Essential addenda to the definition

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. 394-395

Section 3.3.2.2. Essential addenda to the definition

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

3.3.2.2. Essential addenda to the definition

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(a) The orientation relation between two partners is defined as crystallographic and, hence, the corresponding intergrowth is a twin, if the following two minimal conditions are simultaneously obeyed:

(i) at least one lattice row (crystal edge) [uvw] is `common' to both partners I and II, either parallel or antiparallel, i.e. $[[uvw]_{\rm I}]$ is parallel to $[\pm[uvw]_{\rm II}]$ ;
(ii) at least two lattice planes (crystal faces) $[(hkl)_{\rm I}]$ and $[\pm(hkl)_{\rm II}]$ , one from each partner, are `parallel', but not necessarily `common' (see below). This condition implies a binary twin operation (twofold rotation, reflection, inversion).

Both conditions taken together define the minimal geometric requirement for a twin (at least one common row and one pair of parallel planes), as originally pronounced by several classical authors (Tschermak, 1884, 1905; 1904; Tschermak & Becke, 1915; Mügge, 1911, p. 39; Niggli, 1920/1924/1941; Tertsch, 1936) and taken up later by Menzer (1955) and Hartman (1956). It is obvious that these crystallographic conditions apply even more to twins with two- and three-dimensional lattice coincidences, as described in Section 3.3.8. Other orientation relations, as they occur, for instance, in arbitrary intergrowths or bicrystals, are called `noncrystallographic'.

The terms `common edge' and `common face', as used in this section, are derived from the original morphological consideration of twins. Example: a re-entrant edge of a twin is common to both twin partners. In lattice considerations, the terms `common lattice row', `common lattice plane' and `common lattice' require a somewhat finer definition, in view of a possible twin displacement vector t of the twin boundary , as introduced in Note (8) of Section 3.3.2.4 and in Section 3.3.10.3. For this distinction the terms `parallel', `common' and `coincident' are used as follows:

Two lattice rows $[[uvw]_{\rm I}]$ and $[[uvw]_{\rm II}]$ :

Common : rows parallel or antiparallel, with their lattice points possibly displaced with respect to each other parallel to the row by a vector $[{\bf t} \ne {\bf 0}]$ .

Coincident : common rows with pointwise coincidence of their lattice points, i.e. $[{\bf t} = {\bf 0}]$ .
Two lattice planes $[(hkl)_{\rm I}]$ and $[(hkl)_{\rm II}]$ :

Parallel : `only' the planes as such, but not all corresponding lattice rows in the planes, are mutually parallel or antiparallel.

Common : parallel planes with all corresponding lattice rows mutually parallel or antiparallel, but possibly displaced with respect to each other parallel to the plane by a vector $[{\bf t} \ne {\bf 0}]$ .

Coincident : common planes with pointwise coincidence of their lattice points, i.e. $[{\bf t} = {\bf 0}]$ .
Two point lattices I and II:

Parallel or common: all corresponding lattice rows are mutually parallel or antiparallel, but the lattices are possibly displaced with respect to each other by a vector $[{\bf t} \ne {\bf 0}]$ .

Coincident : parallel lattices with pointwise coincidence of their lattice points, i.e. $[{\bf t}= {\bf 0}]$ .

Note that for lattice rows and point lattices only two cases have to be distinguished, whereas lattice planes require three terms.

(b) A twinned crystal may consist of more than two individuals. All individuals that have the same orientation and handedness belong to the same orientation state (component state, domain state, domain variant). The term `twin' for a crystal aggregate requires the presence of at least two orientation states.
(c) The orientation and chirality relation between two twin partners is expressed by the twin law. It comprises the set of all twin operations that transform the two orientation states into each other. A twin operation cannot be a symmetry operation of either one of the two twin components. The combination of a twin operation and the geometric element to which it is attached is called a twin element (e.g. twin mirror plane, twofold twin axis, twin inversion centre).
(d) An orientation relation between two individuals deserves the name `twin law' only if it occurs frequently, is reproducible and represents an inherent feature of the crystal species.
(e) One feature which facilitates the formation of twins is pseudosymmetry, apparent either in the crystal structure, or in special lattice-parameter ratios or lattice angles.
(f) In general, the twin interfaces are low-energy boundaries with good structural fit; very often they are low-index lattice planes.

References

Hartman, P. (1956). On the morphology of growth twins. Z. Kristallogr. 107, 225–237.Google Scholar

Menzer, G. (1955). Über Kristallzwillingsgesetze. Z. Kristallogr. 106, 193–198.Google Scholar

Mügge, O. (1911). Über die Zwillingsbildung der Kristalle. Fortschr. Mineral. Kristallogr. Petrogr. 1, 18–47.Google Scholar

Niggli, P. (1920/1924/1941). Lehrbuch der Mineralogie und Kristallchemie, 1st edition 1920, 2nd edition 1924, 3rd edition, Part I, 1941, especially pp. 136–153, 401–414. Berlin: Gebrüder Borntraeger.Google Scholar

Tertsch, H. (1936). Bemerkungen zur Frage der Verbreitung und zur Geometrie der Zwillingsbildungen. Z. Kristallogr. 94, 461–490.Google Scholar

Tschermak, G. (1884, 1905). Lehrbuch der Mineralogie, 1st ed. 1884, 6th ed. 1905. Wien: Alfred Hölder.Google Scholar

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Tschermak, G. & Becke, F. (1915). Lehrbuch der Mineralogie, 7th edition, pp. 93–114. Wien: Alfred Hölder.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 394-395