Neutron skin thickness from the measured electric dipole polarizability in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$

The information on the symmetry energy and its density dependence is deduced by comparing the available data on the electric dipole polarizability ${\alpha }_D$ of ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ with the predictions of the random-phase approximation, using a representative set of nuclear energy density functionals. The calculated values of ${\alpha }_D$ are used to validate different correlations involving ${\alpha }_D$, the symmetry energy at the saturation density $J$, the corresponding slope parameter $L$, and the neutron skin thickness $\mathrm{\Delta }{r}_{np}$, as suggested by the droplet model. A subset of models that reproduce simultaneously the measured polarizabilities in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ are employed to predict the values of the symmetry energy parameters at saturation density and $\mathrm{\Delta }{r}_{np}$. The resulting intervals are $J=30–35$ MeV, $L=20–66$ MeV; and the values for $\mathrm{\Delta }{r}_{np}$ in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ are in the ranges 0.15–0.19, 0.12–0.16, and 0.13–0.19 fm, respectively. The strong correlation between the electric dipole polarizabilities of two nuclei is instrumental to predict the values of electric dipole polarizabilities in other nuclei.

Figure 1

Plots for the (a) dipole polarizability and (b) product of dipole polarizability times the symmetry energy at saturation $J$ as a function of the neutron skin thickness for ${}^{68}\text{Ni}$ calculated using a large representative set of the EDFs [19, 43]. Values of $r=0.65$ and $r=0.94$ for the respective correlation coefficients are also displayed. The linear fit to the predictions in panel (b) gives ${\alpha }_{D}J=(27\pm 15)+(570\pm 33)\mathrm{\Delta }{r}_{np}$ and the inner (outer) shadowed regions depict the loci of the 95% confidence (prediction) bands of regression (see, e.g., Chap. 3 of Ref. [50]). The symbols that are circled (in red [gray]) correspond to those models that are compatible with experiments on the dipole polarizability of both ${}^{68}\text{Ni}$ and ${}^{208}\text{Pb}$.

Figure 2

(a) Dipole polarizability times the symmetry energy at saturation $J$ of each EDF against the neutron skin thickness in ${}^{120}\text{Sn}$ predicted by nuclear EDFs [19, 43]. The correlation coefficient is $r=0.95$. The linear fit gives ${\alpha }_{D}J=(115\pm 36)+(1234\pm 93)\mathrm{\Delta }{r}_{np}$ and the inner (outer) shadowed regions depict the loci of the 95% confidence (prediction) bands of the regression (see, e.g., Chap. 3 of Ref. [50]). The symbols that are circled (in red [gray]) correspond to the models that are compatible with experiments on the dipole polarizability in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$. (b) Dipole polarizability in the even tin isotopes of $A=118–130$ times $J$ as a function of the mass number. The empty (full) symbols correspond to calculations that include (neglect) pairing correlations. In this panel ${\alpha }_D$ is multiplied by $J$ with the purpose of separating the predictions of the different models.

Figure 3

(a) The product ${\alpha }_{D}J$ in ${}^{208}\text{Pb}$ against the same product in ${}^{68}\text{Ni}$ and ${}^{120}\text{Sn}$; in both cases the resulting correlation coefficients are exceptionally high $(r=0.99)$. The deduced linear fits give ${\alpha }_D({}^{208}\text{Pb})J=(16\pm 2)+(4.7\pm 0.1){\alpha }_D({}^{68}\text{Ni})J$ and ${\alpha }_D({}^{208}\text{Pb})J=(-42\pm 4)+(2.4\pm 0.1){\alpha }_D({}^{120}\text{Sn})J$. (b) Same as for panel (a) but for the pair ${}^{120}\text{Sn}-{}^{68}\text{Ni}$ with a correlation coefficient of $r=0.98$. The linear fit gives ${\alpha }_D\left({}^{120}\text{Sn}\right)J=(16\pm 2)+(2.1\pm 0.1){\alpha }_D\left({}^{68}\text{Ni}\right)J$.

Figure 4

Comparison of the theoretical results for the dipole polarizability with the experimental data. (a) ${}^{68}\text{Ni}\left(3.88\pm 0.31{\text{fm}}^3\right)$ and ${}^{208}\text{Pb}(19.6\pm 0.6{\text{fm}}^3$, taking into account the subtraction of the quasideuteron excitations $0.51\pm 0.15{\text{fm}}^3)$. The linear fit gives ${\alpha }_D\left({}^{208}\text{Pb}\right)=(-0.5\pm 0.5)+(5.0\pm 0.2){\alpha }_D({}^{68}\text{Ni}$) and a correlation coefficient $r=0.96$. (b) ${}^{120}\text{Sn}(8.59\pm 0.37{\text{fm}}^3$, taking into account the subtraction of the quasideuteron excitations $0.34\pm 0.08{\text{fm}}^3)$ and ${}^{208}\text{Pb}$. The linear fit gives ${\alpha }_D\left({}^{208}\text{Pb}\right)=(0.1\pm 0.5)+(2.2\pm 0.1){\alpha }_D({}^{120}\text{Sn}$) and a correlation coefficient $r=0.96$. The symbols that are circled in red (gray) correspond to the models that are compatible with experiments on the dipole polarizability in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$.

Figure 5

$J$ vs $L$ plot showing the constraints obtained in Eqs. (12, 13, 14). We also display the predictions of the EDFs employed in this work. We highlight the models that reproduce the experimental ${\alpha }_D$ in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ by using red (gray) circles.

Figure 6

Dipole polarizability (a) in ${}^{48}\text{Ca}$ and (b) in ${}^{90}\text{Zr}$ as a function of the dipole polarizability in ${}^{208}\text{Pb}$. The linear fits are (a) ${\alpha }_D({}^{48}\text{Ca})=(0.36\pm 0.07)+(0.10\pm 0.01){\alpha }_D({}^{208}\text{Pb}$) with a correlation coefficient $r=0.82$ and (b) ${\alpha }_D({}^{90}\text{Zr})=(1.1\pm 0.1)+(0.24\pm 0.02){\alpha }_D({}^{208}\text{Pb})$ with a correlation coefficient $r=0.91$. Dipole polarizability (c) in ${}^{48}\text{Ca}$ and (d) in ${}^{90}\text{Zr}$ times the symmetry energy at saturation as a function of the neutron skin thickness $\mathrm{\Delta }{r}_{np}$ for the corresponding nuclei predicted by the selected EDFs. The linear fits are (c) ${\alpha }_{D}J=12\pm 19+(355\pm 44)\mathrm{\Delta }{r}_{np}$ with a correlation coefficient $r=0.84$ and (d) ${\alpha }_{D}J=101\pm 26+(1130\pm 90)\mathrm{\Delta }{r}_{np}$ with a correlation coefficient $r=0.92$. The red (gray) circles highlight the interactions that reproduce the experimental data in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$. The inner (outer) colored regions depict the loci of the 95% confidence (prediction) bands of the regression.