# 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 defined as follows:

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

- <math> F_{\Delta\mathbf{k}}=\sum_{j} f_j e^{-i\Delta\mathbf{k} \cdot \mathbf{r}_j}</math>

where the sum over j is a sum over the constituent particles of the structure in question. These can be atoms, molecules, or larger entities like polystyrene spheres periodically arranged in a photonic crystal.

This quantity is very useful. A closer look at the formula reveals that it is the fourier transform of the density distribution; if a structure has any periodicity at one or more lengthscales, this will show up as a peak in the structure factor. Therefore, it is a useful quantity for characterising a bulk structure and picking out spatial correlations in it.

Moreover, it is closely related to the scattering pattern from a material. In the case of single elastic scattering, this relation can be phrased mathematically as follows:

<formula>

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 obtain the fourier transform of a structure, it can be seen that these two explanations of S(q) are synonymous.

Crystallography and materials science studies rely a lot on S(q); all scattering experiments, such as XS, NS, SANS measure the S(q), which can then be inverted (nontrivially) to obtain the density distribution. It is also convenient to use in the study of photonic crystals: any peak in the structure factor corresponds to strong scattering at the corresponding momentum vector, and may thus signal the onset of a photonic stop band or band gap.

## Keyword in references:

Photonic Properties of Strongly Correlated Colloidal Liquids