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X-RAY SPECTRA AS A METHOD FOR STUDYING THE DISTRIBUTION OF ELECTRONS OVER STATES
M. Blokhin
I. Experimental methods.
II. Emission spectra. 1. General survey. 2. The influence of chemical bonds on the principal lines of the spectrum. 3. The width of bands and the determination of the number of outer electrons. 4. The shape of bands and the determination of the distribution of outer electrons over states. 5. The theory of satellites.
III. Absorption spectra. 1. General information. 2. Monatomic gases. 3. Diatomic gases. 4. Polyatomic gases. 5. Liquids. 6. Solids: a) The principal edge. General theory. b) The position of the principal edge. c) The Kossel structure of absorption. d) The Kronig structure of absorption.
X-ray emission series of spectral lines are formed in transitions of the electrons of an atom from higher-lying levels to one of the lower inner levels. Absorption bands of X-ray absorption spectra arise in transitions of electrons from one of the inner levels of the atom to its periphery.
In emission spectra, lines formed in transitions of the peripheral electrons of the atom to one of its inner levels are of special interest. In this case, as also in absorption spectra, the inner level is fixed. Therefore such emission lines (bands), together with absorption spectra, make it possible to judge the distribution of the allowed energy levels at the periphery of the atom and the filling of these levels with electrons.
This makes it possible to draw important conclusions about the chemical bonding of atoms and ions in gas molecules and in the crystal lattice of solids. Herein lies the exceptional advantage precisely of X-ray spectra, since visible and ultraviolet spectra appear in transitions of valence electrons between two peripheral levels, each of which is part of broad, diffuse energy bands. Therefore these spectra only in individual cases (ab-
combinational spectra of rare earths and alkali halides, where one of the levels is fixed and sufficiently narrow) can give analogous results in the study of the distribution of electrons and of the free levels of the atom and the crystal lattice.
I. EXPERIMENTAL METHODS
The usual conception of X-ray spectra as arising in the depths of the atom, and therefore as not depending on the environment of the atom, proves untenable in the study of spectra associated with the peripheral levels of the atom. Such spectra not only depend on the environment of the atom, but also make it possible to determine in detail the type of chemical bond in each particular case. Therefore experimental work in this field should be carried out with great precautions, so that, in exciting X-rays, the substance under investigation is not destroyed.
The most widely used instrument for exciting X-ray emission spectra is the demountable X-ray tube, the various designs of which have been sufficiently described in our literature¹.
The substance under study is placed on the anode of the X-ray tube. In order to avoid chemical reactions on the anode (oxidation, decomposition) and volatilization of the substance, it is necessary, first, to have a good vacuum, of the order of \(10^{-6}\) mm Hg, and, secondly, to reduce as far as possible the specific loading of the anode, i.e. the number of milliamperes of anode current per \(1\ \mathrm{mm}^2\) of the focal-spot surface. The latter can be achieved by reducing the anode current and increasing the exposure time, or by using a Geiger counter as an especially sensitive instrument for recording X-rays.
Good results can also be obtained at a considerable anode current by increasing the size of the focal spot to \(100\text{—}150\ \mathrm{mm}^2\), which makes it possible to use, for obtaining the spectrum, a bent-crystal spectrograph², giving a considerable shortening of the exposure.
The most radical method of eliminating heating of the substance under study is the method of secondary excitation of spectra, i.e. obtaining X-ray fluorescence¹˒³ under the action of primary X-rays. This method gives the best results, but requires either very long exposures or the use of a Geiger counter.
To obtain the greatest resolving power there are two ways. One of them consists in using a double crystal spectrometer¹. The rays from the source pass through a collimator to the first crystal, which selects a comparatively narrow interval of the spectrum. After reflection from the first crystal, which remains stationary during the study—
In studying this narrow interval of the spectrum, the rays fall on a second crystal, which during the study is rotated within the range of Bragg angles corresponding to the chosen spectral interval. In each of its positions the second crystal selects an extremely narrow portion of the spectrum, “one line.” This portion is recorded by a Geiger counter. The well-developed theory of this method makes it possible to take into account the distortions introduced by the crystals.^4 This method has the disadvantage that, with the existing technique of working with a Geiger counter, only a few thousand quanta per minute can be counted. For a desirable counting accuracy of \(1\%\), it is necessary to count 10,000 quanta, and, taking into account possible fluctuations in the operating regime of the X-ray tube, even 20,000 quanta, which on average takes about eight minutes for each point. Thus, recording the intensity-distribution curve of a single line requires several hours.
Another route, giving high resolving power with photographic recording and short exposures, consists in the use of ultralong-wavelength X-ray spectra, in the region from 25 to several hundred angstroms. To obtain spectra in this region, high-vacuum spectrographs with concave diffraction gratings^1 of radius \(1\text{–}2\ m\), with as many as 1200 rulings per mm, are used. With increasing wavelength the dispersion also increases, i.e. the number of electron-volts (eV) per mm of spectral length decreases, owing to which ultralong-wavelength X-ray spectra are especially suitable for studying the distribution of valence electrons.
In studying absorption spectra, the absorber under investigation is placed between the X-ray tube and the spectrograph. To obtain the greatest contrast, the thickness of the absorber must be calculated in advance. The absorption spectrum consists of the principal sharp jump of absorption, called the main absorption edge, on both sides of which the absorption coefficient \(\mu\) has substantially different values; on the short-wavelength side of the main edge there is observed the so-called fine structure of absorption, consisting of small fluctuations of the absorption coefficient.
To obtain the greatest contrast of the main absorption edge, the thickness \(s\) of the specimen is chosen so that the difference \(Q\) of the intensities \(I_2\) and \(I_1\) on the two sides of the edge is greatest:
\[ Q = I_2 - I_1 = I_0 e^{-\mu_2 s} - I_0 e^{-\mu_1 s}, \]
where \(\mu_2\) and \(\mu_1\) are the values of the absorption coefficient on the two sides of the main edge. Equating \(\dfrac{dQ}{ds}\) to zero, we find:
\[ s = \frac{\lg \mu_2 - \lg \mu_1}{\mu_2 - \mu_1}. \tag{1} \]
In the usual region of spectra this thickness reaches from several microns to several tens of microns.
To obtain the greatest contrast of fluctuations of the fine structure⁵, one seeks the conditions for the maximum of the small fluctuation of intensity \(I\) corresponding to the small fluctuation of the absorption coefficient \(\mu\). If
\[ I=I_0 e^{-\mu s}, \]
then:
\[ \Delta I=-\Delta\mu\cdot s\cdot I_0 e^{-\mu s}. \]
From the condition:
\[ \frac{\partial(\Delta I)}{\partial s}=0 \]
we find the condition for the maximum contrast of the intensity fluctuation:
\[ s=\frac{1}{\mu}, \tag{2} \]
where the value of \(\mu\) should be taken on the short-wavelength side of the principal edge. The thickness \(s\), determined by this formula, is approximately half the thickness \(s\) determined by formula (1).
When photographing ultrasoft absorption spectra there is almost no continuous spectrum against whose background the absorption spectra stand out. In this case one has to create a surrogate continuous spectrum in the form of a dense line spectrum of a vacuum high-voltage spark⁶. The intensities of the individual lines of this spectrum are measured without the absorber, \(I_0\), and with it, \(I\), after which the absorption coefficient can be calculated from the formula:
\[ \mu=\frac{1}{s}\cdot \lg\frac{I_0}{I}, \]
where \(s\) is the thickness of the absorber. This work is very laborious, especially with photographic recording of spectra, when each point must be recalculated according to the calibration curve giving the dependence of photographic density on intensity for the given plate. Absorber thicknesses in the ultrasoft region are of the order of from 0.1 to 1 micron and in many cases require extremely careful handling in order to prevent oxidation or recrystallization. Thin metallic films are obtained by evaporation in a high vacuum. Some foils, for example lithium foils, cannot be brought out into the air; therefore, after their preparation and throughout the experiment they must remain in a high vacuum⁶.
For the study of ultrasoft spectra of binary alloys, apparently the best method for preparing foils is the method of S. A. Vekshinsky⁷.
II. EMISSION SPECTRA
1. General Survey
The inner levels of the atom have received the designations: \(K\) (or \(1s\)), \(L(2s)\), \(L_{\mathrm{II}}, L_{\mathrm{III}}(2p)\), \(M_{\mathrm{I}}(3s)\), \(M_{\mathrm{II}}, M_{\mathrm{III}}(3p)\), \(M_{\mathrm{IV}}, M_{\mathrm{V}}(3d)\), \(N_{\mathrm{I}}(4s)\), \(N_{\mathrm{II}}, N_{\mathrm{III}}(4p)\), etc. By the energy of a level one should understand the energy of an ionized atom with the electron absent from the given level. The difference of the energies of two levels, between which the electron makes its transition, gives the energy of the emitted quantum. It is customary to express the energy of a quantum in rydbergs, i.e. in units \(\nu/R\), where \(\nu=1/\lambda\) is the wave number, and \(R=109737\ \mathrm{cm}^{-1}\) is the Rydberg constant. The letters \(K\), \(L_{\mathrm{II}}\), \(M_{\mathrm{III}}\), etc. also denote the energy (terms) of the corresponding levels, expressing this energy likewise in rydbergs, or in electron-volts (\(1\) rydberg \(=13.547\) e.v.). The energy of a quantum, for example, of the line \(K\alpha_1\), emitted in the transition of the atom from the state of \(K\)-ionization to the state of \(L_{\mathrm{III}}\)-ionization (i.e. in the transition of an electron from the level \(L_{\mathrm{III}}\) to the vacant level \(K\)), is equal to:
\[ (\nu/R)_{K\to L_{\mathrm{III}}}=K-L_{\mathrm{III}}. \]
Lines formed in this way are called diagram lines, in contrast to non-diagram weak lines, which do not fit into the diagram of atomic states.
Diagram lines are obtained only under certain selection rules, first established empirically, according to which only transitions are allowed between atomic levels differing in the principal quantum numbers \(n\), while the subsidiary quantum numbers \(l\) must differ by unity. These are the so-called dipole selection rules.
In addition, considerably weaker lines may appear in the spectrum under the so-called quadrupole selection rules, when the subsidiary quantum numbers \(l\) of the levels between which the transition takes place differ by two. Such are the lines \(\beta_4(K\to N_{\mathrm{IV,V}})\) and \(\beta_5(K\to M_{\mathrm{IV,V}})\) in the \(K\)-series, \(\beta_{10}(L_{\mathrm{I}}\to M_{\mathrm{IV}})\), \(\beta_9(L_{\mathrm{I}}\to M_{\mathrm{V}})\) and \(\beta_7(L_{\mathrm{III}}\to N_{\mathrm{VI,VII}})\) in the \(L\)-series.
There also exists a series of very weak lines whose transitions satisfy neither the dipole nor the quadrupole selection rules. Nevertheless, these lines fit well into the diagram of atomic states. Such are the lines \(t\) \((L_{\mathrm{III}}\to M_{\mathrm{II}})\) and \(s\) \((L_{\mathrm{III}}\to M_{\mathrm{III}})\) in the \(L\)-series, the line \((K\to M_{\mathrm{I}})\) in the \(K\)-series of chlorine\(^8\), \((M_{\mathrm{I}}\to M_{\mathrm{II}})\), \((M_{\mathrm{I}}\to M_{\mathrm{III}})\), \((M_{\mathrm{II}}\to M_{\mathrm{IV}})\) and \((M_{\mathrm{III}}\to M_{\mathrm{IV,V}})\) in the \(M\)-series, \((N_{\mathrm{IV}}\to N_{\mathrm{VI}})\) and \((N_{\mathrm{V}}\to N_{\mathrm{VI,VII}})\) in the \(N\)-series. The presence of all these lines shows that the “prohibition” by a selection rule is only a question of intensity, and not an actual prohibition\(^9\).
