Thomas--Fermi--Dirac calculations of incoherent-scattering factors

The incoherent-scattering factor ($S$) derivation of Heisenberg is extended to include the Thomas-Fermi-Dirac (TFD) model of the atom. TFD neutral-atom calculations of the electron densities and potentials are made with the use of a slope-iteration scheme. The calculations of $S$ show that the TFD values of $S$ are much smaller than the Thomas-Fermi values for small momentum transfers and, therefore, are in better agreement with Hartree-Fock values. The improvement is much greater for small-$Z$ atoms. A comparison of Hartree-Fock and TFD $S$ values is given over a wide range of momentum transfers for the rare gases.