Time-dependent Hartree-Fock calculations for multinucleon transfer processes in ${}^{40,48}$Ca+${}^{124}$Sn, ${}^{40}$Ca+${}^{208}$Pb, and ${}^{58}$Ni+${}^{208}$Pb reactions

Multinucleon transfer processes in heavy-ion reactions at energies slightly above the Coulomb barrier are investigated in a fully microscopic framework of the time-dependent Hartree-Fock (TDHF) theory. Transfer probabilities are calculated from the TDHF wave function after collision using the projection operator method which has recently been proposed by Simenel [Phys. Rev. Lett. 105, 192701 (2010)]. We show results of the TDHF calculations for transfer cross sections of the reactions of ${}^{40,48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=170$, 174 MeV, ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{lab}}=235$, 249 MeV, and ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{lab}}=328.4$ MeV, for which measurements are available. We find the transfer processes show different behaviors depending on the $N/Z$ ratios of the projectile and the target and the product of the charge numbers, $Z_{\mathrm{P}}{Z}_{\mathrm{T}}$. When the projectile and the target have different $N/Z$ ratios, fast transfer processes of a few nucleons towards the charge equilibrium of the initial system occur in reactions at large impact parameters. As the impact parameter decreases, a neck formation is responsible for the transfer. A number of nucleons are transferred by the neck breaking when two nuclei dissociate, leading to transfers of protons and neutrons in the same direction. Comparing cross sections by theory and measurements, we find the TDHF theory describes the transfer cross sections of a few nucleons reasonably. As the number of transferred nucleons increases, the agreement becomes less accurate. The TDHF calculation overestimates transfer cross sections accompanying a large number of neutrons when more than one proton are transferred. Comparing our results with those by other theories, we find the TDHF calculations give qualitatively similar results to those of direct reaction models such as grazing and Complex WKB.

Figure 1

Deflection function (a) and total kinetic energy loss (b) as functions of impact parameter $b$ for the reactions of ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=170$ MeV and ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=174$ MeV. Results for the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reactions are denoted by red filled triangles connected with solid lines, while results for the ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reactions are denoted by green open circles connected with dashed lines. In (a), we also show deflection functions for the pure Coulomb trajectories by a red dotted line for the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reactions and by a green dot-dot line for the ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reactions.

Figure 2

Differential cross sections of representative transfer channels as functions of scattering angle in the laboratory frame for the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=170$ MeV. The Coulomb rainbow angle obtained from the TDHF trajectories is denoted by blue solid vertical lines and is compared with measured differential cross sections, red filled circles, which have been reported in Ref. [10].

Figure 3

Snapshots of density distribution of the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=170$ MeV and $b=3.96$ fm, just outside the fusion critical impact parameter.

Figure 4

Left panels for ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=170$ MeV and right panels for ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=174$ MeV. (a),(e) Average number of transferred nucleons from the target to the projectile. (b),(f) Neutron-to-proton ratios, $N/Z$, of the PLF and the TLF after collision. (c),(g) Average number of nucleons emitted to the continuum. (d),(h) Fluctuation of transferred nucleon number. The horizontal axis is the impact parameter $b$. In (b) and (f), the equilibrium $N/Z$ value of the total system is indicated by a horizontal dashed line.

Figure 5

Comparison of calculated results for different initial orientations of ${}^{124}$Sn in the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=170$ MeV. (a) Average number of transferred nucleons from the target to the projectile. (b) Neutron-to-proton ratios, $N/Z$, of the PLF and the TLF after collision. The horizontal axis is the impact parameter $b$. The initial orientations of ${}^{124}$Sn are indicated in the legends.

Figure 6

Neutron (left panels) and proton (right panels) transfer probabilities as functions of impact parameter $b$. (a),(b) Results for the reactions of ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=170$ MeV. (c),(d) Results for the reactions of ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ at $E_{\mathrm{lab}}=174$ MeV. The positive (negative) number of transferred nucleons represents the number of nucleons added to (removed from) the projectile. Shaded regions at small impact parameter ($b\le 3.95$ fm for ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ and $b\le 3.93$ fm for ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$) correspond to the fusion reactions.

