International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.2, pp. 528530

The various analytical expressions given in Section 4.2.5 are mostly rather complex, so their application is often restricted to relatively simple systems. Even in these cases analytical solutions are often not available. For larger displacements the approximations that use an expansion of the exponentials [e.g. equation (4.2.3.27)] are no longer valid. Hence there is a need for alternative approaches to tackling more complicated disorder models. In the past, optical transforms (e.g. Harburn et al., 1974, 1975) and the videographic method (Rahman, 1993) were developed for this purpose. In the first method, a 2D mask with holes with sizes that represent the scattering power of the atoms is generated, which is then subjected to coherent light (from a laser) to produce a diffraction image. Problems arise for strong scatterers requiring very large holes. These problems were overcome in the second method, where the mask is replaced by a computer image with intensities proportional to the scattering power for each pixel. With the advent of more and more powerful computers these methods are now replaced by complete computer simulations, both to set up the disorder model and to calculate the diffraction pattern.
Having established a disordered crystal with the types and positions of all atoms involved (a configuration), e.g. by using one of the methods described below, computer programs employing fastFouriertransform techniques can be used to calculate the diffraction pattern, which may be compared with the observation. It has to be borne in mind, however, that there is still a large gap between a real crystal with its ~10^{23} atoms and the one simulated by the computer with only several thousand atoms. This means that very long range correlations can not be included and have to be treated in an average manner. Furthermore, the limited size of the simulated crystal leads to termination effects, giving rise to considerable noise in the calculated diffraction pattern. Butler & Welberry (1992) have introduced a technique to avoid this problem in their program DIFFUSE by dividing the simulated crystal into smaller `lots'. For each lot the intensity is calculated and then the intensities of all lots are summed up incoherently, which finally results in a smooth intensity distribution. Note, however, that in this case longrange correlations are restricted to even smaller values. To overcome this problem Boysen (1997) has proposed a method for suppressing the subsidiary maxima by multiplying the scattering density of the model crystal by a suitably designed weighting function simulating the effect of the instrumental resolution function.
A very versatile computer program, DISCUS, which allows not only the calculation of the scattering intensities but also allows the model structures to be built up in various ways, has been designed by Proffen & Neder (1999). It contains modules for reverse Monte Carlo (RMC, see below) simulation, the calculation of powder patterns, RMCtype refinement of pairdistribution functions (PDFs, see below) and many other useful tools for analysing disorder diffuse scattering.
Other computer programs have been developed to calculate diffuse scattering, some for specific tasks, such as SERENA (Micu & Smith, 1995), which uses a collection of atomic configurations calculated from a molecular dynamics simulation of molecular crystals.
Several well established methods can be used to create the simulated crystal on the computer. With such a crystal at hand, it is possible to calculate various thermodynamical properties and study the effect of specific parameters of the underlying model. By Fourier transformation, one obtains the diffraction pattern of the total scattering, i.e. the diffuse intensities and the Bragg peaks. The latter may lead to difficulties due to the termination effects mentioned in Section 4.2.7.2. These may be circumvented by excluding regions around the Bragg peaks (note, however, that in this case valuable information about the disorder may be lost), by subtracting the average structure or by using the approximation of Boysen (1997) mentioned above. A major advantage of such modelling procedures is that realistic physical models are introduced at the beginning, providing further insight into the pair interactions of the system, which can only be obtained a posteriori from the correlation parameters or fluctuation wave amplitudes derived from one of the methods described in Section 4.2.5.
Molecular dynamics (MD) techniques have been developed to study the dynamics of a system. They may also be used to study static disorder problems (by taking time averages or snapshots), but they are particularly useful in the case of dynamic disorder, e.g. diffusing atoms in superionic conductors. The principle is to set up a certain configuration of atoms with assumed interatomic potentials Φ_{ij}(r_{ij}) and subject them to Newton's equation of motion,where the force F_{i}(t) is calculated from the gradient of Φ. The equations are solved approximately by replacing the differential dt by a small but finite time step Δt to find new positions r_{i}(t + Δt). This is repeated until an equilibrium configuration is found. MD techniques are quite useful if only shortrange interactions are effective, even allowing the transfer of potential parameters between different systems, but are less reliable in the presence of significant longrange interactions.
