Pair Transfer and Reaction Dynamics in $^{40,48}$Ca + $^{96}$Zr Collisions Below the Coulomb Barrier

Abstract

Sub-barrier fusion reactions are ideal for probing the effects of pairing correlations on simultaneous neutron transfer. Previous calculations using the BCS approximation showed an enhancement of pair transfer, relative to treatments with no pairing, but failed to reproduce the observed enhancement factor between one- and two-neutron transfer probabilities. This work aims to microscopically investigate the dynamics of $^{40,48}$Ca + $^{96}$Zr head-on collisions below the Coulomb barrier, focusing on the role of pairing correlations in neutron transfer. We employ time-dependent energy density functional theory extended to superfluid systems, TDSLDA. Transfer probabilities, including contributions to specific $K$-angular momentum projections, are extracted using projection operators and compared to results from calculations without pairing. Our calculations show that pairing is correlated to the dynamic deformability of the nucleus, which influences mean neutron transfer in sub-barrier reactions. We also show that TDSLDA reproduces the experimentally observed enhancement factor by significantly increasing the probability of transferring a neutron pair in the $K = 0$ spin channel. These results confirm the strong influence of pairing and structure on sub-barrier multi-nucleon transfer, and demonstrate that TDSLDA provides a reliable microscopic framework for describing the interplay between nuclear superfluidity and reaction dynamics.

Figures (5)

• Figure 1: The deformation parameter $\beta_{20}$ of $^{96}$Zr is shown as a function of the separation distance $d = \bm{r}_{cm,R} - \bm{r}_{cm,L}$ between the two nuclei minus the minimum separation distance reached during the collision $d_{\mathrm{min}}$. The separation distance is multiplied by sign($v_F$), which is -1 when the nuclei are moving towards each other, and +1 when they are moving away from each other. The inset shows the density slices in fm$^{-3}$ of a TDSLDA simulation of $^{40}$Ca + $^{96}$Zr at the minimum separation distance. All collisions are shown at $E_{\mathrm{cm}}$ = 95 MeV.
• Figure 2: (Top/bottom panel) The mean transfer of (neutrons/protons) to the Ca isotopes is shown as a function of the collision energy with respect to the threshold energy provided in Table \ref{['tbl:ethres']}. At the same (unshifted) collision energy, the mean neutron transfer is surprisingly similar for both Ca isotopes. This is likely the result of two competing trends; equilibration of the N/Z ratio, which enhances transfer for the $^{40}$Ca + $^{96}$Zr reaction, while the larger neutron skin of $^{48}$Ca, relative to $^{40}$Ca, enhances transfer for the $^{48}$Ca + $^{96}$Zr reaction. This is further supported by the average proton transfer, which is transferred from $^{40}$Ca to $^{96}$Zr. In this case, there are significantly more protons transferred than for the $^{48}$Ca + $^{96}$Zr reaction at the same (unshifted) collision energy. The inset shows $n_{\mathrm{min}}$, defined by Eq. \ref{['eqn:nmin']}, as a function of $t$ with respect to $t_0$, which is defined at the time when $n_{\mathrm{min}}$ reaches its maximum value. The results in the inset correspond to collisions at an energy $E_{\mathrm{cm}}$ = 95 MeV, and the colors correspond to the same collisions labeled in the legend in the bottom panel
• Figure 3: The one- and two-neutron transfer probabilities are shown as a function of the collision energy. The transfer is calculated after particle projection, and goes from $^{96}$Zr to the Ca isotopes. The black square and triangle markers show results from Corradi:2011 (experiment). The black triangles represent $x=1$, the one particle transfer probability, while the black squares represent $x=2$, the two particle transfer probability. The experimental results are at a finite impact parameter, which we have converted into an effective energy for central collisions, assuming a Rutherford trajectory, to perform the comparison. For backward scattering, this approximation is expected to be reasonable.
• Figure 4: (Top panel) The enhancement ratio $P_{2n} / P_{1n}^2$ is shown as a function of the collision energy. The reaction type, $^{48,40}$Ca + $^{96}$Zr, is labeled by the Ca isotope. The black squares show results from Corradi:2011 (experiment). The experimental error is estimated by treating $P_{2n},P_{1n}$ as independent observables. The black line represents the value $P_{2n} / P_{1n}^2 = 3$. (Bottom panel) The enhancement ratio is shown for $^{40}$Ca+$^{96}$Zr for different pairing strengths, characterized by the initial pairing gap in $^{96}$Zr. The default value for TDSLDA is given by the red circles.
• Figure 5: (Top panel) $K$-distribution in one of the fragments for each transfer channel with and without pairing at a center-of-mass energy $E_{\rm c.m.} = 94$ MeV. In the case of no transfer, n = 0, both nuclei remain exclusively in the $K_F = 0$ channel. (Bottom panel) probability of $|K_F| = 1/2$ and $|K_F| = 0$ spin modes are shown as a function of the collision energy for the 1 and 2 neutron transfer channels.