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. 469477
Section 3.3.11. Effect of twinning in reciprocal space^{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 twinning phenomena were considered in direct space, in particular the orientation relations of the twin components (twin domains), i.e. the twin laws, and the contact relations, i.e. the interfaces between twin partners (twin boundaries). The present section extends these considerations to reciprocal space, i.e. to Xray, neutron and electron diffraction of simple and multiple twins, including structure determinations on twinned crystals.
It should be emphasized that for these investigations only `singlecrystal' methods are applicable. Powder diffraction is not capable of revealing an existing twin, but is very useful for characterizing a crystal aggregate independent of twinning. In particular, powder patterns can reveal hightolowsymmetry phase transitions, e.g. by splitting of diffraction peaks, which are the prerequisites of transformation twins and domain structures (cf. Section 3.3.7.2 and Chapter 3.4 ).
In many cases twins are difficult to detect. Methods for identifying twins and their twin laws in direct space (morphology, optics, Xray topography etc.) are given in Section 3.3.6.1. Tests for twinning in the diffraction records, particularly statistical tests, and `warning signs' of twinning are contained in Sections 7.5–7.7 of HerbstIrmer (2006) and in the publication by Kahlenberg (1999).
In the following, the basic features of the diffraction patterns of twins are summarized. They hold for any type of twin and even for intergrowths with any orientation relation, e.g. bicrystals or irregular grain assemblies.
Classification of twinning in reciprocal space. Whereas in direct space the twin elements and the morphological features are the main criteria of twin classification (cf. Sections 3.3.3 and 3.3.4), in reciprocal space the degree of overlapping of the various diffraction patterns is the dominant criterion and leads to the following subdivisions:
This case refers to the most general type of twin, where the lattices of the two twin partners have one of the two minimal elements in common which are required by the definition of a `crystallographic orientation relation' in Section 3.3.2.2(a):
For the diffraction record this has the following consequences:
It is possible that other layer lines of the twin diffraction record also show coincidence features, due to metrical `accidents' in the lattice constants of the crystal [illustration: Massa (2004), Fig. 11.8]. In general, however, the large number of nonoverlapped reflections permits easy determination of the twin law, as well as determination of the volume ratio of the twins (comparison of the intensities of pairs of nonoverlapped symmetryequivalent reflections hkl_{I} and hkl_{II}). Also, the structure determination is usually unproblematic since it can be done with the standard methods using the many nonoverlapped reflections of the twin partner with the larger volume. If needed, even the overlapped reflections can be split into the two twin partners with the help of the previously determined volume fraction. Most twins belong to this type of `general twins'.
For their investigation it is highly advisable to use twodimensional Weissenberg or precession films or a twodimensional detector which shows the splitting of the twin reflections in the reciprocallattice plane.
Note that actual twins may have – in addition to the minimum requirements given above – accidental reflection coincidences which depend on the twin law, the eigensymmetry and the metric of the crystal under consideration, as shown in the following examples.
Examples:
Descriptions of structure determinations (with diagrams) on general (nonmerohedral) twins are contained in the following publications: HerbstIrmer (2006, Sections 7.8.5 and 7.8.6), Massa (2004, Chapter 11), Dunitz (1964, pseudoorthorhombic monoclinic twins) and Herbstein (1965, molecular crystal with albite twin law); many further examples exist in the literature.
In direct space, Σ1 twins by merohedry (cf. Sections 3.3.8.2 and 3.3.9) are characterized by exact parallelism of the lattices of all twin domains, i.e. the coincidencesite lattice (CSL) or twin lattice is identical with the lattice of the untwinned crystal. The twin element, on the other hand, is a symmetry element of the point group of the lattice (holohedry) but not of the crystal, i.e. the (merohedral) crystal point group is a proper subgroup of its holohedry. This implies that only binary twin elements are possible: inversion, twofold rotation or reflection twins of index [2]. There are in total 63 possible merohedral twin laws in the 35 structural settings of the 26 merohedral crystal classes. They are completely listed and characterized in Appendices A and D (Tables 7–9) of Klapper & Hahn (2010). Note the large number of twin laws in the hexagonal crystal family.^{8}
In reciprocal space the counterpart to lattice parallelism, described above, is the complete and exact superposition, with a common origin, of the diffraction patterns of all twin domains, i.e. each reflection in the diffraction record of the twin is the superposition of the twinrelated intensities of all twindomain states, according to their volume fraction.
