Categories of composite symmetries

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. 401-402

Section 3.3.4.4. Categories of composite symmetries

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

3.3.4.4. Categories of composite symmetries

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After this preparatory introduction, the three categories of composite symmetry are treated.

(i) Crystallographic composite symmetry. According to case (I) above, only the following three types of twins have crystallographic composite symmetry $[{\cal K}]$ with two orientation states , one coset and, hence, one twin law:

(a) all merohedral twins (cf. Section 3.3.9);
(b) twins of `monoaxial' eigensymmetry $[{\cal H}]$ that have either a twin reflection plane parallel or a twofold twin axis normal to the single eigensymmetry axis. Monoaxial eigensymmetries are 2, $[m = {\bar 2}]$ , , 3, $[{\bar 3}]$ , 4, $[{\bar 4}]$ , , 6, $[{\bar 6=3/m}]$ , ;
(c) the triclinic eigensymmetry groups 1 and $[{\bar 1}]$ ; here any binary twin element leads to a crystallographic composite symmetry $[{\cal K}]$ .

Examples, including some special cases of trigonal crystals, are given in Section 3.3.6.

(ii) Noncrystallographic composite symmetry. As shown below, a noncrystallographic composite symmetry $[{\cal K}]$ results if the conditions of case (II) apply. Twins of this type are rather complicated because more than one twin law and more than two orientation states are involved. This case is illustrated in Figs. 3.3.3.1(c) and 3.3.4.2, where the twinning of an orthorhombic crystal with eigensymmetry $[{\cal H} = 2/m\,2/m\,2/m]$ and twin mirror plane (110) is considered. In case (II) above and in Fig. 3.3.4.2, domains 2 and 3 are generated from the starting domain 1 by the application of the equivalent twin elements and $[m_1({\bar 1}10]$ ). By applying the two twin elements and $[m_2({\bar 1}10)]$ of domain 2, a new domain 4 is obtained and, at the same time, domain 1 is reproduced. Similarly, the twin elements and $[m_3({\bar 1}10)]$ of domain 3 generate a further new domain 5, and domain 1 is reproduced again.

The continuation of this construction leads in the limit to a circular arrangement with an infinitely large number of domain states . The group-theoretical treatment of this process, based on the full eigensymmetry, results in the infinite composite symmetry group $[{\cal K} = \infty/mm]$ , with the rotation axis parallel to the twofold axis of the intersection symmetry , common to all these infinitely many domains. In an even more general case, for example an orthorhombic crystal with twin reflection plane (111), the infinite sphere group $[{\cal K} = m{\overline{ \infty}}]$ would result as composite symmetry. Neither of these cases is physically meaningful and thus they are not considered further here. It is emphasized, however, that the reduced composite symmetry $[{\cal K}^\ast]$ for any pair of domains in contact, as derived in case (II) above, is finite and crystallographic and, thus, of practical use.

(iii) Pseudo-crystallographic composite symmetry. Among twins with noncrystallographic composite symmetry , described above, those exhibiting structural or at least metrical pseudosymmetries are of special significance. Again we consider an orthorhombic crystal with eigensymmetry $[{\cal H} = 2/m\,2/m\,2/m]$ and equivalent twin reflection planes [(110)] and $[(1{\bar 1}0]$ ), but now with a special axial ratio $[b/a \approx \left\vert \tan(360^\circ/n)\right\vert]$ (n = 3, 4 or 6).

The procedure described above in (ii) leads to three different orientation states for n = 3 and 6 and to two different orientation states for n = 4, forming a cyclic arrangement of sector domains (for cyclic and sector twins see Section 3.3.3). The intersection group $[{\cal H}^\ast]$ of all these domain states is [112/m] , with the twofold axis along the c axis. The reduced composite symmetry of any pair of domains in contact is orthorhombic of type $[{\cal K}^\ast = 2'/m'\,2'/m'\,2/m]$ .

These multiple cyclic twins can be described in two ways (cf. Section 3.3.2.3.2):

(a) by repeated application of equivalent binary twin operations (reflections or twofold rotations) to a pseudosymmetrical crystal, as proposed by Hartman (1960) and Curien (1960). Note that each one of these binary twin operations is `exact', whereas the closure of the cycle of sectors is only approximate; the deviation from $[360^\circ/n]$ depends on the (metrical) pseudosymmetry of the lattice;
(b) by successive application of pseudo n-fold twin rotations around the zone axis of the equivalent twin reflection planes. Note that the individual rotation angles are not exactly $[360^\circ/n]$ , due to the pseudosymmetry of the lattice. This alternative description corresponds to the approach by Friedel (1926, p. 435) and Buerger (1960b).

