Hyperfine interaction

An interaction between a nucleus's magnetic or electric multipole moments and its electronic surroundings. Magnetic contact and dipolar terms can couple nuclear spin to electron spin or orbital magnetic moments. The resulting small energy shifts contribute to hyperfine structure; the contact term can also mediate conduction-electron-mediated exchange between nuclear moments in a metal.

Coupling of the nuclear electric-quadrupole moment to the electronic electric-field gradient. Its first-order splitting requires appropriate nonzero nuclear and electronic tensor moments; the usual I and J must each be at least one. It can produce deviations from a pure magnetic-dipole Landé interval rule.

The interaction of a nuclear magnetic moment with the electronic magnetic field at the nucleus. Within an isolated electronic J manifold, rotational symmetry gives the displayed effective Hamiltonian for angular momenta in hbar units. Its adjacent F intervals obey the Landé interval rule.

A departure of ratios of magnetic-dipole hyperfine structure constants from ratios of nuclear gyromagnetic ratios. Finite spatial distributions of nuclear charge and magnetization alter electronic contact fields, so even the same nominal electronic energy level does not provide an exactly isotope-independent proportionality constant.

The isotropic magnetic interaction of a nuclear magnetic dipole moment with electron spin density at the atomic nucleus. A nonrelativistic penetrating $s$ orbital can have $|{\psi }_s(0){|}^2\ne 0$ and a substantial contact contribution to hyperfine structure. A filled paired shell cancels its net contact spin density in the independent-particle approximation; unpaired occupation and core polarization can restore it. Relativistic and finite nuclear-size effects modify the point-contact description.

For electronic $J=1/2$, nuclear spin $I$ and conserved angular momentum component $m$, define ${\Delta }_{\mathrm{hf}}=A(I+1/2)$ and the dimensionless field $x$ shown in the title. The hyperfine-plus-magnetic energies areThe factors $g_I,g_J$ use the same Hamiltonian convention and Bohr-magneton units. The two stretched states require the branch consistent with their unique spin configuration. The formula follows by diagonalizing each two-dimensional fixed-$m$ block; it connects weak-field coupled-spin levels to decoupled high-field states. The name refers to Gregory Breit and Isidor Isaac Rabi.^[ref]

For electronic $J=1/2$, an extremal total projection has only one product state rather than a two-dimensional spin block. With dimensionless angular-momentum operators and nuclear Zeeman coefficient $g_I$ in Bohr-magneton units, its energy is $AI/2\pm {\mu }_{B}B(g_J/2+g_{I}I)$. Extending both signs of a two-state Breit-Rabi formula square root to each stretched projection would invent an extra state.