Entropy of a finite magnetic multiplet ($s_J/k_B=\ln Z_J-x{\partial }_x\ln Z_J$)

For magnetic quantum numbers $m=-J,\dots ,J$, energies $E_m=-mg{\mu }_{B}B$ and $x=g{\mu }_{B}B/(k_{B}T)$, the partition function is $Z_J={\sum }_m{e}^{mx}=\sinh [(2J+1)x/2]/\sinh (x/2)$. Its entropy is the title expression, a function of $B/T$ alone. At small $J|x|$, $s_J/k_B=\ln (2J+1)-J(J+1)x^2/6+O(x^4)$. The maximal entropy is $k_B\ln (2J+1)$ at zero splitting. A fixed independent-spin entropy consequently fixes $B/T$; an attached heat load changes the combined isentropic path.