Chapter 3.3. Twinning of crystals

(a) The orientation relation between two partners is defined as crystallographic and, hence, the corresponding intergrowth is a twin, if the following two minimal conditions are simultaneously obeyed:

Both conditions taken together define the minimal geometric requirement for a twin (at least one common row and one pair of parallel planes), as originally pronounced by several classical authors (Tschermak, 1884, 1905; 1904; Tschermak & Becke, 1915; Mügge, 1911, p. 39; Niggli, 1920/1924/1941; Tertsch, 1936) and taken up later by Menzer (1955) and Hartman (1956). It is obvious that these crystallographic conditions apply even more to twins with two- and three-dimensional lattice coincidences, as described in Section 3.3.8. Other orientation relations, as they occur, for instance, in arbitrary intergrowths or bicrystals, are called `noncrystallographic'.

The terms `common edge' and `common face', as used in this section, are derived from the original morphological consideration of twins. Example: a re-entrant edge of a twin is common to both twin partners. In lattice considerations, the terms `common lattice row', `common lattice plane' and `common lattice' require a somewhat finer definition, in view of a possible twin displacement vector t of the twin boundary, as introduced in Note (8) of Section 3.3.2.4 and in Section 3.3.10.3. For this distinction the terms `parallel', `common' and `coincident' are used as follows:

Note that for lattice rows and point lattices only two cases have to be distinguished, whereas lattice planes require three terms.

(b) A twinned crystal may consist of more than two individuals. All individuals that have the same orientation and handedness belong to the same orientation state (component state, domain state, domain variant). The term `twin' for a crystal aggregate requires the presence of at least two orientation states.

(c) The orientation and chirality relation between two twin partners is expressed by the twin law. It comprises the set of all twin operations that transform the two orientation states into each other. A twin operation cannot be a symmetry operation of either one of the two twin components. The combination of a twin operation and the geometric element to which it is attached is called a twin element (e.g. twin mirror plane, twofold twin axis, twin inversion centre).

(d) An orientation relation between two individuals deserves the name `twin law' only if it occurs frequently, is reproducible and represents an inherent feature of the crystal species.

(e) One feature which facilitates the formation of twins is pseudosymmetry, apparent either in the crystal structure, or in special lattice-parameter ratios or lattice angles.

(f) In general, the twin interfaces are low-energy boundaries with good structural fit; very often they are low-index lattice planes.

(1) The above definition of twinning covers twins with a size range from decimetres for large mineral specimens to fractions of microns of polydomain twins. The lower limit for a reasonable application of the twin concept lies in the nanometre range of 100–1000 Å. Unit-cell twinning is a limiting case closely related to superstructures or positional disorder and is not treated in this chapter. An extensive monograph on this topic under the name Tropochemical cell-twinning was recently published by Takeuchi (1997).

(2) Rational twin elements are designated by integer Miller indices (hkl) and integer direction indices [uvw] for twin mirror planes and twin rotation axes, respectively; the values of these indices usually are small numbers, . Larger values should not be accepted without critical assessment. Irrational twin elements are described either by their rational `counterparts' (perpendicular planes or lines) or are approximated by high integer values of the indices.

(3) For a twin, the crystallographic orientation relation is a property of the crystal structure, in contrast to bicrystals, where the given orientation relation is either accidental or enforced by the experiment. If such an orientation relation happens to be crystallographic, the bicrystal could formally be considered to be a twin. It is not recommended, however, to accept such a bicrystal as a twin because its orientation relation does not form `spontaneously' as a property of the structure.

(4) There are some peculiarities about the contact relations between two twin components. Often quite different twin boundaries occur for one and the same orientation relation. The boundaries are either irregular (frequently in penetration twins) or planar interfaces. Even though crystallographic boundaries are the most frequent interfaces, the geometry of a contact relation is not suitable as part of the twin definition. Contact relations, however, play an important role for the morphological classification of twins (cf. Section 3.3.3.1). Frequently used alternative names for twin boundaries are twin interfaces, contact planes, composition planes or domain boundaries.

(5) Frequently, the term `twin plane (hkl)' is used for the characterization of a reflection twin. This term is justified only if the twin mirror plane and the composition plane coincide. It is, however, ambiguous if the twin mirror plane and the twin interface have different orientations. In twins of hexagonal KLiSO<sub>4</sub>, for example, the prominent composition plane is normal to the twin mirror plane (cf. Klapper et al., 1987). The short term `twin plane' should be avoided in such cases and substituted by twin mirror plane or twin reflection plane. The frequently used term `twinning on (hkl)' [German: `Zwillinge nach (hkl)'] refers to (hkl) as a twin mirror plane and not as a contact plane.

(6) There exist twins in which the twin operations can be regarded as fivefold or tenfold rotations (Ellner & Burkhardt, 1993; Ellner, 1995). These twins are due to pseudo-pentagonal or pseudo-decagonal metrical features of the lattice [ ]. They can be treated in the same way as the three-, four- and sixfold rotation twins mentioned above. This includes the alternative (`exact') description of the twinning by a cyclic sequence of symmetrically equivalent twin reflection planes or twofold twin axes (cf. aragonite, Fig. 3.3.2.4). For this reason, we recommend that these intergrowths are accepted as (pseudo) n-fold rotation twins, even though the value of n is noncrystallographic.

