Quark-like fractional parentage for the atomic f shell

Abstract

The states of the atomic f shell are constructed by coupling together four quark-like objects belonging to the eight-dimensional spinor irreducible representation (irrep) (1/2 1/2 1/2) of SO(7). Two parity labels are also required. Transformations among the eight components of each quark lead to the groups U(8) and SO(8), which can be used to augment the labels provided by the groups SO(7) and G2 that Racah (1943,1949) used in his classic analysis. Coefficients of fractional parentage (CFP) have been calculated for the quark configurations q2, q3 and q4. To allow for the automorphisms of SO(8), the groups SO(7)' and SO(7)" of Labarthe (1980) are used as alternative intermediaries to SO(7) between SO(8) and G2. Parallel tables of CFP are provided. As an example of the usefulness of the quark model, a correspondence is established between a three-electron operator t4 used in configuration-interaction studies and a two-quark operator. Some matrix elements of the latter in q4 are calculated by means of the quark CFP, thereby giving the corresponding matrix elements of t4 in the electronic configurations f6 and f7. By using the irreps of SO(7)', an unexpected proportionality between the sets of matrix elements can be accounted for.