International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 3.3, pp. 403412
Section 3.3.6. Examples of twinned crystals^{a}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D52056 Aachen, Germany, and ^{b}MineralogischPetrologisches Institut, Universität Bonn, D53113 Bonn, Germany 
In order to illustrate the foregoing rather abstract deliberations, an extensive set of examples of twins occurring either in nature or in the laboratory is presented below. In each case, the twin law is described in two ways: by the coset of alternative twin operations and by the black–white symmetry symbol of the composite symmetry , as described in Sections 3.3.4 and 3.3.5.
For the description of a twin, the conventional crystallographic coordinate system of the crystal and its eigensymmetry group are used in general; exceptions are specifically stated. To indicate the orientation of the twin elements (both rational and irrational) and the composition planes, no specific convention has been adopted; rather a variety of intuitively understandable simple symbols are chosen for each particular case, with the additional remark `rational' or `irrational' where necessary. Thus, for twin reflection planes and (planar) twin boundaries symbols such as , , or are used, whereas twin rotation axes are designated by , , , , , , etc.
The (polar) 180° twin domains in a (potentially ferroelectric) crystal of eigensymmetry () and composite symmetry (e.g. in KTiOPO_{4}, NH_{4}LiSO_{4}, Liformate monohydrate) result from a group–subgroup relation of index with invariance of the symmetry framework (merohedral twins), but antiparallel orientation of the polar axes. The orientation relation between the two domain states is described by the coset of twin operations shown in Table 3.3.6.1, whereby the reflection in (001), , is considered as the `representative' twin operation.

Hence, these twins can be regarded not only as reflection, but also as rotation or inversion twins. The composite symmetry, in black–white symmetry notation, is whereby the primed symbols designate the (alternative) twin operations (cf. Section 3.3.5).
The dovetail twin of gypsum [eigensymmetry , with twin reflection plane ], coset of twin operations and composite symmetry , was treated in Section 3.3.4. Gypsum exhibits an independent additional kind of growth twinning, the Montmartre twin with twin reflection plane . 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 and of both twins are referred, in addition to the monoclinic crystal axes, also to orthorhombic axes for dovetail twins and 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:

In both cases, the (eigensymmetry) framework is invariant under all twin operations; hence, the composite symmetries and are crystallographic of type (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 groups: gypsum is neither structurally nor metrically pseudoorthorhombic, the monoclinic angle being 128°. The two 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 in the zone [010] (monoclinic axis), the oriented eigensymmetry would be the same for all domain states, i.e. the intersection symmetry is identical with the oriented eigensymmetry and, thus, the composite symmetry would be always crystallographic.
For a more general twin reflection plane not belonging to the zone , such as , however, the oriented eigensymmetry would not be invariant under the twin operation. Consequently, an additional twin reflection plane , equivalent with respect to the eigensymmetry , 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 of a domain pair, the reduced composite symmetry with and (irrational) would result. Note that for these (hypothetical) twins the reduced composite symmetry and the eigensymmetry are isomorphic groups, but that their orientations are quite different.
Remark. In the domainstructure 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 to a monoclinic daughter phase of symmetry , with a very strong metrical distortion of 38° from to (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.
Quartz is a mineral which is particularly rich in twinning. It has the noncentrosymmetric trigonal point group 32 with three polar twofold axes and a nonpolar trigonal axis. The crystals exhibit enantiomorphism (right and lefthanded quartz), piezoelectricity and optical activity. The lattice of quartz is hexagonal with holohedral (lattice) point group . Many types of twin laws have been found (cf. Frondel, 1962), but only the four most important ones are discussed here:
The first three types are merohedral (parallellattice) twins and their composite symmetries belong to category (i) in Section 3.3.4.2, whereas the nonmerohedral Japanese twins (twins with inclined lattices or inclined axes) belong to category (ii).
This twinning is commonly described by a twofold twin rotation around the threefold symmetry axis [001]. The two orientation states are of equal handedness but their polar axes are reversed (`electrical twins'). Dauphiné twins can be transformation or growth or mechanical (ferrobielastic) twins. The composite symmetry is , the point group of hightemperature quartz (quartz). The coset decomposition of with respect to the eigensymmetry (index [2]) contains the operations listed in Table 3.3.6.3.

