Paramagnetism

Paramagnetism is a property of matter in which an applied magnetic field produces magnetization in the same direction as the field. A paramagnetic substance is therefore attracted toward regions of stronger magnetic field. Unlike materials exhibiting ferromagnetism, an ordinary paramagnet has no spontaneous magnetization in equilibrium when the field is absent. Its response commonly arises from microscopic magnetic moments associated with unpaired electrons, although conducting electrons provide another important mechanism. (vrchemistry.chem.ox.ac.uk)

Microscopic origin

An electron possesses a magnetic moment associated with its spin angular momentum; orbital motion can also contribute to the moment of an atom. In many atoms, ions, and molecules, paired electrons have opposing spin contributions. An incomplete electron configuration can leave a net moment, with its magnitude depending on the electronic state rather than simply on the total number of electrons. (vrchemistry.chem.ox.ac.uk)

Without an external field, the moments of an ideal collection of independent magnetic particles have no preferred direction. A field changes their energy according to

where $\boldsymbol{\mu}$ is the particle’s magnetic moment and $\mathbf B$ is the magnetic flux density. States with a moment component parallel to the field have lower energy and become more populated. Thermal fluctuations oppose this polarization, so ordinary fields generally produce only partial alignment. This competition is described by statistical mechanics and the Boltzmann distribution. (farside.ph.utexas.edu)

The microscopic description requires quantum mechanics: angular-momentum projections take discrete allowed values. For a localized moment with angular-momentum quantum number $J$, there are $2J+1$ projections. Summing their equilibrium populations gives a field-dependent magnetization that approaches saturation when magnetic energy dominates thermal energy. (farside.ph.utexas.edu)

Susceptibility and temperature dependence

In a weak field, an isotropic paramagnet approximately obeys

where $\mathbf M$ is magnetic moment per unit volume, $\mathbf H$ is the internal magnetic field strength, and $\chi$ is the volume magnetic susceptibility. In the International System of Units, this susceptibility is dimensionless. A paramagnetic contribution has positive $\chi$, whereas diamagnetism contributes a negative response. Measured susceptibility can contain both contributions, so identifying a microscopic paramagnetic mechanism does not automatically establish the sign of the total response. (www-thphys.physics.ox.ac.uk)

For independent localized moments in the weak-field limit, Curie’s law states

where $T$ is absolute temperature and $C$ is the Curie constant. In SI units, a simple quantum model gives

Here $n$ is the number of magnetic particles per unit volume, $g$ is their effective magnetic factor, $\mu_B$ is the Bohr magneton, $k_B$ is the Boltzmann constant, and $\mu_0$ is the vacuum permeability. Curie’s law is named after Pierre Curie, who established the inverse-temperature relationship experimentally. It fails at sufficiently strong fields or low temperatures, where magnetization becomes nonlinear and eventually saturates. (farside.ph.utexas.edu)

Interactions between moments can instead produce an approximate Curie–Weiss law,

The Weiss temperature $\theta$ reflects interactions within the model; it is not universally identical to an actual ordering temperature. Interpreting susceptibility therefore requires attention to the applicable temperature range, background contributions, and the assumptions behind a fit. (arxiv.org)

Localized and conducting-electron responses

Localized-moment paramagnetism occurs when magnetic electrons remain associated primarily with particular atoms, ions, or molecules. Its characteristic inverse-temperature susceptibility differs from Pauli paramagnetism, which arises from the spin polarization of mobile electrons in a metal. For an ideal conducting-electron system at temperatures small compared with its characteristic electronic energy scale, the Pauli contribution is approximately temperature-independent rather than proportional to $1/T$. (tcd.ie)

The distinction matters because the magnetic behavior of a bulk solid cannot generally be inferred from isolated-atom electron configurations. Electronic bands, orbital diamagnetism, and interactions alter the overall response. A material may contain a positive conducting-electron contribution while nevertheless having negative total susceptibility, as illustrated by copper. (www-thphys.physics.ox.ac.uk)

Paramagnetism can also describe a high-temperature phase of a material that orders magnetically upon cooling. Above its Curie temperature, a ferromagnet loses spontaneous magnetization, although interactions between its moments remain relevant. Such a paramagnetic phase is not equivalent to a collection of completely noninteracting spins. (farside.ph.utexas.edu)

Examples and experimental uses

Molecular oxygen is a familiar example. Its ground-state electronic structure contains two unpaired electrons in antibonding orbitals, as described by molecular orbital theory. Liquid oxygen can consequently remain suspended between the poles of a sufficiently strong magnet, whereas liquid nitrogen does not show the same attraction. (chem-textbook.ucalgary.ca)

Magnetic measurements can reveal information about unpaired electrons. Where orbital contributions are small, the spin-only effective moment is approximately

with $u$ the number of unpaired electrons. This relationship connects measured magnetic moments with electronic structure, but its assumptions limit its applicability. (vrchemistry.chem.ox.ac.uk)

Electron paramagnetic resonance uses a static magnetic field and resonant microwave radiation to investigate paramagnetic species. This form of spectroscopy provides information about spin environments, molecular structure, dynamics, and interactions. (chem.ox.ac.uk)

Paramagnetic salts also serve as refrigerants in adiabatic demagnetization. After magnetization and removal of the associated heat, reducing the field under thermally isolated conditions lowers the temperature. The method exploits the relationship between magnetic polarization, entropy, and temperature to reach very low temperatures. (nvlpubs.nist.gov)