# Breit-Rabi Diagram

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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.

Contributed by: Antoine Weis (University of Fribourg) (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

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

with

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.

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