Figure 7

Transfer probabilities in Fig. 6 are shown in logarithmic scale. Nucleon transfer probabilities opposite to the direction of the charge equilibrium, which are not included in Fig. 6, are shown as well.

Figure 8

Cross sections for transfer channels classified according to the change of the proton number of the PLF from ${}^{40}$Ca, as functions of neutron number of the PLF for the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=170$ MeV. Red filled circles denote measured cross sections and red solid lines denote results of the TDHF calculations. The number of transferred protons is indicated as ($xp$) ($-6\le x\le +1$). The measured cross sections have been reported in Ref. [10].

Figure 9

Cross sections for transfer channels classified according to the change of the proton number of the PLF from ${}^{48}$Ca as functions of neutron number of the PLF for the ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=174$ MeV. Red filled circles denote measured cross sections and red solid lines denote results of the TDHF calculations. The number of transferred protons is indicated as ($xp$) ($-2\le x\le +2$). The measured cross sections have been reported in Ref. [13].

Figure 10

Deflection function (a) and TKEL (b) as functions of impact parameter $b$ for the reactions of ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{lab}}=235$ and 249 MeV. Results for the reactions at $E_{\mathrm{lab}}=235$ MeV are denoted by red filled triangles connected with solid lines, while results for the reactions at $E_{\mathrm{lab}}=249$ MeV are denoted by green open circles connected with dashed lines. In (a), we also show deflection functions for the pure Coulomb trajectories at $E_{\mathrm{lab}}=235$ MeV by a red dotted line and at $E_{\mathrm{lab}}=249$ MeV by a green two-dot chain line.

Figure 11

Snapshots of density distribution of the ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=249$ MeV and $b=4.56$ fm, just outside the fusion critical impact parameter.

Figure 12

Differential cross sections of representative transfer channels as functions of scattering angle in the center-of-mass frame for the ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ reactions at $E_{\mathrm{lab}}=235$ and 249 MeV. The Coulomb rainbow angle obtained from the TDHF trajectories is denoted by red solid (green dotted) vertical lines for $E_{\mathrm{lab}}=235$ (249) MeV. They are compared with measured differential cross sections, red filled triangles (green open circles) for $E_{\mathrm{lab}}=235$ (249) MeV, which have been reported in Ref. [19].

Figure 13

The ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ reactions at $E_{\mathrm{lab}}=235$ and 249 MeV. (a) Average number of transferred nucleons from the target to the projectile. (b) Neutron-to-proton ratios, $N/Z$, of the PLF and the TLF after collision. (c) Average number of nucleons emitted to the continuum. (d) Fluctuation of transferred nucleon number. The horizontal axis is the impact parameter $b$. Results for the reactions at $E_{\mathrm{lab}}=235$ MeV are denoted by triangles, while results for the reactions at $E_{\mathrm{lab}}=249$ MeV are denoted by circles. In (b), the equilibrium $N/Z$ value of the total system, 1.43, is indicated by a horizontal dashed line.

Figure 14

Neutron and proton transfer probabilities as functions of impact parameter $b$ for the ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ reactions. (a),(b),(c),(d) Results at $E_{\mathrm{lab}}=235$ MeV. (e),(f),(g),(h) Results at $E_{\mathrm{lab}}=249$ MeV. The positive (negative) number of transferred nucleons represents the number of nucleons added to (removed from) the projectile. Note that horizontal scales are different between the left and the right panels. Shaded regions at small impact parameter ($b\le 3.81$ fm for $E_{\mathrm{lab}}=235$ MeV and $b\le 4.55$ fm for $E_{\mathrm{lab}}=249$ MeV) correspond to the fusion reactions.

Figure 15

Transfer cross sections for the ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ reactions at $E_{\mathrm{lab}}=235$ and 249 MeV. Red filled triangles (green open circles) denote measured cross sections at $E_{\mathrm{lab}}=235$ (249) MeV. Red solid (green dotted) lines denote results of the TDHF calculations at $E_{\mathrm{lab}}=235$ (249) MeV. The number of transferred protons (positive number for the transfer from ${}^{208}$Pb to ${}^{40}$Ca) is indicated as ($xp$) ($-6\le x\le +5$). The measured cross sections have been reported in Ref. [19].

