# 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:

- <math> S_{\mathbf{q}}=\sum_{i,j} e^{-i\mathbf{q} \cdot {(\mathbf{r}_i-\mathbf{r}_j)}}</math>

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; :<math> S_{\mathbf{q}}</math> 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, this relation can be phrased mathematically as follows:

- <math> S_{\mathbf{q}}=\sum_{i,j} e^{-i\mathbf{q} \cdot {(\mathbf{r}_i-\mathbf{r}_j)}}</math>

where () is the form factor, a quantity representing the way each constituent particle scatters: i.e. the scattered light will be <...>. From this expression an alternate interpretation of the structure factor arises: it is the quantity representing how waves scattered from each structural feature interfere with each other. Considering that single elastic scattering is a 'physical' way to see the Fourier transform of a structure, it can be seen that these two explanations of S(q) are synonymous.

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.

## Keyword in references:

Photonic Properties of Strongly Correlated Colloidal Liquids