Structure vernier

In vernier structures the modulation in the composition occurs on the basic stmcture thus combining variable composition with ordering (Anderson 1973). The essential feature of this structure is that the crystal unit can be resolved into two independent sublattices, one with a constant periodicity (based on that of the 

A crystal structure which is composed of more than two sublattices, which are independent, though modulated, is named a vernier or chimney and ladder structure. If the dimension of each sublattice (named A, B, C.) along the c-axis is supposed to be c, Cg, Cg. the unit length of the whole structure (supercell structure) along the c-axis is expressed as 

X-ray analysis/ in which figure (a) is the projection on (001), corresponding to Fig. 2.34. It is to be noted that in the real crystal two excess anions in half of a unit cell are placed at x = 0 and 1, instead of at x = and I as in the ideal structure model. This causes a vernier structure, i.e. the anion along [010] at x = is seven times the unit length, while that at x = 0 is eight times the length. To confirm this relation, the projection on (100) is shown in Fig. 2.35(b), where the dotted net plane Aj (anion plane, 3 net) 

Therefore the ideal structure of Y7O6F9 can be depicted as a stacking of. .. AjBiAjBj. .. along [100], as shown in Fig. 2.36. In the real crystal, shown in Fig. 2.35, mutual correlations among these atomic planes exist. These result in structural modulation, i.e. the movement of not only anions but also cations from the ideal positions, although the dimensions of the subcells of each plane (Aj., Bj, Aj, B2) are assumed to be constant in the ideal structure, and, moreover, the site occupancy of anions is differentiated. 

Oxygen ions occupy the sites labelled 1,2,3,5 and fluorine ions the sites labelled 4,6,7,8,9 on the Aj and A, planes, which indicates clustering of anions. 

Observed EDPs for Bai+,Fe2S4 are classified into the following two groups  

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Some of the earliest examples of modulated structures to be unraveled were the fluorite-related vernier structures. These structures occur in a number of anion-excess fluorite-related phases and use a modulation to accommodate composition variation. They can be illustrated by the orthorhombic phases formed when the oxyfluoride YOF reacts with small amounts of YF3 to give composition YOxF3 with x in the range 0.78-0.87, but similar phases occur in the Zr(N, O, F) system with x taking values of 2.12-2.25 and other systems in which the Zr is replaced by a variety of lanthanides. 

Figure 4.36 Y-O-F vernier structures (a) idealized square anion net in Y706F9 (b) idealized hexagonal anion net in Y7OgF9 and (c) the two nets superimposed to show that seven square cells fit with eight triangular cells, which represents the idealized anion nets in Y7C>6F9.

Composite crystals, in which the structure can be described as resulting from two or more substructures (related to two or more sets of three dimensional lattices) having different periodicity along at least one direction (chimney-ladder structures, vernier structures, misfit-layer structures, etc.). See the scheme presented in Fig. 3.42. 

A group of crystals show diffraction patterns in which two or more 3D lattices having periods commensurate or incommensurate to each other may be recognized. In other words, the crystal consists of two or more interpenetrating substructures (two or more different atom sets) with different periods at least along one direction (see Fig. 3.42). Names such as composite crystals, vernier structures, misfit-layer structures, and chimney-ladder structures have been used for this group of structures. 

Ca and Cg) for simplicity. In this case two types of vernier structure are considered as shown in Fig. 2.32. Structure [a] is composed of A- and B-planes, which are stacked alternatively along the h-axis as. .. ABAB. .. Due to the difference between the dimensions of the unit cells of each plane, the unit length of the supercell structure along the c-axis is expressed by eqn (2.7). In structure [b], the structures A and B are mutually interpenetrated. The dimension of the super structure along the c-axis is also expressed by eqn (2.7). In each case, the relation between structures A and B is very similar to that between the main scale and vernier scale in vernier calipers. 

Fig. 2.32 Schematic drawing of the vernier structure, [a] Sublattices A and B are stacked as. .. ABAB. .. along the fo-axis. [b] Sub-lattices A and B are mutually interpenetrated.

In the Y O iF + 2 system, compounds n = 4,5,6,7,8 have been confirmed, the structures of which can be understood in terms of the vernier structure, similar to structure model (II) (Fig. 2.36). Closely related compounds have also been observed in the Nb-Zr-O and Zr-N-F systems. Figure 2.37 shows the structure of ZriogNggFijg drawn in the same way as YjOgFg in Fig. 2.35. This structure is also an example of a vernier structure based on a Fluorite-type structure. A vernier structure with layer type [a] (as in Fig. 2.32) has been critically and systematically reviewed by Makovicky and Hyde. °... 

Thus we have studied the structure of the Y O iF +2 and Ba-Fe-S systems as typical examples of the vernier structure. In the vernier structure, the deviation from stoichiometry originates from the coexistence of different unit dimensions of the subcells, i.e. Aj and Aj sheets for the Y-O-F system, and Fe2S4 and Ba subcells for the Ba-Fe-S system. The structural principle of the vernier structure can be extended to a more general one, called adaptive structure , in which independent, though mutually related, structures for all the compositions can exist in a limited composition range (see Section 2.6). 

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 . ... 

Natural specimen typical of the homologous series. A complex vernier structure, a composite of 2 X (17-T) and 1 x (12-T) approximate 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... 

It is a complex vernier structure in which Matzat recognised two approximate verniers (pseudo-translations) one unit vernier and one of higher order (cf. Chap. 3 above), which are always combined in the two examples that were studied in detail. If the two... 

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). 

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