chi s}, \tag{86} \]

where

\[ D_{\chi s}=\int D_s(\mathbf r')\frac{\chi}{|\chi|}\chi_\chi(\mathbf r')\,d\tau', \tag{87} \]

and

\[ \mathbf D_s=\mathbf D[\mathbf r',\psi_s] =e\sum_i \frac{|\psi_s(\mathbf r)|^2(\mathbf r'-\mathbf r_i)} {|\mathbf r'-\mathbf r_i|^3}\, d\tau_1\ldots d\tau_i\ldots d\tau_N . \tag{88} \]

The vector \(\mathbf D_s\) is the electrostatic induction created by the \(|\psi_s|^2\)-cloud of optical electrons. The scalar \(D_{\chi s}\) is the coefficient in the expansion of \(\mathbf D_s\) in the trigonometric series of the form (80).

Substitution of (86) into formula (28) gives

\[ J[\psi_s]= \]

\[ =\int \psi_s^*(r) \left[-\frac{\hbar^2}{2m}\sum_i \Delta_i+V_0(r)\right]\psi_s(r)\,dr -\frac{1}{8\pi}\sum_\chi c_\chi D_{\chi s}^2 . \tag{89} \]

The wave function of the optical electrons \(\psi_s\) is determined by extremizing the functional (89).

In the case when the effective radius of the \(|\psi_s|^2\)-cloud of optical electrons is sufficiently large (for example, equal to the lattice constant or exceeding it), the following simplifications may be made in the functional (89): using the method of the effective electron mass in a crystal\(^{20,9}\), one may omit the periodic potential \(W(r_i)\) in the Hamiltonian, simultaneously replacing the electron mass \(m\) by a certain effective mass \(\mu\). Further, since for large radii of the states the induction \(\mathbf D_s\) is a smooth function of the coordinates, its Fourier coefficient \(D_{\chi s}\) is significant only for long-wave harmonics, i.e. for small \(|\chi|\). Thus, in the last term of expression (89), the terms with small \(|\chi|\) dominate in the sum. In this case the coefficient \(c_\chi\) may be taken outside the summation sign at its limiting value \(c_0\). As a result

Under these simplifications, the functional (89) can be rewritten in the following form:

\[ J[\psi_s]=\int \psi_s^*(r)\left[-\frac{\hbar^2}{2\mu}\sum_i \Delta_i+\sum_i W'(r_i)+ \right. \]

\[ \left. +\frac{e^2}{n^2}\sum_{i>j}\frac{1}{r_{ij}}\right]\psi_s(r)\,dr -\frac{a_0}{8\pi}\int \mathbf{D}^2[r'\psi_s]\,d\tau', \tag{90} \]

After \(\psi_s(r)\) has been found by extremizing the functional (90), one must compute, by formulas (88), (87), and (86), respectively, \(\mathbf{D}\), \(D_{zs}\), and \(V_{zs}\). These calculations should be carried out both for the initial and for the final states of the phototransition. Then, by formula (17), the equilibrium positions of the ions \(q_{xs}\) and \(q_{xs_2}\) are calculated, and by formulas (53), (54), (57), and (58) the principal parameters of the absorption band are calculated: the position of the maximum, the maximum value of the absorption coefficient \(\tau_m\), the half-width of the band \(\delta\omega\), the Stokes shift \(\Delta\omega\), and the heat release in the phototransition, calculated by formula (70). In these calculations \(\omega_x\) may be replaced approximately by \(\omega_0\), for in ionic crystals of the type considered there is only one branch of longitudinal polarization vibrations, and the dispersion of the frequencies \(\omega_x\) within this branch is usually small. In addition, for large radii of the electronic states, in all the formulas given above the terms with small \(|\chi|\) play the main role.

The frequency of the purely electronic transition \(\omega^0\) and the energy of thermal excitation \(W_{12}\) are calculated by formulas (31) and (73).

The program of calculations outlined above is general for all types of local “centers.” Only the form of the potential \(W'(r_i)\) and the number of optical electrons depend on the type of center, i.e., on its model. At present, three types of local centers have been calculated:

1) \(F\)-centers \(^{13,22}\). The justification of the model of \(F\)-centers is given in \(^{21}\). In this case there is one optical electron:

\[ W'(r_1)=-\frac{ze^2}{\varepsilon |r_1|}; \tag{91} \]

2) \(F'\)-centers were considered by the author jointly with O. F. Tomasevich \(^{23,24}\). In this case there are two optical electrons. \(W'\), as in the preceding case, is the potential of an attractive Coulomb center:

\[ W'(r_i)=-\frac{ze^2}{\varepsilon |r_i|};\quad i=1,2. \tag{92} \]

3) \(F_2\)-centers, i.e., paired \(F\)-centers interacting with one another and forming in the crystal something like a diatomic molecule, were considered by M. F. Deigen \(^{25}\). In this case the optical

two electrons:

\[ W'(r_i)=-\frac{ze^2}{\varepsilon}\left(\frac{1}{|r_i|}+\frac{1}{|r_i-a|}\right), \tag{93} \]

where \(a\) is the distance between the positive central charges of the \(F\)-centers.

For brevity, we shall not consider all three of the above-mentioned types of centers, but shall consider, as a typical example, only the theory of \(F\)-centers.

7. THE THEORY OF \(F\)-CENTERS AND ITS COMPARISON WITH EXPERIMENT

A variant of the theory of \(F\)-centers in which the motion of the electron was treated quantum-mechanically, and the motion of the ions classically, was developed by the author jointly with M. F. Deigen\(^{21,22}\). The energy levels and wave functions of the electron in the \(F\)-center were determined; however, it was not possible to obtain the shape of the \(F\)-band of light absorption, its temperature dependence, and its half-width. This version of the theory did not fully correspond to the general computational scheme that was outlined above. Therefore we shall not dwell on it.

We shall briefly present the results of a later work by the author\(^{13,9}\), in which \(F\)-centers were considered precisely according to the general computational scheme developed in the preceding section. The ground state of the electron was determined by absolute minimization of the functional (90), (91) by the direct variational method. The electron wave function was approximated by the expression

\[ \psi_{s_1}=\frac{\alpha^{3/2}}{\sqrt{7\pi}}(1+\alpha r)e^{-\alpha r}. \tag{94} \]

The parameter \(\alpha\) was determined so that \(J[\psi_{s_1}]\) would be minimal. As a result one obtains

\[ \alpha=\frac{\mu e^2}{2\hbar^2}\left(\frac{3z}{\varepsilon}+c_0\right), \tag{95} \]

\[ J_{s_1}=-\frac{\mu e^4}{2\hbar^2}\left(\frac{z}{\varepsilon}+\frac{c_0}{3}\right)^2 . \tag{96} \]

Other approximations for \(\psi_{s_1}\)\(^{22}\) were also tried; however, approximation (94) proved to be the most successful of them: it leads to a lower value of the functional \(J[\psi_{s_1}]\). Of the excited states of the electron, that one was calculated in which the optical transition of the electron is most probable. In this state the wave function was approximated by the expression

\[ \psi_{s_2}=\frac{8}{\sqrt{2\pi}}(\alpha\beta)^{5/2} r e^{-2\alpha\beta r}\cos\vartheta . \tag{97} \]

This function is orthogonal to the function of the ground state (94), as it must be in the direct variational method. The parameter \(\beta\) was determined from the condition of an absolute minimum of \(J[\psi_{s_1}]\). As a result of G. E. Zil’berman’s calculations\({}^{26}\), for the case \(z=1\) one obtains:

\[ 2\beta=\frac{1+0.3914\varepsilon_0}{3+\varepsilon_0}, \tag{98} \]

\[ J_{s_1}=-\frac{\mu e^4}{8\hbar^2}\left(\frac{1}{\varepsilon}+0.3914\varepsilon_0\right)^2 . \tag{99} \]

The subsequent calculations were carried out without any new approximations, using the general formulas obtained in the preceding paragraphs.

