Low-lying electric-dipole strengths of Ca, Ni, and Sn isotopes imprinted on total reaction cross sections

Low-lying electric-dipole ($E1$) strength of a neutron-rich nucleus contains information on neutron-skin thickness, deformation, and shell evolution. We discuss the possibility of making use of total reaction cross sections on ${}^{40}\mathrm{Ca}, {}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets to probe the $E1$ strength of neutron-rich Ca, Ni, and Sn isotopes. They exhibit large enhancement of the $E1$ strength at neutron number $N>28$, 50, and 82, respectively, due to a change of the single-particle orbits near the Fermi surface participating in the transitions. The density distributions and the electric-multipole strength functions of those isotopes are calculated by the Hartree-Fock+BCS and the canonical-basis-time-dependent-Hartree-Fock-Bogoliubov methods, respectively, using three kinds of Skyrme-type effective interaction. The nuclear and Coulomb breakup processes are respectively described with the Glauber model and the equivalent photon method in which the effect of finite-charge distribution is taken into account. The three Skyrme interactions give different results for the total reaction cross sections because of different Coulomb breakup contributions. The contribution of the low-lying $E1$ strength is amplified when the low-incident energy is chosen. With an appropriate choice of the incident energy and target nucleus, the total reaction cross section can be complementary to the Coulomb excitation for analyzing the low-lying $E1$ strength of unstable nuclei.

Figure 1

Ratio of the photon numbers $r(b=20\mathrm{fm},\omega )$ of finite- and point-charge distributions of ${}^{40}\mathrm{Ca}, {}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets as a function of $\hslash \omega$. See Eq. (17).

Figure 2

Total reaction (${\sigma }_R$), nuclear breakup (${\sigma }_N$), Coulomb breakup (${\sigma }_C$) cross sections of Sn isotopes, ${}^{100–140}\mathrm{Sn}$, incident on (a1)–(a4) ${}^{40}\mathrm{Ca}$, (b1)–(b4) ${}^{120}\mathrm{Sn}$, and (c1)–(c4) ${}^{208}\mathrm{Pb}$ targets at the incident energies of 100, 200, 550, and $1000A$ MeV. The SkM*, SLy4, and SkI3 interactions are employed.

Figure 3

Percentages of the Coulomb breakup cross sections with electric multipoles, $E1,E2$, and $E3$, in the total reaction cross sections of Sn isotopes, ${}^{100–140}\mathrm{Sn}$, incident on (a1)–(a4) ${}^{40}\mathrm{Ca}$, (b1)–(b4) ${}^{120}\mathrm{Sn}$, and (c1)–(c4) ${}^{208}\mathrm{Pb}$ targets at the incident energies of 100, 200, 550, and $1000A\mathrm{MeV}$. The SkM*, SLy4, and SkI3 interactions are employed.

Figure 4

Contributions of the electric-multipole strengths of ${}^{134}\mathrm{Sn}$ to the Coulomb breakup cross section (${\sigma }_C$) by ${}^{208}\mathrm{Pb}$ target as a function of the excitation energy: (a) $E1$, (b) $E2$, and (c) $E3$. See Eqs. (20) and (21). The weight function $F\left(E\lambda \right)$ is plotted at the incident energies of 100, 200, 550, and $1000A\mathrm{MeV}$. The SkI3 interaction is employed.

Figure 5

Comparison of the electric-dipole ($E1$) contributions of Sn isotopes, ${}^{100,110,120,132,134}\mathrm{Sn}$, to the Coulomb breakup cross sections by ${}^{208}\mathrm{Pb}$ target. The $E1$ strength functions $S\left(E1\right)$ are plotted as a function of the excitation energy for three Skyrme interactions, (a) SkM*, (b) SLy4, and (c) SkI3, and for each case two incident energies of $100A$ and $1000A\mathrm{MeV}$ are chosen to draw the Coulomb breakup cross sections per unit energy $F\left(E1\right)S\left(E1\right)$.

Figure 6

Same as Fig. 2 but for (a1)–(a4) ${}^{40–60}\mathrm{Ca}$ and (b1)–(b4) ${}^{56–84}\mathrm{Ni}$ incident on ${}^{208}\mathrm{Pb}$ target.