 Solution

Using the contact model of (b), $r_a=a/4$ and $r_c=(\sqrt{3}-1)a/4$. Divide the volume of four cation spheres and eight anion spheres by $a^3$ to obtain the packing fraction:For one-radius cubic close-packed structure, face-diagonal contact gives $4r=\sqrt{2}a$, hence ${\eta }_{\mathrm{ccp}}=4(4\pi r^3/3)/a^3=\pi /(3\sqrt{2})\simeq 0.7405$. The limiting fluorite value is lower than the one-radius close-packed value. Its two sizes, stoichiometry and eight-to-four coordination impose a different geometry: the fluoride sites form a simple-cubic array, and the calcium sites are too far apart for calcium–calcium contact. A face-centred arrangement of one species is therefore not sufficient to infer a close-packed whole crystal.