Twins with three-dimensional twin lattices (`triperiodic' twins)

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. 418-419

Section 3.3.8.3. Twins with three-dimensional twin lattices (`triperiodic' twins)

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

3.3.8.3. Twins with three-dimensional twin lattices (`triperiodic' twins)

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The following cases of exact superposition are distinguished:

(i) Twins with ( $[\Sigma 1]$ twins). Here, the crystal lattice and the twin lattice are identical, i.e. the coincidence (parallelism) of the two oriented crystal lattices is complete. Hence, any twin operation must be a symmetry operation of the point group of the lattice (holohedry), but not of the point group of the crystal. Consequently, this twinning can occur in `merohedral' point groups only. This twinning by merohedry (parallel-lattice twins, twins with parallel axes) will be treated extensively in Section 3.3.9.
(ii) Twins with ( $[\Sigma 2]$ twins). This twinning does not occur systematically among the cases listed in Table 3.3.8.1, except for special metrical relations. Example: a primitive orthorhombic lattice with $[b/a = \sqrt{3}]$ and twin reflection plane or $[({\bar 1}10)]$ . The coincidence lattice is hexagonal with $[a_{\rm hex} = 2a]$ and .

(iii) Twins with [[j] = 3] ( $[\Sigma 3]$ twins). Twins with are very common among rhombohedral and cubic crystals (`spinel law') with the following two representative twin operations:

(a) twofold rotation around a threefold symmetry axis [111] (cubic or rhombohedral coordinate axes) or [001] (hexagonal axes);
(b) reflection across the plane (111) or (0001) normal to a threefold symmetry axis.

Both twin operations belong to the same twin law if the crystal is centrosymmetric. Well known examples are the (0001) contact twins of calcite, the penetration twins of iron borate, FeBO₃, with the calcite structure, and the spinel twins of cubic crystals (cf. Examples 3.3.6.5, 3.3.6.6 and Figs. 3.3.6.4–3.3.6.6 ). For crystals with a rhombohedral (R) lattice, the coincidence lattice is the primitive hexagonal (P) sublattice (whose unit cell is commonly used for the hexagonal description of rhombohedral crystals). Here, the two centring points inside the triple hexagonal R cell do not belong to the coincidence sublattice which is, hence, of index [[j] = 3] . The same holds for the spinel twins of cubic crystals, provided only one of the four threefold axes is involved in the twinning.

(iv) Twins with ( $[\Sigma> 3]$ twins). Whereas twins with are very common and of high importance among minerals and metals, twins with higher lattice indices occur hardly at all. All these `high-index' twins can occur systematically only in tetragonal, hexagonal, rhombohedral and cubic crystals, due to the geometric perpendicularity relations set out in Table 3.3.8.1. Note that for special lattice metrics (axial ratios and angles) they can occur, of course, in any crystal system. These special metrics, however, are not enforced by the crystal symmetry and hence the coincidences are not strict, but only `pseudo-coincidences'.

Examples

(1) Tetragonal twins with twin reflection planes {210} or {130}, or twofold twin axes $[\langle 210\rangle]$ or $[\langle 130\rangle]$ lead to [[j] = 5] , the largest value of [[j]] that has been found so far for tetragonal twins. The coincidence lattice is again tetragonal with $[{\bf a}' = 2{\bf a} + {\bf b}]$ , $[{\bf b}' = - {\bf a }+ 2{\bf b}]$ , $[{\bf c}' = {\bf c}]$ and is shown in Fig. 3.3.8.1. An actual example, SmS_1.9 (Tamazyan et al., 2000b), is discussed in Section 3.3.9.2.4.

Figure 3.3.8.1 | top | pdf |

Lattice relations of $[\Sigma 5]$ twins of tetragonal crystals with primitive lattice: twin mirror plane and composition plane (120) with twin displacement vector t = 0. Small dots: lattice points of domain 1; small x: lattice points of domain 2; large black dots: $[\Sigma 5]$ coincidence lattice.

