= Solution
Using the contact model of (b), $r_a=a/4$ and $r_c=(\sqrt3-1)a/4$. Divide the <volume> of four <cation> <spheres> and eight <anion> <spheres> by $a^3$ to obtain the <packing fraction>:
$$
\eta=\frac{4\pi}{3a^3}(4r_c^3+8r_a^3)=\frac\pi{12}\left[(\sqrt3-1)^3+2\right]=\boxed{\frac\pi6(3\sqrt3-4)\simeq0.6263.}
$$
For one-radius <cubic close-packed structure>, face-diagonal contact gives $4r=\sqrt2a$, hence $\eta_{\rm ccp}=4(4\pi r^3/3)/a^3=\pi/(3\sqrt2)\simeq0.7405$. The limiting fluorite value is \b[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.
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