The left coset constitutes the twin law. Note that this coset contains four twofold rotations of which the first one, , is the standard description of Dauphiné twinning. In addition, the coset contains two sixfold rotations, and . The black–white symmetry symbol of the composite symmetry is (supergroup of index [2] of the eigensymmetry group ).
This coset decomposition was first listed and applied to quartz by Janovec (1972, p. 993).
This twinning is commonly described by a twin reflection across a plane normal to a twofold symmetry axis. The two orientation states are of opposite handedness (i.e. the sense of the optical activity is reversed: optical twins) and the polar axes are reversed as well. The coset representing the twin law consists of the following six operations:
The coset shows that Brazil twins can equally well be described as reflection or inversion twins. The composite symmetry is a supergroup of index [2] of the eigensymmetry group 32.
Twins of this type can be described by a twin reflection across the plane (0001), normal to the threefold axis [001]. The two orientation states of this twin are of opposite handedness (i.e. the optical activity is reversed, optical twin), but the polar axes are not reversed. The coset representing the twin law consists of the following six operations:
The composite symmetry is again a supergroup of index [2] of the eigensymmetry group 32. This twin law is usually described as a combination of the Dauphiné and Brazil twin laws, i.e. as the twofold Dauphiné twin rotation followed by the Brazil twin reflection or, alternatively, by the inversion . The product results in a particularly simple description of the combined law as a reflection twin on .
Twin domains of the Leydolt type are very rarely intergrown in direct contact, i.e. with a common boundary. If, however, a quartz crystal contains inserts of Dauphiné and Brazil twins, the domains of these two types, even though not in contact, are related by the Leydolt law. In this sense, Leydolt twinning is rather common in lowtemperature quartz. In contrast, GaPO_{4}, a quartz homeotype with the berlinite structure, frequently contains Leydolt twin domains in direct contact, i.e. with a common boundary (Engel et al., 1989).
In conclusion, the three merohedral twin laws of quartz described above imply four domain states with different orientations of important physical properties. These relations are shown in Fig. 3.3.6.2 for electrical polarity, optical activity and the orientation of etch pits on (0001). It is noteworthy that these three twin laws are the only possible merohedral twins of quartz, and that all three are realized in nature. Combined, they lead to the composite symmetry (`complete twin': Curien & Donnay, 1959).
In the three twin laws (cosets) above, only odd powers of 6, and (rotations and rotoinversions) occur as twin operations, whereas the even powers are part of the eigensymmetry 32. Consequently, repetition of any oddpower twin operation restores the original orientation state, i.e. each of these operations has the nature of a `binary' twin operation and leads to a pair of transposable orientation states.
Among the quartz twins with `inclined axes' (`inclined lattices'), the Japanese twins are the most frequent and important ones. They are contact twins of two individuals with composition plane . This results in an angle of between the two threefold axes. One pair of prism faces is parallel (coplanar) in both partners.
There exist four orientation relations, depending on
These four variants are illustrated in Fig. 3.3.6.3 and listed in Table 3.3.6.4. The twin interface for all four twin laws is the same, , but only in type III do twin mirror plane and composition plane coincide.
^{†}The line is the edge between the faces and and is parallel to the composition plane . It is parallel or normal to the four twin elements. Transformation formulae between the threeindex and the fourindex direction symbols, UVW and uvtw, are given by Barrett & Massalski (1966, p. 13).