(2) There exist several old and still unsubstantiated indications for a cubic garnet twin with twin reflection plane (210), cf. Arzruni (1887); Tschermak & Becke (1915, p. 594).
(3) Klockmannite, CuSe (Taylor & Underwood, 1960; Takeda & Donnay, 1965). This hexagonal mineral seems to be the only example for a hexagonal twin with . X-ray diffraction experiments indicate a reflection twin on $[(13{\bar 4}0)]$ , corresponding to . Later structural studies, however, suggest the possibility of disorder instead of twinning.
(4) Galena, PbS (NaCl structure). Galena crystals from various localities often exhibit lamellae parallel to the planes {441} which are interpreted as (441) reflection twins with ( $[\Sigma 33]$ twin) (cf. Niggli, 1926, Fig. 9k on p. 53). These natural twins are deformation and not growth twins. In laboratory deformation experiments, however, these twins could not be generated. A detailed analysis of twinning in PbS with respect to plastic deformation is given by Seifert (1928).

(5) For cubic metals and alloys annealing twins (recrystallization twins) with [[j]> 3] are common. Among them high-order twins (high-generation twins) are particularly frequent. They are based on the $[\Sigma 3]$ (spinel) twins (first generation) which may coalesce and form `new twins' with $[\Sigma 9 = 3^2]$ [second generation, with twin reflection plane (221)], $[\Sigma 27 = 3^3]$ [third generation, twin reflection plane (115)], $[\Sigma 81 = 3^4]$ [fourth generation, twin reflection plane (447)] etc. Every step to a higher generation increases $[\Sigma]$ by a factor of three (Gottstein, 1984). An interesting and actual example is the artificial silicon tricrystal shown in Fig. 3.3.8.2, which contains three components related by two (111) reflection planes (first generation, two $[\Sigma 3]$ boundaries) and one (221) reflection plane (second generation, one $[\Sigma 9]$ boundary).

Figure 3.3.8.2 | top | pdf |

(a) A (110) silicon slice (10 cm diameter, 0.3 mm thick), cut from a Czochralski-grown tricrystal for solar-cell applications. As seed crystal, a cylinder of three coalesced Si single-crystal sectors in (111) and (221) reflection-twin positions was used. Pulling direction [110] (Courtesy of M. Krühler, Siemens AG, München). (b) Sketch of the tricystal wafer showing the twin relations [twin laws [m(111)] and [m(221)] ] and the $[\Sigma]$ characters of the three domain pairs. The atomic structures of these (111) and (221) twin boundaries are discussed by Kohn (1956, 1958), Hornstra (1959, 1960) and Queisser (1963).

(6) The same type of tricrystal has been found in cubic magnetite (Fe₃O₄) nanocrystals grown from the biogenic action of magnetotactic bacteria in an aquatic environment (Devouard et al., 1998). Here, HRTEM micrographs (Fig. 6 of the paper) show the same triple-twin arrangement as in the Si tricrystal above. The authors illustrate this triple twin by (111) spinel-type intergrowth of three octahedra exhibiting two $[\Sigma 3]$ and one $[\Sigma 9]$ domain pairs. The two $[\Sigma 3]$ interfaces are (111) twin reflection planes, whereas the $[\Sigma 9]$ boundary is very irregular and not a compatible planar (221) interface (i.e. not a twin reflection plane).
(7) A third instructive example is provided by the fivefold cyclic `cozonal' twins (zone axis $[[1{\bar 1}0]]$ ) of Ge nanocrystals (Neumann et al., 1996; Hofmeister, 1998), which are treated in Section 3.3.10.6.5 and Fig. 3.3.10.11. All five boundaries between neighbouring domains (sector angles 70.5°) are of the $[\Sigma3(111)]$ type. Second nearest ( $[2\times 70.5^\circ]$ ), third nearest ( $[3\times 70.5^\circ]$ ) and fourth nearest ( $[4\times70.5^\circ]$ ) neighbours exhibit $[\Sigma9]$ , $[\Sigma27]$ and $[\Sigma81]$ coincidence relations (second, third and fourth $[\Sigma]$ generation), respectively, as introduced above in (5). These relations can be described by the `cozonal' twin reflection planes (111), (221), (115) and (447). Since $[5\times70.5^\circ=352.5^\circ]$ , an angular gap of 7.5° would result. In actual crystals this gap is compensated by stacking faults as shown in Fig. 3.3.10.11. A detailed treatment of all these cases, including structural models of the interfaces, is given by Neumann et al. (1996).
(8) Examples of (hypothetical) twins with due to metrical specialization of the lattice are presented by Koch (2004).

References

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 418-419