An interesting group of lines is represented by the so-called semi-optical lines. These are ordinary diagram lines, arising in transitions of electrons from upper, peripheral levels that are not occupied by electrons in the normal state of the given atom. For these lines to appear, it is necessary first to excite lower-lying electrons to the upper optical levels, i.e., the same mechanism that is needed for the excitation of optical spectra. This excitation may have different origins. First, it may be caused by the impact of an extraneous electron or quantum (in a gas—by collision with another atom). Second, there may be a constant “excitation” owing to the action of neighboring atoms or ions in a molecule or solid, which deform the electron shell of the atom. The semi-optical lines include, for example, \(K\beta_1(K \to M_{\mathrm{II,III}})\) in \(11\,\mathrm{Na}\) and \(12\,\mathrm{Mg}\), since in normal unexcited atoms the level \(M_{\mathrm{II,III}}\) begins to be filled only with \(13\,\mathrm{Al}\); \(K\alpha(K \to L_{\mathrm{II,III}})\) in \(3\,\mathrm{Li}\) and \(4\,\mathrm{Be}\), since the level \(L_{\mathrm{II,III}}\) is filled only with \(5\,\mathrm{B}\); \(K\beta_5(K \to M_{\mathrm{IV,V}})\) in \(11\,\mathrm{Na}\)—\(20\,\mathrm{Ca}\), since the level \(M_{\mathrm{IV,V}}\) begins to be filled only with \(21\,\mathrm{Sc}\), and so on. Obviously, such a definition of semi-optical lines is rather artificial, since in exciting an x-ray spectrum (usually, moreover, in a solid) we are not dealing with a normal, unexcited atom. Therefore another criterion was proposed for defining semi-optical lines. Since absorption transitions take place to the first peripheral level having vacancies, then, when this level is partially filled, the wavelength of the absorption edge will be equal to the wavelength of the last emission line of the given series. When this level is filled, absorption transitions already take place to the next vacant level. Therefore the difference \(\Delta\nu/R\) between the terms of the edge and of the corresponding line undergoes a jump in the element for which the given level is being filled. Graphs of the values of \(\Delta\nu/R\) as a function of atomic number \(Z\) were constructed by Sandström\(^{10,11}\) for the interpretation of absorption spectra. Similar graphs were used by Maitre\(^{12}\) as a criterion for semi-optical lines. Maitre considers that a break in the curve on such a graph corresponds to the element for which the given line is already semi-optical. According to this criterion, for example, the line \(K\beta_1(K \to M_{\mathrm{II,III}})\) is semi-optical for elements with atomic number below \(18\,\mathrm{Ar}\) (Fig. 1), the line \(K\beta_5(K \to M_{\mathrm{IV,V}})\) for elements with atomic number below \(36\,\mathrm{Kr}\), the line \(M\alpha(M_{\mathrm{V}} \to N_{\mathrm{VII}})\) for elements with atomic number below \(68\,\mathrm{Er}\), etc. However,
Fig. 1. Difference of the terms of the K-edge and the \(K\beta_1\) line as a function of atomic number \(Z\).
and this criterion is quite artificial, since it indicates when the given level is completely filled, whereas what is essential for the determination of semiooptic lines is not the filling but the liberation of the given level from electrons. This latter circumstance, for atoms of a solid, can be determined only by indirect methods, which will be discussed below.
On the basis of the calculated energies of the levels of binary compounds, Valasek\(^{55}\) gave a new interpretation of certain lines that had previously been regarded as semiooptic. Thus, for example, the \(K\beta_5\) line of calcium in a CaS crystal arises, according to Valasek, in cross transitions of electrons from the sulfur level \(S-3p^6\) to the calcium \(K\)-level; the \(K\beta_5\) line of potassium in a KCl crystal arises in the transition \(K^+1s \to Cl^-3p\), etc.
With the gradual filling of atomic levels by electrons, with increasing atomic number, the intensity of the emission lines arising from these levels also increases. In Fig. 2, for comparison, are shown the course of the curves for the filling of the \(4p\) level (the number of electrons \(n_{4p}\)) and the relative intensity \(I_{K\beta_2}/I_{K\beta_1}\) of the \(K\beta_2\) line, arising from the \(4p\) level, relative to the \(K\beta_1\) line, arising from the \(3p\) level, which is completely filled throughout the entire series of elements under consideration. The curve for the number of electrons refers to free atoms, whereas the curve for the relative intensity refers to solid metals, in which, owing to the action of the potential field of the metal lattice, part of the electrons from the \(4p\) level is excited to higher levels, as a result of which the \(4p\) level is filled considerably later than in the free atom, namely at \(Z=44\), instead of \(Z=36\) in the free atom. Thus, the true course of the filling of the \(4p\) level in metals can be traced from the given curve of the relative intensity of the \(K\beta_2\) line.
Fig. 2. Relative intensity of the lines \(IK\beta_2/IK\beta_1\) and the number of electrons \(n_{4p}\) as a function of atomic number \(Z\).
An analogous phenomenon can also be traced for the \(L\beta_2\) line (Fig. 3). The corresponding curves show that the filling of the \(4d\) level
occurs in metals approximately at \(Z=50\), whereas in free atoms it occurs at \(Z=46\).
In addition to dipole, quadrupole, and semioptical lines, numerous satellites are also observed in the X-ray spectrum; by these we shall mean nondiagram lines in general, irrespective of their origin*). Satellites are usually located in the immediate vicinity of some diagram line, for the most part on the long-wavelength side. The origin of a satellite is closely connected with the mechanism of formation of this diagram line, which we shall therefore call the “parent” line with respect to the satellites adjacent to it. For the most part, in a group of neighboring elements one observes the same satellites, which can be correlated according to Moseley’s law. Usually, however, such a correlation is replaced by a graph expressing the dependence of \(\sqrt{\Delta \nu/R}\) on the atomic number \(Z\), where \(\Delta \nu\) is the difference between the wave numbers of the satellite and of the parent line. Such a graph, giving a linear dependence, is called Moseley’s “pseudodiagram.” Satellites of neighboring elements that fall on the same straight line of the graph receive identical designations. Only a few satellites are observed for elements over a considerable interval of atomic numbers. Thus, for example, in the K-series one of the brightest satellites, \(a_3\), is observed for the elements \(11\mathrm{Na}\)—\(32\mathrm{Ge}\); the satellite \(\eta\) is observed for the elements \(23\mathrm{V}\)—\(32\mathrm{Ge}\); the satellite \(\beta_8\), for the elements \(32\mathrm{Ge}\)—\(47\mathrm{Ag}\). In the L-series the satellite \(\beta'_1\) is known for the elements \(38\mathrm{Sr}\)—\(68\mathrm{Er}\), and the satellite \(\alpha'\), for the elements \(28\mathrm{Ni}\)—\(38\mathrm{Br}\), etc.
Fig. 3. Relative intensity of the lines \(IL\beta_2/IL\beta_1\) and the number of electrons \(n_{4d}\) as functions of the atomic number \(Z\).
Short-wavelength satellites are thin, weak lines; long-wavelength satellites usually have the form of a comparatively broad band, often beginning at the parent line itself. Theories of the origin of satellites will be considered below.
*) Some authors also classify quadrupole and semioptical lines as satellites; but, in order not to create confusion in terminology, we shall retain for satellites the definition given in the text.
2. Influence of the Chemical Bond on the Principal Lines of the Spectrum
When an atom enters into various chemical compounds, the wavelengths, width, and shape of the lines change, especially those which are associated with the outer levels. These changes have been observed chiefly in the region of relatively long wavelengths and are explained by the action upon the outer levels of the atom of its different surroundings.
a) Wavelength of the lines. We shall not cite the numerous data available in the literature which show the change in the wavelengths of the lines of the X-ray spectrum of various elements when they enter into different chemical compounds, since the theory at present is insufficiently developed to explain in detail all the observed changes. We shall give only a few characteristic examples.
Björckson’s investigations of sulfur vapors showed that the wavelength of the \(K\beta_1\) line is equal to 5016.0 X.U. This line appears as the result of the transition of one of the four outer electrons of the sulfur atom from the \(M_{\mathrm{II,III}}\) level to the \(K\) level. In solid sulfur a part of these four outer electrons passes to the higher-lying \(M_{\mathrm{IV,V}}\) level. As a result, sulfur gives a double line \(K\beta_1\) and \(K\beta_x\), with both lines shifted; their wavelengths are 5021.56 X.U. and 5012.76 X.U., respectively; the first of these lines, corresponding to the \(M_{\mathrm{II,III}}\) level, shows that this level has been lowered. The second line, corresponding to the \(M_{\mathrm{IV,V}}\) level, is a typical semioptical line. In solid cuprous sulfide, CuS, sulfur gives one \(K\beta_x\) line, whose wavelength is 5019.4 X.U.
According to the author’s investigation \(^{13}\), the wavelength of the short-wavelength edge of the chromium \(K\beta_5\) line is, in the metal, 2065.83 X.U. This edge of the \(K\beta_5\) line corresponds to the upper boundary of the \(M_{\mathrm{IV,V}}\) level of the chromium atom. The \(M_{\mathrm{IV,V}}\) level in metallic chromium is a broad conduction band. According to present views, in a metal the atoms lose only part of their conduction electrons, often retaining a considerable part of them near themselves. When the chromium atom enters an ionic compound, for example the trivalent oxide \(\mathrm{Cr_2O_3}\), chromium loses three of its valence electrons completely, and the upper boundary of the filled \(M_{\mathrm{IV,V}}\) band is lowered, owing to which the wavelength of the short-wavelength edge of the \(K\beta_5\) line increases and is equal to 2066.50 X.U.; i.e., the energy of the upper boundary of the \(M_{\mathrm{IV,V}}\) level increases by 2.3 eV. When chromium enters an ionic compound of higher valence, for example hexavalent \(\mathrm{K_2CrO_4}\), the chromium atom loses all six of its valence electrons. The \(M_{\mathrm{IV,V}}\) level would be completely emptied if the action of the surroundings in the crystal lattice did not lead to some redistribution of the lower-lying electrons of the \(M_{\mathrm{II,III}}\) level, which pass partly to the \(M_{\mathrm{IV,V}}\) level and give, in this case,
in the case of the typical doublet line \(K\beta_5^0\). This process leads to a decrease in the shielding, by the inner electrons, of the nuclear charge with respect to the level \(M_{\mathrm{IV,V}}\); the effective charge of the atomic residue increases, the energy of the level \(M_{\mathrm{IV,V}}\) increases, and as a result the wavelength of \(K\beta_5\), equal for \(K_2CrO_4\) to 2065.80 XU (the short-wavelength edge), decreases, i.e. approaches the corresponding value for the metal. Simultaneously with the change in wavelength there also occurs a change in the relative intensity of \(K\beta_5\) as compared with the line \(K\beta_1\): on going from the metal to the oxide the ratio of the intensities of these lines decreases almost by a factor of 1.5; on going to hexavalent compounds there is a further decrease in the intensity of \(K\beta_5\). This phenomenon is also in good agreement with the gradual emptying of the level \(M_{\mathrm{IV,V}}\).
b) Line width. The width \(W\) of a spectral line is determined by the sum of the energy widths of the upper \(W_i\) and lower \(W_k\) levels of the corresponding transition \(^{14}\). The width of the level \(W_k\) is inversely proportional to the mean “lifetime” \(\tau_k\) of the atom in the state corresponding to this level. Let us recall that, for example, by the \(K\)-state of an atom is meant an atom with one missing electron in the \(K\)-level. The mean lifetime of an atom in such a state is, obviously, inversely proportional to the total probability \(\Gamma_k\) for the atom to leave this state by any of the possible means, i.e. with the filling of the \(K\)-level by an electron falling from one of the higher levels. Let us denote the probability of a transition from a higher level \(i\) to a lower level \(k\) by \(\gamma_{ik}\). Then the total probability \(\Gamma_k\) is equal to:
\[ \Gamma_k=\sum_{i>k}\gamma_{ik}. \tag{3} \]
Since the width of the level is
\[ W_k \sim \frac{1}{\tau_k}, \]
and the mean lifetime is
\[ \tau_k \sim \frac{1}{\Gamma_k}, \]
the width of the level is
\[ W_k \sim \Gamma_k. \]
In expression (3) let us split the sum into two sums: over transitions giving radiation (radiative) and over transitions not giving radiation (nonradiative). As is known, a whole series of transitions between the possible states of an atom is accompanied by the ejection of an electron, and not of a quantum (Auger effect). The two sums mentioned give the so-called “radiation” width and “Auger width” of the levels. As an example, we give the data of Ramberg and Richtmyer \(^{15}\) on the width of the levels of a gold atom:
Table 1
| Level | Theoretical width in eV: radiative | Theoretical width in eV: Auger width | Theoretical width in eV: total | Experimental width in eV |
|---|---|---|---|---|
| \(K\) | 66.38 | 0.742 | 67.15 | 54 |
| \(L_{\mathrm{I}}\) | 1.0—1.78 | 5.5—11.91 | 6.5—13.69 | 8.7 |
| \(L_{\mathrm{II}}\) | 0.9 | 2.2 | 3.1 | 3.7 |
| \(L_{\mathrm{III}}\) | 1.6 | 2.6 | 4.2 | 4.4 |
| \(M_{\mathrm{I}}\) | 0.078 | 10.234 | 10.31 | 15.5 |
| \(M_{\mathrm{II}}\) | 0.09 | 11.49 | 11.58 | 10.7 |
| \(M_{\mathrm{III}}\) | 0.05 | 4.45 | 4.50 | 12.1 |
| \(N_{\mathrm{I}}\) | 0.003 | 13.601 | 13.60 | 11.7 |
From the table it is evident that the width of the \(K\)-level depends mainly on radiative transitions, whereas the width of levels more remote from the nucleus is determined mainly by Auger transitions. The influence of the latter can be clearly traced in the well-studied radiationless transitions from the state \(L_{\mathrm{I}}\) to the state \(L_{\mathrm{III}}\). As is known, the energy of this transition \(L_{\mathrm{III}}—L_{\mathrm{I}}\) is sufficient to eject from the atom either an electron from the level \(M_{\mathrm{IV}}\) in elements with atomic number \(Z<50\) or \(Z>77\), or from the level \(M_{\mathrm{V}}\) in elements with \(Z<50\) or \(Z>73\). The probability of this effect is so great that the intensity of the lines of the indicated elements, appearing as a result of transitions of higher-lying electrons to the level \(L_{\mathrm{I}}\), is noticeably reduced. This is illustrated\(^ {16}\) in Fig. 4, where the ordinate gives the ratios of the intensities of the lines \(L\beta_3\) and \(L\beta_4\), associated with the level \(L_{\mathrm{I}}\), to the intensity of the line \(L\beta_2\), associated with the level \(L_{\mathrm{III}}\), and the abscissa gives the atomic numbers \(Z\).
