X-ray scattering by metals and the charge distribution in body-centred cubic iron and chromium

Abstract

The representation of a periodic density by a sum of localized distributions centred on every lattice site in the crystal is shown to greatly facilitate the calculation of X-ray scattering from crystal electron densities. Unfortunately, the localized distributions are not unique when the charge clouds overlap and therefore cannot be claimed to have direct physical significance. However, it appears that the most favourable choice will usually correspond to that localized distribution for which the angularity is reduced to a minimum. Experiments on single crystals, in which reflexions are examined which correspond to different sets of Miller indices hkl with the same value of h 2 + k 2 + l 2 , can decide in a given case whether there is irreducible angularity in the localized densities. The present method is then employed to analyse the X-ray data of Batterman and his co-workers on body-centred cubic iron, and that of Cooper on chromium. Experimental errors are very large for our purposes, the number of reflexions examined is quite limited and the experiments are on powders, and each of these factors must introduce some uncertainties. With these reservations, however, it is shown that the experimental results for both iron and chromium may be interpreted consistently in terms of spherical distributions on the lattice sites. Several forms of localized densities were employed, all compatible with the experimental data, and from these densities, by summation over sites, the charge density in the unit cell may be calculated. In particular, for these choices of localized densities, the s and g terms of the density in the unit cell are displayed. The results show that the changes from the s density given by the superposition of Hartree–Fock atoms on the lattice sites are quite substantial, increasing the boundary density by a factor between 1·5 and 2 from the superposition value for both metals. The magnitude of the correction to the superposition value for the g term is less certain than for the s term , but it is also much smaller, and the ambiguity is therefore unimportant. Finally, the present results for iron, derived from the X-ray experiments, are compared with a calculation of the s component due originally to Slater & Krutter, and also with the g term calculated by Hum. In each case, the agreement is found to be quite reasonable.