Minimal nuclear energy density functional

We present a minimal nuclear energy density functional (NEDF) called “SeaLL1” that has the smallest number of possible phenomenological parameters to date. SeaLL1 is defined by seven significant phenomenological parameters, each related to a specific nuclear property. It describes the nuclear masses of even-even nuclei with a mean energy error of $0.97\mathrm{MeV}$ and a standard deviation of $1.46\mathrm{MeV}$, two-neutron and two-proton separation energies with rms errors of $0.69\mathrm{MeV}$ and $0.59\mathrm{MeV}$ respectively, and the charge radii of 345 even-even nuclei with a mean error ${\epsilon }_r=0.022\mathrm{fm}$ and a standard deviation ${\sigma }_r=0.025\mathrm{fm}$. SeaLL1 incorporates constraints on the equation of state (EoS) of pure neutron matter from quantum Monte Carlo calculations with chiral effective field theory two-body ($\mathit{NN}$) interactions at the next-to-next-to-next-to leading order (N3LO) level and three-body ($\mathit{NNN}$) interactions at the next-to-next-to leading order (N2LO) level. Two of the seven parameters are related to the saturation density and the energy per particle of the homogeneous symmetric nuclear matter, one is related to the nuclear surface tension, two are related to the symmetry energy and its density dependence, one is related to the strength of the spin-orbit interaction, and one is the coupling constant of the pairing interaction. We identify additional phenomenological parameters that have little effect on ground-state properties but can be used to fine-tune features such as the Thomas-Reiche-Kuhn sum rule, the excitation energy of the giant dipole and Gamow-Teller resonances, the static dipole electric polarizability, and the neutron skin thickness.

Figure 1

The differences $E_{\text{exp}}-E_{\mathrm{th}}$ in MeVs between the evaluated ground state energies $E_{\text{exp}}(N,Z)$ [18, 19] of 2375 nuclei with $A\ge 16$ and fitted with the six-parameter mass formula $E_{\mathrm{th}}=E(N,Z)$ [Eq. (2)] with $\mathrm{\Delta }\equiv 0$. One can easily identify the location of closed shells (the blue regions) for protons and neutrons.

Figure 2

The binding energy per nucleon $B/A=\left|E(N,Z)\right|/A$ and the Coulomb, surface and symmetry energy per nucleon in Eq. (2) for the measured 2375 nuclei with $A\ge 16$ [18, 19].

Figure 3

The QMC results of Wlazłowski et al. [131] for the interaction energy per neutron displayed as the ratio ${\mathcal{E}}_{\text{int}}/{\mathcal{E}}_{\text{FG}}$ defined in Eq. (15b) (with $\beta =1$), where ${\mathcal{E}}_{\text{FG}}=3{\hslash }^2{\left(3{\pi }^2{n}_n\right)}^{2/3}{n}_n/\left(10m_n\right)$. If $a_n=0$ in Eq. (15b), the ratio ${\mathcal{E}}_{\text{int}}/{\mathcal{E}}_{\text{FG}}$ would tend to 0 for $n_n\to 0$. For densities $n_n^{1/3}\left|a_{nn}\right|<1$ (where $a_{nn}=-18.9\mathrm{fm}$ is the $s$-wave neutron-neutron scattering length), the leading-order correction to the kinetic energy density per particle contribution would be instead linear in density $4\pi {\hslash }^2{a}_{nn}{n}_n/m_n$.

Figure 4

The contribution to the ground-state energies of the terms quartic in isospin density $\delta E_{I4}=\int d^3\mathit{r}{\mathcal{E}}_2\left(n\right){\beta }^4$, evaluated perturbatively with NEDF-1 (see Table 4). In the lower panel, we display the ratio $(N-Z)/A$ for the nuclei we have considered. Among the 2375 nuclei we have considered, there are 33 nuclei with $N=Z$, 78 nuclei with $Z>N$, and 70 nuclei with $\left|N-Z\right|/A>1/4$.

