PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table
Deriving the Spectrum, Ch Many-electron terms, levels and Hund's rules
Problem
Lead's ground configuration 6p2 has five levels, at 0, 7819, 10650, 21458 and 29467cm−1, with J=0,1,2,2,0 in that order. Take the spin-orbit interaction first: each 6p electron then has a good j, and its spin-orbit energy is 21ζ6p[j(j+1)−l(l+1)−s(s+1)].
((a)) List the Pauli-allowed states (j1,j2)J of two equivalent 6p electrons. For two electrons both in p3/2, find the allowed J by listing the MJ of every Pauli-allowed pair of mj values, and check that the total number of states is (26).
((b)) Ignoring the residual repulsion, find the energies of the three groups (21,21), (21,23) and (23,23) in terms of ζ6p, and show that they are equally spaced.
((c)) Assign lead's five levels to the three groups. Using (2J+1)-weighted centroids, find the two spacings between the groups and the value of ζ6p each implies. Say what the difference between the two values measures.
((d)) Levels of the same J keep their order as the coupling moves from LS to jj. Use this to give each of lead's levels its LS label, and say which of the two J=2 levels carries the label 1D2.