International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.3, pp. 413487
doi: 10.1107/97809553602060000917 Chapter 3.3. Twinning of crystals^{a}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D52056 Aachen, Germany, and ^{b}MineralogischPetrologisches Institut, Universität Bonn, D53113 Bonn, Germany 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, pseudomerohedry 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 crystalstructure determinations, is treated in Section 3.3.11. In the literature, the concept of twinning is very often used in a nonprecise 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.
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 (twodimensional oriented intergrowth on a surface), topotaxy (threedimensional oriented precipitation or exsolution) or the paragenesis of different minerals in a rock or in a technical product are not treated in this chapter.
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) and by Grimmer & Nespolo (2006)]. 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). 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) and Fig. 3.3.1.2] 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 symmetrynonequivalent 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,NH_{4})alum, cf. Fig. 3.3.1.2; Shtukenberg et al., 2001]. Optical anomalies may occur also in symmetryequivalent 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)^{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 meltgrown 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). 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 principle^{2} (Curie, 1894; Chalmers, 1970) 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 pointgroup symmetry G_{CF} of the crystal under the external influence (field) F is the intersection of the symmetry G_{C} of the crystal without field and the symmetry G_{F} of the influence without crystal:i.e. G_{CF} is a (proper or improper) subgroup of both groups G_{C} and G_{F}. In the example of the optical anomaly of the {111} growth sectors of (K,NH_{4})alum (Fig. 3.3.1.2) the crystal point group is and the symmetry of the cause `growth in direction [111]' is (symmetry of a stationary cone, cf. ITA, Table 10.1.4.2 ). The intersection symmetry [i.e. the symmetry of the (111) growth sector] is G_{CF} = 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. An actual review, treating the `historical' observations and various interpretations (starting with Brewster, 1818) as well as the modern aspects of optical anomalies, is presented by Kahr & McBride (1992). A similar, very comprehensive review is contained in the monograph of Shtukenberg et al. (2007).
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; 1926, 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 welldefined laws.
These laws, as formulated by Friedel, are specified in his book (Friedel, 1926). His `lattice theory of twinning' is discussed in Sections 3.3.8 and 3.3.9 of the present chapter.
Paul Niggli (1919, 1920/1924/1941): 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 (`twinplane law'), or they appear rotated by 180° around a (common) direction (`zoneaxis 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, 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.
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).

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 nfold twin rotations (nfold twin axes), i.e. operations of order .
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):

All these binary twin elements – no matter whether rational or irrational – lead to crystallographic orientation relations, as defined in Section 3.3.2.2, because the following lattice items belong to both twin partners:

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) and (vi): only a onedimensional lattice row is `common', two lattice planes are `parallel' and all twin elements are irrational (Fig. 3.3.2.3). The `maximal' condition, a `common' threedimensional lattice, occurs for inversion twins (`merohedral' or `parallellattice twins'), case (vii); for displacement vector t = 0, the threedimensional 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) + (ii), (iii) + (iv) and (v) + (vi) represent the same orientation relation. Case (vii) 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 ) with twin mirror plane (100) can be described by the four alternative twin elements (i), (ii), (iii), (iv) (cf. Section 3.3.4, Fig. 3.3.4.1). Furthermore, with increasing symmetry, the twin elements (i) and (iii) may become even more special, and the nature of the twin type may change as follows:
In both cases, the threedimensional lattice (or a sublattice of it) is now common to both twin partners, i.e. a `merohedral' twin results (cf. Section 3.3.9).
There is one more binary twin type which seems to reduce even further the abovementioned `minimal' condition for a crystallographic orientation relation, the socalled `median law' (German: Mediangesetz) of Brögger (1890), described by Tschermak & Becke (1915, 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 3.3.6.11.
There is a longlasting controversy in the literature, e.g. Hartman (1956, 1960), Buerger (1960b), Curien (1960), 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 highersymmetry lattice. A well known example is the triple twin (German: Drilling) of orthorhombic aragonite, where the rotation angle (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. 3.3.2.4). Another case of n = 3 with a very small metrical deviation is provided by ammonium lithium sulfate (γ = 119.6°).

(a) Triple growth twin of orthorhombic aragonite, CaCO_{3}, with pseudothreefold twin axis. The gap angle is 11.4° (= 360° − 3 × 116.2°). The exact description of the twin aggregate by means of two symmetryequivalent twin mirror planes (110) and () is indicated. In actual crystals, the gap is usually closed as shown in (b). 
All these (pseudo) nfold 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. 3.3.2.4. This possibility of describing cyclic twins by `exact' binary twin operations is the reason why Hartman (1956, 1960) and Curien (1960) do not consider `nonexact' three, four and sixfold rotations as proper twin operations.
The crystals forming twins with pseudo nfold rotation axes always exhibit metrical pseudosymmetries. In the case of transformation twins and domain structures, the metrical pseudosymmetries of the lowsymmetry (deformed) phase result from the true structural symmetry of the parent phase (cf. Section 3.3.7.2). This aspect caused several authors [e.g. Friedel, 1926, pp. 435 and 464; Donnay (cf. Hurst et al., 1956); Buerger, 1960b] 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 and in Section 3.4.3 . For a further extension of this concept to fivefold and tenfold multiple growth twins, see Note (6) below and Example 3.3.6.9.

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. 3.3.3.1(b). Thus, only one orientation relation (one twin law) exists. Contact twins and polysynthetic twins (see Section 3.3.3.1 below) are simple twins, as are the spinel penetration twins.

Schematic illustration of simple (polysynthetic) and multiple (cyclic) twins. (a) Equivalent twin mirror planes (110) and of an orthorhombic crystal. (b) Simple (polysynthetic) twin with two orientation states due to parallel repetition of the same twin mirror plane ; 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:

The distinction of simple and multiple twins is important for the following morphological classification. Further examples are given in Section 3.3.6.
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 growthtwinned crystals is the occurrence of reentrant 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 3.3.7.1). Detailed studies of the morphology of twins versus untwinned crystals were carried out as early as 1911 by Becke (1911). As a general observation, twinned crystals grow larger than untwinned crystals in the same batch.
The following classification of twins is in use:

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.
In this section we turn our attention to the symmetry relations in twinning. The starting point of all symmetry considerations is the eigensymmetry 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 . The term eigensymmetry is introduced here in order to provide a short and crisp distinction between the symmetry of the untwinned crystal (singledomain state) and the composite symmetry 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, pp. 764 and 791).
For a comprehensive characterization of the symmetry of a twinned crystal, we introduce the important concept of composite symmetry . This symmetry is defined as the extension of the eigensymmetry group by a twin operation k. This extension involves, by means of left (or right) coset composition , the generation of further twin operations until a supergroup is obtained. This supergroup is the composite symmetry group .
In the language of group theory, the relation between the composite symmetry group and the eigensymmetry group can be expressed by a (left) coset decomposition of the supergroup with respect to the subgroup : where is the identity operation; .
The number i of cosets, including the subgroup , is the index [i] of in ; this index corresponds to the number of different orientation states in the twinned crystal. If is a normal subgroup of , which is always the case if i = 2, then , 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 in , i.e. , was first expressed by Zheludev & Shuvalov (1956, p. 540) for ferroelectric phase transitions.
These grouptheoretical considerations can be translated into the language of twinning as follows: although the eigensymmetry and the composite symmetry 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) in Section 3.3.2.4, as well as Section 3.3.10.4]. With this restriction, the number of twin operations in each coset equals the order of the eigensymmetry point group . 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. Many examples are given in Section 3.3.6.
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. reentrant 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, pp. 23–25); Tschermak & Becke (1915, p. 97); Hurst et al. (1956, p. 150); Raaz & Tertsch (1958, p. 119); Takano & Sakurai (1971); Takano (1972); Van Tendeloo & Amelinckx (1974); Donnay & Donnay (1983); Zikmund (1984); Wadhawan (1997, 2000); Nespolo et al. (2000). A systematic application of left and double coset decomposition to twinning and domain structures has been presented by Janovec (1972, 1976) in a key theoretical paper. An extensive grouptheoretical treatment with practical examples is provided by Flack (1987).
Example: dovetail twin of gypsum (Fig. 3.3.4.1). Eigensymmetry:Twin reflection plane (100):Composite symmetry group (orthorhombic):given in orthorhombic axes, . The coset contains all four alternative twin operations (Table 3.3.4.1) and, hence, represents the twin law. This is clearly visible in Fig. 3.3.6.3(a). In the symbol of the orthorhombic composite group, the primed operations indicate the coset of alternative twin operations. The above blackandwhite symmetry symbol of the (orthorhombic) composite group is another expression of the twin law. Its notation is explained in Section 3.3.5. The twinning of gypsum is treated in more detail in Example 3.3.6.3.


