= Solution
For weak fields in a linear isotropic material, <magnetization> is $\mathbf M=\chi_m\mathbf H$, with <magnetic susceptibility> $\chi_m$. \b[<Paramagnetism> has $\chi_m>0$], while \b[<diamagnetism> has $\chi_m<0$]. In either case, $\mathbf B=\mu_0(\mathbf H+\mathbf M)=\mu_0(1+\chi_m)\mathbf H$. These weak responses generally disappear when the applied field is removed, unlike hysteretic <ferromagnetism>.
A paramagnet can contain permanent <magnetic moments> from unpaired <Electrons> or atomic <angular momentum>. Alignment lowers the <energy> $-\boldsymbol\mu\cdot\mathbf B$, while thermal disorder opposes it. For independent classical moments of magnitude $\mu$, their weak-field average projection is $\mu^2B/(3k_BT)$, so
$$
\chi_m\simeq\frac{\mu_0n\mu^2}{3k_BT}.
$$
This is <Curie's law>; it assumes weak <magnetization> and no important interactions between moments. With quantum <angular momentum> $J$, replace $\mu^2$ by $(g\mu_B)^2J(J+1)$. Strong-field <magnetization> saturates instead of growing linearly without limit. Conduction <Electrons> also exhibit <Pauli paramagnetism>: only occupations near the <Fermi surface> readily redistribute between <spin> states, giving a nearly temperature-independent susceptibility when $T\ll T_F$.
<Diamagnetism> is an induced orbital response opposing the applied <magnetic field>. For bound <Electrons>, a weak field changes orbital motion and induces an opposing <magnetic moment>. For an isotropic collection of atoms, the elementary orbital estimate is
$$
\chi_{\rm dia}=-\frac{\mu_0ne^2}{6m_e}\sum_j\langle r_j^2\rangle,
$$
where the sum runs over the <Electrons> of one atom. It is usually small and weakly <temperature> dependent. Delocalized <Electrons> also have orbital Landau <diamagnetism>; for an ideal three-dimensional free <Electron> <gas> with a quadratic isotropic dispersion and <electron spin> with $g=2$, $\chi_L=-\chi_P/3$. Both <spin> and orbital contributions must be included to determine the total response. <Diamagnetism> is not a consequence of permanent dipoles choosing to align against the field; that orientation would increase their <energy>.
Back to article page
