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

International Tables for Crystallography (2013). Vol. D, ch. 3.3, pp. 446-450

Section 3.3.9. Twinning by merohedry and pseudo-merohedry

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:

3.3.9. Twinning by merohedry and pseudo-merohedry

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We now resume the discussion of Section 3.3.8[link] on three-dimensional coincidence lattices and pseudo-coincidence 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 pseudo-merohedry (`macles par pseudo-mériédrie'), both for lattice index [[j] = 1] and [[j]> 1], as introduced by Friedel (1926[link], p. 434). Often (strict) merohedral twins are called `parallel-lattice twins' or `twins with parallel axes'. Donnay & Donnay (1974[link]) have introduced the terms twinning by twin-lattice symmetry (TLS) for merohedral twinning and twinning by twin-lattice quasi-symmetry (TLQS) for pseudo-merohedral twinning, but we shall use here the original terms introduced by Friedel. Definitions of merohedry

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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:

  • Case (1): `Merohedry' of point groups

    A merohedral point group is a subgroup of the holohedral point group (lattice point group) of a given Bravais lattice in a crystal family, i.e. group and subgroup belong to the same Bravais system. This is the original sense of the term merohedry, which has the morphological meaning of reduction of the number of faces of a given crystal form as compared with a holohedral crystal form. The degree of merohedry is given by the subgroup index [i]. For point groups within the same crystal family, possible indices [i] are 2 (hemihedry), 4 (tetartohedry) and 8 (ogdohedry). The only example for [[i] = 8] is the point group 3 in the hexagonal holohedry [6/m\,2/m\,2/m].

    If the point group of a crystal is reduced to such an extent that the subgroup belongs to a crystal family of lower symmetry, this subgroup is called a pseudo-merohedral point group, provided that the structural differences and, hence, also the metrical changes of the lattice (axial ratios) are small. Twinning by merohedry corresponds to non-ferroelastic phase transitions, twinning by pseudo-merohedry to ferroelastic phase transitions.

    Both merohedral and pseudo-merohedral subgroups of point groups are listed in Section 10.1.3[link] and Fig.[link] of Volume A of this series (Hahn & Klapper, 2005[link]); cf. also Koch (2004[link]), Table[link]

  • Case (2): `Merohedry' of translation groups (lattices)

    The term `reticular' or `lattice merohedry' designates the relation between a lattice and its `diluted' sublattice (without consideration of their lattice point groups). A sublattice6 is a three-dimensional subset of lattice points of a given lattice and corresponds to a subgroup of index [[j]> 1] of the original translation group. This kind of group–subgroup relation has been called `reticular merohedry' (`mériédrie réticulaire') by Friedel (1926[link], p. 444). Note that the lattice and its sublattice may belong to different crystal systems, and that the lattice point groups (holohedries) of lattice and sublattice generally do not obey a group–subgroup relation. This is illustrated by a cubic P lattice (lattice point group [4/m\,{\bar 3}\,2/m]) and one of its monoclinic sublattices (lattice point group [2/m]) defined by a general lattice plane (hkl) and the lattice row [hkl] normal to it. The symmetry direction [hkl] of the monoclinic sublattice does not coincide with any of the symmetry directions of the cubic lattice, i.e. there is no group–subgroup relation of the lattice point groups. The sub­group common to both (the intersection group) is only [{\bar 1}]. A somewhat more complicated example is the [[j] = 5] ([\Sigma]5) sublattice obtained by a (210) twin reflection of a tetragonal crystal lattice; cf. Fig.[link] and Section[link]. Both lattice and sublattice are tetragonal, [4/mmm], with common c axes, but the intersection group of their holohedries is only [4/m], the further symmetry elements are oriented differently.

