Results

In this section, we present the results of calculations performed for all particle-bound even-even nuclei with

$\leq$ 108 and

$\leq$ 188. The THO basis was implemented according to the prescription developed in the previous section. The

value used in the procedure was obtained in the following way. From the starting HFB+HO calculation, we determined

values separately for neutrons and protons, using Eq. (19). We then associated the

value for the transformation with the smaller of

and

. In this way, the THO basis is always adapted to the less-bound type of particle. The calculations were performed by building THO basis states from spherical HO bases with $N_{\mbox{\rm\scriptsize {sh}}}$ =20 HO shells and with oscillator frequencies of $\hbar\omega_0=1.2\times41$ MeV $/A^{1/3}$ .

In order to meaningfully test predictions of nuclear masses for neutron-rich nuclei, we used the SLy4 Skyrme force parameterization [23], as this was adjusted with special emphasis on the properties of neutron matter. At present, there also exist Skyrme forces that were adjusted exclusively to nuclear masses [35]. These forces were used within a calculation scheme that was not focused on weakly-bound nuclei. In the pairing channel, we used a pure volume contact pairing force $V^{\delta }({\mbox{{\boldmath {$r$}}}},{\mbox{{\boldmath {$r$}}}}^{\prime })=V_{0}\delta ({\mbox{{\boldmath {$r$}}}}-{ \mbox{{\boldmath {$r$}}}}^{\prime })$ with strength

167.35MeVfm

and acting within a phase space limited by a cut-off parameter [19] of $\bar{e}_{\max}$ =60MeV.

Figure 2 summarizes the systematic results of our calculations, both for ground state quadrupole deformations (upper panel) and for two-neutron separation energies (lower panel). For this figure, calculations for a given mass number

were carried out for increasing (decreasing)

, up to the nucleus with positive neutron (proton) Fermi energy. Furthermore, for each nuclide, three independent sets of HFB+THO+LN calculations were performed, for initial wave functions corresponding to oblate, spherical, and prolate shapes, respectively. Depending on properties of a given nucleus, we could therefore obtain one, two, or three solutions with different shapes. For each obtained solution we performed a PNP calculation of the total energy. The lowest of these energies for a given nucleus was then identified with the ground-state solution.

Calculations of a microscopic mass table are greatly helped by taking advantage of parallel computing. We have used two IBM-SP computers at ORNL: Eagle, a 1 Tflop machine, and Cheetah, a 4 Tflop machine (1 Tflop = $1 \times 10^{12}$ operations/second). The code performs at 350 Mflop/processor on Eagle. We created a simple load-balancing routine that allows us to scale the problem to 200 processors. We are able to calculate the entire deformed even-even mass table in a single 24 wall-clock hour run (or approximately 4,800 processor hours). A complete calculated mass table is available online in Ref. [36].

The ground-state quadrupole deformations $\beta$ displayed in Fig. 2 (upper panel) were estimated from the HFB+THO+LN total quadrupole moments and rms radii through a simple first-order expression [30]. In that panel, all even-even nuclei with negative Fermi energies, $\lambda_n$

0 and $\lambda_p$

0, are shown. In the lower panel, showing two-neutron separation energies $S_{2n}$ , results are shown for those

and

values for which the nuclides with both

and

have $\lambda_n$

0. Note that on the proton-rich side the lighter of them may have $\lambda_p$

0; nevertheless, we show these points to make the proton drip line in the $S_{2n}$ panel identical to that of the quadrupole deformation panel. Of course, on the proton drip line values of $S_{2n}$ are large and not very illuminating.

Table 1 summarizes our results for even-even nuclei along the two-particle drip lines. More specifically, for each value of

, the results for the lightest isotope with $\lambda_p$

0, and the heaviest isotope with $\lambda_n$

0 are presented.

