Composite symmetry

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 399-400

Section 3.3.4.1. Composite symmetry

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.4.1. Composite symmetry

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For a comprehensive characterization of the symmetry of a twinned crystal, we introduce the important concept of composite symmetry $[{\cal K}]$ . This symmetry is defined as the extension of the eigensymmetry group $[{\cal H}]$ by a twin operation k. This extension involves, by means of left (or right) coset composition $[k\times {\cal H}]$ , the generation of further twin operations until a supergroup is obtained. This supergroup is the composite symmetry group $[{\cal K}]$ .

In the language of group theory, the relation between the composite symmetry group $[{\cal K}]$ and the eigensymmetry group $[{\cal H}]$ can be expressed by a (left) coset decomposition of the supergroup $[{\cal K}]$ with respect to the subgroup $[{\cal H}]$ : $[{\cal K} = k_1\times {\cal H} \cup k_2\times {\cal H} \cup k_3\times {\cal H} \cup \ldots \cup k_i\times {\cal H},]$ where [k_1] is the identity operation; $[k_1\times {\cal H} = {\cal H}\times k_1 = {\cal H}]$ .

The number i of cosets, including the subgroup $[{\cal H}]$ , is the index [i] of $[{\cal H}]$ in $[{\cal K}]$ ; this index corresponds to the number of different orientation states in the twinned crystal. If $[{\cal H}]$ is a normal subgroup of $[{\cal K}]$ , which is always the case if [i = 2] , then $[k\times {\cal H} = {\cal H}\times k ]$ , 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 $[{\cal H}]$ in $[{\cal K}]$ , i.e. $[n = [i] = \left\vert {\cal K}\right\vert: \left\vert {\cal H}\right\vert]$ , was first expressed by Zheludev & Shuvalov (1956, p. 540) for ferroelectric phase transitions.

These group-theoretical considerations can be translated into the language of twinning as follows: although the eigensymmetry $[{\cal H}]$ and the composite symmetry $[{\cal K}]$ 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 $[\left\vert{\cal H}\right\vert]$ of the eigensymmetry point group $[{\cal H}]$ . 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. re-entrant 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 group-theoretical treatment with practical examples is provided by Flack (1987).

Example: dovetail twin of gypsum (Fig. 3.3.4.1). Eigensymmetry: $[{\cal H} = 1 {2_y\over m_y} 1.]$ Twin reflection plane (100): [k_2 = k = m_x.] Composite symmetry group $[{\cal K}_D]$ (orthorhombic): $[ {\cal K} = {\cal H} \cup k \times {\cal H},]$ given in orthorhombic axes, [x, y, z] . The coset $[k\times {\cal H} ]$ contains all four alternative twin operations (Table 3.3.4.1) and, hence, represents the twin law. This is clearly visible in Fig. 3.3.6.1(a). In the symbol of the orthorhombic composite group, $[{\cal K} = {2'_x\over m'_x}{2_y\over m_y}{2'_z\over m'_z},]$ the primed operations indicate the coset of alternative twin operations. The above black-and-white symmetry symbol of the (orthorhombic) composite group $[{\cal K}]$ 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 3.3.6.2.

Table 3.3.4.1 | top | pdf |
Gypsum, dovetail twins: coset of alternative twin operations (twin law), given in orthorhombic axes of the composite symmetry $[{\cal K}_D]$

$[{\cal H}]$	$[k\times {\cal H}]$
1	$[m_x \times 1 = m_x]$
	$[m_x \times 2_y = m_z]$
	$[m_x \times m_y = 2_z]$
$[{\bar 1}]$	$[m_x \times {\bar 1} = 2_x]$

Figure 3.3.4.1 | top | pdf |

Gypsum dovetail twin: schematic illustration of the coset of alternative twin operations. The two domain states I and II are represented by oriented parallelograms of eigensymmetry [2_y/m_y] . The subscripts x and z of the twin operations refer to the coordinate system of the orthorhombic composite symmetry $[{\cal K}_D]$ of this twin; a and c are the monoclinic coordinate axes.

It should be noted that among the four twin operations of the coset $[k\times {\cal H} ]$ two are rational, [m_x] and [2_z] , and two are irrational, [m_z] and [2_x] (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 [m_x = (100)] , as the representative twin element.

The concept of composite symmetry $[{\cal K}]$ is not only a theoretical tool for the extension of the twin law but has also practical aspects:

(i) Morphology of growth twins. In general, the volume fractions of the various twin domains are different and their distribution is irregular. Hence, most twins do not exhibit regular morphological symmetry. If, however, the twin aggregate consists of p components of equal volumes and shapes ( `length' of the coset of alternative twin operations ) and if these components show a regular symmetrical distribution, the morphology of the twinned crystal displays the composite symmetry. In minerals, this is frequently very well approximated, as can be inferred from Fig. 3.3.6.1 for gypsum.
(ii) Diffraction pattern. The `single-crystal diffraction pattern' of a twinned crystal exhibits its composite symmetry $[{\cal K}]$ if the volume fractions of all domain states are (approximately) equal.
(iii) Permissible twin boundaries. The composite symmetry $[{\cal K}]$ in its black–white notation permits immediate recognition of the `permissible twin boundaries' (W-type composition planes), as explained in Section 3.3.10.2.1.

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 399-400