    Friedel (1926[link], p. 449) also introduced the term `reticular pseudo-merohedry' (`pseudo-mériédrie réticulaire'). This notion, however, can not be applied to a single lattice and its sublattice (a single lattice can be truly diluted but not pseudo-diluted), but requires pseudo-coincidence of two or more superimposed lattices, which form a `pseudo-sublattice' of index [[j]> 1], as described in Section[link]

    Because of this complicating and confusing situation we avoid here the term merohedry in connection with lattices and translation groups. Instead, the terms coincidence(-site) lattice, twin lattice or sublattice of index [j] are preferred, as explained in Section[link](iv)[link]. Note that we also use two different symbols [i] and [j] to distinguish the subgroup indices of point groups and of lattices. Types of twins by merohedry and pseudo-merohedry

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Both kinds of merohedries and pseudo-merohedries were used by Mallard (1879[link]) and especially by Friedel (1904[link], 1926[link]) and the French School in their treatment of twinning. Based on the concepts of exact coincidence (merohedry, [j] = 1), approximate coincidence (pseudo-merohedry) and partial coincidence (twin lattice index [[j]> 1]), four major categories of `triperiodic' twins were distinguished by Friedel and are explained below. Merohedral twins of lattice index [[j] = 1]

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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 single-crystal X-ray 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:

  • (1) Quartz: Dauphiné, Brazil and Leydolt twins (cf. Example[link]).

  • (2) Pyrite, iron-cross twins: crystals of cubic eigensymmetry [2/m{\bar 3}] form penetration twins of peculiar morphology by reflection on (110), with [[i] = 2].

  • (3) KLiSO4: the room-temperature phase III of eigensymmetry 6 exhibits four domain states related by three merohedral twin laws. These growth twins of index [[i] = 4] have been characterized in detail by optical activity, pyroelectricity and X-ray topography (Klapper et al., 1987[link]).

  • (4) Potassium titanyl phosphate, KTiOPO4: polar point group [mm2], forms inversion twins (ferroelectric domains) below its Curie temperature of 1209 K. Pseudo-merohedral twins of lattice index [[j] = 1]

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These twins are characterized by pseudo-merohedry of point groups, Case (1)[link] in Section[link]. 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:

  • (1) Transformation twins of Rochelle salt (Jona & Shirane, 1962[link]): this ferroelastic/ferroelectric transformation at about 295 K follows the group–subgroup relation orthorhombic [2'22'] [ \longleftrightarrow] monoclinic 121 (index [[i] = 2]) with [\beta \approx 90^\circ]. The primed operations form the coset of the group–subgroup relation and thus the twin law. Owing to the small deviation of the angle [\beta] from [90^\circ], the lattices of both twin partners nearly coincide. Note that this group–subgroup relation involves both an orthorhombic merohedral and a monoclinic merohedral point group, viz 222 and 2.

  • (2) Transformation twins orthorhombic [2'/m'\,2/m\,2'/m' \longleftrightarrow] monoclinic [12/m1] with [\beta \approx 90^\circ]. This is a case analogous to that of Rochelle salt, except that the point groups involved are the holohedries of the orthorhombic and of the monoclinic crystal system, mmm and [2/m] [example: KH3(SeO3)2; Shuvalov et al. (1967[link])].

  • (3) Pseudo-hexagonal growth twins of an orthorhombic C-centred crystal with [ b/a \approx \sqrt{3}] and twin reflection planes [m'(110)] and [m'({\bar 1}10)]. The lattices of the three domain states nearly coincide and form a `pseudo-coincidence lattice' of lattice index [[j] = 1], but of point-group index [[i] = 3], with subgroup [{\cal H} = 2/m\,2/m\,2/m] and supergroup [{\cal K}(6) =] [6(2)/m\,2/m\,2/m] (cf. Example[link]). Here, in contrast to exact merohedry, in single-crystal X-ray diffraction patterns most reflection spots will be split into three. Note that the term `index' appears twice, first as the subgroup index [[i] = 3] of the point groups and second as the lattice index [[j] = 1] of the twin lattice: further details are given in Section[link]. Twinning with partial lattice coincidence (lattice index [[j]> 1])

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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)[link] above. Typical examples with [[j] = 3] and [[j]> 3] were described in Section[link] In addition to the sublattice relations, it is reasonable to include the point-group relations as well. Four examples are presented:

  • (1) Twinning of rhombohedral crystals (lattice index [[j] = 3], example FeBO3). The eigensymmetry point groups of the structure and of the R lattice (of the untwinned crystal) are both [{\cal H} = {\bar 3}\,2/m]. The extension of the eigensymmetry by the (binary) twin operation [2_z], as described in Example[link], leads to the composite symmetry [{\cal K} =6'/m'({\bar 3})\,2/m\,2'/m'], i.e. the point-group index is [[i] = 2]. The sublattice index is [[j] = 3], because of the elimination of the centring points of the original triple R lattice in forming the hexagonal P twin lattice; cf. Section[link].

