Fine Structure of X-ray Absorption Spectra

A. Venderovich, Tomsk

Questions connected with the fine structure of X-ray absorption spectra, which until very recently interested only a small number of specialists working in this field, have unexpectedly attracted the attention of a broad circle of physicists.

The considerable number of works on the absorption spectra of X-rays that have appeared in recent times (Kronig, Coster, Lindsay, Hanawalt, and others) are undoubtedly a reflection of the increased interest in the study of this circle of phenomena.

Dependence of X-ray Absorption Spectra on Chemical Combination

Bergengren¹ and Lind² first showed that the absorption spectrum of the X-rays of an element changes depending on whether we take this element in pure form or whether it is present in some chemical compound. In other words, the absorption spectrum is characteristic not only of the atom, but also of the medium in which this atom is found.

Since the study of X-ray absorption spectra could make it possible to establish certain energetic relations in the formation of chemical compounds, a large number of works by various authors³ were devoted to the question of changes in absorption spectra.

The main results of these works may be considered the following:

1. The shift of the edge of the absorption band, which in almost all investigated cases was displaced toward shorter wavelengths in comparison with the wavelength of the edge of the absorption band of the pure element. An exception to this rule was observed only in some sulfur compounds.

2. The magnitude of the shift of the absorption-band edge proved to depend on the valence with which the element under study enters into the given compound; moreover, the greater valence corresponds to the greater displacement of the edge.

3. The displacement of the edge differs depending on which atoms are directly bonded to the absorbing atom. Of interest is the series (established by Stelling for phosphorus compounds), in which the atoms are arranged according to their influence on the change in the position of the edge of the absorption band of the element studied:

\[ \mathrm{O} \to \mathrm{N} \to \mathrm{H} \to \mathrm{C} \to (\mathrm{Cl})? \]

The displacement is greatest when the oxygen atom is directly bonded to the atom under study.

1. The position of the edge is also affected by the physical conditions in which the absorbing atoms are found. Thus, different crystallographic modifications of one and the same element gave an edge of different wavelength.

The possible nature of the displacement of the edge of the absorption band remained unexplained. The explanation proposed by Stelling³, which consisted in the idea that the observed displacement of the edge corresponds to a direct change in the energy of the \(K\)-level under the influence of neighboring atoms, can hardly be regarded as satisfactory, since it is obvious that an analogous change should also take place for other levels, which would require excessively large energies.

for the formation of chemical compounds (the change of the \(K\)-level, Mn then, would be equal to 20 V, whereas the change in the energy of the whole atom would be at least 500 V).

More acceptable might be the point of view of Kossel\(^4\), according to which the frequency of the edge of the absorption band corresponds to the energy required to remove the electron to the first unoccupied possible orbit of the atom. In this case the influence of neighboring atoms is reduced to a considerable extent to deformations of the outer orbits, and the displacement of the edge will now correspond to the change in the energy of the outer orbits.

Fine structure in X-ray absorption spectra

The next complication in the picture of the X-ray absorption spectrum proved to be the discovery of the fine structure of the edge of the absorption band, made as a result of detailed investigations by Stenström\(^5\), Hertz\(^6\), Fricke\(^7\), and others. The essence of this discovery consisted in the fact that, in photographs of absorption spectra, on the short-wavelength side of the edge one could observe alternating dark and light narrow bands, i.e., near the edge of the absorption band the absorption coefficient for different wavelengths of the spectrum changes nonuniformly. The fine structure, discovered by various authors under various conditions, was of an entirely different character, which raised doubts as to the reality of the observed phenomenon. Thus, Koster\(^8\) found in oxygen compounds Ti, V, Cr, and Mn in the \(K\)-series and Zn, As, Fe, and J in the \(L\)-series white lines on the soft side and dark lines on the hard side of the edge of the absorption band. Koster’s work was checked by Lindh\(^9\), who succeeded in showing that the white lines observed by Koster are obtained as the result of partial reduction of oxygen compounds of the element being studied under the action of X-rays during exposure.

Fig. 1a. \(K\)-edge of the absorption band of Ca from a CaCO\(_3\) crystal.
Fig. 1b. \(K\)-edge of the absorption band of Ca from a CaF\(_2\) crystal.

