X-RAY SPECTRA AS A METHOD FOR STUDYING THE DISTRIBUTION OF ELECTRONS AMONG STATES*

M. A. Blokhin

III. ABSORPTION SPECTRA

1. General information

When X-rays pass through a substance, the rays are weakened owing to the scattering of quanta in collisions with electrons (apparent absorption) and owing to the photoelectric effect—the ejection of electrons from atoms from their inner levels (photoelectric, or true, absorption). At wavelengths greater than one angstrom, scattering plays a negligible role, and the attenuation coefficient $\mu$ practically coincides with the coefficient of true absorption $\tau$, determined from measurements of the intensity of X-rays before $(I_0)$ and after $(I)$ passage through a substance of thickness $d$:

\[ I = I_0 e^{-\mu d} \simeq I_0 e^{-\tau d}. \]

In order for true absorption to occur, the energy of the quantum must be equal to the change in the energy of the atom as a result of the photoelectric effect. The electrons ejected by the quantum from inner levels may pass to upper levels of the atom or of the solid-state lattice that are free of electrons. The magnitude of the coefficient of true absorption will depend on the probability of these transitions.

As Ziegban established, the probability of such absorption transitions of electrons from below upward is expressed by the same selection rules that had earlier been established for emission line spectra and were considered at the beginning of Section II of the present article.

Thus, absorption spectra depend on precisely where the electron will be expelled from the atom. The coefficient $\tau$ depends both on the transition probability $p(E)$, i.e., on the quantum numbers of the initial and final levels of the transitions, and on the density $n(E)$ of free allowed states outside the atom, for example in the conduction band of metals:

\[ \tau(E) = p(E) \cdot n(E), \tag{1} \]

* Conclusion. See Uspekhi Fizicheskikh Nauk, Vol. XXVIII, No. 4.

where \(E\) is the change in the energy of the atom as a result of the transition, coinciding with the magnitude of the energy of the absorbed quantum \(h\nu\). By measuring the dependence of \(\tau\) on \(\nu\) in the corresponding region of the spectrum, one can judge the distribution of the free allowed states of the electrons outside the atom.

In order to understand all the phenomena of the fine structure of X-ray absorption spectra, let us first consider absorption by free atoms.

2. Monatomic gases

The clearest interpretation of the absorption spectrum of free atoms was given by Parratt,\(^1\) who studied the \(K\)-edge absorption of argon (Fig. 1). Parratt’s experimental results were subsequently confirmed by Lindh and Nilsson.\(^2\) This spectrum has sharply expressed “white” resonance lines, representing absorption as a result of the transfer of an electron from the \(1s\) level to one of the free \(np\) levels, where \(n\) is the principal quantum number. In argon,

Fig. 1. \(K\)-absorption spectrum of argon.

the last of the occupied levels—\(3p\). Thus, according to the selection rules, absorption gives resonance lines in the transitions \(1s — 4p\), \(1s — 5p\), \(1s — 6p,\ldots,\ 1s — \infty\), i.e. a series of increasingly converging lines passing beyond this series into continuous absorption. The width of these lines is determined by the width of the K-level of argon, since the width of the optical levels may be neglected. The longest-wavelength line is the \(1s — 4p\) transition. Its experimental width is 0.58 eV. Parratt adopted this width also for all the remaining lines, which, owing to their width and their closeness to one another, cease to be resolved as \(n\) increases. Parratt decomposed the absorption curve into its constituent lines, using the position of the \(1s — 4p\) line in the spectrum and the distances between the levels \(4p — 5p\), \(5p — 6p\), \(6p — 7p,\ldots\), and also the distance \(4p — \infty\), according to the data of the optical spectra of the element \(19K\), which follows argon, since the effective charge of the atomic core in ionized argon increases by one. The relative intensity of the individual absorption lines was chosen so that the superposition of all the lines gave the experimental curve. The relative intensities of the first four lines (transitions to \(4p\), \(5p\), \(6p\), \(7p\)) turned out to be: \(100:34:18:8.5\). These quantities give the relative probability of the corresponding transitions. In Fig. 1 the abscissa gives the energies in eV, measured from zero, corresponding to the beginning of continuous absorption.

The intensity of selective absorption lines was calculated by Weinstein and Harbut\(^{51}\) for atoms of the noble gases and for certain ions in gas molecules, if these ions have the structure of a noble-gas atom. The formula obtained by the authors,

\[ I \simeq 340 \cdot \frac{\eta^5}{Z^7 \cdot n^3} \]

gives the dependence of the intensity \(I\) of selective absorption lines on the atomic number \(Z\), the principal quantum number \(n\) of the outer level of the absorption transition, and the effective nuclear charge \(\eta\) of the same level in the atom with an inner electron removed, i.e. after the absorption process. The relative intensities of the first four absorption lines for argon, calculated by this formula, are: \(100:51:30:19\), which more or less agrees with experiment.

An important consequence of this formula is the sharp dependence of \(I\) on \(\eta\). As the authors indicate, in heteropolar molecules for an anion \(\eta\) is close to zero, while for a cation \(\eta > 1\). Therefore, for example, in the K-spectrum of absorption of \(\mathrm{Cl}^{-}\) in the HCl molecule, \(I = 0\), and selective lines should not appear, which corresponds to the experimental data. This effect is especially interesting in view of the fact that the \(\mathrm{Cl}^{-}\) ion has the same structure as the Ar atom.

The resonance absorption described here, associated with transitions of electrons to the optical resonance levels of atoms, was

for the first time examined by Kossel and was given the name “Kossel absorption structure.” This structure extends over only a few eV in the spectrum.

Prins[^50] investigated the \(L_{\mathrm{II,III}}\) absorption edge of argon (Fig. 2). This spectrum arises, according to the selection rules, in transitions of electrons from the \(2p\) level to the levels \(ns\) \((n \geq 4)\) and \(nd\) \((n \geq 3)\). Comparison with the optical levels of 19 K makes it possible here also to give an interpretation of the individual white lines of the argon absorption spectrum; however, the resolving power in this work was considerably lower than in Parratt’s work.

Shaw[^3] investigated the absorption spectrum of the \(K\)-edge of krypton; however, owing to the considerable width of the \(K\)-level of 36 Kr, the fine structure is not resolved, is smeared out over the spectrum, and, as a result, the white lines of resonance absorption do not appear.

Fig. 2. The \(L_{\mathrm{II,III}}\) edge of argon.

The same applies also to the \(K\)-absorption edge of bromine, investigated by Cioffari[^4] from the spectrum of HBr vapor. In this molecule the hydrogen ion is deprived of electrons, and therefore all absorption by the molecule is limited to absorption by the bromine ion. The width of the \(K\)-level of bromine is 7.2 eV, owing to which the Kossel structure is not observed.

3. Diatomic Gases

The problem of absorption by an atom that is part of a diatomic molecule was considered by Kronig[^5]. Usually the absorption coefficient is considered as a function of the wavelength \(\lambda\) or of the frequency \(\nu\), for example \(\mu(\nu)\). If, in the process of absorption of a quantum \(h\nu\), an electron is torn from a free atom, the binding energy of which in the atom is \(E_0 < h\nu\), then the electron acquires the kinetic energy \(w\):

\[ h\nu = E_0 + w. \]

The absorption coefficient for an isolated atom can be represented as a function of \(w\):

\[ \mu(\nu) = k(w). \]

If an atom \(A\) is part of a molecule \(AB\), then the ejected electron moves in the field of both atoms. The absorption coefficient in this

in this case we shall denote it by \(K(w)\). The problem reduces to determining the ratio:

\[ \frac{K(w)}{k(w)}=\chi. \]

At equilibrium, in one second as many atoms recombine as are ionized as a result of the true absorption of X-rays. The probability of absorption by an atom of a quantum \(h\nu\), with the ejection of a photoelectron with kinetic energy \(w\), is equal to the probability of return to the same level of an electron with kinetic energy \(w\), with emission of a quantum \(h\nu\). Taking the equality of these probabilities into account, Kronig solves the inverse problem, determining the probability of the second of the processes mentioned.

The quantity \(k(w)\) is proportional to the probability of transition of an atom from state \(j\) (an electron in the \(K\)-shell) to state \(j'\) (the electron is ejected from the atom). This probability is equal to

\[ |x(j,j')|^2+|y(j,j')|^2+|z(j,j')|^2, \]

where

\[ x(j,j')=\int \Psi_j^* x\Psi_{j'}\,dx\,dy\,dz. \]

and so on. If in an isolated atom the probability of transition of the electronic system from the state \(j\) to a higher energy state \(j'\) is equal to \((x_1^2+y_1^2+z_1^2)\), while when this atom is part of a diatomic molecule the probability of the same transition is equal to \((x_2^2+y_2^2+z_2^2)\), then:

\[ \chi=\frac{x_2^2+y_2^2+z_2^2}{x_1^2+y_1^2+z_1^2}. \tag{2} \]

The function \(\Psi_j=\Psi_0(x,y,z)\) represents the wave function of the \(K\)-electron, independent of the entry of atom \(A\) into the molecule. In the case of an isolated atom, if the electron moves along the \(x\)-axis, then for \(x=\pm\infty\), the function \(\Psi_{j'}\) represents a plane electron wave

\[ \Psi_{j'}=Ne^{2\pi iax}, \tag{3} \]

where \(a=\sqrt{2\mu w}/h\) is the wave number, \(\mu\) is the mass of the electron, and \(N\) is the normalizing factor. Near the atom \(A\), this plane wave is deformed in some way, being transformed into a certain function \(\Psi_1(x,y,z)\). Then:

\[ x_1=\int \Psi_0^*(x,y,z)x\Psi_1(x,y,z)\,dx\,dy\,dz, \]

\[ y_1=z_1=0. \]

In the presence of atom \(B\), when the plane wave (3) reaches this atom, it will produce an elastically scattered spherical wave with decreasing amplitude \(c(\chi)/r\) (where \(\chi\) is the scattering angle), with the quantities \(a\) and \(w\) conserved.

When the plane wave reaches \(A\) (Fig. 3), the spherical wave will have traveled, with the same velocity, a path \(\rho \cos \vartheta\), where \(\rho\) is the distance between the atoms, \(\vartheta\) is the angle between the direction \(BA\) (which we take as the new \(x'\)-axis) and the \(x\)-axis. When the plane wave beyond atom \(A\) has traveled a further distance \(x\), the spherical wave will reach atom \(A\) and beyond it will travel a further segment \(x'\), i.e.:

\[ \rho-\rho\cos\vartheta+x'=\rho(1-\cos\vartheta)+x'. \]

If the dimensions of the molecule \(\rho\) are considerably larger than the electron wavelength \(\lambda\), then at a distance \(\rho\) from atom \(B\) the spherical wave may be replaced by a plane wave:

\[ N\frac{c(\vartheta)}{\rho} e^{2\pi i\alpha[\rho(1-\cos\vartheta)+x']} = Nq e^{2\pi i\alpha x'}, \]

where

\[ q=\frac{c(\vartheta)}{\rho} e^{2\pi i\alpha\rho(1-\cos\vartheta)}. \]

Fig. 3. Scheme of absorption by a diatomic molecule.