In all four types of Japanese twins, the intersection symmetry (reduced eigensymmetry) of a pair of twin partners is 1. Consequently, the twin laws (cosets) consist of only one twin operation and the reduced composite symmetry is a group of order 2, represented by the twin element listed in Table 3.3.6.4. If one were to use the full eigensymmetry , the infinite sphere group would result as composite symmetry .
Many further quartz twins with inclined axes are described by Frondel (1962). A detailed study of these inclinedaxis twins in terms of coincidencesite lattices (CSLs) is provided by McLaren (1986).
Upon heating quartz into the hexagonal hightemperature phase (point group 622) above 846 K, the Dauphiné twinning disappears, because the composite symmetry of the twinned lowtemperature phase now becomes the eigensymmetry of the hightemperature phase. For Brazil twins, however, their nature as reflection or inversion twins is preserved during the transformation.
The eigensymmetry of hightemperature quartz is 622 (order 12). Hence, the coset of the Brazil twin law contains 12 twin operations, as follows:
The composite symmetry is a supergroup of index [2] of the eigensymmetry 622.
In hightemperature quartz, the combined Dauphiné–Brazil twins (Leydolt twins) are identical with Brazil twins, because the Dauphiné twin operation has become part of the eigensymmetry 622. Accordingly, both kinds of twins of lowtemperature quartz merge into one upon heating above 846 K. We recommend that these twins are called `Brazil twins', independent of their type of twinning in the lowtemperature phase. Upon cooling below 846 K, transformation Dauphiné twin domains may appear in both Brazil growth domains, leading to four orientation states as shown in Fig. 3.3.6.2. Among these four orientation states, two Leydolt pairs occur. Such Leydolt domains, however, are not necessarily in contact (cf. Example 3.3.6.3.3 above).
In addition to these twins with `parallel axes' (merohedral twins), several kinds of growth twins with `inclined axes' occur in hightemperature quartz. They are not treated here, but additional information is provided by Frondel (1962).
In some rhombohedral crystals such as corundum Al_{2}O_{3} (Wallace & White, 1967), calcite CaCO_{3} or FeBO_{3} (calcite structure) (Kotrbova et al., 1985; Klapper, 1987), growth twinning with a `twofold twin rotation around the threefold symmetry axis [001]' (similar to the Dauphiné twins in lowtemperature quartz described above) is common. Owing to the eigensymmetry (order 12), the following 12 twin operations form the coset (twin law). They are described here in hexagonal axes:
Some of these twin elements are shown in Fig. 3.3.6.4. They include the particularly conspicuous twin reflection plane perpendicular to the threefold axis [001]. The composite symmetry is
It is of interest that for FeBO_{3} crystals this twin law always, without exception, forms penetration twins (Fig. 3.3.6.4), whereas for the isotypic calcite CaCO_{3} only (0001) contact twins are found (Fig. 3.3.6.5). This aspect is discussed further in Section 3.3.8.6.
The twinning of rhombohedral crystals described above also occurs for cubic crystals as the spinel law (spinel, CaF_{2}, PbS, diamond, sphaleritetype structures such as ZnS, GaAs, CdTe, cubic face and bodycentred metals). In principle, all four threefold axes of the cube, which are equivalent with respect to the eigensymmetry , can be active in twinning. We restrict our considerations to the case where only one threefold axis, [111], is involved. The most obvious twin operations are the twofold rotation around [111] or the reflection across (111). For centrosymmetric crystals, they are alternative twin operations and belong to the same twin law. For noncentrosymmetric crystals, however, the two operations represent different twin laws. Both cases are covered by the term `spinel law'.
The orientation relation defined by the spinel law corresponds to the `obverse' and `reverse' positions of two rhombohedra (cubes), as shown in Fig. 3.3.6.6. For the two (differently) oriented eigensymmetries of the domain states and , the intersection symmetry (order 12) results. With this `reduced eigensymmetry' , the coset of 12 alternative twin operations is the same as the one derived for twinning of rhombohedral crystals in Example 3.3.6.5.
In the following, we treat the spinel twins with the twin axis [111] or the twin reflection plane (111) for the five cubic point groups (eigensymmetries) , , , , in detail. The intersection groups are , , , and , respectively. For these `reduced eigensymmetries', the cosets of the alternative twin operations are listed below with reference to cubic axes.
The restriction to only one of the four spinel twin axes combined with the application of the coset expansion to the reduced eigensymmetry always leads to a crystallographic composite symmetry . The supergroup generated from the full eigensymmetry, however, would automatically include the other three spinel twin axes and thus would lead to the infinite sphere group , i.e. would imply infinitely many cosets and (equivalent) twin laws. Higherorder spinel twins are discussed in Section 3.3.8.3.
K_{2}SO_{4} has an orthorhombic pseudohexagonal roomtemperature phase with point group and axial ratio , and a hexagonal hightemperature phase ( K) with supergroup . It develops pseudohexagonal growthsector twins with equivalent twin reflection planes and which are also composition planes, as shown in Fig. 3.3.6.7. As discussed in Sections 3.3.2.3.2 and 3.3.4.4 under (iii), this corresponds to a pseudothreefold twin axis which, in combination with the twofold eigensymmetry axis, is also a pseudohexagonal twin axis. The extended composite symmetry is

