= Curie's law
{c}
{title2=$\chi=C/T$}
{wiki}
= Curie law
{c}
{synonym}
<Curie's law> is the inverse-temperature magnetic response of independent <magnetic moments> in the weak-polarization regime. Let each moment have <angular momentum> quantum number $J$, <Landé g-factor> $g_J$, and magnetic projection $g_J\mu_Bm$, where $m=-J,\ldots,J$. Its <Zeeman energy> is $-g_J\mu_BBm$. At zero field the projections are equally likely, so $\langle m\rangle=0$ and $\langle m^2\rangle=J(J+1)/3$. Differentiating their <canonical partition function> gives the total-moment response of $N$ independent moments,
$$
\left.\frac{\partial m_{\rm tot}}{\partial B}\right|_{B=0}=\frac{N(g_J\mu_B)^2J(J+1)}{3k_BT}.
$$
For a dilute weakly magnetized sample with <number density> $n=N/V$ and $B\simeq\mu_0H$, the dimensionless SI <volume magnetic susceptibility> is instead
$$
\chi_v=\frac{\partial M}{\partial H}\simeq\frac{\mu_0n(g_J\mu_B)^2J(J+1)}{3k_BT},\qquad M=m_{\rm tot}/V.
$$
These conventions differ by volume, field and $g_J$ factors. A fixed nonzero field eventually saturates the moments as <temperature> falls, so the linear <Curie's law> regime requires $g_J\mu_B|B|J\ll k_BT$.<Statistical Physics by David Tong/3 Quantum Gases>[magnetic response and Curie's law; Tong defines susceptibility using total moment differentiated with respect to B]{ref}
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