This plane wave undergoes near atom \(A\) the same distortion as the plane wave (3) considered above, and may be represented by the same function \(\Psi_1(x')\), only multiplied by \(q\). Thus, two identically distorted waves act simultaneously on atom \(A\):

\[ \Psi_1(x)+q\cdot \Psi_1(x')=\Psi_2(x). \]

It is now possible to determine the matrix elements of the coordinates of the electron scattered by the molecule:

\[ x_2=\int \Psi_0^*(x,y,z)\cdot x\cdot \Psi_2(x,y,z)\,dx\,dy\,dz. \]

Transforming \(x'\) to \(x\), we obtain:

\[ x_2=x_1(1+q\cdot \cos\vartheta);\quad x_2=q\cdot x_1\cdot \sin\vartheta;\quad z_2=0. \]

Hence, from (2), we find:

\[ \chi=1+(q+q^*)\cos\vartheta+|q|^2. \]

Averaging over \(\vartheta\), we obtain:

\[ \chi=1+\frac{1}{2}\int_0^\pi \left[(q+q^*)\cdot\cos\vartheta+|q|^2\right]\sin\vartheta\,d\vartheta. \]

Petersen\(^6\) expanded the wave scattered by atom \(B\) in angular momenta \(l\) of the electrons relative to this atom and took into account the change of phase upon scattering by \(\delta_l\). As a result, he obtained an expression for \(\chi(w)\) in terms of the quantities \(\vartheta\), \(\rho\), and \(w\). The change of phase \(\delta_l\) is calculated from the ...

by the power of the Hartree potential field. Petersen applied the formula obtained to the calculation of absorption by a chlorine molecule and found the distance of the first absorption minimum from the onset of absorption to be \(6.5\) eV. Lind’s experimental results give \(7.5\) eV.

Subsequently, several other diatomic molecules were investigated. Schneider and Shaw\(^{7}\) studied the absorption of the \(Br_2\) molecule, taking as the atomic absorption of bromine the absorption spectrum of the HBr molecule. The course of the experimentally found dependence is very similar to the theoretical curve constructed by the authors, but differs from the latter by the absence of a second maximum and by a considerably greater amplitude of the first maximum and minimum. The character of the absorption spectrum near the \(K\)-edge of bromine is presented in Fig. 4 after the work of Chiofari\(^{4}\). In the latter work iodine vapors were also investigated, which did not give any noticeable structure near the \(K\)-edge of absorption, owing to the considerable width of the iodine \(K\)-level.

Fig. 4. \(K\)-edge of bromine vapors.

Bromine iodide vapors gave a well-pronounced structure of the bromine spectrum; however, decomposition of this compound into \(J_2\) and \(Br_2\) was found.

Prince\(^{8}\) investigated the \(K\)-edge of nitrogen absorption; he found a bright resonance white line, removed from the end of the edge by approximately \(10\) eV. Prince gives the following interpretation of this spectrum. The end of the edge corresponds to the ionization of the \(N_2\) molecule, i.e., to its transition into a molecule with one normal nitrogen atom and a second atom ionized from the \(K\)-level. The latter atom is equivalent to the oxygen atom following nitrogen, i.e., the whole molecule after ionization is similar to an NO molecule, in which, however, one of the four \(2p\) electrons of oxygen is absent, since nitrogen has only three of them. Consequently, the end of the edge corresponds to the transition from the ground state of the \(N_2({}^{1}S)\) molecule to the ionized NO molecule.

The white line is interpreted as the transition of an electron from the \(K\)-level of the nitrogen atom to the \(2p\) level, as a result of which the \(N_2\) molecule passes from its ground state \(({}^{1}S)\) to a normal NO molecule in its ground state \(({}^{2}P)\). The difference in energy of the two indicated transitions gives the ionization energy of the NO molecule from the \(2p\) level, i.e., \(9.4\) eV, which is quite close to the observed separation.

We see that the absorption of diatomic molecules gives several well-pronounced fluctuations extending over \(10\)—\(15\) eV. The discrepancy between theoretical calculations and experimental

the results may be explained, first, by multiple mutual scatterings of electron waves between the atoms of the molecule and, second, by the smearing of the distance $\rho$ between atoms due to thermal vibrations of the molecule.

4. Polyatomic gases

The theory of symmetric polyatomic molecules was given by Hartree, Kronig, and Petersen$^9$. In this work the K-edge absorption of

Fig. 5. K-edge absorption of germanium in GeCl$_4$ vapors. Theoretical curve.

Ge in the molecule GeCl$_4$, which is a tetrahedron with a Ge atom at the center and Cl atoms at the vertices, is considered. The Ge—Cl distance is equal to 2.10 Å, according to electron-diffraction analysis.

If there are several scattering atoms in the molecule, the fine structure of absorption due to each of them is superposed linearly:

\[ \chi - 1 = \sum_s (\chi_s - 1). \]

As a result of the calculation carried out, the dependence of \((\chi - 1)\) on the quantities \(f_l\) was obtained, where

\[ f_l = \sqrt{\frac{\pi}{2\tau}} \cdot I_{l+1/2}(\tau). \]

Here \(I\) is the Bessel function, \(\tau = \sqrt{2w}\cdot \rho\). To each maximum and minimum of the curve \(\chi(w)\) there corresponds a definite value of \(\tau\), i.e., when the distance \(\rho\) between atoms changes, the distances along the spectrum of the individual fluctuations of the absorption curve from the main edge will also change, according to the formula

\[ w \sim \rho^{-2}. \tag{4} \]

Fig. 6. K-edge of absorption of germanium in GeCl\(_4\) vapors. Experimental curve.

The curve obtained by the authors for the dependence of the quantity \((\chi - 1)\) on \(w\) is presented in Fig. 5. As can be seen, the fine structure of the absorption spectrum of polyatomic molecules extends from the absorption edge toward the short-wavelength side for hundreds of eV. In comparison with the experimental absorption curve of GeCl\(_4\), obtained by Koster and Kleimer\(^ {10}\) (Fig. 6), we see that the first two maxima \(A'\) and \(A'\) and the two minima \(\alpha'\) and \(\alpha'\) of the theoretical curve did not appear on the experimental curve, which may be explained by the considerable narrowness of these fluctuations, the insufficient resolving power of the experimental conditions (of the order of 15 eV), and the natural width of the X-ray terms.

Table 1 gives a comparison of the theoretical and experimental distances \(w\) of individual fluctuations from the main absorption edge.

The systematic deviations of the data given in this table may be explained, on the basis of (4), by an inaccurate value of the molecular dimensions determined by Wierl electronographically. The correction introduced according to (4) gives the value \(\rho = 2.07\) Å.

Koster and Kleimer\(^ {10}\) also investigated the K-absorption spectrum of the AsCl\(_3\) molecule, which proved to be quite similar to the spectrum of GeCl\(_4\). This made it possible to apply formula (4) to the calculation of the dimensions of the AsCl\(_3\) molecule, in which the As—Cl distance proved to be equal to 2.20 Å.

Table 1

Fluctuations	\(a\)	\(B\)	\(\beta\)	\(C\)	\(\gamma\)
\(w\) exp., eV	50	86	120	160	203
\(w\) theor., eV	59	85	117	155	196

Prince\(^8\) investigated the L\(_{\mathrm{II,III}}\)-absorption spectrum of chlorine in the CCl\(_4\) molecule. However, in this case no fine structure of the kind found in GeCl\(_4\), extending over hundreds of eV, was observed. The spectrum of CCl\(_4\) gives two sharp white lines at a distance of several eV from the main edge and represents a characteristic Kossel structure, explained by transitions to the optical levels of the ionized molecule.

5. Liquids

There is no special theory of the absorption spectrum of liquids. Chioffari\(^4\) investigated the K-spectrum of liquid bromine, which gave no fine structure, as well as the K-spectra of bromine in bromine water and in a KBr solution, where the fine structure of the Kossel type is only very weakly expressed. This is explained by the considerable width of the K-level of bromine.

Fig. 7. K-edge of germanium.

Prinskii and Smolukhovskii\(^11\) investigated the K-spectrum in liquid and strongly supercooled GeCl\(_4\) (Fig. 7). In both cases a sharply expressed fine structure was obtained, quite similar to the structure of gaseous GeCl\(_4\) described above, but only situated somewhat closer to the main edge. This indicates that in the liquid state the GeCl\(_4\) molecule retains its rigid tetrahedral

structure. Owing to fluctuations in the liquid, the distances between individual molecules change statistically; as a result, one can expect only a weak influence of neighbors on the distribution of electronic levels over energies, which also explains the indicated similarity between the spectra of liquids and of gases.

Biemann and Birden[^12] investigated the K-absorption spectra of ions in aqueous solutions. The ion Cu++ was studied in solutions of CuSO$_4\cdot 5$H$_2$O, Cu(NO$_3$)$_2\cdot 6$H$_2$O, and CuCl$_2\cdot 2$H$_2$O. In all three cases identical spectra were obtained, which indicates complete dissociation of the indicated molecules. The absorption curve of the Cu++ ion is shown in Fig. 8, where the ordinate gives $\log I_0/I$—a quantity proportional to the absorption coefficient $\mu$. The steep rise of the curve represents the so-called principal absorption edge. The maximum $A$ corresponds to the transition of a K-electron to the first unoccupied level of the Cu++ ion allowed by the selection rules, i.e. to the 4p level. The second maximum $B$ corresponds to the transition of a K-electron to the next level allowed by the selection rules, i.e. to the 5p level. The distance in the spectrum between the points $A$ and $B$ is equal to 14–16 eV. If the indicated interpretation is correct, then this distance should be equal to the distance between the optical levels 4p and 5p of the Zn III ion, which is analogous to the Cu++ ion with the K-electron removed. According to optical spectra data, this distance is equal to 10.9–12.6 eV. It should, however, be taken into account that, owing to the superposition of individual maxima caused by transitions of the K-electron to the 5p, 6p, 7p, … levels, these maxima merge into one, which displaces it from $A$ by approximately another 2.5 eV. This already gives good agreement with experiment. The analogous distance between the first two maxima of the K-absorption spectrum curve of the Zn++ ion is equal to 13.5–14.5 eV. The distance between the 4p and 5p levels of the ion of the next element, Ge III, according to optical spectra data, is 11.8–12.1 eV, i.e. good agreement is again obtained if one takes into account the merging of effects from the 5p, 6p, 7p, … levels, shifting this maximum by another 2.5 eV.

Fig. 8. K-edge of copper.

The ion \(\mathrm{Cu(NH_3)_4}^{++}\), in which the bonds of the \(\mathrm{NH_3}\) groups with \(\mathrm{Cu}^{++}\) are to a considerable extent ionic, gives a completely different type of absorption spectrum (Fig. 8). Finally, a third type of spectrum is given by the ion \(\mathrm{Cu_2(CN)_4}^{--}\), which has to a considerable extent homopolar bonds of the same type as the bonds in the diatomic molecules \(\mathrm{Cl_2}\), \(\mathrm{Br_2}\); this explains the identical character of these spectra: the ion \(\mathrm{Cu_2(CN)_4}^{--}\) gives several sharp fluctuations of the absorption coefficient in the region of \(30\) e. v., on the short-wavelength side of the main absorption edge.

We see that the \(\mathrm{Cu}^{++}\) ion in solution gives the characteristic Kossel structure of free atoms, while complex ions give a structure characteristic of molecular gases. Thus, the state of ions in solution approaches the gaseous state; however, owing to hydration and the influence of the perturbing field of the attached water molecules, the principal states of the ions are split, which increases the width of the absorption lines and maxima; as a result, in solutions the elements of fine structure situated close to one another merge, as occurs for the ions \(\mathrm{Cu}^{++}\) and \(\mathrm{Zn}^{++}\).