The theory contains four parameters: \(\varepsilon\), \(n\), \(\omega_0\), and \(\mu\). These parameters are not specific only to \(F\)-centers, polarons (see below), and other local centers, but characterize also an ideal crystal containing no local electronic states. Therefore the values of the parameters can be determined independently of the phenomena considered above, by studying various properties of ideal crystals. In particular, the parameters \(\varepsilon\), \(n\), and \(\omega_0\) enter essentially into the formula expressing the law of light dispersion in an ideal crystal, and may be determined from experimental dispersion curves, if these are known. As for the parameter \(\mu\), until recently there have been no sufficiently reliable methods for the independent determination of this parameter. Therefore the effective electron mass remains the only unknown parameter of the theory. Below, \(\mu\) is determined by matching the calculated abscissa (frequency) of the maximum of the \(F\)-absorption band, \(\omega_m\), with the corresponding experimentally measured quantity.

Table II gives a comparison of the calculated thermal dissociation energies of \(F\)-centers with experimentally measured ones. The latter

Table II

	\(\varepsilon\)	\(n^2\)	\(\hbar\omega_m^{\mathrm{calc}}=\hbar\omega_m^{\mathrm{meas}}\), eV	\(\mu/m\)	\(W^{\mathrm{calc}}\), eV	\(W^{\mathrm{meas}}\), eV
Cu\(_2\)O	9	4	0.62	1.81	0.51	0.5–0.6
NaCl	5.8	2.33	2.65	2.78	2.2	1.9
NaBr	6.39	2.6	2.29	2.96	1.9	1.6
NaJ	6.6	2.91	2.1	3.25	1.8	1.5
KCl	4.78	2.175	2.19	1.85	2.0	2.0
KBr	4.81	2.36	1.96	1.87	1.8	1.7
KJ	5.2	2.65	1.8	2.11	1.7	1.7
RbCl	5.2	2.18	1.98	1.78	1.7	1.7
RbBr	5.16	2.36	1.71	1.70	1.5	1.4
RbJ	5.58	2.6	1.59	1.89	1.4	1.3

were determined from the temperature dependence of the electrical conductivity of crystals in which the donors of current carriers were \(F\)-centers. Evidence that in these crystals the donors of current carriers are indeed \(F\)-centers is provided by the fact that removal of \(F\)-centers from the crystal leads to a decrease in electrical conductivity by many orders of magnitude. In the crystals under consideration the temperature dependence of the electrical conductivity is well represented by the formula

\[ \sigma=\sigma_0 e^{-\frac{W}{2kT}} . \tag{100} \]

The parameter \(W\) appearing here is easily determined experimentally. It is precisely this that we have in mind when speaking of the experimental value of the thermal dissociation energy of \(F\)-centers. Table II uses the values of \(W\) obtained for alkali-halide crystals by Smakula[^27].

Fig. 8. Cuprous oxide. Spectral distribution of photoconductivity associated with excess oxygen.

In the first row of Table II, data are given for cuprous oxide. Here the \(F\)-center is a positive “conduction hole,” localized near a vacant copper lattice site. The spectrum of light absorption by the \(F\)-center of cuprous oxide lies in the infrared region. This absorption is accompanied by photoconductivity. Schenwald[^28] studied the spectral distribution of this photoconductivity; his results are presented in Fig. 8.

The maximum of the curve corresponds to a light quantum energy of 0.62 eV. The curve has a typical bell-shaped form, characteristic of \(F\)-centers also in alkali-halide crystals \(^{29}\).

The temperature dependence of the electrical conductivity in copper oxide has been studied by many authors. There is a certain scatter in the obtained values of the thermal dissociation energy. In Table II the data of Engelhard \(^{30}\) are used, since they were obtained on the same samples of \(\mathrm{Cu}_2\mathrm{O}\) and refer to the same centers as Schenwald’s spectral measurements.

As for the calculated values of the thermal dissociation energy given in Table II, they were obtained under the assumption that, as a result of the dissociation of an \(F\)-center, a free polaron is formed (see below), and that it is the current carrier \(^{9,13}\). The values of the parameters \(\varepsilon\), \(n\), and \(\omega_0\) needed for the calculation were obtained from experimental curves of light dispersion in crystals. For alkali-halide crystals these parameters were determined rather accurately by K. B. Tolpygo \(^{18}\). For copper oxide, unfortunately, the experimental dispersion curve is available over an insufficiently wide spectral interval. Therefore \(\varepsilon\) and \(n\) in this case are determined less accurately, while \(\omega_0\) was determined from the temperature dependence of the mobility of current carriers (see below).

From a comparison of the last two columns of Table II it is seen that the calculated values of the thermal dissociation energy agree well with the measured ones. The agreement is worse for three sodium salts, for which the radius of the \(|\psi|^2\)-cloud of the electron at the \(F\)-center is too small, as a result of which the criteria for applicability of the theory are violated.

The most characteristic feature of the present theory is that the photoexcitation energy \(\hbar\omega_m\) exceeds the thermal dissociation energy \(W\), despite the fact that in the optical transition the electron is transferred to a discrete level, whereas in thermal dissociation it is transferred into a continuous spectrum. This is explained by the fact that in the optical transition, in addition to excitation of the electron, a considerable part of the energy of the light quantum is spent on heat release. From Table II it is seen that the above-mentioned excess of the optical-transition energy over the thermal-dissociation energy is indeed observed experimentally.

In § 5 of the present article it has already been shown that the calculated form and temperature dependence of the \(F\)-absorption bands agree well with experiment. However, the absolute values of the half-widths of the absorption bands were not calculated there, since for this it is necessary to have an explicit expression for the electronic wave function \(\psi_s(\mathbf r)\). In this paragraph, after the explicit expressions for \(\psi_{s_1}\) and \(\psi_{s_2}\) have been obtained, it is possible to calculate the absolute values of the half-widths and of the mean heat releases in the optical transition.

In Table III a comparison with experiment is given for the calculated half-widths of the \(F\)-absorption bands at room temperature \(^{13,9}\). The parameter \(a\) is determined by formula (50). The form of the \(F\)-absorption band was determined ...

