Giant resonances in ${}^{40,48}\mathrm{Ca}, {}^{68}\mathrm{Ni}, {}^{90}\mathrm{Zr}, {}^{116}\mathrm{Sn}, {}^{144}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$

We present results of centroid energies $E_{\mathrm{CEN}}$, of the isoscalar ($T=0$) and isovector ($T=1$) giant resonances of multipolarities $L=0–3$ in ${}^{40,48}\mathrm{Ca}, {}^{68}\mathrm{Ni}, {}^{90}\mathrm{Zr}, {}^{116}\mathrm{Sn}, {}^{144}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$, calculated within the fully self-consistent Hartree-Fock-based random phase approximation theory, using 33 different Skyrme-type effective nucleon-nucleon interactions of the standard form commonly adopted in the literature. We compare the results of our theoretical calculations with the available experimental data. We also study the sensitivity of the calculated $E_{\mathrm{CEN}}$ to physical properties of nuclear matter (NM), such as effective mass m*/m, nuclear matter incompressibility coefficient $K_{\mathrm{NM}}$, enhancement coefficient κ of the energy weighted sum rule for the isovector giant dipole resonance and symmetry energy at saturation density, associated with the Skyrme interactions used in the calculations. This is done by determining the Pearson linear correlation coefficient between the calculated $E_{\mathrm{CEN}}$ and a certain NM property. Constraining the values of the NM properties, by comparing the calculated values of $E_{\mathrm{CEN}}$ to the experimental data, we find that interactions associated with the values of $K_{\mathrm{NM}}=210–240\mathrm{MeV}$ and $\kappa =0.25–0.70$ best reproduce the experimental data.

Figure 1

Various NM properties of the Skyrme interactions are plotted against the incompressibility coefficient $K_{\mathrm{NM}}$. In each panel, from top left to bottom, we have the effective mass ${\mathrm{m}}^{*}/\mathrm{m}$, the total binding energy per nucleon $E/\mathrm{A}$, the Landau parameter $\mathrm{G}{0}^{\prime }$, the saturation density ${\rho }_0$, the symmetry energy at saturation density $J$, the first derivative of the symmetry energy $L=3{\rho }_0\frac{\partial E_{\mathrm{sym}}}{\partial \rho }{|}_{{\rho }_0}$, the second derivative of the symmetry energy $K_{\mathrm{sym}}=9{{\rho }_0}^2\frac{{\partial }^2{E}_{\mathrm{sym}}}{\partial {\rho }^2}{|}_{{\rho }_0}$ and the enhancement coefficient κ of the IVGDR EWSR. We see no strong dependence for any of these parameters and $K_{\mathrm{NM}}$, although a weak relation with ${\mathrm{m}}^{*}/\mathrm{m}$ and ${\rho }_0$ is still present.

Figure 2

Calculated centroid energies $E_{\mathrm{CEN}}$ in MeV (full circle) of the isoscalar giant monopole resonances (ISGMR) for the different interactions, as a function of the incompressibility coefficient $K_{\mathrm{NM}}$. Each nucleus has its own panel and the experimental uncertainties are contained by the dashed lines. As expected we find strong correlation between the calculated values of $E_{\mathrm{CEN}}$ and $K_{\mathrm{NM}}$ with a Pearson linear correlation coefficient $C\sim 0.87$.

Figure 3

Similar to Fig. 2, for the effective mass m*/m. We find a weak correlation between the calculated values of $E_{\mathrm{CEN}}$ and m*/m, with a Pearson linear correlation coefficient $C\sim -0.51$.

Figure 4

Similar to Fig. 2 as a function of the second derivative of the symmetry energy coefficient $K_{\mathrm{sym}}$. We find weak correlation between the calculated values of $E_{\mathrm{CEN}}$ and $K_{\mathrm{sym}}$ with a Pearson linear correlation coefficient $C\sim 0.45$.

Figure 5

The centroid energies [MeV] are plotted against the mass A of each nucleus. Each panel is a different isoscalar multipolarity, (a) $L=0$, (b) $L=1$, (c) $L=2$, and (d) $L=3$. The experimental error bars are shown by the solid vertical lines and are available for all nuclei in $L=0$ and $L=2$, for all but ${}^{68}\mathrm{Ni}$ in $L=1$, and only for the heavier nuclei, ${}^{90}\mathrm{Zr}, {}^{116}\mathrm{Sn}, {}^{144}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$ for $L=3$. The theoretical calculations are shown as dots and are connected by lines meant to guide the eye.

Figure 6

Similar to Fig. 2, for the isoscalar giant dipole resonance (ISGDR) as a function of $K_{\mathrm{NM}}$. We find a weak correlation between the calculated values of $K_{\mathrm{NM}}$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C\sim 0.52$.

