Curie's law

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

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 ec{mu}. Energy of a magnetic moment in a magnetic field is given by

Related Topics:
Paramagnet - Magnetic moment

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:E=-ec{mu}cdotec{B}

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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

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:E_0 = mu B

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and

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:E_1 =- mu B

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

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:Z = sum_{n=0}^{infty} e^{-E_neta} = e^{-mu Beta} + e^{mu Beta} = 2 coshleft(mu Beta ight)

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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 mu.

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:leftlanglemu ight angle = sum_{n=0}^{infty} mu_n Pleft(mu_n ight) = sum_{n=0}^{infty} mu_n {e^{-mu_n Beta}over Z} = {1over Z}sum_{n=0}^{infty}{partial_{eta}e^{-mu_n Beta}over B} = {1over B}{1over Z} partial_{eta} Z

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:leftlanglemu ight angle = {1over 2 B coshleft(mu Beta ight)} 2 mu B sinhleft(mu Beta ight) = mu anhleft(mu Beta ight)

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

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M = Nleftlanglemu ight angle = N mu anhleft({mu Bover k T} ight)

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Formula above is not quite Curie's Law. The reason Pierre Curie was able to correctly guess the law is because his experiments were done at reasonably high temperatures and low magnetic fields. 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

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:left({mu Bover k T} ight)

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this is sometimes called the Curie regime. We also know that if |x|

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: anh x pprox x

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so

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:mathbf{M}(T ightarrowinfty)={Nmu^2over k}{mathbf{B}over T}

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Q.E.D.

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