Mott–Bethe formula

The Mott–Bethe formula is used to calculate electron scattering form factors, ${\displaystyle f_{e}(q,Z)}$, from X-ray scattering form factors, ${\displaystyle f_{x}(q,Z)}$.[1] Following the first Born approximation,

${\displaystyle f_{e}(q,Z)={\frac {me^{2}}{32\pi ^{3}\hbar ^{2}\epsilon _{0}}}{\Bigg (}{\frac {Z-f_{x}(q,Z)}{q^{2}}}{\Bigg )}}$

Here, ${\displaystyle q}$ is the magnitude of the scattering vector of momentum transfer cross section in reciprocal space (in units of inverse distance), ${\displaystyle Z}$ the atomic number of the atom, ${\displaystyle \hbar }$ is Planck's constant, ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity, and ${\displaystyle m_{0}}$ is the electron rest mass.

The units of the scattering factor are ${\displaystyle \mathrm {Potential_{electric}} /\mathrm {distance} ^{3}}$. The accuracy of this formula is best for large values of ${\displaystyle q}$ (${\displaystyle q>0.5\mathrm {\AA} }$). For smaller values of ${\displaystyle q}$, ${\displaystyle f_{e}(q,Z)}$ can be determined from tabulated values, such as those in the International Tables for Crystallography. [2]

References

1. Advanced Computing in Electron Microscopy. ISBN 1475744064.
2. L. M. Peng, S. L. Dudarev, M. J. Whalen (2004). High-Energy Electron Diffraction and Microscopy. New York, NY: Oxford University Press.CS1 maint: uses authors parameter (link)