6. Solid body

The absorption spectrum of a solid body may be divided into three regions: the main edge, representing a steep rise of the absorption coefficient with decreasing wavelength; next, on the short-wavelength side, there follow separate fluctuations of the absorption coefficient, sometimes reaching a considerable amplitude comparable in magnitude with the rise of the main edge, and extending over several electron-volts (up to \(30\) e. v.) from the main edge. These fluctuations sometimes begin already at the main edge itself. They constitute the Kossel structure and are connected with atomic levels, though more or less perturbed by the periodic field of the lattice. Finally, still farther to the short-wavelength side, at several hundred electron-volts from the main edge, small fluctuations of the absorption coefficient are observed, called the Kronig fine structure and depending entirely on the environment of the atom in the crystalline lattice and on its periodic potential field.

a) Main edge. General theory

The theory of the shape and width of the main absorption edge of metals was given by Ritchmaier, Barnes, and Ramberg \(^{13}\). In the process of absorption the metal atom ejects an electron from the inner level \(A\) into the conduction band beyond the Fermi surface. The electron-free part of the conduction band may be represented as an aggregate of narrow resonance levels \(B\), situated close to one another. The absorption curve, expressing the dependence

the absorption coefficient \(\tau\) as a function of the energy of quanta \(E=h\nu\), is a superposition of individual absorption lines and thus depends on the shape of these lines, the density \(n(E_B)\) of their distribution over the spectrum, and the transition probability \(\gamma_{AB}\). The shape of the individual absorption lines is determined by the function \(f(E_A)\) of the distribution of electrons of the lower level \(A\) with respect to the energy \(E_A\), since the width of the upper resonance level \(B\) may be neglected. This is the usual dispersion curve:

\[ f(E_A)=\frac{N_A \Gamma_A/2\pi}{(\Gamma_A/2)^2+(E_{A0}-E_A)^2}, \tag{5} \]

where \(N_A\) is the number of electrons of the lower level, \(\Gamma_A=\Delta E_A\) is the width of level \(A\) at the middle of the maximum ordinate \(f(E_A)_{\max}\), and \(E_{A0}\) is the abscissa of this point. If \(E_B\) is the energy of the upper resonance

a) absorption of a quantum \(E\) by one resonance level \(B\)
b) partial absorption coefficient by one resonance level \(B\)
c) total absorption coefficient by all resonance levels
d) properties of the absorption curve

Fig. 9. Main edge of metals.

level \(B\), then, in transitions of one electron of level \(A\) with energy \(E_A\) to an individual level \(B\) (Fig. 9, a), a quantum with energy

\[ h\nu=E=E_A-E_B \tag{6} \]

is absorbed.

Since at level \(A\) there are \(f(E_A)\) electrons with energy \(E_A\), in their transitions to \(B\) they all absorb the energy

\[ \varepsilon_B=E\cdot f(E_A). \tag{7} \]

Let us denote the number of metal atoms in \(1 \text{ cm}^{3}\) by \(\mathfrak{N}\). Then the number of atoms encountered by an X-ray beam of cross section \(1 \text{ cm}^{2}\) over the path \(dx\) will be equal to \(\mathfrak{N}dx\). Let us denote the number of quanta of the X-ray beam incident in \(1\) sec on one metal atom at depth \(x\) by \(Q_x(E)\). In the case of absorption of quanta with energy \(E\) in transitions from level \(A\) to a resonance level \(B\), the energy absorbed in \(1\) sec along the path \(dx\) from an X-ray beam of cross section \(1 \text{ cm}^{2}\) is equal, with opposite sign, to the change of intensity along the path \(dx\), i.e.:

\[ \varepsilon_B \cdot \mathfrak{N}dx \cdot Q_x(E)=-dI_x(E), \tag{8} \]

where \(I_x(E)\) is the intensity at depth \(x\) of quanta with energy \(E=h\nu\). Obviously:

\[ I_x(E)=Q_x(E)\mathfrak{N}^{2/3}\cdot f(E), \tag{9} \]

where \(\mathfrak{N}^{2/3}\) is the number of atoms situated on an area of \(1 \text{ cm}^{2}\), positioned perpendicular to the direction of the X-ray beam. On the other hand:

\[ I_x(E)=I_0(E)\cdot e^{-\tau_B x}, \]

where \(\tau_B\) is the partial coefficient of absorption by one resonance level. Hence:

\[ dI_x(E)=-\tau_B I_x(E)\,dx. \tag{10} \]

From (7), (8), (9), and (10) we obtain:

\[ \tau_B=\mathfrak{N}^{2/3} f(E_A). \tag{11} \]

Let us denote the energy of the quantum absorbed in the transition to level \(B\) from the maximum of the distribution of the lower level \(A\) by:

\[ E_{AB}=E_{AO}-E_B. \tag{12} \]

From (5), (6), (11), and (12) we obtain:

\[ \tau_B=\frac{\mathfrak{N}^{2/3}N_A\Gamma_A/2\pi}{(\Gamma_A/2)^2+(E_{AB}-E)^2}. \tag{13} \]

The course of the partial absorption coefficient as a function of the quantum energy \(E\) is shown in Fig. 9, б.

A quantum with energy \(E\) may be absorbed by all the resonance levels \(B\) of the conduction band. The resulting effect can be taken into account by integrating (13) over \(E_{AB}\). We shall assume that the probability \(\gamma_{AB}(E)\) of a transition from level \(A\) to various levels \(B\) does not depend on the transition energy \(E\), i.e., on the position of level \(B\):

\[ \gamma_{AB}(E)=\text{const}. \]

This is valid for p-levels\({}^{14}\), i.e., for example, for absorption by electrons of the \(L_{\mathrm{II,III}}\) level, whereas for s-levels, for example, \(K\), \(L_{\mathrm{I}}, \ldots\), the transition probability is proportional to the quantum energy \(E\):

\[ \gamma_{AB}(E)\sim E. \]

In addition, for simplicity we shall first assume that the states \(B\) are distributed in energy with uniform density:

\[ n(E_B)=\mathrm{const}. \]

Then the absorption coefficient \(\tau(E)\) for transitions from level \(A\) to any resonant levels is equal to:

\[ \tau(E)=\int_{E_{AB_0}}^{\infty}\tau_B(E_{AB})\gamma_{AB}(E)n(E_B)dE_{AB}, \]

where \(B_0\) is the Fermi surface and \(E_{AB_0}\) corresponds to the transition from the maximum of the distribution of level \(A\) to the Fermi surface. Consequently:

\[ \tau(E)=\frac{\mathfrak{R}^{1/3}N_A\Gamma_A}{2\pi} \int_{E_{AB_0}}^{\infty} \gamma_{AB}(E)\cdot n(E_B)\cdot \frac{dE_{AB}}{(\Gamma_A/2)^2+(E_{AB}-E)^2}. \tag{14} \]

This gives:

\[ \tau(E)=C\left[\frac{1}{2}-\frac{1}{\pi}\operatorname{arctg}\frac{E_{AB_0}-E}{\Gamma_A/2}\right], \tag{15} \]

where

\[ C=\mathfrak{R}^{1/3}N\gamma_{AB}(E)\cdot n(E_B)=\mathrm{const}. \]

Thus, under the assumptions made, the shape of the principal absorption edge is an arctangent curve.

The decomposition of the absorption curve into individual resonant absorption lines is shown in Fig. 9, в.

Let us denote the asymptotic value of \(\tau(E)\) at \(E=\infty\) by \(\tau_\infty\) (Fig. 9, г). It is easy to show that the abscissa corresponding to \(\frac{1}{2}\tau_\infty\) is equal to \(E_{AB_0}\); this point of the curve is the inflection point of the arctangent. In addition, it can be shown that the ordinates \(\frac{3}{4}\tau_\infty\) and \(\frac{1}{4}\tau_\infty\) correspond to the abscissae \(E_{3/4}\) and \(E_{1/4}\), the difference between which is:

\[ E_{3/4}-E_{1/4}=\Gamma_A. \tag{16} \]

This makes it possible, from the absorption edge, to find the width \(\Gamma_A\) of the lower level \(A\) at half the maximum ordinate of the distribution of electrons of this level.

Further, Richtmyer, Barnes, and Ramberg, instead of the uniform distribution of conduction-band electrons over states arbitrarily adopted in the preceding derivation, introduced their distribution according to Fermi–Sommerfeld, i.e., proportional to the square root of the kinetic energy of the electrons. As a result of the integration, the authors obtained—

yielded a rather complicated expression for $\tau(E)$, which, however, gives a curve fairly close to an arctangent, with a somewhat smaller curvature of the short-wavelength branch of the curve and a somewhat larger curvature of the long-wavelength branch.

Using formula (16), the authors determined, from the main edge $L_{\mathrm{III}}$ of gold, the width of this level, which turned out to be equal to 4.4 eV. Having measured the widths of various lines associated with the level $L_{\mathrm{III}}$, the authors found the widths of all the other levels of gold. Comparing the widths of some level obtained from different lines with one another and with its width from the corresponding absorption edge according to (16), one may note discrepancies exceeding the experimental errors.

In determinations of level widths of this kind one should always take into account the approximate nature of the method itself and the various possible distorting influences. Often the main edge does not, over its whole extent, have the form of an arctangent. Various fluctuations occur in the main edge itself (see below). In these cases it is necessary, on the basis of various theoretical considerations, to establish which part of the curve of the main edge can be brought into coincidence with the theoretical arctangent. The matching is carried out using two or three points of the curve.

First of all, the absorption curve is plotted in coordinates whose abscissa is expressed in eV, while along the ordinate are plotted $\log I_0/I$, where $I_0$ is the intensity before the absorber and $I$ after the absorber. This logarithm is proportional to $\tau$, which is usually not calculated, since this would require exact knowledge of the thickness of the specimen. Since $\log I_0/I=\tau d$, from (15) we obtain:

$$ \log I_0/I=C'\left[\frac{1}{2}-\frac{1}{\pi}\operatorname{arctg}\frac{E_{AB_0}-E}{\Gamma_A/2}\right], \tag{17} $$

where $C'=Cd$.

Having chosen the correct portion of the absorption curve, one finds the abscissa of the inflection point, i.e. $E_{AB_0}$. Then, for two suitable points of the curve (not too close to one another and located on the middle portion of the curve), the ordinates and abscissas $E$ are determined; these are substituted pairwise into (17), and thus the parameters $C'$ and $\Gamma_A$ of this equation are found.

The question of the proper choice of a suitable portion of the absorption curve will be considered further below.

Formula (14) may be regarded as an integral equation. Using the experimental curve of the dependence $\tau(E)$, one can solve equation (14) numerically and construct a graph of the function $[\gamma(E)n(E)]$. Neglecting the dependence of the transition probability $\gamma$ on the energy $E$, or assuming some special form of this function,^14 one can find $n(E)$, i.e. the distribution of unoccupied possible electron states in the crystal lattice.

A simpler, but less accurate, method of determining $n(E)$ was applied by Skinner,^30 who used equation (1) for $\tau(E)$.

The transition probability may, in a first approximation, be regarded as constant within a single absorption edge or a single emission band. The \(K\) bands give the distribution of occupied states of the \(p\)-electrons; the corresponding absorption edges give the distribution of unoccupied states of the \(p\)-electrons. The emission bands \(L_{\mathrm{III}}\) and the absorption edge \(L_{\mathrm{III}}\) give the total distribution of occupied and unoccupied states of the \((s+d)\)-electrons. Thus, X-ray spectra make it possible to find separately the distribution curves of \(p\)- and \((s+d)\)-states and to indicate which of them are occupied and which are free.

Fig. 10. Density of distribution of \(p\)- and \((s+d)\)-states of 12 Mg.

Using the fact that in metals the end of emission coincides with the beginning of absorption, the distributions of occupied and free states (determined separately from the emission and absorption spectra) can be joined into one common curve by making the adjacent ordinates of the two parts of the curve equal. Since the emission intensity is proportional to the cube of the frequency, \(\nu^3\), when joining the two curves into one it is also necessary to divide all ordinates of the emission part of the curve by \(\nu^3\). The results obtained in this way by Skinner\(^{30}\) for magnesium are shown in Fig. 10.

The further development of such a method for finding the distribution of electron states by energy promises to be of great help in the investigation of matter.

The form of the main edge in crystalline ionic compounds has a more complex character. Beeman, Fross, and Humphrey\(^{37}\) indicate that in ionic compounds the main edge is not an arctangent curve, since the initial absorption is explained by transitions to comparatively sparsely situated optical levels (Fig. 15), which, gradually becoming denser, pass into a continuum. Therefore the absorption curve of ionic compounds (as also of monatomic gases, Fig. 1) may be represented by a superposition of an arctangent curve (transitions into the continuum) and several absorption lines (transitions to optical levels).