(7) The classical treatment of twins considers only rotations, reflections and inversions as twin operations. In domain structures, relations between domain states exist that involve only translations (cf. Section 3.4.3 ), specifically those that are suppressed during a phase transition. Since every domain structure can be considered as a transformation twin, it seems legitimate to accept these translations as twin operations and speak of translation twins (T-twins according to Wadhawan, 1997, 2000). The translation vector of this twin operation is also known as the fault vector of the translation boundary (often called the antiphase boundary) between two translation domains [cf. Section 3.3.1(v)]. It must be realized, however, that the acceptance of translation domains as twins would classify all stacking faults in metals, in diamond and in semiconductors as twin boundaries.

(8) The structural consideration of twin boundaries cannot be performed by employing only point-group twin elements as is sufficient for the description of the orientation relation. In structural discussions, in addition, a translational displacement of the two structures with respect to each other by a shift vector, which is called here the twin displacement vector t, has to be taken into account. This displacement vector leads to a minimization of the twin-boundary energy. The components of this vector can have values between 0 and 1 of the basis vectors. In many cases, especially of transformation twins, glide planes or screw axes can occur as twin elements, whereby the glide or screw components may be relaxed, i.e. may deviate from their ideal values. The twin displacement vectors, as introduced here, include these glide or screw components and their relaxations. The role of the twin displacement vectors in the structure of the twin boundary is discussed in Section 3.3.10.4.

(9) In some cases the term `twin' is used for systematic oriented intergrowths of crystals which are not twins as defined in this chapter. The following cases should be mentioned:

(i) Contact twins. Two twin partners are in contact across a single composition plane , the Miller indices of which have the same values for both partners (`common' plane). The contact plane usually has low indices. For reflection twins, the composition plane is frequently parallel to the twin mirror plane [see, however, Note (4) in Section 3.3.2.4]. Examples are shown in Fig. 3.3.6.1 for gypsum, in Fig. 3.3.6.5 for calcite and in Fig. 3.3.6.6(a) for a spinel (111) twin. In most cases, contact twins are growth twins.

(ii) Polysynthetic (lamellar) twins are formed by repetition of contact twins and consist of a linear sequence of domains with two alternating orientation states. The contact planes are parallel (lamellar twinning). This is typical for reflection twins if the twin mirror plane and the composition plane coincide. An illustrative example is the albite growth twin shown in Fig. 3.3.6.11. Polysynthetic twins may occur in growth, transformation and deformation twinning.

(iii) Penetration twins. The name `penetration twin' results from the apparent penetration of two or more (idiomorphic) single crystals. The most prominent examples are twins of the spinel law in cubic crystals (e.g. spinels, fluorite, diamond). The spinel law is a reflection across (111) or a twofold rotation around [111]. Ideally these twins appear as two interpenetrating cubes with a common threefold axis, each cube representing one domain state (Fig. 3.3.6.6b). In reality, these twins usually consist of 12 pyramid-shaped domains, six of each domain state, all originating from a common point in the centre of the twinned crystal (as shown in Figs. 3.3.6.4 and 3.3.6.6). Another famous example is the Carlsbad twin of orthoclase feldspar with [001] as a twofold twin axis (Fig. 3.3.7.1). Penetration twins are always growth twins.

(iv) Cyclic twins and sector twins. In contrast to the linear sequence of domains in polysynthetic twins, cyclic twins form a circular arrangement of domains of suitable shape. They are always multiple twins (three or more orientations states) which are (formally) generated by successive application of equivalent twin laws. The twin aggregate may form a full circle or a fraction of a circle (see Figs. 3.3.3.1c, 3.3.6.7 and 3.3.6.10). Impressive examples are the `sixlings' and `eightlings' of rutile (Fig. 3.3.6.9, cf. Example 3.3.6.9).

A special case of cyclic twins is provided by sector twins. Three or more domains of nearly triangular shape (angular sectors) extend from a common centre to form a twinned crystal with a more or less regular polygonal outline. The boundaries between two sector domains are usually planar and low-indexed. Such twins can be interpreted in two ways:

Prominent examples are the growth twins of NH₄LiSO₄ (Fig. 3.3.7.2), aragonite (Fig. 3.3.2.4), K₂SO₄ (Fig. 3.3.6.7) and certain alloys with pseudo-fivefold twin axes (Fig. 3.3.6.8). Cyclic and sector twins are always growth twins.

(v) Mimetic twins. The term `mimetic' is often applied to growth twins which, by their morphology, simulate a higher crystal symmetry. Regular penetration twins and sector twins are frequently also `mimetic' twins. Particularly impressive examples are the harmotome and phillipsite twins, where monoclinic crystals, by multiple twinning, simulate higher symmetries up to a cubic rhomb-dodecahedron.

