Formula of a Compound and Number of Voids Filled

When particles close-pack into a ccp or hcp structure, two kinds of voids are created. The number of octahedral voids equals the number of close-packed particles ($N$), while the number of tetrahedral voids is twice that ($2N$).

In ionic solids the larger ions (usually the anions) build the close-packed framework, and the smaller ions (usually the cations) sit in the voids — tetrahedral voids if the cation is small enough, octahedral voids if it is larger. Importantly, not all voids are filled: the fraction of octahedral or tetrahedral voids that is occupied depends on the chemical formula of the compound. This connection between voids filled and formula is used in the worked examples that follow.

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Locating the voids in a ccp (fcc) structure

Tetrahedral voids. Imagine the fcc unit cell divided into eight small cubes. Each small cube has atoms at its alternate corners (4 atoms), and joining them gives a regular tetrahedron — so each small cube holds one tetrahedral void. With eight small cubes there are eight tetrahedral voids per unit cell. Since a ccp unit cell has 4 atoms, the number of tetrahedral voids ($8$) is twice the number of atoms.

Octahedral voids. Consider the fcc cell again. The body centre $C$ is empty but is surrounded by the six face-centre atoms; joining these gives an octahedron, so there is one octahedral void at the body centre. In addition there is an octahedral void at the centre of each of the 12 edges. Each edge is shared between four adjacent unit cells, so each edge void contributes only $\tfrac{1}{4}$ to a given cell: …