# Structure factor

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The (static) structure factor is a mathematical expression unique to every structure related to the density distribution of its constituents. Mathematically, it is commonly defined as follows:

$S_{\mathbf{q}}= 1/V \sum_{i,j} e^{-i\mathbf{q} \cdot {(\mathbf{r}_i-\mathbf{r}_j)}}$

where V is the volume, the sum over i,j is a sum over the constituent particles of the structure in question, and q has units of inverse length (wavenumber). The particles can be atoms, molecules, or larger entities like polystyrene spheres periodically arranged in a photonic crystal. Note that this quantity only depends on the positions of the constituent particles and does not depend on their nature or interactions; $S_{\mathbf{q}}$ is a purely structural quantity.

This quantity is very useful. A closer look at the formula reveals that it is the Fourier transform of the density distribution. As such, information regarding some general properties of a material, such as the existence of characteristic length scales, periodicity along some axis, and symmetries, can be easier to identify from the structure factor than from the structure itself.

Moreover, it is closely related to the scattering intensity from a material. In the case of single elastic scattering, for a material comprised of only one type of particles, this relation can be phrased mathematically as follows:

$\sigma=V \int d\Omega S_{\mathbf{q}} F_{\mathbf{q}}$

where $F_{\mathbf{q}}$ is the form factor, a quantity representing the way each constituent particle scatters. This expression can be generalized for arbitrary number or particle types by a simple summation over all of them, where each term has the corresponding form factor. Note that the notions of form factor and structure factor are somewhat relative and depend on the length scale that is considered "fundamental": for example, in order to study scattering from a structure made of 1μm polystyrene spheres is may be convenient to use the form factor of the sphere, however each such sphere is made of atoms and its form factor depends, in turn, on the form factors of its constituent atoms and the way they arranged in space, i.e. the sphere's structure factor.

From this expression for scattering an alternate interpretation of the structure factor arises: it is the quantity representing how waves scattered from each structural feature interfere with each other. Adopting the Rayleigh principle, during elastic scattering of radiation from a material each scattering element acts as a point source of a spherical wave and the resulting pattern is the sum of all these waves originating from each scatterer. Within this picture, the interpretation of S(q) as the Fourier transform of the density becomes synonymous to the interpretation arising from the formula for the cross-section.

Crystallography and materials science rely heavily on S(q); all scattering experiments, such as (small-angle) X-ray scattering and small-angle neutron scattering, measure S(q). This data can either be used directly or inverted to obtain the density distribution. It is also of fundamental importance in the study of photonic crystals: any peak in the structure factor corresponds to strong scattering of light carrying momentum corresponding to the momentum vector q where the peak is located, and may thus signal the onset of a photonic stop band or band gap.

## Reference

Principles of Condensed Matter Physics, Chaikin and Lubensky, Cambridge University Press (1995)