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 . The subscripts x and z of the twin operations refer to the coordinate system of the orthorhombic composite symmetry of this twin; a and c are the monoclinic coordinate axes. 
It should be noted that among the four twin operations of the coset two are rational, and , and two are irrational, and (Fig. 3.3.4.1). 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 , as the representative twin element.
The concept of composite symmetry is not only a theoretical tool for the extension of the twin law but has also practical aspects:

In the example of the dovetail twin of gypsum above, the twin operation is of a special nature in that it maps the entire eigensymmetry onto itself and, hence, generates a single coset, a single twin law and a finite composite group 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 3.3.6.3. In this case, extension of the complete group 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 which are independent (nonalternative) but symmetry equivalent with respect to the eigensymmetry , 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, . 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 which is left invariant by the twin operation k. This subgroup is the `intersection symmetry' of the two `oriented eigensymmetries' and of the domains 1 and 2 (shown in Fig. 3.3.4.2): . This group is now extended by k and leads to the `reduced composite symmetry' of the domain pair (1, 2): , which is a finite supergroup of of index [2]. In this way, the complete coset of the eigensymmetry is split into two (or more) smaller cosets , etc., where are symmetryequivalent twin operations in . Correspondingly, the differently oriented `reduced composite symmetries' , etc. of the domain pairs (1, 2), (1, 3) etc. are generated by the representative twin operations , etc. These cosets are considered as the twin laws for the corresponding domain pairs.

Twinning of an orthorhombic crystal (eigensymmetry 2/m 2/m 2/m) with equivalent twin mirror planes (110) and . Three twin domains 1, 2 and 3, bound by {110} contact planes, are shown. The oriented eigensymmetries , , and the reduced composite symmetries and of each domain pair are given in stereographic projection. The intersection symmetry of all domains is . 
As an example, an orthorhombic crystal of eigensymmetry with equivalent twin reflection planes and is shown in Fig. 3.3.4.2. From the `starting' domain 1, the two domains 2 and 3 are generated by the two twin mirror planes and , symmetry equivalent with respect to the oriented eigensymmetry of domain 1. The intersection symmetries of the two pairs of oriented eigensymmetries & and & are identical: . The three oriented eigensymmetries , , , as well as the two differently oriented reduced composite symmetries and of the domain pairs (1, 2) and (1, 3), are all isomorphic of type , but exhibit different orientations.
The discussions and examples briefly presented in the previous section are now extended in a more general way. For the classification of composite symmetries we introduce the notion of oriented eigensymmetry 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 but exhibit different oriented eigensymmetries , 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 spacegroup diagram (cf. Hahn, 2005, Parts 6 , 7 and 10 ).
Similarly, we also consider the intersection group of the oriented eigensymmetries and and its geometric representation, the framework of intersection symmetry. Two cases of intersection symmetries have to be distinguished:

The orthorhombic example given in Section 3.3.4.1 (Fig. 3.3.4.2) is now extended as follows:
Eigensymmetry , intersection symmetry , identity, , , . The two cosets and are listed in Table 3.3.4.2. From these cosets the two different reduced composite symmetries and of type 2/m 2/m 2/m are derived as follows:

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 ( or ) were applied to the full eigensymmetry (twice as long as ): . 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 and , can be expressed by a black–white symmetry symbol of type with . The frameworks of these two groups, however, are differently oriented (cf. Fig. 3.3.4.2).
In the limiting case, the intersection group consists of the identity alone (index [i] = order 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 of the eigensymmetry group, and each coset consists of one twin operation only.
After this preparatory introduction, the three categories of composite symmetry are treated.

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 higherorder rotation axes that is accompanied by a small metrical deformation of the lattice. As a result, the extended composite symmetries of the transformation twins resemble the symmetry of their parent phase. The occurrence of both multiple growth and multiple transformation twins of orthorhombic pseudohexagonal K_{2}SO_{4} is described in Example 3.3.6.8.
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 nfold rotation or rotoinversion (). The resulting coset again contains the alternative twin operations, but in this case only for the orientation relation , and not for (`nontransposable' 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 of supergroup index with respect to the eigensymmetry . This case can occur only for the following pairs: , , , , , , , , , (monoaxial point groups), as well as for the two cubic pairs , . For the pairs , , , and the two cubic pairs , , the relations are of index [3] and imply three nontransposable domain states. For the pairs , , , as well as and , four or six different domain states occur. Among them, however, domain pairs related by the second powers of 4 and as well as by the third powers of 6 and operations are transposable, because these twin operations correspond to twofold rotations or, for , to m.
No growth twins of this type are known so far. As a transformation twin, langbeinite () is the only known example.
An alternative description of twinning employs the symbolism of colour symmetry. This method was introduced by Curien & Le Corre (1958) and by Curien & Donnay (1959). 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 `antisymmetry' groups) and multiple twins with more than two colours (i.e. `polychromatic' symmetry groups) have to be considered. Two kinds of operations are distinguished:

For simple twins, all colourchanging (twin) operations are binary, hence the two domain states are transposable. The composite symmetry of these twins thus can be described by a `blackandwhite' symmetry group. The coset, which defines the twin law, contains only colourchanging (primed) operations. This notation has been used already in previous sections.
It should be noted that symbols such as and , despite appearance to the contrary, represent binary blackandwhite operations, because contains , and contains 3 and , with being the twin operation. For this reason, these symbols are written here as and , whereby the unprimed symbol in parentheses refers to the eigensymmetry part of the twin axis. In contrast, would designate a (polychromatic) twin axis which relates three domain states (three colours), each of eigensymmetry 2. Twin centres of symmetry 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), the black–white symbols were only used for twinning by merohedry. In the present chapter, the symbols are also applied to nonmerohedral twins, as is customary for (ferroelastic) domain structures. This has the consequence, however, that the eigensymmetries or and the composite symmetries or may belong to different crystal systems and, thus, are referred to different coordinate systems, as shown for the composite symmetry of gypsum in Section 3.3.4.1.
For the treatment of multiple twins, `polychromatic' composite groups are required. These contain colourchanging 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 (). For this reason, the symbols for the composite symmetry of multiple twins are written without primes; see the examples in Section 3.3.4.4(iii). An extension of the dichromatic twin descriptions to polychromatic symbols for multiple twins was recently presented by Nespolo (2004).
In order to illustrate the foregoing rather abstract deliberations, an extensive set of examples of twins occurring either in nature or in the laboratory is presented below. In each case, the twin law is described in two ways: by the coset of alternative twin operations and by the black–white symmetry symbol of the composite symmetry , as described in Sections 3.3.4 and 3.3.5.
For the description of a twin, the conventional crystallographic coordinate system of the crystal and its eigensymmetry group are used in general; exceptions are specifically stated. To indicate the orientation of the twin elements (both rational and irrational) and the composition planes, no specific convention has been adopted; rather a variety of intuitively understandable simple symbols are chosen for each particular case, with the additional remark `rational' or `irrational' where necessary. Thus, for twin reflection planes and (planar) twin boundaries symbols such as , , or are used, whereas twin rotation axes are designated by , , , , , , etc.
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 Xray diffraction topography. Diffraction effects of twins are treated in Section 3.3.11.

The (polar) 180° twin domains in a (potentially ferroelectric) crystal of eigensymmetry () and composite symmetry (e.g. in KTiOPO_{4}, NH_{4}LiSO_{4}, lithium formate monohydrate) result from a group–subgroup relation of index with invariance of the symmetry framework (merohedral twins), but antiparallel orientation of the polar axes. The orientation relation between the two domain states is described by the coset of twin operations shown in Table 3.3.6.2, whereby the reflection in (001), , is considered as the `representative' twin operation.

Hence, these twins can be regarded not only as reflection, but also as rotation or inversion twins. The composite symmetry, in black–white symmetry notation, is whereby the primed symbols designate the (alternative) twin operations (cf. Section 3.3.5).
The dovetail twin of gypsum [eigensymmetry , with twin reflection plane ], coset of twin operations and composite symmetry , was treated in Section 3.3.4. Gypsum exhibits an independent additional kind of growth twinning, the Montmartre twin with twin reflection plane . These two twin laws are depicted in Fig. 3.3.6.3. Many colour photographs are presented by Boulliard (2010), pp. 266–271. The two cosets of twin operations in Table 3.3.6.3 and the symbols of the composite symmetries and of both twins are referred, in addition to the monoclinic crystal axes, also to orthorhombic axes for dovetail twins and for Montmartre twins. This procedure brings out for each case the perpendicularity of the rational and irrational twin elements, clearly visible in Fig. 3.3.6.3, as follows:


