Algebraic solutions for o(12) <-> u(2) circle times u(10) quantum phase transitions in the proton-neutron interacting boson model

A simple systematic procedure to construct the proton-neutron unitary, u(sd)(pi nu)(12), orthogonal, o(sd)(pi nu)(12), and quasi-spin su(sd)(pi nu)(1, 1) algebras of the sd bosonic system is presented. New algebraic substructures of these algebras are discussed and the explicit formulae for their generators and Casimir operators are given in the spherical tensor form. The complementarity relationship of the Casimir operators of the s(sd)(pi nu)(1, 1) and o(sd)(pi nu)( 12) is derived. Theexact algebraic solutions of the quantum phase transition Hamiltonian between the o(sd)(pi nu)( 12) and u(sd)(pi nu)(2) circle times u(d)(pi nu)(10) limits have been considered, for the first time, in the framework of affine su(sd)(pi nu)(1, 1) Lie algebra. The low lying energy spectra of the Ge-70, Se76-78, Mo96-98, and Ru100-102 isotopes are calculated using the o(sd)(pi nu)(12) <-> u(sd)(pi nu)(2) circle times u(d)(pi nu)( 10) transition Hamiltonian. The good agreement of our computation with empirical result in these isotopes emphasizes the importance of u(s)(pi nu)(2) circle times u(d)(pi nu)(10) limit. With this addition, symmetry can be extended to many nuclei.