Editing Curie's law

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

:<math>\mathbf{M} = C \cdot \frac{\mathbf{B}}{T}</math>

:<math>\mathbf{M}</math> is the resulting magnetisation
:<math>\mathbf{B}</math> is the magnetic flux density of the applied field, measured in [[tesla (unit)|teslas]]
:<math>T</math> is absolute [[temperature]], measured in Kelvin.
:<math>C</math> is a material-specific [[Curie constant]]

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

== Simple Derivation (Statistical Mechanics) ==

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 :<math>\vec{\mu}</math>.  Energy of a magnetic moment in a magnetic field is given by

:<math>E=-\vec{\mu}\cdot\vec{B}</math>

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, <math>\mu</math> and <math>-\mu</math>.  If so, then such particle has only two possible energies

:<math>E_0 = - \mu B</math>

and

:<math>E_1 = \mu B</math>

With this information we can construct the [[Partition function (statistical mechanics)|partition function]] of one paramagneton

:<math>Z = \sum_{n=0,1} e^{-E_n\beta} = e^{ \mu B\beta} + e^{-\mu B\beta} = 2 \cosh\left(\mu B\beta\right)</math>

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 <math>\mu</math>.

:<math>\left\langle\mu\right\rangle =  \mu P\left(\mu_n\right) -  \mu P\left(\mu_n\right) 
 = {1 \over Z} \left( \mu e^{ \mu B\beta} -  \mu e^{  - \mu B\beta} \right), </math>
where the probability of a configuration is given by its Boltzmann factor, and 
the partition function provides the necessary normalization for probabilities
(so that the sum of all of them is unity.) A standard procedure is to express
this as a derivative with respect to <math>\beta</math>
:<math>\left\langle\mu\right\rangle  = {1 \over B Z} \partial_{\beta} ( e^{ \mu B\beta} + e^{  - \mu B\beta} ). </math>

But now what is differenciated is nothing but the partition function again. As
<math>\partial_x \log y= (1/y)  \partial_x y </math>, we can write
:<math>\left\langle\mu\right\rangle  = {1 \over B } \partial_{\beta} \log Z = 
\mu \tanh\left(\mu B\beta\right)</math>

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

<blockquote style="border: 1px solid black; padding:10px;">
<math>M = N\left\langle\mu\right\rangle = N \mu \tanh\left({\mu B\over k T}\right)</math></blockquote>
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 <math>T</math> and small <math>B</math>.  As temperature increases and magnetic field decreases, the argument of hyperbolic tangent decreases.  Another way to say this is

:<math>\left({\mu B\over k T}\right) << 1</math>

this is sometimes called the '''Curie regime'''.  We also know that if <math>|x|<<1</math>, then
:<math>\tanh x \approx x</math>
so
<blockquote style="border: 1px solid black; padding:10px;">
:<math>\mathbf{M}(T\rightarrow\infty)={N\mu^2\over k}{\mathbf{B}\over T}</math></blockquote>

Q.E.D.


== More Involved Derivation (Statistical Mechanics) ==

A more involved treatment applies when the paramagnetons are supposed to rotate
freely. In this case, their position will be determined by their angles in [[spherical coordinates]], and the energy for one of them will be:

:<math>E =  - \mu B\cos\phi, </math>

where <math>\phi</math> is the angle between the magnetic moment and
the magnetic field (which we take to be pointing in the <math>z</math>
coordinate.) The corresponding partition function is

:<math>Z = \int_0^{2\theta} d\theta \int_0^{\pi}d\phi \sin\phi \exp( \mu B\beta \cos\phi).</math>

We seeing there is no dependence on the <math>\theta</math> angle, and we can
change variables to <math>y=\cos\phi</math> to obtain

:<math>Z = 2\pi  \int_{-1}^ 1 d y \exp( \mu B\beta y) =
2\pi{\exp( \mu B\beta )-\exp(-\mu B\beta ) \over \mu B\beta }=
{4\pi\sinh( \mu B\beta ) \over  \mu B\beta .}
</math>

Now, the expected value of the <math>z</math> component of the magnetization (the other two are seen to be null, as the should) will be given by

:<math>\left\langle\mu_z \right\rangle = {1 \over Z} \int_0^{2\theta} d\theta \int_0^{\pi}d\phi \sin\theta \mu\cos\phi \exp( \mu B\beta \cos\phi).</math>

Again, we see this can be written as a differenciation of <math>Z</math>:

:<math>\left\langle\mu_z\right\rangle = {1 \over Z B} \partial_\beta Z.</math>

Carrying out the derivation we find

:<math>\left\langle\mu_z\right\rangle = \mu L(\mu B\beta), </math>

where <math>L</math> is the [[Langevin function]]:

:<math> L(x)= \coth x -{1 \over x}.</math>

This function is not singular for small <math>x</math>, since the two singular
terms cancel each other. In fact, its behavior for small arguments is the
same as the <math>\tanh</math> function, so the limit above also applies in this case.


==Applications==
It is the basis of operation of [[magnetic thermometer]]s, that are used to measure very low temperatures.


This entry contains material taken from  [http://en.wikipedia.org/w/index.php?title=Curie%27s_law&oldid=103844905 this wikipedia entry].
[[Category:Magnetism]]
[[Category:Statistical mechanics]]