Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3n grain boundaries

Multiple twinning in cubic crystals is represented geometrically by three-dimensional fractals and algebraically by groupoids. The groupoid composition table can be used to identify the Σ3n grain boundaries in EBSD maps.


Introduction
'A twin is a complex crystalline edifice built up of two or more homogeneous parts of the same crystal species that are in contact and oriented with respect to each other according to well defined laws' (Friedel, 1904). The different origins of the twins (growth, recrystallization, mechanical deformation) are detailed by . In this paper, only the AE3 twins in cubic materials will be studied. These twins, also called in mineralogy sphalerite twins, spinel twins or diamond twins, belong to the class of 'twins by reticular merohedry' (Friedel, 1904), i.e. there is a partial but exact coincidence between the lattices of each individual crystal. AE3 means that this coincidence occurs only for one third of the lattices. More generally, two identical but misoriented cubic lattices have some points in coincidence that constitute a coincidence site lattice (CSL) if and only if they are linked by a transformation matrix T of the form where a ij are integers and AE is the ratio of the unit volume of the CSL referred to the unit volume of the crystal lattice Grimmer, 1976). Very often in metallurgy, metals with low stacking-fault energy form twins by annealing and recrystallization (Kronberg & Wilson, 1949;Kopezky et al., 1991). In this case, a crystal, which we will call the primary crystal, can form twins (twins of first generation), which can themselves form twins (twins of second generation), and so on. Each crystal in this assembly is linked to another by a transformation matrix T with an associated AE = 3 n with n 2 N. These crystals are connected by a chain of AE3 twins (i.e. by AE3 n operators) and form a microstructural entity usually referred to as a twin-related domain (TRD) (Reed & Kumar, 2006). Grain design engineering is an idea introduced by Watanabe (1985). It suggests that microstructures with a high fraction of 'special' grain boundaries (which can be obtained by optimizing the elaboration process or the thermomechanical treatments) have better mechanical properties, such as improved corrosion resistance, creep resistance or weldability. Since the special AE3 grain boundaries are considered to be the 'strongest' ones, many engineers are trying to create microstructures that have a high density of AE3 grain boundaries and large TRDs . These multiply twinned materials can be characterized using electron back-scatter diffraction (EBSD 1 ) in a scanning electron microscope (SEM) (for examples, see Randle & Brown, 1989;Kumar et al., 2000;Gertsman & Henager, 2003). Although the AE3 n grain boundaries are easy to identify for n 4, there is no method that can automatically identify them for higher twinning orders. Indeed, the distribution of the orientations of multiply twinned crystals is very dense and nearly isotropic for n ! 5, and consequently it is very difficult to distinguish a random misorientation from a special AE3 n [this has been shown by Wilbrandt (1980) and will be discussed in detail later]. A similar problem has been encountered for phasetransformed materials during the identification of the operators and the reconstruction of parent grains from the EBSD data obtained from the daughter grains. A solution has been proposed by Cayron et al. (2006) based on groupoid composition tables introduced in Cayron (2006). However, a phase transition corresponds to only half a cycle and we need to generalize the theory to cycled-transformed materials. This paper is a geometric/algebraic study on the particular type of series of cycles of transformations known as multiple twinning. 2 The theoretical results allow the identification of the AE3 n grain boundaries and the reconstruction of the TRDs encountered in some metallurgical problems.
In x2, the approach of Reed et al. (2004) on multiple twinning in metallurgy is discussed. A simple introduction to groupoids is given in x3 so as to facilitate the understanding of the further sections. In x4, multiple twinning is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. It is shown that the AE3 ! free group introduced by Reed et al. (2004) is a possible substructure of this groupoid. Another equivalent but not isomorphic substructure, the AE3 semigroup, is introduced in x5. In x6, the AE3 n operators are encoded by sets of equivalent strings, and a general method to determine their composition is proposed. We prove that the composition is multivalued. The composition table (called groupoid composition table) is given for twinning orders up to n = 4. Finally, in x7, we study some engineering cases (local crystallographic environment of voids or hillocks in copper films, reconstruction of the TRDs in narrow copper lines) in order to show how this table can be used to identify the AE3 n grain boundaries in some EBSD maps.
To begin, we must explain some notations. The point group G of a crystal will be considered as the group of matrices representing its orientational symmetries. |G| is the cardinality of the group G (i.e. the number of matrices). If H is a subgroup of G, the expression gH means a left coset of matrices based on the subgroup H, it is the set {gh, h 2 H}. The expression GTG, where T is a matrix, is the set {g i Tg j , (g i , g j ) 2 G 2 }.

The different algebraic approaches of simple and multiple twinning
Theoretically, the first crystallographic studies on textures generated by multiple twinning date back to the 1940s (Kronberg & Wilson, 1949;Wilbrandt, 1980;Gottstein, 1984). Many authors briefly mention that the algebraic structure associated with multiple twinning is a group. However, this group is rarely defined completely and its properties are not always demonstrated. Its definition appears to vary in different publications. When no detail is given, it can be assumed that that group is O(3) (the group of orthogonal matrices), but it could be also the subgroup constituted by all the matrices verifying equation (1), or a subgroup of this last group defined by imposing the condition AE = 3 n . However, these groups are too large and do not represent the true nature of multiple twinning. In fact, in these groups two equivalent transformation matrices, corresponding to the same misorientation between two crystals but differing due to a different choice of bases in those crystals, would be treated as two distinct elements whereas it would be more appropriate to consider them as the same element.
2.1. The approach of Reed et al. (2004) to multiple twinning An impressive study has recently been published by Reed et al. (2004) on how multiply twinned structures, and more generally the AEX m AEY n structures with X and Y integers, can be constructed from algebraic manipulations on quaternions, simplified by a string representation and illustrated using network graphs. This theoretical work has been recently summarized and applied to simulate multiply twinned microstructures (Reed & Kumar, 2006). Since we will often refer to this work, we will give a brief description of the part of their approach that deals with multiple twinning. The misorientation from a crystal 1 to a homophase crystal 2 can be expressed by a set of equivalent matrices GR 12 G, where R 12 ¼ R À1 1 R 2 is a rotation from crystal 1 to crystal 2 and G is the point group of the cubic crystals. This set is called a 'subtype' and the order of the crystals in the pair (1, 2) is important. The misorientation between two crystals 1 and 2 is expressed by a set of equivalent matrices GR 12 G [ GR 21 G, with R 21 ¼ R À1 12 . This set is called a 'type' and the order of the crystals in the pair (1, 2) is not important. The 'types' define matrices that are 'cubically' equivalent. 3 In their paper, Reed et al. (2004) Reed & Kumar, 2006), and each AE3 n operation was expressed by a string constituted of these four letters. The strings can be composed using simple concatenation and by taking into consideration a rule that eliminates the pairs of equal consecutive letters (for example abbcd = acd). All the operations can be represented by a graph. With this representation, the set of the AE3 n operations form a free group 4 called AE3 ! . The work of Reed et al. (2004) brings a new understanding to multiple twinning and introduces a rule on the AE numbers that can help metallurgists to analyse structures with AE3 n grain boundaries. However, many questions can be raised. The decomposition of the set of equivalent quaternions into four cosets is not clearly explained. It is intuitive (because the [111], [1 " 1 1 " 1 1], [ " 1 1 " 1 11] and [ " 1 11 " 1 1] axes are equivalent), but what are the algebraic justifications for this decomposition? Are the cosets left or right? What is the subgroup on which they are based? The authors assume that a representative can be 'arbitrarily' chosen in the cosets, and they have built the group AE3 ! , not with the entire cosets but with the four quaternions a = [0111], b = [0 " 1 1 " 1 11], c = [01 " 1 1 " 1 1] and d = [0 " 1 11 " 1 1]. What could be the structure if other representatives were considered? Would we obtain the same group with the same elimination rule? There are also other limitations discussed by the authors, such as 'without cubic symmetry, the natural mapping between integer quaternions and CSL rotations is lost'. Is it possible to introduce another approach without quaternions (which could therefore be applied to noncubic materials)? Moreover, because the four elements a, b, c, d generate an infinity of new operations, the group AE3 ! is infinite. Why is it not possible to build a finite algebraic structure to describe finite cases such as the one of a crystal and its four twinned variants? A last but not least question is also raised: we know that the composition of two AE3 operators produces either a AE1 operator or a AE9 operator. However, in a group the composition is a 'classical' mathematical application (the composition of two elements gives only one element), so a group structure cannot explain the multivalued aspect of the composition of the AE3 n operators. What could be the algebraic structure describing the multiple twinning and its multivalued composition? We will try to answer these questions in this paper.

