Near cubic structures

The nearly cubic, rhombohedral unit cell of Mo6PbS8 is shown in Fig. 4.30. The bonding between different clusters in the crystal structure of the Chevrel phase Mo6PbS8 is indicated in Fig. 4.31. 

According to Pearson (1972) the rhombohedral structure of these elements can be considered a distortion of a simple cubic structure in which the d2/d ratio would be 1. The decrease of the ratio on passing from As to Bi, and the corresponding relative increase of the strength of the X-X interlayer bond (passing from a coordination nearly 3, as for the 8 — eat rule, to a coordination closer to 6) can be related to an increasing metallic character. 

Fig. 24. Spectra of three compounds with rock salt structure and one nearly cubic compound (TiO), showing variations of extended fine structures, general similarity of spectra, and special low energy absorption of the metallic TiC.

Monoxides of 3d transition metals, TiO to NiO, possess the rocksalt structure and exhibit properties shown in Table 6.3. While TiO and VO exhibit properties characteristic of itinerant (or nearly itinerant) d electrons, MnO, FeO, CoO and NiO show localized electron properties. The properties can be understood in terms of the possible cation-cation and cation-anion-cation interactions in the rocksalt structure (Fig. 6.12(a)). Direct cation-cation interaction can occur through the overlap of cationic t2g orbitals across the face diagonal of the cubic structure. When this interaction is strong R < and b > b, cationic t2g orbitals are transformed into a cation sublattice t%g band if this band is partly occupied, the material would be... 

An example of such order is shown by the hexagonal symmetry of SBS as revealed by LAXD, electron microscopy and mechanical measurements. In composite materials the choice of phase is at the disposal of the material designer and the phase lattice and phase geometry may be chosen to optimise desired properties of the material. The reinforcing phase is usually regarded elastically as an inclusion in a matrix of the material to be reinforced. In most cases the inclusions do not occupy exactly periodic positions in the host phase so that quasi-hexagonal or quasi-cubic structure is obtained rather than, as in the copolymers, a nearly perfect ordered structure. 

The X-ray analysis confirmed the cubic structure of 18 with Sn—Sn bond lengths of 2.839(2)-2.864(2) A and all Sn—Sn—Sn bond angles nearly 90° (Figure 10). These structural parameters are in close agreement with those calculated for the parent SnsFR octastannacubane (Sn—Sn bond length 2.887 A and Sn—Sn—Sn bond angles 90°)8. 

Chemical stability indicates that in the cubic, metallic perovskites the interstitial C and N are probably neutral. They represent, therefore, an M atom with half filled p or s-p orbitals, and in the cubic structure the metal-M-metal interaction is defined by Figure 86. [This is to be contrasted with low-temperature CrN, which has considerable ionic character and an ordering of its covalent character along a given axis.] In contrast to the M atoms of the Heusler alloys, the p electrons of C and N correlate with, and therefore spin pair, the near-neighbor eg electrons. The metal- -metal interactions are determined by the Ug electron-spin correlations since Hu 2.76 A < Rc. 

Figure 12 A comparison of the two forms of the salt [CrI(r)6-QH5COOH)2][PF6]. Both forms contain chains of cations held together by hydrogen bonds between the carboxyl groups. The structure of the a-form can be ideally converted in that of the (i-form by sliding in opposite directions two out of every four layers formed by cationic chains and anions. Furthermore, whereas the (i-form adopts a nearly cubic NaCl-like structure, the a-form shows the presence of twin rows of cations and anions.

For lithium with a body-centred cubic structure (CN 8) with 8 nearest neighbours at 3.032 A and 6 more at a slightly larger distance of 3.502 A one calculates with a total bond order of 1, a bond order of n = 0.111 and n = 0.018 for both categories with rx — 1.220 A (comp. Li2 r — 1.33 A, but for a nearly pure s bond). 

Let us now turn to the structure factors of Eq. (16-5), to determine them first for the perfect crystal. What we do here is formulate the diffraction theory for crystal lattices, since the interaction of the electron waves with the crystal is a diffraction phenomenon. A perfect crystal is characterized by a set of lattice translations T that, if applied to the crystal, take every ion (except those near the surface) to a position previously occupied by an equivalent ion. The three shortest such translations that arc not coplanar are called pihuitive translations, t, Tj, and Tj, as indicated in Section 3-C. For the face-centered cubic structure, described also in Section 3-A, such a set is [011]a/2, [101]u/2, [ll0]a/2. The nearest-neighbor distance is d = n 2/2. Replacing one of these by, for example, [0lT]a/2, would give an equivalent set. For a body-centered cubic lattice, such a set is [Tll]u/2, [lTl]a/2, and [11 l]ti/2, and the nearest-neighhor distance is For each of these struc-... 

We saw in detail in Section 16-D, and in Fig. 16-6 in particular, how nearly-free-electron bands are constructed. The diamond structure has the translational symmetry of the face-centered cubic structure, so the wave number lattice, the Brillouin Zone, and therefore the nearly-free-electron bands for the diamond structure are identical to the face-centered cubic nearly-free-electron bands and are those shown in Fig. 3-8,c. The bands that are of concern now are redrawn in Fig. 18-l,b. The lowest energy at X, relative to the lowest energy at F, is (fi /2m) x InjaY as shown clearly it is to be identified with the lower level... 

---