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 is invariant under all twin operations; hence, the composite symmetries and are crystallographic of type (supergroup index [2]) but differently oriented, as shown in Fig. 3.3.6.3. There is no physical reality behind the orthorhombic symmetry of the two groups: gypsum is neither structurally nor metrically pseudoorthorhombic, the monoclinic angle being 128°. The two groups and their orthorhombic symbols, however, clearly reveal the two different twin symmetries and, for each case, the perpendicular orientations of the four twin elements, two rational and two irrational. The two twin types originate from independent nucleation in aqueous solutions.
It should be noted that for all (potential) twin reflection planes in the zone [010] (monoclinic axis), the oriented eigensymmetry would be the same for all domain states, i.e. the intersection symmetry is identical with the oriented eigensymmetry and, thus, the composite symmetry would be always crystallographic.
For a more general twin reflection plane not belonging to the zone , such as , however, the oriented eigensymmetry would not be invariant under the twin operation. Consequently, an additional twin reflection plane , equivalent with respect to the eigensymmetry , exists. This (hypothetical) twin would belong to category (ii) in Section 3.3.4.4 and would formally lead to a noncrystallographic composite symmetry of infinite order. If, however, we restrict our considerations to the intersection symmetry of a domain pair, the reduced composite symmetry with and (irrational) would result. Note that for these (hypothetical) twins the reduced composite symmetry and the eigensymmetry are isomorphic groups, but that their orientations are quite different.
Remark. In the domainstructure approach, presented by Janovec & Přívratská in Chapter 3.4 of this volume, both gypsum twins, dovetail and Montmartre, can be derived together as a result of a single (hypothetical) ferroelastic phase transition from a (nonexistent) orthorhombic parent phase of symmetry to a monoclinic daughter phase of symmetry , with a very strong metrical distortion of 38° from to (Janovec, 2003). In this (hypothetical) transition the two mirror planes, (100) and (001), 90° apart in the orthorhombic form, become twin reflection planes of monoclinic gypsum, (100) for the dovetail, (001) for the Montmartre twin law, with an angle of 128°. It must be realized, however, that neither the orthorhombic parent phase nor the ferroelastic phase transition are real.
Quartz is a mineral which is particularly rich in twinning. It has the noncentrosymmetric trigonal point group 32 with three polar twofold axes and a nonpolar trigonal axis. The crystals exhibit enantiomorphism (right and lefthanded quartz), piezoelectricity and optical activity. The lattice of quartz is hexagonal with holohedral (lattice) point group . Many types of twin laws have been found (cf. Frondel, 1962), but only the four most important ones are discussed here:
The first three types are merohedral (parallellattice) Σ1 twins and their composite symmetries belong to category (i) in Section 3.3.4.2, whereas the nonmerohedral Japanese twins (twins with inclined lattices or inclined axes) belong to category (ii).
This twinning is commonly described by a twofold twin rotation around the threefold symmetry axis [001]. The two orientation states are of equal handedness but their polar twofold axes are reversed (`electrical twins'). Dauphiné twins can be transformation or growth or mechanical (ferrobielastic) twins. The composite symmetry is , the point group of hightemperature quartz (quartz). The coset decomposition of with respect to the eigensymmetry (index [2]) contains the operations listed in Table 3.3.6.4.

The left coset constitutes the twin law. Note that this coset contains four twofold rotations of which the first one, , is the standard description of Dauphiné twinning. In addition, the coset contains two sixfold rotations, and . The black–white symmetry symbol of the composite symmetry is (supergroup of index [2] of the eigensymmetry group ).
This coset decomposition was first listed and applied to quartz by Janovec (1972, p. 993).
This twinning is commonly described by a twin reflection across a plane normal to a twofold symmetry axis. The two orientation states are of opposite handedness (i.e. the sense of the optical activity is reversed: optical twins) and the polar axes are reversed as well. The coset representing the twin law consists of the following six operations:
The coset shows that Brazil twins can equally well be described as reflection or inversion twins. The composite symmetry is a supergroup of index [2] of the eigensymmetry group 321.
Twins of this type can be described by a twin reflection across the plane (0001), normal to the threefold axis [001]. The two orientation states of this twin are of opposite handedness (i.e. the optical activity is reversed, optical twin), but the polar axes are not reversed. The coset representing the twin law consists of the following six operations:
The composite symmetry is again a supergroup of index [2] of the eigensymmetry group 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 followed by the Brazil twin reflection or, alternatively, by the inversion . The product results in a particularly simple description of the combined law as a reflection twin on .
Twin domains of the Leydolt type are very rarely intergrown in direct contact, i.e. with a common boundary. If, however, a quartz crystal contains inserts of Dauphiné and Brazil twins, the domains of these two types, even though not in contact, are related by the Leydolt law. In this sense, Leydolt twinning is rather common in lowtemperature quartz. In contrast, GaPO_{4}, a quartz homeotype with the berlinite structure, frequently contains Leydolt twin domains in direct contact, i.e. with a common boundary (Engel et al., 1989).
In conclusion, the three merohedral twin laws of quartz described above imply four domain states with different orientations of important physical properties. These relations are shown in Fig. 3.3.6.4 for electrical polarity, optical activity and the orientation of etch pits on (0001). It is noteworthy that these three twin laws are the only possible merohedral twins of quartz, and that all three are realized in nature. Combined, they lead to the composite symmetry (`complete twin': Curien & Donnay, 1959).

Distinction of the four different domain states generated by the three merohedral twin laws of lowquartz and of quartz homeotypes such as GaPO_{4} (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 (): Dauphiné law; B (): Brazil law; C (): Leydolt law]. For GaPO_{4}, see Engel et al. (1989). 
In the three twin laws (cosets) above, only odd powers of 6, and (rotations and rotoinversions) occur as twin operations, whereas the even powers are part of the eigensymmetry 32. Consequently, repetition of any oddpower twin operation restores the original orientation state, i.e. each of these operations has the nature of a `binary' twin operation and leads to a pair of transposable orientation states.
Among the quartz twins with `inclined axes' (`inclined lattices'), the Japanese twins are the most frequent and important ones. They are contact twins of two individuals with composition plane . This results in an angle of between the two threefold axes. One pair of prism faces is parallel (coplanar) in both partners.
There exist four orientation relations, depending on
These four variants are illustrated in Fig. 3.3.6.5 and listed in Table 3.3.6.5. The twin interface for all four twin laws is the same, , but only in type III do twin mirror plane and composition plane coincide.
^{†}The line is the edge between the faces and and is parallel to the composition plane . It is parallel or normal to the four twin elements. Transformation formulae between the threeindex and the fourindex direction symbols, UVW and uvtw, are given by Barrett & Massalski (1966, p. 13).


The four variants of Japanese twins of quartz (after Frondel, 1962; cf. Heide, 1928). 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, lefthanded 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 for the two rhombohedral faces r and z, and is parallel to the twin and composition plane () and the twin axis in variant II. 
In all four types of Japanese twins, the intersection symmetry (reduced eigensymmetry) of a pair of twin partners is 1. Consequently, the twin laws (cosets) consist of only one twin operation and the reduced composite symmetry is a group of order 2, represented by the twin element listed in Table 3.3.6.5. If one were to use the full eigensymmetry , the infinite sphere group would result as composite symmetry .
Many further quartz twins with inclined axes are described by Frondel (1962). A detailed study of these inclinedaxis twins in terms of coincidencesite lattices (CSLs) is provided by McLaren (1986) and Grimmer (2006).
Upon heating quartz into the hexagonal hightemperature phase (point group 622) above 846 K, the Dauphiné twinning disappears, because the composite symmetry of the twinned lowtemperature phase now becomes the eigensymmetry of the hightemperature phase. For Brazil twins, however, their nature as reflection or inversion twins is preserved during the transformation.
The eigensymmetry of hightemperature quartz is 622 (order 12). Hence, the coset of the Brazil twin law contains 12 twin operations, as follows:
The composite symmetry is a supergroup of index [2] of the eigensymmetry 622.
In hightemperature quartz, the combined Dauphiné–Brazil twins (Leydolt twins) are identical with Brazil twins, because the Dauphiné twin operation has become part of the eigensymmetry 622. Accordingly, both kinds of twins of lowtemperature quartz merge into one upon heating above 846 K. We recommend that these twins are called `Brazil twins', independent of their type of twinning in the lowtemperature phase. Upon cooling below 846 K, transformation Dauphiné twin domains may appear in both Brazil growth domains, leading to the four domain states shown in Fig. 3.3.6.4. Among these four orientation states, two Leydolt pairs occur. Such Leydolt domains, however, are not necessarily in contact (cf. Example 3.3.6.4.3 above).
In addition to these twins with `parallel axes' (merohedral twins), several kinds of growth twins with `inclined axes' occur in hightemperature quartz. They are not treated here, but additional information is provided by Frondel (1962).
In some rhombohedral crystals such as corundum Al_{2}O_{3} (Wallace & White, 1967), calcite CaCO_{3} or FeBO_{3} (calcite structure) (Kotrbova et al., 1985; Klapper, 1987), growth twinning with a `twofold twin rotation around the threefold symmetry axis [001]' (similar to the Dauphiné twins in lowtemperature quartz described above) is common. Owing to the eigensymmetry (order 12), the following 12 twin operations form the coset (twin law). They are described here in hexagonal axes:
Some of these twin elements are shown in Fig. 3.3.6.6. They include the particularly conspicuous twin reflection plane perpendicular to the threefold axis [001]. The composite symmetry is

Twin intergrowth of `obverse' and `reverse' rhombohedra of rhombohedral FeBO_{3}. (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 FeBO_{3}. (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 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 FeBO_{3} crystals this twin law always, without exception, forms penetration twins (Fig. 3.3.6.6), whereas for the isotypic calcite CaCO_{3} only (0001) contact twins are found (Fig. 3.3.6.7). This aspect is discussed further in Section 3.3.8.6. Colour photographs of rhombohedral twins, especially calcite, are provided by Boulliard (2010), pp. 226–238.
The twinning of rhombohedral crystals described above also occurs for cubic crystals as the spinel law (spinel, CaF_{2}, PbS, diamond, sphaleritetype structures such as ZnS, GaAs, CdTe, cubic face and bodycentred metals). In principle, all four threefold axes of the cube, which are equivalent with respect to the eigensymmetry , can be active in twinning. We restrict our considerations to the case where only one threefold axis, [111], is involved. The most obvious twin operations are the twofold rotation around [111] or the reflection across (111). For centrosymmetric crystals, they are alternative twin operations and belong to the same twin law. For noncentrosymmetric crystals, however, the two operations represent different twin laws. Both cases are covered by the term `spinel law'.
The orientation relation defined by the spinel law corresponds to the `obverse' and `reverse' positions of two rhombohedra (cubes), as shown in Fig. 3.3.6.8. For the two (differently) oriented eigensymmetries of the domain states and , the intersection symmetry (order 12) results. With this `reduced eigensymmetry' , the coset of 12 alternative twin operations is the same as the one derived for twinning of rhombohedral crystals in Example 3.3.6.6.

