Since the discovery of the Compton effect it has been evident that the old calculation by Debye of the diffuse scattering of x-rays by a crystal should be amended to take account of the modified radiation, which is generally presumed to be incoherent. Extending the idea suggested by the writer in his theory of x-ray scattering by gases, it is postulated that, in considering the scattering of x-rays by a crystal, only the coherent radiation from the different lattice units will interfere with each other, whereas the incoherent radiation will be simply added up. An expression for the intensity of the diffuse scattering is thence developed which consists of two parts, one taking care of the coherent scattering as originally worked out by Debye and the other accounting for the incoherent scattering. The mathematical formulation is based on the theoretical investigation by Raman and A. H. Compton on the scattering of x-rays by a dynamic atom. The theory is compared with the absolute measurements of the scattering from rocksalt reported by Jauncey and May and the agreement seems to be very close if the temperature factor is taken to be $e^{-2M}$ as calculated by Debye and Waller. The comparison also seems to suggest the presence of the zero-point energy. These are in accord with the conclusions recently drawn by James, Waller and others from a study of the temperature effect on the intensity of the x-rays regularly reflected. The theory therefore seems to account for the so-called excess scattering, for the small scattering in the region of 0° scattering angle, for the position of the maximum scattering as well as the shift of this position with the wave-length of the primary x-rays, for the occurrence of the minimum scattering at about 100° instead of at 90° and finally for the general departure of the scattering curve from that predicted by Thomson's theory. A comparison is also made with Jauncey's experiment on the variation of the scattering by rocksalt with temperature and the result indicates that, owing to the presence of the incoherent term, measurements of this type will not test the present theory. It is pointed out that, being developed for a single crystal in which the atoms are supposed to be arranged with perfect regularity, the theory will not hold for the scattering of x-rays by the so-called amorphous substances. In these substances, the arrangement of atoms evidently introduces a type of irregularity which results in diffusely scattered rays in addition to those due to the thermal agitation. Thus, under certain conditions, only a negligible portion of the coherent intensity of the x-rays diffusely scattered by an amorphous substance is influenced by the temperature. This explains the experimental fact recently reported by Jauncey and Bauer that there is no effect of temperature on the ratio of modified to unmodified rays in the Compton effect. Theoretical predictions for the crystal sylvine are given and absolute measurements performed with homogeneous x-rays should easily test these results.

• Received 16 May 1931

DOI:https://doi.org/10.1103/PhysRev.38.6