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Close-Packed Structures

The crystal structures of a large number of metals, alloys and inoganic compounds can be described geometrically in terms of a close-packing of equal spheres, held together by interatomic forces. Frequently, the positions of one kind of atoms of ions in inorganic structures correspond approximately to those of equal spheres in a close-packing with the other atoms distributed among the voids. All such structures will be referred to as close-packed structures though they may not be ideally close-packed. The close-packed arrangement of equal spheres in a plane is shown in Fig. 1 where each sphere is in contact with six other spheres. Since the symmetry of this layer is 6mm , such a layer is called a hexagonal close-packed layer. Let this layer be called an A layer. It contains two types of triangular voids, one with the apex of the triangle upwards in the diagram and labelled B, and the other with the apex downwards and labelled C. In the two-dimensional unit cell indicated in the figure (a=b, $\gamma$=120$^{\circ}$) the three positions, A, B and C have coordinates 00, $\frac{1}{3}$ $\frac{2}{3}$ and $\frac{2}{3}$ $\frac{1}{3}$.

In a three-dimensional packing the next hexagonal close-packed layer of spheres can occupy either the sites B or C, but not both. Similarly the layer above a B layer can be either C or A and that above a C layer either A or B. No two successive layers can be alike. The positions B and C are displaced with respect to A by vectors +S and -S respectively where S = a/3 $\langle$$\overline{1}$010$\rangle$ in the Miller-Bravais notation.

Any sequence of the letters, A. B and C with no two successive letters alike represents a possible manner of close-packing equal spheres. In such a three- dimensional close-packing, each sphere is surrounded by and touches 12 other spheres. This is the maximum number of spheres that can be arranged to touch a given sphere and it provides the maximum packing density for an infinite lattice arrangement. (There are however other arrangements of a finite number of equal spheres which have a higher packing density1.) It is evident from the foregoing that the number of different close-packed structures that are possible in three dimensions is infinite. The identity period or c dimension of the hexagonal unit cell in a three-dimensional close-packed structure is determined by the number of layers after which the stacking sequences repeats itself. The two most common close-packed structures which occur in nature are: (i) the hexagonal close-packing (hcp) with a layer stacking ABAB.. and (ii) the cubic close-packing (ccp) with a layer stacking ABCABC.. They have identity periods of two and three layers respectively. In addition to the hcp and ccp modifications, a number of materials, like SiC, ZnS, CdI2, PbI2, AgI and GaSe are known2,3,4 to crystallize in a large variety of close-packed structures, called polytypes, with larger identity periods. The different polytype structures of the same material have identical a and b dimensions of their hexagonal unit cell but differ along c . Even for the same identity period of n layers, a number of different close-packed structures are possible with different arrangements of the n layers. The extent to which a real crystal structure approximates to a close- packing can be determined from the h /a ratio, where h is the separation between successive close-packed layers and a is the diameter of the spheres. For an ideally close-packed structure, this ratio must be $\sqrt\frac{2}{3}$ = 0.81652,5. Table 1 lists the h /a ratio for some metals and inorganic materials with hcp structure.

Table 1: Table 1
Material h/a Material h/a
Cd 0.943 AgI 0.815
Zn 0.928 BeO 0.815
He 0.8165 CdSe 0.815
Co 0.814 ZnO 0.800
Mg 0.812 AlN 0.800
Sc 0.797 CdS 0.810

Figure 1: The close-packing of spheres.
\includegraphics {fig1.ps}

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