Simple cubic packing

The low c.n. and low density of this structure make it unsuitable for most metals this structure has been assigned to a-Po. Mercury does, however, crystallize with a closely related structure which can be derived from the simple cubic packing 

The structures of crystalline As (Sb and Bi) and of black P illustrate two different ways of distorting the simple cubic packing so that each atom has three nearest and three more distant neighbours. Both are layer structures which may alternatively be described as forms of the plane hexagon net. They are described and illustrated in the section on the plane hexagon net (p. 88). 

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The fluorite structure may also be regarded as being composed of a simple cubic packing of anions, and the number of cubic holes is the same as the... 

Fig. 2.1 Packing of ions (a) simple cubic packing showing an interstice with eightfold coordination (b) hexagonal close packing (c) cubic close packing showing a face-centred cubic cell.

Cubic Zr02 has the fluorite structure with the O2- ions arranged in simple cubic packing and half the interstices in this lattice occupied by Zr4+ ions (Fig. 4.29) the substitution of lower-valence cations leads to O2- ion vacancies as indicated. The vacancies which stabilize the structure also lead to high mobility in the oxygen sub-lattice and to behaviour as a fast-ion conductor. 

For hexagonal packing of particles in dispersion f = 1.81, and for simple cubic packing f = 1.61. 

The structure of crystalline arsenic provides an example of somewhat distorted simple cubic packing. It is illustrated in Figure 9-26b. The atoms are in the positions of the cubic structure. Each has three nearest and three more distant neighbors. The layers formed by the nearest bonded atoms may also be derived from a plane of hexagons. These layers buckle as the bond angle decreases from 120°. 

However, the VL F value for a spherical foam is substantial and amounts to 48% of the total foam volume for a simple cubic packing and to 26% for a hexagonal close packing. For the latter the foam expansion ratio varies from 2.01 to 3.85 which may introduce large errors into the calculation of the //IIlln value. In a polyhedral foam the liquid volume can be neglected with respect to the foam volume but for the determination of //min( ) more detailed information on the structure of the foam is needed. 

Figure 2.9. Representation of (a) simple cubic packing and (b, c) close-packing. The two sites that may be occupied by subsequent layers in close-packing, B and C sites, are shown. This results in two types of packing ABABAB... (hexagonal close-packing, hep), and ABCABC... (cubic close-packing, ccp).

In contrast, for simple cubic packing (spheres stacked on top of each other in successive layers) the spheres occupy only 52.4% of the space (verify this for yourself). 

Figure 6.36 Shear rate at which the shear viscosity suddenly jumps dis-continuously upward as a function of the volume fraction of poly(vinyI chloride) (PVC) spheres of various diameters in dioctyl phthalate. Point A denotes the volume fraction at which two-dimensional hexagonal close-packed layers of spheres first touch. Point B is the volume fraction for three-dimensional simple cubic packing. (From Hoffman, reprinted from Trans. Soc. Rheol. 16 155, Copyright 1972 American Institute of Physics.)...

With simple cubic packing, the volume of pores amounts to 48% of total volume, and for close-packed hexagonal array 26%. 

From these results and Example 3.3, it becomes clear that closed-packed cubic lattice has the best space economy (best packing, least empty space), followed by the body-centred lattice, whereas the simple cubic packing has the lowest space economy with the highest fraction of unoccupied space. 

A detailed analysis of the behavior of hollow-sphere foams is available in [18,19], The theoretical performance of hollow-sphere foams is on par with that of closed-cell foams. Since hollow-sphere foams can be produced with fewer defects, they have the potential to perform up to three times better than existing closed-cell foams at a relative density of 10% and ten times better below a relative density of 5% [17], The behavior of simple cubic packed (SC) and face-centered cubic packed (FCC) hollow-sphere foams is shown in Figure 1. The FCC hollow-sphere foam generally represents the best performance of optimally bonded hollow spheres that was measured in this work and SC hollow-sphere foam generally represents the performance of non-optimally bonded, random packed hollow spheres [17]... 

The simplest lattice structure is the so-called simple cubic packing (sc). This structure consists of identical layers of atoms placed exactly above and below each other. The structure is sketched in Figure 2- 20. 

The side length in the unit cell is denoted b and the radius of the atoms is denoted r. SC simple cubic packing, BCC Body-centered cubic packing, FCC Face-centered cubic packing, HCP Hexagonal closest... 

The ideal size of the cation relative to the size of the anion for each of the voids can be determined by trigonometry. For example, a cation will fit snugly into an octahedral void if it is 0.41 times the size of the anion. If it is significantly larger than that, it may not be possible to fit the cation into the closest-packed array. This is the case for compounds such as cesium chloride, where the cation has an ionic radius of 1.67 A and the anion has a radius of 1.84 A (thus, the ratio is 0.91). In this solid, the chloride ions open up to form the simple cubic packing shown in Figure 16. 

Can you provide at least one unit cell for the simple cubic packing shown previously ... 

Detailed numerical calculations have been carried out by Bohlen et al. (57) for the transition of mono-disperse spheres in simple cubic packing (Oq= 0.5236) to cubes (d> = 1), for both zero and finite contact angles. Unfortunately, although the results are interesting, this kind of packing is not realistic for foams and emulsions, and will not be discussed further. 

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