Extended Fayans energy density functional: optimization and analysis

Abstract

The Fayans energy density functional (EDF) has been very successful in describing global nuclear properties (binding energies, charge radii, and especially differences of radii) within nuclear density functional theory. In a recent study, supervised machine learning methods were used to calibrate the Fayans EDF. Building on this experience, in this work we explore the effect of adding isovector pairing terms, which are responsible for different proton and neutron pairing fields, by comparing a 13D model without the isovector pairing term against the extended 14D model. At the heart of the calibration is a carefully selected heterogeneous dataset of experimental observables representing ground-state properties of spherical even-even nuclei. To quantify the impact of the calibration dataset on model parameters and the importance of the new terms, we carry out advanced sensitivity and correlation analysis on both models. The extension to 14D improves the overall quality of the model by about 30%. The enhanced degrees of freedom of the 14D model reduce correlations between model parameters and enhance sensitivity.

Figures (8)

• Figure 1: (top) Residual values for the 13D and 14D solutions. (middle) Change in residual values between the 13D and 14D solutions. (bottom) Change in residual magnitude between the 13D and 14D solutions. A negative value indicates that the magnitude of the associated residual decreased as a result of freeing $f_{\mathrm{ex},-}^{\xi}$. The elements are grouped in observable classes of tab:observableClasses with an ordering, from left to right, of $E_{\rm B}$, $R_{\mathrm{box}}$, $\sigma$, $r_{\mathrm{ch}}$, $\epsilon_{\mathrm{ls,p}}$, $\epsilon_{\mathrm{ls,n}}$, $\delta\langle r^2\rangle$, $\Delta^{(3)} E_n$, and $\Delta^{(3)} E_p$.
• Figure 2: Breakdown of the contributions to the total objective function $f$ by observable class (see tab:observableClasses) for the Fy($\Delta r$,13D) and Fy($\Delta r$,14D) parameterizations (see tab:PointEstimates). Presented are the (bottom) summed contribution $f_\mathrm{class}$ within a class and the (top) average contribution per data point $f_\mathrm{class}/N_\mathrm{class}$, where $N_\mathrm{class}$ is the number of data points in the given class.
• Figure 3: Coefficients of determination $R_{\alpha \beta}^2$ for the 13D (lower triangle) and 14D (upper triangle) calibrations. The parameters are ordered to highlight their correlations.
• Figure 4: Relative sensitivities per data class (\ref{['eq:rel_sens_class']}) for the model parameters of the 13D Fy($\Delta r$,13D) (bottom) and 14D Fy($\Delta r$,14D) (top) EDFs. The data classes are represented by colors as indicated. The $\delta\langle r^2\rangle$ represent the isotopic radius$^2$ differences and the $\Delta^{(3)}E$ the odd-even staggerings of energies.
• Figure 5: Total impact of a data point $i$ on the parameters of the Fy($\Delta r$,14D) (a) and Fy($\Delta r$,13D) (b) EDFs. The data classes are separated by dashed vertical lines as in fig:residual_changes. The data points having the largest impact on calibration results are indicated.