Cubic arrangement

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.

Figure 8.7 (a) The low-temperature, ordered, simple cubic arrangement of Qo molecules as revealed by neu-... 

Many metals have close-packed structures, with the atoms stacked in either a hexagonal or a cubic arrangement close-packed atoms have a coordination number of 12. Close-packed structures have one octahedral and tivo tetrahedral holes per atom. 

These are the only cubic arrangements provided by the theory of space groups in which every atom occupies an invariant position, and all of the atomic positions are those corresponding to the A1 arrangement. Very recently we have found that the cubic phase Pt3Cu is highly probably also based upon the cubic structure ABCa (Tang, 1951). 

Fig. 9.7 Two cubic arrangements for Li2ZnGe centrosymmetric Fm3m (a), non-centrosymmetric F43m (b).

Fig. 9.8 Band structure and densities of states calculated for the compound Li2ZnGe in the non-centrosymmetric F43m cubic arrangement. Zn 3d inert orbitals (flat levels at -8 eV in the band structure) are not represented in the DOS.

Yet another common crystal lattice based on the simple cubic arrangement is known as the face-centered cubic structure. When four atoms form a square, there is open space at the center of the square. A fifth atom can fit into this space by moving the other four atoms away from one another. Stacking together two of these five-atom sets creates a cube. When we do this, additional atoms can be placed in the centers of the four faces along the sides of the cube, as Figure 11-28 shows. 

For compounds of the composition MX (M = cation, X = anion) the CsCl type has the largest Madelung constant. In this structure type a Cs+ ion is in contact with eight Cl-ions in a cubic arrangement (Fig. 7.1). The Cl- ions have no contact with one another. With cations smaller than Cs+ the Cl- ions come closer together and when the radius ratio has the value of rM/rx = 0.732, the Cl- ions are in contact with each other. When rM/rx < 0.732, the Cl- ions remain in contact, but there is no more contact between anions and cations. Now another structure type is favored its Madelung constant is indeed smaller, but it again allows contact of cations with anions. This is achieved by the smaller coordination number 6 of the ions that is fulfilled in the NaCl type (Fig. 7.1). When the radius ratio becomes even smaller, the zinc blende (sphalerite) or the wurtzite type should occur, in which the ions only have the coordination number 4 (Fig. 7.1 zinc blende and wurtzite are two modifications of ZnS). 

In crystalline C60 the molecules have a face-centered cubic arrangement, i.e. they are packed as in a cubic closest-packing of spheres as they are nearly spherical, the molecules spin in the crystal. The crystals are as soft as graphite. Similar to the intercalation com-... 

Figure 5.18.1 The NaCl crystal structure consisting of two interpenetrating face-centered cubic lattices. The face-centered cubic arrangement of sodium cations (the smaller spheres) is readily apparent with the larger spheres (representing chloride anions) filling what are known as the octahedral holes of the lattice. Calcium oxide also crystallizes in the sodium chloride structure.

Horse-spleen apoferritin crystallizes in a face-centred, close-packed, cubic arrangement, in the space group F432, with molecules at the 432 symmetry points of the crystal lattice (Harrison, 1959). This publication was the logical extension of the DPhil thesis of the Oxford chemist Pauline M. Cowan (as she was before her marriage to Roy Harrison), and represented the first publication in what was to be a long and distinguished series of contributions on ferritin from the undisputed Iron Lady of iron metabolism. ... 

Figure 10.7 Crystal structure of Lanthanum Hexaboride (prototypre hexaboride). The black circles represent boron octahedra. They form a simple cubic arrangement surrounding the central metal atom.

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...

Obviously, Equation 12b has to be modified somewhat for different floe packings. However, in this approximate treatment, we only wish to investigate trends, and for that purpose a simple cubic arrangement suffices. 

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... 

Figure 6.9 Structure of the high-temperature form of Agl (a-Agl) (a) the body-centered cubic arrangement of iodide (I-) ions the unit cell is outlined (b) two (of four) tetrahedral sites on a cube face, indicated by filled circles and (c) the four tetrahedral sites found on each cube face, indicted by filled circles. Ag+ ions continuously jump between all of the tetrahedral sites.

Even though qualitative bonding descriptions of metal atom clusters up to six or seven atoms can be derived and in some cases correlated with structural detail, it is clear that most structures observed for higher clusters cannot be treated thus. Nor do the structures observed correlate with those observed for borane derivatives with the same number of vertices. Much of borane chemistry is dominated by the tendency to form structures derived from the icosahedron found in elemental boron. However, elemental transition metals possess either a close-packed or body-centered cubic arrangement. In this connection, one can find the vast majority of metal polyhedra in carbonyl cluster compounds within close-packed geometries, particularly hexagonal close-packing. 

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