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

International Tables for Crystallography (2013). Vol. D, ch. 3.3, pp. 469-477

Section 3.3.11. Effect of twinning in reciprocal space

Th. Hahna* and H. Klapperb

aInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail:  hahn@xtal.rwth-aachen.de

3.3.11. Effect of twinning in reciprocal space

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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 X-ray, neutron and electron diffraction of simple and multiple twins, including structure determinations on twinned crystals.

It should be emphasized that for these investigations only `single-crystal' 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 high-to-low-symmetry 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[link] and Chapter 3.4[link] ).

In many cases twins are difficult to detect. Methods for identifying twins and their twin laws in direct space (morphology, optics, X-ray topography etc.) are given in Section 3.3.6.1[link]. Tests for twinning in the diffraction records, particularly statistical tests, and `warning signs' of twinning are contained in Sections 7.5–7.7 of Herbst-Irmer (2006)[link] and in the publication by Kahlenberg (1999)[link].

3.3.11.1. Basic features of twin diffraction records

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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.

  • (i) Whereas in direct space the various twin components are spatially separated, in reciprocal space all their diffraction records are superimposed with a common origin.

  • (ii) The orientation relations of the twin-related diffraction patterns in reciprocal space and of the corresponding domains in real space (`twin laws') are the same.

  • (iii) For a multiple twin with n orientation states [cf. Fig. 3.3.3.1[link](c)] all individual diffraction patterns are superimposed, whereby all twin components of the same orientation state contribute to the same diffraction pattern [e.g. polysynthetic twins, cf. Figs. 3.3.3.1[link](b) and 3.3.6.13[link]].

  • (iv) The diffraction pattern of a twin is – apart from the intensities of reflections – independent of the size, distribution and shape of the twin domains, as well as of their boundaries (contact relations). The intensities are governed by the volume fractions of the orientation states.

  • (v) Friedel's rule also applies to twins, i.e. the diffraction record of a twinned crystal is centrosymmetric, provided anomalous scattering is negligible. This implies that `inversion twins' can only be detected by diffraction experiments if anomalous scattering is sufficiently high.

  • (vi) Since the twin partners are `macroscopic' individuals (cf. the definition of twin in Section 3.3.2.1[link]), the superposition of two or more twin-related diffraction patterns involves the addition of intensities, not of structure factors, as would be the case in superstructures, in so-called `cell twinning' (cf. Takeuchi, 1997[link]), or in `modular' crystal structures [cf. Ferraris (2004)[link] and Note (1) in Section 3.3.2.4[link]].

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[link] and 3.3.4[link]), in reciprocal space the degree of overlapping of the various diffraction patterns is the dominant criterion and leads to the following subdivisions:

  • (a) General (non-merohedral, inclined-lattice) twins: no general overlap of the twin-related diffraction patterns, hence `split reflections'; Section 3.3.11.2[link].

  • (b) Twins by (strict) merohedry (parallel-lattice Σ1 twins): complete and exact overlap (coincidence) of the individual twin-related diffraction patterns, no `split reflections'; Section 3.3.11.3[link].

  • (c) Twins by reticular merohedry (partially parallel-lattice twins, Σ > 1 twins): exact but partial overlap of twin-related diffraction patterns, no `split reflections' but also `single reflections'; Section 3.3.11.4[link].

  • (d) Twins by pseudo(-reticular) merohedry (pseudo-parallel-lattice twins): approximate complete or partial overlap of the twin-related diffraction patterns. Special problem: partially overlapped (partially split) reflections; Section 3.3.11.5[link].

3.3.11.2. General (non-merohedral, inclined-lattice) twins

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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[link](a):

  • (i) at least one lattice row (crystal edge) [uvw] common to both partners I and II, either parallel or antiparallel;

  • (ii) at least two lattice planes (crystal faces) (hkl)I and ±(hkl)II, one from each partner, parallel.

For the diffraction record this has the following consequences:

  • (i) If the common lattice row is a (rational) twofold twin axis [uvw] (binary twin), all lattice planes (hkl) belonging to the zone [uvw] are mapped upon themselves by the twin operation. They fulfil the zone condition uh + vk + wl = 0. In reciprocal space these planes form a single reciprocal plane (hkl) normal to the twofold twin axis in direct space. In the diffraction pattern of the twin this (zero-layer) reciprocal plane is common (coincident) to the diffraction patterns of the two twin components [for an illustration see Massa (2004)[link], Fig. 11.7].

    If the common lattice row is not a rational twofold twin axis, e.g. as in the case of twinning by the Kantennormalengesetz (complex twin, irrational twofold twin axis, cf. Fig. 3.3.2.3[link]), reciprocal-lattice points (diffraction spots) common to the two individual diffraction patterns do not occur. This underlines the limiting character of this intergrowth as a kind of twinning.

  • (ii) If the common lattice plane is a twin reflection plane (hkl), this plane is mapped upon itself by the twin operation, and so is the perpendicular reciprocal lattice row nh, nk, nl (representing all orders of the reflection hkl). If the common lattice plane (hkl) results from an – in general irrational – twofold twin axis normal to it, exactly the same reciprocal row nh, nk, nl is common to the diffraction records of both twin individuals. This is due to the centrosymmetry of the direct and the reciprocal lattices, which leads to the equivalence of the twin mirror plane (hkl) and the twofold twin axis normal to it.

    For the twinning by the Kantennormalengesetz with irrational twin reflection plane (cf. Fig. 3.3.2.3[link]) there are again no diffraction spots (reciprocal-lattice points) common to the two individual diffraction patterns.

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)[link], Fig. 11.8]. In general, however, the large number of non-overlapped 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 non-overlapped symmetry-equivalent reflections hklI and hklII). Also, the structure determination is usually unproblematic since it can be done with the standard methods using the many non-overlapped 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 two-dimensional Weissenberg or precession films or a two-dimensional detector which shows the splitting of the twin reflections in the reciprocal-lattice 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:

  • (i) Dovetail twins of gypsum, eigensymmetry 2y/my, twin reflection plane (100), cf. Section 3.3.4.1[link] and Fig. 3.3.4.1[link]. Here the reciprocal-lattice row [h00] is common to both diffraction patterns. Moreover, since the twin operation also maps the lattice plane (010) upon itself, the reciprocal row [0k0] is also common to both patterns, i.e. both rows exhibit coincidence of the reflections of both partners. All other reflections are `split reflections'.

  • (ii) A similar coincidence behaviour occurs for the frequent (110) reflection twins of orthorhombic crystals (cf. Fig. 3.3.4.2[link], twin domains 1 and 2). They map the direct-lattice planes (110) and (001) upon themselves, and thus the reciprocal-lattice rows hh0 and 00l are common to both diffraction patterns, with coinciding reflections.

