Distances, polyhedrons

Ta-F distances in the TaF72 polyhedron are unequal and vary in the range of 1.976-1.919 A. Two types of fluorine polyhedrons form around the potassium atoms. The first polyhedron is characterized by K-F distances in the range of 2.905—2.646 A, while K-F distances in the second polyhedron are in the range of2.956-2.651 A [144],... 

The structure of KNbF6 consists of potassium ions and isolated NbF6 complex ions that were shown by Bode and Dohren to occur in the lattice in a configuration similar to that of a-CsCl [165]. The complex anion Nb(Ta)F6 has a configuration of a distorted bi-pyramid (four fluorine atoms are shifted in pairs from their positions in the basic plane, towards the vertexes). The structure of KNb(Ta)F6 compounds and of the Nb(Ta)F6 polyhedron are shown in Fig. 26. Nb/Ta-F distances are equal to 2.13 and 2.15 A, respectively, and F-F distances are 2.61, 3.03, 3.22 and 3.55 A. Each potassium atom is surrounded by 12 fluorine atoms that are at unequal distances from each other 8 of them are 2.50 A apart and four others are 2.94 A apart. 

The polyhedron NbF72 is more similar to a pentagonal bi-pyramid but is distorted due to a strong shift of F6 towards F3 (F6F3 and FiF distances are 2.39 and 3.08 A, respectively). This distortion renders the polyhedron structure closer to an Archimedes antiprism with a truncated comer, as shown in Fig. 31. 

In the papers referred to above it is pointed out that the mechanical properties of the transition elements and the distances between atoms in metals and intermetallic compounds are well accounted for by these considerations. In the following sections of the present paper a discussion is given of the number of valence electrons by the Brillouin polyhedron method, and it is shown that the calculations for the filled-zone alloys such as the 7-alloys provide further support for the new system of metallic valences. 

The uncertainties given are calculated standard deviations. Analysis of the interatomic distances yields a selfconsistent interpretation in which Zni is assumed to be quinquevalent and Znn quadrivalent, while Na may have a valence of unity or one as high as lj, the excess over unity being suggested by the interatomic distances and being, if real, presumably a consequence of electron transfer. A valence electron number of approximately 432 per unit cell is obtained, which is in good agreement with the value 428-48 predicted on the basis of a filled Brillouin polyhedron defined by the forms 444, 640, and 800. ... 

The construction relies on Wulff s assumption that the distance from the surface of a specific plane to the center of the crystallite is proportional to the surface energy i.e. hi <=<= Yi. Thus, if we have a surface plane of small surface energy, its distance from the center of the crystallite will be small and this plane will then cut of all others and dominate the polyhedron. 

A coordination polyhedron of anions is formed around every cation. The cation-anion distances are determined by the sum of the ionic radii, and the coordination number of the cation by the radius ratio. 

The distances d(MX) within the coordination polyhedron of a cation M vary in the same... 

For a given pair of ions the average value of the distances d(MX) within a coordination polyhedron, d( MX), is approximately constant and independent of the sum of the p values received by all the anions in the polyhedron. The deviation of an individual bond length from the average value is proportional to Ap = pj -p (p = mean value of the p for the polyhedron). Therefore, the bond lengths can be predicted from the equation ... 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

Change of the coordination polyhedron of a silicon atom at increasing pressures same perspective as for the atom in the dashed octant of Fig. 12.3. Si-Si distances in pm... 

Limits of the mutual rotation of vertex-sharing tetrahedra and of vertex-sharing octahedra and the resulting bond angles at the bridging atoms. The minimum distance between vertices of different polyhedra (dotted) was taken to be equal to the polyhedron edge... 

Table 16.1 Bond angles at the bridging atoms and distances between the central atoms M of linked tetrahedra and octahedra (disregarding possible distortions). The distances are given as multiples of the polyhedron edge length...

Five points can be arranged on the surface of a sphere such that they are all equivalent, only in a planar pentagonal arrangement, which does not maximize the distance between the points. In other words, there is no regular polyhedron with five equivalent vertices. There are two... 

The 12 RP fragments cap alternately the Cu4 faces of the Cu24 polyhedron, resulting in fivefold-coordinated phosphorus atoms. This structure resembles that of the recently described [Cu24(NPh)i4]4 anionic cluster (40). The Cu-P and Si-P distances are unremarkable. The construction principle of parallel Cu layers to form a metal-like package has also been observed for other Cu clusters (41). The main reason for the different structures of Cu2PR and Li2PR clusters is the covalent character of the Cu-P bond, with the additional involvement of favorable Cu-Cu interactions. The latter are probably due to relativistic d10-d10 interactions (dispersion-type of interaction) (42, 43). 

The effect of ligand-field symmetries is less easily understood. In Fe3(CO)12, two Fe-Fe distances of 2.56 A and 2.68 A are observed. It may be argued that these distances reflect the packing of a triangle within the icosahedron, each polyhedron undergoing distortion until a common C2v symmetry is reached. 

---