- 1. Introduction
- 2. A simple model for twinning
- 3. Twinning in crystals
- 4. Diffraction patterns from twinned crystals
- 5. Inversion, merohedral and pseudo-merohedral twins
- 6. Derivation of twin laws
- 7. Examples of twinned crystal structures
- 8. Non-merohedral twinning
- 9. The derivation of non-merohedral twin laws
- 10. Non-merohedral twinning: an example
- References

- 1. Introduction
- 2. A simple model for twinning
- 3. Twinning in crystals
- 4. Diffraction patterns from twinned crystals
- 5. Inversion, merohedral and pseudo-merohedral twins
- 6. Derivation of twin laws
- 7. Examples of twinned crystal structures
- 8. Non-merohedral twinning
- 9. The derivation of non-merohedral twin laws
- 10. Non-merohedral twinning: an example
- References

## CCP4 study weekend

## Introduction to twinning

^{a}School of Chemistry, The University of Edinburgh, King's Buildings, West Mains Road, Edinburgh EH9 3JJ, Scotland^{*}Correspondence e-mail: s.parsons@ed.ac.uk

A twinned crystal is an aggregate in which different domains are joined together according to a specific

the The diffraction patterns derived from different domains are rotated, reflected or inverted with respect to each other, depending on the nature of the relationship between the different domains, and weighted according to the quantity of a particular domain present in the crystal. The diffraction pattern measured during data collection is a superposition of all of these. Reflections from different domains may overlap and twinned crystals fall broadly into two categories in which either all reflections or only certain zones of reflections are affected by overlap. The former occurs when a belongs to a higher than the itself; the latter frequently occurs when the is a belonging to a higher symmetry supercell.Keywords: twinning.

### 1. Introduction

https://www.cryst.chem.uu.nl/lutz/twin/gen_twin.html . The review by Yeates (1997) will be of particular interest to macromolecular crystallographers.

is not an uncommon effect in crystallography, although it has long been considered to be one of the most serious potential obstacles to The use of area detectors has much facilitated the detection of and the treatment of diffraction patterns from twinned crystals. Computer software has also now been developed to such an extent that previously intractable problems have yielded results of comparable precision to those obtained with untwinned samples. Structure determinations from twinned crystals are therefore becoming more common and the aim of this article is to present an introduction to the phenomenon of An extensive database of papers describing has been assembled by Spek and Lutz (Utecht University, The Netherlands) and is available on the internet at### 2. A simple model for twinning

*P*2_{1}/*c* where the unique angle, β, is equal or very close to 90°. In this case, the has 2/*m*, but the lattice has *mmm*. The elements of these point groups are

The important issue is that *mmm* contains symmetry elements which do not occur in 2/*m*. Under these conditions, `mistakes' can occur during crystal growth such that different regions of the crystal (domains) have their unit cells related by symmetry operations which are elements of *mmm* but not 2/*m*: a twofold axis about **a**, for example.

This idea can be illustrated by building up a stack of bricks. The overall shape or outline of a brick is *mmm*, but if we consider the `dent' (brick-layers call this the *frog*) on one side plus the words `London Brick', the is only 2 (the `space group' of this stack of bricks would be *P*2) (Fig. 1*a*). The most obvious way to build a stack of bricks is to place all the bricks in the same orientation, such as in Fig. 1(*b*): notice that the bricks are related to each other by the twofold axes perpendicular to the page or simple translation: both are elements of the However, it is also possible to stack the bricks in such a way that some of the bricks are placed upside down (Fig. 1*c*). The overall shape of the brick, with the 90° angles between the edges, allows this to happen without compromising the stacking of the bricks in any way. In turning some of the bricks upside down, we have used a twofold axis which is a of *mmm* but not 2.

Fig. 1(*b*) is similar to a single crystal; Fig. 1(*c*) resembles a twinned crystal. In Fig. 1(*c*) there are two *domain variants*: bricks (which correspond to unit cells) within the same *domain* are related to each other by translation; bricks in different domains are related by a translation plus a rotation which occurs in the of the outline or overall shape of the brick. This extra corresponds in crystallography to the *twin law*. Had the extra element been chosen to be a mirror plane the mirror image of the words `London Brick' would have appeared in the second domain and it is important to bear this in mind during the analysis of crystals of chiral compounds (such as proteins). In protein crystallography the only possible twin laws are rotation axes. The fraction of the bricks in the alternative orientation corresponds to the *twin scale factor*, which in this example is 0.5.

