Extension to non-merohedral growth and mechanical twins

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 428-429

Section 3.3.10.2.2. Extension to non-merohedral growth and mechanical twins

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

3.3.10.2.2. Extension to non-merohedral growth and mechanical twins

| top | pdf |

The treatment by Sapriel (1975) was directed to (switchable) ferroelastics with a real structural phase transition from a parent phase $[{\cal G}]$ to a deformed daughter phase $[{\cal H}]$ . This procedure can be extended to those non-merohedral twins that lack a (real or hypothetical) parent phase, in particular to growth twins as well as to mechanical twins in the traditional sense [cf. Section 3.3.7.3(i)]. Here, the missing supergroup $[\cal G]$ formally has to be replaced by the `full' or `reduced' composite symmetry $[{\cal K}]$ or $[{\cal K}^\ast]$ of the twin, as defined in Section 3.3.4. Furthermore, we replace the spontaneous shear strain by one half of the imaginary shear deformation which would be required to transform the first orientation state into the second via a hypothetical intermediate (zero-strain) reference state. Note that this is a formal procedure only and does not occur in reality, except in mechanical twinning (cf. Section 3.3.7.3). With respect to this intermediate reference state, the two twin orientations possess equal but opposite `spontaneous' strain. With these definitions, the Sapriel treatment can be applied to non-merohedral twins in general. This extension even permits the generalization of the Aizu notation of ferroelastic species to $[{\cal K}F{\cal H}]$ and $[{\cal K}^\ast F{\cal H}^\ast]$ (e.g. [mmmF 2/m] ), whereby now $[{\cal H}]$ and $[{\cal H}^\ast]$ represent the [eigensymmetry] and the intersection symmetry, and $[{\cal K}]$ and $[{\cal K}^\ast]$ the (possibly reduced) composite symmetry of the domain pair. With these modifications, the tables of Sapriel (1975) can be used to derive the permissible boundaries W and [W'] for general non-merohedral twins.

It should be emphasized that this extension of the Sapriel treatment requires a modification of the definition of the W boundary as given above in Section 3.3.10.2.1: The (rational) symmetry operations of the parent phase, becoming F operations in the phase transformation, have to be replaced by the (growth) twin operations contained in the coset of the twin law. These twin operations now correspond to either rational or irrational twin elements. Consequently, the W boundaries defined by these twin elements can be either rational or irrational, whereas by Sapriel they are defined as rational. The Sapriel definition of the [W'] boundaries, on the other hand, is not modified: [W'] boundaries depend on the direction of the spontaneous shear strain and are always irrational. They cannot be derived from the twin operations in the coset and, hence, do not appear as primed twin elements in the black–white symmetry symbol of the composite symmetry $[{\cal K}]$ or $[{\cal K}^\ast]$ .

In many cases, the derivation of the permissible twin boundaries W can be simplified by application of the following rules:

(i) any twin mirror plane, rational or irrational, is a permissible composition plane W;
(ii) the plane perpendicular to any twofold twin axis, rational or irrational, is a permissible composition plane W;
(iii) all these twin mirror planes and twofold twin axes can be identified in the coset of any twin law, for example by the primed twin elements in the black–white symmetry symbol of the composite symmetry $[{\cal K}]$ (cf. Section 3.3.5).

In conclusion, the following differences in philosophy between the Sapriel approach in Section 3.3.10.2.1 and its extension in the present section are noted: Sapriel starts from the supergroup $[{\cal G}]$ of the parent phase and determines all permissible domain walls at once by means of the symmetry reduction $[{\cal G} \rightarrow {\cal H}]$ during the phase transition. This includes group–subgroup relations of index [[i]> 2] . The present extension to general twins takes the opposite direction: starting from the eigensymmetry $[{\cal H}]$ of a twin component and the twin law $[k \times {\cal H}]$ , a symmetry increase to the composite symmetry $[{\cal K}]$ of a twin domain pair is obtained. From this composite symmetry, which is always a supergroup of $[{\cal H}]$ of index [[i]=2] , the two permissible boundaries between the two twin domains are derived. Repetition of this process, using further twin laws one by one, determines the permissible boundaries in multiple twins of index $[[i]\,\gt \,2]$ .

