Cadmium's configuration gives the same term scheme as mercury's : a triplet ,

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

Cadmium's 5s5p5s5p configuration gives the same term scheme as mercury's 6s6p6s6p: a triplet 3P0{}^3P_0, 3P1{}^3P_1, 3P2{}^3P_2 and a singlet 1P1{}^1P_1. Measured relative to the ground state 5s2 1S05s^2\,{}^1S_0 at 0 cm−10\,\mathrm{cm^{-1}}, the levels lie at3P0=30114.7 cm−1,3P1=30656.1 cm−1,3P2=31826.7 cm−1,1P1=43692.2 cm−1.\begin{aligned}{}^3P_0 &= 30114.7\,\mathrm{cm^{-1}}, &\quad {}^3P_1 &= 30656.1\,\mathrm{cm^{-1}},\\ {}^3P_2 &= 31826.7\,\mathrm{cm^{-1}}, &\quad {}^1P_1 &= 43692.2\,\mathrm{cm^{-1}}.\end{aligned}As for mercury's 6p6p electron, treat only the 5p5p electron as carrying spin-orbit coupling, so that the interval rule constant of the triplet term is ζLS=ζ5p/(2S)\zeta_{LS} = \zeta_{5p}/(2S) with S=1S=1 (Hund's third rule for the highest-spin term), and the off-diagonal spin-orbit matrix element mixing 3P1{}^3P_1 and 1P1{}^1P_1 (both J=1J=1) is ζ5p/2\zeta_{5p}/\sqrt2. ((a)) Use 3P2−3P0{}^3P_2 - {}^3P_0 to find ζ5p\zeta_{5p}. ((b)) Find the numeric value of the off-diagonal matrix element W=ζ5p/2W = \zeta_{5p}/\sqrt2. ((c)) Estimate the singlet admixture sin⁡2θ\sin^2\theta mixed into the physical 3P1{}^3P_1 level, using the small-mixing estimate sin⁡2θ≈(W/ΔE)2\sin^2\theta \approx (W/\Delta E)^2, with ΔE\Delta E the 1P1−3P1{}^1P_1 - {}^3P_1 separation. ((d)) Cadmium's intercombination line (3P1→1S0{}^3P_1 \to {}^1S_0) is observed at 326.1 nm326.1\,\mathrm{nm}, alongside its fully allowed resonance line (1P1→1S0{}^1P_1 \to {}^1S_0) at 228.8 nm228.8\,\mathrm{nm}. Given that the intercombination rate scales roughly as sin⁡2θ\sin^2\theta times the fully allowed rate, say whether cadmium's intercombination line should be relatively stronger or weaker than helium's (sin⁡2θ∼10−7\sin^2\theta \sim 10^{-7}) and than mercury's (sin⁡2θ≈0.04\sin^2\theta \approx 0.04), and connect the trend to the growth of spin-orbit coupling with atomic number.

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