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

International Tables for Crystallography (2013). Vol. D, ch. 3.3, pp. 413-487
doi: 10.1107/97809553602060000917

Chapter 3.3. Twinning of crystals

Th. Hahna* and H. Klapperb

aInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail:

This second chapter in Part 3 on twinning and domain structures deals with the twinning of crystals in all of its forms: growth twins, transformation twins and deformation twins. The treatment ranges from macroscopic considerations of the geometric orientation relations (twin laws) and the morphology of twins to the microscopic (atomistic) structures of the twin boundaries. Each of the following topics is accompanied by illustrative examples of actual twins and many figures: basic concepts and definitions: twinning, crystallographic orientation relations, composite (twin) symmetry, twin law; morphology of twins, description of twins by black–white symmetry; origin of twins and genetic classification; lattice classification of twinning: twinning by merohedry, pseudo-merohedry and `reticular' merohedry; twin boundaries: mechanical (strain) and electrical compatibility of interfaces; extension of the Sapriel approach to growth and deformation twins; twin boundaries: twin displacement and fault vectors; twin boundaries: atomistic structural models and HRTEM investigations of twin interfaces, twin textures, twinning dislocations, coherency of twin interfaces.

In this chapter, the basic concepts and definitions of twinning, as well as the morphological, genetic and lattice classifications of twins, are presented. Furthermore, twin boundaries are discussed extensively. The effect of twinning in reciprocal space, i.e. on diffraction and crystal-structure determinations, is treated in Section 3.3.11[link]. In the literature, the concept of twinning is very often used in a non-precise or ambiguous way. In order to clarify the terminology, this chapter begins with a section on the various kinds of crystal aggregates and intergrowths; in this context twinning appears as a special intergrowth of two or more crystals with well defined crystallographic orientation relations.

3.3.1. Crystal aggregates and intergrowths

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Minerals in nature and synthetic solid materials display different kinds of aggregations, in mineralogy often called intergrowths. In this chapter, we consider only aggregates of crystal grains of the same species, i.e. of the same (or nearly the same) chemical composition and crystal structure (homophase aggregates). Intergrowths of grains of different species (heterophase aggregates), e.g. heterophase bicrystals, epitaxy (two-dimensional oriented intergrowth on a surface), topotaxy (three-dimensional oriented precipitation or exsolution) or the paragenesis of different minerals in a rock or in a technical product are not treated in this chapter.

  • (i) Arbitrary intergrowth: Aggregation of two or more crystal grains with arbitrary orientation, i.e. without any systematic regularity. Examples are irregular aggregates of quartz crystals (Bergkristall) in a geode and intergrown single crystals precipitated from a solution. To this category also belong untextured polycrystalline materials and ceramics, as well as sandstone and quartzite.

  • (ii) Parallel intergrowth: Combination of two or more crystals with parallel (or nearly parallel) orientation of all edges and faces. Examples are dendritic intergrowths as well as parallel intergrowths of spinel octahedra (Fig.[link]a) and of quartz prisms (Fig.[link]b). Parallel intergrowths frequently exhibit re-entrant angles and are, therefore, easily misinterpreted as twins.


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    Parallel intergrowth (a) of spinel octahedra and (b) of hexagonal quartz prisms. Part (a) after Phillips (1971[link], p. 172), part (b) after Tschermak & Becke (1915[link], p. 94).

    Two possible reasons for the formation of parallel intergrowths are mentioned:

    (a) A smaller crystal has set down in parallel orientation on a growth face of an already existing crystal of the same species and has further grown together with its host. Fig.[link] suggests such a mechanism.

    (b) The growth of one or several faces of a crystal is inhibited by a layer of impurities or by foreign particles. By a local `breaking down' of these obstacles, several parallel individual crystals may appear and grow together during further growth. This mechanism is suggested for Fig.[link].

    In this context the term mosaic crystal must be mentioned. It was introduced in the early years of X-ray diffraction in order to characterize the perfection of a crystal. A mosaic crystal consists of small blocks (size typically in the micron range) with orientations deviating only slightly from the average orientation of the crystal; the term `lineage structure' is also used for very small scale parallel intergrowths (Buerger, 1934[link], 1960a[link], pp. 69–73).

  • (iii) Bicrystals: This term is mainly used in metallurgy. It refers to the (usually synthetic) intergrowth of two single crystals with a well defined orientation relation. A bicrystal contains a grain boundary, which in general is also well defined. Usually, homophase bicrystals are synthesized in order to study the structure and properties of grain boundaries. An important tool for the theoretical treatment of bicrystals and their interfaces is the coincidence-site lattice (CSL). A brief survey of bicrystals is given in Section 3.2.2[link] ; a comparison with twins and domain structures is provided by Hahn et al. (1999[link]).

  • (iv) Growth sectors and optical anomaly: Crystals grown with planar faces (habit faces), e.g. from vapour, supercooled melt or solution, consist of regions crystallized on different growth faces (Fig.[link]). These growth sectors usually have the shapes of pyramids with their apices pointing toward the nucleus or the seed crystal. They are separated by growth-sector boundaries, which represent inner surfaces swept by the crystal edges during growth. In many cases, these boundaries are imperfections of the crystal.


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    (a) Optical anomaly of a cubic mixed (K,NH4)-alum crystal grown from aqueous solution, as revealed by polarized light between crossed polarizers: (110) plate, 1 mm thick, horizontal dimension about 4 cm. (b) Sketch of growth sectors and their boundaries of the crystal plate shown in (a). The {111} growth sectors are optically negative and approximately uniaxial with their optical axes parallel to their growth directions [\langle 111\rangle] [birefringence [\Delta n] up to [5 \times 10^{-5}]; Shtukenberg et al. (2001[link])]. The (001) growth sector is nearly isotropic ([\Delta n \,\lt\, 10^{-6}]). Along the boundaries A between {111} sectors a few small {110} growth sectors (resulting from small {110} facets) have formed during growth. S: seed crystal.

    Frequently, the various growth sectors of one crystal exhibit slightly different chemical and physical properties. Of particular interest is a different optical birefringence in different growth sectors (optical anomaly) because this may simulate twinning. A typical example of this optical anomaly is shown in Fig.[link]. Since this phenomenon has sometimes been misinterpreted as twinning, it is treated in detail in the Extended note below.

  • (v) Translation domains: Translation domains are homogeneous crystal regions that exhibit exact parallel orientations, but are displaced with respect to each other by a vector (frequently called a fault vector), which is a fraction of a lattice translation vector. The interface between adjoining translation domains is called the `translation boundary'. Often the terms antiphase domains and antiphase boundaries are used. Special cases of translation boundaries are stacking faults. Translation domains are defined on an atomic scale, whereas the term parallel intergrowth [see item (ii)[link] above] refers to macroscopic (morphological) phenomena; cf. Note (7)[link] in Section[link]

  • (vi) Twins: A frequently occurring intergrowth of two or more crystals of the same species with well defined crystallographic orientation relations is called a twin (German: Zwilling; French: macle). Twins form the subject of the present chapter. The closely related topic of Domain structures is treated in Chapter 3.4[link] .

In 1975, J. D. H. Donnay and H. Takeda even proposed a new name for the `science of twinning': geminography [as reported by Nespolo & Ferraris (2003)[link] and by Grimmer & Nespolo (2006)[link]]. A complete review of the history and the various theories of twinning, together with an extensive list of references, is provided in a recent monograph (in French) by Boulliard (2010)[link]. In addition, it contains an extraordinarily large set of beautiful colour photographs of many natural twins.

Extended note: Optical anomaly, Curie's principle and dis­symmetry

The phenomenon of optical anomaly [cf. Section 3.3.1(iv)[link] and Fig.[link]] can be explained as follows: as a rule, impurities (and dopants) present in the solution are incorporated into the crystal during growth. Usually, the impurity concentrations differ in symmetry-non-equivalent growth sectors (which belong to different crystal forms), leading to slightly changed lattice parameters and physical properties of these sectors. In mixed crystals these changes often arise from a partial ordering of the mixing components parallel to the growth face [example: (K,NH4)-alum, cf. Fig.[link]; Shtukenberg et al., 2001[link]]. Optical anomalies may occur also in symmetry-equivalent growth sectors (which belong to the same crystal form) owing to their different growth directions: as a consequence of growth fluctuations, layers of varying impurity content are formed parallel to the growth face of the sector (`growth striations'). This causes a slight change of the interplanar spacing normal to the growth face. For example, a cubic NaCl crystal grown on the {100} cube faces from an aqueous solution containing Mn ions consists of three pairs of (opposite) growth sectors exhibiting a slight tetragonal distortion with tetragonality 10−5 along their <100> growth directions and hence is optically uniaxial (Ikeno et al., 1968[link])1.

An analogous effect of optical anomaly may be observed in crystals grown from the melt on rounded interfaces with planar facets of prominent habit faces (e.g. of melt-grown synthetic garnets). Owing to different growth mechanisms on round and facet interfaces (rough growth and growth by supercooling, respectively), the incorporation of impurities or dopants is usually different on the two types of interfaces (e.g. Hurle & Rudolph, 2004[link]). The regions crystallized on the rounded faces and on the different facets correspond to different growth sectors and may exhibit optical anomalies.

Although the phenomenon of optical anomaly closely resembles all features of twinning, it does not belong to the category `twinning' because it is not an intrinsic property of the crystal species, but rather the result of different growth conditions (or growth mechanisms) on different faces of the same crystal (`growth anisotropy'). It is the consequence of the well known Curie principle2 (Curie, 1894[link]; Chalmers, 1970[link]) which describes (as an `effect') the reduction of the symmetry (`dissymmetry') of an object (crystal) under an external influence (`cause') which itself exhibits a symmetry. It says, in terms of group theory, that the point-group symmetry GCF of the crystal under the external influence (field) F is the intersection of the symmetry GC of the crystal without field and the symmetry GF of the influence without crystal:[G_{CF}=G_C \cap G_F,]i.e. GCF is a (proper or improper) subgroup of both groups GC and GF. In the example of the optical anomaly of the {111} growth sectors of (K,NH4)-alum (Fig.[link]) the crystal point group is [G_C=2/m\bar3] and the symmetry of the cause `growth in direction [111]' is [G_F=\infty m] (symmetry of a stationary cone, cf. ITA, Table[link] ). The intersection symmetry [i.e. the sym­metry of the (111) growth sector] is GCF = 3m (`dissymmetry') with the threefold axis along the growth direction [111] of this sector. This leads to a reduction of the isotropic optical birefringence of the `undisturbed' cubic alum crystal to an uniaxial birefringence of its {111} growth sectors.

A very early review of the optical anomaly of crystals with many examples was published in 1891 by von Brauns[link]. An actual review, treating the `historical' observations and various interpretations (starting with Brewster, 1818[link]) as well as the modern aspects of optical anomalies, is presented by Kahr & McBride (1992)[link]. A similar, very comprehensive review is contained in the monograph of Shtukenberg et al. (2007)[link].

3.3.2. Basic concepts and definitions of twinning

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Because twinning is a rather complex and widespread phenomenon, several definitions have been presented in the literature. Two of them are quoted here because of the particular engagement of their authors in this topic.

George Friedel (1904[link]; 1926[link], p. 421): A twin is a complex crystalline edifice built up of two or more homogeneous portions of the same crystal species in contact (juxtaposition) and oriented with respect to each other according to well-defined laws.

These laws, as formulated by Friedel, are specified in his book (Friedel, 1926[link]). His `lattice theory of twinning' is discussed in Sections 3.3.8[link] and 3.3.9[link] of the present chapter.

Paul Niggli (1919[link], 1920/1924/1941[link]): If several crystal individuals of the same species are intergrown in such a way that all analogous faces and edges are parallel, then one speaks of parallel intergrowth. If for two crystal individuals not all but only some of the (morphological) elements (edges or faces), at least two independent ones, are parallel or antiparallel, and if such an intergrowth due to its frequent occurrence is not `accidental', then one speaks of twins or twin formation. The individual partners of typical twins are either mirror images with respect to a common plane (`twin-plane law'), or they appear rotated by 180° around a (common) direction (`zone-axis law', `hemitropic twins'), or both features occur together. These planes or axes, or both, for all frequently occurring twins turn out to be elements with relatively simple indices (referred to the growth morphology). (Niggli, 1941[link], p. 137.)

Both definitions are geometric. They agree in the essential fact that the `well defined' laws, i.e. the orientation relations between two twin partners, refer to rational planes and directions. Morphologically, these relations find their expression in the parallelism of some crystal edges and crystal faces. In these and other classical definitions of twins, the structure and energy of twin boundaries were not included. This aspect was first introduced by Buerger in 1945[link]. Definition of a twin

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In a more extended fashion we define twinning as follows:

An intergrowth of two or more macroscopic, congruent or enantiomorphic, individuals of the same crystal species is called a twin, if the orientation relations between the individuals occur frequently and are `crystallographic'. The individuals are called twin components, twin partners or twin domains. A twin is characterized by the twin law, i.e. by the orientation and chirality relation of two twin partners, as well as by their contact relation (twin interface, composition plane, domain boundary). Essential addenda to the definition

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  • (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:

    • (i) at least one lattice row (crystal edge) [uvw] is `common' to both partners I and II, either parallel or antiparallel, i.e. [[uvw]_{\rm I}] is parallel to [\pm[uvw]_{\rm II}];

    • (ii) at least two lattice planes (crystal faces) [(hkl)_{\rm I}] and [\pm(hkl)_{\rm II}], one from each partner, are `parallel', but not necessarily `common' (see below). This condition implies a binary twin operation (twofold rotation, reflection, inversion).

    Both conditions taken together define the minimal geometric requirement for a twin (at least one common lattice row and one pair of parallel lattice planes), as originally pronounced by several classical authors (Tschermak, 1884, 1905[link]; 1904[link]; Tschermak & Becke, 1915[link]; Mügge, 1911[link], p. 39; Niggli, 1920/1924/1941[link]; Tertsch, 1936[link]) and taken up later by Menzer (1955[link]) and Hartman (1956[link]). 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.[link] 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)[link] of Section[link] and in Section[link] For this distinction the terms `parallel', `common' and `coincident' are used as follows:

    • Two lattice rows [[uvw]_{\rm I}] and [[uvw]_{\rm II}]:

      Common: rows parallel or antiparallel, with their lattice points possibly displaced with respect to each other parallel to the row by a vector [{\bf t} \ne {\bf 0}].

      Coincident: common rows with pointwise coincidence of their lattice points, i.e. [{\bf t} = {\bf 0}].

    • Two lattice planes [(hkl)_{\rm I}] and [(hkl)_{\rm II}]:

      Parallel: `only' the planes as such, but not all corresponding lattice rows in the planes, are mutually parallel or antiparallel (i.e. the two planes are rotated with respect to each other).

      Common: parallel planes with all corresponding lattice rows mutually parallel or antiparallel (i.e. no rotation), but possibly displaced with respect to each other parallel to the plane by a vector [{\bf t} \ne {\bf 0}].

      Coincident: common planes with pointwise coincidence of their lattice points, i.e. [{\bf t} = {\bf 0}].

    • Two point lattices I and II:

      Parallel or common: all corresponding lattice rows are mutually parallel or antiparallel, but the lattices are possibly displaced with respect to each other by a vector [{\bf t} \ne {\bf 0}].

      Coincident: parallel lattices with pointwise coincidence of their lattice points, i.e. [{\bf t}= {\bf 0}].

    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. Specifications and extensions of the orientation relations

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In the following, the crystallographic orientation and chirality relations of two or more twin components, only briefly mentioned in the definition, are explained in detail. Two categories of orientation relations have to be distinguished: those arising from binary twin operations (binary twin elements), i.e. operations of order 2, and those arising from pseudo n-fold twin rotations (n-fold twin axes), i.e. operations of order [\geq 3]. Binary twin operations (twin elements)

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The (crystallographic) orientation relation of two twin partners can be expressed either by a twin operation or by its corresponding twin element. Binary twin elements can be either twin mirror planes or twofold twin axes or twin inversion centres. The former two twin elements must be parallel or normal to (possible) crystal faces and edges (macroscopic description) or, equivalently, parallel or normal to lattice planes and lattice rows (microscopic lattice description). Twin elements may be either rational (integer indices) or irrational (irrational indices which, however, can always be approximated by sufficiently large integer indices). Twin reflection planes and twin axes parallel to lattice planes or lattice rows are always rational. Twin axes and twin mirror planes normal to lattice planes or lattice rows are either rational or irrational. In addition to planes and axes, points can also occur as twin elements: twin inversion centres.

There exist seven kinds of binary twin elements that define the seven general twin laws possible for noncentrosymmetric triclinic crystals (crystal class 1):

  • (i) Rational twin mirror plane (hkl) normal to an irrational line: reflection twin (Fig.[link]a). The lattice plane hkl is common to both twin partners.


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    Schematic illustration of the orientation relations of noncentrosymmetric triclinic twin partners, see Section[link], (a) for twin element (i) `rational twin mirror plane' and (b) for twin element (ii) `irrational twofold twin axis' (see text); common lattice plane (hkl) for both cases. The noncentrosymmetry of the crystal is indicated by arrows. The sloping up and sloping down of the arrows is indicated by the tapering of their images. For centrosymmetry, both cases (a) and (b) represent the same orientation relation.

  • (ii) Irrational twofold twin axis normal to a rational lattice plane hkl: rotation twin (Fig.[link]b). The lattice plane hkl is common to both twin partners.

  • (iii) Rational twofold twin axis [uvw] normal to an irrational plane: rotation twin (Fig.[link]a). The lattice row [uvw] is common to both twin partners.


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    As Fig., (a) for twin element (iii) `rational twofold twin axis' and (b) for twin element (iv) `irrational twin mirror plane' (see text); common lattice row [uvw] for both cases. For centrosymmetry, both cases represent the same orientation relation.

  • (iv) Irrational twin mirror plane normal to a rational lattice row [uvw]: reflection twin (Fig.[link]b). The lattice row [uvw] is common to both twin partners.

  • (v) Irrational twofold twin axis normal to a rational lattice row [uvw], both located in a rational lattice plane (hkl); perpendicular to the irrational twin axis is an irrational plane: complex twin; German: Kantennormalengesetz (Fig.[link]). The lattice row [uvw] is `common' to both twin partners; the planes [(hkl)_{\rm I}] and [(hkl)_{\rm II}] are `parallel' but not `common' (cf. Tschermak & Becke, 1915[link], p. 98; Niggli, 1941[link], p. 138; Bloss, 1971[link], pp. 328–330; Phillips, 1971[link], p. 178).


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    Illustration of the Kantennormalengesetz (complex twin) for twin elements (v) and (vi) (see text); common lattice row [uvw] for both cases. Note that both twin elements transform the net plane [(hkl)_{\rm I}] into its parallel but not pointwise coincident counterpart [(hkl)_{\rm II}]. For centrosymmetry, both cases represent the same orientation relation.

  • (vi) Irrational twin mirror plane containing a rational lattice row [uvw]; perpendicular to the twin plane is an irrational direction; the row [uvw] and the perpendicular direction span a rational lattice plane (hkl): complex twin; this `inverted Kantennormalengesetz' is described here for the first time (Fig.[link]). The row [uvw] is `common' to both twin partners; the planes [(hkl)_{\rm I}] and [(hkl)_{\rm II}] are `parallel' but not `common'.

  • (vii) Twin inversion centre: inversion twin. The three-dimensional lattice is common to both twin partners. Inversion twins are always merohedral (parallel-lattice) twins (cf. Section 3.3.9[link]).

All these binary twin elements – no matter whether rational or irrational – lead to crystallographic orientation relations, as defined in Section[link], because the following lattice items belong to both twin partners:

  • (a) The rational lattice planes [(hkl)_{\rm I}] and [(hkl)_{\rm II}] are `common' for cases (i)[link] and (ii)[link] (Fig.[link]).

  • (b) The rational lattice rows [[uvw]_{\rm I}] and [[uvw]_{\rm II}] are `common' and furthermore lattice planes [(hkl)_{\rm I}/(hkl)_{\rm II}] in the zone [uvw] are `parallel', but not `common' for cases (iii)[link], (iv)[link], (v)[link] and (vi)[link]. Note that for cases (iii)[link] and (iv)[link] any two planes [(hkl)_{\rm I}/(hkl)_{\rm II}] of the zone [uvw] are parallel, whereas for cases (v)[link] and (vi)[link] only a single pair of parallel planes exists (cf. Figs.[link] and[link]).

  • (c) The entire three-dimensional lattice is `common' for case (vii)[link].

In this context one realizes which wide range of twinning is covered by the requirement of a crystallographic orientation relation: the `minimal' condition is provided by the complex twins (v)[link] and (vi)[link]: only a one-dimensional lattice row is `common', two lattice planes are `parallel' and all twin elements are irrational (Fig.[link]). The `maximal' condition, a `common' three-dimensional lattice, occurs for inversion twins (`merohedral' or `parallel-lattice twins'), case (vii)[link]; for displacement vector t = 0, the three-dimensional lattice is even `coincident'.

In noncentrosymmetric triclinic crystals, the above twin elements define seven different twin laws, but for centrosymmetric crystals only three of them represent different orientation relations, because both in lattices and in centrosymmetric crystals a twin mirror plane defines the same orientation relation as the twofold twin axis normal to it, and vice versa. Consequently, the twin elements of the three pairs (i)[link] + (ii)[link], (iii)[link] + (iv)[link] and (v)[link] + (vi)[link] represent the same orientation relation. Case (vii)[link] does not apply to centrosymmetric crystals, since here the inversion centre already belongs to the symmetry of the crystal.

For symmetries higher than triclinic, even more twin elements may define the same orientation relation, i.e. form the same twin law. Example: the dovetail twin of gypsum (point group [12/m1]) with twin mirror plane (100) can be described by the four alternative twin elements (i)[link], (ii)[link], (iii)[link], (iv)[link] (cf. Section 3.3.4[link], Fig.[link]). Furthermore, with increasing symmetry, the twin elements (i)[link] and (iii)[link] may become even more special, and the nature of the twin type may change as follows:

  • (i) the line normal to a rational twin mirror plane (hkl) may become a rational line [uvw];

  • (ii) the plane normal to a rational twofold twin axis [uvw] may become a rational plane (hkl).

In both cases, the three-dimensional lattice (or a sublattice of it) is now common to both twin partners, i.e. a `merohedral' twin results (cf. Section 3.3.9[link]).

There is one more binary twin type which seems to reduce even further the above-mentioned `minimal' condition for a crystallographic orientation relation, the so-called `median law' (German: Mediangesetz) of Brögger (1890[link]), described by Tschermak & Becke (1915[link], p. 99). So far, it has been found in one mineral only: hydrargillite (modern name gibbsite), Al(OH)3. The acceptability of this orientation relation as a twin law is questionable; see Section[link]. Pseudo n-fold twin rotations (twin axes) with [n\ge 3]

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There is a long-lasting controversy in the literature, e.g. Hartman (1956[link], 1960[link]), Buerger (1960b[link]), Curien (1960[link]), about the acceptance of three-, four- and sixfold rotation axes as twin elements, for the following reason:

Twin operations of order two (reflection, twofold rotation, inversion) are `exact', i.e. in a component pair they transform the orientation state of one component exactly into that of the other and vice versa. There occur, in addition, many cases of multiple twins, which can be described by three-, four- and sixfold twin axes. These axes, however, are pseudo axes because their rotation angles are close to but not exactly equal to 120, 90 or 60°, due to metrical deviations (no matter how small) from a higher-symmetry lattice. A well known example is the triple twin (German: Drilling) of orthorhombic aragonite, where the rotation angle [\gamma=] [2\arctan b/a = 116.2^\circ] (which transforms the orientation state of one component exactly into that of the other) deviates significantly from the 120° angle of a proper threefold rotation (Fig.[link]). Another case of n = 3 with a very small metrical deviation is provided by ammonium lithium sulfate (γ = 119.6°).


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(a) Triple growth twin of orthorhombic aragonite, CaCO3, with pseudo-threefold twin axis. The gap angle is 11.4° (= 360° − 3 × 116.2°). The exact description of the twin aggregate by means of two symmetry-equivalent twin mirror planes (110) and ([{\bar 1}10]) is indicated. In actual crystals, the gap is usually closed as shown in (b).

All these (pseudo) n-fold rotation twins, however, can also be described by (exact) binary twin elements, viz by a cyclic sequence of twin mirror planes or twofold twin axes. This is also illustrated and explained in Fig.[link]. This possibility of describing cyclic twins by `exact' binary twin operations is the reason why Hartman (1956[link], 1960[link]) and Curien (1960[link]) do not consider `non-exact' three-, four- and sixfold rotations as proper twin operations.

The crystals forming twins with pseudo n-fold rotation axes always exhibit metrical pseudosymmetries. In the case of transformation twins and domain structures, the metrical pseudosymmetries of the low-symmetry (deformed) phase [\cal{H}] result from the true structural symmetry [{\cal G}] of the parent phase (cf. Section[link]). This aspect caused several authors [e.g. Friedel, 1926[link], pp. 435 and 464; Donnay (cf. Hurst et al., 1956[link]); Buerger, 1960b[link]] to accept these pseudo axes for the treatment of twinning. The present authors also recommend including three-, four- and sixfold rotations as permissible twin operations. The consequences for the definition of the twin law will be discussed in Section 3.3.4[link] and in Section 3.4.3[link] . For a further extension of this concept to fivefold and tenfold multiple growth twins, see Note (6)[link] below and Example[link]. Notes on the definition of twinning

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  • (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 published by Takeuchi (1997[link]).

  • (2) Rational twin elements are designated by integer Miller indices (hkl) or (hkil) and integer direction indices [uvw] for twin mirror planes and twin rotation axes, respectively; the values of these indices usually are small numbers, [\lt\, 6]. 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 an inherent 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[link]). 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 KLiSO4, for example, the prominent composition plane [(0001)] is normal to the twin mirror plane [(10{\bar 1}0)] (cf. Klapper et al., 1987[link]). 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[link]; Ellner, 1995[link]). These twins are due to pseudo-pentagonal or pseudo-decagonal metrical features of the lattice [[\gamma=] [\arctan(c/a)\approx72^\circ]]. They can be treated in the same way as the three-, four- and sixfold rotation twins mentioned in Section[link]. This includes the alternative (`exact') description of the twinning by a cyclic sequence of symmetry-equivalent twin reflection planes or twofold twin axes (cf. aragonite, Fig.[link]). 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[link] ), 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[link], 2000[link]). 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[link](v)[link]]. 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 possible 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 as well as their relaxations. The role of the twin displacement vectors in the structure of the twin boundary is discussed in Section[link].

  • (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) Allotwins (Nespolo, Kogure & Ferraris, 1999[link]): oriented crystal associations of different polytypes of the same compound. Different polytypes have the same chemistry and similar but nevertheless different structures. Well known examples are the micas, where twins and allotwins occur together (Nespolo et al., 2000[link]).

    • (ii) The oriented associations of two polymorphs (modifications) of the same compound across a phase boundary also do not deserve the name twin because of the different structures of their components. They are often called `syntaxy'. Note that this case is more general than the `allo­twins' of polytypes mentioned above. Interesting examples are the oriented intergrowths of the TiO2 polymorphs rutile, anatase and brookite. In this respect, the high-resolution transmission electron microscopy (HRTEM) studies of hydrothermally coars­ened TiO2 nanoparticles by Penn & Banfield (1998[link]), showing the structures of anatase [\{121\}] reflection twin boundaries and of {112} anatase–(100) brookite interfaces on an atomic scale, are noteworthy.

    • (iii) Plesiotwins (Nespolo, Ferraris et al., 1999[link]): oriented crystal associations based on a large coincidence-site lattice (CSL). The composition plane has a low degree of restoration of lattice nodes and the relative rotations between individuals are noncrystallographic. In the phlogopite from Mutsure-jima, Japan, both ordinary twins and plesiotwins are reported (Sunagawa & Tomura, 1976[link]).

3.3.3. Morphological classification, simple and multiple twinning

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Before discussing the symmetry features of twinning in detail, it is useful to introduce the terms `simple' and `multiple' twins, which are sometimes grouped under the heading `repetitive or repeated twins'. This is followed by some morphological aspects of twinning.

Simple twins are aggregates that consist of domains of only two orientation states, irrespective of the number, size and shape of the individual domains, Fig.[link](b). Thus, only one orientation relation (one twin law) exists. Contact twins and polysynthetic twins (see Section[link] below) are simple twins, as are the spinel penetration twins.


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Schematic illustration of simple (polysynthetic) and multiple (cyclic) twins. (a) Equivalent twin mirror planes (110) and [(1{\bar 1}0)] of an orthorhombic crystal. (b) Simple (polysynthetic) twin with two orientation states due to parallel repetition of the same twin mirror plane [(1{\bar 1}0)]; the twin components are represented by {110} rhombs. (c) Multiple (cyclic) twin with several (more than two) orientation states due to cyclic repetition of equivalent twin mirror planes of type {110}.

Multiple twins are aggregates that contain domains of three or more orientation states, i.e. at least two twin laws are involved. Two cases have to be distinguished:

  • (i) The twin elements are symmetry-equivalent with respect to the eigensymmetry group [{\cal H}] of the crystal (cf. Section[link]). A typical example is provided by the equivalent (110) and ([1{\bar 1}0]) twin mirror planes of an orthorhombic crystal (e.g. aragonite), which frequently lead to cyclic twins (cf. Figs.[link]a and c).

  • (ii) The twin elements are not equivalent with respect to the eigensymmetry of the crystal, i.e. several independent twin laws occur simultaneously in the twinned crystal. A typical example is provided by a Brazil twin of quartz, with each Brazil domain containing Dauphiné twins. This results in four domain states and three twin laws; cf. Example[link].

The distinction of simple and multiple twins is important for the following morphological classification. Further examples are given in Section 3.3.6.[link] Morphological classification

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The morphology of twinned crystals, even for the same species and the same orientation relation, can be quite variable. For a given orientation relation the morphology depends on the geometry of the twin boundary as well as on the number of twin partners. A typical morphological feature of growth-twinned crystals is the occurrence of re-entrant angles. These angles are responsible for an increased growth velocity parallel to the twin boundary. This is the reason why twinned crystals often grow as platelets parallel to the composition plane (cf. Section[link]). Detailed studies of the morphology of twins versus untwinned crystals were carried out as early as 1911 by Becke (1911[link]). As a general observation, twinned crystals grow larger than untwinned crystals in the same batch.

The following classification of twins is in use:

  • (i) Contact twins. Two twin partners are in contact across a single composition plane [\pm(hkl)], the Miller indices [h,k,l] 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)[link] in Section[link]]. Examples are shown in Fig.[link] for gypsum, in Fig.[link] for calcite and in Fig.[link](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 (Fig.[link]b). 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.[link]. 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.[link](b). 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.[link] and[link]). Another famous example is the Carlsbad twin of orthoclase feldspar with [001] as a twofold twin axis (Fig.[link]). 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.[link]c,[link] and[link]). Impressive examples are the `sixlings' and `eightlings' of rutile (Fig.[link], cf. Example[link]).

    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:

    • (a) they can be described by repeated action of (equivalent) reflection planes or twofold twin axes with suitable angular spacings;

    • (b) they can be described by approximate twin rotation axes of order three or more (including noncrystallographic axes such as fivefold); cf. Section[link] and Note (6)[link] in Section[link]

    Prominent examples are the growth twins of NH4LiSO4 (Fig.[link]), aragonite (Fig.[link]), K2SO4 (Fig.[link]) and certain alloys with pseudo-fivefold twin axes (Fig.[link]). 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, described in Section[link](iii).

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

    • (a) dovetail twins (prominent example: gypsum, cf. Figs.[link] and[link]);

    • (b) elbow twins (rutile, cassiterite, cf. Fig.[link]a);

    • (c) arrowhead twins (diamond);

    • (d) iron-cross twins (pyrite);

    • (e) butterfly twins (perovskite).

It is obvious from the morphological features of twins, described in this section, that crystals – by means of twinning – strive to simulate higher symmetries than they actually have. This will be even more apparent in the following section, which deals with the composite symmetry of twins and the twin law.

3.3.4. Composite symmetry and the twin law

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In this section we turn our attention to the symmetry relations in twinning. The starting point of all symmetry considerations is the eigensymmetry [{\cal H}] of the untwinned crystal, i.e. the point group or space group of the single crystal, irrespective of its orientation and location in space. All domain states of a twinned crystal have the same (or the enantiomorphic) eigensymmetry but may exhibit different orientations. The orientation states of each two twin components are related by a twin operation k which cannot be part of the eigensymmetry [{\cal H}]. The term eigensymmetry is introduced here in order to provide a short and crisp distinction between the symmetry of the untwinned crystal (single-domain state) and the composite symmetry [{\cal K}] of a twinned crystal, as defined below. It should be noted that in crystal morphology the term eigensymmetry is also used, but with another meaning, in connection with the symmetry of face forms of crystals (Hahn & Klapper, 2005[link], pp. 764 and 791). Composite symmetry

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For a comprehensive characterization of the symmetry of a twinned crystal, we introduce the important concept of composite symmetry [{\cal K}]. This symmetry is defined as the extension of the eigensymmetry group [{\cal H}] by a twin operation k. This extension involves, by means of left (or right) coset composition [k\times {\cal H}], the generation of further twin operations until a supergroup is obtained. This supergroup is the composite symmetry group [{\cal K}].

In the language of group theory, the relation between the composite symmetry group [{\cal K}] and the eigensymmetry group [{\cal H}] can be expressed by a (left) coset decomposition of the supergroup [{\cal K}] with respect to the subgroup [{\cal H}]: [{\cal K} = k_1\times {\cal H} \cup k_2\times {\cal H} \cup k_3\times {\cal H} \cup \ldots \cup k_i\times {\cal H},]where [k_1] is the identity operation; [k_1\times {\cal H} = {\cal H}\times k_1 = {\cal H}].

The number i of cosets, including the subgroup [{\cal H}], is the index [i] of [{\cal H}] in [{\cal K}]; this index corresponds to the number of different orientation states in the twinned crystal. If [{\cal H}] is a normal subgroup of [{\cal K}], which is always the case if i = 2, then [k\times {\cal H} = {\cal H}\times k ], i.e. left and right coset decomposition leads to the same coset. The relation that the number of different orientation states n equals the index [i] of [{\cal H}] in [{\cal K}], i.e. [n = [i] = \left\vert {\cal K}\right\vert: \left\vert {\cal H}\right\vert], was first expressed by Zheludev & Shuvalov (1956[link], p. 540) for ferroelectric phase transitions.

