Microscopic study of the ${}^{132,124}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ reactions: Dynamic excitation energy, energy-dependent heavy-ion potential, and capture cross section

We study reactions between neutron-rich ${}^{132}\mathrm{Sn}$ nucleus and ${}^{96}\mathrm{Zr}$ within a dynamic microscopic theory at energies in the vicinity of the ion-ion potential barrier peak, and we compare the properties to those of the stable system ${}^{124}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$. The calculations are carried out on a three-dimensional lattice using the density-constrained time-dependent Hartree-Fock method. In particular, we calculate the dynamic excitation energy $E^{*}(t)$ during the initial stages of the collision. The barrier heights and widths of the heavy-ion potential increase substantially with $E_{\mathrm{c}.\mathrm{m}.}$ energy. The capture cross sections for the two reactions are of similar magnitude, but the interaction barrier for the neutron-rich system is found to be significantly (9 MeV) lower. A comparison with recently measured data is given.

Figure 1

TDHF calculations for ${}^{132}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$. Contour plots of the mass density at the distance of closest approach in a central collision, calculated at three different $E_{\mathrm{c}.\mathrm{m}.}$ energies.

Figure 2

Intrinsic mass quadrupole moment as a function of time.

Figure 3

DC-TDHF calculations for the neutron-rich system ${}^{132}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$. The potential barriers $V(R)$ at four $E_{\mathrm{c}.\mathrm{m}.}$ energies are obtained using Eq. (1). Also shown is the point Coulomb potential. The phenomenological double-folding potential (dashed black curve) is given for comparison.

Figure 4

Comparison of the heavy-ion barriers for the neutron-rich system ${}^{132}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ (solid lines) and the stable system ${}^{124}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ (dashed lines). The potential barriers are obtained with the DC-TDHF method at five $E_{\mathrm{c}.\mathrm{m}.}$ energies.

Figure 5

Precompound excitation energy as a function of the internuclear distance $R$, calculated at zero impact parameter with the DC-TDHF method at four $E_{\mathrm{c}.\mathrm{m}.}$ energies. Compared are the neutron-rich system ${}^{132}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ (solid lines) and the stable system ${}^{124}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ (dashed lines).

Figure 6

Precompound excitation energy at the capture point, $E_c^{*}$, as function of the c.m. energy, as predicted by DC-TDHF for a central collision.

Figure 7

Total cross section for capture (solid line) and for deep-inelastic reactions (dashed line) for the neutron-rich system ${}^{132}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ calculated with the DC-TDHF method as function of $E_{\mathrm{c}.\mathrm{m}.}$. Total capture cross sections predicted by unrestricted TDHF calculations are also given (red dots with error bars). For details, see text.

Figure 8

Total cross section for capture (solid line) and for deep-inelastic reactions (dashed line) for the neutron-rich system ${}^{132}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ calculated with the DC-TDHF method as function of $E_{\mathrm{c}.\mathrm{m}.}$. The experimental capture-fission cross sections are taken from Ref. [3].

Figure 9

Total cross section for capture (solid line) and for deep-inelastic reactions (dashed line) for the stable system ${}^{124}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ calculated with the DC-TDHF method as function of $E_{\mathrm{c}.\mathrm{m}.}$. The red dot represents the total capture cross section at $E_{\mathrm{c}.\mathrm{m}.}=250$ MeV predicted by an unrestricted TDHF run. For details, see text.

Figure 10

Total cross section for capture (solid line) and for deep-inelastic reactions (dashed line) for the stable system ${}^{124}\mathrm{Sn}$$+$${}^{96}\mathrm{Zr}$ calculated with the DC-TDHF method as function of $E_{\mathrm{c}.\mathrm{m}.}$. The experimental capture-fission cross sections are taken from Ref. [3].