Past exam of the physics B course of the University of Cambridge / 2014 / Paper 2 / Section C / C12

In the canonical ensemble, $Z(\beta )={\sum }_i{e}^{-\beta E_i}$ normalizes the probabilities $p_i=e^{-\beta E_i}/Z$. For temperature-independent levels,The derivative holds external parameters, including magnetic field, fixed.

For one independent symmetric three-level system put $ϵ=\mu B$ and $x=\beta ϵ$. Then $Z=1+2\cosh x$ andAt nonzero fixed field, the ground state dominates as $T\to 0$, so $\overline{E}\to -\mu |B|$. At high temperature all three levels become nearly equally populated and their mean tends to zero from below. At fixed positive temperature the energy is even in $B$, is quadratic near zero field, and approaches $-\mu |B|$ at large $|B|$ as the lowest level becomes occupied. The heat capacity vanishes both when excitations are frozen out and when populations are almost equal, with a positive Schottky anomaly between those limits.

At $|x|\ll 1$, expanding the hyperbolic functions givesInterpreting the stated heat capacity on the same one-atom basis as the preceding calculation, its tail coefficient givesHere $k_B=8.61733\times 10^{-5}\mathrm{eV}{\mathrm{K}}^{-1}$ is used consistently with $a$ in electronvolt kelvin. For a sample of $N$ independent atoms, the total coefficient would be $N$ times the one-atom coefficient; it must first be divided by $N$.

Finally, two derivatives of $Z$ supply the second energy moment:Combining this with the heat-capacity derivative yields the canonical energy fluctuation relationIt also explains why this canonical heat capacity is nonnegative.^[ref]