X-RAY SPECTROSCOPIC METHODS FOR STUDYING THE MOSAIC STRUCTURE OF NATURAL AND DEFORMED CRYSTALS

Major recent advances in the field of X-ray spectroscopy make it possible to apply this method successfully to elucidating a number of questions connected with the structure of real crystalline bodies and the mechanism of their deformation.

As is known, the mosaic structure of real crystals leads to a broadening of the region in which they reflect X-rays, so that the actually observed value of the half-width of the reflection maximum considerably exceeds the value calculated under the assumption of an ideally perfect crystal. Bakovskii and Daleyjek, in a number of works, attempted to estimate the influence of the mosaic structure of crystals on the degree of resolution of the \(K_{\alpha}\) doublet. In these studies, various ratios were chosen between the distances from the crystal to the cassette and to the slit of the spectrograph.

In his last work Bakovskii\(^1\), assuming that the deviation of the orientation of individual blocks from the normal follows the Gaussian error curve, obtained (in the case of observance of the Bragg focusing condition) for the magnitude of line broadening associated with the mosaicity of the crystal the value \(\varepsilon\), calculated by the formula:

\[ \varepsilon = 2s \sin \Delta \sigma \cos \theta . \tag{1} \]

In doing so, Bakovskii neglects the depth of penetration of X-rays into the crystal in comparison with the length of the reflecting region on its surface.

In formula (1), \(\varepsilon\) is the line broadening; \(s\) is the distance of the reflecting block from the center of the crystal, measured in the plane of the crystal; \(\Delta \sigma\) is the angle of mosaicity; \(\theta\) is the Bragg angle.

The quantity \(s\), for values of \(\theta \gg \Delta \sigma\), is related to the radius of curvature of the crystal and the angle of incidence of the X-rays on it by the relation

\[ s = \frac{R \sin \Delta \sigma}{\sin \theta} \tag{2} \]

and, consequently,

\[ \varepsilon = 2R(\sin \Delta \sigma)^2 \operatorname{ctg}\theta \cong 2R(\Delta \sigma)^2 \operatorname{ctg}\theta . \tag{3} \]

Calculation shows that if the distances from the crystal to the cassette and to the slit are equal to each other, then the value of \(\varepsilon\) for NaCl does not exceed 2% of the intrinsic line width, equal, according to Allison’s data\(^2\), to 0.58 XE. Thus, with a symmetrical arrangement of the cassette and the slit of the spectrograph relative to the crystal, the mosaic structure of the latter manifests itself very weakly. In the case of an asymmetrical arrangement, realized, for example, in Seemann spectrographs or in those operating on the wedge principle, the situation is different. Under these conditions it is possible to obtain X-ray photographs whose appearance depends strongly on the degree of perfection of the crystal. Bakovskii points out that, when the crystal is placed at a distance of 3 m from the slit and the cassette is in its immediate vicinity, it is possible to obtain an X-ray photograph of the surface.

Using the asymmetrical Seemann method, Daleyjek and Klein\(^3\) measured the width of the Ag \(K_{\alpha}\) line after reflection from calcite. The total line width was found to be 11″. Subtracting from this the width of Ag \(K_{\alpha}\) (0.28 XE = 9.5″), measured by Allison, the authors, following Zachariasen\(^4\), calculated the line broadening arising owing to the mosaicity (\(W_c\)) of the calcite structure; this broadening is equal to 1.5″ (5.6″ according to Gauss). At the same time, the geometrical conditions of the experiment allowed X-rays to be reflected from regions of the crystal surface not exceeding \(1.5 \cdot 10^{-2}\) mm.

In Table 1 data are given for a number of other substances, obtained by the authors mentioned by the same method. The value of the intrinsic width of the \(K_\alpha\) Cu line was taken equal to 0.3 XE (0.28 according to Allison).

The method described can be applied to the study of the consequences of plastic deformation of the corresponding crystals. A study of the comparative perfection of the surface of the planes of freshly cleaved NaCl crystals and crystals subjected to bending around a cylinder of radius 20 cm was carried out in the work of Bosorth and Havort\(^5\). The authors reflected

\(^1\) Cleaved perpendicular to the electric axis.