Figure 16

The same transfer cross sections for the ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ reactions as those in Fig. 14. The number of transferred neutrons is indicated as ($xn$) ($-5\le x\le +9$).

Figure 17

Deflection function (a) and TKEL (b) as functions of impact parameter $b$ for the reactions of ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{lab}}=328.4$ MeV. In (a), we show a deflection function for the pure Coulomb trajectory by a dotted line.

Figure 18

Differential cross sections of representative transfer channels as functions of scattering angle in the center-of-mass frame for the ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=328.4$ MeV. The Coulomb rainbow angle obtained from the TDHF trajectories is denoted by blue solid vertical lines and is compared with measured differential cross sections, red filled circles, which have been reported in Ref. [17].

Figure 19

Snapshots of density distribution of the ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=328.4$ MeV and $b=1.39$ fm, just outside the fusion critical impact parameter.

Figure 20

The ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=328.4$ MeV. (a) Average number of transferred nucleons from the target to the projectile. (b) Neutron-to-proton ratios, $N/Z$, of the PLF and the TLF after collision. (c) Average number of nucleons emitted to the continuum. (d) Fluctuation of transferred nucleon number. The horizontal axis is the impact parameter $b$. In (b), the equilibrium $N/Z$ value of the total system, 1.42, is indicated by a horizontal dashed line.

Figure 21

Neutron and proton transfer probabilities as functions of impact parameter $b$ for the reactions of ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{lab}}=328.4$ MeV. Panels (a) and (b) show probabilities of neutrons, while panels (c) and (d) show those of protons. The positive (negative) number of transferred nucleons represents the number of nucleons added to (removed from) the projectile. Note that horizontal scales are different between the left and the right panels. A shaded region at small impact parameter ($b\le 1.38$ fm) corresponds to the fusion reactions.

Figure 22

Transfer cross sections for the ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=328.4$ MeV. Red filled circles denote measured cross sections and red solid lines denote results of the TDHF calculations. The number of transferred protons (positive number for the transfer from ${}^{208}$Pb to ${}^{58}$Ni) is indicated as ($xp$) ($-6\le x\le +5$). The measured cross sections have been reported in Ref. [17].

Figure 23

The same transfer cross sections for the ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction as those in Fig. 21. The number of transferred neutrons is indicated as ($xn$) ($-4\le x\le +10$).

Figure 24

Transfer cross sections for the ${}^{40}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=170$ MeV. Red filled circles denote measured cross sections, red solid lines denote results of the TDHF calculations, and green crosses (blue open diamonds) connected with dotted lines denote calculated results using the grazing code without (with) the neutron evaporation effect. The number of transferred protons is indicated as ($xp$) ($-6\le x\le +1$). The measured cross sections and the grazing results have been reported in Ref. [10].

Figure 25

Transfer cross sections for the ${}^{48}\mathrm{Ca}+{}^{124}\mathrm{Sn}$ reaction at $E_{\mathrm{lab}}=174$ MeV. Red filled circles denote measured cross sections, red solid lines denote results of the TDHF calculations, and green crosses (blue open diamonds) connected with dotted lines denote calculated results using the grazing code without (with) the neutron evaporation effect. The number of transferred protons is indicated as ($xp$) ($-2\le x\le +2$). The measured cross sections and the grazing results have been reported in Ref. [13].

Figure 26

Cross sections for transfer channels of pure proton stripping without neutron transfer (left) and pure neutron pickup without proton transfer (right) for the ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=328.4$ MeV. Red filled circles denote measured cross sections [17], red solid lines denote results of the TDHF calculations, blue dotted lines denote results of the Langevin calculation [27], and green open triangles connected with dotted lines denote results of the CWKB calculation [17].

Figure 27

Cross sections for transfer channels of ($xp$) ($-3\le x\le -1$) for the ${}^{58}\mathrm{Ni}+{}^{208}\mathrm{Pb}$ reaction at $E_{\mathrm{lab}}=328.4$ MeV. The horizontal axis is the number of neutrons in the PLF. Red filled circles denote measured cross sections [17], red solid lines denote results of the TDHF calculations, blue dotted lines denote results of the Langevin calculation [27], and green open triangles connected with dotted lines denote results of the CWKB calculation [17].