STUDY OF THE STRUCTURE OF ATOMS IN CRYSTALS BY THE PROBE METHOD

S. T. Konobeevsky

Does a crystal consist of atoms? Of course it does, if by this one means only that the crystal is built from individual atoms arranged into a space lattice. But the question can also be posed differently. Do atoms in a crystal retain their individuality; can individual atoms be distinguished in a crystal? This question deserves more detailed consideration.

If one depicts an atom of some metal, for example Al, as modern theory represents it—in the form of a continuous cloud of electronic charge around a positively charged nucleus—then the “dimensions” of such an atom will, generally speaking, turn out to be larger than the space allotted to it in the crystal. The electron clouds of neighboring atoms must overlap. For example, one can calculate that in an Al atom, outside a sphere drawn with a radius equal to half the nearest distance between two neighboring atoms in the crystal, there will be a charge equal to about 1.6 electrons; this is a rather considerable part of the total charge of the atom (3 electrons). Under such conditions, if atoms form a crystal, the interaction of the electrons with one another and with neighboring nuclei must obviously lead to their redistribution. The theory of the metal in its present form, unfortunately, is not able to predict what this new distribution will be in any given case. It is undoubtedly, however, that it must be subject to the symmetry of the space lattice formed by the atomic nuclei. Thus, in a crystal one must distinguish not individual atoms, but rather a continuous electron continuum whose density obeys the law of three-dimensional periodicity.

Diffraction of X-rays in a crystal makes it possible to determine directly the coefficients of the Fourier series by means of which this periodic distribution can be represented. Having carried out the expansion of the electron density

\(A(\mathbf r)\) into the integral

\[ A(\mathbf r)=\int S(\mathbf H)e^{-2\pi i(\mathbf H,\mathbf r)}\,d\tau_r;\qquad d\tau_r=dx\,dy\,dz \tag{1} \]

or, for a periodic function, into a three-dimensional Fourier series:

\[ A(\mathbf r)=\sum S_n e^{-2\pi i(\mathbf H,\mathbf r)}, \tag{2} \]

from the theory of diffraction in crystals we find that the function of the index vector \(S(\mathbf H)\), which is the Fourier transform of the electron-density function \(A(\mathbf r)\), is proportional (in modulus) to the square root of the relative intensity of the X-ray beam “reflected” by the plane represented by the vector \((\mathbf H)\), both in direction (\(\mathbf H\) is perpendicular to this plane) and in magnitude:

\[ \frac{1}{|H|}=d_{h_1h_2h_3}, \tag{3} \]

where \(d_{h_1h_2h_3}\) is the interplanar spacing of the face with indices \(h_1, h_2, h_3\). By specifying the system of coefficients \(S_n\) in (2), the periodic distribution of electrons in the crystal is completely described, and the task is to realize such experimental conditions under which a complete system of coefficients \(S_n\) can be obtained and measured with sufficient accuracy. The latter, however, as is known, encounters very serious difficulties. The practical necessity of prematurely truncating the series, caused by the limited extent of the X-ray diffraction spectrum when its attenuation is weak, has led to the idea of introducing into the series an artificial damping factor (increasing with \(H\)), which is equivalent to introducing a certain uncertainty into the positions of the diffraction centers of the lattice1,2. This, apparently, distorts the true distribution so strongly3 that the method becomes questionable. However, even if the summation (2) could be carried out quite correctly, the result obtained still could not by itself be satisfactory, since in order to extract from it any conclusion having physical meaning it is nevertheless necessary, in one way or another, to single out individual atoms from the distribution obtained. There are no theoretically justified rules for this procedure. Many authors, guided by qualitative considerations, try to construct graphically the boundaries between the atoms (ions) proper in metals and the region of constant charge density, regarding the latter as an indication of the existence of “free electrons.” In drawing these boundaries, considerable arbitrariness is often allowed; therefore it is not surprising that different authors find in one

and in the same metal different numbers of “free electrons” and different degrees of ionization of the atoms. Instead of using artificial devices to separate the continuous function \(A(\mathbf r)\) into a part belonging to the ions and a part belonging to the lattice, one may assume from the very beginning that the function \(A(\mathbf r)\) is formed as a sum over sources constituted by the individual atoms, i.e., set:

\[ A(\mathbf r)=\int P(\mathbf r')\rho(\mathbf r-\mathbf r')\,d\tau_{\mathbf r'}, \tag{4} \]

where \(P(\mathbf r')\) is the function describing the distribution of the atoms, and \(\rho(\mathbf r-\mathbf r')=\rho(\mathbf R)\) is the distribution of electrons inside an atom (see Fig. 1). For simplicity, in what follows we assume that all atoms are identical. As is known from the theory of Fourier integrals (the convolution theorem), the transform of the function (4) will be the simple product of the Fourier transforms of \(P\) and \(\rho\), i.e.