(vi) Common names of twins. In addition to the morphological description of twins mentioned above, some further shape-related names are in use:

(i) Morphology of growth twins. In general, the volume fractions of the various twin domains are different and their distribution is irregular. Hence, most twins do not exhibit regular morphological symmetry. If, however, the twin aggregate consists of p components of equal volumes and shapes ( `length' of the coset of alternative twin operations) and if these components show a regular symmetrical distribution, the morphology of the twinned crystal displays the composite symmetry. In minerals, this is frequently very well approximated, as can be inferred from Fig. 3.3.6.1 for gypsum.

(ii) Diffraction pattern. The `single-crystal diffraction pattern' of a twinned crystal exhibits its composite symmetry if the volume fractions of all domain states are (approximately) equal.

(iii) Permissible twin boundaries. The composite symmetry in its black–white notation permits immediate recognition of the `permissible twin boundaries' (W-type composition planes), as explained in Section 3.3.10.2.1.

Case (I) : . Here, all twin operations map the complete oriented frameworks of the two domain states 1 and 2 onto each other, i.e. the oriented eigensymmetries and and their intersection group coincide. Hence, for binary twin operations there is only one coset and one twin law. The composite symmetry is crystallographic. An example is provided by the dovetail twins of gypsum, described above (cf. Table 3.3.4.1).

Case (II) : . Here, the twin operations map only a fraction of the oriented symmetry elements of domain states 1 and 2 onto each other. Hence, the intersection group of the two oriented eigensymmetries and is a proper subgroup of index of both and . The coset leads to the crystallographic reduced composite symmetry as for case (I) above. The number of twin laws, different but equivalent with respect to the `starting' eigensymmetry of the first domain state 1, equals the index [i]. This implies i differently oriented domain pairs . The composite symmetry of such a domain pair is now defined by and is called the reduced composite symmetry . All twin laws can be expressed by the black–white symbol of the reduced composite symmetry , as described in Section 3.3.5.

(i) Crystallographic composite symmetry. According to case (I) above, only the following three types of twins have crystallographic composite symmetry with two orientation states, one coset and, hence, one twin law:

Examples, including some special cases of trigonal crystals, are given in Section 3.3.6.

(ii) Noncrystallographic composite symmetry. As shown below, a noncrystallographic composite symmetry results if the conditions of case (II) apply. Twins of this type are rather complicated because more than one twin law and more than two orientation states are involved. This case is illustrated in Figs. 3.3.3.1(c) and 3.3.4.2, where the twinning of an orthorhombic crystal with eigensymmetry and twin mirror plane (110) is considered. In case (II) above and in Fig. 3.3.4.2, domains 2 and 3 are generated from the starting domain 1 by the application of the equivalent twin elements and ). By applying the two twin elements and of domain 2, a new domain 4 is obtained and, at the same time, domain 1 is reproduced. Similarly, the twin elements and of domain 3 generate a further new domain 5, and domain 1 is reproduced again.

The continuation of this construction leads in the limit to a circular arrangement with an infinitely large number of domain states. The group-theoretical treatment of this process, based on the full eigensymmetry, results in the infinite composite symmetry group , with the rotation axis parallel to the twofold axis of the intersection symmetry , common to all these infinitely many domains. In an even more general case, for example an orthorhombic crystal with twin reflection plane (111), the infinite sphere group would result as composite symmetry. Neither of these cases is physically meaningful and thus they are not considered further here. It is emphasized, however, that the reduced composite symmetry for any pair of domains in contact, as derived in case (II) above, is finite and crystallographic and, thus, of practical use.

(iii) Pseudo-crystallographic composite symmetry. Among twins with noncrystallographic composite symmetry, described above, those exhibiting structural or at least metrical pseudosymmetries are of special significance. Again we consider an orthorhombic crystal with eigensymmetry and equivalent twin reflection planes and ), but now with a special axial ratio (n = 3, 4 or 6).

The procedure described above in (ii) leads to three different orientation states for n = 3 and 6 and to two different orientation states for n = 4, forming a cyclic arrangement of sector domains (for cyclic and sector twins see Section 3.3.3). The intersection group of all these domain states is , with the twofold axis along the c axis. The reduced composite symmetry of any pair of domains in contact is orthorhombic of type .

These multiple cyclic twins can be described in two ways (cf. Section 3.3.2.3.2):

It is now reasonable to define an extended composite symmetry by adding the n-fold rotation as a further generator to the reduced composite symmetry of a domain pair. This results in the composite symmetry of the complete twin aggregate, in the present case in a modification of the symmetry to:

(three orientation states, two twin laws) for and ;

(two orientation states, one twin law) for .

The eigensymmetry component of the main twin axis is given in parentheses.

This construction can also be applied to noncrystallographic twin rotations etc. (cf. Section 3.3.6.8):

(five orientation states, four twin laws) for and .

The above examples are based on a twofold eigensymmetry component along the n-fold twin axis. An example of a pseudo-hexagonal twin, monoclinic gibbsite, Al(OH)₃, without a twofold eigensymmetry component along [001], is treated as Example 3.3.6.10 and Fig. 3.3.6.10.

(a) Dauphiné twins;

(b) Brazil twins;

(c) Combined-law (Leydolt, Liebisch) twins;

(d) Japanese twins.

