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

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