Whilst measurements of x-ray diffuse scattering in reciprocal space are readily obtainable using the experimental techniques described in this thesis, the extrapolation needed to visualise the implication of these results for the real space structure is less unequivocable. This has remained an intractable problem ever since the advent of diffuse scattering, spawning various ingenious methods (such as optical transforms) which attempt to aid the interpretation of experimental data. With the availability in recent years of increasingly powerful computing resources the most favorable, but previously impractical, approach has become feasible: the direct numerical calculation of the diffraction pattern. Although this has been limited in the past to very simple systems, the viability of simulation on computer of scattering from even very complex structural models has recently been demonstrated [184], and it now has the potential to be a valuable complement in the analysis of almost any experiment. In Bi-2212, the task of calculating the scattered intensity is rendered non-trivial by its incommensurate nature, which brands every unit cell as unique by way of the modulated displacements. Amongst the results presented in this thesis, those of particular interest are the diffuse streaks, which were shown to be of importance in Chapter 5, but whose origin cannot be defined with any certainty from the experimental results alone. In an effort to judge this problem more clearly, this chapter details the neccessary steps to simulate the scattering from a model of the Bi-2212 structure, and presents some preliminary results.
The equations neccessary for the calculation of diffuse scattering
for various types of disorder were introduced in Chapter 2. In
equation 2.14 it was demonstrated that the form of the
diffuse scattering depends strongly upon the correlation of the
disorder between unit cells. So, for the simplest form of
disorder, a point defect without correlation, only the evaluation
of the structure factor for a single unit cell would be required.
Where correlations exist, it must be possible to incorporate them
within the description of the structure to create a realistic model
of the disorder. Analytical approaches to describing the diffuse
intensity have included the use of either correlation parameters or
modulation waves (equation 2.20) to express the disorder.
But the complexity of the equations necessary to describe any but
the simplest of disordered systems in this way limits their
usefulness. An alternative to this analytical approach
has been developed by Welberry and Butler [32]
involving the direct numerical calculation of the scattering from a computer
model of the structure. This method can be applied regardless of
the complexity of the disordered system, as it is ultimately only
limited by the computing resources at hand. It is also desirable
to be able to generate the calculated intensity on a scale in
reciprocal space comparable with the resolution of the
experimentally obtained measurements; typically this can mean
of the order of 10
data points in an intensity map. The task
of any simulation method therefore is to find a means to
accommodate both the handling of a suitably large model
crystal and the generation of results of a desired resolution, and at the same time,
present a calculation of a size no bigger than can be feasibly executed by
the available computing resources.
With access at Edinburgh to some of the most powerful parallel processing facilities possible, this direct numerical method of the large scale computation of the scattered intensity from a computer model of the structure was considered the most amenable. The next section describes how a model of the structure was constructed, incorporating the incommensurate modulation, whilst the following section details the route taken to developing an efficient parallel code for the calculation of the diffraction pattern. In the final section, the results of the simulation are presented.