(i) the six twin operations of a Brazil twin in low-temperature quartz, as listed above in Example 3.3.6.3.2;

(ii) three further reflections across planes , which bisect the three Brazil twin planes of low-temperature quartz;

(iii) three further rotoinversions around [001]: , , .

(i) three rotations around the threefold axis : , , ;

(ii) three twofold rotations around the axes , , ;

(iii) three reflections across the planes , , ;

(iv) three rotoinversions around the threefold axis : , and .

(a) Eigensymmetry (order 48), reduced eigensymmetry (order 12).

Alternative twin operations:

Reduced composite symmetry (order 24).

(b) Eigensymmetry (order 24), reduced eigensymmetry (order 6).

Two different twin laws are possible:

(c) Eigensymmetry (order 24), reduced eigensymmetry (order 6).

Again, two different twin laws are possible:

(d) Eigensymmetry (order 24), reduced eigensymmetry (order 6).

Two different twin laws:

(e) Eigensymmetry (order 12), reduced eigensymmetry (order 3).

Four different twin laws are possible:

(i) (001) reflection twin: the cleavage plane (001) acts both as twin mirror and composition plane. The pseudo-hexagonal axes [001] of both partners are inclined to each other by 9.0°. This twin law is quite common in natural and synthetic gibbsite.

(ii) (100) reflection twin: the twin mirror plane (100) is also the composition plane. The angle between the (001) planes of both partners is 9.0°, as in (i); the pseudo-hexagonal axes [001] of both partners are parallel. This twin law is not common.

(iii) (110) reflection twin: again, twin mirror plane and composition plane coincide. The two (001) planes span an angle of 4.6°. This twin law is very rare in nature, but is often observed in synthetic materials. A sixfold sector twin of synthetic gibbsite, formed by cyclic repetition of {110} twin reflection planes 60.8° apart, is shown in Fig. 3.3.6.10. The pseudo-hexagonal axis [001] is common to all domains. Since the (001) plane is inclined towards this axis at 94.5°, the six (001) facets of the twinned crystal form a kind of `umbrella' with [001] as umbrella axis (Fig. 3.3.6.10a). This (001) umbrella faceting was recently observed in twinned synthetic gibbsite crystals by Sweegers et al. (1999).

Figure 3.3.6.10 | top | pdf | Sixfold reflection twin of gibbsite, Al(OH)3, with equivalent (110) and ), both as twin mirror and composition planes. (a) Perspective view of a tabular sixfold sector twin with pseudo-hexagonal twin axis c. In each sector the monoclinic b axis is normal to the twin axis c, whereas the a axis slopes slightly down by about 4.5° (), leading to an umbrella-like shape of the twin. (b) Polarization micrograph of a sixfold twinned hexagon (six orientation states) of the shape shown in (a). Pairs of opposite twin components have the same optical extinction position. Courtesy of Ch. Sweegers, PhD thesis, University of Nijmegen, 2001.

In contrast to orthorhombic aragonite with only three pseudo-hexagonal orientation states, these gibbsite twins exhibit six different orientation states. This is due to the absence of any eigensymmetry element along the pseudo-hexagonal axis [001]. The intersection symmetry of all orientation states is . The reduced composite symmetry of a domain pair is , with the twin mirror plane (110).

(iv) `Median law': According to Brögger (1890), this twin law implies exact parallelism of non-equivalent edges and , and vice versa, of partners I and II. The twin element is an irrational twofold axis parallel to (001), bisecting exactly the angle between [110] and [010], or alternatively, an irrational twin reflection plane normal to this axis. This interesting orientation relation, which has been observed so far only for gibbsite, does not obey the minimum condition for twinning as set out in Section 3.3.2.2. An alternative interpretation, treating these twins as rational [130] rotation twins, is given by Johnsen (1907), cf. Tertsch (1936), pp. 483–484. Interestingly, this strange `twin law' is the most abundant one among natural gibbsite twins.

Albite law : reflection twin on (010); composition plane (010) rational (Fig. 3.3.6.11, Table 3.3.6.5). with rational

Table 3.3.6.5 | top | pdf | Plagioclase: albite and pericline twins (albite) (pericline) 1 rational rational irrational irrational

Figure 3.3.6.11 | top | pdf | Polysynthetic albite twin aggregate of triclinic feldspar, twin reflection and composition plane (010).

Pericline law : twofold rotation twin along [010]; composition plane irrational : `rhombic section' (Fig. 3.3.6.12, Table 3.3.6.5). with rational

Figure 3.3.6.12 | top | pdf | Pericline twin of triclinic feldspar. Twofold twin axis [010]. (a) Twin with rational composition plane (001), exhibiting clearly the misfit (exaggerated) of the two adjacent (001) contact planes, as indicated by the crossing of lines and . (b) The same (exaggerated) twin as in (a) but with irrational boundary along the `rhombic section': fitting of contact planes from both sides ( and coincide and form a flat ridge). (c) Sketch of a real pericline twin with irrational interface (`rhombic section') containing the twin axis.

