In atomic physics, the Landé interval rule [1] 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).
[2][3] 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.
[4] The rule was first stated in 1923 by German-American physicist Alfred Landé.
[1] As an example,[2] 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
Thus, the total orbital angular momentum is
Then the coupling in the LS-scheme gives rise to a Hamiltonian:
encode the strength of the coupling.
acts as a perturbation to the state
The coupling would cause the total orbital
angular momentums to change directions, but the total angular momentum
would also remain constant, since there is no external torque acting on the system.
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
precesses around the total orbital angular momentum
averages to zero over time, and thus only the component along
Applying this change to all the terms in the Hamiltonian, we can rewrite it as
Consequently, the energy interval between adjacent
This is the Landé interval rule.
As for the spin-spin interaction responsible for the hyperfine structure, because the Hamiltonian of the hyperfine interaction can be written as
is the total angular momentum, we also have an interval rule:
The derivation is essentially the same, but with nuclear spin
The interval rule holds when the coupling is weak.
In the LS-coupling scheme, a weak coupling means the energy of spin-orbit coupling
is smaller than residual electrostatic interaction:
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