These group-theoretical considerations can be translated into the language of twinning as follows: although the eigensymmetry [{\cal H}] and the composite symmetry [{\cal K}] can be treated either as point groups (finite order) or space groups (infinite order), in this and the subsequent sections twinning is considered only in terms of point groups [see, however, Note (8)[link] in Section[link], as well as Section[link]]. With this restriction, the number of twin operations in each coset equals the order [\left\vert{\cal H}\right\vert] of the eigensymmetry point group [{\cal H}]. All twin operations in a coset represent the same orientation relation, i.e. each one of them transforms orientation state 1 into orientation state 2. Thus, the complete coset characterizes the orientation relation comprehensively and is, therefore, defined here as the twin law. The different operations in a coset are called alternative twin operations. A further formulation of the twin law in terms of black–white symmetry will be presented in Section 3.3.5.[link] Many examples are given in Section 3.3.6.[link]

This extension of the `classical' definition of a twin law from a single twin operation to a complete coset of alternative twin operations does not conflict with the traditional description of a twin by the one morphologically most prominent twin operation. In many cases, the morphology of the twin, e.g. re-entrant angles or the preferred orientation of a composition plane, suggests a particular choice for the `representative' among the alternative twin operations. If possible, twin mirror planes are preferred over twin rotation axes or twin inversion centres.

The concept of the twin law as a coset of alternative twin operations, defined above, has been used in more or less complete form before. The following authors may be quoted: Mügge (1911[link], pp. 23–25); Tschermak & Becke (1915[link], p. 97); Hurst et al. (1956[link], p. 150); Raaz & Tertsch (1958[link], p. 119); Takano & Sakurai (1971[link]); Takano (1972[link]); Van Tendeloo & Amelinckx (1974[link]); Donnay & Donnay (1983[link]); Zikmund (1984[link]); Wadhawan (1997[link], 2000[link]); Nespolo et al. (2000[link]). A systematic application of left and double coset decomposition to twinning and domain structures has been presented by Janovec (1972[link], 1976[link]) in a key theoretical paper. An extensive group-theoretical treatment with practical examples is provided by Flack (1987[link]).

Example: dovetail twin of gypsum (Fig.[link]).  Eigensymmetry:[{\cal H} = 1 {2_y\over m_y} 1.]Twin reflection plane (100):[k_2 = k = m_x.]Composite symmetry group [{\cal K}_D] (orthorhombic):[ {\cal K} = {\cal H} \cup k \times {\cal H},]given in orthorhombic axes, [x, y, z]. The coset [k\times {\cal H} ] contains all four alternative twin operations (Table[link]) and, hence, represents the twin law. This is clearly visible in Fig.[link](a). In the symbol of the orthorhombic composite group, [{\cal K} = {2'_x\over m'_x}{2_y\over m_y}{2'_z\over m'_z},]the primed operations indicate the coset of alternative twin operations. The above black-and-white symmetry symbol of the (orthorhombic) composite group [{\cal K}] is another expression of the twin law. Its notation is explained in Section 3.3.5[link]. The twinning of gypsum is treated in more detail in Example[link].

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Gypsum, dovetail twins: coset of alternative twin operations (twin law), given in orthorhombic axes of the composite symmetry [{\cal K}_D]

[{\cal H}][k\times {\cal H}]
1 [m_x \times 1 = m_x]
[2_y] [m_x \times 2_y = m_z]
[m_y] [m_x \times m_y = 2_z]
[{\bar 1}] [m_x \times {\bar 1} = 2_x]

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Gypsum dovetail twin: schematic illustration of the coset of alternative twin operations. The two domain states I and II are represented by oriented parallelograms of eigensymmetry [2_y/m_y]. The subscripts x and z of the twin operations refer to the coordinate system of the orthorhombic composite symmetry [{\cal K}_D] of this twin; a and c are the monoclinic coordinate axes.

It should be noted that among the four twin operations of the coset [k\times {\cal H} ] two are rational, [m_x] and [2_z], and two are irrational, [m_z] and [2_x] (Fig.[link]). All four are equally correct descriptions of the same orientation relation. From morphology, however, preference is given to the most conspicuous one, the twin mirror plane [m_x = (100)], as the representative twin element.

The concept of composite symmetry [{\cal K}] is not only a theoretical tool for the extension of the twin law but has also practical aspects:

  • (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 (p = `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.[link] for gypsum. It is emphasized that, apart from this rare and idealized case, the composite symmetry does not give the symmetry of a twin specimen (twin symmetry), but rather a group-theoretical representation of the twin law including the coset of all alternative twin elements.

  • (ii) Diffraction pattern. The `single-crystal diffraction pattern' of a twinned crystal exhibits its composite symmetry [{\cal K}] if the volume fractions of all domain states are (approximately) equal.

  • (iii) Permissible twin boundaries. The composite symmetry [{\cal K}] in its black–white notation permits immediate recognition of the `permissible twin boundaries' (W-type composition planes), as explained in Section[link] Equivalent twin laws

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In the example of the dovetail twin of gypsum above, the twin operation [k = m_x = m(100)] is of a special nature in that it maps the entire eigensymmetry [{\cal H}=12/m1] onto itself and, hence, generates a single coset, a single twin law and a finite composite group [{\cal K}] of index [2] (simple twin). There are other twin operations, however, which do not leave the entire eigensymmetry invariant, but only a part (subgroup) of it, as shown for the hypothetical (111) twin reflection plane of gypsum in Example[link]. In this case, extension of the complete group [\cal H] by such a twin operation k does not lead to a single twin law and a finite composite group, but rather generates in the same coset two or more twin operations [k_2,k_3,\ldots,] which are independent (non-alternative) but symmetry equivalent with respect to the eigensymmetry [\cal H], each representing a different but equivalent twin law. If applied to the `starting' orientation state 1, they generate two or more new orientation states 2, 3, 4, [\ldots]. In the general case, continuation of this procedure would lead to an infinite set of domain states and to a composite group of infinite order (e.g. cylinder or sphere group). Specialized metrics of a crystal can, of course, lead to a `multiple twin' of small finite order.

In order to overcome this problem of the `infinite sets' and to ensure a finite composite group (of index [2]) for a pair of adjacent domains, we consider only that subgroup of the eigensymmetry [{\cal H}] which is left invariant by the twin operation k. This subgroup is the `intersection symmetry' [{\cal H}^*] of the two `oriented eigensymmetries' [{\cal H}_1] and [{\cal H}_2] of the domains 1 and 2 (shown in Fig.[link]): [{\cal H}^* = {\cal H}_1\cap {\cal H}_2]. This group [{\cal H}^*] is now extended by k and leads to the `reduced composite symmetry' [{\cal K}^*] of the domain pair (1, 2): [{\cal K}^*(1,2)={\cal H}^*\cup k_2\times {\cal H}^*], which is a finite supergroup of [{\cal H}^*] of index [2]. In this way, the complete coset [k \times {\cal H}] of the eigensymmetry [\cal H] is split into two (or more) smaller cosets [k_2 \times {\cal H}^*], [k_3\times{\cal H}^*] etc., where [k_2, k_3,\ldots,] are symmetry-equivalent twin operations in [\cal H]. Correspondingly, the differently oriented `reduced composite symmetries' [{\cal K}^*(1,2) =] [{\cal H}^*\cup k_2\times{\cal H}^*], [{\cal K}^*(1,3)={\cal H}^*\cup k_3\times{\cal H}^*] etc. of the domain pairs (1, 2), (1, 3) etc. are generated by the representative twin operations [k_2], [k_3] etc. These cosets [k_i\times{\cal H}^*] are considered as the twin laws for the corresponding domain pairs.


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Twinning of an orthorhombic crystal (eigensymmetry 2/m 2/m 2/m) with equivalent twin mirror planes (110) and [({\bar 1}10)]. Three twin domains 1, 2 and 3, bound by {110} contact planes, are shown. The oriented eigensymmetries [{\cal H}_1], [{\cal H}_2], [{\cal H}_3] and the reduced composite symmetries [{\cal K}^\star (1,2) = {\cal K}^\star (110)] and [{\cal K}^\star (1,3) = {\cal K}^\star({\bar 1}10)] of each domain pair are given in stereographic projection. The intersection symmetry of all domains is [{\cal H}^\star = 112/m].

As an example, an orthorhombic crystal of eigensymmetry [{\cal H} = 2/m\,2/m\,2/m] with equivalent twin reflection planes [k_2 = m(110)] and [k_3 = m({\bar 1}10)] is shown in Fig.[link]. From the `starting' domain 1, the two domains 2 and 3 are generated by the two twin mirror planes [(110)] and [({\bar 1}10)], symmetry equivalent with respect to the oriented eigensymmetry [{\cal H}_1] of domain 1. The intersection symmetries of the two pairs of oriented eigensymmetries [{\cal H}_1] & [{\cal H}_2] and [{\cal H}_1] & [{\cal H}_3] are identical: [{\cal H}^\ast = 112/m]. The three oriented eigensymmetries [{\cal H}_1], [{\cal H}_2], [{\cal H}_3], as well as the two differently oriented reduced composite symmetries [{\cal K}^\ast(1,2) =] [{\cal K}^\ast(110)] and [{\cal K}^\ast(1,3) = {\cal K}^\ast({\bar 1}10)] of the domain pairs (1, 2) and (1, 3), are all isomorphic of type [2/m\,2/m\,2/m], but exhibit different orientations. Classification of composite symmetries

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The discussions and examples briefly presented in the previous section are now extended in a more general way. For the classification of composite symmetries [{\cal K}] we introduce the notion of oriented eigensymmetry [{\cal H}_j] of an orientation state j and attach to it its geometric representation, the framework of oriented eigensymmetry elements, for short framework of oriented eigensymmetry. Twin partners of different orientation states have the same eigensymmetry [{\cal H}] but exhibit different oriented eigensymmetries [{\cal H}_j], which are geometrically represented by their frameworks of oriented eigensymmetry. The well known crystallographic term `framework of symmetry' designates the spatial arrangement of the symmetry elements (planes, axes, points) of a point group or a space group, as represented by a stereographic projection or by a space-group diagram (cf. Hahn, 2005[link], Parts 6[link] , 7[link] and 10[link] ).

Similarly, we also consider the intersection group [{\cal H}^\ast =] [{\cal H}_1 \cap {\cal H}_2] of the oriented eigensymmetries [{\cal H}_1] and [{\cal H}_2] and its geometric representation, the framework of intersection symmetry. Two cases of intersection symmetries have to be distinguished:

  • Case (I): [ {\cal H}^\ast = {\cal H}_1 \cap {\cal H}_2 = {\cal H}]. 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 [{\cal H}_1] and [{\cal H}_2] and their intersection group [{\cal H}^\ast] coincide. Hence, for binary twin operations there is only one coset [k \times {\cal H} = k \times {\cal H}^\ast] and one twin law. The composite symmetry [{\cal K} = {\cal H}^\ast \cup k \times {\cal H}^\ast] is crystallographic. An example is provided by the dovetail twins of gypsum, described above: [{\cal H}^* = {\cal H}=12/m1] (cf. Table[link]).

  • Case (II): [{\cal H}^\ast ] = [{\cal H}_1 \cap {\cal H}_2 \,\lt\, {\cal H}_1 \hbox{ and } \lt\, {\cal H}_2 \,(\hbox{index }[i]\geq 2)]. Here, the twin operations map only a fraction [1/i] of the oriented symmetry elements of domain states 1 and 2 onto each other. Hence, the intersection group [{\cal H}^\ast] of the two oriented eigensymmetries [{\cal H}_1] and [{\cal H}_2] is a proper subgroup of index [[i]\geq 2] of both [{\cal H}_1] and [{\cal H}_2]. The coset [k \times {\cal H}^\ast] leads to the crystallographic reduced composite symmetry [{\cal K}^\ast = {\cal H}^\ast \cup k \times {\cal H}^\ast,]which is crystallographic as for case (I)[link] above. The number of twin laws, which are different but equivalent with respect to the `starting' eigensymmetry [{\cal H}_1] of the first domain state 1, equals the index [i]. This implies i differently oriented domain pairs [(1, j)\,\, (j = 1,2,\ldots, i)]. The composite symmetry of such a domain pair is now defined by [{\cal K}_{1,j}^\ast = {\cal H}^\ast \cup k_j \times {\cal H}^\ast] and is called the reduced composite symmetry [{\cal K}^*]. All twin laws can also be expressed by the black–white symbol of the reduced composite symmetry [{\cal K}_{1,j}^\ast], as described in Section 3.3.5.[link]

The orthorhombic example given in Section[link] (Fig.[link]) is now extended as follows:

Eigensymmetry [{\cal H} = 2/m\,2/m\,2/m], intersection symmetry [{\cal H}^\ast = 112/m], [k_1=] identity, [k_2 = m(110)], [k_3 = m({\bar 1}10)], [[i]= 2]. The two cosets [k_2 \times {\cal H}^\ast] and [k_3 \times {\cal H}^\ast] are listed in Table[link]. From these cosets the two different reduced composite symmetries [{\cal K}^\ast(1,2)] and [{\cal K}^\ast(1,3)] of type 2/m 2/m 2/m are derived as follows: [{\cal K}^\ast(1,2) = {\cal H}^\ast \cup k_2 \times {\cal H}^\ast \ \ {\rm and} \ \ {\cal K}^\ast(1,3) = {\cal H}^\ast \cup k_3 \times {\cal H}^\ast.]

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Reduced composite symmetries [{\cal K}^*(1,2)={\cal H}^*\cup k_2\times{\cal H}^*] and [{\cal K}^*(1,3)={\cal H}^*\cup k_3 \times {\cal H}^*] for the orthorhombic example in Fig.[link]

[{\cal H}^\ast][k_2 \times {\cal H}^\ast ][k_3 \times {\cal H}^\ast]
1 [m(110)] [m({\bar 1}10)]
[2_z] [m\perp[{\bar 1}10]] [m\perp[110]]
[m_z] [2\parallel [{\bar 1}10]] [2\parallel [110]]
[{\bar 1}] [2\perp (110)] [2\perp ({\bar 1}10)]

These groups of reduced composite symmetry are always crystallographic and finite.

Note that the twin operations in these two reduced cosets would form one coset if one of the operations ([k_2] or [k_3]) were applied to the full eigensymmetry [{\cal H}] (twice as long as [{\cal H}^\ast]): [k_2 \times {\cal H} = k_3 \times {\cal H} = k_2 \times {\cal H}^\ast \cup k_3 \times {\cal H}^\ast]. This process, however, would not result in a finite group, whereas the two reduced cosets lead to groups of finite order.

The two twin laws, based on [k_2(110)] and [k_3({\bar 1}10)], can be expressed by a black–white symmetry symbol of type [{\cal K}^\ast = 2'/m'\,2'/m'\,2/m] with [{\cal H}^\ast = 112/m]. The frameworks of these two groups, however, are differently oriented (cf. Fig.[link]).

In the limiting case, the intersection group [{\cal H}^\ast] consists of the identity alone (index [i] = order [\vert {\cal H}\vert] of the eigensymmetry group), i.e. the two frameworks of oriented eigensymmetry have no symmetry element in common. The number of equivalent twin laws then equals the order [\vert {\cal H}\vert] of the eigensymmetry group, and each coset consists of one twin operation only. Categories of composite symmetries

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After this preparatory introduction, the three categories of composite symmetry are treated.

  • (i) Crystallographic composite symmetry. According to case (I)[link] above, only the following three types of twins have crystallographic composite symmetry [{\cal K}] with two orientation states, one coset and, hence, one twin law:

    • (a) all merohedral twins, both with Σ = 1 and Z > 1 (reticular merohedry) (cf. Section 3.3.9[link]);

    • (b) twins of `monoaxial' eigensymmetry [{\cal H}] that have either a twin reflection plane parallel or a twofold twin axis normal to the single eigensymmetry axis. Monoaxial eigensymmetries are 2, [m = {\bar 2}], [2/m], 3, [{\bar 3}], 4, [{\bar 4}], [4/m], 6, [{\bar 6=3/m}], [6/m];

    • (c) the triclinic eigensymmetry groups 1 and [{\bar 1}]; here any binary twin element leads to a crystallographic composite symmetry [{\cal K}].

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

  • (ii) Noncrystallographic composite symmetry. As shown below, a noncrystallographic composite symmetry [{\cal K}] results if the conditions of case (II)[link] 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.[link](c) and[link], where the twinning of an orthorhombic crystal with eigensymmetry [{\cal H} = 2/m\,2/m\,2/m] and twin mirror plane (110) is considered. In case (II)[link] above and in Fig.[link], domains 2 and 3 are generated from the starting domain 1 by the application of the equivalent twin elements [m_1(110)] and [m_1({\bar 1}10]). By applying the two twin elements [m_2(110)] and [m_2({\bar 1}10)] of domain 2, a new domain 4 is obtained and, at the same time, domain 1 is reproduced. Similarly, the twin elements [m_3(110)] and [m_3({\bar 1}10)] 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 [{\cal K} = \infty/mm], with the rotation axis parallel to the twofold axis of the intersection symmetry [112/m], 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 [{\cal K} = m{\overline{ \infty}}] 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 [{\cal K}^\ast] for any pair of domains in contact, as derived in case (II)[link] 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 [{\cal H} = 2/m\,2/m\,2/m] and equivalent twin reflection planes [(110)] and [(1{\bar 1}0]), but now with a special axial ratio [b/a \approx \left\vert \tan(180^\circ/n)\right\vert] (n = 3, 4 or 6).

    The procedure described above in (ii)[link] 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[link]). The intersection group [{\cal H}^\ast] of all these domain states is [112/m], with the twofold axis along the c axis. The reduced composite symmetry of any pair of domains in contact is orthorhombic of type [{\cal K}^\ast = 2'/m'\,2'/m'\,2/m]. If the only twin element is an n-fold rotation or roto-inversion axis [(m \ge 3)], no restriction on the b/a ratio is required. This case is discussed below under `Remark'.

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

    • (a) by repeated application of equivalent binary twin operations (reflections or twofold rotations) to a pseudosymmetrical crystal, as proposed by Hartman (1960[link]) and Curien (1960[link]). Note that each one of these binary twin operations is `exact', whereas the closure of the cycle of sectors is only approximate; the deviation from [360^\circ/n] depends on the (metrical) pseudosymmetry of the lattice;

    • (b) by successive application of pseudo n-fold twin rotations around the zone axis of the equivalent twin reflection planes. Note that the individual rotation angles are not exactly [360^\circ/n], due to the pseudosymmetry of the lattice. This alternative description corresponds to the approach by Friedel (1926[link], pp. 435 and 464) and Buerger (1960b[link]).

    It is now reasonable to define an extended composite symmetry [{\cal K}(n)] by adding the n-fold rotation as a further generator to the reduced composite symmetry [{\cal K}^\ast] of a domain pair. This results in the composite symmetry [{\cal K}(n)] of the complete twin aggregate, in the present case in a modification of the symmetry [{\cal K}^\ast =] [2'/m'\,2'/m'\,2/m] to:

    [{\cal K}(6) = {\cal K}(3) = 6(2)/m\,2/m\,2/m] (three orientation states, two twin laws) for [n = 3] and [n = 6];

    [{\cal K}(4) = 4(2)/m\,2/m\,2/m] (two orientation states, one twin law) for [n = 4].

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

    This construction can also be applied to noncrystallographic twin rotations [n = 5, 7, 8] etc. (cf. Section[link]), e.g.:

    [ {\cal K}(10) = {\cal K}(5) = 10(2)/m\,2/m\,2/m] (five orientation states, four twin laws) for [n = 5] and [n = 10] (Fig.[link]).

    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[link] and Fig.[link].

It is emphasized that the considerations of this section apply not only to the particularly complicated cases of multiple growth twins but also to transformation twins resulting from the loss of higher-order rotation axes that is accompanied by a small metrical deformation of the lattice. As a result, the extended composite symmetries [{\cal K}(n)] of the transformation twins resemble the symmetry [{\cal G}] of their parent phase. The occurrence of both multiple growth and multiple transformation twins of ortho­rhombic pseudo-hexagonal K2SO4 is described in Example[link].

Remark. It is possible to construct multiple twins that cannot be treated as a cyclic sequence of binary twin elements. This case occurs if a pair of domain states 1 and 2 are related only by an n-fold rotation or roto-inversion ([n \geq 3]). The resulting coset again contains the alternative twin operations, but in this case only for the orientation relation [1 \Rightarrow 2], and not for [2 \Rightarrow 1] (`non-transposable' domain pair). This coset procedure thus does not result in a composite group for a domain pair. In order to obtain the composite group, further cosets have to be constructed by means of the higher powers of the twin rotation under consideration. Each new power corresponds to a further domain state and twin law.

This construction leads to a composite symmetry [{\cal K}(n)] of supergroup index [[i]\geq 3] with respect to the eigensymmetry [{\cal H}]. This case can occur only for the following [{\cal H}\Rightarrow{\cal K}] pairs: [1 \Rightarrow 3], [{\bar 1} \Rightarrow {\bar 3}], [1 \Rightarrow 4], [1 \Rightarrow {\bar 4}], [m \Rightarrow 4/m], [1 \Rightarrow 6], [2 \Rightarrow 6], [m \Rightarrow {\bar 6} = 3/m], [m \Rightarrow 6/m], [2/m \Rightarrow 6/m] (monoaxial point groups), as well as for the two cubic pairs [222 \Rightarrow 23], [mmm \Rightarrow 2/m\,{\bar 3}]. For the pairs [1 \Rightarrow 3], [{\bar 1} \Rightarrow {\bar 3}], [m \Rightarrow {\bar 6} = 3/m], [2/m \Rightarrow 6/m] and the two cubic pairs [222 \Rightarrow 23], [mmm \Rightarrow 2/m\,{\bar 3}], the [{\cal K}] relations are of index [3] and imply three non-transposable domain states. For the pairs [1 \Rightarrow 4], [1 \Rightarrow {\bar 4}], [m \Rightarrow 4/m], as well as [1 \Rightarrow 6] and [m \Rightarrow 6/m], four or six different domain states occur. Among them, however, domain pairs related by the second powers of 4 and [{\bar 4}] as well as by the third powers of 6 and [{\bar 6}] operations are transposable, because these twin operations correspond to twofold rotations or, for [\bar 6], to m.

No growth twins of this type are known so far. As a trans­formation twin, langbeinite ([23 \Longleftrightarrow 222]) is the only known example.

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

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

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

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

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

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

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

3.3.6. Examples of twinned crystals

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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 [{\cal K}], as described in Sections 3.3.4[link] and 3.3.5[link].

For the description of a twin, the conventional crystallographic coordinate system of the crystal and its eigensymmetry group [{\cal H}] 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 [m_x], [m(100)], [m\parallel (100)] or [m\perp[100]] are used, whereas twin rotation axes are designated by [2_z], [2_{[001]}], [2\parallel [001]], [2\perp(001)], [3_z], [3_{[111]}], [4_{[001]}] etc. Macroscopic identification of twins and of twin laws

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As an introduction to the subsequent examples, this section shows how to recognize and identify twinning in a crystal, either by morphological features and observations in polarized light, or by etching, decoration and X-ray diffraction topography. Diffraction effects of twins are treated in Section 3.3.11[link].

  • (i) Re-entrant angles and twin striations

    The most prominent and easily recognizable morphological features are exhibited by penetration and contact twins with their re-entrant angles (edges). Re-entrant edges are typical for twins with non-parallel lattices (non-merohedral twins) and reticular merohedral twins with a Σ > 1 coincidence lattice (e.g. Σ3 spinel twins). Merohedral Σ1 twins with full lattice parallelism in general do not exhibit re-entrant edges but can often be identified by twin-related faces appearing in the morphology of a twinned crystal (e.g. Σ1 twins of quartz). The re-entrant edges mark the outcrop of the twin boundaries, which are defined by the path of this edge during crystal growth. Note that re-entrant edges occur in any arbitrary intergrowth of crystals, which can sometimes be misinterpreted as twinning.

    Illustrations of penetration twins are Figs.[link] (FeBO3),[link] (spinel law, diamond),[link] (staurolite) and[link] (orthoclase feldspar). Contact growth twins with re-entrant angles are presented in Fig.[link] (gypsum) and[link] (calcite). In repeated growth twins (e.g. albite, Fig.[link]) the parallel re-entrant edges form `polysynthetic twins'. If the width of the twin lamellae gets smaller and smaller, twin striations develop as a characteristic feature of these twins. For further examples see the sketches in Figs. 335–342 in Phillips (1971)[link], pp. 180–181. A rare case of a merohedral Σ1 twin with re-entrant edges is provided by the inversion growth twins of sodium chlorate NaClO3, which exhibit intergrowths of two opposite {111} tetrahedra (Lan et al., 2014[link]).

    A prominent example for the morphological identification of twins without re-entrant edges is quartz SiO2 with its merohedral Σ1 Dauphiné and Brazil twins (cf. Section[link]). The crucial feature is provided by the trapezohedral facet `x' (cf. Fig.[link]), which appears on the twin partners either in the same (Dauphiné) or mirrored (Brazil) orientation, cf. Frondel (1962)[link].

  • (ii) Polarized light

    • (a) Optical birefringence. The twinned crystal is placed and rotated between crossed polarizers (e.g. on a polarizing microscope). The twin domains are easily recognized by their different optical extinction positions. In favourable cases it is possible to derive orientation relations and the twin law from the angle of rotation between these positions. This method, however, is applicable only for optically anisotropic crystals (i.e. not for cubic crystals) and for `twins with inclined axes'. In twins with parallel lattices (Σ1 merohedral twins) the refractive-index ellipsoids are parallel: hence, all twin components exhibit the same optical extinction position and cannot be distinguished. For Σ1 inversion or reflection twins of enantiomorphic crystals, however, the twin domains may be distinguished by their opposite sense of optical rotation (optical activity, see below). Note that any arbitrary intergrowth of two crystals shows the same birefringence features and should not be misinterpreted as a twin.

      Examples of visualization of twin domains by optical birefringence are shown in Figs.[link],[link] and[link] (growth-sector twins), as well as in Figs.[link] and[link] (lamellar transformation twins). Further examples are presented in Figs.[link] and Figs.[link] of the following chapter on domain structures (Chapter 3.4 by Janovec & Přívratská). Another illustrative example is given by Niggli (1942[link], p. 802, Figs. 583 & 584).

    • (b) Optical activity. This method is applicable only to enantiomorphic crystals exhibiting optical activity. The crystal is again placed between crossed polarizers, but now one of the polarizers is rotated alternately in clockwise and anticlockwise directions, whereby the optical extinction positions of the right- and left-handed twin components are (alternately) adopted. Owing to the usually high dispersion of the optical rotation, the use of monochromatic light, which provides a sharply defined extinction, is advised. This method works very well for inversion and reflection twins of optically isotropic crystals (i.e. cubic crystals with point groups 23 and 432). An example is the inversion growth twinning of NaClO3 (Lan et al., 2014[link]). For optically anisotropic (i.e. non-cubic) crystals the effect of birefringence usually dominates and obscures the effect of optical activity, which is recognizable only if the light propagates closely along an optical axis of the crystal. In this way, Brazil twinning in quartz (Frondel, 1962[link]; Gordon, 1945[link]) and gallium phosphate GaPO4 (Engel et al., 1989[link]) has been visualized. Another example is shown in Fig.[link] of the following chapter on domain structures (Chapter 3.4 by Janovec & Přívratská). In favourable cases, domains of opposite optical activity can also be distinguished in polarized light propagating perpendicular to an optic axis, e.g. in KLiSO4 [point group 6: Klapper et al. (1987)[link]; Scherf et al., 1997[link]; Klapper et al. (2008)[link]]. Further examples are given by Shuvalov & Ivanov (1964)[link] and Koňák et al. (1978)[link].

  • (iii) Selective surface etching

    Selective chemical etching is an old and powerful tool for revealing the outcrops of crystal defects, such as dislocations, grain boundaries, twin domains and twin boundaries, at the surface of a crystal. The surface is subjected to attack by a suitable liquid (solvent, acid, flux) or gas, which generates an etch pattern that allows one to recognize and identify the defects emerging at the surface. From chemical and physical principles, two different effects are used: (a) the selective attack of defects due to the lattice deformations associated with them, and (b) the different `rate of dissolution' of opposite polar faces in pyroelectric (ferroelectric) crystals, which allows the distinction of domains of opposite polarity.

    • (a) Selective attack of defects

      The etch markings (e.g. etch pits of dislocations) on a crystallographic (or artificial) face exhibit shapes and orientations typical for the defect and the face under consideration. If a twin boundary emerges on the face, the etch markings on the two sides of the boundary show different orientations, which – in favourable cases – allows one to identify the twin law. The boundary itself is either visible by an etch groove (if it is accompanied by lattice distortions) or by the line separating regions (domains) with differently oriented etch markings. The most prominent and frequently studied crystal is natural quartz with its Dauphiné (`electrical') and Brazil (`optical') twins (cf. Sections[link] and[link]): on artificial (0001) surfaces, etched with hydrofluoric acid, regular triangular etch pits are formed. On both sides of a (usually irregular) Dauphiné boundary these pits are mutually rotated by 180°, according to the twofold twin axis along [001]. On both sides of a Brazil boundary the triangular pits are mirrored across the planes [\{11 \bar 2 0\}] (i.e. the twin elements of the Brazil twin; cf. Frondel, 1962[link]; Gordon, 1945[link]; Gault, 1949[link]), whereas they are mirrored across [\{10 \bar 1 0\}] for the `combined law' (Leydolt) twin, cf. Section[link] and Fig.[link]. On (artificial) faces [\{11 \bar 2 0\}] (normal to the twofold eigensymmetry axes) the etch pits are essentially diagonal (twofold symmetry), on all other faces they are non-symmetrical. Their orientation is again changed according to the twin law involved. A comprehensive review of the grading of natural quartz crystals for application as radio-frequency stabilizers, using etching and optical activity with particular emphasis on twinning, is presented by Gordon (1945)[link]. A study of the growth twinning of quartz-homeotype GaPO4 by etching is given by Engel et al. (1989)[link] and Grassl et al. (2000)[link].

    • (b) Etching of antipolar domains

      This method is applicable to pyroelectric crystals (having a unique polar axis), independent of whether they are also ferroelectric or not. These crystals have different growth and, accordingly, different dissolution rates in opposite directions of their polar axis. For example, for ferroelectric lithium niobate LiNbO3 (point group 3m) the rate of etching with a mixture of hot HF + HNO3 is higher on the negative polar side than on the positive one (Niizeki et al., 1967[link]; Nassau et al., 1965[link]). Thus, on natural or artificial polar faces (i.e. not parallel to the polar axis) a typical surface relief is generated by dissolution (etching), revealing the 180° domains emerging from that surface. It consists of elevated and depressed flat regions (`hill-and-valley relief') associated with the polarity of the domains. In addition, etch markings (etch pits) may occur in these regions, which are different in shape and differently oriented in the two regions. Examples of lamellar 180° domains in crystals grown from aqueous solution, revealed by etching with water, are provided by ferroelectric NH4LiSO4 (Hildmann, 1980[link]; Jennissen, 1990[link]) and lithium formate monohydrate LiCOO·H2O (Klapper, 1973[link]), both point group mm2, and triglycine sulfate (point group 2; Sawada & Abe, 1967[link]). A review of the etching of antipolar domains of various ferroelectric crystals (barium titanate, lithium niobate, triglycine sulphate etc.) with their suitable etchants is presented in Tagantsev et al. (2010)[link].

  • (iv) Surface decoration

    This method reveals the distribution of positive and negative charges on the surface of a body. The first report about this technique goes back to Kundt (1883)[link] and Kundt & Blasius (1886)[link], who dusted a mixture of fine powders of sulphur (yellow) and minium (Pb3O4, red) through a small-meshed cotton sieve onto pyroelectric crystals that had previously been heated or cooled. By this process the sulphur particles are negatively charged and the minium particles are positively charged, and collect on the positive and negative regions of the crystal surface, respectively, thus providing a yellow-and-red picture of the charge distribution. This method also visualizes head-to-head and tail-to-tail boundaries (Fig.[link]) emerging from a face parallel to the axis of polarization. Pearson & Feldmann (1958)[link] used a colloidal suspension of sulphur and minium powder in an insulating liquid (hexane) and obtained a higher spatial resolution of ferroelectric domain structures by briefly dipping the crystals into the warmed suspension (in situ decoration). Liquid toners for copying apparatus and nematic liquid crystals are also very effective decoration media.

    A study of antipolar domains and their boundaries of pyroelectric KLiSO4 (point group 6) using both Kundt's method and a Nashua liquid toner is reported by Klapper et al. (1987)[link]. Studies of domains in several pyroelectric nitrate crystals with a Xerox toner are presented by Bury and McLaren (1969)[link]. A liquid-crystal decoration of ferroelectric domains of triglycine sulfate is shown in Fig.[link] of the following chapter on domain structures (Chapter 3.4 by Janovec & Přívratská). A comprehensive review of the various decoration techniques and their application to the visualization of pyroelectric domain structures is given in Tagantsev et al. (2010)[link].

  • (v) X-ray diffraction topography

    X-ray diffraction topography with its various techniques is a powerful method for the visualization and characterization of all kinds of crystal defects, among them twins. Two kinds of X-rays are usually applied: monochromatic radiation [preferably characteristic Kα lines of conventional X-ray tubes, e.g. in the technique by Lang (1959)[link]], and polychromatic (`white') radiation in the Laue method. The latter has gained worldwide application with the development of synchroton-radiation sources (Tanner & Bowen, 1992[link]; Moore, 1995[link]). X-ray topographs can be obtained by reflections from plane surfaces of bulk crystals (Bragg or reflection case), as well as by transmission through crystal plates with not too high absorption (Laue or transmission case). In the latter method all defects inside the plate are imaged. It is a non-magnifying method, which requires crystals of rather low defect density, allowing mainly dynamical diffraction. Reviews of X-ray topography and its application are given, among others, by Lang (1978[link], 2004[link]), Tanner (1976)[link], Authier (2001[link], pp. 513–570) and Baruchel (2004[link]). Pioneering work for the X-ray topographic characterization of twins was done on quartz by Lang (1965[link], 1967a[link],b[link]), Lang & Miuskov (1969[link]) and Phakey (1969)[link]. Overviews particularly devoted to twinning are presented by Klapper et al. (1983)[link] and Klapper (1987)[link].

