In the notation of Sec. 3.9, the potential energy of a magnetic dipole in the presence of a magnetic field \(\boldsymbol{B}=(0,0, B)\) is given by the expression \(-\left(g \mu_{B} m\right) B\), where \(m=-J, \ldots,+J\). The total energy \(\varepsilon\) of the dipole is then given by \(\varepsilon=\left(p^{2} / 2 m^{\prime} \right)-g \mu_{B} m B, m^{\prime}\) being the (effective) mass of the particle; the momentum of the particle may then be written as

\[
p=\left\{2 m^{\prime}\left(\varepsilon+g \mu_{B} m B \right) \right\}^{1 / 2} .
\]

At \(T=0\), the number of such particles in the gas will be

\[
N_{m}=\frac{4 \pi V}{3 h^{3}}\left\{2 m^{\prime}\left(\varepsilon_{F}+g \mu_{B} m B \right) \right\}^{3 / 2}
\]

and hence the net magnetic moment of the gas will be given by

\[
M=\sum_{m}\left(g \mu_{B} m \right) N_{m}=\frac{4 \pi g \mu_{B} V}{3 h^{3}}\left(2 m^{\prime} \right)^{3 / 2} \sum_{m} m\left(\varepsilon_{F}+g \mu_{B} m B \right)^{3 / 2} .
\]

We thus obtain for the low-field susceptibility (per unit volume) of the system

\[
\begin{align*}
\chi_{0} & =\operatorname{Lim}_{B \rightarrow 0}\left(\frac{M}{V B} \right)=\frac{4 \pi g \mu_{B}}{3 h^{3}}\left(2 m^{\prime} \right)^{3 / 2} \cdot \frac{3}{2} g \mu_{B} \varepsilon_{F}^{1 / 2} \sum_{m=-J}^{J} m^{2} \\
& =\frac{2 \pi g^{2} \mu_{B}^{2}}{3 h^{3}}\left(2 m^{\prime} \right)^{3 / 2} \varepsilon_{F}^{1 / 2} J(J+1)(2 J+1) . \tag{1}
\end{align*}
\]

By eqn. (8.1.24),

\[
\begin{equation*}
\varepsilon_{F}^{3 / 2}=\frac{3 n}{4 \pi(2 J+1)} \frac{h^{3}}{\left(2 m^{\prime} \right)^{3 / 2}} \quad\left(n=\frac{N}{V} \right) . \tag{2}
\end{equation*}
\]

Substituting (2) into (1), we obtain the desired result

\[
\chi_{0}=\frac{1}{2} n \mu^{* 2} / \varepsilon_{F} \quad\left\{\mu^{* 2}=g^{2} \mu_{B}^{2} J(J+1) \right\} .
\]

With \(g=2\) and \(J=1 / 2\), we obtain: \(\chi_{0}=(3 / 2) n \mu_{B}^{2} / \varepsilon_{F}\), in agreement with eqn. (8.2.6).

The corresponding result in the limit \(T \rightarrow \infty\) is given by

\[
\chi_{\infty}=\frac{1}{2} n \mu^{* 2} / k T
\]

see eqn. (3.9.26). We note that the ratio \(\chi_{0} / \chi_{\infty}=3 k T / 2 \varepsilon_{F}\), valid for all \(J\).