  • (iii) A particularly illustrative example with both types of overlapping reflections are the triclinic plagioclase feldspars, with two famous twin laws (cf. Section 3.3.6.12[link]):

    Albite law: rational twin reflection plane (010); hence, the reflections of the (one-dimensional) reciprocal row [0k0] of both twin partners are superimposed in the twin diffraction record.

    Pericline law: rational twofold twin rotation axis [010]; the lattice planes (h0l) belong to this zone. Hence, the reflections of the two-dimensional reciprocal layer h0l of both twin partners are superimposed in the diffraction set.

Descriptions of structure determinations (with diagrams) on general (non-merohedral) twins are contained in the following publications: Herbst-Irmer (2006[link], Sections 7.8.5 and 7.8.6), Massa (2004[link], Chapter 11), Dunitz (1964[link], pseudo-orthorhombic monoclinic twins) and Herbstein (1965[link], molecular crystal with albite twin law); many further examples exist in the literature.

3.3.11.3. Twinning by (strict) merohedry (parallel-lattice twins, Σ1 merohedral twins)

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In direct space, Σ1 twins by merohedry (cf. Sections 3.3.8.2[link] and 3.3.9[link]) are characterized by exact parallelism of the lattices of all twin domains, i.e. the coincidence-site 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)[link]. 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 twin-related intensities of all twin-domain states, according to their volume fraction.

For diffraction experiments and structure determinations on merohedral Σ1 twins the above relations have the following consequences:

  • (i) Because of the complete overlap of the diffraction patterns and the parallelism of the symmetry elements, twinning by merohedry cannot be identified by diffraction experiments. In many cases structure refinements of merohedrally twinned crystals can simulate disorder. Hence, chemically or physically `strange' disorder can be a hint for twinning, as are refinements with unusually large R values.

  • (ii) There are no `strange' space-group extinctions as a clue to (strict) merohedral twinning, except for the following four cases:

    • (a) In space group [P2_1/a\bar 3] (No. 205), merohedral twinning causes `violations' of the extinctions in the {0kl} reflection sets, cf. Klapper & Hahn (2012)[link], p. 83. Note that this `violation' occurs for all volume ratios of the twins.

    • (b) In contrast, only for volume ratios exact or close to 1:1 (i.e. by simulation of the higher Laue class), the extinction rules of the following three space groups do not exist in the higher-symmetry Laue class: P42/n (86), I41/a (88) and [I2_1/a\bar 3] (206) [cf. IT A, Part 3, Table 3.1.4.1[link] and Massa (2004)[link], p. 154]. These four cases are strong clues for twins by merohedry.

  • (iii) A strong indication for merohedral twinning can often be obtained if in several specimens of the crystal under investigation some groups of reflections vary in intensity whereas other stay constant. This is due to the three `twin diffraction cases' A, B1 and B2, which differ in the way their twin-related reflection sets are `affected' by the given twin law.

    • Case A: the twin-related reflection sets (face forms) are symmetry equivalent, i.e. their superimposed intensities are not affected by the twinning;

    • Case B2: the twin-related face forms are `opposites' {hkl} and [\{\bar h\bar k\bar l\}], i.e. they differ only in their anomalous-scattering contributions, usually small;

    • Case B1: the twin-related reflections sets are neither symmetry equivalent nor opposite, i.e. their intensities are (often quite strongly) different and their superposition varies with the volume ratio. B1 reflection sets will vary from one specimen to the next, whereas the intensities of A and (in many cases) B2 reflections are relatively unchanged. This opens the possibility of attempting a structure determination only with A and B2 (if anomalous scattering is small) reflection sets. A detailed description of these reflection sets (face forms) and their use, with examples, is presented by Klapper & Hahn (2010)[link]. Their Table 9 lists the three `twin diffraction cases' A, B1 and B2 for the seven reflection sets of all 63 Σ1 merohedral twin laws.

  • (iv) There is another useful subdivision of merohedral twins: Types I and II by Catti & Ferraris (1976)[link].

    Type I twins: the twin element belongs to the Laue symmetry of a noncentrosymmetric crystal (but, of course, not to its eigen­symmetry point group). These twins can always be described as inversion twins, since the inversion is always part of their twin coset. Note that triclinic, monoclinic and orthorhombic crystals can only form merohedral type I twins, because these crystal systems have only one Laue class, whereas the higher-symmetry systems have two Laue classes. For structure determinations type I twins present no problems because, according to Friedel's rule, every diffraction record is centrosymmetric (neglecting anomalous scattering). Determinations of the `absolute structure' and the `absolute polarity' (cf. Klapper & Hahn, 2010[link]), however, are not possible because inversion twins exhibit superposition of `opposite' reflection sets (case B2 sets). Comparison of the measured and calculated (for an untwinned crystal) anomalous-scattering contributions often permits determination of the `Flack factor' (Flack, 1983[link]) and, hence, of the volume ratio of the two enantiomorphic or antipolar twin states. Type I merohedral twins can occur in any of the 21 noncentrosymmetric point groups (28 possible twin laws, cf. Table 9 in Klapper & Hahn, 2010[link]).

    Type II twins: These twins can only occur in all (centrosymmetric and noncentrosymmetric) point groups of the lower-symmetry Laue classes of the tetragonal (4/m), hexagonal ([\bar 3], [\bar 32/m], 6/m) and cubic ([2/m\bar 3]) crystal family (13 point groups, 35 possible twin laws). Here the twin element is a symmetry element of the higher-symmetry Laue class (4/mmm, [\bar 32/m],9 6/mmm, [m\bar 3m]), thus producing superposition of non-equivalent face forms, i.e. diffraction case B1, with volume-fraction dependent intensities, even for negligible anomalous scattering. If these intensities are used for structure refinement, disorder or possibly a hypothetical new modification is simulated. For volume fractions near 1:1 a higher-symmetry space group can even be found [for exceptions see topic (ii) above]. The best procedure – apart from finding an untwinned crystal – is to use refinement programs which include the refinement of the twin volume ratio.