### 3. in crystals

Monoclinic crystal structures sometimes have β very close to 90°. If occurs, the unit cells in one domain may be rotated by 180° about the *a* or *c* axes relative to those in the other domain in exactly the fashion described above for bricks. However, not all monoclinic crystal structures with β ≃ 90° form twinned crystals: will only be observed if intermolecular interactions across a twin boundary are energetically competitive with those that would have been formed in a single crystal. For this reason, very commonly occurs if a high-symmetry phase of a material undergoes a transition to a lower-symmetry form upon change of temperature or pressure: a `lost' which made certain interactions equivalent in the high-symmetry form can act as a in the low-symmetry form. Layered structures, such as the one shown in Fig. 2 (see also §7), are also often susceptible to if the interactions between layers are rather weak and non-specific; alternative orientations of successive layers are then energetically similar. The total energy difference between intermolecular interactions which occur in a single, as opposed to a twinned, form of a crystal is one factor which controls the value of the domain scale factor, although in practice this may also be controlled kinetically by, for example, the rate of crystal growth.

In the foregoing discussion the impression might have been given that a twinned crystal consists of just two domains. A monoclinic crystal with β ≃ 90° twinned *via* a twofold rotation about **a**, though it contains two domain variants, may actually consist of very many domains. The orientations of the unit cells in any pair of domains will be related either by the identity operator or by the Further examples have been illustrated by Giacovazzo (1992) and other lead references can be found in the article by Koch (1992); an illustrative study of the domain structure of KLiSO_{4} has been given by Klapper *et al.* (1987).

Although the properties of a material (*e.g.* mechanical and optical properties) can depend strongly on domain structure, it is usually not necessary to characterize this for the purposes of ordinary structure analysis. However, the twin scale factor may appear to vary when different regions of a crystal are sampled during data collection. This can give rise to powerful non-isomorphism effects, as discussed by Terwisscha van Scheltinga *et al.* (2003).

### 4. Diffraction patterns from twinned crystals

Each domain of a twinned crystal gives rise to its own diffraction pattern; what is measured on a diffractometer is a superposition of all these patterns with intensities weighted according to the domain scale factors. The relative orientations of the diffraction patterns from different domains are the same as the relative orientations of the domains, so that if they are related by a 180° rotation about **a**, then so too are their diffraction patterns. Fig. 3 shows this for a twinned monoclinic for which β = 90°. is a problem in crystallography because it causes superposition or overlap between reflections which are not related by symmetry. In Fig. 3(*c*), the reflection which would have been measured with indices 102 is actually a superposition of the 102 reflection from domain 1 (Fig. 3*a*) and the 02 reflection from domain 2 (Fig. 3*b*). During structure analysis of a twinned crystal, it is important to define exactly which reflections contribute to a given intensity measurement: this is the role of the twin law.

In order to treat , the twofold axis about the *a* axis will transform **a** into **a**, **b** into −**b** and **c** into −**c**. This is the transformation between the cells in different domains of the crystal; written as a matrix this is

The same matrix relates the indices of pairs of overlapping reflections,^{1}

This two-component twin can be modelled using a quantity |*F*_{twin,calc}|^{2}, which is a linear combination (equation 1, Pratt *et al.*, 1971) consisting of |*F*|^{2} terms for each component reflection weighted according to the twin scale factor, *x*, which can be refined,

In the single-crystal reciprocal-lattice plots shown in Figs. 3(*a*) and 3(*b*), although twofold axes and mirror planes about the *a** and *c** axes relate the *positions* of the spots, this symmetry is not expressed in the *intensities* of the spots (for example, the 02 and 102 reflections have different intensities in Fig. 3*a*). However, in the composite twinned pattern (Fig. 3*c*), both the positions *and* the intensities of the spots exhibit the same mirror or twofold symmetry with respect to both of these axes. The composite pattern with equal domain volumes (that is *x* = 0.5; Fig. 3*c*) appears to have orthorhombic diffraction symmetry even though the is monoclinic. In general, for a two-component twin, if *x* is near 0.5 then merging statistics will appear to imply higher than that possessed by the As *x* deviates from 0.5, then the merging in the higher-symmetry gradually becomes poorer relative to merging in the which corresponds to the of the (Fig. 3*d*). Although it is impossible to give a definitive range, merging in the higher-symmetry may yield a merging residual of 0.10–0.35; values of around 0.60 might be expected for untwinned samples (although in, for example, heavy-atom positions can give rise to a similar effect).

