The Effects of Static Disorder

The scattering from a real crystal can, of course, only ever be an approximation to that described by equation 2.10. All real crystals will inevitably be susceptible to defects and deviations from the ideal periodic structure due to the preponderance of stresses and strains which are frozen in during crystallisation. The terms of equation 2.10 suggest two distinct ways in which disorder may be introduced. Through either a variation of the phase, $({\bf k}_s - {\bf k}_o).{\bf r}_n$, or a variation of the amplitude, $f_n$, such that the structure of all unit cells can no longer be considered as identical. The result is an interruption to the periodicity of the structure.

A variation in the atomic form factor, $f_n$, occurs if the occupation of the lattice site $n$ is changed chemically. The site may be unoccupied due to a vacancy, or it may be occupied by a different atomic species such as an impurity. This case is said to be occupancy or compositional disorder. The variation of the phase involves a deviation of the atomic positions away from their lattice site. This is called displacement disorder and requires the inclusion of an additional displacement term, $\Delta$r, such that equation 2.9 now becomes

$${\bf R} = {\bf m} + {\bf r}_n + \Delta{\bf r}_{n,m}$$ (4.12)

The disorder terms, such as equation 2.12, can be incorporated into the scattering equation. For the simplest case of a monatomic lattice, and ${\bf k}_s - {\bf k}_o$ shortened to ${\bf k}$, then the intensity now becomes

$$I({\bf k}) = \sum_{m}^N \sum_{m'}^{N'} f^2 \exp(i{\bf k}.({\... ...bf m'})) \exp(i{\bf k}.(\Delta{\bf r}_m - \Delta{\bf r}_{m'}))$$ (4.13)

It is possible to expand the argument and write it as a sum of component intensities. This is straightforward, if somewhat lengthy, and a complete and general derivation along with much useful discussion is given by Welberry [32]. Here the intensity is given in the simplest terms, for the specific case of displacement disorder

$$\begin{eqnarray*} I({\bf k}) = \sum_{m}^{N} \sum_{m'}^{N'} &f^2& \exp(i{\bf k}.(... ...m - m'}))<({\bf k}.\Delta{\bf r}_m)({\bf k}.\Delta{\bf r}_{m'})> \end{eqnarray*}$$

The first term describes the fundamental reflections due to the average structure. In the discussion it now becomes meaningful to discuss an average structure, which is the resultant average over the whole crystal including the disorder effects, and a basic lattice which is the original ideal crystal. The second term is due solely to the disorder fluctuations and is dependent upon the correlations which may or may not exist between cells $m$ and $m'$. It will be dependent upon the specific nature of the disorder. This additional term contributes what is known as diffuse scattering, the weak, poorly-defined distribution of intensity which lies between fundamental reflections.

If the disorder fluctuations show no correlations, due to defects or impurities of a sufficiently low density and wide separation so as to have no interaction, then the diffuse term is due solely to the self-correlation of each cell and the diffuse intensity will be widely distributed. If correlations do exist, perhaps in connected defect structures, then the diffuse intensity will display some more localised structure in its distribution.