Computer algebra removes much of the drudgery from mathematics; it allows users to formulate models by using the language of mathematics and to have those models evaluated with little effort. This symbolic form of representation is often thought of as being separate to dedicated computational programs such as Rietveld refinement. These dedicated programs are often written in low level languages; they are relatively inflexible in what they do and modifying them to change functionality in a small manner is often a major programming task. This paper describes a symbolic system that is integrated into the dedicated Rietveld refinement program called TOPAS. The symbolic component allows large functional changes to be made at run time and with a relatively small amount of effort. In addition, the system as a whole reduces the programming complexity at the developmental stage.
The presence of amorphous materials in crystalline samples is an increasingly important issue for diffractionists. Traditional phase quantification via the Rietveld method fails to take into account the occurrence of amorphous material in the sample and without careful attention on behalf of the operator its presence would remain undetected. Awareness of this issue is increasing in importance with the advent of nanotechnology and the blurring of the boundaries between amorphous and crystalline species.
The methodology of a number of different approaches to the determination of amorphous content via X-ray diffraction and an assessment of their performance, is described. Laboratory-based, X-ray diffraction data from a suite of synthetic samples, with amorphous content rangäing from 0.0 to 50 wt%, has been analysed using both direct (in which the contribution of the amorphous component to the pattern is used to obtain an estimate of concentration) and indirect (where the absolute abundances of the crystalline components are used to estimate the amorphous content by difference) methodologies. In addition, both single peak and whole pattern methodologies have been assessed.
All methods produce reasonable results, however the study highlights some of the strengths, deficiencies and applicability of each of the approaches.
A fast method for calculating the atomic pair distribution function is described in the context of performing refinements of structural models. Central to the speed of synthesis is the approximation of Gaussian functions of varying full widths at half-maximum using a narrower Gaussian with a fixed full width at half-maximum. The initial Gaussians are first laid down as delta functions which are then convoluted with the narrower Gaussian to form the final pattern. The net result is an algorithm, which has been included in the Rietveld refinement computer program TOPAS, that synthesizes and refines structural parameters a factor of 300-1000 times faster than alternative algorithms/programs, with speed advantages increasing as the number of atomic pairs increases.
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