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

- Introduction
- The warning signs for twinning
- Examples
- Frequently encounted twin laws
- Likely twinning operators

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 I_{hkl} from a single crystal, the experiment
yields

k_{m} I_{hkl}(crystal_{1}) +
(1-k_{m}) I_{h'k'l'}(crystal_{2})

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 **non-merohedral** or
**epitaxial** 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.

- The metric symmetry is higher than the Laue symmetry.
- The R
_{merge}-value for the higher symmetry Laue group is only slightly higher than for the lower symmetry Laue group. - The mean value for
`|E`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.^{2}-1| - The space group appears to be trigonal or hexagonal.
- There are impossible or unusual systematic absences.
- Although the data appear to be in order, the structure cannot be solved.
- The Patterson function is physically impossible.
- There appear to be one or more unusually long axes, but also many absent reflections.
- There are problems with the cell refinement.
- Some reflections are sharp, others split.
- K=mean(
*F*_{o}^{2})/mean(*F*_{c}^{2}) is systematically high for the reflections with low intensity. - For all of the 'most disagreeable' reflections,
*F*_{o}is much greater than*F*_{c}.

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:

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 |

The following cases are relatively common:

- 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)
- Orthorhombic with
**a**and**b**approximately equal in length may emulate tetragonal - Monoclinic with beta approximately 90° may emulate orthorhombic:
- Monoclinic with
**a**and**c**approximately equal and beta approximately 120° may emulate hexagonal [P2_{1}/c would give absences and possibly also intensity statistics corresponding to P6_{3}]. - Monoclinic with
or`na + nc ~ a`can be twinned. See HIPIP examples.`na + nc ~ c`

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
a*new* = a*old*
+ nc*old*. 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.

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) |

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.

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.

- All
**P4i**and related**4i**space groups:

(h,k,l) equivalent to (-h,-k,l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) - Twinning possible with this operator - apparent Laue symmetry for perfect twin would be P422
- For all
**P4i2i2**and related**4i2i2**space groups:

(h,k,l) is equivalent to all of (-h,-k,l)*,*(k,h,-l)*and*(-k,-h,-l) so all axial pairs are already equivalent as a result of the crystal symmetry. - No twinning possible but a perfect twin for the Laue group P4 might appear to have this symmetry.
- All
**P3i**and**H3**:

(h,k,l)*neither*equivalent to (-h,-k,l)*nor*(k,h,-l)*nor*(-k,-h,-l) so we need to check all 4 possibilities. These are the only cases where tetratohedral twinning can occur:real axes: (a,b,c) and (-a,-b,c) and (b,a,-c) and (-b,-a,c) reciprocal axes: (a*,b*,c*) and (-a*,-b*,c*) and (b*,a*,-c*) and (-b*,-a*,c*) *i.e.*For P3, consider reindexing (h,k,l) to (-h,-k,l)*or*(k,h,-l)*or*(-k,-h,-l).

For

**H3**the indices must satisfy the relationship*-h +k+l =3n*so it is only possible to reindex as ( k, h,-l). Note that the latter is a symmetry operator of**H32**, so that twinning is not possible in H32. However, twinning in H3 may give apparent H32 symmetry.*For trigonal space groups, symmetry equivalents do not seem as "natural" as in other systems. Replacing the 4 basic sets with other symmetry equivalents gives a bewildering range of apparent possibilities, but all are equivalent to one of the above.*Two-fold twinning possible with this operator - apparent Laue symmetry for two fold perfect twin could be P321 (operator k,h,-l) or P312 (operator -k,-h,-l) or P6 (operator -h,-k,l) Four-fold twinning with these operators could generate apparent Laue symmetry of P622

space group number space group point group possible twin operators 143 P3 PG3 -h,-k,l; k,h,-l; -k,-h,-l 144 P31 PG3 -h,-k,l; k,h,-l; -k,-h,-l 145 P32 PG3 -h,-k,l; k,h,-l; -k,-h,-l 146 H3 PG3 k,h,-l

- All
**P3i12**:

(h,k,l) already equivalent to (-k,-h,-l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*reindex (h,k,l) to (k,h,-l) [or its equivalent operator (-h,-k,l)]. - Twinning possible with this operator - apparent symmetry for two fold perfect twin would be P622 (operator -h,-k,l)
- All
**P3i21**:

