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Extra Spots on X-ray Photographs
On strongly exposed X-ray photographs of single crystals obtained with white radiation, diffraction spots have repeatedly been observed that are sharply different in appearance from ordinary diffraction spots. These extra spots could not be indexed by integers, and they had to be regarded either as reflections from a system of atomic planes with a wavelength \(\lambda\) not corresponding to their interplanar spacing \(d\), or as reflections from a system of some special planes not coinciding with crystallographic planes. Thus, or otherwise, for extra spots Bragg’s law \(n\lambda = 2d \sin\theta\) (\(\theta\) is the glancing angle) does not hold.
Faxén\(^1\) was the first, in 1923, to try to explain this phenomenon. He assumed that extra spots should be regarded as reflections from extraplanes formed by thermal waves passing along the principal planes of the crystal.
Interest in the phenomenon described unexpectedly reappeared after a 17-year interval, in 1939 and 1940. In 1939 a paper by Preston\(^2\) appeared, and in 1940 papers by Zachariasen\(^3\), Raman\(^4\), and Bragg\(^5\), giving different explanations of the occurrence of extra spots on X-ray photographs.
Preston, and also Bragg, assumes that the thermal motion of the lattice breaks the crystal into groups consisting of an atom and its nearest neighbors (8 or 12), so that the bond between atoms of a group is stronger than the bond between groups. The interatomic distances in different groups of atoms differ slightly from one another. Therefore, alongside the coherent scattering of all the atoms of the crystal, there arises coherent scattering by small groups. The total intensity of the former radiation is, as is known, the square of the sum of the amplitudes scattered by all atoms. The total intensity of the extra radiation is the sum of the intensities scattered by the small groups. The smaller the number of atoms forming a “coherent” group, and the greater the number of these groups, the more intense the extra spots should be.
The X-ray photograph of potassium chloride, on which a series of extra spots is clearly visible, is calculated by Bragg in the following way. As a “coherent” group he takes eight atoms situated at the vertices of a cube. The amplitude of the ray is proportional to
\[ \cos \frac{\pi a}{\lambda}\{h \cos \theta + (1 + l)\sin \theta\} \cos \frac{\pi a k}{\lambda} \cos \frac{\pi a}{\lambda}\{h \sin \theta - (1 - l)\cos \theta\}. \]
The maximum of this function is compared with the centers of the extra spots. The agreement between the calculated and experimental X-ray photographs is quite good, if one only notes that the size of the extra spots is considerably smaller than would follow from the calculation. The latter circumstance can be explained by the fact that the coherent group consists of more than eight atoms. Of course, with an increase in the number of atoms in the group, the maxima will become sharper and ultimately disappear; only diffraction maxima satisfying Bragg’s law will become possible.
The appearance of extra spots on an X-ray photograph was investigated experimentally and theoretically by Raman and his students. Raman indicates that, in accordance with the principles of the quantum theory, X-rays incident on a crystal can lead to pulsations of the electron density in the crystal, i.e., to oscillations of the structural amplitude. In this case the acoustic vibrations of the lattice lead to diffuse scattering, while the optical vibrations of the lattice give regular reflections, which we observe on X-ray photographs as extra spots. The directions in which the extra spots arise are determined by the geometrical reflection not from the atomic planes of the crystal lattice, but from the planes of constant phase of the oscillations of the structural amplitude. The inclination of these planes is determined solely by the conditions of reflection. The geometrical law of reflection has the form
\[ 2d \sin \frac{1}{2}(\theta + \varphi) = n\lambda, \]
where \(\theta, \varphi\) are the glancing angles of the incident and secondary rays. Each extra spot corresponds to an ordinary spot; in the case when white radiation is incident on the crystal, there is a continuous set of extra spots. This is also observed on Laue photographs in the form of regular “tails” with a sharp maximum for the characteristic wavelength. The law of reflection shows that the angle between the incident and reflected rays does not depend on the interplanar spacing. Theory shows, however, that the intensity of the extra spot does depend on the interplanar spacing.
The authors’ experiments with calcite and rock salt confirm his theory. Namely, the interplanar spacings calculated from the coordinate values of the extra spots for \(K_{\alpha}\)- and \(K_{\beta}\)-radiation (Mo anticathode) give the correct value with an accuracy of no less than \(0.02\ \text{Å}\).
A. I. Kitaigorodskii, Moscow
REFERENCES
- Faxen, Z. Physik, 17, 277, 1923.
- Preston, Proc. Roy. Soc., July, 1939.
- Zachariasen, Phys. Rev., 57, 597, 795, 1940.
- Raman and Nilakantan, Nature, 46, 523, 1940; 145, 860, 1940; Proc. Ind. Acad., 11, 398, 1940; 12, 141, 1940; Raman and Nath, Proc. Ind. Acad., 12, 83, 1940.
- Bragg, Nature, 146, 509, 1940.