  • (v) A special, interesting case exists for some naturally occurring amino acids, proteins, nucleic acids and related molecules. They are chiral and occur only with one handedness, i.e. these molecules are `enantiomerically pure'. As a consequence crystals of these molecules are `enantiomorphically pure' and occur with one handedness only. These crystals can, of course, occur only in one of the 11 enantiomorphic crystal classes and their twin laws can only be rotation twins. This combination, however, restricts the occurrence of merohedral Σ1 twins of these crystals still further: merohedral Σ1 rotation twins can only occur in tetragonal, trigonal, hexagonal and cubic crystals with the following seven point groups and structural settings (P = hexagonal P lattice, R = rhombohedral R lattice):[3\ (P),\ 321\ (P),\ 312\ (P),\ 3\ (R) ,\ 4,\ 6\ {\rm and}\ 23.]The resulting number of Σ1 twin laws is reduced from 63 (cf. Table 9 in Klapper & Hahn, 2010[link]) to nine:[\eqalign{&3 (P)\rightarrow321(P),\ 3(P)\rightarrow312(P),\ 3(P) \rightarrow6(P),\cr &3(R) \rightarrow32(R), \ 321(P) \rightarrow622(P),\ 312(P) \rightarrow622(P),\cr &4\rightarrow422,\ 6\rightarrow622,\ 23\rightarrow432.}]The dominant role of trigonal P and R Σ1 twins is apparent. Note that the three twin laws 3(R) [\rightarrow] 6(P), 3(R) [\rightarrow] 312(P) and 32(R) [\rightarrow] 622(P) are Σ3 obverse/reverse twin laws (cf. Table 13 in Klapper & Hahn, 2012[link]).

  • (vi) Structure determinations of merohedrally twinned crystals are described, with examples and diagrams, in publications by Buerger (1960a)[link], Herbst-Irmer & Sheldrick (1998)[link], Herbst-Irmer (2006[link], Section 7.8.1), Sheldrick (2008)[link], Kahlenberg (1999)[link], Massa (2004)[link], Ferraris (2004)[link] and Klapper & Hahn (2010)[link]. Relevant computer programs are listed in Section 3.3.11.6[link].

3.3.11.4. Twinning by reticular merohedry (partially-parallel-lattice twins, Σ > 1 merohedral twins)

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3.3.11.4.1. General survey

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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 twin-related 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[link] and 3.3.9.2.3[link], 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 `two-dimensional', i.e. the same reciprocal-lattice 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[link],b[link], 2003[link]).

In the following, all reciprocal lattices of the Σm twins will be referred to the reciprocal coincidence lattice with basis vectors am*, bm*, cm* (cf. Figs. 3.3.11.1[link]–6) and not to the lattice of one twin component. This coordinate system has the great advantage that all reciprocal-lattice points of both domain states D(I) and D(II) appear with integral indices HKL.

[Figure 3.3.11.1]

Figure 3.3.11.1 | top | pdf |

Rhombohedral Σ3 (reverse/obverse) twins in direct space, described with hexagonal axes, viewed down the common c axis. (a) Obverse rhombohedral R lattice, with lattice points 0, 0, 0; 2/3, 1/3, 1/3; 1/3, 2/3, 2/3. (b) Reverse R lattice, with lattice points 0, 0, 0; 1/3, 2/3, 1/3; 2/3, 1/3, 2/3; dashed lines indicate the lower edges of the primitive rhombohedron. (c) Composite symmetry of the twin (stereographic projection), consisting of the eigensymmetry elements of the rhombohedron (red) and the set of alternative Σ3 twin elements (green); any one of the latter describes the superposition of the two R lattices in the Σ3 twin. The coincidence (twin) lattice is a `diluted' hexagonal P lattice of index 3 and formed by points marked `0' [cf. Arnold (2005)[link]; this article also contains further diagrams and transformation matrices].

In the diffraction record of Σm twins, four types of reflections (`coincidence cases') can be distinguished:

  • (i) reflection HKL is `doubly coincident', i.e. both twin-related D(I) and D(II) reflections are non-extinct and superimposed;

  • (ii) reflection HKL is present in D(I) and extinct in D(II), i.e HKL is a `single' D(I) reflection;

  • (iii) reflection HKL is present in D(II) and extinct in D(I), i.e. HKL is a `single' D(II) reflection;

  • (iv) reflection HKL is `doubly extinct', i.e. absent in D(I) as well as in D(II).

The relative frequencies of these four types as a function of the twin index Σm are presented in Table 3.3.11.1[link]. 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 `non-space-group 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[link]), 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.

Table 3.3.11.1| top | pdf |
Relative frequencies of the four coincidence cases (i)–(iv) for the general Σm twins and the specific twins Σ3, Σ5 and Σ7 treated in this chapter.

For each twin case the sum of all fractions is 1.

Coincidence casesΣmΣ1Σ3Σ5Σ7
(i) Coincidence pair 1/m2 1 1/9 1/25 1/49
(ii) Single reflections of domain D(I) (m − 1)/m2 0 2/9 4/25 6/49
(iii) Single reflections of domain D(II) (m − 1)/m2 0 2/9 4/25 6/49
(iv) Doubly extinct reflections (m − 1)2/m2 0 4/9 16/25 36/49

3.3.11.4.2. The four Σm merohedral twin families

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It is sensible to subdivide the Σm merohedral twins into four families as follows:

  • (i) The 63 Σ1 twins: complete and exact coincidence of all reflections of the two twin components, treated in Section 3.3.11.3[link].

  • (ii) The 11 rhombohedral and 11 cubic Σ3 twins (obverse/reverse twins, spinel twins) with the four twin laws, represented by (hexagonal/cubic axes)[\displaylines{2'[001]/[111],\quad m'(0001)/(111),\cr 2'[210]/[2 \bar 1 \bar 1],\quad m'(10 \bar 1 0)/(2 \bar 1\bar 1)\semi}](for the full cosets of these twin laws see Tables 12 and 13 of Klapper & Hahn, 2012[link]). All these twin laws transform an obverse rhombohedron with integral reflection conditions −h + k + l = 3n into a reverse rhombohedron with reflection conditions hk + l = 3n and vice versa. These four twin laws are different in point groups 3 (rhombohedral) and 23 (cubic), appear in different pairs in point groups [\bar 3], 32, 3m and [2/m\bar 3], 432, [\bar 43m] and form one twin law in groups [\bar 32/m] and [4/m\bar 32/m]. These Σ3 twins are by far the most frequent among the Σm twins by reticular merohedry. In particular, the cubic spinel twins occur very often in minerals and inorganic compounds.

    Illustrations of the direct lattices of the two twin partners and the l = 3n, l = 3n + 1, l = 3n + 2 layers (hexagonal axes) of the reciprocal twin lattice are given in Figs. 3.3.11.1[link] and 3.3.11.2[link]. One realizes that for l = 3n only doubly coincident and doubly extinct [types (i) and (iv)] but no single reflections [types (ii) and (iii)] occur, whereas for l = 3n + 1 and l = 3n + 2 only single and doubly extinct reflections occur.

