Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: effect of triaxial shape

Abstract

The modelling of nuclear reactions and radioactive decays in astrophysical or earth-based conditions requires detailed knowledge of the masses of essentially all nuclei. Microscopic mass models based on nuclear energy density functionals (EDFs) can be descriptive and used to provide this information. The concept of intrinsic symmetry breaking is central to the predictive power of EDF approaches, yet is generally not exploited to the utmost by mass models because of the computational demands of adjusting up to about two dozen parameters to thousands of nuclear masses. We report on a first step to bridge the gap between what is presently feasible for studies of individual nuclei and large-scale models: we present a new Skyrme-EDF-based model that was adjusted using a three-dimensional coordinate-space representation, for the first time allowing for both axial and triaxial deformations during the adjustment process. To compensate for the substantial increase in computational cost brought by the latter, we have employed a committee of multilayer neural networks to model the objective function in parameter space and guide us towards the overall best fit. The resulting mass model BSkG1 is computed with the EDF model independently of the neural network. It yields a root mean square (rms) deviation on the 2457 known masses of 741 keV and an rms deviation on the 884 measured charge radii of 0.024 fm.

Appendices

Appendix A: Coupling constants of $E_{\mathrm{Sk}}$

The Skyrme energy density $\mathcal {E}$ of Eq. (5) is determined by ten coupling constants, which are determined by the model parameters $t_{0-3}, x_{0-3}, W_0$ and $W_0'$ as follows:

$$\begin{aligned}&C^{\rho \rho }_0 = \tfrac{3}{8}t_0 \, , \end{aligned}$$

(A.1a)

$$\begin{aligned}&C^{\rho \rho }_1 = - \tfrac{1}{4}t_0 \left( \tfrac{1}{2} + x_0\right) \, , \end{aligned}$$

(A.1b)

$$\begin{aligned}&C^{\rho \rho \rho ^{\gamma }}_0 = \tfrac{3}{48} t_3 \, , \end{aligned}$$

(A.1c)

$$\begin{aligned}&C^{\rho \rho \rho ^{\gamma }}_1 = - \tfrac{1}{24} t_3 \left( \tfrac{1}{2} + x_3\right) \, , \end{aligned}$$

(A.1d)

$$\begin{aligned}&C^{\rho \tau }_0 = \tfrac{3}{16} t_1 + \tfrac{1}{4} t_2 \left( \tfrac{5}{4} + x_2\right) \, , \end{aligned}$$

(A.1e)

$$\begin{aligned}&C^{\rho \tau }_1 = - \tfrac{1}{8} t_1\left( \tfrac{1}{2} + x_1\right) - t_2 \left( \tfrac{1}{2} + x_2 \right) \, , \end{aligned}$$

(A.1f)

$$\begin{aligned}&C^{\rho \varDelta \rho }_0 = - \tfrac{9}{64} t_1 + \tfrac{1}{16} t_2 \left( \tfrac{5}{4} + x_2\right) \, , \end{aligned}$$

(A.1g)

$$\begin{aligned}&C^{\rho \varDelta \rho }_1 = \tfrac{3}{32} t_1 (\tfrac{1}{2} + x_1) + \tfrac{1}{32} t_2 (\tfrac{1}{2} + x_2) \, , \end{aligned}$$

(A.1h)

$$\begin{aligned}&C^{\rho \nabla \cdot J}_0 = - \frac{W_0}{2} - \frac{W_0'}{4} \, , \end{aligned}$$

(A.1i)

$$\begin{aligned}&C^{\rho \nabla \cdot J}_1 = - \frac{W_0'}{4} \, . \end{aligned}$$

(A.1j)

Appendix B: Further details on the rotational correction and the MOI

The rotational correction, Eq. (12a), depends on the calculation of $\langle {\hat{J}}_{\mu }^2 \rangle$ and $\mathcal {I}_{\mu }$ for all three principal axes of the nucleus. Formulas are available in the literature for even-even nuclei (see, e.g. [56, 61]). However, naively utilizing these expressions in our calculations is problematic for two reasons.