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 and three 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) , , , , in detail. The intersection groups are , , , and , respectively. For these `reduced eigensymmetries', the cosets of the alternative twin operations are listed below with reference to cubic axes.

The restriction to only one of the four spinel twin axes combined with the application of the coset expansion to the reduced eigensymmetry always leads to a crystallographic composite symmetry . The supergroup generated from the full eigensymmetry, however, would automatically include the other three spinel twin axes and thus would lead to the infinite sphere group , i.e. would imply infinitely many cosets and (equivalent) twin laws. Higherorder spinel twins are discussed in Section 3.3.8.3. Further details can be found in Klapper & Hahn (2012) Sections 3 and 4, and in Section 3.3.11.4 below.
K_{2}SO_{4} has an orthorhombic pseudohexagonal roomtemperature phase with point group and axial ratio , and a hexagonal hightemperature phase ( K) with supergroup . It develops pseudohexagonal growthsector twins with equivalent twin reflection planes and which are also composition planes, as shown in Fig. 3.3.6.9. As discussed in Sections 3.3.2.3.2 and 3.3.4.4 under (iii), this corresponds to a pseudothreefold twin axis which, in combination with the twofold eigensymmetry axis, is also a pseudohexagonal twin axis. The extended composite symmetry is

Pseudohexagonal growth twin of K_{2}SO_{4} showing six sector domains in three orientation states. (001) plate, about 1 mm thick and 5 mm in diameter, between polarizers deviating by 45° from crossed position for optimal contrast of all domains. The crystal was precipitated from aqueous K_{2}SO_{4} solution containing 5% S_{2}O_{3} ions. Courtesy of M. Moret, Milano. 
Upon heating above 853 K, the growthsector twinning disappears. On cooling back into the lowtemperature phase, transformation twinning (`domain structure') with three systems of lamellar domains appears. The three orientation states are identical for growth and transformation twins, but the morphology of the twins is not: sectors versus lamellae. The composite symmetry of the twins at room temperature is the true structural symmetry of the `parent' phase at high temperatures.
As was pointed out in Note (6) of Section 3.3.2.4 and in part (iii) of Section 3.3.4.4, there exist twin axes with noncrystallographic multiplicities etc. Twins with five or tenfold rotations are frequent in intermetallic compounds. As an example, FeAl_{4} is treated here (Ellner & Burkhardt, 1993; Ellner, 1995). This compound is orthorhombic, , with an axial ratio close to , corresponding to a pseudofivefold axis along and equivalent twin mirror planes and , which are about 36° apart. In an ideal intergrowth, this leads to a cyclic pseudopentagonal or pseudodecagonal sector twin (Fig. 3.3.6.10). All features of this twinning are analogous to those of pseudohexagonal aragonite, treated in Section 3.3.2.3.2, and of K_{2}SO_{4}, described above as Example 3.3.6.8.

Pentagonal–decagonal twins. (a) Decagonal twins in the shape of tenfold stars on the surface of a bulk alloy, formed during the solidification of a melt of composition Ru_{8}Ni_{15}Al_{77}. Scanning electron microscopy picture. Typical diameter of stars ca. 200 µm. The arms of the stars show parallel intergrowth. (b) Pentagonal twin aggregate of Fe_{4}Al_{13} with morphology as grown in the orthorhombic hightemperature phase, showing several typical 72° angles between neighbouring twin partners (diameter of aggregate ca. 200 µm). Orthorhombic lattice parameters , , Å, space group . The parameters c and a approximate the relation ; the pseudopentagonal twin axis is [010]. On cooling, the monoclinic lowtemperature phase is obtained. The twin reflection planes in the orthorhombic unit cell are (101) and , in the monoclinic unit cell (100) and ; cf. Ellner & Burkhardt (1993, Fig. 10) and Ellner (1995). Both figures courtesy of M. Ellner, Stuttgart. 
The intersection symmetry of all twin partners is ; the reduced composite symmetry of a domain pair in contact is . The extended composite symmetry of the ideal pentagonal sector twin is .
Rutile with eigensymmetry develops growth twins with coinciding twin reflection and composition plane {011}. Owing to its axial ratio , the tetragonal c axes of the two twin partners form an angle of 114.4°. The intersection symmetry of the two domains is along the common direction [100]. The reduced composite symmetry of the domain pair is , with the primed twin elements parallel and normal to the plane (011). A twin of this type, consisting of two domains, is called an `elbow twin' or a `knee twin', and is shown in Fig. 3.3.6.11(a).

Various forms of rutile (TiO_{2}) 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 . (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) (heartshaped 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, p. 512. Parts (g) and (h) courtesy of S. HertingAgthe (1999), Mineralogical Collections, Technical University Berlin. Photograph (i) courtesy of M. Medenbach, Bochum. 
In point group , there exist four equivalent twin reflection planes {011} (four different twin laws) with angles of 65.6° between and and 45° between and , leading to a variety of multiple twins. They may be linear polysynthetic or multiple elbow twins, or any combination thereof (Fig. 3.3.6.11). Very rare are complete cyclic sixfold twins with a large angular excess of (corresponding formally to a `5.5fold' twin axis) and extended composite pseudosymmetry , or cyclic eightfold twins with a nearly exact fit of the sectors and a morphological pseudo twin axis. In the `sixling', the tetragonal axes of the twin components are coplanar, whereas in the `eightling' they alternate `up and down', exhibiting in ideal development the morphological symmetry of the twin aggregate. The extended composite symmetry is with eight twin components, each of different orientation state. These cyclic twins are depicted in Figs. 3.3.6.11(d), (e), (f) and (i).
The sketch of the 'eightling' in Fig. 3.3.6.11(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 3.3.7.1.1). Fig. 3.3.6.11(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. 3.3.6.11(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 halfrings, 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, p. 381), and Fig. 10.12 in Berry & Mason (1959, 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. 3.3.6.11g) 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 pseudohexagonal triangular plane grid of intergrown prismatic [001] rods usually twinned on (011), Fig. 3.3.6.11(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 (HertingAgthe, 2009) 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; Force et al., 1996). 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 3.3.10.6.2).
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 pseudohexagonal, and thus epitactic growth with orientations and (`structural coherence': Armbruster, 1981; Force et al., 1996) 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.
Gibbsite (older name: hydrargillite) forms a pronounced layer structure with a perfect cleavage plane . It is monoclinic with eigensymmetry , but strongly pseudohexagonal in (001) with an axial ratio . In contrast to most other pseudohexagonal crystals, the twofold eigensymmetry axis b is not parallel but normal to the pseudohexagonal c axis. The normal to the cleavage plane is inclined by against [001]. Owing to the pseudohexagonal metrics of the plane , the lattice planes and , equivalent with respect to the eigensymmetry , form an angle of 60.8°.
The following four significant twin laws have been observed by Brögger (1890):

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

Both twin laws resemble closely the monoclinic pseudosymmetry in two slightly different but distinct fashions: each twin law uses one rational twin element from , the other one is irrational. The two frameworks of twin symmetry are inclined with respect to each other by about , corresponding to the angle between b (direct lattice) and (reciprocal lattice).
Both twins occur as growth and transformation twins: they appear together in the characteristic lamellar `transformation microclines'. Details of feldspar twins are given in Smith (1974).
The mineral staurolite, which has the approximate formula Fe_{2}Al_{9}[O_{6}(O,OH)_{2}/(SiO_{4})_{4}], has `remained an enigma' (Smith, 1968) to date with respect to the subtle details of symmetry, twinning, structure and chemical composition. A lively account of these problems is provided by Donnay & Donnay (1983). Staurolite is strongly pseudoorthorhombic, , and only detailed optical, morphological and Xray experiments reveal monoclinic symmetry, , with , , Å and within experimental errors (Hurst et al., 1956; Smith, 1968; and especially Hawthorne et al., 1993).
Staurolite exhibits two quite different kinds of twins:

Notes

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

The beautiful polysynthetic twin structure of BaTiO_{3} is shown in the colour micrograph Fig. 3.4.1.1 in Chapter 3.4 of this volume.
Crystals of pentaerythrite (PE), C(CH_{2}OH)_{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 bipyramidal morphology (Ernst, 1928), which is not compatible with the point group (space group ) of the single crystal. The point group allows three merohedral twin laws:
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), all three twin elements generate the tetragonal bipyramid {111} (pseudocubic octahedron) in their composite symmetry, as is actually observed.
A detailed analysis of the growth twinning of PE has been reported by Renninger (1957). 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 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 axis. Twin laws (b) or (c) were identified by the intensity interchange of twinrelated 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 structurefactor moduli) on both sides of a (b) or (c) twin boundary. The Xray 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 3.3.6.1(iii)(a)].
This term is due to Henke (2003) and refers to the simultaneous occurrence (superposition) of two or more different twin types (twin laws) in one and the same crystal. In twins of twins, one `generation' of twin domains is superimposed upon the other, each with its own twin law. This may occur as a result of:
Typical examples are:

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

In Section 3.3.3, 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) 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).
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:
In many cases, twins are formed during the first stages of spontaneous nucleation, possibly before the subcritical nucleus reaches the critical size necessary for stable growth. This idea was originally proposed by Buerger (1945, 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 FeBO_{3} (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 3.3.6.6 and Fig. 3.3.6.6). Experimental details are presented by Klapper (1987) and Kotrbova et al. (1985). Further examples are the penetration twins of the spinel law (Example 3.3.6.7 and Fig. 3.3.6.8), the very interesting and complex [001] penetration twin of the monoclinic feldspar orthoclase (Fig. 3.3.7.1) and the sector twins of ammonium lithium sulfate with three orientation states (Fig. 3.3.7.2).