    [Figure 3.3.11.2]

    Figure 3.3.11.2 | top | pdf |

    Reciprocal lattice of the rhombohedral Σ3 twins for the layers l = 3n (a), l = 3n + 1 (b), l = 3n + 2 (c), described with hexagonal axes. The orientation corresponds to that of the direct lattice in Fig. 3.3.11.1[link]. Circles: obverse twin domain I with reflection condition −h + k + l = 3n; crosses: reverse twin domain II with reflection condition hk + l = 3n. In (a) all reflections of the two domains coincide; in (b) and (c) no coincidences, only `single' reflections occur. In the large cell 00l, 30l, 33l, 03l with l = 0 and 3, which contains 27 points of the reciprocal twin lattice, three `doubly coincident' points 000, 110, 220, six `single' points of domains I and II each, and 12 `doubly extinct' points occur, in accordance with Table 3.3.11.1[link]; the latter are characteristic `non-space-group' extinctions.

    Full details, references and examples, particularly of the somewhat complicated cubic Σ3 twins, are presented in Sections 3 and 4, Tables 4 and 5, and Appendices A and B of Klapper & Hahn (2012)[link]. A survey of structure determinations and refinements of Σ3 obverse/reverse twins is provided by Herbst-Irmer (2006)[link] and in Section 3.6 of Klapper & Hahn (2012)[link]. Relevant computer programs are listed in Section 3.3.11.6[link] below.

  • (iii) The 12 tetragonal Σ5 twins. Among the tetragonal twins with parallel c axes the smallest possible lattice index is Σ = h2 + k2 = 5 or Σ = (h2 + k2)/2 = 10/2 = 5 (similarly for Σ = u2 + v2), with twin symmetry (reduced oriented composite symmetry) 4/m 2′/m′ 2′/m′ and the following twin elements (cf. Fig. 3.3.11.3[link]):

    [Figure 3.3.11.3]

    Figure 3.3.11.3 | top | pdf |

    Tetragonal lattices (ab planes, common c axis pointing upwards) of twin domain I (start domain, lattice points small circles, right-handed green unit cell a1, b1, c1), of the Σ5 twin-related domain II (small crosses, left-handed blue unit cell a2, b2, c2) and the Σ5 coincidence lattice (large black points, right-handed red unit cell aT, bT, cT). The four alternative twin reflection planes m′(120), [m'(2\overline 10)], m′(310) and [m'(\overline 130)] are indicated by dashed lines. The coordinate axes a2, b2, c2 of domain II (blue) are defined by the reflection plane m′(120). The right-handed yellow unit cell a3, b3, c3 of domain II is obtained from a1, b1, c1 by a clockwise rotation of ϕ = 2 arctan  (1/2) = 53.13° around the tetragonal c axis. This cell is commonly used in structure determinations. (From Hahn & Klapper, 2012[link].)

    m′(120), [m'( 2 \bar10)], 2′[120], [2'[ 2 \bar10]] (second position of the twin point-group symbol);

    m′(310), [m'( \bar 1 30)], 2′[310], [2'[ \bar 130]] (third position of the twin point-group symbol).

    In each of the two centrosymmetric tetragonal point groups 4/m and 4/m 2/m 2/m these eight elements form one twin law, whereas in the five noncentrosymmetric groups they split in various fashions into two twin laws, resulting in a total of 12 tetragonal Σ5 twin laws (see Table 8 in Klapper & Hahn, 2012[link]).

    The hk0 plane of the reciprocal lattice of the Σ5 twins is shown in Fig. 3.3.11.4[link]. One recognizes that in the 5 × 5 cell formed by the points 000, 500, 550, 050 (referred to aT* and bT*) one doubly coincident reflection 000 (large black circles), four `single' D(I) (open circles), four `single' D(II) (crosses) and 16 `doubly extinct' reciprocal lattice points (reflections) occur, thus confirming the entries in Table 3.3.11.1[link].

    [Figure 3.3.11.4]

    Figure 3.3.11.4 | top | pdf |

    Reciprocal tetragonal lattices (hk0 lattice planes) of twin domain I (start domain, lattice points small circles) and of the Σ5 twin-related domain II (small crosses). The reciprocal lattice of the (direct-space) Σ5 coincidence lattice is represented by the grid of small squares. The unit cells, their handedness and their colours correspond to those of the direct lattices in Fig. 3.3.11.3[link]. In the square formed by the four reciprocal coincidence points 000, [2\overline 10], 310, 120 (in terms of a1*, b1*) or 000, 500, 550, 050 (in terms of aT*, bT*) there are four `single' points of twin domains I and II each, one `coincident' point 000 and, with reference to aT*, bT*, 16 `extinct' reciprocal points (cf. Table 3.3.11.1[link]). These strange `non-space-group extinctions' are characteristic of the Σ5 twin law. (From Hahn & Klapper, 2012[link].)

    Full details of the tetragonal Σ5 twins and their treatment in X-ray diffraction work can be found in Section 5 and Appendix C1 of Klapper & Hahn (2012)[link]. Twins of this type are very rare; not more than six structure determinations are known. These are also quoted in the above mentioned paper.

    All 12 twin laws of a tetragonal Σm family follow the rule m = h2 + k2 = u2 + v2:

    Σ5 with twin elements m′(120) and 2′[120],

    Σ13 with twin elements m′(230) and 2′[230],

    Σ17 with twin elements m′(140) and 2′[140],

    Σ25 with twin elements m′(340) and 2′[340] etc.

    Further Σm values, up to 50, are listed by Grimmer (2003)[link], Table 1. Concrete tetragonal twin cases with Σm higher than 5 are not known.

  • (iv) The 14 hexagonal and 26 trigonal Σ7 twins. The Σ7 twins of the hexagonal crystal family (hexagonal and trigonal crystal systems, hexagonal P and rhombohedral R lattices) are the hexagonal equivalents of the tetragonal Σ5 twins treated above. Hence, many features agree: the Σ7 twins are also parallel c-axes twins, i.e. they preserve the hexagonal or trigonal axis and, thus, the twinning is `two-dimensional' (Fig. 3.3.11.5[link]). Twins with inclined main axes have been derived by Grimmer (1989a[link]), but real examples have not yet been observed.

