Trial Methods

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Trial Methods

In practice most powder diffraction patterns are auto-indexed using computer programs, which require as input data the positions of at least 20 peaks from the diffraction pattern. How do these computer programs work? They all make use of the general equation:

where the factor of 10⁴ is an arbitrary scalar used to increase the size of the reciprocal space metric tensor terms A to F.

The methods fall into the following categories: deductive, semi-exhaustive, and exhaustive. The terms refer to various degrees of certainty of finding the correct solution! They can also be classified by the methods they use, e.g. parameter space or index space. The actual methods include zone-indexing suggested by Ito and developed by de Wolff and Visser, trial index methods as used by Taupin, Werner, and Kohlbeck, successive dichotomy as programmed by Louër, and grid search as suggested by Shirley§.

An example of the trial index method is that developed by Taupin. If n is the number of unknown metric tensor parameters in the equation:

then the first n lines are assigned computer generated hkl values to give n simultaneous linear equations. Considering the case of orthorhombic symmetry with n = 3, then we have three equations:

which in matrix form may be written as:

The n × n matrix is then inverted and multiplied by the Q matrix to give the unknown parameters A, B, C. If the values are sensible (e.g. A > 0, etc.) then they are used to calculate all possible Q values using all possible hkl triplets and the results are then compared with the remaining lines. If a match is found, it can be listed for further investigation together with a figure of merit. The program then continues with the next computer generated solution.

The method is exhaustive but has several weaknesses: Firstly, if one of the n initial lines is spurious, e.g. due to an impurity reflection, then no correct solution will ever be found. Secondly, for large values of n, i.e. for the monoclinic and especially the triclinic crystal systems, the computer time involved is enormous even on modern computers compared to other indexing methods. This is because the number of trial permutations, P, is given by N! / (N-n)! where N is the number of hkl triplets generated for use in the matrix and n is the number of unknown metric tensor parameters. For N = 20 and n = 3, then P = 6840, but if n = 6, then P = 2.8 × 10⁷. In practice, the number of hkl triplets doubles in number on decreasing the symmetry from orthorhombic to monoclinic and from monoclinic to triclinic, while at the same time n increases from 3 to 4 and then to 6, respectively. This implies that the computational time will increase rapidly when lower-symmetry crystal systems are tried out. In practice, the method works well for tetragonal, hexagonal/trigonal, and orthorhombic crystal systems, but is best avoided for monoclinic and triclinic systems where other methods work better.

The method can be improved considerably by optimisation of the initial list of reflections from which the trial indices are permuted. Thus it is more efficient to test for primitive and centred unit cells using separate sets of trial indices: for example, to attempt indexing assuming primitive orthorhombic symmetry, one could limit the triplets such that:

this limits the number of initial triplets, N, to 19. By contrast the first 19 triplets for C-centred orthorhombic has the conditions:

thus allowing a greater range of hkl values to be considered while avoiding a massive increase in the computer time required for the trial search.

The weakness concerning impurity lines can be overcome partly by an intelligent crystallographer and to some extent by computer programming, though at the expense of computer time. For example, with the orthorhombic system it is possible to pick permutations of three lines from, say, the first four lines; this allows one of the first four to be spurious. The crystallographer can do likewise, but with the advantage that extra parameters can be taken into consideration. Thus if one of the first few lines is suspected of being an impurity line (e.g. different width, shape, or simply very low intensity), then it can be left out of the initial set of reflections used for indexing. This strategy applies to all computer indexing methods, but is especially pertinent to any computer program that assumes that all of the input data is from the same phase.

§ A good review of the computer techniques is the article by Shirley in Computing in Crystallography - Proceedings of an International Summer School on Crystallographic Computing held in Twente, The Netherlands, eds. H. Schenk et al. 1978, 221-234. See the links page for the software package by Shirley on the CCP14 web site.

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