    X-ray topography is not suitable for twins with inclined axes (non-merohedral twins) because their domains have different lattice orientations and, thus, reflection conditions are usually fulfilled in domains of only one orientation state. These twins are more easily and better studied by optical birefringence [cf. Section[link](ii)]. For twins by merohedry (Σ = 1) and reticular merohedry (Σ > 1), however, all domains of all orientation states are exactly in the same reflection position and, thus, simultaneously imaged in one exposure. Two kinds of `topographic contrast' are used for depicting and characterizing the twin domains: `structure-factor contrast' (or `domain contrast') of twin domains and `boundary contrast' of twin boundaries.

    • (a) Imaging of domains. Structure-factor (domain) contrast of twin domains occurs if the twin-related reflections hklI and hklII of domains I and II have different structure-factor moduli F, i.e. if they belong to twin diffraction case B1 or B2 (cf. Klapper & Hahn, 2010[link]): the domains of different orientation states are distinguished by different intensities (different film blackening). Twin-related reflections with equal F moduli (diffraction case A) do not generate domain contrast and are suitable for imaging and characterizing other defects, such as dislocations. Thus it is important to know which reflections belong to diffraction cases A and B1, B2. If the twin law is known or suspected, these cases are easily found as follows: generate the twin composite symmetry by combining the point group of the untwinned crystal and the twin operation, and compare the seven diffraction sets (`face forms') {hkl}, {hhl}, {h0l} etc. of the untwinned and the composite group in Table[link] of ITA (Hahn & Klapper, 2005[link]). If the (names of the) face forms are the same for both groups, the reflection sets belong to diffraction case A, if they are different, to diffraction case B1.

      An illustrative example is given by KLiSO4 (room-temperature phase III, point group 6), which exhibits, among others, growth twins with twin law [m\parallel [001]] and composite point group 6mm. A comparison of the face forms (reflection sets) of groups 6 and 6mm reveals the characteristics of diffraction cases A, B1, B2 of the sets, shown in Table[link]. X-ray topographs of this twin are presented in Fig.[link] (Klapper et al., 1987[link]). Another illustrative example is provided by the X-ray topographic study of the [{\rm III}\rightarrow {\rm IV}] phase transition on cooling of KLiSO4: it shows the vanishing of the growth twinning with [m\parallel [001]] described above and the appearance of a new twinning with twofold twin axis 2[001], thus proving the symmetry change from point group 6 of phase III to point group 31m of the low-temperature phase IV (Klapper et al., 2008[link]).

      Table| top | pdf |
      Types of X-ray reflections generating (`yes') or not generating (`no') X-ray topographic domain contrast (yes/no) for the [6\rightarrow 6mm] growth twins of KLiSO4

      There are no B2 diffraction cases, because the twin operation does not invert the polarity.

      Type of X-ray reflectionsDomain contrast (twin diffraction case)
      {hkil} Yes (B1)
      [\{hh \overline {2h} l\}] No (A)
      [\{h0 \bar h l\}] No (A)
      {hki0} Yes (B1)
      [\{hh \overline {2h} 0\}] No (A)
      [\{h0 \bar h 0\}] No (A)
      {000l} No (A)

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      X-ray topographs (Mo Kα radiation) of a [(11 \bar 2 0)] plate (about 12 × 12 mm, about 1 mm thick) cut from a hexagonal KLiSO4 (phase III) crystal (point group 6; cf. Klapper et al., 1987[link]), grown from aqueous solution, exhibiting growth reflection twinning of law [m\parallel [001]] with two domain states I and II. Arrows: diffraction vectors. The twin-related reflections [21 \bar 3 0({\rm I})/12 \bar 3 0({\rm II})] (a) and [12 \bar 3 0({\rm I})/21 \bar 3 0({\rm II})] (b) (diffraction case B1), with very different moduli F[(12 \bar 3 0) = 1.6] and [F(21 \bar 3 0) = 18.2] (Chung, 1972[link]), map the domain states I and II with very strong domain contrast, which is reversed in (b) when the two twin-related reflections are interchanged. The imaging reflections 0003 (c) and [30 \bar 3 0] (d), both twin diffraction case A, do not reveal the domain states I and II (no domain contrast), but exhibit other growth defects, such as dislocations, faulted growth-sector boundaries etc. Note that these defects are not visible in the faintly diffracting domains of (a) and (b) owing to their small F modulus, which leads to kinematical (i.e. extinction-free) diffraction (after Klapper et al., 1987[link]; Klapper & Hahn, 2010[link]).

    • (b) Imaging of domain boundaries. This is essential in such twins by merohedry for which domain contrast does not occur for any of the seven reflection sets. This occurs only for inversion twins of noncentrosymmetric crystals (only B2 diffraction cases are possible), when anomalous scattering is negligible and the F moduli of opposite reflections hkl and [\bar h\,\bar k\,\bar l] have equal or nearly equal values (obeying Friedel's rule). In such cases, however, the twin boundaries appear by dynamical fringe contrast (corresponding to the stacking-fault contrast), provided that the crystal perfection is sufficiently high. An example is shown in Fig.[link].


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      X-ray transmission topograph of a (110) crystal plate (2.4 mm thick, width about 15 mm) cut from a very perfect crystal of polar lithium formate monohydrate, HCOOLi·H2O, point group mm2, grown from aqueous solution (reflection 111, Mo Kα radiation). The two contrast bands Tw with dynamical interference fringes arise from two (100) inversion-twin lamellae parallel to the polar axis [001]. They have opposite polarity with respect to the bulk crystal and are inclined by 48.3° to the plate surface. The dynamical fringe contrast (stacking-fault contrast) arises from the twin boundaries only, not from the volume of the lamellae (after Klapper, 1973[link]). The interference fringes at the inclined top edge are Pendellösung fringes arising from the tapering thickness of the plate.

      The dynamical fringe contrast originates from the phase jump of the dynamical X-ray wavefields (i.e. of the structure factor) upon crossing the twin boundary. This phase jump may be due to a twin displacement (fault) vector t (cf. Section[link]). The contrast depends on the component of t parallel to the diffraction vector g of the reflection used in the topograph. Thus, by imaging the boundary in several different reflections and analysing the variation of its contrast, the displacement vector can be determined. Details and examples are given in Sections[link] and[link]. Inversion twins in orthorhombic crystals

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The (polar) 180° twin domains in a (potentially ferroelectric) crystal of eigensymmetry [{\cal H} = mm2] ([m_xm_y2_z]) and composite symmetry [{\cal K} = 2/m\,2/m\,2/m] (e.g. in KTiOPO4, NH4LiSO4, lithium formate monohydrate) result from a group–subgroup relation of index [[i] = 2] 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 [k\times {\cal H}] of twin operations shown in Table[link], whereby the reflection in (001), [m_z], is considered as the `representative' twin operation.

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Orthorhombic inversion twins: coset of alternative twin operations (twin law)

[{\cal H}][k \times {\cal H} = m_z \times {\cal H} ]
1 [m_z ] (normal to the polar axis [001])
[m_x] [2_x] (normal to the polar axis)
[m_y] [2_y] (normal to the polar axis)
[2_z] [{\bar 1}] (inversion)

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 [{\cal K} = {2_x'\over m_x}{2_y'\over m_y}{2_z\over m_z'}({\bar 1}{^\prime}),]whereby the primed symbols designate the (alternative) twin operations (cf. Section 3.3.5[link]). Twinning of gypsum, CaSO4·2H2O

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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.[link] 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.[link]. Many colour photographs are presented by Boulliard (2010)[link], pp. 266–271. The two cosets of twin operations in Table[link] 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.[link], 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)}.}]

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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)

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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.[link]. 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)[link] in Section[link] 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 by Janovec & Přívratská in Chapter 3.4[link] 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[link]). 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. Twinning of low-temperature quartz (α-quartz, SiO2)

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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 non-polar trigonal axis. The crystals exhibit enantiomorphism (right- and left-handed quartz), piezoelectricity and optical activity. The lattice of quartz is hexagonal with holohedral (lattice) point group [6/m\,2/m\,2/m]. Many types of twin laws have been found (cf. Frondel, 1962[link]), but only the four most important ones are discussed here:

  • (a) Dauphiné twins;

  • (b) Brazil twins;

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

  • (d) Japanese twins.

The first three types are merohedral (parallel-lattice) Σ1 twins and their composite symmetries belong to category (i)[link] in Section[link], whereas the non-merohedral Japanese twins (twins with inclined lattices or inclined axes) belong to category (ii)[link]. Dauphiné twins

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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 twofold axes are reversed (`electrical twins'). Dauphiné twins can be transformation or growth or mechanical (ferrobielastic) twins. The composite symmetry is [{\cal K} = 622], the point group of high-temperature quartz ([\beta]-quartz). The coset decomposition of [{\cal K}] with respect to the eigensymmetry [{\cal H} = 32] (index [2]) contains the operations listed in Table[link].

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Dauphiné twins of α-quartz: coset of alternative twin operations (twin law)

[{\cal H}][2_z \times {\cal H} ]
1 [2_z = 6^3 ]
[3^1] [6^5\ (= 6^{-1})]
[3^2] [6^1]
[2_{[100]}] [2_z\times 2_{[100]} = 2_{[120]} ]
[2_{[010]}] [2_z\times 2_{[010]} = 2_{[210]}]
[2_{[{\bar 1}{\bar 1}0]}] [2_z\times 2_{[{\bar 1}{\bar 1}0]} = 2_{[1{\bar 1}0]}]

The left coset [2_z \times {\cal H}] constitutes the twin law. Note that this coset contains four twofold rotations of which the first one, [2_z], is the standard description of Dauphiné twinning. In addition, the coset contains two sixfold rotations, [6^1] and [6^5 = 6^{-1}]. The black–white symmetry symbol of the composite symmetry is [{\cal K} =] [6'(3)22'] (supergroup of index [2] of the eigensymmetry group [{\cal H} = 321]).

This coset decomposition [622 \Rightarrow 32] was first listed and applied to quartz by Janovec (1972[link], p. 993). Brazil twins

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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:

  • (i) three reflections across planes [\{11{\bar 2}0\}], normal to the three twofold axes;

  • (ii) three rotoinversions [{\bar 3}] around [001]: [{\bar 3}{^1}], [{\bar 3}{^3} = {\bar 1}], [{\bar 3}{^5} = {\bar 3}{^{-1}}].

The coset shows that Brazil twins can equally well be described as reflection or inversion twins. The composite symmetry [{\cal K} = {\bar 3}{^\prime}(3){2\over m'}1({\bar 1}{^\prime})]is a supergroup of index [2] of the eigensymmetry group 321. Combined Dauphiné–Brazil (Leydolt, Liebisch) twins

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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:

  • (i) three twin reflections across planes [\{10{\bar 1}0\}], parallel to the three twofold axes;

  • (ii) three rotoinversions [{\bar 6}] around [001]: [{\bar 6}{^1}], [{\bar 6}{^3} = m_z], [{\bar 6}{^5} = {\bar 6}{^{-1}}].

The composite symmetry [{\cal K} = {\bar 6}{^\prime}(3)2 m' = {3\over m'} 2m']is again a supergroup of index [2] of the eigensymmetry group 321. This twin law is usually described as a combination of the Dauphiné and Brazil twin laws, i.e. as the twofold Dauphiné twin rotation [2_z] followed by the Brazil twin reflection [m(11{\bar 2}0)] or, alternatively, by the inversion [{\bar 1}]. The product [2_z\times {\bar 1} = m_z] results in a particularly simple description of the combined law as a reflection twin on [m_z].

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 low-temperature quartz. In contrast, GaPO4, 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[link]).

In conclusion, the three merohedral twin laws of [\alpha]-quartz described above imply four domain states with different orientations of important physical properties. These relations are shown in Fig.[link] 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 [{\cal K} = 6/m\,2/m\,2/m] (`complete twin': Curien & Donnay, 1959[link]).


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Distinction of the four different domain states generated by the three merohedral twin laws of low-quartz and of quartz homeotypes such as GaPO4 (Dauphiné, Brazil and Leydolt twins) by means of three properties: orientation of the three electrical axes (triangle of arrows), orientation of etch pits on (001) (solid triangle) and sense of the optical rotation (circular arrow). The twin laws relating two different domain states are indicated by arrows [D ([2_z]): Dauphiné law; B ([{\bar 1}]): Brazil law; C ([m_z]): Leydolt law]. For GaPO4, see Engel et al. (1989[link]).

In the three twin laws (cosets) above, only odd powers of 6, [{\bar 3}] and [{\bar 6}] (rotations and rotoinversions) occur as twin operations, whereas the even powers are part of the eigensymmetry 32. Consequently, repetition of any odd-power 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. Japanese twins (or La Gardette twins)

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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 [(11{\bar 2}2)]. This results in an angle of [84^\circ 33^\prime] between the two threefold axes. One pair of prism faces is parallel (coplanar) in both partners.

There exist four orientation relations, depending on

  • (i) the handedness of the two twin partners (equal or different);

  • (ii) the azimuthal difference (0 or [180^\circ]) around the threefold axis of the two partners.

These four variants are illustrated in Fig.[link] and listed in Table[link]. The twin interface for all four twin laws is the same, [(11{\bar 2}2)], but only in type III do twin mirror plane and composition plane coincide.

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The four different variants of Japanese twins according to Frondel (1962[link])

Handedness of twin partnersAzimuthal difference (°)Twin element = twin lawLabel in Fig. 65 of Frondel (1962[link])
L–L or R–R 0 Irrational twofold twin axis normal to plane [(11{\bar 2}2)] I(R), I(L)
180 Rational twofold twin axis [[11{\bar 1}]\equiv[11{\bar 2}{\bar 3}]] parallel to plane [(11{\bar 2}2)] II(R), II(L)
L–R or R–L 0 Rational twin mirror plane [(11{\bar 2}2)] III
180 Irrational twin mirror plane normal to direction [[11{\bar 1}]\equiv[11{\bar 2}{\bar 3}]] IV
The line [[11{\bar 1}]\equiv[11{\bar 2}{\bar 3}]] is the edge between the faces [z(01{\bar 1}1)] and [r(10{\bar 1}1)] and is parallel to the composition plane [(11{\bar 2}2)]. It is parallel or normal to the four twin elements. Transformation formulae between the three-index and the four-index direction symbols, UVW and uvtw, are given by Barrett & Massalski (1966[link], p. 13).

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The four variants of Japanese twins of quartz (after Frondel, 1962[link]; cf. Heide, 1928[link]). The twin elements 2 and m and their orientations are shown. In actual twins, either the upper or the lower part of each figure is realized. The lower part has been added for better understanding of the orientation relation. R, L: right-, left-handed quartz. The polarity of the twofold axis parallel to the plane of the drawing is indicated by an arrow. In addition to the cases I(R) and II(R) , I(L) and II(L) also exist, but are not included in the figure. Note that a vertical line in the plane of the figure is the zone axis [[11{\bar 1}]] for the two rhombohedral faces r and z, and is parallel to the twin and composition plane ([11{\bar 2}2]) and the twin axis in variant II.

In all four types of Japanese twins, the intersection symmetry (reduced eigensymmetry) [{\cal H}^\ast] of a pair of twin partners is 1. Consequently, the twin laws (cosets) consist of only one twin operation and the reduced composite symmetry [{\cal K}^\ast] is a group of order 2, represented by the twin element listed in Table[link]. If one were to use the full eigensymmetry [{\cal H} = 32], the infinite sphere group would result as composite symmetry [{\cal K}].

Many further quartz twins with inclined axes are described by Frondel (1962[link]). A detailed study of these inclined-axis twins in terms of coincidence-site lattices (CSLs) is provided by McLaren (1986[link]) and Grimmer (2006[link]). Twinning of high-temperature quartz (β-quartz)

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Upon heating quartz into the hexagonal high-temperature phase (point group 622) above 846 K, the Dauphiné twinning disappears, because the composite symmetry [{\cal K}] of the twinned low-temperature phase now becomes the eigensymmetry [{\cal H}] of the high-temperature phase. For Brazil twins, however, their nature as reflection or inversion twins is preserved during the transformation.

The eigensymmetry of high-temperature quartz is 622 (order 12). Hence, the coset of the Brazil twin law contains 12 twin operations, as follows:

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

  • (ii) three further reflections across planes [\{10{\bar 1}0\}], which bisect the three Brazil twin planes [\{11{\bar 2}0\}] of low-temperature quartz;

  • (iii) three further rotoinversions around [001]: [{\bar 6}{^1}], [{\bar 6}{^3} =] [ m_z], [{\bar 6}{^5} = {\bar 6}{^{-1}}].

The composite symmetry is [ {\cal K} = {6\over m'}{2\over m'}{2\over m'}({\bar 1}{^\prime}),]a supergroup of index [2] of the eigensymmetry 622.

In high-temperature 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 low-temperature 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 low-temperature phase. Upon cooling below 846 K, transformation Dauphiné twin domains may appear in both Brazil growth domains, leading to the four domain states shown in Fig.[link]. Among these four orientation states, two Leydolt pairs occur. Such Leydolt domains, however, are not necessarily in contact (cf. Example[link] above).

In addition to these twins with `parallel axes' (merohedral twins), several kinds of growth twins with `inclined axes' occur in high-temperature quartz. They are not treated here, but additional information is provided by Frondel (1962[link]). Twinning of rhombohedral crystals

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In some rhombohedral crystals such as corundum Al2O3 (Wallace & White, 1967[link]), calcite CaCO3 or FeBO3 (calcite structure) (Kotrbova et al., 1985[link]; Klapper, 1987[link]), growth twinning with a `twofold twin rotation around the threefold symmetry axis [001]' (similar to the Dauphiné twins in low-temperature quartz described above) is common. Owing to the eigensymmetry [{\bar 3}2/m] (order 12), the following 12 twin operations form the coset (twin law). They are described here in hexagonal axes:

  • (i) three rotations around the threefold axis [[001]]: [6^1], [6^3 = 2_z], [6^5 = 6^{-1}];

  • (ii) three twofold rotations around the axes [[120]], [[210]], [[1{\bar 1}0]];

  • (iii) three reflections across the planes [(10{\bar 1}0)], [(1{\bar 1}00)], [(01{\bar 1}0)];

  • (iv) three rotoinversions around the threefold axis [[001]]: [{\bar 6}{^1}], [{\bar 6}{^3} = m_z] and [{\bar 6}{^5} = {\bar 6}{^{-1}}].

Some of these twin elements are shown in Fig.[link]. They include the particularly conspicuous twin reflection plane [m_z] perpendicular to the threefold axis [001]. The composite symmetry is [{\cal K} = {6'\over m'} ({\bar 3}) {2\over m} {2'\over m'} \ \ ({\rm order} \ \ 24).]


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Twin intergrowth of `obverse' and `reverse' rhombohedra of rhombohedral FeBO3. (a) `Obverse' rhombohedron with four of the 12 alternative twin elements. (b) `Reverse' rhombohedron (twin orientation). (c) Interpenetration of both rhombohedra, as observed in penetration twins of FeBO3. (d) Idealized skeleton of the six components (exploded along [001] for better recognition) of the `obverse' orientation state shown in (a). The components are connected at the edges along the threefold and the twofold eigensymmetry axes. The shaded faces are [\{10{\bar 1}0\}] and (0001) coinciding twin reflection and contact planes with the twin components of the `reverse' orientation state. Parts (a) to (c) courtesy of R. Diehl, Freiburg.

It is of interest that for FeBO3 crystals this twin law always, without exception, forms penetration twins (Fig.[link]), whereas for the isotypic calcite CaCO3 only (0001) contact twins are found (Fig.[link]). This aspect is discussed further in Section[link]. Colour photographs of rhombohedral twins, especially calcite, are provided by Boulliard (2010)[link], pp. 226–238.


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Contact growth twin of calcite with the same twin law as FeBO3 in Fig.[link]. Conspicuous twin element: twin reflection plane (0001), coinciding with the composition plane (0001). Spinel twins

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The twinning of rhombohedral crystals described above also occurs for cubic crystals as the spinel law (spinel, CaF2, PbS, diamond, sphalerite-type structures such as ZnS, GaAs, CdTe, cubic face- and body-centred metals). In principle, all four threefold axes of the cube, which are equivalent with respect to the eigensymmetry [{\cal H}], 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.[link]. For the two (differently) oriented eigensymmetries [4/m\,{\bar 3}\,2/m] of the domain states [{\cal H}_1] and [{\cal H}_2], the intersection symmetry [{\cal H}^\ast = {\bar 3}\,2/m] (order 12) results. With this `reduced eigensymmetry' [{\cal H}^\ast], the coset of 12 alternative twin operations is the same as the one derived for twinning of rhombohedral crystals in Example[link].


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Spinel (111) twins of cubic crystals (two orientation states). (a) Contact twin with (111) composition plane (two twin components). (b) and (c) Penetration twin (idealized) with one [(111)] and three [\{11{\bar 2}\}] composition planes (twelve twin components, six of each orientation state) in two different views, (b) with one [001] axis vertical, (c) with the twin axis [111] vertical.

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) [{\cal H} = m{\bar 3}m], [{\bar 4}3m], [432], [m{\bar 3}], [32] in detail. The intersection groups are [{\cal H}^\ast = {\bar 3}2/m], [3m], [32], [{\bar 3}] and [3], respectively. For these `reduced eigensymmetries', the cosets of the alternative twin operations are listed below with reference to cubic axes.

  • (a) Eigensymmetry [{\cal H} = 4/m\,{\bar 3}\,2/m] (order 48), reduced eigensymmetry [{\cal H}^\ast = {\bar 3}2/m1] (order 12).

    Alternative twin operations:

    • (1) three rotations [6^1], [6^3 = 2], [6^5 = 6^{-1}] around the axis [111];

    • (2) three twofold rotations around the axes [[11{\bar 2}]], [[{\bar 2}11]], [[1{\bar 2}1]];

    • (3) three reflections across the planes [(11{\bar 2})], [({\bar 2}11)], [(1{\bar 2}1)];

    • (4) three rotoinversions around the axis [111]: [{\bar 6}{^1}], [{\bar 6}{^3} = m_z], [{\bar 6}{^5} = {\bar 6}{^{-1}}].

    Reduced composite symmetry [{\cal K}^\ast = 6'/m'\,({\bar 3})\,2/m\,2'/m'] (order 24).

  • (b) Eigensymmetry [{\cal H} = {\bar 4}3m] (order 24), reduced eigensymmetry [{\cal H}^\ast = 3m1] (order 6).

    Two different twin laws are possible:

    • (1) Twin law representative: `twofold rotation around [111]';

      Alternative twin operations: lines (1)[link] and (3)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast = 6'(3)mm'] (order 12).

    • (2) Twin law representative: `reflection across (111)';

      Alternative twin operations: lines (2)[link] and (4)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast \ =\ {\bar 6}{^\prime}(3)m2'\ =\ 3/m'\,m2'] (order 12).

  • (c) Eigensymmetry [{\cal H} = 432] (order 24), reduced eigensymmetry [{\cal H}^\ast = 321] (order 6).

    Again, two different twin laws are possible:

    • (1) Twin law representative: `twofold rotation around [111]';

      Alternative twin operations: lines (1)[link] and (2)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast = 6'(3)22'] (order 12).

    • (2) Twin law representative: `reflection across (111)';

      Alternative twin operations: lines (3)[link] and (4)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast\ =\ {\bar 6}{^\prime}(3)2m'\ = \ 3/m'2m'] (order 12).

  • (d) Eigensymmetry [{\cal H} = 2/m\,{\bar 3}] (order 24), reduced eigen­symmetry [{\cal H}^\ast = {\bar 3}] (order 6).

    Two different twin laws:

    • (1) Twin law representative: `twofold rotation around [111]' or `reflection across (111)';

      Alternative twin operations: lines (1)[link] and (4)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast = 6'/m'({\bar 3})] (order 12).

    • (2) Twin law representative: `reflection across [(11{\bar 2})]' or `twofold rotation around [[11{\bar 2}]]';

      Alternative twin operations: lines (2)[link] and (3)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast = {\bar 3} 1 2'/m'] (order 12).

  • (e) Eigensymmetry [{\cal H} = 23] (order 12), reduced eigensymmetry [{\cal H}^\ast = 3] (order 3).

    Four different twin laws are possible:

    • (1) Twin law representative: `twofold rotation around [111]';

      Alternative twin operations: line (1)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast = 6'(3)] (order 6).

    • (2) Twin law representative: `reflection across (111)'.

      Alternative twin operations: line (4)[link] of case (a)[link] above.

      Reduced composite symmetry: [{\cal K}^\ast = {\bar 6}{^\prime}(3) = 3/m' ] (order 6).

    • (3) Twin law representative: `twofold rotation around [[11{\bar 2}]]';

      Alternative twin operations: line (2)[link] of case (a)[link] above;

      Reduced composite symmetry [{\cal K}^\ast = 312'] (order 6).

    • (4) Twin law representative: `reflection across [(11{\bar 2})]';

      Alternative twin operations: line (3)[link] of case (a)[link] above;

      Reduced composite symmetry: [{\cal K}^\ast = 31m'] (order 6).

The restriction to only one of the four spinel twin axes [\langle111\rangle] combined with the application of the coset expansion to the reduced eigensymmetry [{\cal H}^\ast] always leads to a crystallographic composite symmetry [{\cal K}^\ast]. 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 [m{\overline \infty}], i.e. would imply infinitely many cosets and (equivalent) twin laws. Higher-order spinel twins are discussed in Section[link]. Further details can be found in Klapper & Hahn (2012)[link] Sections 3 and 4, and in Section[link] below. Growth and transformation twins of K2SO4

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K2SO4 has an orthorhombic pseudo-hexagonal room-temperature phase with point group [{\cal H} = mmm] and axial ratio [b/a = \tan 60.18^\circ], and a hexagonal high-temperature phase ([> 853] K) with supergroup [{\cal G} = 6/m\,2/m\,2/m]. It develops pseudo-hexagonal growth-sector twins with equivalent twin reflection planes [(110)] and [(1{\bar 1}0)] which are also composition planes, as shown in Fig.[link]. As discussed in Sections[link] and[link] under (iii)[link], this corresponds to a pseudo-threefold twin axis which, in combination with the twofold eigensymmetry axis, is also a pseudo-hexagonal twin axis. The extended composite symmetry is [{\cal K}(6) = {\cal K}(3) = 6(2)/m \,2/m\,2/m.]


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Pseudo-hexagonal growth twin of K2SO4 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 K2SO4 solution containing 5% S2O3 ions. Courtesy of M. Moret, Milano.

Upon heating above 853 K, the growth-sector twinning disappears. On cooling back into the low-temperature 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 [{\cal K}] of the twins at room temperature is the true structural symmetry [{\cal G}] of the `parent' phase at high temperatures. Pentagonal–decagonal twins

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As was pointed out in Note (6)[link] of Section[link] and in part (iii)[link] of Section[link], there exist twin axes with noncrystallographic multiplicities [n = 5, 7, 8] etc. Twins with five- or tenfold rotations are frequent in intermetallic compounds. As an example, FeAl4 is treated here (Ellner & Burkhardt, 1993[link]; Ellner, 1995[link]). This compound is orthorhombic, [2/m\,2/m\,2/m], with an axial ratio close to [ c/a = \tan 72^\circ], corresponding to a pseudo-fivefold axis along [[010]] and equivalent twin mirror planes [(101)] and [({\bar 1}01)], which are about 36° apart. In an ideal intergrowth, this leads to a cyclic pseudo-pentagonal or pseudo-decagonal sector twin (Fig.[link]). All features of this twinning are analogous to those of pseudo-hexagonal aragonite, treated in Section[link], and of K2SO4, described above as Example[link].


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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 Ru8Ni15Al77. Scanning electron microscopy picture. Typical diameter of stars ca. 200 µm. The arms of the stars show parallel intergrowth. (b) Pentagonal twin aggregate of Fe4Al13 with morphology as grown in the orthorhombic high-temperature phase, showing several typical 72° angles between neighbouring twin partners (diameter of aggregate ca. 200 µm). Orthorhombic lattice parameters [a = 7.7510], [b = 4.0336], [c = 23.771] Å, space group [Bmmm]. The parameters c and a approximate the relation [c/a = \tan 72^\circ]; the pseudo-pentagonal twin axis is [010]. On cooling, the monoclinic low-temperature phase is obtained. The twin reflection planes in the orthorhombic unit cell are (101) and [(10{\bar 1})], in the monoclinic unit cell (100) and [({\bar 2}01)]; cf. Ellner & Burkhardt (1993[link], Fig. 10) and Ellner (1995[link]). Both figures courtesy of M. Ellner, Stuttgart.

The intersection symmetry of all twin partners is [{\cal H}^\ast = 12/m1]; the reduced composite symmetry [{\cal K}^\ast] of a domain pair in contact is [ 2'/m'\,2/m\,2'/m']. The extended composite symmetry of the ideal pentagonal sector twin is [{\cal K}(10)={\cal K}(5) = 10'(2')/m'\,2/m\,2'/m']. Multiple twins of rutile, TiO2

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Rutile with eigensymmetry [4/m\,2/m\,2/m] develops growth twins with coinciding twin reflection and composition plane {011}. Owing to its axial ratio [a/c = \tan 57.2^\circ], the tetragonal c axes of the two twin partners form an angle of 114.4°. The intersection symmetry of the two domains is [{\cal H}^\ast = 2/m] along the common direction [100]. The reduced composite symmetry of the domain pair is [{\cal K}^\ast = 2/m\,2'/m'\,2'/m'], 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.[link](a).


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Various forms of rutile (TiO2) twins with one or several equivalent twin reflection planes {011}. (a) Elbow twin (two orientation states). (b) Triple twin (three orientation states) with twin reflection planes (011) and [(0 \bar 1 1)]. (c) Triple twin with twin reflection planes (011) and (101). (d) Cyclic eightfold twin with eight orientation states. (e) Cyclic sixfold twin with six orientation states. Two sectors appear strongly distorted due to the large angular excess of 33.6°. (f) Perspective view of the cyclic twin of (e). (g) Twin with reflection plane (031) (heart-shaped twin). (h) Sagenite, an intergrowth of (011) twinned rutile (001) prisms. (i) Photograph of a rutile eightling (ca. 15 mm diameter) from Magnet Cove, Arkansas (Geologisk Museum, Kopenhagen). Parts (a) to (f) courtesy of H. Strunz, Unterwössen, cf. Ramdohr & Strunz, 1967[link], p. 512. Parts (g) and (h) courtesy of S. Herting-Agthe (1999)[link], Mineralogical Collections, Technical University Berlin. Photograph (i) courtesy of M. Medenbach, Bochum.

In point group [4/m\,2/m\,2/m], there exist four equivalent twin reflection planes {011} (four different twin laws) with angles of 65.6° between [(001)] and [(0{\bar 1}1)] and 45° between [(011)] and [(101)], leading to a variety of multiple twins. They may be linear polysynthetic or multiple elbow twins, or any combination thereof (Fig.[link]). Very rare are complete cyclic sixfold twins with a large angular excess of [6 \times 5.6^\circ = 33.6^\circ] (corresponding formally to a `5.5-fold' twin axis) and extended composite pseudosymmetry [{\cal K}(6) = 6(2)/m\,2/m\,2/m], or cyclic eightfold twins with a nearly exact fit of the sectors and a morphological pseudo-[{\bar 8}] 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 [{\bar 8}2m] of the twin aggregate. The extended composite symmetry is [{\cal K}(8) = 8(1)/m\,2/m\,2/m] with eight twin components, each of different orientation state. These cyclic twins are depicted in Figs.[link](d), (e), (f) and (i).

The sketch of the 'eightling' in Fig.[link](d) shows 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 any offset? Moreover, the coinciding twin and composition planes {011} in the figures are not the growth planes {111} of rutile, i.e. all twin boundaries must have been formed already in the nucleus of the twin (cf. Section[link]). Fig.[link](i) convincingly shows that the ring is closed at its flat back side (on which the crystal was obviously lying during growth), i.e. the growth of all domains started from a central point (nucleus).3 The open ring in Fig.[link](d) is idealized in order to bring out the mutual orientation of the twin components. This seems also to be the case for the other sketches (b) and (c): here the different domains and the boundaries between them must have started from a common nucleus located in the centre of the half-rings, whereas the sixling in parts (e) and (f) shows the common nucleus clearly [as well as Fig. 11.13 in the textbook by Klein & Hurlbut (1993[link], p. 381), and Fig. 10.12 in Berry & Mason (1959[link], p. 373)].

The (011) rutile twinning treated above is the most frequent one. Another twin, the less frequent (031) reflection twin, usually appears as an intergrowth of two tetragonal prisms in the form of a `V' (Fig.[link]g) or of an `X' with an acute angle of 54.7° and with contact plane (031). `V'-shaped twins with short `arms' are often called `heart'- or `kite'-shaped twins.

Growth twinning of rutile exhibits a vast variety of complicated (often multiply twinned) forms, the nucleation and growth development of which are in many cases not clear. A well known example is sagenite (`reticulated rutile'), a usually pseudo-hexagonal triangular plane grid of intergrown prismatic [001] rods usually twinned on (011), Fig.[link](h). The meshes of the grids are often triangles, with two angles of 65.6° [(011) twins] and one angle of 48.8°. The latter angle is enforced by the two prismatic rods in exact (011) twin orientation and does not represent a twin in the strict sense. Rarely also (031) twins occur at a corner of the triangular meshes.

The nucleation and growth history of sagenite is usually based on two observations: free hydrothermal growth, often in the close neighbourhood of quartz (Herting-Agthe, 2009[link]) and by the epitaxial nucleation and growth of prismatic rutile rods with their pseudohexagonal (100) planes on the (0001) planes of hematite or ilmenite (Armbruster, 1981[link]; Force et al., 1996[link]). Interestingly, triangular grids with exactly 60° angles, following the symmetry of the substrate, as well as with angles corresponding to the (011) rutile twin are observed (see also Section[link]).