Table III

\(T = 293^\circ\mathrm{K}\)

	NaCl	KCl	KBr	KI	Cu\(_2\)O
\(\hbar\omega_0\) (eV)	0,032	0,026	0,020	0,017	0,031
\(\hbar\omega_0/kT\)	1,28	1,04	0,795	0,675	1,24
\(\sqrt{\bar n_0(\bar n_0+1)}\)	0,73	0,92	1,22	1,45	0,76
\(a\)	52	47,6	51,9	53,7	11,6
\(\hbar\Delta\omega\) (eV)	1,66	1,24	1,04	0,913	—
\(\hbar\delta\omega^{\mathrm{calc}}\) (eV)	0,47	0,40	0,37	0,35	0,22
\(\hbar\delta\omega^{\mathrm{meas}}\) (eV)	0,47	0,35	0,37	0,35	0,35

by formulas (47) and (51). From the table it is seen that, for alkali-halide crystals, the theoretical values of the half-widths agree well with experiment. In cuprous oxide the measured half-width is larger by 0,1–0,15 eV than the calculated one, which can be explained as follows: the above-noted spread of the experimental values of the thermal dissociation energy of \(F\)-centers in cuprous oxide is connected with the nonideality of the crystals and with a certain nonidentity of the \(F\)-centers. If it is assumed that in the above-mentioned cuprous-oxide specimens the energy levels of the \(F\)-centers had a spread of the order of 0,1–0,15 eV, then this explains at the same time both the spread of the thermal-dissociation energies and the observed additional broadening of the \(F\)-band.

In the limiting case of low temperatures, the form of the absorption band and, in particular, its half-width must not depend on temperature. This conclusion of the theory is fully confirmed by experiment.

Table IV

	KBr	NaF
\(\delta\omega^{\mathrm{calc}}\) (eV)	0,24	0,61
\(\delta\omega^{\mathrm{meas}}\) (eV)	0,22	0,62

Table IV gives the calculated and measured half-widths of the \(F\)-absorption bands in the limiting case of low temperatures.

The limited scope of the present article compels us to discontinue the further exposition of the comparison of theory with experiment and to refer the reader to the appropriate literature.

The consideration of the recombination of conduction electrons (polarons) at \(F\)-centers with the formation of \(F'\)-centers, the calculation of the corresponding mean free path of trapping of the electron, and its comparison with experiment are set forth in \(^{31,32}\) and \(^{9}\). An explanation of the shift of the maximum of the \(F\)-absorption band toward the red upon heating of the crystal, and a comparison of theory with experiment, are given in \(^{13}\) and \(^{9}\).

The quantum yield of photodissociation of \(F\)- and \(F'\)-centers, its dependence on temperature, a comparison of theory with experiment, and a number of other questions are considered in \({}^{9}\) and \({}^{33}\).

8. POLARONS

We now turn to the consideration of the quantum states of the conduction electron in an ideal ionic crystal, in which there are no violations of periodicity, impurities, etc. In reality the original investigations proceeded in the reverse order: first the conduction electron in an ideal crystal was considered; in doing so, the mathematical apparatus and system of approximations were developed which were then used to consider electrons localized near crystal defects and impurities. It is precisely this sequence that the author retained in the monograph \({}^{9}\), devoted to a detailed exposition of the original investigations. Here, however, we have preferred another order of presentation: the “electron in an ideal crystal” is considered as a special case of the problem of “electrons in a crystal with defects,” considered in the preceding paragraphs. In this way greater brevity of exposition is achieved, but some sacrifice must be made in rigor and persuasiveness.

Since in the case under consideration there are no violations of periodicity, and there is also only one electron, the second and third terms should be omitted from formula (74). Assuming that the radius of the electron state is sufficiently large, and applying the effective-mass method, one may in formula (74) also discard the first term, assigning to the electron an effective mass \(\mu^{20,9}\). As a result, the potential energy of interaction of the electron with the vibrations of the ions is written as follows:

\[ V(\mathbf r, q)=-e\int \frac{\mathbf P(\mathbf r')(\mathbf r'-\mathbf r)}{|\mathbf r'-\mathbf r|^3}\,d\tau'=\sum_{\varkappa} V_{\varkappa}(\mathbf r)q_{\varkappa}. \tag{101} \]

Here \(\mathbf P(\mathbf r')\) is the inertial part of the specific polarization of the dielectric, associated with longitudinal polarization vibrations. \(V_{\varkappa}\) are determined by formulas (80), (83), and (85). The Hamiltonian of the system (6) can now be rewritten in the following form:

\[ \hat H=-\frac{\hbar^2}{2\mu}\Delta+\sum_{\varkappa}V_{\varkappa}(\mathbf r)q_{\varkappa} +\frac{1}{2}\sum_{\varkappa}\hbar\omega_{\varkappa} \left(q_{\varkappa}^{2}-\frac{\partial^2}{\partial q_{\varkappa}^{2}}\right). \tag{102} \]

In the usual band theory the second term of the Hamiltonian—the interaction of the electron with vibrations—is regarded as a small perturbation and, in the zeroth approximation, is omitted. As a result the system splits into two noninteracting subsystems: a free electron and a system of independent harmonic oscillators, while the wave function of the system is represented as the product of the corresponding two multi-

particles. Then the interaction term is introduced as a small perturbation, and in the first approximation the scattering of free electrons by ion oscillations is calculated. This scattering determines the mean free path of the conduction electron and its mobility. Consideration of higher approximations of perturbation theory should give only small corrections. Qualitatively, however, the character of the solution should remain the same as in the zeroth approximation. Otherwise the perturbation method itself would not be justified.

There are weighty objections to the application of the perturbation method to the problem under consideration. We shall not, however, dwell on them, but shall pass to the exposition of another method, in which the interaction of the electron with the oscillations is not assumed to be small.

For a preliminary orientation and an estimate of the quantities, let us apply to the problem under consideration directly the apparatus developed in §§ 2 and 6. We replace the wave equation of the system by an equivalent variational principle and first solve the problem by the direct variational method. Approximating the wave function of the system by the multiplicative form (25), we obtain for the oscillatory part of the wave function expression (22). The electronic part of the wave function is determined as the extremal of the functional (90), which in the case under consideration can be rewritten as follows:

\[ J[\psi_s]=\frac{\hbar^2}{2\mu}\int |\nabla\psi_s|^2\,d\tau -\frac{c_0}{8\pi}\int \mathbf{D}^2[\mathbf{r}',\psi_s]\,d\tau'. \tag{103} \]

\(\mathbf{D}\) is defined by formula (88) and, in the case of one electron, is equal to

\[ \mathbf{D}[\mathbf{r}',\psi_s] = e\int \frac{|\psi_s(\mathbf{r})|^2(\mathbf{r}'-\mathbf{r})} {|\mathbf{r}'-\mathbf{r}|^3}\,d\tau . \tag{104} \]

The ground state of the system \(\psi_0\) is the absolute minimum of the functional \(J[\psi_s]\). It was determined in \(^{34,35}\). As a result of the calculations, one obtains

\[ \psi_0(r)\simeq 0.1229\,a_0^{3/2} \left(1+a_0r+0.4516\,a_0^2r^2\right)e^{-a_0r}, \tag{105} \]

where

\[ a_0=0.6585\,\frac{\mu e^2}{\hbar^2}\,c_0 . \tag{106} \]

