Parameterized scattering factors
A computational tool which (a) enables Gaussian parameterization of elastic electron scattering factors [International Tables for Crystallography, Volume C, pp. 263-281 (2004)] into six analytical approximations and (b) derives fractionally charged scattering factors by computing weighted sums of adjacent integral neighbors. Curve-fitting is treated as a nonlinear least-squares optimization problem and implemented using a Levenberg-Marquardt algorithm in MATLAB.
Supported approximations include:
where j = 4 or 5. Inclusion of the charge-correction term given by Peng [Acta Cryst. A54, 481-485 (1998)] yields a superior statistical fit for ionic electron scattering factors; however, this term is unsupported by crystallographic refinement programs like PHENIX, REFMAC, and SHELXL. For PHENIX, select 5 Gaussians + c; for REFMAC, 5 Gaussians; and for SHELXL, 4 Gaussians.
Elastic and inelastic cross-sections
In crystallography, we typically express elastic X-ray or electron scattering amplitudes f(s) as a function of the vector s, where s = sin(θ)/λ. Transmission electron microscopy employs a different convention, where each scattering factor is explicitly expressed as a function of the scattering angle θ in lieu of s. Once refactored into this form, we can derive the total elastic cross-section σe directly from f(θ) by integrating the differential cross-section dθ/dΩ over all scattering angles from 0 to π:
This exploits the fact that the differential cross-section is equivalent to the squared magnitude of f(θ). f(θ) is simply taken as the 5 Gaussian approximation from FAES. Within the range of accelerating voltages relevant to TEM (80 to 400 kV), elastic cross-sections scale linearly with β-2 and nonlinearly with incident energy. We can estimate σi, the corresponding inelastic cross-section, by multiplying its elastic counterpart by a scalar factor of 20.2/Z [J. Microsc. 159, 143–160 (1990)]. For elements lighter than Ar, σe < σi.