Figure 5

Saturation density $n_0$ dependence of the energy residual ${\chi }_E$ and charge radii residual ${\chi }_r$ of the SeaLL1 functional. After holding $n_0$ fixed (through the parameter $c_0$), the remaining five shaded parameters in Table 2 were fit by minimizing only ${\chi }_E^2=\sum {\left|E_{N,Z}-E(N,Z)\right|}^2/N_E$ over the $N_E=196$ spherical even-even nuclei with $A\ge 16$ measured (not extrapolated) from Audi et al. [18], Wang et al. [19]. The value $n_0=0.154{\mathrm{fm}}^{-3}$ fixed in the SeaLL1 functional represents a compromise between these residuals here both ${\chi }_E$ and ${\chi }_r$ increase by about 10%.

Figure 6

The principal component analysis of the SeaLL1 NEDF in the case of the orbital-free (a) and orbital-based (b) approach.

Figure 7

The changes in ${\chi }_E$ and ${\chi }_r$ for the $N_E=196$ even-even spherical nuclei with $A\ge 16$, similarly to Fig. 5 as a function of the fixed parameter $a_0$, while the rest of the seven parameters of SeaLL1 specified in Table 2 are optimized.

Figure 8

Mass residuals between SeaLL1 and measured masses for 606 even-even nuclei, of which 410 are deformed nuclei and 196 are spherical nuclei, plotted with red squares and blue bullets respectively as a function of proton number $Z$ (a) and neutron number $N$ (b).

Figure 9

The histogram of the mass residuals between SeaLL1 and experiment for 606 even-even nuclei.

Figure 10

The residual of the two-nucleon separation energies between SeaLL1 and experiment for 606 even-even nuclei: $S_{2p}\left(Z\right)$ for constant $N$ (a) and $S_{2n}\left(N\right)$ for constant $Z$ (b) chains connected by lines.

Figure 11

The calculated proton $n_p\left(r\right)$ (dashed) and charge $n_{\text{ch}}\left(r\right)$ (dotted) densities for ${}^{48}\mathrm{Ca}$ (red) and ${}^{208}\mathrm{Pb}$ (blue), calculated with SeaLL1 compared to charge densities (solid) extracted from electron scattering experiments [157].

Figure 12

Radii residuals between SeaLL1 and experiment for 345 even-even nuclei. Isotonic (a) and isotopic (b) chains are connected by lines.

Figure 13

Single particle energies in ${}^{48}\mathrm{Ca}$ (a) and ${}^{208}\mathrm{Pb}$ (b) for a variety of functionals UNEDF0-2 [73, 74, 75] and SeaLL1 (calculated using the hfbtho DFT solver [154]).

Figure 14

Two-dimensional potential energy surface of ${}^{240}\mathrm{Pu}$ with SeaLL1 for $0\le Q_{20}\le 200\mathrm{b},0\le Q_{30}\le 40{\mathrm{b}}^{3/2}$. The least-energy fission path is marked as white dashed line.

Figure 15

Fission pathway for ${}^{240}\mathrm{Pu}$ along the mass quadrupole moment $Q_{20}$ calculated using hfbtho with SeaLL1, SkM*, and UNEDF1-HFB.

Figure 16

Fully self-consistent calculations of the proton and neutron driplines for the SeaLL1 NEDF (thick blue line) compared with predictions of the functionals SLy4 and UNEDF1 extracted from Ref. [169], and FRLDM [66]. The vertical axis is shifted by the approximate $\beta$-stability line $Z_{\beta }\left(N\right)$ which minimizes Eq. (1) at constant $A$ with parameters from Table 1: ${\partial }_Z{E(A-Z,Z)|}_{Z=Z_{\beta }}=0, Z_{\beta }=A/(2+a_C{A}^{2/3}/2a_I)$. The inset shows the usual $Z$ vs $N$ plot, with the $Z=Z_{\beta }\left(N\right)$ curve as a solid (yellow) line. The 2375 nuclear masses from Refs. [18, 19] are displayed as dots. We have plotted possible $r$-process trajectories predicted to be realized in the case of two neutron star mergers [16, 17] (red circles), in a classical hot $(n,\gamma )\leftrightarrow (\gamma ,n)$ in equilibrium $r$-process [170] (green circles) with the FRDM model [66] and neutron star merger with the UNEDF1 functional [74] (blue circles). With pink and green bands we display the $r$-process paths obtained by Mendoza-Temis et al. [171] under various conditions using the FRDM model [66] and the Duflo-Zuker model [172].

