Curie's law

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In a paramagnetic material Curie's law relates the magnetization of the material to the applied magnetic field and temperature.

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

$\mathbf{M}$ is the resulting magnetisation

$\mathbf{B}$ is the magnetic flux density of the applied field, measured in teslas

T

is absolute temperature, measured in kelvins

C

is a material-specific Curie constant

This relation was discovered experimentally (by fitting the results to a correctly guessed model) by Pierre Curie.

[edit] Derivation (Statistical Mechanics)

Magnetization of a paramagnet as a function of inverse temperature.

A simple model of a paramagnet concentrates on the particles which compose it, call them paramagnetons. Assume that each paramagneton has a magnetic moment given by $\vec{\mu}$ . Energy of a magnetic moment in a magnetic field is given by

Failed to parse (unknown error\c): E=-\vec{\mu}\c\vec{B}

To simplify the calculation, we are going to work with a 2-state paramagnet, that is, the particle can either align its magnetic moment with the magnetic field, or against it. No other orientations are possible. If so, then such particle has only two possible energies

E 0 = μ B

and

E 1 = - μ B

With this information we can construct the partition function of one paramagneton

$Z = \sum_{n=0}^{\infty} e^{-E_n\beta} = e^{-\mu B\beta} + e^{\mu B\beta} = 2 \cosh\left(\mu B\beta\right)$

When one seeks the magnetization of a paramagnet, one is interested in the likelihood of a paramagneton to align itself with the field. In other words, one seeks the expectation value of orientation $μ$ .

$\left\langle\mu\right\rangle = \sum_{n=0}^{\infty} \mu_n P\left(\mu_n\right) = \sum_{n=0}^{\infty} \mu_n {e^{-\mu_n B\beta}\over Z} = {1\over Z}\sum_{n=0}^{\infty}{\partial_{\beta}e^{-\mu_n B\beta}\over B} = {1\over B}{1\over Z} \partial_{\beta} Z$

$\left\langle\mu\right\rangle = {1\over 2 B \cosh\left(\mu B\beta\right)} 2 \mu B \sinh\left(\mu B\beta\right) = \mu \tanh\left(\mu B\beta\right)$

This is magnetization of one paramagneton, total magnetization of the solid is given by

$M = N\left\langle\mu\right\rangle = N \mu \tanh\left({\mu B\over k T}\right)$

The formula above is known as the Langevin Paramagnetic equation. Pierre Curie found an approximation to this law that applies to the reasonably high temperatures and low magnetic fields used in his experiments. Let's see what happens to the magnetization as we specialize it to large $T$ and small $B$ . As temperature increases and magnetic field decreases, the argument of hyperbolic tangent decreases. Another way to say this is

$\left({\mu B\over k T}\right) << 1$