Isovector splitting of nucleon effective masses, ab initio benchmarks and extended stability criteria for Skyrme energy functionals

We study the effect of the splitting of neutron and proton effective masses with isospin asymmetry on the properties of the Skyrme energy density functional. We discuss the ability of the latter to predict observables of infinite matter and finite nuclei, paying particular attention to controlling the agreement with ab initio predictions of the spin-isospin content of the nuclear equation of state, as well as diagnosing the onset of finite size instabilities, which we find to be of critical importance. We show that these various constraints cannot be simultaneously fulfilled by the standard Skyrme force, calling at least for an extension of its $P$-wave part.

Figure 1

SNM and PNM EOS as given by the Skyrme functionals presently discussed (see text) compared with VCS results by Akmal et al. [51] (crosses, PNM; plusses, SNM).

Figure 2

Single-particle energies (MeV) in ${}^{132}\mathrm{Sn}$ and ${}^{208}\mathrm{Pb}$ computed with the indicated forces. Thick lines indicate the Fermi level ${\varepsilon }_F$.

Figure 3

Same as Fig. 2 in ${}^{78}\mathrm{Ni}$ and ${}^{156}\mathrm{Sn}$.

Figure 4

Neutron spectral gaps computed in Sn (bottom panel) and Pb (top panel) series with $f_{-},f_0$, and $f_{+}$, as a function of asymmetry. Experimental ${\Delta }^{(5)}$ gaps extracted from masses [57] are plotted with error bars.

Figure 5

Binding energy residuals computed with forces $f_{-},f_0$, and $f_{+}$ for semi-magic series of nuclei, as indicated.

Figure 6

Energy per particle in each $(S,T)$ channel for SNM, as a function of density. Crosses refer to the BHF calculations [62].

Figure 7

Neutron and proton densities in central regions of ${}^{56}\mathrm{Ni}$ (bottom) and ${}^{40}\mathrm{Ca}$ (top) plotted for a fully converged computation using the SLy5 force (solid line; relative variation of energy between iterations less than ${10}^{-14}$) and along a series of iterations done with SkP (bottom panels) and LNS (top panels). Number of iterations as indicated in key. In both cases the collapse happens after a seemingly converged situation ($~{10}^{-9}$ relative energy variation, steady over a large number of iterations indicating a nearly linear evolution of the energy), which can be mistaken for an energy minimum if too loose a convergence criterion is used.

Figure 8

The lowest density of occurrence of a pole in ${\chi }^{\mathrm{sv}}(\omega =0,\mathbf{q})$ is plotted against the wave number $q$ of the scalar-isovector perturbation. The curves end at $q=2k_F$ because the ground state cannot couple to excitations with $\omega =0,q>2k_F$.

Figure 9

Same as Fig. 8, for all spin-isospin channels. The lower-left panel shows the region of spinodal instabilities below ${\rho }_{\mathrm{sat}}$. The domain of $q$ covered in this case determines the size of structures formed, while the region between 0.1 and $0.16\hspace{1em}{\mathrm{fm}}^{-1}$ appears to be metastable.