Full Text
FROM THE CURRENT LITERATURE
REDUCTION OF X-RAY ABSORPTION BY A CRYSTAL UNDER INCIDENCE AT THE REFLECTION ANGLE
In considering the reflection of X-rays by a crystal, we always assume that the attenuation of the primary beam is due to two factors: ordinary absorption and the screening of the lower layers of the crystal by the upper ones. An increase in the attenuation coefficient of X-rays upon incidence on a crystal at an angle \(\theta\) \(\left(\sin \theta = \frac{n\lambda}{2d}\right)\) seems at first sight to be an entirely necessary occurrence, since in this position an additional fraction of the energy of the primary beam goes into reflection.
In this connection, recently published experiments\(^1\) are of interest; they distinctly show that, in a number of similar cases, instead of the expected increase in attenuation, there is a decrease in the attenuation of the primary beam.
The experiments were carried out using copper \(K_{\beta}\)-radiation. The behavior of crystals of calcite, mica, and hexamethylenetetramine was investigated.
A parallel beam of strictly monochromatic rays passed through a crystal, which had been cut out and mounted in such a way that the paths of the transmitted and reflected rays in the crystal were exactly the same. The intensities of both rays were measured with counters. The crystal was rotated through the reflecting position (within \(\pm 2\) minutes for calcite, \(\pm 4\) minutes for hexamethylenetetramine, and \(\pm 20\) minutes for mica). Curves of intensity as a function of the angle of rotation were constructed, examples of which are given in the figure.
As can be seen from the graphs, at the reflection angle the intensity of the transmitted beam increases. In the case of Fig. 1, б, the intensity of the transmitted beam outside the region of reflection is zero; in the diffraction interval the attenuation decreased so much that the transmitted beam gave as distinct an intensity curve as the reflected one.
However, the indicated effect is not universal. In the case of hexamethylenetetramine the intensity of the transmitted beam does not change when the crystal passes through the reflecting position. A weak increase in intensity was observed for mica.
Apparently, the effect described is scarcely noticeable in mosaic crystals, since it is smaller the wider the interval of reflection. What, then, is the explanation of this effect?
The dynamical theory of the interference of X-rays (strictly speaking, applicable only to an ideal crystal) was extended
for the case of an absorbing crystal as early as 1933. However, the question of the influence of absorption on the intensity of the transmitted beam in the case of the reflecting position of the crystal was not considered.
Fig. 1. a) calcite. Reflection \((10 — 11)\). Etched crystal approximately \(1\ \mathrm{mm}\) thick; b) calcite. Reflection \((10 — 11)\). Polished crystal \(0.75\ \mathrm{mm}\) thick. Incident beam \(1.2\cdot 10^{6}\) quanta/min.
In both cases the vertical axis gives the intensity in 1000 quanta/min., and the abscissa gives the angle of rotation in minutes of arc. The dashed curve is the transmitted beam. The solid curve is the reflected beam.
To investigate this problem, Laue² considers a complex function \(\chi(xyz)\)—a function of the coordinates of the crystal space, whose real part is the electron density of the crystal and whose imaginary part describes absorption. If \(\chi\) is expanded in a Fourier series, the coefficients in the expansion of the real part give the amplitudes of the scattered radiation, while the coefficients in the expansion of the imaginary part give the attenuation factors. This essentially assumes that the absorption coefficient of the rays has physical meaning only when it is specified which diffraction ray \(hkl\) is meant.
By replacing in the well-known formulas of the dynamical theory of interference (see, for example, P. P. Ewald, Optical Principles of the Diffraction of X-rays, IL, 1950) the real function—the electron density—by a complex one, it is possible to show what features absorption introduces into the phenomenon of X-ray diffraction by crystals. A whole series of these features (for example, the asymmetry of the interference curve) had been established earlier.
In the cited work Laue calculates the absorption coefficients of the transmitted and diffracted waves and shows: 1) that these absorption coefficients are different; 2) that they depend on the quantity characteristic of the given interference,
\[ \Psi_{hkl}=\chi'_{hkl}\chi'_{\bar h\bar k\bar l}+\chi^i_{hkl}\chi^i_{\bar h\bar k\bar l}, \]
where \(\chi'\) and \(\chi^i\) are the real and imaginary parts of the coefficient in the Fourier expansion of the function \(\chi(xyz)\); 3) that for \(\Psi_h>0\) the absorption is greater than normal (i.e., that occurring in the absence of diffraction) for the reflected ray, while for \(\Psi_h<0\) it is greater for the transmitted ray.
Thus, apparently, the decrease in absorption of the transmitted ray is formally explained by Laue’s theory; it must be accompanied by enhanced absorption of the diffraction ray.
It should be noted that, quantitatively, there are substantial discrepancies between the calculations and experiment. Be that as it may, it is clear that the absorption of a plane wave passing through a crystal differs substantially from the absorption of a wave field in the case when the crystal is in the reflecting position.
In a number of cases the screening phenomenon is opposed by another, considerably stronger effect.
A. K.
CITED LITERATURE
- H. N. Campbell, J. Appl. Phys. 22, 1139 (1951).
- M. Laue, Acta Cryst. 2, 126 (1949).