**Figure 2:** Quadrupole deformations $\beta$ (upper panel) and two-neutron separation energies $S_{2n}$ in MeV (lower panel) of particle-bound even-even nuclei calculated within the HFB+THO method with Lipkin-Nogami correction followed by exact particle number projection. The Skyrme SLy4 interaction and volume contact pairing were used.
$\begin{figure}\centerline{\epsfig{width=\columnwidth,file=fig2-75.eps}}\end{figure}$

**Table 1:** Results of the HFB+THO calculations for drip-line nuclei with the SLy4 Skyrme force and volume delta pairing force. The left and right columns show results for proton and neutron drip-line isotopes from He to Pb. For both drip lines we show deformations $\beta$ , Fermi energies $\lambda$ (in MeV), two-particle separation energies (in MeV), and neutron and proton pairing gaps (in MeV).
Two-proton drip line						Two-neutron drip line

Nucleus	$\beta$	$\lambda_{p}$	$S_{2p}$	$\bar{\Delta}_{n}+\lambda_{2n}$	$\bar{\Delta}_{p}+\lambda_{2p}$	Nucleus	$\beta$	$\lambda_{n}$	$S_{2n}$	$\bar{\Delta}_{n}+\lambda_{2n}$	$\bar{\Delta}_{p}+\lambda_{2p}$