The origin of the planar sagenite grid is explained by the epitaxial nucleation and growth of prismatic rutile rods with their (100) planes on the (0001) planes of hematite or ilmenite. The (0001) oxygen layer of hematite is exactly hexagonal, the (100) (undulated) oxygen layer of rutile pseudo-hexagonal, and thus epitactic growth with orientations [(100)_{\rm rut}\parallel(0001)_{\rm hem}] and [[001]_{\rm rut} \parallel \langle100\rangle_{\rm hem}] (`structural coherence': Armbruster, 1981[link]; Force et al., 1996[link]) is suggested. Thus, the rutile rods are nucleated in three orientations and grow together in a triangular arrangement. The hematite substrate is supposed to be dissolved by a later geological process, leaving the sagenite grid. This explanation, however, does not explain the angles of 65.6° and 54.7° between the rutile rods, since the epitaxial nucleation on (0001) hematite would suggest exact 60° angles. This matter deserves further investigation. Variety of twinning in gibbsite, Al(OH)3

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Gibbsite (older name: hydrargillite) forms a pronounced layer structure with a perfect cleavage plane [(001)]. It is monoclinic with eigensymmetry [{\cal H} = 12/m1], but strongly pseudo-hexagonal in (001) with an axial ratio [b/a = \tan 30.4^\circ]. In contrast to most other pseudo-hexagonal crystals, the twofold eigensymmetry axis b is not parallel but normal to the pseudo-hexagonal c axis. The normal to the cleavage plane [(001)] is inclined by [\beta - 90^\circ = 4.5^\circ] against [001]. Owing to the pseudo-hexagonal metrics of the plane [(001)], the lattice planes [(110)] and [({\bar 1}10)], equivalent with respect to the eigensymmetry [{\cal H} = 2/m], form an angle of 60.8°.

The following four significant twin laws have been observed by Brögger (1890[link]):

  • (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)[link]; 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.[link]. 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.[link]a). This (001) umbrella faceting was recently observed in twinned synthetic gibbsite crystals by Sweegers et al. (1999[link]).


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    Sixfold reflection twin of gibbsite, Al(OH)3, with equivalent (110) and [({\bar 1}10]), 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° ([\beta = 94.5^\circ]), 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 [\bar 1]. The reduced composite symmetry of a domain pair is [{\cal K}^*=12'/m'1], with [m'] the twin mirror plane (110).

  • (iv) `Median law': According to Brögger (1890[link]), this twin law implies exact parallelism of non-equivalent edges [[110]_{\rm I}] and [[010]_{\rm II}], 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[link]. An alternative interpretation, treating these twins as rational [130] rotation twins, is given by Johnsen (1907[link]), cf. Tertsch (1936[link]), pp. 483–484. Interestingly, this strange `twin law' is the most abundant one among natural gibbsite twins. Plagioclase twins

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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 [{\bar 1}]) albite NaAlSi3O8 and anorthite CaAl2Si2O8 (also microcline, triclinic KAlSi3O8), 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 triclinic twins can be formulated as follows:

  • Albite law: reflection twin on (010); composition plane (010) rational (Fig.[link], Table[link]). [{\cal K}_A = 2'/m'({\bar 1})] with rational [m'\parallel(010).]

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    Plagioclase: albite and pericline twins

    [{\cal H}][k \times{\cal H}] (albite)[k \times{\cal H}] (pericline)
    1 [m\parallel (010)] rational [2\parallel [010]] rational
    [{\bar 1}] [2\perp (010)] irrational [m\perp [010]] irrational

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    Polysynthetic albite twin aggregate of triclinic feldspar, twin reflection and composition plane (010).

  • Pericline law: twofold rotation twin along [010]; composition plane irrational [\parallel[010]]: `rhombic section' (Fig.[link], Table[link]). [{\cal K}_P = 2'/m'({\bar 1})] with rational [2'\parallel[010].]


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    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 [{\bf a}] and [{\bf a}']. (b) The same (exaggerated) twin as in (a) but with irrational boundary along the `rhombic section': fitting of contact planes from both sides ([{\bf a}] and [{\bf a}'] coincide and form a flat ridge). (c) Sketch of a real pericline twin with irrational interface (`rhombic section') containing the twin axis.

Both twin laws resemble closely the monoclinic pseudosymmetry [2/m] in two slightly different but distinct fashions: each twin law [{\cal K}] uses one rational twin element from [2/m], the other one is irrational. The two frameworks of twin symmetry [ 2'/m'] are inclined with respect to each other by about [4^\circ], corresponding to the angle between b (direct lattice) and [b^\ast] (reciprocal lattice).

Both twins occur as growth and transformation twins: they appear together in the characteristic lamellar `transformation micro­clines'. Details of feldspar twins are given in Smith (1974)[link]. Staurolite

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The mineral staurolite, which has the approximate formula Fe2Al9[O6(O,OH)2/(SiO4)4], has `remained an enigma' (Smith, 1968[link]) 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[link]). Staurolite is strongly pseudo-orthorhombic, [Ccmm], and only detailed optical, morphological and X-ray experiments reveal monoclinic symmetry, [C12/m1], with [a = 7.87], [b = 16.62], [c = 5.65] Å and [\beta = 90^\circ] within experimental errors (Hurst et al., 1956[link]; Smith, 1968[link]; and especially Hawthorne et al., 1993[link]).

Staurolite exhibits two quite different kinds of twins:

  • (i) Twinning by high-order merohedry (after Friedel, 1926[link], p. 56) was predicted by Hurst et al. (1956[link]) in their detailed study of staurolite twinning. Staurolite crystals are supposed to consist of very fine scale monoclinic ([{\cal H} = 12/m1]) microtwins on [m(001)], which yield a twin aggregate of orthorhombic composite symmetry [{\cal K} = 2'/m'\,2/m\,2'/m']. The coset consists of [m'(001)], [m'(100)], [2'\parallel [100]] and [2'\parallel [001]]. 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[link]), so far it has never been unambiguously proven. In particular, electron-microscopy investigations by Fitzpatrick (1976, quoted in Bringhurst & Griffin, 1986[link], 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[link], p. 461). For pictures see Boulliard (2010)[link], p. 264.

    • (a) The 90° cross (Greek cross) with twin reflection and composition plane (031) is illustrated in Fig.[link](a) [cf. also the figures on p. 151 of Hurst et al. (1956[link]) for less idealized drawings]. Plane (031) generates two twin components with an angle of [2 \arctan (b/3c) = 2 \arctan 0.9805 = 88.9^\circ], very close to 90°, between their c axes. The equivalent twin reflection plane [(0{\bar 3}1)] leads to the same angle, and both twin planes intersect along the lattice row [100].


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      Twinning of staurolite. (a) 90° cross (`Greek cross') with twin reflection and composition planes (031) and [(03{\bar 1})]. (b) 60° cross (`St Andrew's cross') with twin reflection and composition plane (231).

      With eigensymmetry [{\cal H} = 12/m1], the intersection symmetry of the domain pair is [{\cal H}^\ast = {\bar 1}] and the reduced composite symmetry is [{\cal K}^\ast = 2'/m'] [[m' = (031)]]. Owing to the special axial ratio [b/3c \approx 1] mentioned above, the 90° cross is an excellent example of a pseudo-tetragonal twin. The extended composite symmetry of this twin is oriented along [100]:[ {\cal K}(4) = 4(2)/m\,2/m\,2/m][cf. Section[link](iii)[link]] with two domain states and all twin operations binary.

    • (b) The 60° cross (St Andrew's cross) with twin reflection plane (231) is illustrated in Fig.[link](b). It is the more abundant of the two crosses, with a ratio of [60^\circ{:}90^\circ] twins [\approx 9{:}1] in one Georgia, USA, locality (cf. Hurst et al., 1956[link], p. 152). Two equivalent twin mirror planes, [(231)] and [(2{\bar 3}1)], intersecting in lattice row [[10{\bar 2}]] exist. They include an angle of 60.4°. The action of one of these twin reflection planes leads to the 60° cross with an angle of 60° between the two c axes. The reduced composite symmetry of this twin pair is [{\cal K}^\ast = 2'/m'] [[m' = (231)]].

      In rare cases, penetration trillings occur by the action of both equivalent mirror planes, [(231)] and [(2{\bar 3}1)], leading to three interpenetrating twin components with angles of about 60° between neighbouring arms.


  • (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[link]) and Hurst et al. (1956[link]) 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[link] as a characteristic example of `twinning by reticular pseudo-merohedry'.

  • (3) A very similar twinning situation, including the use of a pseudo-cube, has been reported for the monoclinic (P21/n) pseudo-cubic mineral cryolite, Na3AlF6, by Donnay (1952)[link] and Wrinch (1952)[link]. BaTiO3 transformation twins

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The perovskite family, represented by its well known member BaTiO3, is one of the technically most important groups of dielectric materials, characterized by polar structures which exhibit piezoelectricity, pyroelectricity and, most of all, ferro­electricity.

BaTiO3 is cubic and centrosymmetric (paraelectric) above 393 K. Upon cooling below this temperature it transforms in one step (first-order transformation with small [\Delta H]) into the ferroelectric tetragonal phase with polar space group [P4mm]. This transition is translationengleich of index [[i]=6]. Hence there are domains of six possible orientation states at room temperature. The transformation can be theoretically divided into two steps:

  • (i) Translationengleiche symmetry reduction cubic [Pm{\bar 3}m\,\,\longrightarrow] tetragonal [P4/mmm] of index [[i_1]=3], 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 [P4/mmm] is translationsgleich reduced to [P4mm] of index [[i_2]=2]. The total index is: [[i]=[i_1]\cdot [i_2]=6].

The beautiful polysynthetic twin structure of BaTiO3 is shown in the colour micrograph Fig.[link] in Chapter 3.4[link] of this volume. Merohedral growth twinning of pentaerythrite

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Crystals of pentaerythrite (PE), C(CH2OH)4, grown from solutions in water or a water/ethanol mixture, exhibit nearly without exception twinning by merohedry. This is already indicated by its predominantly tetragonal bi-pyramidal morphology (Ernst, 1928[link]), which is not compatible with the point group [\bar 4] (space group [I\bar 4]) of the single crystal. The point group [\bar 4] allows three merohedral twin laws:

  • (a) m′(001), composite symmetry 4/m′ (inversion twin [\bar 1']);

  • (b) m′(100), composite symmetry [\bar 4m'2'] ;

  • (c) m′(110), composite symmetry [\bar 42'm'].

Of special interest are here the laws (b) and (c), which lead to two different structural settings and are, thus, different twin laws. Since the predominant growth face of PE is the tetragonal sphenoid (tetragonal tetrahedron) (111) (Groth, 1910[link]), all three twin elements generate the tetragonal bi-pyramid {111} (pseudo-cubic octahedron) in their composite symmetry, as is actually observed.

A detailed analysis of the growth twinning of PE has been reported by Renninger (1957)[link]. He identified the inversion twinning by studying (001) cleavage plates by polarization optics. These generally exhibit four regions, corresponding to the four pyramid growth sectors, of different optical behaviour. This, however, is due to a pronounced `optical anomaly' (cf. `Extended note' in Section 3.3.1[link] above), here by a biaxial splitting of the optical axis of up to 8°, and not by a reversal of optical rotation, which does not exist along a [\bar 4] axis. Twin laws (b) or (c) were identified by the intensity interchange of twin-related reflections (hkl) and (khl) on Laue diffraction patterns. This was confirmed by a rather unusual method, by rocking curves of (130)–(310) reflections (having very different structure-factor moduli) on both sides of a (b) or (c) twin boundary. The X-ray methods, however, could not distinguish between twin laws (b) and (c). This is possible by the different mutual orientations of etch pits on (001) cleavage planes on both sides of the twin boundaries [cf. Section[link](iii)(a)]. Twins of twins

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This term is due to Henke (2003[link]) 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:

  • (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.

Typical examples are:

  • (i) the cubic–tetragonal ([m{\bar 3}m \Longleftrightarrow 4mm]) 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[link]), (NO)2VCl6;

  • (iii) ammonium lithium sulfate exhibits pseudo-hexagonal growth-sector twins upon which lamellae of ferroelectric 180° domains are superimposed (Hildmann, 1980[link]; Jennissen, 1990[link]).

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

  • (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:[\eqalign{\hbox{Dauphin\'e twin law: } 321 &\Rightarrow 6'(3)22'\cr\hbox{Brazil twin law: } 321 &\Rightarrow {\bar 3}{^\prime}(3)2/m'1({\bar 1}{^\prime})\cr\hbox{Leydolt twin law: } 321 &\Rightarrow {\bar 6}{^\prime}(3)2m' = 3/m'2m'.}]Combination = `complete twin': [6'(3)/m'\,2/m'\,2'/m'({\bar 1}{^\prime})]; this sym­metry corresponds to the hexagonal holohedral point group [6/m\,2/m\,2/m], cf. Example[link].

  • (ii) Another example is provided by KLiSO4 (crystal class 6), extensively investigated by Klapper et al. (1987[link]):[\eqalign{\hbox{Inversion twins: }6 &\Rightarrow 6/m'({\bar 1}{^\prime})\cr\hbox{Reflection twins: }6 &\Rightarrow 6m'm'\cr\hbox{Rotation twins: }6 &\Rightarrow 62'2'.}]Combination = `complete twin': [ 6/m'2'/m'2'/m'({\bar 1}{^\prime})]; this sym­metry is isomorphic to the complete-twin symmetry of quartz, given above, and to the hexagonal holohedral point group [6/m\,2/m\,2/m].

  • (iii) An example of a very extensive series of pseudo-merohedral growth twins is provided by the zeolite minerals phillipsite and harmotome (Hoffmann et al., 1973[link]). Both minerals are monoclinic, space group P12/m1 (Rinaldi et al., 1974[link]). The crystals occur in series of pseudo-merohedral growth twins as follows:[1\,2/m\,1 \rightarrow 2'/m'\,2/m\,2'/m' \rightarrow 4'/m'\,2/m\,2'/m' \rightarrow 4'/m'\, \bar 3'\, 2/m.]The total subgroup index [i] from the monoclinic to the cubic twin symmetry is stepwise 2 × 2 × 3 = 12. These steps are illustrated in the sketches of Fig.[link]. If the gaps between the interpenetrating domains are filled, as in Fig.[link](e), the twin appears as a pseudo-cubic rhomb-dodecahedron.


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    The various steps (idealized) of phillipsite growth twins: (a) monoclinic untwinned, (b) orthorhombic, (c) tetragonal, (d) cubic, and (e) `filled-in' pseudo rhomb-dodecahedron (after Ramdohr & Strunz, 1967[link], p. 754).

3.3.7. Genetic classification of twins

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In Section 3.3.3[link], a classification of twins based on their morphological appearance was given. In the present section, twins are classified according to their origin. Genetic terms such as growth twins, transformation twins and mechanical twins were introduced by Buerger (1945[link]) and are in widespread use. They refer to the physical origin of a given twin in contrast to its geometrical description in terms of a twin law. The latter can be the same for twins of different origin, but it will be seen that the generation of a twin has a strong influence on the shape and distribution of the twin domains. An extensive survey of the genesis of all possible twins is given by Cahn (1954[link]). Growth twinning

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Growth twins can occur in nature (minerals), in technical processes or in the laboratory during growth from vapour, melt or solution. Two mechanisms of generation are possible for growth twins:

  • (i) formation during nucleation of the crystal;

  • (ii) formation during crystal growth. Twinning by nucleation

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In many cases, twins are formed during the first stages of spontaneous nucleation, possibly before the sub-critical nucleus reaches the critical size necessary for stable growth. This idea was originally proposed by Buerger (1945[link], p. 476) under the name supersaturation twins. There is strong evidence for twin formation during nucleation for penetration and sector twins, where all domains originate from one common well defined `point' in the centre of the twinned crystal, which marks the location of the spontaneous nucleus.

Typical examples are the penetration twins of iron borate FeBO3 (calcite structure), which are intergrowths of two rhombohedra, a reverse and an obverse one, and consist of 12 alternating twin domains belonging to two orientation states (see Example[link] and Fig.[link]). Experimental details are presented by Klapper (1987[link]) and Kotrbova et al. (1985[link]). Further examples are the penetration twins of the spinel law (Example[link] and Fig.[link]), the very interesting and complex [001] penetration twin of the monoclinic feldspar orthoclase (Fig.[link]) and the sector twins of ammonium lithium sulfate with three orientation states (Fig.[link]).


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Orthoclase (monoclinic K-feldspar). Two views, (a) and (b), of Carlsbad penetration twins (twofold twin axis [001]).


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Photographs of (001) plates ([\approx] 20 mm diameter, [\approx] 1 mm thick) of NH4LiSO4 between crossed polarizers, showing sector growth twins due to metric hexagonal pseudosymmetry of the orthorhombic lattice. (a) Nearly regular threefold sector twin (three orientation states, three twin components). (b) Irregular sector twin (three orientation states, but five twin components). After Hildmann (1980)[link].

It should be emphasized that all iron borate crystals that are nucleated from flux or from vapour (chemical transport) exhibit penetration twinning. The occurrence of untwinned crystals has not been observed so far. Crystals of isostructural calcite and NaNO3, on the other hand, do not exhibit penetration twins at all. In contrast, for ammonium lithium sulfate, NH4LiSO4, both sector-twinned and untwinned crystals occur in the same batch. In this case, the frequency of twin formation increases with higher supersaturation of the aqueous solution.

The formation of contact twins (such as the dovetail twins of gypsum) during nucleation also occurs frequently. This origin must always be assumed if both partners of the final twin have roughly the same size or if all spontaneously nucleated crystals in one batch are twinned. For example, all crystals of monoclinic lithium hydrogen succinate precipitated from aqueous solution form dovetail twins without exception.

The process of twin formation during nucleation, as well as the occurrence of twins only for specific members of isostructural series (cf. Section[link]), are not yet clearly understood. A hypothesis advanced by Senechal (1980[link]) proposes that the nucleus first formed has a symmetry that is not compatible with the lattice of the (macroscopic) crystal. This symmetry may even be noncrystallographic. It is assumed that, after the nucleus has reached a critical size beyond which the translation symmetry becomes decisive, the nucleus collapses into a twinned crystal with domains of lower symmetry. This theory implies that for nucleation-twinned crystals, a metastable modification with a structure different from that of the stable macroscopic state may exist for very small dimensions. For this interesting theoretical model no experimental proof is yet available, but it appears rather reasonable; as a possible candidate of this kind of genesis, the rutile `eightling' in Example[link] (Figs.[link]d and i) may be considered.

Recently, the ideas on twin nucleation have been experimentally substantiated by HRTEM investigations of multiple twins. The formation of these twins in nanocrystalline f.c.c. and diamond-type cubic materials, such as Ge, Ag and Ni, is explained by the postulation of various kinds of noncrystallographic nuclei, which subsequently `collapse' into multiply twinned nanocrystals, e.g. fivefold twins of Ge; cf. Section[link]. An extensive review is provided by Hofmeister (1998[link]). Twinning during crystal growth

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  • (a) An alternative theory of twinning postulates the formation of a two-dimensional nucleus in twin position on a growth face of an existing macroscopic (previously untwinned) crystal. Such a mechanism was extensively described by Buerger (1945[link], pp. 472–475) and followed up by Menzer (1955[link]) and Holser (1960[link]). Obviously, this process is favoured by defects (inclusions, impurities) in the growth face. If the twin nucleus spreads out over the entire growth face, the twin boundary coincides with the growth face. This mechanism is generally assumed for the generation of large-area lamellar polysynthetic growth twins as observed for albite (Example[link] and Fig.[link]). For a critical discussion of the origin of irrational twin interfaces in rotation twins such as the pericline twins see Cahn (1954[link], p. 408). It should be noted that this mechanism is possible only for twin boundaries of very low energy, since the boundary energy of the large interface has to be supplied in one step, i.e. during spreading out of one growth layer in twin position. It is obvious that this kind of twin formation can only occur if the twin boundary coincides with a prominent growth face (F-face, rarely S-face, according to Hartman, 1956[link]).

  • (b) In the majority of growth twins, the twin boundary does not coincide with the growth face. This is the rule for merohedral twins, where the twin domains appear as `inserts' in the shape of pyramids or lamellae extending from the initiating defects (mostly inclusions) into the direction of growth of the face on which the twin has started. Examples are the pyramid-shaped Brazil-twin inserts of quartz (Frondel, 1962[link], Fig. 61 on p. 87) and the lamellar stripes of growth twins of KLiSO4 (Klapper et al., 1987[link], especially Fig. 5). Similar pyramidal twin inserts are observed for Dauphiné growth twins in natural and synthetic quartz. These twin morphologies in quartz, however, are often considerably modified after growth by (partial) ferrobielastic switching of the domains, which is easily induced by stress at elevated temperatures [cf. Section[link](iii)[link]]. Illustrations of such Dauphiné twins are given by Frondel (1962[link], Fig. 49 on p. 78).

    The growth-twin inserts as described above appear improbable for non-merohedral twins because unfavourable high-energy boundaries would be involved. As a consequence, it must be concluded that non-merohedral twins with boundaries not coinciding with a (prominent) growth face (e.g. dovetail twins of gypsum) must form during the nucleation stage of the crystal (Section[link] above).

  • (c) Another model of twin formation has been suggested by Schaskolsky & Schubnikow (1933[link]). It is based on the idea that in the melt or solution the pre-existing small crystals make accidental contact with analogous faces [(hkl)_1] and [(hkl)_2] parallel, rotate and agglutinate in twin position, and continue to grow as a twin. This concept is also favoured by Buerger (1960b[link]). The model of Schaskolsky & Schubnikow is based on their interesting experiments with many ([\approx 1400]) K-alum crystals (up to 0.5 mm in size), which sediment in solution on horizontal octahedron (111) and cube (100) faces of large alum crystals (20–30 mm in size). A statistical analysis of the orientation distribution of the sedimented crystals reveals a significantly increased frequency of (111)/(111) parallel intergrowths, of regular (001)/(111) intergrowths and of (111) spinel twins. The authors interpret this result as a rotation of the small crystals around the contact-face normal after deposition on the large crystal. This initial contact plane (ICP) model of twin formation was critically discussed by Senechal (1980[link]) and considered as questionable, an opinion which is shared by the present authors.

  • (d) Finally, it is pointed out that twinning may drastically modify the regular growth morphology of (untwinned) crystals. A prominent example is the tabular shape of (111)-twinned cubic crystals with the large face parallel to the (111) contact plane This is due to the increased lateral growth rate of the faces meeting in re-entrant edges (re-entrant corner effect; Hartman, 1956[link]; Ming & Sunagawa, 1988[link]). The (111) tabular shape of twinned cubic crystals plays an important role for photographic materials such as silver bromide, AgBr (Buerger, 1960b[link]; Bögels et al., 1997[link], 1998[link]). A more extreme habit modification is exhibited by the [\langle110\rangle] growth needles of cubic AgBr, which contain two [\{111\}] twin planes intersecting along [\langle110\rangle] (Bögels et al., 1999[link]).

    The phenomenon of habit modification by twinning has been developed further by Senechal (1976[link], 1980[link]), who presents an alternative model of the genesis of penetration twins (cf. Section[link] above): initial cubic (111) contact twins consisting of two octahedra change their habit during growth so as to form two interpenetrating cubes of the spinel law. As a further example, chabasite is cited.

  • (e) During melt growth of the important cubic semiconductors with the diamond structure (Si, Ge) and sphalerite (zinc sulfide) structure (e.g. indium phosphide, InP), twins of the spinel law [twin mirror plane (111) or twofold twin axis [111], cf. Section[link]] are frequently formed. Whereas this twinning is relatively rare and can easily be avoided for Si and Ge, it is a persistent problem for the III–V and II–VI compound semiconductors, especially for InP and CdTe crystals, which have a particularly low {111} stacking-fault energy (Gottschalk et al., 1978[link]). For Czochralsky growth, these twins are usually nucleated at `edge facets' forming at the surface of the `shoulder' (or `cone region') where the growing crystal widens from the seed rod to its final diameter. Once nucleated, they proceed during further growth as bulk twins or, more frequently, as twin lamellae with sharp [\{111\}] contact planes. For a [111] pulling direction, the three equivalent {111} twin planes with inclination of 19.5° against the pull axis [111] are usually activated, whereas the perpendicular (111) twin plane does not occur, or only rarely occurs (Bonner, 1981[link]; Tohno & Katsui, 1986[link]). These twins can be avoided by optimizing the growth conditions, in particular by the choice of a proper cone angle, which is the most crucial parameter. A mechanism of the {111} twin formation of III–V compound semiconductors was suggested by Hurle (1995[link]) and experimentally confirmed for InP, using synchroton-radiation topography combined with chemical etching and Nomarski microscopy, by Chung et al. (1998[link]) and Dudley et al. (1998[link]). A comprehensive X-ray topographic study of (111) twinning in indium phosphide crystals, grown by the liquid-encapsulated Czochralski technique, and its interaction with dislocations is presented by Tohno & Katsui (1986[link]).

  • (f) It should be noted lastly that `annealing twins' (which are an important subject in metallurgy) are not treated in this section, because they are considered to be part of bicrystallography. These twins are formed during recrystallization and grain growth in annealed polycrystalline materials (cf. Cahn, 1954[link], pp. 399–401). Transformation twinning

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A solid-to-solid (polymorphic) phase transition is – as a rule – accompanied by a symmetry change. For displacive and order–disorder transitions, the symmetries of the `parent phase' (prototype phase) [{\cal G}] and of the `daughter phase' (deformed phase) [{\cal H}] exhibit frequently, but not always, a group–subgroup relation. During the transition to the low-symmetry phase the crystal usually splits into different domains. Three cases of transformation-twin domains are distinguished:

  • (i) The symmetry operations suppressed during the transition belong to the point group [{\cal G}] 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 [[i] = \left\vert {\cal G} \right\vert/ \left\vert {\cal H} \right\vert] of the group–subgroup relation, i.e. to the number of cosets of [{\cal G}] with respect to [{\cal H}], including [{\cal H}] itself; cf. Section[link]. 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[link]). A further well known example is the α–β phase transformation of quartz at 846 K. On cooling from the hexagonal [\beta] phase (point group 622) to the trigonal [\alpha] phase (point group 32), the twofold rotation [2_z = 6^3], contained in the sixfold axis of [\beta]-quartz, is suppressed, and so are the other five rotations of the coset (cf. Example[link]). 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[link]). 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[link], 2000[link]), cf. Section[link], Note (7)[link].

  • (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.

Well known examples of ferroelastic transformation twins are K2SO4 (Example[link]) and various perovskites (Example[link]). Characteristic for non-merohedral (ferroelastic) transformation twins are their planar twin boundaries and the many parallel (lamellar) twin domains of nearly equal size. In contrast, the twin boundaries of merohedral (non-ferroelastic) transformation twins, e.g. Dauphiné twins of quartz, often are curved, irregular and non-parallel.

Transformation twins are closely related to the topic of Domain structures, which is extensively treated by Janovec & Přívratská in Chapter 3.4[link] of this volume.

A generalization of the concept of transformation twins includes twinning due to structural relationships in a family of related compounds (`structural twins'). Here the parent phase is formed by the high-symmetry `basic structure' (`aristotype') from which the `deformed structures' and their twin laws, occurring in other compounds, can be derived by subgroup considerations similar to those for actual transformation twins. Well known families are ABX3 (`perovskite tree', now often generalized as `Bärnighausen tree'; Bärnighausen, 1980[link]) and A2BX4 (Na2SO4- and K2SO4-type compounds). In Example (3)[link] of Section[link], growth twins among MeX2 dichalcogenides are described in detail. Mechanical twinning

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Under mechanical load, some crystals can be `switched' – partly or completely – from one orientation state into another. This change frequently proceeds in steps by the switching of domains. As a rule, the new orientation is related to the original one by an operation that obeys the definition of a twin operation (cf. Section[link]). In many cases, the formation of mechanical twins (German: Druckzwillinge) is an essential feature of the plasticity of crystals. The deformation connected with the switching is described by a homogeneous shear. The domain arrangement induced by mechanical switching is preserved after the mechanical load is released. In order to re-switch the domains, a mechanical stress of opposite sign (coercive stress) has to be applied. This leads to a hysteresis of the stress–strain relation. In many cases, however, switching cannot be repeated because the crystal is shattered.

All aspects of mechanical twinning are reviewed by Cahn (1954[link], Section 3). A comprehensive treatment is presented in the monograph Mechanical Twinning of Crystals by Klassen-Neklyudova (1964[link]). A brief survey of mechanical twinning in metals is given by Barrett & Massalski (1966[link]).

With respect to symmetry, three categories of mechanical twins are distinguished in this chapter:

  • (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[link](ii)[link]]. This case is illustrated by the famous deformation twins of calcite (Fig.[link]): The eigensymmetry [{\cal H}] of calcite is [{\bar 3}2/m], and the most conspicuous twin element is the twin reflection plane [(01{\bar 1}2)] which is parallel to an edge of the cleavage rhombohedron [\{10{\bar 1}1\}]. The extension of the eigensymmetry by this twin operation does not lead to a crystallographic composite symmetry, but the reduced composite symmetry is crystallographic, [{\cal K}^\ast = 2/m\,2/m\,2/m].


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    Mechanical twins of calcite, CaCO3. All indices refer to the standard morphological cell [cf. Section[link], Example (5)[link]]. (a) Generation of a deformation twin by a knife-edge impact (after Baumhauer, 1879[link]). (b) Description as a glide process (`twin glide') on plane ([01{\bar 1}2]). (c), (d) Generation of deformation twins by compression of the cleavage rhombohedron [\{10{\bar 1}1\}] (Mügge, 1883[link]). 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 [(10{\bar 1}2)] and the glide direction [[21{\bar 1}]]. The direction of the twofold axis is shown in the left-hand figure. Original size of the sample: ca. [3\times 3\times\sim9.5] mm. Part (e) courtesy of H. E. Hoefer, PhD thesis, University of Cologne, 1989.

    Another famous case is that of the [\Sigma 3] 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 [4/m\,{\bar 3}\,2/m] by the twin operation leads to a noncrystallographic composite symmetry. The reduced composite symmetry [{\cal K}^\ast], which is constructed from the intersection symmetry [{\cal H}^\ast], however, is crystallographic (cf. Example[link]).

    A description of the (plastic) deformation by twinning in terms of strain ellipsoids is presented in Section[link]

  • (ii) Ferroelastic twinning. In 1970, a special category of mechanical twins was introduced and characterized by Aizu (1970a[link]), 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[link] 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. 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[link]); a review is provided by Abrahams (1994[link]).

  • (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[link]). 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 3.3.9[link]). 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[link]) and described in detail by Schubnikow & Zinserling (1932[link]) and Zinserling & Schubnikow (1933[link]). 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[link]). Aizu (1973[link]) 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 modulus4 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[link]).

    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[link] and[link]). The switching from a cubic obverse rhombohedron into the reverse one actually takes place in the `[\Sigma 3] deformation twins' of cubic metals [cf. part (i)[link] 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[link]; Klassen-Neklyudova, 1964[link], 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[link]; Iliescu et al., 1997[link]). Brazil twins of quartz, which also entail the reversal of the electric axes (cf. Fig.[link]) cannot be detwinned. Mechanical detwinning by appropriate stress is also used to obtain single-domain crystals of the ferroelastic YBa2Cu3O7−δ high-Tc 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 [222\longleftrightarrow2] at about 297 K with coupled ferroelectricity and ferroelasticity in the monoclinic phase (cf. Zheludev, 1971[link], 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[link], 1955[link]; quoted after Klassen-Neklyudova (1964[link]), pp. 75–78 and Fig. 100].

  • (v) Non-ferroelastic and co-elastic twins. Phase transitions with symmetry changes within the same crystal family 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[link]) uses the term co-elastic. In crystallography, twins resulting from this kind of phase transition are grouped under twins by merohedry (cf. Section 3.3.9[link]). Typical examples of non-ferroelastic and co-elastic materials are quartz (merohedral Dauphiné twins, phase transition [P3_121 \longleftrightarrow P6_422] at 846 K) and calcite (transition [R{\bar 3}c \longleftrightarrow R{\bar 3}m] at about 1523 K, cf. Salje, 1993[link], Chapter 2).

In conclusion, it is pointed out that twins with one and the same twin law can be generated in different ways. In addition to the twins of potassium sulfate mentioned above [growth twins, transformation twins and mechanical (ferroelastic) twins], the Dauphiné twinning of quartz is an example: it can be formed during crystal growth, by a phase transition and by mechanical stress [ferrobielasticity, cf. part (iii)[link] above]. As a rule, the domain textures of a twinned crystal are quite different for growth twins, transformation twins and mechanical twins.

3.3.8. Lattice aspects of twinning

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In the previous sections of this chapter, the symmetry relations and the morphological classification of twins have been presented on a macroscopic level, i.e. in terms of point groups. It would be ideal if this treatment could be extended to atomic dimensions, i.e. if twinning could be explained and even predicted in terms of space groups, crystal structures, interface structures and structural defects. This approach is presently only possible for a few specific crystals; for the majority of twins, however, only general rules are known and qualitative predictions can be made.

An early and very significant step towards this goal was the introduction of the lattice concept in the treatment of twinning (three-periodic twins). This was first done about a hundred years ago – based on the lattice analysis of Bravais – by Mallard (1879[link]) and especially by Friedel (1904[link], 1926[link]), in part before the advent of X-ray diffraction. The book by Friedel (1926[link]), particularly Chapter 15, is the most frequently cited reference in this field. Later, Friedel (1933[link]) sharpened his theories to include two further types of twins: `macles monopériodiques' and `macles dipériodiques', in addition to the previous `macles tripériodiques', see Section[link] below. These concepts were further developed by Niggli (1919[link], 1920/1924/1941[link]).5

The lattice aspects of twinning (triperiodic twins) are discussed in this section and in Section 3.3.9.[link] An important concept in this field is the coincidence-site sublattice of the twin in direct space and its counterpart in reciprocal space. Extensive use of the notion of coincidence-site lattices (CSLs) is made in bicrystallography for the study of grain boundaries, as briefly explained in Section 3.2.2[link] .

The coincidence-site lattice and further related lattices (O- and DSC-lattices) were introduced into the study of bicrystals by Bollmann (1970[link], 1982[link]) and were theoretically thoroughly developed by Grimmer (1989a[link],b[link], 2003[link]). Their applications to grain boundaries are contained in the works by Sutton & Balluffi (1995[link]) and Gottstein & Shvindlerman (1999[link]). Basic concepts of Friedel's lattice theory

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The basis of Friedel's (1904[link], 1926[link]) lattice theory of twinning is the postulate that the coincidence-site sublattice common to the two twin partners (twin lattice) suffers no deviation (strict condition) or at most a slight deviation (approximate condition) in crossing the boundary between the two twin components (composition plane). This purely geometrical condition is often expressed as `three-dimensional lattice control' (Santoro, 1974[link], p. 225), which is supposed to be favourable to the formation of twins.

In order to define the coincidence sublattice (twin lattice) of the two twin partners, it is assumed that their oriented point lattices are infinitely extended and interpenetrate each other. The lattice classification of twins is based on the degree of coincidence of these two lattices. The criterion applied is the dimension of the coincidence-site subset of the two interpenetrating lattices, which is defined as the set of all lattice points common to both lattices, provided that two initial points, one from each lattice, are brought to coincidence (common origin). This common origin has the immediate consequence that the concept of the twin displacement vector t – as introduced in Note (8)[link] of Section[link] – does not apply here. The existence of the coincidence subset of a twin results from the crystallographic orientation relation (Section[link]), which is a prerequisite for twinning. This subset is one-, two- or three-dimensional (monoperiodic, diperiodic or triperiodic twins).