    [Figure 3.3.11.5]

    Figure 3.3.11.5 | top | pdf |

    Hexagonal lattices (ab planes, common c axis pointing upwards) of twin domain I (start domain, lattice points small circles, right-handed green unit cell a1, b1, c1), of the Σ7 twin-related domain II (small crosses, left-handed blue unit cell a2, b2, c2) and of the Σ7 coincidence lattice (large black points, right-handed red unit cell aT, bT, cT). The six alternative twin reflection planes [m'(12\bar 30)], [m'(\overline 3120)], [m'(2\overline 310)], [m'(\overline 5410)], [m'(1\overline 540)] and [m'(41\overline 50)] are indicated by dashed lines. The coordinate axes a2, b2, c2 of domain II (blue) are defined by the reflection plane [m'(12\overline 30)]. The right-handed yellow unit cell a3, b3, c3 of domain II is obtained from a1, b1, c1 by a clockwise rotation of ϕ = 120° + 2 arcsin [(1/2)(3/7)1/2] = 120° + 38.2° = 158.2° around the hexagonal c axis. This cell is commonly used in structure determinations. (From Hahn & Klapper, 2012[link].)

    The smallest possible lattice index is Σ = h2 + hk + k2 = 7 or Σ = (h2 + hk + k2)/3 = 21/3 = 7 (similarly for Σ = u2uv + v2) for a twin with twin symmetry (reduced oriented composite symmetry) 6/m 2′/m′ 2′/m′ and the following four twin laws, represented by:

    [m'\{12\bar 30\}], 2′<450> (second position of the twin point-group symbol),

    [m'\{\bar 5410\}], [2'\langle2\bar 10\rangle] (third position of the twin point-group symbol).

    In each of the two hexagonal centrosymmetric point groups 6/m and 6/m 2/m 2/m these four twin laws form one twin law, whereas in the six noncentrosymmetric point groups (structural settings) they combine in different ways into two twin laws.

    In the three trigonal centrosymmetric point groups (structural settings) [\bar 3], [\bar 32/m1] and [\bar 3 12/m], they combine into two twin laws each, whereas in the remaining five trigonal structural settings all four twin laws are different, leading to 14 hexagonal and 26 trigonal possible Σ7 twins. Details of these twin cases are presented in Table 10 of Klapper & Hahn (2012)[link].

    The reciprocal lattice of the hexagonal Σ7 twins is shown in Fig. 3.3.11.6[link]. As for the Σ5 twins, it confirms the data in Table 3.3.11.1[link]: the 7 × 7 cell formed by the coincident lattice points 000, 700, 770, 070 contains one `doubly coincident' point 000, six `single' points of twin domains D(I) and D(II) each and, if referred to aT* and bT*, 36 `doubly extinct' points.

    [Figure 3.3.11.6]

    Figure 3.3.11.6 | top | pdf |

    Reciprocal hexagonal lattices (hk0 lattice planes) of twin domain I (start domain, lattice points small circles) and of the Σ7 twin-related domain II (small crosses). The reciprocal lattice of the (direct-space) Σ7 coincidence lattice is represented by the grid of small rhombuses. The unit cells, their handedness and their colours correspond to those of the direct lattices in Fig. 3.3.11.5[link]. In the large cell formed by the four reciprocal coincidence points 000, [3\overline 10], 410, 120 (in terms of a1*, b1*) or 000, 700, 770, 070 (in terms of aT*, bT*) there are six `single' points of twin domains I and II each, one `coincident' point 000 and, with reference to aT*, bT*, 36 `extinct' reciprocal points (cf. Table 3.3.11.1[link]). These strange `non-space-group extinctions' are characteristic of the Σ7 twin law. (From Hahn & Klapper, 2012[link].)

    Section 6 and Appendix C2 of the paper by Klapper & Hahn (2012)[link] contains full details of the hexagonal Σ7 twins and their treatment in diffraction and structure work. Twins of this kind have not been found so far. This also applies to the Σ7 twins of crystals with a rhombohedral R lattice [the frequent rhombohedral Σ3 twins are treated above under (ii)]. Their (somewhat complicated) twin phenomena are also described in the above-mentioned paper.

    Beyond the `starting type' Σ7, the following twins also belong to this family:

    Σ13 with [m'\{31 \bar 40\}] or 2′<140> (second position of the point-group symbol),

    Σ19 with [m'\{32 \bar 5 0\}] or 2′<250> (second position of the point-group symbol), similarly for higher Σ values.

    Twins of this hexagonal/trigonal Σm `family' are not known.

3.3.11.5. Pseudo-merohedral twins

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3.3.11.5.1. General remarks

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This type of twins includes both pseudo-merohedral Σ1 and Σm > 1 twins. It refers to small deviations from strict full or partial lattice coincidence of the twin partners. Pseudo-merohedral 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 B-centred orthorhombic crystal), orthorhombic crystals with nearly b/a[\sqrt 3] (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 twin-related reciprocal-lattice points (reflections) are not symmetry equivalent and have different structure-factor 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[link] and 3.3.8.5[link], introducing the terms `twin obliquity' ω and lattice index [j] as given by Friedel (1926)[link]. Further treatments are contained in Sections 3.3.9.2[link] and 3.3.9.3[link]. The number of possible pseudo-merohedral 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 pseudo-merohedral twins are important:

  • (i) The splitting of the reflections in the entire accessible twin diffraction record is so small that they appear as `untwinned' or, at least, as unresolvable reflections, possibly with wrong unit cell and symmetry. In these cases very often twinning will not even be recognized and, as a result, the structure determination may fail. Even if the twinning is suspected as `strictly merohedral', the structure determination may fail or present severe problems because the twin law is of a non-merohedral and difficult type.

  • (ii) The opposite case is a diffraction record in which many reflections (those with higher 2θ values) are clearly split, even if reflections with small 2θ values are not resolved. The resolved pairs of reflections can then be used to determine the twin law, but not the volume fraction of the twin because their F moduli are different. With the intensity data of the stronger reflection set (i.e. of the larger twin partner), however, the structure can often be solved, as described in Section 3.3.11.2[link].

  • (iii) A problem is also provided by twins in which all (or nearly all) reflections of the diffraction record overlap, but are not fully `split'. Here, twinning is usually recognizable but the main problem is the separation of the many overlapped unresolved reflections in order to arrive at the two `true' sets of intensities with which a successful structure determination can be performed. There are several computer programs which are designed to split overlapped reflections, quoted in Section 3.3.11.6[link], but in many cases in addition intuition by the researcher and examination of several twinned crystals, with different volume fractions, is required.

