Quark Rotation Groups for the Atomic f Shell

Abstract

It is standard practice for theoretical spectroscopists working with rare-earth elements to use the group-theoretical scheme that Racah1 devised in 1949 to cope with the Coulomb interaction between the f electrons. In addition to SO(7), the rotation group that transforms the seven orbital functions of an f electron among themselves, the exceptional Lie group G2 is used to classify operators and states. This scheme forms the basis for the computer programs that have been developed to calculate coefficients of fractional parentage2 and the matrix elements of various n-electron operators. In particular, it was used by Hannah Crosswhite to calculate the matrix elements of the six three-electron operators t i that are required to represent the effects of distant configurations of the type f N±1 l ±1 on the ground configuration f N of the tripositive rare-earth ions.3 This work has been used by Carnall and others in their analyses of rare-earth and actinide spectra.4 It also stimulated us to check Carnall’s computer output for unexpected simplifications. The many surprises led us to introduce a radically different basis in which to carry out shell-model calculations.5 Instead of the 15 configurations f N that form the 16384 states of the f shell, we take four quarks, each of dimension 8 and belonging to the irreducible representation (irrep) (1/2 1/2 1/2) of SO(7), together with two parity labels that characterize the number of electrons in the spin-up and spin-down spaces. This four-fold multiplicity, taken with the 84 quark states, provides an alternative source for the 16384 states of the f shell.