Thus, the true arctangent curve in this case lies comparatively far from the main edge, whose form cannot be explained by formula (15). Consequently, to determine the width

of the lower level by formula (16) for ionic compounds. In the case when the first absorption lines are located far from one another, one may assume that the initial rise of the absorption curve has the form of the dispersion curve of a separate absorption line, i.e. is determined by the equation:

\[ y=\frac{A}{x^{2}+B^{2}}, \]

where \(y\) is proportional to the absorption coefficient, \(B\) is the half-width at half the maximum ordinate. For this equation the following relation is valid:

\[ |y'/y|=|2x/(x^{2}+B^{2})|, \]

where \(y'\) is the first derivative. The maximum of the function \(|y'/y|\) occurs at \(x=B\). At this point:

\[ |y'/y|_{\max}=1/B. \]

On the rise of an experimental curve it is easy to find the point at which \(|y'/y|\) has a maximum, and consequently to determine the width \(2B\) of the lower level:

\[ 2B=2|y/y'|_{\min}. \tag{18} \]

b) Position of the main edge

In what follows, by the position of the main absorption edge we shall understand the abscissa of its inflection point, i.e. the energy of the quantum whose absorption ejects an electron of an inner level of the atom to the first free level beyond the Fermi surface, provided that the selection rules permit such a transition. As was indicated in § 1 of Chapter II of the present paper, Sandström constructed graphs expressing, as a function of the atomic number \(Z\), the difference between the energies of the absorption edge and of the shortest-wavelength line of the corresponding emission series, for example, \([(\nu/R)_{K}-(\nu/R)_{K\beta_{2}}]\), for elements from \(31\,\mathrm{Ga}\) to \(47\,\mathrm{Ag}\). These graphs show that, with partial filling of the upper level, the electrons of this level, in passing to an inner level of the atom, give an emission line, while in the process of absorption an electron of the inner level makes the reverse transition to the same upper level. In the example mentioned, the transitions occur between the levels \(1s\) and \(4p\). With increasing \(Z\), when the \(4p\) level is filled, the absorption transition occurs to the next level permitted by the selection rules, i.e. to \(5p\). An energy gap appears between the short-wavelength end of the emission and the long-wavelength beginning of absorption.

Absorption transitions from different inner levels occur to different upper free levels, depending on the selection rules. Thus, for example, as Prince and Tejens \(^{15}\) showed, the positions of the main absorption edges \(M_{\mathrm{IV}}\) and \(M_{\mathrm{V}}\) (\(3d\)) of the elements

47 Ag, 48 Cd, and 50 Sn can be calculated from the absorption edge \(L_{\mathrm{III}}\) \((2p)\) (transition of an electron from the \(2p\) level to the partially free \(5s\) level in metals in the solid state) and the emission line \(L\alpha\) (transition of an electron from \(3d\) to \(2p\)). Indeed, the difference of the energies corresponding to the \(L_{\mathrm{III}}\) edge and the \(L\alpha\) line is equal to:

\[ (E_{2p} - E_{5s}) - (E_{2p} - E_{3d}) = E_{3d} - E_{5s}, \]

i.e., it is equal to the energy necessary to eject an electron from the \(3d\) level \((M_{\mathrm{IV,V}})\) to the first free level beyond the Fermi surface. However, this energy proved to be smaller than the energy corresponding to the principal absorption edge \(M_{\mathrm{IV,V}}\). According to the selection rules, in the absorption process an electron from the \(3d\) level can pass only to the \(5p\) level. In this case the inflection point of the principal \(M_{\mathrm{IV,V}}\) edge will no longer correspond to the Fermi surface, since the latter lies in the conduction band consisting of \(5s\) levels. The inflection point of this edge will correspond to the beginning of the next band, consisting of \(5p\) levels. In this case, too, a gap is observed between the end of emission and the beginning of absorption.

Such a gap is by no means always observed in metals, since, owing to the overlap of two or three broad outer bands, mixing of the electron states of these bands occurs, and transitions become possible directly beyond the Fermi surface, although the corresponding transitions in gases may be forbidden. Thus, for example, in the transition elements of the iron group the Fermi surface lies in the \(4s\) band. According to the selection rules, \(K\)-electrons in the absorption process can pass only to \(4p\), which should give a gap between the \(K\)-edge and the last line of the \(K\)-series permitted by the selection rules, i.e. \(K\beta_1\) \((1s — 3p)\). However, with decreasing atomic number \(Z\) in the region of the elements \(32\mathrm{Ge} — 26\mathrm{Fe}\), the \(3d\), \(4s\), and \(4p\) levels broaden more and more, and the indicated mixing of states occurs. As a result, on the one hand, the intensity of the quadrupole line \(K\beta_5\) \((1s — 3d)^{16}\) increases more and more, and emission transitions to the \(K\)-level of electrons located immediately below the Fermi surface become possible; on the other hand, absorption transitions immediately beyond the Fermi surface become possible. The gap between the end of emission and the beginning of absorption disappears: the wavelengths of the inflection points of the principal \(K\) edge and of the short-wavelength branch of the last emission line \(K\beta_5\) coincide\({}^{17}\).

An analogous comparison of the last emission lines of the \(L\)-series with the edges \(L_{\mathrm{I}}, L_{\mathrm{II}}\), and \(L_{\mathrm{III}}\) of tungsten was made in the work of Birden and Snyder\({}^{18}\). In Fig. 11, by way of example, are presented the intensity curves \(I\) of the emission line \(L\gamma_6\) and \(\log I_0/I\) (i.e., a quantity proportional to the absorption coefficient) for the edge \(L_{\mathrm{II}}\).

Such a coincidence of the end of emission and the beginning of absorption can be expected only in metals, since in them the Fermi surface is located...

divides the conduction band into an occupied and a free part. In nonconductors the band of valence electrons is filled, and absorption is possible only into the next allowed band, which leads to a gap between the end of emission and the beginning of absorption. This gap is small in semiconductors and considerably larger in dielectrics. Thus, for example, in the cited work of Bird and Schneider[^18] the spectra of tungsten in the oxide WO$_3$ were also investigated. It turned out that the emission lines did not change their position in the spectrum, whereas the absorption edges shifted by 2.5 eV toward the short-wavelength side, which led to the appearance of the above-mentioned gap.

Fig. 11. The L$_\gamma$ line and the L$_{\mathrm{II}}$ edge of tungsten.

Measuring the distance between the end of emission and the beginning of absorption is a good method for determining the width of the energy gap between the corresponding two energy bands of semiconductors and dielectrics.

This method was applied by Fogel,[^52] who investigated the fluorescence spectra of the Kβ group of lines and the K-edge of absorption of elemental sulfur and of sulfur in several compounds. The gap $\Delta E$ between the short-wavelength edge of Kβ$_x$ and the beginning of K-absorption in elemental sulfur is equal to 1.15 eV, which corresponds to the fact that sulfur is an insulator. In the compound CuS this gap is $\Delta E = 0.25$ eV, i.e., within the limits of experimental error the end of emission coincides with the beginning of absorption; this corresponds to the known metallic electrical conductivity of CuS. In MoS$_2$ the gap is $\Delta E = 1.33$ eV, which corresponds to an insulator, although for this compound conductivity of the semiconductor type has been experimentally established; apparently, the electrical conductivity of MoS is explained by impurities, so that molybdenite is an impurity semiconductor. The gap $\Delta E$ in stibnite, Sb$_2$S$_3$, is equal to 1.18 eV, which also corresponds to an insulator. The electronic conductivity found by some investigators in stibnite can be explained, as in molybdenite, by impurities. In ZnS the gap is $\Delta E = 4.48$ eV, which corresponds to an insulator, which ZnS in fact is.

Fogel compared the data given above with determinations of the gap $\Delta E$ by other methods, through the study of the photoelectric, optical, and electrical properties of the same substances. These properties are highly sensitive to various kinds of

impurities; discrepancies in the data in some cases fall within the experimental error, while in other cases they exceed the possible errors. In the case of ZnS, instead of the value indicated above, \(\Delta E = 4.48\) eV, from the threshold of the internal photoelectric effect one obtains the value \(\Delta E = 2.43\) eV. Fogel explains this discrepancy by the fact that within the gap between the filled and free bands of sulfur there is a band of states of the zinc ion, into which photoelectrons from the sulfur \(3p\) level are transferred.

Numerous studies have established the influence of chemical bonding on the position of the principal absorption edge. Let us consider the main causes producing these shifts.

1. In metals and alloys the position of the point of inflection of the principal edge is determined by the Fermi surface energy \(W_{\mathrm{F}}\), which can be determined by the following formula:

\[ W_{\mathrm{F}}=\frac{h^{2}}{8m}\left(\frac{3n_{0}}{\pi}\right)^{2/3}, \tag{19} \]

where \(n_{0}\) is the number of free electrons per unit volume of the alloy. According to Weinstein\(^{19}\), with a change in the number \(n_{0}\) of free electrons, the Fermi surface energy changes, and consequently the principal edge is shifted by the amount (on the frequency scale):

\[ \Delta \nu=\left[\frac{h}{12m}\left(\frac{3}{\pi}\right)^{2/3}\cdot n_{0}^{-1/3}\right]\cdot \Delta n_{0}. \tag{20} \]

Thus, with an increase in \(n_{0}\), the edge shifts toward the short-wave side, since with the increase in the kinetic energy of the electrons their total energy decreases (the potential energy of the electrons is negative), and the energy difference corresponding to the absorption transition increases. To verify this theory it is necessary to know the number \(n_{0}\) of free electrons of the alloy. The corresponding data can be obtained by studying the width of the emission bands of the alloy, as was shown in the first part of the present article. However, up to the present time such work has not been carried out.

A whole series of studies has established that in binary alloys (for example, Ni—Cu, Mn—Al, Cu—Zn, etc.) an exchange of electrons takes place between the atoms of the alloy. Thus, for example, in Ni—Cu alloys, with increasing copper content the \(3d\) band of nickel is gradually filled. In connection with this, at 60% copper (constantan), when the filling of the band is completed, the ferromagnetism of the alloys disappears. As experiment has shown\(^{20}\), in these alloys, with increasing copper content, the width of the emission band \(K\beta_{5}\) of nickel increases. Consequently, the number of free electrons also increases, and according to (20) one may expect a shift of the absorption edge toward the short-wave side. Filling of the \(3d\) band of nickel leads to an increase in the screening of the nuclear charge and to a decrease in its effective charge. This changes

energy of the atom in the final state after absorption and, consequently, also shifts the edge; however, it is difficult to take this influence into account.

Finally, it must be taken into account that, with increasing copper content, the environment of nickel changes; but this influence too is theoretically difficult to allow for. It is not yet possible to predict the final result of the action of all the causes considered above, and in order to clarify this question in complicated cases one has to turn to experiment.

The alloys considered above, Ni—Cu, were investigated by Friedman and Beeman^20, who found no displacement of the K-edges of Ni and Cu. Likewise, no shifts of the K-edges of Zn and Cu were found in Zn—Cu alloys in the work of Bearden and Friedman^21. In Zn—Ni alloys, according to Bearden and Beeman^22, no displacement of the Ni K-edge was detected either, but the Zn K-edge, on going from pure Zn to the alloys, is shifted to the long-wavelength side by 1.3 eV. In this last case, apparently, zinc gives the alloy more electrons than nickel; therefore, when zinc is diluted with nickel, the number of free electrons per atom of the alloy decreases, the Fermi surface energy decreases, and according to (20) the edge is shifted to the long-wavelength side.