Examples

(1) Gypsum dovetail twin, eigensymmetry $[{\cal H} = 2/m]$ , twin element: reflection plane (100) (cf. Example 3.3.6.2, Fig. 3.3.6.1).

Intersection symmetry of two twin domains: $[{\cal H}^\ast = 1\,2/m\,1]$ (= eigensymmetry $[{\cal H}]$ ); composite symmetry: $[{\cal K} = 2'/m'\,2/m\,2'/m']$ (referred to orthorhombic axes); corresponding Aizu notation: $[ 2/m\,2/m\,2/m\,F 12/m1]$ ; alternative twin elements in the coset: rational twin reflection plane $[m'\parallel(100)]$ and irrational plane normal to [001], as well as the twofold axes normal to these planes, all referred to monoclinic axes. The two alternative twin reflection planes are at the same time permissible W boundaries.

In most dovetail and Montmartre twins of gypsum only the rational (100) or (001) twin boundary is observed. In some cases, however, both permissible W boundaries occur, whereby the irrational interface is usually distinctly smaller and less perfect than the rational one, cf. Fig. 3.3.6.1.
(2) Multiple twins with orthorhombic eigensymmetry $[ {\cal H} = 2/m\,2/m\,2/m]$ and equivalent twin mirror planes (110) and $[(1{\bar 1}0)]$ .

Intersection symmetry of two or more domain states : $[{\cal H}^\ast =]$ ; reduced composite symmetry: $[{\cal K}^\ast =]$ $[2'/m'\,2'/m'\,2/m]$ .

Reference is made to Fig. 3.3.4.2 in Section 3.3.4.2, where the complete cosets for both twin laws (110) and $[(1{\bar 1}0)]$ are shown. For each twin law, the two perpendicular twin mirror planes are at the same time the two permissible W twin boundaries. The (110) boundary is rational, the second permissible boundary, perpendicular to (110), is irrational; similarly for $[(1{\bar 1}0)]$ . The rational boundary is always observed. This rule remains valid for multiple twins, in particular for the spectacular cyclic twins with pseudo n-fold twin axes : $[\arctan b/a \approx 60^\circ]$ (aragonite), $[72^\circ]$ (AlMn alloy), $[90^\circ]$ (staurolite $[90^\circ]$ cross), $[\ldots\,, 360^\circ/n]$ . A pentagonal twin is shown in Fig. 3.3.6.8.

(3) Twins of triclinic feldspars, eigensymmetry $[{\cal H} = {\bar 1}]$ (cf. Example 3.3.6.11, Figs. 3.3.6.11 and 3.3.6.12).

(a) Albite law: twin reflection plane (010) (referred to triclinic, pseudo-monoclinic axes), Fig. 3.3.6.11.

Intersection symmetry: $[{\cal H}^\ast = {\cal H} = {\bar 1}]$ ; composite symmetry: $[{\cal K} = 2'/m'({\bar 1})]$ .

The two permissible twin boundaries are the W twin plane (010) (fixed and rational) and a plane perpendicular to the first in the zone of the reciprocal axis $[{\bf b^*} = [010]^*]$ , but `floating' with respect to its azimuth. The rational W plane (010) is always observed in the form of large-area, polysynthetic twin aggregates.

(b) Pericline law: twofold twin rotation axis [010] (referred to triclinic, pseudo-monoclinic axes), Fig. 3.3.6.12.

Intersection symmetry: $[{\cal H}^\ast = {\cal H} = {\bar 1}]$ ; composite symmetry: $[{\cal K} = 2'/m'({\bar 1})]$ .

