 Curie's law

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 $g_J{\mu }_{B}m$, where $m=-J,\dots ,J$. Its Zeeman energy is $-g_J{\mu }_{B}Bm$. At zero field the projections are equally likely, so $⟨m⟩=0$ and $⟨m^2⟩=J(J+1)/3$. Differentiating their canonical partition function gives the total-moment response of $N$ independent moments,For a dilute weakly magnetized sample with number density $n=N/V$ and $B\simeq {\mu }_{0}H$, the dimensionless SI volume magnetic susceptibility is insteadThese 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 $g_J{\mu }_B|B|J\ll k_{B}T$.^[ref]

Curie's law, named after the French physicist Pierre Curie, describes the magnetic properties of paramagnetic materials. It states that the magnetization \( M \) of a paramagnetic material is directly proportional to the applied magnetic field \( H \) and inversely proportional to the absolute temperature \( T \).