Fig. 1.

\[ \left. \begin{aligned} A(\mathbf r)&=\int L(\mathbf H)\,F(\mathbf H)\,e^{-2\pi i(\mathbf H,\mathbf r)}\,d\tau_{\mathbf h};\\ L(\mathbf H)&=\int P(\mathbf r)\,e^{2\pi i(\mathbf H,\mathbf r)}\,d\tau_{\mathbf r};\\ F(\mathbf H)&=\int \rho(\mathbf R)\,e^{2\pi i(\mathbf H,\mathbf R)}\,d\tau_{\mathbf R}. \end{aligned} \right\} \tag{5} \]

But the transform \(\rho(\mathbf R)\), \(F(\mathbf H)\), represents the diffraction image of an atom located at the point \(\mathbf r\), while the transform of \(P(\mathbf r)\), \(L(\mathbf H)\), is, for a periodic infinite lattice (\(\mathbf a_1,\mathbf a_2,\mathbf a_3\) are translation vectors), the well-known three-dimensional delta function, different from zero only at the points \(\mathbf H\) corresponding to

\[ \mathbf H=h_1\mathbf b_1+h_2\mathbf b_2+h_3\mathbf b_3, \]

\[ \mathbf b_1=\frac{[\mathbf a_2\mathbf a_3]}{(\mathbf a_1[\mathbf a_2\mathbf a_3])},\quad \text{etc.} \tag{6} \]

Since diffraction experiments in a crystal lattice are equivalent to finding the Fourier transform of the function \(A(\mathbf r)\), which, as is evident above, is always obtained in the form of the product

\[ L(\mathbf H)F(\mathbf H), \]

then instead of the continuous field \(F(\mathbf{H})\), necessary for a complete determination of the electron distribution in the atom, we always obtain only separate points of this field, just as a typographic plate consists of separate dots—the so-called screen. Unfortunately, in the case of diffraction in a periodic lattice this screen, which limits information about the function \(F(\mathbf{H})\), turns out to be much sparser than the screen of a typographic plate, and the more so the closer together the atoms are in the crystal.

If some curve \(F(\mathbf{H})\) is known only at separate points, then the corresponding Fourier transform \(\rho(\mathbf{R})\), generally speaking, remains undetermined until one or another method of interpolation has been chosen. Usually we carry out such interpolation by choosing the “smoothest” curve. Nevertheless, it is obvious that, since the choice is not unique, one and the same actually existing distribution of electron density may be decomposed into “atoms” in various ways.

It is easy to see that this ambiguity especially concerns the most peripheral part of the atom, i.e. that where the atoms overlap one another considerably, and where it would be especially interesting to know the behavior of the function \(\rho(\mathbf{R})\). Indeed, for a more accurate representation of this region one must know the behavior of the function \(F(\mathbf{H})\) at small values of \(\mathbf{H}\). But precisely in this region \(F(\mathbf{H})\) has a large curvature and few determining points \(\mathbf{H}\), not to mention that for

\[ 0<|\mathbf{H}|<|\mathbf{H}|_{\min}\approx \frac{1}{d}, \]

where \(d\) is the distance between two neighboring atoms, it has no points at all, which makes consideration of the atom beyond these limits meaningless.

Although the functions \(\rho(\mathbf{R})\) and \(F(\mathbf{H})\) seem, in a mathematical sense, to be quite reciprocally convertible, in reality they differ substantially from one another in many respects.