The first is purely technical: the calculation of $\mathcal {I}_{\mu }$ involves a summation over all possible two-quasiparticle excitations in the model space, weighted by the inverse of the sum of their quasiparticle energies. Unlike any other quantity discussed here, this sum is not naturally cut by the single-particle occupation factors. As our numerical implementation can only represent a fraction of the entire quasiparticle spectrum, we have introduced an additional cutoff for the rotational correction. We replace the matrix elements of the single-particle angular momentum operator ${\hat{\jmath }}$ that figure into the calculation of both $\langle {\hat{J}}^2_{\mu } \rangle$ and $\mathcal {I}_{\mu }$ as

$$\begin{aligned} \langle k | {\hat{\jmath }}_{\mu } | l\rangle&\rightarrow f_{q, k}^\mathrm{MOI} f_{q, l}^{\mathrm{MOI}} \langle k | {\hat{\jmath }}_{\mu } | l\rangle \, , \end{aligned}$$

(B.2)

$$\begin{aligned} f_{q, k}^{\mathrm{MOI}}&= \left[ 1 + e^{(\epsilon _k - \lambda _q - E_\mathrm{cut})/\mu _{\mathrm{MOI}}}\right] ^{-1/4} \, , \end{aligned}$$

(B.3)

where the cut-off energy $E_{\mathrm{cut}}$ is identical to the one used in the pairing channel (Eq. 7), but $\mu _{\mathrm{MOI}} = 1$ MeV.

The second problem affects the calculation of both quantities for odd-A and odd-odd nuclei. For the ground states of even-even nuclei, the expectation value of ${\hat{J}}_{\mu }^2$ and the Belyaev MOI are purely collective in nature, i.e. non-zero values are generated by many nucleons as a result of the nuclear deformation. This picture is modified significantly by the presence of blocked quasiparticles, whose individual contributions to both ${\hat{J}}_{\mu }^2$ and $\mathcal {I}_B$ are generally sizeable and cannot be considered as collective. Furthermore, the Belyaev MOI is fundamentally a quantity obtained from second-order perturbation theory of an HFB minimum. While its calculation can be generalized to include the possibility of blocked quasiparticles along the lines of Ref. [182], the validity of such an approach can be questioned. As a purely practical recipe to sidestep these issues, we calculate both ${\hat{J}}_{\mu }^2$ and $\mathcal {I}_B$ for odd-A and odd-odd nuclei by omitting the contributions from all blocked quasiparticles, mirroring the approach of Ref. [183].

Finally, we comment on the comparison of calculated (Belyaev) MOI with experimental data, as we do in Fig. 10. For axial configurations, the Belyaev MOI along the symmetry axis vanishes, while the two remaining values are equal; comparison to experiment is then straightforward. For triaxial nuclear configurations, we obtain however three non-zero, distinct values for the MOI. In those cases, we have chosen systematically the largest among the three values as the one to be compared to experiment. We have made this rather ad-hoc choice motivated by a naive semi-classical model of rotation, where the largest MOI produces the lowest-lying rotational excitations. As the experimental data is extracted from the excitation energy of the first $2^+$ state in rotational nuclei, this seems to be the most appropriate choice.

Table 4 Contents of the Mass_Table_BSkG1.dat file

Explanation of the supplementary material

We provide as supplementary material the file Mass_Table_BSkG1.dat, which contains the calculated ground state properties of all nuclei with $8 \le Z \le 110$ lying between the proton and neutron drip lines. Its content is summarized and explained in Table  4. A few additional remarks are in order:

• Column 11/12: A unique definition of the pairing gap exists only for HFB calculations with schematic interactions. To extract some information on the overall importance of the pairing correlations for a given nucleus, we use the uv-weighted average pairing gaps $\langle \varDelta \rangle _{n/p}$ of Ref. [184].

• Column 16: we report only the largest MOI among all three directions, i.e. the file contains $\max _{\mu = x,y,z} \{ \mathcal {I}_{\mu } \}$.

• Column 17/18: as discussed in Sect. 2.2.3, we construct auxiliary states for odd-A and odd-odd nuclei through a self-consistent blocking procedure. To make these calculations reproducible, we provide for such nuclei the parity quantum number of the blocked quasiparticles.