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

Photographs of (001) plates ( 20 mm diameter, 1 mm thick) of NH_{4}LiSO_{4} 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). 
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 NaNO_{3}, on the other hand, do not exhibit penetration twins at all. In contrast, for ammonium lithium sulfate, NH_{4}LiSO_{4}, both sectortwinned 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 3.3.8.6), are not yet clearly understood. A hypothesis advanced by Senechal (1980) 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 nucleationtwinned 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 3.3.6.10 (Figs. 3.3.6.11d 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 diamondtype 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 3.3.10.6.6. An extensive review is provided by Hofmeister (1998).

A solidtosolid (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) and of the `daughter phase' (deformed phase) exhibit frequently, but not always, a group–subgroup relation. During the transition to the lowsymmetry phase the crystal usually splits into different domains. Three cases of transformationtwin domains are distinguished:

Well known examples of ferroelastic transformation twins are K_{2}SO_{4} (Example 3.3.6.8) and various perovskites (Example 3.3.6.14). Characteristic for nonmerohedral (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 (nonferroelastic) transformation twins, e.g. Dauphiné twins of quartz, often are curved, irregular and nonparallel.
Transformation twins are closely related to the topic of Domain structures, which is extensively treated by Janovec & Přívratská in Chapter 3.4 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 highsymmetry `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 ABX_{3} (`perovskite tree', now often generalized as `Bärnighausen tree'; Bärnighausen, 1980) and A_{2}BX_{4} (Na_{2}SO_{4} and K_{2}SO_{4}type compounds). In Example (3) of Section 3.3.9.2.4, growth twins among MeX_{2} dichalcogenides are described in detail.
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 3.3.2.3). 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 reswitch 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, Section 3). A comprehensive treatment is presented in the monograph Mechanical Twinning of Crystals by KlassenNeklyudova (1964). A brief survey of mechanical twinning in metals is given by Barrett & Massalski (1966).
With respect to symmetry, three categories of mechanical twins are distinguished in this chapter:

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) above]. As a rule, the domain textures of a twinned crystal are quite different for growth twins, transformation twins and mechanical twins.
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 (threeperiodic twins). This was first done about a hundred years ago – based on the lattice analysis of Bravais – by Mallard (1879) and especially by Friedel (1904, 1926), in part before the advent of Xray diffraction. The book by Friedel (1926), particularly Chapter 15, is the most frequently cited reference in this field. Later, Friedel (1933) sharpened his theories to include two further types of twins: `macles monopériodiques' and `macles dipériodiques', in addition to the previous `macles tripériodiques', see Section 3.3.8.2 below. These concepts were further developed by Niggli (1919, 1920/1924/1941).^{5}
The lattice aspects of twinning (triperiodic twins) are discussed in this section and in Section 3.3.9. An important concept in this field is the coincidencesite sublattice of the twin in direct space and its counterpart in reciprocal space. Extensive use of the notion of coincidencesite lattices (CSLs) is made in bicrystallography for the study of grain boundaries, as briefly explained in Section 3.2.2 .
The coincidencesite lattice and further related lattices (O and DSClattices) were introduced into the study of bicrystals by Bollmann (1970, 1982) and were theoretically thoroughly developed by Grimmer (1989a,b, 2003). Their applications to grain boundaries are contained in the works by Sutton & Balluffi (1995) and Gottstein & Shvindlerman (1999).
The basis of Friedel's (1904, 1926) lattice theory of twinning is the postulate that the coincidencesite 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 `threedimensional lattice control' (Santoro, 1974, 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 coincidencesite 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) of Section 3.3.2.4 – does not apply here. The existence of the coincidence subset of a twin results from the crystallographic orientation relation (Section 3.3.2.2), which is a prerequisite for twinning. This subset is one, two or threedimensional (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 below.
Four types of (exact) lattice coincidences have to be distinguished in twinning:

After these general considerations of coincidencesite and twin lattices and their lattice index, specific cases of `triperiodic twins' are treated in Section 3.3.8.3. In addition to the characterization of the twin lattice by its index , the notation used in metallurgy is included.
The following cases of exact superposition are distinguished:

Examples

In part (iv) of Section 3.3.8.2, threedimensional lattice coincidences and twin lattices (sublattices) were considered under two restrictions:
In the present section these two conditions are relaxed as follows:

The concept of twin obliquity has been introduced by Friedel (1926, p. 436) to characterize (metrical) pseudosymmetries of lattices and their relation to twinning. The obliquity 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 3.3.8.1).
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 or [cf. Section 3.3.8.2(iv)]. The quantities and j can be calculated for any lattice and any (hkl)/[uvw] combination by elementary formulae, as given by Friedel (1926, pp. 249–252) and by Donnay & Donnay (1972). A computer program has been written by Le Page (1999, 2002) which calculates for a given lattice all (hkl)/[uvw]//j combinations up to given limits of 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 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 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, p. 437) quotes frequently observed twin obliquities of 3–4° (albite , aragonite ) with `rare exceptions' of 5–6°. In a paper devoted to the quartz twins with `inclined axes', Friedel (1923, pp. 84 and 86) accepts the La Gardette (Japanese) and the Esterel twins, both with large obliquities of and , as pseudomerohedral twins only because their lattice indices and 3 are (`en revanche') remarkably small. He considers as a limit of acceptance [`limite prohibitive'; Friedel (1923, p. 88)].
Lattice indices are very common (in cubic and rhombohedral crystals), twins are rare and seems to be the maximal value encountered in twinning (Friedel, 1926, pp. 449, 457–464; Donnay & Donnay, 1974, Table 1). In his quartz paper, Friedel (1923, p. 92) rejects all pseudomerohedral quartz twins with despite small values, and he points out, as proof that high j values are particularly unfavourable for twinning, that strictly merohedral quartz twins with do not occur, i.e. that cannot `compensate' for high j values.
In agreement with all these results and later experiences (e.g. Le Page, 1999, 2002), we consider in Table 3.3.8.2 only lattice pseudosymmetries with and , preferably . (It should be noted that, on purely mathematical grounds, arbitrarily small values can always be obtained for sufficiently large values of and , which would be meaningless for twinning.) The program by Le Page (1999, 2002) 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 3.3.8.2, some examples are presented that bring out both the merits and the problems of lattice geometry for the theory of twinning. The `permissibility criteria' and , mentioned above, are observed for most cases.

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 and [j] are listed in the table; similarly for the less common (001) Montmartre twin. In addition, two cases of lowindex (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 , 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 , 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 and (cf. Fig. 3.3.8.1).
Quartz: The various quartz twins with inclined axes were studied extensively by Friedel (1923). The two most frequent cases, the Japanese twin (called La Gardette twin by Friedel) and the 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 and does not occur [cf. Section 3.3.9.2.3, Example (2)].
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 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 3.3.9.2 under the aspect of `reticular pseudomerohedry'.
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 , etwin after Bueble & Schmahl (1999), cf. Section 3.3.10.2.2, 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 , rtwin, the situation is similar. The planes and permit small obliquities and lattice indices , but do not appear as twin planes.
The discussion of the examples in Table 3.3.8.2 shows that, with one exception [staurolite (231) twin], the obliquities and lattice indices of common twins fall within the limits accepted for lattice pseudosymmetry. Three aspects, however, have to be critically evaluated:

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, 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).
In the present section, the critical discussion of the lattice theory of twinning is extended from the individual crystal species, treated in Section 3.3.8.5, 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:

These examples corroborate the early observations of Cahn (1954, 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'.
In conclusion, the lattice theory of twinning, presented in this section, can be summarized as follows:

We now resume the discussion of Section 3.3.8 on threedimensional coincidence lattices and pseudocoincidence 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 pseudomerohedry (`macles par pseudomériédrie'), both for lattice index and , as introduced by Friedel (1926, p. 434). Often (strict) merohedral twins are called `parallellattice twins' or `twins with parallel axes'. Donnay & Donnay (1974) have introduced the terms twinning by twinlattice symmetry (TLS) for merohedral twinning and twinning by twinlattice quasisymmetry (TLQS) for pseudomerohedral twinning, but we shall use here the original terms introduced by Friedel.
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:

