Time-dependent mean-field determination of the excitation energy in transfer reactions: Application to the reaction ${}^{238}\mathrm{U}$ on ${}^{12}\mathrm{C}$ at 6.14 MeV/nucleon

The internal excitation of nuclei after multinucleon transfer is estimated by using the time-dependent mean-field theory. Transfer probabilities for each channel as well as the energy loss after reseparation are calculated. By combining these two pieces of information, we show that the excitation energy distribution of the transfer fragments can be obtained separately for the different transfer channels. The method is applied to the reaction involving a ${}^{238}\mathrm{U}$ beam on a ${}^{12}\mathrm{C}$ target, which has recently been measured at GANIL. It is shown that the excitation energy calculated with the microscopic theory compares well with the experimental observation, provided that the competition with fusion is properly taken into account. The reliability of the excitation energy is further confirmed by the comparison with the phenomenological heavy-ion phase-space model at higher center-of-mass energies.

Figure 1

Snapshots of the density profile $\rho (x,y,0)$ in the center of mass at different times for the collision ${}^{238}\mathrm{U}$ on ${}^{12}\mathrm{C}$. The results shown in the left- and right-hand plots correspond to the perpendicular ($\theta =\pi /2$) and parallel ($\theta =0$) orientations, respectively. The center-of-mass energies are $E_{\mathrm{c}.\mathrm{m}.}^{\prime }=63.75$ MeV (for $\theta =\pi /2$) and $E_{\mathrm{c}.\mathrm{m}.}^{\prime }=56.25$ (for $\theta =0$).

Figure 2

Excitation energy as a function of the center-of-mass energy for the reaction ${}^{238}\mathrm{U}+{}^{12}\mathrm{C}$. Blue crosses and dots correspond to perpendicular and parallel orientations of the uranium, respectively. In both cases, the dotted red line corresponds to the function (2), which is adjusted to reproduce the $\mathrm{TDHF}+\mathrm{BCS}$ results.

Figure 3

Final neutron (a),(b) and proton (c),(d) transfer probability distributions obtained for the ${}^{238}\mathrm{U}+{}^{12}\mathrm{C}$ reaction, for $\theta =\pi /2$ (a),(c) and $\theta =0$ (b),(d). The number of transferred protons and neutrons are represented on the horizontal axis. Positive values correspond to transfer of nucleons from ${}^{238}\mathrm{U}$ to ${}^{12}\mathrm{C}$. Calculations are done for three different center-of-mass energies: $E_{\mathrm{c}.\mathrm{m}.}-B_{\mathrm{fus}}\left(\theta \right)=-0.1$ MeV/nucleon (green triangles), −0.4 MeV/nucleon (blue dots), and −1.4 MeV/nucleon (red crosses).

Figure 4

Transfer probability leading to ${}^{10}\mathrm{Be}$ as a function of the center-of-mass energy for the $\theta =0$ (blue dots) and $\theta =\pi /2$ (red crosses) orientations of ${}^{238}\mathrm{U}$. The black dotted lines correspond to function (3), which was adjusted to reproduce the $\mathrm{TDHF}+\mathrm{BCS}$ results.

Figure 5

Experimental fission probability of ${}^{240}\mathrm{Pu}$ as a function of the excitation energy (blue cross) [33]. The solid line is fit to the experimental data with Eq. (6).

Figure 6

Transfer-induced fission cross section for the reaction ${}^{12}\mathrm{C}({}^{238}\mathrm{U},{}^{240}\mathrm{Pu}){}^{10}\mathrm{Be}$ as a function of the excitation energy for the two orientations of ${}^{238}\mathrm{U}, \theta =0$ (blue dotted line) and $\theta =\pi /2$ (red solid line). Here, the simulated center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}=6.14$ MeV/nucleon is the same for the two orientations.

Figure 7

${\sigma }_{\mathrm{tr},\mathrm{fis}}(N,Z,E^{*})$ as a function of excitation energy for the main transfer channels observed experimentally at a center-of-mass energy of 59 MeV.

Figure 8

Average excitation energy as a function of the center-of-mass energy for the main channels observed experimentally: experimental data (blue dots), $\mathrm{TDHF}+\mathrm{BCS}$ results (black triangles), $\mathrm{TDHF}+\mathrm{BCS}$ results where the excitation energy has been shifted by 3 MeV (red squares), and the HIPSE results (orange down triangles). In the latter case, error bars correspond to the widths of the calculated distributions. The superimposed blue (gray) areas correspond to the experimental event-by-event distributions of the excitation energy.