The two permissible contact planes are:

(i) the irrational W plane normal to the twin axis [010] [parallel to the reciprocal $[(010)^\ast]$ plane];
(ii) the irrational plane, normal to the first W plane, in the zone of the [010] twin axis, but `floating' with respect to its azimuth.

The latter composition plane is the famous `rhombic section' which is always observed. The azimuthal angle of the rhombic section around [010] depends on the Na/Ca ratio of the plagioclase crystal and is used for the determination of its chemical composition.

Remark. Both twin laws (albite and pericline ) occur simultaneously in microcline, KAlSi₃O₈ (`transformation microcline') as the result of a slow Si/Al order–disorder phase transition from monoclinic sanidine to triclinic microcline, forming crosshatched lamellae of albite and pericline twins (Aizu species $[2/m\,F{\bar 1}]$ ).

(4) Carlsbad twins of monoclinic orthoclase KAlSi₃O₈ (cf. Fig. 3.3.7.1).

Eigensymmetry : $[{\cal H} = 12/m1]$ ; twin element: twofold axis [001] (referred to monoclinic axes); intersection symmetry: $[{\cal H}^\ast = {\cal H} = 12/m1]$ ; composite symmetry: $[{\cal K} = 2'/m'\,2/m\,2'/m']$ (referred to orthorhombic axes).

Permissible W twin boundaries (referred to monoclinic axes):

(i) $[ m \perp [001]]$ (irrational),
(ii) $[m \parallel (100)]$ (rational).

Carlsbad twins are penetration twins. The twin boundaries are more or less irregular, as is indicated by the re-entrant edges on the surface of the crystals. From some of these edges, it can be concluded that boundary segments parallel to the permissible (100) planes as well as parallel to the non-permissible (010) planes (which are symmetry planes of the crystal) occur. This is possibly due to complications arising from the penetration morphology.

(5) Calcite deformation twins (e-twins) [cf. Section 3.3.7.3(i) and Fig. 3.3.7.3].

The deformation twinning in calcite has been extensively studied by Barber & Wenk (1979). Recently, these twins were discussed by Bueble & Schmahl (1999) from the viewpoint of Sapriel's strain compatibility theory of domain walls.

For calcite (space group $[R{\bar 3}c]$ ) three unit cells are in use:

(i) Structural triple hexagonal R-centred cell (`X-ray cell'): a_hex = 4.99, c_hex = 17.06 Å. This cell is used by both Barber & Wenk and Bueble & Schmahl.
(ii) Morphological cell: a_morph = a_hex = 4.99, c_morph = 1/4c_hex = 4.26 Å. This cell is used in many mineralogical textbooks for the description of the calcite morphology and twinning.
(iii) Rhombohedral (pseudo-cubic) cell, F-centred, corresponding to the cleavage rhombohedron and the cell of the cubic NaCl structure: a_pc = 3.21 Å, α_pc = 101.90°.

Eigensymmetry: $[{\cal H} = {\bar 3}2/m1]$ ; twin reflection and interface plane: $[(01{\bar1}8)_{\rm hex} = (01{\bar 1}2)_{\rm morph} = (110)_{\rm pc}]$ (similar for the two other equivalent planes); intersection symmetry: $[{\cal H}^\ast = 2/m]$ along $[[010]_{\rm hex} = [10{\bar 1}]_{\rm pc}]$ ; reduced composite symmetry: $[{\cal K}^\ast = 2/m\,2'/m_1'\,2'/m_2']$ , with [m_1] [ =] $[(01{\bar 1}8)_{\rm hex} =]$ $[(01{\bar 1}2)_{\rm morph} =]$ $[(110)_{\rm pc}]$ rational and [m_2] an irrational plane normal to the edge of the cleavage rhombohedron (cf. Fig. 3.3.7.3). Planes [m_1] and [m_2] are compatible W twin boundaries, of which the rational plane [m_1] is the only one observed.