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

These twins are characterized by pseudomerohedry of point groups, Case (1) in Section 3.3.9.1. 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:

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) above. Typical examples with and were described in Section 3.3.8.3. In addition to the sublattice relations, it is reasonable to include the pointgroup relations as well. Four examples are presented:

This type can be derived from the category in Section 3.3.9.2.3 above by relaxation of the condition of exact lattice coincidence, resulting in two nearly, but not exactly, coinciding lattices (pseudocoincidence, cf. Section 3.3.8.4). In this sense, the two Sections 3.3.9.2.3 and 3.3.9.2.4 are analogous to the two Sections 3.3.9.2.1 and 3.3.9.2.2.
The following four examples are characteristic of this group:

In conclusion, it is pointed out that the above four categories of twins, described in Sections 3.3.9.2.1 to 3.3.9.2.4, refer only to cases with exact or approximate threedimensional lattice coincidence (triperiodic twins). Twins with only two or onedimensional 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 3.3.8.2. 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.
The large group of pseudomerohedral 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 pseudocoincidence of the lattices of the twin partners and thus to pseudomerohedral 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 3.3.9.2.2, Example (1). This topic is extensively treated in Chapter 3.4 on domain structures.
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 nonoccurrence 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, 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. 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). 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) for the reflection twins of monoclinic saccharin crystals. In this study, the 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 3.3.10.4.1).
Calculations of interface energies, as performed by Fleming et al. (1997) and Lieberman et al. (1998), 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, highresolution 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 lowenergy 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) for ferroelastic crystals. His treatment assumes a phase transition from a real or hypothetical parent phase (supergroup ) to a `distorted' (daughter) phase of lower eigensymmetry (subgroup ), leading to two (or more) domain states of equal but opposite shear strain. The subgroup must belong to a lowersymmetry crystal system than the supergroup , as explained in Section 3.3.7.3(ii). Similar criteria, but restricted to ferroelectric materials, had previously been devised in 1969 by Fousek & Janovec (1969). A review of ferroelastic domains and domain walls is provided by Boulesteix (1984) 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).
For a simple derivation of stressfree 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. 3.3.10.1(a) (cf. Liebisch, 1891; Niggli, 1941; KlassenNeklyudova, 1964). 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 and . The undeformed state of the body and the deformed states of its two parts are represented by a sphere () and by two ellipsoids and , as shown in Fig. 3.3.10.1(b). We now look for stressfree 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)(; are Cartesian coordinates) which has as solutions the two planes (plane BB in Fig. 3.3.10.1b) and (plane AA). These planes are called `planes of strain compatibility' or `permissible' planes. From the solutions of the above equation and Fig. 3.3.10.1(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.

(a) Classical description of mechanical twinning by homogeneous shear deformation (Liebisch, 1891, pp. 104–118; Niggli, 1941, pp. 145–149; KlassenNeklyudova, 1964, 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 . (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 and of two orientation states, referred to the (real or hypothetic) intermediate (prototypic) state with (sphere). The switching of orientation state into state through the shear angle is, analogous to (a), indicated by arrows. The shear ellipsoids and have common cross sections along the perpendicular planes AA and BB which are both, therefore, mechanically compatible contact planes of the and twin domains. 
It is noted that during a shear deformation induced by the (horizontal) translations shown in Fig. 3.3.10.1(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 and 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 and . Both contact planes often occur simultaneously in growth twins, see for example the dovetail and the Montmartre twins of gypsum (Fig. 3.3.6.3). 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' lowenergy boundaries. Other physical properties, such as electrical polarization, may reduce the number of mechanically permissible boundaries, e.g. due to energetically unfavourable headtohead or tailtotail orientation of the axis of spontaneous polarization in polar crystals. The mechanical compatibility criterion is, however, always applicable to centrosymmetric materials.
3.3.10.2.1. Sapriel approach to permissible (compatible) boundaries in ferroelastic (nonmerohedral) transformation twins
The general approach to strain compatibility, as given above, can be employed to derive the permissible composition planes for twins with inclined axes (nonmerohedral twins; for merohedral twins see Section 3.3.10.2.3 below). This concept was applied by Sapriel (1975) to the 94 Aizu species of ferroelastic transformation twins. According to Aizu (1969, 1970a,b), each species is represented by a pair of symmetry groups, separated by the letter F (= ferroic) in the form , e.g. or . The parent phase with symmetry represents the undeformed (zerostrain) reference state (the sphere in Fig. 3.3.10.1b), whereas the spontaneous strain of the two orientation states of phase 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).
Two kinds of permissible boundaries are distinguished by Sapriel:

Example. The distinction between these two types of boundaries is illustrated by the example of the (triclinic) Aizu species . 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 composition plane in the zone of the direction normal to triclinic (010), i.e. of the triclinic reciprocal 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) 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 plane, or of two planes. Examples are the Aizu species , and , respectively. There are even four Aizu cases without any permissible boundaries: , , , . An example is langbeinite (), which was discussed at the end of Section 3.3.4.4.
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 orientation have been interpreted as relaxation of the perpendicularity condition in the deformed phase, resulting from the ferroelastic phase transition (cf. Sapriel, 1975, p. 5138). This, however, is not the reason for the deviation from , but rather a splitting of the two (exactly) perpendicular symmetry planes in the parent phase into two pairs of compatible twin boundaries (i.e. two independent twin laws) in the deformed phase , 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.
Examples

The treatment by Sapriel (1975) was directed to (switchable) ferroelastics with a real structural phase transition from a parent phase to a deformed daughter phase . This procedure can be extended to those nonmerohedral twins that lack a (real or hypothetical) parent phase, in particular to growth twins as well as to mechanical twins in the traditional sense [cf. Section 3.3.7.3(i)]. Here, the missing supergroup formally has to be replaced by the `full' or `reduced' composite symmetry or of the twin, as defined in Section 3.3.4. Furthermore, we replace the spontaneous shear strain by one half of the imaginary shear deformation which would be required to transform the first orientation state into the second via a hypothetical intermediate (zerostrain) reference state. Note that this is a formal procedure only and does not occur in reality, except in mechanical twinning (cf. Section 3.3.7.3). With respect to this intermediate reference state, the two twin orientations possess equal but opposite `spontaneous' strain. With these definitions, the Sapriel treatment can be applied to nonmerohedral twins in general. This extension even permits the generalization of the Aizu notation of ferroelastic species to and (e.g. ), whereby now and represent the eigensymmetry and the intersection symmetry, and and the (possibly reduced) composite symmetry of the domain pair. With these modifications, the tables of Sapriel (1975) can be used to derive the permissible boundaries W and for general nonmerohedral twins.
It should be emphasized that this extension of the Sapriel treatment requires a modification of the definition of the W boundary as given above in Section 3.3.10.2.1: The (rational) symmetry operations of the parent phase, becoming F operations in the phase transformation, have to be replaced by the (growth) twin operations contained in the coset of the twin law. These twin operations now correspond to either rational or irrational twin elements. Consequently, the W boundaries defined by these twin elements can be either rational or irrational, whereas by Sapriel they are defined as rational. The Sapriel definition of the boundaries, on the other hand, is not modified: 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 or .
In many cases, the derivation of the permissible twin boundaries W can be simplified by application of the following rules:

In conclusion, the following differences in philosophy between the Sapriel approach in Section 3.3.10.2.1 and its extension in the present section are noted: Sapriel starts from the supergroup of the parent phase and determines all permissible domain walls at once by means of the symmetry reduction during the phase transition. This includes group–subgroup relations of index . The present extension to general twins takes the opposite direction: starting from the eigensymmetry of a twin component and the twin law , a symmetry increase to the composite symmetry of a twin domain pair is obtained. From this composite symmetry, which is always a supergroup of of index , 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 .
Examples

In merohedral twins (lattice index ), 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:

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.
In contrast to the mechanical compatibility of any composition plane in merohedral twins (lattice index ), twins of higher lattice index 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 nonmerohedral twins described in Section 3.3.10.2.2 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 3.3.10.2.2 is the fact that by this deformation only a subset of lattice points is `restored'. This subset forms the sublattice of index common to both domains (coincidencesite 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 deformation twins (spinel law) of cubic metals which are essential elements of the plasticity of these metals. During the deformation, the {100} cube (a rhombohedron) is switched from its `reverse' into its `obverse' orientation and vice versa, whereby the hexagonal P sublattice of index 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 nfold symmetry axes of order , i.e. by three, four and sixfold axes. In other words: twins of lattice index occur systematically in trigonal, hexagonal, tetragonal and cubic crystals. This may lead to trigonal, tetragonal and hexagonal intersection symmetries (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 3.3.10.1.
^{†}The existence of this deformation twin is still in doubt (cf. Seifert, 1928).
^{‡}The intersection symmetry and the permissible boundaries are referred to the coordinate system of the eigensymmetry; the reduced composite symmetries are based on their own conventional coordinate system derived from the intersection symmetry plus the twin law (cf. Section 3.3.4). 
For the cubic and rhombohedral 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), , , are indeed observed in the spineltype penetration twins, recognizable by their reentrant edges (Fig. 3.3.6.8). They also occur as twin glide planes of cubic metals. For the tetragonal twin, two pairs of perpendicular permissible W composition planes result, (120) & ( and (310) & (), one pair bisecting the other pair under 45°. For the cubic twin [galena PbS, cf. Section 3.3.8.3, example (4)], due to the low intersection symmetry, only one pair of permissible W boundaries results.
As mentioned before, the mechanical compatibility of twin boundaries is a necessary but not a sufficient criterion for the occurrence of stressfree lowenergy 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'.
Of particular significance are merohedral twins with polar domains of antiparallel spontaneous polarization (180° domains). The charge density at a boundary between two twin domains is given by where is the component of the polarization normal to the boundary. The interfaces with positive charge are called `headtohead' boundaries, those with negative charge `tailtotail' boundaries. Interfaces parallel to the polarization direction are uncharged () (Fig. 3.3.10.2).