Fig. 4. Relative intensities of the lines \(IL\beta_3\) and \(IL\beta_4\) as functions of the atomic number \(Z\).
As can be seen, the relative intensities of the lines \(L\beta_3\) and \(L\beta_4\) decrease with increasing \(Z\) after \(Z=73\) and decrease still more after \(Z=77\). At the same time the width of the level \(L_{\mathrm{I}}\) also increases, since with the appearance of Auger transitions the probability that the atom will leave the \(L_{\mathrm{I}}\)-state increases. This is illustrated\(^ {16}\) in Fig. 5, where the ordinate gives the width of the lines \(L\beta_3\), \(L\beta_4\), \(L\gamma_2\), and \(L\gamma_3\) (initial level \(L_{\mathrm{I}}\)), as well as of the lines \(L\beta_1\) and \(L\gamma_1\) (initial level \(L_{\mathrm{II}}\)) and the line \(L\beta_2\) (initial level \(L_{\mathrm{III}}\)). As can be seen, after \(Z=73\) the width of the lines associated with the level \(L_{\mathrm{I}}\) begins to increase, whereas the width of the other lines does not change.*)
*) An analogous effect of a sharp weakening of the \(L_{\mathrm{I}}\) emission band by several tens of times in comparison with the \(L_{\mathrm{III}}\)-band was found by Skinner\(^ {56}\) in the light elements 11 Na, 12 Mg, 13 Al.
Thus, we see that the line width ultimately depends not only on the given level, but on the probabilities of all transitions of the atom from the given state to all higher-lying ones and, consequently, must depend on chemical bonds. Exact calculations of these effects are not yet possible. We shall cite only some experimental results. According to Rosberry and Birdenu \(^{17}\), upon the transition of metallic titanium to the oxide \(\mathrm{TiO_2}\), the width of \(K\alpha_1\) decreases by \(3\%\), \(K\alpha_2\) increases by \(3\%\), and \(K\beta_1\) increases by \(11\%\). Upon the transition of metallic
Fig. 5. Width of lines of the L-series as a function of atomic number \(Z\).
chromium to the oxide \(\mathrm{Cr_2O_3}\), the width of \(K\alpha_1\) increases by \(35\%\), \(K\alpha_2\)—by \(27\%\), \(K\beta_1\)—by \(34\%\). Upon the transition of metallic manganese to the oxides \(\mathrm{MnO}\) and \(\mathrm{Mn_3O_4}\), its lines broaden: \(K\alpha_1\) by \(22\%\) and \(30\%\) (respectively), \(K\alpha_2\)—by \(9\%\) and by \(16\%\); \(K\beta_1\) does not change. Upon the transition of metallic iron to the oxide \(\mathrm{Fe_2O_3}\) and \(\mathrm{FeS}\), its lines broaden: \(K\alpha_1\) by \(25\%\) and \(16\%\) (respectively), \(K\alpha_2\)—by \(6\%\) and \(2\%\), \(K\beta_1\)—by \(16\%\) and \(14\%\).
Below the width of lines will be considered in more detail.
b) Shape of the lines. Let us now turn to the change in the shape of the lines. The shape of lines is determined by the superposition of two levels of the atom. In free atoms, according to theoretical concepts \(^{14}\), the distribution of electrons over the width of a level follows a dispersion curve:
\[ f(E)=\frac{n\Gamma/2\pi}{(E_0-E)^2+(\Gamma/2)^2}, \]
where \(E_0\) is the energy of the middle of the level, \(f(E)\) is the number of electrons with energy \(E\), \(n\) is the total number of electrons on the level, and \(\Gamma\) is the total probability of the atom’s departure from the given state. It is not difficult to see that the width \(W\) of the level at half the maximum value of the distribution function \(f(E_0)\) is equal to \(\Gamma\). The theoretical shape of the levels, and consequently—
tels and lines—is completely symmetrical. In reality, however, significant deviations from the symmetrical form are observed. For a numerical characterization of these deviations, an “asymmetry index” \(a\) was introduced, representing the ratio of the fraction of the line width from the maximum ordinate to the long-wavelength boundary of the line to the fraction of its width from the same ordinate to the short-wavelength boundary of the line, both portions of the line width being measured at a height equal to half the maximum ordinate. Usually \(a>1\). The asymmetry index depends markedly on chemical bonds. Thus\({}^{17}\), in the transition from metallic titanium to the oxide \(\mathrm{TiO}_2\), the asymmetry of the \(K\alpha_1\) line decreases by 21%, while that of \(K\alpha_2\) increases by 15%. Such a sharp influence of the surroundings of the given atom on lines arising from deep inner levels requires a special explanation.
It was observed that in the transition group of elements of the iron family, with increasing atomic number the asymmetry indices of the \(K\alpha_1\) and \(K\alpha_2\) lines change, and these changes are observed only in elements with incompletely filled 3d shells, whereas in other elements these lines are quite symmetrical.
Fig. 6. Comparison of the curves of the asymmetry index of the \(K\alpha_1\) line and of the magnetic moment \(\mu\) as functions of atomic number \(Z\).
E. Weinstein\({}^{18}\) explains this phenomenon in the following way. The \(K\alpha_1\) and \(K\alpha_2\) lines arise as a result of the transition of an atom from the state \(1s\) to the state \(2p\), which splits into two close levels \(L_{II}\) and \(L_{III}\). The energy of each of these levels depends on the interaction of the electrons of these levels with all the other electrons of the atom. Every filled shell of the noble-gas type has a magnetic moment equal to zero, i.e. the spins of half of the electrons are directed in one direction, and the spins of the other half in the other. The energy of the “exchange” interaction of the electrons of the 2p and 3d levels is determined only by those electrons of these levels whose spin directions coincide, whereas the energy of electrostatic repulsion does not depend on the spin directions. In an unfilled 3d shell of the elements of the transition group, the number \(n_1\) of electrons with one spin direction is not equal to the number \(n_2\) of electrons with the opposite spin direction. As a result, each of the levels \(L_{II}\) and \(L_{III}\) splits into two. If the transition of an electron from the K-shell
states for each of the two component levels \(L_{II}\) and \(L_{III}\) gives lines of different intensity, then each of the lines \(K\alpha_1\) and \(K\alpha_2\) is an extremely close doublet, whose components partially overlap one another, and their superposition gives a single asymmetric line. The larger the number \(|n_2-n_1|\) of unpaired electrons in the 3d shell of the atom, the greater will be the asymmetry index of these lines. Since this same number of unpaired electrons determines the magnetic moment \(\mu\) of the ions, the course of the change of asymmetry and of magnetic moment with atomic number \(Z\) should be the same. In Fig. 6 both corresponding graphs are presented for comparison. The asymmetry is characterized by the deviation of the index \(a\) from unity, i.e. by the quantity \((a-1)\). The magnetic moment \(\mu\) is chosen for divalent ions, since for these ions there is an almost continuous series of data for the elements of the iron group. The magnetic moment is plotted on the ordinate in Bohr magnetons.
The theory developed explains the influence of chemical bonds on the asymmetry index. If the number of unpaired electrons in a metal and its oxide is the same, one should expect equal values of this index. Indeed, the asymmetry index of iron\(^{17}\) is equal to 1.6, while for \(\mathrm{Fe_2O_3}\) \(a=1.61\); for metallic chromium \(a=1.37\), for \(\mathrm{Cr_2O_3}\) \(a=1.36\); for metallic zinc \(a=1.12\), for \(\mathrm{ZnO}\) \(a=1.12\). On the other hand, the asymmetric line of metallic titanium \((a=1.22)\) becomes symmetric in the oxide \(\mathrm{TiO_2}\), in which there are no unpaired electrons. In metallic copper the 3d shell is filled, and the \(K\alpha\) line has only a small asymmetry \((a=1.17)^{19}\), whereas in the ion \(\mathrm{Cu}^{++}\) in the compound \(\mathrm{CuSO_4}\) the asymmetry of this line increases to the value \(a=1.26\), in accordance with the unfilled 3d shell of this ion (9 electrons instead of 10).
The form of the lines will be considered in more detail below.
3. Band Width and Determination of the Number of Outer Electrons
The width of the levels of a free atom is determined theoretically by the probability that the atom will leave the given state, as was explained above. The inner levels of an atom in a solid metal are subjected to only insignificant effects from the surrounding atoms, owing to which the width of these levels changes little, as is seen from Table 1, where the theoretical calculation carried out for free atoms gives the correct order of magnitude for the width of the levels of solid gold. The situation is different with the outer electrons, which are subjected to so strong an influence from neighboring atoms that their distribution over possible energies must be described by Fermi–Dirac statistics. As is known, the outer electrons of atoms of solids form bands that are broader the farther they are from the nucleus. If \(E_0\) is the lowest value of the band energy, i.e. the potential energy of the band electrons, then the remaining electrons
the band possess kinetic energy \(E-E_0\), where \(E\) is their total energy. The maximum kinetic energy of the band electrons \((E_{\max}-E_0)\) determines the energy width of the band. Let us find the distribution of these electrons according to their energies.
In phase space, by the uncertainty principle, each pair of electrons corresponds to one cell of volume:
\[ (\Delta p)^3 \cdot \Delta v = h^3, \tag{4} \]
where, for brevity, we have denoted:
\[ \Delta p_x \cdot \Delta p_y \cdot \Delta p_z = (\Delta p)^3,\qquad \Delta v = \Delta x \cdot \Delta y \cdot \Delta z. \]
Let us replace the momentum \(\mathbf{p}\) by the wave vector \(\mathbf{k}\):
\[ \mathbf{p} = h \cdot \mathbf{k};\qquad \Delta p = h \cdot \Delta k, \]
and assume that the uncertainty in the electron position is limited by the volume of the atom \(v_a\), which can be determined by dividing the volume \(v_{\mathrm{э}}\) of the elementary cell by the number \(N\) of atoms in the cell:
\[ \Delta v = v_a = \frac{v_{\mathrm{э}}}{N}. \]
Then (4) can be represented in the form:
\[ (\Delta k)^3 = \frac{1}{v_a}, \tag{5} \]
where, for brevity, we have denoted:
\[ (\Delta k)^3 = \Delta k_x \cdot \Delta k_y \cdot \Delta k_z. \]
Let us pass from phase space to the space of wave vectors, i.e., along the coordinate axes we shall plot \(k_x\), \(k_y\), and \(k_z\) in reciprocal centimeters. Each pair of electrons occupies in this space a cell of volume:
\[ (\Delta k)^3 = \frac{1}{v_a}. \]
If there are \(n\) valence electrons per atom, which may be regarded as nearly free, then these \(n\) electrons occupy in \(k\)-space the volume:
\[ \frac{n}{2}\cdot(\Delta k)^3 = \frac{n}{2v_a}. \]
These electrons, striving toward the minimum of energy, occupy in \(k\)-space a sphere with its center at the origin of coordinates*). The radius of this sphere
*) The correct application of Fermi statistics requires the simultaneous consideration of the electrons of a large number of atoms, usually of \(G^3\) elementary cells, where \(G\) is a large number. However, for simplification, the derivations have been carried out for the electrons of one atom, which does not change the final results.
will give the largest wave vector \(k_{\max}\) of the electrons (at temperature \(T=0\)):
\[ \frac{n}{2v_a}=\frac{4}{3}\pi k_{\max}^3 . \tag{6} \]
The greatest kinetic energy of these electrons is related to the wave vector by the formula:
\[ (E_{\max}-E_0)=\frac{h^2 k_{\max}^2}{2m}. \]
Determining from this \(k_{\max}\) and substituting in (6), we find:
\[ n=\frac{8\pi(2m)^{3/2}v_a}{3h^3}(E_{\max}-E_0)^{3/2}. \tag{7} \]
When electrons pass from the external band of width \((E_{\max}-E_0)\) to an inner level of width \(W_i\), an X-ray emission line of width \(W\) will be radiated. Let us put:
\[ E_{\max}-E_0=W-W_i. \]
If the energy is expressed in eV, and the atomic volume in cubic ångströms \((\text{\AA})^3\), then from (7) we obtain:
\[ n=0.00453\cdot v_a\cdot (W-W_i)^{3/2}. \tag{8} \]
Let us give several results obtained with the aid of formula (8). Birden and Friedman\(^{20}\) investigated the \(K\beta_5\) lines of copper and zinc in brass with 30% (atomic) zinc. The \(K\beta_5\) lines appear as a result of the transition of electrons from the common conduction band of brass to the \(K\)-levels of Zn and Cu. The mean number of free electrons per atom of the alloy is equal to:
\[ n=\frac{30\cdot 2+70\cdot 1}{100}=1.3, \]
if one assumes that each Zn atom gives 2 valence electrons, and a copper atom—1 valence electron. The elementary cell of the brass studied is cubic, face-centered, with constant \(a=3.67\,\text{\AA}\), and contains 4 atoms. Hence the mean atomic volume is
\[ v_a=\frac{a^3}{4}=12.4\,\text{\AA}^3. \]
By formula (8) the width of the conduction band is obtained as \(W=8.0\) eV. To find the width of the emission band \(K\beta_5\), to this value one must add the width of the \(K\)-level of the corresponding atom. The necessary data were determined in the work of Beeman and Friedman\(^{21}\): the wi-
of the \(K\)-level of Zn turned out to be \(1.5 \pm 0.5\) eV, and for Cu \(1.3 \pm 0.5\) eV. Introducing these corrections, we find the width of the \(K\beta_1\) line of zinc to be 9.5 eV, and that of copper 9.3 eV, whereas the experiment gave the width of the \(K\beta_5\) line of zinc as 9.0 eV, and of copper as 8.8 eV, i.e. agreement within the limits of possible experimental error. The experimental data are smaller than the theoretical ones, since Beeman and Friedman measured the line width at the middle of the maximum intensity, instead of measuring its full width.
