Beyond-mean-field effects on the symmetry energy and its slope from the low-lying dipole response of ${}^{68}\mathrm{Ni}$

We study low-energy dipole excitations in the unstable nucleus ${}^{68}\mathrm{Ni}$ with the beyond-mean-field (BMF) subtracted second random-phase approximation (SSRPA) model based on Skyrme interactions. First, strength distributions are compared with available experimental data and transition densities of some selected peaks are analyzed. The so-called isospin splitting is also discussed by studying the isoscalar-isovector character of such excitations. We estimate then in an indirect way BMF effects on the symmetry energy of infinite matter and on its slope, starting from the BMF SSRPA low-lying strength distribution. For this, several linear correlations are used, the first one being a correlation existing between the contribution (associated with the low-energy strength) to the total energy-weighted sum rule (EWSR) and the slope of the symmetry energy. BMF estimates for the slope of the symmetry energy can be extracted in this way. Correlations between such a slope and the neutron-skin thickness of ${}^{68}\mathrm{Ni}$ and correlations between the neutron-skin thickness of ${}^{68}\mathrm{Ni}$ and the electric-dipole polarizability times the symmetry energy are then used to deduce BMF effects on the symmetry energy.

Figure 1

(a) SSRPA dipole strength distribution obtained for ${}^{68}\mathrm{Ni}$ with the parametrization SGII. (b) RPA dipole strength distribution obtained for ${}^{68}\mathrm{Ni}$ with the parametrization SGII. The vertical black line represents the neutron threshold.

Figure 2

(a) SSRPA dipole strength distribution obtained for ${}^{68}\mathrm{Ni}$ with the parametrization SGII. (b) Photoabsorption cross section extracted from Fig. 3 of Ref. [12]. (c) $E1$ strength distribution extracted from Fig. 3 of Ref. [13]. The vertical black line represents the neutron threshold. The vertical magenta dotted lines correspond to the energy values of the two experimental low-energy centroids measured in Ref. [12] [panel (b)] and in Ref. [13] [panel (c)].

Figure 3

(a), (c), (e), (g), (i) Neutron and proton transition densities and (b), (d), (f), (h), (j) isoscalar and isovector transition densities. Transition densities are calculated for the peaks shown in Fig. 2 and located at 9.14 MeV [panels (a) and (b)], 9.70 MeV [panels (c) and (d)], 10.25 MeV [panels (e) and (f)], 11.10 MeV [panels (g) and (h)], and 11.31 MeV [panels (i) and (j)]. The Skyrme parametrization used is SGII. The transition densities are multiplied by $r^2$ and are thus expressed in units of ${\mathrm{fm}}^{-1}$.

Figure 4

(a) Isovector and (b) isoscalar low-lying strength distributions, corresponding respectively to the operators (2) and (3). Full lines represent the projected results whereas the dotted lines are obtained without correcting for spurious components.

Figure 5

(a) Mean-field EOSs of pure neutron matter computed with the four Skyrme parametrizations SIII, SGII, SkI3, and SkI4. (b) Mean-field symmetry-energy coefficient associated with the four Skyrme parametrizations used in panel (a).

Figure 6

(a) Percentage of the EWSR calculated up to 12.5 MeV for the nucleus ${}^{68}\mathrm{Ni}$ with the RPA model with the four Skyrme parametrizations SIII, SGII, SkI3, and SkI4 (red squares) as a function of the slope $L$ (mean-field values) of the symmetry energy. A linear fit is carried out on the four points (dotted line). The gray area represents the associated uncertainty band (see text). The BMF SSRPA% EWSR value is included for the case SGII (horizontal green line) and a BMF SGII value of $L$ is correspondingly extracted (vertical green line) with an associated uncertainty (orange area). (b) Same as in panel (a), where the EWSR percentage is computed up to 12 MeV.

Figure 7

Neutron and proton transition densities for the state located at 12.17 MeV. The transition densities are multiplied by $r^2$ and are thus expressed in units of ${\mathrm{fm}}^{-1}$.

Figure 8

Neutron-skin thickness computed for the nucleus ${}^{68}\mathrm{Ni}$ with mean-field calculations using the four Skyrme parametrizations SIII, SGII, SkI3, and SkI4 as a function of the slope $L$ (mean-field values) of the symmetry energy (red squares). A linear fit is carried out on the four points (dotted line) and an uncertainty band (gray area) is also displayed (see text). The BMF $L$ value for SGII is included (vertical green line) with its uncertainty band which is represented by vertical orange dashed lines, and the corresponding BMF value for $\mathrm{\Delta }{r}_{np}$ is extracted (horizontal green line) with an associated uncertainty (orange area).

Figure 9

Electric-dipole polarizability computed with the RPA model for the nucleus ${}^{68}\mathrm{Ni}$ times the mean-field value of $J$ versus the mean-field value of $\mathrm{\Delta }{r}_{np}$, using the four Skyrme parametrizations SIII, SGII, SkI3, and SkI4 (red squares). A linear fit is carried out on the four points (dotted line) and an uncertainty band (gray area) is also displayed (see text). The BMF value for $\mathrm{\Delta }{r}_{np}$ is included for SGII (vertical green line) with its uncertainty band represented by vertical orange dashed lines. The corresponding BMF value of $\alpha J$ is extracted (horizontal green line) with its uncertainty (orange area). Since $\alpha$ is the same in RPA and in SSRPA, this procedure provides a BMF value for $J$.

Figure 10

Experimental constraints coming from the measurement of the electric-dipole polarizability in ${}^{208}\mathrm{Pb}$ extracted from Refs. [35] (region between the blue solid lines) and [42] (region between the indigo dashed lines). Experimental constraints provided by heavy-ion collisions are also reported (yellow area), which are extracted from Ref. [42]. The mean-field and the BMF points are indicated for the Skyrme parametrization SGII as a red square and a dark-green diamond, respectively. The uncertainty associated with the BMF value is also reported as a light-green area.