Letters to the Editor

On the Theory of the Fine Structure of X-ray Absorption Spectra of Solids

(Several comments on M. A. Blokhin’s article “X-ray Spectra as a Method for Studying the Distribution of Electrons over States.”)

In issues 1–2 of volume XXIX of the journal Uspekhi Fizicheskikh Nauk there appeared an article by M. A. Blokhin, “X-ray Spectra as a Method for Studying the Distribution of Electrons over States” (part 3—Absorption spectra). The article, in general, constitutes a very good and quite up-to-date review both of the experimental data and of the basic theoretical concepts relating to X-ray absorption spectra.

However, it is impossible to agree with some of the statements made by the author of that article. This applies chiefly to the positive evaluation which the author gives to Kronig’s theory,^1 which claims to explain the fine structure of the X-ray absorption spectra of “solid metals and alloys.”

On p. 133, beginning his exposition of this theory, the author evaluates it as a whole as a “successful theory.” However, contrary to this evaluation, a number of theoretical^2,3 and experimental^4,9 investigations of recent years have already shown the complete inadequacy of Kronig’s theory, its illusory character. Although the author passes over in silence the greater part of these works, nevertheless, contrary to his own positive assessment of Kronig’s theory, he is forced on p. 143 to acknowledge its extreme limitation and insufficiency.

What, then, are the facts that refute Kronig’s theory?

On p. 142 the author writes: “Kronig’s theory explains experimental data well in all cases in which the theory of almost free electrons is applicable, i.e., for metals and alloys far from the main absorption edge, at a distance of more than 50–100 eV.” Let us add, incidentally, that the region of fine structure of the X-ray absorption spectrum of a solid that begins there extends at most to 300–400 eV from the main edge.

However, as has been shown in two of my papers^2,3 (see also Heitler^10), precisely in this region of the spectrum the approximations of free or almost free electrons are inapplicable. Indeed, the theoretical absorption curves (Fig. 1 in paper^2 and Fig. 2 in paper^3) have almost nothing in common with the experimental ones. This conclusion is in natural agreement with Slater’s data^11 and with other authors, which establish an extraordinarily slow convergence of the expansion of the Bloch wave function of the X-ray electron (in the given energy interval) in plane waves near atomic nuclei in the lattice of a solid.

Therefore it is natural that, contrary to Blokhin’s opinion, Kronig’s theory not only cannot explain a whole series of experimental facts, but even simply contradicts them.

Let us give several examples.

1) Kronig’s theory is powerless to explain the influence of the immediate environment of the absorbing atom on the fine structure of the X-ray absorption spectrum. This is especially clearly seen in attempts to interpret the similarity of the fine structures of absorption, for example, in cubic (Ca) and hexagonal (Ti) close-packed lattices (Koster⁴, see also ⁵).

On p. 140 Blokhin asserts that, proceeding from the similarity of short-range order in both crystals, on the basis of Kronig’s theory one should expect a similarity of the fine absorption structures. But it is impossible to agree with this, since Kronig’s theory, by its very nature, is a theory of long-range order, which, incidentally, the author himself notes on p. 143. Thus, contrary to Blokhin’s opinion, on the basis of the fact of a noticeable difference in long-range order in the two types of lattices, according to Kronig’s theory one should expect differences, and not similarities, in the fine absorption structures.

2) As shown in my work³, Kronig’s theory inevitably leads to the requirement that there exist anisotropy of the fine structure of the X-ray absorption spectrum of single crystals, i.e. to a dependence of the form of the fine structure on the orientation of the ray with respect to the axes of the single crystal. However, experimental searches for this effect for single crystals of Zn by Levitskii⁹ gave a negative result. Undoubtedly, this fact also testifies to the inadequacy of Kronig’s theory.

Blokhin’s remark (p. 142) about the insufficient resolving power of the spectrograph in Levitskii’s experiments is unfounded, since the latter was not inferior to that, for example, in the sufficiently fine work of Biman and Frolimann¹⁵, and, as follows from my calculations³, was quite sufficient for detecting the sought effect.

3) It would have been quite natural to suppose, as Kronig himself did¹², that his theory should also be valid for nonmetallic crystals. This is all the more natural because in them (for example, in ionic crystals) the forces of influence of the surrounding atoms on the electrons decrease with distance much more slowly than in metals. However, experimental investigations by Koster and Knaper⁷, Stephenson⁸, and others refute the applicability of Kronig’s theory in this field as well, to which Blokhin also points on p. 143. He sees the reason for this in the fact that in nonmetallic crystals “there are no generalized electrons, and band theory cannot be applicable.”

Such an explanation is wholly unconvincing, since it is known that modern band theory (see, for example, Seitz¹³) embraces all kinds of solids, and electron conduction bands exist also in nonmetallic bodies, not differing in principle from those for metals. In the elementary acts of absorption precisely the throwing of an electron, torn from an atom, into the conduction band occurs, as is seen from the fact of the appearance of electronic conductivity in nonmetallic crystals subjected to the action of X-rays.

