Twin laws

Aquilano, D. and Franchini-Angela, M., 1985. Twin laws of calcium oxalate trihydrate (COT). Journal of Crystal Growth, 73, 558-562. 

T. Yasuda and 1. Sunagatva, X-ray topographic study of quartz crystals twinned according to Japan twin law, Phys. Chem. Min., 8,1982,121-7... 

Sunagawa and T. Yasuda, Apparent re-entrant corner effect upon the morphologies of twinned crystals A case study of quartz twinned according to Japanese twin law,... 

Figure 8.12(a) is a two-dimensional representation showing the origin of a fault vector R associated with the presence of a slab of twin one unit cell thick. In the albite twin law, the two parts of the twin are related by a rotation of 180° about b. From Figure 8.12(b), we see that Rj is defined as... 

Sanidine is monoclinic (space group C2/m), and there is complete disorder in the occupation of the tetrahedral (T) sites by the A1 and Si atoms. Over geological time, ordering takes place. In low (or maximum) micro-cline, the ordering is complete (all A1 in TiO sites), and the symmetry is reduced to triclinic (CT). There are four main orientational variants in this structure two orientations related by the albite twin law (rotation of 180° about b ) and two orientations related by the pericline twin law (rotation of 180° about b). The composition planes of these two twins are, respectively, (010) and the rhombic section which is parallel to b and approximately normal to (001). Thus, the characteristic cross-hatched pattern observed in (001) sections between crossed-polarizers in the optical microscope has, for many years, been simply interpreted as intersecting sets of albite and pericline twin lamellae formed at the monoclinic-to-triclinic transformation. However, TEM observations indicate that this model is too simple. Because these observations, collectively, also constitute an excellent example of the application of the principal modes of operation of TEM to a specific mineralogical problem, we discuss them in some detail. 

Microstructures in deformed dolomite. The deformation characteristics of dolomite are markedly different from those of calcite and have been studied in detail by Barber, Heard, and Wenk (1981). Not only are the twin laws different, but twinning in dolomite occurs only at temperatures above about 250°C. The lower dislocation densities observed in twinned dolomite and at twin intersections is perhaps due to the greater ease of stress relaxation at the higher temperatures required for twinning. 

Baronnet A (1997) Equilibrirrm and kinetic processes for polytype and polysome generation. In S Merlino (ed) Modular Aspects ofMinerals. Errr Mineral Union Notes in Mineralogy 1 119-152 Belov NV (1949) The twin laws of micas and micaceous minerals. Minerd sb LVovsk geol obva pri rmiv 3 29-40 (in Russian)... 

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 ... 

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 =... 

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 ... 

If the lattice of the individual is oC, the first two cosets in Equation (5) and the first four cosets [Eqn. (6)] correspond to metric merohedry, whereas the others correspond to pseudo-merohedry (ra = ra 0, raj = 0). If the lattice of the individual is mC Class a, the twin laws in Equations (5) and (6) correspond to reticular pseudo-merohedry. The hP twin lattice is a sublattice for the individual, with subgroup of translation 3 the twin index is thus 3. 

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. 

Syngony of the individual Lattice of the individual Twin lattice Kind of twinning Twin laws Twin index Rotation between pairs of individuals Polytypes... 

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... 

Table 12a. Classification of diffraction patterns for N = 3K+L. For the correspondence between the relative rotations of twinned individuals and the twin laws see Table 11 (after Nespolo 1999). 

For polytypes in which layers are related only by proper motions, like 37, two twins operations with the same rotational part and differing only for the proper/improper character of the motion produce the same twin lattice. The corresponding two twin laws are however different, and thus an orientation produced by an improper motion is hereafter distinguished by a small black circle ( ) after the Zt symbol. 

M2 polytype. Being a Class b polytype, IM2 has a markedly pseudo-rhombohedral lattice and two of the five pairs of twin laws, namely those corresponding to 120° rotation about c, correspond to pseudo-merohedry, whereas the remaining three correspond to reticular pseudo-merohedry. Each of the six x60° rotations belong to the point group of the family structure (subfamily B), and thus the family sublattice of the... 

The allotwin laws include the twin laws for each of the individuals, as well as the symmetry operations of the crystal(s) point group(s). The six rotations about c now must be considered. By indicating the first individual with a superscript and the second one with a subscript, the allotwin Zt = 3 must be considered also, whereas the Zt = 33 twin simply corresponds to a parallel growth. Therefore, the number of possible laws increases and depends upon the number of different polytypes undergoing allotwinning. 

Our theoretical analysis of the twin structure of LSGMO allows to correlate each twin law with a transformation matrix connected to the orthorhombic or trigonal basis vectors of neighbouring domains. Using twin models and the analytical geometry approach, the relationship of the basis vectors was obtained for all symmetry allowed domain pairs of the trigonal... 

If we transform this crystal by the mirror that is only fulfilled by the metric symmetry of the cell, but not by its contents, we obtain the crystal of Figure 7.1B. If both crystals grow together we have a twin (see Figure 7.2). The twin operation of this twin—the so-called twin law—is the mirror plane that transforms one domain into the other. As both domains in Figure 7.2 are equal in size, the fractional contributions of both domains are 0.5 and this twin is a perfect twin. In Figure 7.3 the fractional contributions are 0.67 0.33, corresponding to a partial twin. 

These two features— the twin law and the fractional contribution—are necessary for the description of a twin. The twin law can be expressed as a matrix that transforms the hkl indices of one species into the other. If x is going down and y to right, the transformation for the cell (and the hk indices) in the above example would be... 

Fig. 7.3 Partial twin (ratio is 2 1) following the same twin-law as in Figine 7.2.

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. 

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