  • (2) Reflection twinning across [\{21{\bar 3}0\}] or [\{14{\bar 5}0\}], or twofold rotation twinning around [\langle 540\rangle] or [\langle 230\rangle] of a hexagonal crystal with a P lattice (lattice symmetry [6/m\,2/m\,2/m]). The twin generates a hexagonal coincidence lattice of index [[j] = 7] ([\Sigma 7]) with [{\bf a}' = 3{\bf a} + 2{\bf b}], [{\bf b}' = -2{\bf a} + {\bf b}], [{\bf c}' = {\bf c}]. The hexagonal axes [{\bf a}'] and [{\bf b}'] are rotated around [001] by an angle of 40.9° with respect to a and b. The intersection lattice point group of both twin partners is [6/m]. The extension of this group by the twin operation `reflection across [\{21{\bar 3}0\}]' leads to the point group of the coincidence lattice [6/m\,2'/m'\,2'/m'] (referred to the coordinate axes [{\bf a}', {\bf b}', {\bf c}']). The primed operations define the coset (twin law). For hexagonal lattices rotated around [001], the [\Sigma 7] coincidence lattice ([[j] = 7]) is the smallest sublattice with lattice index [[j]> 1] (least-diluted hexagonal sublattice). No example of a hexagonal [\Sigma7] twin seems to be known; for details see Section[link].

  • (3) Tetragonal growth twins with [[j] = 5] ([\Sigma5] twins) in SmS1.9 (Tamazyan et al., 2000b[link]). This rare twin is illustrated in Fig.[link] and is described, together with the twins of the related phase PrS2, in Example (3)[link] of Section[link] below.

  • (4) Reflection twins across a general net plane (hkl) of a cubic lattice. This example has been treated already in Section[link], Case (2)[link]. Twinning with partial lattice pseudo-coincidence (lattice index [[j]>1])

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This type can be derived from the category in Section[link] above by relaxation of the condition of exact lattice coincidence, resulting in two nearly, but not exactly, coinciding lattices (pseudo-coincidence, cf. Section[link]). In this sense, the two Sections[link] and[link] are analogous to the two Sections[link] and[link].

The following four examples are characteristic of this group:

  • (1) (110) reflection twins of a pseudo-hexagonal orthorhombic crystal with a P lattice: If the axial ratio [b/a = \sqrt{3}] were exact, the lattices of both twin partners would coincide exactly on a sublattice of index [[j] = 2] (due to the absence of the C centring); cf. Koch (2004[link]), Fig.[link] . If [b/a] deviates slightly from [\sqrt{3}], the exact coincidence lattice changes to a pseudo-coincidence lattice of lattice index [[j] = 2]. Examples are ammonium lithium sulfate, NH4LiSO4 (Hildmann, 1980[link]), many members of the K2SO4-type series (cf. Docherty et al., 1988[link]) and aragonite, CaCO3; for further details see Section[link].

  • (2) Staurolite twinning: This topic has been extensively treated as Example[link]. The famous 90°- and 60°-twin `crosses' are a complicated and widely discussed example for Friedel's notion of `twinning by reticular merohedry' (Friedel, 1926[link], p. 461). It was followed up by an extensive analysis by Hurst et al. (1956[link]). Both twin laws (90° and 60° crosses) can be geometrically derived from a multiple pseudo-cubic cell [{\bf a}_c'], [{\bf b}_c'], [{\bf c}_c'] (so-called `Mallard's pseudo-cube') which is derived from the structural monoclinic C-centred cell [{\bf a}_m], [{\bf b}_m], [{\bf c}_m] as follows, involving a rotation of [\sim 45^\circ] around [100]: [{\bf a}_c' = {\bf b}_m + 3{\bf c}_m,\quad {\bf b}_c' = - {\bf b}_m + 3 {\bf c}_m, \quad {\bf c}_c' = 3{\bf a}_m.]