Chamberlain’s attempt\(^ {10}\) likewise to explain the dark lines on the hard side of the spectrum by partial reduction of compounds under the influence of radiation can hardly be considered satisfactory, since less oxidized compounds should possess a softer edge.

In subsequent work by Koster and his collaborators\(^ {11}\), fine structure was found also for argon in the form of a kink on the absorption curve at a distance of 1.7 Å from the main edge. Explanation of the presence of fine structure in argon by any chemical considerations is, of course, of little probability.

At the same time, the absorption spectra of the \(K\)-series of potassium (potash) and chlorine (ammonium chloride) were studied by him. In the photograph, clearly expressed secondary edges were visible, situated from the main edge at distances of 25.8 V for potassium and 14.6 V for chlorine.

A number of authors who studied the fine structure of the edge of the absorption band found a much more complex structure of the edge.

Thus, Lindh and van Deik\(^ {12}\), studying the absorption spectrum of Ca in crystals CaCO\(_3\), CaSO\(_4\), and CaF\(_2\), which simultaneously served both as the absorbing filter and as the reflecting grating, found in all cases, on the hard side,

three secondary edges. In Fig. 1 are shown the photographs of absorption spectra obtained by them. The distances of the secondary edges from the principal edge ($A$) for all crystals are given in Table 1.

From these data it is seen that the fine structure extends over a considerable distance from the principal edge and that it, like the principal edge, depends on the chemical compound. Similar results were obtained by Nattol[^13], who studied the potassium absorption spectrum in the compounds KCl (sylvine), $(K,H)_2(Mg,Fe)_2(Al,Fe)_2(SiO_4)_3$ (lepidomelane), $(K,H)_2(Mg_3AlSiO_4)_3$ (phlogopite) and $KSiO_4$ (orthoclase). In Table 2 are given the distances, obtained by him, of the secondary edges from the principal edge ($A$).

TABLE 1

We see here too a complex structure, differing for different chemical compounds.

Lindh and Buryis[^14] studied the absorption spectrum of iron in various compounds. In addition to seven secondary absorption edges situated from the principal edge at a distance of 200 V, they also found a broad absorption line beginning immediately after the principal edge and extending over a distance from 3 to 7 V, depending on the compound.

A distinct and richly expressed fine structure was obtained by Lindh and Kwiet[^15], who studied the absorption spectra of elements from Ca to Zn in the metallic state. As a result of all the works cited above, it was no longer possible to doubt the reality of the existence of fine structure.

TABLE 2

But if greater difficulties were encountered in establishing the very fact of the existence of fine structure, still greater difficulties arose in attempting to explain this structure.

According to Kossel, it may be assumed that the principal edge corresponds to the energy required to remove an electron to the first possible unoccupied orbit; then, on the short-wave side of the edge, the appearance of secondary bands corresponding to the energy of removal of an electron to the next possible orbits is possible.

It is not difficult, however, to see that all the structure which could be obtained in this way should be situated at distances from the principal edge not exceeding the first ionization potential of the element with atomic number one greater than that of the given element. Indeed, the difference between the works of removing an electron beyond the atom and to the first free possible orbit may be regarded as the energy of removal of an electron from this orbit to infinity in the absence of a neighboring internal electron, i.e. under conditions of an increase of the effective nuclear charge of the atom by one unit.

The ionization potential of most elements, however, does not exceed 20 V. Thus this theory can explain only an insignificant part of the entire observed fine structure. In many cases no such structure could be detected.

Rey and Maranti[^16] believe that such Kossel structure in solids cannot in general be obtained. They proceed from the fact that in metals the outer electrons may be regarded as free; consequently, they may possess any energy, and the removal of inner electrons into these

for the formation of chemical compounds (a change of the \(K\)-level; Mn would then be equal to 20 V, while the change in the energy of the whole atom is at least 500 V).

A more acceptable point of view might be that of Kossel\(^4\), according to which the frequency of the edge of the absorption band corresponds to the energy required to remove an electron to the first unoccupied possible orbit of the atom. In this action the states of the atom relate to a considerable extent to deformations of the outer orbits, and the displacement of the edge will now correspond to the change in the energy of the outer orbits.

