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

Technical Summary: Pair Transfer and Reaction Dynamics in 40,48Ca + 96Zr Collisions Below the Coulomb Barrier

Problem Statement
Sub-barrier fusion reactions serve as a sensitive probe for nuclear structure and correlations, particularly the role of pairing. While experimental observations confirm that pairing enhances two-neutron transfer cross-sections (e.g., in $^{40}\text{Ca} + ^{96}\text{Zr}$ and $^{60}\text{Ni} + ^{116}\text{Sn}$), previous theoretical treatments using the BCS approximation within time-dependent frameworks failed to reproduce the experimentally observed enhancement factor between one- and two-neutron transfer probabilities ($P_{2n}/P_{1n}^2$). Theoretical models without pairing predict a ratio between 0.25 and 0.5, whereas experimental values are significantly larger (approx. 3 for $^{40}\text{Ca} + ^{96}\text{Zr}$). Furthermore, the interplay between pairing correlations, dynamic nuclear deformability, and the specific angular momentum channels involved in transfer remains to be fully understood microscopically.

Methodology
This study employs Time-Dependent Superfluid Local Density Approximation (TDSLDA), an extension of Time-Dependent Density Functional Theory (TDDFT) to superfluid systems, to investigate head-on collisions of $^{40,48}\text{Ca} + ^{96}\text{Zr}$ below the fusion threshold. The calculations utilize the nuclear energy density functional (NEDF) SeaLL1. To isolate the effects of pairing, the authors perform parallel simulations using standard Time-Dependent Hartree-Fock (TDHF) theory (without pairing) as reference states, using the same code and functional.

Key technical components include:

• Projection Operators: To extract transfer probabilities microscopically, the study applies particle-number projection operators to the neutron sector after the reaction. This method, originally proposed for TDHF by Simenel and generalized for superfluid systems, determines the probability of the system being in a state with a specific number of nucleons in the spatial region of an outgoing fragment.
• Angular Momentum Projection: An additional projector on the angular momentum $K$ is employed to characterize the $K$-distribution of neutrons in transfer channels, distinguishing between sequential transfer and correlated pair transfer.
• Energy Referencing: Due to the sensitivity of the fusion threshold to NEDF parameter tuning (where SeaLL1 predicts a threshold $\approx 4$ MeV higher than experiment), all observables are recorded relative to the calculated threshold energy to ensure consistent comparison with experimental data.

Key Results

1. Dynamic Deformability and Mean Transfer: Pairing correlations significantly alter the dynamic response of the nucleus. In TDSLDA, $^{96}\text{Zr}$ develops an oblate quadrupole deformation at large separation distances induced by the Coulomb field, stabilizing at $\beta_{20} \approx -0.15$ to $-0.2$. In contrast, TDHF predicts $^{96}\text{Zr}$ remains spherical until contact, after which it develops a prolate deformation. The prolate shape in TDHF allows for greater density overlap and longer interaction times, leading to a larger mean neutron transfer compared to TDSLDA. This demonstrates that pairing enhances dynamic deformability (adiabaticity) but suppresses the mean transfer magnitude in these specific systems.
2. Transfer Probabilities and Enhancement Ratio:
   • One-Neutron Transfer: The probability of one-neutron transfer is smaller when pairing is included, consistent with the reduced mean transfer.
   • Two-Neutron Transfer: The probability of two-neutron transfer is significantly enhanced by pairing, particularly for the $^{48}\text{Ca} + ^{96}\text{Zr}$ reaction.
   • Enhancement Ratio ($P_{2n}/P_{1n}^2$): TDSLDA calculations reproduce the experimentally observed enhancement factor (approx. 3 for $^{40}\text{Ca} + ^{96}\text{Zr}$), whereas previous BCS-based approaches underestimated this ratio. The study predicts an even larger enhancement ratio for $^{48}\text{Ca} + ^{96}\text{Zr}$, correlating with larger final pairing gaps in that system.
3. $K$-Angular Momentum Distribution: The analysis of angular momentum projections reveals that while one-neutron transfer predominantly involves states with $|K|=1/2$ (associated with wave functions extending along the collision axis), two-neutron transfer is dominated by the $K=0$ channel. This $K=0$ dominance signifies the transfer of correlated Cooper pairs, providing a microscopic explanation for the enhancement ratio. The pair-breaking mechanism is less significant here than in fission due to the lower excitation energy in sub-barrier collisions.

Significance and Claims
The paper claims to provide the first microscopic investigation of one- and two-neutron transfer probabilities in low-energy heavy-ion collisions using the full TDSLDA framework. The primary significance lies in demonstrating that TDSLDA successfully reproduces the experimental enhancement factor for two-neutron transfer, a quantity that previous approximations (like TDHF+BCS) failed to capture accurately.

The authors conclude that pairing correlations are strongly linked to the dynamic deformability of the nucleus, which dictates mean transfer, while the enhancement of pair transfer is driven by the population of the $K=0$ channel. These results confirm that TDSLDA offers a reliable microscopic framework for describing the interplay between nuclear superfluidity and reaction dynamics. The enhancement ratio $P_{2n}/P_{1n}^2$ is identified as a critical observable for constraining both the strength and the theoretical treatment of pairing correlations in nuclear models.