$^{ 4}$ He	0.00	10.49		5.59	5.50	$^{ 8}$ He	0.00	1.26	2.69	2.71	5.35
$^{ 6}$ Be	0.00	2.13	1.79	5.51	2.89	$^{ 12}$ Be	0.00	2.70	6.92	2.70	2.76
$^{ 10}$ C	0.00	4.38	11.44	3.03	3.15	$^{ 22}$ C	0.00	0.34	2.97	2.03	2.69
$^{ 14}$ O	0.00	3.76	10.80	3.17	2.86	$^{ 26}$ O	0.00	0.97	0.53	1.53	2.87
$^{ 18}$ Ne	0.00	3.46	7.26	2.96	1.85	$^{ 34}$ Ne	0.28	0.39	0.50	1.40	1.76
$^{ 20}$ Mg	0.00	1.64	2.76	2.98	1.84	$^{ 42}$ Mg	0.18	0.29	0.44	1.09	1.64
$^{ 24}$ Si	0.07	2.65	5.63	1.85	1.87	$^{ 46}$ Si	0.00	0.99	1.71	1.07	1.86
$^{ 28}$ S	0.00	2.08	6.10	1.92	1.92	$^{ 52}$ S	0.00	0.05	0.96	1.00	1.49
$^{ 32}$ Ar	0.00	1.85	4.50	2.15	1.48	$^{ 58}$ Ar	0.00	0.39	2.37	1.31	1.39
$^{ 36}$ Ca	0.00	1.49	5.24	1.77	1.76	$^{ 68}$ Ca	0.00	0.11	0.40	1.10	1.73
$^{ 40}$ Ti	0.00	0.95	2.31	1.74	1.26	$^{ 72}$ Ti	0.00	0.63	2.59	1.15	1.05
$^{ 44}$ Cr	0.00	1.57	3.58	1.94	1.30	$^{ 80}$ Cr	0.00	0.07	0.01	0.72	1.14
$^{ 46}$ Fe	0.00	0.25	1.07	1.94	1.31	$^{ 84}$ Fe	0.00	0.12	0.60	0.80	1.15
$^{ 52}$ Ni	0.03	1.45	3.74	1.37	1.56	$^{ 88}$ Ni	0.00	0.19	0.09	0.91	1.53
$^{ 56}$ Zn	0.13	0.57	2.45	1.39	1.24	$^{100}$ Zn	0.24	0.02	0.29	0.90	1.10
$^{ 60}$ Ge	0.09	0.17	0.63	1.67	1.22	$^{108}$ Ge	0.16	0.13	0.12	0.93	1.07
$^{ 64}$ Se	0.17	0.15	0.83	1.25	1.27	$^{114}$ Se	0.08	0.27	0.69	0.91	1.08
$^{ 70}$ Kr	0.22	1.10	2.67	1.38	1.10	$^{118}$ Kr	0.00	0.23	3.29	1.20	1.08
$^{ 72}$ Sr	0.36	0.16	1.74	1.26	1.18	$^{120}$ Sr	0.00	0.86	4.61	1.23	1.06
$^{ 76}$ Zr	0.00	0.19	0.89	1.37	1.25	$^{124}$ Zr	0.00	0.04	0.74	0.60	1.05
$^{ 82}$ Mo	0.00	0.83	2.09	1.37	0.98	$^{132}$ Mo	0.00	0.05	0.14	0.66	0.87
$^{ 86}$ Ru	0.00	0.83	2.27	1.13	0.98	$^{142}$ Ru	0.27	0.02	0.23	0.84	0.89
$^{ 90}$ Pd	0.07	0.90	2.57	1.11	0.93	$^{150}$ Pd	0.22	0.02	0.44	0.84	0.82
$^{ 94}$ Cd	0.00	0.88	1.72	1.08	0.89	$^{168}$ Cd	0.02	0.01	0.62	0.82	0.75
$^{102}$ Sn	0.00	0.80	6.03	0.99	1.54	$^{174}$ Sn	0.00	0.27	1.11	0.76	1.16
$^{108}$ Te	0.16	1.00	2.39	1.13	0.89	$^{176}$ Te	0.00	0.83	1.90	0.78	0.77
$^{112}$ Xe	0.22	0.83	2.54	1.10	0.88	$^{178}$ Xe	0.00	1.37	2.82	0.80	0.83
$^{116}$ Ba	0.32	1.02	2.60	1.07	0.87	$^{182}$ Ba	0.00	0.28	4.36	1.26	0.87
$^{118}$ Ce	0.37	0.19	1.71	1.12	0.87	$^{186}$ Ce	0.43	0.11	16.29	0.72	0.88
$^{124}$ Nd	0.38	0.33	1.98	0.98	0.93	$^{188}$ Nd	0.44	0.51	15.32	0.75	0.71
$^{130}$ Sm	0.36	0.64	2.09	1.00	0.83	$^{204}$ Sm	0.28	0.01	0.11	0.69	0.75
$^{134}$ Gd	0.36	0.44	1.60	0.99	0.82	$^{208}$ Gd	0.29	0.20	0.84	0.73	0.74
$^{138}$ Dy	0.36	0.12	0.78	0.98	0.82	$^{216}$ Dy	0.22	0.02	4.70	0.73	0.71
$^{144}$ Er	0.19	0.41	1.64	0.89	0.89	$^{222}$ Er	0.28	0.08	0.16	0.65	0.70
$^{148}$ Yb	0.16	0.11	0.85	0.88	0.86	$^{230}$ Yb	0.21	0.00	0.06	0.70	0.71
$^{152}$ Hf	0.10	0.05	0.59	0.82	0.92	$^{254}$ Hf	0.00	0.02		0.72	0.86
$^{158}$ W	0.06	0.50	1.36	0.84	0.94	$^{256}$ W	0.00	0.30		0.70	0.83
$^{162}$ Os	0.11	0.09	0.57	0.84	0.78	$^{258}$ Os	0.00	0.57	0.51	0.67	0.79
$^{168}$ Pt	0.14	0.04	0.43	0.96	0.66	$^{260}$ Pt	0.00	0.83	1.19	0.65	0.73
$^{172}$ Hg	0.08	0.04	1.13	1.14	0.69	$^{262}$ Hg	0.00	1.09	2.37	0.62	0.69
$^{182}$ Pb	0.00	0.11	1.65	1.24	1.38	$^{266}$ Pb	0.00	0.03	3.21	1.06	0.98