Another striking feature of the twinned diffraction pattern shown in Fig. 3(*c*) is that it appears to have a more acentric intensity distribution than the component patterns. The superposition of the diffraction patterns arising from the different domains tends to average out intensities because strong and weak reflections sometimes overlap. The quantity |*E*^{2} − 1|, which adopts values of 0.97 and 0.74 for ideal centric and acentric distributions, respectively, may assume a value in the range 0.4–0.7 for twinned crystal structures. Intensity statistics can therefore be a valuable tool for the diagnosis of although it is important to bear in mind all the usual caveats relating to the assumption of a random distribution of atoms, which is not appropriate, for example, in the presence of heavy atoms or (Alexeev, 2003). Rees (1980) has shown that an estimate of the twin scale factor, *x*, can be derived from the value of |*E*^{2} − 1|. Other procedures have been developed by Britton (1972) and Yeates (1988) and these have been compared by Kahlenberg (1999). The latter statistical tests will fail, however, for twins with *x* near 0.5. If the value of *x* is known and is not near 0.5, (equation 1) can be used to `detwin' a data set. This procedure may be useful for the purposes of structure solution, although it is generally preferable to refine against the original twinned data set.

Common signs of , 2002). Additional signs to those described above include an inability to solve a structure even though the data appear to be of good quality or, if a structure can be solved, a high *R* factor or a noisy inexplicable difference electron-density map. also reveals itself in the and this is discussed by Dauter (2003) and Yeates (1997). Since reflections from one domain may overlap with from another, the observed may either not be consistent with any known or appear to imply a very rare Low-symmetry trigonal and hexagonal crystal structures appear to be particularly susceptible to It was pointed out by a referee to this paper that because makes the appear higher than it actually is, there seem to be more molecules in the than is actually the case, so that an unreasonably high packing density can also be taken to be a useful warning sign of in macromolecular crystallography.

### 5. Inversion, and pseudo-merohedral twins

*P*2_{1} may contain enantiomorphic domains (Flack, 2003). This type of does not occur for an compound and it can therefore be ruled out in protein crystallography. The in this case is the inversion operator,

This kind of ). The domain scale factor in this case is referred to as the Flack parameter.

does not hamper structure solution and is most commonly encountered in Flack's method for determination (Flack, 1983Twinning may also occur in lower symmetry tetragonal, trigonal and cubic systems. Thus, a tetragonal structure in *m* may twin about the twofold axis along [110], which is a of the higher-symmetry tetragonal 4/*mmm*. The in this case is

this matrix may also be used in the treatment of low-symmetry trigonal, hexagonal and cubic crystal structures, producing diffraction patterns with apparent *m*1, 6/*mmm* and *m**m* symmetry, respectively, when the domain scale factor *x* is 0.5.

Two further twin laws need to be considered in low-symmetry trigonal crystals. A twofold rotation about [10], mimicking *m* when *x* = 0.5, is expressed by the matrix

By *via* a twofold axis about [001], a trigonal crystal may also appear from merging statistics to be hexagonal if *x* = 0.5. The in this case is

In rhombohedral crystal structures,

of this type leads to obverse–reverse twinning.The point groups of the crystal lattices ( for triclinic, 2/*m* for monoclinic, *mmm* for orthorhombic, 4/*mmm* for tetragonal, *m* for rhombohedral, 6/*mmm* for hexagonal and *m**m* for cubic) are referred to as the *holohedral* point groups. Those point groups which belong to the same but which are subgroups of relevant holohedral are referred to as *merohedral* point groups (this classification is discussed in detail by Hahn & Klapper, 1996). Thus, 4/*m* is a of 4/*mmm*. With the exception of obverse–reverse (see below), in all the cases described in the previous paragraphs in this section the was a of the relevant (*i.e.* of the crystal lattice) which was not expressed in the corresponding to the For this reason, this type of phenomenon is referred to as *twinning by merohedry*. Such twins are often described as *merohedral* and although this usage is occasionally criticized in the literature (Catti & Ferraris, 1976), it appears to have stuck.^{2} Though it is quite rare in molecular crystals, twins containing more than two domain variants are sometimes observed (see §6); more commonly only two are present, however, and such twins are also described as *hemihedral* twins.