With gradual dilution of copper by zinc, the mean number \(n\) of electrons per atom of the alloy increases from 1 for pure copper to 2 for pure zinc. Accordingly, the width of the \(K\beta_2\) line of both elements in their binary alloys also increases (Table 2). It is interesting to note that, owing to the common conduction band of the alloy for both its components, the width of both lines in one alloy is almost identical.
Table 2
| Content in %/% | Width of \(K\beta_2\) in eV | Width of \(K\beta_2\) in eV |
|---|---|---|
| Cu | Zn | |
| 0 | \(8.2 \pm 0.2\) | — |
| 5 | 8.3 | \(8.4 \pm 0.3\) |
| 10 | 8.5 | 8.5 |
| 20 | 8.6 | 9.0 |
| 30 | 8.8 | 9.0 |
| 48 | 9.8 | 9.0 |
| 67 | 9.0 | 9.2 |
| 80 | 9.1 | 9.2 |
| 100 | — | 9.4 |
The following Table 3 gives data on measurements of the width of the emission band associated with the outer electrons, for light atoms, for which the width \(W_i\) of the inner level, because of the smallness of this quantity, may be neglected. The data given refer to the lines of the K- and L-series of these elements. The width of \(K\beta_x\) of Al and Mg in the binary alloys \(Al_2Mg_3\) and \(Al_3Mg_2\) is the same.
Table 3
| Substance | Series | Band width, experimental | Band width, theoretical | Source |
|---|---|---|---|---|
| Li | K | 4.2 | 4.6 | 22 |
| Be | K | 13.5 | 13.8 | 22 |
| Na | L | 3.5 | 3.2 | 22 |
| Mg | L | 9.0 | 7.2 | 22 |
| Mg | K | \(7.7 \pm 1\) | 7.2 | 23 |
| Al | L | 16.0 | 11.6 | 22 |
| Al | K | \(13 \pm 1\) | 11.6 | 23 |
| Si | L | 19.2 | 13.5 | 22 |
| Si | K | \(18 \pm 2\) | 13.5 | 23 |
| \(Al_2Mg_3\) | K | \(11.5 \pm 1.5\) | 9 | 23 |
| \(Al_3Mg_2\) | K | \(11.5 \pm 1.5\) | 9.4 | 23 |
We see fairly good agreement for metals and alloys, but fairly poor agreement for the metalloid Si, whose valence electrons cannot be regarded as nearly free.
Using formula (8), Farineau determined \(^{23}\) the distribution of the outer electrons of Co and Ni, for which this was of particular interest in connection with their ferromagnetism. Table 4 gives the corresponding data. In the column \(W\) is given, in eV, the width of the \(L_{\alpha}\) lines; in the column \(W_i\), the width of the lower level of this line (\(L_{\mathrm{III}}\)) in eV according to the data of Beeman and Friedman \(^{21}\); in the column \(v_a\), the atomic volume in \((\text{\AA})^3\); in the column \(n_{4s}\), the number of outer electrons of the \(4s\) level obtained from formula (8); in the column \(n_m\), the number of electrons at the \(4s\) level according to magnetic data; in the column \(n_{3d}\) is given the difference between the total number of \(3d\) and \(4s\) electrons of Co and Ni and their number at the \(4s\) level (the total number of electrons at the \(3d\) and \(4s\) levels is 9 for cobalt and 10 for nickel).
Table 4
| Element | \(W\) | \(W_i\) | \(v_a\) | \(n_{4s}\) | \(n_m\) | \(n_{3d}\) |
|---|---|---|---|---|---|---|
| \(\alpha\) Co | 6 eV | 1.3 eV | \(11.24\,(\text{\AA})^3\) | 0.52 | 0.7 | 8.5 |
| \(\alpha\) Ni | 6 » » | 0.7 » » | 10.8 | 0.56 | 0.6 | 9.4 |
As can be seen, the agreement between the magnetic and X-ray spectral data is excellent. It is interesting to note that when nickel is heated above the Curie point, the width of the band does not change.
4. Shape of the Bands and Determination of the Distribution of Outer Electrons over States
In a solid crystalline body, the shape of the X-ray emission bands associated with the outer valence electrons depends on the distribution of the latter over states and is determined by the periodic field of the crystal lattice. This distribution is well described by Brillouin’s zone theory. In order to clarify better the influence of the zones on the shape of the bands, let us turn to the derivation of the Brillouin zones, specially adapted to solving the question posed.
Let us determine the possible values of the kinetic energy of electrons moving in a crystal lattice. When an electron wave with wavelength \(\lambda\) propagates through the crystal lattice, Bragg reflection takes place in certain directions, owing to which electrons with a definite energy cannot propagate in the corresponding directions. However, if Bragg’s law is satisfied but the structural factor \(\Sigma\) of the corresponding plane is equal to zero, then no scattering occurs, and such planes do not za-
hinder the propagation of the electron wave. The structural factor \(\Sigma\) for electron waves is related to the atomic scattering factor of these waves by a formula identical to the formula for the scattering of X-rays. Therefore, if the well-known structural factor for X-rays is equal to zero, then \(\Sigma = 0\).
Thus, if \(\Sigma \ne 0\) and Bragg’s law is satisfied:
\[ \lambda = 2d \sin \vartheta, \tag{9} \]
then, in the direction determined by the angle \(\vartheta\), electrons with wavelength \(\lambda\) cannot propagate. In (9), \(d\) is related to the dimensions of the elementary cell by the usual crystallographic formula, but the Miller indices \(h, k, l\) are assumed to be multiplied by the order of reflection \(n\). Let us replace the glancing angle \(\vartheta\) by the angle
\[ \varphi = \left(\frac{\pi}{2} - \vartheta\right) \]
between the normal to the plane \((h, k, l)\) and the direction of propagation of the wave. Let us also replace the wavelength \(\lambda\) by the wave number \(k = 1/\lambda\). Then formula (9) can be represented in the form:
\[ k \cdot \cos \varphi = \frac{1}{2}\left(\frac{1}{d}\right). \]
It follows from this formula that the projection of the vector \(\mathbf{k}\) onto the direction of the normal to the plane \((h, k, l)\) is equal to one half of the quantity \(1/d\), i.e. Bragg reflection occurs in cases when the end of the vector \(\mathbf{k}\) lies in a plane perpendicular to the mentioned normal, this plane being removed from the origin by the segment \(\frac{1}{2}(1/d)\), i.e. this plane bisects the segment of length \(1/d\).
If each plane is characterized by a point removed from the origin by the distance \(1/d\) and lying on the normal to this plane drawn from the origin, then the coordinates of such points will be
\[ h\left(\frac{1}{a}\right), \quad k\left(\frac{1}{b}\right), \quad l\left(\frac{1}{c}\right), \]
where \(h, k, l\) are Miller indices, and \(a, b, c\) are the dimensions of the elementary cell of the crystal. Since the indices \(h, k, l\) are integers, the set of these points forms a regular space lattice in the coordinate system whose axes carry the unit vectors \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\), measured in reciprocal centimeters. With such a choice of scales along the coordinate axes, the plane \((h, k, l)\) of the crystal is repre-
is represented by a point with coordinates \(h, k, l\). From the totality of such points let us discard all points corresponding to planes for which the structure factor \(\Sigma=0\). The remaining points form the so-called reciprocal lattice.
In a crystal there can be no electrons whose wave-vector endpoint lies in a plane perpendicular to the radius vector of a point of the reciprocal lattice and bisecting this radius vector. Constructing such planes for all points of the reciprocal lattice, we obtain an infinite set of polyhedra. The polyhedron nearest to the origin is called the first Brillouin zone; the space between this polyhedron and the next nearest one is called the second zone, etc. Since along the axes of the reciprocal lattice reciprocal lengths \(\mathrm{cm}^{-1}\) are laid off, in this same coordinate system one can also lay off the wave vectors of electrons moving in the lattice. We saw above that in \(k\)-space each valence electron occupies a volume determined by formula (5). If the volume \(v_B\) of the first Brillouin zone exceeds the volume occupied by all valence electrons of the atom:
\[ v_B>\frac{n}{2v_a}, \]
then, within the range of values of the wave vector from \(0\leq k\leq k_{\max}\), the energy of the electrons can assume all possible values determined by the parabolic dependence:
\[ E-E_0=\frac{\hbar^2 k^2}{2m}. \tag{10} \]
If
\[ v_B=\frac{n}{2v_a}, \]
then the zone is completely filled with electrons, which occurs, therefore, when
\[ n=2v_a\cdot v_B. \tag{11} \]
If
\[ v_B<\frac{n}{2v_a}, \]
then the electrons fill the first zone and pass into the second. With a gradual increase of the vector \(\mathbf{k}\) within \(0\leq k\leq k_{\max}\), for each chosen direction there will exist a vector \(\mathbf{k}\) whose end lies on the surface of the zone. There are no electrons with the corresponding energy in the crystal. Near the surface of the zone the wave vectors \(\mathbf{k}\) fall into a region of values where a strong perturbation of the electron wave by the periodic field of the lattice is manifested. Such electrons cannot be regarded as free; their energy is no longer deter-
is determined by (10). In Fig. 7 the course of the electron energy is presented as a function of their wave vector. The distortions at \(k=\pm k_B\) lead to a discontinuity \(\Delta E\) in the smooth increase of the electron energy with increasing wave vector. Here \(k_B\) is the distance from the origin to the surface of the zone.
In a face-centered cubic lattice the structure factor \(\Sigma \ne 0\) if the indices \(h, k, l\) are either all even or all odd. We construct the reciprocal lattice. Points with even coordinates \(h, k, l\) form a simple cubic lattice with constant \(2 \cdot \left(\frac{1}{a}\right)\), since the scales along all axes are equal to \(\frac{1}{a}\). Points with odd coordinates \(h, k, l\) are located at the centers of the cubes of the preceding lattice. The totality of these and the other points gives the reciprocal lattice. Thus, a face-centered cubic lattice with constant \(a\) has as its reciprocal lattice a body-centered cubic lattice with constant \(2 \cdot \left(\frac{1}{a}\right)\).
Fig. 7. Dependence of the electron energy on the wave vector \(k\).
In a body-centered cubic lattice \(\Sigma \ne 0\) if \((h+k+l)\) is equal to an even number. This can occur in two cases: if \(h, k, l\) are all even numbers, and if two of them are odd and the third is even. It is easy to see that in the first case the points of the reciprocal lattice form a simple cubic lattice with constant \(2 \cdot \left(\frac{1}{a}\right)\). In the second case the points fall at the centers of the faces of the cubes of the first lattice. Thus, a body-centered cubic lattice with constant \(a\) has as its reciprocal lattice a face-centered cubic lattice with constant \(2 \cdot \left(\frac{1}{a}\right)\).
Let us now consider several of the simplest, frequently encountered Brillouin zones.
a) Face-centered cubic lattice with constant \(a\). The lattice has 4 atoms in the cell. The first zone is a tetrakaidecahedron having 8 hexagons and 6 squares. In Fig. 8 this zone is shown, and its position relative to the coordinate axes is indicated. The principal dimensions of this zone are as follows: the lengths of the sides of the squares and hexagons are equal to \(1/a\sqrt{2}\), the surfac-
Fig. 8. First zone of a face-centered cubic lattice.
zone surface is \(S = 13.4 a^{-2}\). All vertices are at a distance \(\sqrt{5}/2a\) from the origin, the center of the square is at a distance \(1/a\) from the origin, and the center of the hexagon at \(\sqrt{3}/2a\). The volume of the zone is \(v_B = 4/a^3\). For complete filling of the zone, according to (11), 2 electrons from each atom are required, since \(v_a = a^3/4\).
The following zones are bounded by polyhedra similar to the first, but their linear dimensions are \(m\) times larger, if \(m\) is the number of the zone.
b) Body-centered cubic lattice with constant \(a\). The lattice has 2 atoms in the cell. The first zone is a rhombic dodecahedron consisting of 12 rhombi with side \(\sqrt{3}/2a\). Its shape and position relative to the coordinate axes are shown in Fig. 9. The surface of the zone is \(S = 6\sqrt{2}/a^2\). Its vertices are at distances \(1/a\) from the origin (the farthest vertex of the acute angle of the rhombus) and at \(\sqrt{3}/2a\) (the vertex of the obtuse angle of the rhombus). The centers of the rhombi are at a distance \(\sqrt{2}/2a\) from the origin. The volume of the zone is \(v_B = 2/a^3\). For complete filling of the zone, according to (11), two electrons from each atom are required, since \(v_a = a^3/2\).
