Some simple R 5 Wigner coefficients and their application

Abstract

Generalized R 5 Wigner coefficients are calculated in algebraic form for the Kronecker products of an arbitrary irreducible representation with the 4, 5 and 10-dimensional irreducible representations of R 5, the group of rotations in 5-dimensional space. These are expressed in terms of the mathematically natural quantum numbers associated with the group chain R 5 ⊃ R 4 ≡ R 3×R 3. The transformation to physically interesting quantum numbers is discussed for two applications. The first involves the seniority force for nuclei with both protons and neutrons (see a companion paper by J. C. Parikh) and the possibility of extracting the N and T dependence of nuclear matrix elements in the seniority scheme. ( N and T are the nucleon number and the total isospin). The second application involves the calculation of fractional parentage coefficients for spin-2 phonons in the seniority scheme, for large phonon number N and seniority ν. Properties of the generalized Wigner coefficients are used to relate all such fractional parentage coefficients to those of type 〈 N−1 = ν−1|{ N = ν〉. A general prescription is given for the calculation of this type of coefficient from the R 5 Wigner coefficients, and numerical tables are given for ν = 5 and 6.