Effect of the Particle-Number Symmetry Restoration on Root Mean Square Radii of Even–even Neutron-Deficient Nuclei in the Isovector Pairing Case

Abstract

The effect of the particle-number symmetry restoration on the root mean square (rms) proton and neutron radii of neutron-deficient nuclei is studied in the isovector pairing case. As a first step, an expression of the nuclear radii which includes the neutron-proton pairing effects and which strictly conserves the particle-number has been established using the SBCS (Sharp BCS) method. It is shown that this expression generalizes the one obtained in the pairing between like-particles case. As a second step, the proton and neutron rms radii are numerically evaluated for even-even nuclei such as 16 Z 56 and 0 (N − Z) 4 using the single-particle energies of a Woods-Saxon mean-field. The results are compared with experimental data when available and with the results obtained when one considers only the pairing between like-particles.Finally, the projection effect, when including or not the isovector pairing effect, has been studied by evaluating the relative discrepancies: δr n-proj and δr n-proj-np . This effect is roughly the same in both cases since the average values of these latter are respectively 0.70% and 0.74%. These values are also very close to that of the proton system.However, one notices that, in the neutron case, the np pairing effect seems to be more important than that of the projection. The particle-fluctuation effects in the BCS states being, by definition, nonphysical it is difficult to explain why Figs. 1 and 2 show very different trends.

Number projected isovector neutron–proton pairing effect in odd-mass nuclei

“…However, the particle-number symmetry may be restored using a particlenumber projection method. In the present work, we use the sharp-BCS (SBCS) one [37][38][39][40].…”

“…In the isovector pairing case, the ground-state (9) is simultaneously projected on the good neutron and proton numbers, i.e. [38][39][40]:…”

“…Several methods have been already proposed in the np pairing case, as the quasiparticle random phase approximation (QRPA) [23][24][25][26][27][28][29][30][31], the LipkinNogami method [32], the generator coordinate method [33], and the PBCS-type projection methods [34], of FBCS-type [35] or the isospin and particle-number projection one [36]. In previous papers [37][38][39][40], we proposed and applied a generalization of the SBCS (sharp-BCS) projection method [41][42][43]. However, this generalization is valid only for even-even nuclei and has not been yet extended to odd-mass systems.…”

Particle-number projected electric quadrupole moment of even-even proton-rich nuclei in the isovector pairing case

“…The neutrons and the protons are assumed to occupy the same energy levels. The total Hamiltonian of the system is given, in the isovector case, by [7][8][9][10]: …”

“…It is thus projected using the method described in Refs. [7][8][9][10]. The corresponding wave-function is given by:…”