As can be seen from Fig. 2 and Table 1, our calculations produce several particle-bound even-even nuclei (i.e., nuclei with negative Fermi energies) that at the same time have negative two-proton (or two-neutron) separation energies. Such an effect was already noticed in light nuclei in Ref. [19]. The current calculations suggest it may be generic, occurring near both the two-neutron and two-proton drip lines and for nuclei as light as $^{ 42}$ Mg and as heavy as $^{216}$ Dy. It seems to be related to the fact that the Fermi energies pertain to stability with respect to particle emission of a given configuration or shape, namely that of the ground state. In many of the cases in which we observe this phenomenon, (a) the neighboring even-even nucleus, the one to which it would decay by two-nucleon emission, has two distinct shapes, each with negative Fermi energies, (b) the ground state of that neighboring nucleus has a shape that is different than that of the parent nucleus, (c) the shape of the excited bound configuration is the same as that of the parent nucleus, and (d) decay to the excited configuration is energetically forbidden.

The precise results of course depend sensitively on properties of the interaction, both in the particle-hole and particle-particle channels. Despite its many good features, the force we use is far from perfect. For example, the positions we obtain for the two-neutron drip lines in the Be and O isotopes are not correct. In addition, the method itself has limitations, as it leaves out potentially important effects beyond mean field. Despite these limitations, we feel it is nevertheless worthwhile to point out some of the interesting new physical situations that are predicted in these calculations and which may therefore occur in weakly-bound systems. The above example of nuclei that are formally beyond one of the two-particle drip lines but nevertheless are localized and do not spontaneously spill off a nucleon is just one of several. We will now discuss in greater detail some specific isotopic chains to see how this and other interesting exotic new features emerge.

We focus our discussion on the heaviest isotopes of four isotopic chains; neon, magnesium, sulfur, and zinc (see Figs. 3-6, respectively). The figures show the Fermi energies $\lambda_n$ , $\lambda'_n$ , and $\lambda''_n$ [see Eqs. (17) and (18)], and the total binding energies, obtained in constrained HFB+THO+LN+PNP calculations as functions of the quadrupole deformation $\beta$ for the last three particle-bound isotopes of the respective chains. In each figure, the binding energies of the last three isotopes are shown on a common energy scale. As a reminder, two neutron separation energies can be readily obtained from the binding energies according to $S_{2n}=E(Z,N-2)-E(Z,N)$ .

We should note that the minima of the constrained energies need not exactly correspond to the PNP of the HFB+THO+LN minima, which were used in Fig. 2 and Table 1. Indeed, in the constrained calculations the deformation serves as an additional variational parameter for the variation after PNP. Optimally, the full variation after projection should be performed, which, however, requires a much larger numerical effort, and is left to future work. Such an optimal method will also remove the ambiguities related to the definition of the Fermi energy, discussed in Sec. 2. At present, we illustrate these ambiguities by showing in Figs. 3-6 the three possible values of the Fermi energy, $\lambda_n$ , $\lambda'_n$ , and $\lambda''_n$ .

**Figure 3:** Neutron Fermi energies $\lambda$ (upper panels) and the total binding energies (lower panels) calculated for $^{30}$ Ne, $^{ 32}$ Ne, and $^{ 34}$ Ne as functions of the quadrupole deformation $\beta$ .
$\begin{figure}\centerline{\epsfig{width=\columnwidth,file=fig3.eps}}\end{figure}$

**Figure 4:** Same as in Fig. 3 but for $^{38}$ Mg, $^{ 40}$ Mg, and $^{ 42}$ Mg.
$\begin{figure}\centerline{\epsfig{width=\columnwidth,file=fig4.eps}}\end{figure}$

Consider first the Ne isotopes, for which the results are shown in Fig. 3. For the SLy4 interaction that we use, a strong shell gap at

=20 persists up to the heaviest isotopes of Ne, and this produces a stiff spherical minimum for $^{30}$ Ne. Adding two neutrons gives rise to the nucleus $^{ 32}$ Ne, which is particle-bound ( $\lambda_n$

0), but at the same time two-neutron unstable ( $S_{2n}$

0). [Note that this nucleus does not exactly fit into the picture given earlier for such nuclei.] Interestingly, when we add two more neutrons, we obtain a strongly (prolate) deformed particle-bound ground configuration in $^{ 34}$ Ne, which is again two-neutron stable ( $S_{2n}$

0).

