Number 6 Octahedron

In hep arrays, the plane parallel with the close-packed layers (i.e. parallel with the basal faces of the octahedra) is the preferred possibility for ordering of the 

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The truncated octahedron and the rhombic dodecahedron provide periodic cells that are approximately spherical and so may be more appropriate for simulations of spherical molecules. The distance between adjacent cells in the truncated octahedron or the rhombic df)decahedron is larger than the conventional cube for a system with a given number of particles and so a simulation using one of the spherical cells will require fewer particles than a comparable simulation using a cubic cell. Of the two approximately spherical cells, the truncated octahedron is often preferred as it is somewhat easier to program. The hexagonal prism can be used to simulate molecules with a cylindrical shape such as DNA. 

Structures of heteropolytungstate and isopolytungstate compounds have been determined by x-ray diffraction. The anion stmctures are represented by polyhedra that share corners and edges with one another. Each W is at the center of an octahedron, and an O atom is located in each vertex of the octahedron. The central atom is similarly located at the center of an XO tetrahedron or XO octahedron. Each such polyhedron containing the central atom is generally surrounded by octahedra, which share corners, edges, or both with it and with one another. Thus, the correct total number of... 

Equations (6-236) to (6-239) are based on experiments on cube-oc tahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity f ranges from 0.906 to 0.670, respectively. See also Chft, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with y as a parameter may be found in Brown, et al. (Unit Operations, Whey, New York, 1950) and in Govier and Aziz. 

An extreme example of hybidization is the structure proposed for sulphur hexafluoride, SFe. The six S-F bonds are dhected to the apices of a regular octahedron. An aiTangement which would satisfy this number of covalent bonds is sp d hybridization. The ground state of the sulphur atom is s p° and... 

Suppose you have six balls with three different colors, three red, two blue, one yellow. Balls of the same color cannot be distinguished. In how many ways can you assign the six balls to the six vertices of an octahedron which moves freely in spacel If the octahedron is fixed in space in such a way that the vertices are designated as upper, lower, front, back, left, and right vertex, then the number is determined by basic permutation principles as... 

Octahedral Having the symmetry of a regular octahedron. In an octahedral species, a central atom is surrounded by six other atoms, one above, one below, and four at the comers of a square, 176 complex in transitional metals, 418-420 geometric isomerism, 415 Octane number, 584... 

The structure of LiTa02F2, as reported by Vlasse et al. [218], is similar to a ReC>3 type structure and consists of triple layers of octahedrons linked together through their vertexes. The layers are perpendicular to the c axis, and each layer is shifted, relative to the layer below, by half a cell in the direction (110). Lithium atoms are situated in the centers of the tetragonal pyramids (coordination number = 5). The other lithium atoms are statistically distributed along with tantalum atoms (coordination number = 6) at a ratio of 1 3. The sequence of the metal atoms in alternating layers is (Ta-Li) - Ta - (Ta-Li). Positions of oxygen and fluorine atoms were not determined. The main interatomic distances are (in A) Ta-(0, F) - 1.845-2.114 Li-(0, F) - 2.087-2.048 (O, F)-(0,F) - 2.717-2.844. 

The lowest coordination number of tantalum or niobium permitted by crystal chemistry formalism is 6, which corresponds to an octahedral configuration. X Me ratios that equal 3, 2 or 1 can, therefore, be obtained by corresponding substitutions in the cationic sub-lattice. A condition for such substitution is no doubt steric similarity between the second cation and the tantalum or niobium ion so as to enable its replacement in the octahedral polyhedron. In such cases, the structure of the compound consists of oxyfluoride octahedrons that are linked by their vertexes, sides or faces, according to the compound type, MeX3, MeX2 or MeX respectively. Table 37 lists compounds that have a coordination-type structure [259-261]. 

Reduction of X Me from 8 to 6 leads to a reduction of the metal coordination number from 8 to 6, respectively, and to the formation of octahedrons (distorted octahedrons) as a basic unit. 