In Dekhtyar’s work^23 the cobalt absorption K-edges in Co—Cr alloys were investigated. In these alloys, apparently, chromium gives more electrons than cobalt; therefore, with increasing chromium content, the Co K-edge, according to (20), should shift to the short-wavelength side, as is indeed observed for $\alpha$-phase alloys with contents of 100, 95, 90, and 75% Co. At a cobalt content of 40% the alloy has another structure. The $\beta$-phase at a Co content of 90%, obtained by quenching from $1200^\circ$C, was also investigated. In both of the latter alloys, with the change of the reciprocal lattice (as compared with the $\alpha$-phase), a rearrangement of the outer electrons occurs, and at present it is difficult to predict the result theoretically. As experiment has shown, when the Co content is decreased from 75 to 40%, the edge shifts in the opposite direction (toward longer wavelengths). On going from the $\alpha$-phase containing 90% cobalt to the $\beta$-phase with the same content, the edge shifts to the short-wavelength side by 2.5–3 eV.

In Mn—Al alloys (according to unpublished work by I. V. Sorokina) the K-edges of manganese were investigated. In these alloys, apparently, aluminum gives more electrons than manganese, so that with increasing aluminum content the manganese K-edge, according to (20), should shift to the short-wavelength side. It is true that one may suppose that part of the aluminum electrons pass in the alloys into the $3d$ band of manganese. As experiment showed, on going from pure manganese to alloys containing 78 and 64% of it, there occurs a gradual shift of the edge to the short-wavelength side, reaching 3.75 eV. With a further increase in the aluminum content in alloys with 34 and 20.3% manganese, the wavelength of the edge begins to increase,

which may be connected with a change in the lattice of the alloys and, consequently, also of the reciprocal-lattice zones.

1. In passing from metals to ionic compounds, the energy band of the valence electrons is filled, and absorption transitions become possible only into the next allowed band, in which the electrons possess lower energy. Consequently, the displacement of the edge must be toward the short-wavelength side. This is indeed confirmed by experiment. Thus, for example, the L-edge of tungsten shifts on passing to the oxide, as was already indicated above. The wavelength of the L_{\mathrm{III}}-edge of aluminum\(^{24}\) in the pure metal is equal to 170.56 Å; in the oxide Al\(_2\)O\(_3\) it is equal to 161.96 Å. In the same direction, by 2 eV, the K-edge of Na shifts on passing from the metal to the oxide Na\(_2\)O\(^{38}\). Lind and Stelling investigated numerous compounds of potassium, calcium, titanium, vanadium, chromium, manganese, and iron (see the summary in Korsunsky\(^{25}\)). In all these compounds the wavelength of the K-edge is smaller than that of the corresponding pure metal.

2. As the same investigations of Lind and Stelling showed, with increasing valence of the atoms in compounds the wavelength of the absorption edge of anions and cations shifts toward the short-wavelength side. According to Kuntzl’s experimental rule\(^{26}\), the magnitude of this displacement increases linearly with increasing valence. This law, however, is only very approximate and has so far received no theoretical justification.

3. In passing from pure metalloids to their compounds with a homeopolar bond, depending on the change in the character of the bond the edge may shift to one side or the other, which is also observed experimentally. Thus, for example, according to Stelling\(^{27}\), the wavelength of the K-edge of rhombic sulfur is 5008.6 XE, of monoclinic sulfur 5009.0 XE, on average 5008.8 XE. Table 2 gives the wavelengths of the K-edges of sulfur in various compounds and their displacements in comparison with the cited mean value for pure sulfur.

Table 2

Compound	\(\lambda\) XE	\(\Delta\lambda\) XE	Compound	\(\lambda\) XE	\(\Delta\lambda\) XE
Sulfur	5008.8	0.0	MoS	5009.9	+1.1
Cr\(_2\)S\(_3\)	5011.7	+2.9	Sb\(_2\)S\(_3\)	5009.9	+1.1
CS\(_2\)	5011.4	+2.6	Na\(_2\)S	5009.6	+0.8
FeS	5011.4	+2.6	CdS	5007.5	−1.3
CuS	5011.3	+2.5	BaS	5007.5	−1.3
SnS\(_2\)	5011.3	+2.5	H\(_2\)S	5007.1	−1.7
CoS	5010.9	+2.1	CaS	5006.6	−2.2
Sb\(_2\)S\(_5\)	5010.8	+2.0	MgS	5005.6	−3.2
Bi\(_2\)S\(_3\)	5010.3	+1.5	ZnS	5005.3	−3.5
Ag\(_2\)S	5010.1	+1.3	SO\(_2\)	5004.5	−4.3

As is evident from this table, the shifts reach a considerable magnitude and in different compounds have different signs. Theory is as yet powerless to deal with these questions in detail.

1. In ionic compounds of the same valence and the same crystal structure there is observed a systematic shift of the absorption edge of the anion toward the short-wavelength side with decreasing ionic radius of the cation, i.e., with increasing polarization of the anion (the cations are polarized comparatively little). Weinstein \(^{28}\) assumed this shift (on an energy or frequency scale) to be proportional to the polarization energy, which is inversely proportional to the fourth power of the interatomic distance \(r\) and reduces the binding energy of the electron with the atom, as a result of which the energy of the outer level decreases and the edge is shifted toward shorter wavelengths.

If the frequency of the absorption edge at \(r=\infty\), i.e., of the “free anion,” is denoted by \(\nu_\infty\), then in compounds the edge frequency may be expressed by the formula:

\[ \nu=\nu_\infty + k\cdot r^{-4}, \]

where \(k\) is a certain coefficient of proportionality. Hence the shift of the wavelength is:

\[ \Delta\lambda=\lambda-\lambda_\infty=-\frac{k\lambda_\infty^2}{r^4}, \]

i.e., for a given series of compounds at constant \(\lambda_\infty\), \(\Delta\lambda\) is inversely proportional to \(r^4\). The graphs constructed by Weinstein of the dependence of \(\lambda\) on \(r^{-4}\) (for compounds of sulfur and chlorine of the same valence and structure) gave a linear dependence.

1. In the work of Kurilenko \(^{29}\) the influence of a magnetic field of strength 10,000 gauss on the K-edge of \(\alpha\)-iron was investigated. It was found that the edge is shifted to the long-wavelength side by 3 eV.

c) Kossel Structure of Absorption

As already indicated, the fine structure of the principal absorption edge itself and of the region of the absorption spectrum nearest to it (5–30 eV toward the short-wavelength side) depends mainly on atomic levels and their deformation in the lattice. This region of the absorption spectrum is called the Kossel structure. Let us dwell on individual works devoted to this part of the spectrum.

Beeman and Friedman \(^{17}\) investigated the K-edges of the transition group of elements from 26 Fe to 32 Ge. To explain the interpretation of these edges proposed by them, Fig. 12 presents the distribution of possible electron states by energy according to Slater’s calculations for copper. The states occupied by electrons are shaded. The point \(L\) corresponds to the Fermi surface. Near the Fermi surface there is an overlap of the 3d, 4s, and 4p bands, owing to which transitions of a K-electron in the absorption process are possible directly beyond the surf-

ness of the Fermi level. The behavior of the absorption coefficient as a function of energy is determined by the behavior of the total density of states of these three bands, however, taking into account the small probability of transition into the \(3d\)- and \(4s\)-states and the large probability of transition into the \(4p\)-states. The authors consider that the \(3d\) band extends from \(-13\) eV to \(-5.5\)

Fig. 12. Distribution of electron states in copper.

eV, the \(4s\) band extends from \(-13\) eV to \(-0.5\) eV, and \(4p\) begins at the point \(M\) (see Fig. 12). This point on the absorption curve should correspond to the beginning of a sharp rise of the absorption coefficient.

Fig. 13. The \(K_{\beta_{2,5}}\) line and the \(K\) edge of copper.

The absorption curve of the copper \(K\) edge, as well as the emission band \(K\beta_{2,5}\) of copper, are presented in Fig. 13. As can be seen, the inflection point \(L\) of the absorption curve and the inflection point of the short-wavelength branch of the emission band lie at the same ordinate, i.e. emission ends where absorption begins. This point corresponds to the Fermi surface. Near the point \(M\) the density of states has a minimum (see Fig. 12), which is reproduced by the absorption curve (Fig. 13). Farther on, the high probability of transition into \(4p\)-states gives a considerable rise of the absorption curve, after which oscillations of this curve are observed, the maxima \(A\) and \(B\) of which correspond to the maxima \(A\) and \(B\) of the density-of-states distribution curve according to Slater (Fig. 12).

I. Borovsky^31 investigated the absorption spectra of elements of the same group from \(22\mathrm{Ti}\) to \(29\mathrm{Cu}\), both in pure metals and in ionic compounds of different valence. It was observed that the absorption maximum \(m\) (the white line, see Fig. 13) is found not only in metals, but also in certain compounds. Its width from point \(m\) to the minimum \(M\) is, in metals, \(0.7\)—\(1.2\) eV, and in cations \(4\)—\(7\) eV. This indicates that the segment of the curve from the beginning of the edge to point \(M\) corresponds to absorption by \(3d\)-states, i.e. to a comparatively narrow band, under a small perturbation. Since in metals the \(3d\) band overlaps with the \(4s\) and \(p\) bands, the distance \(mM\) is small. In ionic compounds there is a gap between the \(3d\) bands and the higher-lying bands, owing to which the distance \(mM\) increases sharply. In metals the inflection point \(L\) corresponds to the Fermi surface. In cations giving a white line, the latter appears as a result of transitions of electrons from the \(K\)-level to the unfilled part of the narrow \(3d\)-band; this is resonant line absorption by an atomic level. Further, after the gap, absorption begins in the broad \(4p\) band, which gives the arctangentoid on the segment \(MA\); the inflection point of this segment should be used when applying Kuntz’s rule. Finally, at the edges of cations giving white lines, in applying Kuntz’s rule the abscissa of the inflection point of the entire edge must be taken.

The interpretation set forth for the white line of certain cations, according to Borovsky, leaves open the question of so large a magnitude of absorption of the \(K\)-electron in \(3d\)-states. Indeed, in ionic compounds the \(3d\) and \(4p\) bands do not overlap, and the transition of an electron from \(1s\) to \(3d\) is forbidden by the selection rules. However, in the emission spectra of these same cations a comparatively bright quadrupole line \(K\beta_5\) \((1s \to 3d)\) is observed, the presence of which indicates a sufficient probability of such transitions.

Birden, Beeman, and Friedman^20,^21,^22 investigated the absorption edge of Zn, Cu, and Ni over an interval up to \(25\)—\(30\) eV from the beginning of the edge in various binary alloys of these metals. In alloys of zinc with copper or nickel, the initial absorption of copper or nickel increases with increasing content of the other component. This indicates that in these alloys a redistribution of valence electrons occurs: the outer electrons of zinc partially pass into the \(3d\)-bands of nickel or copper. In alloys of copper with nickel, the absorption spectra of both components in the region up to \(25\) eV do not change noticeably in comparison with the pure metals and give different absorption curves, which indicates poor sharing of conduction electrons. In Cu—Zn alloys the copper \(K\)-edge, with zinc content up to \(30\%\), does not change noticeably in the region up to \(30\) eV; the zinc \(K\)-edge, with copper content up to \(80\%\), changes, but does not resemble the absorption curve of copper. This also indicates poor sharing of electrons in this alloy. Similar results were obtained by the same authors also in inter-

pretation of emission spectra (see the first part of the present article).

Feldkamp \(^{32}\) investigated the \(L\)-absorption edges of tantalum, tungsten, platinum, and gold. The outer electrons of these metals are situated at the \(5d\) and \(6s\) levels; \(5d\) is a comparatively narrow atomic level, whereas \(6s\) is a broad, generalized conduction band. In \(73\mathrm{Ta}\) and \(74\mathrm{W}\) the \(5d\) level is not filled with electrons, and absorption with a transition of inner electrons to this level is possible. The selection rules permit electrons \(L_{II}\) or \(L_{III}\) to pass to this level, whereas from \(L_I\) the transition is forbidden. Therefore the initial absorption

Fig. 14. \(L\)-edge absorption of Au and W.

in the \(L_{II}\) and \(L_{III}\)-edges of tantalum and tungsten (see the tungsten edge in Fig. 14) is characterized by a bright white narrow line, absent at the \(L_I\) edge. In \(78\mathrm{Pt}\) and \(79\mathrm{Au}\) the \(5d\) level is filled; absorption is possible only into the \(6s\) band and higher; here the white line is absent at all \(L\)-edges (see the gold edges, Fig. 14).

