International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.3, pp. 419423
Section 3.3.4. Composite symmetry and the twin law^{a}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D52056 Aachen, Germany, and ^{b}MineralogischPetrologisches Institut, Universität Bonn, D53113 Bonn, Germany 
In this section we turn our attention to the symmetry relations in twinning. The starting point of all symmetry considerations is the eigensymmetry of the untwinned crystal, i.e. the point group or space group of the single crystal, irrespective of its orientation and location in space. All domain states of a twinned crystal have the same (or the enantiomorphic) eigensymmetry but may exhibit different orientations. The orientation states of each two twin components are related by a twin operation k which cannot be part of the eigensymmetry . The term eigensymmetry is introduced here in order to provide a short and crisp distinction between the symmetry of the untwinned crystal (singledomain state) and the composite symmetry of a twinned crystal, as defined below. It should be noted that in crystal morphology the term eigensymmetry is also used, but with another meaning, in connection with the symmetry of face forms of crystals (Hahn & Klapper, 2005, pp. 764 and 791).
For a comprehensive characterization of the symmetry of a twinned crystal, we introduce the important concept of composite symmetry . This symmetry is defined as the extension of the eigensymmetry group by a twin operation k. This extension involves, by means of left (or right) coset composition , the generation of further twin operations until a supergroup is obtained. This supergroup is the composite symmetry group .
In the language of group theory, the relation between the composite symmetry group and the eigensymmetry group can be expressed by a (left) coset decomposition of the supergroup with respect to the subgroup : where is the identity operation; .
The number i of cosets, including the subgroup , is the index [i] of in ; this index corresponds to the number of different orientation states in the twinned crystal. If is a normal subgroup of , which is always the case if i = 2, then , i.e. left and right coset decomposition leads to the same coset. The relation that the number of different orientation states n equals the index [i] of in , i.e. , was first expressed by Zheludev & Shuvalov (1956, p. 540) for ferroelectric phase transitions.
These grouptheoretical considerations can be translated into the language of twinning as follows: although the eigensymmetry and the composite symmetry can be treated either as point groups (finite order) or space groups (infinite order), in this and the subsequent sections twinning is considered only in terms of point groups [see, however, Note (8) in Section 3.3.2.4, as well as Section 3.3.10.4]. With this restriction, the number of twin operations in each coset equals the order of the eigensymmetry point group . All twin operations in a coset represent the same orientation relation, i.e. each one of them transforms orientation state 1 into orientation state 2. Thus, the complete coset characterizes the orientation relation comprehensively and is, therefore, defined here as the twin law. The different operations in a coset are called alternative twin operations. A further formulation of the twin law in terms of black–white symmetry will be presented in Section 3.3.5. Many examples are given in Section 3.3.6.
This extension of the `classical' definition of a twin law from a single twin operation to a complete coset of alternative twin operations does not conflict with the traditional description of a twin by the one morphologically most prominent twin operation. In many cases, the morphology of the twin, e.g. reentrant angles or the preferred orientation of a composition plane, suggests a particular choice for the `representative' among the alternative twin operations. If possible, twin mirror planes are preferred over twin rotation axes or twin inversion centres.
The concept of the twin law as a coset of alternative twin operations, defined above, has been used in more or less complete form before. The following authors may be quoted: Mügge (1911, pp. 23–25); Tschermak & Becke (1915, p. 97); Hurst et al. (1956, p. 150); Raaz & Tertsch (1958, p. 119); Takano & Sakurai (1971); Takano (1972); Van Tendeloo & Amelinckx (1974); Donnay & Donnay (1983); Zikmund (1984); Wadhawan (1997, 2000); Nespolo et al. (2000). A systematic application of left and double coset decomposition to twinning and domain structures has been presented by Janovec (1972, 1976) in a key theoretical paper. An extensive grouptheoretical treatment with practical examples is provided by Flack (1987).
Example: dovetail twin of gypsum (Fig. 3.3.4.1). Eigensymmetry:Twin reflection plane (100):Composite symmetry group (orthorhombic):given in orthorhombic axes, . The coset contains all four alternative twin operations (Table 3.3.4.1) and, hence, represents the twin law. This is clearly visible in Fig. 3.3.6.3(a). In the symbol of the orthorhombic composite group, the primed operations indicate the coset of alternative twin operations. The above blackandwhite symmetry symbol of the (orthorhombic) composite group is another expression of the twin law. Its notation is explained in Section 3.3.5. The twinning of gypsum is treated in more detail in Example 3.3.6.3.

It should be noted that among the four twin operations of the coset two are rational, and , and two are irrational, and (Fig. 3.3.4.1). All four are equally correct descriptions of the same orientation relation. From morphology, however, preference is given to the most conspicuous one, the twin mirror plane , as the representative twin element.
The concept of composite symmetry is not only a theoretical tool for the extension of the twin law but has also practical aspects:
In the example of the dovetail twin of gypsum above, the twin operation is of a special nature in that it maps the entire eigensymmetry onto itself and, hence, generates a single coset, a single twin law and a finite composite group of index [2] (simple twin). There are other twin operations, however, which do not leave the entire eigensymmetry invariant, but only a part (subgroup) of it, as shown for the hypothetical (111) twin reflection plane of gypsum in Example 3.3.6.3. In this case, extension of the complete group by such a twin operation k does not lead to a single twin law and a finite composite group, but rather generates in the same coset two or more twin operations which are independent (nonalternative) but symmetry equivalent with respect to the eigensymmetry , each representing a different but equivalent twin law. If applied to the `starting' orientation state 1, they generate two or more new orientation states 2, 3, 4, . In the general case, continuation of this procedure would lead to an infinite set of domain states and to a composite group of infinite order (e.g. cylinder or sphere group). Specialized metrics of a crystal can, of course, lead to a `multiple twin' of small finite order.