(h,k,l) already equivalent to (k,h,-l) so we only need to check:real axes: (a,b,c) and (-a,-b,-c) reciprocal axes: (a*,b*,c*) and (-a*,-b*,-c*) *i.e.*reindex (h,k,l) to (-h,-k,l) [or its equivalent operator (-k,-h,-l)]. - Twinning possible with this operator - apparent symmetry for two fold perfect twin would be P622 (operator -h,-k,l)
- All
**P6i**:

(h,k,l) already equivalent to (-h,-k,l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*reindex (h,k,l) to (k,h,-l). - Twinning possible with this operator - apparent symmetry for two fold perfect twin would be
P622 (operator k,k,-l)
space group number space group point group possible twinning operator 168 P6 PG6 k,h,-l 169 P61 PG6 k,h,-l 170 P65 PG6 k,h,-l 171 P62 PG6 k,h,-l 172 P64 PG6 k,h,-l 173 P63 PG6 k,h,-l

- All
**P6i22**:

(h,k,l) already equivalent to (-h,-k,l)*and*(k,h,-l)*and*(-k,-h,-l) so no twinning possible. However a perfect twin for the Laue group, P312, P321 or P6 might appear to have this symmetry. - All
**P2i3**and related**2i3**space groups:

(h,k,l) already equivalent to (-h,-k,l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*reindex (h,k,l) to (k,h,-l). - Twinning possible with this operator - apparent symmetry for two fold perfect twin would be
P43 (operator k,h,-l)
space group number space group point group possible twinning operator 195 P23 PG23 k,h,-l 196 F23 PG23 k,h,-l 197 I23 PG23 k,h,-l 198 P213 PG23 k,h,-l 199 I213 PG23 k,h,-l

- All
**P4i32**and related**4i32**space groups:

(h,k,l) already equivalent to (-h,-k,l)*and*(k,h,-l)*and*(-k,-h,-l) so we do not need to check.

space group number | space group | point group | possible twin operator |
---|---|---|---|

75 | P4 | PG4 | k,h,-l |

76 | P41 | PG4 | k,h,-l |

77 | P42 | PG4 | k,h,-l |

78 | P43 | PG4 | k,h,-l |

79 | I4 | PG4 | k,h,-l |

80 | I41 | PG4 | k,h,-l |

space group number | space group | point group | no twin operators |
---|---|---|---|

89 | P422 | PG422 | none |

90 | P4212 | PG422 | none |

91 | P4122 | PG422 | none |

92 | P41212 | PG422 | none |

93 | P4222 | PG422 | none |

94 | P42212 | PG422 | none |

95 | P4322 | PG422 | none |

96 | P43212 | PG422 | none |

97 | I422 | PG422 | none |

98 | I4122 | PG422 | none |

space group number | space group | point group | possible twinning operator |
---|---|---|---|

149 | P312 | PG312 | -h,-k,l or k,h,-l |

151 | P3112 | PG312 | -h,-k,l or k,h,-l |

153 | P3212 | PG312 | -h,-k,l or k,h,-l |

space group number | space group | point group | possible twinning operator |
---|---|---|---|

150 | P321 | PG321 | -h,-k,l or -k,-h,-l |

152 | P3121 | PG321 | -h,-k,l or -k,-h,-l |

154 | P3221 | PG321 | -h,-k,l or -k,-h,-l |

space group number | space group | point group | no twinning operator |
---|---|---|---|

177 | P622 | PG622 | none |

178 | P6122 | PG622 | none |

179 | P6522 | PG622 | none |

180 | P6222 | PG622 | none |

181 | P6422 | PG622 | none |

182 | P6322 | PG622 | none |

space group number | space group | point group | no twinning operator |
---|---|---|---|

207 | P432 | PG432 | none |

208 | P4232 | PG432 | none |

209 | F432 | PG432 | none |

210 | F4132 | PG432 | none |

211 | I432 | PG432 | none |

212 | P4332 | PG432 | none |

213 | P4132 | PG432 | none |

214 | I4132 | PG432 | none |

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