Research School of Chemistry **Machine** VP

**Co-Investigators** Sheridan C Mayo, Andrew G Christy and Aidan P
Heerdegen

Research School of Chemistry

**Computation of X-ray Diffraction Patterns for 3D Model Systems**

The aim of our project is modelling the disorder that occurs in crystals of some organic molecules, inorganic materials and mineral systems, which we observe in our diffuse X-ray diffraction experiments. Conventional crystal structure analysis of disordered materials using the sharp Bragg diffraction data reveals only average one-body structural information, such as atomic positions, thermal ellipsoids and site occupancies. Diffuse scattering, on the other hand, gives two-body information and is thus potentially a rich source of information of how atoms and molecules interact with each other.

Traditionally there have been two major mathematical approaches employed to
describe and understand this diffuse part of the diffraction pattern--a
reciprocal space (modulation-wave) approach and a real-space (correlation)
approach. Either of these approaches may be employed to obtain sets of
physical parameters which describe quite accurately the observed diffuse
intensities. The ultimate goal of such studies, however, is not simply to find
a set of mathematical correlation parameters that can describe the diffraction
pattern but to obtain a realistic model of the how the material is organized
and that is consistent with the observed scattering. Although such models can
be derived *a posteriori* from the mathematical fit to the data, this
approach can all too often lead to misinterpretation and unrealistic models for
the disorder. This possibility can often be reduced if the normal order of
analysis is reversed such that the investigation begins not with a mathematical
description but with the models themselves. The idea is to develop a model
which first satisfies any known physical/chemical constraints and then
iteratively adjust the simulation until the diffraction pattern from the
computer model matches the measured intensities. By turning the usual analysis
procedure around the physical organization of the material is given the
greatest emphasis and many possible, but unlikely or non-physical,
configurations that are consistent with the diffraction data will thus be
eliminated from consideration.

**What are the basic questions addressed?**

Can we, by using a detailed potential model of the systems under investigation, describe the short-range order properties of the materials sufficiently well that we may obtain computed diffuse diffraction patterns which are in substantive agreement with observed X-ray diffraction patterns?

**What computational techniques are used and why is a supercomputer
required?**

We use mainly Monte Carlo simulation for the modelling. This part of the work is not generally performed on the VP since it does not vectorise well. The main part of the work carried out on the VP is obtaining the three dimensional diffraction patterns from the simulation results. This involves direct Fourier transformation for which we have developed a highly vectorised algorithm. For disordered crystals, large sample sizes are necessary in order to obtain statistically useful spatial information. In addition the diffraction patterns need to be computed on a fine grid of points in three dimensions. Because of the large amounts of CPU time which would be required, lesser classes of computer allow only the crudest models to be explored.

**What are the results to date and future of the work?**

The method has been used to study disorder in a number of quite diverse systems. One major on-going project is involved with trying to understand the disorder in cubic stabilized zirconias (CSZ's) which have commercial importance as "cubic zirconia" gems. A second system is that of Mullite which is a major component of nearly all aluminosilicate ceramics. For both of these systems three dimensional models of the way in which oxygen vacancies order have been established, and the way in which the rest of the structure relaxes around the vacancies. A new study that has commenced recently is on the material wüstite, Fe1-xO, which is thought to be a major constituent of the Earth's lower mantle.

The methods have also been applied to a number of organic molecular crystal systems which exhibit disorder. This area possibly presents the most promise of realising a quantitative interpretation of observed diffuse scattering, thereby yielding valuable detailed information about intermolecular interactions. A new study that has recently commenced in this field is on the urea/hexadecane inclusion compound.

Our methods are now established as a viable means of interpreting and studying disorder in a whole range of different materials. The diffraction calculation algorithm will continue to be used as a routine tool in the process.

**Publications**

*Lead Article: Interpretation of Diffuse X-ray Scattering via Models
of Disorder,* T R Welberry and B D Butler, J Appl. Cryst., **27**,
205-231 (1994).

*Analysis of Diffuse Scattering from the Mineral Mullite,* B D Butler and
T R Welberry, J. Appl. Cryst., **27**, 741-754 (1994).

*Diffuse X-ray Scattering in KLiSO4, *T R Welberry and A M Glazer, J.
Appl. Cryst., **27**, 733-741 (1994).

*A paracrystalline description of defect distributions in wüstite,
Fe1-xO*, T R Welberry and A G Christy, Journal of Solid State Chemistry
(1995), in press.

*A Modulation Wave Approach to Understanding the Disordered Structure of
Cubic Stabilised Zirconias (CSZ's), *T R Welberry, R L Withers and S C Mayo,
J.Solid State Chem., (1995), in press.