Close-Packed Sphere structure

While not overcrowded, the polyethylene structure uses space with admirable efficiency, the atoms filling the available space with 73% efficiency. For contrast, recall that close-packed spheres fill space with 74% efficiency, so polyethylene does about as well as is possible in its utilization of space. 

The corresponding unit cells are shown in Figure 1.1 and an examination of simple ball-and-stick models (which the reader is strongly urged to carry out) shows that the face-centred cubic (fee) and hexagonal close-packed (hep) structures correspond to the only two possible ways of close-packing spheres, in which each sphere has twelve nearest neighbours. 

Several cubic structures, therefore, in which (besides 0, 0, 0 0, K, M M, 0, M M, M, 0) one or more of the reported coordinate groups are occupied could be considered as filled-up derivatives of the cubic close-packed structures. The NaCl, CaF2, ZnS (sphalerite), AgMgAs and Li3Bi-type structures could, therefore, be included in this family of derivative structures. For this purpose, however, it may be useful to note that the radii of small spheres which fit exactly into tetrahedral and octahedral holes are, respectively, 0.225. and 0.414... if the radius of the close-packed spheres is 1.0. For a given phase pertaining to one of the aforementioned types (NaCl, ZnS, etc.) if the stated dimensional conditions are not fulfilled, alternative descriptions of the structure may be more convenient than the reported derivation schemes. 

It is not hard to understand why many metals favor an fee crystal structure there is no packing of hard spheres in space that creates a higher density than the fee structure. (A mathematical proof of this fact, known as the Kepler conjecture, has only been discovered in the past few years.) There is, however, one other packing that has exactly the same density as the fee packing, namely the hexagonal close-packed (hep) structure. As our third example of applying DFT to a periodic crystal structure, we will now consider the hep metals. 

The factor 1.091 in equation (6.2) arises from the characteristics of close-packed hexagonal structures. If D is the distance between centers of adjacent spheres, the spacing between the centers of adjacent rows in a plane is 3 D/2. The spacing between centers of adjacent planes is (2/3) Allowing iV and N to represent the number of spheres along the x and y axes of a plane of spheres, the planar area, A, is given by... 

When we determined the crystalline structure of solids in Chapter 4, we noted that most transitional metals form crystals with atoms in a close-packed hexagonal structure, face-centered cubic structure, or body-centered cubic arrangement. In the body-centered cubic structure, the spheres take up almost as much space as in the close-packed hexagonal structure. Many of the metals used to make alloys used for jewelry, such as nickel, copper, zinc, silver, gold, platinum, and lead, have face-centered cubic crystalline structures. Perhaps their similar crystalline structures promote an ease in forming alloys. In sterling silver, an atom of copper can fit nicely beside an atom of silver in the crystalline structure. 

Iversen et al. [6] found that for a polymer strucmre similar to the interstices between closely packed spheres (phase inversion membrane), Equation 38.4 is able to well describe the tortuosity-porosity relationship whereas for a polymer structure similar to random spheres or clusters (stretched membrane), Equation 38.5 has to be used. 

Pilot plant smdied have also been performed by Larsen et al. [37], who obtained stable operation and more than 95% SO2 removal from flue gas streams with a gas-side pressure drop of less than 1000 Pa. The importance of the membrane structure on the SO2 removal has been studied by Iversen et al. [6], who calculated the influence of the membrane resistance on the estimated membrane area required for 95% SO2 removal from a coal-fired power plant. Authors performed experiments on different hydrophobic membranes with sodium sulfite as absorbent to measure the SO2 flux and the overall mass-transfer coefficient. The gas mixture contained 1000 ppm of SO2 in N2. For the same thickness, porosity, and pore size, membranes with a structure similar to random spheres (typical of stretched membranes) showed a better performance than those with a closely packed spheres stmcture. 

The surface of a crystalline solid is strongly correlated to its bulk structure. The atoms of a crystal are arranged in a periodical sequence forming the crystal lattice. Most frequently metals and metal alloys tend to form close-packed sphere (cps) arrangements reflecting the isotropy of the forces of atomic interaction. 

Coordination number of 8 ( ot close-packed) Spheres touch along the body diagonal of a cube. Can be converted into close-packed structures under high pressure... 

Figure 1. Phase diagram for a structurally symmetric coil—coil block copolymer (Lam = lamellae, Hex = hexago-najly packed cylinders, Q/a3d = bicontinuous cubic with laid symmetry, Q/m3m = body-centered cubic, CPS = close packed sphere).

There are four copper atoms in the unit cell, (Figure 1.7). Besides some metals, the noble gases, Ne(s), Ar(s), Kr(s), Xe(s), also adopt this structure in the solid state. This structure is often called the face-centred cubic (fee) structure or the cubic close-packed (cep) structure, but the Strukturbericht symbol, Al is the most compact notation. Each atom has 12 nearest neighbours, and if the atoms are supposed to be hard touching spheres, the fraction of the volume occupied is... 

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