Fine Structure in the Absorption Spectra of X-rays

The next complication in the picture of the X-ray absorption spectrum proved to be the discovery of the fine structure of the edge of the absorption band, carried out as a result of detailed investigations by Stenström\(^5\), Hertz\(^6\), Fricke\(^7\), and others. The essence of this discovery was that, in photographs of absorption spectra, on the short-wavelength side of the edge one could observe alternating dark and light narrow bands, i.e. near the edge of the absorption band there is a sequence of bands, or, for different spectra, in some cases uncountable bands. The fine structure, discovered by various authors under various conditions, at first had a completely different character, which raised doubts as to the reality of the phenomenon discovered. Thus, Koster\(^8\) found, for oxygen compounds Ti, V, Cr and Mn in the \(K\)-series and Zn, As, Fe and J in the \(L\)-series, white lines on the soft side and dark lines on the hard side of the edge of the absorption band. Koster’s work was checked by Lindh\(^9\), who succeeded in showing that the white lines observed by Koster are obtained as a result of partial reduction of oxygen compounds by the action of X-rays during exposure. Chamberlain’s attempt\(^ {10}\) to explain also the dark lines on the hard side of the spectrum by partial reduction of compounds under the influence of radiation can hardly be regarded as satisfactory, since less oxidized compounds should have softer edges.

Fig. 1a. \(K\)-edge of the absorption band of Ca from a CaCO\(_3\) crystal.

Fig. 1b. \(K\)-edge of the absorption band of Ca from a CaF\(_2\) crystal.

In Koster’s subsequent works and those of his collaborators\(^ {11}\), a fine structure was also discovered in argon, in the form of a kink in the absorption curve at a distance of \(1.7\) Å from the main edge. An explanation of the presence of fine structure in argon by any chemical considerations is, of course, offered only with little probability.

At the same time, they also studied the absorption spectra of the \(K\)-series of potassium (iodate) and chlorine (potassium chlorate). In the photograph, clearly expressed secondary edges were visible, located from the main edge at distances of 25.8 V for potassium and 14.6 V for chlorine.

A number of authors who investigated the fine structure of absorption-band edges discovered a much more complex structure of the edge.

Thus, Lindsay and Van-Den\(^ {12}\), studying the absorption spectrum of Ca in crystals of CaCO\(_3\), CaSO\(_4\), and CaF\(_2\), which simultaneously served both as the absorbing filter and as the reflecting grating, found in all cases, on the hard side,

FINE STRUCTURE OF ABSORPTION SPECTRA

places cannot create any structure. In most solids, according to their calculations, the distance between the shells of atoms in the crystal lattice is such that there is no room left for further possible orbits.

Without dwelling in detail on these considerations, we shall only note that, in any case, in gases a fine structure has been observed, and calculations show the possibility of a satisfactory explanation of it from the point of view of the theory of Kossel’s representations. As for the solid state, here too Rydberg’s considerations are hardly always valid. In many cases broad white lines have been observed here at distances of several volts from the main edge. Exact calculations are impossible here, since the differences in the energies of the corresponding transitions to the various possible orbits are so small in comparison with the energy of the transition itself that it is not possible to resolve these broad white lines into separate absorption lines corresponding to individual transitions.

In this respect the work of Neufeld^17 is of interest; he studied the absorption spectra of the oxygen and nitrogen \(K\)-series (in cellulose) and the chromium \(L\)-series. Here he succeeded in obtaining completely separate and well-pronounced absorption lines, exactly coinciding with emission lines.

Thus, part of the structure of the absorption spectrum can be understood on the basis of Kossel’s theory, but it is impossible to regard it as a general explanation of the fine structure, if only for the simple reason that the latter extends over several hundred volts. It was necessary to seek other paths that might explain the formation of complex X-ray absorption spectra.

Such a possibility is provided by the theory of multiple ionization, according to which a quantum of light can be absorbed simultaneously by two or more electrons of an atom. On the basis of these ideas Lindsay and Knipp^15 calculated absorption spectra for a number of elements they investigated, and obtained good agreement with experimental data. However, the unsystematic character and extreme diversity in the behavior of different elements that result in this case make all the authors’ results doubtful.

