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. 446450
Section 3.3.9. Twinning by merohedry and pseudomerohedry^{a}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D52056 Aachen, Germany, and ^{b}MineralogischPetrologisches Institut, Universität Bonn, D53113 Bonn, Germany 
We now resume the discussion of Section 3.3.8 on threedimensional coincidence lattices and pseudocoincidence lattices and apply it to actual cases of twinning, i.e. we treat in the present section twinning by merohedry (`macles par mériédrie') and twinning by pseudomerohedry (`macles par pseudomériédrie'), both for lattice index and , as introduced by Friedel (1926, p. 434). Often (strict) merohedral twins are called `parallellattice twins' or `twins with parallel axes'. Donnay & Donnay (1974) have introduced the terms twinning by twinlattice symmetry (TLS) for merohedral twinning and twinning by twinlattice quasisymmetry (TLQS) for pseudomerohedral twinning, but we shall use here the original terms introduced by Friedel.
In the context of twinning, the term `merohedry' is applied with two different meanings which should be clearly distinguished in order to avoid confusion. The two cases are:

Both kinds of merohedries and pseudomerohedries were used by Mallard (1879) and especially by Friedel (1904, 1926) and the French School in their treatment of twinning. Based on the concepts of exact coincidence (merohedry, [j] = 1), approximate coincidence (pseudomerohedry) and partial coincidence (twin lattice index ), four major categories of `triperiodic' twins were distinguished by Friedel and are explained below.
Here the lattices of all twin partners are parallel and coincide exactly. Consequently, all twin operations are symmetry operations of the lattice point symmetry (holohedral point group), but not of the point group of the structure. Here the term `merohedry' refers to point groups only, i.e. to Case (1) above. Experimentally, in singlecrystal Xray diffraction diagrams all reflections coincide exactly, and tensorial properties of second rank (e.g. birefingence, dielectricity, electrical conductivity) are not influenced by this kind of twinning.
Typical examples of merohedral twins are:
These twins are characterized by pseudomerohedry of point groups, Case (1) in Section 3.3.9.1. The following examples are based on structural pseudosymmetry and consequently also on lattice pseudosymmetry, either as the result of phase transformations or of structural relationships:
For these twins with partial but exact coincidence Friedel has coined the terms `twinning by reticular merohedry' or `by lattice merohedry'. Here the term merohedry refers only to the sublattice, i.e. to Case (2) above. Typical examples with and were described in Section 3.3.8.3. In addition to the sublattice relations, it is reasonable to include the pointgroup relations as well. Four examples are presented:
This type can be derived from the category in Section 3.3.9.2.3 above by relaxation of the condition of exact lattice coincidence, resulting in two nearly, but not exactly, coinciding lattices (pseudocoincidence, cf. Section 3.3.8.4). In this sense, the two Sections 3.3.9.2.3 and 3.3.9.2.4 are analogous to the two Sections 3.3.9.2.1 and 3.3.9.2.2.
The following four examples are characteristic of this group:

In conclusion, it is pointed out that the above four categories of twins, described in Sections 3.3.9.2.1 to 3.3.9.2.4, refer only to cases with exact or approximate threedimensional lattice coincidence (triperiodic twins). Twins with only two or onedimensional lattice coincidence (diperiodic or monoperiodic twins) [e.g. the (100) reflection twins of gypsum and the (101) rutile twins] belong to other categories, cf. Section 3.3.8.2. The examples above have shown that for triperiodic twins structural pseudosymmetries are an essential feature, whereas purely metrical (lattice) pseudosymmetries are not a sufficient tool in explaining and predicting twinning, as is evidenced in particular by the case of staurolite, discussed above in detail.
The large group of pseudomerohedral twins (irrespective of their lattice index) contains a very important subset which is characterized by the physical property ferroelasticity. Ferroelastic twins result from a real or virtual phase transition involving a change of the crystal family (crystal system). These transitions are displacive, i.e. they are accompanied by only small structural distortions and small changes of lattice parameters. The structural symmetries lost in the phase transition are preserved as pseudosymmetries and are thus candidates for twin elements. This leads to a pseudocoincidence of the lattices of the twin partners and thus to pseudomerohedral twinning. Because of the small structural changes involved in the transformation, domains usually switch under mechanical stress, i.e. they are ferroelastic. A typical example for switchable ferroelastic domains is Rochelle salt, the first thoroughly investigated ferroelastic transformation twin, discussed in Section 3.3.9.2.2, Example (1). This topic is extensively treated in Chapter 3.4 on domain structures.
