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 ${\displaystyle F_{hkl}}$ 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 ${\displaystyle (hkl)}$ planes of the crystal (${\displaystyle (hkl)}$ are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell. ${\displaystyle F_{hkl}}$ is thus not a special case of ${\displaystyle S(\mathbf {q} )}$; the observed scattering intensity depends on ${\displaystyle S(\mathbf {q} )}$, but on the squared modulus ${\displaystyle |F_{hkl}|^{2}}$. ${\displaystyle F_{hkl}}$ is defined in terms of 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.

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