3.3.11.5.2. Example: pseudohexagonal (cyclic) twins of ortho­rhombic crystals (pseudo-coincident Σ3 twins)

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This kind of twinning with m′(110) and [m'(1 \bar 10)] or pseudo-threefold twin axis is often observed in orthorhombic crystals with nearly ortho-hexagonal metric, i.e. with an axial ratio b/a[\sqrt 3] = tan 60° (pseudo-hexagonal axis c). Prominent examples are aragonite (Fig. 3.3.2.4[link]; b/a = tan 58.1°), K2SO4 (Fig. 3.3.6.9[link]; b/a = tan 60.18°), NH4LiSO4 (Fig. 3.3.7.2[link]; b/a = tan 59.99°) and (NH4)2SO4 (b/a = tan 60.85°), all having a primitive (not a C-centred, see below) lattice. They often appear as `growth-sector twins' with three sector domains (Figs. 3.3.2.4[link] and 3.3.7.2[link]a), six sector domains (Fig. 3.3.6.9[link] with equal orientation of opposite sectors) or with a more-or-less irregular distribution of the sectors (Fig. 3.3.7.2[link]b). A morphologically idealized triple-sector twin is shown in Fig. 3.3.11.7[link](a). Apart from the reflection intensities, the diffraction patterns of these triple twins, showing a pseudo-hexagonal 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 [m'(1 \bar 10)], occur.

  • (i) Pseudo-hexagonal (cyclic) twins based on an orthorhombic P lattice. For an easier understanding of the superposition and coincidence behaviour of twin-related reflections of the domains 1, 2 and 3 shown in Fig. 3.3.11.7[link](a), the basis vectors a2, b2, c2 and a3, b3, c3 are generated from the basis vectors a1, b1, c1 of start domain 1 for exact hexagonal metric, as follows:[\eqalignno{{\bf a}_2&=-\textstyle{1\over 2}({\bf a}_1 +{\bf b}_1),\quad {\bf b}_2=\textstyle{1\over 2}(-3{\bf a}_1+{\bf b}_1),\quad {\bf c}_2={\bf c}_1,&(1)\cr {\bf a}_3 &= \textstyle{1\over 2}(-{\bf a}_1+{\bf b}_1),\quad {\bf b}_3=\textstyle{1\over 2}(3{\bf a}_1+ {\bf b}_1), \quad {\bf c}_3={\bf c}_1.&(2)}]Correspondingly for Miller indices:[\eqalignno{h_2 &= -\textstyle{1\over 2}(h_1 +k_1),\quad k_2=\textstyle{1\over 2}(-3h_1+k_1),\quad l_2=l_1,&(3)\cr h_3&=\textstyle{1\over 2}(-h_1+k_1),\quad k_3=\textstyle{1\over 2}(3h_1+k_1),\quad l_3=l_1.&(4)}]For a pseudo-hexagonal metric the coefficients of the transformation matrices deviate more or less from their ideal values ±1/2 and ±3/2. It is easily recognized that coincidence of reflections is obtained only if all indices h2k2l2 and h3k3l3 are integer, i.e. if the condition h1 + k1 = 2N (and correspondingly h2 + k2 = 2N and h3 + k3 = 2N) is fulfilled. Thus, for pseudo-hexagonal twins with a primitive lattice (no lattice extinctions) and parallel c axes, two sets of reflections are distinguished:

    • (a) Reflections hkl with h + k = 2N: for exact hexagonal metric with [b/a=\sqrt 3=\tan60^\circ] these reflections of all domains coincide exactly, as shown in Fig. 3.3.11.7[link](b). For a pseudo-hexagonal metric these reflections are, depending on the deviation from exact hexagonality, more-or-less split into three spots (for three domains states) or two spots (for two domain states).

      [Figure 3.3.11.7]

      Figure 3.3.11.7 | top | pdf |

      (a) Morphologically idealized pseudo-hexagonal threefold sector twin of an orthorhombic crystal with b/a = tan 58.5°, generated by the two symmetry-equivalent twin mirror planes (110) and [(1 \bar 1 0)]. In real growth twins the gap (here about 5°) is closed with an irregular twin boundary (cf. Fig. 3.3.2.4[link]). For b/a > tan 60° there is a (formal) overlap of domains 2 and 3. (b) Reciprocal-lattice plane hkl (l = 0, 1, 2, …) of the threefold pseudo-hexagonal twin shown in (a), viewed along the pseudo-hexagonal c* axis, but drawn with exactly b/a = tan 60°. It shows the superposition of the three lattices of domain 1 (black dots), domain 2 (crosses) and domain 3 (open circles). Lattice points with h + k = 2N are threefold coincident, the others are single. This holds for all layers l = 0, ±1, ±2, …. Note that the twin reflection planes m′(110) and [m'(1 \bar 1 0)] in direct space correspond to the planes m*(310) and [m^*( \bar 3 10)] in reciprocal space, respectively.

    • (b) Reflections hkl with h + k = 2N + 1: these are `single' for each domain state, forming an exact hexagonal twin diffraction pattern for b/a = [\sqrt 3] and a more-or-less distorted hexagonal pattern for [b/a\ne \sqrt3].

    The occurrence of both single and (pseudo-)coincident reflections in the diffraction pattern classifies the pseudo-hexagonal twins of orthorhombic crystals with a P lattice as twins by reticular pseudo-merohedry of index 2. Note that the entire diffraction record is pseudo-hexagonal without any `non-space-group' extinctions (cf. Section 3.3.11.4.1[link]).

  • (ii) Special reflection sets, their coincidences and diffraction cases for pseudo-hexagonal orthorhombic centrosymmetry. Since the twin elements m′(110) and [m'(1 \bar 1 0) ] do not belong to the eigensymmetry of the untwinned crystal, the twin-related reflections (of domains 1 + 2, 1 + 3, as well as 2 + 3) are in general not symmetry-equivalent and have different F moduli (diffraction cases B1). There are, however, two special types of reflections which are equivalent and coincident or pseudo-coincident and have equal F moduli (diffraction cases A):

    • (a) Twin-related reflection pairs [(hhl)_1/(\bar h\bar hl)_2] (twin element m′(110), equation (3) above, common zone axis [[1\bar10]]) and [(h\bar hl)_1/(\bar hhl)_3] (twin element [m'(1 \bar 1 0)], equation (4) above, common zone axis [110]): The reflections of each of these pairs coincide exactly (for any b/a ratio) and have equal F moduli. The twin-related reflections of the third domain, [(0_\prime 2h_\prime l)_3] for m′(110) and [(0_\prime \overline {2h} _\prime l)_2] for [m'(1 \bar 1 0)], are more-or-less separated from the coincident reflections of the pairs [(hhl)_1/ (\bar h\bar h l)_2] and [(h \bar h l)_1/(\bar hhl)_3] and have different F moduli. This is also the case for (hh0)1 and [(h\bar {h}0)_1], i.e. the reflecting planes parallel to the twin mirror planes. Furthermore, for reflections 00l, the diffraction spots of all three domains coincide and have equal F moduli. Thus, if both m′(110) and [m'(1 \bar 10)] are active and form a triple twin, the triple diffraction spot consists of reflections of type hhl, [\bar h\bar hl] (coincident, equal F moduli) and [0_\prime2h_\prime l] (split, different F modulus).

