Landé interval rule
In atomic physics, the Landé interval rule states that, due to weak angular momentum coupling (either spin-orbit or spin-spin coupling), the energy splitting between successive sub-levels are proportional to the total angular momentum quantum number (J or F) of the sub-level with the larger of their total angular momentum value (J or F). Background. The rule assumes the Russell–Saunders coupling and that interactions between spin magnetic moments can be ignored. The latter is an incorrect assumption for light atoms. As a result of this, the rule is optimally followed by atoms with  medium atomic numbers. The rule was first stated in 1923 by German-American physicist Alfred Landé. Derivation. As an example, consider an atom with two valence electrons and their fine structures in the LS-coupling scheme. We will derive heuristically the interval rule for the LS-coupling scheme and will remark on the similarity that leads to the interval rule for the hyperfine structure. The interactions between electrons couple their orbital and spin angular momentums. Let's denote the spin and orbital angular momentum as formula_1 and formula_2 for each electrons. Thus, the total orbital angular momentum is formula_3 and  total spin momentum is formula_4. Then the coupling in the LS-scheme gives rise to a Hamiltonian: formula_5 where formula_6 and formula_7 encode the strength of the coupling. The Hamiltonian formula_8 acts as a perturbation to the state formula_9 . The coupling would cause the total orbital formula_10 and spin formula_11 angular momentums to change directions, but the total angular momentum formula_12 would remain constant. Its z-component formula_13 would also remain constant, since there is no external torque acting on the system. Therefore, we shall change the state to formula_14, which is a linear combination of various formula_15. The exact  linear combination, however, is unnecessary to determine the energy shift. To study this perturbation, we consider the vector model where we treat each formula_2 as a vector. formula_17 and formula_18 precesses around the total orbital angular momentum formula_10 . Consequently, the component perpendicular to formula_10 averages to zero over time, and thus only the component along formula_10 needs to be considered. That is, formula_22 . We replace formula_23 by formula_24 and formula_25 by the expectation value formula_26. Applying this change to all the terms in the Hamiltonian, we can rewrite it as formula_27 The energy shift is then formula_28 Now we can  apply the substitution formula_29 to write the energy as formula_30 Consequently, the energy interval between adjacent formula_31 sub-levels is: formula_32 This is the Landé interval rule. As an example, consider a formula_33 term, which has 3 sub-levels formula_34 . The separation between formula_35 and formula_36 is formula_37 , twice as the separation between formula_36 and formula_39 is formula_40 . As for the spin-spin interaction responsible for the hyperfine structure, because the Hamiltonian of the hyperfine interaction can be written as formula_41 where formula_42 is the nuclear spin and formula_43 is the total angular momentum, we also have an interval rule : formula_44 where formula_45 is the total angular momentum formula_46 . The derivation is essentially the same, but with nuclear spin formula_42 , angular momentum formula_43 and total angular momentum formula_49 . Limitations. The interval rule holds when the coupling is weak. In the LS-coupling scheme, a weak coupling means the energy of spin-orbit coupling formula_50 is smaller than residual electrostatic interaction: formula_51. Here the residual electrostatic interaction refers to the term including electron-electron interaction after we employ the central field approximation to the Hamiltonian of the atom. For the hyperfine structure, the interval rule for two magnetic moments can be  disrupted by magnetic quadruple interaction between them, so we want formula_52. For example, in helium, the spin-spin interactions and spin-other-orbit interaction have an energy comparable to that of the spin-orbit interaction.