Mass Tables

The LST employed in Ref. [3] was based on HO densities corrected in the asymptotic region by the contribution from the lowest-energy quasiparticle. Since a common LST has to be carried out for both neutrons and protons, for each nucleus one is forced to make a decision whether the LST is to be based on neutron or proton density. In Ref. [3] we used a prescription (referred to as LAM) that the neutron densities were used for $\lambda_n$$\geq$$\lambda_p$ and vice versa. In this work, we use the condition $\rho_n(R_{{min}})$$\geq$ $\rho_p(R_{{min}})$, where $R_{{min}}$ is the point where the neutron or proton logarithmic density has a minimum as a function of $r$. In practice, the above condition, dubbed RHO, does not depend on whether the neutron or proton $R_{{min}}$ is considered.

Figure 1: (Color online) Left: differences in $E_{{tot}}$ obtained in HFBTHO by using LST based on RHO or LAM conditions. Right: differences between $E_{{tot}}$ obtained in THO and HO bases. Calculations were performed using the SLy4 interaction with volume pairing and 20 oscillator shells. Lipkin-Nogami method followed by the exact particle-number projection was used to correct for the particle number nonconservation in HFB.

In Fig. 1 (left panel) we show the differences in $E_{{tot}}$ obtained in HFBTHO by using the LST condition employing the Fermi energies (LAM) [3] or the densities (RHO). One can see that in the majority of neutron-rich nuclei both prescriptions lead to identical results. However, in many proton-rich nuclei the new prescription decreases binding up to about 500keV, and for some medium-mass proton-rich nuclei the RHO method decreases binding by up to 100keV. This latter effect is due to a better description of asymptotics in the pairing channel, which leads to extended pairing fields and reduced pairing energies [10]. The right panel of Fig. 1 shows differences in $E_{{tot}}$ obtained in THO and HO bases. In most nuclei, by using the THO basis, one obtains a small energy gain of up to 10keV. This grows to $\sim$500keV for the very neutron-rich systems. Again, in lighter nuclei, a better asymptotics may lead to a reduced binding. In fact, our results show that improvements in density profiles at large distances cannot be treated variationally. First, $E_{{tot}}$ is quite insensitive to the precise description of nucleonic densities in outer nuclear regions. Second, due to the pairing-space cutoff, the pairing energy is not reacting variationally on the improvement of the wave function.

Figures 2 and 3 present HFBTHO results obtained with the SLy4 and SkP Skyrme forces. It is obvious that without further improvements these traditional Skyrme forces describe nuclear masses rather poorly. The rms deviations between calculated and measured masses are as large as 3.14MeV for SkP and 5.10MeV for SLy4, respectively, as compared to about 0.70MeV deviations obtained for forces fitted specifically to masses (see Ref. [11] for a review). Moreover, pronounced kinks obtained at magic numbers suggest that the quality of the description of (semi)magic and open-shell systems is not the same. This may point to a need to systematically include dynamical zero-point corrections [12]. Work in this direction is in progress.

Figure 2: (Color online) Ground-state deformations $\beta$ (left) and two-neutron separation energies $S_{2n}$ (right) obtained within HFBTHO using SkP (top) and SLy4 (bottom) interactions.

Figure 3: (Color online) Deviations of ground-state HFBTHO energies from experiment [13] for SkP (left) and SLy4 (right) interactions. Positive values correspond to underbound nuclei. No corrections beyond mean field were included.

This work was supported in part by the Polish Committee for Scientific Research (KBN); by the Foundation for Polish Science (FNP); by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-AC05-00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory), and DE-FG05-87ER40361 (Joint Institute for Heavy Ion Research); and by the National Nuclear Security Administration under the Stewardship Science Academic Alliances program through DOE Research Grant DE-FG03-03NA00083.