In order to overcome this problem of the `infinite sets' and to ensure a finite composite group (of index [2]) for a pair of adjacent domains, we consider only that subgroup of the eigensymmetry which is left invariant by the twin operation k. This subgroup is the `intersection symmetry' of the two `oriented eigensymmetries' and of the domains 1 and 2 (shown in Fig. 3.3.4.2): . This group is now extended by k and leads to the `reduced composite symmetry' of the domain pair (1, 2): , which is a finite supergroup of of index [2]. In this way, the complete coset of the eigensymmetry is split into two (or more) smaller cosets , etc., where are symmetryequivalent twin operations in . Correspondingly, the differently oriented `reduced composite symmetries' , etc. of the domain pairs (1, 2), (1, 3) etc. are generated by the representative twin operations , etc. These cosets are considered as the twin laws for the corresponding domain pairs.
As an example, an orthorhombic crystal of eigensymmetry with equivalent twin reflection planes and is shown in Fig. 3.3.4.2. From the `starting' domain 1, the two domains 2 and 3 are generated by the two twin mirror planes and , symmetry equivalent with respect to the oriented eigensymmetry of domain 1. The intersection symmetries of the two pairs of oriented eigensymmetries & and & are identical: . The three oriented eigensymmetries , , , as well as the two differently oriented reduced composite symmetries and of the domain pairs (1, 2) and (1, 3), are all isomorphic of type , but exhibit different orientations.
The discussions and examples briefly presented in the previous section are now extended in a more general way. For the classification of composite symmetries we introduce the notion of oriented eigensymmetry of an orientation state j and attach to it its geometric representation, the framework of oriented eigensymmetry elements, for short framework of oriented eigensymmetry. Twin partners of different orientation states have the same eigensymmetry but exhibit different oriented eigensymmetries , which are geometrically represented by their frameworks of oriented eigensymmetry. The well known crystallographic term `framework of symmetry' designates the spatial arrangement of the symmetry elements (planes, axes, points) of a point group or a space group, as represented by a stereographic projection or by a spacegroup diagram (cf. Hahn, 2005, Parts 6 , 7 and 10 ).
Similarly, we also consider the intersection group of the oriented eigensymmetries and and its geometric representation, the framework of intersection symmetry. Two cases of intersection symmetries have to be distinguished:
The orthorhombic example given in Section 3.3.4.1 (Fig. 3.3.4.2) is now extended as follows:
Eigensymmetry , intersection symmetry , identity, , , . The two cosets and are listed in Table 3.3.4.2. From these cosets the two different reduced composite symmetries and of type 2/m 2/m 2/m are derived as follows:

These groups of reduced composite symmetry are always crystallographic and finite.
Note that the twin operations in these two reduced cosets would form one coset if one of the operations ( or ) were applied to the full eigensymmetry (twice as long as ): . This process, however, would not result in a finite group, whereas the two reduced cosets lead to groups of finite order.
The two twin laws, based on and , can be expressed by a black–white symmetry symbol of type with . The frameworks of these two groups, however, are differently oriented (cf. Fig. 3.3.4.2).
In the limiting case, the intersection group consists of the identity alone (index [i] = order of the eigensymmetry group), i.e. the two frameworks of oriented eigensymmetry have no symmetry element in common. The number of equivalent twin laws then equals the order of the eigensymmetry group, and each coset consists of one twin operation only.
After this preparatory introduction, the three categories of composite symmetry are treated.
It is emphasized that the considerations of this section apply not only to the particularly complicated cases of multiple growth twins but also to transformation twins resulting from the loss of higherorder rotation axes that is accompanied by a small metrical deformation of the lattice. As a result, the extended composite symmetries of the transformation twins resemble the symmetry of their parent phase. The occurrence of both multiple growth and multiple transformation twins of orthorhombic pseudohexagonal K_{2}SO_{4} is described in Example 3.3.6.8.
Remark. It is possible to construct multiple twins that cannot be treated as a cyclic sequence of binary twin elements. This case occurs if a pair of domain states 1 and 2 are related only by an nfold rotation or rotoinversion (). The resulting coset again contains the alternative twin operations, but in this case only for the orientation relation , and not for (`nontransposable' domain pair). This coset procedure thus does not result in a composite group for a domain pair. In order to obtain the composite group, further cosets have to be constructed by means of the higher powers of the twin rotation under consideration. Each new power corresponds to a further domain state and twin law.
This construction leads to a composite symmetry of supergroup index with respect to the eigensymmetry . This case can occur only for the following pairs: , , , , , , , , , (monoaxial point groups), as well as for the two cubic pairs , . For the pairs , , , and the two cubic pairs , , the relations are of index [3] and imply three nontransposable domain states. For the pairs , , , as well as and , four or six different domain states occur. Among them, however, domain pairs related by the second powers of 4 and as well as by the third powers of 6 and operations are transposable, because these twin operations correspond to twofold rotations or, for , to m.
No growth twins of this type are known so far. As a transformation twin, langbeinite () is the only known example.
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