= Solution
Write the two allowed <Spin-state population and net polarization>[moment projections] as $\pm\mu$, where $\mu=g\mu_B/2$ for <spin one-half>. Their <Zeeman effect>[Zeeman energies] in the prescribed <Magnetic flux density>[flux density] are $\mp\mu B$. The single-particle <partition function> and the mean moment along the field are
$$
z=e^{\beta\mu B}+e^{-\beta\mu B}=2\cosh x,\qquad
\langle\mu_z\rangle=\mu\frac{e^x-e^{-x}}{e^x+e^{-x}}=\mu\tanh x,
\quad x=\frac{\mu B}{k_BT}.
$$
For noninteracting moments, <magnetization> is moment per volume:
$$
\boxed{M=M_s\tanh(\eta B/T),\qquad M_s=\frac{N\mu}{V},\quad
\eta=\frac\mu{k_B}.}
$$
For electron-like <Spin-only magnetic moment>[spin-only] moments $g\simeq2$, these become $M_s=N\mu_B/V$ and $\eta=\mu_B/k_B$.
<Derivative>[Differentiate] with respect to the specified $B$ to obtain
$$
\boxed{\chi_B=\left(\frac{\partial M}{\partial B}\right)_T
=\frac{N\mu^2}{Vk_BT}\operatorname{sech}^2x.}
$$
This $B$ <derivative> is not a dimensionless SI <Volume magnetic susceptibility>[volume susceptibility]. In the weak-feedback approximation $B\simeq\mu_0H$, the usual dimensionless <Differential magnetic susceptibility>[differential susceptibility] is $\chi_{\rm diff}\simeq\mu_0\chi_B$. The <Linear response>[small-field limit] gives Curie's law $\chi_0=\mu_0N\mu^2/(Vk_BT)$. At finite bias a secant response $M/H\simeq\mu_0M/B=\chi_0\tanh(x)/x$ differs from the differential slope $\chi_0\operatorname{sech}^2x$. For an internal $B=\mu_0(H+M)$ retained self-consistently, the exact local conversion is $\chi_H=\mu_0\chi_B/(1-\mu_0\chi_B)$; <Demagnetizing field>[demagnetizing] geometry can further affect an applied-field measurement. Stating the field convention avoids a hidden factor of $\mu_0$ or confusion between a slope and a ratio.
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