(i) Twinning by high-order merohedry (after Friedel, 1926, p. 56) was predicted by Hurst et al. (1956) in their detailed study of staurolite twinning. Staurolite crystals are supposed to consist of very fine scale monoclinic () microtwins on , which yield a twin aggregate of orthorhombic composite symmetry . The coset consists of , , and . Even though this twinning appears highly probable due to the pronounced structural pseudosymmetry (`high-order merohedry') of staurolite and has been mentioned by several authors (e.g. Smith, 1968), so far it has never been unambiguously proven. In particular, electron-microscopy investigations by Fitzpatrick (1976, quoted in Bringhurst & Griffin, 1986, p. 1470) have failed to detect the submicroscopic twins.

(ii) Superimposed upon this first generation of microtwins very often occurs one or the other of two spectacular `macroscopic' growth penetration twins in the shape of a cross, from which in 1792 the name `stauros' of the mineral was given by Delamétherié. The first detailed analysis of these twins was provided by Friedel (1926, p. 461).

(1) In many books, the twin reflection planes for the 90° cross and the 60° cross are given as (032) and (232) instead of (031) and (231). The former Miller indices refer to the morphological cell, which has a double c axis compared with the structural X-ray cell, used here.

(2) Friedel (1926) and Hurst et al. (1956) have derived both twin laws (031) and (231), mentioned above, from a multiple cubic pseudo-cell, the `Mallard pseudo-cube'. This derivation will be presented in Section 3.3.9.2.4 as a characteristic example of `twinning by reticular pseudo-merohedry'.

(i) Translationengleiche symmetry reduction cubic tetragonal of index , leading to three sets of ferroelastic `90° domains', related by the (lost) cubic {110} twin mirror planes or the (lost) cubic threefold axes. These three pseudo-merohedral orientation states point with their tetragonal c axes along the three former cube axes [100], [010] and [001], thus including angles of nearly 90°.

(ii) Each of these centrosymmetric domains splits into two antiparallel polar ferroelectric `180° domains', whereby the space group is translationsgleich reduced to of index . The total index is: .

(1) two successive phase transitions, each with its own twinning scheme, or

(2) one phase transition with loss of two kinds of symmetry elements, or

(3) a phase transition superimposed on an existing growth twin.

(i) the cubic–tetragonal () phase transition of BaTiO₃, described above. Here, 90° domains (due to the loss of the diagonal mirror planes) are superimposed by 180° domains (due to the loss of the inversion centres);

(ii) a similar case (tetragonal–monoclinic) is provided by the `type case' of Henke (2003), (NO)₂VCl₆;

(iii) ammonium lithium sulfate exhibits pseudo-hexagonal growth-sector twins upon which lamellae of ferroelectric 180° domains are superimposed.

(i) The complete merohedral `twins of twins' of quartz, i.e. the superposition of the Dauphiné, Brazil and Leydolt twins, can be formulated as follows:Combination = `complete twin': ; this symmetry corresponds to the hexagonal holohedral point group , cf. Example 3.3.6.3.

(ii) Another example is provided by KLiSO₄ (crystal class 6), extensively investigated by Klapper et al. (1987):Combination = `complete twin': ; this symmetry is isomorphic to the complete-twin symmetry of quartz, given above, and to the hexagonal holohedral point group .

(i) formation during nucleation of the crystal;

(ii) formation during crystal growth.

(i) The symmetry operations suppressed during the transition belong to the point group of the high-symmetry (prototype) phase, whereas the lattice, except for a small affine deformation, is unchanged (translationengleiche subgroup). In this case, the structures of the domains have different orientations and/or different handedness, both of which are related by the suppressed symmetry elements. Thus, the transition induces twins with the suppressed symmetry elements acting as twin elements (twin law). The number of orientation states is equal to the index of the group–subgroup relation, i.e. to the number of cosets of with respect to , including itself; cf. Section 3.3.4.1. If, for example, a threefold symmetry axis is suppressed, three domain states related by approximate 120° rotations will occur (for the problems of pseudo-n-fold twin axes, see Section 3.3.2.3.2). A further well known example is the α–β phase transformation of quartz at 846 K. On cooling from the hexagonal phase (point group 622) to the trigonal phase (point group 32), the twofold rotation , contained in the sixfold axis of -quartz, is suppressed, and so are the other five rotations of the coset [cf. Example 3.3.6.3.1]. Consequently, two domain states appear (Dauphiné twins). These twins are usually described with the twofold axis along [001] as twin element.

(ii) If a lattice translation is suppressed without change of the point-group symmetry (klassengleiche subgroup), i.e. due to loss of cell centring or to doubling (tripling etc.) of a lattice parameter, translation domains (antiphase domains) are formed (cf. Wondratschek & Jeitschko, 1976). The suppressed translation appears as the fault vector of the translation boundary (antiphase boundary) between the domains. Recently, translation domains were called `translation twins' (T-twins, Wadhawan, 1997, 2000), cf. Section 3.3.2.4, Note (7).

(iii) The two cases can occur together, i.e. point-group symmetry and translation symmetry are both reduced in one phase transition (general subgroup). Here caution in the counting of the number of domain states is advisable since now orientation states and translation states occur together.

