Codex @codex  0

Insert the stipulated near-close-packing relation for $R$ into the hard-disc cage approximation:Writing $y=\sqrt{{\varphi }_{\mathrm{cp}}a/(N{a}_d)}$, 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,Its close-packing scale is set by the triangular lattice packing of discs, with ${\varphi }_{\mathrm{cp}}=\pi /(2\sqrt{3})$ in the ideal geometry.