Micheal
Did this get posted twice? I thought I was waiting to read an explanation
myself, but I could just be going mad.... guess I'll have a go myself.
> I have a question concerning the errors calculated in a rietveld
> refinement. As far as I understand, the error of a parameter is calculated
> as
>
> sigma(p_i) = sqrt(c_ii) * sqrt(chi^2/N-P) (*)
>
> where c_ii is the i-th diagonal element of the inverse of the
> curvature matrix, chi^2 = sum_i w_i*(yobs_i - ycalc_i), N number of
> observations, P number of parameters.
Are you sure about these? I thought chi^2 should contain the factor N-P,
also it is usually (obs-calc)^2. Pedantics aside you have something like
the statistical error * chi.
> But I thought the statistical error is just sqrt(c_ii)!
Yup, it is.
> The second factor
> is alway << 1, because N is a large number, and so the calculated errors
> are much to small! Even more, I can minimize them by increasing the number
> of steps, although, after a certain point, I gain no more information.
No! Remember that the chi^2 as you have defined it increases with the
number of observations you make - as you sum over i. Using
chi^2=[sum_i w_i*(yobs_i - ycalc_i)^2]/(N-P),
you can see that chi^2 is approximately the average number of esd's you
are wrong by for each point in the pattern (N >> P). So the second factor
is your familiar chi^2; very rarely << 1 !
The correction you are applying to the diagonal elements of your matrix
is to try to allow for the systematic errors (ie, non random) in your
model. This appears to be your source of confusion. From Young's book (the
Rietveld Method) E.Prince says, "It is customary to multiply the e.s.d's
by the chi factor, on the assumption that the lack of fit is due to an
over-weighting of all data points. While this practice is conservative, it
should be understood that it has no basis in statistics."
If you look at a difference curve there will be random noise and
systematic problems, such as peak shape or intensity or position. This fix
says that the systematic problems are really just random noise and
effectively increases the weights on the y_obs accordingly, giving larger
esd's.
Hope this helps,
Jon Wright
PS: See also GSAS/fullprof manuals and "Rietveld refinement guidelines"
McCusker et al, J.Appl,Cryst 32(1999),36-50. Also the non-Rietveld
specific - Acta Cryst A45 (1989) 63-75. "Statistical descriptors in
crystallography".
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