X-rays first from the surface of a flat crystal and observed the splitting of the \(K_\alpha\) doublet at a distance of 1 m from the crystal. The resulting reflex was microphotometered across the band. The dip in the intensity curve between the maxima \(\alpha_1\) and \(\alpha_2\) reached 67% of the maximum value of the intensity \(\alpha_1\). On the basis of these curves the authors estimate the angular mosaic spread at 30″. This indicates a considerable degree of perfection of the surface of the cleaved pieces of NaCl subjected to investigation. Photometry of the reflex of the bent crystal, obtained at the focus, showed that the intensity of reflection from the bent crystal exceeds the corresponding values for the flat crystal by 30 times, while the integral reflection value is 2.2 times. Consideration of the reflected line makes it possible to establish the inhomogeneity of the bending. According to the authors’ data, the observed intensity is only 60% of that which would have been obtained if all points of the bent crystal reflected equally the maximum value of the intensity.

The author of the present article, together with Gogoberidze\(^1\), made an attempt to carry out a study of the phenomena accompanying the bending of real crystals in a spectrograph specially constructed for this purpose and operating according to Johann. The radius of curvature of the bent crystal was 470 mm. This makes it possible to bend a number of crystals, including rock salt\(^2\), without gross disturbance of the surface. The crystal was bent in a special device consisting of two plates ground to a prescribed radius and pressed against each other by four screws. The bending of the crystals was carried out in the spectrograph itself.

At the focus one succeeds in observing a well-split \(K_\alpha\) Cu doublet upon reflection from sufficiently perfect crystals. However, if the radiograph is taken beyond or in front of the focus of the apparatus, then, as experience has shown, no continuous blurring of the image occurs. The photograph assumes a specific appearance, representing a band stratified into a large

\(^1\) Communication at the scientific colloquium, June 1938, LPI (being prepared for publication).

\(^2\) Bosorth and Havort bent their crystals under water.

number of sharply delineated lines in the case of mica and NaCl crystals, and a somewhat different appearance for gypsum. In view of the great resolving power of such an apparatus and the high intensity of the image, which sometimes, according to Bozorth, exceeds the intensity of reflection from a flat crystal by a factor of 30, it proves possible to judge from the change in the crystal connected with various mechanical actions upon it. In photographs obtained from bent mica, we were able to observe splitting along the band that differed in magnitude. In this case, individual observed reflexes were sometimes displaced relative to one another vertically. Such a picture is easily explained if one assumes that, when bent, blocks are formed in the crystal that are rotated relative to one another through small angles. It is important to note that Lauegrams taken from such crystals in the straightened and bent states practically coincide; thus, apparently, bending does not produce any noticeable change in the crystal lattice of the specimens.

Accepting the proposition of the block structure of bent crystals, in accordance with the views of Kossel, Bakovsky, and a number of others, one can characterize the mutual position of the blocks that are formed by angles in the vertical and horizontal planes. The position of the plane of a block is determined unambiguously by these data only in the case where the surfaces of the blocks and, after bending, remain flat. Apparently, for the bending of some crystals this assumption is not justified. The mutual displacement of blocks in the horizontal plane determines the splitting of the band; that in the vertical plane determines the magnitude of the vertical displacement of the strokes relative to one another. The magnitude of the horizontal splitting \(l\) is related to the angle \(\psi\) by the relation

\[ l = \frac{2r}{\sin \theta}\operatorname{ctg}\frac{\psi}{2}. \]

The angle characterizing the degree of non-coplanarity of the individual blocks, \(\alpha\), is calculated from the formula

\[ \operatorname{tg}\alpha \sin \theta = 2\sin\frac{\varphi}{2}, \]

where \(\theta\) is the Bragg angle, and \(\varphi\) is the angle between the rays reflected from two displaced blocks in the plane containing both these rays.

At the same time the following relation holds between the sines of the reflection angles of two rays lying in planes the angle between which is \(\alpha\):

\[ \sin \theta = \sin \theta' \cos \alpha . \]

The magnitude \(\varphi\) is found from the vertical displacement of the lines on the X-ray photograph and the distance from the crystal to the image. According to calculations, the value \(180-\psi\) ranges from \(5'\) to \(40'\), while the angle \(\alpha\) in some cases reaches \(1^\circ 30'\). At very high orders (up to the eighth), it is sometimes possible to observe finer splitting of the band, which is apparently due to the natural mosaic structure of the crystal.

Parallel optical and radiographic photographs of the crystal surface confirm the assumption of the presence of rotations accompanied by a strong disturbance of the lattice at the boundary between two rotated blocks.

E. E. Weinstein, Leningrad

Literature

1. Backovsky, J. d. Physique, 11, 471, 1938.
2. Allison, Phys. Rev., 44, 63, 1933.
3. Doleyssek, Nature, 139, 886, 1937.
4. Hoyt, Phys. Rev., 32, 477, 1932.
5. Bozorth et Haworth, Phys. Rev., 53, 538, 1938.