Equivalent twin laws

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. 400-401

Section 3.3.4.2. Equivalent twin laws

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

3.3.4.2. Equivalent twin laws

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In the example of the dovetail twin of gypsum above, the twin operation [k = m_x = m(100)] is of a special nature in that it maps the entire eigensymmetry $[{\cal H}=12/m1]$ onto itself and, hence, generates a single coset, a single twin law and a finite composite group $[{\cal K}]$ of index [2] (simple twins). 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 3.3.6.2. In this case, extension of the complete group $[\cal H]$ 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 $[k_2,k_3,\ldots,]$ which are independent (non-alternative) but symmetrically equivalent with respect to the eigensymmetry $[\cal H]$ , 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, $[\ldots]$ . 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 $[{\cal H}]$ which is left invariant by the twin operation k. This subgroup is the `intersection symmetry' $[{\cal H}^*]$ of the two `oriented eigensymmetries' $[{\cal H}_1]$ and $[{\cal H}_2]$ of the domains 1 and 2 (shown in Fig. 3.3.4.2): $[{\cal H}^* = {\cal H}_1\cap {\cal H}_2]$ . This group $[{\cal H}^*]$ is now extended by k and leads to the `reduced composite symmetry' $[{\cal K}^*]$ of the domain pair (1, 2): $[{\cal K}^*(1,2)={\cal H}^*\cup k_2\times {\cal H}^*]$ , which is a finite supergroup of $[{\cal H}]$ of index [2]. In this way, the complete coset $[k \times {\cal H}]$ of the eigensymmetry $[\cal H]$ is split into two (or more) smaller cosets $[k_2 \times {\cal H}^*]$ , $[k_3\times{\cal H}^*]$ etc., where $[k_2, k_3,\ldots,]$ are symmetrically equivalent twin operations in $[\cal H]$ . Correspondingly, the differently oriented `reduced composite symmetries' $[{\cal K}^*(1,2) =]$ $[{\cal H}^*\cup k_2\times{\cal H}^*]$ , $[{\cal K}^*(1,3)={\cal H}^*\cup k_3\times{\cal H}^*]$ etc. of the domain pairs (1, 2), (1, 3) etc. are generated by the representative twin operations [k_2] , [k_3] etc. These cosets $[k_i\times{\cal H}^*]$ are considered as the twin laws for the corresponding domain pairs.

Figure 3.3.4.2 | top | pdf |

Twinning of an orthorhombic crystal with equivalent twin mirror planes (110) and $[({\bar 1}10)]$ . Three twin domains 1, 2 and 3, bound by {110} contact planes, are shown. The oriented eigensymmetries $[{\cal H}_1]$ , $[{\cal H}_2]$ , $[{\cal H}_3]$ and the reduced composite symmetries $[{\cal K}^\star (1,2) = {\cal K}^\star (110)]$ and $[{\cal K}^\star (1,3) = {\cal K}^\star({\bar 1}10)]$ of each domain pair are given in stereographic projection. The intersection symmetry of all domains is $[{\cal H}^\star = 112/m]$ .

As an example, an orthorhombic crystal of eigensymmetry $[{\cal H} = 2/m\,2/m\,2/m]$ with equivalent twin reflection planes [k_2 = m(110)] and $[k_3 = m({\bar 1}10)]$ 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 [(110)] and $[({\bar 1}10)]$ , symmetrically equivalent with respect to the oriented eigensymmetry $[{\cal H}_1]$ of domain 1. The intersection symmetries of the two pairs of oriented eigensymmetries $[{\cal H}_1]$ & $[{\cal H}_2]$ and $[{\cal H}_1]$ & $[{\cal H}_3]$ are identical: $[{\cal H}^\ast = 112/m]$ . The three oriented eigensymmetries $[{\cal H}_1]$ , $[{\cal H}_2]$ , $[{\cal H}_3]$ , as well as the two differently oriented reduced composite symmetries $[{\cal K}^\ast(1,2) =]$ $[{\cal K}^\ast(110)]$ and $[{\cal K}^\ast(1,3) = {\cal K}^\ast({\bar 1}10)]$ of the domain pairs (1, 2) and (1, 3), are all isomorphic of type $[2/m\,2/m\,2/m]$ , but exhibit different orientations.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 400-401