A brief overview on simple twinning
The work of Reed et al. (2004) is mainly based on metallurgical tools (such as the CSL rotations). The crystallographic developments of twinning applied in other fields of material science can be used advantageously to complete their approach. The colour group introduced by Shubnikov & Koptsik (1974) (see also Senechal, 1983) is now integrated in the modern theories of twinning. Crystallographers working in mineralogy (Wadhawan, 1997; and those working in physics of ferroelectric domains (Janovec, 1976) have made a synthesis of their respective approaches. This synthesis uses mathematical tools based on modern group theory such as orbits, stabilisers, coset partitioning etc. (Hahn et al., 1999;. Some crystallographers interested in grain boundaries are following the work of Pond & Vlachavas (1983) and are trying to integrate the CSL rotations in that synthesis (Grimmer & Nespolo, 2006).
Let us briefly explain the principle of coset partitioning in the case of twinning. We call G 0 the point group of crystal 0 and T a twin operation. We call G 1 the point group of crystal 1 which is the twin of crystal 0 by the operation T. If the matrices that constitute G 1 are expressed in the same base as those of G 0 , the two groups are linked by G 1 = TG 0 T À1 . The two crystals have some symmetries in common that constitute the intersection group H = G 0 \ G 1 . Moreover, owing to the symmetries of the crystal 0, more than one twin crystal might be created and each element g of G 0 that does not belong to H creates a new variant. Thus the group G 0 can be partitioned into left cosets that represent the distinct variants: with g 0 = e the neutral element (i.e. the identity matrix). The number of variants is given by the Lagrange formula N = |G 0 |/|H| and their orientations are given by the sets g i HT. Now, we can explain the idea of Reed et al. (2004), the 'cosets' of their decomposition are in fact the sets of type g i HT (which are not strictly speaking cosets because only g i H are cosets). The set of cosets in the decomposition (2) represents the assembly of twinned variants. This is sometimes called 'the reduced composite group'. However, this name may be confusing. One must distinguish the group G 0 of crystal 0 that has generated the set of variants from the set of variants itself (i.e. the set of cosets) given by In general, G 0 /H does not have a group structure. It is a group if and only if H is a normal subgroup of G 0 . If this condition is fulfilled, G 0 /H is a group and each coset of this group can be represented by one matrix arbitrarily chosen in each coset. For example, H is a normal subgroup when there are only two cosets in the set (3). However, we stress here that the condition of normality of H is not fulfilled in the case of AE3 twinning as discussed in Cayron (2006). Consequently, although one matrix (or one quaternion) can be arbitrarily chosen in each coset for numerical calculations (Reed et al., 2004), a method based on arbitrary choices of representatives cannot be used to justify the algebraic structure of multiple twinning. The misorientations from a variant represented by the coset g i H to a variant represented by the coset g j H are isomorphic to the double cosets Hg ij H, where g ij = g À1 i g j Cayron, 2006). Therefore, the distinct types of misorientations between ordered pairs of variants are given by the partition of G 0 into double cosets, These types of misorientations are also the orbits of the action of G on the sets of the ordered pairs of variants (Janovec, Acta Cryst. (2007). A63, 11-29 Cyril Cayron Multiple twinning in cubic crystals 13 research papers 4 If S is a set, the expressions s " 1 1 . . . s " n n with s i 2 S and " i 2 fÀ1; 1g are called the strings (or words) of S. A string is said to be reduced if it does not contain two adjacent terms of the form s 1 i s À1 i or s À1 i s 1 i . A free group F is a group generated by a set S in which the distinct elements are represented by distinct reduced strings in S. 1972). Double cosets are not limited to crystallography; they are also widely used in physics and chemistry 5 (Ruch & Klein, 1983). Since double cosets can be viewed as types of actions that act on the variants, we call them operators. Their set is written The primes in g 0 i are introduced to distinguish the elements g 0 i of (5) from the elements g i of (3). The number of operators, N 0 , is given by the Burnside formula or it can be obtained by a class equation (Cayron, 2006). In general, it is possible to distinguish the operator transforming the variant i into the variant j from the operator transforming the variant j into the variant i because in general Hg ij H and Hg À1 ij H are two distinct double cosets. The operators are then called polar. If it is not the case, i.e. Hg ij H = Hg À1 ij H, the operators are called ambivalent (Janovec & Přívratská , 2003). Now, it can be realized that the 'subtypes' (see x2.1) correspond to the operators, and that the 'types' correspond to an 'artificially forced' union of complementary operators Hg ij H [ Hg À1 ij H (always ambivalent).
In Cayron (2006), all these ideas have been used to show that the set of variants associated with the set of operations that link them form an algebraic structure called groupoid of orientational variants. Is it possible to generalize the concept of groupoid to multiply twinned variants? Before going further, we would like to give some explanations on groupoids and a possible way to use them in crystallography.

A brief introduction to groupoids and operators
Groupoids are very useful because they are 'the ideal tool for describing symmetries that apply only to parts of systems. Groupoids are more flexible and often more appropriate than the better-known groups . . . ' (Stewart, 2004). Groupoids were first introduced in mathematics by Brandt (1926) and they now play a key role in the category and homotopy theories. For an exact definition and mathematical details, the reader may refer to Brown (1987) or Weinstein (1996). Groupoids were used in material science to represent polytypic structures, also known as order-disorder structures (Dornberger-Schiff & Grell-Niemann, 1961;Sadanaga, 1978;Fichtner, 1986). Their practical applications however have remained mainly limited to the problem of diffraction enhancement of symmetry (Sadanaga & Ohsumi, 1979). Since the definition of groupoids given by mathematicians may be difficult to understand for non-mathematicians, we would like to give here our personal geometrical and simple vision on this algebraic structure.
The most important point to understand is the groupoid composition law. It says that two pairs of objects (i, j) and (j, k) can be composed and the result is (i, k). This condition is classical and everyone has used it to add geometrically two vectors U and V. Each vector is written as a pair of points such that U = (P 1 , P 2 ), V = (P 2 , P 3 ) and the result is W = U + V = (P 1 , P 3 ). Such geometrical construction has been forgotten in Cartesian geometry because the addition of vectors is commutative and can be simply resumed to additions of two numbers. The decomposition into pairs of objects is also possible for invertible matrices if they are expressed as transformation matrices from a base i to a base j: Then, the composition XY can be written X In addition, the analytical result of the product of two matrices is also a groupoid composition law as pointed out by Connes (1990): The decomposition of matrices into pairs of bases explains why the order of the matrices is important when calculating their product (i.e. XY is in general different from YX). The groupoid composition law can be imagined spatially as a 'head/tail' condition and also temporally as a 'before/after' condition. It can be used in quantum physics, 6 biology, computer science and for any system that has connections such as graphs and networks (Stewart, 2004).
In these examples, it may be noticed that the groupoid law is based on the existence of two complementary entities: the objects (the points, the bases, the energies 6 ) and the operations between these objects (the vectors, the matrices, the frequencies 6 ). These operations can be represented by arrows that link the objects. 7 In addition to its composition law, a groupoid must have the following properties: the composition between the arrows is associative and each arrow has an inverse. Groupoids are more general than groups because in the latter the objects and the arrows are two isomorphic entities that cannot be distinguished. Indeed, an element g of a group G is also an arrow referenced to the neutral element e because g.e (g is an object transformed by the arrow e) = g (g is an arrow).
A new idea known as 'operator' was also introduced by Cayron (2006). This term has a meaning different from the term 'operation'. An operator is a type of operation and can be expressed by a set of equivalent operations. 8 Moreover, situations or figures that have partial iterative symmetries can often be represented by graphs. Since the vertices and the edges of a graph can be viewed respectively as the objects and the operations of a groupoid, an operator can also be viewed as a type of path in a graph and expressed by a set of equivalent paths. In Appendix A, a non-crystallographic example is described in order to familiarize the reader with the idea of operator (as a type of path in a graph or as a type of arrow in a groupoid).   Connes (1990) makes a clear parallel between the matrix mechanics of Heisenberg in quantum physics and the groupoid composition law of the spectral frequencies of hydrogen given by ðE i ;E j Þ þ ðE j ;E k Þ = ðE i ;E k Þ , where ðE i ;E j Þ is the frequency of the spectral line between the energy levels E i and E j . 7 In this paper, we will use equivalently the terms 'operation' and 'arrow'.