Boundaries B–B between 180° domains (merohedral twins) of pyroelectric crystals. (a) Tailtotail boundary. (b) Headtohead boundary. (c) Uncharged boundary (). (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 and contain grownin 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 headtohead or tailtotail 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 headtohead and tailtotail character, even if they are uncharged due to charge compensation.
Examples

Charged and uncharged boundaries may also occur in nonmerohedral twins of pyroelectric crystals. In this case, the polar axes of the two twin domains 1 and 2 are not parallel. The charge density of the boundary is given by with and 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 . These crystals exhibit, besides the inversion twinning mentioned above, growthsector 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. 3.3.10.3, 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 reflectiontwin 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.

Charged and uncharged boundaries B–B of nonmerohedral twins of pseudohexagonal NH_{4}LiSO_{4}. Point group , spontaneous polarization P along twofold axis [010]. (a) Twin element mirror plane (110): electrically charged boundary of headtohead character. (b) Twin element twofold twin axis normal to plane (110): uncharged twin boundary (`headtotail' boundary). 
Finally, it is pointed out that electrical constraints of twin boundaries do not occur for nonpyroelectric 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 . Two twin laws, different due to the lack of the symmetry centre, are possible:
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).
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 crystalchemical 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.
As a first step of the structural elucidation of a reflectiontwin 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. 3.3.10.4 [see also Section 3.3.2.4, Note (8)].

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 () but also `twin glide planes' (, where 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 αquartz resembles the structure of βquartz; the ironcross (110) twin interface of pyrite, FeS_{2}, resembles the structure of marcasite, another polymorph of FeS_{2}.]
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 as twin reflection and composition plane (Lieberman et al., 1998). The authors calculated the boundary energy as a function of the lattice displacement vector t, which was varied within the mesh of the 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 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) growthtwin boundary of aragonite, suggested by Bragg (1924) (cf. Section 3.3.10.5 below). An even more instructive model is presented by Bragg (1937, pp. 246–248) and Bragg & Claringbull (1965, 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 was reported by Black (1955) for the (110) twin reflection boundary of the alloy Fe_{4}Al_{13}.
In their interesting theoretical study of the morphology and twinning of gypsum, Bartels & Follner (1989, 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' parallel to the twin reflection plane . [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 3.3.6.3.]
The occurrence of twin displacement vectors can be visualized by highresolution transmission electron microscopy (HRTEM) studies of twin boundaries. Fig. 3.3.10.5 shows an HRTEM micrograph of a (112) twin reflection boundary of anatase TiO_{2}, viewed edge on (arrows) along (Penn & Banfield, 1998). 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 . Twin displacement vectors have also been observed on HRTEM micrographs of sputtered Fe_{4}Al_{13} alloys by Tsuchimori et al. (1992).
Twin displacement vectors can occur in twin boundaries of both nonmerohedral (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 3.3.2.4, Note 7). In contrast to nonmerohedral twins, for merohedral twins these displacement vectors can be determined by imaging the twin boundaries by means of electron or Xray 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 Xray diffraction conditions are always simultaneously fulfilled for both partners. Thus, in transmission electron microscopy and Xray 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 (`stackingfault contrast', cf. Fig. 3.3.6.2).
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 , with f the fault vector of the boundary and the diffraction vector of the reflection used for imaging. For (merohedral) twin boundaries the total phase jump is composed of two parts, with the phase change due to the twin operation and the phase change resulting from the lattice displacement vector f. The boundary contrast is strongest if the phase jump is an odd integer multiple of , and it is zero if is an integer multiple of . By imaging the boundary in various reflections hkl and analysing the boundary contrast, taking into account the known phase change (calculated from the structurefactor phases of the reflections and 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, 1969) and into Xray topography by Lang (1967a,b) and McLaren & Phakey (1969).
In the equation given above, for each reflection hkl the total phase jump is independent of the origin of the unit cell. The individual quantities and , however, vary with the choice of the origin but are coupled in such a way that (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 and . Their structure factors are (assuming Friedel's rule to be valid)
The phase difference of the two structure factors is and depends on the choice of the origin. If the origin is chosen at the twin inversion centre (superscript ^{o}), the phase jump at the boundary is given by
This is the total phase jump occurring for reflection pairs at the twin boundary.
If the origin is not located at the twin inversion centre but is displaced from it by a vector , the phases of the structure factors of reflections and are From these equations the phase difference of the structure factors is calculated as and the total phase jump at the boundary is
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 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 latticetranslation vector lost in a phase transition.

Until the rather recent advent of highresolution 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).
Some examples of twinboundary 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 twininterface structures by HRTEM follow in Section 3.3.10.6.
The earliest structural model of a twin boundary was derived for aragonite by Bragg (1924), reviewed in Bragg (1937, pp. 119–121) and Bragg & Claringbull (1965, pp. 131–133). Aragonite is orthorhombic with space group Pmcn. It exhibits a pronounced hexagonal pseudosymmetry, corresponding to a (hypothetical) parent phase of symmetry , in which the Ca ions form a hexagonal closepacked structure with the CO_{3} groups filling the octahedral voids along the axes. By eliminating the threefold axis and the Ccentring 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 and the three cglide planes parallel to , one of each set is retained in the orthorhombic structure, whereas the other two appear as possible twin mirror planes and . It is noted that predominantly planes of type are observed as twin boundaries, but less frequently those of type .
From this structural pseudosymmetry the atomic structure of the twin interface was easily derived by Bragg. It is shown in Fig. 3.3.10.6. 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 (= twin displacement vector t).

Structural model of the (110) twin boundary of aragonite (after Bragg, 1924), projected along the pseudohexagonal 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 ). The model is based on a hexagonal cell with , the true angle is . The origin of the orthorhombic cell is chosen at the inversion centre halfway between two CO_{3} groups along c. 
For this merohedral twin (eigensymmetry 32) a real parent phase, hexagonal quartz (622), exists. The structural relation between the two Dauphiné twin partners of quartz is best seen in projection along [001], as shown in Fig. 3.3.10.7 and in Figure 3 of McLaren & Phakey (1966), assuming a fault vector 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 . It is noteworthy that the two models of the boundary structure by KlassenNeklyudova (1964) and McLaren & Phakey (1966) imply a slab with the quartz structure in the centre of the transition zone (Fig. 3.3.10.7b). This is in agreement with the assumption voiced by several authors, first by Aminoff & Broomé (1931), that the central zone of a twin interface often exhibits the structure of a different (real or hypothetical) polymorph of the crystal.

Simplified structural model of a Dauphiné twin boundary in quartz (after KlassenNeklyudova, 1964). Only Si atoms are shown. (a) Arrangement of Si atoms in the lowtemperature 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 hightemperature phase of quartz which is shown in (c). 
There are, however, Xray topographic studies by Lang (1967a,b) and Lang & Miuskov (1969) which show that curved Dauphiné boundaries may be finestepped on a scale of a few tens of microns and exhibit a pronounced change of the Xray 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) and requires further experimental elucidation.
The roomtemperature phase of KLiSO_{4} is hexagonal with space group . It forms a `stuffed' tridymite structure, consisting of a framework of alternating SO_{4} and LiO_{4} tetrahedra with the K ions `stuffed' into the framework cavities. Crystals grown from aqueous solutions exhibit merohedral growth twins with twin reflection planes (alternatively ) with extended and sharply defined (0001) twin boundaries. The twins consist of left and righthanded partners with the same polarity. The left and righthanded structures, projected along the polar hexagonal c axis, are shown in Figs. 3.3.10.8(a) and (b) (Klapper et al., 1987). 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. 3.3.10.8(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 lowenergy fit of the two partner structures. Note that all six (alternative) twin reflection planes and are normal to the twin boundary. It is not possible to establish a similar lowenergy structural model of a boundary which is parallel to one of these twin mirror planes (Klapper et al., 1987).

