Absorptive form factors for high-energy electron diffraction

Abstract

The thermal diffuse scattering contribution to the absorptive potential in high-energy electron diffraction is calculated in the form of an absorptive contribution to the atomic form factor. To do this, the Einstein model of lattice vibrations is used, with isotropic Debye-Waller factors. The absorptive form factors are calculated as a function of scattering vector s and temperature factor M on a grid which enables polynomial interpolation of the results to be accurate to better than 2% for much of the ranges 0-< Ms 2<-6 and 0_< M_<2/~2. The computed values, together with an interpolation routine, have been incorporated into a Fortran subroutine which calculates both the real and absorptive form factors for 54 atomic species.

Robust Parameterization of Elastic and Absorptive Electron Atomic Scattering Factors

“…The numerical values of the absorptive structure factors are calculated based on the original formula of Hall & Hirsch (Hall & Hirsch, 1965;Bird & King, 1990):…”

“…The imaginary part of the complex scattering factor is called the absorptive scattering factor. It has been shown by several authors that, for all g ~ 0, the atomic contribution to the absorptive scattering factor is dominated by thermal diffuse scattering (TDS) (Whelan, 1965;Humphreys & Hirsch, 1968;Radi, 1970) and the TDS contribution can be calculated readily following the original formulation of Hall & Hirsch (1965) assuming isotropic Debye-Waller factors and an Einstein model for TDS scattering (Bird & King, 1990;Weickenmeier & Kohl, 1991). While the numerical values of the scattering factors can be incorporated directly into computer programs for dynamical electron diffraction calculations, there exist many situations where an analytic expression for the scattering factors is desired, as in the case of reflection high-energy electron diffraction (RHEED) (Dudarev, Peng & Whelan, 1995) and the development of tensor methods for the direct determination of structures using high-energy electrons (Peng & Dudarev, 1994a,b;Peng & Zuo, 1995;Peng, 1995).…”

Transmission low-energy electron diffraction (TLEED) and its application to the low-voltage point-projection microscope

“…This scattering is localized within the crystal unit cell and so causes 'anomalous' absorption of LEED beams. Several calculations of the Fourier coefficients V~ of the contribution to the absorption potential V'(r) describing excitation of phonons at different temperatures have been published for THEED (Bird & King, 1990;Radi, 1970). These are based on the original formulation of Hall & Hirsch (1965), which adopts an Einstein model for lattice vibrations.…”

“…Unlike the THEED case, it is important to include backscattered beams to account for all multiple-scattering processes. The absorption coefficients used to describe the depletion of the elastic wavefield by inelastic scattering must be investigated, since the coefficients used for high-energy electrons (Bird & King, 1990) may not be applicable. Exchange and virtual inelastic scattering effects cannot be ignored, as in THEED, and their corrections to the crystal potential must be estimated.…”

Energy-filtered HREM images of valence-loss electrons

“…This is equivalent to introducing an absorption potential in the calculation of elastic wave. This absorption factor can be calculated either in real [24,25] or reciprocal [26] space. The phonon scattered electrons falling inside the objective aperture angular range form incoherent atomic resolution image.…”