Simplified components of the spin-other-orbit interaction near the middle of the atomic f shell

Abstract

To honour the memory of Brian Garner Wybourne, an analysis is presented of three components of the spin-other-orbit interaction for f electrons using the kind of Lie groups he would have been familiar with. The components have been named z 6, z 8 and z 10. They all belong to the irreducible representation (IR) (30) of Racah’s group G2. Near the middle of the f shell it is often found that fewer independent blocks of numbers are needed to express their matrix elements than the Wigner–Eckart theorem, generalized to the IRs U of G2, would indicate. Each block corresponds to a given U and U ′, and possesses rows and columns labelled by the angular momenta L and L′. The number of independent blocks would be expected to be given by Racah’s multiplicity function c(UU ′ (30)); but near the middle of the shell the number c(UU ′ (20)) (or less) often suffices. For this to occur, z 8 and z 10 have to be replaced by linear combinations corresponding to IRs of the types (20)×(10) and (21)×(10) of the direct product group G2A×G2B, where A and B refer to electrons with their spins up (A) and spins down (B). A detailed example is provided by the IR (31) of G2, which occurs in the configurations f 5 through f 9. In addition, two antiHermitian operators (z a6 and z a7) that also belong to the IR (30) of G2 are discussed.