Fig. 9. First zone of a body-centered cubic lattice.
The second zone is bounded by a cube (with edge \(2/a\)), parallel to the coordinate planes. The volume of this zone is equal to \(6/a^3\); it can accommodate 6 electrons per atom.
The following zones alternate, repeating the first two in shape.
Fig. 10. First zone of a hexagonal close-packed lattice.
c) Hexagonal close-packed lattice with edges \(a\) and \(c\). The lattice has 2 atoms in the cell. The first zone is a hexagonal icositetrahedron, shown in Fig. 10. This zone is bounded by planes parallel to the planes of the direct lattice \((1\overline{1}0,0)\), \((1\overline{1}0,1)\), and \((000,2)\). The distance from the origin to the center \(A\) of the face of the middle belt (the point of the zone nearest to the origin) is equal to \(1/a\sqrt{3}\); the distances from the origin to the centers of the upper and lower faces (point \(B\)) are \(1/c\); the distances from the origin to the vertices of the upper or lower faces (point \(C\)—the point of the zone farthest from the origin) are:
\[ \sqrt{\frac{a^2}{4}\left(\frac{4}{3a^2}-\frac{1}{c^2}\right)^2+\frac{1}{c^2}}. \]
The volume of the zone is equal to:
\[ v_B=\frac{4}{ca^2\sqrt{3}}-\frac{3}{2c^3\sqrt{3}}\left[1-\frac14\left(\frac{a}{c}\right)^2\right]. \]
The atomic volume is \(v_a=ca^2\sqrt{3}/4\); hence from (11), to fill the zone, 1.745 electrons per atom are required.
Fig. 11. First zone of a cubic lattice of diamond type.
d) Cubic lattice of diamond type with constant \(a\). The lattice has 8 atoms in the cell. The first zone (Fig. 11) is bounded by planes parallel to the planes \((111)\). It is a double pyramid with a square base. The smallest radius vector of points on the surface of this zone corresponds to the center of a face and is equal to \(\sqrt{3}/2a\); the largest, corresponding to the vertices, is equal to \(3/2a\). The volume of the zone is \(v_B=9/2a^3\). The atomic volume is \(v_a=a^3/8\); to fill the zone, \(9/8\) electron per atom is required.
The second zone is formed by the planes \((220)\). It is a dodecahedron shown in Fig. 12. The point of the zone farthest from the origin (one of the vertices \(A\)) has radius vector \(2/a\); another vertex \(B\) has radius vector \(\sqrt{3}/a\). The center \(C\) of one of the faces has radius vector \(\sqrt{2}/a\). The volume of this zone is \(v_B=16/a^3\). To fill it, \(23/8\) electron per atom is required. Both zones contain 4 electrons per atom.
Fig. 12. Second zone of a cubic lattice of diamond type.
Fig. 13. Influence of zones on the shape of emission bands.
Let us now turn to the question of the influence of zones on the shape of emission bands given by valence electrons. We shall explain this influence on
Fig. 13, where the planar case is shown for clarity. The first zone is shown in the figure as a square with sides parallel to the axes \(k_x\) and \(k_y\), and the second zone as a square rotated by \(45^\circ\) relative to the first, with a side equal to the diagonal of the first.
Let us see how the number of possible electron states with a given energy changes as the wave vector is gradually increased from zero to the boundaries of the region of \(k\)-space occupied by electrons, i.e. to the Fermi surface. Electrons with a given energy are located (in the three-dimensional problem) in a spherical layer between two spheres with radii \(k\) and \(k+\delta k\). The number of states \(\delta_n\) in the interval \(\delta k\) is equal to:
\[ \delta_n=\frac{4\pi k^2}{(\Delta k)^3}\cdot \delta k \]
or, according to (5), substituting \(k\) from the condition
\[ E-E_0=\frac{h^2 k^2}{2m}, \]
we have
\[ \delta_n=\frac{4\pi v_a m\sqrt{2m}}{h^3}(E-E_0)^{\frac12}\delta E. \]
The increase in the density of states \(n(E)=\delta n/\delta E\) with increasing kinetic energy \((E-E_0)\) is determined by the formula:
\[ n(E)=\frac{4\pi v_a m\sqrt{2m}}{h^3}(E-E_0)^{\frac12}. \tag{12} \]
This dependence is illustrated in Fig. 14, a.
If, within the electron-filled states, there are Brillouin zones, then in each direction of the vector \(k\) a discontinuity of the possible energy values occurs. Let us plot on the \(k\)-axis (Fig. 15) a series of values of \(k\) at equal distances from one another. Then, with a monotonic increase of the chosen values of \(k\), the corresponding free-electron energies also increase. The density of states grows according to law (12). In the presence of a plane of discontinuity of the energy, the states are concentrated near the upper and lower boundaries of the gap \(\Delta E\), as shown in Fig. 15. As a result, the course of the density-of-states curve will have the form shown in Fig. 14, b. Integrating this effect
Fig. 14. Influence of zones on the distribution of electrons by energies.
a) Free electrons.
b) Influence of a gap in a given direction.
c) Influence of a gap; integral effect, electrons fill one first zone.
d) Influence of a gap; integral effect, electrons in all zones.
over all possible directions of \(\mathbf{k}\) from \(k_{\min}\) (and the corresponding \(E_{\min}\)) to \(k_{\max}\) (and the corresponding \(E_{\max}\)), determined by the geometrical dimensions of the zones, we obtain the distribution shown in Fig. 14, c for complete filling of one first zone, and in Fig. 14, d—when there are electron-filled states in both zones.
The influence of the zone boundary on the intensity of the X-ray emission bands may be summarized as follows. The density \(n(E)\) of electron-filled states with energy \(E\) determines the intensity of the point of the band whose distance from the beginning of the band (i.e., from its long-wavelength edge) is equal to the value of the kinetic energy of the electrons \((E-E_0)\). As \(k\) increases from zero to \(k_{\min}\), i.e., to the smallest radius vector of points on the zone surface, \(n(E)\) increases monotonically according to (12). When the ring shown in Figure 13 goes beyond the boundary of the filled zone, the number of states inside the ring begins to fall to zero at \(k=k_{\max}\), i.e., at the value of the radius vector of the point of the zone surface farthest from the origin. If the second zone is also filled, then the emission of electrons from this zone appears at \(k_{\min}\) of this zone and then increases according to law (12), where the value \((E=E_0)\) now corresponds not to \(k=0\), but to \(k=k_{\min}\) for the second zone. Thus, the special points of the zone must affect the distribution of intensity along the emission band.
Fig. 15. Condensation of states near the discontinuity boundary.
The radius vectors of these points \(k_B\) are related to the distances \((E-E_0)\) of the corresponding points of the emission band from its long-wavelength edge by the relation:
\[ E - E_0 = \frac{h^2}{2m} k_B^2 = 154 k_B^2, \tag{13} \]
where \(k_B\) is measured in reciprocal ångströms \((\text{\AA})^{-1}\), and \((E-E_0)\) in eV. The values of \(k_B\) for special points of the zone surface of frequently encountered lattices were given above and make it possible to investigate the shape of emission bands.
However, the density of states \(n(E)\) of the electrons is not the only factor determining the distribution of intensity \(I(E)\) of an emission band. Another such factor is the transition probability \(p(E)\):
\[ I(E) \sim n(E)\cdot p(E). \tag{14} \]
The influence of the transition probability was investigated by Jones, Mott, and Skinner \({}^{24}\). The electrons of the conduction band of a metal, according to Bloch, may be
are described by the wave function:
\[ \psi_k(x,y,z)=u_k(x,y,z)\cdot e^{2\pi i(\mathbf{k}\cdot\mathbf{r})} =\sum_n a_n\Phi_n(x,y,z), \]
where \(\Phi_n\) is the wave function of an electron in a free atom. The coefficients \(a_n\) take the values \(a_s, a_p,\ldots\) and correspond to the lowest \(s\)-, \(p\)-, ... states of the conduction band (for \(4\mathrm{Be}\)—\(2s, 2p\), for \(12\mathrm{Mg}\)—\(3s, 3p\), etc.).
Let us consider transitions of these electrons to one of the lower atomic levels, whose electrons are described by the wave function \(\Phi(x,y,z)\). The probability \(p(E)\) of such transitions depends on the symmetry of the lower state, i.e. on whether this level is an \(s\)- or a \(p\)-state.
\[ p(E)=\nu^3\left|\int \Phi^*(x,y,z)\,[d\psi_k(x,y,z)/dx]\,dx\,dy\,dz\right|^2 . \]
Over the width of the x-ray emission bands (of the order of 10 eV) the factor \(\nu^3\) (where \(\nu\) is the radiation frequency) changes by no more than a factor of 2. Since this change occurs monotonically, one may expect to obtain a good description of the band shape without taking the factor \(\nu^3\) into account at all.
At the beginning of the band (its long-wavelength part) the wave vector \(\mathbf{k}\) of the conduction electrons is small; in this case:
\[ |a_s|^2\sim \mathrm{const.}, \tag{15} \]
\[ |a_p|^2\sim k^2\sim (E-E_0). \tag{15a} \]
If the electrons of the conduction band make transitions to \(K\), \(L_I\), etc. levels of symmetry \(s\), then, according to the selection rules, the transition probability of the \(s\)-electrons of the conduction band is equal to zero and the transitions are made only by the \(p\)-electrons of the band. In this case:
\[ p(E)\sim |a_p|^2 . \tag{16} \]
According to (12), (14), and (15a), we obtain the shape of the long-wavelength part of the emission band \(K\), \(L_I\), etc.:
\[ I_s(E)\sim (E-E_0)^{3/2}. \tag{17} \]
Fig. 16. Shape of the emission bands of the \(K\)-series of Be and Li.
The graph of this function is convex downward. Such a shape should be possessed, for example, by the emission bands of the \(K\)-series of lithium and beryllium.
Figure 16 presents experimental results on determining the shape of these lines in the work of O’Bryan and Skinner \(^{25}\). The direction of the convexity is clearly visible and agrees with theoretical predictions.
If the conduction-band electrons make transitions to the \(L_{\mathrm{II,III}}, M_{\mathrm{II,III}}\), etc. levels of symmetry \(p\), then, by the selection rules, the transition probability of the \(p\)-electrons of the conduction band is equal to zero, and the transitions are made only by \(s\)-electrons. In this case
\[ p(E)\sim |a_s|^2 . \tag{18} \]
From (12), (14), and (15) we obtain the form of the long-wavelength portion of the emission band \(L_{\mathrm{II,III}}, M_{\mathrm{II,III}}\), etc.:
\[ I_p(E)\sim (E-E_0)^{\frac12}, \tag{19} \]
whose convexity is directed upward. Such a form should be possessed, for example, by the emission bands \(L_{\mathrm{II,III}}\) of magnesium and aluminum. In Fig. 17 the corresponding experimental results of O’Bryan and Skinner\({}^{25}\) are presented. The convexity, especially for Al, is clearly noticeable and is directed upward. The observed dragging-out of the line toward the long-wavelength side (the so-called “long-wavelength tail”) is explained, according to Skinner\({}^{56}\), by a special type of Auger effect in the lower levels of the valence-electron band, in which an accidentally formed free level in the lower part of the band is filled by a radiationless transition of one of the higher-lying electrons of the same band, with the emission of another electron of this band.
Fig. 17. Form of the emission bands of the \(L\)-series of Al and Mg.
Let us now turn to the short-wavelength end of the band. Here zone surfaces come into play. Jones, Mott, and Skinner\({}^{24}\) consider this question using as an example the \(K\)-band of beryllium and the \(L_{\mathrm{II,III}}\)-band of magnesium. Both metals have 2 valence electrons, which would just fill the first zone, i.e. all the valence electrons would be located between two discontinuities of the possible energy values, filling one energy band. This would deprive Be and Mg of their metallic electrical conductivity, whence it follows that part of the electrons passes into the second zone. In this case the Fermi surface must pass near the end of the first zone, inside it, and near the beginning of the second zone, also inside it, i.e. the two zones must partially overlap, as is shown in Fig. 18, \(a\), where the distribution of the density of states in the first two zones is presented (the result of integration over all possible directions). \(E_{\min}\) is the beginning of the second zone, \(E_{\max}\) is the end of the first zone, \(E_{\Phi}\) is the energy corresponding to the Fermi surface.
In each zone there are \(s\)- and \(p\)-states; the corresponding densities of distribution of states are \(n(E)\cdot |a_s|^2\) and \(n(E)\cdot |a_p|^2\).
According to (14), (16), and (18), these quantities precisely determine the intensities of the emission bands \(L_{\mathrm{II,III}}\) and \(K\). As calculation shows, in each zone \(|a_s|^2\) changes from 1 to 0 as the wave vector increases from zero to the boundary of the zone, while \(|a_p|^2\) simultaneously changes from 0 to 1.