KLiSO_{4}: Bulk tetrahedral framework structures and models of (0001) twin boundary structures of phases III and IV. Small tetrahedra: SO_{4}; large tetrahedra: LiO_{4}; black spheres: K. All three figures play a double role, both as bulk structure and as (0001) twinboundary structures. (a) and (b) Left and righthanded bulk structures of phase III (), as well as possible structures of the (0001) twin boundary in phase IV. (c) Bulk structure of phase IV (), as well as possible structure of the (0001) twin boundary in phase III. The SO_{4} tetrahedra covered by the LiO_{4} tetrahedra are shown by thin lines. Dotted line: cglide 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. 3.3.10.8(c) shows that the tetrahedra related by the twin reflection plane (one representative plane is indicated by the dotted line) are shifted with respect to each other by a twin displacement vector . Thus, on an atomic scale, these twin reflection planes are in reality twin cglide planes, bringing the right and lefthand partner structures into coincidence.
Interestingly, upon cooling below 233 K, KLiSO_{4} undergoes a (very) sluggish phase transition from the phase III into the trigonal phase IV with space group by suppression of the twofold axis parallel [001] and by addition of a cglide plane. Structure determinations show that the bulk structure of IV is exactly the atomic arrangement of the grownin twin boundary of phase III, as presented in Fig. 3.3.10.8(c). Moreover, Xray topography reveals transformation twins III IV, exhibiting extended and sharply defined polysynthetic (0001) twin lamellae in IV. From the Xray topographic domain contrast, it is proven that the twin element is the twofold rotation axis parallel to [001] (Klapper et al., 2008). The structural model of the (0001) twin interfaces is given in Figs. 3.3.10.8(a) 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. 3.3.10.8c). It is immediately recognized that the two tetrahedral layers, one above and one below the (0001) twin boundary in Fig. 3.3.10.8(a) or (b), are related by screw axes.
Thus, the (idealized) (0001) twin boundary of the transformation twins of phase IV is represented by the bulk structure of the hexagonal roomtemperature phase III, whereas the twin boundary of the growth twins of the hexagonal phase III is represented by the bulk structure of the trigonal lowtemperature phase IV. Upon cooling from (phase III) to (phase IV), the axes are suppressed as symmetry elements, but they now act as twin elements. In the model they are located as in space group , one type being contained in the axes, the other type halfway in between. Upon heating, the retransformation IV III restores the 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).
Thus, KLiSO_{4} is another particularly striking example of the phenomenon, mentioned above for the Dauphiné twins of quartz, that the twininterface 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 . This may be explained by the compensation of the glide component of the cglide plane in phase IV by the screw component of the screw axis in phase III and vice versa.
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, 1996, 1997 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 latticeenergy 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, 1,2,4,5tetrabromobenzene, maleic acid and 3,5dimethylbenzoic acid.
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 3.3.10.2 and 3.3.10.3), followed by structural aspects, viz by displacement and fault vectors (Section 3.3.10.4), as well as atomistic models of twin boundaries (Section 3.3.10.5), 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 highresolution 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), especially Sections 2.3 and 5; Amelinckx et al. (1978), especially pp. 107–151 and 217–314; McLaren (1991); Buseck et al. (1992), especially Chapter 11; and Putnis (1992), 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.
This investigation has been presented already in Section 3.3.10.4.1 and Fig. 3.3.10.5 as an example of the occurrence of a twin displacement vector, leading to , where 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 TiO_{2} 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). A detailed structural model for this anatasetorutile phase transition is proposed by the authors, from which a sluggish nucleation of rutile followed by rapid growth of this phase was concluded.
The two reflection twins of rutile, (011) and (031), have been treated geometrically and with several figures in Section 3.3.6.10. 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:

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.
Twin interfaces (011) of the closely related tetragonal SnO_{2} (cassiterite) were investigated by Smith et al. (1983). A very close agreement between HRTEM images and corresponding computer simulations was obtained for . This twin is termed a `glide twin' by the authors, because the twin operation is a reflection across (011) followed by a displacement vector parallel to the twin plane (011).
In cubic crystals, twins of the (111) spinel type are by far the most common. A technologically very important phase, BaTiO_{3} perovskite, was investigated by Rečnik et al. (1994) employing HRTEM, computer simulations and EELS (spatially resolved electronenergyloss spectroscopy). The samples were prepared by sintering at 1523 K, i.e. in the cubic phase, whereby (111) growth twins were formed. These twins are preserved during the transition into the tetragonal phase upon cooling below T_{c} = 398 K. Note that these (now tetragonal) (111) twins are not transformation twins, as are the (110) ferroelastic twins.
Fig. 3.3.10.9(a) shows an HRTEM micrograph and Fig. 3.3.10.9(b) the structural model of the (111) twin boundary, both projected along . The main results of this study can be summarized as follows.

A very interesting structural feature of the BaTiO_{3} (111) twin interface was discovered by Jia & Thust (1999), applying sophisticated HRTEM methods to thin films of nanometre thickness (grown by the pulsedlaser deposition technique). The distance of the nearest Ti plane on either side of the (111) twin reflection plane (which is formed by a BaO_{3} layer, see above) from this twin plane is increased by 0.19 Å, i.e. the distance between the Ti atoms in the Ti_{2}O_{9} groups, mentioned above under (2), 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 Ti_{2}O_{9} groups. A similar expansion of the Ti–Ti distance in the Ti_{2}O_{9} 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 BaTiO_{3} (Burbank & Evans, 1948).
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 parallel to the (111) twin interface, were observed.
Note that BaTiO_{3} is treated again in Section 3.3.10.7.5 below, with respect to its twin texture in polycrystalline aggregates.
Differently oriented interfaces in bicrystals of Cu and Ag were elucidated by Hoffmann & Ernst (1994) and Ernst et al. (1996). They prepared bicrystals of fixed 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 and [both parallel to the (111) twin reflection plane] in the range 0–90°, where corresponds to the (111) `coherent twin plane', as illustrated in Figs. 3.3.10.10(a) and (b).

(a) Schematic block diagram of a bicrystal (spinel twin) for , i.e. for coinciding (111) twin reflection and composition plane. (b) Schematic block diagram of the bicrystal for . (c) HRTEM micrograph of the boundary of Cu for , projected along . The black spots coincide with the Cuatom columns. The micrograph reveals a thin ( 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). 
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 for rotations around both and . This corresponds to the coherent (111) twin boundary and is to be expected. It is surprising, however, that a second, very shallow energy minimum occurs in both cases at high angles: and , rather than at the compatible (112) contact plane for [the contact plane for is not compatible]. For these two angular inclinations, the boundaries, as determined by HRTEM and computer modelling, exhibit complex threedimensional boundary structures with thin slabs of unusual Cu arrangements: the slab has a rhombohedral structure of nine closepacked 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. 3.3.10.10(c). Similarly, for the slab a b.c.c. structure (as for Fe) was found, again with a thickness of Å. For further results see Sutton & Balluffi (1995), Section 4.3.1.9, p. 305.
Multiply twinned particles occur frequently in nanocrystalline (spherelike or rodshaped) particles and amorphous thin films (deposited on crystalline substrates) of cubic facecentred metals, diamondtype semiconductors (C, Si, Ge) and alloys. Hofmeister & Junghans (1993) and Hofmeister (1998) 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. 3.3.10.11: The five different {111} twin boundaries are perpendicular to the image plane () 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. 3.3.10.11). The junction line of the five sectors can be considered as a pseudofivefold twin axis (similar to the pseudotrigonal twin axis of aragonite, cf. Fig. 3.3.2.4; see also the fivefold twins in the alloy FeAl_{4}, described in Example 3.3.6.9 and Fig. 3.3.6.10).

HRTEM micrograph of a fivefoldtwinned Ge nanocrystal (right) in an amorphous Ge film formed by vapour deposition on an NaCl cleavage plane. Projection along a [] lattice row that is the junction of the five twin sectors; plane of the image: . 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); Hofmeister (1998). 
For the formation of fivefold twins, different mechanisms have been suggested by Hofmeister (1998): 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).
The fivefold multiple twins provide an instructive example of a twin texture, a subject which is treated in the following section.
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 spacefilling 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 3.3.10.7.1 and 3.3.10.7.2 below, have to be distinguished. Discussions of the most important twin cases follow in Sections 3.3.10.7.3 to 3.3.10.7.5.
In these twins, the lattices of all domains are exactly parallel (`parallellattice 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 3.3.10.7.5 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 3.3.7.3(iii)].
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:

Transformation and deformation twins are extensively treated in the following Section 3.3.10.7.3.
The real problem of spaceconstrained twin textures, however, is provided by nonmerohedral (ferroelastic) transformation and deformation twins (including the cubic deformation twins of the spinel law). This is schematically illustrated in Fig. 3.3.10.12 for the very common case of orthorhombicmonoclinic transformation twins (.

Illustration of spacefilling problems of domains for a (ferroelastic) orthorhombicmonoclinic phase transition with an angle (exaggerated) of spontaneous shear. (a) Orthorhombic parent crystal with symmetry . (b) Domain pairs , and of the monoclinic daughter phase () with independent twin reflection planes (100) and (001). (c) The combination of domain pairs and leads to a gap with angle , whereas the combination of the three domain pairs , and generates a wedgeshaped overlap (hatched) of domains 3 and 4 with angle . (d) Twin lamellae systems of domain pairs (left) and (or ) (right) with lowenergy contact planes (100) and (001). Depending on the value of , 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 . 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 . 
Figs. 3.3.10.12(a) and (b) show the `splitting' of two mirror planes (100) and (001) of parent symmetry mmm, as a result of a phase transition , into the two independent and symmetrynonequivalent 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 . Note that in transformation twins the angle