= Entropy of a finite magnetic multiplet
{title2=$s_J/k_B=\ln Z_J-x\partial_x\ln Z_J$}
For magnetic quantum numbers $m=-J,\ldots,J$, energies $E_m=-mg\mu_BB$ and $x=g\mu_BB/(k_BT)$, 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.
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