Structure factor
; it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit.
For partially ordered systems such as crystalline polymers there is obviously overlap, and experts will switch from one expression to the other as needed.
; this may correspond, for instance, to a mass or charge distribution or to the refractive index of an inhomogeneous medium.
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).
To take this into account we can rewrite Equation (3) as: In a crystal, the constitutive particles are arranged periodically, with translational symmetry forming a lattice.
Equation (1) can be written as The structure factor is then simply the squared modulus of the Fourier transform of the lattice, and shows the directions in which scattering can have non-zero intensity.
The value of the structure factor is the same for all these reciprocal lattice points, and the intensity varies only due to changes in
may appear and disappear, and care to maintain consistent quantities is required to get correct numerical results.
which determines the amplitude and phase of the diffracted beams: where the sum is over all atoms in the unit cell,
Equation (8) becomes with the result The most intense diffraction peak from a material that crystallizes in the FCC structure is typically the (111).
Films of FCC materials like gold tend to grow in a (111) orientation with a triangular surface symmetry.
, the resulting structure factor is Cesium chloride is a simple cubic crystal lattice with a basis of Cs at (0,0,0) and Cl at (1/2, 1/2, 1/2) (or the other way around, it makes no difference).
, and from there consider the modulus squared so hence This leads us to the following conditions for the structure factor: The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period
apply with a scattering vector of limited dimensionality and a crystallographic structure factor can be defined in 2-D as
The Figure shows the construction of one vector of a 2-D reciprocal lattice and its relation to a scattering experiment.
In one-dimension for simplicity and with N planes, we then start with the expression above for a perfect finite lattice, and then this disorder only changes
by a multiplicative factor, to give[1] where the disorder is measured by the mean-square displacement of the positions
However, fluctuations that cause the correlations between pairs of atoms to decrease as their separation increases, causes the Bragg peaks in the structure factor of a crystal to broaden.
To see how this works, we consider a one-dimensional toy model: a stack of plates with mean spacing
In contrast with crystals, liquids have no long-range order (in particular, there is no regular lattice), so the structure factor does not exhibit sharp peaks.
They do however show a certain degree of short-range order, depending on their density and on the strength of the interaction between particles.
Liquids are isotropic, so that, after the averaging operation in Equation (4), the structure factor only depends on the absolute magnitude of the scattering vector
in the double sum, whose phase is identically zero, and therefore each contribute a unit constant: One can obtain an alternative expression for
:[8] In the limiting case of no interaction, the system is an ideal gas and the structure factor is completely featureless:
of different particles (they are independent random variables), so the off-diagonal terms in Equation (9) average to zero:
This reasoning does not hold for a perfect crystal, where the distribution function exhibits infinitely sharp peaks.
limit, as the system is probed over large length scales, the structure factor contains thermodynamic information, being related to the isothermal compressibility
Although highly simplified, it provides a good description for systems ranging from liquid metals[10] to colloidal suspensions.
[11] In an illustration, the structure factor for a hard-sphere fluid is shown in the Figure, for volume fractions
In polymer systems, the general definition (4) holds; the elementary constituents are now the monomers making up the chains.
