= Solution
The <microcanonical ensemble> assigns equal <probability> to accessible quantum states in a narrow <energy> shell of an isolated system with fixed <energy>, particle number and external parameters. To derive the <canonical ensemble>, put a small system in weak thermal contact with a much larger isolated bath. If the small-system state has <energy> $E_i$, its <probability> is proportional to the number $\Omega_b(E_{\rm total}-E_i)$ of bath states. Expanding the bath <entropy> gives $S_b(E_{\rm total}-E_i)=S_b(E_{\rm total})-E_i/T+\cdots$, since $\partial S_b/\partial E=1/T$. Negligible interaction <energy> and an effectively constant bath temperature therefore give $p_i=e^{-E_i/(k_BT)}/Z$. The <microcanonical ensemble> suits an isolated fixed-energy system; the <canonical ensemble> suits a system exchanging <energy> with a thermostat at fixed temperature, with particle number still fixed.
For one atom, summing the finite geometric <series> over $m=-J,\ldots,J$ gives the <partition function>
$$
\boxed{Z_1=\sum_{m=-J}^Je^{xm}=\frac{\sinh((J+1/2)x)}{\sinh(x/2)},\qquad x=\frac{\mu_BB}{k_BT}}.
$$
At $x=0$ the ratio is understood by <continuity> as $2J+1$. Logarithmic differentiation gives the mean <magnetic dipole moment>
$$
\boxed{\langle\mu_z\rangle=\mu_B\left[(J+\tfrac12)\coth((J+\tfrac12)x)-\tfrac12\coth(x/2)\right]}.
$$
This is odd in $x$, has small-$x$ expansion $\mu_BJ(J+1)x/3+O(x^3)$, and saturates at $\pm J\mu_B$. The requested common graph is plotted in units of $\mu_B$: its ordinate is $\langle\mu_z\rangle/(J\mu_B)$, so multiplication by $\mu_B$ gives $\langle\mu_z\rangle/J$. The normalized slopes at zero are $(J+1)/3$ and the limits are $\pm1$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-1-magnetization.png]
{title=Magnetization as a function of field and temperature}
The <magnetic susceptibility> is
$$
\boxed{\chi=\frac{N\mu_B^2}{k_BT}\left[\frac14\operatorname{csch}^2(x/2)-(J+\tfrac12)^2\operatorname{csch}^2((J+\tfrac12)x)\right]}.
$$
Equivalently it is $N\mu_B^2\operatorname{Var}(m)/(k_BT)\ge0$. For $k_BT\gg\mu_B|B|$, the apparent singular terms cancel, yielding <Curie law>
$$
\boxed{\chi\sim\frac{N\mu_B^2J(J+1)}{3k_BT}}.
$$
For fixed nonzero $B$, $T\to0$ and integer $J\ge1$, the first excited magnetic level is separated from the aligned <ground state> by $\mu_B|B|$. Its exponentially small occupation gives
$$
\boxed{\chi\sim\frac{N\mu_B^2}{k_BT}e^{-\mu_B|B|/(k_BT)}\longrightarrow0}.
$$
The moments are saturated, so changing the <magnetic field> hardly changes their alignment. This is a fixed-nonzero-field low-temperature limit. At exactly $B=0$, differentiating first gives the Curie expression for every $T>0$, which diverges as $T\to0$; the two limits do not commute in this noninteracting model. For $J=0$ both moment and susceptibility vanish identically.
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