Let us consider the influence of this effect on the shape of the emission bands. As indicated above, the shape of the beryllium \(K\) band is determined by the distribution of the \(p\)-electrons of the conduction band. At the beginning of the first zone the factor \(|a_p|^2\) strongly lowers the values of \(I(E)\) in comparison with the values of \(n(E)\), while at the end of the zone it changes these values only slightly. As a result, the maximum of the distribution is shifted toward higher energies. The second zone is filled only at the beginning, where the factor \(|a_p|^2\) is close to zero, and therefore gives no noticeable emission at all. Therefore the intensity of the band drops off abruptly at \(E_{\Phi}\), corresponding to the Fermi surface (Fig. 18, b).
The shape of the \(L_{\mathrm{II,III}}Mg\) band is determined by the distribution of the \(s\)-electrons of the conduction band. The factor \(|a_s|^2\) at the beginning of the first zone is close to unity and changes the shape of the curve \(n(E)\) only slightly; at the end of this zone the product \(n(E)\cdot |a_s|^2\) falls to zero, but precisely in this region emission from the second zone begins to appear; at the beginning of that zone the product \(n(E)|a_s|^2\) is close to the value \(n(E)\) of this zone, since \(|a_s|^2\) is close to unity. As a result of the superposition of the emission of both zones, in this region the curve \(I(E)\) begins to rise, dropping sharply at the Fermi surface when \(E = E_{\Phi}\) (Fig. 18, c).
Fig. 18. Influence of two overlapping zones on the shape of emission bands.
The indicated rise of the intensity curve at the end of the band, characteristic of the \(L_{\mathrm{II,III}}\)-bands and absent in the \(K\)-band, is readily observed on the experimental curves of O’Bryan and Skinner\({}^{25}\) (Figs. 16 and 17).
On the short-wavelength side of the steep decline of the intensity of the emission band of metals, a prolongation of the band is also observed (the so-called “short-wavelength tail”), which Skinner\({}^{56}\) explains by the appearance of satellites. These satellites arise when an electron of the conduction band passes to the lower x-ray level of the atom, if in the conduction band another electron is simultaneously absent.
Let us turn to the description of some experimental results of the study of the shape of emission bands. In Fig. 19 the distribution of intensity over the \(K\beta_x\) band of magnesium is presented, according to the work of Farineau\(^{23}\). Magnesium has a compact hexagonal lattice with two atoms in the cell, with parameters \(a=3.2\,\text{\AA}\), \(c=5.2\,\text{\AA}\). The width of the Mg \(K\beta_x\) band is \(7.7\pm1\) eV. The first zone of the Mg lattice, shown in Fig. 10 and described above, has its most distant points \(C\)
Fig. 19. Shape of the magnesium \(K\beta_x\) band.
at a distance of 9 eV from the origin, i.e. the zone is not filled, which ensures the metallic electrical conductivity of magnesium. In this zone there are altogether 1.75 electrons. Since Mg has 2 valence electrons, on the average 0.25 electron per atom goes beyond the first zone into the second zone, being located near the points \(A\) and \(B\) of the first zone nearest to the origin (Fig. 10), to which, by formula (13), the energies 4.9 and 5.7 eV may be assigned, respectively. The sharp break of the intensity curve on the short-wavelength side (Fig. 19) is characteristic of metals and indicates that the zone is not filled.
In Fig. 19 the theoretical course of the intensity according to formula (17) is drawn with a dotted line. As can be seen, the approximation of almost free electrons is justified fairly well almost up to the boundaries of the zone. As we saw above, the \(L_{\mathrm{II,III}}\) band of magnesium (Fig. 17) follows formula (19) and has a hump on the short-wavelength side, absent in the \(K\beta_x\) band, as it should be according to the band theory.
In Fig. 20 the \(K\beta_x\) band of solid aluminum is presented, from the same work of Farineau\(^{23}\). We see at the beginning of the band an increase in intensity characteristic of a \(K\)-band; the theoretical curve according to formula (17) is drawn with a dotted line. As can be seen, this section of the curve is bent in the direction opposite to the curve of the \(L_{\mathrm{II,III}}\) band of Al (Fig. 17), which follows formula (19). Further on the curve two humps are observed, which, according to Farineau’s interpretation, are a consequence of the distribution of the valence electrons of Al over two zones; the corresponding contribution to the emission from the separate zones is indicated in Fig. 20 by dashed lines.
Fig. 20. Shape of the \(K\beta_x\) band of solid aluminum.
Aluminum crystallizes in a face-centered cubic lattice with edge \(a = 4.06\) Å. The first zone of this lattice is shown in Fig. 8 and described above. Since this zone is filled, it gives a gradual decrease of the intensity curve on the short-wavelength side. The decrease begins as soon as, with increasing wave vector \(k\), it begins to pass beyond the limits of the zone. The point of the zone closest to the origin (the center of the square) corresponds, by formula (13), to an energy of 7.2 eV. As is seen from Fig. 20, the first hump is located precisely at this distance from the long-wavelength beginning of the band. The most distant point of the first zone (the corner) corresponds to the end of the decrease in the emission intensity of the first zone, after which a rise in intensity is observed owing to an increase in the integral density of states of the second zone. The corner of the first zone, by formula (13), may be assigned an energy of 11.8 eV. We see that the second hump of the \(K\beta_x\) curve of aluminum (Fig. 20) is removed from the beginning of the band by 12 eV, i.e., it is also well explained by the zone theory. Since the second zone is not filled, on the short-wavelength side \(K\beta_x\mathrm{Al}\) has a sharp break.
Fig. 21. Shape of the \(K\beta_x\) band of liquid aluminum.
In order to demonstrate especially clearly the influence of zones on the shape of emission bands, Farineau placed on the anode a graphite block, in a recess of which there was a piece of aluminum. When heated by the cathode beam, the aluminum melted, and its drop served as the anode. In order that aluminum vapors should not spoil the vacuum, the heating was carried out only a few degrees above the melting point. Figure 21 presents the \(K\beta_x\) of liquid Al. As can be seen, the humps have been considerably smoothed out, but have not disappeared entirely. This apparently indicates that, with slight overheating, in any case in liquid Al, there is some order, which is in agreement with modern ideas about the nature of the liquid state.
Fig. 22. Shape of the \(K\beta_x\) band of silicon.
In Fig. 22 the \(K\beta_x\) of silicon is presented according to Farineau’s work\({}^{23}\). The absence of a sharp short-wavelength boundary at once indicates the filling of the zone and the absence of metallic electrical conductivity. Silicon
crystallizes in a cubic lattice of the diamond type with constant \(a=5.42\ \text{Å}\). The influence of the first Si zone (Fig. 11) on the shape of the band is not apparent; the surface points of this zone correspond, according to (13), to energies over very wide limits: from 4 to 11.8 eV. The influence of this zone is as if smeared out. The second zone (Fig. 12) gives a noticeable effect. The maximum intensity of the curve in Fig. 22 is 13 eV from the beginning of the band. This point corresponds to the shortest distance to the boundary of the second zone, i.e., to point \(C\) in Fig. 12. To this point, by formula (13), an energy of 10.5 eV can be assigned. There is no strict agreement here, since the zone theory can be successfully applied only to conductors with almost free electrons. An acceleration of the further fall of the intensity curve should be expected at point \(B\) in Fig. 12, which corresponds to an energy of 15.7 eV. In fact (see Fig. 22), the increase in the rate of decline of the intensity occurs at a distance of 16.5 eV from the beginning of the band.
Further, Farineau investigated the \(K\beta_x\) bands of aluminum and magnesium in the alloys \(\mathrm{Al_3Mg_2}\) (Fig. 23) and \(\mathrm{Al_2Mg_3}\) (Fig. 24). The dashed lines show the curves of the bands of the corresponding pure metals. It is interesting to note,
Fig. 23. Shape of the \(K\beta_x\) bands of aluminum and magnesium in the alloy \(\mathrm{Al_3Mg_2}\).
Fig. 24. Shape of the \(K\beta_x\) bands of aluminum and magnesium in the alloy \(\mathrm{Al_2Mg_3}\).
that all four curves differ sharply from the curves of the pure metals, and the curves of one alloy differ from the curves of the other alloy; however, in each alloy the curves of both metals are almost identical. This indicates a good sharing of the valence electrons in the conduction band of the alloys, from which these electrons make transitions to the free \(K\)-levels of one metal or the other. True, such a result is by no means always observed in binary alloys.
In the work of Birden and Friedman ^20, brasses of various compositions were investigated. Figure 25 presents the intensity-distribution curves of the \( \mathrm{Cu}K\beta_{2,5} \) and \( \mathrm{Zn}K\beta_{2,5} \) lines at different atomic contents of the components. As can be seen, the components of a single alloy give lines identical in shape and width. As the copper content increases, the intensity of the short-wavelength part of the zinc band decreases, which indicates the departure from zinc of electrons with the greatest kinetic energy. On the other hand, as the zinc content increases, the intensity of the short-wavelength part of the copper band grows, which indicates that the electrons that have left zinc have gone over to copper.
Fig. 25. Shape of the \(K\beta_{2,5}\) lines of copper and zinc in binary alloys.
An attempt to estimate the magnitude of the redistributed charge from the displacement of the lines leads the authors to a value of \(0.1\) electron (on average per atom).
These data are in agreement with the theoretical work of Mott, who investigated the electrical conductivity of \(\beta\)-brasses. Mott found that, of the two valence electrons of zinc, the latter retains about \(1.925\) near itself, and gives \(0.075\) electrons (on average per atom) to the copper atom. As a result, the copper atom is surrounded, on average, by \(1.075\) electrons. Thus, the resulting charge of the polyhedral cell of zinc proves to be equal to \(+0.075\) electrons, and that of copper to \(-0.075\). There is no complete sharing of the valence electrons.
In alloys of various composition Al—Ni, according to Farineau’s data ^26, the \(K\beta_x\mathrm{Al}\) band is considerably broader than the LaNi band. If all valence electrons were shared and free, the width of the conduction band-
would be 12 eV. Meanwhile, the width of the LaNi band is half as large; whence it follows that not all the conduction electrons surround Ni; some of them are located only near Al.
At 50% (atomic) Al, the Al—Ni alloy has a cubic body-centered lattice with constant $a = 2.82$ Å. The shortest distance to the boundary of the first zone (Fig. 9) corresponds to an energy of 9.7 eV. Consequently, one may expect that the maximum intensity of the band is displaced from its long-wavelength onset by 9.7 eV. Experiment gives 8.5 eV.
In the same work, Farineau$^{26}$ investigated Al—Cu alloys of different composition. In these alloys one can readily trace the redistribution of the valence electrons: as the Cu content increases, the short-wavelength part of the $K\beta_x$Al band decreases in intensity, while the corresponding part of the LaCu band increases in intensity with increasing Al content. Thus, the Al electrons with the greatest kinetic energy pass to the Cu atoms. However, electrons with not very great kinetic energy are not collectivized and are retained near the Al atoms.
At 19% (atomic) Al the Al—Cu alloy has a cubic face-centered lattice with constant $a = 3.6$ Å. The first zone (Fig. 8) is bounded by the planes (111) and (200), the shortest distances to which correspond to energies of 8.6 eV and 11.5 eV. The first of these distances corresponds to the maximum intensity of the $K\beta_x$Al band, the second to a minimum, after which begins the rise of the intensity curve to a second maximum, due to the second zone. The alloy has 1.38 conduction electrons on the average per atom. The first zone can be filled by two electrons. The form of $K\beta_x$Al with two maxima shows that the conduction electrons are distributed over two zones, each of which is only partially filled. At 66% (atomic) Al both bands, $K\beta_x$Al and LaCu, have three maxima each, which are explained quite well by the peculiarity of the form of the first three zones of this alloy.
It is interesting to cite an entirely different interpretation of the form of the $K\beta_x$Al band in Al—Cu alloys, given in the work of Skinner and Johnston$^{27}$. These authors point out that at 20% and 50% (atomic) Al two phases exist simultaneously in the alloys, and this gives two maxima on the intensity curve of $K\beta_x$Al.
The same authors investigated$^{27}$ the K band of Be in metallic Be and in its dilute solid solutions in Cu and Al at Be contents of 1–2–3% (Fig. 26). The authors draw attention to the fact that in such dilute solid solutions there occurs a sharp redistribution of the intensity along the KBe band: the intensity of the long-wavelength part increases in comparison with the intensity of the short-wavelength part. This indicates that into the vacant K shell of the atom of the minor impurity there penetrate, chiefly, electrons of the conduction band with the smallest kinetic energy, i.e. here too
complete collectivization of the electrons. Only the electrons with the greatest kinetic energy, which leave the atoms of the small impurity, become collectivized. The latter are surrounded in the lattice only by electrons with small kinetic energies.
This leads the authors to the following interpretation of the long-wavelength hump of the \(K\mathrm{Be}\) band in the pure metal: owing to thermal vibrations, some Be atoms temporarily drop out of the lattice and play the role of a small impurity. These atoms give rise to the hump on the long-wavelength side of the \(K\) band of beryllium.
