Summary of the article ‘Analysis of the structural continuity in twinned crystals in terms of pseudo-eigensymmetry of crystallographic orbits’

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The effects of twinning on the properties of a crystal

Individual crystals in a twin are separated by an interface called composition surface which represents a discontinuity for at least a sub-structure. The heterogeneous character of a twin represents an obstacle to the structural investigation and the technological applications of the material under investigation. In particular:
· The potential technological applications are hindered by the presence of twinning (e.g. the piezoelectric effect is reduced or annihilated).
· The presence of twinning reduces the amount of details that can be obtained from a structural study by diffraction experiments, especially for samples with large unit cells (as, for example, for macromolecules) for which the resolution that can be achieved is already limited by the size of the unit cell.
Material scientists growing crystals with targeted properties aim at avoiding the formation of twins. Understanding the conditions under which they are likely to form is an important prerequisite to develop a synthesis protocol capable of reducing, if not eliminating, the occurrence of twins. Furthermore, as already remarked, even nowadays twinned crystals still form an obstacle in the automatic solution and refinement of crystal structures.
In this thesis, we take a novel approach to investigate the possible structural basis for the formation of twins. As of today, the only systematic approach to twinning has been through the empirical rules of twinning enunciated by the reticular theory of twinning (“French school”: Friedel, 1904).
From the reticular point of view, twinning can occur when the operation mapping the orientation of the individuals overlaps a substantial amount of the nodes of the individual lattices (restored nodes). As a heuristic rule, at least one sixth of the lattice nodes should be restored in order to make the formation of the twin likely (Friedelian twin). Although the reticular theory ignores the actual contents of the unit cell, a high restoration of lattice nodes indicates the possible existence of a common sub-structure across the interface between separated individuals. The definite criterion, however, is the restoration of the atoms within the structure under a mapping known as the twin operation, which is the object of the structural theory of twinning: its aim is to identify the atomic substructure invariant under the twin operation (restored substructure). The underlying lattice common to both individuals (twin lattice) reduces the analysis to the atoms in a finite volume of the structure. In this thesis, two approaches towards a structural theory of twinning are considered, the crystallographic orbit approach and the layer group approach.
The crystallographic orbit approach: As its name suggests, the crystallographic orbit approach is based on the analysis of the twin structure via its crystallographic orbits in order to detect the subset of atoms which crosses the interface unperturbed or almost unperturbed. Under the action of the space group G, each atom in a crystal is repeated in space to form a crystallographic orbit O, i.e. O is the set of all atoms obtained from a single atom under the symmetry operations of the space group G. Each point of a crystallographic orbit defines uniquely a largest subgroup of G which maps that point onto itself and is called its site-symmetry group (or stabilizer group).
The site-symmetry groups belonging to different points in the same crystallographic orbit are conjugate subgroups of G and all points X for which the site-symmetry groups are conjugate subgroups of G form a single Wyckoff position. Under the action of a subgroup of G, an orbit O can be split into suborbits, since positions which are symmetrically equivalent under G may be no longer equivalent under the subgroup. The common sub-structure across the interface between separated individuals (the composition surface), consists of split suborbits which are invariant under the twin operation t.

Basic definitions and classifications of twins

A twinned crystal (twin) is a heterogeneous crystalline edifice composed of two or more homogeneous crystals of the same phase with different orientation related by a twin operation, i.e. a crystallographic operation mapping the orientation of one individual onto that of the other(s) (Friedel, 1904, 1926, 1933) (Figure 2.1). The atomic structure itself is mapped by a space group operation having the twin operation as its linear part. If we want to emphasize that combining the twin operation with different vector parts gives rise to symmetry operations of different types, we will use the term restoration operation for the space group operations. However, it will usually be clear from the context whether a twin operation (mapping the orientation) or a restoration operation (mapping the atom positions) is meant. A twin element is the geometric element in direct space (plane, line, centre) about which the twin operation is performed.
The result of a twin operation is an over- or intergrowth of two or more separate crystals sharing a common substructure. This phenomenon is called crystal twinning.

