## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

## Curie's law

For low levels of magnetization, the magnetization of paramagnets follows what is known as Curie's law, at least approximately. This law indicates that the susceptibility χ of paramagnetic materials is inversely proportional to their temperature, i.e. that materials become more magnetic at lower temperatures. The mathematical expression is:

$oldsymbol{M} = chioldsymbol{H} = frac{C}{T}oldsymbol{H}$

where:

M is the resulting magnetization
χ is the magnetic susceptibility
H is the auxiliary magnetic field, measured in amperes/meter
T is absolute temperature, measured in kelvins
C is a material-specific Curie constant

Curie's law is valid under the commonly encountered conditions of low magnetization (μBH ≲ kBT), but does not apply in the high-field/low-temperature regime where saturation of magnetization occurs (μBH ≳ kBT) and magnetic dipoles are all aligned with the applied field. When the dipoles are aligned, increasing the external field will not increase the total magnetization since there can be no further alignment.

For a paramagnetic ion with noninteracting magnetic moments with angular momentum J, the Curie constant is related the individual ions' magnetic moments,

$C=frac{N_{A}}{3k_{B}}mu_{mathrm{eff}}^{2} ext{ where }mu_{mathrm{eff}} = g_{J}mu_{B}sqrt{J(J+1)}$.

The parameter μeff is interpreted as the effective magnetic moment per paramagnetic ion. If one uses a classical treatment with molecular magnetic moments represented as discrete magnetic dipoles, μ, a Curie Law expression of the same form will emerge with μ appearing in place of μeff.

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