(i) Mechanical twinning in the `traditional' sense. This kind of twinning has been studied by mineralogists and metallurgists under the name deformation twins for a long time. Well known examples are the deformation twins of calcite, galena, chalcopyrite and cubic metals. The characteristic feature is the non-existence of a real or virtual parent phase with a crystallographic supergroup. From a symmetry point of view, this means that the composite symmetry of the twin is noncrystallographic [cf. Section 3.3.4.4(ii)]. This case is illustrated by the famous deformation twins of calcite (Fig. 3.3.7.3): The eigensymmetry of calcite is , and the most conspicuous twin element is the twin reflection plane which is parallel to an edge of the cleavage rhombohedron . The extension of the eigensymmetry by this twin operation does not lead to a crystallographic composite symmetry, but the reduced composite symmetry is crystallographic, .

Figure 3.3.7.3 | top | pdf | Mechanical twins of calcite, CaCO3. All indices refer to the standard morphological cell [cf. Section 3.3.10.2.2, Example (5)]. (a) Generation of a deformation twin by a knife-edge impact (after Baumhauer, 1879). (b) Description as a glide process (`twin glide') on plane (). (c), (d) Generation of deformation twins by compression of the cleavage rhombohedron (Mügge, 1883). The shaded twin components have the same orientation. (e) Successive stages of deformation twinning of NaNO3 (isotypic with calcite) by uniaxial compression. The compression axis (vertical) is chosen with an angle of 45° against the twin glide plane and the glide direction . The direction of the twofold axis is shown in the left-hand figure. Original size of the sample: ca.  mm. Part (e) courtesy of H. E. Hoefer, PhD thesis, University of Cologne, 1989.

Another famous case is that of the deformation twins of cubic metals that obey the spinel law of mineralogy [most conspicuous twin element: reflection plane parallel to (111)]. The extension of the eigensymmetry by the twin operation leads to a noncrystallographic composite symmetry. The reduced composite symmetry , which is constructed from the intersection symmetry , however, is crystallographic (cf. Example 3.3.6.6).

A description of the (plastic) deformation by twinning in terms of strain ellipsoids is presented in Section 3.3.10.1.

(ii) Ferroelastic twinning. In 1970, a special category of mechanical twins was introduced and characterized by Aizu (1970a), who also coined the term ferroelasticity. This group of twins had already been treated, as part of the mechanical twins, by Klassen-Neklyudova in her 1964 monograph mentioned above. The defining property of a ferroelastic crystal is the existence of a displacive (group–subgroup) transition, either real or virtual, from the parent phase into the ferroelastic phase with the essential requirement that parent and daughter phase belong to different crystal families (crystal systems). Only this symmetry feature allows for a spontaneous shear strain of the twin domains. The spontaneous deformations in a pair of domains have the same magnitude, but opposite signs. If the domain states can actually be switched into each other by a mechanical stress, the phase is called ferroelastic, otherwise it is called here potentially ferroelastic.

Ferroelastic twinning is not necessarily the result of a real phase transition. Switchable ferroelastic twins are frequently formed during growth. An example is orthorhombic ammonium sulfate, which can only be grown from aqueous solution and which frequently develops pseudohexagonal growth twins with three orientation states. Above about 353 K, the grown-in domains can easily be switched stepwise from one domain state into another by an appropriate shear stress, without the sample ever undergoing a phase transition. Ammonium sulfate exhibits a virtual phase transition into a hexagonal prototype phase. It decomposes, however, at about 473 K, well before reaching the phase transition.

There are many examples (e.g. Rochelle salt) in which a ferroelastic domain structure can be generated by a real phase transition as well as by growth below the transition temperature. As a rule, the domain textures of growth and transformation twins are quite different. A detailed account of ferroelastic crystals is given by Salje (1993); a recent review is provided by Abrahams (1994).

(iii) Ferrobielastic twinning. Ferroelastic twinning implies a switchable spontaneous strain, i.e. a change of the unit-cell orientation in the different domains (twinning with change of form; Klassen-Neklyudova, 1964). In some species of crystals, however, mechanical twinning without change of the unit-cell orientation is possible (twinning without change of form). This can occur, for example, in trigonal crystals with a hexagonal P lattice. Here, the shape and orientation of the unit cell does not change from one domain to the other, and the twin is always merohedral (cf. Section 3.3.8). The atomic structure within the unit cell, however, is altered by the switching. The most famous example is the Dauphiné twinning of quartz, which can be induced by uniaxial stress along an appropriate direction. This effect was observed a long time ago by Judd (1888) and described in detail by Schubnikow & Zinserling (1932) and Zinserling & Schubnikow (1933). The `critical stress' for the Dauphiné switching decreases with increasing temperature and becomes zero at the transition to the hexagonal phase at 846 K.