Twinning by 4 where a monoclinic accidentally had a β angle near 90°; for example, there is nothing `accidental' about a low-symmetry tetragonal structure having a lattice with symmetry 4/*mmm*: all low-symmetry tetragonal structures have this property. Put another way, the of the tetragonal lattice is 4/*mmm*; the low-symmetry tetragonal structure might belong to 4/*m*, 4 or , which are all nevertheless still tetragonal point groups; this is what would make this by merohedry.

A trigonal *via* a twofold axis about the [001] direction (parallel to the threefold axis), because this is a of the 6/*mmm* However, the rhombohedral lattice is *m* and this does not contain a twofold axis parallel to the threefold axis. Although *via* a twofold axis in this direction can certainly occur for rhombohedral crystal structures, it is not Instead, it is referred to as obverse–reverse or by *reticular merohedry*; this is an important distinction because overlap between reflections from different domain variants in obverse–reverse twins only affects a third of the intensity data. This has recently been discussed in detail by Herbst-Irmer & Sheldrick (2002).

A monoclinic β ≃ 90° has a lattice with, at least approximately, the *mmm* symmetry characteristic of the orthorhombic If occurs by a twofold axis about **a** or **c**, the crystal is not merohedrally twinned, since monoclinic and orthorhombic are two different crystal families. This type of effect is instead referred to as by *pseudo-merohedry*. A further example might occur in an orthorhombic crystal where two sides (*b* and *c*, say) are of equal length (pseudo-tetragonal). The in this case could be a fourfold axis about **a**,

A monoclinic crystal where *a* ≃ *c* and β ≃ 120° may be twinned by a threefold axes about **b**. The clockwise and anticlockwise threefold rotations (3^{+} and 3^{−}) about this direction are

potentially yielding a three-component pseudo-merohedral twin appearing from the diffraction symmetry to be hexagonal.

Note that higher symmetry may be `hidden' in a centred setting of a ) have described two illustrations of this].

and not be immediately obvious from the cell dimensions and it is necessary to inspect carefully the output from whichever program has been used to check the metric symmetry of the [Herbst-Irmer & Sheldrick (1998### 6. Derivation of twin laws

In §4 the case of a monoclinic crystal where β ≃ 90° was examined and it was shown that could occur about a twofold axis about **a**. This leads to overlap between reflections with indices *hkl* and *h*. *via* a twofold axis about **c** would lead to overlap between reflections with indices *hkl* and *l*. However, since reflections *h* and *l* are related by the monoclinic twofold axis about **b*** which must be present if the crystal is 2 or 2/*m*, these twin laws are equivalent. However, in the about two threefold axes described in §5 for a monoclinic crystal with *a* ≃ *c* and β ≃ 120°, the rotations are not equivalent because they are not related by any of the symmetry operations of 2/*m*.

It is usually the case that several equivalent descriptions may be used to describe a particular twin. However, several distinct twin laws may be possible and they can be expressed simultaneously. There clearly exists a potential for possible twin laws to be overlooked during structure analysis. Flack (1987) has described the application of *coset decomposition* to this problem, enabling this danger to be systematically avoided. The procedure has been incorporated by Litvin into the computer program *TWINLAWS* (Schlessman & Litvin, 1995).^{3}

Suppose that a *G* crystallizes in a lattice with a higher point-group symmetry *H*. The number of possible twin laws is given by (*h _{H}*/

*h*) − 1, where

_{G}*h*

_{G}and

*h*

_{H}are the respective orders of point groups

*G*and

*H*(that is, the number of symmetry operations they contain). For example, in a protein crystallizing in 2 (space group

*P*2,

*C*2 or

*P*2

_{1}) with a with parameters

*a*= 30.5,

*b*= 30.5,

*c*= 44.9 Å, β = 90.02°,

*G*is 2 and

*H*is effectively 422 (4/

*mmm*in principle, but mirror symmetry is not permitted for an protein crystal). The orders of

*G*and

*H*are 2 and 8, respectively, and so this crystal may suffer from up to three twin laws to form, at most, a twin with four domain variants (the reference domain plus three others).

Coset decomposition yields the symmetry elements which must be added to *G* to form the higher *H*. Table 1 shows the output of the program *TWINLAWS*, listing decomposition of 422 into cosets with 2. Possible twin laws are twofold axes about the [100], [10] and [110] directions. However, the twofold rotation about [110] is an equivalent to the 4^{−} (*i.e.* the 4^{3}) rotation about [001] and the twofold axis about [100] is equivalent to that about [001].