Corkum^18 repeated these calculations. He considered incorrect the earlier notion that the electron knocked out of the atom enters a potential equal to zero, and determined it from Richardson’s work-function effect. His calculations likewise led to good agreement with the experimental data of Lindsay and Knipp, and in this case the behavior of different elements already shows a certain systematic character.

However, against the theory of multiple ionization as a whole one may put forward a number of fundamental objections; in particular, it has proved completely unacceptable for explaining new facts established by Hanawalt^19. He studied the absorption spectra of a number of substances in the solid, liquid, and gaseous states. It was found that polyatomic vapors, although they give secondary absorption, have an edge structure less complex than that of these substances in the solid state (\(\mathrm{As}_4\), \(\mathrm{AsCl}_3\), \(\mathrm{As}_2\mathrm{O}_3\), \(\mathrm{SeO}_2\)). Similarly, liquid substances give a less complex structure (\(\mathrm{NaBrO}_3\)).

Monatomic vapors, however, do not show secondary absorption at all (vapors of Hg, Zn, Xe, and Kr). The spectra of some of them (Zn and Xe) show a fine structure that is entirely explicable from the point of view of Kossel’s representations.

On the basis of his experiments Hanawalt comes to the conclusion that secondary absorption is conditioned by the structure of the crystal lattice or of individual molecules.

Namely, he proposed that secondary edges are connected with the existence of structural electrons, first introduced into consideration by Richardson^20 as a result of studies of the emission of X-rays and secondary electron emission when various substances were bombarded by electrons of variable velocity. These experiments showed that on the curve representing the dependence of electron emission (or of the energy of X-radiation) on

of the primary electrons, besides jumps corresponding to the usual Bohr levels, there are also very numerous breaks in the region from 30 to 500 V. These breaks occur at the same places both in the curve of electronic radiation and in that of X-ray radiation and, consequently, are not accidental in character. The position of the breaks on the curves proved to depend on the state of the material of which the anticathode is made. Curves obtained with an anticathode consisting of a single crystal show breaks which are then repeated also on the curves from an anticathode made of a fine-crystalline substance. But in the latter case additional breaks are observed. Analysis of the velocities of the secondary electronic radiation shows that it consists of three groups of electrons: 1) primary electrons reflected from the anticathode without loss of velocity, 2) electrons which have lost some part of their energy as the result of some inelastic collision with the anticathode, and 3) electrons of very low velocity, which constitute the “true” secondary electrons.

The electrons of the second and third groups are closely connected with one another; they appear together and disappear together when the energy of the primary electrons becomes too small. From this one may conclude that secondary electrons appear as the result of an inelastic collision of primary electrons with the electrons of the anticathode.

The three groups of electrons on the velocity-distribution curve correspond to three maxima. From the fact that, between the maxima corresponding to electrons of the first and second groups, there is a clearly expressed minimum, Richardson concludes that the electrons falling on the anticathode undergo inelastic collisions not with free electrons, since this could give only a smooth and continuous change in the initial velocity of the electron.

Richardson considers it possible to explain the phenomenon he observed and its associated pattern of the existence, in the crystalline lattice of electrons, of “standing” electrons, which are not free, i.e., which cannot have arbitrary kinetic energy, but are also not bound to any definite atoms. He proposes to call these electrons “structural,” relating them to the crystalline lattice.

In colliding with them, the primary electrons lose that part of their energy which is necessary to transfer a structural electron to a higher energy state: thus Richardson assumes the existence of quantum states also for electrons moving outside atoms in the crystalline lattice. In the reverse transition to the normal state, the “structural” electrons emit soft X-rays, which partly leave the anticathode and are partly absorbed in it, giving rise to the secondary electrons of the third group.

Such, in brief outline, is the explanation of the extremely interesting phenomena observed by Richardson. Between these phenomena and the secondary structure in the absorption spectra of X-rays there is a clear parallelism.

The fact that the breaks in the curves of secondary electron emission are encountered in the range 30–500 V corresponds exactly to the width of the energy interval in which secondary absorption is observed. On this basis secondary structure can be interpreted only as the result of the ejection by atoms of inner electrons to various energy levels of “structural” electrons. From this point of view the important fact of the absence of secondary structure in monatomic vapors becomes comprehensible, as do other features of the absorption spectra of X-rays.