If a coincidence relation exists between lattices in direct space, a complementary superposition relation occurs for their reciprocal lattices. This superposition can often, but not always, be detected in the diffraction patterns of twinned crystals; cf. Section 3.3.11[link] below. Lattice coincidences, twin lattice, twin lattice index

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Four types of (exact) lattice coincidences have to be distinguished in twinning:

  • (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[link], the following three generate one-dimensional lattice coincidence:

    • (a) twofold rotation around a (rational) lattice row normal to an irrational plane [twin operation (iii)[link]];

    • (b) reflection across an irrational plane normal to a (rational) lattice row (note that the coincidence would be three-dimensional if this plane were rational) [twin operation (iv)[link]];

    • (c) twofold rotation around an irrational axis normal to a (rational) lattice row (complex twin, Kantennormalengesetz) [twin operations (v)[link] and (vi)[link]].

    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:

    • (a) reflection across a (rational) lattice plane [twin operation (i)[link]];

    • (b) twofold rotation around an irrational axis normal to a (rational) lattice plane (note that the coincidence would be three-dimensional if this axis were rational) [twin operation (ii)[link]].

    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 [\Sigma], especially in metallurgy. It is the volume ratio of the primitive cells of the twin lattice and of the (original) crystal lattice (i.e. [1/j] is the `degree of dilution' of the twin lattice with respect to the crystal lattice): [[j] = \Sigma = V_{\rm twin} / V_{\rm crystal}.]

    The lattice index is always an integer: [j = 1] means complete coincidence (parallelism), [j> 1] partial coincidence of the two lattices. The index [j] can also be interpreted as elimination of the fraction [(j - 1)/j] 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 [[j] = 1] has been called by Friedel (1926[link], p. 427) twinning by merohedry (`macles par mériédrie') (for short: merohedral twinning), whereas twinning with [[j]> 1] is called twinning by lattice merohedry or twinning by reticular merohedry (`macles par mériédrie réticulaire') (Friedel, 1926[link], p. 444). The terms for [[j] = 1] are easily comprehensible and in common use. The terms for [[j]> 1], however, are somewhat ambiguous (the term `merohedral' refers to point groups, not lattices). In the present section, therefore, the terms sublattice, coincidence lattice or twin lattice of index [[j]] are preferred. Merohedral twinning is treated in detail in Section 3.3.9.[link]

    Complete and exact three-dimensional lattice coincidence ([[j] = 1]) always exists for inversion twins (of noncentrosymmetric crystals) [twin operation (vii)[link]]. 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[link]. 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 [[001]_{\rm tet}] and the [[001]_{\rm hex}] (or [[111]_{\rm rhomb}]) zones, i.e. for planes parallel and rows perpendicular to these directions, in addition to the trivial cases [[001]\perp(001)] or [[111]\perp(111)]. In cubic lattices, every lattice plane (hkl) is perpendicular to a lattice row [uvw] (with [h = u], [k = v], [l = w]). More general coincidence relations were derived by Grimmer (1989a[link],b[link], 2003[link]).

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    Lattice planes (hkl) and lattice rows [uvw] that are mutually perpendicular (after Koch, 2004[link])

    LatticeLattice plane (hkl)Lattice row [uvw]Perpendicularity condition and quantity [j\,' = hu + kv + lw]
    Monoclinic (unique axis b) (010) [010]
    Monoclinic (unique axis c) (001) [001]
    Orthorhombic (100) [100]
    (010) [010]
    (001) [001]
    Hexagonal and rhombohedral (hexagonal axes) ([hki0]) [[uv0]] [u = 2h + k], [v = h + 2k], [j\,' = 2 h^2 + 2k^2 + 2 hk]
    (0001) [001]
    Rhombohedral (rhombohedral axes) ([h, k, -h-k]) [[u, v, -u-v]] [u = h], [v = k], [j\,' = 2 h^2 + 2k^2 + 2 hk]
    (111) [111]
    Tetragonal ([hk0]) [[uv0]] [u = h], [v = k], [j\,' = h^2 + k^2]
    ([001]) [[001]]
    Cubic ([hkl]) [[uvw]] [u = h], [v = k], [w = l]; [j\,' = h^2 + k^2 + l^2]

    The index [[j]] of a coincidence or twin lattice can often be obtained by inspection; it can be calculated by using a formula for the auxiliary quantity [j\,'] as follows: [j\,' = hu + kv + lw \ \ ({\rm scalar\ product}\ {\bf r}^\star _{hkl} \cdot {\bf t}_{uvw})]with sublattice index [\eqalign{ [j] &= \vert j\,'\vert \hbox{ for }j\,' = 2n + 1 \cr&= \vert j\,'\vert /2 \hbox{ for }j\,' = 2n. }]

    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[link]). Formulae and tables are presented by Friedel (1926[link], pp. 245–252) and by Donnay & Donnay (1972[link]). The various equations for the quantity [j\,'] are also listed in the last column of Table[link].

    Note that in the tetragonal system for any ([hk0]) reflection twin and any [[uv0]] twofold rotation twin, the coincidence lattices are also tetragonal and have the same lattice parameter c. Further details are given by Grimmer (2003[link]). An analogous relation applies to the hexagonal crystal family for ([hki0]) and [[uv0]] twins. In the cubic system, the following types of twin lattices occur:

    • (111) and [111] twins: hexagonal P lattice (e.g. spinel twins);

    • [(hk0)] and [[uv0]] twins: tetragonal lattice;

    • [(hhl)] and [[uuw]] twins: orthorhombic lattice;

    • [(hkl)] and [[uvw]] twins: monoclinic lattice.

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

After these general considerations of coincidence-site and twin lattices and their lattice index, specific cases of `triperiodic twins' are treated in Section[link] In addition to the characterization of the twin lattice by its index [[j]], the [\Sigma] notation used in metallurgy is included. Twins with three-dimensional twin lattices (`triperiodic' twins)

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The following cases of exact superposition are distinguished:

  • (i) Twins with [[j] = 1] ([\Sigma 1] 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 Sections 3.3.9[link] and[link]; cf. Klapper & Hahn (2010[link]).

  • (ii) Twins with [[j] = 2] ([\Sigma 2] twins). This twinning does not occur systematically among the cases listed in Table[link], except for special metrical relations. Example: a primitive orthorhombic lattice with [b/a = \sqrt{3}] and twin reflection plane [(110)] or [({\bar 1}10)]. The coincidence lattice is hexagonal with [a_{\rm hex} = 2a] and [ [j] = 2].

  • (iii) Twins with [[j] = 3] ([\Sigma 3] twins). Twins with [[j] = 3] are very common among rhombohedral and cubic crystals (`spinel law') with the following two representative twin operations:

    • (a) twofold rotation around a threefold symmetry axis [111] (cubic or rhombohedral coordinate axes) or [001] (hexagonal axes);

    • (b) reflection across the plane (111) or (0001) normal to a threefold symmetry axis.

    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[link],[link] and Figs.[link]–[link][link]). 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 [[j] = 3]. The same holds for the spinel twins of cubic crystals, provided only one of the four threefold axes is involved in the twinning. For further details see Section[link].

  • (iv) Twins with [[j]> 3] ([\Sigma> 3] twins). Whereas twins with [[j] = 3] are very common and of high importance among minerals and metals, twins with higher lattice indices occur hardly at all. All these `high-index' twins can occur systematically only in tetragonal, hexagonal, rhombohedral and cubic crystals, due to the geometric perpendicularity relations set out in Table[link]. Note that for special lattice metrics (axial ratios and angles) they can occur, of course, in any crystal system. These special metrics, however, are not enforced by the crystal symmetry and hence the coincidences are not strict, but only `pseudo-coincidences'. Σ5 and Σ7 twins are treated in Section[link] and in Klapper & Hahn (2012[link]).


  • (1) Tetragonal twins with twin reflection planes {120} or {310}, or twofold twin axes [\langle 120\rangle] or [\langle 310\rangle] lead to [[j] = 5], the largest value of [[j]] that has been found so far for tetragonal twins. The coincidence lattice is again tetragonal with [{\bf a}' = 2{\bf a} + {\bf b}], [{\bf b}' = - {\bf a }+ 2{\bf b}], [{\bf c}' = {\bf c}] and is shown in Fig.[link]. An actual example, SmS1.9 (Tamazyan et al., 2000b[link]), is discussed in Section[link].


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    Lattice relations of [\Sigma 5] twins of tetragonal crystals with primitive lattice: twin mirror plane and composition plane (120) with twin displacement vector t = 0. Small circles: lattice points of domain 1; small x: lattice points of domain 2; large black dots: [\Sigma 5] coincidence sublattice.

  • (2) There exist several old and still unsubstantiated indications for a [[j] = 5] cubic garnet twin with twin reflection plane (210), cf. Arzruni (1887[link]); Tschermak & Becke (1915[link], p. 594).

  • (3) Klockmannite, CuSe (Taylor & Underwood, 1960[link]; Takeda & Donnay, 1965[link]). This hexagonal mineral seems to be the only example for a hexagonal twin with [[j]> 3]. X-ray diffraction experiments indicate a reflection twin on [(13{\bar 4}0)], corresponding to [[j] = 13]. Later structural studies, however, suggest the possibility of disorder instead of twinning.

  • (4) Galena, PbS (NaCl structure). Galena crystals from various localities often exhibit lamellae parallel to the planes {441} which are interpreted as (441) reflection twins with [[j] = 33] ([\Sigma 33] twin). These natural twins are deformation and not growth twins. In laboratory deformation experiments, however, these twins could not be generated. A detailed analysis of twinning in PbS with respect to plastic deformation is given by Seifert (1928[link]).

  • (5) For cubic metals and alloys annealing twins (recrystallization twins) with [[j]> 3] are common. Among them high-order twins (high-generation twins) are particularly frequent. They are based on the [\Sigma 3] (spinel) twins (first generation) which may coalesce and form `new twins' with [\Sigma 9 = 3^2] [second generation, with twin reflection plane (221)], [\Sigma 27 = 3^3] [third generation, twin reflection plane (115)], [\Sigma 81 = 3^4] [fourth generation, twin reflection plane (447)] etc. Every step to a higher generation increases [\Sigma] by a factor of three (Gottstein, 1984[link]). An interesting and actual example is the artificial silicon tricrystal shown in Fig.[link], which contains three components related by two (111) reflection planes (first generation, two [\Sigma 3] boundaries) and one (221) reflection plane (second generation, one [\Sigma 9] boundary).


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    (a) A (110) silicon slice (10 cm diameter, 0.3 mm thick), cut from a Czochralski-grown tricrystal for solar-cell applications. As seed crystal, a cylinder of three coalesced Si single-crystal sectors in (111) and (221) reflection-twin positions was used. Pulling direction [110] (Courtesy of M. Krühler, Siemens AG, München). (b) Sketch of the tricystal wafer showing the twin relations [twin laws [m(111)] and [m(221)]] and the [\Sigma] characters of the three domain pairs. The atomic structures of these (111) and (221) twin boundaries are discussed by Kohn (1956[link], 1958[link]), Hornstra (1959[link], 1960[link]) and Queisser (1963[link]).

  • (6) The same type of tricrystal has been found in cubic magnetite (Fe3O4) nanocrystals grown from the biogenic action of magnetotactic bacteria in an aquatic environment (Devouard et al., 1998[link]). Here, HRTEM micrographs (Fig. 6 of the paper) show the same triple-twin arrangement as in the Si tricrystal above. The authors illustrate this triple twin by (111) spinel-type intergrowth of three octahedra exhibiting two [\Sigma 3] and one [\Sigma 9] domain pairs. The two [\Sigma 3] interfaces are (111) twin reflection planes, whereas the [\Sigma 9] boundary is very irregular and not a compatible planar (221) interface (i.e. not a twin reflection plane).

  • (7) A third instructive example is provided by the five­fold cyclic `cozonal' twins (zone axis [[1{\bar 1}0]]) of Ge nanocrystals (Neumann et al., 1996[link]; Hofmeister, 1998[link]), which are treated in Section[link] and Fig.[link]. All five boundaries between neighbouring domains (sector angles 70.5°) are of the [\Sigma3(111)] type. Second nearest ([2\times 70.5^\circ]), third nearest ([3\times 70.5^\circ]) and fourth nearest ([4\times70.5^\circ]) neighbours exhibit [\Sigma9], [\Sigma27] and [\Sigma81] coincidence relations (second, third and fourth [\Sigma] generation), respectively, as introduced above in (5[link]). These relations can be described by the `cozonal' twin reflection planes (111), (221), (115) and (447). Since [5\times70.5^\circ=352.5^\circ], an angular gap of 7.5° would result. In actual crystals this gap is compensated by stacking faults as shown in Fig.[link]. A detailed treatment of all these cases, including structural models of the interfaces, is given by Neumann et al. (1996[link]).

  • (8) Examples of (hypothetical) twins with [[j]> 1] due to metrical specialization of the lattice are presented by Koch (2004[link]). Approximate (pseudo-)coincidences of two or more lattices

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In part (iv)[link] of Section[link], three-dimensional lattice coincidences and twin lattices (sublattices) were considered under two restrictions:

  • (a) the lattice coincidences (according to the twin lattice index [j]) are exact (not approximate);

  • (b) only two lattices are superimposed to form the twin lattice.

In the present section these two conditions are relaxed as follows:

  • (1) In addition to exact lattice coincidences (as they occur for all merohedral twins) approximate lattice coincidences (pseudo-coincidences) are taken into account.

    In this context, it is important to explain the meaning of the terms approximate lattice coincidences or pseudo lattice coincidences as used in this section. Superposition of two or more equal lattices (with a common origin) that are slightly misoriented with respect to each other leads to a three-dimensional moiré pattern of coincidences and anti-coincidences. The beat period of this pattern increases with decreasing misorientation. It appears sensible to use the term approximate or pseudo-coincidences only if the `splitting' of lattice points is small within a sufficiently large region around the common origin of the two lattices. Special cases occur for reflection twins and rotation twins of pseudosymmetrical lattices. For the former, exact two-dimensional coincidences exist parallel to the (rational) twin reflection plane and the moiré pattern is only one-dimensional in the direction normal to this plane. Hence, the region of `small splitting' is a two-dimensional (infinitely extended) thin layer of the twin lattice on both sides of the twin reflection plane [example: pseudo-monoclinic albite (010) reflection twins]. For rotation twins, the region of `small splitting' is an (infinitely long) cylinder around the twin axis. On the axis the lattice points coincide exactly.

    In general, a typical measure of this region, in terms of the reciprocal lattice, could be the size of a conventional X-ray diffraction photograph or a two-dimensional detector. Whereas the slightest deviations from exact coincidence lead to pseudo-coincidences, the `upper limit of the splitting', up to which two lattices are considered as pseudo-coincident, is not definable on physical grounds and thus is a matter of convention and personal preference. As an angular measure of the splitting the twin obliquity has been introduced by Friedel (1926[link]). This concept and its use in twinning will be discussed below in Section[link]

  • (2) The previous treatment of superposition of only two lattices is extended to multiple twins with several interpenetrating lattices which are related by a pseudo n-fold twin axis. Such a twin axis cannot be `exact', no matter how close its rotation angle comes to the exact angular value. For this reason, twin axes of order [n> 2] necessarily lead to pseudo lattice coincidences.

    Here it is assumed that such pseudo-coincidences exist for any pair of neighbouring twin domains. As a consequence, pseudo-coincidences occur for all n domains. For this case, the following rules exist:

    • (i) Only n-fold twin axes with the crystallographic values n = 3, 4 and 6 lead to pseudo lattice coincidences of all domains. Example: cyclic triplets of aragonite (Fig.[link]).

    • (ii) The number of (interpenetrating) lattices equals the number of different domain states [cf. Section[link](iii)[link]], viz.[\eqalign{\hbox{6, 3 or 2 lattices for }n &= 6,\cr\hbox{3 lattices for }n&=3,\cr\hbox{4 or 2 lattices for }n&=4,}]whereby the case `2 lattices' for [n = 6] leads to exact lattice coincidence (merohedral twinning, e.g. Dauphiné twins of quartz).

    • (iii) There always exists exact (one-dimensional) coincidence of all lattice rows along the twin axis.

    • (iv) If there is a (rational) lattice plane normal to the twin axis, the splitting of the lattice points occurs only parallel to this plane. If, however, this lattice plane is pseudo-normal (i.e. slightly inclined) to the twin axis, the splitting of lattice points also has a small component along the twin axis. Twin obliquity and lattice pseudosymmetry

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The concept of twin obliquity has been introduced by Friedel (1926[link], p. 436) to characterize (metrical) pseudosymmetries of lattices and their relation to twinning. The obliquity [\omega] is defined as the angle between the normal to a given lattice plane (hkl) and a lattice row [uvw] that is not parallel to (hkl) and, vice versa, as the angle between a given lattice row [uvw] and the normal to a lattice plane (hkl) that is not perpendicular to [uvw]. The twin obliquity is thus a quantitative (angular) measure of the pseudosymmetry of a lattice and, hence, of the deviation which the twin lattice suffers in crossing the composition plane (cf. Section[link]).

The smallest mesh of the net plane (hkl) together with the shortest translation period along [uvw] define a unit cell of a sublattice of lattice index [j]; j may be [=1] or [>1] [cf. Section[link](iv)[link]]. The quantities [\omega] and j can be calculated for any lattice and any (hkl)/[uvw] combination by elementary formulae, as given by Friedel (1926[link], pp. 249–252) and by Donnay & Donnay (1972[link]). A computer program has been written by Le Page (1999[link], 2002[link]) which calculates for a given lattice all (hkl)/[uvw]/[\omega]/j combinations up to given limits of [\omega] and j. In the theory of Friedel and the French School, a (metrical) pseudosymmetry of a lattice or sublattice is assumed to exist if the twin obliquity [\omega] as well as the twin lattice index j are `small'. This in turn means that the pair lattice plane (hkl)/lattice row [uvw] is the better suited as twin elements (twin reflection plane/twofold twin axis) the smaller [\omega] and j are.

The term `small' obviously cannot be defined in physical terms. Its meaning rather depends on conventions and actual analyses of triperiodic twins. In his textbook, Friedel (1926[link], p. 437) quotes frequently observed twin obliquities of 3–4° (albite [4^\circ 3'], aragonite [3^\circ 44']) with `rare exceptions' of 5–6°. In a paper devoted to the quartz twins with `inclined axes', Friedel (1923[link], pp. 84 and 86) accepts the La Gardette (Japanese) and the Esterel twins, both with large obliquities of [\omega = 5^\circ 27'] and [\omega = 5^\circ 48'], as pseudo-merohedral twins only because their lattice indices [[j] = 2] and 3 are (`en revanche') remarkably small. He considers [\omega = 6^\circ] as a limit of acceptance [`limite prohibitive'; Friedel (1923[link], p. 88)].

Lattice indices [[j] = 3] are very common (in cubic and rhombo­hedral crystals), [[j] = 5] twins are rare and [[j] = 6] seems to be the maximal value encountered in twinning (Friedel, 1926[link], pp. 449, 457–464; Donnay & Donnay, 1974[link], Table 1). In his quartz paper, Friedel (1923[link], p. 92) rejects all pseudo-merohedral quartz twins with [[j]\geq 4] despite small [\omega] values, and he points out, as proof that high j values are particularly unfavourable for twinning, that strictly merohedral quartz twins with [[j] = 7] do not occur, i.e. that [\omega = 0] cannot `compensate' for high j values.

In agreement with all these results and later experiences (e.g. Le Page, 1999[link], 2002[link]), we consider in Table[link] only lattice pseudosymmetries with [\omega \le 6^\circ] and [[j] \le 6], preferably [[j] \le 3]. (It should be noted that, on purely mathematical grounds, arbitrarily small [\omega] values can always be obtained for sufficiently large values of [h,k,l] and [u,v,w], which would be meaningless for twinning.) The program by Le Page (1999[link], 2002[link]) enables for the first time systematic calculations of many (`all possible') (hkl)/[uvw] combinations for a given lattice and, hence, statistical and geometrical evaluations of existing and particularly of (geometrically) `permissible' but not observed twin laws. In Table[link], some examples are presented that bring out both the merits and the problems of lattice geometry for the theory of twinning. The `permissibility criteria' [\omega \le 6^\circ] and [[j] \le 6], mentioned above, are observed for most cases.

Table| top | pdf |
Examples of calculated obliquities [\omega] and lattice indices [j] for selected (hkl)[/][uvw] combinations and their relation to twinning

Calculations were performed with the program OBLIQUE written by Le Page (1999[link], 2002[link]).

Crystal(hkl)Pseudo-normal [uvw]Obliquity [[^\circ]]Lattice index [j]Remark
[a = 6.51], [b = 15.15], [c = 6.28] Å
[\beta = 127.5^\circ]
(100) [302] 2.47 3 Dovetail twin (very frequent)
  [805] 0.42 4
(001) [203] 5.92 3 Montmartre twin (less frequent)
  [305] 0.95 5
(101) [101] 2.60 2 No twin
[(11{\bar 1})] [[31{\bar 4}]] 1.35 4 No twin
[a = 4.5933], [c = 2.9592] Å
(101) [102] 5.02 3 Frequent twin
  [307] 0.84 5
(301) [101] 5.43 2 Rare twin
(201) [304] 2.85 5 No twin
(210) or (130) [210] or [130] 0 5 No twin
[a = 4.9031], [c = 5.3967] Å
[(11{\bar 2}2)] [111] 5.49 2 Japanese twin (La Gardette) (rare)
[(10{\bar 1}1)] [211] 5.76 3 Esterel twin (rare)
[(10{\bar 1}2)] [212] 5.76 3 Sardinia twin (very rare)
[(21{\bar 3}0)] or [(14{\bar 5}0)] [540] or [230] 0 7 No twin
[a = 7.871], [b = 16.620], [c = 5.656] Å
[\beta = 90.00^\circ]
(031) [013] 1.19 6 90° twin (rare)
(231) [313] 0.90 12 60° twin (frequent)
(201) [101] 0.87 3 No twin
(101) [102] 0.87 3 No twin
[R{\bar 3}c]
[a = 4.989], [c = 17.062] Å
[hexagonal axes, structural
X-ray cell; cf. Section[link],
Example (5)[link]]
[(01{\bar 1}2)] [5.10.1] 5.31 2 No twin
  [7.14.2] 2.57 3  
  [481] 0.59 5  
[(10{\bar 1}4)] [421] 0.74 4 Rare deformation twin (r-twin)
[(01{\bar 1}8)] [121] 0.59 5 Frequent deformation twin (e-twin)
[(10{\bar 1}1)] [14.7.1] 1.54 5 No twin

The following comments on these data should be made.

Gypsum: The calculations result in about 70 `permissible' (hkl)/[uvw] combinations. For the very common (100) dovetail twin, four (100)/[uvw] combinations are obtained. Only the two combinations with smallest [\omega] and [j] are listed in the table; similarly for the less common (001) Montmartre twin. In addition, two cases of low-index (hkl) planes with small obliquities and small lattice indices are listed, for which twinning has never been observed.

Rutile: Here nearly twenty `permissible' (hkl)/[uvw] combinations with [\omega \le 6^\circ], [[j] \le 6] occur. For the frequent (101) reflection twins, five permissible cases are calculated, of which two are given in the table. For the rare (301) reflection twins, only the one case listed, with high obliquity [\omega = 5.4^\circ], is permissible. For the further two cases of low obliquity and lattice index [5], twins are not known. Among them is one case of (strict) `reticular merohedry', (210) or (130), with [\omega = 0] and [[j] = 5] (cf. Fig.[link]).

Quartz: The various quartz twins with inclined axes were studied extensively by Friedel (1923[link]). The two most frequent cases, the Japanese [(11{\bar 2}2)] twin (called La Gardette twin by Friedel) and the [(10{\bar 1}1)] Esterel twin, are considered here. In both cases, several lattice pseudosymmetries occur. Following Friedel, those with the smallest lattice index but relatively high obliquity close to 6° are listed in the table. Again, a twin of (strict) `reticular merohedry' with [\omega = 0] and [[j] = 7] does not occur [cf. Section[link], Example (2)[link]].

Staurolite: Both twin laws occurring in nature, (031) and (231), exhibit small obliquities but rather high lattice indices [6] and [12]. The frequent (231) 60° twin with [[j] = 12] falls far outside the `permissible' range. The further two planes listed in the table, (201) and (101), exhibit favourably small obliquities and lattice indices, but do not form twins. The existing (031) and (231) twins of staurolite are discussed again in Section[link] under the aspect of `reticular pseudo-merohedry'.

Calcite: For calcite, 19 lattice pseudosymmetries obeying Friedel's `permissible criteria' are calculated. Again, only a few are mentioned here (indices referred to the structural cell). For the primary deformation twin [(01{\bar 1}8)], e-twin after Bueble & Schmahl (1999[link]), cf. Section[link], Example (5), one permissible lattice pseudosymmetry with small obliquity 0.59 but high lattice index [5] is found. For the less frequent secondary deformation twin [(10{\bar 1}4)], r-twin, the situation is similar. The planes [(01{\bar 1}2)] and [(10{\bar 1}1)] permit small obliquities and lattice indices [\le [5]], but do not appear as twin planes.

The discussion of the examples in Table[link] shows that, with one exception [staurolite (231) twin], the obliquities and lattice indices of common twins fall within the [\omega/[j]] limits accepted for lattice pseudosymmetry. Three aspects, however, have to be critically evaluated:

  • (i) For most of the lattice planes (hkl), several pseudo-normal rows [uvw] with different values of [\omega] and [j] within the 6°/[6] limit occur, and vice versa. Friedel (1923[link]) discussed this in his theory of quartz twinning. He considers the (hkl)/[uvw] combination with the smallest lattice index as responsible for the observed twinning.

  • (ii) Among the examples given in the table, low-index (hkl)/[uvw] combinations with more favourable [\omega/[j]] values than for the existing twins can be found that never form twins. A prediction of twins on the basis of `lattice control' alone, characterized by low [\omega] and [j] values, would fail in these cases.

  • (iii) All examples in the table were derived solely from lattice geometry, none from structural relations or other physical factors.

Note. As a mathematical alternative to the term `obliquity', another more general measure of the deviation suffered by the twin lattice in crossing the twin boundary was presented by Santoro (1974[link], equation 36). This measure is the difference between the metric tensors of lattice 1 and of lattice 2, the latter after retransformation by the existing or assumed twin operation (or more general orientation operation). Twinning of isostructural crystals

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In the present section, the critical discussion of the lattice theory of twinning is extended from the individual crystal species, treated in Section, to the occurrence of merohedral twinning in series of isotypic and homeotypic crystals. The crystals in each series have the same (or closely related) structure, space group, lattice type and lattice coincidences. The following cases are of interest here:

  • (i) Quartz (SiO2), quartz-homeotypic gallium phosphate (GaPO4) and benzil [(C6H5CO)2, so-called `organic quartz'] crystallize under normal conditions in the enantiomorphic space groups [P3_121] and [P3_221]. In quartz, merohedral Dauphiné and Brazil [\Sigma 1] twins are very frequent, whereas twins of the Leydolt (or `combined') law are very rare (cf. Example[link]). In gallium phosphate, Leydolt twins occur as frequently as Dauphiné and Brazil twins (Engel et al., 1989[link]). In benzil crystals, however, these twins are never observed, although the same space-group symmetries and conditions for systematic lattice coincidences as in quartz and in gallium phosphate exist. The reason is the completely different structure and chemical bonding of benzil, which is not capable of forming low-energy boundaries for these three twin laws.

  • (ii) Iron borate FeBO3, calcite CaCO3 and sodium nitrate NaNO3 crystallize under normal conditions in the calcite structure with space group [R{\bar 3}2/c]. The rhombohedral lattice allows twinning with a hexagonal [\Sigma3] coincidence lattice (cf. Example[link]). Practically all spontaneously nucleated FeBO3 crystals grown from vapour (chemical transport) or solution (flux) are [\Sigma 3]-twinned and form intergrowths of reverse and obverse rhombohedra (penetration twins). This kind of twinning is comparatively rare in calcite, where the twins usually appear with another morphology [contact twins on (0001)]. Interestingly, this [\Sigma3] twinning does not occur (or is extremely rare) in sodium nitrate. This shows that even for isotypic crystals, the tendency to form [\Sigma 3] twins is extremely different. This can also be observed for crystals with the sodium chloride structure. Crystals of the silver halogenides AgCl and AgBr, precipitated from aqueous solution, develop multiple [\Sigma3] twins with high frequency (Bögels et al., 1999[link]), and so does galena PbS, whereas the isotypic alkali halogenides (e.g. NaCl, LiF) practically never (or only extremely rarely) form [\Sigma3] twins.

  • (iii) Another instructive example is provided by the [\Sigma3] (111) spinel twins in the sphalerite (ZnS) structure of III–V and II–VI semiconductor crystals (cf. Example[link]). In some of these compounds this kind of twinning is quite rare (e.g. in GaAs), but in others (e.g. InP, CdTe) it is very frequent. Gottschalk et al. (1978[link]) have quantitatively shown that the ease and frequency of twin formation is governed by the (111) stacking-fault energy [which is the energy of the (111) twin boundary]. They have calculated the (111) stacking-fault energies of various III–V semiconductors, taking into account the different ionicities of the bonds. The results prove quantitatively that the frequency of the [\Sigma 3] twin formation is correlated with the (111) boundary energy.

These examples corroborate the early observations of Cahn (1954[link], pp. 387–388). The present authors agree with his elegantly formulated conclusion, `that the fact that two substances are isostructural is but a slender guide to a possible similarity in their twinning behaviour'. Conclusions

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In conclusion, the lattice theory of twinning, presented in this section, can be summarized as follows:

  • (i) The lattice theory represents one of the first systematic theories of twinning; it is based on a clear and well defined concept and thus has found widespread acceptance, especially for the description, characterization and classification of `triperiodic' (merohedral and pseudo-merohedral) twins.

  • (ii) The concept, however, is purely geometrical and has as its object a mathematical, not a physical, item, the lattice. It takes into account neither the crystal structure nor the orientation and energy of the twin interface. This deficit has been pointed out and critically discussed as early as 1905 by Mügge[link], and later by Burgers (1927[link], 1932[link]), Heide (1928)[link], Buerger (1945)[link], Cahn (1954[link], Section 1.3), Hartman (1956)[link] and Holser (1958[link], 1960[link]); it is the major reason for the limitations of the theory and its low power of prediction for actual cases of twinning.

  • (iii) The relations between twinning and lattice (pseudo-)symmetries, however, become immediately obvious and are proven by many observations as soon as structural pseudosymmetries exist. Twinning is always facilitated if a real or hypothetical `parent structure' exists from which the twin law and the interface can be derived. Here, the lattice pseudosymmetry appears as a necessary consequence of the structural pseudosymmetry, which usually involves only small deformations of the parent structure, resulting in small obliquities of twin planes and twin axes (which are symmetry elements of the parent structure) and, hence, in twin interfaces of low energy. These structural pseudosymmetries are the result either of actual or hypothetical phase transitions (domain structures, cf. Chapter 3.4[link] ) or of structural relationships to a high-symmetry `prototype' structure, as explained in Section[link].

    The importance of structural pseudosymmetries was emphasized as early as 1927 by Burgers[link]. His statement (`the possibility of the twinning must, therefore, be connected with a definite pseudo-symmetrical character of the structure of the untwinned crystal'), however, was seriously attacked by Friedel (1928)[link]: La formation de la macle est totalement indifférente à la symétrie ou à la pseudo-symétrie de l'édifice cristallin ou des groupes d'atomes qui le constiuent. Elle est due uniquement à la rencontre des symétries ou de pseudo-symétries dans les périodes (mailles, simples ou multiples) de cet édifice (`The formation of a twin is totally independent of the symmetry or pseudosymmetry of the crystal structure or the groups of atoms which build it. It is uniquely due to the presence of symmetries or pseudosymmetries of the lattice [simple or multiple unit cells] of the crystal'). In a reply to this paper, Burgers (1932)[link] corroborates his previous criticism by the statement `that the fundamental cause of twinning must be looked for in the force relations in the immediate neighbourhood of the last-formed layer of the growing crystal'. In his view the (pseudo)symmetry of the simple or multiple lattice of a twinned crystal `is only another manifestation of a pseudo-symmetrical character of the structure of the untwinned crystal as a whole'.

  • (iv) On the other hand, twinning quite often occurs without recognizable structural pseudosymmetry, e.g. the (100) dovetail twins and the (001) Montmartre twins of gypsum, as well as the (101) and (301) reflection twins of rutile and some further examples listed in Table[link]. In all these cases, it can be concluded that the lattice theory of twinning is not the suitable tool for the characterization and prediction of the twins; in the terminology of Friedel: the twins are not `triperiodic' but only `diperiodic' or `monoperiodic'.

3.3.9. Twinning by merohedry and pseudo-merohedry

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We now resume the discussion of Section 3.3.8[link] on three-dimensional coincidence lattices and pseudo-coincidence lattices and apply it to actual cases of twinning, i.e. we treat in the present section twinning by merohedry (`macles par mériédrie') and twinning by pseudo-merohedry (`macles par pseudo-mériédrie'), both for lattice index [[j] = 1] and [[j]> 1], as introduced by Friedel (1926[link], p. 434). Often (strict) merohedral twins are called `parallel-lattice twins' or `twins with parallel axes'. Donnay & Donnay (1974[link]) have introduced the terms twinning by twin-lattice symmetry (TLS) for merohedral twinning and twinning by twin-lattice quasi-symmetry (TLQS) for pseudo-merohedral twinning, but we shall use here the original terms introduced by Friedel. Definitions of merohedry

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In the context of twinning, the term `merohedry' is applied with two different meanings which should be clearly distinguished in order to avoid confusion. The two cases are:

  • Case (1): `Merohedry' of point groups

    A merohedral point group is a subgroup of the holohedral point group (lattice point group) of a given Bravais lattice in a crystal family, i.e. group and subgroup belong to the same Bravais system. This is the original sense of the term merohedry, which has the morphological meaning of reduction of the number of faces of a given crystal form as compared with a holohedral crystal form. The degree of merohedry is given by the subgroup index [i]. For point groups within the same crystal family, possible indices [i] are 2 (hemihedry), 4 (tetartohedry) and 8 (ogdohedry). The only example for [[i] = 8] is the point group 3 in the hexagonal holohedry [6/m\,2/m\,2/m].

