Authors: Dieter Schwarzenbach and Geoff King
Contact : Geoff King, Fysica-Chemische Geologie, K.U. Leuven, B-3001 Heverlee, Belgium
LATCON is a rewrite of Dieter Schwarzenbach's program of the same name in the XRAY76 system. It determines the 'best' lattice parameters and their standard deviations from the 2 diffraction angles of a set of indexed reflections, taking into account any constraints imposed by the crystal system and allowing for systematic errors in the zero values.
Given a set of indexed reflections with their diffraction angles 2 , the program uses the least-squares method to determine the best values of the reciprocal and direct metric tensors and lattice parameters with their estimated standard deviations (esd's) together with the variance-covariance and correlation matrices. The 2 values may have been obtained by powder or single-crystal measurements. Different schemes are available for allowing for systematic errors.
In principle there are two methods of refining lattice constants by minimizing
The first is to express 2 as a non-linear function of the lattice constants. This leads to a non-linear least-squares procedure which requires starting values and several cycles of refinement. The second, used here, takes advantage of the fact that is a linear function of the reciprocal metric tensor giving a linear least-squares problem which can be solved directly without the need for starting values and without iteration. The much more complicated task consists then in computing the esd's of the best lattice constants from those of the reciprocal metric tensor elements.
The following concepts are useful:
All symbols containing a "*" refer to reciprocal space. The matrix M * is the inverse of the matrix M. The vector v * is the row vector (h k l), i.e. v * transposed. Therefore,
The variance-covariance matrix COV (see e.g. ref. 1) is calculated from the inverse N of the normal-equation matrix N.
With R defined above, n = number of observations, k = number of parameters refined, n-k is the number of degrees of freedom and e the variance of an observation with unit weight, so , the error of fit, is the corresponding . If the weights are the reciprocals of real variances, the expectation value for the error of fit is unity and R should be distributed as . The diagonal terms of COV are the variances whose square roots are the standard deviations (esd's), . The ij th term of the correlation matrix COR is given by
The law of propagation of errors expresses the variance-covariance matrix COVB of derived quantities = function1( , ,...), = function2( , ...), etc in terms of COVA of , ,... Denoting the derivative of with respect to by and the matrix of derivatives as D, we obtain
COVB = D COVA D
The program proceeds by the following steps:
Two mutually exclusive schemes for taking account of systematic errors may be used in the refinement. Both involve refinement of an additional parameter .
LATCON should give exactly the same results as a non-linear least-squares procedure. The errors in the variances and covariances derived from the error propagation law, which is due to the series expansion involved, are very small for 's of a few percent or less.
LATCON can also use the angle data and the crystal system stored on the archive bdf. The option line skip enables reflections to be omitted. Options also exist for outputting the refined cell dimensions.