The interpretation of the shape of bands according to Skinner and Johnston presented here dispenses with band theory. This interpretation cannot explain all the details of band shapes and therefore is unpromising. However, the authors’ considerations should be borne in mind, since in individual cases they may prove significant. In particular, they explain the disappearance, characteristic of the metal, of the sharp short-wavelength boundary of the emission band of atoms of a small impurity.
[Figure labels: “\(K\)-band of Be in Be–Cu alloys”; “pure Be”; “\(\sim 3\%\) Be”; “\(\sim 1\%\) Be”; axes \(\lambda\) and \(E\), with marks 130, 120, 110 Å and 100, 110, 120 eV. “\(K\)-band of Be in Be–Al alloys”; “pure Be”; “\(\sim 2\%\) Be”; “\(\sim 1\%\) Be”; axes \(\lambda\) and \(E\), with marks 130, 120, 110 Å and 100, 110, 120 eV.]
Fig. 26. Shape of the beryllium \(K\)-band in dilute solid solutions of Cu and Al.
The spectra of atoms of dilute solid solutions have yet another interesting feature, observed both in the cases cited above and in a number of others, for example, in Ni—Cu alloys according to the work of Farine\(^{28}\): as dilution proceeds, the width of the band decreases.
As is known, as the atoms of a solid are separated and the interaction between them decreases, the width of the outer band also decreases; in gases this band is narrowest. In Ni—Cu alloys, the \(3d\) shells of Ni and Cu interact only weakly with one another; the wave functions of these electrons do not overlap. When the Ni concentration in Ni—Cu alloys is decreased, the average distance between nickel atoms increases, i.e., the interaction between them decreases, while the action of copper atoms on nickel atoms is not great. Therefore the width of the \(L_{\alpha}\mathrm{Ni}\) band also decreases when it is diluted in a solid solution with copper. At 20% nickel in the alloy, the width of \(L_{\alpha}\mathrm{Ni}\) decreases by a factor of 2. An analogous effect is also observed on
... the LaCu band, which narrows noticeably when copper is diluted in a solid solution with nickel.
In Ni—Cu alloys the shapes of the La bands of Ni and Cu are entirely different, which also indicates the non-growing-together of the wave functions of the 3d shells of Ni and Cu and the absence of complete collectivization of the valence electrons of these atoms in binary alloys.
O’Brien and Skinner\(^{57}\) found a considerable similarity between the oxygen K band and the L\(_{\mathrm{II,III}}\) band of the metal in the following oxides: B\(_2\)O\(_3\), BeO, MgO, and Al\(_2\)O\(_3\). This indicates that in these compounds there is a strong exchange interaction between the valence electrons of both components, as a result of which the wave function of the valence-electron band of these oxides has symmetry of the s type near the lattice site occupied by the metal ion (2s in the case of B or Be and 3s in the case of Mg or Al), and symmetry of the p type near the oxygen ion (2p).
Fig. 27. Thermal broadening of the short-wavelength edge of the L\(_{\mathrm{II,III}}\) emission band of aluminum.
Let us now dwell on one more interesting application of the emission bands of the x-ray spectrum to the study of the thermal smearing of the Fermi surface. As is known, at absolute zero the Fermi distribution function \(f\) has a sharp boundary on the side of large values of the energy \(E\), and falls to zero at the Fermi surface when \(E = E_{\Phi}\). At temperature \(T\) the Fermi distribution loses its sharp boundary and is smeared over all energy values, according to the formula
\[ f=\frac{1}{e^{(E-E_{\Phi})/kT}+1}. \tag{20} \]
The sharp short-wavelength edge of the emission bands of metals is explained by the break in the occupied states at the Fermi surface. With increasing temperature this edge of the band should become more sloping, and its width increases. In Fig. 27 are presented the results of Skinner’s\(^{29,56}\) corresponding investigation of the edge of the Al L\(_{\mathrm{II,III}}\) band at temperatures 110°K, 300°K, and 680°K. The wavelength of this edge, 171 Å, provides sufficient resolving power in eV. Assuming that experimentally one can detect an effect only from 95% of the total number of electrons, we obtain, starting from formula (20), a smearing width equal to \(6kT\). Neglecting the correction for the width of the lower level (L\(_{\mathrm{III}}\)) and introducing a correction for the slit width, using the width of the line x-ray spectrum in the same wavelength region, Skinner obtained the following Table 5, in which the corrected width \(W\) of the edge in eV is compared with the quantity \(6kT\) in eV.
As is evident, the theory is very well justified at \(110^\circ\mathrm{K}\) and \(300^\circ\mathrm{K}\), but at a temperature of \(680^\circ\mathrm{K}\), close to the melting point, owing to the incipient destruction of the lattice, an additional broadening is observed.
Table 5
| \(T^\circ\mathrm{K}\) | \(W\) | \(6kT\) |
|---|---|---|
| 110 | \(0.06 \pm 0.03\) | 0.057 |
| 300 | \(0.17 \pm 0.03\) | 0.156 |
| 680 | \(0.44 \pm 0.05\) | 0.354 |
From the considerations presented here, and also from the theory of the principal edge of the absorption spectra of metals (see below), it follows that the Fermi surface corresponds to the point of inflection of the short-wavelength, steeply descending branch of the emission band of metals. Therefore (the width of the band should be determined from the long-wavelength beginning of it, cutting off the “long-wavelength tail”) to the indicated point of inflection.
5. Theory of satellites
There are several theories of satellites. It may be assumed that each of these theories is valid for some group of satellites. The theory of multiple ionization of Wentzel\(^{30}\) and Druyvesteyn\(^{31}\) is at present the most widely used. According to this theory, satellites arise in the same transitions in which the neighboring principal, “parent” lines are formed, but with the additional absence of one or two more electrons from the atom, i.e., with multiple ionization of the atom. This somewhat decreases the wavelength of the parent line, which gives the short-wavelength satellite. Thus, for example, according to Druyvesteyn the satellite \(K\beta''\) appears with simultaneous double ionization \(KL_{\mathrm{III}}\).
By this symbol is denoted the energy of the atom in the absence of one \(K\)-electron and one \(L_{\mathrm{III}}\)-electron. The energy is assumed to be expressed in rydbergs. The satellite \(K\beta''\) is characterized by the transition \(KL_{\mathrm{III}}\to L_{\mathrm{III}}M_{\mathrm{II,III}}\), since the parent line \(K\beta_1\) arises in the transition \(K\to M_{\mathrm{II,III}}\). The energy of the state of double ionization may be approximately calculated according to the following scheme (taking the screening constant equal to 1):
\[ (KL_{\mathrm{III}})_z = K_z + (L_{\mathrm{III}})_{z+1}; \qquad (L_{\mathrm{III}}M_{\mathrm{II,III}})_z = (L_{\mathrm{III}})_z + (M_{\mathrm{II,III}})_{z+1}, \]
where \(Z\) is the atomic number. The difference of these terms gives the quantum energy of the satellite \(K\beta''\), whereas the energy \(K\beta_1\) is equal to \(K_z-(M_{\mathrm{II,III}})_z\). Thus, the energy difference between the satellite and the parent line is equal to:
\[ \frac{\Delta \nu}{R} = (L_{\mathrm{III}}-M_{\mathrm{II,III}})_{z+1} - (L_{\mathrm{III}}-M_{\mathrm{II,III}})_z . \tag{21} \]
This quantity depends linearly on the atomic number \(Z\). Indeed, the differences \((L_{\mathrm{III}}-M_{\mathrm{II,III}})\) give the energy (in rydbergs) of some-
of the same series, although also forbidden by the selection rules, lines. According to Moseley’s law:
\[ (L_{\mathrm{III}}-M_{\mathrm{II,III}})_Z=A(Z-a)^2, \]
whence, by (21):
\[ \frac{\Delta \nu}{R}=A[(Z+1)-a]^2-A(Z-a)^2=2AZ-A(2a-1). \tag{22} \]
The relation obtained differs from Moseley’s law, according to which \(\sqrt{\nu/R}\) depends linearly on the atomic number. Unfortunately, the available experimental material is insufficient to verify expression (22).
Double ionization of an atom may occur as a result of the following phenomena:
1) successive ionization by two electrons;
2) simultaneous ionization of the atom by one electron of the cathode beam;
3) internal conversion of the atom after single ionization.
The first of these phenomena is improbable, owing to the short lifetime of the metastable state of the atom. Moreover, since the probability of ionization of an atom by electron impact depends linearly on the current in the X-ray tube, the intensity of the satellites in successive ionization would depend on the square of the current. However, according to experimental data, the intensity of the satellites depends linearly on the current. Thus, the first of the indicated possibilities must be rejected.
In simultaneous ionization by the impact of a single electron, the energy of this electron must be sufficient to ionize two inner levels of the atom. Therefore the excitation voltage of the satellites must be higher than the excitation voltage of the parent lines. Verification of this is very difficult, since the intensity of the satellites near the excitation potential is very small. Nevertheless, a whole series of experiments undoubtedly shows that some satellites of the K-series appear at a higher excitation potential than do the parent lines.
As applied to satellites, the internal conversion of the atom, leading to double ionization, was first considered in the work of Coster and Kronig \(^{32}\). These authors observed that, although the excitation probabilities of all three levels of the L-series are identical, the total intensity of all lines arising from the \(L_{\mathrm{I}}\)-level is considerably smaller than the total intensity of the lines arising from the levels \(L_{\mathrm{II}}\) and \(L_{\mathrm{III}}\). This may be explained by the Auger effect, according to which, upon ionization of the \(L_{\mathrm{I}}\) level, an electron passes from \(L_{\mathrm{III}}\) to \(L_{\mathrm{I}}\), filling the \(L_{\mathrm{I}}\)-level. The energy released in such a transition in elements with atomic number \(Z<50\) proves sufficient for ionization of the \(M_{\mathrm{IV,V}}\) level. As a result, the transition of an electron from the \(L_{\mathrm{III}}\) level to \(L_{\mathrm{I}}\) proves to be radiationless; the atom emits, instead of
an electron of the quantum, passing into the state of double ionization \(L_{\mathrm{III}}M_{\mathrm{IV,V}}\). Thus, the Auger effect described prepares the atom for the emission of satellites. Indeed, according to experimental data, the intensity of the short-wavelength satellites of the \(L\alpha_1\) line decreases sharply from 45 Rh to 49 In.
With an increase in atomic number above \(Z=73\), the Auger effect described again becomes possible; the energy released in the transition of an electron from \(L_{\mathrm{III}}\) to \(L_{\mathrm{I}}\) is sufficient for ionization of the \(M_{\mathrm{V}}\) level, and for \(Z>77\), also of the \(M_{\mathrm{IV}}\) level. The intensity of the \(L\alpha\) satellites as a function of atomic number was investigated by Schröder. The data obtained were published in the work of Cooper\(^{16}\). In Fig. 28 the solid curve gives the course of the integral intensity of the \(L\alpha\)-satellites; the dotted lines show the decomposition into groups of \(L_{\mathrm{III}}M_{\mathrm{V}}\)- and \(L_{\mathrm{III}}M_{\mathrm{IV}}\)-satellites. The sharp increase in intensity at \(Z>73\) and \(Z>77\) is clearly noticeable.
Fig. 28. Dependence of the relative intensity of \(L\alpha\)-satellites on atomic number \(Z\).
An analogous phenomenon occurs with the satellites of the \(L\beta_1\) line. According to Hirsh\(^{33}\), the integral intensity of the short-wavelength satellites of this line decreases sharply near atomic number 40. The \(L\beta_1\) line appears as a result of the transition \(L_{\mathrm{II}}\to M_{\mathrm{IV}}\). Since \((L_{\mathrm{I}}-L_{\mathrm{II}})_z>(M_{\mathrm{IV,V}})_{z+1}\) only for elements with atomic number \(Z\leqslant 40\), the indicated decrease in the brightness of the satellites of the \(L\beta_1\) line is connected with the Auger effect, as a result of which the atom finds itself in the state of double ionization \(L_{\mathrm{II}}M_{\mathrm{IV,V}}\). If after this an electron transition occurs from the \(M_{\mathrm{IV}}\) level to the \(L_{\mathrm{II}}\) level, then instead of the \(L\beta_1\) line the atom will emit a short-wavelength satellite of this line.
In the \(M\)-series the Auger effect has been established for the short-wavelength satellite of the \(M\alpha_1\) line; its intensity increases for elements with atomic numbers from 78 (Pt) to 82 (Pb) and drops sharply at 90 (Th) and 92 (U). This is well explained if one assumes that ionization of the \(M_{\mathrm{III}}\) level gives a radiationless transition \(M_{\mathrm{III}}\to M_{\mathrm{V}}\) with the simultaneous emission of an electron from the \(N_{\mathrm{IV,V}}\) level. Then a satellite can arise in the transition \(M_{\mathrm{V}}N_{\mathrm{IV,V}}\to N_{\mathrm{VII}}N_{\mathrm{IV,V}}\).
