Recursive calculation of matrix elements for the generalized seniority shell model

Abstract

A recursive calculational scheme is developed for matrix elements in the generalized seniority scheme for the nuclear shell model. Recurrence relations are derived which permit straightforward and efficient computation of matrix elements of one-body and two-body operators and basis state overlaps.

Generalized seniority with realistic interactions in open-shell nuclei

“…Several approaches [4, 13, 28, 30, 31] have been developed for evaluating matrix elements of one-body and twobody operators in the generalized seniority basis. The present calculations make use of recurrence relations derived in [32], where the notation and methods used in the present work are also established in detail. A generalized seniority basis can be defined for nuclei with valence particles of both types via a proton-neutron scheme, that is, by taking all possible products of proton and neutron generalized seniority states, with generalized seniorities v p and v n [4,19].…”

“…Unlike conventional shell-model basis states, the generalized seniority basis states do not have definite occupation for each orbital, rather, involving a BCS-like distribution of occupations. Therefore, the occupation n a of an orbital a = (n a l a j a ) in an eigenstate represented in this basis cannot be obtained as a simple average over the contributing basis states, but rather must be evaluated as the expectation value of a one-body operator, the number operator for the orbital [n a = − a (C † a Ca ) (0) ], by the process described in [23,32]. Occupations of each of the pf -shell orbitals in the 0 + ground state are shown in figure 5, for the Ca, Ti, and Cr isotopes, both for neutron orbitals [figure 5 (left)] and proton orbitals [figure 5 (right)].…”

Practical calculation scheme for generalized seniority

“…We count the number of solutions (n α i , n β i , n γ i ) of Eqs. ( 16) and (17) to get the number of different t's at given µ, p, and r.…”

Generalized seniority for the shell model with realistic interactions

“…The present calculations have made use of the recurrence relations derived in Ref. [31]. Matrix elements are first calculated with respect to the original nonorthogonal, unnormalized, and overcomplete generalized seniority basis.…”