# Curie constant

The Curie constant is a material-dependent property that relates a material's magnetic susceptibility to its temperature.

The Curie constant, when expressed in SI units, is given by

${\displaystyle C={\frac {\mu _{0}\mu _{B}^{2}}{3k_{B}}}ng^{2}J(J+1)}$[1]

where ${\displaystyle n}$ is the number of magnetic atoms (or molecules) per unit volume, ${\displaystyle g}$ is the Landé g-factor, ${\displaystyle \mu _{B}}$ is the Bohr magneton, ${\displaystyle J}$ is the angular momentum quantum number and ${\displaystyle k_{B}}$ is Boltzmann's constant. For a two-level system with magnetic moment ${\displaystyle \mu }$, the formula reduces to

${\displaystyle C={\frac {1}{k_{B}}}n\mu _{0}\mu ^{2}}$

while the corresponding expressions in Gaussian units are

${\displaystyle C={\frac {\mu _{B}^{2}}{3k_{B}}}ng^{2}J(J+1)}$
${\displaystyle C={\frac {1}{k_{B}}}n\mu ^{2}}$

The constant is used in Curie's Law, which states that for a fixed value of a magnetic field, the magnetization of a material is (approximately) inversely proportional to temperature.

${\displaystyle \mathbf {M} ={\frac {C}{T}}\mathbf {B} }$

This equation was first derived by Pierre Curie.

Because of the relationship between magnetic susceptibility ${\displaystyle \chi }$, magnetization ${\displaystyle \scriptstyle \mathbf {M} }$ and applied magnetic field ${\displaystyle \scriptstyle \mathbf {H} }$ is almost linear at low fields, then

${\displaystyle \chi ={\frac {\mathrm {d} \mathbf {M} }{\mathrm {d} \mathbf {H} }}\approx {\frac {\mathbf {M} }{\mathbf {H} }}}$,

this shows that for a paramagnetic system of non-interacting magnetic moments, magnetization ${\displaystyle \scriptstyle \mathbf {M} }$ is inversely related to temperature ${\displaystyle T}$ (see Curie's Law).