Packing of atoms

Figure 1.4 (a) Close packing of atoms in a cubic structure, showing six... 

The general geometrical problem of the packing of spheres has not been solved. An example of closest packing of atoms with some variation in effective radius is the icosahedral packing found (13) in the intermetallic compound Mg3B(Al,Zn) (Fig. 1). The successive layers in this structure contain 1, 12, 32, and 117 spheres. These numbers are reproduced (to within 1) by the empirical equation (12)... 

Figure 1.4 (a) Close packing of atoms in a cubic structure, showing six in-plane neighbours for each atom (b) An expanded diagram of the packing of atoms above and below the plane. A above and A below represents the location of atoms in the hexagonal structure, and A above with B below, the face-centred cubic structure... 

Figure 3.11 Cubic close-packed structure of face-centered cubic crystals such as copper as a packing of atom layers (a) a single close-packed layer of copper atoms (b) two identical layers, layer B sits in dimples in layer A (c) three identical layers, layer C sits in dimples in layer B that are not over atoms in layer A. The direction normal to these layers is the cubic [111] direction.

The same atom-centered polyhedra can be used to describe interstitial diffusion in all the many metal structures derived from both face-centered cubic and hexagonal closest packing of atoms. In these cases the polyhedra are centered upon a metal atom and all the tetrahedral and octahedral interstitial sites are empty. The hardening of metals by incorporation of nitrogen or carbon into the surface layers of the material via interstitial diffusion will use these pathways. 

Again we may note that the density of surface packing of atoms... 

Notice that the 111 planes are the close-packed layers of the structure and so have the densest packing of atoms, as we would expect. 

Many systems of notation and classification have been proposed. The well-known books by R. W. G. Wyckoff, A. F. Wells, F. C. Phillips, L. Bragg, M. J. Buerger, L. V. Azakoff, D. M Adams, and W. B. Pearson (Appendix A, Further Reading) have discussed these proposals. These proposals include close packing of atoms, nets, or prism connections, stacking of coordination polyhedra and even a crystal-algebra method. Application of most of these proposals requires familiarity with the features of many structures. Only specialists can be expected to have... 

The degree of short-range order in an amorphous material can be characterized by a hard sphere model if the basic structure of an amorphous material is approximated by spheres. The density of packing of atoms around a reference atom is described by the number of atom centers per volume that lie in a spherical shell of thickness, dr, and radius about the reference atom. In a hard sphere model, the number, n, of neighboring spheres with centers between r and dr is measured as a function of r. 

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