    If the point group of a crystal is reduced to such an extent that the subgroup belongs to a crystal family of lower symmetry, this subgroup is called a pseudo-merohedral point group, provided that the structural differences and, hence, also the metrical changes of the lattice (axial ratios) are small. Twinning by merohedry corresponds to non-ferroelastic phase transitions, twinning by pseudo-merohedry to ferroelastic phase transitions.

    Both merohedral and pseudo-merohedral subgroups of point groups are listed in Section 10.1.3[link] and Fig.[link] of Volume A of this series (Hahn & Klapper, 2005[link]); cf. also Koch (2004[link]), Table[link]

  • Case (2): `Merohedry' of translation groups (lattices)

    The term `reticular' or `lattice merohedry' designates the relation between a lattice and its `diluted' sublattice (without consideration of their lattice point groups). A sublattice6 is a three-dimensional subset of lattice points of a given lattice and corresponds to a subgroup of index [[j]> 1] of the original translation group. This kind of group–subgroup relation has been called `reticular merohedry' (`mériédrie réticulaire') by Friedel (1926[link], p. 444). Note that the lattice and its sublattice may belong to different crystal systems, and that the lattice point groups (holohedries) of lattice and sublattice generally do not obey a group–subgroup relation. This is illustrated by a cubic P lattice (lattice point group [4/m\,{\bar 3}\,2/m]) and one of its monoclinic sublattices (lattice point group [2/m]) defined by a general lattice plane (hkl) and the lattice row [hkl] normal to it. The symmetry direction [hkl] of the monoclinic sublattice does not coincide with any of the symmetry directions of the cubic lattice, i.e. there is no group–subgroup relation of the lattice point groups. The sub­group common to both (the intersection group) is only [{\bar 1}]. A somewhat more complicated example is the [[j] = 5] ([\Sigma]5) sublattice obtained by a (210) twin reflection of a tetragonal crystal lattice; cf. Fig.[link] and Section[link]. Both lattice and sublattice are tetragonal, [4/mmm], with common c axes, but the intersection group of their holohedries is only [4/m], the further symmetry elements are oriented differently.

    Friedel (1926[link], p. 449) also introduced the term `reticular pseudo-merohedry' (`pseudo-mériédrie réticulaire'). This notion, however, can not be applied to a single lattice and its sublattice (a single lattice can be truly diluted but not pseudo-diluted), but requires pseudo-coincidence of two or more superimposed lattices, which form a `pseudo-sublattice' of index [[j]> 1], as described in Section[link]

    Because of this complicating and confusing situation we avoid here the term merohedry in connection with lattices and translation groups. Instead, the terms coincidence(-site) lattice, twin lattice or sublattice of index [j] are preferred, as explained in Section[link](iv)[link]. Note that we also use two different symbols [i] and [j] to distinguish the subgroup indices of point groups and of lattices. Types of twins by merohedry and pseudo-merohedry

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Both kinds of merohedries and pseudo-merohedries were used by Mallard (1879[link]) and especially by Friedel (1904[link], 1926[link]) and the French School in their treatment of twinning. Based on the concepts of exact coincidence (merohedry, [j] = 1), approximate coincidence (pseudo-merohedry) and partial coincidence (twin lattice index [[j]> 1]), four major categories of `triperiodic' twins were distinguished by Friedel and are explained below. Merohedral twins of lattice index [[j] = 1]

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Here the lattices of all twin partners are parallel and coincide exactly. Consequently, all twin operations are symmetry operations of the lattice point symmetry (holohedral point group), but not of the point group of the structure. Here the term `merohedry' refers to point groups only, i.e. to Case (1) above. Experimentally, in single-crystal X-ray diffraction diagrams all reflections coincide exactly, and tensorial properties of second rank (e.g. birefingence, dielectricity, electrical conductivity) are not influenced by this kind of twinning.

Typical examples of merohedral twins are:

  • (1) Quartz: Dauphiné, Brazil and Leydolt twins (cf. Example[link]).

  • (2) Pyrite, iron-cross twins: crystals of cubic eigensymmetry [2/m{\bar 3}] form penetration twins of peculiar morphology by reflection on (110), with [[i] = 2].

  • (3) KLiSO4: the room-temperature phase III of eigensymmetry 6 exhibits four domain states related by three merohedral twin laws. These growth twins of index [[i] = 4] have been characterized in detail by optical activity, pyroelectricity and X-ray topography (Klapper et al., 1987[link]).

  • (4) Potassium titanyl phosphate, KTiOPO4: polar point group [mm2], forms inversion twins (ferroelectric domains) below its Curie temperature of 1209 K. Pseudo-merohedral twins of lattice index [[j] = 1]

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These twins are characterized by pseudo-merohedry of point groups, Case (1)[link] in Section[link]. The following examples are based on structural pseudosymmetry and consequently also on lattice pseudosymmetry, either as the result of phase transformations or of structural relationships:

  • (1) Transformation twins of Rochelle salt (Jona & Shirane, 1962[link]): this ferroelastic/ferroelectric transformation at about 295 K follows the group–subgroup relation orthorhombic [2'22'] [ \longleftrightarrow] monoclinic 121 (index [[i] = 2]) with [\beta \approx 90^\circ]. The primed operations form the coset of the group–subgroup relation and thus the twin law. Owing to the small deviation of the angle [\beta] from [90^\circ], the lattices of both twin partners nearly coincide. Note that this group–subgroup relation involves both an orthorhombic merohedral and a monoclinic merohedral point group, viz 222 and 2.

  • (2) Transformation twins orthorhombic [2'/m'\,2/m\,2'/m' \longleftrightarrow] monoclinic [12/m1] with [\beta \approx 90^\circ]. This is a case analogous to that of Rochelle salt, except that the point groups involved are the holohedries of the orthorhombic and of the monoclinic crystal system, mmm and [2/m] [example: KH3(SeO3)2; Shuvalov et al. (1967[link])].

  • (3) Pseudo-hexagonal growth twins of an orthorhombic C-centred crystal with [ b/a \approx \sqrt{3}] and twin reflection planes [m'(110)] and [m'({\bar 1}10)]. The lattices of the three domain states nearly coincide and form a `pseudo-coincidence lattice' of lattice index [[j] = 1], but of point-group index [[i] = 3], with subgroup [{\cal H} = 2/m\,2/m\,2/m] and supergroup [{\cal K}(6) =] [6(2)/m\,2/m\,2/m] (cf. Example[link]). Here, in contrast to exact merohedry, in single-crystal X-ray diffraction patterns most reflection spots will be split into three. Note that the term `index' appears twice, first as the subgroup index [[i] = 3] of the point groups and second as the lattice index [[j] = 1] of the twin lattice: further details are given in Section[link]. Twinning with partial lattice coincidence (lattice index [[j]> 1])

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For these twins with partial but exact coincidence Friedel has coined the terms `twinning by reticular merohedry' or `by lattice merohedry'. Here the term merohedry refers only to the sublattice, i.e. to Case (2)[link] above. Typical examples with [[j] = 3] and [[j]> 3] were described in Section[link] In addition to the sublattice relations, it is reasonable to include the point-group relations as well. Four examples are presented:

  • (1) Twinning of rhombohedral crystals (lattice index [[j] = 3], example FeBO3). The eigensymmetry point groups of the structure and of the R lattice (of the untwinned crystal) are both [{\cal H} = {\bar 3}\,2/m]. The extension of the eigensymmetry by the (binary) twin operation [2_z], as described in Example[link], leads to the composite symmetry [{\cal K} =6'/m'({\bar 3})\,2/m\,2'/m'], i.e. the point-group index is [[i] = 2]. The sublattice index is [[j] = 3], because of the elimination of the centring points of the original triple R lattice in forming the hexagonal P twin lattice; cf. Section[link].

  • (2) Reflection twinning across [\{21{\bar 3}0\}] or [\{14{\bar 5}0\}], or twofold rotation twinning around [\langle 540\rangle] or [\langle 230\rangle] of a hexagonal crystal with a P lattice (lattice symmetry [6/m\,2/m\,2/m]). The twin generates a hexagonal coincidence lattice of index [[j] = 7] ([\Sigma 7]) with [{\bf a}' = 3{\bf a} + 2{\bf b}], [{\bf b}' = -2{\bf a} + {\bf b}], [{\bf c}' = {\bf c}]. The hexagonal axes [{\bf a}'] and [{\bf b}'] are rotated around [001] by an angle of 40.9° with respect to a and b. The intersection lattice point group of both twin partners is [6/m]. The extension of this group by the twin operation `reflection across [\{21{\bar 3}0\}]' leads to the point group of the coincidence lattice [6/m\,2'/m'\,2'/m'] (referred to the coordinate axes [{\bf a}', {\bf b}', {\bf c}']). The primed operations define the coset (twin law). For hexagonal lattices rotated around [001], the [\Sigma 7] coincidence lattice ([[j] = 7]) is the smallest sublattice with lattice index [[j]> 1] (least-diluted hexagonal sublattice). No example of a hexagonal [\Sigma7] twin seems to be known; for details see Section[link].

  • (3) Tetragonal growth twins with [[j] = 5] ([\Sigma5] twins) in SmS1.9 (Tamazyan et al., 2000b[link]). This rare twin is illustrated in Fig.[link] and is described, together with the twins of the related phase PrS2, in Example (3)[link] of Section[link] below.

  • (4) Reflection twins across a general net plane (hkl) of a cubic lattice. This example has been treated already in Section[link], Case (2)[link]. Twinning with partial lattice pseudo-coincidence (lattice index [[j]>1])

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This type can be derived from the category in Section[link] above by relaxation of the condition of exact lattice coincidence, resulting in two nearly, but not exactly, coinciding lattices (pseudo-coincidence, cf. Section[link]). In this sense, the two Sections[link] and[link] are analogous to the two Sections[link] and[link].

The following four examples are characteristic of this group:

  • (1) (110) reflection twins of a pseudo-hexagonal orthorhombic crystal with a P lattice: If the axial ratio [b/a = \sqrt{3}] were exact, the lattices of both twin partners would coincide exactly on a sublattice of index [[j] = 2] (due to the absence of the C centring); cf. Koch (2004[link]), Fig.[link] . If [b/a] deviates slightly from [\sqrt{3}], the exact coincidence lattice changes to a pseudo-coincidence lattice of lattice index [[j] = 2]. Examples are ammonium lithium sulfate, NH4LiSO4 (Hildmann, 1980[link]), many members of the K2SO4-type series (cf. Docherty et al., 1988[link]) and aragonite, CaCO3; for further details see Section[link].

  • (2) Staurolite twinning: This topic has been extensively treated as Example[link]. The famous 90°- and 60°-twin `crosses' are a complicated and widely discussed example for Friedel's notion of `twinning by reticular merohedry' (Friedel, 1926[link], p. 461). It was followed up by an extensive analysis by Hurst et al. (1956[link]). Both twin laws (90° and 60° crosses) can be geometrically derived from a multiple pseudo-cubic cell [{\bf a}_c'], [{\bf b}_c'], [{\bf c}_c'] (so-called `Mallard's pseudo-cube') which is derived from the structural monoclinic C-centred cell [{\bf a}_m], [{\bf b}_m], [{\bf c}_m] as follows, involving a rotation of [\sim 45^\circ] around [100]: [{\bf a}_c' = {\bf b}_m + 3{\bf c}_m,\quad {\bf b}_c' = - {\bf b}_m + 3 {\bf c}_m, \quad {\bf c}_c' = 3{\bf a}_m.]

    Using Smith's (1968[link]) lattice constants for the structural monoclinic cell with space group C2/m and a = 7.871, b = 16.620, c = 5.656 Å, β = 90° (within the limits of error), Vm = 740 Å3, the pseudo-cube has the following lattice constants:[\matrix{a'_c = 23.753\hfill &b'_c = 23.753\hfill & c'_c = 23.613\,\,\hbox{\AA}\hfill &\cr \alpha_c = 90\hfill & \beta_c = 90\hfill & \gamma_c = 88.81^\circ\hfill & V'_c = 13323\,\,\hbox{\AA}^3.\hfill}]

    The volume ratio [V'_c/V_m] of the two cells is 18, i.e. the sublattice index is [[j] = 18]. If, however, the primitive monoclinic unit cell is used, the volume ratio doubles and the sublattice index used in the twin analysis increases to [[j] = 36]. The (metrical) eigensymmetry of the pseudo-cube is orthorhombic (due to [\beta_c = 90^\circ]), [(2/m)_{[001]}(2/m)_{[110]}(2/m)_{[1{\bar 1}0]}], referred to [{\bf a}'_c], [{\bf b}'_c], [{\bf c}'_c].

    Note, however, that this pseudo-cube in reality is C-centred because the C-centring vector [1/2({\bf a}'_c + {\bf b}'_c) = 3{\bf c}_m] is a lattice vector of the monoclinic lattice. This C-centring has not been considered by Friedel, Hurst and Donnay, who have based their analysis on the primitive pseudo-cube.

    According to Friedel, the `symmetry elements' of the pseudo-cube are potential twin elements of staurolite, except for [(2/m)_{[1{\bar 1}0]}], which is the monoclinic symmetry direction of the structure. In Table[link], the twin operations of the [90^\circ] and [60^\circ] twins are compared with the `symmetry operations' of the pseudo-cube with respect to obliquities [\omega] and lattice indices [j], referred to both sets of axes, pseudo-cubic [{\bf a}_c'], [{\bf b}'_c], [{\bf c}'_c] and monoclinic (but metrically orthorhombic) [{\bf a}_m], [{\bf b}_m], [{\bf c}_m]. The calculations were again performed with the program OBLIQUE by Le Page (1999[link], 2002[link]). In order to keep agreement with the interpretation of Friedel and Hurst et al., the pseudo-cube is treated as primitive, with [[j] = 36].

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    Staurolite, 60° and 90° twins

    Comparison of the twin operations with the `symmetry operations' of the primitive pseudo-cube with respect to obliquity [\omega] and lattice index [[j]], referred both to the pseudo-cubic axes, [{\bf a}'_c], [{\bf b}'_c], [{\bf c}'_c], and the monoclinic (metrically orthorhombic) axes, [{\bf a}_m], [{\bf b}_m], [{\bf c}_m]. The calculations were performed with the program OBLIQUE by Le Page (1999[link], 2002[link]).

    (a) 90° cross (eight twin operations).

    Twin operations referred toObliquity [\omega] [[^\circ]]Lattice index [j] referred toRemarks
    [{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m][{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m]
    [4[001]_c] [4[100]_m] 0 1 1 Four collinear twin operations [4^1], [4^3], [{\bar 4}{^1}], [{\bar 4}{^3}]
    [2[100]_c] [2[013]_m] 1.19 1 6 Four `diagonal' (with respect to the monoclinic unit cell) twin operations intersecting in [[001]_c=[100]_m]
    [m(100)_c] [m(031)_m] 1.19 1 6
    [2[010]_c] [2[0{\bar 1}3]_m] 1.19 1 6
    [m(010)_c] [m(0{\bar 3}1)_m] 1.19 1 6

    (b) 60° cross.

    Twin operations referred toObliquity [\omega] [[^\circ]]Lattice index [j] referred toEquivalent directions
    [{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m][{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m]
    [3[111]_c] ([\pm120^\circ]) [3[102]_m] 0.87 3 3 [[11{\bar 1}]_c = [{\bar 1}02]_m ]
    [3[{\bar 1}11]_c] ([\pm120^\circ]) [3[320]_m] 0.25 3 9 [[1{\bar 1}1]_c = [3{\bar 2}0]_m ]
    [4[100]_c] ([\pm 90^\circ]) [4[013]_m] 1.19 1 6 [[010]_c = [0{\bar 1}3]_m ]
    [m(100)_c] [m(031)_m] 1.19 1 6  
    [2[101]_c] [2[313]_m] 0.90 1 12 [[011]_c = [3{\bar 1}3]_m ]
    [m(101)_c] [m(231)_m] 0.90 1 12 [[10{\bar 1}]_c = [{\bar 3}13]_m ]
              [[01{\bar 1}]_c = [31{\bar 3}]_m ]

    The following interpretations can be given (cf. Fig. 13 in Hurst et al., 1956[link]):

    • (a) 90° cross (Table[link]a, Fig.[link]a):

      • (i) The pseudo-tetragonal 90° cross can be explained and visualized very well with eight twin operations, a fourfold twin axis along [[100]_m =[001]_c] with operations [4^1], [4^3], [{\bar 4}{^1}], [{\bar 4}{^3}] and two pairs of `diagonal' twin operations 2 and m. They form the coset of the (metrically) `orthorhombic' ([\beta = 90^\circ]) eigensymmetry [{\cal H} = mmm] which results in the composite symmetry [{\cal K} = 4'(2)/m\,2/m\,2'/m'].

      • (ii) The obliquities for all twin operations are at most 1.2°, the lattice index is [[j]_m = 1] for the twin axis, but for the `diagonal' twin elements it is [[j]_m = 6], which is at the limit of the permissible range. Because of these facts, Friedel prefers to consider the 90° cross as a 90° rotation twin around [[100]_m] rather than as a (diagonal) reflection twin across [(031)_m] or [(0{\bar 3}1)_m].

      • (iii) Note that for the interpretation of the 90° cross the complete pseudo-cube with lattice index [[j] = 36] is not required. Because [{\bf c}'_c = 3{\bf a}_m], a pseudo-tetragonal unit cell with axes [{\bf a}_c'], [{\bf b}'_c], [(1/3){\bf c}'_c] and [[j] = 12] is sufficient.

    • (b) 60° cross (Table[link]b, Fig.[link]b):

      • (iv) The widespread 60° cross is much more difficult to interpret and visualize. The four threefold twin axes around [\langle 111\rangle] of the pseudo-cube split into two pairs, both with very small obliquities [ \,\lt\, 1^\circ]. One pair, [[102]_m] and [[{\bar 1}02]_m], has a favourable index [[j]_m = 3]; however, the other one, [[320]_m] and [[3{\bar 2}0]_m] is with [[j]_m = 9] unacceptably high. According to Friedel's theory, this makes [[102]_m] the best choice as threefold twin axis.

      • (v) There is a further [\pm 90^\circ] twin rotation around [[100]_c] or [[013]_m] with small obliquity, [\omega = 1.2^\circ], but very high lattice index, [[j]_m = 6]. Note that this is the same axis that has been used already for the 90° twin, but with a 180° rotation.

      • (vi) The greatest deviation from the `permissibility' criterion is exhibited by the twin axes [2[101]_c = 2[313]_m] and [2[011]_c =] [2[3{\bar 1}3]_m] and the twin planes, pseudo-normal to them, [(231)_m] and [(2{\bar 3}1)_m]. The obliquity [\omega = 0.9^\circ] is very good but the twin index is [[j] = 12], a value far outside Friedel's `limite prohibitive'. These operations, however, are the `standard' twin operations that are always quoted for the 60° twins. Following Friedel (1926[link], p. 462), the best definition of the 60° twin are the [\pm 120^\circ] rotations around [[102]_m] with [\omega = 0.87^\circ] and [[j]_m = 3].

      • (vii) If the (true) C-centring of the pseudo-cube is taken into account, however, no [ \langle 111\rangle] pseudo-threefold axes remain; hence, the 60° cross cannot be explained by the lattice construction of the pseudo-cube.

    Nespolo and Ferraris' description of the staurolite twins. Nespolo & Ferraris (2005[link], 2006[link], 2007a[link],b[link]) have formulated extensions of Mallard's and Friedel's lattice theory of twinning. They introduce `hybrid twinning' (Nespolo & Ferraris, 2005[link], 2006[link]), which involves more than one sublattice contributing to the `overall lattice restoration', a new calculation of the twin index for `manifold twins' (`multiple twins' in the present chapter, cf. Section[link]) and a `twin misfit δ' (Nespolo & Ferraris, 2007b[link]), defined as `the distance between the first nodes along the shortest direction in the plane quasi-perpendicular to the twin axis that are quasi-restored by the twin operation'.

    Based on these considerations, the twin lattices of staurolite have been reconsidered (Nespolo & Ferraris, 2007a[link],b[link]), with the following results in comparison to the discussion above by the present authors:

    • (i) For the `Greek cross twins' (90° cross), the two treatments agree and result in the twin index [j] = 6, just at the limit of Friedel's rule for the occurrence of twinning.

    • (ii) The much more frequent `St Andrews cross twins' (60° cross) have a twin index 12, far too high according to Friedel's standard. This discrepancy is eliminated by Nespolo & Ferraris (2007a[link]) by considering this twin as a hybrid twin with two concurring monoclinic sublattices, generated by twin axis/plane [313]/(231) with index 12 (obliquity 0.90°) and [313]/(352) with index 10 (obliquity 6.21°). Since two of the nodes of the [313]/(231) sublattice are quasi-coincident with nodes of the sublattice [313]/(352), the index 12 is reduced by a factor 2 to an effective index 6, comparable to index 6 of the 90° twin and, thus, just acceptable by Friedel's criterion, but with one large obliquity of 6.21°.

    • (iii) The last paper in the series (Nespolo & Ferraris, 2007b[link]) is a critical discussion of the twin-lattice treatment of previous authors, including the present ones. Its essential statement is that the Friedel criterion is applicable only to binary twin operations, not to `manifold twins' (`multiple twins', see above), for which a new twin index is defined. This index is now the same for all operations in the coset of the manifold twin. For the Greek cross (90° twin), Nespolo & Ferraris arrive at the same result (index 6) as the present authors. For the `St Andrews cross' (60° twin) there is a significant difference in the two approaches: the present authors believe that the 60° cross cannot be explained by the lattice construction of `Mallard's pseudo-cube' because the C-centred cubic twin lattice, which is actually tetragonal P with [c/a=1{:}\sqrt2/2], has no pseudo-threefold axis along any <111> direction which could act as a threefold twin rotation axis. Nespolo & Ferraris, on the other hand, disregard the C-centring of the Mallard's pseudo-cube and explain the 60° twin by the threefold axes of the cubic P cell (see their Fig. 1 and the trichromatic twin point group given on their page 285), as was done by Mallard (1879)[link], Friedel (1926)[link] and Donnay & Donnay (1983)[link].

  • (3) Growth twins of monoclinic PrS2 and of tetragonal SmS1.9: These two rather complicated examples belong to the structural family of MeX2 dichalcogenides which is rich in structural relationships and different kinds of twins. The `basic structure' and `aristotype' of this family is the tetragonal ZrSSi structure with axes [a_b = b_b \approx 3.8], [c_b \approx 7.9\ \hbox{\AA}], [V_b \approx114\ \hbox{\AA}^3], space group [P4/nmm] (b stands for basic). The crystal chemistry of this structural family is discussed by Böttcher et al. (2000[link]).

    • (a) PrS2 (Tamazyan et al., 2000[link]a)

      PrS2 is a monoclinic member of this series with space group [P2_1/b11] (unique axis a!) and axes [a \approx 4.1], [b \approx 8.1], [c \approx 8.1\ \hbox{\AA}], [\alpha \approx 90.08^\circ], [V \approx 269\ \hbox{\AA}^3]. The structure is strongly pseudo-tetragonal along [001] (with cell a, b/2, c) and is a `derivative structure' of ZrSSi. Hence pseudo-merohedral twinning that makes use of this structural tetragonal pseudosymmetry would be expected, with twin elements 4[001] or [m(210)] or 2[120] etc. and [[j] = 2] because [b \approx 2a], but, surprisingly, this twinning has not been observed so far. It may occur in other PrS2 samples or in other isostructural crystals of this series.

      Instead, the monoclinic crystal uses another structural pseudo­symmetry, the approximate orthorhombic symmetry along [100] with [\alpha \approx 90^\circ], to twin on [2_y], [2_z], [m_y] or [m_z] (coset of [2_x/m_x]) with composite symmetry [{\cal K} = 2/m\,2'/m'\,2'/m'], [[j] = 1] and [[i] = 2] (cf. Fig. 4 of the paper).

      The monoclinic PrS2 cell has a third kind of pseudosymmetry that is not structural, only metrical. The cell is pseudo-tetragonal along [100] due to [ b \approx c] and [\alpha \approx 90^\circ]. This pure lattice pseudosymmetry, not surprisingly, is not used for twinning, e.g. via 4[100] or [m(011)] or [m(0{\bar 1}1)] or 2[011] or [2[0{\bar 1}1]].

    • (b) SmS1.9 (Tamazyan et al., 2000b[link])

      This structure is (strictly) tetragonal with axes [a = b \approx 8.8], [c\approx 15.9\ \hbox{\AA}], [V \approx 1238\ \hbox{\AA}^3] and space group [P4_2/n]. It is a tenfold superstructure of ZrSSi with the following basis-vector relations: [ {\bf a} = 2{\bf a}_b + {\bf b}_b,\quad {\bf b} = -{\bf a}_b + 2{\bf b}_b,\quad {\bf c} = 2{\bf c}_b,]leading to lattice constants [a \approx \sqrt{5}a_b], [b \approx \sqrt{5}b_b], [c \approx 2c_b]. This well ordered tetragonal supercell now twins on [m(210)] or 2[210] or [m(130)] or 2[130] (which is equivalent to a rotation around [001] of 36.87°) to form a [\Sigma 5] twin by `reticular merohedry' ([[j] =] [5]) with lattice constants [a' = a \sqrt{5} = 19.72], [b' =] [b \sqrt{5} = 19.72], c′ = c = 15.93 Å, V = 6192 Å3. This is illustrated in Fig.[link].

      SmS1.9 represents the first thoroughly investigated and documented tetragonal [[j] = 5] ([\Sigma 5]) twin known to us. The sublattice of this twin is the tetragonal coincidence lattice with smallest lattice index [[j]> 1], i.e. the `least-diluted' systematic tetragonal sublattice; for further details see Section[link].

  • (4) Growth twins of micas: A rich selection of different twin types, both merohedral and pseudo-merohedral, with [[j]=1] and 3, is provided by the mineral family of micas, which includes several polytypes. A review of these complicated and interesting twinning phenomena is presented by Nespolo et al. (1997[link]). Detailed theoretical derivations of mica twins and allotwins, both in direct and reciprocal space, are published by Nespolo et al. (2000[link]).

In conclusion, it is pointed out that the above four categories of twins, described in Sections[link] to[link], refer only to cases with exact or approximate three-dimensional lattice coincidence (triperiodic twins). Twins with only two- or one-dimensional lattice coincidence (diperiodic or monoperiodic twins) [e.g. the (100) reflection twins of gypsum and the (101) rutile twins] belong to other categories, cf. Section[link] The examples above have shown that for triperiodic twins structural pseudosymmetries are an essential feature, whereas purely metrical (lattice) pseudosymmetries are not a sufficient tool in explaining and predicting twinning, as is evidenced in particular by the case of staurolite, discussed above in detail. Pseudo-merohedry and ferroelasticity

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The large group of pseudo-merohedral twins (irrespective of their lattice index) contains a very important subset which is characterized by the physical property ferroelasticity. Ferroelastic twins result from a real or virtual phase transition involving a change of the crystal family (crystal system). These transitions are displacive, i.e. they are accompanied by only small structural distortions and small changes of lattice parameters. The structural symmetries lost in the phase transition are preserved as pseudosymmetries and are thus candidates for twin elements. This leads to a pseudo-coincidence of the lattices of the twin partners and thus to pseudo-merohedral twinning. Because of the small structural changes involved in the transformation, domains usually switch under mechanical stress, i.e. they are ferroelastic. A typical example for switchable ferroelastic domains is Rochelle salt, the first thoroughly investigated ferroelastic transformation twin, discussed in Section[link], Example (1)[link]. This topic is extensively treated in Chapter 3.4[link] on domain structures.

3.3.10. Twin boundaries

| top | pdf | Contact relations in twinning

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So far, twinning has been discussed only in terms of symmetry and orientation relations of the (bulk) twin components. In this chapter, the very important aspect of contact relations is discussed. This topic concerns the orientation and the structure of the twin boundary, which is also called twin interface, composition plane, contact plane, domain boundary or domain wall. It is the twin boundary and its structure and energy which determine the occurrence or non-occurrence of twinning. In principle, for each crystal species an infinite number of orientation relations obey the requirements for twinning, as set out in Section 3.3.2[link], because any rational lattice plane (hkl), as well as any rational lattice row [uvw], common to both partners would lead to a legitimate reflection or rotation twin. Nevertheless, only a relatively small number of crystal species exhibit twinning at all, and, if so, with only a few twin laws. This wide discrepancy between theory and reality shows that a permissible crystallographic orientation relation (twin law) is a necessary, but not at all a sufficient, condition for twinning. In other words, the contact relations play the decisive role: a permissible orientation relation can only lead to actual twinning if a twin interface of good structural fit and low energy is available.

In principle, a twin boundary is a special kind of grain boundary connecting two `homophase' component crystals which exhibit a crystallographic orientation relation, as defined in Section 3.3.2.[link] For a given orientation relation of the twin partners, crystallographic or general, the interface energy depends on the orientation of their boundary. It is intuitively clear that crystallographic orientation relations lead to energetically more favourable boundaries than noncrystallographic ones. As a rule, twin boundaries are planar (at least in segments), but for certain types of twins curved and irregular interfaces have been observed. This is discussed later in this section.

In order to determine theoretically for a given twin law the optimal interface, the interface energy has to be calculated or at least estimated for various boundary orientations. This problem has not been solved for the general case so far. The special situation of reflection twins with coinciding twin mirror and composition planes has recently been treated by Fleming et al. (1997[link]). These authors calculated the interface energies for three possible reflection twin laws in each of aragonite, gibbsite, corundum, rutile and sodium oxalate, and they compared the results with the observed twinning. In all cases, the twin law with lowest boundary energy corresponds to the twin law actually observed. Another calculation of the twin interface energy has been performed by Lieberman et al. (1998[link]) for the [(10{\bar 2})] reflection twins of monoclinic saccharin crystals. In this study, the [(10{\bar 2})] boundary energy was calculated for different shifts of the two twin components with respect to each other. It was shown that a minimum of the boundary energy is achieved for a particular `twin displacement vector' (cf. Section[link]).

Calculations of interface energies, as performed by Fleming et al. (1997[link]) and Lieberman et al. (1998[link]), however, require knowledge of the atomic potentials and their parameters for each pair of bonded atoms. They are, therefore, restricted to specific crystals for which these parameters are known. Similarly, high-resolution electron microscopy (HRTEM) images of twin boundaries have been obtained so far for only a small number of crystals.

It is possible, however, to predict for a given twin law low-energy twin boundaries on the basis of symmetry considerations, even without knowledge of the crystal structure, as discussed in the following section. This prediction has been carried out by Sapriel (1975[link]) for ferroelastic crystals. His treatment assumes a phase transition from a real or hypothetical parent phase (supergroup [{\cal G}]) to a `distorted' (daughter) phase of lower eigensymmetry (subgroup [{\cal H}]), leading to two (or more) domain states of equal but opposite shear strain. The subgroup [{\cal H}] must belong to a lower-symmetry crystal system than the supergroup [{\cal G}], as explained in Section[link](ii). Similar criteria, but restricted to ferroelectric materials, had previously been devised in 1969 by Fousek & Janovec (1969[link]). A review of ferroelastic domains and domain walls is provided by Boulesteix (1984[link]) and an extension of the Sapriel procedure to phase boundaries between a ferro­elastic and its `prototypic' (parent) phase is given by Boulesteix et al. (1986[link]). Strain compatibility of interfaces

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For a simple derivation of stress-free contact planes, we go back to the classical description of mechanical twinning by a homogeneous shear, which is illustrated by a deformation ellipsoid as shown in Fig.[link](a) (cf. Liebisch, 1891[link]; Niggli, 1941[link]; Klassen-Neklyudova, 1964[link]). In a modification of this approach, we consider two parts of a homogeneous, crystalline or noncrystalline, solid body, which are subjected to equal but opposite shear deformations [-\varepsilon] and [+\varepsilon]. The undeformed state of the body and the deformed states of its two parts are represented by a sphere ([\varepsilon = 0]) and by two ellipsoids [-\varepsilon] and [+\varepsilon], as shown in Fig.[link](b). We now look for stress-free contact planes between the two deformed parts, i.e. planes for which line segments of any direction parallel to the planes experience the same length change in both parts during the shear. This criterion is obeyed by those planes that exhibit identical cross sections through both ellipsoids. Mathematically, this is expressed by the equation (Sapriel, 1975[link])[(\varepsilon^{\rm I}_{ij}-\varepsilon^{\rm II}_{ij})x_ix_j=2\varepsilon x_1x_2=0]([\varepsilon^{\rm I}_{ij}=\varepsilon^{\rm I}_{12}=+\varepsilon, \varepsilon^{\rm II}_{ij}=\varepsilon^{\rm II}_{12}=-\varepsilon]; [x_1,x_2,x_3] are Cartesian coordinates) which has as solutions the two planes [x_1=0] (plane BB in Fig.[link]b) and [x_2=0] (plane AA). These planes are called `planes of strain compatibility' or `permissible' planes. From the solutions of the above equation and Fig.[link](b) it is apparent that two such planes, AA and BB, normal to each other exist. The intersection line of the two compatible planes is called the shear axis of the shear deformation.


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(a) Classical description of mechanical twinning by homogeneous shear deformation (Liebisch, 1891[link], pp. 104–118; Niggli, 1941[link], pp. 145–149; Klassen-Neklyudova, 1964[link], pp. 4–10). The shear deforms a sphere into an ellipsoid of equal volume by translations (arrows) parallel to the twin (glide) plane AA. The translations are proportional to the distance from the plane AA. Shear angle [2\varepsilon]. (Only the translations in the upper half of the diagram are shown, in the lower half they are oppositely directed.) (b) Ellipsoids representing the (spontaneous) shear deformations [-\varepsilon] and [+\varepsilon] of two orientation states, referred to the (real or hypothetic) intermediate (prototypic) state with [\varepsilon = 0] (sphere). The switching of orientation state [-\varepsilon] into state [+\varepsilon] through the shear angle [2\varepsilon] is, analogous to (a), indicated by arrows. The shear ellipsoids [-\varepsilon] and [+\varepsilon] have common cross sections along the perpendicular planes AA and BB which are both, therefore, mechanically compatible contact planes of the [+\varepsilon] and [-\varepsilon] twin domains.