References
Böttcher, P., Doert, Th., Arnold, H. & Tamazyan, R. (2000). Contributions to the crystal chemistry of rareearth chalcogenides. I. The compounds with layer structures LnX_{2}. Z. Kristallogr. 215, 246–253.Docherty, R., ElKorashy, A., Jennissen, H.D., Klapper, H., Roberts, K. J. & ScheffenLauenroth, T. (1988). Synchrotron Laue topography studies of pseudohexagonal twinning. J. Appl. Cryst. 21, 406–415.
Donnay, G. & Donnay, J. D. H. (1974). Classification of triperiodic twins. Can. Mineral. 12, 422–425.
Donnay, J. D. H. & Donnay, G. (1983). The staurolite story. Tschermaks Mineral. Petrogr. Mitt. 31, 1–15.
Friedel, G. (1904). Etude sur les groupements cristallins. Extrait du Bulletin de la Société d' Industrie Minérale, Quatrième Série, Tomes III et IV. Saint Etienne: Imprimerie Théolier J. et Cie.
Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: BergerLevrault. [Reprinted (1964). Paris: Blanchard].
Hahn, Th. & Klapper, H. (2005). Point groups and crystal classes. Part 10 in International Tables for Crystallography, Vol. A, SpaceGroup Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.
Hildmann, B. O. (1980). Ferroelektrische–ferroelastische Eigenschaften, Phaseumwandlungen und Kristallstrukturen von NH_{4}LiSO_{4}. PhD Thesis, RWTH Aachen University, Germany.
Hurst, V. J., Donnay, J. D. H. & Donnay, G. (1956). Staurolite twinning. Mineral. Mag. 31, 145–163.
Jona, F. & Shirane, G. (1962). Ferrolectric Crystals, ch. VII, pp. 280–317. Oxford: Pergamon Press.
Klapper, H., Hahn, Th. & Chung, S. J. (1987). Optical, pyroelectric and Xray topographic studies of twin domains and twin boundaries in KLiSO_{4}. Acta Cryst. B43, 147–159. Erratum: Acta Cryst. B43, >406.
Koch, E. (2004). Twinning. In International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, edited by E. Prince, 3rd ed., ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Le Page, Y. (1999). Low obliquity in pseudosymmetry of lattices and structures, and in twinning by pseudomerohedry. Acta Cryst. A55, Supplement. Abstract M12.CC001.
Le Page, Y. (2002). Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry. J. Appl. Cryst. 35, 175–181.
Mallard, E. (1879). Traité de cristallographie, géométrie et physique. Vol. I. Paris: Dunod.
Nespolo, M. & Ferraris, G. (2005). Hybrid twinning – a cooperative type of crystal association. Z. Kristallogr. 220, 317–323.
Nespolo, M. & Ferraris, G. (2006). The derivation of twin laws in nonmerohedric twins. Application to the analysis of hybrid twins. Acta Cryst. A62, 336–349.
Nespolo, M. & Ferraris, G. (2007a). Hybrid twinning in staurolite. Acta Cryst. A63, s267–s268.
Nespolo, M. & Ferraris, G. (2007b). Overlooked problems in manifold twins: twin misfit in zeroobliquity TLQS twinning and twin index calculation. Acta Cryst. A63, 278–286.
Nespolo, M., Ferraris, G. & Takeda, H. (2000). Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space. Acta Cryst. A56, 132–148.
Nespolo, M., Takeda, H. & Ferraris, G. (1997). Crystallography of mica polytypes. In EMU Notes in Mineralogy, edited by St. Merlino, Vol. 1, ch. 2, pp. 81–118. Budapest: Eötvos University Press.
Shuvalov, L. A., Ivanov, N. R. & Sitnik, T. K. (1967). KH_{3}(SeO_{3})_{2} crystal, a new representative of the hydroselenite family. Dielectric anomalies and twinned structures. Sov. Phys. Crystallogr. 12, 315–318.
Smith, J. V. (1968). The crystal structure of staurolite. Am. Mineral. 53, 1139–1155.
Tamazyan, R., Arnold, H., Molchanov, V. N., Kuzmicheva, G. M. & Vasileva, I. G. (2000a). Contribution to the crystal chemistry of rareearth chalcogenides. II. The crystal structure and twinning of rareearth disulfide PrS_{2}. Z. Kristallogr. 215, 272–277.
Tamazyan, R., Arnold, H., Molchanov, V. N., Kuzmicheva, G. M. & Vasileva, I. G. (2000b). Contribution to the crystal chemistry of rareearth chalcogenides. III. The crystal structure and twinning of SmS_{1.9}. Z. Kristallogr. 215, 346–351.