When a system of two spins (here and ) whose associated magnetic moments are coupled by the hyperfine interaction is exposed to a magnetic field , the energies of the magnetic sublevels depend in a nonlinear manner on The diagram representing the magnetic field dependence of the sublevel energies is known as a Breit-Rabi diagram. It is shown here in a universal form by using dimensionless field and energy scales.
The magnetic moment associated with the total electronic angular momentum of an atom and its nuclear magnetic moment associated with its nuclear spin ( is the Bohr magneton and , are the electronic and nuclear -factors, respectively) are coupled by the hyperfine Hamiltonian . If the system has groups of degenerate energy eigenstates, labeled by the total atomic angular momentum , which takes values . Each group consists of degenerate sublevels, yielding a total of substates, labeled and .
When placed in an external magnetic field the combined hyperfine-Zeeman Hamiltonian reads
and its energy eigenvalues are found by diagonalization. If one of the spins, say , is 1/2, the diagonalization yields an algebraic expression
for the energies of the states , known as the Breit-Rabi formula. is the hyperfine splitting for , that is, the energy difference between the states belonging to the manifolds .
By introducing the dimensionless parameter
the energies can be written in the dimensionless form
which have an explicit dependence (given by and ) on the specific atom considered.
Considering that is in general on the order of a few times of , one can neglect the second term to obtain a universal expression
that is valid for all Zeeman-hyperfine problems in which one of the spins is 1/2. It is the latter equation that is displayed in the present Demonstration.