This structure is named after the mineral fluorite, or CaF2. In this solid, the valency of the cation is twice that of the anion, that is, MX2. This structure can be described in two ways. Figure 2.22 shows the view of a CaF2-type unit cell where the FCC cation sublattice is interlaced with a simple cubic anion sublattice. That means all the tetrahedral interstitial positions of the FCC cation sublattice are filled with anions. Thus, each fluorite unit cell contains four cations and eight anions or four CaF2 molecules. This type of structure has the following relation between the lattice constant and ionic radii: a = (4Д/3) (R+ + R~), which can be derived from purely geometrical aspects noting that the F ion is at a/4, a/4, a/4 position, thus, R+ + R~ becomes the body diagonal of a cube with side equal to a/4. Some examples of the fluorite-type crystal structure are ZrO2, HfO2, UO2, ThO2, PuO2, and CeO2. UO2 is the most widely used nuclear fuel in most com­mercial power reactors. One of the beneficial effects of using UO2 is its high melting point (2860 °C), thus giving it excellent stability. UO2 structure has more empty space than a UC structure due to the simple cubic structure of the interior F~ sublattice. Even though this feature aids in providing more space for the fission products to accumulate inside the structure, it also reduces the fissile atom density. Notably, when the cations and anions exchange their positions in the fluorite structure (i. e., becomes M2X type), the resulting structure is called an “antifluorite” structure. Examples of such crystal structure are alkali metal oxides, that is, Li2O, Na2O, K2O, and Rb2O.

■ Example 2.3

Calculate the density of uranium dioxide (UO2) from the first principles given: atomic weights of uranium and oxygen are 238 and 16, respectively. The atomic radius data are U4+ = 0.105 nm and O2~ = 0.132 nm.

Solution

The theoretical density of UO2 can be obtained from the mass of the atoms in the unit cell divided by the volume of the unit cell.

The lattice constant of a UO2 unit cell (a) is given by (4/^/3) (R+ + R~) = (4/P3) (0.105 + 0.132) nm = 0.547 nm.

The volume of the UO2 unit cell (a3) is given by (0.547 nm)3 = 0.164 nm3 = 0.164 x (10~9)3 m3 = 0.164 x 10~27m3.

UO2 has a fluorite-type crystal structure, that is, the effective number of U4+ ions is 4 and of O2~ ions is 8.

The atomic weight of the uranium is 238. This means the mass of each U4+ ion is 238 amu or (238 x 1.66 x 10~27)kg (since 1 amu = 1.66 x 10~27 kg). There are four U4+ ions of a combined mass of (4 x 238 x 1.66 x 10~27) kg = 1.58 x 10~24kg.

Similarly, the mass of eight oxygen ions in the unit cell is given by 0.213 x 10~24kg.

The total mass of the UO2 unit cell is (1.58 + 0.212) x 10~24kg = 1.792 x 10~24kg.

The density of UO2 is then given by (1.792 x 10~24kg/0.164 x 10-27 m3) « 10 926kg m~3 « 10.93 g cm~3.