Twinning (CCP4: General)


twinning - dealing with data from twinned crystals

PLEASE NOTE: Most of this document has been taken directly from chapter 6 of the SHELX-97 Manual.



A 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 Ihkl from a single crystal, the experiment yields
km Ihkl(crystal1) + (1-km) Ih'k'l'(crystal2)

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 km. 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 non-merohedral or epitaxial twinning.

The warning signs for twinning

Experience 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.

  1. The metric symmetry is higher than the Laue symmetry.
  2. The Rmerge-value for the higher symmetry Laue group is only slightly higher than for the lower symmetry Laue group.
  3. The mean value for |E2-1| is much lower than the expected value of 0.736 for the non-centrosymmetric case. 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 intensities will be distributed so that there are fewer extreme values. This can be seen by plotting the output of TRUNCATE or ECALC.
  4. The space group appears to be trigonal or hexagonal.
  5. There are impossible or unusual systematic absences.
  6. Although the data appear to be in order, the structure cannot be solved.
  7. The Patterson function is physically impossible.
  8. The following points are typical for non-merohedral twins, where the reciprocal lattices do not overlap exactly and only some of the reflections are affected by the twinning:

  9. There appear to be one or more unusually long axes, but also many absent reflections.
  10. There are problems with the cell refinement.
  11. Some reflections are sharp, others split.
  12. K=mean(Fo2)/mean(Fc2) is systematically high for the reflections with low intensity.
  13. For all of the 'most disagreeable' reflections, Fo is much greater than Fc.


Example 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.)

Cumulative intensity distribution for twin Cumulative intensity distribution for twin

Frequently encountered twin laws

The following cases are relatively common:

  1. Twinning by merohedry. The lower symmetry trigonal, rhombohedral, tetragonal, hexagonal or cubic Laue groups may be twinned so that they look (more) like the corresponding higher symmetry Laue groups (assuming the c-axis unique except for cubic)
  2. Orthorhombic with a and b approximately equal in length may emulate tetragonal
  3. Monoclinic with beta approximately 90° may emulate orthorhombic:
  4. Monoclinic with a and c approximately equal and beta approximately 120° may emulate hexagonal [P21/c would give absences and possibly also intensity statistics corresponding to P63].
  5. Monoclinic with na + nc ~ a or na + nc ~ c can be twinned. See HIPIP examples.

Likely twinning operators

Data 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 Remarks

A crystal is a 3-dimensional 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 2-fold). All angles 90; a = b.
Trigonal - one 3 fold axis of rotation (plus possible perpendicular 2-folds). Alpha and Beta = 90, Gamma = 120 ; a = b (hexagonal setting).
Hexagonal - one 6 fold axis of rotation (plus possible perpendicular 2-fold). Alpha and Beta = 90, Gamma = 120 ; a = b.
Cubic - all axes equal and equivalent, related by a diagonal 3-fold; also 2-fold, or 4-fold 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 b axes 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.


It 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|/2|a|, or, if |a|>|c|, cos(Beta*) = |na|/2|c|.


For orthorhombic crystal forms the only possibility for twinning is if there are two axes with nearly the same length.

tetragonal, trigonal, hexagonal, cubic

For tetragonal, trigonal, hexagonal or cubic systems it is a requirement of the symmetry that two cell axes are equal. Assuming the lengths of a and b to be equal, and maintaining a right-handed axial system, we find:

For these spacegroups the real axial system could be:(a,b,c)or(-a,-b,c)or (b,a,-c)or(-b,-a,-c)
with corresponding reciprocal axes:(a*,b*,c*)or(-a*,-b*,c*)or (b*,a*,-c*)or(-b*,-a*,-c*)
Corresponding indexing systems:(h,k,l)or(-h,-k,l)or (k,h,-l)or(-k,-h,-l)

N.B. There may be alternatives where other pairs of symmetry operators are paired, but this is the simplest and most general set of operators and has the added advantage that the transformation matrices in real and reciprocal space are the same. For example: in P3i (-a,-b,c) is a equivalent of (-b,a+b,c) but the corresponding reciprocal space conversion matches (a*,b*,c*) to (a*-b*,a*,c*)

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. Different domains may have different definitions of axes, which lead to different diffraction intensities superimposed on the same lattice.

Lookup tables for tetragonal, trigonal, hexagonal, cubic

Here 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. While twinning involves more than one indexing possibility within a single dataset, these operators are also relevant for ensuring the same indexing between multiple datasets when there is no twinning.


More information on twinning can be found at: Fam and Yeates' Introduction to Hemihedral Twinning, which includes a Twinning test.


Acknowledgement in SHELX manual:

"I should like to thank Regine Herbst-Irmer
            who wrote most of this chapter."

Prepared for CCP4 by Maria Turkenburg, University of York, England