Spin-orbit matrix elements in the atomic f shell from automorphisms of SO(8)

A scrutiny of the matrix elements within the f shell of the spin-other-orbit interaction H-soo has revealed many unexpected proportionalities that go beyond an application of the Wigner-Eckart theorem to Racah's groups G(2) and SO(7). An explanation is sought by using the automorphisms of SO(8) by means of which the states of the atomic f shell are generated by two alternative SO(7) bases of the type (1/2 1/2 1/2)(4), each augmented by two parity labels. Transformations between the bases can be made by reversing the relative phases of the two angular-momentum states 3 and 0 that comprise Luny one of the four SO(7) spinors. The method is exemplified by the single-electron spin-orbit interaction H-so, for which a component can be found that is invariant under the phase reversal. The extension from H-so to H-soo is described for the component z(6) of H-soo, and several examples drawn from the comparatively inaccessible configurations near the middle of the f shell an presented.