Twinning

Definition

"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.

Classification

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 overlap exactly, while others are non-overlapped. A typical example is an obverse/reverse twin in case of a rhombohedral crystal.

Non-Merohedral Twins

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.

References

Giacovazzo, C. ed. (2002). Fundamentals in Crystallography, I.U.Cr. & O.U.P.: Oxford, UK.