There are three basic methodologies for applying absorption corrections to reflection data. These are laid out in the menu scheme in WinGX, in decreasing order of theoretical rigour.
It is generally agreed that the best absorption corrections are provided by the analytical  or Gaussian quadrature  methods. These two methods require that the crystal faces are well defined and can be accurately indexed and measured. These conditions are not often met. It can be time consuming to index a specimen with numerous faces and difficult to measure accurately the distances between faces. For these reasons, numerical methods are used less often than those which are easier to implement experimentally. In cases of severe absorption however, they are the only really effective methods. The spherical and cyclindrical corrections also provide numerically accurate corrections for crystals adopting the requisite morphology, but are less useful, since grown crystals are not usually spheres or cylinders. Crystals may be ground to spheres, but many materials do not survive such treatment, The program XtalView is provided in WinGX as an aid to face-indexing.
The semi-empirical methods rely on further intensity measurements. The multiscan method of Blessing  is of most use when there is a large redundancy in the data-set, as is usually the case for area-detector data. Equivalent intensities are analysed in terms of a multipolar spherical harmonic expansion and the method is implemented in the programs PLATON , SORTAV  and SADABS . A somewhat similar method called CAMEL-JOCKEY  uses a trigonometric series expansion of the diffractometer angles, but is little used since so many more experimental measurements are required. The most commonly used method is the azimuthal scan or psi-scan method of North et al . This is the simplest method, involving the measurement of the intensities of a (usually small) number of reflections with chi values close to 90o at different psi(phi) values. An averaged absorption surface is thus computed and used to calculate the transmission factors. It works remarkably well, but is unsuitable for crystals with large muR values.
The final type of absorption corrections, the so-called refined
corrections DIFABS , XABS2  and SHELXA  have fallen
out of favour in the recent past. For comments on the use of DIFABS see
the web site
A data-set for sodium tungstate dihydrate (Na2WO4.2H2O) is provided (download NAWO4 here) for WinGX as test data for absorption corrections. The compound was chosen because :
Crystal data for Na2WO4.2H2O : a = 8.4797(5) b = 10.5930(5) c = 13.8527 (10) Å, V = 1244.33(1) Å3, orthorhombic, space group Pbca, Z = 8, Mr = 329.8, T = 295 K, mu(Mo-K-alpha) = 18.664 mm-1, crystal size 0.391x0.344x0.109 mm. 7543 reflections were measured, yielding 1811 unique reflections.
All refinements were carried out using SHELXL-93 and a number of criteria for assessing the results are given. The transmission factors obtained are given in Table 1 and the results of refinement are summarised in Tables 2 and 3. As expected, all methods offer a significant improvement over the uncorrected data in terms of the residuals. The best methods are clearly the analytical and Gaussian quadrature, which give virtually identical results. For all other methods, (except XABS2) the range of transmission factors is smaller, suggesting they may be under-correcting the data.
The method which gives the second best set of figures in Table 2 is DIFABS. At this stage it is also useful to compare the adp's especially the degree of anisotropy given by the three principal mean-square displacements. The eigenvalues of the Uij tensors of the W and Na atoms are listed in Table 3. The accepted wisdom is that absorption errors cause the adp's to be somewhat smaller and more anisotropic than the "true" values (in this context the "true" value is assumed to be that obtained from the analytical correction.). Examination of the values obtained with no absorption correction bear this out. It can be seen that the adp for the W atom calculated using DIFABS is slightly smaller and more isotropic than the "true" value, but the agreement is quite respectable. Moreover the adp's for the light Na atoms are also quite similar to their "true" value. These results suggest that the oft quoted problems with DIFABS are not always manifest. Rather unexpectedly in view of the large muR value, the psi-scan method gives values in Table 2 quite similar to DIFABS. However, the resultant adp's are much less acceptable.
None of the remaining methods provide a satisfactory correction to the NAWO4 data. The next best correction is probably SHELXA (though it does give a large +ve residual), followed by multiscan and XABS2. The multiscan method of Blessing  surprisingly performs rather poorly for the NAWO4 data, even though there is a large degree of redundancy. It may be that the transmission paths have not been sampled adequately in this example.
The analytical and DIFABS corrections are further compared in Table 4. The W-O bond lengths can be seen to be insensitive to the method of correction, in particular the values using DIFABS and the analytical correction differ by 2xsigma at most.
The ORTEP views of the WO4 anion are shown below (click on thumb-nail to view full size picture).
|ORTEP view of WO4 anion with NO absorption correction.|
|ORTEP view of WO4 anion with analytical absorption correction, showing the principal axes of the libration and translation tensors T and L.|
|ORTEP view of WO4 anion with DIFABS correction, showing the principal axes of the libration and translation tensors T and L.|
|Table 1 Transmission factors|
|METHOD||T minimum||T maximum||Ratio|
|Table 2 Refinement results|
* NIEQ = number of inconsistent equivalent reflections flagged
No theta correction, unit weights
|Table 3 Eigenvalues (Å2) of the Uij tensors for the W and Na atoms*|
* The standard uncertainties on the eigenvalues are ca 0.0002 for W and ~0.001 for Na
|Table 4 Bond lengths (Å) for the WO4 anion.|