= Solution
{c}
Let the dimensionful <angular momentum operators> obey $\mathbf F=\mathbf I+\mathbf J$ and take the interaction <Hamiltonian> as $H'=A\mathbf I\cdot\mathbf J$. Since
$$
\mathbf I\cdot\mathbf J=\tfrac12(\mathbf F^2-\mathbf I^2-\mathbf J^2),
$$
the <energy eigenvalues> at fixed $I,J$ are
$$
E_F=\frac{a}{2}\{F(F+1)-I(I+1)-J(J+1)\},\qquad a=A\hbar^2,
$$
where $F=|I-J|,\ldots,I+J$. Subtraction gives the <Landé interval rule>
$$
\boxed{E_F-E_{F-1}=aF.}
$$
Many atomic conventions use dimensionless <angular momentum operators> and call the energy-valued constant itself $A$; then the right-hand side is $AF$. Each <energy level> has $2F+1$ <magnetic sublevels> with the same <energy> in zero field.
For <fine structure>, the effective interaction is $a_{LS}\mathbf L\cdot\mathbf S$ with dimensionless <angular momentum operators>. It originates in the <spin-orbit interaction>, including the magnetic interaction of <electron spin> with orbital motion and its relativistic correction. Within an isolated <LS coupling> <atomic term>, the labels are the <electron configuration>, $L,S,J,M_J$, and the interval between adjacent $J$ <energy levels> is $a_{LS}J$. This is smaller than the electrostatic separation of <atomic terms>.
For magnetic-dipole <hyperfine structure>, the <nuclear magnetic dipole moment> interacts with the magnetic field generated by the <electrons>. Contributions include the <Fermi contact interaction> of penetrating $s$ <electrons>, orbital magnetic fields, and the dipolar field of <electron spin>. Projected onto one electronic $J$ <energy level>, their rotationally invariant magnetic-dipole part has the form $a_{IJ}\mathbf I\cdot\mathbf J$. The labels are $I,J,F,M_F$, together with the approximate electronic labels such as $L,S$. This splitting is generally smaller than <fine structure>, reflecting the much smaller nuclear <magnetic dipole moment>. An electric <nuclear quadrupole moment> can add a different interaction and violate the simple <Landé interval rule>.
At the larger electrostatic <energy> scale, an effective two-<spin> exchange interaction can also be written $a_s\mathbf s_1\cdot\mathbf s_2$ within fixed <orbitals>. Its origin is the <Coulomb interaction> combined with the <Pauli exclusion principle>, rather than a large direct magnetic interaction. It separates <singlet state> and <triplet state> <atomic terms>; before <spin-orbit coupling>, useful labels include the <electron configuration>, $L,M_L,S,M_S$. General electrostatic <atomic term> separations need not obey a single <Landé interval rule> as $L$ varies.
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