The corresponding minimum value of the functional is equal to

\[ J_0=-0.0544\,\frac{\mu e^4}{\hbar^2}\,c_0^2 . \tag{107} \]

For cruder calculations, in minimizing the functional \(J[\psi_s]\), one may use not the three-term but the two-term approximation

\[ \psi_0(r)\simeq \frac{\alpha^{3/2}}{\sqrt{7\pi}}\,(1+\alpha r)e^{-\alpha r}. \tag{108} \]

This yields

\[ \alpha=\frac{\mu e^{2}}{2\hbar^{2}}\,c_{0}, \tag{109} \]

\[ J_{0}=-0.0536\,\frac{\mu e^{4}}{\hbar^{2}}\,c_{0}^{2}. \tag{110} \]

Thus, the electron wave function is obtained of the damped type, which corresponds to the discrete negative energy level of the electron \(E_{0}\). If, using formula (17), one calculates the equilibrium positions of the ions \(q_{x0}\) and the corresponding equilibrium polarization of the crystal \(\mathbf{P}_{0}(\mathbf{r})\), then one obtains

\[ \mathbf{P}_{0}(\mathbf{r})=\frac{c_{0}}{4\pi}\,\mathbf{D}\,[\mathbf{r},\psi_{0}]. \tag{111} \]

Comparing this formula with formula (77), it is easy to see that \(\mathbf{P}_{0}\) is precisely the polarization that would be produced by an external electrostatic field possessing induction \(\mathbf{D}\).

The solution obtained may be given the following visual interpretation: a crystal with local polarization \(\mathbf{P}_{0}(\mathbf{r})\) represents for the electron a potential well \(V(\mathbf{r},q_{0})\), whose form can be calculated by formula (101). In this well there exist discrete energy levels for the electron. Having become localized on one of these levels, the electron creates an electric field whose mean induction is equal to \(\mathbf{D}\) and is determined by formula (104). This induction stationarily maintains the local polarization of the crystal \(\mathbf{P}_{0}(\mathbf{r})\). The electron \(\psi\)-function and the polarization of the crystal mutually determine one another and are self-consistent. Such self-consistent states of an electron in an inertially polarizing dielectric medium were called “polarons” by the author. \(^{34,35}\)

A shortcoming of the above solution is the “forced” approximation of the system wave function in the form of the product (25). As a consequence, the energy level of the system, calculated by the direct variational method, is somewhat overestimated. This shortcoming will be largely eliminated below.

In addition to the ground state of the electron considered above, in the polarization potential well \(V(\mathbf{r},q_{0})\) there is an infinite number of excited discrete levels. Phototransitions of the electron from the ground state to excited states are possible. If one assumes that during the short time of the phototransition the heavy ions do not have time to shift noticeably, then the phototransition should be considered under the conditions of a prescribed, fixed potential well corresponding to the polarization of the initial state of the electron \(\mathbf{P}_{0}(\mathbf{r})\). Such absorption of light by polarons was considered in \(^{22,9}\). Formula (18) for the polaron state can be rewritten in the form

\[ J_{s}=E_{s}(q_{s})+\frac{2\pi}{c_{0}}\int \mathbf{P}_{s}^{2}(\mathbf{r})\,d\tau . \tag{112} \]

Here \(E_s(q_s)\) is the discrete energy level of the electron appearing in equation (8). The second term in (112), according to formula (79), is the potential energy of the polarized crystal (or the work of polarization) \(U_p\). Both in the ground state and in any other self-consistent state of the polaron\(^9\)

\[ U_{ps}=-2J_s \tag{113} \]

and from (112) one obtains

\[ E_s=3J_s . \tag{114} \]

The total energy of the entire system is determined by formula (21).

Considering the solution of the polaron problem obtained above as a rough guide, we now turn to a more exact solution. From the form of the functional (103) it follows that, if \(\psi_s(\mathbf r)\) is its extremal, then the function \(\psi_s(\mathbf r-\boldsymbol{\xi})\), where \(\boldsymbol{\xi}\) is an arbitrary constant vector, is also an extremal and corresponds to the same system energy as \(\psi_s(\mathbf r)\). Physically this corresponds to the fact that the polaron can be localized with equal success at any point of a homogeneous dielectric medium, and the energy of the system will then always be one and the same. Hence it follows that the polaron can move progressively in the dielectric and that no work need be expended on such motion; it will take place “by inertia.” We shall call such motion translational. In such motion the polarization potential well is displaced wave-like in the dielectric, while the \(\psi\)-function of the electron, adiabatically following the motion of the potential well, will likewise undergo translational motion.

The principal shortcoming of the multiplicative approximation (25) consists in the fact that it is unable to take into account the correlation between the vibrations of the ions and the motion of the electron. The point is that during ion vibrations the form of the polarization potential well varies near some equilibrium form \(V(\mathbf r-\boldsymbol{\xi}, q_0)\). Adiabatically following the potential well, the \(\psi\)-function of the electron is also deformed. This means that the form of the electron \(\psi\)-function must depend on the coordinates of the ions. It is precisely this dependence that is absent in approximation (25).

Therefore we shall now abandon the multiplicative approximation (25), but retain that part of the apparatus of § 2 which is based on the adiabatic approximation. According to this approximation\(^ {11}\), the problem should be divided into two stages. First one must determine the energy and wave function of the electron for an arbitrary fixed configuration of the ions. The second stage is the consideration of the motion of the ions, with the total energy of the electron \(E_s\) entering as a term in the potential energy of the ions, as is clear from formula (16).

Proceeding to the first stage of the solution, one may replace the wave equation determining the motion of the electron by the equivalent variational

... principle. Then one obtains:

\[ E_s(\ldots q_x \ldots)=\min_{\psi_s}\left\{\frac{\hbar^2}{2\mu}\int |\nabla\psi_s|^2\,d\tau+\int V(\mathbf r,q)|\psi_s|^2\,d\tau\right\}= \]

\[ =\min_{\psi_s}\left\{\frac{\hbar^2}{2\mu}\int |\nabla\psi_s|^2\,d\tau-\int \mathbf P(\mathbf r)\mathbf D[\mathbf r,\psi_s]\,d\tau\right\}. \tag{115} \]

This functional must be minimized by varying \(\psi_s\) at fixed polarization, i.e. at fixed \(q_x\). It can be shown\(^9\) (§ 16) that if the “binding energy” of the polaron is considerably greater than the mean energy of thermal vibration, more precisely, if

\[ 1.8\,|J_s|\gg 3\hbar\omega_0\left(\bar n_0+\frac12\right), \tag{116} \]

then the polarization of the crystal in the region of the polaron varies only slightly near the equilibrium polarization \(\mathbf P_0(\mathbf r-\boldsymbol\xi)\), and the electron wave function differs only slightly from \(\psi_0(\mathbf r-\boldsymbol\xi)\). Thus the polarization may be represented in the form

\[ \mathbf P(\mathbf r)=\mathbf P_0(\mathbf r-\boldsymbol\xi)+\mathbf P'(\mathbf r), \tag{117} \]

where \(\mathbf P'(\mathbf r)\) is a small perturbation, provided only that \(\boldsymbol\xi\) is chosen in a proper way. Using the usual perturbation theory, and regarding \(\psi_0(\mathbf r-\boldsymbol\xi)\) as the solution of the zeroth approximation, one can calculate the electron energy in the first approximation. For the ground state of the electron one obtains

\[ E_0(\ldots q_x \ldots)=\frac{\hbar^2}{2\mu}\int |\nabla\psi_0|^2\,d\tau-\int \mathbf P(\mathbf r)\mathbf D_0(\mathbf r-\boldsymbol\xi)\,d\tau. \tag{118} \]

Here \(\mathbf D_0(\mathbf r-\boldsymbol\xi)=\mathbf D[\mathbf r,\psi_0(\mathbf r-\boldsymbol\xi)]\).

