= Solution
For two <valence electrons>, the central-field shell structure should be the largest scale, residual electrostatic interactions should establish a reasonably definite configuration and <atomic term>, and <spin-orbit coupling> should be weaker than the separations between the relevant terms. Then $L$ and $S$ remain approximately good quantum numbers. Within a well-isolated term the leading <fine structure> interaction can be represented as $A\mathbf L\cdot\mathbf S$. Its coefficient must be effectively constant for all $J$ levels in that term. Mixing of neighboring terms, <spin-spin interactions>, higher-order corrections or strong applied fields can spoil the simple interval rule; <hyperfine structure> must be negligible or separately resolved.
A suitable approximate example is the <magnesium> $3s3p\,{}^3P_J$ term, with $L=1$, $S=1$ and $J=0,1,2$. The interval-rule prediction is
$$
\boxed{E_0=-2A,\qquad E_1=-A,\qquad E_2=A,\qquad
E_1-E_0=A,\quad E_2-E_1=2A.}
$$
An arbitrary common term <energy> may be added. For $A>0$ the levels ascend in the order ${}^3P_0,{}^3P_1,{}^3P_2$, with gap ratio one to two. Each level still contains $2J+1$ degenerate <magnetic sublevels> in zero external field. The superscript three labels the <spin multiplicity> $2S+1$, while the different subscripts identify <fine structure> components of that same term. This is an approximate example of <LS coupling>, not a claim that every real two-electron atom has exact one-to-two fine splitting.
\Image[/past-exam-of-the-quantum-atomic-and-molecular-physics-course-of-the-university-of-oxford/2016-angular-momentum-interval-ladders.png]
{title=Fine-structure and magnetic-dipole hyperfine interval ladders}
{description=Original angular-momentum eigenvalues for positive coupling coefficients. The fine-structure gaps are A and2A; the two hyperfine examples have different lowest total angular momenta. Measured positive gaps alone do not fix the coupling signs.}
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