One of these differences is that the function \(\rho(\mathbf{R})\) is rather “sensitive” to changes in \(F(\mathbf{H})\), but, conversely, \(F(\mathbf{H})\) is comparatively little affected by small variations of \(\rho(\mathbf{R})\). We shall show this in the following example. The function \(\rho(\mathbf{R})\) can be found on the basis of a theoretical conception of the structure of atoms, using the V. A. Fock–Hartree method, which conveys the essential features in the structure of the atom. The radial distributions of electron density in atoms obtained by this method show well-pronounced oscillations of \(\rho(\mathbf{R})\), corresponding to the \(K, L, M,\ldots\) levels of the atom (see, for example, \(^{4}\)). On the other hand,

having \(\rho(R)\), one can compute \(F(H)\), for example, by the formula

\[ F(H)=\int_0^\infty 4\pi R^2\rho(R)\frac{\sin 2\pi HR}{2\pi HR}\,dR. \tag{7} \]

Table 1 gives the values of \(F(H)\) at the points \(2\pi H=10^8\cdot n\) (\(n\) from 0 to 17), computed in this way for the aluminum atom.

Table 1

As is easy to see from the table, the theoretically calculated values are, with sufficient approximation, described by the interpolation formula[^5]

\[ F(H)=13\cdot e^{-aH} \tag{8a} \]

\[ \text{for } a=0.879\cdot 10^{-8}\ \text{cm} \]

(at least for the values \(H<2\cdot 10^8\)). If now, using formula (8a), one attempts to pass back to the function \(\rho(R)\), then a simple calculation will lead us to the dependence:

\[ \rho(R)=\frac{2Z}{R}\int He^{-aH}\sin 2\pi HR\,dH = \frac{8\pi Z}{a^3\left(1+\frac{4\pi^2R^2}{a^2}\right)^2}. \tag{8} \]

The graph of this function is a monotonic curve, devoid of any maxima except the trivial one (\(R=0\)).

Both this example and all that has been said above show that it is hardly advisable to seek the actual radial distribution of electron density in an atom (whether free or located in a crystal) by proceeding from a direct measurement of the diffraction spectrum and passing from it to the space of the atom.

A more justified method is the trial method, consisting in the theoretical construction of the function \(\rho(\mathbf{R})\), the calculation of the diffraction spectrum \(F(\mathbf{H})\), and comparison of the latter with experiment. This trial method can be recommended also because, as we shall see below, it permits one to investigate the function \(\rho(\mathbf{R})\), so to speak, “in parts.” For this purpose the distribution in the atom can first be expressed in a fairly general form, by introducing several parameters; in doing so, of course, the symmetry conditions must be used. On passing to the function \(F(\mathbf{H})\), the latter proves to depend on the same parameters. However, in order to determine these parameters it is not necessary to know exactly the entire course of the function \(F(\mathbf{H})\); it is sufficient to use only a few of its points, comparing individual interferences with one another.

The indicated method was applied in the work of K. P. Mamedov and the author of the present article; the work was devoted to the investigation of the anisotropy of aluminum and carbon atoms in crystals of aluminum and diamond.^6

In the crystal lattice of aluminum the atom occupies a position of high symmetry. The angular distribution of the electrons in it must in any case be no lower than the symmetry of the position. If, to begin with, the electron distribution in an Al atom situated at a lattice node is represented as the product of a radial function and an angular one:

\[ \rho(\mathbf{R})=\rho(R)\,\theta(\vartheta,\varphi), \tag{9} \]

then, expanding the latter in spherical functions, we must discard a series of higher terms which do not satisfy the symmetry conditions. As a result we obtain:

\[ \theta=1+x_1\left(a_1^2a_2^2+a_2^2a_3^2+a_3^2a_1^2\right)+x_2a_1^2a_2^2a_3^2 \tag{10} \]

(\(a_1, a_2, a_3\) are direction cosines; \(x_1, x_2\) are anisotropy coefficients).

The essential features characterizing the anisotropy of the atoms are already determined by these first two terms. The sign and magnitude of \(x_1\) determine the difference of the electron density in the directions of the cube, the rhombic dodecahedron \([110]\), and the octahedron \([111]\); \(x_2\) chiefly increases or decreases the value of the density in the direction of the octahedron \([111]\). Thus, three principal directions are determined.

If, as in the present work, it is assumed that \(\theta\) depends only on one of the anisotropy coefficients, \(x_1\), then in order to find it one may try to compare with one another reflections from two planes with one and the same value of the interplanar spacing

\[ d=\frac{1}{|\mathbf{H}|} \]

and, consequently, of the modulus of the vector \(|\mathbf{H}|\), but differing-

...in direction. As such for a face-centered cubic crystal the following interferences were chosen:

\[ 333 \text{ and } 511; \]

the values of \(\theta\) for these interferences will be:

\[ \begin{aligned} &1 + 0.33 x_1 \quad (333),\\ &1 + 0.07 x_1 \quad (511), \end{aligned} \]

that is, the value of the electron-distribution density for positive \(x_1\) will be greater in the direction \([333]\) than in the direction \([511]\), and consequently the intensity of the 333 interference is less than that of 511. For negative \(x_1\) the relations will be reversed. Reflections from the octahedral plane will be stronger than from the \((511)\) planes.

