Fermi contact interaction

Not to be confused with Fermi's interaction .

The Fermi contact interaction is the magnetic interaction between an electron and an atomic nucleus when the electron is inside that nucleus. It is of magnitude

$U = - \frac{8}{3} \pi \left \langle \boldsymbol{\mu}_n \cdot \boldsymbol{\mu}_e \right \rangle |\Psi (0)|^2\qquad \mbox{(c.g.i)}$

and

$U = - \frac{2}{3} \mu_0 \left \langle \boldsymbol{\mu}_n \cdot \boldsymbol{\mu}_e \right \rangle |\Psi(0)|^2 \qquad \mbox{(S.I.)}$

where U is the energy of the interaction, μ_n is the nuclear magnetic moment, μ_e is the electron magnetic dipole moment, and Ψ(0) is the electron wavefunction at the nucleus.

Within an atom, only s-orbitals have non-zero electron density at the nucleus, so the contact interaction can only occur for s-electrons. Its major manifestation is in electron paramagnetic resonance and nuclear magnetic resonance spectroscopies, where it is responsible for the appearance of isotropic hyperfine coupling.

The interaction was first derived by Enrico Fermi in 1930. A classical derivation of this term is contained in J.D. Jackson. In short the classical energy U may be written in terms of the energy of one magnetic dipole moment in the magnetic field B (r) of another. This field acquires a simple expression when the distance r between the two moments goes to zero, since:

$\int_{S(r)} B(r) d^3r = -\frac 2 3 \mu_0 \boldsymbol{\mu}$