Tetrahedral lattice sites

Magnetite exists in the spinel structure which can be represented by the formula (Fe " ") [Fe ,Fe " ]0, where the parentheses denote cations in tetrahedral lattice sites, and the brackets denote cations in octahedral lattice sites (J ). Figure 1 is a representation of the idealized spinel structure (note that the structure has been extended in the [001] direction for clarity). The oxygen anions form a cubic close-packed framework in which there are 2 tetrahedral vacancies and 1 octahedral vacancy per oxygen anion. From the above formula, it can be seen that one-eighth of the tetrahedral sites and one-half of the octahedral sites are occupied by iron cations. The ordered occupation of octahedral sites shown in Figure 1 facilitates electron hopping between ferrous and ferric cations at temperatures above 119 K( ). As a result, the oxidation state of these octahedral cations can be considered to be +2.5. 

Figure 30 Schematic representation of the zeolite ZSM-39 lattice framework. The three crystallographically inequivalent tetrahedral lattice sites are indicated by T, T2, and T3 (inside circles), and in each case the identities of the four nearest neighbors are shown.

With HRTEM, dissociations in a completely stoichiometric garnet were observed [310], contrary to the notion that dissociation is linked to traces of impurities. Parallel and narrow <111> partial dislocations were separated by stacking faults that corresponded to a low-energy configuration resulting from the occupancy of previously vacant dodecahedral and tetrahedral lattice sites. 

Cation Octahedral Lattice Site Tetrahedral Lattice Site Net Magnetic Moment... 

Convincing evidence for the incorporation of aluminum into tetrahedral lattice sites comes from Al magic angle spinning (MAS) NMR, and from indirect evidence such as enhanced catalytic activity [12,13] and infrared spectroscopy. According to the best documented... 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

Fig. 8.39 Octahedral (a) and tetrahedral (b) sites in the bcc lattice Table 8.3 Classification of trap types...

The structure of alumina on NiAl(l 1 0) was the subject of a surface X-ray diffraction study by Stierle et al. [46]. The model derived by Stierle et al. from the analysis of the X-ray diffraction data was based on a strongly distorted double layer of hexagonal oxygen ions, where the Al ions are hosted with equal probability on octahedral- and tetrahedral-coordinated sites the resulting film structure was closely related to bulk k-A1203. An attractive feature of Stierle s model was that it provided a natural explanation of the domain structure of the alumina overlayer, which is induced by a periodic row matching between film and substrate lattices. However, as pointed out recently by Kresse et al. [47], this structure model has two bonds with... 

Spinels have a crystal structure in which there is a face-centered cubic arrangement of O2 ions. There are two types of structures in which cations have octahedral or tetrahedral arrangements of anions surrounding them. In the spinel structure, it is found that the +3 ions are located in octahedral holes and the tetrahedral holes are occupied by the +2 ions. A different structure is possible for these ions. That structure has half of the +3 metal ions located in the tetrahedral holes while the other half of these ions and the +2 ions are located in the octahedral holes. In order to indicate the population of the two types of lattice sites, the formula for the compound is grouped with the tetrahedral hole population indicated first (the position normally occupied by the +2 ion, A) followed by the groups populating the octahedral holes. Thus, the formula AB204 becomes B(AB)04 in order to correctly... 

Fig. 9. Calculated angular scans for 700 keV 3He ions in (110) silicon for different 2H lattice sites the tetrahedral interstitial site (T), a back-bonded site 1.5 A from a lattice site (BB), and a bond-centered site (BC). From Marwick et al. (1988).

In Eq. (6.1) 1 is the unit operator, there is one state /> associated with each lattice site, and l and l A A = 1, 2, 3, 4) label a molecule and its nearest neighbors in the tetrahedral lattice. Weare and Alben show also that the theorem remains valid when small distortions away from tetrahedrality exist, hence it can be used to describe a random amorphous solid derived from a tetrahedral parent lattice. Basically, the density of states of the amorphous solid is a somewhat washed out version of that of the parent lattice. The general shape of the frequency spectrum is not much altered by the inclusion of a non zero bond-bending force constant provided the ratio of it to the bond stretching force constant is small relative to unity. 

Interstitial sites are defined as those that would usually be empty in an ideal structure. Occasionally in real structures, ions may be displaced from their lattice sites into interstitial sites Frenkel defect formation). Once this happens, the ions in interstitial sites can often hop into adjacent interstitial sites. These hops may be one stage in a long range conduction process. A schematic example is shown in Fig. 2.1(h) a small number of Na ions are displaced into the tetrahedral interstitial sites and can subsequently hop into adjacent tetrahedral sites. It should be noted, however, that while a small number of Frenkel defects may form in NaCl, conduction is primarily by means of vacancies whereas in some other structures, e.g. AgCl, Frenkel defects do predominate. 

Virtually all of the reported structural data on titanium alloy hydrides and deuterides indicate that the solute atoms occupy tetrahedral interstitial sites in the metal lattice. Neutron diffraction data obtained for deuterium in Ti/34 atom % Zr and in Ti/34 atom % Nb (17) indicate tetrahedral site occupancy in the bcc /3-phase. Similarly, data reported for deuterium in Ti/19 atom % V and in Ti/67 atom % Nb (18) indicate tetrahedral site occupancy in the fee 7-phase. Crystallographic examination of the 7-phase Ti-Nb-H system (19) reveals that increasing niobium content linearly increases the lattice parameter of the fee 7-phase for Nb contents ranging from 0 to 70.2 atom %. Vanadium, on the other hand, exerts the opposite effect (6) at H/M = 1.85, the 7-phase lattice parameter decreases with increasing vanadium contents. 

Point Defects. Point defects are defined as atomic defects. Atomic defects such as metal ions can diffuse through the lattice without involving themselves with lattice atoms or vacancies (Figure 9), in contrast to atomic defects such as self-interstitials. The silicon self-interstitial is a silicon atom that is bonded in a tetrahedral interstitial site. Examples of point defects are shown in Figure 9. 

The need for high chemical and physical stability in the substances that will hold nuclear waste ions in an immobilized state has stimulated extensive materials research. In support of the candidacy of the cation-linked tetrahedral compounds having the monazite, scheelite or zircon crystal structure, studies have sought to determine the extent to which various elements occurring in the waste can occupy stable substitutional positions on the host lattice sites.(l)... 

As this takes place, hydrogen atoms can occupy quite definite positions in the lattice of metal (octahedral, tetrahedral, or both octa- and tetrahedral interstitial sites). 

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