= Solution
Insert the stipulated near-close-packing relation for $R$ into the <hard-disc cage approximation>:
$$
\mathcal A=4a_d\left(\sqrt{\frac{\phi_{\rm cp}}\phi}-1\right)^2
=4a_d\left(\sqrt{\frac{\phi_{\rm cp}a}{Na_d}}-1\right)^2.
$$
Writing $y=\sqrt{\phi_{\rm cp}a/(Na_d)}$, we have $\partial_a y=y/(2a)$ and therefore $\partial_a\ln\mathcal A=y/[a(y-1)]$. From the supplied crystalline <canonical partition function>,
$$
\boxed{\Pi_{\rm xtl}=Nk_BT\partial_a\ln\mathcal A
=\frac{k_BT}{a_d}\frac{\phi}{1-\sqrt{\phi/\phi_{\rm cp}}}}.
$$
Its close-packing scale is set by the <triangular lattice packing of discs>, with $\phi_{\rm cp}=\pi/(2\sqrt3)$ in the ideal geometry.
Back to article page
