The energy levels of an imperfect Fermi gas in the presence of an external magnetic field \(\boldsymbol{H}\), to the first order in \(a\), may be written as

\[E_{n}=\sum_{\boldsymbol{p}}\left(n_{\boldsymbol{p}}^{+}+n_{\boldsymbol{p}}^{-}\right) \frac{p^{2}}{2 m}+\frac{4 \pi a \hbar^{2}}{m V} N^{+} N^{-}-\mu^{*} \mu_{0} H\left(N^{+}-N^{-}\right)\]

see equations (8.2.8) and (11.7.12). Using this expression for \(E_{n}\) and following the procedure adopted in Section 8.2.A, study the magnetic behavior of this gas - in particular, the zero-field susceptibility \(\chi(T)\). Also examine the possibility of spontaneous magnetization arising from the interaction term with \(a>0\).