The size of this note does not permit us to cite the many other facts contradicting Kronig’s theory [say, the preservation of the form of the fine structure of absorption of a polyatomic gas upon its condensation into a molecular lattice (for example GeI₄)—Drinskii and Smolukhovskii⁶, and others]. Finally, the author himself, on p. 143, writes that “Kronig’s theory in many cases cannot claim to explain the observed facts. Moreover, the determination of the energetic position of the elements of the fine structure from columnar diagrams (Kronig’s) is connected with a very arbitrary division of these columns into groups. One may even doubt the possibility of such a division without previously known

…of the results of the experiment.” These, undoubtedly correct, conclusions of the author are truly devastating.

What has been said above is quite sufficient to draw the final conclusion that Kronig’s theory is completely untenable. It can be regarded only as an already passed stage in the development of views on the nature of the X-ray absorption spectrum of a solid. It is therefore hardly expedient to devote so much space to it in a modern review article.

At first glance, however, it might seem that Kronig’s theory could have been preserved by abandoning only its weakest point—the desire to regard the electron as free or almost free (the wave function being one or two plane waves), while retaining the assumption that the fine structure of the absorption band is caused only by the distribution of the electron states over an energy scale (Jones and Mott ¹⁴, Biman and Friedman ¹⁵, etc.). But in the present case, as is easy to see, nothing in essence remains of “Kronig’s theory” as such, since the “Kronig” fine structure reduces to the “Kossel” one, which, as experiment shows, is situated on the ascending main edge (or otherwise, on the long-wavelength side) of the absorption band. Thus it is in reality.

As for the allegedly “Kronig” fine structure, i.e. the most sharply expressed fine structure situated on the already descending crest (or short-wavelength side) of the absorption band, its explanation requires a different conception. This latter must explain not only the distant (as in Kronig’s theory), but also the closer order of the form of the fine structure, i.e. it must in essence be a theory of “closer order.” Such a theory must encompass a greater number of facts than the theory of distant order, for the existence of a definite distant order presupposes the existence also of a definite closer order, but not conversely. Therefore, where experimental facts would seem to confirm the theory of distant order, they will also confirm the theory of closer order, but not conversely.

Distances of the maxima and minima of the fine structure of the K-bands from the edge of the X-ray absorption spectrum (in eV)

Such a theory was proposed by me[^16] in 1941. Although it is also set forth by the author of the article under consideration on p. 144, in view of the untenability of Kronig’s theory it seemed to me necessary to give a fuller account of its physical foundations in the present article.

In conclusion, let us note that the method used by me in constructing this theory, and which is a development of Petersen’s method,[^17] applied by him in treating molecular X-ray absorption, might have appeared unreliable because of the poor agreement of Petersen’s calculations with experiment. However, I was able to identify the presence of errors in Petersen’s work. In order to test the method, I carried out a new recalculation[^18] of the fine structure of the X-ray absorption spectrum of gaseous $\mathrm{GeCl}_4$, and also, for the first time, calculated the fine structure for gaseous $\mathrm{AsCl}_3$. The results are given in the table.

They agree no worse, and perhaps even better, with experiment than the results of the exact, but very cumbersome, Hartree–Kronig–Petersen method.[^20] Thus, this method of calculating the influence of the near environment on fine structure may be considered reliably justified. It is regrettable that this last work did not find its proper place in Blokhin’s article.

A. I. Kostarev

References

1. R. L. Kronig, Zschr. f. Physik, 70, 317, 1931; 75, 191 (1932).
2. A. I. Kostarev, ZhETF, 9, 267 (1939).
3. A. I. Kostarev, ZhETF, 16, 739 (1946).
4. D. Coster, Physica, 2, 606 (1935).
5. J. H. Minier, J. A. Bearden, C. H. Shaw, Phys. Rev., 58, 537 (1940).
6. T. Drunski a. R. Smoluchowski, Physica, 6, 929 (1939).
7. D. Coster a. G. H. Klamer, Physica, 1, 145 (1934).
8. S. Stephenson, Phys. Rev., 58, 873 (1940).
9. B. M. Levitskii, Dissertation, Gorky (1940).
10. V. Heitler, The Quantum Theory of Radiation.
11. J. C. Slater, Rev. Mod. Phys., 6, 209 (1934).
12. R. L. Kronig, Phys. Zschr., 36, 729 (1935).
13. F. Seitz, Modern Theory of Solids.
14. H. Jones a. N. F. Mott, Proc. Roy. Soc., A 162, 49 (1937).
15. W. Beeman a. H. Friedman, Phys. Rev., 56, 392 (1939).
16. A. I. Kostarev, ZhETF, 11, 60 (1941).
17. H. Petersen, Zschr. f. Physik, 98, 569 (1936).
18. A. I. Kostarev, ZhFKh, 20, 1 (1946).
19. D. Coster a. G. H. Klamer, Physica, 1, 889 (1934).
20. D. R. Hartree, R. L. Kronig a. H Petersen, Physica, 1, 895 (1934).