Degeneracies when T=0 Two Body Matrix Elements are Set Equal to Zero and Regge's 6j Symmetry Relations

Abstract

The effects of setting all T=0 two body interaction matrix elements equal to a constant (or zero) in shell model calculations (designated as $<T=0>=0$) are investigated. Despite the apparent severity of such a procedure, one gets fairly reasonable spectra. We find that using $<T=0>=0$ in single j shell calculations degeneracies appear e.g. the $I={1/2} ^{-}$ and ${13/2}^{-}$ states in $^{43}$Sc are at the same excitation energies; likewise the I=$3_{2}^{+}$,$7_{2}^{+}$,9$^{+}_{1}$ and 10$^{+}_{1}$ states in $^{44}$Ti. The above degeneracies involve the vanishing of certain 6j and 9j symbols. The symmetry relations of Regge are used to explain why these vanishings are not accidental. Thus for these states the actual deviation from degeneracy are good indicators of the effects of the T=0 matrix elements. A further indicator of the effects of the T=0 interaction in an even - even nucleus is to compare the energies of states with odd angular momentum with those that are even.