Structure factor

The Fourier-space measure of spatial correlations (static structure factor), central to scattering and to diagnosing order and criticality.

Structure factor

Definition (static structure factor)

Let $A_x$ be a local observable on a lattice or continuum (e.g., spin component, particle density). The connected two-point correlation function is

C(x-y)=\langle A_x A_y\rangle - \langle A_x\rangle\langle A_y\rangle.

When the state is translation-invariant, the static structure factor is the Fourier transform of $C$ :

S(k)=\sum_{r} e^{ik\cdot r}\,C(r)

(on a lattice, sum over $r$ ; in the continuum, replace by an integral with the appropriate convention). In finite volume with $N$ lattice sites, an equivalent normalization often used is

S(k)=\frac{1}{N}\sum_{x,y} e^{ik\cdot(x-y)}\Big(\langle A_x A_y\rangle - \langle A\rangle^2\Big).

Interpretation and key uses

Scattering: For density observables, $S(k)$ is (up to conventions) proportional to the intensity measured in X-ray/neutron scattering at momentum transfer $k$ .
Order detection: Long-range order produces sharp features (Bragg peaks) in $S(k)$ at ordering wavevectors.
Criticality: Near a critical point, growth of long-range correlations appears as a strong enhancement of $S(k)$ at small $|k|$ , controlled by the correlation length and the anomalous dimension $\eta$ .

Ornstein–Zernike form (common approximation)

If correlations decay roughly as $C(r)\sim e^{-r/\xi}$ away from criticality, then for small $|k|$ one often has the Ornstein–Zernike-like behavior

S(k)\approx \frac{\chi}{1+(|k|\xi)^2},

where $\chi=S(0)$ is the susceptibility (again up to conventions). See Ornstein–Zernike form for context and refinements.

Definition (static structure factor)

Interpretation and key uses

Ornstein–Zernike form (common approximation)

Prerequisites / cross-links