Twin obliquity and lattice pseudosymmetry

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. 420-421

Section 3.3.8.5. Twin obliquity and lattice pseudosymmetry

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

3.3.8.5. Twin obliquity and lattice pseudosymmetry

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The concept of twin obliquity has been introduced by Friedel (1926, p. 436) to characterize (metrical) pseudosymmetries of lattices and their relation to twinning. The obliquity $[\omega]$ is defined as the angle between the normal to a given lattice plane (hkl) and a lattice row [uvw] that is not parallel to (hkl) and, vice versa, as the angle between a given lattice row [uvw] and the normal to a lattice plane (hkl) that is not parallel to [uvw]. The twin obliquity is thus a quantitative (angular) measure of the pseudosymmetry of a lattice and, hence, of the deviation which the twin lattice suffers in crossing the composition plane (cf. Section 3.3.8.1).

The smallest mesh of the net plane (hkl) together with the shortest translation period along [uvw] define a unit cell of a sublattice of lattice index [j]; j may be [=1] or [>1] [cf. Section 3.3.8.2(iv)]. The quantities $[\omega]$ and j can be calculated for any lattice and any (hkl)/[uvw] combination by elementary formulae, as given by Friedel (1926, pp. 249–252) and by Donnay & Donnay (1972). Recently, a computer program has been written by Le Page (1999, 2002) which calculates for a given lattice all (hkl)/[uvw]/ $[\omega]$ /j combinations up to given limits of $[\omega]$ and j. In the theory of Friedel and the French School, a (metrical) pseudosymmetry of a lattice or sublattice is assumed to exist if the twin obliquity $[\omega]$ as well as the twin lattice index j are `small'. This in turn means that the pair lattice plane (hkl)/lattice row [uvw] is the better suited as twin elements (twin reflection plane/twofold twin axis) the smaller $[\omega]$ and j are.

The term `small' obviously cannot be defined in physical terms. Its meaning rather depends on conventions and actual analyses of triperiodic twins. In his textbook, Friedel (1926, p. 437) quotes frequently observed twin obliquities of 3–4° (albite $[4^\circ 3']$ , aragonite $[3^\circ 44']$ ) with `rare exceptions' of 5–6°. In a paper devoted to the quartz twins with `inclined axes', Friedel (1923, pp. 84 and 86) accepts the La Gardette (Japanese) and the Esterel twins, both with large obliquities of $[\omega = 5^\circ 27']$ and $[\omega = 5^\circ 48']$ , as pseudo-merohedral twins only because their lattice indices [[j] = 2] and 3 are (`en revanche') remarkably small. He considers $[\omega = 6^\circ]$ as a limit of acceptance [`limite prohibitive'; Friedel (1923, p. 88)].

Lattice indices [[j] = 3] are very common (in cubic and rhombohedral crystals), [[j] = 5] twins are rare and [[j] = 6] seems to be the maximal value encountered in twinning (Friedel, 1926, pp. 449, 457–464; Donnay & Donnay, 1974, Table 1). In his quartz paper, Friedel (1923, p. 92) rejects all pseudo-merohedral quartz twins with $[[j]\geq 4]$ despite small $[\omega]$ values, and he points out, as proof that high j values are particularly unfavourable for twinning, that strictly merohedral quartz twins with [[j] = 7] do not occur, i.e. that $[\omega = 0]$ cannot `compensate' for high j values.

In agreement with all these results and later experiences (e.g. Le Page, 1999, 2002), we consider in Table 3.3.8.2 only lattice pseudosymmetries with $[\omega \le 6^\circ]$ and $[[j] \le 6]$ , preferably $[[j] \le 3]$ . (It should be noted that, on purely mathematical grounds, arbitrarily small $[\omega]$ values can always be obtained for sufficiently large values of [h,k,l] and [u,v,w] , which would be meaningless for twinning.) The program by Le Page (1999, 2002) enables for the first time systematic calculations of many (`all possible') (hkl)/[uvw] combinations for a given lattice and, hence, statistical and geometrical evaluations of existing and particularly of (geometrically) `permissible' but not observed twin laws. In Table 3.3.8.2, some examples are presented that bring out both the merits and the problems of lattice geometry for the theory of twinning. The `permissibility criteria' $[\omega \le 6^\circ]$ and $[[j] \le 6]$ , mentioned above, are observed for most cases.

