The Landé interval rule is an empirical relationship in atomic spectroscopy that describes the proportionality between the energy separations of adjacent fine‑structure levels within a multiplet and the total angular momentum quantum number $J$. Formulated by the German physicist Alfred Landé in 1923, the rule states that the energy difference $\Delta E$ between two successive levels with quantum numbers $J$ and $J-1$ in a given term is proportional to $J$:
$$ \Delta E_{J,J-1} \propto J . $$
Consequently, the spacing of the fine‑structure components increases linearly with $J$. In practice, the rule is often expressed as
$$ E(J) - E(J-1) = A,J, $$
where $A$ is a constant characteristic of the particular electronic configuration and term.
Historical Context
Alfred Landé introduced the rule while analyzing the fine‑structure splittings observed in the spectra of alkali and alkaline‑earth elements. The rule provided a simple quantitative guideline that complemented the more detailed theoretical treatments based on spin–orbit coupling and the quantum mechanical coupling schemes (LS coupling, jj coupling).
Theoretical Basis
The Landé interval rule emerges naturally from the LS coupling approximation, in which the total Hamiltonian for spin–orbit interaction can be written as
$$ H_{\text{SO}} = \zeta , \mathbf{L}\cdot\mathbf{S}, $$
with $\zeta$ the spin‑orbit coupling constant, $\mathbf{L}$ the total orbital angular momentum, and $\mathbf{S}$ the total spin angular momentum. The eigenvalues of $\mathbf{L}\cdot\mathbf{S}$ are
$$ \frac{1}{2}\left[J(J+1)-L(L+1)-S(S+1)\right], $$
which yields energy separations that are linear in $J$ for a given term ${}^{2S+1}L$.
Applications
- Spectroscopic Identification: The rule assists in assigning observed spectral lines to specific fine‑structure transitions, especially when multiple multiplets are present.
- Atomic Structure Calculations: It serves as a benchmark for testing the accuracy of theoretical models that compute spin‑orbit splittings.
- Educational Context: The Landé interval rule is commonly taught in undergraduate quantum mechanics and atomic physics courses as an illustration of angular momentum coupling.
Limitations
The rule holds accurately when LS coupling is a good approximation, i.e., for light atoms where spin‑orbit interaction is relatively weak compared to electrostatic interactions. In heavier atoms, where jj coupling or intermediate coupling dominates, deviations from the linear $J$ dependence become significant, and the rule must be applied with caution.
Related Concepts
- Landé g‑factor: A related quantity, also introduced by Landé, that gives the magnetic moment of an atomic level in terms of $L$, $S$, and $J$.
- Fine Structure: The splitting of atomic energy levels due to relativistic corrections and spin‑orbit coupling, of which the Landé interval rule describes the pattern of level spacings within a multiplet.
- Russell–Saunders (LS) Coupling: The coupling scheme under which the Landé interval rule is derived.