Crystal symmetries sphere packing

The English physicist William Barlow began as a London business man later he became interested in crystal structures and devoted his life to that study. In 1894, he published his findings of the 230 space groups. It is amazing that from consideration of symmetry three scientists in different countries arrived at the 230 space groups of crystals at about this time. Barlow then worked with ideas of close packing. He pictured the atoms in a crystal as spheres, which, under the influence... 

The symmetry of the crystal structure is a direct consequence of dense packing. The densest packing is when each building element makes the maximum number of contacts in the structure. First, the packing of equal spheres in atomic and ionic systems will be discussed. Then molecular packing will be considered. Only characteristic features and examples will be dealt with here since systematic treatises on crystal symmetries are available for consultation [9-17, 9-22, 9-26]. 

Table 18.1 Crystal data of structures mentioned in Fig. 18.6. The ideal coordinates would apply to an undistorted packing of spheres. Coordinate values fixed by symmetry are stated as 0 or fractional numbers, otherwise as decimal numbers...

When all the rotations are possible in the solid state the symmetry increases to hexagonal. This form corresponds to the close packing of spheres or cylinders and the molecule is in a rotational crystalline state, characterized by rigorous order in the arrangement of the center (axes) of the molecules and by disordered azimuthal rotations [118]. If the chain molecules are azimuthally chaotic (they rotate freely around their axes), their average cross sections are circular and, for this reason, they choose hexagonal packing. The ease of rotation of molecules in the crystal depends merely on the molecular shape, as in molecules of an almost spherical shape like methane and ethane derivatives with small substituents, or molecules of a shape close to that of a cylinder (e.g., paraffin-like molecules). 

An interesting experiment on the oxidation of a single crystal of cobalt, which has a phase change at 420°C, was carried out by Kehrer and Leidheiser (22). Below 420°C cobalt exists in the hexagonal close-packed structure, and above 420°C in the face-centered cubic structure. After electropolishing in orthophosphoric acid a cobalt sphere 5/16 inches in diameter, it was oxidized in air in separate experiments both below and above 420°C. At 400°C the symmetry of the oxidation pattern, which indicates the variation of rate of oxidation with face, followed the hexagonal structure, while at 450°C it followed the symmetry of the face-centered cubic structure. 

Special interest attaches to molecules or crystals that possess symmetry which is lower or higher than might be expected. This statement calls for a brief explanation. The packing of equal spheres is discussed in Chapter 4 where we shall see that the majority of metals adopt certain highly symmetrical structures. Some metals, however, crystallize with less symmetrical variants of these structures, and these are obviously of interest since the lower symmetry presumably indicates some peculiarity in the bonding in such crystals. Some crystals containing only highly... 

In crystals where bonding is largely ionic (see Section 2.3.2), the densest possible packing of equal-sized anions (represented by spheres) is achieved by stacks of regular planar layers, as shown in Fig. 4.3. Spheres in a single layer have hexagonal symmetry, i.e. they are in symmetrical contact with six spheres. The layers are stacked such that each sphere fits into the depression between three other spheres in the layer below. 

As indicated in the hard-sphere model in Fig. 33 a hemispherical emitter surface is made up of a variety of crystal planes. The closer packed of these, such as the 110, 211, and 100 in the body centered cubic lattice, have a higher work function they therefore appear as dark spots in the more brightly emitting areas corresponding to stepped regions. With the aid of a standard orthographic projection (Fig. 34). the orientation of the emitter and the identity of the planes can thus be deduced from the symmetry of the pattern. 

FIGURE t48. Ice crystals viewed by Hooke under the microscope. He understood that their sixfold symmetry derived from packing of spheres (Figure 147). Hooke found that the ice crystals derived from urine were pure water (they lacked the urinous taste). 

The first crystal structures to be solved were those of inorganic compounds, for which great help was found through considerations of how one could pack spheres into structures that minimized the magnitude of interstitial space [8]. In this approach, the crystal is built up from, spherical structural units that make contact with one another. Alternatively, one could consider the points of a lattice network to be inflated into spheres. As a result, all individual symmetry is removed from the individual particles, and the symmetry of the lattice structure follows from the fashion in which the spheres are arranged. 

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