In an exclusive measurement of 7Li+65Cu^{7}\mathrm{Li}+{}^{65}\mathrm{Cu} at 25 MeV, Shrivastava et al. (Phys. Lett. B 633 (2006) 463) showed that the dominant α+d\alpha+d yield arises from one-neutron transfer populating the unbound 2.186 MeV 3+3^+ resonance of 6Li^{6}\mathrm{Li} (direct transfer to the continuum), followed by breakup. Their coupled-channels Born approximation (CCBA) analysis had to truncate the residual 66Cu^{66}\mathrm{Cu} spectrum at 1.43 MeV because spins above 1.5 MeV are uncertain and levels are unresolved, and the transfer step was treated perturbatively, so the contribution of higher-lying and continuum states and a fully coupled (non-perturbative) treatment of transfer to the continuum remained open. The problem is to give a quantitatively reliable, internally consistent quantum coupled-channel description of this two-step transfer-breakup mechanism — treating elastic, inelastic, transfer-to-bound, and transfer-to-continuum channels with continuum-continuum couplings and the dense 66Cu^{66}\mathrm{Cu} spectrum on the same footing — and thereby to determine the still-unquantified transfer-breakup strength to higher-lying and unresolved 66Cu^{66}\mathrm{Cu} states, validated against the published exclusive α+d\alpha+d data.

Background

The nuclei 6,7Li^{6,7}\mathrm{Li} are weakly bound: 7Li^{7}\mathrm{Li} is held together by only 2.467 MeV in the α+t\alpha+t cluster channel, so at energies near the Coulomb barrier its breakup and nucleon-transfer channels compete strongly with elastic scattering and fusion. Exclusive coincidence measurements, which detect the two breakup fragments in coincidence and reconstruct their relative energy, are the cleanest way to separate direct projectile breakup from multi-step processes. The standard quantum framework for direct breakup is CDCC (continuum-discretized coupled channels), in which the continuum of the weakly bound projectile is discretized into bins and all couplings between the bound state, the bins, and the target are solved simultaneously. For transfer followed by breakup, the workhorse is CCBA (coupled-channels Born approximation): CDCC is used in the entrance and exit partitions but the nucleon-transfer step itself is computed in DWBA (distorted-wave Born approximation), i.e. to first order. In the experiment of Shrivastava et al. (Phys. Lett. B 633 (2006) 463; arXiv:nucl-ex/0512032), 7Li+65Cu^{7}\mathrm{Li}+{}^{65}\mathrm{Cu} at 25 MeV was measured with α\alpha–dd and α\alpha–tt coincidences. The α+d\alpha+d channel, absent for direct 7Li^{7}\mathrm{Li} breakup, was shown to proceed by one-neutron transfer, 65Cu(7Li,6Li∗(3+, 2.186 MeV))66Cu^{65}\mathrm{Cu}(^{7}\mathrm{Li},{}^{6}\mathrm{Li}^{*}(3^+,\,2.186\ \mathrm{MeV})){}^{66}\mathrm{Cu}, followed by breakup of the unbound 3+3^+ resonance, with a cross section about twice the direct α+t\alpha+t breakup; CCBA calculations established direct transfer to the continuum, rather than final-state interaction, as the mechanism. The calculations were performed with FRESCO, a standard coupled-channels reaction code, but were restricted to 66Cu^{66}\mathrm{Cu} levels up to 1.43 MeV, partly because spin assignments above about 1.5 MeV are uncertain and the spectrum contains unresolved doublets; the summed calculation reproduced the shape of the measured angular distribution but underpredicted its magnitude, and a normalization to the data was needed to extract a total 6Li(3+)^{6}\mathrm{Li}(3^+) production cross section of about 9 mb. In such calculations the population of each residual level is weighted by its spectroscopic factor — the overlap strength of that level with the 65Cu^{65}\mathrm{Cu}+neutron single-particle configuration — so an uncertain level scheme translates directly into an uncertain predicted yield. Three later methodological strands are relevant. First, exact few-body methods: momentum-space Faddeev–Alt-Grassberger–Sandhas (AGS) equations treat elastic, transfer, and breakup channels of deuteron-like two-body projectiles on inert-core targets in one consistent model space (Deltuva and Fonseca, Phys. Rev. C 79, 014606 (2009)), but no such exact treatment exists for a three-cluster projectile on a medium-mass target with a dense residual spectrum. Second, extensions of CDCC: four-body CDCC handles three-body projectiles such as 6He^{6}\mathrm{He} (Rodríguez-Gallardo et al., Phys. Rev. C 77, 064609), and the extended-CDCC method (XCDCC) adds excitation of the projectile's core (Summers, Nunes, and Thompson, Nucl. Phys. A 788 (2007) 325), but neither couples transfer into a dense residual-nucleus spectrum; the role of continuum-continuum couplings between continuum bins has been studied for elastic breakup and found non-negligible near the barrier (Mukeru and Lekala, Nucl. Phys. A 965 (2017) 1). Third, semiclassical tools: classical dynamical models with stochastic breakup (Diaz-Torres, Comput. Phys. Commun. 182 (2011) 1100, the PLATYPUS code) and classical trajectory models have been used to quantify transfer-triggered breakup (Luong et al., Phys. Lett. B 695 (2011) 105; Carnelli et al., Nucl. Phys. A 969 (2018) 94), at the price of abandoning a fully quantum treatment. Reviews (Canto et al., Phys. Rep. 596 (2015); Yang et al., Fundamental Research 5 (2025)) confirm that transfer-triggered breakup is a significant mode for 6,7Li^{6,7}\mathrm{Li} and 9Be^{9}\mathrm{Be} near the barrier and that its quantitative description still relies on CCBA-type and semiclassical approximations.