Figure 7

Similar to Fig. 2, for the ISGDR as a function of ${\mathrm{m}}^{*}/\mathrm{m}$. We find strong correlation between the calculated values of $E_{\mathrm{CEN}}$ and ${\mathrm{m}}^{*}/\mathrm{m}$ the with a Pearson linear correlation coefficient $C=-0.88$.

Figure 8

Similar to Fig. 2, for the isoscalar giant quadrupole resonance (ISGQR) as a function of the effective mass m*/m. We find strong correlation between the calculated values of m*/m and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C$ close to $-0.93$ in all cases.

Figure 9

Similar to Fig. 2, for the ISGQR as a function of the incompressibility coefficient. We find a weak correlation between the calculated values of $K_{\mathrm{NM}}$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient close to $C=0.41$ for all isotopes.

Figure 10

Similar to Fig. 2, for the isoscalar giant octupole resonance (ISGOR) as a function of the effective mass m*/m. We find strong correlation between the calculated values of m*/m and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=-0.96$ in all cases.

Figure 11

Similar to Fig. 2, for the ISGOR as a function of the incompressibility coefficient $K_{\mathrm{NM}}$. We find a weak correlation between the calculated values of $K_{\mathrm{NM}}$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=0.42$.

Figure 12

Similar to Fig. 2, for the isovector giant monopole resonance (IVGMR) as a function of the incompressibility coefficient. We do not find any correlation between the calculated values of $E_{\mathrm{CEN}}$ and $K_{\mathrm{NM}}$ with a Pearson linear correlation coefficient $C=0.23$ in most cases.

Figure 13

Similar to Fig. 2, for the IVGMR as a function of the effective mass. We find medium correlation between the calculated values of $E_{\mathrm{CEN}}$ and m*/m with a Pearson linear correlation coefficient $C=-0.70$.

Figure 14

Similar to Fig. 2, for the IVGMR as a function of the symmetry energy at saturation density, $J$. We don't find any correlation between the calculated values of $J$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C\sim -0.26$.

Figure 15

Similar to Fig. 2, for the IVGMR as a function of the enhancement coefficient, κ, of the EWSR of the IVGDR. We find strong correlation between the calculated values of κ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=0.86$ for all nuclei considered.

Figure 16

The centroid energy [MeV] is plotted against the mass $A$ of each nucleus. Each panel is a different multipolarity, (a) $L=0$, (b) $L=1$, (c) $L=2$, and (d) $L=3$. The experimental error bars are shown by the solid vertical lines and are available only for ${}^{40}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ in $L=0$, for all nuclei in $L=1$, for ${}^{40}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ for $L=2$ and unavailable for all nuclei for $L=3$. The theoretical calculations are shown as dots and are connected by lines meant to guide the eye.

Figure 17

Similar to Fig. 2, for the isovector giant dipole resonance (IVGDR) as a function of $J$. We find a weak correlation between the calculated values of $J$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C\sim -0.37$.

Figure 18

Similar to Fig. 2, for the IVGDR as a function of the enhancement coefficient, κ, of the EWSR for the IVGDR. We find a strong correlation between the calculated values of κ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=0.84$ for all nuclei considered.

Figure 19

Similar to Fig. 2, for the IVGDR as a function of the effective mass. We find a weak correlation between the calculated values of m*/m and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient close to $C=-0.60$ for all the nuclei considered here.

Figure 20

Similar to Fig. 2, for the isovector giant quadrupole resonance (IVGQR) as a function of the symmetry energy coefficient $J$. We find a weak correlation between the calculated values of $J$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C\sim -0.35$.

Figure 21

Similar to Fig. 2, for the IVGQR as a function of the enhancement coefficient, κ, of the EWSR of the IVGDR. We find medium correlation between the calculated values of κ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=0.80$ for all nuclei considered.

Figure 22

Similar to Fig. 2, for the IVGQR as a function of the effective mass m*/m. We find medium correlation between the calculated values of m*/m and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient of $C=-0.74$ for all the nuclei considered here.

Figure 23

Similar to Fig. 2, for the isovector giant octupole resonance (IVGOR) as a function of the symmetry energy coefficient $J$. We don't find any correlation between the calculated values of $J$ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C\sim -0.32$.

Figure 24

Similar to Fig. 2, for the IVGOR as a function of the enhancement coefficient, κ, for the EWSR of the ISGDR. We find strong correlation between the calculated values of κ and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=0.81$ for all nuclei considered.

Figure 25

Similar to Fig. 2, for the calculated values or the IVGOR as a function of the effective mass m*/m. We find strong correlation between the calculated values of m*/m and $E_{\mathrm{CEN}}$ with a Pearson linear correlation coefficient $C=-0.83$ for all nuclei considered.