Figure 1 illustrates the calculated evolution of intrinsic deformation with increasing neutron number in the Radium isotopes. It plots the intrinsic ground-state mass-density contours predicted by SkO'. The mean-field ground states go from having a spherical shape at the magic number to a quadrupole deformed (reflection-symmetric) shape at , then to quadrupole+octupole deformed (reflection-asymmetric) shapes for , and finally back to quadrupole deformed shapes at higher . Because the ground states are obtained from a variational principle, all shape moments higher than octupole are also optimized (the isoscalar dipole moment is constrained to be zero). The nucleus Ra, with , will clearly be well deformed in both the quadrupole and octupole coordinates. The structures at small radii visible for reflect small oscillations of the density distribution around the saturation value (for a given neutron excess) caused by shell effects.
We must note that the octupole-deformed minima are not equally pronounced for all forces and isotopes. In addition, in some of the isotopes with reflection-symmetric minima, some of the Skyrme forces predict an excited octupole-deformed minimum separated by a small barrier from the ground-state minimum. Furthermore, in the transitional nuclei, which have soft potentials in the octupole direction, all parity-breaking intrinsic deformations are subject to collective correlations as discussed in Ref. [Egido and Robledo(1989)]. The influence of correlations will be smallest for the nuclides with the most pronounced octupole-deformed minima, usually Ra and Ra. This fact supports our belief that our mean-field calculations supply a good approximation to the intrinsic structure in Ra.
Figure 2 shows the relative error in the predicted binding energies for all four forces, and the predicted two-neutron separation energies, along with the measured values. All the forces do a good job with binding, which is not surprising given the way their parameters were fit. The fact that the error in binding for SkO' is nearly constant with is reflected in the near perfect agreement in the bottom panel with the measured two-neutron separation energies . The errors in predicted values of around probably reflect the deficiencies of mean-field models in transitional nuclei.
Figure 3 shows three parity-violating intrinsic quantities. In the top panel is the ground-state octupole deformation (where ), as a function of neutron number. The trend mirrors that in the density profiles shown earlier. At , one less neutron than in Ra, the forces all predict almost identical octupole deformation, a result we like. Experimental data for octupole moments are still sparse in this region; we are only aware of for , a value that can be deduced from the given in Ref. [Spear(1989)]. (In Ra and Ra, by the way, we agree fairly well with the quadrupole moments obtained from 's in Ref. [Raman et al.(2001)Raman, Jr., and Tikkanen]. For example, SkO' gives in Ra and experiment gives .)
The second panel in the figure shows the absolute values of intrinsic dipole moments , along with experimental data extracted from transition probabilities [Butler and Nazarewicz(1996)]. The calculated values for change sign from positive to negative between and , reflecting a small change in the location of the center of charge from the ``top" half of the pear-shaped nucleus to the ``bottom" half. This predicted sign change is consistent with the near-zero experimental value for . None of the forces precisely reproduces the trend through all the isotopes, but the comparison has to be taken with a grain of salt because ``data" derive from transitions between excited rotational states, and therefore are not necessarily identical to the ground-state dipole moments. Cranked Skyrme-HF calculations without pairing correlations [Tsvetkov et al.(2002)Tsvetkov, Kvasil, and Nazmitdinov] and cranked HFB calculations with the Gogny force [Garrote et al.(1997)Garrote, Egido, and Robledo] predict that for most Ra isotopes changes significantly with angular momentum. In any event, as thoroughly discussed in Ref. [Butler and Nazarewicz(1996)], the intrinsic dipole moment is a small and delicate quantity.
The intrinsic Schiff moment , the quantity we're really interested in, is more collective and under better control, as the bottom panel of the figure shows. The various predictions are usually within 20 of one another. The octupole deformation and intrinsic dipole moment have been shown to change only slightly with parity projection from the intrinsic states [Garrote et al.(1997)Garrote, Egido, and Robledo], and the same is probably true of the intrinsic Schiff moment.
By turning the pairing force off, we are able to see whether the parity-violating quantities in Fig. 3 are affected by pairing correlations. In Ra, for example, SkO' gives , fm, and fm without pairing, and , fm, and fm when pairing is included. In this nucleus uncertainties related to pairing are very small.
Finally, in Fig. 4, we show the predicted proton and neutron single-particle spectra generated by the ground-state mean-field in Ra. The combination of quadrupole, octupole, and higher deformations reduces the level density around the Fermi surface for both kinds of nucleon, leading to significant deformed and shell closures for all interactions, and a somewhat weaker subshell closure for SIII, SkM* and SkO'. The small level density around the Fermi surface might explain the insensitivity of the deformation to pairing correlations mentioned above. For all the forces except SkM, the first empty neutron level clearly has , implying that in Ra the ground-state parity-doublet bands will be built on states. For SkM the situation is less clear because the and 3/2 states are nearly degenerate, and it is necessary to carry out the calculation in Ra itself to see which becomes the lowest.