Matrix elements of the proton–neutron symplectic model

The tensor properties of the Sp(12,R) algebra generators are determined in respect to the reduction chain Sp(12,R) ⊃ U(6) ⊃ Up(3) ⊗ Un(3) ⊃ U(3) ⊃ SO(3), which defines a shell-model coupling scheme of the proton–neutron symplectic model (PNSM). They are further used to calculate the matrix elements of the basic Sp(12,R) operators of the PNSM in the space of fully symmetric representations in the U(6)-coupled basis using a generalized Wigner–Eckart theorem. The obtained results allow further the matrix elements of any physical operator of interest, such as the relevant transition operators or the collective potential, to be calculated. As an illustration, the matrix elements of the basic irreducible tensor terms which appear in the U(6) decomposition of the long-range full major-shell mixing proton–neutron quadrupole–quadrupole interaction are presented.