In what follows it is sufficient to restrict ourselves to this first approximation of perturbation theory.

The best choice of \(\boldsymbol\xi\) should be regarded as that for which \(E(\ldots q_x \ldots)\) is expressed most accurately by formula (118). According to the variational principle (115), the greatest accuracy in calculating \(E_0(\ldots q_x \ldots)\) is provided by that choice of \(\boldsymbol\xi\) for which the energy (118) is minimal. The minimum conditions have the form:

\[ \frac{\partial E_0(\ldots q_x \ldots \boldsymbol\xi)}{\partial \xi_i} = -\int \mathbf P(\mathbf r)\frac{\partial \mathbf D_0(\mathbf r-\boldsymbol\xi)}{\partial \xi_i}\,d\tau=0 \quad (i=1,2,3). \tag{119} \]

From these three equations the components of the vector \(\boldsymbol\xi\), \(\xi_1,\xi_2\) and \(\xi_3\), are determined as functions of \(\mathbf P(\mathbf r)\). Thus \(\boldsymbol\xi\) is a functional of \(\mathbf P(\mathbf r)\), as was to be expected. In other words, \(\boldsymbol\xi\) is a function of \(q_x\).

The choice of \(\boldsymbol\xi\) set forth above can be interpreted in the following way. Suppose that in an inertially polarizing medium with

with specific polarization \(\mathbf{P}(\mathbf{r}, t)\) there is a free, weightless charge, the charge density being \(e|\psi_0(\mathbf{r}-\boldsymbol{\xi})|^2\) and, consequently, the electrostatic induction of the charge being \(\mathbf{D}_0(\mathbf{r}-\boldsymbol{\xi})\) (\(\boldsymbol{\xi}\) is the coordinate of the center of charge). If the motion of the charge is considered on the basis of classical mechanics, then the position \(\boldsymbol{\xi}\) of such a weightless charge at any instant of time is determined by the condition that the potential energy of interaction of the charge with the polarized medium be a minimum:

\[ -\int \mathbf{P}(\mathbf{r}; t)\mathbf{D}_0(\mathbf{r}-\boldsymbol{\xi})\,d\tau=\min . \tag{120} \]

This condition coincides with (119).

Let us now proceed to the second stage in the solution of the problem in the adiabatic approximation—to the consideration of the motion of the ions. The Hamiltonian of the system determining the motion of the ions in the presence of an electron in the adiabatic approximation is obtained by adding to the Hamiltonian of the natural vibrations of the ions the term \(E_0(\ldots q_x\ldots)_\lambda\), which plays the role of an additional potential energy of the ions (see formula (15)). As a result, the Hamiltonian of the ions may be written in the form

\[ \hat{H}=J_0+\frac{1}{2}\hbar\omega_0\sum_x \left[(q_x-q_{x\xi})^2-\frac{\partial^2}{\partial q_x^2}\right], \tag{121} \]

where

\[ q_{x\xi}=\sqrt{\frac{4\pi}{\hbar\omega_0\varepsilon_0}}\,P_{x\xi}, \tag{122} \]

\[ P_{x\xi}=\int \mathbf{P}_0(\mathbf{r}-\boldsymbol{\xi})\frac{x}{|x|}\chi_x(\mathbf{r})\,d\tau . \tag{123} \]

It should be borne in mind that \(\boldsymbol{\xi}\) is a function of \(q_x\). This function is given implicitly by equations (119).

Let us now turn to the solution of the wave equation with the Hamiltonian (121). Instead of the former variables \(q_x\), one may introduce new variables

\[ \begin{aligned} q'_x&=q_x-q_{x\xi}(\ldots q_x\ldots)\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \end{aligned} \left\}\ \nu-6,\right. \tag{124a} \]

\[ q'_{x_l}=\sqrt{q^2_{x_l}+q^2_{-x_l}}\qquad (l=1,2,3), \tag{124б} \]

\[ \xi_j=\xi_j(\ldots q_x\ldots)\qquad (j=1,2,3). \tag{124в} \]

The new (“primed”) variables are introduced somewhat asymmetrically: from the total arbitrarily large number of coordinates \(q\), \(\nu-6\), but-

... coordinates are determined by formulas (124a), three coordinates are determined by formulas (124b); finally, as the last three new coordinates, three components of the vector \(\xi\) have been chosen. In what follows, the physical results do not depend on which particular six old coordinates enter into the transformation (124) in a special way as the coordinates \(q_{\pm l}\) \((l=1,2,3)\).

In the new coordinates the Hamiltonian (121) has the following form\(^9\):

\[ \hat H = J_0+\frac{1}{2}\hbar\omega_0 \sum_x' \left( q_x'^2-\frac{\partial^2}{\partial q_x'^2} \right) - \frac{\hbar^2}{2M}\sum_{j=1}^{3}\frac{\partial^2}{\partial \xi_j^2} +\hat H'+\hat H'', \tag{125} \]

where

\[ \hat H' = \frac{\hbar^2}{2M} \sum_x'\sum_{j=1}^{3} \frac{\partial^2 q_{x\xi}}{\partial \xi_j^2} \frac{\partial}{\partial q_x'}, \tag{126} \]

\[ \hat H'' = \frac{\hbar^2}{2M} \sum_{xx_1}'\sum_{j=1}^{3} \frac{\partial q_{x\xi}}{\partial \xi_j} \frac{\partial q_{x_1\xi}}{\partial \xi_j} \frac{\partial^2}{\partial q_x'\partial q_{x_1}'}, \tag{127} \]

\[ \sum_x' \]

denotes summation over all values of \(x\), except the three: \(-\chi_l\) \((l=1,2,3)\).

