** Next:** Voids in a Close-Packing

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 6*mm* , 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*,
=120) the three positions, A, B and C have coordinates 00,
and .

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*/3 010 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 density^{1}.) 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, CdI_{2},
PbI_{2}, AgI and GaSe are known^{2,3,4} 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 = 0.8165^{2,5}. Table 1 lists the *h* /*a*
ratio for some metals and inorganic materials with hcp structure.

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 |

**Copyright © 1981, 1997 International Union of
Crystallography**