Semi-commensurate structures

IR-11.6.1 Introduction Several special problems of nomenclature for non-stoichiometric phases have arisen with the improvements in the precision with which their structures can be determined. Thus, there are references to homologous series, non-commensurate and semi-commensurate structures, Vernier structures, crystallographic shear phases, Wadsley defects, chemical twinned phases, infinitely adaptive phases and modulated structures. Many of the phases that fall into these classes have no observable composition ranges although they have complex structures and formulae an example is Mo17047. These phases, despite their complex formulae, are essentially stoichiometric and possession of a complex formula must not be taken as an indication of a non-stoichiometric compound (cf. Section IR-11.1.2). 

The terms incommensurate and semi-commensurate are analogous to incoherent and semi-coherent for interfaces - in grain boundaries, heterophase interfaces and epitaxial layers (cf. also Nabarro - with which layered misfit structures have much in common. In extreme cases noncommensurability may arise by mutual rotation (to varying degrees) of component layers with identical component lattices... 

In the case of semi-commensurability in one or two directions it is likely that, structurally, the two layer sets are not quite independent. Semi-coherent structural and/or compositional modulation is then present i.e. a cooperative periodic variation in the size and/or content of the component subcells. Each modulation vector (one only in Fig. 2a, two in Fig. 2 b) will be equal to (or a multiple or sub-multiple of) that of the coincidence net. For each modulated layer set, A or B, a true-structure (component) lattice and unit cell can then be defined, based on its modulation period or periods plus the basic vectors or vector in the direction(s) in which there is no visible modulation of the basic structure. The longer-range modulation pattern of the two layer sets is imposed on the short range approximate periodicities which, in turn, describe sub-motifs manifested as a subnet (or subcell) of each layer set. If, as may be the case, the separate component unit cells of each of the two sets are identical, then they are also the coincidence cell of the two sets (Fig. 2 b). In the more general case, when this is not so, the vectors of the coincidence net will be multiples of the identity vectors of the unit nets of the two layer sets (or some simple summations of them). 

The modulated SC case is the one originally termed a vernier structure by Hyde et al. due to the analogy between the fit of the two stacks of the SC subcells - which are semi-commensurate along the (one) modulation direction - and the vernier (or nonius) scale. In such cases the commensurate unit cell parameters (in the non-modulated directions) are small, so that the coincidence cell is strongly elongated. In SS cases the structures may be similarly modulated in two directions, so that they may be termed two-dimensional verniers . ... 

Cylindrite is a collective name for a homologous series of compounds with very closely related vernier structures. These have slightly different chemical compositions and consequently different numbers of matching pseudo-tetragonal (T) and pseudo-hexagonal (H) subcells in the modulated, semi-commensurate direction . Some are modulated in two directions and are therefore SS structures, but the majority have a... 

Lattice geometry, crystal morphology and cleavage all confirm that the two component structures are layer-like and semi-commensurate in both intralayer directions b and... 

Before the similar situation in cylindrite can be treated, we have to describe the special type of layer stacking disorder which is inherent to non-commensurate layer structures. When two types of layers are semi-commensurate (or nearly so) and non-modulated in the semi-commensurate intralayer direction (e.g. b), the layer B can be placed on layer A in a number of ways with equivalent layer match, mutually displaced by the translation mb for the primitive mesh A, or mbAl2 + c tl in the more usual case of a centred layer mesh m stays within the range of the vernier repeat (n) of the layer A. Several of these positions will not coincide with each other, depending on the ratio of to ng in the semi-commensurate direction The same will in general be true for the positions of layer A on layer B, but this time the positions will be determined by the vectors of the layer B. 

Lengenbachite (Makovicky and Leonardsen, unpublished) is an intensely disordered structure. Its very compUcated natiu-e can be simplified by projecting on (100) (Rg. 21). In this projection, two A-centred T subcells match exactly with three A-centred H subcells in the unmodulated semi-commensurate b direction whereas 12 T subcells match exactly with 11 H subcells in the modulated c direction. The first match allows two equivalent positions for the H layer after each T layer, spaced bj/3 = byj2 apart, and three equivalent positions of the T layer after each H layer, again spaced at the same intervals. The resulting variation and the superposition subcells are also shown in Fig. 21. 

The structure is of the fluorite type with extra sheets of atoms inserted into the parent YX2 structure. When these are ordered, a homologous series of phases results. When they are disordered, there is a non-commensurate, non-stoichiometric phase, while partial ordering will give a Vernier or semi-commensurate effect. Other layer structures can be treated in the same way. 

Those I and S structures without modulation may also, within certain limits, be infinitesimally adaptive. Atomic substitutions within the two layer sets (which includes the introduction of vacancies ) will change the dimensions of the component unit cells and hence also the (approximate) coincidence cell, as well as the chemical composition and the layer valence balance. And so, once again, the division between semi- and incommensurability - and sometimes even commensurability - may, in these compounds, be rather arbitrary. 

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