The terms \(\hat H'\) and \(\hat H''\) are regarded as a small perturbation and are neglected in the zeroth approximation. After this, the eigenfunctions of the operator (125) have the following form:

\[ \Psi_{k\ldots n_x\ldots} = \frac{1}{L^{3/2}}e^{ik\xi} \prod_x'\Phi_{n_x}(q_x'), \tag{128} \]

and the eigenvalues of the energy of the ions, and therefore of the energy of the whole system\(^ {11}\), are equal to

\[ H_{k\ldots n_x\ldots} = J_0+\hbar\omega_0 \sum_x' \left(n_x+\frac{1}{2}\right) + \frac{\hbar^2 k^2}{2M}. \tag{129} \]

The functions \(\Phi_{n_x}\) are determined by formula (22). The constant \(M\) represents the effective mass of the polaron in its translational motion

\[ M = \frac{c_0}{4\pi\omega_0^2} \int \left(\frac{\partial \mathbf D_0}{\partial x}\right)^2 d\tau \simeq 5.8\cdot 10^{-3} \left(\frac{\mu e^2}{\hbar^2}c_0\right)^3 \frac{e^2 c_0}{\omega_0^2}. \tag{130} \]

$M$ should be understood as the inertial mass of the polaron, which manifests itself, for example, in an attempt to accelerate the polaron by an external electric field. Formulas (128) and (129) show that the center of the polaron $\xi$ moves like a free particle of mass $M$. The effective mass of the polaron may differ substantially from the mass of a free electron, and also from the effective mass of an electron $\mu$, which characterizes, in the given crystal, the inertial mass of a conduction electron in the abstract case of ions fixed immovably at the lattice sites. Table V gives the effective masses of polarons in several crystals.

Table V

	NaCl	KCl	KBr	KJ	Cu$_2$O
$\dfrac{M}{m}$	391	160	155	167	9.69

The function (128) represents a wave of the polaron state. Formula (129) expresses the dependence of the energy of such a wave on the wave vector $\mathbf{k}$. If a wave packet is constructed from these waves, then its group velocity is determined by the usual formula

\[ \mathbf{v}=\frac{1}{\hbar}\nabla_k H_{k_1\ldots n_x\ldots}=\frac{\hbar \mathbf{k}}{M}. \tag{131} \]

The classical velocity of motion of the polaron corresponds to the group velocity of the wave packet $\mathbf{v}$. With the aid of formula (131), the last term in (129) can be written in the form

\[ \frac{\hbar^2 k^2}{2M}=\frac{Mv^2}{2}. \tag{132} \]

This quantity is the kinetic energy of the translational motion of the polaron. Formula (131) may be regarded as the de Broglie relation for the polaron; in this case $M\mathbf{v}$ plays the role of the momentum of the polaron.

In formulas (128) and (129), the primes on the product and summation signs indicate that the system under consideration has three fewer vibrational degrees of freedom than a dielectric without a conduction electron. This is easily explained by the following considerations: the introduction into the crystal of an electron, whose state adiabatically “follows” the motion of the ions, should not change the total number of degrees of freedom of the ions. Before the introduction of the conduction electron all degrees of freedom of the ions were vibrational. After the introduction of the electron and the formation of the polaron, three degrees of freedom of the ions appear

\(\xi_1\), \(\xi_2\), and \(\xi_3\), whose motion is not oscillatory but translational in character. These three new degrees of freedom could appear only at the expense of a reduction by three of the number of vibrational degrees of freedom.

The energy of the ground state of the polaron is obtained from formula (129), if in it one sets \(n_x=0\), \(k=0\). It indeed turns out to be somewhat \(\left(\text{by } \dfrac{3}{2}\hbar\omega_0\right)\) lower than the energy (21) obtained by the direct variational method with the multiplicative approximation (25). This was to be expected, since, as has already been mentioned, the direct variational method with approximation (25) should lead to a somewhat overestimated value of the energy of the system.

The calculation given above refers to the case when the electron in the polarization potential well is in the ground state, at the level \(E_0\). However, an entirely analogous calculation can also be carried out in the case when the electron in the polarization potential well is at some excited level \(E_s\).

For brevity, the criteria for the applicability of the theory are not considered here. The reader will find them in \({}^{9}\) (§§ 7 and 16).

The polaron states of a conduction electron in an ideal crystal obtained above differ radically from the results of band theory, in which the electron possesses a wave function of the nondecaying type and in which there is no correlation between the motion of the ions and the motion of the electron. It should be emphasized that certain qualitative features of the theory of polarons set forth above had already been noted earlier by Soviet theorists in the form of individual remarks. Thus, in 1933 L. D. Landau \({}^{36}\) expressed the important idea of the self-localization of an electron in an ideal crystal as a result of deformation of the lattice by the field of the electron itself. These local states were assumed, however, to be immobile, and L. D. Landau tried to identify them with \(F\)-centers in alkali-halide crystals. In 1936 Ya. I. Frenkel \({}^{37}\) noted that a conduction electron should deform the atoms of the crystal nearest to it and that this local deformation should move through the crystal following the electron. In 1936 D. I. Blokhintsev made an attempt, on the basis of the approximation of strongly bound electrons, to determine in which crystals one should expect the self-localization of electrons indicated earlier by Landau. At that time no way had been found for a quantitative treatment of self-localized states, and therefore it was not possible to prove their existence or to investigate their properties. However, the articles mentioned had a definite influence on the author’s work in this field.

In view of the great importance of the polaron problem for the foundations of the theory of electrical, photoelectric, and optical phenomena in semiconductors and dielectrics, the results of this paragraph were obtained by the author independently by four different mathematical methods, not counting the approximate estimate by the direct variational method,

based on the approximation (25). All calculations led to consistent results. The first method was semiclassical: the motion of the electron was considered on the basis of wave mechanics, and the motion of the ions on the basis of classical mechanics \(^{34,35,21,39}\). The advantage of this method lies in its great clarity. The effective mass of the polaron \(M\) was calculated by this method by the author jointly with L. D. Landau \(^{40}\). Other methods, in which the motion of the ions was also considered quantum-mechanically, the reader will find in \(^{41}\) and \(^{9}\). It should also be noted that our results were obtained by N. N. Bogolyubov \(^{42}\) and S. V. Tyablikov \(^{43}\), who applied their own mathematical method.

9. POLARON CONDUCTIVITY OF CRYSTALS

It can be shown with the aid of ordinary statistics that, in the case of thermal equilibrium, polarons possess a Maxwellian distribution of group velocities \(^{9}\) (§ 19). The waves of polaron states considered in the preceding section are not exact, but approximate stationary states of the system, since they are obtained as a result of neglecting the terms of the Hamiltonian \(\hat H'\) and \(\hat H''\). Taking these terms into account, as small perturbations, in the first approximation leads to scattering of the mentioned waves by thermal vibrations of the ions. The scattering of polarons by optical vibrations of the ions is considered in detail in \(^{9}\). Taking this scattering into account, by the method of the usual kinetic equation, the corresponding mobility of the polaron in an external electric field was calculated. The calculation showed that at low temperatures the scattering associated with the perturbation \(\hat H'\) usually dominates. However, with increasing temperature the relative role of the scattering associated with the term \(\hat H''\) also increases. At sufficiently high temperatures both types of scattering have probabilities comparable in order of magnitude.