The groupoid of orientational variants
It has been shown that the orientational variants formed by a structural phase transition ! and the operations that link them form a groupoid (Cayron, 2006). Geometrically, the daughter variants were identified to the objects of the groupoid and the misorientations were identified to the arrows between these objects. Algebraically, the details of the structure were obtained by unifying the external symmetries of the parent crystal G to the internal symmetries of the daughter crystals G with the help of a transformation matrix T representing the orientation relationship between one daughter crystal and its parent crystal. More precisely, the variants i were identified to the simple cosets g i H, where g i 2 G and H is the intersection group between the parent crystal and a daughter crystal (given by H = TG T À1 \ G ). The types of misorientations between the variants, i.e. the operators, were identified to the double cosets T À1 Hg ij HT. The arrow from the variant i to j and the arrow from the variant j to k can be composed and the result is the arrow and the resulting operators are those containing the arrows ( i . k ). Since many operators may be obtained, the composition is multivalued. The groupoid composition table characterizes the crystallographic aspect of the transition; some tables were given for the Burgers transition by Cayron (2006) and for the martensitic transitions by Cayron et al. (2006).

The groupoids of simple twinning
Simple AE3 twinning in face-centred cubic (f.c.c.) materials may be imagined as a phase transition with G = G = G and T = T À1 the matrix representing the mirror symmetry through the (111) plane (see Table 1). The intersection group contains 12 symmetry operations and in consequence the number of variants is N = |G|/|H| = 4. Let us call 0 the primary crystal (parent crystal), and 1 0 , 1 1 , 1 2 and 1 3 the four twins of this crystal (daughter crystals). More generally, the ith twin of the nth generation will be denoted n i .The five crystals are represented in Fig. 1(b) (this figure is probably closer to the actual algebraic developments of twinning and coset partitioning than the classical representation of Fig. 1a). The four variants 1 0 , 1 1 , 1 2 and 1 3 are linked to the primary crystal 0 by a AE3 operator, between them by a AE9 operator and to themselves by a AE1 operator: . Three groupoids may be defined: Cyril Cayron Multiple twinning in cubic crystals 15 research papers Table 1 Matrices representing the transformations from a primary cubic crystal into its four AE3 twinning variants.
They have been chosen in their respective set of type g i HT to create a 'harmonious' enumeration of the two-dimensional fractal of Fig. 5(a). The geometrical operations corresponding to these matrices are also reported (m for a mirror symmetry and R for a rotation).
AE3 twin representations: (a) a AE3 twin is in general represented by an association of two crystals of equivalent size, but (b) a AE3 twin results in general from a transformation during mechanical experiments or annealing treatments, and four distinct orientations (four variants) may appear after one twinning transformation (eight are represented but can be associated two by two due to the centrosymmetry of the cubes).

Figure 2
Composition table of the groupoid C (0+1) representing an assembly of crystals constituted by a crystal 0 and its four twinned variants 1 0 , 1 1 , 1 2 and 1 3 . The operators AE1 and AE3 are given with reference to the crystal 0 . The composition of the operators appears as a multivalued function, for example AE3AE3 = {AE1, AE9}.
(ii) C (1) is constituted by four objects ( 1 0 , 1 1 , 1 2 and 1 3 ), 16 arrows ( 1 i . 1 j ) with (i, j) 2 [0, 3] 2 partitioned into two operators (AE1 and AE9). This is a groupoid of orientational variants described in x4.1. The four variants are algebraically identified with the four cosets g i H that constitute the set G/H and the two operators expressed in a basis of 0 are the double cosets of the set G\H/G: AE1 = H and AE9 = Hg 0 1 H. ) , which is a union of groupoids is constituted by 5 objects, 25 arrows partitioned into 3 operators (AE1, AE3 and AE9). Its composition table is reported in Fig. 2. It may be checked in this table that the composition of two AE3 can be either a AE1 or a AE9 operator. One may add that the result is AE1 with a probability of 1/4 and is AE9 with a probability of 3/4. Only arguments based on calculations of energies can modify significantly these crystallographic probabilities (but in this study, we will only consider crystallographic arguments).

The groupoids of multiple twinning
By considering simple twinning as a phase transition with = , multiple twinning AE3 n may now be imagined as a series of phase transitions ! ! ! ! etc. In Cayron (2006), we raised the following questions: is there a general formula to calculate the number of variants and the number of operators of the nth generation? Do these numbers increase to infinity with n? We do not know the general solution to these questions but we will answer them in the special case of multiple twinning.
The approach of the previous section can be generalized, the variants of generation n and the operations that link them form a groupoid C (n) . This groupoid can be associated with the groupoids of the previous generations C (i) with i < n to form a groupoid denoted C (0+1+ . . . +n) . This last groupoid and its subgroupoids form a structure similar to a Russian doll. Its algebraic details remain to be fully determined; such a study implies the generalization of the use of cosets and double cosets to multiple cosets linked by a transformation matrix. 9 However, we will see in the following that some basic geometrical considerations are actually sufficient to determine the operators and their composition table.

Three-dimensional fractal representations
We may imagine all the twinning variants in three dimensions: (i) by representing the primary crystal with a tetrahedron constituted by its four {111} planes, (ii) by creating its four variants by applying the {111} mirror symmetries, and (iii) by repeating this process. The topology makes this approach impossible for orders higher than three (it is the well known problem of tetrahedra packing). However, all the variants can be created if one decreases the size of the tetrahedra by a factor of two at each generation of the process. The result is the three-dimensional fractal illustrated in Fig. 3. This fractal is a more complex version than the three-dimensional Kepler fractal in which the tetrahedra are only translated but not rotated. The representation of Fig. 3 is a convenient way to illustrate the orientations of all the multiply twinned crystals on the same drawing. It is particularly illustrative for twinning in " 4 43m crystals (such as sphalerite). It may be noticed that, owing to the absence of centrosymmetricity, the [111] and [ " 1 1 " 1 1 " 1 1] directions are not equivalent (compare Figs. 3a and b). For m3m crystals, such as f.c.c. metals, the tetrahedra can be substituted by cubes; the generated fractal is then illustrated in Fig. 4. This fractal is constituted by interpenetrated cubes, the twinned cubes of the (n + 1)th generation are positioned at the corners of the nth-generation cubes. By construction, the whole fractal has the same symmetries as for the primary   Three-dimensional tetrahedral fractal representation of multiply twinned crystals ( " 4 43m) inherited from the same primary crystal 0 (in green), viewed (a) in the [111] direction and (b) in the [ " 1 1 " 1 1 " 1 1] direction of 0 (the two directions are not equivalent because a tetrahedron is not centrosymmetric). This fractal is built from twinned tetrahedra reduced in size by a factor of two at each generation and lying in the centres of the faces of the tetrahedra of the preceding generation. Distinct colours are associated with the orders n of the fractal. 9 The orientations of four crystals of first generation are given in reference to the orientation of the primary crystal 0 : the orientation of 1 0 is given by HT, the orientation of 1 1 by g 1 HT, 1 2 by g 2 HT and 1 3 by g 3 HT. Each 1 i becomes the parent crystal of the variants of second generation 2 j . The orientations of these variants in reference to the primary crystal 0 are given by the set of matrices g i HTg j HT. More generally, the orientations of the twinned crystals at any order n are given by the sets of type g i HTg j HTg k HT . .
crystal. For example, it may be checked that the cubic fractal viewed in h111i and h100i directions (Figs. 4a and b) exhibits the two-dimensional 3m and 4mm symmetries, respectively.
From these figures, it appears that the number of distinct variants at the nth generation is given by = 4.3 n , for n ! 1 (see also Wilbrandt, 1980;Gottstein, 1984). We have seen that the assembly of grains twinned to the nth generation constitutes the groupoid C (0+1+ . . . +n) and can be represented by a fractal (stopped at the nth generation). Such assembly has its own CSL, which will be denoted CSL n . It is the intersection of the lattices of all these crystals. Therefore, the CSL n has the same symmetry elements as for the primary crystal and it can be referenced in the primary crystal basis with a matrix of type aE, where E is the 3 Â 3 identity matrix and a 2 N. Moreover, the CSL n can also be expressed with integer coordinates in the reference bases of all the twinned crystals, which means that the CSL n matrix multiplied by any transformation matrix given in equation (1) with AE = 3 n should be equal to an integer matrix. Since the a ij coefficients in equation (1) are co-prime (they have no common divisor except 1), such a condition leads to It follows that the volume of the CSL n lattice is AE multi = 3 3n . This result could also probably be derived from the general formula suggested by Gertsman (2001b).

