I'm posting the following abstract, which has been submitted for the BCA 2000 Conference in Edinburgh this April, in case it may be of interest to list subscribers, and to throw open its propositions for criticism. Though brief, I have tried to include in the abstract the main points of my argument. Some additional notes follow the abstract itself, on points that were forced by space considerations to be condensed or omitted from the main text. **************************** THE MAGIC NUMBER 20: HOW MANY LINES ARE REALLY NEEDED TO INDEX A POWDER PATTERN, AND HOW MANY DEGREES OF FREEDOM ARE INVOLVED? R. Shirley, School of Human Sciences, University of Surrey, Guildford, Surrey, UK. The powder-indexing problem is not reversible. For a cubic material, only one constant - the cell side a - is needed to generate the complete powder diffraction pattern, and so this generative process does indeed have only one degree of freedom (df). If that process were reversible, one could determine the cell of an unknown cubic material with a reasonable degree of confidence from a single unindexed powder line, which plainly is not the case. One might debate what number of lines would be regarded as sufficient to be convincing, but it is certainly more than one. In fact, it will be argued that, where the crystal system is not known in advance, the indexing process has as many df for high-symmetry materials as for general triclinic ones. In this context, a description like "orthorhombic" merely becomes a shorthand for saying that three of the df are now known. Hence the indexing process for an unknown material always has 6 metrical df, whatever the crystal system turns out to be. This still does not complete the indexing process, since the diffraction order of a powder line cannot be observed. Each side of the unit cell, that has been proposed to account for a set of powder lines, is thus subject to one of 3 unknown integer multipliers - the lowest common denominators of the true h, k and l indices of those lines. Hence the indexing process also contains 3 order-fixing df. In principle, each order-fixing factor may take an infinite number of positive integers, though in practice these are usually confined to either 1 or 2, with a reasonable upper bound of 4. So, for a perfectly calibrated powder pattern from a single solid phase, the indexing process has a total of 9 df. Since order-fixing reasoning is inductive rather than deductive, no specific number of additional lines can be laid down as being necessary to give confidence that order-fixing is complete, but clearly it will not be less than 3. In practice, there are often reasons (such as sample instability) why powder patterns are not perfectly calibrated. This has the effect of introducing additional instrumental df, of which the most important is usually the zero of two-theta. So, for a single solid phase, the indexing process has 9 and perhaps 10 df. This means that at least 6 powder lines must be observed, provided that they have an ideal set of Miller indices: 100, 010, 001, 110, 101, 011 (or their prime-number-indexed equivalents, such as 130), plus as many other lines (not higher orders of any of these) as are required to give confidence that none of the cell sides needs to be doubled. Assuming that this will total (at least) 9 or 10 lines, only the remaining observed lines become available for the statistical df (whose number is thus often over-estimated?) needed by cell-fitting procedures such as least-squares. Over-determination by a factor of two then leads to the traditional minimum of 20 lines for reliable indexing. And, of course, each unknown impurity phase takes a further 9 df! ************************* Notes: 1) It may seem to fly in the face of experience to propose that the crystal system is irrelevant to the number of metrical degrees of freedom involved in the indexing process. However, remember that the crystal system is assumed here not to be known in advance, but to have been inferred from that same powder pattern. My position is that this process of induction must itself consume the relevant number of degrees of freedom. 2) It's important to remember throughout this argument that, once degrees of freedom have been consumed, in principle you don't then get them back again to carry forward into the next stage. Thus one cannot properly say, for example, that after consuming 3 df in deciding that a pattern is orthorhombic, one can then have them back again for reuse when proceeding to index the pattern on the basis of the resulting orthorhombic model. Any neglect of this principle may well contribute to the shakiness of unit cells determined by indexing from patterns containing a smaller than usual number of powder lines. 3) Maybe I am being a little reckless in seeming to claim that the diffraction order of a powder line cannot be observed in *any* circumstances, but I think that this is a realistic position to take when referring to the context of indexing a powder diffraction pattern. 4) If one is content to accept what may well only be a sub-cell, then the order-fixing df are not required. This is a legitimate position to take, for example, when attempting to index patterns recorded in high-P / high-T experiments, which may unavoidably contain a lower than usual number of diffraction lines. 5) People often claim that their beautiful and expensive instruments do not have calibration problems. Well, perhaps I only get to see pathological cases, but my experience does not seem to bear this out. 6) Note that only an ideal set of powder lines, such as one with actual indices 100, 010, 001, 110, 101, 011 (or an equivalent set that also lacks any linear redundancy), will contribute the required 6 metrical df. Lines that are higher orders of existing lines, and, more insidiously, lines that belong to a zone that has already been fully defined, do not formally contribute any further metrical df. Thus, for example, suppose that in a high-P/high-T experiment one has 12 lines, but 10 of them all belong to the same powder zone such as hk0, as can sometimes happen. In that case it would be a trap to suppose that one is in a position to over-determine all the metrical df, since in the general case such data can only fix 5 metrical df. OK, one can argue that this situation improves in higher symmetry, so that if all 12 of the lines in the example could be explained by a cubic model, then one could still proceed with reasonable confidence. I sympathise with people who are faced with this sort of situation, and concede that in practice there may be something in that approach. However, I still have a bad feeling about the re-use of the df that were involved in arriving at the high-symmetry model, and consider that one should be aware that to do so is (perhaps unavoidably) to offer hostages to fortune. 7) This also raises doubts about the usual practice of assuming that each fresh stage in the process, such as least-squares unit cell refinement, can always start with a clean slate as far as df are concerned. Thus it is not necess
