Global systematics of octupole excitations in even-even nuclei

We present a computational methodology for a theory of the lowest axially symmetric octupole excitations applicable to all even-even nuclei beyond the lightest. The theory is the well-known generator-coordinate extension (GCM) of the Hartree-Fock-Bogoliubov (HFB) self-consistent mean field theory. We use the discrete-basis Hill-Wheeler (HW) method to compute the wave functions with an interaction from the Gogny family of Hamiltonians. Comparing to the compiled experimental data on octupole excitations, we find that the performance of the theory depends on the deformation characteristics of the nucleus. For nondeformed nuclei, the theory reproduces the energies to about $\pm 20$% apart from an overall scale factor of $\approx 1.6$. The performance is somewhat poorer for (quadrupole) deformed nuclei, and for both together the dispersion of the scaled energies about the experimental values is about $\pm 25$%. This compares favorably with the performance of similar theories of the quadrupole excitations. Nuclei having static octupole deformations in HFB theory form a special category. These nuclei have the smallest measured octupole excitation energies as well as the smallest predicted energies. However, in these cases the energies are seriously underpredicted by the theory. We find that a simple two-configuration approximation, the minimization after projection (MAP) method, is almost as accurate as the full HW treatment, provided that the octupole-deformed nuclei are omitted from the comparison. This article is accompanied by a tabulation of the predicted octupole excitations for 818 nuclei extending from drip-line to drip-line, computed with several variants of the Gogny interaction.

Figure 1

Energy of ${}^{208}$Pb as a function of octupole deformation ${\beta }_3$. Open circles: HFB energy of constrained configurations; solid squares: energy $E_e$ of the even-parity projected wave function; solid circles: the odd-parity projected energy $E_o$. See the Appendix for explanation of the fitted lines.

Figure 2

Excitation energy $E_3$ in ${}^{208}$Pb as a function of the configuration space choice. Solid circles: HW using the singular value decomposition to keep $N_{\mathrm{basis}}$ states; solid square: HW with the two MAP states; open square: energy difference of the two MAP states.

Figure 3

Wave function amplitudes. See text for explanation.

Figure 4

Energy of ${}^{158}$Gd as a function of octupole deformation ${\beta }_3$. Open circles: HFB energy of constrained configurations; solid squares: energy $E_e$ of the even-parity projected wave function; solid circles: the odd-parity projected energy $E_o$. The line along the HFB values is the function $E_q=E_0+K_1{\beta }_3^2$ with $K_1=48.8$ MeV fitted to the values ${\beta }_3\le 0.05$. The line along the $E_e$ values is the fit motivated by the Gaussian overlap approximation, $E_e=E_q-K_2{\beta }^2/[1.0+\exp \left(\alpha {\beta }^2\right)]$, with $K_2$ and $\alpha$ fitted.

Figure 5

Energy of ${}^{226}$Ra as a function of octupole deformation ${\beta }_3$ as in Figs. 1, 4g. A very similar plot is shown in Fig. 3 of Ref. [3].

Figure 6

Energy of ${}^{20}$Ne as a function of octupole deformation ${\beta }_3$ as in Figs. 1, 4 and 5.

Figure 7

Nucleon density distribution in ${}^{20}$Ne at ${\beta }_{3p}$ (left) and ${\beta }_{3m}$ (right).

Figure 8

Deformation of the octupole-constrained HFB configurations for ${}^{16}$O and ${}^{20}$Ne.

Figure 9

Chart of the nuclides showing those calculated in the present study. Those in black have static octupole deformations in HFB. Except for the nuclei near $N\sim Z\sim 40$, the nucleon numbers correspond well to the numbers 56, 88, and 136 listed in Ref. [10] as especially favorable for octupole deformation.

Figure 10

Octupole excitation energies as a function of mass number $A$. Circles: experiment; triangles: theory.

Figure 11

Octupole excitation energies, comparing the theory with experiment. Filled circles are excitations with measured $B\left(E3\right)$ strengths; open circles are other identified octupole transitions [25].

Figure 12

Ratio of theoretical octupole transition strength to experimental, with the theoretical strength obtained using Eq. (11). The horizontal axis is the ratio $R_{42}$ from the theory of Ref. [12]. Experimental $B\left(E3\right)$ values are from Ref. [25].

Figure 13

Ratio of MAP deformations ${\beta }_{3p}/{\beta }_{3m}$ for nuclei with measured $E_3$ [25].

Figure 14

Ratio of MAP deformations ${\beta }_{3m}/{\beta }_{3p}$ for nuclei with measured $E_3$ [25].