Nuclear energy density optimization

We carry out state-of-the-art optimization of a nuclear energy density of Skyrme type in the framework of the Hartree-Fock-Bogoliubov theory. The particle-hole and particle-particle channels are optimized simultaneously, and the experimental data set includes both spherical and deformed nuclei. The new model-based, derivative-free optimization algorithm used in this work has been found to be significantly better than standard optimization methods in terms of reliability, speed, accuracy, and precision. The resulting parameter set unedf0 results in good agreement with experimental masses, radii, and deformations and seems to be free of finite-size instabilities. An estimate of the reliability of the obtained parameterization is given, based on standard statistical methods. We discuss new physics insights offered by the advanced covariance analysis.

Figure 1

Experimental set of fit observables used in this work. The set contains data for 11 nuclei with $A<66$ and 61 nuclei with $A>106$.

Figure 2

Convergence of the minimization of Eq. (36) with the standard N-M algorithm (dashed line) and the model-based pounders (solid line).

Figure 3

Dielectric function $D(\omega =0,\mathbf{q})$ for the scalar-isovector channel in SLy4, unedf0, and unedfnb as a function of the transferred momentum $q$ for $k_F=1.33$ fm${}^{-1}$.

Figure 4

Neutron single-particle energies in ${}^{48}\mathrm{Ca}$ obtained from a HF calculation with the functional unedf0. Experimental s.p. levels [96] and SLy4 results are shown for comparison.

Figure 5

Similar as in Fig. 4 but for proton single-particle energies in ${}^{208}\mathrm{Pb}$.

Figure 6

Deformation energy curves as functions of the quadrupole deformation ${\beta }_2$ for selected even-even Zr isotopes calculated in the $\mathrm{HFB}+\mathrm{LN}$ approach with unedf0 Skyrme functional.

Figure 7

Binding energy residuals between theory and experiment for 520 even-even nuclei. The $\mathrm{HFB}+\mathrm{LN}$ results with SLy4 (top) are compared with those of unedf0 (middle) and unedfnb (bottom).

Figure 8

Two-neutron separation energy residuals between theory and experiment for 520 even-even nuclei. The $\mathrm{HFB}+\mathrm{LN}$ results with SLy4 (top) are compared with those of unedf0 (middle) and unedfnb (bottom).

Figure 9

Neutron (left) and proton (right) OES residuals ${\Delta }_{\mathrm{th}}-{\Delta }_{\exp }^{(3)}$ for the nuclei listed in Table . The results with fixed (nonoptimized) values of $V_0^n=V_0^p$ are marked with upside-down triangles. The optimized results are marked with triangles ($V_0^n=V_0^p$), dots ($V_0^n\ne V_0^p$; unedfnb), and squares ($V_0^n\ne V_0^p$; unedf0).

Figure 10

Sensitivity of the parameters of unedf0 to different data types entering ${\chi }^2$. The EDF parameters are labeled as in Table .

Figure 11

Overall change in $\stackrel{\wedge }{\mathbf{x}}$ (51) for the unedfnb and unedf0 parameter sets when data $d_{i,j}$ change by $0.1w_i$ one by one. The labeling of data points is consistent with that in Tables .