Twinning operation

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

Thus, the regularly spaced twinning operation to the HCP lattice produces a new type of structure having trigonal prism interstices in the region of the... 

Fig. 2.100 Twinning operation to UgOg-type structure with (110) twin plane. It is noted that the pentagonal bipyramids on the twin plane are substantially distorted (see also Fig. 2.105).

Fig. 2.101 Micro-twinning operation to UjOg-type structure. The structures are represented by a chain of pentagonal bipyramids for simplicity. The structures with a (/, /)ij,OB typc twinning operation are shown, where I is the number of pentagonal bipyramids between the twin planes (hatched ones), (a) UjOg-type structure (mother structure) (b) / = 0 (same as the mother structure) (c) I = 1 (d) I = 2 (e) / = 3 (f) Z = 4. The chain lines indicate the folding planes.

We refer to Eq. (34) as the asymmetric formulation of the L-space quantum dynamics. Also the expectation value of any TD operator A(rt) in the ordinary H-space is just a matrix element of either of its twin operators A rt) or A rt) in the -space ... 

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 ... 

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. 

Only the two-fold rotation about c of the twin lattice is a correct twin operation, in the sense that it restores the lattice, or a sublattice, of the individuals. If however rotations about c give simply the (approximate) relative rotations between pairs of twinned mica individuals, but are not true twin operations. Similar considerations apply also to the rotoinversion operations, s depends upon the obliquity of the twin but, at least in Li-poor trioctahedral micas, is sufficiently small to be neglected for practical purposes (Donnay et al. 1964 Nespolo et al. 1997a,b, 2000a). 

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).

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. 

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. 

The cell of the true space group must be transformed into the apparent cell to use the description of the twin operation in this Laue group. Then the cell must be re-transformed into the true Laue group. 

The monoxides are known for all of the rare earth metals. They are all crystallizing at low temperatures isostructural to the NaCl-type. EU3O4 crystallizes orthorhombic [45, 46] with space group Pnam. This mixed valent compound contains Eu + and Eu + in the cation sublattice with a distorted octahedral coordination around Eu + and a coordination number of eight around Eu +. Fig. 3-19 shows the projection of the structure down [010]. The octahedral framework around Eu + can be related by a twinning operation (glide reflection) to the structure of ramsdellite (hep). 

Fig. 10. Relative position of the two different (132) reciprocal planes for (313) twinned crystals of B. Filled circles are relative to one crystal, open circles to the other. The indexing is given for one crystal only (for the filled circles). Only one node in three has a homologue in the twinning operation 313, 310, 023, 603,... have homologues, but 201, 402, Til, 112,... have not.

It has been shown previously that B twins have a reticular character for both type I and type II twins. This means that not all the nodes of the reciprocal lattice have homologues in the twin operation (one over three for all the twins of B). This is due to the fact that the unit cell of the primitive lattice of B is three times the unit cell of the lattice of A, so that the reciprocal lattice of B is three times as dense. All B twins are related to the structure of A, so that only the nodes related to A have an homologue in the twin operation. For instance nodes 111, 202, 020, 313, 310,... of the reciprocal lattice that can be related to nodes of A have homologues in each of the twin operation, while nodes Til, 112, 201, 402,... that are not related to nodes of A have no homologues in a twin operation, as shown on fig. 10. So the reticular character of the B twins, nearly superlattice conserving twins (NSLCT), can be related to the complete description of the relationship existing between the A and B structures. 

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