Hyperfine structure

Hyperfine structure is the small splitting of atomic or molecular energy levels caused by interactions involving the atomic nucleus's magnetic dipole moment or electric quadrupole moment. It is finer than the electronic fine structure. A leading magnetic-dipole hyperfine interaction couples the nuclear spin $I$ to the electronic angular momentum $J$; their total angular momentum $F$ labels the resulting hyperfine levels.

With dimensionless angular momentum operators, the model $H_{\mathrm{hf}}=A\mathbf{I}\cdot \mathbf{J}$ gives shifts $A[F(F+1)-I(I+1)-J(J+1)]/2$. A hydrogen atom in its electronic ground state has $I=J=1/2$, giving $F=0,1$. Resolved microwave transitions between hyperfine levels permit precise spectroscopy and provide the working transitions in many atomic clocks. Applied magnetic fields can split each hyperfine level further into magnetic sublevels.

The set of hyperfine levels for fixed nuclear spin I and electronic J. Its total angular momenta run from F=|I-J| to F=I+J; each zero-field F level contains 2F+1 magnetic substates. The multiplet is the collection, rather than a synonym for one member level.

An energy level resolved by coupling the nuclear spin $I$ to electronic total angular momentum $J$. For a rotationally invariant Hamiltonian, a fixed $F$ has $2F+1$ degenerate magnetic sublevels. Spectral component counts identify the number of upper hyperfine levels only when lower hyperfine structure, selection rules, unresolved overlaps and missing lines have been accounted for.

A qubit encoded in two resolved hyperfine levels of the same electronic ground state. Such states avoid optical excited-state decay and can support coherent radiofrequency or microwave control through a magnetic dipole transition. Technical magnetic noise and differential shifts can still dephase them; first-order field insensitivity requires additional choices of states and operating point.

A transition whose frequency is stationary to first order in a specified perturbation at an operating point. For magnetic sensitivity, the derivative shown vanishes, but second and higher derivatives can remain nonzero. A clock transition need not have zero individual energy slopes: equal slopes suffice. Optical light shifts and other perturbations require their own sensitivity analysis.

For the two $m=0$ branches of the Breit-Rabi formula, their splitting is $\sqrt{{\Delta }_{\mathrm{hf}}^2+[(g_J-g_I){\mu }_{B}B{]}^2}$. Expanding at small field gives the quadratic frequency shift displayed above. Vanishing linear frequency sensitivity at zero field therefore does not eliminate field-gradient dephasing at a finite bias field.

The ground-hyperfine magnetic-dipole transition between $F=1,m_F=0$ and $F=2,m_F=0$ has vanishing first-order Zeeman shift near zero field. It obeys $\Delta F=1,\Delta m_F=0$. Its residual quadratic shift follows from the Breit-Rabi formula, rather than from assigning both states permanently zero magnetic dipole moment.

A qubit encoded in the two $m_F=0$ ground state hyperfine levels of rubidium-87. The electronic $J=1/2$ and nuclear spin $I=3/2$ combine to $F=1,2$. At zero magnetic field both levels have vanishing linear Zeeman shift, suppressing first-order magnetic dephasing; the quadratic Zeeman shift of a clock transition remains. Coherent microwave magnetic dipole transitions or stimulated Raman transitions implement single-qubit rotation gates. These are coupled electron-nucleus states, not bare nuclear-spin states.

For electron spin $s=1/2$, first add orbital and spin angular momenta to obtain $j=L\pm 1/2$. For nuclear spin $I=1$, add again to obtain $F=|j-1|,|j-1|+1,\dots ,j+1$. The projection has $2F+1$ possible values. A hydrogenic bound state still requires $L\le n-1$.

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.