Structure Factor - Derivation

Derivation

Let us consider a scalar (real) quantity defined in a volume ; it may correspond, for instance, to a mass or charge distribution or to the refractive index of an inhomogeneous medium. We can define its Fourier transform . Expressing the field in terms of the spatial frequency instead of the point position is very useful, for instance, when interpreting scattering experiments. Indeed, in the Born approximation (weak interaction between the field and the medium), the amplitude of the signal corresponding to the scattering vector is proportional to . Very often, only the intensity of the scattered signal is detectable, so that .

If the system under study is composed of a number of identical constituents (atoms, molecules, colloidal particles, etc.) it is very convenient to explicitly capture the variation in due to the morphology of the individual particles using an auxiliary function, such that:

,

(1)

with the particle positions. In the second equality, the field is decomposed as the convolution product of the function, describing the "form" of the particles, with a sum of Dirac delta functions depending only on their positions. 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, such that:

.

(2)

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. We can finally write:

,

(3)

thus defining the structure factor

.

(4)