# Paramagnetism

Paramagnetism is the tendency of the atomic magnetic dipoles, due to quantum-mechanical spin as well as electron orbital angular momentum, to align with an external magnetic field. Paramagnetic materials attract and repel like normal magnets when subject to a magnetic field. This alignment of the atomic dipoles with the magnetic field tends to strengthen it, and is described by a relative magnetic permeability greater than unity (or, equivalently, a small positive magnetic susceptibility).

Paramagnetism requires that the atoms individually have permanent dipole moments even without an applied field, which typically implies a partially filled electron shell. In pure paramagnetism, these atomic dipoles do not interact with one another and are randomly oriented in the absence of an external field, resulting in zero net moment. If they do interact, they can spontaneously align or anti-align, resulting in ferromagnetism (permanent magnets) or antiferromagnetism, respectively. Paramagnetic behaviour can also be observed in ferromagnetic materials that are above their Curie temperature, and in antiferromagnets above their Neel temperature.

(In atoms with no permanent dipole moment, e.g. for filled electron shells, a weak dipole moment can be induced in a direction anti-parallel to an applied field, an effect called diamagnetism. Paramagnetic materials also exhibit diamagnetism, but the latter effect is typically orders of magnitude weaker.)

Under relatively low magnetic field saturation when the majority of the atomic dipoles are not aligned with the field, paramagnetic materials exhibit magnetisation according to Curie's Law:

[itex] \mathbf{M} = C \cdot \frac{\mathbf{B}}{T} [itex]

where

M is the resulting magnetisation
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 law indicates that paramagnetic materials tend to become increasingly magnetic as the applied magnetic field is increased, but less magnetic as temperature is increased. Curie's law is incomplete because it fails to predict the saturation that occurs when most of the atomic dipoles are aligned (after everything is aligned, increasing the external field will not increase the total magnetization). So the Curie constant, really should be expressed as a function of how much of the material is already aligned.

Paramagnetic materials in magnetic fields will act like magnets but when the field is removed, thermal motion will quickly disrupt the magnetic alignment. In general paramagnetic effects are small (magnetic susceptibility of the order of 10-3 to 10-5).

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## References

• Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).
• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• John David Jackson, Classical Electrodynamics (Wiley: New York, 1999).

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