An interesting type of Auger effect has been found in the \(L\)-series of silver. Studying this series at voltages above the ionization potential of the \(K\)-level of silver, Barbank\(^{35}\) discovered three new weak lines
with wavelengths 4.030 Å, 4.016 Å, and 3.805 Å. These lines appeared in the spectrum only at voltages above the ionization potential of the K-level. In order that such weak lines should be visible against the background of the continuous spectrum, which at a voltage of 80 kV reaches considerable intensity in silver, an aluminum anode was used, coated with only a thin layer of silver. The thickness of the layer was chosen so that the rays excited at its lower boundary with wavelength 4 Å could pass outward through this layer; a thicker layer would have given only the harder part of the continuous spectrum, which in aluminum is considerably weaker than in silver. The origin of the lines found was explained by R. D. Richtmyer^36, who proposed the existence of radiationless transitions in the K-series from one of the levels of the L-series to the K-level. As a result of such a transition the atom emits a second L-electron and is left in a state of double ionization. Richtmyer explained the three lines found by Barbank by the following transitions: \(L_{II}L_{III}\to L_{III}M_{IV,V}\); \(L_{II}L_{III}\to L_{II}M_{IV,V}\), and \(L^{2}_{III}\to L_{III}M_{IV,V}\). The probability of the processes considered, calculated by Richtmyer, agrees with the intensity of these satellites found by Barbank. It should be noted that the experimental detection of such weak lines, drowned by the continuous spectrum, presents great difficulties. Thus, for example, Faith and Kirkpatrick^37, working with Barbank’s apparatus, were unable to detect analogous new lines in the L-series of molybdenum when the ionization potential of its K-level was exceeded. In order to get rid, as far as possible, of the continuous spectrum, this work used total reflection from a layer of magnesium on glass at such an angle that reflections of higher orders were completely cut off.
Satellites of the K-series obviously cannot be explained by the Auger effect. Double K-ionization, in which one K-electron is absent, can, as indicated above, arise only directly as the result of impact by an electron whose energy is sufficient to tear two electrons from the atom, for example, one K-electron and one L-electron.
Another theory of satellites was proposed by F. K. Richtmyer^38. According to this theory, satellites arise upon ionization of an inner level with simultaneous excitation of an outer electron to one of the optical levels of the atom. As Blokh^39 showed, the probability of the simultaneous occurrence of both processes upon impact by an electron of sufficient energy is close to unity. In an atom prepared in this way, two transitions may occur simultaneously: 1) the transition giving the producing (principal) line, and 2) the transition of the electron excited to an optical level to one of the outer levels of the atom.
Since both transitions take place in a single act, the atom emits one quantum, whose energy and frequency differ from the energy and frequency of the producing line by an amount corresponding to the tran-
to the optical-electron transition, characteristic, thus, for a certain optical line. Therefore the difference of terms between the satellite and the parent line obeys Moseley’s law:
\[ \frac{\Delta \nu}{R}=A(Z-a)^2, \]
differing from the analogous dependence in the theory of multiple ionization (see formula 22). However, investigation of this dependence does not make it possible to choose between the two theories, since usually one and the same satellite is not encountered among elements with a considerable interval of atomic numbers. Moreover, as we shall see below, the wavelengths of satellites depend rather strongly on the chemical bond and the crystal lattice.
The difficulty of determining the excitation potentials of satellites also does not allow this criterion to be used to establish one or another mechanism of origin of a given satellite; although it is obvious that for ionization of an inner level and excitation of an outer one (according to the theory of double transitions) considerably less energy is required than for ionization of two inner levels (according to the theory of double ionization for the K-series).
Comparing the probabilities of double ionization and of a double transition, Pincherle^40 comes to the conclusion that the intensity of the satellites formed according to the second of these theories is too small to be accessible to experimental detection. Bloch believes that the first theory explains the satellites of the L-series, whereas the second explains the satellites of the K-series. Apparently, nevertheless, the Kα-satellites and some Kβ-satellites are explained by the theory of multiple ionization^41,42,43.
In comparing the two theories considered above, it is also necessary to note that the influence of the chemical bond and of the environment in the lattice of a solid must have a very strong effect on satellites arising in double transitions, since in this case optical spectra are, as it were, superposed on the ordinary X-ray spectra (a kind of Raman effect). Satellites, however, formed according to the theory of multiple ionization, should be considerably less subject to the influence of the chemical bond, approximately to the same degree as the parent lines.
The influence of the chemical bond on satellites has been investigated in a whole series of works. In studying the group of Kβ satellites of Al and Mg, Karlsson and Siegbahn^52 observed that the form of the satellites Kβ^III and Kβ^IV reproduces the form of the parent line Kβ_x for the metal, and Kβ_1 for the oxide. As already indicated, Kβ_x of metals has a sharp short-wavelength edge, in accordance with an unfilled conduction band. The satellites β^III and β^IV have the same form. In the oxides of Al and Mg the band is filled, and Kβ_1 is a symmetric line. The satellites β^III and β^IV are also symmetric. This confirms the commonality of the origin
of satellites and parent lines in transitions between the same levels; in the present case there is a transition \(K \to M_{\mathrm{II,III}}\), with an additional absence of an electron in the \(L_{\mathrm{II,III}}\) level in the case of \(\beta^{\mathrm{III}}\), or in the \(L_{\mathrm{I}}\) level in the case of \(\beta^{\mathrm{IV}}\).
An analogous effect was found by M. Blokhin\(^{13}\) for the same satellites of chromium. In the same work the following changes of satellites, depending on the nature of the chemical bonds, were also discovered: the \(K\beta^{\mathrm{III}}\) satellite of chromium in the metal is a single line, while in compounds it splits. With an increase in the valence of chromium in compounds, \(\beta^{\mathrm{III}}\) and \(\beta^{\mathrm{IV}}\) shift to the long-wavelength side; for example, \(\beta^{\mathrm{III}}\) in trivalent compounds is shifted by 0.8 eV, and in hexavalent compounds by 1.9 eV, in comparison with metallic chromium. There are also sharp changes in the intensity of the satellites. Thus, for example, the intensity of \(\beta^{\mathrm{III}}\), compared with the intensity of the diagram line \(\beta_{5}\), increases by a factor of 1.5 in the transition from the metal to the oxide \(\mathrm{Cr}_{2}\mathrm{O}_{3}\), and in hexavalent compounds by almost a factor of two. At the same time the relative intensity of \(\beta^{\mathrm{IV}}\) in trivalent compounds also increases by a factor of 1.5, while in hexavalent compounds it becomes almost 2 times smaller than in the metal. The relative intensity of \(\beta^{\mathrm{II}}\) (in comparison with \(\beta_{5}\)) in hexavalent compounds is twice as great as in trivalent compounds, while in metallic chromium this satellite is entirely absent. The indicated changes are so sharp that they can serve as a criterion for establishing the valence of chromium in compounds.
There are satellites that appear on the long-wavelength side of the parent lines. Such are, for example, \(K\beta'\), \(K\eta\), and the long-wavelength satellite \(K\beta_{5}^{\prime}\) of molybdenum. The latter was examined by Bloch and Ross\(^{44,39}\), who propose the following mechanism for the formation of this satellite: the transition that gives the parent line (in the present case, the quadrupole \(K\beta_{5}\)) is accompanied by the ejection of one of the outer electrons from the atom. The energy released in the first of these transitions is distributed between the emitted quantum and the electron. As a result, a long-wavelength satellite appears, whose short-wavelength limit must be sharp and must correspond to ionization of the outer electron without imparting kinetic energy to it. The considerable width of the long-wavelength satellites is explained by the fact that the electron leaving the atom may receive a greater or lesser kinetic energy. It is obvious that such satellites must be subject, to a considerable extent, to the influence of chemical bonds.
An analogous mechanism was proposed by M. Blokhin\(^{13}\) to explain the satellites \(K\eta\) and \(K\beta'\). For the elements \(23\mathrm{V}—29\mathrm{Cr}\), the position of \(K\eta\) in the spectrum is very well explained by the forbidden transition \(K \to M_{\mathrm{I}}\) with simultaneous emission of an electron from the \(M_{\mathrm{II,III}}\) level. The \(K\beta'\) satellite of chromium is well explained by the transition \(K \to M_{\mathrm{II,III}}\) (giving the line \(K\beta'\)) with simultaneous ejection into the conduction band of an electron from the \(M_{\mathrm{IV,V}}\) level of the atom. Depending on the co-
...the finite energy of the ejected electron, the wavelength of the emitted quantum will be greater or smaller. Therefore the energy distribution of the intensity of this broad band gives the distribution of the density of states of the conduction band in metals. The described mechanism of the formation of \(K\beta'\) in chromium proved to be quite applicable to iron\(^{45}\) and to manganese (according to an unpublished work by I. Sorokin).
Long-wavelength satellites depend especially sharply on chemical bonds. Thus, \(K\beta'\) of Al and Mg\(^{52}\) is almost imperceptible in metals and very bright in oxides. The \(\eta\) satellite of chromium\(^{13}\), in going from the metal to the oxide, is shifted to the long-wavelength side by 2 eV. The \(K\beta'\) satellite of chromium, in going from the metal to tri- and hexavalent compounds, sharply changes its form and relative intensity.
In recent years a whole series of very weak satellites has been discovered which have so far received no explanation. There are especially many such satellites in the \(K\beta\)-group\(^{9,46,47,48,49}\). The available experimental material on these lines is still very insufficient. In the future it is necessary to investigate these satellites for as large a number of elements as possible. Numerous satellites have also been found near the \(\alpha\) lines of the L-series for elements from \(42\mathrm{Mo}\) to \(56\mathrm{Ba}\)\(^{50}\). The satellites of the M-series of tungsten were investigated by Mönch, Birden, and Shaw\(^{51}\).
Besides the principal theories of satellites considered above, attempts are encountered in the literature to explain certain satellites by other mechanisms. Thus, Valasek\(^{8}\) explains some short-wavelength satellites of chlorine in the \(K\beta\)-group in rock salt by transitions of electrons from optical levels, and one long-wavelength satellite by a transition from the \(L_{\mathrm{II,III}}\) level of the sodium ion. Gulobey\(^{46}\) explains the \(K\beta_6\) satellite in \(42\mathrm{Mo}\)—\(45\mathrm{Rh}\) by a double transition \((K \to M_{\mathrm{III}}) + (M_{\mathrm{I}} \to M_{\mathrm{II,III}})\), arising upon simultaneous double ionization of two inner levels \(K\) and \(M_{\mathrm{I}}\).
Let us now turn to attempts to explain the form of the complex “satellite structure” observed near the bright lines of the vacuum region of the spectrum. Despite the comparatively large dispersion in this region, the individual satellites are so broad and so close to one another that they cannot be separated, and they merge into a single band of nonuniform intensity. In the works of Richtmyer and Ramberg\(^{53}\) for the \(L\alpha\) and \(L\beta_2\) satellites of gold, and of Pincherle\(^{54}\) for the \(L\alpha\) satellites of \(37\mathrm{Rb}\), \(42\mathrm{Mo}\), \(47\mathrm{Ag}\), \(51\mathrm{Sb}\), and \(56\mathrm{Ba}\), the \(L\beta_1\) satellites of \(46\mathrm{Pd}\), and the \(L\beta_2\) satellites of \(55\mathrm{Cs}\), the positions and intensities of all possible cases of transitions between two states of double ionization were calculated, taking into account the various possible states of the outer electrons. The results obtained reproduce only in general outline the actually observed form of the satellite structure, and therefore we shall not dwell on these works in greater detail.
A special class of satellites was investigated by O’Bryan and Skinner[^57] in the ultra-long-wavelength region of the spectrum. Emission bands of halides in alkali halides were obtained: the K-bands of fluorides, the L$_{II,III}$-bands of chlorides, the M$_{IV,V}$-bands of bromides, and the N$_{IV,V}$-band of CsJ. For RbBr the spectrum of the rubidium M$_{IV,V}$ band was also obtained. In all cases, on the short-wavelength side of the principal emission band, at a distance of 1 to 4 eV, a satellite of exceptional brightness is observed. An analogous satellite was also found for the K-bands of metals in the compounds B$_2$O$_3$, BeO, MgO, Al$_2$O$_3$, BN; the oxygen K-band in the same compounds has no such satellite.
The authors believe that in compounds with mixed ionic and covalent bonding there actually exist ions and neutral atoms, which explains this doubling of the spectra. Each lattice site of such compounds may alternately be an ion or a neutral atom, with a change of state on average every $10^{-15}$ sec. In the halides the admixture of neutral atoms amounts to only a small percentage, since the satellites under consideration are relatively weak. The oxides mentioned above consist of positive metal ions and negative oxygen ions, with a considerable admixture of neutral metal atoms. The typical homopolar compound SiC gives no such satellites either in the K-spectrum of carbon or in the L$_{II,III}$-spectrum of Si. The spectra of SiO$_2$ give the same results, which indicates the covalent character of the bond in quartz.
One may, however, doubt the correctness of the authors’ interpretation of the spectra of the alkali halides, since it is known that in some of them neutral atoms appear only under the influence of the action of X-rays.
An analogous effect of band splitting was investigated by Fogel[^58] for the K$_\beta$ band of sulfur in elemental sulfur and in several compounds. This splitting, of the order of 3.5 eV, may also be explained by the presence of different valence states of sulfur atoms at the lattice sites.
(To be continued in the next issue.)
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