For diffraction experiments and structure determinations on merohedral Σ1 twins the above relations have the following consequences:

3.3.11.4. Twinning by reticular merohedry (partiallyparallellattice twins, Σ > 1 merohedral twins)
The Σ3, Σ5 and Σ7 `twin families', described in the subsequent sections, can be considered as an extension of the Σ1 merohedral twins, treated in the previous section: instead of the complete coincidence of the twinrelated lattices, only partial coincidence exists, i.e. the coincidence lattice is a diluted sublattice of the untwinned crystal lattice. As explained in detail in Sections 3.3.8.2 and 3.3.9.2.3, the sublattice index Σm is the volume ratio of the primitive cells of the twin lattice and the untwinned lattice. Hence 1/m is the degree of `dilution' of the original crystal lattice. In reciprocal space the index Σm is the volume ratio of the primitive cells of the untwinned and the twin lattice, i.e. the degree of `densification' of the reciprocal twin lattice with respect to the reciprocal crystal lattice.
Twins by reticular merohedry, as discussed here, include only Σm twins with `parallel main axes', which can occur for all lattice parameters and all axial ratios c/a (or rhombohedral angles α) of a crystal. In these twins all twin elements preserve the orientation of the three, four or sixfold main symmetry axis of the lattice. This implies that only twin reflections planes m′ parallel and twofold twin axes 2′ perpendicular to this main axis are possible; for rhombohedral and cubic Σ3 twins a plane normal and a twofold twin axis parallel to the (odd!) threefold axis are also possible. Hence, the twinning is always `twodimensional', i.e. the same reciprocallattice layers of the twin partners are superimposed in the reciprocal lattice of the twin.
In contrast to the twins with parallel axes, twins with `inclined axes' can exist which exhibit coincidences only for special values of the axial ratio c/a or of the rhombohedral angle α. Cases of this type have been derived by Grimmer (1989a,b, 2003).
In the following, all reciprocal lattices of the Σm twins will be referred to the reciprocal coincidence lattice with basis vectors a_{m}*, b_{m}*, c_{m}* (cf. Figs. 3.3.11.1–6) and not to the lattice of one twin component. This coordinate system has the great advantage that all reciprocallattice points of both domain states D(I) and D(II) appear with integral indices HKL.
In the diffraction record of Σm twins, four types of reflections (`coincidence cases') can be distinguished:
The relative frequencies of these four types as a function of the twin index Σm are presented in Table 3.3.11.1. Noteworthy features are the strong reduction with m of the `doubly coincident' case (i) and the strong increase of the `doubly extinct' reflections (iv). The latter case (iv) represents strange `nonspacegroup extinctions'. They are an indication of the presence of twinning by reticular merohedry and often a help in determining the twin law [cf. Buerger (1960a), ch. 5]. The `single' reflections (ii) and (iii) can be used to determine the volume fractions of the two domain states and, if numerous enough, the crystal structure can even be determined with the stronger of the two sets.

