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 and for each electrons. Thus, the total orbital angular momentum is and total spin momentum is . Then the coupling in the LS-scheme gives rise to a Hamiltonian: where and encode the strength of the coupling. The Hamiltonian acts as a perturbation to the state . The coupling would cause the total orbital and spin angular momentums to change directions, but the total angular momentum would remain constant. Its z-component would also remain constant, since there is no external torque acting on the system. Therefore, we shall change the state to , which is a linear combination of various . 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 as a vector. and precesses around the total orbital angular momentum . Consequently, the component perpendicular to averages to zero over time, and thus only the component along needs to be considered. That is, . We replace by and by the expectation value . Applying this change to all the terms in the Hamiltonian, we can rewrite it as The energy shift is then Now we can apply the substitution to write the energy as Consequently, the energy interval between adjacent sub-levels is: This is the Landé interval rule. As an example, consider a term, which has 3 sub-levels . The separation