Macro/microscopic examples of three-dimensional fractal shapes
We recall that the fractal representation is just a convenient way that will help us to visualize the symmetries and to simplify the calculations. However, we may wonder if such shapes can exist in nature. Indeed, with perfect isotropic growth conditions, the macroscopic shape of an assembly of multiply twinned crystals could be close to Fig. 3 for " 4 43m structures or to Fig. 4 for m3m structures. Even if isotropic conditions are rarely found in mineralogy, multiple twins of diamond and sphalerite crystals can sometimes look like Fig.  3(b) -the reader is invited to look at the photographs reported by Palache (1932). The star polyhedral gold nanoparticles recently discovered by Burt et al. (2005) also exhibit shapes close to Fig. 3 (with n = 2). Another case is probably the fractal structure of a dislocation-free bicrystal silicon ribbon studied by Cheng (1994). He reported angles of reentrant corners (141 and 109.5 ) that correspond respectively to some rotation angles of the AE9 operator (180 À 141 ' 38.94 ) and of the AE3 operator (180 À 109.5 = 70.5 = angle between two {111} planes). An EBSD study of this ribbon could be interesting to confirm that the orientations of the crystals are of type AE3 n . Fractal structures are also obtained by phase transformations very similar to the twinning transformation. The hyperbranched structures of CdTe or CdSe nanocrystals are constituted of branched tetrapods (Milliron et al., 2004) resulting from the alternating transitions between the cubic sphalerite phase and the hexagonal wurtzite phase: sphalerite transforms into four wurtzite branches in the h111i directions (as in the twinning case), and each wurtzite branch transforms into two sphalerite branches in the two +c and Àc directions.

The R3 n semigroup of multiple twinning
The whole three-dimensional fractal represents the whole twinning groupoid C 1 = C (0+1+ . . . +n) with n = 1. What is the relation between the free group AE3 ! introduced by Reed et al. (2004) and C 1 ? In the group AE3 ! , the four strings that begin with the letters a, b, c and d correspond to the four branches of the fractal. Each string of the free group AE3 ! encodes a path on the fractal. It must be remembered that the four letters represent 180 rotation matrices that were chosen in the sets HT, g 1 HT, g 2 HT and g 3 HT. The relative simplicity of the AE3 ! structure results from this 'not so arbitrary' choice. Other choices of representatives in those sets lead to different algebraic structures that are all substructures of C 1 . In order to convince the reader, we are creating a structure denoted AE3 , which will be proved to be not isomorphic to AE3 ! , but that can also be used to encode the AE3 n operators. The representatives are chosen in the sets HT, g 1 HT, g 2 HT and g 3 HT such that the faces of the tetrahedra in the developed representation of the fractal of Fig. 3 (Fig. 5a) are 'harmoniously' enumerated. These matrices T 0 , T 1 , T 2 , T 3 and their corresponding geometrical meaning are reported in Table 1.

Calculation of the operators
The whole fractal and the associated AE3 n operators can be constructed from the four T i matrices of Table 1 by calculating the matrix products T i T j T k . . . T l . In all the following, these products will be written as strings Enumeration of the faces of the tetrahedra. (a) Developed representation of the fractal of Fig. 3 (in fact, to avoid overlapping of the faces, the scaling factor is not 1/2 as for the three-dimensional fractal but 1/4). This two-dimensional fractal is usually called a Sierpinsky fractal. The faces have been enumerated in such a way that the arrangement of the numbers in this figure respects a simple 'harmonious' rule. (b) The corresponding numbers are reproduced in the three-dimensional fractal limited to n = 2. Table 2 AE3 n operators (reproduced here up to n = 6).
For space reasons, the CSL rotation matrices are given only up to n = 4 with only the integer coefficients a ij of expression (1). For n > 4, for more visibility, ny should be read AE3 n y (for example 5a is AE3 5 a). The polar operators are marked by +/À signs, the others are ambivalent operators. The string coding i j . . . l corresponds to the matrix product given by the formula (8)  The operations are encoded by the strings and the operators are encoded by sets of equivalent strings. The calculations can be reduced to the minimum if the symmetries of the fractal (i.e. the symmetries of the primary crystal) are taken into consideration. Indeed, the fractal is constituted by four main branches on the four threefold axes of 0 , which are obtained by taking as the first matrix in the product (8) T 0 for branch 0, T 1 for branch 1, T 2 for branch 2, and T 3 for branch 3. Since these branches are all geometrically equivalent, all the operations can be done with only one branch. We choose branch 1, i.e. the one starting with T 1 . These geometrical considerations explain why there is only one AE3 operator. The first variant on this branch is 1 1 . From this variant, three new variants 2 1 , 2 2 and 2 3 can be created, and they are all equivalent due to the threefold symmetry of this branch. Therefore, to calculate the other operators, we can limit ourselves to calculate the matrix products that start with T 1 T 1 , i.e. the strings 11 . . . These geometrical considerations explain why there is only one AE9 operator. From 2 1 , three new variants can be created, but two are equivalent due to the mirror symmetry on the plane (011). Therefore, there are only two AE27 operators: one containing the string 111 (= AE27a) and another one containing to the string 112 (= AE27b). From this step (n = 3), each variant of generation n will generate three new variants of generation n + 1, with the exception of the one that keeps the (011) plane of 0 as a mirror plane. Consequently, for the orders n > 3, all the operators can be recursively generated with the help of five distinct types of operations ( n i . nþ1 j ). (a) If n i keeps having a mirror symmetry through the initial mirror plane (011), the operation ( 0 . n i ) is of type T n 1 = 111 . . . 1 (n terms). Three new variants and two new operators can be created from n i : the first operator contains the string 111 . . . 11 (n + 1 terms) = T n 1 T 1 , and the second one contains the string 111 . . . 12 (n + 1 terms) = T n 1 T 2 . Both are illustrated in Fig. 6 (in the case of n = 4).
(b) If n i is not symmetric through the initial mirror plane (011), the operation ð 0 . n i Þ is of type M = 11 . . . 2 . . . i (n terms). Three new variants and three new operators can be created from n i : they contain the strings 11 . . . 2 . . . ij (n + 1 terms) = MT j , j 2 {1, 2, 3}. If j = i, the connection is a 'forward' connection; if j 6 ¼ i, the first index h before i with h 6 ¼ i must be determined and if j = h the connection is a 'circular' one, and if j 6 ¼ h the connection is a 'zigzag' one.
These symmetry considerations allow the creation of the minimum number of variants required to compute the construction of the fractal, all the other variants are deduced by the symmetries of the primary crystal 0 (i.e. the point group G). At the nth generation, the number of distinct operators is easily deduced from this geometrical approach. Indeed, it respects the arithmetic geometric sequence N op nþ1 = 3N op n À 1 with N op 2 = 1. In consequence, By construction, AE3 , the set of strings based on the four indices (0, 1, 2, 3) is a semigroup. 10 AE3 is not a group because the indices 1, 2 and 3 have no inverse. Of course, AE3 is not isomorphic to AE3 ! .

