Electric Dipole (E1) Selection Rules in LS Coupling
The strongest atomic lines correspond to electric dipole (E1) transitions between Russell–Saunders term states \(\;{}^{2S+1}L_J \rightarrow {}^{2S'+1}L'_{J'}\;\). In LS coupling, the dipole operator acts on the spatial (orbital) part of the electronic wavefunction and does not flip electron spins, so only certain changes in quantum numbers are allowed.
- Spin rule: \(\Delta S = 0\) (multiplicity \((2S+1)\) is conserved).
- Orbital angular momentum rule: \(\Delta L = 0,\pm 1\), but \(L=0 \nleftrightarrow L'=0\).
- Total angular momentum rule: \(\Delta J = 0,\pm 1\), but \(J=0 \nleftrightarrow J'=0\).
- Parity rule: the electronic parity must change (even \(\leftrightarrow\) odd). In practice, this commonly corresponds to a one-unit change in orbital type (e.g., \(s \leftrightarrow p\), \(p \leftrightarrow d\)).
Transitions that violate these rules are electric-dipole forbidden (they may still occur weakly via magnetic dipole or electric quadrupole mechanisms, but with much smaller intensity).
Worked Example: \(\;{}^3D \rightarrow {}^3P\;\)
Test whether the term-to-term transition \(\;{}^3D \rightarrow {}^3P\;\) is E1-allowed.
- Spin: \({}^3D\) and \({}^3P\) both have \(2S+1=3\Rightarrow S=1\), so \(\Delta S = 0\) ✅
- Orbital angular momentum: \(D \Rightarrow L=2\), \(P \Rightarrow L'=1\), so \(\Delta L = -1\) ✅
- Total angular momentum: For \({}^3D\), allowed \(J = 3,2,1\); for \({}^3P\), allowed \(J' = 2,1,0\). Since \(\Delta J=0,\pm1\), several fine-structure branches are allowed, e.g. \({}^3D_3\rightarrow{}^3P_2\), \({}^3D_2\rightarrow{}^3P_2,{}^3P_1\), \({}^3D_1\rightarrow{}^3P_2,{}^3P_1,{}^3P_0\) ✅
- Parity: An E1 transition requires a parity change. A \(D\leftrightarrow P\) change is consistent with opposite parity (typical for \(\Delta l=\pm 1\) behavior in the underlying one-electron promotion), so the parity requirement can be satisfied ✅
Conclusion: \(\boxed{{}^3D \rightarrow {}^3P \text{ is E1-allowed in LS coupling.}}\) In observed spectra, this usually appears as a multiplet of several closely spaced lines due to spin–orbit splitting of the \(J\) levels.
Landé Interval Rule (Spin–Orbit Splittings in LS Coupling)
In Russell–Saunders (LS) coupling, a single term \(\;{}^{2S+1}L\;\) is split by spin–orbit coupling into several fine-structure levels \(\;{}^{2S+1}L_J\;\) with \(J = L+S,\,L+S-1,\,\ldots,\,|L-S|\). The Landé interval rule predicts how the energy spacing between adjacent \(J\) levels scales across the multiplet.
A convenient LS-coupling model for the spin–orbit energy shift is \[ E_J = E_0 + \frac{A}{2}\Big(J(J+1)-L(L+1)-S(S+1)\Big), \] where \(E_0\) is the term center-of-gravity energy and \(A\) (often called the spin–orbit constant) depends on the specific atom and term.
Taking the difference between adjacent \(J\) values gives the Landé interval rule: \[ E_{J+1}-E_{J} = A\,(J+1). \] So the separations between consecutive fine-structure levels are in the ratio \(\;1:2:3:\cdots\;\) as \(J\) increases (within a given term).
- Key prediction: adjacent level spacings grow linearly with \(J\).
- How it’s used: if you measure two adjacent splittings, you can solve for \(A\) and check whether a multiplet is well-described by LS coupling.
- Sign of \(A\): if \(A>0\), larger \(J\) lies higher in energy; if \(A<0\), larger \(J\) lies lower (the ordering flips).
The Landé interval rule is most accurate when LS coupling is a good approximation; significant deviations can indicate intermediate coupling or strong configuration interaction.
Interactive: Fine-Structure Splittings & Allowed E1 Lines (Landé Interval Rule)
Enter spin–orbit constants \(A'\) (upper term) and \(A''\) (lower term), then click Update to compute: (1) the fine-structure energy level diagram and (2) the allowed E1 transition wavenumbers and a stick spectrum.
Model used: \(E_J = E_{\mathrm{cg}} + \frac{A}{2}\big(J(J+1)-L(L+1)-S(S+1)\big)\). The spectrum uses E1 selection rules: \(\Delta S=0\), \(\Delta L=0,\pm1\) (no \(0\leftrightarrow0\)), \(\Delta J=0,\pm1\) (no \(0\leftrightarrow0\)).
(1) Energy level diagram
(2) Allowed transitions & stick spectrum
Tip: try changing the sign of \(A\) to see how the ordering of \(J\) levels flips.