Next we turn to the Mg isotopes, for which results are presented in Fig. 4. In $^{ 40}$ Mg the neutron Fermi energies $\lambda_n$ have negative values for all deformations, so that the configurations for all deformations are particle-bound, with the prolate minimum being slightly lowest. The same is also true for the next nucleus $^{ 42}$ Mg where the ground state deformation changes from prolate to oblate. It is clear from Fig. 4 that in $^{ 42}$ Mg the two-neutron separation energy is negative; however, since $^{ 42}$ Mg and $^{ 40}$ Mg have different shapes in their ground states, the real process of emitting two neutrons may occur towards the shape isomer in $^{ 40}$ Mg. (The situation will be even more complicated if the oblate minimum in $^{ 42}$ Mg is unstable to triaxial deformations, i.e., it is a saddle point.)

**Figure 5:** Same as in Fig. 3 but for $^{48}$ S, $^{50}$ S, and $^{ 52}$ S.
$\begin{figure}\centerline{\epsfig{width=\columnwidth,file=fig5.eps}}\end{figure}$

**Figure 6:** Same as in Fig. 3 but for $^{96}$ Zn, $^{98}$ Zn, and $^{100}$ Zn.
$\begin{figure}\centerline{\epsfig{width=\columnwidth,file=fig6.eps}}\end{figure}$

The results for the S isotopes are given in Fig. 5. Here, the spherical HFB+THO+LN minimum in $^{ 52}$ S is shifted in the constrained PNP calculations towards a small oblate deformation. All shapes appear to be very weakly particle-bound, and have negative two-neutron separation energies at the same time. It is obvious that in the case of so poorly defined a minimum, its precise location is not relevant and full configuration mixing, e.g., within the generator coordinate method (GCM) [30,37,38], should be applied. This complication specific to weakly-bound nuclei is related to the fact that it is not clear how to take into account in the GCM the regions of the collective coordinate corresponding to $\lambda$

0, hence to particle-unbound states.

In the results for the zinc isotopes (Fig. 6), we see strong competition between oblate, prolate, and spherical shapes. In $^{96}$ Zn, all shapes are particle-bound and the ground state is oblate. The situation changes in $^{98}$ Zn, where the oblate configuration, though lowest in energy, becomes particle-unbound and the prolate minimum becomes the ground-state configuration. Though this ground state is two-neutron unstable ( $S_{2n}$

0), its decay to the ground state of $^{96}$ Zn may be hindered by the shape change. Finally, in $^{100}$ Zn the particle-stable prolate ground state is also two-neutron unstable. Hence in this isotopic chain the last two even isotopes are unstable with respect to two-neutron emission.

In heavier nuclei near the neutron drip line, we often obtain particle-stable and two-neutron-unstable isotopes right after closed neutron magic shells. As in Ne, this reflects the fact that strong shell gaps persist up to the heaviest isotopes in a chain when the calculations are based on the SLy4 interaction. In the

=126 isotopes of Ce and Nd, for example, the ground-state configurations are strongly spherical. In the neighboring

=128 isotopes, these spherical configurations become particle unbound. However, in these same isotopes, there are strongly prolate particle-bound configurations with very large negative two-neutron separation energies (see Table 1). An analogous situation occurs in the

=186 and 188 drip-line nuclei, where the last two even isotopes may have particle-bound prolate states with unbound spherical configurations.

Strong SLy4 neutron magic numbers also result in the characteristic non-monotonic behavior of the $S_{2n}$ values (Fig. 2). Indeed, lines of constant $S_{2n}$ often follow decreasing

with increasing

, which is particularly conspicuous near

=126. This effect even creates a small peninsula of stability near

=140. Such strong neutron closed shells could create the well-known deficiencies in the r-process abundances [39].