Names of the operators
Once the matrices M representing the operators AE3 n are determined (with their string code), the operators can be named according to the following method: (a) calculate for each matrix M the set GMG of polarly cubically equivalent matrices (see footnote 3), (b) choose in this set the rotation with the minimum angle as representative and (c) order the operators according to these minimum angles. For example, the rotations with minimum angles of 31.58 and 35.43 are associated with the operators AE27a and AE27b, respectively. This way of ordering is equivalent to comparing the norms of the quaternions. However, as already noticed by Reed et al. (2004), many distinct operators can have the same representative minimum angle. A solution was proposed by these authors for quaternions, however, here, since only the matrix expressions are used, we have decided to choose another ordering rule. For each rotation matrix representative of the operator (i.e. with the minimum angle), we also consider the orientation of its rotation axis, a second ordering is then realized (if necessary) by calculating the minimum scalar product between this axis and the h111i axes. Since this ordering sometimes is not enough because two distinct operators can have the same minimum rotation angle and the same angle between the rotation axis and the h111i axes, a third ordering rule is sometimes required by calculating the minimum scalar product between the rotation axis and the h100i axes. Once the operators have been ordered, they are identified with letters. Since the alphabet is not large enough for twinning orders n > 6, it has been extended according to the rule: a, . . . , z, aa, ab, . . . , az, ba, bb, . . . , bz, . . . etc. The naming and ordering of the operators takes most of the computing time (one minute for order n = 8) because it implies matrix calculations. The ordered operators are reported in Table 2 for twinning order n 6 (the list for higher twinning order is available on demand). It may be noted that the lists given in some previous studies (Gottstein, 1984;Andreeva & Firsova, 1996) are not quite complete. We also would like to stress that it is possible to distinguish the complementary polar Three-dimensional graphical representation of two AE81 operators: (a) AE81d containing the string 1111, (b) AE81+a containing 1112. operators (see footnote 3). For example, the two AE81a operators with a minimum angle of 38.376 and rotation axes of type h135i were distinguished; we called them AE3 4 +a and AE3 4 Àa. Indeed, if we denote T 4 þa = 1112 the matrix associated with AE3 4 +a and T 4 Àa = 1122 the matrix associated with AE3 4 Àa (given in Table 2), the reader may check (by computing) that the associated set of polarly cubically equivalent matrices forming the AE3 4 +a and AE3 4 Àa operators, i.e. GT 4 a G and GT 4 Àa G, with G the m3m point group, do not intersect: T 4 Àa = 2 fg i T 4 þa g j ; ðg i ; g j Þ 2 G 2 g but T 4 Àa 2 fg i T 4 þa À1 g j , ðg i ; g j Þ 2 G 2 g. The operator AE3 4 +a and AE3 4 Àa are complementary polar operators.

Two-dimensional graph and pole-figure representations
The different operators corresponding to the AE3 semigroup can be represented on a graph similar to the one introduced by Reed et al. (2004) (the only slight difference is the distinction we have made in the labelling between the polar and ambivalent operators). This graph can be redrawn to obtain a two-dimensional fractal graph based on Templar-style crosses, as shown in Fig. 7 (limited here to n 5 for space reasons). In this figure, the operators are the centres of the crosses and the variants are the tips. This graph is called a Cayley graph (used by mathematicians to encode the free group based on two generators). One may observe that the types of connections between the operations such as the circular or zigzag connections also appear in this two-dimensional fractal scheme.
The twinned variants can also be represented by drawing their orientations on a pole figure in one of the reference bases of the primary crystal, as illustrated for twinning orders n 5 in Fig. 8 (for higher orders there are too many points and only the densities could be represented). Such figures were already presented by Gottstein (1984) but were incomplete because of some missing operators. As already calculated by Wilbrandt (1980), it is very difficult to distinguish a random misorientation from a AE3 n operator for twinning orders n higher than 5. Some special patterns can also be noticed in these pole figures (such as the circles). These come from operators that have very close minimum rotation angles (such as the AE81+a, AE81Àa and AE81b operators). As the order n increases, the number of operators with close or even exact minimum rotation angles increases and the number of singular patterns increases (as confirmed by simulations of pole figures of operators with equal rotation angles and random rotation axes, not presented here).  (a) Two-dimensional fractal graph of the multiply twinned variants and AE3 n operators (up to order n = 5). The fractal is built from a Templar-style cross represented in (b) reduced in size by a factor of two at each generation step; the crosses are joined at the tips of those of the preceding generation. The variants are the tips of the crosses (not labelled) and the AE3 n operators are the centres of the crosses. This two-dimensional graph is usually called a Cayley graph.

Comparison of the R3 n semigroup with the R3 x group
The AE3 semigroup (based on the indices 0, 1, 2, 3) and the AE3 ! free group (based on the letters a, b, c, d) are represented by the same graph although these two structures are not isomorphic.
In the AE3 ! group, all of the letters play the same role in the strings, and they can be interchanged. Moreover, the simplification rule aa = bb = cc = dd = ø with ø the empty string (coding the identity matrix) is based on the choice of 180 rotation matrices in the four sets g i HT that define the orientations of the variants with i 2 [0, 3]. This rule makes the string representation unique and consequently makes AE3 ! free.
In the AE3 semigroup, only the indices 1, 2 and 3 are equivalent. The index 0 appears only one time in the whole AE3 graph (it begins the strings that codes the branch 0). In fact, the index 0 (and the associated matrix T 0 ) is 'special' and cannot be treated as the three other indices. Since 0 represents a mirror symmetry, it obeys the simplification rule 00 = ø. Consequently, 000 . . . (n times) is equal to 0 if n is odd, and to ø if n is even, whereas the strings 111 . . . , 222 . . . or 333 . . . (n times) cannot be reduced. The asymmetry between the index 0 and the three others has a temporal meaning if twinning is considered as a temporal process. Let us imagine a progressive twinning by annealing and recrystallization from a variant of order n, the indices 1, 2 and 3 produce three new crystals of order n + 1, whereas the index 0 corresponds to a crystal of generation n À 1 that was already produced in the past. The composition with the index 0 means that there is a unique way to go back to the past, and the composition with the indices 1, 2, 3 means that there are three equivalent (locally symmetric) ways to go to the future. There is no simplification rule between the indices (1, 2, 3). The simplification is realized only between the index 0 and the three indices (1, 2, 3). What is the rule? For example, how could we simplify the string that corresponds to 112.012 = 112012? More generally, how could we simplify the string h . . . i.0j . . . k = h . . . i0j . . . k? The specific matrices T i for i 2 [1, 3] that we have chosen in the sets g i HT to harmoniously enumerate the faces of the tetrahedra on the two-dimensional developed fractal (see x4.4) give the following rule: This property comes from the fact that 101 = 0, 102 = 3, 103 = 2, which can be checked by calculating the matrix products (8) with the matrices given in Table 1 (the other combinations are true due to the equivalence of the 1, 2, 3 indices).

Composition of the R3 n operators
The idea of CSL was initially introduced to characterize the grain boundaries between two crystals. When metallurgists became interested in triple junctions, a method to compose the CSL rotations and to determine the CSL of an assembly of grains was developed. Most of the studies were restricted to finding a rule for the composition of the AE numbers that appear in expression (1), and few studies treated the composition of the operators.

