Indexing powder patterns from mixed samples using joint-probability heuristics: Hmap and Peurist

Robin Shirley

School of Human Sciences, University of Surrey, Guildford, Surrey GU2 7XH, U.K.

The ab initio lattice reconstruction and unit cell determination for single-component powder diffraction data is already a difficult computational problem, even for well-corrected data with only six unknown cell parameters [1], so it is not surprising that even quite small numbers of impurity lines can make indexing uncertain. A second unknown major phase brings a further six parameters, so that the prospects of successfully indexing mixed samples are generally regarded as poor, despite recent progress using global optimisation programs like McMaille [2] and AUTOX [3].

Apart from the computational demands of searching the resulting greatly enlarged solution space, a major problem is the poor ability of the usual indexing figures of merit (FOM) like M20, M1 and FN to identify solutions in the presence of extraneous lines. All such FOM are based on the discrepancies between observed and calculated lines. They work well when all observed lines are explained by the model, but the signal from a correct cell is easily lost in the large noise term from even a few unindexed lines. A traditional answer is to exclude "unindexed" lines from the FOM calculation, but this just introduces new problems, through making wrong cells appear to have good FOM by excluding their worst discrepancies.

This property of reporting overall misfit rather than fit prevents traditional discrepancy-based FOM from performing well when the model is incomplete, which also limits their usefulness for parameter-separation heuristics, which can greatly speed computations by avoiding the need to search many unknown parameters concurrently.

These disadvantages would disappear for FOM that reported the amount of fit between data and model, while being broadly transparent to any remaining misfit. Such properties are offered by a joint-probability approach like the PM measure originally proposed by Ishida and Watanabe [4]. PM itself is not well behaved mathematically, due to arbitrary discontinuities, but the present study will report new better-behaved measures – in particular Ir, which reports the "indexedness" of a pattern for a particular trial cell, as a non-integral effective number of indexed lines.

Tests using the new program Hmap running under the latest release of the Crysfire system [5] have shown that Ir has the desired property of being relatively transparent to unexplained lines, routinely disregarding 10 or more impurity lines. It will be demonstrated that in non-pathological cases there are now reasonable hopes of indexing all the components of mixtures of two and sometimes three unknown low-symmetry phases, even with only (good) laboratory data. Hmap is not able to perform full ab initio indexing unaided, since it requires a 4-parameter SIW basis set obtained, for example, with a program like Lzon [5]. This limitation will be removed in a new program Peurist, currently under development, which will carry out full ab initio indexing using joint probability criteria at al stages. Progress with Hmap and Peurist will be reported and demonstrated.

[1] R. Shirley, IUCr Computing Commission Newsletter 2 (2003) 48-54.

[2] A. Le Bail, Powder Diffraction (2004) in press.

[3] V. B. Zlokazov, J. Appl. Crystallogr 25 (1992) 69-72.

[4] T. Ishida & Y. Watanabe, Z. Krist. 160 (1982) 19-32.

[5] R. Shirley, The Crysfire 2003 System for Automatic Powder Indexing: User's Manual, Lattice Press: Guildford, U.K. (2003).