"Twins are regular aggregates consisting of crystals of the same species joined together in some definite mutual orientation" (Giacovazzo, 2002). So for the description of a twin two things are necessary: a description of the orientation of the different species relative to each other (the twin law) and the fractional contribution of each component. The twin law can be expressed as a matrix that transforms the hkl indices of one species into the other.
Depending on the twin law four types of twins can be distinguished:
Twinning by Merohedry
In a merohedral twin, the twin law is a symmetry operator of the crystal system, but not of the point group of the crystal. This means that the reciprocal lattices of the different twin domains superimpose exactly and the twinning is not directly detectable from the reflection pattern. This type is possible in the trigonal, tetragonal, hexagonal and cubic crystal systems, which have more than one Laue group. The twin law corresponds to the two-fold operation that is present in the apparent Laue group, but not in the true space group. Only for trigonal crystals is there more than one possible twin law.
Twinning by Pseudo-Merohedry
In a pseudo-merohedral twin, the twin operator belongs to a higher crystal system than the structure. This may happen if the metric symmetry is higher than the symmetry of the structure. Depending on how well the higher metric symmetry is fulfilled, it may happen that the reciprocal lattices overlap exactly and the twinning is not detectable from the diffraction pattern. But, compared to merohedral twins,the number of possible twin laws is much higher.
Twinning by Reticular Merohedry
Part of the reflections overlaps exactly, while others are non-overlapped. A typical example is an obverse/reverse twin in case of a rhombohedral crystal.
For non-merohedral twins, the twin law does not belong to the crystal class of the structure nor to the metric symmetry of the cell. Therefore the different reciprocal lattices do not overlap exactly. There are three types of reflections, non-overlapped, partially overlapped and exactly overlapped reflections. Here the problems start in the data collection. If both twin domains are similar in size, there are often problems with the cell determination and usual automatic indexing programs fail. More than one orientation matrix is needed to index all reflections. In the integration process the information of all matrices should be used.
Tests for Twinning
Refinement of merohedral twinned data is possible using CNS (for input files, see CNS homepage - main menu - input files - x ray - twinning) . Pseudo-merohedrally twinned data can be refined with SHELXL (Twin-Refinement with SHELXL) and PHENIX (Structure refinement in PHENIX).
Warning Signs for Twinning
Experience shows that there are a number of characteristic warning signs of twinning, as given in the following list. Of course not all of them can be present in any particular example, but if one or several apply, the possibility of twinning should be given serious consideration.
a) The metric symmetry is higher than the Laue symmetry.
b) The Rint-value for the higher symmetry Laue group is only slightly higher than for the lower symmetry Laue group.
c) If different crystals of the same compound show significantly different Rint values for the higher symmetry Laue group, this clearly shows that the lower symmetry Laue group is correct and indicates different extents of twinning.
d) The mean value for |E^2-1| is much lower than the expected value of 0.736 for the non-centrosymmetric case (see also Intensity statistics). If we have two twin domains and every reflection has contributions from both, it is unlikely that both contributions will have very high or that both will have very low intensities, so the combined intensities are distributed to give fewer extreme values.
e) The space group appears to be trigonal or hexagonal.
f) The apparent systematic absences are not consistent with any known space group.
g) Although the data appear to be in order, the structure cannot be solved. This may of course also happen if the cell is wrong, for example with an halved axis
h) The Patterson function is physically impossible.
The following features are typical of non-merohedral twins, where the reciprocal lattices do not overlap exactly and only some of the reflections are affected by the twinning:
i) There appear to be one or more unusually long axes.
j) There are problems with the unit cell refinement.
k) Some reflections are sharp, others split.
l) K = mean(Fo^2)/mean(Fc^2) is systematically high for reflections with low intensity. This may also indicate a wrong choice of space group in the absence of twinning.
Giacovazzo, C. ed. (2002). Fundamentals in Crystallography, I.U.Cr. & O.U.P.: Oxford, UK.