Density-constrained time-dependent Hartree-Fock-Bogoliubov method

Background: The density-constrained time-dependent Hartree-Fock method is currently a tool of choice to predict fusion cross sections. However, it does not include pairing correlations, which have been found recently to play an important role.

Purpose: To describe the fusion cross section with a method that includes the superfluidity and to understand the impact of pairing on both the fusion barrier and cross section.

Method: The density-constrained method is tested first on the following reactions without pairing, ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$. A new method is developed, the density-constrained time-dependent Hartree-Fock-Bogoliubov method. Using the Gogny-TDHFB code, it is applied to the reactions ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ and ${}^{44}\mathrm{Ca}+{}^{44}\mathrm{Ca}$.

Results: The Gogny approach reproduces the experimental data well for reaction ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$. The DC-TDHFB method is coherent with the TDHFB fusion threshold. The effect of the phase-lock mechanism is shown for those reactions.

Conclusions: The DC-TDHFB method is a useful new tool to determine the fusion potential between superfluid systems and to deduce their fusion cross sections.

Figure 1

Comparison between the TDHF (black solid line) and DC-TDHF densities (green dashed line) as a function of time for the system ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ at center of mass energy $E_{\mathrm{c}.\mathrm{m}.}=55$ MeV. The contour lines are computed at the densities 0.04, 0.08, 0.12, and 0.16 ${\mathrm{fm}}^{-1}$. (a)–(d) correspond to the times 0, 180, 360, and 540 $\mathrm{fm}{\mathrm{c}}^{-1}$, respectively.

Figure 2

Nucleus-nucleus potential obtained with the DC-TDHF method for the reaction ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ at $E_{\mathrm{c}.\mathrm{m}.}=11$ MeV. The calculation is done with $N_{\mathrm{shell}}=2$ (green solid line), $N_{\mathrm{shell}}=3$ (blue dotted line), $N_{\mathrm{shell}}=4$ (red dashed line), $N_{\mathrm{shell}}=5$ (black thin line). The position of the TDHF threshold barrier is shown, respectively, by a square, triangle, cross, and dot. The point-Coulomb potential is shown by a dotted line.

Figure 3

Nucleus-nucleus potential obtained with the DC-TDHF method for the reaction ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ at $E_{\mathrm{c}.\mathrm{m}.}=55$ MeV. The calculation is done with number of shell, $N_{\mathrm{shell}}=3$ (green solid line), $N_{\mathrm{shell}}=4$ (blue dotted line), $N_{\mathrm{shell}}=5$ (red dashed line), $N_{\mathrm{shell}}=6$ (black thin line). The point-Coulomb potential is shown by a thin dotted line.

Figure 4

Top: Nucleus-nucleus potential obtained with the DC-TDHF theory for the reaction ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ with $N_{\mathrm{shell}}=5$ for different TDHF energies. The dotted line represents the point coulomb potential. Bottom: Fusion cross-section predicted by the DC-TDHF method compared to the experimental values from Ref. [39].

Figure 5

Same as Fig. 4 for the reaction ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ with $N_{\mathrm{shell}}=5$. The experimental data are obtained from Ref. [40].

Figure 6

${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ nucleus-nucleus potential with the DC-TDHFB method for different initial gauge angle $\phi$ at energies just above the barrier $E_{\mathrm{cm}}=9.254$ MeV, $E_{\mathrm{cm}}=9.451$ MeV, and $E_{\mathrm{cm}}=9.800$ MeV respectively for $\phi =0$ (solid green line), $\phi =\pi /4$ (dotted blue line), and $\phi =\pi /2$ (dashed red line). The point-Coulomb potential is shown by a thin dotted line. The threshold barrier obtained with the TDHFB method is shown by green triangle ($\phi =\pi /4$), blue dot ($\phi =0$), and red square ($\phi =\pi /2$). Each calculation was done with $N_{\mathrm{shell}}=5$.

Figure 7

Threshold barrier for the ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ reaction as a function of the $N_{\mathrm{shell}}$ value for different values of $\phi$ (the symbols are the same as in Fig. 6).

Figure 8

Neutron pairing energy as a function of the distance between the fragments for the reaction ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ at the same center of mass energies of Fig. 6. The energy obtained from the TDHFB evolution (solid green line for $\phi =0$, dotted blue line for $\phi =\pi /4$ and dashed red line for $\phi =\pi /2$) is compared to the DCTDHFB pairing energy (respectively, green triangles, blue dots, and red squares).

Figure 9

Neutron pairing energy as a function of the distance between the fragments for the reaction ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ for $\phi =0$ at different TDHFB center of mass energies.

Figure 10

Top: ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ fusion cross section with $\phi =0$ (solid green line) and $\phi$ = $\pi /2$ (dashed red line). The gauge angle averaged cross section [Eq. (15)] is shown with dotted lines. The DCTDHFB potentials are evaluated at the same center of mass energies of Fig. 6 and with $N_{\mathrm{shell}}=5$. Bottom: Dependence of the cross section with the TDHFB energy.

Figure 11

${}^{44}\mathrm{Ca}+{}^{44}\mathrm{Ca}$ nucleus-nucleus potential obtained with the DC-TDHFB method obtained for different initial gauge angle $\phi$ at energies just above the barrier $E_{\mathrm{cm}}=53$ MeV for $\phi =0$ (solid green line), $\phi =\pi /4$ (dotted blue line), and $\phi =\pi /2$ (dashed red line). The point-Coulomb potential is shown by the thin dotted line. Each calculation was done with $N_{\mathrm{shell}}=6$.