W. Jones

Lattice dynamics, X-ray scattering and one-body potential theory

One-body potential theory, which includes the effect of exchange and correlation forces, is used to calculate the change in the electron density due to small displacements of the ions. The final result contains a Dirac density matrix for the perfect crystal, the diagonal element being the exact ground state density ρ0(r). The basic quantity R(r) determining the electronic contribution to the dynamical matrix is such that the gradient of ρ0(r) is obtained by superposition of R(r - l) on each lattice site l. An integral equation is obtained which gives R(r) uniquely once the exchange and correlation energy is known. The Fourier transform Rk of R(r) is given in term s of the Fourier components ρKn of the charge density, which are known from X-ray scattering, by RKn = iρKnKn the reciprocal lattice vectors K n. This is the same result as the rigid-ion model at the Kns, which makes the assumption that this is true for all k. Deviations from rigid ions can be evaluated quantitatively from the integral equation obtained here. Such deviations reflect the role of many-body forces in lattice dynamics and the present theory provides a systematic basis for their calculation.

X-ray Scattering by Metals and the Charge Distribution in Body-Centred Cubic Iron and Chromium

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 h2+ k2+l2, 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.

The energy and the Dirac density matrix of a non-uniform electron gas

S>Diracs density matrix is found in a self-consistent framework. The problem of electrons moving in a periodic lattice of protons is considered, and difficulties of introducing Hartree fields associated with inner electrons are circumvented. For the conduction electrons, a Hartree description is adopted essentially. Physical significance of the results is enhanced by including screening from the outset. (L.N.N.)