X-ray atomic form factors of oxygen (blue), chlorine (green), Cl⁻ (magenta), and K⁺ (red); smaller charge distributions have a wider form factor.

In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident radiation, typically X-ray, electron or neutron. The common feature of all form factors is that they involve a Fourier transform of a spatial density distribution of the scattering object from real space to momentum space (also known as reciprocal space). For an object with spatial density distribution, ${\displaystyle \rho (\mathbf {r} )}$, the form factor, ${\displaystyle f(\mathbf {Q} )}$, is defined as

${\displaystyle f(\mathbf {Q} )=\int \rho (\mathbf {r} )e^{i\mathbf {Q} \cdot \mathbf {r} }\mathrm {d} ^{3}\mathbf {r} }$,

where ${\displaystyle \rho (\mathbf {r} )}$ is the spatial density of the scatterer about its center of mass (${\displaystyle \mathbf {r} =0}$), and ${\displaystyle \mathbf {Q} }$ is the momentum transfer. As a result of the nature of the Fourier transform, the broader the distribution of the scatterer ${\displaystyle \rho }$ in real space ${\displaystyle \mathbf {r} }$, the narrower the distribution of ${\displaystyle f}$ in ${\displaystyle \mathbf {Q} }$; i.e., the faster the decay of the form factor.

For crystals, atomic form factors are used to calculate the structure factor for a given Bragg peak of a crystal.

X-ray form factors

The energy dependence of the real part of the atomic scattering factor of chlorine.

X-rays are scattered by the electron cloud of the atom and hence the scattering amplitude of X-rays increases with the atomic number, ${\displaystyle Z}$, of the atoms in a sample. As a result, X-rays are not very sensitive to light atoms, such as hydrogen and helium, and there is very little contrast between elements adjacent to each other in the periodic table. For X-ray scattering, ${\displaystyle \rho (r)}$ in the above equation is the electron charge density about the nucleus, and the form factor the Fourier transform of this quantity. The assumption of a spherical distribution is usually good enough for X-ray crystallography.^[1]

In general the X-ray form factor is complex but the imaginary components only become large near an absorption edge. Anomalous X-ray scattering makes use of the variation of the form factor close to an absorption edge to vary the scattering power of specific atoms in the sample by changing the energy of the incident x-rays hence enabling the extraction of more detailed structural information.

Atomic form factor patterns are often represented as a function of the magnitude of the scattering vector ${\displaystyle Q=2k\sin(\theta )}$. Herein ${\displaystyle k=2\pi /\lambda }$ is the wavenumber and ${\displaystyle 2\theta }$ is the scattering angle between the incident x-ray beam and the detector measuring the scattered intensity, while ${\displaystyle \lambda }$ is the wavelength of the X-rays. One interpretation of the scattering vector is that it is the resolution or yardstick with which the sample is observed. In the range of scattering vectors between ${\displaystyle 0<Q<25}$ Å⁻¹, the atomic form factor is well approximated by a sum of Gaussians of the form

${\displaystyle f(Q)=\sum _{i=1}^{4}a_{i}\exp \left(-b_{i}\left({\frac {Q}{4\pi }}\right)^{2}\right)+c}$

where the values of a_i, b_i, and c are tabulated here.^[2]

Electron form factor

The relevant distribution, ${\displaystyle \rho (r)}$ is the potential distribution of the atom, and the electron form factor is the Fourier transform of this.^[3] The electron form factors are normally calculated from X-ray form factors using the Mott–Bethe formula.^[4] This formula takes into account both elastic electron-cloud scattering and elastic nuclear scattering.

Neutron form factor

There are two distinct scattering interactions of neutrons by nuclei. Both are used in the investigation structure and dynamics of condensed matter: they are termed nuclear (sometimes also termed chemical) and magnetic scattering.

References