= Breit-Rabi formula
{c}
{title2=$x=(g_J-g_I)\mu_BB/\Delta_{\rm hf}$}
For electronic $J=1/2$, nuclear <spin> $I$ and conserved <angular momentum> component $m$, define $\Delta_{\rm hf}=A(I+1/2)$ and the dimensionless field $x$ shown in the title. The hyperfine-plus-magnetic energies are
$$
E_\pm(m)=-\frac{\Delta_{\rm hf}}{2(2I+1)}+g_I\mu_BmB
\pm\frac{\Delta_{\rm hf}}2\sqrt{1+\frac{4mx}{2I+1}+x^2}.
$$
The 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>.<Rubidium 87 D Line Data by Daniel A. Steck>[equations (26)–(28) and reference 37]{ref}
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