The property of a crystal to form `twins without change of form' under mechanical stress was called ferrobielasticity by Newnham (1975). Aizu (1973) speaks of second-order (ferro­elastic) state shifts. It implies a change in the orientation of some tensorial properties. For Dauphiné twins of quartz, it is the elastic (fourth-rank) tensor that is responsible for the switching of the structure. Under uniaxial stress, a direction of high Young's modulus¹ is transformed into a (compatible) direction of smaller Young's modulus for which the material responds with a higher elastic yield. Note that this switching is induced by both compressive and tensile stress. A derivation of all crystal species capable of second-order ferroic state shifts by electric fields and mechanical stress, including a series of photographs showing the development of Dauphiné twins of quartz under stress, is presented by Aizu (1973).

For trigonal crystals with a rhombohedral (R) lattice, on the other hand, this switching implies the change of the obverse into the reverse rhombohedron and vice versa. In this case, the orientation of the primitive rhombohedral unit cell is changed, leading to `twinning with change of form' (i.e. not to ferrobi­elasticity), even though the orientation of the triple hexagonal cell is not changed. This kind of twinning corresponds to the (0001) reflection twins of rhombohedral crystals and the (111) spinel twins of cubic crystals (cf. Examples 3.3.6.5 and 3.3.6.6). The switching from a cubic obverse rhombohedron into the reverse one actually takes place in the ` deformation twins' of cubic metals [cf. part (i) above].

(iv) Detwinning. The generation of twins by mechanical stress allows, in reverse, the detwinning of crystals by the application of appropriate stress. This method has been extensively used for the elimination of Dauphiné twins in quartz (Thomas & Wooster, 1951; Klassen-Neklyudova, 1964, pp. 75–86). The presence of these `electrical' twins impairs the function of piezoelectric devices, such as piezoelectric resonators, made from these crystals (Iliescu & Chirila, 1995; Iliescu et al., 1997). Brazil twins of quartz, which also entail the reversal of the electric axes (cf. Fig. 3.3.6.2) cannot be detwinned. Mechanical detwinning by appropriate stress is also used to obtain single-domain crystals of the ferroelastic YBa₂Cu₃O_7−δ high-T_c superconductor. In most cases, elevated temperatures reduce the critical stress required for domain switching.

It is characteristic of ferroelectric crystals that they can be switched into a single-domain state (i.e. `detwinned' or `poled') by a sufficiently strong (coercive) electric field of proper direction. It is, however, also possible to detwin ferroelastic domains by the application of electric fields. This occurs in ferroelectric–ferroelastic crystals, where ferroelectricity and ferroelasticity are coupled, i.e. where the reversal of the electric polarity is accompanied by (mechanical) switching of the ferroelastic domains into the other deformation state and vice versa. An outstanding and well known example is Rochelle salt, which undergoes an orthorhombic–monoclinic phase transition at about 297 K with coupled ferroelectricity and ferroelasticity in the monoclinic phase (cf. Zheludev, 1971, pp. 143, 226). An extensive crystal-optical study of the ferroelastic domain switching and detwinning in Rochelle salt by electric fields, including film records of the domain movements, was presented by Chernysheva [1951, 1955; quoted after Klassen-Neklyudova (1964), pp. 75–78 and Fig. 100].

(v) Non-ferroelastic and co-elastic twins. Phase transitions with symmetry changes within the same crystal family (crystal system) also exhibit a spontaneous deformation of the unit cell, but all orientation states have the same deformation, both in magnitude and orientation. Hence, a domain switching is not possible (except for ferrobielastic crystals treated above). Therefore, this kind of phase transition and its associated domain structure are called non-ferroelastic. Salje (1993) uses the term co-elastic. In crystallography, twins resulting from this kind of phase transition are grouped under twins by merohedry (cf. Section 3.3.8). Typical examples of non-ferroelastic and co-elastic materials are quartz (merohedral Dauphiné twins, phase transition at 846 K) and calcite (transition at about 1523 K, cf. Salje, 1993, Chapter 2).

(i) No coincidence of lattice points (except, of course, for the initial pair). This case corresponds to arbitrary intergrowth of two crystals or to a general bicrystal.

(ii) One-dimensional coincidence: Both lattices have only one lattice row in common. Of the seven binary twin operations listed in Section 3.3.2.3.1, the following three generate one-dimensional lattice coincidence:

Lattices are always centrosymmetric; hence, for lattices, as well as for centrosymmetric crystals, the first two twin operations above belong to the same twin law. For noncentrosymmetric crystals, however, the two twin operations define different twin laws.

(iii) Two-dimensional coincidence: Both lattices have only one lattice plane in common. The following two (of the seven) twin operations lead to two-dimensional lattice coincidence:

Again, for lattices and centrosymmetric crystals both twin operations belong to the same twin law.

(iv) Three-dimensional coincidence: Here the coincidence subset is a three-dimensional lattice, the coincidence-site lattice or twin lattice. It is the three-dimensional sublattice common to the (equally or differently) oriented lattices of the two twin partners. The degree of three-dimensional lattice coincidence is defined by the coincidence-site lattice index, twin lattice index or sublattice index [j], for short: lattice index. This index is often called , especially in metallurgy. It is the volume ratio of the primitive cells of the twin lattice and of the (original) crystal lattice (i.e. is the `degree of dilution' of the twin lattice with respect to the crystal lattice):

The lattice index is always an integer: means complete coincidence (parallelism), partial coincidence of the two lattices. The index [j] can also be interpreted as elimination of the fraction of the lattice points, or as index of the translation group of the twin lattice in the translation group of the crystal lattice. The coincidence lattice, thus, is the intersection of the oriented lattices of the two twin partners.

