Lattice coincidences, twin lattice, twin 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. 417-418

Section 3.3.8.2. Lattice coincidences, twin lattice, twin 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

3.3.8.2. Lattice coincidences, twin lattice, twin lattice index

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Four types of (exact) lattice coincidences have to be distinguished in twinning:

(i) No coincidence of lattice points (except, of course, for the initial pair). This case corresponds to arbitrary intergrowth of two crystals or to a general bicrystal.

(ii) One-dimensional coincidence: Both lattices have only one lattice row in common. Of the seven binary twin operations listed in Section 3.3.2.3.1, the following three generate one-dimensional lattice coincidence:

(a) twofold rotation around a (rational) lattice row [twin operation (iii) in Section 3.3.2.3.1];
(b) reflection across an irrational plane normal to a (rational) lattice row (note that the coincidence would be three-dimensional if this plane were rational) [twin operation (iv)];
(c) twofold rotation around an irrational axis normal to a (rational) lattice row (complex twin, Kantennormalengesetz) [twin operations (v) and (vi)].

Lattices are always centrosymmetric; hence, for lattices, as well as for centrosymmetric crystals, the first two twin operations above belong to the same twin law. For noncentrosymmetric crystals, however, the two twin operations define different twin laws.

(iii) Two-dimensional coincidence: Both lattices have only one lattice plane in common. The following two (of the seven) twin operations lead to two-dimensional lattice coincidence:

(a) reflection across a (rational) lattice plane [twin operation (i)];
(b) twofold rotation around an irrational axis normal to a (rational) lattice plane (note that the coincidence would be three-dimensional if this axis were rational) [twin operation (ii)].

Again, for lattices and centrosymmetric crystals both twin operations belong to the same twin law.

(iv) Three-dimensional coincidence: Here the coincidence subset is a three-dimensional lattice, the coincidence-site lattice or twin lattice. It is the three-dimensional sublattice common to the (equally or differently) oriented lattices of the two twin partners . The degree of three-dimensional lattice coincidence is defined by the coincidence-site lattice index, twin lattice index or sublattice index [j], for short: lattice index. This index is often called $[\Sigma]$ , especially in metallurgy. It is the volume ratio of the primitive cells of the twin lattice and of the (original) crystal lattice (i.e. [1/j] is the `degree of dilution' of the twin lattice with respect to the crystal lattice): $[[j] = \Sigma = V_{\rm twin} / V_{\rm crystal}.]$

The lattice index is always an integer: [j = 1] means complete coincidence (parallelism), [j> 1] partial coincidence of the two lattices. The index [j] can also be interpreted as elimination of the fraction [(j - 1)/j] of the lattice points, or as index of the translation group of the twin lattice in the translation group of the crystal lattice. The coincidence lattice, thus, is the intersection of the oriented lattices of the two twin partners .

Twinning with [[j] = 1] has been called by Friedel (1926, p. 427) twinning by merohedry (`macles par mériédrie') (for short: merohedral twinning), whereas twinning with [[j]> 1] is called twinning by lattice merohedry or twinning by reticular merohedry (`macles par mériédrie réticulaire') (Friedel, 1926, p. 444). The terms for [[j] = 1] are easily comprehensible and in common use. The terms for , however, are somewhat ambiguous. In the present section, therefore, the terms sublattice, coincidence lattice or twin lattice of index [[j]] are preferred. Merohedral twinning is treated in detail in Section 3.3.9.

Complete and exact three-dimensional lattice coincidence ( [[j] = 1] ) always exists for inversion twins (of noncentrosymmetric crystals) [twin operation (vii)]. For reflection twins, complete or partial coincidence occurs if a (rational) lattice row [uvw] is (exactly) perpendicular to the (rational) twin reflection plane (hkl); similarly for rotation twins if a (rational) lattice plane (hkl) is (exactly) perpendicular to the (rational) twofold twin axis [uvw].

The systematic perpendicularity relations (i.e. relations valid independent of the axial ratios) for lattice planes (hkl) and lattice rows [uvw] in the various crystal systems are collected in Table 3.3.8.1. No perpendicularity occurs for triclinic lattices (except for metrical accidents). The perpendicularity cases for monoclinic and orthorhombic lattices are trivial. For tetragonal (tet), hexagonal (hex) and rhombohedral (rhomb) lattices, systematic perpendicularity of planes and rows occurs only for the $[[001]_{\rm tet}]$ and the $[[001]_{\rm hex}]$ (or $[[111]_{\rm rhomb}]$ ) zones, i.e. for planes parallel and rows perpendicular to these directions, in addition to the trivial cases $[[001]\perp(001)]$ or $[[111]\perp(111)]$ . In cubic lattices, every lattice plane (hkl) is perpendicular to a lattice row [uvw] (with [h = u] , [k = v] , [l = w] ). More general coincidence relations were derived by Grimmer (1989, 2003).

Table 3.3.8.1 | top | pdf |
Lattice planes (hkl) and lattice rows [uvw] that are mutually perpendicular (after Koch, 2004)

Lattice	Lattice plane (hkl)	Lattice row [uvw]	Perpendicularity condition and quantity
Triclinic	—	—	—
Monoclinic (unique axis b)	(010)	[010]	—
Monoclinic (unique axis c)	(001)	[001]	—
Orthorhombic	(100)	[100]	—
	(010)	[010]	—
	(001)	[001]	—
Hexagonal and rhombohedral (hexagonal axes)	()	[]	, ,
Hexagonal and rhombohedral (hexagonal axes)	(0001)	[001]	—
Rhombohedral (rhombohedral axes)	()	[]	, ,
Rhombohedral (rhombohedral axes)	(111)	[111]	—
Tetragonal	()	[]	, ,
Tetragonal	()	[]	—
Cubic	()	[]	, , ;

The index [[j]] of a coincidence or twin lattice can often be obtained by inspection; it can be calculated by using a formula for the auxiliary quantity [j'] as follows: $[j' = hu + kv + lw \ \ ({\rm scalar\ product}\ {\bf r}^\star _{hkl} \cdot {\bf t}_{uvw})]$ with sublattice index $[\eqalign{ [j] &= \vert j'\vert \hbox{ for }j' = 2n + 1 \cr&= \vert j'\vert /2 \hbox{ for }j' = 2n. }]$

Here, the indices of the plane (hkl) and of the perpendicular row [uvw] are referred to a primitive lattice basis (primitive cell). For centred lattices, described by conventional bases, modifications are required; these and further examples are given by Koch (2004). Formulae and tables are presented by Friedel (1926, pp. 245–252) and by Donnay & Donnay (1972). The various equations for the quantity [j'] are also listed in the last column of Table 3.3.8.1.

Note that in the tetragonal system for any ( [hk0] ) reflection twin and any [ [uv0] ] twofold rotation twin, the coincidence lattices are also tetragonal and have the same lattice parameter c. Further details are given by Grimmer (2003). An analogous relation applies to the hexagonal crystal family for ( [hki0] ) and [ [uv0] ] twins. In the cubic system, the following types of twin lattices occur:

(111) and [111] twins: hexagonal P lattice (e.g. spinel twins);
and twins: tetragonal lattice;
and twins: orthorhombic lattice;
and twins: monoclinic lattice.

Note that triclinic twin lattices are not possible for a cubic lattice.

After these general considerations of coincidence-site and twin lattices and their lattice index, specific cases of `triperiodic twins' are treated in Section 3.3.8.3. In addition to the characterization of the twin lattice by its index [[j]] , the $[\Sigma]$ notation used in metallurgy is included.

References

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

Grimmer, H. (1989). Systematic determination of coincidence orientations for all hexagonal lattices with axial ratio c/a in a given interval. Acta Cryst. A45, 320–325.Google Scholar

Grimmer, H. (2003). Determination of all misorientations of tetragonal lattices with low multiplicity; connection with Mallard's rule of twinning. Acta Cryst. A59, 287–296.Google Scholar

Koch, E. (2004). Twinning. In International tables for crystallography, Vol. C. Mathematical, physical and chemical tables, edited by E. Prince, 3rd ed., ch. 1.3. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 417-418