It is noted that during a shear deformation induced by the (horizontal) translations shown in Fig.[link](b), only the plane AA, parallel to the arrows, can be generated as a contact plane between the two domains. A contact plane BB, normal to the arrows, cannot be formed by this process, because this would lead to a gap on one side and a penetration of the material on the other side. Plane BB, however, could be formed during a (virtual) switching between [+\varepsilon] and [-\varepsilon] with `vertical' translations, parallel to BB, which would formally result in the same mutual arrangement of the ellipsoids. The compatibility criterion, as expressed by the equation above (which applies to elastic continua), does not distinguish between these two cases. Note that the planes AA and BB are mirror planes relating the deformations [+\varepsilon] and [-\varepsilon]. Both contact planes often occur simultaneously in growth twins, see for example the dovetail and the Montmartre twins of gypsum (Fig.[link]). In general, each interface coinciding with a twin mirror plane or a plane normal to a twin axis is a (mechanically) compatible contact plane.

It should be emphasized that the criterion `strain compatibility' is a purely mechanical one for which only stress and strain are considered. It leads to `mechanical' low-energy boundaries. Other physical properties, such as electrical polarization, may reduce the number of mechanically permissible boundaries, e.g. due to energetically unfavourable head-to-head or tail-to-tail orientation of the axis of spontaneous polarization in polar crystals. The mechanical compatibility criterion is, however, always applicable to centrosymmetric materials. Sapriel approach to permissible (compatible) boundaries in ferroelastic (non-merohedral) transformation twins

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The general approach to strain compatibility, as given above, can be employed to derive the permissible composition planes for twins with inclined axes (non-merohedral twins; for merohedral twins see Section[link] below). This concept was applied by Sapriel (1975[link]) to the 94 Aizu species of ferroelastic transformation twins. According to Aizu (1969[link], 1970[link][link]a,b), each species is represented by a pair of symmetry groups, separated by the letter F (= ferroic) in the form [{\cal K}F{\cal H}], e.g. [2/m\,2/m\,2/mF 12/m1] or [m{\bar 3}mF {\bar 3}m]. The parent phase with symmetry [{\cal G}] represents the undeformed (zero-strain) reference state (the sphere in Fig.[link]b), whereas the spontaneous strain of the two orientation states of phase [{\cal H}] is represented by the two ellipsoids. Details of the calculation of the permissible domain boundaries for all ferroelastic transformation twins are given in the paper by Sapriel (1975[link]).

Two kinds of permissible boundaries are distinguished by Sapriel:

  • (a) W boundaries. These interfaces are parallel to symmetry planes of the parent phase (supergroup [{\cal K}]), which are lost in the transition and have become twin reflection planes (F operations) of the deformed phase (subgroup [{\cal H}]), i.e. they are `crystallographically prominent planes of fixed indices' (Sapriel, 1975[link], p. 5129), which are fixed by the symmetry of the parent phase. A rational lattice plane perpendicular to a lost twofold symmetry axis of the parent phase is also a W boundary. W boundaries are crystallographically invariant with respect to temperature and pressure.

  • (b) [W'] boundaries. In contrast to W boundaries, [W'] interfaces are not fixed by the symmetry of the parent phase, i.e. they do not correspond to lost symmetry elements. Their orientation depends on the direction of the spontaneous shear strain and thus changes with temperature and pressure. In general, [W'] boundaries are irrational planes.

Example.  The distinction between these two types of boundaries is illustrated by the example of the (triclinic) Aizu species [2/mF{\bar 1}]. Here, the lost mirror plane of the monoclinic parent phase (F operation) yields the permissible prominent W twin boundary (010). The second permissible boundary, perpendicular to the first, is an irrational [W'] composition plane in the zone of the direction normal to triclinic (010), i.e. of the triclinic reciprocal [{\bf b}^\ast] axis. The azimuthal orientation of this boundary around the zone axis is not determined by symmetry but depends on the direction of the spontaneous shear strain of the deformed triclinic phase.

Sapriel (1975[link]) has shown that for ferroelastic crystals the pair of perpendicular permissible domain boundaries can consist either of two W planes, or of one W and one [W'] plane, or of two [W'] planes. Examples are the Aizu species [2/m\,2/m\,2/mF12/m1], [2/mF{\bar 1}] and [4F2], respectively. There are even four Aizu cases without any permissible boundaries: [3F1], [{\bar 3}F{\bar 1}], [23F222], [m{\bar 3}Fmmm]. An example is langbeinite ([23F222]), which was discussed at the end of Section[link].

Note. The two members of a pair of permissible twin boundaries are always exactly perpendicular to each other. Frequently observed slight deviations from the strict [90^\circ] orientation have been interpreted as relaxation of the perpendicularity condition in the deformed phase, resulting from the ferroelastic phase transition (cf. Sapriel, 1975[link], p. 5138). This, however, is not the reason for the deviation from [90^\circ], but rather a splitting of the two (exactly) perpendicular symmetry planes in the parent phase [{\cal G}] into two pairs of compatible twin boundaries (i.e. two independent twin laws) in the deformed phase [{\cal H}], whereby the pairs are nearly perpendicular to each other. From each pair only one interface (usually the rational one) is realized in the twin, whereas the other compatible twin boundary (usually the irrational one) is suppressed because of its unfavourable energetic situation.


  • (1) Phase transition orthorhombic ([{\cal G} = 2/m\,2/m\,2/m]) [\Rightarrow] monoclinic ([{\cal H} = 1\,2/m\,1], [\beta \approx 90^\circ]), in Aizu notation [2/m2/m2/mF12/m1]. Whereas in [{\cal G}] the mirror planes [m_x] and [m_z] are exactly perpendicular, these planes deviate slightly (by [\beta - 90^\circ]) from perpendicularity in [{\cal H}] and split into two different twin laws (cosets), the first one containing twin plane [m_x], the second one [m_z]. Each twin law contributes its rational twin boundary, [m_x] or [m_z], to the observed twin aggregate, whereas in each pair the perpendicular irrational plane is suppressed.

  • (2) In a tetragonal–orthorhombic phase transition, the two exactly perpendicular mirror planes [(110)] and [({\bar 1}10)] of the tetragonal prototype phase [{\cal G}] split into two independent (rational) twin boundaries in the deformed orthorhombic phase [{\cal H}], which are now nearly perpendicular to each other.

    The most famous example of this type of twinning is the 1023 K ferroelastic phase transition of the high-Tc superconductor YBa2Cu3O7−δ. The twinning on {110} in this compound was first extensively studied by Roth et al. (1987[link]), both in direct space (TEM) and in reciprocal space (electron and X-ray diffraction), and by Schmid et al. (1988[link]); see also Shektman (1993[link]). Extension to non-merohedral growth and mechanical twins

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The treatment by Sapriel (1975[link]) 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[link](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.[link] 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[link]). 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[link]) 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[link]: 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[link]).

In conclusion, the following differences in philosophy between the Sapriel approach in Section[link] 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]\ge 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].


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

    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 [m'] 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.[link].

  • (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 =] [112/m]; reduced composite symmetry: [{\cal K}^\ast =] [2'/m'\,2'/m'\,2/m].

    Reference is made to Fig.[link] in Section[link], 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: [2\arctan( b/a) \approx 120^\circ] (aragonite), [72^\circ] (AlMn alloy), [90^\circ] (staurolite [90^\circ] cross), [\ldots, 360^\circ/n]. A pentagonal twin is shown in Fig.[link].

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

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

      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 [W'] 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.[link].

      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 [W'] 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.[link]).

    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[link](i)[link] and Fig.[link]].

    The deformation twinning in calcite has been extensively studied by Barber & Wenk (1979[link]). Recently, these twins were discussed by Bueble & Schmahl (1999[link]) 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'): ahex = 4.99, chex = 17.06 Å. This cell is used by both Barber & Wenk and Bueble & Schmahl.

    • (ii) Morphological cell: amorph = ahex = 4.99, cmorph = 1/4chex = 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: apc = 3.21 Å, αpc = 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.[link]). 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[link]) 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)[link] 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[link]) or structural channels along crystallographic directions (`Rose channels'; Rose, 1868[link]). Twinning dislocations and cracks (Barber & Wenk, 1979[link]) also relax high stress. In `real' ferroelastic crystals with their small shear (usually below 1°) these stress-relaxing phenomena usually do not occur. Permissible boundaries in merohedral twins (lattice index [j] = 1)

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In merohedral twins (lattice index [[j] = 1]), the twin elements map the entire lattice exactly upon itself. Hence there is no spontaneous strain, in which the twin domains would differ. The mechanical compatibility criterion means in this case that any orientation of a twin boundary is permissible, because interfaces of any orientation obey the mechanical compatibility criterion, no matter whether the planes are rational, irrational or even curved interfaces. This variety of interfaces is brought out by many actual cases, as shown by the following examples:

  • (1) Quartz.

    The boundaries of Dauphiné transformation twins ([622 \Rightarrow 32]) are usually irregular, curved interfaces without macroscopically flat parts (segments) (Frondel, 1962[link]). The boundaries of Dauphiné growth twins are usually curved too, but sometimes they exhibit large segments roughly parallel to the rhombo­hedron faces [r\{10{\bar 1}1\}] and [z\{{\bar 1}011\}]. Inserts of growth-twin domains often have the shape of rounded cones with apices located at the nucleating perturbation (usually an inclusion). X-ray topography has shown that Dauphiné boundaries are sometimes stepped on a fine scale just above the topographic resolution of a few µm (Lang, 1967[link][link]a,b).

    In contrast to Dauphiné boundaries, the contact faces of Brazil twins (which are always growth twins) strictly adopt low-index lattice planes, preferentially those of the major rhombohedron [r\{10{\bar 1}1\}], less frequently of the minor rhombohedron [z\{{\bar 1}011\}]. More rarely observed are boundary segments parallel to [\{10{\bar 1}0\}], [\{11{\bar 2}1\}] and [\{0001\}] (Frondel, 1962[link]). A special case are the differently dyed [r\{10{\bar 1}1\}] Brazil-twin lamellae of amethyst (Frondel, 1962[link]).

  • (2) Triglycine sulfate (TGS).

    This crystal exhibits a ferroelectric transition [2/m \Leftrightarrow 2] with antipolar (merohedral) inversion twins ([180^\circ] domains). The domains usually have the shape of irregular cylinders parallel to the polar axis. All boundaries parallel to the polar axis are observed, whereas boundaries inclined or normal to the polar axis are `electrically forbidden' (they would be head-to-head and tail-to-tail boundaries, cf. Section[link]), even though they are mechanically permissible.

  • (3) Lithium formate monohydrate (polar point group [mm2]).

    The crystals grown from aqueous solutions exhibit inversion twins in the shape of sharply defined (010) lamellae (non-switchable [180^\circ] domains). Other boundaries of these growth twins have not been observed (Klapper, 1973[link]).

  • (4) KLiSO4 (polar point group 6).

    Among the three different merohedral twin laws, one type, with twin reflection plane m parallel to the hexagonal axis [001], stands out. The numerous grown-in twin lamellae are bounded by large planar (0001) interfaces (normal to the polar axis). It should be noted that these very prominent twin boundaries are perpendicular to the twin reflection plane. This is a rather rare case of boundary orientation. These planes are oriented normal to the polar axis but are not electrically forbidden, due to the twin law [m \parallel [001]] which preserves the polar direction (Klapper et al., 1987[link]).

These examples demonstrate that in many merohedral twins only a small number of rational, well defined boundaries occur, even though any boundary is permitted by the mechanical compatibility criterion. This shows that the latter criterion is a necessary, but not a sufficient, condition and that further influences, in particular electrical or structural ones, are effective. Permissible twin boundaries in twins with lattice index [[j]\, \gt\, 1]

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In contrast to the mechanical compatibility of any composition plane in merohedral twins (lattice index [[j] = 1]), twins of higher lattice index [[j]> 1] are more restricted with respect to the orientation of permissible twin boundaries. In fact, these special twins can be treated in the same way as the general non-merohedral twins described in Section[link] above. Again, we attribute equal but opposite spontaneous shear strain to the two twin domains 1 and 2. This `spontaneous' shear strain (referred to an intermediate state of zero strain) is half the shear deformation necessary to transform the orientation of domain 1 into that of domain 2. This also means that the lattice of domain 1 is transformed into the lattice of domain 2. The essential difference to the case in Section[link] is the fact that by this deformation only a subset of lattice points is `restored'. This subset forms the sublattice of index [[j] \ge 2] common to both domains (coincidence-site sublattice, twin lattice). With this analogy, the Sapriel formalism can be applied to the derivation of the mechanically compatible (permissible) twin boundaries. Again, the easiest way to find the permissible planes is the construction of the black–white symmetry symbol of the twin law, in which planes parallel to primed mirror planes or normal to primed twofold axes constitute the permissible W interfaces.

It is emphasized that the concept of a deformation from domain state 1 to domain state 2 is not always a mere mental construction, as it is for growth twins. It is physical reality in some deformation twins, for example in the famous [\Sigma 3] deformation twins (spinel law) of cubic metals which are essential elements of the plasticity of these metals. During the [\Sigma 3] deformation, the {100} cube (a [90^\circ] rhombohedron) is switched from its `reverse' into its `obverse' orientation and vice versa, whereby the hexagonal P sublattice of index [[j] = 3] is restored and, thus, is common to both twin domains.

Exact lattice coincidences of twin domains result from special symmetry relations of the lattice. Such relations are systematically provided by n-fold symmetry axes of order [n> 2], i.e. by three-, four- and sixfold axes. In other words: twins of lattice index [[j]> 2] occur systematically in trigonal, hexagonal, tetragonal and cubic crystals. This may lead to trigonal, tetragonal and hexagonal intersection symmetries [{\cal H}^\ast] (reduced eigensym­metries) of domain pairs. Consequently, if there exists one pair of permissible composition planes, all pairs of planes equivalent to the first one with respect to the intersection symmetry are permissible twin boundaries as well. This is illustrated by three examples in Table[link].

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Examples of permissible twin boundaries for higher-order merohedral twins ([j]> 1)

 [\Sigma 3] growth and deformation twins of cubic crystals, twin mirror plane (111) (spinel law)[\Sigma 5] growth twins of tetragonal rare-earth sulfides (SmS1.9), twin mirror plane (120)[\Sigma 33] deformation twins of cubic galena (PbS), twin mirror plane (441)
Eigensymmetry [{\cal H}] [4/m{\bar 3}2/m] [4/m\,2/m\,2/m] [4/m{\bar 3}2/m]
Intersection symmetry [{\cal H}^\ast] [{\bar 3}2/m] parallel to [111] [4/m] parallel to [001] [2/m] parallel to [[1{\bar 1}0]]
Reduced composite symmetry [{\cal K}^\ast] [6'/m'_1({\bar 3})2/m\,2'/m'_3] [4/m\,2'/m'_1\,2'/m'_2] [2'/m'_1\,2/m\,2'/m'_2]
Permissible twin boundaries Three pairs of perpendicular planes Two pairs of perpendicular planes One pair of permissible planes
[m_1 = (111)] & [m_3 = (11{\bar 2})] [m_1 = (120)] & [({\bar 2}10)] [m_1 \,= \,(441)] & [m_2 \,= \,(11{\bar 8})]
[m_1 = (111)] & [m_3 = ({\bar 2}11)] [m_2 = (310)] & [({\bar 1}30)]  
[m_1 = (111)] & [m_3 = (1{\bar 2}1)]    
Reference system Cubic axes Tetragonal axes Cubic axes
The existence of this deformation twin is still in doubt (cf. Seifert, 1928[link]).
The intersection symmetry [{\cal H}^\ast] and the permissible boundaries are referred to the coordinate system of the eigensymmetry; the reduced composite symmetries [{\cal K}^\ast] are based on their own conventional coordinate system derived from the intersection symmetry [{\cal H}^\ast] plus the twin law (cf. Section 3.3.4[link]).

For the cubic and rhombohedral [\Sigma 3] twins (spinel law), due to the threefold axis of the intersection symmetry, three pairs of permissible planes occur. The plane (111), normal to this threefold axis, is common to the three pairs of boundaries (threefold degeneracy), i.e. in total four different permissible W twin boundaries occur. These composition planes (111), [(11{\bar 2})], [({\bar 2}11)], [(1{\bar 2}1)] are indeed observed in the [\Sigma 3] spinel-type penetration twins, recognizable by their re-entrant edges (Fig.[link]). They also occur as twin glide planes of cubic metals. For the tetragonal [\Sigma 5] twin, two pairs of perpendicular permissible W composition planes result, (120) & ([{\bar 2}10)] and (310) & ([{\bar 1}30]), one pair bisecting the other pair under 45°. For the cubic [\Sigma 33] twin [galena PbS, cf. Section[link], example (4)], due to the low intersection symmetry, only one pair of permissible W boundaries results. Electrical constraints of twin interfaces

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As mentioned before, the mechanical compatibility of twin boundaries is a necessary but not a sufficient criterion for the occurrence of stress-free low-energy twin interfaces. An additional restriction occurs in materials with a permanent (spontaneous) electrical polarization, i.e. in crystals belonging to one of the ten pyroelectric crystal classes which include all ferroelectric materials. In these crystals, domains with different directions of the spontaneous polarization may occur and lead to `electrically charged boundaries'. Merohedral twins

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Of particular significance are merohedral twins with polar domains of antiparallel spontaneous polarization [\pm {\bf P}] (180° domains). The charge density [\rho] at a boundary between two twin domains is given by [\rho = \pm 2 P_n,]where [P_n] is the component of the polarization normal to the boundary. The interfaces with positive charge are called `head-to-head' boundaries, those with negative charge `tail-to-tail' boundaries. Interfaces parallel to the polarization direction are uncharged ([P_n = 0]) (Fig.[link]).


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Boundaries B–B between 180° domains (merohedral twins) of pyroelectric crystals. (a) Tail-to-tail boundary. (b) Head-to-head boundary. (c) Uncharged boundary ([P_n = 0]). (d) Charged zigzag boundary, with average orientation normal to the polar axis. The charge density is significantly reduced. Note that the charges at the boundaries are usually compensated by stray charges of opposite sign.

The electrical charges on a twin boundary constitute an additional (now electrostatic) energy of the twin boundary and are `electrically forbidden'. Only boundaries parallel to the polar axes are `permitted'. This is in fact mostly observed: practically all 180° domains originating during a phase transition from a paraelectric parent phase to the polar (usually ferroelectric) daughter phase exhibit uncharged boundaries parallel to the spontaneous polarization. Uncharged boundaries have also been found in inversion growth twins obtained from aqueous solutions, such as lithium formate monohydrate and ammonium lithium sulfate. Both crystals possess the polar eigensymmetry [mm2] and contain grown-in inversion twin lamellae (180° domains) parallel to their polar axis.

`Charged' boundaries, however, may occur in crystals that are electrical conductors. In such cases, the polarization charges accumulating along head-to-head or tail-to-tail boundaries are compensated by opposite charges obtained through the electrical conductivity. This compensation may lead to a considerable reduction of the interface energy. Note that the term `charged' is often used for boundaries of head-to-head and tail-to-tail character, even if they are uncharged due to charge compensation.


  • (1) Lithium niobate, LiNbO3, exhibits a phase transition from [{\bar 3}2/m] to [3m] between 1323 and 1473 K (depending on the Li/Nb stoichiometry). Crystals are grown from the melt ([T_m = 1538] K) by Czochralski pulling along the trigonal axis [001] in the paraelectric phase. They transform into the ferroelectric polar phase when cooled below the Curie temperature. The crystals are electrically conductive at high temperatures and can be poled by an electric field parallel to the polar axis. By applying an alternating rectangular voltage between seed crystal and melt, a sequence of 180° domains is formed during the subsequent transition. The domain boundaries follow the curved growth front (crystal–melt interface) and have alternating head-to-head and tail-to-tail character (Räuber, 1978[link]).

  • (2) Orthorhombic polar potassium titanyl phosphate, KTiOPO4 (KTP), exhibits a para- to ferroelectric phase transition ([mmm \Longleftrightarrow mm2]) and a considerable conductivity of potassium ions. In this material, head-to-head and tail-to-tail boundaries are common. Sometimes strongly folded, charged zigzag boundaries occur, which contain large segments of faces nearly parallel to the spontaneous polarization (Scherf et al., 1999[link]). The average orientation of these boundaries is roughly normal to the polar axis (Fig.[link]d), but their charge density is considerably reduced by the zigzag geometry.

  • (3) Head-to-head and tail-to-tail twin boundaries are also found in crystals grown from aqueous solutions. In such cases, the polarization charges are compensated by the opposite charges present in the electrolytic solution. An interesting example is hexagonal potassium lithium sulfate KLiSO4 (point group 6) which exhibits, among other types of twins, anti-polar domains of inversion twins. The twin boundaries often have head-to-head or tail-to-tail character and frequently coincide with the growth-sector boundaries (Klapper et al., 1987[link]). Non-merohedral twins

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Charged and uncharged boundaries may also occur in non-merohedral twins of pyroelectric crystals. In this case, the polar axes of the two twin domains 1 and 2 are not parallel. The charge density [\rho] of the boundary is given by [\rho = P_n(2) - P_n(1),]with [P_n(1)] and [P_n(2)] the components of the spontaneous polarization normal to the boundary. An example of both charged and uncharged boundaries is provided by the growth twins of ammonium lithium sulfate with eigensymmetry [m2m]. These crystals exhibit, besides the inversion twinning mentioned above, growth-sector twins with twin laws `reflection plane (110)' and `twofold twin axis normal to (110)'. (Both twin elements would constitute the same twin law if the crystal were centrosymmetric.) The observed and permissible composition plane for both laws is (110) itself. As is shown in Fig.[link], the (110) boundary is charged for the reflection twin and uncharged for the rotation twin. Both cases are realized for ammonium lithium sulfate. The charges of the reflection-twin boundary are compensated by the charges contained in the electrolytic aqueous solution from which the crystal is grown. On heating (cooling), however, positive (negative) charges appear along the twin boundary.


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Charged and uncharged boundaries B–B of non-merohedral twins of pseudo-hexagonal NH4LiSO4. Point group [m2m], spontaneous polarization P along twofold axis [010]. (a) Twin element mirror plane (110): electrically charged boundary of head-to-head character. (b) Twin element twofold twin axis normal to plane (110): uncharged twin boundary (`head-to-tail' boundary). Non-pyroelectric noncentrosymmetric crystals

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Finally, it is pointed out that electrical constraints of twin boundaries do not occur for non-pyroelectric acentric crystals. This is due to the absence of spontaneous polarization and, consequently, of electrical boundary charges. This fact is apparent for the Dauphiné and Brazil twins of quartz: they exhibit boundaries normal to the polar twofold axes which are reversed by the twin operations.

Nevertheless, it seems that among possible twin laws those leading to opposite directions of the polar axes are avoided. This can be explained for spinel twins of cubic crystals with the sphalerite structure and eigensymmetry [{\bar 4}3m]. Two twin laws, different due to the lack of the symmetry centre, are possible:

  • (i) twofold twin rotation around [111],

  • (ii) twin reflection across the plane (111).

In the first case, the sense of the polar axis [111] is not reversed, in the second case it is reversed. All publications on this kind of twinning, common in III–V and II–VI compound semiconductors (GaAs, InP, ZnS, CdTe etc.), report the twofold axis along [111] as the true twin element, not the mirror plane (111); this was discussed very early on in a significant paper by Aminoff & Broomé (1931[link]). Displacement and fault vectors of twin boundaries

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The statements of the preceding sections on the permissibility of twin boundaries are very general and derived without any regard to the crystal structure. For example, any arbitrary reflection plane relating two partners of a crystal aggregate or even of an anisotropic continuous elastic medium represents a mechanically permissible boundary. For twin boundaries in crystals, however, additional aspects have to be taken into account, viz the atomic structure of the twin interface, i.e. the geometrical configuration of atoms, ions and molecules and their crystal-chemical interactions (bonding topology) in the transition region between the two twin partners. Only if the configurations and interactions of the atoms lead to boundaries of good structural fit and, consequently, of low energy, will the interfaces occur with the reproducibility and frequency that are prerequisites for a twin. In this respect, the mechanical and electrical permissibility conditions given in the preceding sections are necessary but not sufficient conditions for the occurrence of a twin boundary and – in the end – of the twin itself. In the following considerations, all twin boundaries are assumed to be permissible in the sense discussed above. Twin displacement vector [{\bf t}]

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As a first step of the structural elucidation of a reflection-twin boundary, the mutual relation of the two lattices of the twin partners 1 and 2 at the boundary is considered. It is assumed that the unit cells of both lattices have the same origin with respect to their crystal structure, i.e. that the lattice points are located in the same structural sites of both partners. Three cases of lattice relations across the rational composition plane (hkl) (assumed to be parallel to the twin reflection plane) are considered, as outlined in Fig.[link] [see also Section[link], Note (8)].

  • (a) The composition lattice planes [(hkl)_1] and [(hkl)_2] of domains 1 and 2 coincide pointwise (Fig.[link]a), i.e. the lattices of the two twin partners coincide in the twin boundary.


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    Lattice representation of twin displacement vectors. (a) Lattice representation of a `pure' twin reflection plane (t = 0). (b) Twin reflection plane with parallel displacement vector t (generalized twin glide plane). (c) Twin reflection plane with a general twin displacement vector with parallel and normal components: [{\bf t} = {\bf t}_n + {\bf t}_p]. By choosing a suitable new lattice point x (origin shift), the normal component [{\bf t}_n] disappears, preserving the parallel component [{\bf t}_p] as the true twin displacement vector and leading to case (b), as shown in (d).

  • (b) The composition lattice planes [(hkl)_1] and [(hkl)_2] are common but not pointwise coincident, i.e. the two lattices are displaced by a non-integer vector t (twin displacement vector) parallel to the composition plane (Fig.[link]b).

  • (c) The composition lattice planes [(hkl)_1] and [(hkl)_2] are displaced – in addition to the parallel component – by a component normal to the composition plane (Fig.[link]c). By an appropriate choice of the lattice points with respect to the structure, this normal component vanishes and, hence, this general case reduces to case (b)[link] (Fig.[link]d).

This shows that for the characterization of a twin with coinciding twin reflection and contact plane only the component of a twin displacement vector parallel to the twin boundary is significant. Thus, on an atomic scale, not only twin reflection planes ([{\bf t} \approx {\bf 0}]) but also `twin glide planes' ([{\bf t} \approx 1/2 {\bf v}_L], where [{\bf v}_L] is a lattice translation vector), as well as all intermediate cases, have to be considered. In principle, these considerations also apply to irrational twin reflection and composition planes. Moreover, twin displacement vectors also have to be admitted for the other types of twins, viz rotation and inversion twins. Examples are given below.

So far, the considerations about twin boundaries are based on the idealized concept that the bulk structure extends without any deformation up to the twin boundary. In reality, however, near the interface the structures are more or less deformed (relaxed), and so are their lattices. This transition region may even contain a central slab exhibiting a different structure, which is often close to a real or hypothetical polymorph or to the parent structure. [Examples: the Dauphiné twin boundary of α-quartz resembles the structure of β-quartz; the iron-cross (110) twin interface of pyrite, FeS2, resembles the structure of marcasite, another polymorph of FeS2.]

Whereas the twin displacement vector keeps its significance for small distortions of the boundary region, it loses its usefulness for large structural deformations. It should be noted that rational twin interfaces are usually observed as `good', whereas irrational twin boundaries, despite mechanical compatibility, usually exhibit irregular features and macroscopically visible deformations.

Twin displacement vectors are a consequence of the minimization of the boundary energy. This has been proven by a theoretical study of the boundary energy of reflection twins of monoclinic saccharine crystals with [(10{\bar 2})] as twin reflection and composition plane (Lieberman et al., 1998[link]). The authors calculated the boundary energy as a function of the lattice displacement vector t, which was varied within the mesh of the [(10{\bar 2})] composition plane, admitting also a component normal to the composition plane. The calculations were based on a combination of Lennard–Jones and Coulomb potentials and result in a flat energy minimum for a displacement vector t = [0.05/0.71/0.5] (referred to the monoclinic axes). The calculations were carried out for the undistorted bulk structure. The actual deformation of the structure near the twin boundary is not known and, hence, cannot be taken into account. Nevertheless, this model calculation shows that in general twin displacement vectors [{\bf t} \ne {\bf 0}] are required for the minimization of the boundary energy.

Twin displacement vectors have been considered as long as structural models of boundaries have been derived. One of the oldest examples is the model of the (110) growth-twin boundary of aragonite, suggested by Bragg (1924[link]) (cf. Section[link] below). An even more instructive model is presented by Bragg (1937[link], pp. 246–248) and Bragg & Claringbull (1965[link], pp. 302–303) for the Baveno (021) twin reflection and interface plane of feldspars. It shows that the tetrahedral framework can be continued without interruption across the twin boundary only if the twin reflection plane is a glide plane parallel to (021). A model of a twin boundary requiring a displacement vector [{\bf t} \ne {\bf 0}] was reported by Black (1955[link]) for the (110) twin reflection boundary of the alloy Fe4Al13.

In their interesting theoretical study of the morphology and twinning of gypsum, Bartels & Follner (1989[link], especially Fig. 4) conclude that the (100) twin interface of Montmartre twins is a pure twin reflection plane without displacement vector, whereas the dovetail twins exhibit a `twin glide component' [{\bf a}/2 + {\bf b}/2 + {\bf c}/2] parallel to the twin reflection plane [({\bar 1}01)]. [Note that in the present chapter, due to a different choice of coordinate system, the Montmartre twins are given as (001) and the dovetail twins as (100), cf. Example[link].]

The occurrence of twin displacement vectors can be visualized by high-resolution transmission electron microscopy (HRTEM) studies of twin boundaries. Fig.[link] shows an HRTEM micrograph of a (112) twin reflection boundary of anatase TiO2, viewed edge on (arrows) along [[1{\bar 3}1]] (Penn & Banfield, 1998[link]). The offset of the lattices along the twin boundary is clearly visible. This result is confirmed by the structural model presented by the authors, which indicates a parallel displacement vector [{\bf t} \approx 1/2 {\bf v}_L]. Twin displacement vectors have also been observed on HRTEM micrographs of sputtered Fe4Al13 alloys by Tsuchimori et al. (1992[link]).


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HRTEM micrograph of anatase, TiO2, with a (112) reflection twin boundary (arrows), viewed edge-on along [[1{\bar 3}1]]. The twin displacement vector t = 1/2 of the boundary translation period is clearly visible. Courtesy of R. L. Penn, Madison, Wisconsin; cf. Penn & Banfield, 1998[link]. Fault vectors of twin boundaries in merohedral twins

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Twin displacement vectors can occur in twin boundaries of both non-merohedral (see above) and merohedral twins. For merohedral twins, the displacement vector is usually called the `fault vector', because of the close similarity of these twin boundaries with antiphase boundaries and stacking faults (cf. Section[link], Note 7[link]). In contrast to non-merohedral twins, for merohedral twins these displacement vectors can be determined by imaging the twin boundaries by means of electron or X-ray diffraction methods. The essential reason for this possibility is the exact parallelism of the lattices of the two twin partners 1 and 2, so that for any reflection hkl the electron and X-ray diffraction conditions are always simultaneously fulfilled for both partners. Thus, in transmission electron microscopy and X-ray topography, both domains 1 and 2 are simultaneously imaged under the same excitation conditions. By a proper choice of imaging reflections, both twin domains exhibit the same diffracted intensity (no `domain contrast'), and the twin boundary is imaged by fringe contrast analogously to the imaging of stacking faults and antiphase boundaries (`stacking-fault contrast', cf. Fig.[link]).

This contrast results from the `phase jump' of the structure factor upon crossing the boundary. For stacking faults and antiphase boundaries this phase jump is [2\pi {\bf f} \cdot {\bf g}_{hkl}], with f the fault vector of the boundary and [{\bf g}_{hkl}] the diffraction vector of the reflection used for imaging. For (merohedral) twin boundaries the total phase jump [\Psi_{hkl}] is composed of two parts, [\Psi_{hkl} = \phi_{hkl} + 2\pi {\bf f} \cdot {\bf g}_{hkl},]with [\phi_{hkl}] the phase change due to the twin operation and [2\pi {\bf f }\cdot {\bf g}_{hkl}] the phase change resulting from the lattice displacement vector f. The boundary contrast is strongest if the phase jump [\Psi_{hkl}] is an odd integer multiple of [\pi], and it is zero if [\Psi_{hkl}] is an integer multiple of [2\pi]. By imaging the boundary in various reflections hkl and analysing the boundary contrast, taking into account the known phase change [\phi_{hkl}] (calculated from the structure-factor phases of the reflections [hkl_1] and [hkl_2] related by the twin operation), the fault vector f can be determined (see the examples below). This procedure has been introduced into transmission electron microscopy by McLaren & Phakey (1966[link], 1969[link]) and into X-ray topography by Lang (1967[link][link]a,b) and McLaren & Phakey (1969[link]).

In the equation given above, for each reflection hkl the total phase jump [\Psi_{hkl}] is independent of the origin of the unit cell. The individual quantities [\phi_{hkl}] and [2\pi {\bf f} \cdot {\bf g}_{hkl}], however, vary with the choice of the origin but are coupled in such a way that [\Psi_{hkl}] (which alone has a physical meaning) remains constant. This is illustrated by the following simple example of an inversion twin.

The twin operation relates reflections [hkl_1] and [hkl_2 =] [{\bar h}{\bar k}{\bar l}_1]. Their structure factors are (assuming Friedel's rule to be valid)[F_1 = \vert F\vert \exp(-{i}\varphi)\ \ {\rm and}\ \ F_2 = \vert F \vert \exp(+ {i}\varphi).]

The phase difference of the two structure factors is [\phi_{hkl} = 2 \varphi] and depends on the choice of the origin. If the origin is chosen at the twin inversion centre (superscript o), the phase jump [\Psi_{hkl}] at the boundary is given by[ \Psi_{hkl} = \phi^o_{hkl} = 2\varphi^o.]

This is the total phase jump occurring for reflection pairs [hkl_1/hkl_2 = hkl/{\bar h}{\bar k}{\bar l}] at the twin boundary.

If the origin is not located at the twin inversion centre but is displaced from it by a vector [\textstyle{1\over 2} {\bf f}], the phases of the structure factors of reflections [hkl_1] and [hkl_2] are [\eqalign{\varphi_1 &= \varphi^o - 2\pi (\textstyle{1\over 2}{\bf f}) \cdot{\bf g}_{hkl} \ \ {\rm and}\cr \varphi_2 &= -\varphi_1 = - [\varphi^o - 2\pi (\textstyle{1\over 2}{\bf f}) \cdot {\bf g}_{hkl}].}]From these equations the phase difference of the structure factors is calculated as [\phi_{hkl} = \varphi_1 - \varphi_2 = 2\varphi^o - 2\pi {\bf f} \cdot {\bf g}_{hkl},]and the total phase jump at the boundary is [\Psi_{hkl} = 2\varphi^o = \phi_{hkl} + 2\pi {\bf f} \cdot {\bf g}_{hkl}.]