    Using Smith's (1968[link]) lattice constants for the structural monoclinic cell with space group C2/m and a = 7.871, b = 16.620, c = 5.656 Å, β = 90° (within the limits of error), Vm = 740 Å3, the pseudo-cube has the following lattice constants:[\matrix{a'_c = 23.753\hfill &b'_c = 23.753\hfill & c'_c = 23.613\,\,\hbox{\AA}\hfill &\cr \alpha_c = 90\hfill & \beta_c = 90\hfill & \gamma_c = 88.81^\circ\hfill & V'_c = 13323\,\,\hbox{\AA}^3.\hfill}]

    The volume ratio [V'_c/V_m] of the two cells is 18, i.e. the sublattice index is [[j] = 18]. If, however, the primitive monoclinic unit cell is used, the volume ratio doubles and the sublattice index used in the twin analysis increases to [[j] = 36]. The (metrical) eigensymmetry of the pseudo-cube is orthorhombic (due to [\beta_c = 90^\circ]), [(2/m)_{[001]}(2/m)_{[110]}(2/m)_{[1{\bar 1}0]}], referred to [{\bf a}'_c], [{\bf b}'_c], [{\bf c}'_c].

    Note, however, that this pseudo-cube in reality is C-centred because the C-centring vector [1/2({\bf a}'_c + {\bf b}'_c) = 3{\bf c}_m] is a lattice vector of the monoclinic lattice. This C-centring has not been considered by Friedel, Hurst and Donnay, who have based their analysis on the primitive pseudo-cube.

    According to Friedel, the `symmetry elements' of the pseudo-cube are potential twin elements of staurolite, except for [(2/m)_{[1{\bar 1}0]}], which is the monoclinic symmetry direction of the structure. In Table[link], the twin operations of the [90^\circ] and [60^\circ] twins are compared with the `symmetry operations' of the pseudo-cube with respect to obliquities [\omega] and lattice indices [j], referred to both sets of axes, pseudo-cubic [{\bf a}_c'], [{\bf b}'_c], [{\bf c}'_c] and monoclinic (but metrically orthorhombic) [{\bf a}_m], [{\bf b}_m], [{\bf c}_m]. The calculations were again performed with the program OBLIQUE by Le Page (1999[link], 2002[link]). In order to keep agreement with the interpretation of Friedel and Hurst et al., the pseudo-cube is treated as primitive, with [[j] = 36].

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    Staurolite, 60° and 90° twins

    Comparison of the twin operations with the `symmetry operations' of the primitive pseudo-cube with respect to obliquity [\omega] and lattice index [[j]], referred both to the pseudo-cubic axes, [{\bf a}'_c], [{\bf b}'_c], [{\bf c}'_c], and the monoclinic (metrically orthorhombic) axes, [{\bf a}_m], [{\bf b}_m], [{\bf c}_m]. The calculations were performed with the program OBLIQUE by Le Page (1999[link], 2002[link]).

    (a) 90° cross (eight twin operations).

    Twin operations referred toObliquity [\omega] [[^\circ]]Lattice index [j] referred toRemarks
    [{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m][{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m]
    [4[001]_c] [4[100]_m] 0 1 1 Four collinear twin operations [4^1], [4^3], [{\bar 4}{^1}], [{\bar 4}{^3}]
    [2[100]_c] [2[013]_m] 1.19 1 6 Four `diagonal' (with respect to the monoclinic unit cell) twin operations intersecting in [[001]_c=[100]_m]
    [m(100)_c] [m(031)_m] 1.19 1 6
    [2[010]_c] [2[0{\bar 1}3]_m] 1.19 1 6
    [m(010)_c] [m(0{\bar 3}1)_m] 1.19 1 6

    (b) 60° cross.

