Specifications and extensions of the orientation relations

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 395-397

Section 3.3.2.3. Specifications and extensions of the orientation relations

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

3.3.2.3. Specifications and extensions of the orientation relations

| top | pdf |

In the following, the orientation and chirality relations of two or more twin components, only briefly mentioned in the definition, are explained in detail. Two categories of orientation relations have to be distinguished: those arising from binary twin operations (binary twin elements), i.e. operations of order 2, and those arising from pseudo n-fold twin rotations (n-fold twin axes), i.e. operations of order $[\geq 3]$ .

3.3.2.3.1. Binary twin operations (twin elements)

| top | pdf |

The (crystallographic) orientation relation of two twin partners can be expressed either by a twin operation or by its corresponding twin element. Binary twin elements can be either twin mirror planes or twofold twin axes or twin inversion centres. The former two twin elements must be parallel or normal to (possible) crystal faces and edges (macroscopic description) or, equivalently, parallel or normal to lattice planes and lattice rows (microscopic lattice description). Twin elements may be either rational (integer indices) or irrational (irrational indices which, however, can always be approximated by sufficiently large integer indices). Twin reflection planes and twin axes parallel to lattice planes or lattice rows are always rational. Twin axes and twin mirror planes normal to lattice planes or lattice rows are either rational or irrational. In addition to planes and axes, points can also occur as twin elements: twin inversion centres.

There exist seven kinds of binary twin elements that define the seven general twin laws possible for noncentrosymmetric triclinic crystals (crystal class 1):

(i) Rational twin mirror plane (hkl) normal to an irrational line: reflection twin (Fig. 3.3.2.1a). The lattice plane hkl is common to both twin partners.

Figure 3.3.2.1 | top | pdf |

Schematic illustration of the orientation relations of triclinic twin partners, see Section 3.3.2.3.1, (a) for twin element (i) `rational twin mirror plane' and (b) for twin element (ii) `irrational twofold twin axis' (see text); common lattice plane (hkl) for both cases. The noncentrosymmetry of the crystal is indicated by arrows. The sloping up and sloping down of the arrows is indicated by the tapering of their images. For centrosymmetry, both cases (a) and (b) represent the same orientation relation.

(ii) Irrational twofold twin axis normal to a rational lattice plane hkl: rotation twin (Fig. 3.3.2.1b). The lattice plane hkl is common to both twin partners.

(iii) Rational twofold twin axis [uvw] normal to an irrational plane: rotation twin (Fig. 3.3.2.2a). The lattice row [uvw] is common to both twin partners.

Figure 3.3.2.2 | top | pdf |

As Fig. 3.3.2.1, (a) for twin element (iii) `rational twofold twin axis' and (b) for twin element (iv) `irrational twin mirror plane' (see text); common lattice row [uvw] for both cases. For centrosymmetry, both cases represent the same orientation relation.

(iv) Irrational twin mirror plane normal to a rational lattice row [uvw]: reflection twin (Fig. 3.3.2.2b). The lattice row [uvw] is common to both twin partners.

(v) Irrational twofold twin axis normal to a rational lattice row [uvw], both located in a rational lattice plane (hkl); perpendicular to the irrational twin axis is an irrational plane: complex twin; German: Kantennormalengesetz (Fig. 3.3.2.3). The lattice row [uvw] is `common' to both twin partners; the planes $[(hkl)_{\rm I}]$ and $[(hkl)_{\rm II}]$ are `parallel' but not `common' (cf. Tschermak & Becke, 1915, p. 98; Niggli, 1941, p. 138; Bloss, 1971, pp. 228–230; Phillips, 1971, p. 178).

Figure 3.3.2.3 | top | pdf |

Illustration of the Kantennormalengesetz (complex twin) for twin elements (v) and (vi) (see text); common lattice row [uvw] for both cases. Note that both twin elements transform the net plane $[(hkl)_{\rm I}]$ into its parallel but not pointwise coincident counterpart $[(hkl)_{\rm II}]$ . For centrosymmetry, both cases represent the same orientation relation.

