Atomic form factor
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.
As a result of the nature of the Fourier transform, the broader the distribution of the scatterer
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.
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
One interpretation of the scattering vector is that it is the resolution or yardstick with which the sample is observed.
is the potential distribution of the atom, and the electron form factor is the Fourier transform of this.
The wavelength of thermal (several ångströms) and cold neutrons (up to tens of Angstroms) typically used for such investigations is 4-5 orders of magnitude larger than the dimension of the nucleus (femtometres).
The free neutrons in a beam travel in a plane wave; for those that undergo nuclear scattering from a nucleus, the nucleus acts as a secondary point source, and radiates scattered neutrons as a spherical wave.
(Although a quantum phenomenon, this can be visualized in simple classical terms by the Huygens–Fresnel principle.)
is the spatial density distribution of the nucleus, which is an infinitesimal point (delta function), with respect to the neutron wavelength.
The delta function forms part of the Fermi pseudopotential, by which the free neutron and the nuclei interact.
The Fourier transform of a delta function is unity; therefore, it is commonly said that neutrons "do not have a form factor;" i.e., the scattered amplitude,
This Fourier transform is scaled by the amplitude of the spherical wave, which has dimensions of length.
Hence, the amplitude of scattering that characterizes the interaction of a neutron with a given isotope is termed the scattering length, b. Neutron scattering lengths vary erratically between neighbouring elements in the periodic table and between isotopes of the same element.
They may only be determined experimentally, since the theory of nuclear forces is not adequate to calculate or predict b from other properties of the nucleus.
In neutron scattering from condensed matter, magnetic scattering refers to the interaction of this moment with the magnetic moments arising from unpaired electrons in the outer orbitals of certain atoms.
Since these orbitals are typically of a comparable size to the wavelength of the free neutrons, the resulting form factor resembles that of the X-ray form factor.