    Twin operations referred toObliquity [\omega] [[^\circ]]Lattice index [j] referred toEquivalent directions
    [{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m][{\bf a}'_c, {\bf b}'_c, {\bf c}'_c][{\bf a}_m, {\bf b}_m, {\bf c}_m]
    [3[111]_c] ([\pm120^\circ]) [3[102]_m] 0.87 3 3 [[11{\bar 1}]_c = [{\bar 1}02]_m ]
    [3[{\bar 1}11]_c] ([\pm120^\circ]) [3[320]_m] 0.25 3 9 [[1{\bar 1}1]_c = [3{\bar 2}0]_m ]
    [4[100]_c] ([\pm 90^\circ]) [4[013]_m] 1.19 1 6 [[010]_c = [0{\bar 1}3]_m ]
    [m(100)_c] [m(031)_m] 1.19 1 6  
    [2[101]_c] [2[313]_m] 0.90 1 12 [[011]_c = [3{\bar 1}3]_m ]
    [m(101)_c] [m(231)_m] 0.90 1 12 [[10{\bar 1}]_c = [{\bar 3}13]_m ]
              [[01{\bar 1}]_c = [31{\bar 3}]_m ]

    The following interpretations can be given (cf. Fig. 13 in Hurst et al., 1956[link]):

    • (a) 90° cross (Table[link]a, Fig.[link]a):

      • (i) The pseudo-tetragonal 90° cross can be explained and visualized very well with eight twin operations, a fourfold twin axis along [[100]_m =[001]_c] with operations [4^1], [4^3], [{\bar 4}{^1}], [{\bar 4}{^3}] and two pairs of `diagonal' twin operations 2 and m. They form the coset of the (metrically) `orthorhombic' ([\beta = 90^\circ]) eigensymmetry [{\cal H} = mmm] which results in the composite symmetry [{\cal K} = 4'(2)/m\,2/m\,2'/m'].

      • (ii) The obliquities for all twin operations are at most 1.2°, the lattice index is [[j]_m = 1] for the twin axis, but for the `diagonal' twin elements it is [[j]_m = 6], which is at the limit of the permissible range. Because of these facts, Friedel prefers to consider the 90° cross as a 90° rotation twin around [[100]_m] rather than as a (diagonal) reflection twin across [(031)_m] or [(0{\bar 3}1)_m].

      • (iii) Note that for the interpretation of the 90° cross the complete pseudo-cube with lattice index [[j] = 36] is not required. Because [{\bf c}'_c = 3{\bf a}_m], a pseudo-tetragonal unit cell with axes [{\bf a}_c'], [{\bf b}'_c], [(1/3){\bf c}'_c] and [[j] = 12] is sufficient.

    • (b) 60° cross (Table[link]b, Fig.[link]b):

      • (iv) The widespread 60° cross is much more difficult to interpret and visualize. The four threefold twin axes around [\langle 111\rangle] of the pseudo-cube split into two pairs, both with very small obliquities [ \,\lt\, 1^\circ]. One pair, [[102]_m] and [[{\bar 1}02]_m], has a favourable index [[j]_m = 3]; however, the other one, [[320]_m] and [[3{\bar 2}0]_m] is with [[j]_m = 9] unacceptably high. According to Friedel's theory, this makes [[102]_m] the best choice as threefold twin axis.

      • (v) There is a further [\pm 90^\circ] twin rotation around [[100]_c] or [[013]_m] with small obliquity, [\omega = 1.2^\circ], but very high lattice index, [[j]_m = 6]. Note that this is the same axis that has been used already for the 90° twin, but with a 180° rotation.

      • (vi) The greatest deviation from the `permissibility' criterion is exhibited by the twin axes [2[101]_c = 2[313]_m] and [2[011]_c =] [2[3{\bar 1}3]_m] and the twin planes, pseudo-normal to them, [(231)_m] and [(2{\bar 3}1)_m]. The obliquity [\omega = 0.9^\circ] is very good but the twin index is [[j] = 12], a value far outside Friedel's `limite prohibitive'. These operations, however, are the `standard' twin operations that are always quoted for the 60° twins. Following Friedel (1926[link], p. 462), the best definition of the 60° twin are the [\pm 120^\circ] rotations around [[102]_m] with [\omega = 0.87^\circ] and [[j]_m = 3].

