Mott–Bethe formula explained

The Mott–Bethe formula is an approximation used to calculate atomic electron scattering form factors,

fe(q,Z)

, from atomic X-ray scattering form factors,

fx(q,Z)

.[1] [2] The formula was derived independently by Hans Bethe and Neville Mott both in 1930,[3] [4] 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,
f(
e(q,Z)=me2
3\hbar
32\pi
2\epsilon
0
Z-fx(q,Z)
q2

)=

1(
8\pi2a0
Z-fx(q,Z)
q2

)(0.2393~rm{nm}-1)(

Z-fx(q,Z)
q2

)

Here,

q

is the magnitude of the scattering vector of momentum-transfer cross section in reciprocal space (in units of inverse distance),

Z

the atomic number of the atom,

\hbar

is the Planck constant,

\epsilon0

is the vacuum permittivity, and

m0

is the electron rest mass,

a0

is the Bohr Radius, and

fx(q,Z)

is the dimensionless X-ray scattering form factor for the electron density.

The electron scattering factor

fe(q,Z)

has units of length, as is typical for the scattering factor, unlike the X-ray form factor

fx(q,Z)

, 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

\sqrt{\sigmaT

} = r_\text, where

re

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

q

(

q>0.5

Å) because the distribution of the charge density at smaller

q

(i.e. long distances) can deviate from the atomic distribution of electrons due the chemical bonds between atoms in a solid.[5] For smaller values of

q

,

fe(q,Z)

can be determined from tabulated values, such as those in the International Tables for Crystallography using (non)relativistic Hartree–Fock calculations,[6] or other numerical parameterizations of the calculated charge distribution of atoms.

Notes and References

  1. Cowley. J. M.. 2006. Electron diffraction and electron microscopy in structure determination. International Tables for Crystallography. B. 276–345. 10.1107/97809553602060000558. 978-0-7923-6592-1.
  2. Book: Advanced Computing in Electron Microscopy. 978-1-4757-4406-4. Kirkland. Earl J.. 17 April 2013. Springer .
  3. Mott. Nevill Francis. Bragg. William Lawrence. 1930-06-02. The scattering of electrons by atoms. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 127. 806. 658–665. 10.1098/rspa.1930.0082. 1930RSPSA.127..658M . free.
  4. Bethe. H.. 1930. Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie. Annalen der Physik. en. 397. 3. 325–400. 10.1002/andp.19303970303. 1930AnP...397..325B . 1521-3889.
  5. Lobato. I.. Van Dyck. D.. 2014-11-01. An accurate parameterization for scattering factors, electron densities and electrostatic potentials for neutral atoms that obey all physical constraints. Acta Crystallographica Section A. en. 70. 6. 636–649. 10.1107/S205327331401643X. 2053-2733. 10067/1221030151162165141. free.
  6. Book: High-Energy Electron Diffraction and Microscopy. 2004. Oxford University Press. New York, NY. registration. L. M. Peng . S. L. Dudarev . M. J. Whalen . 978-0-19-850074-2.