Bueble & Schmahl (1999) treated the mechanical twinning of calcite by using the Sapriel formalism for ferroelastic crystals. The authors devised a virtual prototypic phase of cubic $[m{\bar 3}m]$ symmetry with the NaCl unit cell (iii) mentioned above. From a virtual ferroelastic phase transition $[m{\bar 3}m \Rightarrow {\bar 3}2/m]$ , they derived four orientation states (corresponding to compression axes along the four cube diagonals). The W boundaries are of type $[\{110\}_{\rm pc} =]$ $[\{01{\bar 1}8\}_{\rm hex} =]$ $[\{01{\bar 1}2\}_{\rm morph}]$ and $[\{001\}_{\rm pc} =]$ $[\{0{\bar 1}14\}_{\rm hex} =]$ $[\{0{\bar 1}11\}_{\rm morph}]$ (cleavage faces). These boundaries are observed. The e-twins (primary twins) with $[\{110\}_{\rm pc}]$ W walls, however, dominate in calcite (primary deformation twin lamellae), whereas the secondary r-twins with $[\{001\}_{\rm pc}]$ W boundaries are relatively rare.

A comparison with the compatible twin boundaries [m_1] and [m_2] derived from the reduced composite symmetry $[{\cal K}^\ast]$ shows that the $[m_1 = \{110\}_{\rm pc} = \{01{\bar 1}8\}_{\rm hex}]$ boundary is predicted by both approaches, whereas [m_2] and $[\{001\}_{\rm pc} = \{0{\bar 1}14\}_{\rm hex}]$ differ by an angle of $[26.2^\circ]$ . The twin reflection planes $[\{110\}_{\rm pc}]$ (e-twin) and $[\{001\}_{\rm pc}]$ (r-twin) represent different twin laws and are not alternative twin elements of the same twin law, as are $[m_1 = \{110\}_{\rm pc}]$ and [m_2] . They would be alternative elements if the rhombohedron (pseudo-cube), keeping its structural $[{\bar 3}2/m]$ symmetry, were re-distorted into an exact cube.

This situation explains the rather complicated deformation twin texture of calcite . Whereas two e-twin components can be stress-free attached to each other along a boundary consisting of compatible $[m_1 = \{110\}_{\rm pc}]$ and [m_2] segments, a boundary of $[\{110\}_{\rm pc}]$ and (incompatible) $[\{001\}_{\rm pc}]$ segments would generate stress, which is extraordinarily high due to the extreme shear angle of $[26.2^\circ]$ . The irrational [m_2] boundary, though mechanically compatible, is not observed in calcite and is obviously suppressed due to bad structural fit. As a consequence, the stress in the boundary regions between the mutually incompatible $[{\bf e}\{110\}_{\rm pc}]$ and $[{\bf r}\{001\}_{\rm pc}]$ twin systems is often buffered by the formation of needle twin lamellae (Salje & Ishibashi, 1996) or structural channels along crystallographic directions (`Rose channels'; Rose, 1868). Twinning dislocations and cracks (Barber & Wenk, 1979) also relax high stress. In `real' ferroelastic crystals with their small shear (usually below 1°) these stress-relaxing phenomena usually do not occur.

References

Barber, D. J. & Wenk, H.-R. (1979). Deformation twinning in calcite, dolomite, and other rhombohedral carbonates. Phys. Chem. Miner. 5, 141–165.Google Scholar

Bueble, S. & Schmahl, W. W. (1999). Mechanical twinning in calcite considered with the concept of ferroelasticity. Phys. Chem. Miner. 26, 668–672.Google Scholar

Rose, G. (1868). Über die im Kalkspath vorkommenden hohlen Canäle. Abh. Königl. Akad. Wiss. Berlin, 23, 57–79.Google Scholar

Salje, E. K. H. & Ishibashi, Y. (1996). Mesoscopic structures in ferroelastic crystals: needle twins and right-angled domains. J. Phys. Condens. Matter, 8, 1–19.Google Scholar

Sapriel, J. (1975). Domain-wall orientations in ferroelastics. Phys. Rev. B, 12, 5128–5140.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 428-429