Below are given formulas for the mobility \(u'\), associated with the perturbation \(\hat H'\), and the mobility \(u''\), associated with the perturbation \(\hat H''\). At low temperatures, i.e. when

\[ kT \ll \hbar\omega_0, \tag{133} \]

\[ u'=\frac{\hbar^{3/2}}{M^{3/2} c_0 e\sqrt{2\omega_0}}\, \frac{(1+x)^8}{\left(1+\frac{1}{7}x\right)^2} \left(e^{\frac{\hbar\omega_0}{kT}}-1\right), \tag{134} \]

\[ x=0.0116\,\frac{\mu e^4 c_0^2}{\hbar^3\omega_0}, \]

\[ u''=\frac{288\,(kT)^{1/2}\left(e^{\frac{\hbar\omega_0}{kT}}-2+\cdots\right)} {(2\pi M)^{1/2}\,\varphi_0^2\,\mu e c_0}. \tag{135} \]

At high temperatures, i.e., when the inequality opposite to (133) holds,

\[ u' = 0.733\,\frac{e^{3}c_0}{(\hbar\omega_0)^2}\sqrt{\frac{kT}{2\pi M}}, \tag{136} \]

\[ u'' = -\frac{288\,\hbar^2}{\mu e c_0(2\pi MkT)^{1/2}}. \tag{136'} \]

The actual mobility of the polaron is determined by the smaller of the quantities \(u'\) and \(u''\). In order to visualize better the magnitudes of the mobilities and their relation, let us consider a numerical example, taking typical mean values of the crystal parameters (see Table VI).

Table VI

\[ \varepsilon = 5,\quad n = 1.58,\quad \mu = m,\quad \omega_0 = 5\cdot 10^{13}\ \mathrm{sec}^{-1},\quad \hbar\omega_0 = 0.033\ \mathrm{eV},\quad c_0 = 0.2 \]

	Low temperatures \(kT = \dfrac{1}{3}\hbar\omega_0\)	High temperatures \(kT = 3\hbar\omega_0\)
\(u'\ \mathrm{cm}^2/\mathrm{sec}\cdot\mathrm{V}\)	63	40
\(u''\ \mathrm{cm}^2/\mathrm{sec}\cdot\mathrm{V}\)	1070	160

The calculated values of the mobilities of polarons given in the table coincide, in order of magnitude, with the mobilities of current carriers in semiconductors, as determined experimentally from electrical conductivity and the Hall constant.

The scattering of polarons by acoustic vibrations of the ions and the corresponding mobility were calculated by Yu. E. Perlin\(^{44}\). It turned out that at room temperatures the principal role is played by scattering not by acoustic, but by polarization vibrations. However, at very low temperatures scattering by acoustic vibrations may dominate. For most crystals this occurs at such low temperatures that scattering of polarons by thermal vibrations is no longer of essential importance in general, and the mobility is determined by scattering by impurities and defects, by the anharmonicity of the vibrations of the ions, etc.\(^{9,39}\)

Alongside the polaron states considered above, in which the electron is in the ground state, at the level \(E_0\), there also exist polarons in which the electron is at the excited level \(E_s\). Furthermore, as a result of a thermal fluctuation of the ion vibrations, the local polarization of the crystal may disappear or even change sign. In this case the polarization potential well of the electron disappears and the latter becomes an ordinary “band” electron. Such a process is called the thermal dissociation of a polaron.

Both polarons in the ground and in excited states, and band electrons, are carriers of current. The question arises which of these current carriers plays the principal role in electrical conductivity. The concentration of polarons with an electron in the ground state overwhelmingly exceeds the concentration of excited polarons and band electrons. The ratio of the concentration of unexcited polarons to the concentration of band electrons can be roughly expressed by the following simple formula, obtained on the basis of equilibrium statistics \({}^{9,39}\):

\[ \frac{N_{\mathrm{pol}}}{N_{\mathrm{band}}} \approx \left(\frac{M}{\mu}\right)^{3/2} e^{\frac{W_p}{kT}}, \tag{137} \]

where \(W_p\) is the energy of thermal dissociation of the polaron; it is equal to

\[ W_p = 0.0544 \frac{\mu e^4}{\hbar^2} c_0^2 + \frac{3}{2}\hbar \omega_0 . \tag{138} \]

In order of magnitude \(W_p\) is equal to several tenths of an electron-volt. Thus, polaron states are energetically more favorable than band states, i.e., dissociated electron states. The ratio

\[ \frac{N_{\mathrm{pol}}}{N_{\mathrm{band}}} \]

at room temperature in most crystals fluctuates between \(10^3\) and \(10^6\). At low temperatures this ratio is still larger. Band electrons could play a substantial role in electrical conductivity only if their mobility were at least this many times greater than the mobility of polarons. Thus, in order for band electrons to be able to compete in electrical conductivity with polarons, one would have to ascribe to them an inadmissibly large mobility, at which even in a not very strong field their velocity would be greater than the speed of light. Such a large mobility would, in order of magnitude, sharply exceed the mobilities of current carriers determined experimentally. If, however, one ascribes to band electrons a mobility of reasonable order, then in electrical conductivity the principal role must be played by polarons, owing to their concentration predominance \({}^{39,9}\). For the same reason, among the various polaron states the principal role is played by unexcited polarons, provided only that the applicability criterion of the theory (116) is satisfied. As was already mentioned, the calculated mobility of unexcited polarons coincides, in order of magnitude, with the experimentally determined mobilities of current carriers.

In Fig. 9 a comparison is given of the temperature dependence of the mobility with experiment. The solid curve represents formula (134) for \(\hbar \omega_0 = 0.031\) ev. The crosses correspond to Enegard’s experimental data for \(\mathrm{Cu_2O}\) \({}^{30}\). At a temperature of \(150^\circ\mathrm{K}\) the theoretical curve has been made to coincide with the experimental point by a choice of the coefficient.

The conclusion that polarons are the principal current carriers is also confirmed in various applications of the theory. Thus, for example, in § 7 the calculated energies of thermal dissociation of \(F\)-centers were given, agreeing with experiment (see Table II). These energies were calculated on the assumption that, as a result of dissociation of an \(F\)-center, it is precisely a polaron, and not a band electron, that is formed. For the formation of a band electron a larger energy would be required.

Fig. 9. Temperature dependence of the mobility of current carriers in copper. The curve is the formula \(u = 27\,(e^{363/T}-1)\), crosses are Engelhard’s experimental data.

Further, the investigation shows that every local state of an electron in a crystal, associated with an impurity or a lattice defect, must possess an infinite number of discrete levels for the electron, these levels becoming infinitely dense as the energy \(E_s = 0\) is approached (the bottom of the conduction band). This assertion becomes clear if one takes into account that for every local state of the electron there exists a polarization potential well which, at large radii, has a Coulomb form and, consequently, possesses a discrete energy spectrum, the upper part of which is similar to the spectrum of the electron in the hydrogen atom. In the absorption of light by such a localized electron, the most probable is a phototransition of the electron to the lowest of the \(p\)-levels; transitions directly into the continuous spectrum are very improbable. If the current carriers were band electrons, then the absorption of light in the region of the maximum of the coeffi-

cient would not lead directly to photoconductivity; for the latter to arise, an additional thermal fluctuation would be necessary, throwing the electron from the discrete \(p\)-level into the conduction band. In this case the quantum yield of photoelectrons would always depend substantially on temperature and, under deep cooling of the crystal, would always tend to zero. However, experiment shows that there are cases in which the quantum yield of photoelectrons as \(T \to 0\) tends not to zero, but to unity\(^9\). In those cases, however, where the quantum yield under deep cooling of the crystal falls exponentially to zero, the corresponding thermal-activation energy proves to be less than that which one should expect if one assumes that the electron must be thrown into the conduction band. The indicated facts show that for dissociation of an impurity center with the formation of a free current carrier, a smaller energy is required than for transferring an electron from the impurity center into the conduction band. This confirms the assumption that the current carriers are not band electrons, but polarons, possessing lower energy. The quantum yield of photodissociation of \(F\)- and \(F'\)-centers is considered in more detail in\(^9\).