The task of the present work was to establish the anisotropy of the atomic factor of Al in the crystal by comparing the intensity of reflection from the internal faces of the crystal \((111\) third order) and \((511)\). The stationary-crystal method (Laue method) was chosen. A single crystal of Al, selected so that the axis of the single-crystal wire was symmetric with respect to the directions \([111]\) and \([511]\), was oriented with one of the \([111]\) axes parallel to the beam of X-rays emitted by a tube with a Mo anode. The continuous spectrum of the tube was limited on the short-wave side by the applied voltage, and on the long-wave side, in addition to absorption in the tube wall, also by Al filters of various thicknesses placed in the path of the beam. On the photograph three pairs of the 511 and 333 interferences were obtained simultaneously. The latter (333) in the general case also include the first and second orders of reflection, which is why the corresponding spots prove to be stronger than 511. By placing Al filters of successively increasing thickness in the path of the primary beam, it was possible to reduce to zero the part representing the first and second orders of reflection, after which the ratio of the intensities 511 and 333 no longer changes with increasing filter thickness. The graph showing the ratio of the intensities 511 and 333 as a function of filter thickness is presented in Fig. 2. From it it is seen that the final intensity of reflection, and consequently also the atomic factor in the direction \([111]\), is greater than in the direction \([511]\) (approaching the cube) by approximately 10%. As follows from the preceding, this means that the atom in the directions \([111]\) and \([110]\) is more rarefied than in the direction of the cube. Applying the corresponding calculations, one can find: \(x_1 = -0.11\). The negative sign of \(x_1\) shows that the electrons of the Al atom in the crystal tend to leave the direction \([111]\) (and also \([110]\)) and pass into \([100]\), i.e., they tend to fill structurally the most...

Change in the intensity ratio
as a function of the thickness of the Al filter

Fig. 2.

Fig. 3.

more free sites (octahedral pores) of the face-centered lattice.

Although this result could have been predicted, taking into account the tendency of the electrons of a metallic crystal to be distributed as uniformly as possible in the space of the lattice, the example considered nevertheless gives perhaps the clearest and simplest experimental proof of the existence of such a distribution.

A second example of the application of the trial method to the study of electron density in crystals concerns diamond.

The position of the carbon atom in diamond indicates its tetrahedral symmetry. A model of such an atom may be a combination of a spherically symmetric distribution with total charge

\[ Z = 6\left(1-\frac{2\Delta}{3}\right)e \]

at the center of the atom and four charges \(\Delta e\), situated at the vertices of a tetrahedron (see Fig. 3) at a distance \(\dfrac{ua\sqrt{3}}{8}\) from the center (\(a\) is the period of the diamond lattice). This model is readily generalized by considering, for example, the charges \(\Delta e\) not as concentrated at points, but as distributed along the directions of the tetrahedron. The parameters of the structure of the C atom will be \(\Delta\) and \(u\). Both can be found by comparing the corresponding interferences. The general expression for the structure amplitude will be:

\[ S = 2A \cos \frac{\pi}{4}\sum_{1}^{3} h_j + \]

\[ +4B\left( \cos \frac{\pi}{4}\sum_{1}^{3} h_j \prod_{1}^{3}\cos \frac{\pi u}{4}h_j - \sin \frac{\pi}{4}\sum_{1}^{3} h_j \prod_{1}^{3}\sin \frac{\pi u}{4}h_j \right). \tag{11} \]

Here

\[ A=Z'f_0,\qquad B=2\Delta f_{\Delta}, \]

\(f_0\) is the normalized atomic factor of the central charge, and \(f_{\Delta}\) is the same for the additional charge \(\Delta\) at the tetrahedron vertex. Taking \(\Delta=0\), we arrive at the structure factor of the diamond lattice for spherically symmetric atoms. The extinction conditions in this case are easily found. For

\[ \sum_{1}^{3} h_j = 4n+2 \]

the structural factor vanishes. If \(\Delta \ne 0\), then forbidden interferences arise:

\[ S_{4n+2}=\pm B \prod_{1}^{3}\sin \frac{\pi u}{4}\,h_j . \tag{12} \]

The parameters \(u\) and \(\Delta\) are thus determined by observation of the intensity of the “forbidden” interferences.

