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Disambiguated: clustering → cluster analysis, classification → classification in machine learning

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In a [[paramagnetic]] material the [[magnetization]] of the material is (approximately) directly proportional to an applied [[magnetic field]]. However, if the material is heated, this proportionality is reduced: for a fixed value of the field, the magnetization is (approximately) inversely proportional to [[temperature]]. This fact is encapsulated by '''Curie's law''':

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

where
:<math>\mathbf{M}</math> is the resulting magnetisation
:<math>\mathbf{B}</math> is the magnetic field, measured in [[tesla (unit)|teslas]]
:<math>T</math> is absolute temperature, measured in [[kelvin]]s
:<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]]. It only holds for high temperatures, or weak magnetic fields. As the derivations below
show, the magnetization saturates in the opposite limit of low temperatures, or strong fields.

== Derivation with quantum mechanics ==
[[Image:magnetization2.jpg|thumb|'''Magnetization''' of a paramagnet as a function of [[Multiplicative inverse|inverse]] temperature.|right|300px]]

A simple [[Mathematical model|model]] of a [[paramagnet]] concentrates on the particles which compose it which do not interact with each other. Each particle has a [[magnetic moment]] given by <math>\vec{\mu}</math>. The [[energy]] of a [[magnetic moment]] in a magnetic field is given by

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

===Two-state (spin-1/2) particles ===
To simplify the [[calculation]], we are going to work with a '''2-state''' particle: it may either align its magnetic moment with the magnetic field, or against it. So the only possible values of magnetic moment are then <math>\mu</math> and <math>-\mu</math>. If so, then such a particle has only two possible energies

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

and

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

When one seeks the magnetization of a paramagnet, one is interested in the likelihood of a particle to align itself with the field. In other words, one seeks the [[expectation value]] of the magnetization <math>\mu</math>:
:<math>\left\langle\mu\right\rangle = \mu P\left(\mu\right) + (-\mu) P\left(-\mu\right)
= {1 \over Z} \left( \mu e^{ \mu B\beta} - \mu e^{ - \mu B\beta} \right)
= {2\mu \over Z} \sinh( \mu B\beta), </math>
where the [[probability]] of a configuration is given by its [[Boltzmann factor]], and
the [[Partition function (statistical mechanics)|partition function]] <math>Z</math> provides the necessary [[Normalizing_constant|normalization]] for probabilities (so that the [[sum]] of all of them is unity.)
The partition function of one particle is:
:<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>

Therefore, in this simple case we have:
:<math>\left\langle\mu\right\rangle = \mu \tanh\left(\mu B\beta\right).</math>

This is magnetization of one particle, the 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]] which applies to the relatively high temperatures and low magnetic fields used in his [[experiment]]s. 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) \ll 1</math>

this is sometimes called the '''Curie regime'''. We also know that if <math>|x| \ll 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>

with a [[Curie constant]] given by <math>C= N\mu^2/k</math>. Also, in the opposite regime
of low temperatures or high fields, <math>M</math> tends to a maximum value of <math>N\mu</math>,
corresponding to all the particles being completely aligned with the field.

=== General case ===

When the particles have an arbitrary spin (any number of spin states), the formula is a bit more complicated.
At low magnetic fields or high temperature, the spin follows Curie's law, with
:<math>C = \frac{\mu_B^2}{3 k_B}N g^2 J(J+1)</math><ref>{{cite book | last = Kittel | first = Charles | title = Introduction to Solid State Physics, 8th Edition | publisher = Wiley | pages = 304 | isbn = 0-471-41526-X}}</ref>
where <math>J</math> is the [[total angular momentum quantum number]] and <math>g</math> is the spin's g-factor (such that <math>\mu = g J \mu_B</math> is the magnetic moment).

For this more general formula and its derivation (including high field, low temperature) see the article: [[Brillouin function]].
As the spin approaches infinity, the formula for the magnetization approaches the classical value derived in the following section.

== Derivation with classical statistical mechanics ==

An alternative treatment applies when the paramagnetons are imagined to be classical, freely-rotating magnetic moments. In this case, their [[position (vector)|position]] will be determined by their [[angles]] in [[spherical coordinates]], and the energy for one of them will be:

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

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

We see there is no dependence on the <math>\phi</math> angle, and also we can
change variables to <math>y=\cos\theta</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 (due to integration over <math>\phi</math>), as they should) will be given by

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

To simplify the calculation, we see this can be written as a differentiation of <math>Z</math>:

:<math>\left\langle\mu_z\right\rangle = {1 \over Z B} \partial_\beta Z.</math>
(This approach can also be used for the model above, but the calculation was so simple this
is not so helpful.)

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 would appear to be singular for small <math>x</math>, but it is not,
since the two singular terms cancel each other. In fact, its behavior for small arguments is
<math>L(x) \approx x/3</math>, so the Curie limit also applies, but with a Curie constant
three times smaller in this case. Similarly, the function saturates at <math>1</math> for large values of its argument, and the opposite limit is likewise recovered.

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

==See also==
*[[Curie-Weiss law]]

==References==
{{reflist}}


[[Category:Electric and magnetic fields in matter]]

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96.58.130.47: /* Bibliography */

en>Pichpich: Disambiguated: clustering → cluster analysis, classification → classification in machine learning