Table 3.3.8.2 | top | pdf |
Examples of calculated obliquities $[\omega]$ and lattice indices [j] for selected (hkl) [/] [uvw] combinations and their relation to twinning

Calculations were performed with the program OBLIQUE written by Le Page (1999, 2002).

Crystal	(hkl)	Pseudo-normal [uvw]	Obliquity $[[^\circ]]$	Lattice index [j]	Remark
Gypsum , , Å $[\beta = 127.5^\circ]$	(100)	[302]	2.47	3	Dovetail twin (very frequent)
		[805]	0.42	4	Dovetail twin (very frequent)
	(001)	[203]	5.92	3	Montmartre twin (less frequent)
		[305]	0.95	5	Montmartre twin (less frequent)
	(101)	[101]	2.60	2	No twin
	$[(11{\bar 1})]$	$[[31{\bar 4}]]$	1.35	4	No twin
Rutile , Å	(101)	[102]	5.02	3	Frequent twin
		[307]	0.84	5	Frequent twin
	(301)	[101]	5.43	2	Rare twin
	(201)	[304]	2.85	5	No twin
	(210) or (130)	[210] or [130]	0	5	No twin
Quartz , Å	$[(11{\bar 2}2)]$	[111]	5.49	2	Japanese twin (La Gardette) (rare)
	$[(10{\bar 1}1)]$	[211]	5.76	3	Esterel twin (rare)
	$[(10{\bar 1}2)]$	[212]	5.76	3	Sardinia twin (very rare)
	$[(21{\bar 3}0)]$ or $[(14{\bar 5}0)]$	[540] or [230]	0	7	No twin
Staurolite , , Å $[\beta = 90.00^\circ]$	(031)	[013]	1.19	6	90° twin (rare)
	(231)	[313]	0.90	12	60° twin (frequent)
	(201)	[101]	0.87	3	No twin
	(101)	[102]	0.87	3	No twin
Calcite $[R{\bar 3}c]$ , Å [hexagonal axes, structural X-ray cell; cf. Section 3.3.10.2.2, Example (5)]	$[(01{\bar 1}2)]$	[ $[5_\prime10_\prime1]$ ]	5.31	2	No twin
		[ $[7_\prime14_\prime2]$ ]	2.57	3
		[481]	0.59	5
	$[(10{\bar 1}4)]$	[421]	0.74	4	Rare deformation twin (r-twin)
	$[(01{\bar 1}8)]$	[121]	0.59	5	Frequent deformation twin (e-twin)
	$[(10{\bar 1}1)]$	[14.7.1]	1.54	5	No twin

The following comments on these data should be made.

Gypsum : The calculations result in nearly 70 `permissible' (hkl)/[uvw] combinations. For the very common (100) dovetail twin, four (100)/[uvw] combinations are obtained. Only the two combinations with smallest $[\omega]$ and [j] are listed in the table; similarly for the less common (001) Montmartre twin. In addition, two cases of low-index (hkl) planes with small obliquities and small lattice indices are listed, for which twinning has never been observed.

Rutile : Here nearly twenty `permissible' (hkl)/[uvw] combinations with $[\omega \le 6^\circ]$ , $[[j] \le 6]$ occur. For the frequent (101) reflection twins, five permissible cases are calculated, of which two are given in the table. For the rare (301) reflection twins, only the one case listed, with high obliquity $[\omega = 5.4^\circ]$ , is permissible. For the further two cases of low obliquity and lattice index [5], twins are not known. Among them is one case of (strict) `reticular merohedry', (210) or (130), with $[\omega = 0]$ and [[j] = 5] (cf. Fig. 3.3.8.1).