From what has been said it is clear that the concept of the polaron is not only the fundamental supporting concept in the theory of electrical conductivity, but also plays an important role in the consideration of many other phenomena.

10. THE CASE OF WEAK INTERACTION OF A CONDUCTION ELECTRON WITH ION VIBRATIONS

After the publication of the basic works on the theory of polarons, based on the adiabatic approximation, attempts began to consider polarons by other mathematical methods. Let us dwell in more detail on one of them. This method consists in the fact that in the Hamiltonian (102) the second term—the interaction of the electron with the vibrations—is assumed to be small, and ordinary perturbation theory is applied to the problem. The zeroth and first approximations were considered long ago and, as is known, constitute the essence of the ordinary “band theory of electrical conductivity.” These approximations still take no account at all of the polarization of the lattice by the electron field and the reverse action of this polarization on the electron. To take the mentioned polarization into account it is necessary to consider the second approximation of perturbation theory, which was done in recent years\(^{45,46,47}\).

It turned out that the second-order correction to the energy of the system converges (is finite) only in the limiting case of low temperatures and only if the kinetic energy of the conduction electron is less than \(\hbar \omega_0\). This energy correction is negative; thus, taking polarization into account leads to a lowering of the energy levels of the system. The calculation also revealed other elements of the theory of polarons. Thus, for example, as S. V. Tyablikov showed\(^{46,47}\), the conduction electron moves accom—

of excitation of the local inertial polarization of the crystal, and therefore the effective mass of the current carrier \(M\) is greater than the effective mass of the electron \(\mu\) in a crystal with immovably fixed ions. To an accuracy up to terms of second order of smallness,

\[ \frac{M}{\mu} = \left( 1-\frac{1}{12}\frac{e^2 c_0}{\hbar} \sqrt{\frac{2\mu}{\hbar\omega_0}} \right)^{-1}. \tag{139} \]

It should be emphasized, however, that quantitatively the results of the perturbation method just mentioned differ substantially from the results of § 8, which are based on the adiabatic approximation. There is nothing surprising in this, since the perturbation method is applicable if

\[ \left. \begin{array}{c} \displaystyle \frac{c_0 e^2}{\hbar} \sqrt{\frac{\mu}{2\hbar\omega_0}} \ll 1,\\[1.2em] \text{i.e.}\\[0.4em] \displaystyle \frac{|J_0|}{\hbar\omega_0} \ll 0.1, \end{array} \right\} \tag{140} \]

whereas the adiabatic approximation is applicable when just the opposite inequality is satisfied (see formula (116), in which one should put \(\bar n_0=0\)). Consequently, the adiabatic approximation and the perturbation method have no common domain of applicability, but are suitable for considering two opposite limiting cases—the cases of strong and weak interaction of the electron with the vibrations of the ions.

In considering a series of alkali-halide crystals and cuprous oxide\(^9\) (addendum), it turned out that in all cases the inequality opposite to (140) was satisfied. Thus, in these crystals the adiabatic approximation is applicable and the perturbation method is not applicable. If crystals with such a small \(c_0\) were found that criterion (140) were satisfied, then in their properties they would come very close to homeopolar crystals and in them, apparently, the interaction of the electron with polarization vibrations would play a secondary role.

11. CRITIQUE OF MOTT AND GURNEY’S PROOF OF ELECTRON SELF-LOCALIZATION

Mott and Gurney\(^48\) (pp. 103–106) proposed a proof of the existence in an ideal crystal of self-consistent self-localized electron states predicted earlier by Landau\(^36\). Their proof, in our notation, reduces to the following. The inertial polarization of the medium is determined by the \(\psi\)-cloud field of the localized electron according to formulas (104) and (77). The corresponding polarization potential well of the electron has the form:

\[ V(\mathbf r)=-c_0 e^2 \int \frac{|\psi_0(\mathbf r')|^2}{|\mathbf r-\mathbf r'|}\,d\tau'. \tag{141} \]

In the limiting case of large \(r\) this potential has the Coulomb form:

\[ r \to \infty \qquad V(r)\to -\frac{C_{0}e^{2}}{r}. \tag{142} \]

Further, according to a well-known theorem of quantum mechanics, if the potential has the asymptotic behavior (142), then there exists an infinite number of discrete energy levels of the electron; moreover, for states with a sufficiently large effective radius the energy spectrum of the electron approaches the spectrum of a hydrogen-like atom. Thus, in the polarization potential well (141), created by the field of the electron itself, there exist discrete levels and, consequently, local states of the electron.

With regard to this proof we would like to make the following remarks. Let \(r_{0}\) be the effective radius of the \(\psi_{0}\)-function of the electron appearing in expression (141). Then it is easy to see that the potential (141) has a Coulomb form only in the region \(r \gg r_{0}\). The above-mentioned theorem of quantum mechanics guarantees the existence only of local states of sufficiently large effective radius \(r_{1}\), more precisely, of states in which the greater part of the electron \(\psi_{1}\)-cloud is situated in the region of the Coulomb potential; for such states one must have \(r_{1} \gg r_{0}\). The theorem by no means guarantees the existence in the potential well (141) of a stationary state \(\psi_{0}\) with radius \(r_{0}\), since in this case the electron \(\psi_{0}\)-cloud would be enclosed in a region where the potential differs substantially from the Coulomb potential.

However large the radius \(r_{0}\) of the function \(\psi_{0}\) appearing in (141) may be, the theorem guarantees only the existence in the given well of local states with much larger radii \(r_{1}\). Thus, the considerations of Mott and Gurney do not prove the existence of self-consistent states.

If one assumes that, as a result of fluctuations of vibrations or of a temporary intervention of external forces, a polarization potential well of the form (141) has been formed, then, according to the theorem mentioned, an electron can be localized in it by entering the state \(\psi_{1}\). However, since \(r_{1} \gg r_{0}\), it is easy to show that the electron \(\psi_{1}\)-cloud will be too “smeared out,” and therefore its field will be insufficiently large for stationary maintenance of the initial polarization of the crystal. Consequently, after the external forces have been removed, depolarization will begin, as a result of which the radius of the function \(\psi_{1}\) will increase still further. The question of whether this depolarization will continue until the polarization potential well disappears completely, or whether a self-consistent local state will be formed, cannot be decided on the basis of the considerations of Mott and Gurney.

Entirely analogous remarks may be made concerning the proof of the possibility of the attachment to an \(F\)-center of a second electron and the formation of a stable \(F'\)-center \(^{48}\) (p. 150).

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