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. 440446
Section 3.3.8. Lattice aspects of twinning^{a}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D52056 Aachen, Germany, and ^{b}MineralogischPetrologisches Institut, Universität Bonn, D53113 Bonn, Germany 
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:

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