    • (b) Twin-related pairs [(h_\prime\overline {3 h}_\prime l)_1/(h_\prime\overline{3h}_\prime l)_2] [twin element m′(110), equation (3) above, slightly different zone axes [310] pseudo-perpendicular to the twin plane] and [(h_\prime 3h_\prime l)_1/(h_\prime 3h_\prime l)_3] [[m'(1 \bar 1 0)], equation (4) above, slightly different zone axes [[3\bar10]] pseudo-perpendicular to the twin plane]: The twin-related reflections of these pairs are split but have equal F moduli (diffraction case A).10 The twin-related reflection of the third domain is [(\overline {2h} 0l)_3] for m′(110) and [(\overline {2h} 0l)_2] for [m'(1 \bar 1 0)]. Thus, if both m′(110) and [m'(1 \bar 1 0)] are active and form a triple twin, the triple diffraction spot consists of three split reflections of type [h_\prime\overline{3h}_\prime l], [h_\prime3h_\prime l] and [\overline{2h}_\prime0_\prime l], the former two having equal F moduli. This case is presented by the Laue-diffraction spot S in Fig. 3.3.11.8[link](b).

      [Figure 3.3.11.8]

      Figure 3.3.11.8 | top | pdf |

      (a) (001) plate of ammonium lithium sulfate, NH4LiSO4, (about 11 mm diameter, 0.8 mm thick) between crossed polarizers, exhibiting sectorial pseudo-hexagonal growth twinning. The circle around the triple point of the three domains marks the area intercepted by the white synchroton beam for recording a topographic Laue pattern. (b) Part of the topographic Laue pattern taken with white-beam synchrotron radiation (SRS Daresbury) along the pseudo-hexagonal axis from the encircled region of the twin shown in (a). (For the whole Laue pattern see Fig. 4 of Docherty et al., 1988[link]). The Laue spots appear as small topographs of the 120° twin sectors hit by the white beam. For reflections with h + k = 2N the three sectors are simultaneously imaged; for all other reflections (h + k = 2N + 1) only single sectors appear. The affiliation to the three domains is recognized from the orientation of their 120° sectors. Owing to the very small metrical deviation b/a = tan 59.95° from exact hexagonality (gap angle 0.15°) there is practically no splitting of the coincident reflections, i.e. no mutual tilt of their lattice planes and, thus, no separation or partial overlapping of their topographic images. Note the vertical mirror plane m. In the triple Laue spot S the reflection types described in Section 3.3.11.5.2[link] paragraph (b) are present, i.e. two reflections of type [h_\prime 3h_\prime l] imaging domains 2 and 3 (symmetry-equivalent, equal F moduli) and reflection of type [2h_\prime 0_\prime l], imaging domain 1 (different F modulus).

    The superposition of the reciprocal P lattices of the three twin domains, corresponding to those shown in Fig. 3.3.11.7[link](a), however with exact b/a = tan 60°, is presented in Fig. 3.3.11.7[link](b). It shows single (h + k = 2N + 1) and triply coincident (h + k = 2N) lattice points, as derived by the above transformations. Figs. 3.3.11.8[link](a,b) present an illustration of the diffraction and coincidence features of triply pseudo-hexagonal growth-twinned orthorhombic NH4LiSO4 (point group 2mm, gap angle ≃ 0.1°). It shows a section of an X-ray topographic Laue pattern of the circular region centred on the meeting point of the domains [shown in Fig. 3.3.11.7[link](a); splitting of the reflections is practically zero]. A similar topographic Laue pattern of a ferroelastically introduced twin (100) lamellae embedded in a (001) plate of pseudo-hexagonal (NH4)2SO4 (point group mmm, overlap angle 5°, very strong splitting) is presented by Docherty et al. (1988)[link]. In this case, the Laue diffraction pattern does not show any symmetry, due to the presence of only two non-symmetrically arranged domain states.

    An illustrative description of the diffraction patterns of pseudo-hexagonal twins of crystals with a primitive orthorhombic P lattice and its application to the X-ray topographic study of twins and their boundaries in NH4LiSO4 is given by Klapper (1987[link], pp. 372–378). Another study of pseudo-hexagonal twinning of NH4LiSO4 and (NH4)2SO4, using synchroton Laue topography, is presented by Docherty et al. (1988)[link].

  • (iii) Pseudo-hexagonal twins of orthorhombic crystals based on a C lattice. In the orthorhombic C lattice, reflections of all domains with h + k = 2N + 1 are extinct, and therefore single reflections do not occur in the superposition of the diffraction patterns of the three domains. Only triply (pseudo-)coincident reflections with h + k = 2N occur. This is an example of a Σ1 twinning by pseudo-merohedry. All other diffraction features are the same as those quoted above for twins of orthorhombic crystals with a P lattice.

  • (iv) Additional remark. The pseudo-hexagonal triple growth twins are morphologically often described by a pseudo-threefold rotation with angle φ′ = 2 arctan(b/a), clockwise (φ′+) for domain pair [1\rightarrow 2] and anticlockwise (φ′−) for domain pair [1\rightarrow3], both together approximately filling the full 360° circle. The basis-vector transformations (1) and (2) given above for m′(110) and [m'(1 \bar 1 0)] have to be modified as follows: The vectors a2 and a3 remain unchanged, whereas b2 and b3 are inverted into their opposites −b2 and −b3, thus leading to right-handed coordinate systems for domains 2 and 3. Similarly for h2, k2, h3 and k3 [equations (3) and (4) above]. Since the reversal of the axis b is part of the eigensymmetry of point group mmm, the effect of φ′+ and φ′− is the same as that of m′(110) and [m'(1 \bar 10)]. Thus each of the three twin elements m′(110), 2′irrat ≃ [310] and φ′+ represents in point group mmm the same orientation relation for domain pair [1\rightarrow2]. Similarly: [m'(1 \bar 1 0)], 2′irrat[[3 \bar 1 0]] and φ′− for domain pair [1\rightarrow3].

    For the hemihedral point groups 222 and mm2, m2m, 2mm these results have to be modified. For group 222 the two reflection twin elements lead to opposite handedness of domains 1 and 2 and 1 and 3, but equal handedness of domains 2 and 3, whereas twin elements 2′irrat and φ′± provide equal handedness of all three domains. For point group mm2 etc. the situation is more complicated due to the different settings with the polar axis along a, b or c, which physically lead to polar domains with different head-to-tail, head-to-head and tail-to-tail domain boundaries. These cases are not further analysed here.

