Twinning with partial lattice coincidence (lattice index )

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 423-424

Section 3.3.9.2.3. Twinning with partial lattice coincidence (lattice index )

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.9.2.3. Twinning with partial lattice coincidence (lattice index )

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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) above. Typical examples with [[j] = 3] and [[j]> 3] were described in Section 3.3.8.3. 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 , 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 , as described in Example 3.3.6.5, leads to the composite symmetry $[{\cal K} =6'/m'({\bar 3})\,2/m\,2'/m']$ , i.e. the point-group index is . The sublattice index is , because of the elimination of the centring points of the original triple R lattice in forming the hexagonal P twin lattice.
(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 ( $[\Sigma 7]$ ) 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 . 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 () is the smallest sublattice with lattice index (least-diluted hexagonal sublattice). No example of a hexagonal $[\Sigma7]$ twin seems to be known.
(3) Tetragonal growth twins with ( $[\Sigma5]$ twins) in SmS_1.9 (Tamazyan et al., 2000b). This rare twin is illustrated in Fig. 3.3.8.1 and is described, together with the twins of the related phase PrS₂, in Example (3) of Section 3.3.9.2.4 below.
(4) Reflection twins across a general net plane (hkl) of a cubic P lattice. This example has been treated already in Section 3.3.9.1, Case (2).

References

Tamazyan, R., Arnold, H., Molchanov, V. N., Kuzmicheva, G. M. & Vasileva, I. G. (2000b). Contribution to the crystal chemistry of rare-earth chalcogenides. III. The crystal structure and twinning of SmS_1.9. Z. Kristallogr. 215, 346–351.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 423-424