To determine the number of tetrahedral and octahedral voids in a three-dimensional close-packing of spheres, we note that a sphere in a hexagonal close-packed layer A is surrounded by three B voids and three C voids (Fig. 1). When the next layer is placed on top of this, the three voids of one kind (say B) are occupied and the other three (say C) are not. Thus the three B voids become tetrahedral voids and the three C voids become octahedral voids. A single sphere in a three-dimensional close-packing will have similar voids on the lower side as well. In addition, the particular sphere being considered covers a triangular void in the layer above it and another in the layer below it. Thus two more tetrahedral voids surround the spheres. This results in 2 3 + 1 + 1 = 8 tetrahedral voids and 2 3 = 6 octahedral voids surrounding the sphere. Since a tetrahedral void is shared by four spheres, there are twice as many tetrahedral voids as there are spheres. Similarly, since an octahedral void is surrounded by six spheres, there are as many octahedral voids as there are spheres
