A figure ( GIF file of 11KB or PS file of 185KB ) presents the corrected total energies Ec of 1029 even-even nuclei calculated with the HF+BCS method with the Skyrme SIII force. Click here (html file of 1KB) for the conversion between the binding energy and the atomic mass (excess). For graphical reason, the smooth (i.e., macroscopic) part Emacro has been subtracted, which is defined by fitting functions of the Bethe-Weizsaecker type,
One can see regions of enhanced stability around double-magic nuclei with (N,Z) = (50,50), (82,50), and (126,82). Another double-magic nucleus (82,82) is outside the two-proton drip line. The super-heavy double-magic nucleus (184,114) does not look like a local minimum of nuclear mass in this result.
The two-proton drip lines of the HF+BCS with SIII (solid line) and the FRDM (dash line) are overlapping in most places. The distance between them is Delta Z=4 for N=40, Delta N=4 for Z=42 and 78, and Delta Z < = 2 and Delta N < = 2 for the other isotope and isotone chains.
The two-neutron drip lines of the two theoretical approaches are also close to each other. The difference looks of the same size as that between the FRDM and the TUYY mass formula [TUY88], both of which are models whose parameters were determined through extensive fittings to the mass data. This fact indicates the quantitative appropriateness of the macroscopic isospin dependence of the SIII force. Indeed, it is what we expected in choosing the SIII force for our first extensive HF+BCS calculation. The r.m.s. deviation of the calculated ground-state masses from the experimental ones of 480 even-even nuclei (the best recommended values of Ref. [AW93] excluding those estimated from systematic trends) is 2.2 MeV. Note that the inaccuracy of calculations due to the finite mesh size remaining after the correction is by far smaller than this deviation.
A figure ( GIF file of 10KB or PS file of 145KB )shows the error of mass for each nucleus. The solid (open) circles are put when the calculated masses are smaller (larger) than the experimental ones, while the diameter of the circles is proportional to the magnitude of the difference. One can see that the calculated masses tend to be overbinding for Z=8 and 20 isotopes, and N=50, 82, and 126 isotones. Unlike spherical nuclei including these semi-magic isotopes and isotones, deformed nuclei have positive errors, which are about 3 MeV rather independently of the size of deformation.
This systematic difference between spherical and deformed solution are more clearly seen in a figure ( GIF file of 6.1KB or PS file of 18KB ). Here, the mass error divided by A2/3 (meaning the nuclear surface area) is plotted versus the increase of the nuclear surface area owing to deformation: Bsurf-1, where Bsurf is the ratio of the surface area of a deformed liquid drop to that of the spherical one. The liquid drop shape parameters are determined in the manner described in a different paper of this paper.
The largest underbinding (about 7 MeV) occurs in 8Be, which is the most strongly deformed nuclei among the 1029 nuclei. An additional binding is expected to arise from the angular momentum projection effect. It is uncertain, however, whether a large deformations means a large angular-momentum projection effect or not, because a large deformation results in a large moment of inertia as well as admixture of large angular momentum components.
AW'93 TUYY FRDM ETFSI EV8C
TUYY 0.52 (480)
FRDM 0.68 (462) 4.31 (1521)
ETFSI 0.74 (430) 4.27 (1472) 2.74 (1742)
EV8C 2.22 (480) 2.59 (977) 2.50 (958) 2.26 (940)
macro 3.55 (480) 17.25 (2228) 16.07 (2246) 8.29 (1895) 5.71 (1029)
In the abobe table, we show the r.m.s. differences of the
masses of even-even nuclei between theoretical models as well as
between the experiments and the models. In the table, AW'93
represents the experimental atomic mass table by Audi and
Wapstra [AW93], TUYY the mass formula of Tachibana et
al. [TUY88], FRDM the finite-range droplet
model [MNM94], ETFSI the extended Thomas-Fermi Strutinsky
integral method with the SkSC4 force [APD92], EV8C the
HF+BCS results using the Skyrme SIII force with the correction,
and macro the Bethe-Weizsaecker type
function fitted to AW'93 (aV =-15.280 MeV, aS =
16.01 MeV, aI = 22.33 MeV, aC = 0.6896 MeV). In
parentheses are the number of nuclei to calculate the difference. The
r.m.s. deviation from AW'93 is 2.2 MeV for EV8C, which is 3-4 times
as large as that of 0.52 MeV for TUYY, 0.68 MeV for FRDM, and 0.74 MeV
for ETFSI. It should be noticed, however, that the parameters of
FRDM, TUYY, and ETFSI were fitted to all the available recent
experimental mass data while the parameters of the SIII force were
determined by fitting to the masses and charge radii of only seven
spherical nuclei. In addition, the number of the fitting parameters
is 275 in TUYY, 19 in FRDM, and 8 in ETFSI, while it is only 6 in the
SIII force.
AW'93 TUYY FRDM ETFSI
TUYY 0.50
FRDM 0.68 1.73
ETFSI 0.68 1.53 2.10
EV8C 1.62 1.80 1.57 1.54
As a further investigation of the macroscopic properties of the mass, we examine the possibility to decrease the r.m.s. deviation by improving the macroscopic part of the mass models. Namely, for each combination of the nuclear mass models and the experimental data, we add the Bethe-Weizsaecker type function to one of them and determine the four coefficients to minimize the r.m.s. difference between them. The resulting r.m.s. differences are tabulated in above table. This simple correction method can decrease the r.m.s. error of EV8C from experiments by 27%. As for the other models, however, the improvements are marginal. On the other hand, the differences between the models are greatly decreased because they come predominantly from nuclei near the neutron drip line. This fact suggests that substantial improvements of the macroscopic part of the nuclear mass models are possible only if new experimental mass data of neutron-rich nuclei are provided.