(vi) Irrational twin mirror plane containing a rational lattice row [uvw]; perpendicular to the twin plane is an irrational direction; the row [uvw] and the perpendicular direction span a rational lattice plane (hkl): complex twin; this `inverted Kantennormalengesetz' is not described in the literature (Fig. 3.3.2.3). The row [uvw] is `common' to both twin partners; the planes $[(hkl)_{\rm I}]$ and $[(hkl)_{\rm II}]$ are `parallel' but not `common'.
(vii) Twin inversion centre: inversion twin. The three-dimensional lattice is common to both twin partners. Inversion twins are always merohedral (parallel-lattice) twins (cf. Section 3.3.8).

All these binary twin elements – no matter whether rational or irrational – lead to crystallographic orientation relations , as defined in Section 3.3.2.2, because the following lattice items belong to both twin partners:

(a) The rational lattice planes $[(hkl)_{\rm I}]$ and $[(hkl)_{\rm II}]$ are `common' for cases (i) and (ii) (Fig. 3.3.2.1).
(b) The rational lattice rows $[[uvw]_{\rm I}]$ and $[[uvw]_{\rm II}]$ are `common' and furthermore lattice planes $[(hkl)_{\rm I}/(hkl)_{\rm II}]$ in the zone [uvw] are `parallel', but not `common' for cases (iii), (iv), (v) and (vi). Note that for cases (iii) and (iv) any two planes $[(hkl)_{\rm I}/(hkl)_{\rm II}]$ of the zone [uvw] are parallel, whereas for cases (v) and (vi) only a single pair of parallel planes exists (cf. Figs. 3.3.2.2 and 3.3.2.3).
(c) The entire three-dimensional lattice is `common' for case (vii).

In this context one realizes which wide range of twinning is covered by the requirement of a crystallographic orientation relation: the `minimal' condition is provided by the complex twins (v) and (vi): only a one-dimensional lattice row is `common', two lattice planes are `parallel' and all twin elements are irrational (Fig. 3.3.2.3). The `maximal' condition, a `common' three-dimensional lattice, occurs for inversion twins (`merohedral' or `parallel-lattice twins'), case (vii).

In noncentrosymmetric triclinic crystals, the above twin elements define seven different twin laws, but for centrosymmetric crystals only three of them represent different orientation relations, because both in lattices and in centrosymmetric crystals a twin mirror plane defines the same orientation relation as the twofold twin axis normal to it, and vice versa. Consequently, the twin elements of the three pairs (i) + (ii), (iii) + (iv) and (v) + (vi) represent the same orientation relation. Case (vii) does not apply to centrosymmetric crystals, since here the inversion centre already belongs to the symmetry of the crystal.

For symmetries higher than triclinic, even more twin elements may define the same orientation relation, i.e. form the same twin law. Example: the dovetail twin of gypsum (point group [12/m1] ) with twin mirror plane (100) can be described by the four alternative twin elements (i), (ii), (iii), (iv) (cf. Section 3.3.4, Fig. 3.3.4.1). Furthermore, with increasing symmetry, the twin elements (i) and (iii) may become even more special, and the nature of the twin type may change as follows:

(i) the line normal to a rational twin mirror plane (hkl) may become a rational line [uvw];
(ii) the plane normal to a rational twofold twin axis [uvw] may become a rational plane (hkl).

In both cases, the three-dimensional lattice (or a sublattice of it) is now common to both twin partners, i.e. a `merohedral' twin results.

There is one more binary twin type which seems to reduce even further the above-mentioned `minimal' condition for a crystallographic orientation relation, the so-called `median law' (German: Mediangesetz) of Brögger (1890), described by Tschermak & Becke (1915, p. 99). So far, it has been found in one mineral only: hydrargillite (modern name gibbsite), Al(OH)₃. The acceptability of this orientation relation as a twin law is questionable; see Section 3.3.6.10.

