The Mott–Bethe formula is an approximation used to calculate atomic

electron scattering Electron scattering occurs when electrons are displaced from their original trajectory. This is due to the electrostatic forces within matter interaction or, if an external magnetic field is present, the electron may be deflected by the Lorentz ...

form factors,

f_\text (q,Z)

, from atomic X-ray scattering form factors,

f_x(q,Z)

. The formula was derived independently by

Hans Bethe Hans Albrecht Eduard Bethe (; ; July 2, 1906 – March 6, 2005) was a German-American physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics and solid-state physics, and received the Nobel Prize in Physi ...

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 Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...

for the charge density of an atom (including both the nucleus and electron cloud) in the Fourier domain. Following the first Born approximation, :

f_\text(q,Z)=\frac\Bigg(\frac\Bigg) =\frac\Bigg(\frac\Bigg) \approx (0.2393~\textrm^)\cdot \Bigg(\frac\Bigg)

Here,

q

is the magnitude of the scattering vector of momentum-transfer cross section in

reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e ...

(in units of inverse distance),

Z

the

atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...

of the atom,

\hbar

is the

Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

\epsilon_0

is the

vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...

, and

m_0

is the electron rest

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

a_0

is the

Bohr Radius The Bohr radius () is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an at ...

, and

f_x(q,Z)

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

f_\text(q,Z)

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

f_x(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 = r_\text

, where

r_\text

is the

classical electron radius The classical electron radius is a combination of fundamental Physical quantity, physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic sel ...

, 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. For smaller values of

q

f_\text(q,Z)

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.

References

{{DEFAULTSORT:Mott-Bethe formula Atomic physics Scattering theory