It is sensible to subdivide the Σm merohedral twins into four families as follows:
This type of twins includes both pseudomerohedral Σ1 and Σm > 1 twins. It refers to small deviations from strict full or partial lattice coincidence of the twin partners. Pseudomerohedral twins occur if metrical (but not necessary structural) pseudosymmetries occur in a crystal and, hence, the twin element belongs to the symmetry of a higher crystal system. Frequently occurring typical examples are monoclinic crystals with the angle β very close to 90° (simulating an orthorhombic crystal) or with approximately a = c (simulating a Bcentred orthorhombic crystal), orthorhombic crystals with nearly b/a ≃ (simulating a hexagonal crystal, cf. examples below), or a tetragonal crystal with c/a ≃ 1 (simulating a cubic crystal). In contrast to twins by strict Σ1 merohedry the twin operation is not a symmetry operation of the holohedry of the untwinned crystal. Thus the twinrelated reciprocallattice points (reflections) are not symmetry equivalent and have different structurefactor moduli (diffraction case B1). Exceptions are those few reflections which are mapped by the twin operation upon themselves or their opposites.
Twins of this type in direct space, with many examples, have been discussed already in Sections 3.3.8.4 and 3.3.8.5, introducing the terms `twin obliquity' ω and lattice index [j] as given by Friedel (1926). Further treatments are contained in Sections 3.3.9.2 and 3.3.9.3. The number of possible pseudomerohedral twin laws is very great (much larger than the `strict' merohedral Σ1 and Σ > 1 twin laws) and, hence, only examples and general rules for the experimental work can be given here.
In reciprocal space, three border cases of pseudomerohedral twins are important:

3.3.11.5.2. Example: pseudohexagonal (cyclic) twins of orthorhombic crystals (pseudocoincident Σ3 twins)
This kind of twinning with m′(110) and or pseudothreefold twin axis is often observed in orthorhombic crystals with nearly orthohexagonal metric, i.e. with an axial ratio b/a ≃ = tan 60° (pseudohexagonal axis c). Prominent examples are aragonite (Fig. 3.3.2.4; b/a = tan 58.1°), K_{2}SO_{4} (Fig. 3.3.6.9; b/a = tan 60.18°), NH_{4}LiSO_{4} (Fig. 3.3.7.2; b/a = tan 59.99°) and (NH_{4})_{2}SO_{4} (b/a = tan 60.85°), all having a primitive (not a Ccentred, see below) lattice. They often appear as `growthsector twins' with three sector domains (Figs. 3.3.2.4 and 3.3.7.2a), six sector domains (Fig. 3.3.6.9 with equal orientation of opposite sectors) or with a moreorless irregular distribution of the sectors (Fig. 3.3.7.2b). A morphologically idealized triplesector twin is shown in Fig. 3.3.11.7(a). Apart from the reflection intensities, the diffraction patterns of these triple twins, showing a pseudohexagonal arrangement of diffraction spots, are independent of the size, shape and distribution of the sector domains. Sometimes twins with only two domain orientations, related by m′(110) or , occur.

Programs for the determination of crystal structures from merohedral and pseudomerohedral twins are, among others, SFLS (Eitel & Bärnighausen, 1986), TWINXLI (Hahn & Massa, 1997), TWIN 3.0 (Kahlenberg & Messner, 2001), CRYSTALS 12 (Betteridge et al., 2003), JANA2006 (Petricek et al., 2006), DIRAX (A. J. M. Van Duisenberg, University of Utrecht, The Netherlands; email: duisenberg@chem.uu.nl) and especially SHELXL (Sheldrick, 1997).
Detailed descriptions of the (widely used) SHELXL program system for structure determinations and for refinements of merohedrally and pseudomerohedrally twinned crystals are provided by HerbstIrmer & Sheldrick (1998, 2002), by Guzei et al. (2012) and, in particular, by HerbstIrmer (2006).
Textbook descriptions of structure determinations of twins are provided by Buerger (1960a), Massa (2004) and Ferraris (2004). Systematic analyses of the diffraction intensities of all Σ1, Σ3, Σ5 and Σ7 merohedral twins are contained in two publications by Klapper & Hahn (2010, 2012).
The following note on domain structures is supplied by V. Janovec. It describes briefly and clearly how strategies used in the study of `domain structures', treated in Chapter 3.4 , can be used for the investigation of twins. Section 3.3.12 thus forms a bridge between the present chapter on Twinning and the following chapter on Domain structures.
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