Composition of the R numbers 11
It was believed for a long time that the AE numbers in the CSL matrix expressions (1) are numbers that can be multiplied without precaution. It was believed for example that the three AE numbers which determine the respective misorientations between three crystals (indexed by 1, 2, 3) follow the rule AE 13 = AE 12 AE 23 , where AE ij is the AE value of the misorientation matrix between the crystal i and the crystal j. This rule was puzzling because it does not respect the symmetry of the problem, i.e. the three crystals do not play similar roles. Actually, it was proved to be wrong. The correct AE composition rule was proposed by Miyazawa et al. (1996), with a demonstration given by Gertsman (2001a). It is derived from the following relationship: where 123 is the greatest common odd divisor of the quaternion produced by the multiplication of the two quaternions describing the two generating CSL misorientations. 12 This property is general; therefore, we also have By multiplying equations (11) and (12), and simplifying the result by using the equality AE 23 = AE 32 (because the inverse of the matrix in equation (1) is its transpose), it follows that AE 23 = AE 32 = 123 132 . Similarly, by using equations (12) and (13), it follows that AE 13 = AE 31 = 132 213 and, by using equations (11) and (13), it follows that AE 21 = AE 12 = 123 213 . By writing 213 = p 1 , 123 = p 2 and 132 = p 3 , i.e. ijk = p j , the three equalities can be summarized by using a simple rule: This formula has been shown by Miyazawa et al. (1996) and Gertsman (2001a), but we have preferred to show a complete demonstration. 12 This rule is illustrated for three crystals in Fig. 9. This approach can also be generalized to four crystals. 13 The rule (14) indicates that two AE numbers AEX and AEY can be composed if and only if they have a common integer p j in their decomposition, i.e. AEX = p i p j and AEY = p j p k and that the result of this composition is AEZ = p i p k . By denoting AEX = AE (p i ,p j ) , AEY = AE (p j ,p k ) and AEZ = AE (p i ,p k ) , one can write the composition rule in the form which is a groupoid composition law (it has the same form as for the hydrogen frequencies, see footnote 6). Since in general the decomposition of two AE numbers into two products of two integers p i p j and p j p k is not uniquely reduced to the case p j = 1, the composition of two AE numbers is a multivalued function.
This rule can be applied to the AE3 n values. It shows that any AE3 m number can be composed with another AE3 n number with m ! n, by writing AE3 m = 3 mÀi 3 i , AE3 n = 3 i 3 nÀi , and the result is 3 m+nÀ2i for any i 2 [0, n]. This rule was already obtained in Reed et al. (2004, equation 14) with a demonstration based on string representations. It may be noted that this rule concerns the AE numbers, which are only one aspect of the AE  Table 3 Composition table of the AE3 n operators for n 4.
The composition is given for the column m and the line n by AE3 m AE3 n = {AE3 q }. It is a multivalued composition. For space reasons, the resulting operators AE3 q z are denoted by qz (for example 5a is AE3 5 a).

Composition of the R operators
As introduced in x3, the operators can be viewed as types of paths in a graph or as types of arrows in a groupoid. Both representations are suitable to calculate their composition (see also Appendix A).
(i) The operators AE3 n can be represented by types of arrows in the groupoid C 1 . They define a specific misorientation between the primary crystal 0 and a set of variants of nth generation, n j , equivalently oriented with reference to 0 . As introduced in x4.1, a AE3 m x operator can be composed with a AE3 n y operator by writing AE3 m x = fð 0 . m i Þg and AE3 n y = fð m i . mþn j Þg and the result is the set of operators that link the variant 0 to the variants mþn j : AE3 m xAE3 n y = fð 0 . m i Þð m i . mþn j Þg = fð 0 . mþn j Þg = {AE3 q z}. This composition is multivalued. By determining the variants associated with each operator, the composition can be easily determined without any matrix calculation. This method has already been applied to calculate the composition of the operators that link the orientational variants generated by a structural phase transition (Cayron, 2006).
(ii) The operators AE3 n can be represented by types of paths in the graph of Fig. 7. The method used to calculate their composition is the same whatever the substructure of C 1 chosen for the coding (AE3 ! or AE3 ). It is based on the fact that each string in AE3 ! (or AE3 ) is equivalent to other strings of AE3 ! (or AE3 ) due to the global symmetry. The operators are elements of the quotient structure AE3 ! =< ! (or AE3 =< ), where < ! (or < ) is the equivalence relation on the strings. The composition of two operators AE3 m x and AE3 n y, both written as sets of equivalent strings, is then easily obtained: (a) by choosing one string in each of the two lists, and concatenating the two strings (respecting the simplification rule of the structure), (b) by identifying the resulting string with an operator (which supposes that all the operators have been previously encoded up to the order m + n), and (c) by repeating the process for all the strings of the two operators. In other words, if AE3 m x = {s 1 , . . . , s i , . . . } and AE3 m y = {t 1 , . . . , t j , . . . }, we chose AE3 m x 3 s i and AE3 n y 3 t j , then s i t j 2 AE3 q z, and by repeating the process AE3 m xAE3 n y = {AE3 q z}. The composition is multivalued. This method is very effective due to the highly symmetric character of the graph. We can give some examples with the two encoding structures: the AE3 ! group and the AE3 semigroup.

Calculation with the free group R3 x
The equivalence of the four fractal branches imposes that the letters a, b, c, d used in the free group AE3 ! are equivalent. These letters can be permuted in the strings. For example, the string aca is equivalent to the string dbd by the permutation (a, b, c, d) ! (d, c, b, a). Both strings belong to the same operator. Each operator is encoded by a set of equivalent strings. For example, the operator AE3 = {a, b, c, d} and the operator AE27a = {aba, aca, ada, bab, bcb, bdb, cac, cbc, cdc, dad, dbd, dcd}. The composition of the operators is obtained by concatenation and by applying the simplification rule ii = ø for i 2 {a, b, c, d}. It gives AE27aAE3 = {ab, ac, ad, abab, acab, adab, abac, acac, adac, abad, acad, adad, . . . } = {AE9, AE81+a, AE81d}.

Calculation with the semigroup R3 n
The threefold symmetry of each fractal branch imposes that the indices 1, 2, 3 in the semigroup AE3 are equivalent and can therefore be permuted in the strings. Moreover, owing to the equivalence of the four branches of the fractal, each string of kind ijk . . . l is equivalent to a string of kind 0jk . . . l. Here again, an operator is a set of equivalent strings. For example, the operator AE3 = {0, 1, 2, 3} and the operator AE27a = {111, 222, 333, 211, 311, 122, 322, 133, 233, 011, 022, 033}. The composition of the operators is obtained by concatenation and by applying the simplification rule (10). It gives AE27a AE3 = {11, 1111, 1112, . . . } = {AE9, AE81+a, AE81d}. One may observe that the result is the same as that obtained with the AE3 ! group.
The AE3 m xAE3 n y composition table for m 4 and n 4 can be determined using the AE3 coding in a few seconds using a modern desktop computer, and is reported in Table 3 (the tables for higher orders are available on demand). The composition is multivalued due to the groupoid structure of C 1 . The table is asymmetric because of the non-commutativity of the operators. It can be forced to be symmetric by ignoring the signs of the polar operators, but this would lead to lost information. We will now explain the practical importance of such a table in metallurgy.

Application for the identification of R3 n grain boundaries
Many defects in metals and alloys are formed in grain boundaries. The study of their local crystallographic environment can bring new understanding of their formation. Are the defects in twinned materials situated in random grain boundaries or in the special AE3 n grain boundaries? The response is not always obvious because it is difficult to distinguish a random misorientation from a AE3 n operator with high order n. Indeed, the AE3 n operators are numerous and Acta Cryst. Schematic representation of the AE composition rule for three crystals 1, 2 and 3 linked by misorientations for which the AE numbers are AE 12 , AE 13 and AE 23 . The rule imposes that AE 12 = p 1 p 2 , AE 23 = p 2 p 3 and AE 13 = p 1 p 3 . nearly homogeneously distributed in space for twinning order n ! 5 (see Fig. 8). The identification of the AE3 n grain boundaries is also a key point used to reconstruct the twinrelated domains. As introduced in x1, the TRD size is an important microstructural parameter and many engineers try to improve the mechanical properties by optimizing the elaboration processes in order to produce structures with large TRDs. The determination of the TRDs by EBSD could be a useful aid for these engineering developments. Very often, the TRDs can be easily identified by following the connected paths (i.e. chains) of AE3 boundaries between the grains. However, such a method is not always possible. Indeed, in some architectured materials with low dimensionality, the AE3 chains that link the grains of the TRDs are not at the sample surface but situated under the surface and their locations make their direct identification impossible with the EBSD technique (which is a surface technique). We give some engineering examples to show how the composition Table 3 helps to solve these metallurgical problems. The EBSD maps have been acquired on a LEO-1530 SEM equipped with a Nordlys II CCD camera and have been analysed using the Channel5 software (HKL Technology).