Twinning with has been called by Friedel (1926, p. 427) twinning by merohedry (`macles par mériédrie') (for short: merohedral twinning), whereas twinning with is called twinning by lattice merohedry or twinning by reticular merohedry (`macles par mériédrie réticulaire') (Friedel, 1926, p. 444). The terms for are easily comprehensible and in common use. The terms for , however, are somewhat ambiguous. In the present section, therefore, the terms sublattice, coincidence lattice or twin lattice of index are preferred. Merohedral twinning is treated in detail in Section 3.3.9.

Complete and exact three-dimensional lattice coincidence () always exists for inversion twins (of noncentrosymmetric crystals) [twin operation (vii)]. For reflection twins, complete or partial coincidence occurs if a (rational) lattice row [uvw] is (exactly) perpendicular to the (rational) twin reflection plane (hkl); similarly for rotation twins if a (rational) lattice plane (hkl) is (exactly) perpendicular to the (rational) twofold twin axis [uvw].

The systematic perpendicularity relations (i.e. relations valid independent of the axial ratios) for lattice planes (hkl) and lattice rows [uvw] in the various crystal systems are collected in Table 3.3.8.1. No perpendicularity occurs for triclinic lattices (except for metrical accidents). The perpendicularity cases for monoclinic and orthorhombic lattices are trivial. For tetragonal (tet), hexagonal (hex) and rhombohedral (rhomb) lattices, systematic perpendicularity of planes and rows occurs only for the and the (or ) zones, i.e. for planes parallel and rows perpendicular to these directions, in addition to the trivial cases or . In cubic lattices, every lattice plane (hkl) is perpendicular to a lattice row [uvw] (with , , ). More general coincidence relations were derived by Grimmer (1989, 2003).

Table 3.3.8.1 | top | pdf | Lattice planes (hkl) and lattice rows [uvw] that are mutually perpendicular (after Koch, 2004) Lattice Lattice plane (hkl) Lattice row [uvw] Perpendicularity condition and quantity Triclinic — — — Monoclinic (unique axis b) (010) [010] — Monoclinic (unique axis c) (001) [001] — Orthorhombic (100) [100] — (010) [010] — (001) [001] — Hexagonal and rhombohedral (hexagonal axes) () [] , , (0001) [001] — Rhombohedral (rhombohedral axes) () [] , , (111) [111] — Tetragonal () [] , , () [] — Cubic () [] , , ;

The index of a coincidence or twin lattice can often be obtained by inspection; it can be calculated by using a formula for the auxiliary quantity as follows: with sublattice index

Here, the indices of the plane (hkl) and of the perpendicular row [uvw] are referred to a primitive lattice basis (primitive cell). For centred lattices, described by conventional bases, modifications are required; these and further examples are given by Koch (2004). Formulae and tables are presented by Friedel (1926, pp. 245–252) and by Donnay & Donnay (1972). The various equations for the quantity are also listed in the last column of Table 3.3.8.1.

Note that in the tetragonal system for any () reflection twin and any [] twofold rotation twin, the coincidence lattices are also tetragonal and have the same lattice parameter c. Further details are given by Grimmer (2003). An analogous relation applies to the hexagonal crystal family for () and [] twins. In the cubic system, the following types of twin lattices occur:

Note that triclinic twin lattices are not possible for a cubic lattice.

(i) Twins with ( twins). Here, the crystal lattice and the twin lattice are identical, i.e. the coincidence (parallelism) of the two oriented crystal lattices is complete. Hence, any twin operation must be a symmetry operation of the point group of the lattice (holohedry), but not of the point group of the crystal. Consequently, this twinning can occur in `merohedral' point groups only. This twinning by merohedry (parallel-lattice twins, twins with parallel axes) will be treated extensively in Section 3.3.9.

(ii) Twins with ( twins). This twinning does not occur systematically among the cases listed in Table 3.3.8.1, except for special metrical relations. Example: a primitive orthorhombic lattice with and twin reflection plane or . The coincidence lattice is hexagonal with and .

(iii) Twins with ( twins). Twins with are very common among rhombohedral and cubic crystals (`spinel law') with the following two representative twin operations:

Both twin operations belong to the same twin law if the crystal is centrosymmetric. Well known examples are the (0001) contact twins of calcite, the penetration twins of iron borate, FeBO₃, with the calcite structure, and the spinel twins of cubic crystals (cf. Examples 3.3.6.5, 3.3.6.6 and Figs. 3.3.6.4–3.3.6.6). For crystals with a rhombohedral (R) lattice, the coincidence lattice is the primitive hexagonal (P) sublattice (whose unit cell is commonly used for the hexagonal description of rhombohedral crystals). Here, the two centring points inside the triple hexagonal R cell do not belong to the coincidence sublattice which is, hence, of index . The same holds for the spinel twins of cubic crystals, provided only one of the four threefold axes is involved in the twinning.