Monte Carlo (MC) methods appear to be more suited to the study of static disorder and many examples of their application can be found in the literature. Different variants allowing simulations and refinements have been applied:
All of the different modelling techniques mentioned in this section have their specific merits and limitations and have contributed much to our understanding of disorder in crystalline materials following the interpretation of the corresponding diffuse scattering. It should be borne in mind, however, that application of these methods is still far from being routine work and it requires a lot of intuition to ensure that the final model is physically and chemically reasonable. In particular, it must always be ensured that the average structure remains consistent with that derived from the Bragg reflections alone. This may be done by keeping the Bragg reflections, i.e. by analysing the total scattering, or by designing special algorithms, e.g. by swapping two atoms at the same time (Proffen & Welberry, 1997). Moreover, possible traps and corrections like local minima, termination errors, instrumental resolution, statistical noise, inelasticity etc. must be carefully considered. All this means that the analytical methods outlined in Section 4.2.5 keep their value and should be preferred wherever possible.
The newly emerging technique of using full quantummechanical ab initio calculations for structure predictions along with MD simulations may also be applied to disordered systems. The limited currently available computer power, however, restricts this possibility to rather simple systems and small simulation box sizes, but, with the expected further increase of computer capacities, this may open up new perspectives for the future.
References
Billinge, S. J. L. & Egami, T. (1993). Shortrange atomic structure of Nd_{2−x}Ce_{x}CuO_{4−y} determined by realspace refinement of neutronpowderdiffraction data. Phys. Rev B, 47, 14386–14406.Boysen, H. (1997). Suppression of subsidiary maxima in computer simulations of diffraction intensities. Z. Kristallogr. 212, 634–636.
Butler, B. D. & Welberry, T. R. (1992). Calculation of diffuse scattering from simulated disordered crystals: a comparison with optical transforms. J. Appl. Cryst. 25, 391–399.
Egami, T. (1990). Atomic correlations in nonperiodic matter. Mater. Trans. JIM, 31, 163–176.
Egami, T. (2004). Local crystallography of crystals with disorder. Z. Kristallogr. 219, 122–129.
Harburn, G., Miller, J. S. & Welberry, T. R. (1974). Opticaldiffraction screens containing large numbers of apertures. J. Appl. Cryst. 7, 36–38.
Harburn, G., Taylor, C. A. & Welberry, T. R. (1975). An Atlas of Optical Transforms. London: Bell.
McGreevy, R. L. (2001). Reverse Monte Carlo modelling. J. Phys. Condens. Matter, 13, R877–R913.
McGreevy, R. L. & Pusztai, L. (1988). Reverse Monte Carlo simulation: a new technique for the determination of disordered structures. Mol. Simul. 1, 359–367.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. (1953). Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092.
Micu, A. M. & Smith, J. C. (1995). SERENA: a program for calculating Xray diffuse scattering intensities from molecular dynamics trajectories. Comput. Phys. Commun. 91, 331–338.
Nield, V. M., Keen, D. A. & McGreevy, R. L. (1995). The interpretation of singlecrystal diffuse scattering using reverse Monte Carlo modelling. Acta Cryst. A51, 763–771.
Proffen, Th. & Neder, R. B. (1999). DISCUS, a program for diffuse scattering and defectstructure simulations – update. J. Appl. Cryst. 32, 838–839.
Proffen, Th. & Welberry, T. R. (1997). Analysis of diffuse scattering via reverse Monte Carlo technique: a systematic investigation. Acta Cryst. A53, 202–216.
Rahman, S. H. (1993). The local domain configuration in partially ordered AuCu_{3}. Acta Cryst. A49, 68–79.
Warren, B. E. (1969). Xray Diffraction. Reading: AddisonWesley.
Weber, T. & Bürgi, H.B. (2002). Determination and refinement of disordered crystal structures using evolutionary algorithms in combination with Monte Carlo methods. Acta Cryst. A58, 526–540.
Weber, Th. (2005). Cooperative evolution – a new algorithm for the investigation of disordered structures via Mont Carlo modelling. Z. Kristallogr. 220, 1099–1107.
Welberry, T. R. & Butler, B. D. (1994). Interpretation of diffuse Xray scattering via models of disorder. J. Appl. Cryst. 27, 205–231.
Welberry, T. R. & Proffen, Th. (1998). Analysis of diffuse scattering from single crystals via the reverse Monte Carlo technique. I. Comparison with direct Monte Carlo. J. Appl. Cryst. 31, 309–317.
Welberry, T. R., Proffen, Th. & Bown, M. (1998). Analysis of singlecrystal diffuse Xray scattering via automatic refinement of a Monte Carlo model. Acta Cryst. A54, 661–674.