Pseudohexagonal growth twin of K_{2}SO_{4} showing six sector domains in three orientation states. (001) plate, about 1 mm thick and 5 mm in diameter, between polarizers deviating by 45° from crossed position for optimal contrast of all domains. The crystal was precipitated from aqueous K_{2}SO_{4} solution containing 5% S_{2}O_{3} ions. Courtesy of M. Moret, Milano. 
Upon heating above 853 K, the growthsector twinning disappears. On cooling back into the lowtemperature phase, transformation twinning (`domain structure') with three systems of lamellar domains appears. The three orientation states are identical for growth and transformation twins, but the morphology of the twins is not: sectors versus lamellae. The composite symmetry of the twins at room temperature is the true structural symmetry of the `parent' phase at high temperatures.
As was pointed out in Note (8) of Section 3.3.2.4 and in part (iii) of Section 3.3.4.4, there exist twin axes with noncrystallographic multiplicities etc. Twins with five or tenfold rotations are frequent in intermetallic compounds. As an example, FeAl_{4} is treated here (Ellner & Burkhardt, 1993; Ellner, 1995). This compound is orthorhombic, , with an axial ratio close to , corresponding to a pseudofivefold axis along and equivalent twin mirror planes and , which are about 36° apart. In an ideal intergrowth, this leads to a cyclic pseudopentagonal or pseudodecagonal sector twin (Fig. 3.3.6.8). All features of this twinning are analogous to those of pseudohexagonal aragonite, treated in Section 3.3.2.3.2, and of K_{2}SO_{4}, described above as Example 3.3.6.7.

Pentagonal–decagonal twins. (a) Decagonal twins in the shape of tenfold stars on the surface of a bulk alloy, formed during the solidification of a melt of composition Ru_{8}Ni_{15}Al_{77}. Scanning electron microscopy picture. Typical diameter of stars ca. 200 µm. The arms of the stars show parallel intergrowth. (b) Pentagonal twin aggregate of Fe_{4}Al_{13} with morphology as grown in the orthorhombic hightemperature phase, showing several typical 72° angles between neighbouring twin partners (diameter of aggregate ca. 200 µm). Orthorhombic lattice parameters , , Å, space group . The parameters c and a approximate the relation ; the pseudopentagonal twin axis is [010]. On cooling, the monoclinic lowtemperature phase is obtained. The twin reflection planes in the orthorhombic unit cell are (101) and , in the monoclinic unit cell (100) and ; cf. Ellner & Burkhardt (1993, Fig. 10), Ellner (1995). Both parts courtesy of M. Ellner, Stuttgart. 
The intersection symmetry of all twin partners is ; the reduced composite symmetry of a domain pair in contact is . The extended composite symmetry of the ideal pentagonal sector twin is .
Rutile with eigensymmetry develops growth twins with coinciding twin reflection and composition plane {011}. Owing to its axial ratio , the tetragonal c axes of the two twin partners form an angle of 114.4°. The intersection symmetry of the two domains is along the common direction [100]. The reduced composite symmetry of the domain pair is , with the primed twin elements parallel and normal to the plane (011). A twin of this type, consisting of two domains, is called an `elbow twin' or a `knee twin', and is shown in Fig. 3.3.6.9(a).

Various forms of rutile (TiO_{2}) twins, with one or several equivalent twin reflection planes {011}. (a) Elbow twin (two orientation states). (b) Twin with two orientation states. One component has the form of an inserted lamella. (c) Triple twin (three orientation states) with twin reflection planes (011) and . (d) Triple twin with twin reflection planes (011) and (101). (e) Cyclic sixfold twin with six orientation states. Two sectors appear strongly distorted due to the large angular excess of 35.6°. (f) Cyclic eightfold twin with eight orientation states. (g) Perspective view of the cyclic twin of (e). (h) Photograph of a rutile eightling (ca. 15 mm diameter) from Magnet Cove, Arkansas (Geologisk Museum, Kopenhagen). Parts (a) to (e) courtesy of H. Strunz, Unterwössen, cf. Ramdohr & Strunz, 1967, p. 512. Photograph (h) courtesy of M. Medenbach, Bochum. 
In point group , there exist four equivalent twin reflection planes {011} (four different twin laws) with angles of 65.6° between and and 45° between and , leading to a variety of multiple twins. They may be linear polysynthetic or multiple elbow twins, or any combination thereof (Fig. 3.3.6.9). Very rare are complete cyclic sixfold twins with a large angular excess of (corresponding formally to a `5.5fold' twin axis) and extended composite pseudosymmetry , or cyclic eightfold twins with a nearly exact fit of the sectors and a morphological pseudo twin axis. In the `sixling', the tetragonal axes of the twin components are coplanar, whereas in the `eightling' they alternate `up and down', exhibiting in ideal development the morphological symmetry of the twin aggregate. The extended composite symmetry is with eight twin components, each of different orientation state. These cyclic twins are depicted in Figs. 3.3.6.9(e), (f), (g) and (h).
The sketch of the `eightling' in Fig. 3.3.6.9(f) suggests a hole in the centre of the ring, a fact which would pose great problems for the interpretation of the origin of the twin: how do the members of the ring `know' when to turn and close the ring without an offset? Fig. 3.3.6.9(h) suggests that the ring is covered at the back, i.e. originates from a common point (nucleus). This was confirmed by a special investigation of another `eightling' from Magnet Cove (Arkansas) by Lieber (2002): the `eightling' started to grow from the nucleus and developed into the shape of a funnel with an opening of increasing diameter in the centre. This proves the nucleation growth of the ring (cf. Section 3.3.7.1.1).
Gibbsite (older name: hydrargillite) forms a pronounced layer structure with a perfect cleavage plane . It is monoclinic with eigensymmetry , but strongly pseudohexagonal with an axial ratio . In contrast to most other pseudohexagonal crystals, the twofold eigensymmetry axis b is not parallel but normal to the pseudohexagonal c axis. The normal to the cleavage plane is inclined by against [001]. Owing to the pseudohexagonal metrics of the plane , the lattice planes and , equivalent with respect to the eigensymmetry , form an angle of 60.8°.
The following four significant twin laws have been observed by Brögger (1890):

