Logo Estimated Standard Deviations
(Standard Uncertainties)

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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, pj, is calculated as follows:

σ (pj) = [ χ2 A-1jj ]1/2
  = [ A-1jj ]1/2 Rwp / Rexp
It is important to realize that this quantity only has any meaning when (a) the model is correct and (b) when the refinement has completely minimized. Most crystallographic programs will calculate the error on the parameter during all stages of the refinement, but the numbers are only useful when the correct model is completely refined. Clearly the poorer the fit as measured by χ-squared, then the bigger will be the error in the value of the refined parameter.

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, Nhkl 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.

Correlation Coefficients

The inverted matrix A also yields other useful values, in particular the covariance and correlation coefficient between parameters pj and pk. Both of these are calculated from the off-diagonal terms of the inverted matrix. The covariance is calculated using:

σ2 (pj pk) = χ2 A-1jk

while the more useful correlation coefficient, cjk, is given by:

cjk = ( A-1jk  / { A-1jj A-1kk }1/2 ) × 100%

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.

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