Interpretation of the Wigner Energy as due to RPA Correlations

In a schematic model with equidistant fourfold degenerate single-nucleon levels, an isovector pairing force acting only on pairs with time-reversed orbits and spins, and a symmetry force, the RPA correlation energy rises almost linearly with the isospin T , thus producing a Wigner term in accordance with the empirical proportionality of the symmetry energy to T (T + 1). Nearly symmetric nuclei have an extra binding, the so-called Wigner energy, that is not described by the quadratic symmetry term in the semi-empirical mass formula [1]. The literature has several explanations of this phenomenon. Thus, counting the even bonds among supermultiplet degenerate nucleon orbitals, Wigner estimates that the isospin-dependent part of the interaction energy in the ground states of even nuclei is proportional to T (T + 4), where T is the isospin [2]. Talmi proves that for seniority-conserving forces such as the pairing force acting in a single j-shell, this part of the interaction energy is proportional to T (T + 1) [3], and Bohr and Mottelson point out that this isospin-dependence arises in general from an interaction proportional to the scalar product of the isospins of the interacting nucleons [4]. A symmetry energy proportional to T (T + 1) also fits the empirical masses well [5]. Myers and Swiatecki attribute the extra binding of nearly symmetric nuclei to the interaction of neutrons and protons in overlapping orbitals [6]. Shell model calculations with realistic forces are succesful in reproducing the measured binding energies [7], whereas with Skyrme forces no Wigner term appears in Hartre-Fock-Bogolyubov calculations and only a small one in Hartree-Fock calculations [8]. However, by invoking a particular isoscalar pairing force that breaks geometric symmetries, Satula and Wyss obtain a significant Wigner energy in approximately number-projected Bogolyubov calculations [9]. The RPA is the leading order correction to the Hartree-Fock-Bogolyubov approximation. I have therefore calculated the RPA correlation energy for the schematic Hamiltonian H = H0 −GP † · P + κ 2 T , H0 = ∑