It is now reasonable to define an extended composite symmetry $[{\cal K}(n)]$ by adding the n-fold rotation as a further generator to the reduced composite symmetry $[{\cal K}^\ast]$ of a domain pair. This results in the composite symmetry $[{\cal K}(n)]$ of the complete twin aggregate , in the present case in a modification of the symmetry $[{\cal K}^\ast =]$ $[2'/m'\,2'/m'\,2/m]$ to:

$[{\cal K}(6) = {\cal K}(3) = 6(2)/m\,2/m\,2/m]$ (three orientation states , two twin laws) for [n = 3] and [n = 6] ;

$[{\cal K}(4) = 4(2)/m\,2/m\,2/m]$ (two orientation states, one twin law) for [n = 4] .

The eigensymmetry component of the main twin axis is given in parentheses.

This construction can also be applied to noncrystallographic twin rotations [n = 5, 7, 8] etc. (cf. Section 3.3.6.8):

$[ {\cal K}(10) = {\cal K}(5) = 10(2)/m\,2/m\,2/m]$ (five orientation states, four twin laws) for [n = 5] and [n = 10] .

The above examples are based on a twofold eigensymmetry component along the n-fold twin axis. An example of a pseudo-hexagonal twin, monoclinic gibbsite, Al(OH)₃, without a twofold eigensymmetry component along [001], is treated as Example 3.3.6.10 and Fig. 3.3.6.10.

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 higher-order rotation axes that is accompanied by a small metrical deformation of the lattice. As a result, the extended composite symmetries $[{\cal K}(n)]$ of the transformation twins resemble the symmetry $[{\cal G}]$ of their parent phase. The occurrence of both multiple growth and multiple transformation twins of orthorhombic pseudo-hexagonal K₂SO₄ is described in Example 3.3.6.7.

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 n-fold rotation or roto-inversion ( $[n \geq 3]$ ). The resulting coset again contains the alternative twin operations , but in this case only for the orientation relation $[1 \Rightarrow 2]$ , and not for $[2 \Rightarrow 1]$ (`non-transposable' 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 $[{\cal K}(n)]$ of supergroup index $[[i]\geq 3]$ with respect to the eigensymmetry $[{\cal H}]$ . This case can occur only for the following $[{\cal H}\Rightarrow{\cal K}]$ pairs: $[1 \Rightarrow 3]$ , $[{\bar 1} \Rightarrow {\bar 3}]$ , $[1 \Rightarrow 4]$ , $[1 \Rightarrow {\bar 4}]$ , $[m \Rightarrow 4/m]$ , $[1 \Rightarrow 6]$ , $[m \Rightarrow {\bar 6} = 3/m]$ , $[m \Rightarrow 6/m]$ , $[2/m \Rightarrow 6/m]$ (monoaxial point groups), as well as for the two cubic pairs $[222 \Rightarrow 23]$ , $[mmm \Rightarrow 2/m\,{\bar 3}]$ . For the pairs $[1 \Rightarrow 3]$ , $[{\bar 1} \Rightarrow {\bar 3}]$ , $[m \Rightarrow {\bar 6} = 3/m]$ , $[2/m \Rightarrow 6/m]$ and the two cubic pairs $[222 \Rightarrow 23]$ , $[mmm \Rightarrow 2/m\,{\bar 3}]$ , the $[{\cal K}]$ relations are of index [3] and imply three non-transposable domain states . For the pairs $[1 \Rightarrow 4]$ , $[1 \Rightarrow {\bar 4}]$ , $[m \Rightarrow 4/m]$ , as well as $[1 \Rightarrow 6]$ and $[m \Rightarrow 6/m]$ , four or six different domain states occur. Among them, however, domain pairs related by the second powers of 4 and $[{\bar 4}]$ as well as by the third powers of 6 and $[{\bar 6}]$ operations are transposable, because these twin operations correspond to twofold rotations.

No growth twins of this type are known so far. As transformation twin, langbeinite ( $[23 \Longleftrightarrow 222]$ ) is the only known example.

References

Buerger, M. J. (1960b). Introductory remarks. Twinning with special regard to coherence. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 3 and 5–7. Madrid: CSIC.Google Scholar

Curien, H. (1960). Sur les axes de macle d'ordre supérieur à deux. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 9–11. Madrid: CSIC.Google Scholar

Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: Berger-Levrault. [Reprinted (1964). Paris: Blanchard].Google Scholar

Hartman, P. (1960). Epitaxial aspects of the atacamite twin. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 15–18. Madrid: CSIC.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 401-402