In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns (interference patterns) obtained in X-ray, electron and neutron diffraction experiments.
Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written ; it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit. The other is usually written or and is only valid for systems with long-range positional order — crystals. This expression relates the amplitude and phase of the beam diffracted by the planes of the crystal ( are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell. is not a special case of ; gives the scattering intensity, but gives the amplitude. It is the modulus squared that gives the scattering intensity. is defined for a perfect crystal, and is used in crystallography, while is most useful for disordered systems. For partially ordered systems such as crystalline polymers there is obviously overlap, and experts will switch from one expression to the other as needed.
The static structure factor is measured without resolving the energy of scattered photons/electrons/neutrons. Energy-resolved measurements yield the dynamic structure factor.
Derivation of
Consider the scattering of a beam of wavelength by an assembly of particles or atoms stationary at positions . Assume that the scattering is weak, so that the amplitude of the incident beam is constant throughout the sample volume (Born approximation), and absorption, refraction and multiple scattering can be neglected (kinematic diffraction). The direction of any scattered wave is defined by its scattering vector . , where and ( ) are the scattered and incident beam wavevectors, and is the angle between them. For elastic scattering, and , limiting the possible range of (see Ewald sphere). The amplitude and phase of this scattered wave will be the vector sum of the scattered waves from all the atoms
For an assembly of atoms, is the atomic form factor of the -th atom. The scattered intensity is obtained by multiplying this function by its complex conjugate
The structure factor is defined as this intensity normalized by
If all the atoms are identical, then Equation () becomes and so
Another useful simplification is if the material is isotropic, like a powder or a simple liquid. In that case, the intensity depends on and . In three dimensions, Equation () then simplifies to the Debye scattering equation:
An alternative derivation gives good insight, but uses Fourier transforms and convolution. To be general, consider a scalar (real) quantity defined in a volume ; this may correspond, for instance, to a mass or charge distribution or to the refractive index of an inhomogeneous medium. If the scalar function is integrable, we can write its Fourier transform as . In the Born approximation the amplitude of the scattered wave corresponding to the scattering vector is proportional to the Fourier transform . When the system under study is composed of a number of identical constituents (atoms, molecules, colloidal particles, etc.) each of which has a distribution of mass or charge then the total distribution can be considered the convolution of this function with a set of delta functions.
with the particle positions as before. Using the property that the Fourier transform of a convolution product is simply the product of the Fourier transforms of the two factors, we have , so that:
This is clearly the same as Equation () with all particles identical, except that here is shown explicitly as a function of .
In general, the particle positions are not fixed and the measurement takes place over a finite exposure time and with a macroscopic sample (much larger than the interparticle distance). The experimentally accessible intensity is thus an averaged one ; we need not specify whether denotes a time or ensemble average. To take this into account we can rewrite Equation () as: