-
(1) In addition to exact lattice coincidences (as they occur for all merohedral twins) approximate lattice coincidences (pseudo-coincidences) are taken into account.
In this context, it is important to explain the meaning of the terms approximate lattice coincidences or pseudo lattice coincidences as used in this section. Superposition of two or more equal lattices (with a common origin) that are slightly misoriented with respect to each other leads to a three-dimensional moiré pattern of coincidences and anti-coincidences. The beat period of this pattern increases with decreasing misorientation. It appears sensible to use the term approximate or pseudo-coincidences only if the `splitting' of lattice points is small within a sufficiently large region around the common origin of the two lattices. Special cases occur for reflection twins and rotation twins of pseudosymmetrical lattices. For the former, exact two-dimensional coincidences exist parallel to the (rational) twin reflection plane and the moiré pattern is only one-dimensional in the direction normal to this plane. Hence, the region of `small splitting' is a two-dimensional (infinitely extended) thin layer of the twin lattice on both sides of the twin reflection plane [example: pseudo-monoclinic albite (010) reflection twins]. For rotation twins, the region of `small splitting' is an (infinitely long) cylinder around the twin axis. On the axis the lattice points coincide exactly.
In general, a typical measure of this region, in terms of the reciprocal lattice, could be the size of a conventional X-ray diffraction photograph. Whereas the slightest deviations from exact coincidence lead to pseudo-coincidences, the `upper limit of the splitting', up to which two lattices are considered as pseudo-coincident, is not definable on physical grounds and thus is a matter of convention and personal preference. As an angular measure of the splitting the twin obliquity has been introduced by Friedel (1926![[link]](/graphics/greenarr.gif) ). This concept and its use in twinning will be discussed below in Section 3.3.8.5.
-
(2) The previous treatment of superposition of only two lattices is extended to multiple twins with several interpenetrating lattices which are related by a pseudo n-fold twin axis. Such a twin axis cannot be `exact', no matter how close its rotation angle comes to the exact angular value. For this reason, twin axes of order ![[n> 2]](/teximages/dach3o3/dach3o3fi676.svg) necessarily lead to pseudo lattice coincidences.
Here it is assumed that such pseudo-coincidences exist for any pair of neighbouring twin domains. As a consequence, pseudo-coincidences occur for all n domains. For this case, the following rules exist:
-
(i) Only n-fold twin axes with the crystallographic values n = 3, 4 and 6 lead to pseudo lattice coincidences of all domains. Example: cyclic triplets of aragonite.
-
(ii) The number of (interpenetrating) lattices equals the number of different domain states [cf. Section 3.3.4.4(iii)], viz.![[\eqalign{\hbox{6, 3 or 2 lattices for }n &= 6,\cr\hbox{3 lattices for }n&=3,\cr\hbox{4 or 2 lattices for }n&=4,}]](/teximages/dach3o3/dach3o3fd21.svg) whereby the case `2 lattices' for ![[n = 6]](/teximages/bach3o4/bach3o4fi146.svg) leads to exact lattice coincidence (merohedral twinning, e.g. Dauphiné twins of quartz).
-
(iii) There always exists exact (one-dimensional) coincidence of all lattice rows along the twin axis.
-
(iv) If there is a (rational) lattice plane normal to the twin axis, the splitting of the lattice points occurs only parallel to this plane. If, however, this lattice plane is pseudo-normal (i.e. slightly inclined) to the twin axis, the splitting of lattice points also has a small component along the twin axis.
|
|
Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: Berger-Levrault. [Reprinted (1964). Paris: Blanchard].Google Scholar