    Concerning the diffraction patterns of pseudo-hexagonal twins of hemihedral orthorhombic crystals: The splitting of diffraction spots is a matter of the lattice metric and independent of the point group. Regarding reflection intensities: among the triply split reflections only reflections of sets {hhl} and [\{h_\prime 3h_\prime l\}] may undergo a change from diffraction case A in point groups mmm to diffraction case B2 in the hemihedral groups.

3.3.11.6. Programs for structure determinations with twinned crystals

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Programs for the determination of crystal structures from merohedral and pseudo-merohedral twins are, among others, SFLS (Eitel & Bärnighausen, 1986[link]), TWINXLI (Hahn & Massa, 1997[link]), TWIN 3.0 (Kahlenberg & Messner, 2001[link]), CRYSTALS 12 (Betteridge et al., 2003[link]), JANA2006 (Petricek et al., 2006[link]), DIRAX (A. J. M. Van Duisenberg, University of Utrecht, The Netherlands; e-mail: duisenberg@chem.uu.nl) and especially SHELXL (Sheldrick, 1997[link]).

Detailed descriptions of the (widely used) SHELXL program system for structure determinations and for refinements of merohedrally and pseudo-merohedrally twinned crystals are provided by Herbst-Irmer & Sheldrick (1998[link], 2002[link]), by Guzei et al. (2012)[link] and, in particular, by Herbst-Irmer (2006)[link].

Textbook descriptions of structure determinations of twins are provided by Buerger (1960a)[link], Massa (2004)[link] and Ferraris (2004)[link]. Systematic analyses of the diffraction intensities of all Σ1, Σ3, Σ5 and Σ7 merohedral twins are contained in two publications by Klapper & Hahn (2010[link], 2012[link]).

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[link] , can be used for the investigation of twins. Section 3.3.12[link] thus forms a bridge between the present chapter on Twinning and the following chapter on Domain structures.

References

Betteridge, P. W., Carruthers, J. R., Cooper, R. I., Prout, K. & Watkin, D. J. (2003). CRYSTALS version 12: software for guided crystal structure analysis. J. Appl. Cryst. 36, 1487.
Buerger, M. J. (1960a). Crystal Structure Analysis, especially ch. 3. New York: Wiley.
Catti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray crystal structure determination. Acta Cryst. A32, 163–165.
Docherty, R., El-Korashy, A., Jennissen, H.-D., Klapper, H., Roberts, K. J. & Scheffen-Lauenroth, T. (1988). Synchrotron Laue topography studies of pseudo-hexagonal twinning. J. Appl. Cryst. 21, 406–415.
Dunitz, J. D. (1964). The interpretation of pseudo-orthorhombic diffraction patterns. Acta Cryst. 17, 1299–1304.
Eitel, M. & Bärnighausen, H. (1968). Programm zur Verfeinerung von Strukturen verzwillingter Kristalle. Universität Karlsruhe, Germany.
Ferraris, G. (2004). Modularity at crystal-scale twinning. In Crystallography of Modular Materials, edited by G. Ferraris, E. Makovicky & St. Merlino, ch. 5, pp. 280–308. Oxford University Press.
Flack, H. D. (1983). On enantiomorph-polarity estimation. Acta Cryst. A39, 876–881.
Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: Berger-Levrault. [Reprinted (1964). Paris: Blanchard].
Grimmer, H. (1989a). Systematic determination of coincidence orientations for all hexagonal lattices with axial ratio c/a in a given interval. Acta Cryst. A45, 320–325.
Grimmer, H. (1989b). Coincidence orientations of grains in rhombo­hedral materials. Acta Cryst. A45, 505–523.
Grimmer, H. (2003). Determination of all misorientations of tetragonal lattices with low multiplicity; connection with Mallard's rule of twinning. Acta Cryst. A59, 287–296.
Guzei, I., Herbst-Irmer, R., Munyaneza, A. & Darkwa, J. (2012). Detailed example of the identification and crystallographic analysis of a pseudo-merohedrally twinned crystal. Acta Cryst. B68, 150–157.
Hahn, F. & Massa, W. (1997). TWINXL, Programm zur Aufbereitung von Datensätzen verzwillingter Kristalle. Phillips-Universität, Germany (e-mail: massa@chemie.uni-marburg.de).
Herbst-Irmer, R. (2006). Twinning. In Crystal Structure Refinement, edited by P. Müller, ch. 7, pp. 106–149. Oxford University Press.
Herbst-Irmer, R. & Sheldrick, G. M. (1998). Refinement of twinned structures with SHELXL97. Acta Cryst. B54, 443–449.
Herbst-Irmer, R. & Sheldrick, G. M. (2002). Refinement of obverse/reverse twins. Acta Cryst. B58, 477–481.
Herbstein, F. H. (1965). Twinned crystals. III. γ-o-Nitroaniline. Acta Cryst. 19, 590–595 (with references to Parts I and II).
Kahlenberg, V. (1999). Application and comparison of different tests on twinning by merohedry. Acta Cryst. B55, 745–751.
Kahlenberg, V. & Messner, T. (2001). TWIN3.0 – a program for testing twinning by merohedry. J. Appl. Cryst. 34, 405.
Klapper, H. (1987). X-ray topography of twinned crystals. In Progress in Crystal Growth and Characterization, Vol. 14, edited by P. Krishna. pp. 367–401. Oxford: Pergamon.
Klapper, H. & Hahn, Th. (2010). The application of eigensymmetries of face forms to anomalous scattering and twinning by merohedry in X-ray diffraction. Acta Cryst. A66, 327–346.
Klapper, H. & Hahn, Th. (2012). The application of eigensymmetries of face forms to X-ray diffraction intensities of crystals twinned by `reticular merohedry'. Acta Cryst. A68, 82–109.
Massa, W. (2004). Crystal Structure Determination, 2nd ed., pp. 148–156. Berlin: Springer.
Petricek, V., Dusek, M. & Palatinus, L. (2006). JANA: Crystallographic Computing System for Standard and Modulated Structures. http://jana.fzu.cz . (This website contains also a number of manuscripts on twinning.)
Sheldrick, G. M. (1997). SHELXL97. Programs for Crystal Structure Analysis (release 97–2). University of Göttingen, Germany.
Sheldrick, G. M. (2008). A short history of SHELX. Acta Cryst. A64, 112–122.
Takeuchi, Y. (1997). Tropochemical Cell-Twinning. Tokyo: Terra Scientific Publishing Company.








































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