Introduction to Hemihedral Twinning

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Introduction to Hemihedral Twinning

Twinning is a crystal growth disorder in which the specimen is composed of distinct domains whose orientations differ but are related in a particular, well-defined way. Sometimes non-convex crystal morphology implicates twinning, but microscopic examination is often uninformative. Twinning usually prevents a successful structure determination unless it is detected and either avoided or corrected. In order to know when one is at risk, it is necessary to understand the symptoms of the different types of twinning and the symmetries where they may arise.

There are two fundamentally different categories of twinning, epitaxial and merohedral. In epitaxial twinning, different domains are typically oriented so that their molecular spacings match at the crystal face where they meet, but their crystal lattices are not superimposable. In other words, their crystal lattices superimpose in fewer than three-dimensions. This leads to distinct, interpenetrating reciprocal lattices in the diffraction pattern. Since epitaxial twinning is recognizable in the diffraction pattern, attempts can be made to integrate reflections from a single lattice, or untwinned crystals can be sought. We do not consider epitaxial twinning further.

Merohedral twinning refers to the special cases where the lattices of the different domains overlap in three dimensions. Domains whose orientations are crystallographically distinct can have superimposable lattices in cases where the rotational symmetry of the lattice exceeds the rotational symmetry of the crystal space group. This situation arises in several point symmetries: 3, 32 (hexagonal but not rhombohedral), 4, 6, and 23. We focus on the most common type of merohedral twinning, hemihedral, which involves just two different domain orientations.

As a result of hemihedral twinning, each observed diffraction intensity is actually a weighted sum of two crystallographically distinct, twin-related, reflections. Two twin-related reflections contribute to an observed diffraction peak with weights determined by the fractional volume of the specimen represented by each domain orientation. This is the twin fraction, alpha. The observed intensity is equal to alpha x I(hkl) + (1-alpha) x I(h'k'l'), where hkl and h'k'l' are twin-related indices.

Depending on the hemihedral twin-fraction, there are two different scenarios that must be understood. If the twin-fraction is nearly equal to one-half, the observed diffraction pattern acquires the additional symmetry imposed by the twinning operation. The data obey an erroneously high symmetry and are processed as such. This has been called 'perfect twinning'. The true crystallographic intensities cannot be recovered from the observed measurements, but it is still possible to solve the structure by molecular replacement (in practice) and by MIR (in theory).

Clues that diffraction data may be from a perfectly twinned specimen include: (a) a unit cell that is too small to contain the known molecule under the apparent space group symmetry, and (b) an intensity distribution that does not follow Wilson statistics. The latter test is generally applicable and is performed by this Web-server. Several related expressions can be computed. One whose expected values are rational (and therefore easy to remember) is <I2> / <I>2. The expected value is 1.5 for (acentric) twinned data and 2.0 for (acentric) untwinned data. The test must be performed on normalized data or in thin shells. Anisotropic diffraction is a complicating factor, as it tends to have the opposite effect from twinning.

The situation in which the hemihedral twin fraction is not one-half has been called 'partial twinning'. The result is not higher apparent symmetry, but observed intensities that contain contributions from distinct reflections. This effect can be reversed to give the true crystallographic intensities if: (1) the twin fraction is significantly different from one-half and (2) the twin-fraction can be estimated accurately. The important problem of estimating alpha has been covered by numerous investigators. On this Web site we estimate alpha by a statistical evaluation of the similarity between twin-related observations. Pseudo- or non-crystallographic symmetry can be a complicating factor, as it tends in special cases to mimic twinning (at least at lower resolution) by leading to similarity between crystallographically distinct reflections. Measurement errors lead to a systematic underestimation of alpha, so the test should be performed over a resolution range where the data are well-measured.

For more information on twinning and how to proceed if twinning is detected, the reader is referred to the review by Yeates (T. O. Yeates (1997), Detecting and Overcoming Crystal Twinning. Methods in Enzymology 276, 344-358), which is a suitable citation for those using this Web-based server. Queries by e-mail are also welcome. I am especially interested in hearing about cases where twinning is detected.

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