Local environment of a void on a copper line
In microelectronic devices, the reduction of the width of the copper interconnection lines and the use of new barrier and capping layers have led to some new reliability problems. The stress concentrations during the elaboration processes sometimes lead to the formation of voids that can grow and completely damage the lines, this type of defect is called stressinduced voiding (SIV) (Børgesen et al., 1992;Shao et al., 2006). Some relations exist between the global texture and the SIV (Nucci, 1997;Sekiguchi et al., 2003), but the coupling mechanisms between the stresses, the atom migration and the local crystallographic environment remain poorly understood. Local EBSD analyses could help in the understanding of these mechanisms.
In Fig. 10, a stress-induced void has been localized in a copper line. It is situated inside a TRD (here clearly identifiable by the AE3 chains between the grains) in a boundary between four grains (denoted 1, 2, 3 and 4). This TRD has been reconstructed by ignoring the AE3 grain boundaries, i.e. the same colour has been attributed to the grains separated by a AE3 grain boundary. The misorientations between these grains are R 21 = (28.4 , [2 " 1 1 " 4 4]), R 23 = (59.8 , [ " 1 1 " 1 11]), R 34 = (35.9 , [2 " 1 10]), R 41 = (38.4 , [ " 1 10 " 1 1]), R 13 = (48.9 , [ " 1 1 " 4 44]) and R 24 = (38.7 , [ " 2 241]). Three of them are easy to index: R 23 = AE3, R 34 = AE27b and R 41 = AE9, but the three others correspond to higher twinning orders and are more difficult to identify. We determine the maximum possible twinning order by counting the number of AE3 in the shorter AE3 chain that links two grains. Then, we identify some possible solutions by looking at Table  2 (rotation angles and axes). Three solutions are possible for R 24 = (AE81+a, AE81Àa or AE81b), two for R 13 = (AE3 5 +h or AE3 5 Àh), and four for R 12 = (AE3 6 +f, AE3 6 Àf, AE3 6 +g or AE3 6 Àg). Moreover, R 21 = R 24 R 41 = (AE81+aAE9, AE81ÀaAE9 or AE81bAE9), and if one looks at the results of these compositions in Table 3 and compares them with the possible solutions already identified from the EBSD map (AE3 6 +f, AE3 6 Àf, AE3 6 +g or AE3 6 Àg),  EBSD map around a void (the white disc) situated at the junction of four grains. In the figure, the AE3 boundaries are in red, the AE9 in yellow, the AE27 in blue and the AE81 in green. The higher-order AE3 n with n ! 5 and the 'random' boundaries are in black. The grains around the void are linked by the following rotations: R 21 = (28.4 , [2 " 1 1 " 4 4]), R 23 = (59.8 , [ " 1 1 " 1 11]), R 34 = (35.9 , [2 " 1 10]), R 41 = (38.4 , [ " 1 10 " 1 1]), R 13 = (48.9 , [ " 1 1 " 4 44]) and R 24 = (38.7 , [ " 2 241]). The rotation angles and axes are given by the Channel5 software (the reported axes are not the true ones but axes with indices lower than 10 and close to the true ones with a tolerance angle of 5 ). The identification of these experimental rotations to some AE3 n operators requires the use of the groupoid composition Table 3.   1 , [434]). These are close to the AE81+a, AE9 and AE81d operators, respectively. Their composition verifies the groupoid composition Table 3. We conclude that the hillock is a twinned branch of the left TRD and not a new nucleated TRD. one may check that the only solution is R 24 = AE81+a and R 21 = AE3 6 +g. This result could also have been obtained by considering R 21 = R 23 R 31 = (AE3AE3 5 +h or AE3AE3 5 Àh) = ({AE3 6 +l, AE3 6 +b, AE3 6 e, AE3 4 c} or {AE3 6 +g, AE3 6 c, AE3 6 Àx, AE3 4 b}) (not presented in Table 3 for space reasons), which leads to the same unique solution R 21 = AE3 6 +g. This example is very interesting because it proves that it is sometimes possible to unambiguously identify the AE3 n grain boundaries for orders n ! 6 and that the distinction between direct and inverse polar operators is important for the identification method.

Local environment of a hillock on a copper film
Hillocks are defects that may appear in the interconnection lines of electronic devices during the elaboration process or by electromigration damage. Wei et al. (2002) believe that these are likely to be new nucleated grains, whereas Gladkikh et al. (1995) believe that they result from the growth of a neighbouring grain. Let us now consider the hillock on a copper film shown in the EBSD map in Fig. 11. The TRDs were reconstructed by neglecting only the AE3 and AE9 special boundaries. At first glance, the hillock appears as a new nucleated TRD. Indeed, there is no AE3 chain between it and any grain of the three TRDs. However, if we consider the AE3 n operators at higher orders, the hillock seems to be linked to the left TRD by two misorientations close to AE81 forming a triple junction R 12 = AE81+a, R 23 = AE9 and R 13 = AE81d. Are these misorientations close to the AE3 n operators 'by accident' and in fact 'random'? A way to be more confident is to check the coherency of the triple junction, i.e. to check that their composition is in agreement with the theoretical composition Table 3. The verification is done because R 12 R 23 = AE81+aAE9 = {AE3 2 , AE3 4 +a, AE3 4 d, AE3 6 +a, AE3 6 +g, AE3 6 Ài, AE3 6 +l, AE3 6 n, AE3 6 p, AE3 6 +s, AE3 6 +t, AE3 6 +u} 3 AE81d = R 13 . This verification reinforces the probability that the hillock is in fact a multiply twinned branch of the left TRD, and its formation should imply a growth mechanism (with twinning) without nucleation. A deeper statistical study is required to quantify these probabilities as a function of the tolerance angles. The calculations could be based on the generation of triplets of randomly oriented crystals following a method introduced in Cayron et al. (2006).

Application to the reconstruction of TRDs in narrow copper lines
The AE3 twins have in general an electrical resistivity one decade lower than that of the conventional high-angle grain boundaries [and the least resistive AE3 boundaries are those with {111} boundary planes, see Sutton & Balluffi (1995)]. Therefore, some engineering teams increase the size of the TRDs in the interconnection Cu lines to reduce their resistivity. 14 In parallel, some EBSD characterization studies try to reconstruct the TRDs in order to correlate their mean size to the electric measurements. However, this reconstruction is usually performed by neglecting only the AE3 boundaries (Mirpuri & Szpunar, 2004), which gives undervalued results for narrow lines. Indeed, when the lines are narrow (<1 mm), they often have a 'bamboo-like' structure (the grains have the same width as the lines), as shown in Fig. 12. Then, although there is always a AE3 chain between two grains of a TRD, this chain is not always situated at the surface of the sample and becomes invisible in the EBSD maps. When this situation occurs, these grains seem to be separated by a AE3 n boundary with n ! 2 in EBSD, but one must remember that they are in fact connected by a AE3 chain of grains located under the surface which acts as a low resistivity path between these two grains. Therefore, all the 'reasonable' twinning orders n of the AE3 n boundaries should be considered to reconstruct satisfactorily the TRDs. And considering the AE9 in addition to the AE3 boundaries is far from enough. On the copper lines represented in Fig. 12, some TRDs have been reconstructed by neglecting only the AE3 and AE9 boundaries. We have identified many grain boundaries separating these partial TRDs that are of type AE3 n with n ! 3 and that verify Table 3. One frontier is presented in the square on the bottom left of the figure. It is constituted of AE3 n boundaries: R 23 = AE3, R 31 = AE81b and R 21 = AE3 5 g. Such a junction is coherent with Table 3 because it respects R 23 R 31 = AE3AE81b = {AE3 3 b, AE3 5 +e, AE3 5 g, AE3 5 Àh} 3 AE3 5 g = R 21 . Therefore, the two TRDs in the square of Fig. 12 are in fact only one TRD. In a first analysis, for narrow copper lines with a 'bamboo-like' structure, we estimate that the TRD mean size calculated by neglecting only the AE3 boundaries (and not the AE3 n ones with n ! 2) is at least 50% undervalued.
Once a TRD is reconstructed, it may be checked that its corresponding pole figure is in agreement with the simulations of   1 1]). These are close to the AE3, AE81b and AE3 5 g operators, respectively. Their composition verifies the groupoid composition Table 3. The same verification was done along the whole boundary. We conclude that these two TRDs are in fact only one TRD. information about a variant selection mechanism during the recrystallization. For instance, the experimental pole figure of a TRD reconstructed in a copper film and oriented in the [111] direction of the primary crystal is reported in Fig. 13(a); its comparison to the theoretical pole Fig. 13(b) proves that the twinning order of this TRD is n = 2 and that one main branch of the fractal is missing [the fractal is constituted of only three branches, such as the ones visible in Fig. 3(a)]. More thorough analysis would be required to know if the fourth branch is missing or if it is situated below the surface.

