Twinning by merohedry

In a merohedral twin, the twin law is a symmetry operator of the crystal system, hut not of the point group of the crystal. This means that the reciprocal lattices of the different twin domains superimpose exactly and the twinning is not directly detectable from the reflection pattern. Two types are possible. 

True Laue group Apparent Laue group Twin law 

Merohedral twinning may, at least in theory, occur simultaneously with racemic twinning. 

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By expressing the twin laws through the Shubnikov s two-color group notation (in which the twin elements are dashed Curien and Le Corre 1958), the three twin laws are 6 2 2 6 m l, 3T2//w. The complete twin [i.e. twin by merohedry or reticular... 

In micas, and more generally in layer compounds, plesiotwinning represents a generalization of the concept of twinning, at least from the lattice viewpoint. In twins the CSL produced in each plane (orthogonal polytypes) or in one plane out of three (non-orthogonal polytypes) has E factor 1, whereas in plesiotwins the CSL has E factor of n > 1 (n > 7 for the hp lattice). The twin/plesiotwin index is thus 1 (twinning by merohedry)... 

The first structure (Herbst-Irmer and Sheldrick, 1998) is an example of twinning by merohedry, which could not be solved by routine methods. The composition of the compound was not known with certainty, but an osmium compound with some triphenylphosphine and chloro ligands was expected. The first problem is to determine the space group. XPREP gives the following output... 

Polytypes of the orthorhombic syngony with a hP lattice may undergo twinning by metric merohedry, the twin lattice coinciding with the lattice of the individual. The coset decomposition gives two twin laws ... 

Polytypes of the monoclinic and triclinic syngony with an hP lattice may undergo twinning by metric merohedry. For the monoclinic syngony the coset decomposition gives five twin laws, each with four equivalent twin operators ... 

Effect of twinning by selective merohedry on the diffraction pattern... 

Figure 21. ZiO/ r.p. (SD family plane) of the ZT polytype twinned by selective merohedry. Black circles family reflections overlapped by the twin operation (common to both individuals). Gray and white circles family reflections from two individuals rotated by (2 +l) x 60°, not overlapped by the twin operation (modified after Nespolo et al. 1999a).

Twinning by complete merohedry (Zt = UU ) by definition produces a diffraction pattern with the same geometrical appearance as the single crystal, which in its turn may be geometrically identical to the pattern of IM twinned as Zt = 351. In contrast, for twinning by selective merohedry (Zt = UE, UE, UU E, UU E, UU EE ), the two D-type Ci correspond to have two reflections at Ij = 2(mod 6) and 4(mod 6). This is the same geometrical appearance of IM twinned as Zt = 3451. The distinction between IM twins and the 3T polytype (twinned or untwinned) requires by very careful examination of the violation of the additional reflection conditions (Nespolo et al. 2000a). 

In contrast to the two first t5rpes of twinning (twiiuiing by merohedry and pseudo-merohedry), in the remaining two t5rpes not every reflection is affected by the twinning. This means that the twinning may be detectable from the diffraction... 

If only part of the reflections have a contribution from the second domain (twinning by reticular merohedry and non-merohedral twins), a special reflection file is necessary, which is read in by the command... 

In the following sections we present examples of how to refine twinned structures with SHELXL. All files you may need in order to perform the refinements yourself are given on the CD-ROM that accompanies this book. The first example is a case of merohedral twinning that will acquaint you with the basics of practical twin refinement. The second example describes a typical pseudo-merohedral twin such as every crystallographer will encounter sooner or later. Two different examples for twinning by reticular merohedry are given next and the chapter ends with two cases of non-merohedral twinning. 

This shows that the structure has metric orthorhombic symmetry to a good approximation. The comparison of the Rjnt values makes clear that the correct Laue group is only monoclinic, but because of the higher metric symmetry there is the possibility of twinning by pseudo-merohedry The additional twofold axis, which is present in the orthorhombic system but not in the monoclinic one, is the twin law. To describe this axis in the monoclinic system three matrices need to be multiplied ... 

All the operators corresponding to the same twin law are equivalent under the action of the symmetry operators of the orthorhombic syngony. If the lattice is only oC, twinning is by pseudo-merohedry. The twin lattice (hP) does not coincide exactly with the lattice of the individual, because for the latter the orthohexagonal relation b =... 

The first two [Eqn. (7)] or four [Eqn. (8)] cosets give the twin laws by metric merohedry, the others give the twin laws by reticular merohedry. Twin operators in each coset are equivalent by the action of the symmetry elements of the syngony. 

Orthogonal plytyps. In the Trigonal model, the lattice is hP (o = 0) in the true structure for orthorhombic polytypes the lattice is normally oC but pseudo-/ / (co 0). For subfamily B and mixed-rotation polytypes the limiting symmetry is hP and there is only one independent orientation of the w.r.l. Twinning is either by complete merohedry or by pseudo-merohedry and does not modify the geometry of the diffraction pattern. 

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