Asymptotic forms of atomic scattering factors and momentum densities

Within the framework of the nonrelativistic Schrödinger equation, the Coulomb field case is used to assess the range of validity of the Thomas–Fermi statistical theory of atoms. In particular, attention is focused on (a) the x‐ray scattering factor, which is the Fourier transform of the electron density; and (b) the momentum density. In each case the predictions of the statistical theory are compared with the exact results for the Coulomb potential. These can conveniently be calculated using earlier work of Fock. Some assessment is also made of the accuracy of the statistical approximation for the kinetic energy density; this is of interest in connection with the density functional approach. Finally, some brief comments are made on the relation between the self‐consistent Thomas–Fermi method and the Hartree theory for atoms.