Rull \(^{53}\) investigated the absorption spectra \(M_{IV}\) and \(M_V\) of \(62\mathrm{Sm}\). As is known, in the rare earths the inner level \(N_{VI,VII}\) is not filled. Transitions from \(M_{IV}\) and \(M_V\) to \(N_{VI}\) and \(N_{VII}\) are allowed by the selection rules. Therefore in this case the absorption spectrum has the form not of an absorption edge, but of sharp intense white lines.

O’Brien \(^{33}\) investigated the long-wavelength \(L\)-absorption spectra of sodium metal and in halide compounds over a range of 10–12 eV from the onset of selective absorption. The external appearance of the spec-

X-RAY SPECTRA AS A METHOD OF INVESTIGATION

the halide compounds differs sharply from the spectrum of metallic sodium. In the compounds NaCl, NaBr, and NaJ, several narrow absorption lines are observed—the result of absorption with transitions to the optical levels of sodium. The width of these lines is of the order of 0.2–0.5 eV at half maximum intensity. In NaF this structure is absent owing to the strong perturbing action of the fluorine ion, which broadens the outer levels of sodium into wide, overlapping bands.

On passing from NaCl to NaBr and then to NaJ, with the increase in the ionic radius of the anion and the decrease in polarization energy, the energy of the absorption transition decreases in accordance with the above-described theory of Weinstein,^28 which leads to a systematic shift of the spectrum toward the long-wavelength side.

Beeman, Forsa, and Hemphrey^37 investigated the K absorption spectra of copper halide compounds. Table 3 gives the positions of the first three absorption maxima A, B, and C, measured from the inflection point of the main edge of metallic copper.

Table 3

Cu⁺ in:	A eV	B eV	C eV
CuCl	7.1	11.2	14.6
CuBr	6.4	10.5	14.2
CuJ	5.6	10.3	13.0

Here, too, with decreasing polarizing action of the anions the spectrum shifts toward the long-wavelength side. The absorption maximum is a quadrupole transition \(1s \to 3d\). The remaining maxima appear as a result of transitions \(1s \to np\) for \(n \geq 4\). However, under the influence of the strong action of the anions surrounding the Cu⁺ ion in the lattice, the optical levels of Cu⁺ are split, and the absorption curve is a superposition of many absorption lines. It is impossible to predict theoretically the positions and intensities of the individual lines; therefore the authors attempted to select such a group of lines whose superposition would reproduce the experimental curve as closely as possible. The width of the lines at half maximum ordinate was taken as constant and equal to the width of the copper K level, 1.22 eV, determined by formula (18). The result of such a decomposition for CuCl is shown in Fig. 15.

Sometimes a white line is observed on the long-wavelength side of the main edge. Such a line is observed, for example, at the K edge of cobalt in the compound \(K_3[\mathrm{Co}(\mathrm{NO}_2)_6]\). Dexter,^23 who investigated this spectrum,

gives an interpretation of this line according to Pauling, according to whose ideas, in the case of ionic bonds of trivalent cobalt, six 3d electrons are distributed over five 3d levels, avoiding pairing, so that on four of these levels there remain unoccupied places into which absorption transitions are possible, giving a white line on the long-wavelength side of the main edge.

Fig. 15. K-edge of copper in CuCl.

In the case of covalent bonds the same 6 electrons are distributed in pairs over three levels and leave no vacant places for absorption. Thus the indicated white line may serve as an indicator of ionic bonds.

It should be pointed out, however, that the interpretation set forth encounters serious difficulties and can be applied only with great caution.

г) Kronig fine structure of absorption

In the absorption spectra of solids, on the short-wavelength side of the main edge, a fine structure is observed—fluctuations of the absorption coefficient. An explanation of this structure was proposed by Kivitt and Lindsay ^34 in 1930. According to the hypothesis of these authors, in absorption transitions of an inner electron to the periphery of the atom, simultaneous transitions of one or two more electrons to the peripheral levels of the atom, or even complete ionization of these electrons, are possible. This hypothesis has now been abandoned, since it does not explain a whole series of observed phenomena and regularities, which will be discussed below.

In 1937 Sato ^35 proposed another hypothesis, according to which, simultaneously with the principal absorption transition, transitions up or down between the optical levels of the atom are possible. To explain the fine structure of zinc in ZnO, Sato admitted, simultaneously with the absorption transition in the zinc atom, transitions between the optical levels of oxygen. This hypothesis has now likewise been abandoned for the same reasons as the preceding one.

In 1932, a successful theory of the fine structure of the absorption spectra of solid metals and alloys was proposed by Kronig^36, proceeding from the band theory set forth in the first part of the present article.

If one follows the changes in the energy of the nearly free electrons of a crystal lattice with a gradual increase of the wave vector \(k\) of the electrons in the reciprocal lattice, then, for the direction of motion of the electrons perpendicular to the plane with indices \((\alpha,\beta,\gamma)\) and interplanar spacing

\[ d_{\alpha\beta\gamma}=a/\sqrt{\alpha^2+\beta^2+\gamma^2} \]

at

\[ k_{0\min}=\frac{1}{2}\left(\frac{1}{d}\right) \]

there will occur a discontinuity in the possible values of the kinetic energy of the electrons; to the wave vector \(k_{0\min}\) there corresponds the kinetic energy:

\[ W_{0\min}=\frac{h^2 k_{0\min}^2}{2m}=\frac{h^2 S^2}{8ma^2}, \tag{21} \]

where \(a\) is the lattice constant (for simplicity a cubic lattice is taken) and \(S^2=\alpha^2+\beta^2+\gamma^2\). Absorption of an x-ray quantum transfers an inner electron of the atom to one of the possible states of the electrons in the lattice. If the electron passes directly beyond the Fermi surface (the energy of which we shall denote by \(W_{\Phi}\)), then the energy of the final state \(W_{\Phi}\) is, according to (19), equal to:

\[ W_{\Phi}=\frac{h^2}{8m}\left(\frac{3n_0}{\pi}\right)^{2/3} =\frac{h^2}{8ma^2}\left(\frac{3n_e}{\pi}\right)^{2/3}, \tag{22} \]

where \(n_e=a^3 n_0\) is the number of free electrons in the elementary cell. Comparing (21) and (22), we see that discontinuities are possible when:

\[ S^2 \geq \left(\frac{3n_e}{\pi}\right)^{2/3}. \]

The plane \((\alpha,\beta,\gamma)\) will cause discontinuities also for other directions of \(k\) (Fig. 16), when

\[ k_0=\frac{k_{0\min}}{\cos\vartheta_0}. \]

Averaging the effect under consideration over all possible directions, we find that the plane \((\alpha,\beta,\gamma)\) causes a certain fluctuation of the absorption coefficient near the energy value determined by (21). If one takes into account the entire set of planes causing discontinuities (with a structural factor different from zero), then one obtains a whole series of successive fluctuations explaining the fine structure of the absorption spectrum.

It follows from formula (21) that metals of identical crystalline structure should have similar fluctuations. The distance from the main edge to the individual corresponding fluctuations

tions should be inversely proportional to the square of the lattice constant. These two propositions were checked on extensive experimental material and are very well satisfied.

Further, Kronig calculated in greater detail the position and form of the fluctuations of the absorption coefficient \(\tau(\nu)\) as a function of the electron energy \(W\). Let us replace the function \(\tau(\nu)\) by a function equal to it:

\[ \chi(W)=\tau(\nu). \]

The energy absorbed in the interval from \(W\) to \((W+dW)\) is proportional to:

\[ \chi(W)\cdot dW. \]

This quantity depends only on the number of absorbing levels falling within the interval \(dW\). The presence of discontinuity planes distorts the distribution of free electrons. It is natural to assume that the energy absorbed by each level does not depend on the perturbation; only the distribution of the number of levels over the interval of values of the wave vector \(dk\) changes.

Fig. 16. Toward Kronig’s derivation of fine structure.

Let, in the unperturbed problem, an energy interval \(dw\) correspond to the interval \(dk\) (Fig. 17), while in the perturbed problem, near the discontinuity planes, the same interval \(dk\) corresponds to an energy interval \(dW\). Near the discontinuity, on both sides of it, in the presence of the perturbing lattice field, a condensation of possible electron states occurs. In what follows, in the unperturbed problem we shall denote the electron energy by \(w\), and the absorption coefficient by \(\chi(w)\); in the perturbed problem the same quantities will be denoted by \(W\) and \(K(W)\).

Fig. 17. Toward Kronig’s derivation of fine structure.

From what has been said above it follows that the energy absorbed in the interval \(dw\) is equal to the energy absorbed in the interval \(dW\), i.e.:

\[ K(W)dW=\chi(w)\,dw. \tag{23} \]

In Fig. 17 the dashed line shows the dependence of the energy \(w\) of free electrons on the magnitude of the wave vector \(\mathbf{k}\). In the presence of pla-

…of the discontinuity plane \((\alpha,\ \beta,\ \gamma)\); the corresponding wave vector is \(\mathbf{k}_0\), and the course of the energy \(W\) is shown in Fig. 17 by the solid curve. To the wave vector \(\mathbf{k}_0\) there corresponds, in the unperturbed problem, the energy \(w_0\), and in the perturbed problem, \(W_0\).

For a certain value of the wave vector \(\mathbf{k}\), the change in energy caused by the perturbation will be:

\[ |w-W|\leq |w_0-W_0|=|V_{\alpha\beta\gamma}|, \]

where \(V_{\alpha\beta\gamma}\) is the coefficient in the expansion of the periodic potential of the lattice in a Fourier series. This quantity is of the order of a few volts. Therefore, without great error one may make the substitution:

\[ \chi(w)\cong \chi(W). \]

Then from (23) we obtain:

\[ K(W)=\chi(W)\cdot \frac{dw}{dW}. \tag{24} \]

If the course of the absorption coefficient \(\chi(W)\) is known in the absence of perturbations, then from (24) one can find the course of this coefficient in the perturbed problem. For this purpose one may use the second-order approximation:

\[ W=W_0+(w-W_0)\sin^2\vartheta_0\pm \sqrt{(w-W_0)^2\cdot \cos^4\vartheta_0+V_{\alpha\beta\gamma}^2}, \]

where \(\vartheta_0\) is the angle between the wave vector and the perpendicular to the discontinuity plane \((\alpha,\ \beta,\ \gamma)\) (Fig. 16).

Let us denote by \(W_{0\min}\) the energy corresponding to the wave vector \(\mathbf{k}_{0\min}\) (perpendicular to the plane and touching this plane with its end). To the wave vector \(\mathbf{k}_0\), touching with its end another point of the plane \(\alpha,\ \beta,\ \gamma\), there corresponds the energy \(W_0\). Let us denote the distribution function of the states \(W_0\) by \(f(W_0)\). This function must satisfy the condition:

\[ \int_{W_{0\min}}^{\infty} f(W_0)\,dW_0=1, \]

whence:

\[ f(W_0)=\frac{1}{2}\sqrt{\frac{W_{0\min}}{W_0^3}}, \]

if the emission of electrons in any direction is regarded as equiprobable, i.e., if one assumes the absence of polarization of the X-rays and of absorption in the polycrystalline body.

