Structure factor

In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns (interference patterns) obtained in X-ray, electron and neutron diffraction experiments.

Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written ${\displaystyle S(\mathbf {q} )}$; it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit. The other is usually written ${\displaystyle F}$ or ${\displaystyle F_{hk\ell }}$ and is only valid for systems with long-range positional order — crystals. This expression relates the amplitude and phase of the beam diffracted by the ${\displaystyle (hk\ell )}$ planes of the crystal (${\displaystyle (hk\ell )}$ are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell. ${\displaystyle F_{hk\ell }}$ is not a special case of ${\displaystyle S(\mathbf {q} )}$; ${\displaystyle S(\mathbf {q} )}$ gives the scattering intensity, but ${\displaystyle F_{hk\ell }}$ gives the amplitude. It is the modulus squared ${\displaystyle |F_{hk\ell }|^{2}}$ that gives the scattering intensity. ${\displaystyle F_{hk\ell }}$ is defined for a perfect crystal, and is used in crystallography, while ${\displaystyle S(\mathbf {q} )}$ is most useful for disordered systems. For partially ordered systems such as crystalline polymers there is obviously overlap, and experts will switch from one expression to the other as needed.

The static structure factor is measured without resolving the energy of scattered photons/electrons/neutrons. Energy-resolved measurements yield the dynamic structure factor.