The twinning parameters and the reticular classification

A prerequisite for the formation of a twin is the existence of a common substructure which crosses, more or less unperturbed, the interface physically separating the individuals (Marzouki et al., 2014a). The crystal structure cannot have a symmetry higher than its Bravais lattice, however, it can be the same (holohedral structure) or a lower symmetry (merohedral structure), thus a common lattice or a sublattice is a pre-requisite for the existence of a common substructure. This represents the necessary condition of the reticular theory of twinning developed by the so-called “French school” (Bravais, 1851; Mallard,1885, Friedel, 1904, 1926). The common (sub-)lattice called the twin lattice LT (Donnay, 1940) is based on the twin element (twin plane or twin axis) and the lattice element (line or plane) that are mutually (quasi-)perpendicular. LT is defined by these two elements, the lattice plane (hkl)T and the lattice row [uvw]T.
When the individual lattice Lind or a sublattice of it crosses the composition surface without any perturbation, the lattice plane (hkl)T and the lattice row [uvw]T are mutually perpendicular and LT coincides precisely with Lind or a sublattice of it. One speaks of Twin Lattice Symmetry or TLS. When (hkl)T and [uvw]T are quasi-perpendicular, this shows a certain mismatch on the two sides of the composition surface. In this case the common sublattice LT is defined as an idealisation which does not take into account this mismatch and one speaks of Twin Lattice Quasi Symmetry or TLQS (Donnay & Donnay, 1974). In other words, in the TLQS case, Lind or a sublattice of it crosses the composition surface with small perturbation which becomes worse the farther one moves from this surface. As a consequence, LT can be, approximately, defined everywhere with a small change of orientation on the composition surface. The degree of pseudo-symmetry corresponds to the deviation from the perpendicularity condition and is measured by the angle called the obliquity: concretely is the angle between the direction perpendicular to the twin plane and the rational direction closest to it (or, for rotation twins, between the plane perpendicular to the twin axis and the rational plane closest to it). The obliquity is the first parameter of twinning, the second being the twin index, see below.
A zero-obliquity TLQS may occur for manifold twins, i.e. twins in which the twin operation is of order higher than two. For example, in a pseudo-tetragonal twin lattice LT with cell parameters a and b numerically close to each other where the twin operation is a fourfold rotation along c, the twin axis is exactly perpendicular to the
(1) plane, yet the overlap of the lattices of the two individuals is only approximate. In a case like this, a linear, rather than angular, measure of the mismatch is necessary, like the twin misfit defined as the distance between the first nodes along the two shortest directions in the plane of LT perpendicular to the twin axis that are quasi-restored by the twin operation (Nespolo & Ferraris, 2007; Nespolo, 2015). The twin misfit is computed as: d = áDuDvDw|GDuDvDwñ.
where DuDvDw is the difference between the uvw indices in LT of the two nodes quasi-restored by the twin operation (for details of the calculation, see Nespolo & Ferraris, 2007). This twin misfit defines the degree of lattice misorientation for the zero-obliquity TLQS, (for details, see Nespolo, 2015).

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Towards a structural theory of twinning

The reticular theory of twinning can only provide partial prerequisites for the formation of twins, which are governed by the structure. More conclusive conditions can only be obtained via an analysis of the atomic contents of the unit cells. Under the action of the space group G of a crystal, each atom in the crystal is repeated in space to form a crystallographic orbit O, i.e. O is the set of all atoms obtained from a single atom under the symmetry operations of the space group G. The eigensymmetry E(O) of the orbit may be a supergroup of G or coincide with it. Accordingly, crystallographic orbits are classified into three types depending on the relation between G and E: Characteristic orbit: G = E.
Extraordinary orbit: TG < TE, a special case of a non-characteristic orbit defining a superlattice (smaller unit cell).
Here, TE and TG are the translation subgroups of E and G, respectively. When G < E, an operation t belonging to E but not to G may map the orientation of crystal 1 onto that of crystal 2 while preserving O and may thus serve as twin operation which fixes the orbit O as a common substructure. As we will see, the symmetry analysis will mainly be applied to orbits under a particular subgroup H of G, namely the largest subgroup of G compatible with the twin lattice LT. With respect to the twin lattice, the continuous substructure across the interface is then given by the split orbits Oij or the union of them: it is this substructure which is restored by the twin operation t:
• a split orbit Oij (obtained under the action of H) is restored by the twin operation t if and only if its eigensymmetry E (Oij) contains t.
• the unionijOij is restored by t if and only if its eigensymmetry E (ijOij) contains t.
For twins in which the composition surface is planar and parallel to the twin plane or, for rotation twins, to the plane (quasi-)perpendicular to the twin axis, one can obtain additional information by restricting the symmetry analysis locally, information which is not independent but complements the evidence derived from the restoration of crystallographic orbits. Let K be a slice through the crystal structure taken around the composition plane of the twin and let L be the symmetry group of K (layer group). The two individual crystals have space groups G1 and G2 (of the same type but with different orientation) and induce layer groups L1 and L2 of the slice K, the intersection of which is included in L, i.e. L1 ∩ L2 ⊂ L. If L contains any operation that maps the orientation of G1 onto that of G2, this operation can explain the formation of the twin. In order to find the twin operation as an element of a layer group, it may be necessary to exclude some atoms from the slice which are not restored by the twin operation. This is analogous to the fact that not all crystallographic orbits under G are invariant under the twin operation, but that only some of the split orbits under H are restored. It is worthwhile to note that the two approaches just sketched have useful interrelations. On the one hand, having a crystallographic orbit (or split orbit) which is restored by the twin operation, restricting it to the slice around the composition plane yields a substructure within the slice which is invariant under the twin operation (note that the twin operation fixes the composition plane). On the other hand, finding the twin operation as an element of a layer group shows which (split) orbits are related by the twin operation by identifying to which orbits the atoms in the slice belong.

Table of contents :

Chapter I. Introduction
Chapter II. Description and analysis of twins
II.1. Basic definitions and classifications of twins
II.2. The twinning parameters and the reticular classification
II.3. Friedelian and hybrid twins
II.4. Towards a structural theory of twinning
Crystallographic orbit approach
Layer group approach
Chapter III. Summaries of articles
III.1. Summary of the article ‘Analysis of the structural continuity in twinned crystals in terms of pseudo-eigensymmetry of crystallographic orbits’
III.2. Summary of the article ‘The staurolite enigma solved’
III.3. Summary of the article ‘Twinning of aragonite – the crystallographic orbit and sectional
layer group approach’
Chapter IV. Discussion
IV.1. The restoration percentage and twinning frequency
IV.2. Examples of studied twins with a high restoration percentage
IV.3. Example of a « negative » twin case
IV.4. « Necessary » vs. « Sufficient » conditions
IV.5. Conclusion and outlook
Chapter V. References

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