Twinning of gypsum

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. 403-404

Section 3.3.6.2. Twinning of gypsum

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

3.3.6.2. Twinning of gypsum

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The dovetail twin of gypsum [eigensymmetry $[{\cal H} = 1\,2/m\,1]$ , with twin reflection plane $[m \parallel (100)]$ ], coset of twin operations $[k \times {\cal H}]$ and composite symmetry $[{\cal K}]$ , was treated in Section 3.3.4. Gypsum exhibits an independent additional kind of growth twinning, the Montmartre twin with twin reflection plane $[m \parallel (001)]$ . These two twin laws are depicted in Fig. 3.3.6.1. The two cosets of twin operations in Table 3.3.6.2 and the symbols of the composite symmetries $[{\cal K_{\rm D}}]$ and $[{\cal K_{\rm M}}]$ of both twins are referred, in addition to the monoclinic crystal axes, also to orthorhombic axes $[x_{\rm D}, y, z_{\rm D}]$ for dovetail twins and $[x_{\rm M}, y, z_{\rm M}]$ for Montmartre twins. This procedure brings out for each case the perpendicularity of the rational and irrational twin elements, clearly visible in Fig. 3.3.6.1, as follows: $[\matrix{{\cal K}_{\rm D} = {\displaystyle{ 2_{x{\rm D}}'\over m_{x{\rm D}}'}{2_y\over m_y}{2_{z{\rm D}}'\over m_{z{\rm D}}'}} &{\cal K}_{\rm M} = {\displaystyle{2_{x{\rm M}}'\over m_{x{\rm M}}'}{2_y\over m_y}{2_{z{\rm M}}'\over m_{z{\rm M}}'}}\cr&&\cr & &\cr x_{\rm D}\hbox{ (ortho)}\perp(100)\hbox{ (mono)}& x_{\rm M}\hbox{ (ortho)}\parallel[100]\hbox{ (mono)}\cr z_{\rm D}\hbox{ (ortho)}\parallel[001]\hbox{ (mono)} & z_{\rm M}\hbox{ (ortho)}\perp(001)\hbox{ (mono)}.}]$

Table 3.3.6.2 | top | pdf |
Gypsum: cosets of alternative twin operations of the dovetail and the Montmartre twins, referred to their specific orthorhombic axes (subscripts D and M)

$[{\cal H}]$	Dovetail twins $[m_{x{\rm D}} \times {\cal H}]$	Montmartre twins $[m_{z{\rm M}} \times {\cal H}]$
1	$[m_{x{\rm D}} = m \parallel(100) ]$ (rational)	$[m_{z{\rm M}} = m\parallel (001) ]$ (rational)
$[2_y = 2 \parallel [010]]$	$[m_{z{\rm D}} = m\perp[001]]$ (irrational)	$[m_{x{\rm M}} = m\perp[100]]$ (irrational)
$[m_y = m \parallel (010)]$	$[2_{z{\rm D}} = 2\parallel[001] ]$ (rational)	$[2_{x{\rm M}} = 2\parallel [100]]$ (rational)
$[{\bar 1}]$	$[ 2_{x{\rm D}} = 2\perp(100)]$ (irrational)	$[2_{z{\rm M}} = 2\perp[001]]$ (irrational)

Figure 3.3.6.1 | top | pdf |

Dovetail twin (a) and Montmartre twin (b) of gypsum. The two orientation states of each twin are distinguished by shading. For each twin type (a) and (b), the following aspects are given: (i) two idealized illustrations of each twin, on the left in the most frequent form with two twin components, on the right in the rare form with four twin components, the morphology of which displays the orthorhombic composite symmetry; (ii) the oriented composite symmetry in stereographic projection (dotted lines indicate monoclinic axes).

In both cases, the (eigensymmetry) framework [2_y /m_y] is invariant under all twin operations; hence, the composite symmetries $[{\cal K}_{\rm D}]$ and $[{\cal K}_{\rm M}]$ are crystallographic of type $[2/m\,2/m\,2/m]$ (supergroup index [2]) but differently oriented, as shown in Fig. 3.3.6.1. There is no physical reality behind the orthorhombic symmetry of the two $[{\cal K}]$ groups: gypsum is neither structurally nor metrically pseudo-orthorhombic, the monoclinic angle being 128°. The two $[{\cal K}]$ groups and their orthorhombic symbols, however, clearly reveal the two different twin symmetries and, for each case, the perpendicular orientations of the four twin elements, two rational and two irrational. The two twin types originate from independent nucleation in aqueous solutions.

It should be noted that for all (potential) twin reflection planes [(h0l)] in the zone [010] (monoclinic axis), the oriented eigensymmetry $[{\cal H} = 1\,2/m\,1]$ would be the same for all domain states , i.e. the intersection symmetry $[{\cal H}^\ast]$ is identical with the oriented eigensymmetry $[{\cal H}]$ and, thus, the composite symmetry would be always crystallographic .

For a more general twin reflection plane not belonging to the zone [(h0l)] , such as [(111)] , however, the oriented eigensymmetry $[{\cal H}]$ would not be invariant under the twin operation. Consequently, an additional twin reflection plane $[(1{\bar 1}1)]$ , equivalent with respect to the eigensymmetry $[1\,2/m\,1]$ , exists. This (hypothetical) twin would belong to category (ii) in Section 3.3.4.4 and would formally lead to a noncrystallographic composite symmetry of infinite order. If, however, we restrict our considerations to the intersection symmetry $[{\cal H}^\ast = {\bar 1}]$ of a domain pair, the reduced composite symmetry $[{\cal K}^\ast = 2'/m']$ with $[m'\parallel (111)]$ and $[2'\perp (111)]$ (irrational) would result. Note that for these (hypothetical) twins the reduced composite symmetry $[{\cal K}^\ast]$ and the eigensymmetry $[{\cal H}]$ are isomorphic groups, but that their orientations are quite different.

Remark . In the domain-structure approach, presented in Chapter 3.4 of this volume, both gypsum twins, dovetail and Montmartre , can be derived together as a result of a single (hypothetical) ferroelastic phase transition from a (nonexistent) orthorhombic parent phase of symmetry $[{\cal G}=2/m2/m2/m]$ to a monoclinic daughter phase of symmetry $[{\cal H}=12/m1]$ , with a very strong metrical distortion of 38° from $[\beta=90^\circ]$ to $[\beta=128^\circ]$ (Janovec, 2003). In this (hypothetical) transition the two mirror planes, (100) and (001), 90° apart in the orthorhombic form, become twin reflection planes of monoclinic gypsum, (100) for the dovetail, (001) for the Montmartre twin law, with an angle of 128°. It must be realized, however, that neither the orthorhombic parent phase nor the ferroelastic phase transition are real.

References

Janovec, V. (2003). Personal communication.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 403-404