Cubic arrangement of atoms

Although correlation between parameters is a function of the data structure and has nothing to do with deficiencies in the model, it has implications for both the choice of the model and the design of the experiment. EVANS described his experiences with the determination of the crystal structure of tetragonal barium titanate (BaTiOa). The problem was ample in that it involved only three atomic positional parameters (one for Ti and two for 0), plus nine thermal parameters. There was considerable interest in the details of the structure because of the ferroelectric properties of the material. The proposed model was essentially a simple cubic arrangement of atoms, but with Ti displaced slightly from the center of an octahedron. By ordinary x-ray standards, this distortion (which was expected to be on the order of 0.15 A) could be measured with a standard error of 0.01-0.02 A if... 

A cubane contains a central cubic (or near-cubic) arrangement of atoms. 

Top row Cubic arrangements of atoms in expanded view. Second row Space-filling views of these cubic arrangements. All atoms are identical but. for clarity, corner atoms are blue, body-centered atoms pink, and face-centered atoms yellow. Third row A unit cell shaded... 

The simple cubic arrangement of atoms. The unit of structure is a cube, with one atom per unit, its coordinates being 0, 0, 0. 

The way in which the structure of crystals was then determined is illustrated in Figure IV-4. Here we show a simple cubic arrangement of atoms, as seen along one of the cube faces. It is evident that there are layers of atoms, shown by their traces in the plane of the paper, with spacings d, do, d, , which are in the ratios 1 2 - 5 . Since... 

Figure A2.5.18. Body-centred cubic arrangement of (3-brass (CiiZn) at low temperature showing two interpenetrating simple cubic superlattices, one all Cu, the other all Zn, and a single lattice of randomly distributed atoms at high temperature. Reproduced from Hildebrand J H and Scott R L 1950 The Solubility of Nonelectrolytes 3rd edn (New York Reinliold) p 342.

There are three possible arrangements of atoms in a layer of SiC crystal, and each type has the same layers but a different stacking sequence (29). Designation (30) is by the number of layers in the sequence, followed by H, R, or C to indicate whether the type belongs to the hexagonal, rhombohedral, or cubic class. 

In the latter the surfactant monolayer (in oil and water mixture) or bilayer (in water only) forms a periodic surface. A periodic surface is one that repeats itself under a unit translation in one, two, or three coordinate directions similarly to the periodic arrangement of atoms in regular crystals. It is still not clear, however, whether the transition between the bicontinuous microemulsion and the ordered bicontinuous cubic phases occurs in nature. When the volume fractions of oil and water are equal, one finds the cubic phases in a narrow window of surfactant concentration around 0.5 weight fraction. However, it is not known whether these phases are bicontinuous. No experimental evidence has been published that there exist bicontinuous cubic phases with the ordered surfactant monolayer, rather than bilayer, forming the periodic surface. 

We have already dlsussed structure factors and symmetry as they relate to the problem of defining a cubic unit-cell and find that still another factor exists if one is to completely define crystal structure of solids. This turns out to be that of the individual arrangement of atoms within the unit-cell. This then gives us a total of three (3) factors are needed to define a given lattice. These can be stipulated as follows ... 

Fig. 26 Skutterudite-type structure in terms of a framework of M-centred octahedra and b cubic arrangement of M atoms with Pn4 rings and dodecahedral cages filled with RE atoms in the ternary variants. Reprinted with permission from [110]. Copyright the American Chemical Society...

In addition to the two structures already discussed, another arrangement of atoms in a cubic unit cell is possible. Atoms of a metal are identical, so the ratio of atomic sizes is 1.000, which allows a coordination number of 12. One structure that has a coordination number of 12 is known as face-centered cubic fee), and it has one atom on each comer of the cube and an atom on each of the six faces of the cube. The atoms on the faces are shared by two cubes, so one-half of each atom belongs in each cube. With there... 

Figure 11.3 Arrangement of atoms in an ionic solid such as NaCl. (a) shows a cubic lattice with alternating Na+ and Cl- ions, (b) is a space-filling model of the same structure, in which the small spheres are Na+ ions, the larger Cl-. The structure is described as two interlocking face-centred cubic lattices of sodium and chlorine ions. 

Inert Gases. The calculation of 7 should be relatively straightforward for crystals of inert gases, in which only one kind of interaction may be expected. These crystals have a face-centered cubic structure. If each atom is treated as a point source of attractive and repulsive forces, only the forces between the nearest pairs of atoms are considered, the zero point energy is neglected, and no re-arrangement of atoms in the surface region is permitted, then the calculated 7 still depends on the equation selected to represent the interatomic potential U. 

Body-centered cubic (bcc) is the lattice symmetry of Fe, for instance (Fig. 16.2c). Bcc here refers to a crystal arrangement of atoms at the corners of a cube and one atom in the center of the cube equidistant from each face. 

Figure 25 The Nd2Cu04 structure. Note the square-planar Cu02 planes and the cubical arrangement of oxygen atoms about the Nd atoms. From Reference 135.

Cuthbert and Linnett80 have suggested that the stability of the cubic closest packed arrangement of atoms in crystals of neon, argon, krypton, and xenon (helium crystallizing instead in hexagonal closest packing)81 is explicable by the tetrahedral electron distribution of the atoms... 

Fig. 11-13.—The arrangement of atoms in the cubic crystal MgCu2 (the cubic Friauf structure).

Cotton has pointed out that a metal in a low oxidation slate can adopt one of two strategies in forming clusters. It can form multiple bonds to another metal, as in [Re,Xj<] . or it can form single bonds to several other metal atoms, as in the octahedral clusters. It is interesting that Mo(II) adopts both methods (Fig. 16.65) and that both structures have u cubic arrangement of chloride ions. 

Figure 1 shows the arrangements of metal atoms on (111), (100), and (110) surfaces of a face-centered cubic (fee) metal. It can readily be seen that different sites on a particular surface have different symmetry properties. For example, the top layer of atoms on sites 1, 2, 3, and 3 on the (111) face (where the number denotes the number of metal atoms associated with the site) exhibit 6-fold, 2-fold, 3-fold, and 3-fold rotation axes of symmetry, respectively. At this level of discrimination the point group symmetries of those sites are C6(, C2 , Cj , and Cj , respectively. However, when the arrangement of atoms in the second layer is taken into account (there is another atom under only half of the 3-fold sites) the point group symmetries of the first two sites... 

The mineral wustite is a nonstoichiometric iron oxide with the empirical formula Fe O, where x is a number slightly less than 1. Wustite can be regarded as an FeO in which some of the Fe sites are vacant. It has a density of 5.75 g/cm3, a cubic unit cell with an edge length of 431 pm, and a face-centered cubic arrangement of oxygen atoms. 

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