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Concept
Curie's law
Summary
For many paramagnetic materials, the magnetization of the material is directly proportional to an applied magnetic field, for sufficiently high temperatures and small fields. However, if the material is heated, this proportionality is reduced. For a fixed value of the field, the magnetic susceptibility is inversely proportional to temperature, that is
where
is the (volume) magnetic susceptibility,
is the magnitude of the resulting magnetization (A/m),
is the magnitude of the applied magnetic field (A/m),
is absolute temperature (K),
is a material-specific Curie constant (K).
Pierre Curie discovered this relation, now known as Curie's law, by fitting data from experiment. It only holds for high temperatures and weak magnetic fields. As the derivations below show, the magnetization saturates in the opposite limit of low temperatures and strong fields. If the Curie constant is null, other magnetic effects dominate, like Langevin diamagnetism or Van Vleck paramagnetism.
A simple 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 . The energy of a magnetic moment in a magnetic field is given by
where is the magnetic field density, measured in teslas (T).
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 and . If so, then such a particle has only two possible energies, when it is aligned with the field and when it is oriented opposite to the field.
The extent to which the magnetic moments are aligned with the field can be calculated from the partition function. For a single particle, this is
The partition function for a set of N such particles, if they do not interact with each other, is
and the free energy is therefore
The magnetization is the negative derivative of the free energy with respect to the applied field, and so the magnetization per unit volume is
where n is the number density of magnetic moments.
Official source
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