‡This is a 4 ^{−} or 4^{3} rotation about [001]. §This is a twofold rotation about [110]. |

The 5, which apply to trigonal crystals. A trigonal in *P*3, *P*3_{1} or *P*3_{2} belongs to 3 (order 3), but is built on a lattice with 6/*mmm* symmetry (order 24). Twins with eight domain variants are possible in this case! The twin laws are formed by twofold rotations about [110], [10] and [001], the inversion operator and mirror planes perpendicular to [110], [10] and [001]. Of course, the last four of these can be ruled out in crystals of chiral compounds and it should be stressed that such complicated multiple is rather uncommon in molecular crystals.

### 7. Examples of twinned crystal structures

Crystals of the compound C_{30}H_{27}N (1) (Fig. 2) diffracted rather weakly. The appeared to be orthorhombic with parameters *a* = 8.28, *b* = 12.92, *c* = 41.67 Å. The volume here fits for *Z* = 8 and the value of |*E*^{2} − 1| was 0.725. None of this is especially unusual, although the assuming orthorhombic symmetry appeared to be *P*22_{1}2, which is rare. Merging statistics (*R*_{int}), were as follows: *mmm*, 0.14; 2/*m*, *a* unique, 0.13; 2/*m*, *b* unique, 0.06; 2/*m*, *c* unique, 0.09. The lowest *R*_{int} indicated monoclinic symmetry with the *b* axis of the orthorhombic cell corresponding to the unique axis of the monoclinic cell. Taken with the space-group information described above this seemed to be a twin. The used was

and *P*2_{1} was assumed. The structure did not solve by but a position and orientation for one molecule (there are four in the asymmetric unit) was obtained by Patterson search methods (*DIRDIF*; Beurskens *et al.*, 1996) using the rigid part of the molecule as a search fragment. The structure was completed by iterative cycles of least squares and Fourier syntheses (*SHELXL*97; Sheldrick, 1997). A search for missed space-group symmetry did not reveal any glide or mirror planes: the final *R* factor was 10%.

Further examples, more relevant to macromolecular ) and Terwisscha van Scheltinga *et al.* (2003). Worked examples for several problems have been assembled by Herbst-Irmer and are available from https://shelx.uni-ac.gwdg.de/~rherbst/twin.html .

### 8. Non-merohedral twinning

In *h*, *k* and/or *l* are affected by twinning.

A non-merohedral *a* ≃ *b* (Fig. 4). A metrically tetragonal can be formed by doubling the length of *a* so that there is a pseudo-fourfold axis about **c**. The diffraction pattern from one domain of the crystal is related to that from the other by a 90° rotation about **c***. Superposition of the two diffraction patterns shows that data from the first domain are only affected by overlap with data from the second domain when *k* is even (Fig. 4*d*). For the purposes of structure analysis, this needs to be expressed with respect to the axes of the true orthorhombic cell. From Fig. 4(*a*), **a**′ = −0.5**b**, **b**′ = 2**a**, **c**′ = **c**, so that the is

The effect of this matrix on the data is

confirming that only data with *k* = 2*n* are affected by the Thus, the 143 reflection from the first domain (green) is overlapped with the −223 reflection from the second (blue) domain. The 413 reflection in the green domain would be unaffected by twinning.

It is likely that the example given here would index readily on the tetragonal (*d*). Zones of unusual are frequently a sign that a crystal is non-merohedrally twinned. This pseudotranslational symmetry should enable the true orthorhombic cell to be inferred and it can be characterized in a Patterson synthesis.

In orthorhombic and higher systems, potential non-merohedral twins laws can often be derived from inspection of the unit-cell parameters. In low-symmetry crystals the *a* or *c* axes. The matrix for a twofold rotation about the *a* axis is

The corresponding rotation about *c* is

Likely twin laws can be derived for monoclinic crystals by evaluating the off-diagonal terms in these matrices; if near-rational values are obtained, the corresponding matrix should be investigated as a possible twin law.

### 9. The derivation of non-merohedral twin laws

Diffraction patterns from non-merohedrally twinned crystals contain many more spots than would be observed for an untwinned sample. Since individual spots may come from different domains of the twin, such diffraction patterns are frequently difficult to index. Overlap between reflections may be imperfect in some or all zones of data affected and integration and data reduction needs to be performed carefully. Software for integrating data sets from non-merohedral twins and performing absorption corrections has recently become available [for example, *SAINT* (Bruker-Nonius, 2002); *EVAL*-14 (Duisenberg *et al.*, 2003) and *TWINABS* (Sheldrick, 2002)].

