Monte Carlo Methods

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Monte Carlo Methods

The use of Monte Carlo methods for indexing powder diffraction patterns is a relatively recent and novel idea, having been described and discussed at the XIX IUCr Congress, Geneva, in 2002. Monte Carlo methods were first developed as part of commercial software packages, but a freely available program, MCMAILLE, has recently been developed and distributed by Armel LeBail.

Monte Carlo indexing methods work by generating unit cells at random. From each unit cell, a set of peak positions may be calculated. These are compared to an observed set of d-spacing values and intensities. Some programs may use the raw profile when comparing the calculated and observed patterns, but in practice it is quicker to use to use a crude peak shape such as a triangle or square-wave function. An agreement index is calculated for each randomly generated pattern and cells may be retained for further examination when the index drops below some pre-defined threshold value.

Once a potential match is obtained, random small changes may be made to the unit cell parameters via the same Monte Carlo process so that better unit cell parameters can be obtained. If the 2θ zero is suspect, then this may be made one of the Monte Carlo parameters also, but at the cost of using more CPU time: however, as for all indexing methods, it is better to obtain the most accurate data permitted by the instrument rather than rely on a variable (or unknown) instrumental parameter.

One of the strongest points in favour of Monte Carlo indexing programs is that they are relatively insensitive to impurities, and especially to impurities with large d-spacing values. This is because the method makes no assumptions about the large d-spacing values, e.g. it does not use these to find zones as in zone indexing. In addition, Monte Carlo programs can make use of integrated peak intensities as well as d spacing values, which makes them less sensitive to weak impurity peaks.

The main disadvantage is probably CPU time, but given the ever increasing speed of computers, this is becoming less of a problem. If the classical methods of indexing fail, then it is probably worth using more CPU time to solve what might be a difficult problem. As with dichotomy indexing, Monte Carlo methods require more CPU time the lower the symmetry of the system, being very fast for cubic (even with impurities), but being quite slow for triclinic even for pure phases.

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