Structure factor

The structure factor is a contribution to measured scattering. The structure factor, $\scriptstyle S(q)$ can be thought of as encoding the structural information about the sample: how the constituents are organized, in relation to one another. This can be contrasted with the form factor, which is the scattering coming from the constituents themselves (i.e. their size, shape, and composition). The structure factor may also be called the interference function, since it describes how the scattering from different objects interferes.

Different structure factors describe different spatial arrangements. Random or disordered materials have diffuse structure factors. Well-ordered materials, such as atomic crystals or nanoscale superlattices, have structure factors with well-defined peaks, which encode the layer-spacing of repeating structures (i.e. $\scriptstyle S(q)$ reflects the Fourier transform of the crystallographic symmetry).

Mathematics

The scattering intensity is frequently divided into the contribution from the form factor (F or P) and structure factor (S):

${\begin{alignedat}{2}I(q)&=\langle |F(\mathbf {q} )|^{2}S(\mathbf {q} )\rangle \\&=P(q)\left\langle {\frac {|F(\mathbf {q} )|^{2}}{P(q)}}S(\mathbf {q} )\right\rangle \\&=P(q)S(q)\end{alignedat}}$

Thus, the structure factor can be obtained by dividing the measured intensity ($\scriptstyle I(q)$) by an independently-measured (or predicted/assumed) isotropic form-factor intensity ($\scriptstyle P(q)$):

${\begin{alignedat}{2}S(q)&={\frac {I(q)}{P(q)}}\end{alignedat}}$

The structure factor thus 'divides out' the contribution to the scattering from the shape of the constituents, thereby highlighting their organization with respect to one another.

Example Structure Factors

• Wikipedia:
  • BCC
  • FCC
  • Diamond Crystal Structure
• NCNR SANS Resources:
  • SquareWell Model
  • HardSphere Model
• Pedersen Review: Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. doi: 10.1016/S0001-8686(97)00312-6
  1. Hard-sphere potential
  2. Sticky hard-sphere potential
  3. Screened Coulomb potential
  4. Hard-sphere potential, polydisperse system
  5. Sticky hard-sphere potential, polydisperse system
  6. Screened Coulomb potential, polydisperse system
  7. Cylinders
  8. Solutions of polymers