Excellent programs such as *DIRAX* (Duisenberg, 1992) and *GEMINI* (Sparks, 1999) have been developed to index diffraction patterns from non-merohedral twins. In many cases a pattern can be completely indexed with two orientation matrices and both these programs offer procedures by which the relationship between these alternative matrices is analysed to suggest a It is usually the case that can be described by a twofold rotation about a direct or reciprocal-lattice direction. Indeed, if two such directions are parallel and the vectors describing them have a dot product of greater than two, then a higher-symmetry can be derived. The program *CREDUC* (Le Page, 1982) is extremely useful for investigating this; it is available in the *Xtal* suite of software (Hall *et al.*, 1992), which can be downloaded from https://www.ccp14.ac.uk ; the algorithm is also used in the *LEPAGE* routine in *PLATON* (Spek, 2003). A related program, called *OBLIQUE*, has recently been described by Le Page (2002).

It is sometimes the case that the first intimation that the analyst has that a crystal is twinned is during *R* factor may indicate that is a problem, while careful analysis of poorly fitting data reveals that they belong predominantly to certain distinct zones in which |*F*_{obs}|^{2} is systematically larger than |*F*_{calc}|^{2}. If is not taken into account, it is likely that these zones are being poorly modelled and that trends in their indices may provide a clue to a possible The computer program *ROTAX* (Cooper *et al.*, 2002; also available from https://www.ccp14.ac.uk ) makes use of this idea to identify possible twins laws. A set of data with the largest values of [|*F*_{obs}|^{2} − |*F*_{calc}|^{2}]/*u*(|*F*_{obs}|^{2}) (where *u* is the of |*F*_{obs}|^{2}) is identified and the indices transformed by twofold rotations or other symmetry operations about possible direct and reciprocal-lattice directions. Matrices which transform the indices of the poorly fitting data to integers are identified as possible twin laws. The analyst then has a set of potential matrices which might explain the source of the problems described above. A related procedure is available in the TwinRotMat routine in *PLATON*, but this uses a set of reflections from planes with similar *d* spacings rather than high values of [|*F*_{obs}|^{2} − |*F*_{calc}|^{2}]/*u*(|*F*_{obs}|^{2}).

### 10. Non-merohedral an example

Crystals of (2) grew as coaxially aligned aggregates of needles (Smith, 2000). The diffraction pattern was completely indexed using*DIRAX*using two orientation matrices. Some reflections in the search list gave integral indices with only one of these matrices, while others could be satisfactorily indexed with both. The was monoclinic

*P*,

*a*= 7.28,

*b*= 9.74,

*c*= 15.23 Å, β = 94.39°,

*P*2

_{1}/

*n*. We describe below how the program

*ROTAX*can be used to evaluate the however, the same matrix could also be obtained with the program 2

*VIEW*, which is part of the

*DIRAX*system. Data were collected using one of the matrices derived by

*DIRAX*. The structure was solved by (

*DIRDIF*) and (

*CRYSTALS*; Watkin

*et al.*, 2002) converged to

*R*= 14.8%, with anisotropic displacement parameters and all H atoms except that attached to the hydroxyl group placed in calculated positions. The difference synthesis maximum of 1 e Å

^{−3}was close to the S atom, but in a chemically implausible position.

The program *ROTAX* has been incorporated into the *CRYSTALS* suite, the application and of a model being facilitated by a GUI. All poorly fitting data had *h* = 3*n* and the was readily identified as the matrix

which corresponds to a twofold axis about the [106] *hkl* with *h* = 3*n* from one domain with reflections −*h*, −*k*, (*h*/3 + *l*) from the other domain. Incorporation of this into the cleaned up the difference map considerably, allowing the hydroxyl H atom to be located. The final *R* factor was 5.16%, with a twin scale factor of 0.437 (3).

In the case of this crystal, a

can be obtained by application of the matrixwhich transforms the true monoclinic cell to a metrically orthorhombic cell with dimensions *a* = 7.29, *b* = 9.74, *c* = 91.12 Å. It is important to attempt to identify the metric symmetry of the because the fact that this cell is orthorhombic and no higher implies that there is no need to search for further twin laws. Decomposition of *mmm* into cosets with 2/*m* yields the identity operator and either a twofold axis about **a** or **c** of the which has, of course, already been identified.

