Physics of Very Heavy Nuclei

The stability of the heaviest and superheavy elements has been a longstanding fundamental question in nuclear science. Theoretically, the mere existence of the heaviest elements with Z>102 is entirely due to quantal shell effects. Indeed, for these nuclei the shape of the classical nuclear droplet, governed by surface tension and Coulomb repulsion, is unstable to surface distortions driving these nuclei to spontaneous fission. That is, if the heaviest nuclei were governed by the classical liquid drop model, they would fission immediately from their ground states due to the large electric charge. However, in the mid-sixties, with the invention of the shell-correction method, it was realized that long-lived superheavy elements (SHE) with very large atomic numbers could exist due to the strong shell stabilization [34,35].

The last three years (1999-2001) have brought a number of experimental surprises which have truly rejuvenated the field (see Ref.[36] for a recent review). Most significant are reports from Dubna announcing the discovery of elements 114 and 116 in hot fusion reactions [37,38,39]. Further experiments with stable beams are planned; they will be extremely helpful for the theoretical modeling of the SHE formation. It is anticipated, however, that future experimental progress in the synthesis of new elements will be possible - thanks to radioactive nuclear beams, especially the doubly magic neutron-rich beam of ¹³²Sn [36].

In spite of an impressive agreement with experimental data for the heaviest elements, theoretical uncertainties are large when extrapolating to unknown regions of the nuclear chart. In particular, there is no consensus among theorists with regard to the center of the shell stability in the superheavy region. Since in these nuclei the single-particle level density is relatively large, small shifts in the position of single-particle levels (e.g., due to the Coulomb or spin-orbit interaction) can be crucial for determining the shell stability of a nucleus.

  

Figure 8: Total shell energy (sum of proton and neutron shell corrections) calculated for spherical even-even nuclei with 40$\le$Z$\le$60 (top) and for superheavy nuclei around the expected island of stability around Z=120, N=180. Dots mark nuclei predicted to be stable with respect to $\beta$-decay (from Ref. [40]).

But does it actually make sense to talk about ``magic superheavy nuclei"? Recent theoretical work [40] sheds new light on this question. According to calculations, the patterns of single-particle levels are significantly modified in the superheavy elements. Firstly, the overall level density grows with mass number A, as $\propto A^{1/3}$. Secondly, no pronounced and uniquely preferred energy gaps appear in the spectrum. This shows that shell closures which are to be associated with large gaps in the spectrum are not robust in superheavy nuclei. Indeed, the theory predicts that beyond Z=82 and N=126 the usual localization of shell effects at magic numbers is basically gone. Instead the theory predicts fairly wide areas of large shell stabilization without magic gaps (see Fig. 8). This is good news for experimentalists: there is a good chance to reach shell-stabilized superheavy nuclei using a range of beam-target combinations.

The Coulomb and nuclear interactions act in opposite ways on the total nucleonic density in the nuclear interior. As a consequence of their saturation properties, nuclear forces favor values of the internal density close to the saturation density of nuclear matter. On the contrary, since the Coulomb interaction tends to increase the average distance between protons, the Coulomb energy is significantly lowered by either the creation of a central depression or by deformation, or both. Based on this general argument, one expects the formation of voids in heavy nuclei in which the Coulomb energy is very large [41,42]. Recently, the subject of exotic (bubble, toroidal, band-like) configurations in nuclei with very large atomic numbers has been revisited by self-consistent calculations. The important question which is asked in this context is: What are the properties of the heaviest nuclei that can be bound (at least, in theory), in spite of the tremendous Coulomb force?

Calculations do predict the existence of bubble nuclei [43,44,45] which are stabilized by shell effects [46]. Figure 9 shows the results of the coordinate-space SLy6+HF calculations with delta pairing [47]. The potential energy surfaces (PES) corresponding to various density distributions are plotted as functions of the mass quadrupole deformation $\beta_2$.

  

Figure 9: Variety of shapes predicted for the superheavy nucleus ⁷⁸⁰254₅₂₆ in the SLy6+HF model with delta pairing. Axial symmetry was assumed. The local minimum at $\beta_2$=0 can be associated with a bubble configuration (solid line). At small deformations, the PES corresponding to the ``normal" density distribution (dashed line) is predicted to lie very high in energy. At small oblate deformations, the unstable ``band"-like structure becomes favored energetically. The single-particle levels corresponding to the spherical bubble minimum are shown in the inset. The contour plots of the corresponding total densities are given in the boxes. The contours are axially symmetric with respect to r=0 and the contour lines are at 0.01, 0.03, 0.06, 0.09, 0.12, and 0.15 fm^-3 (from Ref. [47]).

This figure nicely illustrates the competition between various forms of nuclear matter constituting a finite heavy nucleus possessing a huge electric charge. It is difficult to say at present whether these exotic topologies can occur as metastable states or whether they can form isolated islands of nuclear stability stabilized by shell effects. The interplay between exotic configurations in superheavy nuclei is likely to impact the exact position of the borders of the superheavy and hyperheavy region.