The experimental investigation of diamond was carried out, as before, by methods of X-ray photography of a stationary crystal oriented in the most favorable way for detecting the corresponding spots. The principal difficulty lay in the proper limitation of the continuous spectrum, making it possible to isolate the spectral interval needed for reflection of the given order. For this purpose, in individual cases selective filters (Ni, Sr) were used, and the voltage on the tube was controlled by the absence of the corresponding reflections.

The presence of interferences was established:

\[ 222 \text{ and } 622 \]

and their complete absence:

\[ 420 \text{ and } 442. \]

From this fact (the absence of combinations in which at least one of \(h_1h_2h_3\) is equal to \(4n\)) it follows, as is easily seen from (12), that

\[ u=1. \]

In other words, the additional charges of neighboring atoms coalesce at the midpoint of the distance between them. The intermediate charge represents an electronic maximum connecting two neighboring atoms. It is significant that the above extinction law is compatible only with the conception of a spherically symmetric charge at the midpoint of the distance between atoms. Comparison of the intensities of 222 and 622 makes it possible simply to find the density of the intermediate charge. Indeed, the denser the charge, the smaller the difference between the atomic factors 222 and 622 should be; the more diffuse it is, the more rapidly the intensity should weaken with increasing \(H\). Even a qualitative comparison of the brightness of the spots 222 and 622 shows that the corresponding atomic factors differ comparatively little. The estimate gives:

\[ f_{222}:f_{622}\approx 2. \]

This means that the “dimensions” of the electronic maximum, assuming conditionally a uniform charge density, will be of the order

\[ 2r=0.5\,\text{kX}. \]

The value of the charge \(2\Delta\) can be determined (of course, approximately) by comparing the intensity of the “forbidden” 222 with the normal 333. The calculation gives about 0.12 electron charge. Of course, the calculations presented cannot claim to be more than indicative numerical estimates. However, the qualitative picture that they depict can hardly be disputed, since it is based on simple and easily observable facts. At the same time, it cannot fail to seem somewhat unexpected.

Attempts at a quantum-mechanical calculation of the structure of diamond, carried out by Ewald and Hönl in 1936[^7], also led to the existence of an intermediate charge between the carbon atoms; however, the density of this charge proved to be considerably smaller than what we found from the experimental data. Suffice it to say that the ratio of the atomic factors 622 and 222, calculated according to Ewald, should have been of the order of \(10^{-5}\), whereas experiment gives 0.5.

The appearance of forbidden interferences had also been observed earlier. In the work of Brill, Grimm, and others, carried out in 1940[^2], an attempt was made to construct the projection of the electron density in diamond by the Fourier-synthesis method; however, the cited authors did not find any intermediate electron maximum. This, incidentally, is not surprising, since the method used (as was indicated above in this article) is so crude that the weak “forbidden” interferences were simply discarded by them in constructing the Fourier series. Yet it is precisely these that should, in the main, give the image of the intermediate charges.

The last example clearly shows the considerably greater possibilities of the trial method when applied to the study of electron distribution in crystals. Requiring only a successful choice of the model and of the parameters characterizing it, it subsequently reduces the problem to a comparison of specially selected points of the interference field, which, generally speaking, can be done quite reliably and with great accuracy. It is to be hoped that the development of this method will lead to a fuller knowledge of the electronic structure of crystals, which is important for the study of the chemical bond in crystals.

CITED LITERATURE

1. N. V. Ageev and L. M. Guseva, Izv. AN SSSR, Otd. Khim. Nauk, issue 4, 289 (1945); issue 1, 17 (1948), issue 3, 273 (1948).
2. R. Brill, H. Grimm, K. Hermann, Kl. Peters, Uspekhi khimii, 9, issue 4, 413 (1940).
3. V. K. Kritskaya and B. M. Rovinskii, ZhETF, 18, 785 (1948).

1. A. I. Kitaigorodskii, X-ray Structural Analysis, Gostekhizdat, 1950, p. 220 ff.
2. S. T. Konobeevskii, DAN, 49, 33 (1948).
3. S. T. Konobeevskii, Izv. AN SSSR—Sector of Physico-Chemical Analysis, 19, 19 (1949).
4. P. P. Ewald and H. Höhl, Ann. d. Physik, 25, 281 (1936).