Further examples of non-merohedral ), Choe *et al.* (2000), Colombo *et al.* (2000), Gaudin *et al.* (2000), Guelylah *et al.* (2001), Cooper *et al.* (2002) and Tang *et al.* (2001). A worked example (Herbst-Irmer & Sheldrick, 1998) is available from https://shelx.uni-ac.gwdg.de/~rherbst/twin.html .

*Note added in proof*: Padilla & Yeates (2003) have very recently described a statistical test for the detection of hemihedral twinning.

### Footnotes

^{1}Here, the triple *hkl* is represented as a column vector; if it is treated as a row vector (as it is in some software packages), the twin matrices discussed in this paper should be transposed.

^{2}*Holo* and *mero* are Greek stems meaning whole and part, respectively. This `French School' nomenclature was originally devised to describe crystal morphology and is used here because it is currently popular in the literature. Different nomenclature is also encountered; see, for example, Giacovazzo (1992) or van der Sluis (1989).

^{3}This program is available free of charge to academic users from https://www.bk.psu.edu/faculty/litvin/Download.html or *via* the CCP14 web site (https://www.ccp14.ac.uk ).

### Acknowledgements

It is a pleasure to thank Richard Cooper, Howard Flack, Bob Gould and David Watkin for a very fruitful collaboration on

Howard Flack deserves particular thanks for reading and commenting on the manuscript of this paper. Funding by the EPSRC and The University of Edinburgh is also gratefully acknowledged.### References

Alexeev, D. (2003). *CCP4 Study Weekend on Experimental Phasing*, January 2003, York, England. Google Scholar

Beurskens, P. T., Beurskens, G., Bosman, W. P., de Gelder, R., Garcia-Granda, S., Gould, R. O., Israel, R. & Smits, J. M. M. (1996). *The DIRDIF96 Program System.* Technical Report of the Crystallography Laboratory, University of Nijmegen, The Netherlands. Google Scholar

Britton, D. (1972). *Acta Cryst.* A**28**, 296–297. CrossRef IUCr Journals Web of Science Google Scholar

Bruker-Nonius (2002). *SAINT*, version 7. Bruker-Nonius, Madison, Wisconsin, USA. Google Scholar

Catti, M. & Ferraris, G. (1976). *Acta Cryst.* A**32**, 163–165. CrossRef IUCr Journals Web of Science Google Scholar

Choe, W., Pecharsky, V. K., Pecharsky, A. O., Gschneidner, K. A., Young, V. G. & Miller, G. J. (2000). *Phys. Rev. Lett.* **84**, 4617–4620. Web of Science CrossRef PubMed CAS Google Scholar

Colombo, D. G., Young, V. G. & Gladfelter, W. L. (2000). *Inorg. Chem.* **39**, 4621–4624. Web of Science CrossRef CAS Google Scholar

Cooper, R. I., Gould, R. O., Parsons, S. & Watkin, D. J. (2002). *J. Appl. Cryst.* **35**, 168–174. Web of Science CrossRef CAS IUCr Journals Google Scholar

Dauter, Z. (2003). *Acta Cryst.* D**59**, 2004–2016. Web of Science CrossRef CAS IUCr Journals Google Scholar

Duisenberg, A. J. M. (1992). *J. Appl. Cryst.* **25**, 92–96. CrossRef CAS Web of Science IUCr Journals Google Scholar

Duisenberg, A. J. M., Kroon-Batenburg, L. M. J. & Schreurs, A. M. M. (2003). *J. Appl. Cryst.* **36** 220–229. Web of Science CrossRef CAS IUCr Journals Google Scholar

Flack, H. D. (1983). *Acta Cryst.* A**39**, 876–881. CrossRef CAS Web of Science IUCr Journals Google Scholar

Flack, H. D. (1987). *Acta Cryst.* A**43**, 564–568. CrossRef CAS Web of Science IUCr Journals Google Scholar

Flack, H. D. (2003). *Helv. Chim. Acta*, **86**, 905–921. Web of Science CrossRef CAS Google Scholar

Gaudin, E., Petricek, V., Boucher, F., Taulelle, F. & Evain, M. (2000). *Acta Cryst.* B**56**, 972–979. Web of Science CrossRef CAS IUCr Journals Google Scholar

Giacovazzo, C. (1992). Editor. *Fundamentals of Crystallography.* Oxford University Press. Google Scholar

