Related Observables in Even Isotopes

Intrinsic-parity breaking in even radium isotopes is the subject of several theoretical analyses; see the review in Ref. [Butler and Nazarewicz(1996)] and the more recent studies in Refs. [Garrote et al.(1997)Garrote, Egido, and Robledo,Tsvetkov et al.(2002)Tsvetkov, Kvasil, and Nazmitdinov]. To assess the ability of the Skyrme interactions to handle it, we perform a series of Hartree-Fock (HF) + BCS calculations for even Radium isotopes. We use the Skyrme-HF+BCS code from Ref. [Bender(1998)]; it represents single-particle wave functions on an axially symmetric mesh, and uses Fourier definitions of the derivative, $1/r_\perp$, and $1/r_\perp^2$ operators. We choose 75 grid points in the $z$ direction, and 27 in the $r_\perp$ (perpendicular) direction, with 0.8 fm between them. The code uses a density-independent zero-range pairing interaction with a self-adjusting cutoff as described in Ref. [Bender et al.(2000)Bender, Rutz, Reinhard, and Maruhn]. For each Skyrme force we adjust the pairing strength separately for protons and neutrons [Bender et al.(2000)Bender, Rutz, Reinhard, and Maruhn]. We should note that other self-consistent mean-field models, namely HF+BCS with the nonrelativistic Gogny force [Egido and Robledo(1989)] and the relativistic mean-field model [Rutz et al.(1995)Rutz, Maruhn, Reinhard, and Greiner], yield results that are similar to those we describe now.

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 $N=126$ to a quadrupole deformed (reflection-symmetric) shape at $N=130$, then to quadrupole+octupole deformed (reflection-asymmetric) shapes for $132 \leq N \leq 140$, and finally back to quadrupole deformed shapes at higher $N$. 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 $^{225}$Ra, with $N=137$, will clearly be well deformed in both the quadrupole and octupole coordinates. The structures at small radii visible for $N \geq 132$ reflect small oscillations of the density distribution around the saturation value (for a given neutron excess) caused by shell effects.

Figure 2: Relative error in binding energy (top) and predicted two-neutron separation energies (bottom) for four Skyrme interactions in a series of even-$N$ Radium isotopes. The experimental separation energies are also shown.

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 $^{222}$Ra and $^{224}$Ra. This fact supports our belief that our mean-field calculations supply a good approximation to the intrinsic structure in $^{225}$Ra.

Figure 2 shows the relative error in the predicted binding energies $\delta E = (E_{\text{calc}}-E_{\text{expt}})/E_{\text{expt}}$ 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 $N$ is reflected in the near perfect agreement in the bottom panel with the measured two-neutron separation energies $S_{2n}$. The errors in predicted values of $S_{2n}$ around $N=128$ probably reflect the deficiencies of mean-field models in transitional nuclei.

Figure 3: The predicted first-order [Ring and Schuck(1980)] octupole deformations (top), intrinsic dipole moments (middle) and intrinsic Schiff moments (bottom) for four Skyrme interactions in a series of even-$N$ Radium isotopes. The experimental intrinsic dipole moments are also shown. Where symbols are missing, the corresponding predicted values are zero because the mean field is not asymmetrically deformed.

Figure 3 shows three parity-violating intrinsic quantities. In the top panel is the ground-state octupole deformation $\beta_3 = 4 \pi \langle r^3 Y_{30} \rangle / (3 A R^3)$ (where $R=1.2 A^{1/3}$), as a function of neutron number. The trend mirrors that in the density profiles shown earlier. At $N=136$, one less neutron than in $^{225}$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 $\beta_3 = 0.105 (4)$ for $N=138$, a value that can be deduced from the $B(E3;0^+_1 \to 3^-_1)$ given in Ref. [Spear(1989)]. (In $^{224}$Ra and $^{226}$Ra, by the way, we agree fairly well with the quadrupole moments obtained from $B(E2)$'s in Ref. [Raman et al.(2001)Raman, Jr., and Tikkanen]. For example, SkO' gives $\beta_2 = 0.184$ in $^{224}$Ra and experiment gives $\beta_2 = 0.179(4)$.)

The second panel in the figure shows the absolute values of intrinsic dipole moments $D_0 = e \sum_p \langle z_p \rangle$, along with experimental data extracted from $E1$ transition probabilities [Butler and Nazarewicz(1996)]. The calculated values for $D_0$ change sign from positive to negative between $N=134$ and $N=138$, 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 $N=136$. 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 $D_0$ 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 $\langle S_z \rangle$, 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 $^{224}$Ra, for example, SkO' gives $\beta_3=0.141$, $D_0=-0.103$ $e$fm, and $\langle S_z \rangle =34.4$ $e$fm$^3$ without pairing, and $\beta_3=0.143$, $D_0=-0.093$ $e$fm, and $\langle S_z \rangle =34.3$ $e$fm$^3$ when pairing is included. In this nucleus uncertainties related to pairing are very small.

Figure 4: Single-particle spectra for protons (top) and neutrons (bottom) in $^{224}$Ra, for the four Skyrme interactions.

Finally, in Fig. 4, we show the predicted proton and neutron single-particle spectra generated by the ground-state mean-field in $^{224}$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 $Z=88$ and $N=132$ shell closures for all interactions, and a somewhat weaker $N=136$ 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 $j_z=1/2$, implying that in $^{225}$Ra the ground-state parity-doublet bands will be built on $J^\pi=1/2^\pm$ states. For SkM$^*$ the situation is less clear because the $j_z=1/2$ and 3/2 states are nearly degenerate, and it is necessary to carry out the calculation in $^{225}$Ra itself to see which becomes the lowest.