3.3.2.3.2. Pseudo n-fold twin rotations (twin axes) with $[n\ge 3]$

| top | pdf |

There is a long-lasting controversy in the literature, e.g. Hartman (1956, 1960), Buerger (1960b), Curien (1960), about the acceptance of three-, four- and sixfold rotation axes as twin elements, for the following reason:

Twin operations of order two (reflection, twofold rotation, inversion) are `exact', i.e. in a component pair they transform the orientation state of one component exactly into that of the other. There occur, in addition, many cases of multiple twins, which can be described by three-, four- and sixfold twin axes . These axes, however, are pseudo axes because their rotation angles are close to but not exactly equal to 120, 90 or 60°, due to metrical deviations (no matter how small) from a higher-symmetry lattice. A well known example is the triple twin (German: Drilling) of orthorhombic aragonite, where the rotation angle $[\gamma=]$ $[2\arctan b/a = 116.2^\circ]$ (which transforms the orientation state of one component exactly into that of the other) deviates significantly from the 120° angle of a proper threefold rotation (Fig. 3.3.2.4). Another case of n = 3 with a very small metrical deviation is provided by ammonium lithium sulfate (γ = 119.6°).

Figure 3.3.2.4 | top | pdf |

(a) Triple growth twin of orthorhombic aragonite, CaCO₃, with pseudo-threefold twin axis. The gap angle is 11.4°. The exact description of the twin aggregate by means of two symmetrically equivalent twin mirror planes (110) and ( $[{\bar 1}10]$ ) is indicated. In actual crystals, the gap is usually closed as shown in (b).

All these (pseudo) n-fold rotation twins, however, can also be described by (exact) binary twin elements , viz by a cyclic sequence of twin mirror planes or twofold twin axes . This is also illustrated and explained in Fig. 3.3.2.4. This possibility of describing cyclic twins by `exact' binary twin operations is the reason why Hartman (1956, 1960) and Curien (1960) do not consider `non-exact' three-, four- and sixfold rotations as proper twin operations.

The crystals forming twins with pseudo n-fold rotation axes always exhibit metrical pseudosymmetries. In the case of transformation twins and domain structures, the metrical pseudosymmetries of the low-symmetry (deformed) phase $[\cal{H}]$ result from the true structural symmetry $[{\cal G}]$ of the parent phase (cf. Section 3.3.7.2). This aspect caused several authors [e.g. Friedel, 1926, p. 435; Donnay (cf. Hurst et al., 1956); Buerger, 1960b] to accept these pseudo axes for the treatment of twinning. The present authors also recommend including three-, four- and sixfold rotations as permissible twin operations. The consequences for the definition of the twin law will be discussed in Section 3.3.4 and in Section 3.4.3 . For a further extension of this concept to fivefold and tenfold multiple growth twins, see Note (6) below and Example 3.3.6.8.

References

Bloss, F. D. (1971). Crystallography and crystal chemistry, pp. 324–338. New York: Holt, Rinehart & Winston.Google Scholar

Brögger, W. C. (1890). Hydrargillit. Z. Kristallogr. 16, second part, pp. 16–43, especially pp. 24–43 and Plate 1.Google Scholar

Buerger, M. J. (1960b). Introductory remarks. Twinning with special regard to coherence. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 3 and 5–7. Madrid: CSIC.Google Scholar

Curien, H. (1960). Sur les axes de macle d'ordre supérieur à deux. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 9–11. Madrid: CSIC.Google Scholar

Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: Berger-Levrault. [Reprinted (1964). Paris: Blanchard].Google Scholar

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

Hartman, P. (1960). Epitaxial aspects of the atacamite twin. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 15–18. Madrid: CSIC.Google Scholar

Hurst, V. J., Donnay, J. D. H. & Donnay, G. (1956). Staurolite twinning. Mineral. Mag. 31, 145–163.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

Phillips, F. C. (1971). An introduction to crystallography, 4th ed. London: Longman.Google Scholar

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. 395-397