Mott–Bethe formula

The Mott–Bethe formula is used to calculate electron form factors from X-ray form factors.^[1] The equation uses the first Born approximation.

$f^B(q,Z)=\frac{me^2}{2\pi\hbar^2\epsilon_0}\Bigg(\frac{Z-f_x(q,Z)}{q^2}\Bigg)$

where $f_e (q,Z)$ is the electron scattering cross-section and $f_x(q,Z)$ is the x-ray scattering cross section. This is a function of $q$, the scattering vector of momentum transfer cross section in reciprocal space (in units of inverse distance), and Z is the atomic number of the atom. In the equation $\hbar$ is Planck's constant, $\epsilon_0$ is the vacuum permittivity.

The units of the scattering factor are $\mathrm{Potential_{electric}}/\mathrm{distance}^3$

References

1. Advanced Computing in Electron Microscopy. ISBN 1475744064.