This shows that here the fault vector f has no physical meaning. It merely compensates for the phase contributions that result from an `improper' choice of the origin. If, by the procedures outlined above, a fault vector f is determined, the true twin inversion centre is located at the endpoint of the vector [\textstyle{1\over 2}{\bf f}] attached to the chosen origin.

Similar considerations apply to reflection and twofold rotation twins. In these cases, the components of the fault vectors normal to the twin plane or to the twin axis can also be eliminated by a proper choice of the origin. The parallel components, however, cannot be modified by changes of the origin and have a real physical significance for the structure of the boundary.

Particularly characteristic fault vectors occur in (merohedral) `antiphase domains' (APD). Often the fault vector is the lattice-translation vector lost in a phase transition. Examples of fault-vector determinations

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  • (1) Inversion boundaries (180° domain walls) of ferroelectric ammonium lithium sulfate, NH4LiSO4 (Klapper, 1987[link]), cf. Example[link].

    NH4LiSO4 is ferroelectric at room temperature (point group [m2m]) and transforms into the paraelectric state (point group mmm) at 459 K. Crystals grown from aqueous solution at about 313 K contain grown-in inversion twins with boundaries exactly parallel to (001), appearing on X-ray topographs by stacking-fault fringe contrast. It was found that the boundaries are invisible in reflections of type [h0l] (zone of the polar axis [010]), but show contrast in reflections with [k \ne 0] with some exceptions (e.g. no contrast for reflection 040). The `zero-contrast' reflections are particularly helpful for the determination of the fault vector. Applying the procedure described above, a fault vector [{\bf f} = 1/2[010]] (parallel to the polar axis) was derived for the chosen origin, which is located on the polar twofold symmetry axis. Thus, the true twin inversion centre is located at the endpoint of the vector [\textstyle{1\over 2} {\bf f} = 1/4[010]]. An inspection of the NH4LiSO4 structure shows that this point is the location of the inversion centre of the paraelectric parent phase above 459 K. Thus, during the transition from the para- to the ferroelectric phase the structural inversion centres vanish in the bulk of the domains, but are preserved in the domain boundaries as twin inversion centres.

    For this (001) twin interface, a reasonable structural model without any breaking of the framework of SO4 and LiO4 tetrahedra could be derived easily. The tetrahedra adopt a staggered orientation across the boundary, compared with a nearly eclipsed orientation in the bulk structure.

  • (2) Brazil twin boundaries of quartz.

    Brazil twins are commonly classified as [\{11{\bar 2}0\}] reflection twins but can alternatively be considered as inversion twins, as explained in Section[link]. The twin boundaries are usually strictly planar and mainly parallel to one of the major rhombohedron faces [\{10{\bar 1}1\}] but, less frequently, to one of the minor rhombohedron faces [\{{\bar 1}011\}] or prism faces [\{10{\bar 1}0\}]. From electron microscopy studies of polysynthetic (lamellar) Brazil twins in amethyst (McLaren & Phakey, 1966[link]),7 fault vectors of type [{\bf f} = 1/2 \langle 010\rangle], i.e. one half of one of the three translations along the twofold axes, were obtained for twin boundaries parallel to [\{10{\bar 1}1\}], where f is parallel to the boundary. The same but slightly shorter fault vector [{\bf f} = 0.4 \langle010\rangle] for [\{10{\bar 1}1\}] Brazil boundaries was determined in X-ray topographic studies by Lang (1967[link][link]a,b) and Lang & Miuskov (1969[link]). Another detailed X-ray topographic investigation was carried out by Phakey (1969[link]). He confirmed the existence of the fault vector [{\bf f} = 1/2\langle010\rangle] but proved also the occurrence of further fault vectors of type [{\bf f} = \langle0, \textstyle{1\over 2}, {1\over 3}\rangle] for [\{10{\bar 1}1\}] twin boundaries. Based on these fault vectors, the structures of the Brazil twin boundaries could be derived: it was shown that no Si—O bonds are broken and that the left- and right-handed partner structures join each other with only small distortions of the tetrahedral framework. It is worth mentioning that the structural channels along the threefold axes do not continue smoothly across the boundary but are mutually displaced by the fault-vector component parallel to the basal plane (0001). McLaren, Phakey and Lang, however, did not consider the location of the twin elements (twin inversion centre or twin reflection plane) in the structure, which can be determined from the fault vectors.

    These structural studies of the Brazil twin boundaries have shown that the fault vectors f are different for different orientations of the interface. As a consequence, a `stair-rod' dislocation must occur along the bend of the twin interface from one orientation to the other. Stair-rod dislocations have been observed and characterized in the X-ray topographic study of Brazil boundaries by Phakey (1969[link]). Examples of structural models of twin boundaries

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Until the rather recent advent of high-resolution transmission electron microscopy (HRTEM), no experimental method for the direct elucidation of the atomic structures of twin interfaces existed. Thus, many authors have devised structural models of twin interfaces based upon the (undeformed) bulk structure of the crystals and the experimentally determined orientation and contact relations. The criterion of good structural fit and low energy of a boundary was usually applied in a rather intuitive manner to the specific case in question. The first and classic example is the model of the aragonite (110) boundary by Bragg (1924[link]).

Some examples of twin-boundary models from the literature are given below. They are intended to show the wide variety of substances and kinds of models. Examples for the direct observation of twin-interface structures by HRTEM follow in Section[link] Aragonite, CaCO3

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The earliest structural model of a twin boundary was derived for aragonite by Bragg (1924[link]), reviewed in Bragg (1937[link], pp. 119–121) and Bragg & Claringbull (1965[link], pp. 131–133). Aragonite is orthorhombic with space group Pmcn. It exhibits a pronounced hexagonal pseudosymmetry, corresponding to a (hypothetical) parent phase of symmetry [P6_3/mmc], in which the Ca ions form a hexagonal close-packed structure with the CO3 groups filling the octahedral voids along the [6_3] axes. By eliminating the threefold axis and the C-centring translation of the orthohexagonal unit cell, the above orthorhombic space group results, where the lost centring translation now appears as the glide component n. Of the three mirror planes parallel to [\{11{\bar 2}0\}_{\rm hex}] and the three c-glide planes parallel to [\{10{\bar 1}0\}_{\rm hex}], one of each set is retained in the orthorhombic structure, whereas the other two appear as possible twin mirror planes [\{110\}_{\rm orth}] and [\{130\}_{\rm orth}]. It is noted that predominantly planes of type [\{110\}_{\rm orth}] are observed as twin boundaries, but less frequently those of type [\{130\}_{\rm orth}].

From this structural pseudosymmetry the atomic structure of the twin interface was easily derived by Bragg. It is shown in Fig.[link]. In reality, small relaxations at the twin boundary have to be assumed. It is clearly evident from the figure that the twin operation is a glide reflection with glide component [\textstyle{1\over2}{\bf c}] (= twin displacement vector t).


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Structural model of the (110) twin boundary of aragonite (after Bragg, 1924[link]), projected along the pseudo-hexagonal c axis. The ortho­rhombic unit cells of the two domains with eigensymmetry Pmcn, as well as their glide/reflection planes m and c, are indicated. The slab centred on the (110) interface between the thin lines is common to both partners. The interface coincides with a twin glide plane c and is shown as a dotted line (twin displacement vector [{\bf t} = 1/2 {\bf c}]). The model is based on a hexagonal cell with [\gamma = 120^\circ], the true angle is [\gamma = 116.2^\circ]. The origin of the orthorhombic cell is chosen at the inversion centre halfway between two CO3 groups along c. Dauphiné twins of [\alpha]-quartz

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For this merohedral twin (eigensymmetry 32) a real parent phase, hexagonal [\beta]-quartz (622), exists. The structural relation between the two Dauphiné twin partners of [\alpha]-quartz is best seen in projection along [001], as shown in Fig.[link] and in Figure 3 of McLaren & Phakey (1966[link]), assuming a fault vector [{\bf f} = {\bf 0}] in both cases. These figures reveal that only small deformations occur upon passing from one twin domain to the other, irrespective of the orientation of the boundary. This is in agreement with the general observation that Dauphiné boundaries are usually irregular and curved and can adopt any orientation. The electron microscopy study of Dauphiné boundaries by McLaren & Phakey confirms the fault vector [{\bf f} = {\bf 0}]. It is noteworthy that the two models of the boundary structure by Klassen-Neklyudova (1964[link]) and McLaren & Phakey (1966[link]) imply a slab with the [\beta]-quartz structure in the centre of the transition zone (Fig.[link]b). This is in agreement with the assumption voiced by several authors, first by Aminoff & Broomé (1931[link]), that the central zone of a twin interface often exhibits the structure of a different (real or hypothetical) polymorph of the crystal.


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Simplified structural model of a [\{10{\bar 1}0\}] Dauphiné twin boundary in quartz (after Klassen-Neklyudova, 1964[link]). Only Si atoms are shown. (a) Arrangement of Si atoms in the low-temperature structure of quartz viewed along the trigonal axis [001]. (b) Model of the Dauphiné twin boundary C–D. Note the opposite orientation of the three electrical axes shown in the upper left and lower right corner of part (b). In this model, the structural slab centred along the twin boundary has the structure of the hexagonal high-temperature phase of quartz which is shown in (c).

There are, however, X-ray topographic studies by Lang (1967[link][link]a,b) and Lang & Miuskov (1969[link]) which show that curved Dauphiné boundaries may be fine-stepped on a scale of a few tens of microns and exhibit a pronounced change of the X-ray topographic contrast of one and the same boundary from strong to zero (invisibility), depending on the boundary orientation. This observation indicates a change of the fault vector with the boundary orientation. It is in contradiction to the electron microscopy results of McLaren & Phakey (1966[link]) and requires further experimental elucidation. Potassium lithium sulfate, KLiSO4

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The room-temperature phase of KLiSO4 is hexagonal with space group [P6_3]. It forms a `stuffed' tridymite structure, consisting of a framework of alternating SO4 and LiO4 tetrahedra with the K ions `stuffed' into the framework cavities. Crystals grown from aqueous solutions exhibit merohedral growth twins with twin reflection planes [\{10{\bar 1}0\}] (alternatively [\{11{\bar 2}0\}]) with extended and sharply defined (0001) twin boundaries. The twins consist of left- and right-handed partners with the same polarity. The left- and right-handed structures, projected along the polar hexagonal c axis, are shown in Figs.[link](a) and (b) (Klapper et al., 1987[link]). The tetrahedra of the two tetrahedral layers within one translation period c are in a staggered orientation. A model of the twin boundary is shown in Fig.[link](c): the tetrahedra on both sides of the twin interface (0001), parallel to the plane of the figure, now adopt an eclipsed position, leading to an uninterrupted framework and a conformation change in second coordination across the interface. It is immediately obvious that this (0001) interface permits an excellent low-energy fit of the two partner structures. Note that all six (alternative) twin reflection planes [\{10{\bar 1}0\}] and [\{11{\bar 2}0\}] are normal to the twin boundary. It is not possible to establish a similar low-energy structural model of a boundary which is parallel to one of these twin mirror planes (Klapper et al., 1987[link]).


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KLiSO4: Bulk tetrahedral framework structures and models of (0001) twin boundary structures of phases III and IV. Small tetrahedra: SO4; large tetrahedra: LiO4; black spheres: K. All three figures play a double role, both as bulk structure and as (0001) twin-boundary structures. (a) and (b) Left- and right-handed bulk structures of phase III ([P6_3]), as well as possible structures of the (0001) twin boundary in phase IV. (c) Bulk structure of phase IV ([P31c]), as well as possible structure of the (0001) twin boundary in phase III. The SO4 tetrahedra covered by the LiO4 tetrahedra are shown by thin lines. Dotted line: [\{10{\bar 1}0\}] c-glide plane. In all cases, the (0001) twin boundary is located between the two tetrahedral layers parallel to the plane of the figure.

Inspection of the boundary structure in Fig.[link](c) shows that the tetrahedra related by the twin reflection plane [\{10{\bar 1}0\}] (one representative plane is indicated by the dotted line) are shifted with respect to each other by a twin displacement vector [{\bf t} = 1/2 [001]]. Thus, on an atomic scale, these twin reflection planes are in reality twin c-glide planes, bringing the right- and left-hand partner structures into coincidence.

Interestingly, upon cooling below 233 K, KLiSO4 undergoes a (very) sluggish phase transition from the [P6_3] phase III into the trigonal phase IV with space group [P31c] by suppression of the twofold axis parallel [001] and by addition of a c-glide plane. Structure determinations show that the bulk structure of IV is exactly the atomic arrangement of the grown-in twin boundary of phase III, as presented in Fig.[link](c). Moreover, X-ray topography reveals transformation twins III [\rightarrow] IV, exhibiting extended and sharply defined polysynthetic (0001) twin lamellae in IV. From the X-ray topographic domain contrast, it is proven that the twin element is the twofold rotation axis parallel to [001] (Klapper et al., 2008[link]). The structural model of the (0001) twin interfaces is given in Figs. and (b). They show that across the (0001) twin boundary the tetrahedra are staggered, in contrast to the bulk structure of IV where they are in an eclipsed orientation (Fig.[link]c). It is immediately recognized that the two tetrahedral layers, one above and one below the (0001) twin boundary in Fig.[link](a) or (b), are related by [2_1] screw axes.

Thus, the (idealized) (0001) twin boundary of the transformation twins of phase IV is represented by the bulk structure of the hexagonal room-temperature phase III, whereas the twin boundary of the growth twins of the hexagonal phase III is represented by the bulk structure of the trigonal low-temperature phase IV. Upon cooling from [P6_3] (phase III) to [P31c] (phase IV), the [2_1] axes are suppressed as symmetry elements, but they now act as twin elements. In the model they are located as in space group [P6_3], one type being contained in the [6_3] axes, the other type halfway in between. Upon heating, the re-transformation IV [\rightarrow] III restores the [\{10{\bar 1}0\}/\{11{\bar 2}0\}] reflection twins with the same large (0001) boundaries in the same geometry as existed before the transition cycle, but now as result of a phase transition, not of crystal growth (strong memory effect; Klapper et al., 2008[link]).

Thus, KLiSO4 is another particularly striking example of the phenomenon, mentioned above for the Dauphiné twins of quartz, that the twin-interface structure of one polymorph may resemble the bulk structure of another polymorph.

The structural models of both kinds of twin boundaries do not exhibit a fault vector [{\bf f} \ne {\bf 0}]. This may be explained by the compensation of the glide component [\textstyle{1\over2}c] of the c-glide plane in phase IV by the screw component [\textstyle{1\over2}c] of the [2_1] screw axis in phase III and vice versa. Twin models of molecular crystals

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An explanation for the occurrence of twinning based on the `conflict' between the energetically most favourable (hence stable) crystal structure and the arrangement with the highest possible symmetry was proposed by Krafczyk et al. (1994[link], 1996[link], 1997[link] and references therein) for some molecular crystals. According to this theory, pseudosymmetrical structures exhibit `structural instabilities', i.e. symmetrically favourable structures occur, whereas the energetically more stable structures are not realized, but were theoretically derived by lattice-energy calculations. The differences between the two structures provide the explanation for the occurence of twins. The twin models contain characteristic `shift vectors' (twin displacement vectors). The theory was successfully applied to pentaerythrite, 1,2,4,5-tetrabromo­benzene, maleic acid and 3,5-dimethylbenzoic acid. Observations of twin boundaries by transmission electron microscopy

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In the previous sections of this chapter, twin boundaries have been discussed from two points of view: theoretically in terms of `compatibility relations', i.e. of mechanically and electrically `permissible' interfaces (Sections and, followed by structural aspects, viz by displacement and fault vectors (Section[link]), as well as atomistic models of twin boundaries (Section[link]), in each case accompanied by actual examples.

In the present section, a very powerful method of direct experimental elucidation of the atomistic structure of twin interfaces is summarized, transmission electron microscopy (TEM), in particular high-resolution transmission electron microscopy (HRTEM). This method enjoys wider and wider application because it can provide in principle – if applied with proper caution and criticism – direct evidence for the problems discussed in earlier sections: `good structural fit', `twin displacement vector', `relaxation of the structure' across the boundary etc.

The present chapter is not a suitable place to introduce and explain the methods of TEM and HRTEM and the interpretation of the images obtained. Instead, the following books, containing treatments of the method in connection with materials science, are recommended: Wenk (1976[link]), especially Sections 2.3 and 5; Amelinckx et al. (1978[link]), especially pp. 107–151 and 217–314; McLaren (1991[link]); Buseck et al. (1992[link]), especially Chapter 11; and Putnis (1992[link]), especially pp. 67–80.

The results of HRTEM investigations of twin interfaces are not yet numerous and representative enough to provide a complete and coherent account of this topic. Instead, a selection of typical examples is provided below, from which an impression of the method and its usefulness for twinning can be gained. Anatase, TiO2 (Penn & Banfield, 1998[link], 1999[link])

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This investigation has been presented already in Section[link] and Fig.[link] as an example of the occurrence of a twin displacement vector, leading to [{\bf t}\approx 1/2 {\bf v}_L], where [{\bf v}_L] is a lattice translation vector parallel to the (112) twin reflection plane of anatase. Another interesting result of this HRTEM study by Penn & Banfield is the formation of anatase–brookite intergrowths during the hydrothermal coarsening of TiO2 nanoparticles. The preferred contact plane is (112) of anatase and (100) of brookite, with [131] of anatase parallel to [011] of brookite in the intergrowth plane. Moreover, it is proposed that brookite may nucleate at (112) twin boundaries of anatase and develop into (100) brookite slabs sandwiched between the anatase twin components. Similarly, after hydrothermal treatment at 523 K, nuclei of rutile at the anatase (112) twin boundary were also observed by HRTEM (Penn & Banfield, 1999[link]). A detailed structural model for this anatase-to-rutile phase transition is proposed by the authors, from which a sluggish nucleation of rutile followed by rapid growth of this phase was concluded. Rutile, TiO2

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The two reflection twins of rutile, (011) and (031), have been treated geometrically and with several figures in Section[link]. In the present section the atomistic structure of the twin interfaces, which in both cases coincide with the corresponding twin reflection plane, is discussed for two rutiles of different origin:

  • (i) Twin boundaries of synthetic rutiles, formed during the chemical vapour deposition of TiO2 onto sapphire (α-Al2O3) were investigated by Lee et al. (1993)[link] by HRTEM and ionic-shell model energy calculations, whereby the twin displacement vector t (cf. Section[link]) was varied, with two characteristic results:

    • (a) the (031) twin boundary is a true reflection boundary for both the metal and the oxygen atoms, i.e. there is no twin displacement vector. The calculation of the boundary energy leads to a reasonable energy of 420 mJ m−2 for the model with t = 0 and instability for t = ½[111];

    • (b) the (011) twin boundary is a glide reflection boundary with a twin displacement vector t = ½<111>. Since the Ti sublattice is tetragonal body-centred, this implies a true mirror plane without glide component for the Ti atoms, but a real glide mirror plane for the oxygen substructure. The calculated interface energy is 124 mJ m−2 for t = ½[111] and about 15 times higher, 1853 mJ m−2, for t = 0.

  • (ii) The (031) twin boundary of a natural rutile from Diamantina (Brazil) is more complicated (Daneu et al., 2007[link]). In the twin interface, between the two mirror-related twin components, HRTEM investigations have revealed a slice of coherent interlayers of hexagonal ilmenite, FeTiO3 with some Al content, of several nanometres thickness, which exhibits perfect metrical fit with the two rutile domains. This is interpreted as the nucleation site of the epitaxial growth of the two (031) twin domains. At an even smaller level, ilmenite lamellae contain nanotwins.

These two examples with completely different (031) twin boundary structures of rutile show that the interface structure may be dependent on the genesis of the twin. Cassiterite, SnO2 (rutile structure)

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Twin interfaces (011) of the closely related tetragonal SnO2 (cassiterite) were investigated by Smith et al. (1983[link]). A very close agreement between HRTEM images and corresponding computer simulations was obtained for [{\bf t} = 1/2 [1{\bar 1}1](011)]. This twin is termed a `glide twin' by the authors, because the twin operation is a reflection across (011) followed by a displacement vector [{\bf t} = 1/2 [1{\bar 1}1](011)] parallel to the twin plane (011). Σ3 (111) twin interface in BaTiO3 [cf. Section[link](iii)[link]]

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In cubic crystals, twins of the [\Sigma 3] (111) spinel type are by far the most common. A technologically very important phase, BaTiO3 perovskite, was investigated by Rečnik et al. (1994[link]) employing HRTEM, computer simulations and EELS (spatially resolved electron-energy-loss spectroscopy). The samples were prepared by sintering at 1523 K, i.e. in the cubic phase, whereby [\Sigma 3] (111) growth twins were formed. These twins are preserved during the transition into the tetragonal phase upon cooling below Tc = 398 K. Note that these (now tetragonal) (111) twins are not transformation twins, as are the (110) ferroelastic twins.

Fig.[link](a) shows an HRTEM micrograph and Fig.[link](b) the structural model of the (111) twin boundary, both projected along [[1{\bar 1}0]]. The main results of this study can be summarized as follows.

  • (1) The twin boundary coincides exactly with the twin reflection plane (111). It is very sharp and consists of one atomic layer only, common to both twin components. It is fully `coherent' (cf. Section[link]).


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    (a) HRTEM micrograph of a coherent (111) twin boundary in BaTiO3, projected along [[1{\bar 1}0]]. The intense white spots represent the [[1{\bar 1}0]] Ba–O columns, the small weak spots in between represent the Ti columns. Thickness of specimen 4 nm. (b) Structural model of the (111) twin boundary (arrow), as derived from the micrograph shown in (a) and confirmed by computer simulation. Note that the pure oxygen columns (open circles) are not visible in the micrograph in (a), due to the low scattering power of oxygen. Some slight structural deformations along the twin boundary are discussed in the text. Courtesy of W. Mader, Bonn; cf. Rečnik et al. (1994)[link].

  • (2) The twin boundary is formed by a close-packed BaO3 layer, and the TiO6 octahedra on both sides share faces to form Ti2O9 groups. These groups occur also in the hexagonal high-temperature modification of BaTiO3, i.e. this is a further example of a twin interface having the structure of another polymorph of the same compound.

  • (3) The EELS results suggest a reduction in the valence state of the Ti4+ ions in the interface towards Ti3+ which is compensated by some oxygen vacancies, leading to the composition of the interface layer BaO3−x(VO)x instead of BaO3. This result indicates that the stoichiometry of a boundary, even of a coherent one, may differ from that of the bulk.

A very interesting structural feature of the BaTiO3 (111) twin interface was discovered by Jia & Thust (1999[link]), applying sophisticated HRTEM methods to thin films of nanometre thickness (grown by the pulsed-laser deposition technique). The distance of the nearest Ti plane on either side of the (111) twin reflection plane (which is formed by a BaO3 layer, see above) from this twin plane is increased by 0.19 Å, i.e. the distance between the Ti atoms in the Ti2O9 groups, mentioned above under (2)[link], is increased from the hypothetical value of 2.32 Å for Ti in the ideal octahedral centres to 2.70 Å in the actual interface structure. This expansion is due to the strong repulsion between the two neighbouring Ti ions in the Ti2O9 groups. A similar expansion of the Ti–Ti distance in the Ti2O9 groups (from 2.34 to 2.67 Å), again due to the strong repulsion between the Ti atoms, has been observed in the bulk crystal structure of the hexagonal modification of BaTiO3 (Burbank & Evans, 1948[link]).

In addition, a decrease by 0.16 Å of the distance between the two nearest BaO planes across the twin interface was found, which corresponds to a contraction of this pair of BaO planes from 2.32 Å in the bulk to 2.16 Å at the twin interface (corresponding closely to the value of 2.14 Å in the hexagonal phase).

It is remarkable that no significant (i.e. > 0.05 Å) displacements were found for second and higher pairs of both Ti–Ti and BaO–BaO layers. Moreover, no significant lateral shifts, i.e. no twin displacements vectors [{\bf t}\neq 0] parallel to the (111) twin interface, were observed.

Note that BaTiO3 is treated again in Section[link] below, with respect to its twin texture in polycrystalline aggregates. Σ = 3 bicrystal boundaries in Cu and Ag

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Differently oriented interfaces in [\Sigma 3] bicrystals of Cu and Ag were elucidated by Hoffmann & Ernst (1994[link]) and Ernst et al. (1996[link]). They prepared bicrystals of fixed [\Sigma 3] orientation relationship [corresponding to the (111) spinel twin law] but with different contact planes. The inclinations of these contact planes vary by rotations around the two directions [[{\bar 1}10]] and [[11{\bar 2}]] [both parallel to the (111) twin reflection plane] in the range [\varphi =] 0–90°, where [\varphi = 0^\circ] corresponds to the (111) `coherent twin plane', as illustrated in Figs.[link](a) and (b).


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(a) Schematic block diagram of a [\Sigma = 3] bicrystal (spinel twin) for [\varphi[{\bar 1}10] = 0^\circ], i.e. for coinciding (111) twin reflection and composition plane. (b) Schematic block diagram of the [\Sigma = 3] bicrystal for [\varphi[{\bar 1}10] = 82^\circ]. (c) HRTEM micrograph of the [\Sigma = 3] boundary of Cu for [\varphi[{\bar 1}10] = 82^\circ], projected along [[{\bar 1}10]]. The black spots coincide with the [[{\bar 1}10]] Cu-atom columns. The micrograph reveals a thin ([\approx] 10 Å) interface slab of a rhombohedral 9R structure, which can be derived from the bulk cubic 3C structure by introducing a stacking fault SF on every third (111) plane. The (111) planes are horizontal, the interface is roughly parallel to (4.4.11) and (223), respectively. Courtesy of F. Ernst, Stuttgart; cf. Ernst et al. (1996)[link].

The boundary energies were determined from the surface tension derived from the characteristic angles of surface grooves formed along the boundaries by thermal etching. The theoretical energy values were obtained by molecular statics calculations. The measured and calculated energy curves show a deep and sharp minimum at [\varphi = 0^\circ] for rotations around both [[{\bar 1}10]] and [[11{\bar 2}]]. This corresponds to the coherent (111) [\Sigma 3] twin boundary and is to be expected. It is surprising, however, that a second, very shallow energy minimum occurs in both cases at high [\varphi] angles: [\varphi_{[{\bar 1}10]} \approx 82^\circ] and [\varphi_{[11{\bar 2}]} \approx 84^\circ], rather than at the compatible (112) contact plane for [\varphi_{[{\bar 1}10]} = 90^\circ] [the contact plane [(1{\bar 1}0)] for [\varphi_{[11{\bar 2}]} = 90^\circ] is not compatible]. For these two angular inclinations, the boundaries, as determined by HRTEM and computer modelling, exhibit complex three-dimensional boundary structures with thin slabs of unusual Cu arrangements: the [\varphi_{[{\bar 1}10]} \approx 82^\circ] slab has a rhombohedral structure of nine close-packed layers (9R) with a thickness of about 10 Å (in contrast to the f.c.c bulk structure, which is 3C). This is shown and explained in Fig.[link](c). Similarly, for the [\varphi_{[11{\bar 2}]} \approx 84^\circ] slab a b.c.c. structure (as for [\alpha]-Fe) was found, again with a thickness of [\approx 10] Å. For further results see Sutton & Balluffi (1995)[link], Section, p. 305. Fivefold cyclic twins in nanocrystalline materials

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Multiply twinned particles occur frequently in nanocrystalline (sphere-like or rod-shaped) particles and amorphous thin films (deposited on crystalline substrates) of cubic face-centred metals, diamond-type semiconductors (C, Si, Ge) and alloys. Hofmeister & Junghans (1993[link]) and Hofmeister (1998[link]) have carried out extensive HRTEM investigations of nanocrystalline Ge particles in amorphous Ge films. The particles reveal, among others, fivefold cyclic twins with coinciding (111) twin reflection planes and twin boundaries (spinel type). A typical example of a fivefold twin is presented in Fig.[link]: The five different {111} twin boundaries are perpendicular to the image plane ([1{\bar 1}0]) and should theoretically form dihedral angles of 70.5° (supplement to the tetrahedral angle 109.5°), which would lead to an angular gap of about 7.5°. In reality, the five twin sectors are more or less distorted with angles ranging up to 76°. The stress due to the angular mismatch is often relaxed by defects such as stacking faults (marked by arrows in Fig.[link]). The [[1{\bar 1}0]] junction line of the five sectors can be considered as a pseudo-fivefold twin axis (similar to the pseudo-trigonal twin axis of aragonite, cf. Fig.[link]; see also the fivefold twins in the alloy FeAl4, described in Example[link] and Fig.[link]).



HRTEM micrograph of a fivefold-twinned Ge nanocrystal (right) in an amorphous Ge film formed by vapour deposition on an NaCl cleavage plane. Projection along a [[1{\bar 1}0]] lattice row that is the junction of the five twin sectors; plane of the image: [(1{\bar 1}0)]. The coinciding {111} twin reflection and composition planes (spinel law) are clearly visible. In one twin sector, two pairs of stacking faults (indicated by arrows) occur. They reduce the stress introduced by the angular misfit of the twin sectors. The atomic model (left) shows the structural details of the bulk and of one pair of stacking faults. Courtesy of H. Hofmeister, Halle; cf. Hofmeister & Junghans (1993)[link]; Hofmeister (1998)[link].

For the formation of fivefold twins, different mechanisms have been suggested by Hofmeister (1998[link]): nucleation of noncrystallographic clusters, which during subsequent growth collapse into cyclic twins; successive growth twinning on alternate cozonal (111) twin planes; and deformation twinning (cf. Section 3.3.7[link]).

The fivefold multiple twins provide an instructive example of a twin texture, a subject which is treated in the following section. Twin textures

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So far in Section 3.3.10, `free' twin interfaces have been considered with respect to their mechanical and electrical compatibility, their twin displacement vectors and their structural features, experimentally and by modelling. In the present section, the `textures' of twin domains, both in a `single' twin crystal and in a polycrystalline material or ceramic, are considered. With the term `twin texture', often also called `twin pattern', `domain pattern' or `twin microstructure', the size, shape and spatial distribution of the twin domains in a twinned crystal aggregate is expressed. In a (polycrystalline) ceramic, the interaction of the twin interfaces in each grain with the grain boundary is a further important aspect. Basic factors are the `form changes' and the resulting space-filling problems of the twin domains compared to the untwinned crystal. These interactions can occur during crystal growth, phase transitions or mechanical deformations.

From the point of view of form changes, two categories of twins, described in Sections[link] and[link] below, have to be distinguished. Discussions of the most important twin cases follow in Sections[link] to[link]. Merohedral (non-ferroelastic) twins (see Sections 3.3.9[link] and[link])

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In these twins, the lattices of all domains are exactly parallel (`parallel-lattice twins'). Hence, no spontaneous lattice deformations (spontaneous strain) occur and the development of the domain pattern of the twins is not infringed by spatial constraints. As a result, the twin textures can develop freely, without external restraints (cf. Section[link] below).

It should be noted that these features apply to all merohedral twins, irrespective of origin, i.e. to growth and transformation twins and, among mechanical twins, to ferrobielastic twins [for the latter see Section[link](iii)[link]]. Non-merohedral (ferroelastic) twins

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Here, the lattices of the twin domains are not completely parallel (`twins with inclined axes'). As a result, severe space problems may arise during domain formation. Several different cases have to be considered:

  • (1) Only one twin law, i.e. only two domain states occur which can form two-component twins (e.g. dovetail twins of gypsum, Carlsbad twins of orthoclase) or multi-component twins (e.g. lamellar, polysynthetic twins of albite). For these twins, no spatial constraints are imposed and, hence, the twin crystal can develop freely, without external restraints. Again, this applies to both growth and transformation twins.

  • (2) Two or more twin laws, i.e. three or more domain states coexist. Here, the free development of a twin domain is impeded by the space requirements of its neighbours. For growth twins, typical cases are sector and cyclic twins (e.g. K2SO4 and aragonite). More complicated examples are the famous harmotome and phillipsite growth twins, where the combined action of several twin laws leads to a pseudo-cubic twin texture and twin morphology [see Section[link](iii)].

Transformation and deformation twins are extensively treated in the following Section[link]. Fitting problems of ferroelastic twins

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The real problem of space-constrained twin textures, however, is provided by non-merohedral (ferroelastic) transformation and deformation twins (including the cubic deformation twins of the spinel law). This is schematically illustrated in Fig.[link] for the very common case of orthorhombic[\longrightarrow]monoclinic transformation twins ([\beta = 90^\circ + \varepsilon)].


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Illustration of space-filling problems of domains for a (ferroelastic) orthorhombic[\longrightarrow]monoclinic phase transition with an angle [\varepsilon] (exaggerated) of spontaneous shear. (a) Orthorhombic parent crystal with symmetry [2/m\,2/m\,2/m]. (b) Domain pairs [1+2], [1+3] and [2+4] of the monoclinic daughter phase ([\beta = 90^\circ + \varepsilon]) with independent twin reflection planes (100) and (001). (c) The combination of domain pairs [1+2] and [1+3] leads to a gap with angle [90^\circ - 3\varepsilon], whereas the combination of the three domain pairs [1+2], [1+3] and [2+4] generates a wedge-shaped overlap (hatched) of domains 3 and 4 with angle [4\varepsilon]. (d) Twin lamellae systems of domain pairs [1+2] (left) and [1+3] (or [2+4]) (right) with low-energy contact planes (100) and (001). Depending on the value of [\varepsilon], adaptation problems with more or less strong lattice distortions arise in the boundary region A–A between the two lamellae systems. (e) Stress relaxation and reduction of strain energy in the region A–A by the tapering of domains 2 (`needle domains') on approaching the (nearly perpendicular) boundary of domains [3+1]. The tips of the needle lamellae may impinge on the boundary or may be somewhat withdrawn from it, as indicated in the figure. The angle between the two lamellae systems is [90^\circ - \varepsilon].

Figs.[link](a) and (b) show the `splitting' of two mirror planes (100) and (001) of parent symmetry mmm, as a result of a phase transition [mmmF12/m1], into the two independent and symmetry-non-equivalent twin reflection planes (100) and (001), each one representing a different (monoclinic) twin law. The two orientation states of each domain pair differ by the splitting angle [2\varepsilon]. Note that in transformation twins the angle [\varepsilon]