Fluorite packing fraction ($\eta$)

With anion radius $r_a$, cation radius $r_c$ and unlike-neighbour contact, the conventional cell gives $\eta =(\pi \sqrt{3}/4)(r_c^3+2r_a^3)/(r_c+r_a{)}^3$. At the eight-coordinate limiting contact ratio this is $\pi (3\sqrt{3}-4)/6$. The cation-contact tetrahedral-hole limit is a different geometric ideal.