Twinning
From Media Wiki
Contents |
[edit] Twinning - dealing with data from twinned crystals
Much of this document has been taken from chapter 6 of the SHELX-97 Manual.
[edit] Introduction
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 at least 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 I(twin)hkl and I(twin)h'k'l' where
I(twin)hkl = km Ihkl(crystal1) + (1-km) Ih'k'l'(crystal2) and
I(twin)h'k'l' = (1-km) Ihkl(crystal1) + 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.
When the diffraction patterns from the different domains are completely super-imposable, 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.
[edit] 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.
- The metric symmetry is higher than the Laue symmetry. For example, this occurs for monoclinic crystals where all angles are nearly 90.
- The Rmerge-value for a higher symmetry Laue group is only slightly higher than for the lower symmetry Laue group. For example, the Rmerge-value for Laue group P4/mmm is a little higher than that for P4.
- The mean value for |E2-1| is much lower than the expected value of 0.736 for the non-centrosymmetric case.
- The cumulative intensity distribution appears sigmoidal; ie there are fewer weak reflections than expected. If there are 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.
- Trigonal or hexagonal space group are often twinned.
- There are impossible or unusual systematic absences.
- Although the data appear good, the structure cannot be solved.
- The Patterson function is physically impossible.
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:
- 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(Fo2)/mean(Fc2) is systematically high for the reflections with l
ow intensity.
- For all of the 'most disagreeable' reflections, Fo is much greater than Fc.
[edit] Examples
Example of a cumulative intensity distribution with twinning present, as plotted by TRUNCATE.
[edit] Frequently encountered twin laws
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.
- Orthorhombic with two axes, eg a and b approximately equal in length may emulate tetragonal
- Monoclinic with β approximately 90° may emulate orthorhombic:
- Monoclinic with a and c approximately equal and β approximately 120° may emulate hexagonal [P21/c would give absences and possibly also intensity statistics corresponding to P63].
- Monoclinic with na + nc ~ a or na + nc ~ c can be twinned. See Reindexing.
[edit] Likely twinning operators
Data from a merohedrally twinned crystal can be deconvoluted using the program <a href="sfcheck.html">SFCHECK</a>. This program will assign a likely twinning operator for the spacegroup in question. Commonly occurring operators are listed here.
[edit] 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 α β γ
- Monoclinic - one 2 fold axis of rotation - two angles must be 90; usually α and γ.
- 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).
α and β = 90, γ = 120 ; a = b (hexagonal setting).
- Hexagonal - one 6 fold axis of rotation (plus possible perpendicular 2-fold).
α and β = 90, γ = 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.
[edit] monoclinic
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(β*) = |nc|/2|a|, or, if |a|>|c|, cos(β*) = |na|/2|c|.
[edit] orthorhombic
For orthorhombic crystal forms the only possibility for twinning is if there are two axes with nearly the same length.
[edit] 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.
[edit] 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.
- 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*) |
When more than one dataset, check if reindexing (h,k,l) to (k,h,-l) gives a better match to previous data sets.
- Twinning possible with this operator - apparent Laue symmetry for perfect twin would be P422
| 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 |
- 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.
| 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 |
- 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)
| 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 |
- 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)
| 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 |
- 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.
| 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 |
- 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 | 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 |
[edit] More information on twinning:
Fam and Yeates' Introduction to Hemihedral Twinning, which includes a Twinning test.
[edit] AUTHORS
Prepared for CCP4 by Maria Turkenburg, University of York, England
Acknowledgement in SHELX manual:
" I should like to thank Regine Herbst-Irmer who wrote most of this chapter.