Problem Statement

Provide a quantitatively reliable and internally consistent quantum-mechanical coupled-channel description of the transfer-followed-by-breakup mechanism in 7Li+65Cu^{7}\mathrm{Li}+{}^{65}\mathrm{Cu} near the Coulomb barrier, as benchmarked by the exclusive α+d\alpha+d data of Shrivastava et al. at 25 MeV (integrated over 66Cu^{66}\mathrm{Cu} excitation energy up to 2.5 MeV). Concretely, the required treatment must, within one calculation: (i) include the one-neutron transfer couplings populating both bound and unbound (continuum) states of the outgoing 6Li^{6}\mathrm{Li} — in particular the 2.186 MeV 3+2.186\ \mathrm{MeV}\ 3^+ resonance — non-perturbatively, on the same footing as the elastic and inelastic channels, rather than through a first-order DWBA transfer step; (ii) include the continuum-continuum couplings among the discretized α+d\alpha+d continuum states of the exit partition; and (iii) include the dense spectrum of the residual nucleus 66Cu^{66}\mathrm{Cu}, extending above the 1.43 MeV cutoff of the original analysis, with explicit level assignments where the data permit. The calculation must predict the measured exclusive α+d\alpha+d differential cross section and the total transfer-breakup yield without fitting an arbitrary overall normalization constant; the prediction must agree with the exclusive α+d\alpha+d data of Shrivastava et al. (ref. 1) within their experimental uncertainties and must lie within the theoretical band obtained by varying the uncertain 66Cu^{66}\mathrm{Cu} spin assignments and spectroscopic factors, and must thereby determine the hitherto unquantified contribution of higher-lying and unresolved 66Cu^{66}\mathrm{Cu} states (above about 1.43 MeV; since the published data are integrated up to 2.5 MeV of 66Cu^{66}\mathrm{Cu} excitation, data-based validation directly constrains the 1.43–2.5 MeV window, while the portion above 2.5 MeV is predictive) to the total transfer-breakup yield, including which states dominate and how sensitive the yield is to the uncertain spin assignments and spectroscopic factors of 66Cu^{66}\mathrm{Cu}. The methodology developed should be computationally tractable and stated precisely enough to be applicable to other weakly bound projectiles, including the low-energy radioactive-ion-beam cases that motivated the original work; extending it to that broader class is a desirable outlook, whereas the required target level is the 7Li+65Cu^{7}\mathrm{Li}+{}^{65}\mathrm{Cu} system addressed above.

