Face-centered arrangement

Dawson and coworkers pioneered the application of the OPP model to diamond-type structures (Dawson 1967, Dawson et al. 1967). In the diamond-type structure, common to diamond, silicon, and germanium, the atoms are located at 1/8, 1/8, 1/8, at the center-of-symmetry related position at —1/8, —1/8, —1/8, and repeated in a face-centered arrangement. The tetrahedral symmetry of the atomic sites greatly limits the allowed coefficients in the expansion of Eq. (2.39). With x, y, z expressed relative to the nuclear position, the potential is given by... 

The occupation of lattice sites differing widely in size and coordination, by equally sized A-ions, is the reason, why the ideal cryolite structure appears strained and therefore tends to distort. This distortion, as Bode and Foss (44) described in detail, results in various rotations of the MeFe-octahedra, whereby the fluoride ions always move away from the edges of the unit cell and thus enlarge the space available for the octahedrally coordinated A-ion. In addition a further distortion may shift the atoms from the originally face-centered arrangement, so that structures of lower S3unmetry may be observed. But the cubic cryolite-type is still to be seen as the basic form of these distorted modifications, which appear at lower temperatures only and become cubic if the temperature is raised (J52). 

The X-ray powder patterns of A[MX ] (A = Na, K, Rb, Cs, Tl)65,66,77 showed the low temperature modifications, with the exception of K[NbCl6], to be isotypic with Cs[WCl6], and the high temperature forms to crystallize in a cubic-face-centered arrangement. The Raman... 

The sodium chloride structure. Sodium chloride crystallizes in a face-centered cubic structure (Fig. 4.1a). To visualize the face-centered arrangement, consider only the sodium ions or the chloride ions (this will require extensions of the sketch of the lattice). Eight sodium ions form the comers of a cube and six more are centered on the faces of the cube. The chloride ions are similarly arranged, so that the sodium chloride lattice consists of two interpenetrating face-centered cubic lattices. The coordination number (C.N.) of both ions in the sodium chloride lattice is 6. that is, there are six chloride ions about each sodium ion and six sodium ions about each chloride ion. 

L. Pauling, Icosahedral quasicrystals of intermetallic compounds are icosahedral twins of cubic crystals of three kinds, consisting of large (about 5000 atoms) icosahedral complexes in either a cubic body-centered or a cubic face-centered arrangement or smaller (about 1350 atoms) icosahedral complexes in the /2-tungsten arrangement. Proc. Natl. Acad. Sci. (USA) 86, 8595-8599 (1989). 

Figure 2.1. Ionic model for sodium chloride. This is a face-centered arrangement of chloride ions (white), with sodium ions occupying the octahedral interstitial sites (red). The attractive electrostatic forces, a, between adjacent Na+ and Cl ions, and repulsive forces, r, between Na" " ions are indicated.

A third lyotropic mesophase which occurs frequently in surfactant-water systems is normally designated viscous isotropic. This phase is very viscous, sometimes brittle, but unlike neat and middle it is not bire-fringent. The structure of the viscous isotropic phase is still not known with certainty. In some systems x-ray studies have indicated that the structure consists of spherical units packed in a face-centered arrangement 4, 22). It has been proposed that the polar groups of the molecules cover the outside surfaces of the spherical units and that the hydrocarbon chains are essentially liquid in their arrangement inside the units. In this respect the structure is similar to one of the proposed middle phase structures (4). As in the other lyotropic phases, the solvent probably fills the voids among the spherical units of surfactant. 

Because the spheres are present at the comers and at the center of each face, this is usually referred to as a face-centered arrang ent. 

We return now to the NaCl structure, which contains sodium ions in the voids of a cubic closest-packed array of chloride ions. The cubic closest-packed array of ions was illustrated previously in Figures 14 and 34. The unit cell shown in Figure 38 is based on a face-centered arrangement of sodium ions, which is equivalent to the face-centered cubic structure of the chloride ions. The lines between adjacent ions in Figure 38 are not covalent bonds they merely represent ionic interactions between adjacent ions. 

Figure 49 The face-centered arrangement of calcium ions in fluorite.

You should remember that sphalerite has a cubic closest-packed (face-centered) arrangement of sulfide ions with zinc cations occupying half of the tetrahedral voids. 

Perhaps the most common AX crystal structure is the sodium chloride (NaCl), or rock salt, type. The coordination number for both cations and anions is 6, and therefore the cation-anion radius ratio is between approximately 0.414 and 0.732. A unit cell for this crystal structure (Figure 12.2) is generated from an FCC arrangement of anions with one cation situated at the cube center and one at the center of each of the 12 cube edges. An equivalent crystal structure results from a face-centered arrangement of cations. Thus, the rock salt crystal structure may be thought of as two interpenetrating FCC lattices—one composed of the cations, the other of anions. Some common ceramic materials that form with this crystal structure are NaCl, MgO, MnS, LiF, and FeO. 

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