# Curie constant

In magnetism, the Curie constant is a material-dependent property that relates a material's magnetic susceptibility to its temperature through Curie's law.

The Curie constant when expressed in SI units, is given in kelvins (K),[1] by

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

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

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

while the corresponding expressions in Gaussian units are

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

The constant is used in Curie's law, which states that for a fixed value of an applied magnetic field ${\displaystyle \scriptstyle \mathbf {H} }$, the magnetization of a material is (approximately) inversely proportional to temperature.

${\displaystyle M={\frac {C}{T}}H}$.

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

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

this shows that for a paramagnetic system of non-interacting magnetic moments, magnetization ${\displaystyle \scriptstyle \mathbf {M} }$ is inversely related to temperature ${\displaystyle T}$.

## References

1. ^ Coey, J. M. D.; Coey, J. M. D. (2010-03-25). Magnetism and Magnetic Materials. Cambridge University Press. ISBN 978-0-521-81614-4.
2. ^ Kittel, Charles (11 November 2004). Introduction to Solid State Physics, 8th Edition. Wiley. pp. 304. ISBN 0-471-41526-X.