      • (vii) If the (true) C-centring of the pseudo-cube is taken into account, however, no [ \langle 111\rangle] pseudo-threefold axes remain; hence, the 60° cross cannot be explained by the lattice construction of the pseudo-cube.

    Nespolo and Ferraris' description of the staurolite twins. Nespolo & Ferraris (2005[link], 2006[link], 2007a[link],b[link]) have formulated extensions of Mallard's and Friedel's lattice theory of twinning. They introduce `hybrid twinning' (Nespolo & Ferraris, 2005[link], 2006[link]), which involves more than one sublattice contributing to the `overall lattice restoration', a new calculation of the twin index for `manifold twins' (`multiple twins' in the present chapter, cf. Section[link]) and a `twin misfit δ' (Nespolo & Ferraris, 2007b[link]), defined as `the distance between the first nodes along the shortest direction in the plane quasi-perpendicular to the twin axis that are quasi-restored by the twin operation'.

    Based on these considerations, the twin lattices of staurolite have been reconsidered (Nespolo & Ferraris, 2007a[link],b[link]), with the following results in comparison to the discussion above by the present authors:

    • (i) For the `Greek cross twins' (90° cross), the two treatments agree and result in the twin index [j] = 6, just at the limit of Friedel's rule for the occurrence of twinning.

    • (ii) The much more frequent `St Andrews cross twins' (60° cross) have a twin index 12, far too high according to Friedel's standard. This discrepancy is eliminated by Nespolo & Ferraris (2007a[link]) by considering this twin as a hybrid twin with two concurring monoclinic sublattices, generated by twin axis/plane [313]/(231) with index 12 (obliquity 0.90°) and [313]/(352) with index 10 (obliquity 6.21°). Since two of the nodes of the [313]/(231) sublattice are quasi-coincident with nodes of the sublattice [313]/(352), the index 12 is reduced by a factor 2 to an effective index 6, comparable to index 6 of the 90° twin and, thus, just acceptable by Friedel's criterion, but with one large obliquity of 6.21°.

    • (iii) The last paper in the series (Nespolo & Ferraris, 2007b[link]) is a critical discussion of the twin-lattice treatment of previous authors, including the present ones. Its essential statement is that the Friedel criterion is applicable only to binary twin operations, not to `manifold twins' (`multiple twins', see above), for which a new twin index is defined. This index is now the same for all operations in the coset of the manifold twin. For the Greek cross (90° twin), Nespolo & Ferraris arrive at the same result (index 6) as the present authors. For the `St Andrews cross' (60° twin) there is a significant difference in the two approaches: the present authors believe that the 60° cross cannot be explained by the lattice construction of `Mallard's pseudo-cube' because the C-centred cubic twin lattice, which is actually tetragonal P with [c/a=1{:}\sqrt2/2], has no pseudo-threefold axis along any <111> direction which could act as a threefold twin rotation axis. Nespolo & Ferraris, on the other hand, disregard the C-centring of the Mallard's pseudo-cube and explain the 60° twin by the threefold axes of the cubic P cell (see their Fig. 1 and the trichromatic twin point group given on their page 285), as was done by Mallard (1879)[link], Friedel (1926)[link] and Donnay & Donnay (1983)[link].

  • (3) Growth twins of monoclinic PrS2 and of tetragonal SmS1.9: These two rather complicated examples belong to the structural family of MeX2 dichalcogenides which is rich in structural relationships and different kinds of twins. The `basic structure' and `aristotype' of this family is the tetragonal ZrSSi structure with axes [a_b = b_b \approx 3.8], [c_b \approx 7.9\ \hbox{\AA}], [V_b \approx114\ \hbox{\AA}^3], space group [P4/nmm] (b stands for basic). The crystal chemistry of this structural family is discussed by Böttcher et al. (2000[link]).