Quartz : The various quartz twins with inclined axes were studied extensively by Friedel (1923). The two most frequent cases, the Japanese $[(11{\bar 2}2)]$ twin (called La Gardette twin by Friedel) and the $[(10{\bar 1}1)]$ Esterel twin, are considered here. In both cases, several lattice pseudosymmetries occur. Following Friedel, those with the smallest lattice index, but relatively high obliquity close to 6° are listed in the table. Again, a twin of (strict) `reticular merohedry' with $[\omega = 0]$ and [[j] = 7] does not occur [cf. Section 3.3.9.2.3, Example (2)].

Staurolite : Both twin laws occurring in nature, (031) and (231), exhibit small obliquities but rather high lattice indices [6] and [12]. The frequent (231) 60° twin with [[j] = 12] falls far outside the `permissible' range. The further two planes listed in the table, (201) and (101), exhibit favourably small obliquities and lattice indices, but do not form twins. The existing (031) and (231) twins of staurolite are discussed again in Section 3.3.9.2 under the aspect of `reticular pseudo-merohedry'.

Calcite : For calcite, 19 lattice pseudosymmetries obeying Friedel's `permissible criteria' are calculated. Again, only a few are mentioned here (indices referred to the structural cell). For the primary deformation twin $[(01{\bar 1}8)]$ , e-twin after Bueble & Schmahl (1999), cf. Section 3.3.10.2.2, Example (5), one permissible lattice pseudosymmetry with small obliquity 0.59 but high lattice index [5] is found. For the less frequent secondary deformation twin $[(10{\bar 1}4)]$ , r-twin, the situation is similar. The planes $[(01{\bar 1}2)]$ and $[(10{\bar 1}1)]$ permit small obliquities and lattice indices $[\le [5]]$ , but do not appear as twin planes.

The discussion of the examples in Table 3.3.8.2 shows that, with one exception [staurolite (231) twin], the obliquities and lattice indices of common twins fall within the $[\omega/[j]]$ limits accepted for lattice pseudosymmetry. Three aspects, however, have to be critically evaluated:

(i) For most of the lattice planes (hkl), several pseudo-normal rows [uvw] with different values of $[\omega]$ and [j] within the 6°/[6] limit occur, and vice versa. Friedel (1923) discussed this in his theory of quartz twinning. He considers the (hkl)/[uvw] combination with the smallest lattice index as responsible for the observed twinning.
(ii) Among the examples given in the table, low-index (hkl)/[uvw] combinations with more favourable $[\omega/[j]]$ values than for the existing twins can be found that never form twins. A prediction of twins on the basis of `lattice control' alone, characterized by low $[\omega]$ and [j] values, would fail in these cases.
(iii) All examples in the table were derived solely from lattice geometry, none from structural relations or other physical factors.

Note . As a mathematical alternative to the term `obliquity', another more general measure of the deviation suffered by the twin lattice in crossing the twin boundary was presented by Santoro (1974, equation 36). This measure is the difference between the metric tensors of lattice 1 and of lattice 2, the latter after retransformation by the existing or assumed twin operation (or more general orientation operation).

References

Bueble, S. & Schmahl, W. W. (1999). Mechanical twinning in calcite considered with the concept of ferroelasticity. Phys. Chem. Miner. 26, 668–672.Google Scholar

Donnay, G. & Donnay, J. D. H. (1974). Classification of triperiodic twins. Can. Mineral. 12, 422–425.Google Scholar

Donnay, J. D. H. & Donnay, G. (1972). Crystal geometry, Section 3 (pp. 99–158). In International tables for X-ray crystallography, Vol. II, Mathematical tables, edited by J. C. Kasper & K. Lonsdale. Birmingham: Kynoch Press.Google Scholar

Friedel, G. (1923). Sur les macles du quartz. Bull. Soc. Fr. Minéral. Cristallogr. 46, 79–95.Google Scholar

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

Le Page, Y. (1999). Low obliquity in pseudo-symmetry of lattices and structures, and in twinning by pseudo-merohedry. Acta Cryst. A55, Supplement. Abstract M12.CC001.Google Scholar

Le Page, Y. (2002). Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry. J. Appl. Cryst. 35, 175–181.Google Scholar

Santoro, A. (1974). Characterization of twinning. Acta Cryst. A30, 224–231.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 420-421