The integral effect from the plane \((\alpha,\ \beta,\ \gamma)\) is obtained by integrating (24) over all possible values of \(W_0\):

\[ \overline{K}(W)=\int K(W)\cdot f(W_0)\cdot dW_0. \]

We shall make here the following substitution:

\[ f(W_0)=f(W_{0\min})=\frac{1}{2}W_{\min}, \]

since \(f(W_0)\) is noticeably different from zero only near \(W_0=W_{0\min}\). Then we obtain:

1) for \(W_{0\min}+V_{\alpha\beta\gamma}<W\):

\[ \overline{K}(W)=\varkappa(W)\left\{1-\frac{1}{2W_{0\min}}\left(W-W_{0\min}-\sqrt{(W-W_{0\min})^2-V_{\alpha\beta\gamma}^{\,2}}\right)\right\}; \]

2) for \(W_{0\min}-V_{\alpha\beta\gamma}<W<W_{0\min}+V_{\alpha\beta\gamma}\):

\[ \overline{K}(W)=\varkappa(W)\left[1-\frac{(W-W_{0\min})}{2W_{0\min}}\right]; \]

3) for \(W<W_{0\min}-V_{\alpha\beta\gamma}\):

\[ \overline{K}(W)=\varkappa(W)\left[1+\frac{1}{2W_{0\min}}\left(W_{0\min}-W-\sqrt{(W_{0\min}-W)^2-V_{\alpha\beta\gamma}^{\,2}}\right)\right]. \]

In Fig. 18 the dashed line represents the curve of the course of the absorption coefficient \(\varkappa(W)\) in the absence of perturbations. The solid line represents the course of the absorption \(\overline{K}(W)\) in the presence of a discontinuity plane near \(W_{0\min}\). The distance between the maximum and the minimum of the fluctuation is equal to \(2V_{\alpha\beta\gamma}\), the full width is of the order of \(5V_{\alpha\beta\gamma}\); the height of the fluctuation is:

\[ \Delta \overline{K}=\varkappa(W)\cdot \frac{V_{\alpha\beta\gamma}}{W_{0\min}}. \]

Fig. 18. Fluctuation of the absorption coefficient in the presence of one discontinuity plane \((\alpha,\beta,\gamma)\).

Obviously, it must further be multiplied by the repeatability factor \(H_{\alpha\beta\gamma}\) of the planes \((\alpha,\beta,\gamma)\).

According to Bethe, for a cubic lattice:

\[ V_{\alpha\beta\gamma}\cong \frac{Z}{a\cdot S^2}, \]

where \(S^2=\alpha^2+\beta^2+\gamma^2\). Hence it follows that the fine structure is the brighter, the larger the atomic number \(Z\) of the absorbing atom and the smaller the lattice constant \(a\).

The quantity \(V_{\alpha\beta\gamma}\) for pure metals and in alloys whose components can be neglected is proportional to the structural factor of the plane \((\alpha,\beta,\gamma)\), which is usually well known. Indeed, the coefficients \(V_{\alpha\beta\gamma}\) are determined from the condition:

\[ V=\sum V_{\alpha\beta\gamma}\, e^{2\pi i(\mathbf{g}\cdot\mathbf{r})}, \]

where \(\mathbf{g}(\alpha,\beta,\gamma)\) and \(\mathbf{r}(x,y,z)\) are the vectors of the reciprocal and direct lattices, respectively.

According to Poisson,

\[ \nabla^{2}V=-4\pi^{2}g^{2}\sum V_{\alpha\beta\gamma}e^{2\pi i(\mathbf{g}\cdot\mathbf{r})}=-4\pi\rho, \tag{25} \]

where \(\rho\) is the density of the distribution of electric charges in the lattice. If we denote the numerical density of the charges of the atomic nuclei by \(\rho^{k}\), and the numerical density of the charges of the electrons by \(\rho^{E}\), then:

\[ \rho=(\rho^{k}-\rho^{E})\varepsilon =\sum(\rho^{k}_{\alpha\beta\gamma}-\rho^{E}_{\alpha\beta\gamma})\cdot \varepsilon \cdot e^{2\pi i(\mathbf{g}\cdot\mathbf{r})}, \tag{26} \]

where \(\varepsilon\) is the charge of the electron.

Substituting (26) into (25) and comparing the coefficients of the corresponding terms in the expansions of the left- and right-hand sides of equality (25), we find:

\[ V_{\alpha\beta\gamma} = \frac{\varepsilon}{\pi g^{2}} (\rho^{k}_{\alpha\beta\gamma}-\rho^{E}_{\alpha\beta\gamma}). \tag{27} \]

Let us represent the numerical charge densities by the following expansions:

\[ \rho^{k}_{\alpha\beta\gamma} = \sum_{j} e^{-2\pi i(\alpha x_{j}+\beta y_{j}+\gamma z_{j})}\cdot Z_{j}, \tag{28} \]

\[ \rho^{E}_{\alpha\beta\gamma} = \sum_{j} e^{-2\pi i(\alpha x_{j}+\beta y_{j}+\gamma z_{j})} \cdot \int_{0}^{\infty}\rho_{j}(r)\cdot r\cdot \sin(2\pi g\cdot r)\,dr, \tag{29} \]

where \(x_{j}\), \(y_{j}\), and \(z_{j}\) are the coordinates of atom \(j\) in the unit cell, \(Z_{j}\) is its atomic number, and \(\rho_{j}(r)\) is the electron density (assumed to be spherically symmetric). The integral in the last expression is the atomic factor and can be calculated with the aid of the Hartree or Thomas–Fermi atomic fields. The latter method makes it possible, for heavy atoms, to replace the atomic factor by the product \(Z_{j}F_{j}\), where \(F_{j}\) is a comparatively slowly and monotonically varying function of \(Z_{j}\). Having made this replacement in (29) and substituted (28) and (29) into (27), we obtain:

\[ V_{\alpha\beta\gamma} = \frac{\varepsilon}{\pi g^{2}} \sum_{j}e^{-2\pi i(\alpha x_{j}+\beta y_{j}+\gamma z_{j})} \cdot Z_{j}(1-F_{j}). \]

In metals or alloys with close values of \(Z_{j}\), the individual components can represent \(V_{\alpha\beta\gamma}\) in the following form:

\[ V_{\alpha\beta\gamma} = \frac{\varepsilon}{\pi g^{2}}Z(1-F)\cdot Q_{\alpha\beta\gamma}, \tag{30} \]

where

\[ Q_{\alpha\beta\gamma} = \sum_{j} e^{-2\pi i(\alpha x_{j}+\beta y_{j}+\gamma z_{j})} \]

is the structure factor. According to (30), with increase of the vector \(\mathbf{g}\), \(V_{\alpha\beta\gamma}\) decreases monotonically as \(1/g^{2}\). Thus, the principal factor determining the character of the variation of \(V_{\alpha\beta\gamma}\) is the structure factor \(Q_{\alpha\beta\gamma}\).

The position of the fluctuation is determined by (21). From this formula it follows that lattices of one type must give similar fine

structures. The positions of the corresponding fluctuations in cubic lattices must be inversely proportional to the squares of the constants of the $\alpha$-lattices. This last conclusion may serve to check the correctness of Kronig’s theory.

The distances between the fluctuations caused by individual planes are usually considerably smaller than the width of the fluctuations, as a result of which the latter overlap, and only the result of such a superposition can be observed. Thus, the experimentally observed maxima and minima of the fine structure are the result of groupings of individual fluctuations. This, however, does not exclude the dependence indicated above of the positions of the elements of the fine structure on the square of the constant of the $\alpha$-lattice.

Fig. 19. Kronig bar diagram.

For comparing the theoretical data obtained from Kronig’s theory with experiment, it should be taken into account that the quantities $W$ in formula (21) are measured from the mean potential of the lattice, whereas the experimental distances of the individual maxima and minima of the fine structure are measured from the energy corresponding to the Fermi surface. This makes it necessary to add to the experimental data the appropriate correction, whose magnitude is of the order of 10 eV.

In Fig. 19 there is presented the so-called “bar diagram” of Kronig for the CuBe alloy, according to the work of Smoluchowski[^46]. Along the abscissa is plotted the energy calculated by formula (21). For each of the values $W$ obtained from this formula, a bar is constructed whose height is proportional to the coefficient $V_{\alpha\beta\gamma}$ or simply to the structural factor $Q_{\alpha\beta\gamma}$. The individual groupings of bars constructed in this way must correspond to individual experimental fluctuations of the absorption coefficient in the region of the fine structure. Taking into account the form of an elementary fluctuation according to Kronig’s theory (Fig. 18), it is evident that a grouping must begin with a maximum and end with a minimum of the absorption coefficient. It is customary to denote the first maximum of the experimental curve from the main edge by $A$, and the first following minimum by $\alpha$. The second pair—maximum and minimum—is denoted, respectively, by $B—\beta$, then $C—\gamma$, and so on. Thus, the middles of each pair $A—\alpha$, $B—\beta$, $C—\gamma$, etc., must correspond to the middles of the theoretical groupings of the diagram

Fig. 20. The $K$-edge of copper in CuBe.

Kronig. The corresponding designations are indicated at the top of Fig. 19. In Fig. 20 an experimental curve is presented for the course of the absorption coefficient of the Cu K-edge in CuBe. Table 4 gives the experimental data (to which the above-mentioned correction of 10 eV has been added) and the theoretical data, i.e., the midpoints of the groupings according to Fig. 19.

As is evident from the table, Kronig’s theory is not applicable to the section of the absorption curve near the main edge, but far from it the agreement of the data is good.

Table 4

Positions of the fluctuations of the Cu K-edge in CuBe

	Experimental	Theoretical
$A-\alpha$	36 eV	65 eV
$B-\beta$	92	100
$C-\gamma$	146	143
$D-\delta$	180	184
$E-\varepsilon$	220	223

The region of fine structure near the main edge is due to electrons which, in the process of absorption, enter levels at which the forces of binding with the atom cannot be neglected and the theory of almost free electrons, which underlies the exposition of Kronig’s theory of the spectrum, cannot be applied. Therefore this region is poorly interpreted by Kronig’s theory.

Kurilenko^29 attempted to construct theoretically the shape of the absorption curve, representing the elementary fluctuations by isosceles triangles (instead of Kronig columns), whose height is proportional to the structure factor $Q_{\alpha\beta\gamma}$ of the corresponding planes, while the width of the base was chosen arbitrarily, in order to obtain the best agreement with experiment. As a result of such constructions it proved possible to obtain a very good reproduction, in the shape of the curves, of the experimental course of the absorption coefficient in the region of several hundred eV on the short-wavelength side of the main absorption edge.

Let us turn to a survey of individual works on the fine structure of absorption spectra.

In 1937 Hanawalt^39 observed that, with increasing temperature, the fine structure of the iron K-edge becomes blurred and disappears; moreover, this process begins far from the edge and gradually embraces the entire structure. As was subsequently shown by Kurilenko^29 and Coster and Levi^40, this blurring occurs near the transition point of $\alpha$-iron into $\gamma$-iron; at temperatures above this point (920°C) the fine structure appears again, but has an entirely different character, since $\alpha$-Fe has a body-centered cubic lattice, while $\gamma$-Fe has a face-centered cubic lattice. Thus, near the transition point a gradual rearrangement of the lattice takes place.

F. Gal’perin^41 showed theoretically that the same blurring of the fine structure can be caused by a sufficiently large gradient of the electric field in dielectrics. However, no experimental work in this direction has been carried out.

Koster and Kleimer \(^{42}\) showed that the fine structure of the K-edges of nickel in the pure metal and in the NiFe alloy, and of iron in the NiFe alloy, has the same character (these substances have a face-centered cubic lattice). Iron and chromium in pure metals have fine structures quite similar to each other, but already of another type. These two metals crystallize in a body-centered cubic lattice.