    • (a) PrS2 (Tamazyan et al., 2000[link]a)

      PrS2 is a monoclinic member of this series with space group [P2_1/b11] (unique axis a!) and axes [a \approx 4.1], [b \approx 8.1], [c \approx 8.1\ \hbox{\AA}], [\alpha \approx 90.08^\circ], [V \approx 269\ \hbox{\AA}^3]. The structure is strongly pseudo-tetragonal along [001] (with cell a, b/2, c) and is a `derivative structure' of ZrSSi. Hence pseudo-merohedral twinning that makes use of this structural tetragonal pseudosymmetry would be expected, with twin elements 4[001] or [m(210)] or 2[120] etc. and [[j] = 2] because [b \approx 2a], but, surprisingly, this twinning has not been observed so far. It may occur in other PrS2 samples or in other isostructural crystals of this series.

      Instead, the monoclinic crystal uses another structural pseudo­symmetry, the approximate orthorhombic symmetry along [100] with [\alpha \approx 90^\circ], to twin on [2_y], [2_z], [m_y] or [m_z] (coset of [2_x/m_x]) with composite symmetry [{\cal K} = 2/m\,2'/m'\,2'/m'], [[j] = 1] and [[i] = 2] (cf. Fig. 4 of the paper).

      The monoclinic PrS2 cell has a third kind of pseudosymmetry that is not structural, only metrical. The cell is pseudo-tetragonal along [100] due to [ b \approx c] and [\alpha \approx 90^\circ]. This pure lattice pseudosymmetry, not surprisingly, is not used for twinning, e.g. via 4[100] or [m(011)] or [m(0{\bar 1}1)] or 2[011] or [2[0{\bar 1}1]].

    • (b) SmS1.9 (Tamazyan et al., 2000b[link])

      This structure is (strictly) tetragonal with axes [a = b \approx 8.8], [c\approx 15.9\ \hbox{\AA}], [V \approx 1238\ \hbox{\AA}^3] and space group [P4_2/n]. It is a tenfold superstructure of ZrSSi with the following basis-vector relations: [ {\bf a} = 2{\bf a}_b + {\bf b}_b,\quad {\bf b} = -{\bf a}_b + 2{\bf b}_b,\quad {\bf c} = 2{\bf c}_b,]leading to lattice constants [a \approx \sqrt{5}a_b], [b \approx \sqrt{5}b_b], [c \approx 2c_b]. This well ordered tetragonal supercell now twins on [m(210)] or 2[210] or [m(130)] or 2[130] (which is equivalent to a rotation around [001] of 36.87°) to form a [\Sigma 5] twin by `reticular merohedry' ([[j] =] [5]) with lattice constants [a' = a \sqrt{5} = 19.72], [b' =] [b \sqrt{5} = 19.72], c′ = c = 15.93 Å, V = 6192 Å3. This is illustrated in Fig.[link].

      SmS1.9 represents the first thoroughly investigated and documented tetragonal [[j] = 5] ([\Sigma 5]) twin known to us. The sublattice of this twin is the tetragonal coincidence lattice with smallest lattice index [[j]> 1], i.e. the `least-diluted' systematic tetragonal sublattice; for further details see Section[link].

  • (4) Growth twins of micas: A rich selection of different twin types, both merohedral and pseudo-merohedral, with [[j]=1] and 3, is provided by the mineral family of micas, which includes several polytypes. A review of these complicated and interesting twinning phenomena is presented by Nespolo et al. (1997[link]). Detailed theoretical derivations of mica twins and allotwins, both in direct and reciprocal space, are published by Nespolo et al. (2000[link]).

In conclusion, it is pointed out that the above four categories of twins, described in Sections[link] to[link], refer only to cases with exact or approximate three-dimensional lattice coincidence (triperiodic twins). Twins with only two- or one-dimensional 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[link] 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. Pseudo-merohedry and ferroelasticity

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The large group of pseudo-merohedral 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 pseudo-coincidence of the lattices of the twin partners and thus to pseudo-merohedral 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[link], Example (1)[link]. This topic is extensively treated in Chapter 3.4[link] on domain structures.


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