From the point of view of the relationship between pseudosymmetry and twinning, triclinic crystals are of particular interest. Classical mineralogical examples are the plagioclase feldspars with the `albite' and `pericline' twin laws of triclinic (crystal class ) albite NaAlSi_{3}O_{8} and anorthite CaAl_{2}Si_{2}O_{8} (also microcline, triclinic KAlSi_{3}O_{8}), which all exhibit strong pseudosymmetries to the monoclinic feldspar structure of sanidine. Microcline undergoes a very sluggish monoclinic–triclinic phase transformation involving Si/Al ordering from sanidine to microcline, whereas albite experiences a quick, displacive transformation from monoclinic monalbite to triclinic albite.
The composite symmetries of these twins can be formulated as follows:

Both twin laws resemble closely the monoclinic pseudosymmetry in two slightly different but distinct fashions: each twin law uses one rational twin element from , the other one is irrational. The two frameworks of twin symmetry are inclined with respect to each other by about , corresponding to the angle between b (direct lattice) and (reciprocal lattice).
Both twins occur as growth and transformation twins: they appear together in the characteristic lamellar `transformation microclines'.
The mineral staurolite, approximate formula Fe_{2}Al_{9}[O_{6}(O,OH)_{2}/(SiO_{4})_{4}], has `remained an enigma' (Smith, 1968) to date with respect to the subtle details of symmetry, twinning, structure and chemical composition. A lively account of these problems is provided by Donnay & Donnay (1983). Staurolite is strongly pseudoorthorhombic, , and only detailed optical, morphological and Xray experiments reveal monoclinic symmetry, , with , , Å and within experimental errors (Hurst et al., 1956; Smith, 1968).
Staurolite exhibits two quite different kinds of twins:

Notes

The perovskite family, represented by its well known member BaTiO_{3}, is one of the technically most important groups of dielectric materials, characterized by polar structures which exhibit piezoelectricity, pyroelectricity and, most of all, ferroelectricity.
BaTiO_{3} is cubic and centrosymmetric (paraelectric) above 393 K. Upon cooling below this temperature it transforms in one step (firstorder transformation with small ) into the ferroelectric tetragonal phase with polar space group . This transition is translationengleich of index . Hence there are domains of six possible orientation states at room temperature. The transformation can be theoretically divided into two steps:

The beautiful polysynthetic twin structure of BaTiO_{3} is shown in the colour micrograph Fig. 3.4.1.1 in Chapter 3.4 of this volume.
This term is due to Henke (2003) and refers to the simultaneous occurrence (superposition) of two or more different twin types (twin laws) in one and the same crystal. In twins of twins, one `generation' of twin domains is superimposed upon the other, each with its own twin law. This may occur as a result of:
Typical examples are:

In this context, the term complete twin should be noted. It was coined by Curien & Donnay (1959) for the symmetry description of a crystal containing several merohedral twin laws. Their preferred example was quartz, but there are many relevant cases:

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