Single j-shell studies of cross-conjugate nuclei and isomerism

In the single j-shell with the same interaction cross -conjugate pairs should have identical spectra. There are differences.For the lighter members of cross-conjugate pairs of four nucleons with T=1 The ground state spin is two for the heavier members it is (2j-1). This result can be obtained by using the spectrum of 2 particles as an effective interaction for the lighter member and the spectrum of 2 holes for the heavier member. But what is most new in this work is the observation in many single j-shell calculations and some experiments of criss-cross behavior. The J=2 state in the heavier member, if not the ground state is still low lying and hence isomeric. Likewise the (2j-1) state in the lighter member is also low lying and is isomeric. An exception to the isomers comes when J=(2j-1) differs from J=2 by two units or less. A key point in the difference of the particle-particle and the hole-hole interaction is that in the latter the state with J$^{max}$=2j is much lower than in the former. In $^{44}$Sc the J=2 state is calculated to be the ground state and in $^{52}$Mn J=6. But J=6 is at 0.381 MeV in the former and J=2 at 0.202 MeV in the latter.In ${96}$Ag J=(2j-1)=8 is the ground state and J=2 is at 0.097 MeV. J=15 is also isomeric. We thus a (2j-1) rule.