Figure 17

(a) Energy per baryon in the pasta phase ($E_{\text{past}}$), energy per neutron in pure neutron matter ($E_{\text{pnm}}$), and energy per baryon in uniform nuclear matter ($E_{\text{uni}}$) as a function of average baryon density. (b) Charge ratio of the nuclear pasta as a function of average baryon density. (c) The energy per nucleon difference between the uniform and the inhomogeneous matter configurations in $\beta$ equilibrium as a function of the average baryon density.

Figure 18

The energy per nucleon for pure neutron matter and symmetric neutron matter used in SeaLL1, compared to the corresponding energies used by Fayans [143] and Baldo et al. [211, 212, 213]. For comparison, we have shown with a dashed line the results of the QMC calculation of Wlazłowski et al. [131], with $2N$ and $3N$ interactions as well the result with the $2N$ interactions alone.

Figure 19

The blue pluses show the results obtained using the orbital-free approximation with ${\chi }_E=2.86\mathrm{MeV}$, while the red crosses are the results of the fits using nuclear mass formula Eq. (2), with ${\chi }_E=2.64\mathrm{MeV}$. When compared against each other, the rms energy deviation between the two fits is $\mathrm{\Delta }{\chi }_E=1.10\mathrm{MeV}$. Thus, the orbital-free theory essentially reproduces the nuclear mass formula Eq. (2). The main plot is the same as in Fig. 1, in which one can see clearly the magic numbers separately for neutrons and protons.

Figure 20

Principal component analysis for the NEDF-1 fit (a) and the NEDF-3 fit (b). Plotted are the components of the eigenvectors ${\mathit{v}}_n$ defining the principal component Eq. (25b) as linear combinations of the various dimensionless parameters. From this, we see that for NEDF1 the most significant component $p_0\approx {\tilde{b}}_0+{\tilde{c}}_0$, which fixes the saturation energy to high precision. At the same time, the component $p_4\approx {\tilde{b}}_0-{\tilde{c}}_0$ in NEDF-1 (and similarly in NEDF-3n) is not well constrained. We also see that the least-significant component $p_5\approx {\tilde{a}}_1-{\tilde{b}}_1$ is essentially unconstrained. For NEDF-3, we find three insensitive components, two of which can be used to set the smallest parameters $a_0=c_1=0$. After removing these, one obtains a similar analysis as for NEDF-1 above.

Figure 21

The various ellipses show the region in the $({\varepsilon }_0,n_0)$ plane, in which the NEDF parameters can be changed and to lead to changes in the residual $\delta {\chi }_E<0.2\mathrm{MeV}$. While the equilibrium energy ${\varepsilon }_0$ and density $n_0$ are controlled mainly by the combination ${\tilde{b}}_0+{\tilde{c}}_0$, which is constrained with very high precision, the combination ${\tilde{b}}_0-{\tilde{c}}_0$ has significantly less constraint; see Sec. 3i. This aspect allows us to manipulate to a certain degree the saturation properties, while affecting the overall fit only slightly.

Figure 22

The energy density per nucleon for (a), pure neutron matter for NEDF-0, 1, 1r, 2, 3, E, and Er, which do not constrain the neutron EoS and have only ${\beta }^2$ contributions; and (b), symmetric nuclear matter for all orbital-free functionals, and neutron matter for NEDF-3n, 3nr, En, and Enr, which collapse to the single curve fitting the QMC results [131] (dots).

Figure 23

The contribution to the ground-state energies of the terms quartic in isospin density $\delta E_{I4}=\int {\mathrm{d}}^3\mathit{r}{\mathcal{E}}_2\left(n\right){\beta }^4$, evaluated perturbatively with NEDF-1; see Table 4.