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

International Tables for Crystallography (2013). Vol. D, ch. 3.3, p. 423

Section 3.3.5. Description of the twin law by black–white symmetry

Th. Hahna* and H. Klapperb

aInstitut 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

3.3.5. Description of the twin law by black–white symmetry

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An alternative description of twinning employs the symbolism of colour symmetry. This method was introduced by Curien & Le Corre (1958[link]) and by Curien & Donnay (1959[link]). In this approach, a colour is attributed to each different domain state. Depending on the number of domain states, simple twins with two colours (i.e. `black–white' or `dichromatic' or `anti-symmetry' groups) and multiple twins with more than two colours (i.e. `polychromatic' symmetry groups) have to be considered. Two kinds of operations are distinguished:

  • (i) The symmetry operations of the eigensymmetry (point group) of the crystal. These operations are `colour-preserving' and form the `monochromatic' eigensymmetry group [{\cal H}]. The symbols of these operations are unprimed.

  • (ii) The twin operations, i.e. those operations which transform one orientation state into another, are `colour-changing' operations. Their symbols are designated by a prime if of order 2: [2'], [m'], [{\bar 1}{^\prime}].

For simple twins, all colour-changing (twin) operations are binary, hence the two domain states are transposable. The composite symmetry [\cal K] of these twins thus can be described by a `black-and-white' symmetry group. The coset, which defines the twin law, contains only colour-changing (primed) operations. This notation has been used already in previous sections.

It should be noted that symbols such as [4'] and [6'], despite appearance to the contrary, represent binary black-and-white operations, because [4'] contains [2], and [6'] contains 3 and [2'], with [2'] being the twin operation. For this reason, these symbols are written here as [4'(2)] and [6'(3)], whereby the unprimed symbol in parentheses refers to the eigensymmetry part of the twin axis. In contrast, [6'(2)] would designate a (polychromatic) twin axis which relates three domain states (three colours), each of eigensymmetry 2. Twin centres of symmetry [{\bar 1}{^\prime}] are always added to the symbol in order to bring out an inversion twinning contained in the twin law. In the original version of Curien & Donnay (1959[link]), the black–white symbols were only used for twinning by merohedry. In the present chapter, the symbols are also applied to non-merohedral twins, as is customary for (ferroelastic) domain structures. This has the consequence, however, that the eigensymmetries [{\cal H}] or [{\cal H}^\ast] and the composite symmetries [{\cal K}] or [{\cal K}^\ast] may belong to different crystal systems and, thus, are referred to different coordinate systems, as shown for the composite symmetry of gypsum in Section 3.3.4.1[link].

For the treatment of multiple twins, `polychromatic' composite groups [{\cal K}(n)] are required. These contain colour-changing operations of order higher than 2, i.e. they relate three or more colours (domain states). Consequently, not all pairs of domain states are transposable. This treatment of multiple twins is rather complicated and only sensible if the composite symmetry group is finite and contains twin axes of low order ([n \leq 8]). For this reason, the symbols for the composite symmetry [{\cal K}] of multiple twins are written without primes; see the examples in Section 3.3.4.4[link](iii)[link]. An extension of the dichromatic twin descriptions to polychromatic symbols for multiple twins was recently presented by Nespolo (2004)[link].

References

Curien, H. & Donnay, J. D. H. (1959). The symmetry of the complete twin. Am. Mineral. 44, 1067–1071.
Curien, H. & Le Corre, Y. (1958). Notation des macles à l'aide du symbolisme des groupes de couleurs de Choubnikov. Bull. Soc. Fr. Minéral. Cristallogr. 81, 126–132.
Nespolo, M. (2004). Twin point group and the polychromatic symmetry of twins. Z. Kristallogr. 219, 57–71.








































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