Editing Paramagnetism (section)

=== Pauli paramagnetism===
For some alkali metals and noble metals, conduction electrons are weakly interacting and delocalized in space forming a [[Fermi gas]]. For these materials one contribution to the magnetic response  comes from the interaction between the electron spins and the magnetic field known as Pauli paramagnetism. For a small magnetic field <math>\mathbf{H}</math>, the additional energy per electron from the interaction between an electron spin and the magnetic field is given by:

:<math>\Delta E= -\mu_0\mathbf{H}\cdot\boldsymbol{\mu}_e=- \mu_0\mathbf{H}\cdot\left(-g_e\frac{\mu_\mathrm{B}}{\hbar}\mathbf{S}\right)=\pm \mu_0 \mu_\mathrm{B} H,</math>

where <math>\mu_0</math> is the [[vacuum permeability]], <math>\boldsymbol{\mu}_e</math> is the [[electron magnetic moment]], <math>\mu_{\rm B}</math> is the [[Bohr magneton]], <math>\hbar</math> is the reduced Planck constant, and the [[g-factor (physics)|g-factor]] cancels with the spin <math>\mathbf{S}=\pm\hbar/2</math>. The <math>\pm</math> indicates that the sign is positive (negative) when the electron spin component in the direction of <math>\mathbf{H}</math> is parallel (antiparallel) to the magnetic field.
[[File:Pauli_2bis.jpg|thumb|right|In a metal, the application of an external magnetic field increases the density of electrons with spins antiparallel with the field and lowers the density of the electrons with opposite spin. Note: The arrows in this picture indicate spin direction, not magnetic moment.]]

For low temperatures with respect to the [[Fermi energy|Fermi temperature]] <math>T_{\rm F}</math> (around 10<sup>4</sup> [[kelvin]]s for metals), the [[number density]] of electrons <math>n_{\uparrow}</math> (<math>n_{\downarrow}</math>) pointing parallel (antiparallel) to the magnetic field can be written as:

:<math>n_{\uparrow}\approx\frac{n_e}{2}-\frac{\mu_0\mu_\mathrm{B}}{2}g(E_\mathrm{F})H\quad;\quad \left(n_{\downarrow}\approx\frac{n_e}{2}+\frac{\mu_0\mu_\mathrm{B}}{2}g(E_\mathrm{F})H\right),</math>

with <math>n_e</math> the total free-electrons density and <math>g(E_\mathrm{F})</math> the electronic density of states (number of states per energy per volume) at the [[Fermi energy]] <math>E_\mathrm{F}</math>.

In this approximation the magnetization is given as the magnetic moment of one electron times the difference in densities:

:<math>M=\mu_\mathrm{B}(n_{\downarrow}-n_{\uparrow})=\mu_0\mu_\mathrm{B}^2g(E_\mathrm{F})H,</math>

which yields a positive paramagnetic susceptibility independent of temperature:

:<math>\chi_\mathrm{P}=\mu_0\mu_\mathrm{B}^2g(E_\mathrm{F}).</math>

The Pauli paramagnetic susceptibility is a macroscopic effect and has to be contrasted with [[Landau diamagnetism|Landau diamagnetic susceptibility]] which is equal to minus one third of Pauli's and also comes from delocalized electrons. The Pauli susceptibility comes from the spin interaction with the magnetic field while the Landau susceptibility comes from the spatial motion of the electrons and it is independent of the spin. In doped semiconductors the ratio between Landau's and Pauli's susceptibilities changes as the [[effective mass (solid-state physics)|effective mass]] of the charge carriers <math>m^*</math> can differ from the electron mass <math>m_e</math>.

The magnetic response calculated for a gas of electrons is not the full picture as the magnetic susceptibility coming from the ions has to be included. Additionally, these formulas may break down for confined systems that differ from the bulk, like [[quantum dot]]s, or  for high fields, as demonstrated in the [[De Haas-Van Alphen effect]].

Pauli paramagnetism is named after the physicist [[Wolfgang Pauli]]. Before Pauli's theory, the lack of a strong Curie paramagnetism in metals was an open problem as the [[Drude model|leading Drude model]] could not account for this contribution without the use of [[Fermi–Dirac statistics|quantum statistics]].
Pauli paramagnetism and Landau diamagnetism are essentially applications of the spin and the [[free electron model]], the first is due to intrinsic spin of electrons; the second is due to their orbital motion.<ref>Pauli, Z.Phys. 41, 81, 1927</ref><ref>Landau, Z.Phys. 64, 629, 1930</ref>