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