Koster and Smoluchowski \(^{43}\) studied the K-edges of \(\alpha\)-, \(\beta\)-, \(\gamma\)-, and \(\varepsilon\)-brass. The fine structures of Cu and Zn in this brass proved to be completely identical. The authors compared the fine structures of brasses and of the pure metals Cu and Zn for the same type of lattice. For this comparison formula (21) was used, from which it follows that the energy distances \(W\) of similar elements of the fine structure (maxima and minima) from the main edge in different substances with the same lattice must be inversely proportional to the square of the lattice constant \(a\) (for a cubic lattice):

\[ W_i \cdot a^2 = \mathrm{const}. \tag{31} \]

In the case of hexagonal lattices, a comparison may be made by the formula:

\[ W_i(2a^2 + c^2) = \mathrm{const}. \tag{31'} \]

As Koster and Smoluchowski showed in the work under consideration, these formulas are very well justified far from the edge, where Kronig’s theory should give a good approximation. The same comparison was made for \(\beta\)-brass and \(\alpha\)-iron (both have a body-centered cubic lattice). In all these cases, very good agreement was obtained for the values (31) for similar elements of the fine structure.

Koster \(^{44}\) studied the K-edges of atoms in various close-packed lattices; the edges of calcium (face-centered cube) and titanium (hexagonal lattice) were compared, for which in the first case formula (31) was used, and in the second—(31′). The nearest several atomic layers surrounding calcium and titanium (in the pure metals) are almost identical. Therefore one should expect that the values (31) for calcium and (31′) for titanium will be close for similar elements (maxima and minima) of the fine structure, which was well confirmed by experiment. The same comparison was made for \(\alpha\)- and \(\varepsilon\)-brasses, also with good results.

Feldkamp \(^{32}\) studied the \(L_I\)-, \(L_{II}\)-, and \(L_{III}\)-edges of tantalum, tungsten, platinum, and gold. A comparison was made of the fine-structure plots of all three edges for each of these elements. On the plots, the energy distances of maxima (long strokes) and minima (short strokes) from the main edge are laid off on the energy scale. Fig. 21 shows such a comparison for Au. The plots are arranged so as to obtain the best agreement of the elements of the fine structure.

structure. It turned out that, for all four elements, the zeros of the energy scale (i.e., the positions of the principal edge) for \(L_{\mathrm{II}}\) and \(L_{\mathrm{III}}\) coincide, while for \(L_{\mathrm{I}}\) they are shifted by 15–20 eV toward higher energies. This indicates that, in the process of absorption of \(L_{\mathrm{II,III}}\)-electrons, according to the selection rules, transitions occur to \(5d, 6s\), while \(L_{\mathrm{I}}\)-electrons—to \(6p\). As was indicated above, the investigation of the fine structure of these spectra leads to the same result.

In the same work Feldkamp\({}^{32}\) compared, according to (31), the \(K\)-edge of \(\alpha\)-iron and the \(L_{\mathrm{III}}\)-edge of tungsten; both metals crystallize in a body-centered cubic lattice. The comparison gave good agreement at large distances from the edge (more than 100 eV).

Fig. 21. Scheme of the fine structure of the L-edges of gold.

Smoluchowski\({}^{45}\) compared, according to (31), the fine-structure elements of the \(K\)-edges of copper and zinc in the alloy \(\mathrm{Cu}_{5}\mathrm{Zn}_{8}\) and in \(\beta\)-brass; of zinc and silver in the alloy \(\mathrm{Ag}_{5}\mathrm{Zn}_{8}\), and of iron in pure \(\alpha\)-Fe. In all cases very good agreement was obtained.

In another work Smoluchowski\({}^{46}\) investigated the \(K\)-edge of copper in the alloy CuBe. This alloy is interesting in that the atomic numbers of the components are far apart and, as indicated above, in this case the coefficients \(V_{\alpha 3\gamma}\) cannot, according to (30), be considered proportional to the structural factors \(Q_{\alpha 3\gamma}\) of the corresponding planes. CuBe has a cubic body-centered lattice of the CsCl type. If the components of the alloy under investigation were neighbors, the lattice would be of the \(\beta\)-brass or \(\alpha\)-iron type. However, comparison according to (31) of the fine-structure elements of copper in CuBe with analogous data for \(\beta\)-brass and \(\alpha\)-iron proved impossible.

If, in view of the small scattering power of beryllium atoms, one simply neglects them and regards the CuBe lattice as consisting only of copper atoms, i.e., replaces this lattice by a simple cubic lattice, then agreement with Kronig’s theory is obtained as very good.

In the process of absorption of X-rays, photoelectrons leave the atom in the lattice predominantly in the plane perpendicular to the direction of propagation of the X-rays, since the electric vector is perpendicular to the ray. Therefore the fine structure obtained upon absorption by a single crystal must depend on the orientation of the crystal with respect to the ray. In Kronig’s theory (see above), the absorption has been averaged over all directions, on the assumption that the absorber is polycrystalline. This averaging for a single crystal must be done differently. In order to test the influence of the orientation of a single crystal on the fine ...

structure. Levitskii (unpublished work) investigated the \(K\)-absorption spectrum of zinc in monocrystalline, differently oriented thin foils. However, owing to insufficient resolving power, no changes could be observed.

Kurilenko \(^{29}\) compared the fine structure of the \(K\)-edge of annealed and rolled copper. The individual maxima and minima did not shift appreciably, but their shape changed noticeably. This may be explained by the influence of the predominant orientation of the crystallites of the rolled copper. In addition, disturbances of the lattice levels as a result of rolling may also be significant.

It is interesting to investigate the influence, on the fine structure, of the degree of ordering of the distribution of atoms among the lattice sites in alloys. As is known, with a completely ordered arrangement of atoms in alloys, superstructure lines appear. Consequently, for a number of lattice planes for which, under a statistically uniform (random) distribution of atoms among the sites, the structure factor is zero, with a completely ordered arrangement the structure factor is not zero. In constructing a Kronig bar diagram, in the second case the number of bars will increase, which should affect the fine structure of the absorption spectrum, especially for bright superstructure lines. In addition, as Muto \(^{47}\) showed, the form of the elementary fluctuation, its height and width, should depend on the degree of ordering.

Smoluchowski \(^{46}\) investigated the \(K\)-edge of copper in the alloy \(\mathrm{AuCu}_3\), which at temperatures above \(400^\circ\mathrm{C}\) has a statistically uniform arrangement of atoms, while at a lower temperature, by prolonged annealing, almost complete ordering can be obtained without a change of lattice. As the experiment showed, the maxima and minima did not shift, but the shape of some of them changed noticeably.

From the foregoing review it is evident that Kronig’s theory explains the experimental data well in all cases in which the theory of nearly free electrons is applicable, i.e. for metals and alloys far from the main absorption edge, at distances of more than 50–100 eV.

Let us now turn to a review of studies of the fine structure of the absorption spectra of nonmetallic compounds. Smoluchowski \(^{46}\) investigated the \(K\)-edge of nickel in nickel oxide \(\mathrm{NiO}\). The quantities \(W_i\) of this spectrum, calculated by formula (31), were compared with the corresponding quantities for the \(K\)-spectrum of calcium in \(\mathrm{CaS}\), since both these substances crystallize in identical lattices of the \(\mathrm{NaCl}\) type. In addition, a comparison was made with the \(K\)-spectrum of pure copper, which crystallizes in a simple face-centered cubic lattice. All three series of quantities have very close values of the distances of the corresponding maxima and minima of the fine structure from the main edge. The possibility of comparing the spectra of nickel in \(\mathrm{NiO}\) and of pure copper is explained by the fact that

the atomic number of nickel is much greater than the atomic number of oxygen; consequently, the scattering power of the latter may be neglected in comparison with that of the former. If all the oxygen atoms are removed from the NiO lattice, there remains a simple face-centered cubic lattice—the same as in pure copper. However, Smoluchowski’s result is rather an exception than the rule.

Rule\(^{38}\) compared, according to (31), the K-spectra of sodium and potassium in halide compounds. Although the author finds the comparison successful, a simple glance at the diagrams he presents convinces us of the rather poor agreement of these data.

Koster and Kleimer\(^{42}\) investigated the K-spectra of potassium and chlorine in KCl and KClO\(_3\). All four spectra proved to be entirely different. If in KClO\(_3\) the surroundings of potassium and chlorine are different, then in KCl they are the same. Moreover, the structure of the electron shells K\(^{-}\) and Cl\(^+\) in KCl is the same—of the argon type. Thus, it was natural to expect that the spectra of these ions in KCl would be identical. The experimental result indicated above shows that the energetic role of the cation and anion in the KCl lattice is different.

Stephenson\(^{48}\) investigated the K-edges of two components of various ionic and homopolar compounds. As a result of this investigation Stephenson arrives at the following conclusions:

1. In the case of different surroundings of two components of one ionic lattice, their spectra are different.

2. In the case of identical surroundings (RbBr was investigated), the spectra are identical. This result contradicts the result of Koster and Kleimer for KCl.

3. Homopolar compounds do not give a reproducible fine structure.

From the review presented it follows that Kronig’s theory is not applicable to nonmetallic compounds, since these compounds have no generalized free electrons, and band theory cannot be applied.

We thus see that Kronig’s theory has, in general, a rather limited application to metals and alloys with closely related components, far from the principal edge. In all other cases this theory cannot claim to explain the observed facts. Moreover, the determination of the energetic position of the elements of the fine structure from column diagrams 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 experimental results. Kronig constructed his theory taking into account the general properties of the metallic lattice. This is a theory of “long-range order.” The fine structure is ascribed to changes in the density of electron states in the lattice.

Kostarev\(^{49}\) developed a theory of “short-range order,” taking into account the nearest environment of the absorbing atom. The fine structure is attri-

is ascribed to changes in the transition probability of the electron torn away in the process of absorption. This theory is based on the same principle as the Hartree, Kronig, and Petersen theory of absorption in polyatomic molecular gases.^9 The similarity or difference of the long-range order determines the similarity or difference of the short-range order, but not conversely. Therefore, in cases where Kronig’s theory is applicable, Kostarev’s theory should also give good results. In addition, this latter theory may be expected to explain a whole series of other facts as well, for example, the above-mentioned similarity of the spectra of densely packed hexagonal and cubic lattices (in Koster’s work^44).

The rather complicated formula obtained by Kostarev was applied to the calculation of the fine structure of the copper K-edge, taking into account only the atoms of the nearest coordination shell. Table 5 gives a comparison of the results of this theory with the experimental data.

Table 5

Fine structure of the copper K-spectrum.
Distances from the main edge in eV.

	$A$	$\alpha$	$B$	$\beta$	$C$	$\gamma$	$D$	$\delta$	$E$	$\varepsilon$	$F$	$\zeta$	$G$
Theory	9	23	40	60	84	112	142	176	213	254	297	344	395
Experiment	17	16	40	56	82	108	144	170	220	250	299	340	397

Here, as usual, $A, B, C, \ldots$ are maxima, and $\alpha, \beta, \gamma, \ldots$ are absorption minima. As may be seen, Kostarev’s theory explains the fine structure well, except for the maximum nearest to the main edge, which lies in the region of the Kossel structure.

Kostarev next compares the midpoints of the pairs $A\alpha$, $B\beta$, $C\gamma$ of the copper K-spectrum according to Kronig’s calculations, according to his own theory, and according to the experimental data. The results are presented in Table 6.

Table 6

Fine structure of the copper K-spectrum.
Distances from the main edge in eV.

	$A\alpha$	$B\beta$	$C\gamma$	$D\delta$	$E\varepsilon$	$F\zeta$
According to Kronig	—	62	94	152	215	—
According to Kostarev	16	50	98	159	234	321
Experiment	21	48	95	157	235	320

As may be seen, Kostarev’s theory explains the copper spectrum better than Kronig’s theory.

In conclusion of the present article, it should be pointed out that it is desirable to study jointly the distribution of electrons and the structure of matter by all physical methods: by X-ray spectra, optical spectra, magnetic methods, the internal photoelectric effect, dielectric properties, etc. Only such a comprehensive study can yield reliable results and solve the question of the nature of the chemical bond, as well as explain the various chemical and physical properties of matter.

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