Electronic and lattice heat-capacity crossover in a monovalent metal ($T_{\ast }^2=15k_B{\theta }_D^3/(8{\pi }^4{E}_F)$)

For a monovalent three-dimensional free electron gas, the spin-summed molar density of states is $g(E_F)=3N_A/(2E_F)$. Combining the shell estimate $C_{el}=3k_B^{2}g(E_F)T$ with $C_{la}=(12{\pi }^4/5)R(T/{\theta }_D{)}^3$ gives the displayed nonzero crossover. The controlled Sommerfeld expansion replaces $3$ by ${\pi }^2/3$ and multiplies $T_{\ast }^2$ by ${\pi }^2/9$. A Fermi energy alone does not determine the DOS in an unspecified band structure.