Transfer probabilities for the reactions ${}^{14,20}\mathrm{O}+{}^{20}\mathrm{O}$ in terms of multiple time-dependent Hartree-Fock-Bogoliubov trajectories

The transfer reaction between two nuclei in the superfluid phase is studied with the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. To restore the symmetry of the relative gauge angle, a set of independent TDHFB trajectories is taken into account. Then, the transfer probability is computed using a triple projection method. This method is first tested to determine the transfer probabilities on a toy model and compared to the exact solution. It is then applied to the reactions ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ and ${}^{14}\mathrm{O}+{}^{20}\mathrm{O}$ in a realistic framework with a Gogny interaction.

Figure 1

Pair transfer probability as a function of time in the toy model. Calculations by the present method with (red dashed curve) and without (dotted blue line) an effective interaction are compared to the exact solution (black solid curve).

Figure 2

Pair transfer probability as a function of the strength $V_0$ of the transfer coupling in the Hamiltonian (6). Calculation by the present method with the effective interaction (red dashed curve) is compared to the exact solution (black solid curve). The blue triangles represent the results obtained without the chemical potential and quasiparticle energy correction at an arbitrary time of $50\times {10}^{-22}$ s. The error bars on the red curve represent the standard deviation of the probabilities as a function of time after the reaction.

Figure 3

Superposition of the isodensity surfaces at $\rho =0.8{\mathrm{fm}}^{-3}$ as a function of time for the reaction ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ with the center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}=9.31$ MeV. The red and blue surfaces are respectively computed from the state $|{\varphi }_2\rangle$ and $|{\varphi }_4\rangle$ while the green surface is computed from the states $|{\varphi }_1\rangle$ and $|{\varphi }_3\rangle$.

Figure 4

Norm of the overlap between states 1 and 2 (respectively $\phi =0^{\circ }$ and ${45}^{\circ })$ as a function of time, with (red solid line) and without (purple dashed line) the center-of-mass shift.

Figure 5

Pair transfer probability as a function of time for the reaction ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$. (a) TDHFB calculation without center-of-mass shift (purple dashed line), without the chemical potential and quasi-particle energy correction (solid line), and the correct calculation at the center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}=9.21$ MeV (green dotted line). (b) Corrected calculation at $E_{\mathrm{c}.\mathrm{m}.}=9.31$ MeV (red solid line), $E_{\mathrm{c}.\mathrm{m}.}=9.21$ MeV (green dotted line), and $E_{\mathrm{c}.\mathrm{m}.}=8.91$ MeV (blue dashed line).

Figure 6

Neutron pair transfer probability as a function of the distance of closest approach for the reaction ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$.

Figure 7

Probabilities to obtain $N_L$ neutrons on the left part $\left(Z<0\right)$ of the lattice after the collision computed with pairing (blue bars) and without pairing (red triangles) for the reaction ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ at $E_{\mathrm{c}.\mathrm{m}.}=9.31$ MeV.

Figure 8

Transfer probability of one (red solid line) and two neutrons (blue dotted line) as a function of the distance of closest approach for the reaction ${}^{14}\mathrm{O}+{}^{20}\mathrm{O}$. The dashed black line represents $P_1^2/4$.