Estimated Standard Deviations
(Standard Uncertainties) |

**Estimated Standard Deviations
(Standard Uncertainties)**

In addition to refining the parameters so as to obtain the best fit between the
model and the diffraction data, it is useful to be able to estimate the
precision of the parameters. For example, the cell parameter *a* may
be quoted as 8.9512(3) Å. The number "3" in round brackets
indicates the precision in the last digit of 8.9512, and is referred to
as an estimated standard deviation (esd) or, in more modern nomenclature,
as a standard uncertainty (su).
Apart from providing a measure of precision of refined parameter,
esd's are essential for testing whether two refined
parameters are equal (within statistical error).

Esd's are calculated by most least-squares refinement programs using the
inverted least-squares matrix discussed previously.
An estimated standard deviation for each refined
parameter, *p*_{j}, is calculated as follows:

σ (p_{j}) |
= | [
χ^{2} A^{-1}_{jj}
]^{1/2} |

= | [
A^{-1}_{jj}
]^{1/2} R_{wp} /
R_{exp} |

The expected profile *R*-factor that is used in the above equation
contains the number of observations, *N*, used in the
least-squares refinement.
This number was the subject of much debate during the early years of
the Rietveld method: for parameters that
affect the peak position, e.g. the unit cell constants, *N* is clearly
the number of profile points used in the summation, but for
those that affect peak intensity, then this number is effectively too
large. This is because the number of contributing reflections,
*N*_{hkl} is usually much smaller
than the number of profile points, *N*.
Since the estimated standard deviations (esd's)
on the parameters are determined by
χ-squared, which in turn depends upon *N*,
then "overestimating" the value of *N* results in
an underestimation of the esd's on the structural parameters.
It is generally accepted that the esd's on most structural parameters
from a Rietveld refinement when calculated using the above equation
are often underestimated by a factor of about 3: this factor is usually less
for low-symmetry crystal systems where more extensive overlap of the
diffraction peaks occurs.

The inverted matrix A also yields other useful values, in particular the
covariance and correlation coefficient between parameters
*p*_{j} and *p*_{k}.
Both of these are calculated from the off-diagonal terms
of the inverted matrix. The covariance is calculated using:

while the more useful correlation coefficient, *c*_{jk}, is
given by:

Correlation coefficients can sometimes be used to identify unstable parameters in a refinement. The values always lie in the range -100% to +100%. High absolute values often indicate over-parameterization: this is commonly observed when attempts are made to refine a structure using an incorrect space group of too low a symmetry. Note that while correlations of + or -100% between "independent" parameters will always result in the least-squares matrix failing to invert, values close to + or -100% will normally result in matrix instability, with consequential large and erroneous shifts in the least-squares parameters.

© Copyright 1997-2006. Birkbeck College, University of London. |
Author(s):
Jeremy Karl Cockcroft |