Guelylah, A., Madariaga, G., Petricek, V., Breczewski, T., Aroyo, M. I. & Bocanegra, E. H. (2001). *Acta Cryst.* B**57**, 221–230. Web of Science CrossRef CAS IUCr Journals Google Scholar

Hahn, T. & Klapper, H. (1996). *International Tables for Crystallography*, Vol. *A*, edited by T. Hahn, ch. 10. Dordrecht: Kluwer Academic Publishers. Google Scholar

Hall, S. R., Flack, H. & Stewart, R. F. (1992). *Xtal*3.2. Universities of Western Australia, Australia, Geneva, Switzerland and Maryland, USA. Google Scholar

Herbst-Irmer, R. & Sheldrick, G. M. (1998). *Acta Cryst.* B**54**, 443–449. Web of Science CSD CrossRef CAS IUCr Journals Google Scholar

Herbst-Irmer, R. & Sheldrick, G. M. (2002). *Acta Cryst.* B**58**, 477–481. Web of Science CrossRef CAS IUCr Journals Google Scholar

Kahlenberg, V. (1999). *Acta Cryst.* B**55**, 745–751. Web of Science CrossRef CAS IUCr Journals Google Scholar

Klapper, H., Hahn, T. & Chung, S. J. (1987). *Acta Cryst.* B**43**, 147–159. CrossRef CAS Web of Science IUCr Journals Google Scholar

Koch, E. (1992). *International Tables for Crystallography*, Vol. *C*, edited by A. J. C. Wilson, pp. 10–14. Dordrecht: Kluwer Academic Publishers. Google Scholar

Le Page, Y. (1982). *J. Appl. Cryst.* **15**, 255–259. CrossRef CAS Web of Science IUCr Journals Google Scholar

Le Page, Y. (2002). *J. Appl. Cryst.* **35**, 175–181. Web of Science CrossRef CAS IUCr Journals Google Scholar

Pratt, C. S., Coyle, B. A. & Ibers, J. A. (1971). *J. Chem. Soc.* 2146–2151. Google Scholar

Padilla, J. E. & Yeates, T. O. (2003). *Acta Cryst.* D**59**, 1124–1130. Web of Science CrossRef CAS IUCr Journals Google Scholar

Rees, D. C. (1980). *Acta Cryst.* A**36**, 578–581. CrossRef CAS IUCr Journals Web of Science Google Scholar

Schlessman, J. & Litvin, D. B. (1995). *Acta Cryst.* A**51**, 947–949. CrossRef CAS Web of Science IUCr Journals Google Scholar

Sheldrick, G. M. (1997). *SHELXTL*97. University of Göttingen, Germany. Google Scholar

Sheldrick, G. M. (2001). *XPREP.* Bruker-AXS, Madison, Wisconsin, USA. Google Scholar

Sheldrick, G. M. (2002). *TWINABS.* Bruker-AXS, Madison, Wisconsin, USA. Google Scholar

Smith, A. (2000). PhD thesis, The University of Edinburgh, Edinburgh, Scotland. Google Scholar

Sparks, R. A. (1999). *GEMINI.* Bruker AXS, Madison, Wisconsin, USA. Google Scholar

Spek, A. L. (2003). *PLATON: A Multipurpose Crystallographic Tool.* Utrecht University, Utrecht, The Netherlands. Google Scholar

Tang, C. Y., Coxall, R. A., Downs, A. J., Greene, T. M. & Parsons, S. (2001). *J. Chem. Soc. Dalton Trans.*, pp. 2141–2147. Google Scholar

Terwisscha Van Scheltinga, A. C., Valegård, K., Hadju, J. & Andersson, I. (2003). *Acta Cryst.* D**59**, 2017–2022. Web of Science CrossRef CAS IUCr Journals Google Scholar

Van der Sluis, P. (1989). Thesis, University of Utrecht, Utrecht, The Netherlands. Google Scholar

Watkin, D. J., Prout, C. K., Carruthers, J. R., Betteridge, P. W. & Cooper, R. I. (2002). *CRYSTALS.* Issue 11. Chemical Crystallography Laboratory, University of Oxford, England. Google Scholar

Yeates, T. O. (1988). *Acta Cryst.* A**44**, 142–144. CrossRef CAS Web of Science IUCr Journals Google Scholar

Yeates, T. O. (1997). *Methods Enzymol.* **276**, 344–358. CrossRef CAS PubMed Web of Science Google Scholar

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