For dimensionless angular momentum operators and a Hamiltonian at fixed , addition of angular momentum gives . Subtracting neighbouring energy levels proves the displayed rule. For fine structure substitute and total ; for magnetic-dipole hyperfine structure use electronic , nuclear spin and total . Electric-quadrupole interactions and mixing with nearby energy levels can violate the rule.
Consecutive positive gaps obey for an isolated hyperfine multiplet governed by the magnetic-dipole hyperfine interaction. Their common difference estimates , and the first gap divided by that scale identifies the first higher F. The endpoints of the resulting F ladder constrain I and J. Gap magnitudes alone do not fix the sign of A or an absolute nuclear g-factor.
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The Landé interval rule is a principle used in atomic and molecular physics, specifically in the context of spectroscopic transitions and the determination of energy levels. It provides a way to estimate the intervals between energy levels of complex systems, like atoms with multiple electrons, which can be challenging to analyze due to electron-electron interactions and the configuration of electrons. The rule helps predict the positions of spectral lines in atomic spectra by considering the contributions of different electron configurations to the total angular momentum of the system.