= Solution
In a weak applied field, <magnetic susceptibility> relates <magnetization> to <magnetic field intensity>, $M=\chi H$. Its sign, <temperature> dependence and nonlinear field dependence distinguish several mechanisms.
\b[<Diamagnetism>] is the orbital response opposing an applied <magnetic field>, with $\chi<0$. Closed-shell atoms and filled bands generally give a weak, nearly temperature-independent response. In a simple free <Electron> <metal>, <Landau diamagnetism> contributes $\chi_L=-\chi_P/3$; the factor is specific to the isotropic parabolic-band model and is not universal in crystals.
\b[<Paramagnetism>] gives $\chi>0$. Localized <magnetic moments> align statistically with the field against thermal disorder. For independent moments of <angular momentum> $J$,
$$
\chi=\frac{\mu_0n(g\mu_B)^2J(J+1)}{3k_BT},
$$
which is the <Curie law>. Exchange interactions often lead approximately to the <Curie–Weiss law>, $\chi=C/(T-\Theta)$, sufficiently far above any ordering <temperature>. The sign of $\Theta$ suggests predominantly parallel or antiparallel exchange, though it is not a complete classification. Conduction <Electrons> instead exhibit <Pauli paramagnetism>:
$$
\chi_P=\mu_0\mu_B^2g(E_F),
$$
for $g\simeq2$ and a <spin-summed density of states> per unit volume. Only states close to the <Fermi energy> change occupation, making this response nearly <temperature> independent when $k_BT\ll E_F$.
\b[<Ferromagnetism>] has spontaneous parallel magnetic order below a <Curie temperature>. <Exchange interaction>, not simply classical dipole attraction, provides the dominant alignment mechanism. In the convention $H=-J_{\rm ex}\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j$, positive $J_{\rm ex}$ favours alignment. <Magnetic domains> can make a ferromagnet's total <magnetization> vanish before an external field aligns its domains. Domain motion and anisotropy produce saturation, <hysteresis>, <remanent magnetization> and a reverse field measured by <magnetic coercivity>.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2012-magnetic-response.png]
{title=Qualitative linear magnetic responses and a ferromagnetic <hysteresis> loop}
\b[<Antiferromagnetism>] has oppositely ordered sublattices below a <Néel temperature>; equal opposing moments give zero net <magnetization> while retaining a nonzero staggered <order parameter>. A susceptibility anomaly and <magnetic neutron diffraction> reveal this order even when bulk <magnetization> vanishes. \b[<Ferrimagnetism>] also has opposing sublattices but unequal contributions, giving a nonzero spontaneous net moment. Above their ordering <temperatures> both can become paramagnetic. Measurements with a <magnetometer> as a function of field and <temperature>, supplemented by magnetic <diffraction>, therefore distinguishes a compensated ordered material from an ordinary paramagnet.
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