 Solution

Differentiation of the restricted free energy gives the magnetizationIn the weak-field limit $|\mu B|\ll k_{\mathrm{B}}T$, $\tanh x\simeq x$, soThe printed coefficient is inconsistent with the stated moment $\mu =2{\mu }_{\mathrm{B}}$. The requested expression without the factor four would follow from $\mu ={\mu }_{\mathrm{B}}$ in this restricted model. This discrepancy is present in the original PDF, rather than introduced by its transcription.

For completeness, differentiating the full thermal free energy givesIts weak-field form isAt $k_{\mathrm{B}}T\gg \Delta$ this approaches $2N{\mu }_{\mathrm{B}}^{2}B/(V{k}_{\mathrm{B}}T)$, so restoring the two nonmagnetic ground states does not recover the printed coefficient either. The inverse-temperature Curie law form is valid, but the population factor and moment squared must be retained consistently.