The shortcomings of such a treatment, alongside the indicated positive aspects, are the uncertainty and vagueness of the very notion of a “structural” electron, which leaves a certain dissatisfaction in the endeavor to obtain a complete coverage of all the observed phenomena.

In this respect, the new theory of Kronig has unquestionable advantages, giving a simple and rational explanation of the observed phenomena on the basis of the wave mechanics of the motion of an electron inside a solid body.

Kronig’s Theory

Kronig’s theory is based on ideas developed by Bloch^22 for the interpretation of the electrical conductivity, thermal conductivity, Hall effect, and a number of other properties of metals.

Considering these properties of metals, Bloch came to the conclusion that the action of the individual parts of the crystal lattice on weakly bound electrons can be replaced by the action of a potential, periodically varying with the same period as that of the crystal lattice.

The motion of an electron in this case will be determined by Schrödinger’s equation

\[ \Delta\psi+k^2(W-V)\psi=0, \]

where \(V\) is a function varying as indicated above.

A number of works^23 were devoted to the solution of this problem.

In some simple cases it proved possible to carry this problem through to the end; moreover, it turned out that the motion of an electron in a crystal lattice, when its momentum is parallel to a crystallographic axis, cannot occur with arbitrary values of the energy.

Fig. 2a.          Fig. 2b.

Possible values of the energy are contained in certain intervals—allowed zones—which alternate with forbidden zones.

In Figs. 2a and 2b the possible energy positions of an electron are shown schematically in the case of a free atom and in the case of an atom of a solid body.

Following Kronig and Penney, we shall consider the motion of an electron in a potential field consisting of rectangular barriers separated from one another by a distance \(a\) (Fig. 3).

In this case the wave equation will have the form:

\[ \frac{d^2\psi}{dx^2}+k^2\{W-V(x)\}\psi=0, \tag{1} \]

where

\[ k^2=\frac{8\pi^2 m}{h^2}. \]

The solution of this equation, as Bloch showed, is given in the form:

\[ \psi(x)=u(x)e^{\alpha x};\quad \alpha=\frac{2\pi l}{L}, \tag{2} \]

where

\[ L=G(a+b), \]

and \(G\) is a large number.

A. VENDEROVICH

In accordance with the two different values of the potential inside the lattice, we shall have two different equations for determining the functions \(u\). These equations are obtained by the simultaneous solution of equalities (1) and (2).

For the region

\[ b \le x \le 0 \]

\[ \frac{d^{2}u}{dx^{2}}+2i\alpha\frac{du}{dx}-(\alpha^{2}+\gamma^{2})u=0, \]

and

\[ u=Ae^{(-i\alpha+\gamma)x}+Be^{(-i\alpha-\gamma)x}; \tag{3} \]

Fig. 3.

for the region

\[ 0 \le x \le a \]

\[ \frac{d^{2}u}{dx^{2}}+2i\alpha\frac{du}{dx}-(\alpha^{2}-\beta^{2})u=0 \]

and

\[ u=Ce^{i(-\alpha+\beta)x}+De^{i(-\alpha-\beta)x}; \tag{4} \]

here

\[ \beta=k\sqrt{W},\qquad \gamma=k\sqrt{V_{0}-W}. \tag{5} \]

The conditions of single-valuedness and periodicity require that the function \(U\), determined by equality (8), and its first derivative at \(x=-b\) be equal to the function (4) and its first derivative at \(x=a\).

Moreover, the values of both functions \(u\) and of their first derivatives must be equal at \(x=0\).

From these requirements there follow four equations relating the constants

\[ A+B=C+D,\qquad (-i\alpha+\gamma)A+(-i\alpha-\gamma)B =i(-\alpha+\beta)C+i(-\alpha-\beta)D. \]

\[ Ae^{(-i\alpha-\gamma)b}+Be^{(i\alpha+\gamma)b} =Ce^{i(-\alpha+\beta)a}+De^{i(-\alpha-\beta)a} \]

\[ (-i\alpha+\gamma)Ae^{(i\alpha-\gamma)b} +(-i\alpha-\gamma)Be^{(i\alpha+\gamma)b} =i(-\alpha+\beta)Ce^{i(-\alpha+\beta)a} +i(-\alpha-\beta)De^{(-\alpha-\beta)a}, \]

which can be compatible under the condition

\[ \frac{\gamma^{2}-\beta^{2}}{2\beta\gamma}\sin h\gamma b\,\sin\beta a +\cos h\gamma b\,\cos\beta a =\cos\alpha(a+b). \]

