Structure factor

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Sofia Magkiriadou, AP225, Fall 2011


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

<math> S_{\mathbf{q}}= 1/V \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 all constituent particles and q is spatial frequency measured in units of inverse length. The particles can be atoms, molecules, or larger entities like polystyrene spheres arranged to form a specific structure. 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.


The structure factor is a very useful concept. A closer look at the defining formula reveals that it is the Fourier transform of the density distribution. Information regarding general properties of a material structure, such as the existence of characteristic length scales, periodicity along some axis, and symmetries, can be easier to identify in the structure factor than in the structure itself. For example, the structure factor for a one-dimensional crystal of characteristic spacing <math>d</math> will have a maximum at <math>q = \frac{2 \pi} {d}</math>, the corresponding spatial frequency, and the width of this peak will be a measure of the quality of the crystalline order, as expected from Fourier Theory. While this is a trivial example where the periodic feature is readily identifiable, the structure factor can be a very powerful quantity for studying more complex structures such as quasi-crystals.


Moreover, it is closely related to the scattering intensity from a material <math>\sigma</math>. In the case of single elastic scattering and for a material comprised of only one type of particles, this relation is, up to constants, as follows:

<math>\sigma_{\mathbf{q}} = C \int d\Omega S_{\mathbf{q}} F_{\mathbf{q}}</math>

where C is a constant and <math>F_{\mathbf{q}}</math> is the form factor, a quantity representing the way each constituent particle scatters. This expression can be generalized for an arbitrary number of particle types by a simple summation where each term has the corresponding form factor (see also scattering). 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 it 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 each scatterer acts as a point source of a spherical wave and the resulting pattern is the sum of all these waves originating from different points in space. Within this picture, which emphasizes the analogy between a scattering pattern and a Fourier Transform, the interpretation of S(q) as the Fourier transform of the density becomes synonymous to the interpretation of S(q) as the collective interference term.


The structure factor is a key quantity in crystallography and materials science; 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

Reference

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