Hyperfine splitting and nuclear spin

Denote nuclear spin by I and total electronic angular momentum by J. In zero field, their coupled total is $\mathbf F=\mathbf I+\mathbf J$. If a single electronic J manifold is isolated, F ranges from $|I-J|$ to $I+J$ in unit steps. Each F level contains $2F+1$ substates. Summing those counts gives $(2I+1)(2J+1)$, ensuring that coupling reorganizes rather than creates states. This counting includes nuclear spin degeneracy absent from an electronic term symbol.

An effective magnetic-dipole interaction is proportional to $\mathbf I\cdot\mathbf J$. The angular factor follows from the same squared-vector identity used for spin-orbit coupling. Its coefficient A contains nuclear magnetic moment and electronic matrix elements. For an s electron, a contact contribution samples electron density at the nucleus; orbital and spin-dipole contributions also occur in general. A nonzero nuclear spin need not imply a large measured splitting, because moment size and electronic structure both matter.

Spectroscopic tables often quote $A/h$ in megahertz rather than A in joules. Then the level formula predicts $E_F/h$ directly in the same frequency unit. A table quoting $A/\hbar$ instead would use angular frequency and differ by a factor of $2\pi$. Keeping an explicit h in a calculation prevents a familiar numerical error. The weighted sum of dipole shifts over all F substates vanishes relative to the electronic center of gravity.

For $J=1/2$, only two F values occur when $I\geq1/2$, and their interval is $A(I+1/2)$. Positive A puts the larger F higher; negative A reverses the ordering. Nuclear spin zero removes magnetic-dipole and electric-quadrupole hyperfine splitting, although isotope mass and charge-radius changes still move electronic energies. A spectral isotope shift and a hyperfine interval therefore encode different physical information and should be reported separately.

A nucleus with spin at least one can have an electric quadrupole moment, which couples to the electronic electric-field gradient. A first-order quadrupole splitting within a J manifold additionally requires $J\geq1$. Thus a spin-one nucleus does not by itself produce a first-order quadrupole pattern in a $J=1/2$ state. Higher-order mixing with other electronic levels can create additional small corrections. Fitting every observed interval with a single dipole constant can conceal these effects.

An external magnetic field competes with hyperfine coupling. When the electronic Zeeman energy is much smaller than the hyperfine interval, F remains an approximate good quantum number. At larger fields the nucleus and electronic angular momentum progressively decouple, and a matrix treatment or an appropriate strong-field expansion is required. Optical electric-dipole transitions usually conserve nuclear spin and obey total-F triangle constraints; the ground-state hydrogen hyperfine radio transition instead proceeds magnetically. A Bloch sphere cannot reproduce these multilevel couplings or their transition rates.