Mott–Bethe formula
The Mott–Bethe formula is an approximation used to calculate atomic electron scattering form factors, formula_1, from atomic X-ray scattering form factors, formula_2. The formula was derived independently by Hans Bethe and Neville Mott both in 1930, and simply follows from applying the first Born approximation for the scattering of electrons via the Coulomb interaction together with the Poisson equation for the charge density of an atom (including both the nucleus and electron cloud) in the Fourier domain. Following the first Born approximation, Here, formula_4 is the magnitude of the scattering vector of momentum-transfer cross  section in reciprocal space (in units of inverse distance), formula_5 the atomic number of the atom, formula_6 is Planck's constant, formula_7 is the vacuum permittivity, and formula_8 is the electron rest mass, formula_9 is the Bohr Radius, and formula_2 is the dimensionless X-ray scattering form factor for the electron density. The electron scattering factor formula_11 has units of length, as is typical for the scattering factor, unlike the X-ray form factor formula_2 which is usually presented in dimensionless units. To perform a one-to-one comparison between the electron and X-ray form factors in the same  units, the X-ray form factor should be multiplied by the square root of the Thomson cross section formula_13, where formula_14 is the classical electron radius, to convert it back to a unit of length. The Mott–Bethe formula was originally derived for free atoms, and is rigorously true provided the X-ray scattering form factor is known exactly. However, in solids, the accuracy of the Mott–Bethe formula is best for large values of formula_4 (formula_16 Å) because the distribution of the charge density at smaller formula_4 (i.e. long distances) can deviate from the atomic distribution of electrons due  the chemical bonds between atoms in a solid. For smaller values of formula_4, formula_11 can be determined from tabulated values, such as those in the International Tables for Crystallography using (non)relativistic Hartree–Fock calculations, or other numerical parameterizations of the calculated charge distribution of atoms.