Relation between Wigner energy and proton-neutron pairing

Abstract

The linear term proportional to |N − Z| in the nuclear symmetry energy (Wigner energy) is obtained in a model that uses isovector pairing on single particle levels from a deformed potential combined with a T 2 interaction. The pairing correlations are calculated by numerical diagonalization of the pairing Hamiltonian acting on the six or seven levels nearest the N = Z Fermi surface. The experimental binding energies of nuclei with N ≈ Z are well reproduced. The Wigner energy emerges as a consequence of restoring isospin symmetry. We have found the Wigner energy to be insensitive to the presence of moderate isoscalar pair correlations.

Overview of neutron–proton pairing

“…[57] as long as the nucleus is sufficiently away from the phase transition. They suggested an interpolation for the cross-over region based on the QRPA solutions above and below the transition, which very well approximates the Shell Model results.…”

“…Their investigations of the rotational response will be discussed in Section 4 In a series of papers, Chasman [97][98][99][100] investigated the pair correlations by means of a version of the model Hamiltonian (23). At variance with [57], he did not include the term κ T · T /2, assumed G V = G S , and multiplied all non-diagonal matrix elements by a factor of 0.5. Instead of diagonalization, he optimized the trial wave function…”

“…It is important to note that as soon as ω is finite, the isovector pair field will be ∆ p = ∆ n = 0 and ∆ np = 0. For ω = 0 all directions of ∆ are solutions, and the direction found may depend on the details The average pair gap∆ for N = Z nuclei obtained by HFB calculations using the KB3 effective interaction and three different sets of spherical single particle levels labelled by 57 Ni, Nilsson, KFP (see text). The numbers indicate the isospin of the pair field, where 0+1 means that the isovector and isoscalar pair fields coexist.…”

“…This work focused on the influence of the pair correlations on the rotational response and will be discussed in Section 4. Bentley and Frauendorf [57] studied the competition between isovector and isoscalar pairing starting from a set of single particle energies k in a Nilsson potential, at a deformation calculated by means of the Micro-Macro (Strutinsky) method, and diagonalized the Hamiltonian in a basis generated by all configurations within the seven single particle states closest to the Fermi level. They used an isospin invariant model Hamiltonian…”

“…Bentley and Frauendorf [57] calculated the binding energies of even-A nuclei with T ≤ 5 by numerically diagonalizing the Hamiltonian (23), which avoids the problems that are caused by the small amplitude approximation of the QRPA (see Section 2.2.2). They determined the equilibrium deformation individually for each isobaric chain for 30 ≤ A ≤ 100 by means of the Micro-Macro method in order to properly account for the changing level density near the Fermi surface.…”