Conclusions
The idea of a groupoid has been introduced. Three types of groupoid elements have been described: the objects, the operations (also called arrows) between these objects and the operators that are types of operations (they are written as sets of equivalent operations).
An assembly constituted of one crystal with its four twinned variants can be represented by a groupoid C (0+1) . More generally, an assembly of multiply twinned crystals linked by AE3 n operators with n 2 N can be represented geometrically by a three-dimensional fractal and algebraically by the groupoid C 1 = C (0+1+ . . . +n) with n ! 1. The algebraic details of C 1 were not studied but symmetry considerations on the threedimensional fractal have allowed us to determine some of its properties. For example, the general formulae giving the number of variants and the number of operators as functions of the twinning order n were established. The AE3 ! free group introduced by Reed et al. (2004) is a substructure of C 1 . Other substructures can be used. For example, we have introduced the AE3 semigroup that leads to a different coding (with a different simplification rule), but to a two-dimensional fractal graph similar to the one that could be obtained with the AE3 ! free group. Whatever the substructure we use for the coding (AE3 ! or AE3 ), the AE3 n operators can be written as sets of equivalent strings. The composition of two operators can then be easily determined without any matrix calculation by concatenating all the couples of strings chosen in their respective set and by applying the simplification rule of the substructure. The composition of operators is multivalued. This property can be understood by taking into consideration that the general structure of multiple twinning is a groupoid and not simply a group. The composition table has been reported for orders n 4 (and was determined at higher orders).
Some metallurgical examples were given showing how this table could be used in EBSD for a better identification of the local crystallographic environment of some defects in multiply twinned materials or for improving the reconstruction of the TRDs. We have stressed that the distinction between polar and ambivalent AE3 n operators is important in the identification method. The approach we have followed is general and can be used to compute an automatic recognition of the AE3 n grain boundaries and an automatic reconstruction of the TRDs.
The present study can be applied to treat multiple twinning in non-cubic materials because it is not based on quaternions. However, the general theory treating any series of cycles of phase transitions remains to be established. The answer is probably to be found in algebraic structures more elaborated than groupoids such as the cohomology of groupoids (Connes, 1990). This research will require the help of mathematicians.   Comparison between (a) an experimental pole figure of a TRD observed by EBSD in a copper film with a primary crystal oriented with [111]//z, where z is the normal to the sample surface, and (b) a theoretical pole figure of an assembly of multiply twinned crystals with n 2. In this simulation, only three of the four branches have been considered (branch 0 would give many spots that do not appear in the experimental pattern). Some theoretical variants of order n = 2 are also absent (they correspond to the differences between the theoretical and experimental patterns).
To conclude, we would like to encourage crystallographers interested in phase transitions, quasicrystals or other problems involving partial symmetries to consider groupoids as a very useful algebraic tool that advantageously enlarges the idea of groups.

APPENDIX A
In this Appendix, we present a simple non-crystallographic geometrical case, with local and global symmetries, in order to clarify the idea of 'operator' and its link with the groupoids. Let us imagine a ball moving on a flat surface in a structure constituted by isosceles triangles that have vertical mirror symmetry (Fig. 14). How can we encode the different movements of the ball on such a structure?

A1. Operators viewed as types of paths in a graph
One may notice that the ball has five possibilities at each vertex: left or right while going to the bottom (L or R), left or right while going to the top (L À1 , R À1 ), or coming to rest (denoted ;). This classification of possibilities is due to the local symmetries of the structure. Of course, it can be created a free group G 1 constituted by the letters L, R, L À1 and R À1 and all the infinite combinations of these letters. 15 The corresponding infinite graph would be similar to Fig. 14(a), imagined to be repeated an infinity of times. Any trajectory of the ball is represented by a string of the free group G 1 . But there are two problems with such a representation: (a) such a group is infinite whereas Fig. 14(a) is finite, and (b) in such a graph the local and global symmetries are not represented. Indeed, one may assume that different trajectories are of the same type due to the local symmetries of the structure. From Fig. 14(a), we can effectively notice that two strings are equivalent by global permutation: partial commutativity: partial commutativity: ðL; R; L À1 ; R À1 Þ ! ðR; L; R À1 ; L À1 Þ LR ¼ RL The rule (16) is the algebraic expression of the global vertical mirror symmetry. Conditions (17) and (18) are the algebraic expressions of the local vertical mirror symmetries. 16 These rules allow the creation of an equivalence relation < on the different trajectories of the ball. The distinct types of trajectories are given by the elements of the quotient set G 1 /<. There is no reason for G 1 /< to be a group, and in fact this structure appears to be quite complex. We propose in the following a method to define its elements and their compositions.
Owing to the local vertical mirror symmetry, the actions L and R are equivalent; there is no way to predict that the ball will move to the left or to the right. One may then accept that the action 'moving down' on the structure of Fig. 14 is not a classical application, but a bivalued function that we will denote O 1 1 ðxÞ = {L, R}, for any position x of the ball on the structure. Similarly, the action 'moving up' is denoted O 1 1 À1 = {L À1 , R À1 }. The operator 'coming to rest' is O 1 0 = ;. These three operators will be called operators of first generation. These operators are represented in Fig. 14(b). More generally, one can form the operators of order n, O n j , by forming all the strings with n letters in the set {L, R, L À1 , R À1 } and finding the strings that are equivalent by the conditions (16), (17)  From these examples, the composition of operators (i.e. types of trajectories, i.e. elements of G 1 /<) appears to be multivalued. Some readers may be shocked by the use of a multivalued composition; a way to accept it is to realize that such a composition naturally results from an underlying groupoid structure.

A2. Operators viewed as types of arrows in a groupoid
In the previous paragraph, we have voluntarily ignored the positions x on which the operations were applied (denoted as 0 to 9 in Fig. 14a). A pair of positions, denoted (x . y), can be viewed as an arrow from x to y. The arrows form a pair groupoid. The composition law is (x . y)(y . z) = (x . z) and each arrow has an inverse given by (x . y) À1 = (y . x). The operators appear as types of arrows. Here, the different arrows constituting the operators are geometrically obvious. Generally, they can be found by expressing the local symmetries. In the present example, one could use the vertical mirror symmetry that is the permutation (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ! (0, 2, 1, 5, 4, 3, 9, 8, 7, 6) and the translations that put in correspondence some points of the structure, for example the translation (0 . 1) expressed by the application (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ! (1, 3, 4, 6, 7, 8, ø, ø, ø, ø) on the left and one arrow of type (0 . z) in O n j on the right. This method has the advantage of representing the objects, the operators and their composition in the same table. The geometrical structure of Fig. 14 can then be algebraically written in a unique groupoid composition table (Fig. 15), which can be viewed as a generalization of the composition table of groups.
In this example, we have seen that, due to the local symmetries, there exist some types of paths or arrows that we have called operators. These operators can be written as sets of equivalent strings in a graph or as sets of equivalent arrows in a groupoid. They can be composed by string concatenation (in a graph) or by respecting the composition law between the arrows (in a groupoid). Whatever we choose as the method, this composition is multivalued. Composition tables can be calculated. Graphs and strings are very effective for high symmetric problems; groupoids and arrows are less effective but more general and can be applied to less symmetric problems. For example, it would be the same formalism if some isosceles triangles in Fig. 14 were suppressed or if they were changed by equilateral triangles.