Date Published: November 01, 2019
Publisher: International Union of Crystallography
Author(s): Markus Lentzen.
Relativistic electron diffraction depends on linear and quadratic terms in the electric potential, the latter being neglected in the frequently used relativistically corrected Schrödinger equation. Conventional tabulations for electron scattering and its large-angle extrapolations can be amended in closed form by a universal correction based on the screened Coulomb potential squared.
A frequently used framework for the calculation of high-energy electron diffraction by an atom or ion is the solution of the relativistically corrected Schrödinger equation (Molière, 1947 ▸; Fujiwara, 1961 ▸) with a model for the atomic or ionic electric potential. These model potentials are tabulated for a wide range of atomic numbers and frequently occurring ionic charges in the form of scattering factors (Doyle & Turner, 1968 ▸; Doyle & Cowley, 1974 ▸; Rez et al., 1994 ▸, 1997 ▸) or their parameterizations (Doyle & Turner, 1968 ▸; Doyle & Cowley, 1974 ▸; Fox et al., 1989 ▸; Rez et al., 1994 ▸, 1997 ▸; Waasmaier & Kirfel, 1995 ▸; Weickenmeier & Kohl, 1998 ▸; Peng, 1998 ▸; Lobato & Van Dyck, 2014 ▸); see Kirkland (2010 ▸) for a survey. Conventionally, tables of the scattering factors display the Born scattering amplitude, that is the Fourier transform of the electric potential times an interaction constant. A relativistic correction, dependent on the electron speed, is applied to the tabulated values, which can be directly used to determine scattering cross sections on the first Born approximation.
Born scattering amplitudes [equation (8)] were calculated for carbon (), germanium () and gold () at kinetic energies of 20, 200 and 2000 keV over the full range of scattering angles, . Two different models were used for the scattering potential: the screened Coulomb potential [equation (9)] and the screened Coulomb potential extended by the squared Coulomb potential term [equation (12)]. The scattering amplitudes for both models, and , are displayed in Figs. 1 ▸, 2 ▸ and 3 ▸. The difference between the two scattering amplitudes increases with increasing scattering angle, increasing atomic number and increasing kinetic energy.
Conventional tables of the scattering factors (Doyle & Turner, 1968 ▸; Doyle & Cowley, 1974 ▸; Rez et al., 1994 ▸, 1997 ▸; Kirkland, 2010 ▸) are organized such that the Born scattering amplitude [equation (8)] is only tabulated for a range of scattering vectors where Rutherford scattering is modified by the effects of screening, up to, e.g., s = g/2 = 60.0 nm−1. The amplitudes for larger scattering vectors are understood to be calculated with the Rutherford formula [equation (16)]. In a last step the tabulated values have to be multiplied by γ as the interaction constant used in the tabulations conventionally contains m and not .
The conventional framework of electron scattering by an electric potential is modified by an additional quadratic term in the electric potential, if the correct relativistic energy-momentum relation (1) is considered. The respective modification of atomic scattering amplitudes increases with increasing scattering angle, increasing atomic number and increasing kinetic energy. Conventional tabulations for electron scattering (Doyle & Turner, 1968 ▸; Doyle & Cowley, 1974 ▸; Rez et al., 1994 ▸, 1997 ▸; Kirkland, 2010 ▸) and its large-angle extrapolations can be amended in closed form by a universal correction [equation (13)] based on the screened Coulomb potential squared [equation (12)].