Structure factor

Mathematical description in crystallography
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.

The zincblende structure is similar to the diamond structure except that it is a compound of two distinct interpenetrating fcc lattices, rather than all the same element. Denoting the two elements in the compound by and , the resulting structure factor is

The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period , the reciprocal lattice is an infinite array of points with spacing In two dimensions, there are only five Bravais lattices. The corresponding reciprocal lattices have the same symmetry as the direct lattice. 2 D lattices are excellent for demonstrating simple diffraction geometry on a flat screen, as below. Equations (1)–(7) for structure factor apply with a scattering vector of limited dimensionality and a crystallographic structure factor can be defined in 2 D as
However, recall that real 2 D crystals such as graphene exist in 3 D. The reciprocal lattice of a 2 D hexagonal sheet that exists in 3 D space in the plane is a hexagonal array of lines parallel to the or axis that extend to and intersect any plane of constant in a hexagonal array of points. The Figure shows the construction of one vector of a 2 D reciprocal lattice and its relation to a scattering experiment. A parallel beam, with wave vector is incident on a square lattice of parameter The scattered wave is detected at a certain angle, which defines the wave vector of the outgoing beam, (under the assumption of elastic scattering, ). One can equally define the scattering vector and construct the harmonic pattern In the depicted example, the spacing of this pattern coincides to the distance between particle rows: , so that contributions to the scattering from all particles are in phase (constructive interference). Thus, the total signal in direction is strong, and belongs to the reciprocal lattice. It is easily shown that this configuration fulfills Bragg's law.

Technically a perfect crystal must be infinite, so a finite size is an imperfection. Real crystals always exhibit imperfections of their order besides their finite size, and these imperfections can have profound effects on the properties of the material. André Guinier proposed a widely employed distinction between imperfections that preserve the long range order of the crystal that he called disorder of the first kind and those that destroy it called disorder of the second kind. An example of the first is thermal vibration; an example of the second is some density of dislocations. The generally applicable structure factor can be used to include the effect of any imperfection. In crystallography, these effects are treated as separate from the structure factor , so separate factors for size or thermal effects are introduced into the expressions for scattered intensity, leaving the perfect crystal structure factor unchanged. Therefore, a detailed description of these factors in crystallographic structure modeling and structure determination by diffraction is not appropriate in this article.

For a finite crystal means that the sums in equations 1 7 are now over a finite The effect is most easily demonstrated with a 1 D lattice of points. The sum of the phase factors is a geometric series and the structure factor becomes:

This function is shown in the Figure for different values of When the scattering from every particle is in phase, which is when the scattering is at a reciprocal lattice point , the sum of the amplitudes must be and so the maxima in intensity are Taking the above expression for and estimating the limit using, for instance, L'Hôpital's rule) shows that as seen in the Figure. At the midpoint (by direct evaluation) and the peak width decreases like In the large limit, the peaks become infinitely sharp Dirac delta functions, the reciprocal lattice of the perfect 1 D lattice. In crystallography when is used, is large, and the formal size effect on diffraction is taken as , which is the same as the expression for above near to the reciprocal lattice points, Using convolution, we can describe the finite real crystal structure as [lattice] [basis] rectangular function, where the rectangular function has a value 1 inside the crystal and 0 outside it. Then [crystal structure] = [lattice] [basis] [rectangular function]; that is, scattering [reciprocal lattice] [structure factor] [ sinc function]. Thus the intensity, which is a delta function of position for the perfect crystal, becomes a function around every point with a maximum , a width , area .

In the limiting case of no interaction, the system is an ideal gas and the structure factor is completely featureless: , because there is no correlation between the positions and of different particles (they are independent random variables), so the off diagonal terms in Equation average to zero: .

Even for interacting particles, at high scattering vector the structure factor goes to 1. This result follows from Equation , since is the Fourier transform of the "regular" function and thus goes to zero for high values of the argument This reasoning does not hold for a perfect crystal, where the distribution function exhibits infinitely sharp peaks.

In the hard sphere model, the particles are described as impenetrable spheres with radius ; thus, their center to center distance and they experience no interaction beyond this distance. Their interaction potential can be written as:

This model has an analytical solution in the Percus–Yevick approximation. Although highly simplified, it provides a good description for systems ranging from liquid metals to colloidal suspensions. In an illustration, the structure factor for a hard sphere fluid is shown in the Figure, for volume fractions from 1% to 40%.

In polymer systems, the general definition holds; the elementary constituents are now the monomers making up the chains. However, the structure factor being a measure of the correlation between particle positions, one can reasonably expect that this correlation will be different for monomers belonging to the same chain or to different chains. Let us assume that the volume contains identical molecules, each composed of monomers, such that ( is also known as the degree of polymerization). We can rewrite as:

where indices label the different molecules and the different monomers along each molecule. On the right hand side we separated intramolecular and intermolecular terms. Using the equivalence of the chains, can be simplified:

where is the single chain structure factor.