From: Robin Shirley [R.Shirley@surrey.ac.uk] Organization: Surrey Univ. U.K. To: L.M.D.Cranswick@dl.ac.uk (L. Cranswick) Date: Wed, 28 Jul 1999 20:35:10 GMT Subject: Re: Crysfire Concerning the "only 15 lines" and especially "only *7* lines" examples. The point is that one's data need to have at least 6 degrees of freedom before one can even *define* a triclinic cell, and similarly 3 before one can define an orthorhombic one etc. This is before any question of goodness of fit arises, since it requires 6 degrees of freedom just to *define* an (exact) triclinic cell, before we can consider doing least-squares fits, etc. Note that 6 degrees of freedom *doesn't* just mean 6 lines. This is because several powder lines can be, and usually are, linearly dependent - i.e. they belong to the same zone. At most, only the first 3 lines that belong to one zone provide degrees of freedom for the *cell definition* (fewer if some of its constants have already been defined by other lines). Similarly, second or higher orders of an observed line don't contribute any degrees of freedom to the *cell definition* either (though they may well do so to its least-squares refinement). Thus, for example, if you had 10 observed lines, but the first 6 all belonged to hk0 and another was a second order not from that zone (e.g. 002), then you'd be down to the bare minimum of 6 degrees of freedom to define a general (triclinic) cell geometrically, with nothing remaining to test the results against the quality of the data, distinguish pseudo-solutions, etc., etc. The bottom line here is that *7* lines are way too few to have much real hope of distinguishing a correct cell, except possibly when the line width is very narrow and the symmetry tetragonal or higher. 15 lines are certainly much better, but still not good. The problem is that, usually, when one is short of data, circumstances also dictate that the purity will be uncertain, the lines broader and the dataset more likely to be pathological. For example, all three of the above issues are frequently present with clay mnerals, which are seldom completely pure, usually have poorish texture and and also very often exhibit flat unit cells that lead to a dominant zone and hence to difficulty finding enough lines that don't belong to hk0. 20 lines are barely sufficient in the presence of a dominant zone (30 are safer), and the degree of reliability, such as it is, drops off markedly as we get down to 16 or less. Best wishes Robin