Polyhedral domain

Nicaise S. (1992) About the Lame system in a polygonal or polyhedral domain and coupled problem between the Lame system and the plate equation. 1. Regularity of the solutions. Ann. Scuola Norm. Super. Pisa, Serie IV 19, 327-361. 

A large number of metallacarboranes and polyhedral metallaboranes of s-, p-, d-, and f-block elements are known. In these compounds, the carboranes and boranes act as polyhapto ligands. The domain of metallaboranes and metallacarboranes has grown enormously and engenders a rich structural chemistry. Some new advances in the chemistry of metallacarboranes of f-block elements are described below. 

It may often be convenient to describe the crystal structure in terms of the domains of the atoms [40], The domain is the polyhedron enclosed by planes drawn midway between the atom and each neighbor, these planes being perpendicular to the lines connecting the atoms. The number of faces of the polyhedral domain is the coordination number of the atom and the whole structure is a space-filling arrangement of such polyhedra. 

This surfaceiStouches other surfaces,S, S". . ., which are the domains of the precipitates C, C".. . Thus the possible equilibrium states, under the pressure considered, between the liquid mixture and a single solid precipitate are represaited by the various points of a polyhedral surface with curved faces which has as many faces as there are distinct solid precipitates able to exist within the liquid mixture. 

Figure 12. Polyhedral representation of a fragment of the quartz stracture showing the loeal relationship of the two DauphinD twin orientations of the a-phase (left and right) to the average structure of p-quaitz (center). The orientation of the Si[0]4 tetrahedra in a-quartz results from a rotation about the a-axes from the p-quartz stracture (0 = 16.5° at 25°C). DauphinD twin domains are related by rotations of opposite sense. [Modified after Heaney (1994), Fig. 3, p. 8 and Fig. 5, p. 11.]...

Crystal structures may be described in terms of the coordination polyhedra MX of the atoms or in terms of their duals, that is, the polyhedra enclosed by planes drawn perpendicular to the lines M-X joining each atom to each of its neighbours at the mid-points of these lines. Each atom in the structure is then represented as a polyhedron (polyhedral domain), and the whole structure as a space-filling assembly of polyhedra of one or more kinds. We can visualize these domains as the shapes the atoms (ions) would assume if the structure were uniformly compressed. For example, h.c.p. and c.c.p. spheres would become the polyhedra shown in Fig. 4.29. These polyhedra are the duals of the coordination polyhedra illustrated in Fig. 4.5. These domains provide an alternative way of representing relatively simple c.p. structures (particularly of binary compounds) because the vertices of the domain are the positions of the interstices. The (8) vertices at which three edges meet are the tetrahedral interstices, and those (6) at which four edges meet are the octahedral interstices. Table 4.9 shows the octahedral positions occupied in some simple structures c.p. structures in which tetrahedral or tetrahedral and octahedral sites are occupied may be represented in a similar way. (For examples see JSSC 1970 1 279.)... 

FIG. 4.29. The polyhedral domains for (a) hexagonal and (b) cubic closest packing, showing the six positions of octahedral coordination around a c.p. sphere. The remaining vertices of the domains are tetrahedral sites. 

The change in the direction of the oxygen vacancy ordering implies a perpendicular orientation of the CX1O4 groups from one domain to the adjacent one. Possible models of the junction between the domains, with or without mirror plane (110) are drawn in figure 8 only one layer out of three polyhedral layers has... 

After the introduction of the tetrahedral carbon atom concept into organic chemistry, various polyhedral structures were postulated in the inorganic domain. Work in the field of coordination chemistry, which was devrioped virtually sin e haodedly by Werner [33], led to the idea that... 

Figure 9-26d shows the body-centered packing with eight-coordination. For the central atom, the six next nearest neighbors are at the centers of neighboring unit cells. In terms of polyhedral domains, a truncated octahedron is adopted here. The central atom, in fact, has a coordination number of 14. 

The restriction operator. P X denotes the mathematical restriction of expression X to the (polyhedral) domain P. 

The case operator. If X and Y are two expressions with disjoint (polyhedral) domains domX and domT, then caseX yesac denotes the conditional expression defined as X on domX and Y on domK. 

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