Within a single term the spin-orbit interaction can be replaced by

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table · Many-electron atoms and the periodic table

Deriving the Spectrum, Ch Many-electron terms, levels and Hund's rules

Problem

Within a single term the spin-orbit interaction can be replaced by Hso=ζLS L⃗⋅S⃗H_{\rm so}=\zeta_{LS}\,\vec L\cdot\vec S, and L⃗⋅S⃗=12[J(J+1)−L(L+1)−S(S+1)]\vec L\cdot\vec S=\tfrac12[J(J+1)-L(L+1)-S(S+1)] (quantum numbers only, ℏ\hbar dropped as is standard for this ratio). ((a)) Derive the Lande interval rule EJ−EJ−1=ζLSJE_J-E_{J-1}=\zeta_{LS}J for the fine-structure levels of a term. ((b)) The ground term of atomic vanadium is 3d3 4s2 4F3d^3\,4s^2\ {}^4F (L=3L=3, S=3/2S=3/2), the highest-spin term of the open 3d33d^3 subshell. Its levels lie at 0 cm−10\,\mathrm{cm^{-1}} (J=3/2J=3/2), 137.4 cm−1137.4\,\mathrm{cm^{-1}} (J=5/2J=5/2), 323.5 cm−1323.5\,\mathrm{cm^{-1}} (J=7/2J=7/2) and 552.9 cm−1552.9\,\mathrm{cm^{-1}} (J=9/2J=9/2). Compute the three successive intervals and compare each with the ratio the interval rule predicts. ((c)) Extract ζLS\zeta_{LS} (in cm−1\mathrm{cm^{-1}}) from each interval, state whether the term is normal or inverted, and check the sign against Hund's third rule for a subshell less than half filled.

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