To simplify the calculation, without diminishing the generality of the question under consideration, we pass to the limit

\[ b\to 0 \quad \text{and} \quad V_{0}\to\infty; \]

then, denoting

\[ \lim \frac{\gamma^{2}ab}{2}=P, \]

we obtain:

\[ b\to 0 \quad V_{0}\to\infty \]

\[ \frac{P\sin\beta a}{\beta a}+\cos\beta a=\cos\alpha a. \tag{6} \]

This equation is fundamental for understanding the motion of an electron in a crystal lattice. It is immediately evident from it that the electron cannot possess arbitrary energy, since the expression \(\beta a\), which determines the energy of the electron (5), must have quite definite values.

FINE STRUCTURE OF ABSORPTION SPECTRA

Thus, \(\beta a\) cannot take such values for which

\[ \frac{P\sin \beta a}{\beta a}+\cos \beta a \underset{< -1}{> +1}. \]

To illustrate the results obtained, let us construct the curve of the dependence of this expression on \(\beta a\) for some definite value of \(P\), for example \(\frac{3}{2}\pi\). The possible values of \(\beta a\) satisfying equation (6) lie in certain sections of the \(\beta a\) axis, indicated in Fig. 4 by heavy dashes.

Hence, by virtue of equation (5), we conclude that the energy values which an electron moving through the lattice can possess are arranged in a spectrum consisting of separate intervals of continuous bands.

[Figure: graph of \(\frac{P}{\beta a}\sin \beta a+\cos \beta a\) as a function of \(\beta a\), with the levels \(+1\) and \(-1\) marked and intervals indicated along the axis.]

Fig. 4.

Let us see how this spectrum changes depending on the magnitude of \(P\). At small values of \(P\), i.e. for narrow and low barriers, the whole curve in Fig. 4 may lie within the region bounded by the lines \(\pm 1\). In this case an electron inside the lattice may possess all possible energy values.

If we now pass to the other case, \(P=\infty\), then equation (6) will be satisfied under the condition \(\sin \beta a=0\), i.e. \(\beta a=n\pi\) \((n=2\ldots)\), whence, by virtue of (5) and (1), we obtain:

\[ W=\frac{n^{2}h^{2}}{8ma^{2}}, \tag{7} \]

i.e. discrete energy values.

Thus, between the removal of an electron by a light quantum from isolated atoms (monatomic vapors) and the removal of electrons from atoms constituting a crystalline lattice, there exists a sharp difference.

In the first case the electron may be emitted from the atom with any kinetic energy, and therefore the structure of X-ray absorption spectra must be simple. In the case of removing electrons from atoms inside a crystalline lattice, however, we have in essence a transition of electrons from intra-atomic levels to some higher level—the “allowed zone,” brought about by the very existence of the crystalline lattice. The reflection of these levels is the fine structure of the absorption spectra of X-rays, for it is quite clear that not every quantum will be able to transfer an electron, say, from the \(K\) level into the allowed zone.

It would seem that from this point of view we should have expected, in the course of the partial absorption coefficient of the group \(K\) (the part of the absorption coefficient due to absorption of radiation only by the electrons of group \(K\)), values equal to zero for certain wavelengths. In reality, however, this cannot occur, since the width and position of the allowed zone depend on the period of the lattice and, consequently, will be different along different crystallographic directions; as a result we shall observe the total effect produced by the superposition of a number of fine structures upon one another, which quite naturally leads to a blurring of the observed picture.

The studies carried out recently show that Kronig’s theory provides not only a general explanation of the fine-structure pattern of x-ray absorption spectra, but also indicates a number of details that are quite accessible to verification.

Thus, from Fig. 4 it is directly evident that, with increasing \(\beta a\), the ratio of the width of the forbidden zones to the width of the allowed zones decreases. This agrees exactly with the results of Hanawalt\({}^{24}\) and others, according to which, at the very edge of an absorption band, the distance between secondary edges is smaller than farther from the edge.

