International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 6.3, pp. 607608

Some crystals do not have regular faces, or cannot be measured because these are obscured by the crystal mounting. If corrections based on measurements of the crystal shape are not feasible, absorption measurements may be estimated, either from the intensities of the same reflection at different azimuthal angles ψ (see Subsection 6.3.3.6), or from measurements of equivalent reflections, by empirical methods.
There are variants of the method related to differences in experimental technique. The principles may be illustrated by reference to the procedure for a fourcircle diffractometer (Flack, 1977).
Intensities are measurements for a reflection S at the angular positions , , , . Corrected intensities are to be derived from the measurements by means of a correction factor such that It is assumed that the correction can be written in the form of a rapidly converging Fourier series The form of the geometrical terms may be simplified by taking advantage of the symmetry of the fourcircle diffractometer. If it is assumed that diffraction is invariant to reversal of the incident and diffracted beams, the settings ; ; ; ; ; ; ; are equivalent. In shorthand notation, the series (6.3.3.41) reduces to The range of indices for some terms may be restricted by noting other symmetries in the diffraction experiment. Thus, equation (6.3.3.40) will define the absorption correction for measurements of the incidentbeam intensity, with . Since with this geometry the correction will be invariant to rotation about the χ axis, the coefficients for the function involving must vanish if the χ index, k, is nonzero. By similar reasoning with the axis along the incident beam, one may deduce that coefficients for will vanish unless l = 0.
Because for a given reflection all measurements are made at the same Bragg angle, the dependence of the correction cannot be determined by empirical methods. This factor in A is obtained from the absorption correction for a spherical crystal of equivalent radius.
Since an empirical absorption correction is defined only to within a scale factor, the scale must be specified by applying a constraint such that where is the number of independent reflections. Equation (6.3.3.42) may be expressed in the shorthand notation where is the coefficient in a term such as or and is the corresponding geometrical function. Labelling the constant geometrical term with a value of unity as and rearranging leads to which defines .
Equation (6.3.3.40) is now expressed as in which the coefficients are to be chosen so that the values of for each S are as near equal as possible. Since the values within each set will not be exactly equal, we rewrite (6.3.3.46) as in which the mean intensity and the are chosen to minimize , where and is the weight for that reflection.
If the equation to be solved is written in the shorthand form in which D corresponds to , the and correspond to C, with and corresponding to F, the solution to (6.3.3.50) can be determined from the normal equations where is the transpose of F. This procedure suffers from the disadvantages of requiring a matrix inversion whenever the set of trial functions (i.e. those multiplied by the coefficients ) is modified. The tedious inversion of the normal equations, described by (6.3.3.51), may be replaced by a simple inversion via the Gram–Schmidt orthogonalizing process, i.e. by calculating a matrix W with mutually orthogonal columns such that The minimizing of (D − FC)^{2} is replaced by minimizing (D − WA)^{2}. Differentiating with respect to yields If equation (6.3.3.52) is written as where the upper triangular matrix B is the vector determining the coefficients is in which the inversion of B is straightforward.
In difficult cases, with data affected by errors in addition to absorption, the method described may give physically unreasonable absorption corrections for some reflections. In such cases, it may help to impose the approximate constraints If , this reduces to the M constraint equations where is the square root of the weight for the weighted mean of the equivalent reflections , defined as and the multiplier controls the strength with which the additional constraints are enforced. With the additional constraint equations, the sum of squares to be minimized, corresponding to (6.3.3.48), becomes
A closely related procedure expressing the absorption corrections as Fourier series in polar angles for the incident and diffracted beams is described by Katayama, Sakabe & Sakabe (1972). A similar method minimizing the difference between observed and calculated structure factors is described by Walker & Stuart (1983). Other experimental techniques for measuring data for empirical absorption corrections that could be analysed by the Fourierseries method are described by Kopfmann & Huber (1968), North, Phillips & Mathews (1968), Flack (1974), Stuart & Walker (1979), Lee & Ruble (1977a,b), Schwager, Bartels & Huber (1973), and Santoro & Wlodawer (1980).
References
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