Structure types simple cubic

The remaining trifluorides of the transition elements known so far apparently crystallize rhombohedrally like VFa. Prior to this structure type the cubic ReOs-type will be discussed briefly, though only oxide-fluorides rather than trifluorides seem to adapt this structure. Yet it is te simple basic type of which the others may be derived. 

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

Considering Au in 0, 0, 0 as the reference atom, the next neighbours Au atoms are the six Au shown in Fig. 3.29(a), corresponding to the same Wyckoff position and having, in comparison with the reference atom, the coordinates 0, 0, 1 0, 0, 1 0, 1, 0 0, 1, 0 1, 0, 0 1, 0, 0, all at a distance d = a = 374.8 pm, that is at a reduced distance dr = d/dmin = 1.414. Notice that in the analysis of the structure it may be necessary to consider not only the positions of the atoms in the reference cell but also those in the adjacent cells. Notice also that, in a simple cubic structure without free positional parameters such as the AuCu3 type, the reduced distances are independent of the values of the lattice parameters and are the same for all the isostructural compounds. 

The W body-centred cubic structure can be compared with the simple cubic CsCl-type structure (which can be obtained from the W type by an ordered substitution of the atoms) and with the MnCu2Al-type structure ( ordered superstructure of the CsCl type) see Fig. 3.31 and notice the typical eight (cubic) coordination. 

Well-ordered intermetallic compounds (alloys), when processed in high-energy ball mills exhibit atomic (chemical) disordering in the early stages of ball milling [153]. Let us take as an example ordered AlRu intermetallic crystals that are of B2 type and P-CuZn structure. This structure consists of two simple cubic interpenetrating... 

You now know how to define a supercell for a DFT calculation for a material with the simple cubic crystal structure. We also said at the outset that we assume for the purposes of this chapter that we have a DFT code that can give us the total energy of some collection of atoms. How can we use calculations of this type to determine the lattice constant of our simple cubic metal that would be observed in nature The sensible approach would be to calculate the total energy of our material as a function of the lattice constant, that is, tot(a). A typical result from doing this type of calculation is shown in Fig. 2.1. The details of how these calculations (and the other calculations described in the rest of the chapter) were done are listed in the Appendix at the end of the chapter. 

There is sometimes ambiguity as to which nearest neighbours to assign to the local coordination polyhedron. This is illustrated by the two different puckered layer structure types which are displayed by the most stable polymorphs of phosphorus and arsenic respectively. They may both be thought of as resulting from the breaking of three bonds about each atom on a simple cubic lattice as shown in Fig. 1.5. The layers then distort Black... 

Fig. 1.6 The breaking of three-bonds about each simple cubic site that leads to the black phosphorus and arsenic structure types. After Burdett and Lee (1985).

The ten most commonly occurring structure types in order of frequency are NaCl, CsCl, CrB, FeB, NiAs, CuAu, cubic ZnS, MnP, hexagonal ZnS, and FeSi respectively. Structures cF8 (NaCl) and cP2 (CsCl) are ordered with respect to underlying simple cubic and body-centred cubic lattices respectively, as is clear from Figs 1.10(a) and 1.11(a). The Na, G sites and Cs, Cl sites are, therefore, six-fold octahedrally coordinated and fourteen-fold rhombic dodecahedrally coordinated, respectively, as indicated by the Jensen symbols 6/6 and 14/14. 

Many simple minerals, especially simple salts like halite, NaCl, sulfides, sulfosalts and oxides, have structures based upon cubic or hexagonal closest-packed arrays of either cations or anions. Coordination geometries of metal ions in many of these kinds of minerals are thus confined to more or less regular octahedra and tetrahedra. The occupancy of the two types of sites is dictated by the stoichiometry of the mineral, the radius of the ions involved and their preferred coordination geometries. Coordination of cations in mineral species in terms of bonding and crystal field effects has been extensively reviewed.16-21 Comprehensive lists of ionic radii relevant to cation coordination geometries in minerals have also been compiled.16,21... 

The plumbides R(Ag, Pb)3 with R = Y, Sm, Gd-Tm crystallize with a very simple structure type, i.e. Q13AU, an ordered version of the cubic close packing. The rare earth atoms fill Wyckoff position la (the origin of the unit cell, see Figure 19), while the silver and lead atoms show random distribution on the 3c site. The phase analytical investigations reveal that up to 78% lead can occupy that site. Both sites have cuboctahedral coordination (CN 12). 

Cubic structures can be of three types simple, body centered or face centered. Their low angle X-ray patterns are characterized by the following sequences of reflexions when the diffraction angle increases ... 

For particular lattices with two molecules per unit cell, with simple structure and nearest-neighbor interactions (linear alternating chains, square 2D lattices, or simple cubic lattices of the NaCl type), the absorption occurs for two values of q q = 0 (the b component) and q = Q (the a component) see Section II.B.l.b. In these simple lattices the environment of the site is symmetric and the complete inversion in P, TP, (2.77) is not necessary, because, in the coupling of P, to P, only the completely symmetric state S> intervenes ... 

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