Structure factor

Definition

The structure factor [math]\mathbf{F}_{hkl}[/math] is a mathematical function describing the amplitude and phase of a wave diffracted from crystal lattice planes characterised by Miller indices [math]h, k, l[/math].

The structure factor may be expressed as

[math]\mathbf{F}_{hkl} = F_{hkl}\exp(i\alpha_{hkl}) = \sum_j f_j\exp[2\pi i (hx_j + ky_j + lz_j)] [/math]

[math]\qquad = \sum_j f_j\cos[2\pi (hx_j + ky_j + lz_j)] + i\sum_{j} f_j\sin[2\pi (hx_j + ky_j + lz_j)][/math]

[math]\qquad = A_{hkl} + iB_{hkl}[/math]

where the sum is over all atoms in the unit cell, [math]x_j, y_j, z_j[/math] are the positional coordinates of the [math]j[/math]th atom, [math]f_j[/math] is the scattering factor of the [math]j[/math]th atom, and [math]\alpha_{hkl}[/math] is the phase of the diffracted beam.

The intensity of a diffracted beam is directly related to the amplitude of the structure factor, but the phase must normally be deduced by indirect means. In structure determination, phases are estimated and an initial description of the positions and anisotropic displacements of the scattering atoms is deduced. From this initial model, structure factors are calculated and compared with those experimentally observed. Iterative refinement procedures attempt to minimise the difference between calculation and experiment, until a satisfactory fit has been obtained.

Units

The units of the structure-factor amplitude depend on the incident radiation. For X-ray crystallography they are multiples of the unit of scattering by a single electron ([math]2.82 \times 10^{-15}[/math] m); for neutron scattering by atomic nuclei the unit of scattering length of [math]10^{-14}[/math] m is commonly used.

See also

Introduction to the Calculation of Structure Factors, teaching pamphlet No. 3 of the International Union of Crystallography

The structure factor. P. Coppens. International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 10-24 doi:10.1107/97809553602060000550