The structure of the crystal has a tendency to utilize steric similarity of its component ions and is defined by the number of anions (oxygen and fluorine) per cation in each oxyfluoride octahedron. 

The main source of spontaneous polarization in crystals is the relative freedom of cations that fit loosely into the crystal s octahedral cavities. The number of degrees of freedom of the octahedrons affects the spontaneous polarization value and hence influences the crystal s ferroelectric properties. Abrahams and Keve [389] classified ferroelectric materials into three structural categories according to their atomic displacement mechanisms onedimensional, two-dimensional and three-dimensional. 

S Tantalum and niobium are present in the crystal structure in the form of complex ions. The lowest coordination number, 6, corresponds to the formation of slightly distorted octahedrons. The linking and packaging of the octahedrons depends on the X Me ratio, where X is the total number of oxygen and fluorine atoms, and Me is the total number of tantalum or niobium ions as well as other metals that can replace tantalum or niobium in the octahedral polyhedron. The crystal structure type can be defined based on the X Me ratio, as follows ... 

The holes in the close-packed structure of a metal can be filled with smaller atoms to form alloys (alloys are described in more detail in Section 5.15). If a dip between three atoms is directly covered by another atom, we obtain a tetrahedral hole, because it is formed by four atoms at the corners of a regular tetrahedron (Fig. 5.30a). There are two tetrahedral holes per atom in a close-packed lattice. When a dip in a layer coincides with a dip in the next layer, we obtain an octahedral hole, because it is formed by six atoms at the corners of a regular octahedron (Fig. 5.30b). There is one octahedral hole for each atom in the lattice. Note that, because holes are formed by two adjacent layers and because neighboring close-packed layers have identical arrangements in hep and ccp, the numbers of holes are the same for both close-packed structures. 

The richness of coordination chemistry is enhanced by the variety of shapes that complexes can adopt. The most common complexes have coordination number 6. Almost all these species have their ligands at the vertices of a regular octahedron, with the metal ion at the center, and are called octahedral complexes (1). An example of an octahedral complex is the hexacyanoferrate(ll) ion, [Fe(CN)f, 4. ... 

The structure theory of inorganic chemistry may be said to have been bom only fifty years ago, when Werner, Nobel Laureate in Chemistry in 1913, found that the chemical composition and properties of complex inorganic substances could be explained by assuming that metal atoms often coordinate about themselves a number of atoms different from their valence, usually four atoms at the comers either of a tetrahedron or of a square coplanar with the central atom, or six atoms at the comers of an octahedron. His ideas about the geometry of inorganic complexes were completely verified twenty years later, through the application of the technique of x-ray diffraction. 

The Mg clusters with 6N + 14 (N = 0-2) valence electrons assume regular octahedrons, whereas those with the other numbers of valence electrons do not. 

Table 3 Number of valence electrons and regular octahedrons ...

The octahedron is classified into the c/o o-structure by Wade [3,4]. Closo-structures with n skeletal atoms are stable when they have 4n-i- 2 valence electrons. Wade s rules predict that the 26 (= 4 x 6 + 2) valence electrons could stabilize the regular octahedrons since n is 6 for the octahedron. This prediction is contained in our 6N + 14 (N= 2) valence electron rule. Our rule also predicts the stability of octahedral metal clusters with the other numbers (14 and 20) of valence electrons. 

Solid-state cluster chemistry is dominated by octahedral (M 5L8)L6 and (MsLi2)L units which are the focus of this paper. These two cluster types are different in the way the metal octahedral core is surrounded by the ligands. In (MsLg)L6-type clusters (Fig. 6.1a), typical for molybdenum and rhenium halides, chalcogenides, and chalcohalides, eight innei hgands (L ) cap the octahedron faces and six outer ligands (L ) are located in the apical positions [9]. For metals with a smaller number of valence electrons, the (M6L i2)L -type clusters... 

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