>>>Unlike most existing cell refinement programs, UnitCell does not require
>>>initial estimates for cell constants;
and
>it uses a non-linear least squares ...
O.K. folks, it could have been better expressed, I agree. For those who are
getting hot under the bonnet about this, I amplify:
a) The equation for Q(hkl) is linear in six parameters which may be
determined by linear least-squares. These six parameters may be determined
without the requirement for an initial guess (which is needed in non-linear
least squares) using linear least squares procedures. (Yes, we need to know
h,k,l for each of the observed d(hkl), or at least be reasonably certain of
many of them (deletion diagnostics help decide if you got it wrong), but
that was not the context of the initial statement).
b) However, three of the unknown parameters (b*c*cosalpha* etc.) are
combinations of the cell parameters and are not independent of the other
three parameters (a*^2, b*^2, and c*^2). Furthermore, the equation is only
linear when written in terms of Q(hkl). If we wish to minimise residuals in
another dependent variable, such as two-theta(hkl) or d(hkl), then the
expression becomes non-linear and linear least-squares cannot be used.
c) What UnitCell does is to get first guesses for the parameters using
linear least squares, then employ non-linear least squares to refine the
cell parameters based on minimising on your chosen quantity (Q, two-theta,
d, or energy). This then allows the use of regression and deletion
diagnostics, to assess the quality of each of the observations and their
influence on the fitted parameters. It is this last point that is, I think,
fairly unique to UnitCell in the run of things.
Apologies for sending this to the Reitveld list, when it concerns
techniques far from whole pattern refinement. I am beginning to wish I had
kept the whole matter in the dark!
Yours
Simon Redfern.
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Dr Simon Redfern Tel: +44 1223 333475
Dept Earth Sciences Fax: +44 1223 333450
University of Cambridge
Downing Street
Cambridge, CB2 3EQ, UK www.esc.cam.ac.uk/astaff/redfern
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