Lindeh\({}^{25}\) investigated the absorption spectra of the \(K\)-series of calcium in crystals of KCl, KBr, KI, and found that, for all these three substances, the absorption spectra are analogous, but that the distances between the secondary edges are inversely proportional to the square of the lattice constant of these crystals, as is also required by equation (7).

In exactly the same way, Coster and Feldkamp\({}^{26}\), on the basis of their measurements, came to the conclusion that Cu, Pt, and Au, all of which crystallize in a face-centered cubic lattice, give analogous absorption spectra, whereas Fe (body-centered cubic) and Zn (hexagonal system) give different absorption spectra. As for the spectra of Cu, Pt, and Au, here too the distances between the secondary edges proved to be inversely proportional to the square of the lattice constant.

Coster and Feldkamp also carried out other very interesting experiments.

The absorption spectra of the elements Cu and Au in the pure state were compared with their spectra in a solution of \(50\%\) Cu and \(50\%\) Au. It turned out that the distance between the secondary edges in the case of the solution was exactly the same for both elements, despite the difference in the spectra of the pure elements. This phenomenon fits quite naturally into Kronig’s theoretical scheme, confirming that in the absorption process we are in fact dealing with the transition of an electron from the inner levels of the atom to levels determined by the crystal lattice and, consequently, identical for all the atoms forming the given lattice.

The theory also explains well the temperature dependence of secondary absorption established by Hanawalt\({}^{27}\) for iron. He obtained absorption spectra of the iron \(K\)-series at various temperatures in the range from 20 to \(850^\circ\) C. It turned out that, while the position of the main edge did not change, the secondary edges shifted, by about \(1.8\%\).

If one takes the average coefficient of thermal expansion of iron in the range \(20\text{--}850^\circ\) C and, calculating the increase in the lattice constant, substitutes its new value into equation (7), then for the relative decrease in the distance between the secondary edges one obtains the number 1.6, which is in good agreement with experiment.

The following phenomenon, observed by Hanawalt when heating iron, also seems very interesting.

With increasing temperature, a gradual blurring of the fine structure was observed; it became less and less sharply expressed. On cooling, the structure again assumed a distinctly sharp form. Heating especially strongly smooths out the secondary edges located far from the main edge; the structure situated close to the latter disappears at temperatures close to the melting temperature of iron.

Kronig associated this phenomenon with a distortion of periodicity as a result—

Fine Structure of Absorption Spectra

of the increasing thermal motion of the atoms making up the lattice. This point of view is also confirmed by the experiments of Coster and Veldkamp[^27], who measured the oscillations of the absorption coefficient in the absorption spectra of the \(K\)-series of Cu and Zn. It turned out that whereas the largest oscillations for Cu reached 13%, for Zn they were only 7%. This can be explained by the difference in the melting temperatures of Cu and Zn (1800 and 420° C), as a result of which the latter has significantly reduced thermal motion.

The material presented above, far from complete, shows that Kronig’s theory gives a quite exhaustive explanation of all the observed phenomena in the absorption spectra of X-rays. A number of phenomena predicted by it were confirmed by subsequent experiments and confirmed brilliantly. But the significance of Kronig’s theory is not exhausted by the fact that it explains the fine structure of the edge and predicts new phenomena connected with this structure; the significance of the theory is much greater: it completely changes the view of what absorption spectra give us.

Until now we have assumed that, by studying absorption spectra, we determine the theoretical levels of the atom. It turned out that, using absorption spectra, we can determine not only the energy levels of the atom, but also the energy levels of the solid body, thereby providing a rational basis for the theory of the solid body.

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23. Strutt, Ann. d. Phys. 86, 319, 1928; Morse, Phys. Rev. 35, 1310, 1930; Kronig and Penney, Proc. Roy. Soc. Al. 30, 499, 1931.
24. Hanawalt, Phys. Rev. 37, 715, 1931.
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26. Coster and Veldkamp, Zs. Physik 74, 191, 1932.
27. Hanawalt, Zs. Physik 70, 293, 1921.
28. Coster and Veldkamp, Zs. Physik 70, 306, 1931.