[sdpd] Degrees of Freedom in Powder Indexing

[Robin Shirley <R.Shirley...@surrey.ac.uk>]

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 necessarily correct when estimating esd's to allow only for
the number of df directly consumed in *generating* the calculated pattern,
and ignoring others consumed previously in the indexing process **if the
trial cell was obtained from the same data used for the refinement**. 

8) I have not included the point, since it has long been well known to 
experienced indexers, that further lines beyond the first 20 or 30 contribute 
much less information to the indexing process.  This occurs because in those 
regions of the pattern the *calculated* lines from trial cells become so 
closely spaced and change their affiliation to the nearest observed line so 
readily, that they place increasingly minimal constraints on possible models.

This is the reason for not being able to power one's way through the df 
problem simply by measuring more lines.

In recent years this situation has been eased a little by the availability
of very high resolution patterns, such as those from synchrotron sources, but
not as much as one might wish, because the number of calculated lines out to
some radius d* in reciprocal space increases as the cube of d*, which
quickly overwhelms any gains in instrumental line width.

Also, as instrumental contributions to resolution improve, the inherent line
width produced by sample characteristics becomes more significant.  This is
particularly likely to be important for the kinds of sample, often with
texture problems, that can only be characterised from powder measurements.


Robin Shirley