REGARDING THE LETTER BY M. F. DEIGEN AND K. B. TOLPYGO

ON THE REVIEW OF S. I. PEKAR’S BOOK

“STUDIES IN THE ELECTRONIC THEORY OF CRYSTALS”

Our review of S. I. Pekar’s book Studies in the Electronic Theory of Crystals, published in UFN (Vol. 48, No. 3, 1953), along with noting the achievements of the author of the monograph, also contained certain critical remarks concerning particular questions of the electronic theory of ionic crystals. These critical remarks elicited on the part of the students of S. I. Pekar—M. F. Deigen and K. B. Tolpygo—sharp objections and reproaches concerning errors which, in their opinion, are present

in the review. Since the letter by M. F. Deigen and K. B. Tolpygo incorrectly conveys the content of the review, giving it an interpretation that does not follow from it, we are compelled once again to consider the substance of the questions raised in the letter by M. F. Deigen and K. B. Tolpygo.

1. The authors assert that in the review we insisted on “the continuous and band character of the energy spectrum of the electron... in opposition (emphasized by us—S. T.) to the consideration, carried out in the monograph, according to which one may speak of a discrete spectrum of the electron in a polarization well,” and further note that only in the adiabatic approximation does it make sense to speak of the energy levels of a single electron. In view of this they consider that the question is how applicable the adiabatic approximation is to the problem of studying the motion of an electron in a crystal.

With regard to this point one may note that M. F. Deigen and K. B. Tolpygo evidently understood our remark not in the sense in which it was meant in the review. In the review, in fact, the discussion was not about opposing a continuous spectrum to a point spectrum, but about opposing a band spectrum to a continuous one. The review indicated that the energy spectrum of an electron, even when the polarization of the lattice by the field of the electron is taken into account, retains a band structure and is not continuous, as is obtained in the approximation of the effective-mass method.

As for the question of the suitability of the adiabatic approximation, it has no direct relation to the question of the character of the spectrum.

1. Our next remark, which drew objections from M. F. Deigen and K. B. Tolpygo, was that if one considers the motion of a polaron with direct allowance for the periodic potential, and not in the approximation of the effective-mass method, then its energy will be a periodic function of the quasi-momentum, and not a quadratic one.

This remark meant the following: if we take the same equation for an electron moving in an ionic crystal that was used by S. I. Pekar (Eq. (12.10)) and, without passing to the approximation of the effective-mass method, find the proper energy values precisely for this equation (leaving aside the question of the limits of its applicability), then we shall obtain bands for them. The quadratic dependence of the polaron energy on its momentum is a consequence of using the approximate effective-mass method. It is possible that at “large” polaron velocities this equation is inapplicable because of a violation of certain conditions. But since the latter do not follow from the equation, it is obvious that they have no relation to the question of what the energy spectrum of the polaron will be if one assumes that its motion is described by this equation.

We further noted that for this reason certain features of the ordinary band scheme must also be retained in the polaron theory (band widths, distances between them), and that one cannot say that the polaron theory has completely eliminated all the concepts of the band scheme. But, of course, it should be borne in mind that these will be polaron bands which, owing to their specificity, cannot be treated in exactly the same way as ordinary ones.

It is not clear why this essentially trivial remark provoked such sharp objections from M. F. Deigen and K. B. Tolpygo.

1. Considering the question of the character of the energy spectrum of the polaron as a whole, we noted in passing that here, as it seems to us, the inclusion of temperature terms will be essential and that this question is almost not touched upon in the monograph (for example, the question of the shift, with temperature, of the boundary of the energy spectrum of the polaron). From the text of the review one could see that the discussion concerned polarons, and it is unclear why in this

question, M. F. Deigen and K. B. Tolpygo began to argue on the basis of the temperature dependence of the \(F\)-absorption band and other questions connected with impurity centers, about which, incidentally, the review dealt separately.

1. The authors of the letter consider erroneous our remark that the use of the effective-mass method raises the symmetry of the Hamiltonian. However, it is known*) that, for example, when a field possessing cubic symmetry is imposed, the terms of a hydrogen-like atom split. This difference is not quantitative but qualitative, and attention was drawn to this in the review. Taking into account, further, that in the monograph transition probabilities are calculated with the wave function determined as the solution of Schrödinger’s equation written in the approximation of the effective-mass method (see Ch. II and Ch. IV, for example), we see that our remark is not removed by the authors’ objections.

As for the so-called “\(1:2:3:4\) theorem,” it loses its force not because the conditions of applicability of the effective-mass method may be violated, as M. F. Deigen and K. B. Tolpygo suppose, but because the periodic field of the lattice is simply not taken into account, since in this case it is not enough merely to replace the true mass of the electron by the effective one. One can verify this by considering equation (12.10), and not the approximate equation (12.11) or (6.4).

1. The last remark of M. F. Deigen and K. B. Tolpygo concerns the question that, in our opinion, it might perhaps have been more expedient not to bring all the experimental data together in one chapter, but to present part of them in parallel with the exposition of the theoretical results. But this, in the final analysis, is a matter of the author’s taste, and it is hardly worth saying anything more about it.

S. Tyablikov