
Twinning (CCP4: General)NAMEtwinning  dealing with data from twinned crystalsPLEASE NOTE: Most of this document has been taken directly from chapter 6 of the SHELX97 Manual. Contents
IntroductionA typical definition of a twinned crystal is the following: "Twins are regular
aggregates consisting of crystals of the same species joined together in some definite
mutual orientation" (Giacovazzo, 1992). For this to happen two lattice repeats in the
crystal must be of equal length to allow the array of unit cells to pack compactly.
The result is that the reciprocal lattice diffracted from each component will overlap,
and instead of measuring only I_{hkl} from a single crystal, the experiment
yields For a description of a twin it is necessary to know the matrix that transforms the hkl indices of one crystal into the h'k'l' of the other, and the value of the fractional component k_{m}. Those space groups where it is possible to index the cell along different axes are also very prone to twinning. When the diffraction patterns from the different domains are completely superimposable, the twinning is termed merohedral. The special case of just two distinct domains (typical for macromolecules) is termed hemihedral. When the reciprocal lattices do not superimpose exactly, the diffraction pattern consists of two (or more) interpenetrating lattices, which can in principle be separated. This is termed nonmerohedral or epitaxial twinning. The warning signs for twinningExperience shows that there are a number of characteristic warning signs for twinning. Of course not all of them can be present in any particular example, but if one finds several of them, the possibility of twinning should be given serious consideration.
The following points are typical for nonmerohedral twins, where the reciprocal lattices do not overlap exactly and only some of the reflections are affected by the twinning: ExamplesExample of a cumulative intensity distribution with twinning present, as plotted by TRUNCATE. (A full size version of the example can be viewed by clicking on the small picture.)
Frequently encountered twin lawsThe following cases are relatively common:
Likely twinning operatorsData from a merohedrally twinned crystal can be deconvoluted using the program DETWIN. This program requires a likely twinning operator for the spacegroup in question to be specified. Possible operators are listed here.General RemarksA crystal is a 3dimensiona translational repeat of a structural pattern which may comprise a molecule, part of a symmetric molecule, or several molecules. The repeats which can overlap by simple translation, are called unit cells. Lattice symmetry enforces extra limitations. There are 7 basic symmetry classes possible within a crystal. Triclinic  no rotational symmetry. No restrictions on a b c or alpha beta gamma Monoclinic  one 2 fold axis of rotation  two angles must be 90; usually alpha and Gamma. Orthorhombic  two perpendicular 2 fold axes of rotation (these must generate a 3rd) All angles 90. Tetragonal  one 4 fold axis of rotation (Plus possible perpendicular 2fold) All angles 90; a=b Trigonal  one 3 fold axis of rotation (Plus possible perpendicular 2folds)Alpha and Beta 90, Gamma 120 ; a=b Hexagonal  one 6 fold axis of rotation (Plus possible perpendicular 2fold)Alpha and Beta 90, Gamma 120 ; a=b Cubic  all axes equal and equivalent, related by a diagonal 3fold; also 2fold ,or 4fold axes of rotation along crystal axes. All angles 90 ; a=b = c Problems arise most commonly when two or more crystal axes are the same length, either by accident in the monoclinic and orthorhombic system , or as a requirement of the symmetry as in the tetragonal, trigonal, hexagonal or cubic systems. Although the a and baxes in the tetragonal, trigonal, hexagonal and cubic classes must be equal in length, there can still be ambiguities in their definition, and consequentially in the indexing of the diffraction pattern. It is these classes of crystals which are most prone to twinning. ( Figure) monoclinicIt is possible that in P21 or C2 there are two possible choices of a with anew = aold + ncold. If the magnitude of a is equal to that of a+nc, the cos rule requires that cos(Beta*) = nc/2a, or, if a>c, cos(Beta*) = na/2c. orthorhombicFor orthorhombic crystal forms the only possibility for twinning is if there are two axes with nearly the same length. tetragonal, trigonal, hexagonal, cubicFor tetragonal, trigonal, hexagonal or cubic systems it is a requirement of the symmetry that two cell axes are equal.
In these cases any of the above definitions of axes is equally valid. For many cases the alternative systems are symmetry equivalents, and hence do not generate detectable differences in the diffraction pattern. But for crystals where this is not true, twinning is possible. Lookup tables for tetragonal, trigonal, hexagonal, cubicHere are details for the possible systems. These tables are generated by considering each of the indexing systems above, and eliminating those which correspond to symmetry operators of the spacegroup.
SEE ALSOMore information on twinning can be found at: Fam and Yeates' Introduction to Hemihedral Twinning, which includes a Twinning test. AUTHORSAcknowledgement in SHELX manual: "I should like to thank Regine HerbstIrmer who wrote most of this chapter." Prepared for CCP4 by Maria Turkenburg, University of York, England 