= Solution
The actual <ionic radii> need not equal the limiting ratio, and electronic charge distributions are not rigid <sphere> boundaries. In the same unlike-contact <hard-sphere model>, put $q=r_a/r_c$ and use $a=4(r_a+r_c)/\sqrt3$ to get
$$
\eta(q)=\frac{\pi\sqrt3}{4}\frac{1+2q^3}{(1+q)^3}.
$$
A <cation> somewhat larger relative to the <anion> than the limiting size opens gaps between the <anions>. Starting from $q=1.366$ and reducing $q$ modestly reduces this <packing fraction>; thus \b[a lower value is expected for that departure from the ideal contact construction]. The formula is not monotonic for all imaginable ratios, so no universal inequality should be inferred without specifying <radii>. Coordination-dependent <ionic radii>, <polarizability>, nonionic contributions to bonding, <thermal expansion> and <point defects> also limit the model. No numerical real-crystal <packing fraction> is determined by the supplied data alone.
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