References

  1. A. Shrivastava, A. Navin, N. Keeley, K. Mahata, K. Ramachandran, V. Nanal, V. V. Parkar, A. Chatterjee, S. Kailas, 'Evidence for transfer followed by breakup in 7Li + 65Cu', Phys. Lett. B 633 (2006) 463-468, DOI 10.1016/j.physletb.2005.12.060, https://doi.org/10.1016/j.physletb.2005.12.060 (arXiv:nucl-ex/0512032, https://arxiv.org/abs/nucl-ex/0512032)
  2. A. Deltuva, A. C. Fonseca, 'Three-body Faddeev-Alt-Grassberger-Sandhas approach to direct nuclear reactions', Phys. Rev. C 79 (2009) 014606, DOI 10.1103/PhysRevC.79.014606, https://doi.org/10.1103/PhysRevC.79.014606
  3. M. Rodríguez-Gallardo, J. M. Arias, J. Gómez-Camacho, R. C. Johnson, A. M. Moro, I. J. Thompson, J. A. Tostevin, 'Four-body continuum-discretized coupled-channels calculations using a transformed harmonic oscillator basis', Phys. Rev. C 77 (2008) 064609, DOI 10.1103/PhysRevC.77.064609, https://doi.org/10.1103/PhysRevC.77.064609
  4. N. C. Summers, F. M. Nunes, I. J. Thompson, 'The effects of core excitation in the breakup of exotic nuclei', Nucl. Phys. A 788 (2007) 325-330, DOI 10.1016/j.nuclphysa.2007.01.095, https://doi.org/10.1016/j.nuclphysa.2007.01.095
  5. B. Mukeru, M. Lekala, 'Importance of the continuum-continuum couplings in the 6Li elastic breakup on different target masses', Nucl. Phys. A 965 (2017) 1-12, DOI 10.1016/j.nuclphysa.2017.05.075, https://doi.org/10.1016/j.nuclphysa.2017.05.075
  6. L. F. Canto, P. R. S. Gomes, R. Donangelo, J. Lubian, M. S. Hussein, 'Recent developments in fusion and direct reactions with weakly bound nuclei', Phys. Rep. 596 (2015) 1-86, DOI 10.1016/j.physrep.2015.08.001, https://doi.org/10.1016/j.physrep.2015.08.001
  7. L. Yang, C. J. Lin, N. R. Ma, P. W. Wen, H. M. Jia, F. Yang, 'Breakup dynamics of weakly bound nuclei at energies around the Coulomb barrier', Fundamental Research 5 (2025) 2463-2473, DOI 10.1016/j.fmre.2023.10.006, https://doi.org/10.1016/j.fmre.2023.10.006
  8. D. R. Otomar, J. Lubian, P. R. S. Gomes, T. Correa, 'Breakup following neutron transfer for the 7Li + 144Sm system', J. Phys. G 40 (2013) 125105, DOI 10.1088/0954-3899/40/12/125105, https://doi.org/10.1088/0954-3899/40/12/125105
  9. P. F. F. Carnelli, D. Martinez Heimann, A. J. Pacheco, A. Arazi, O. A. Capurro, J. O. Fernández Niello, M. A. Cardona, E. de Barbará, J. M. Figueira, D. L. Hojman, G. V. Martí, A. E. Negri, 'Inclusive and exclusive measurements of alpha particle production mechanisms in the 7Li + 144Sm system', Nucl. Phys. A 969 (2018) 94-113, DOI 10.1016/j.nuclphysa.2017.08.007, https://doi.org/10.1016/j.nuclphysa.2017.08.007
  10. D. H. Luong, M. Dasgupta, D. J. Hinde, R. du Rietz, R. Rafiei, C. J. Lin, M. Evers, A. Diaz-Torres, 'Insights into the mechanisms and time-scales of breakup of 6,7Li', Phys. Lett. B 695 (2011) 105-109, DOI 10.1016/j.physletb.2010.11.007, https://doi.org/10.1016/j.physletb.2010.11.007
  11. E. C. Simpson, K. J. Cook, D. H. Luong, S. Kalkal, I. P. Carter, M. Dasgupta, D. J. Hinde, E. Williams, 'Disintegration locations in 7Li→8Be transfer-triggered breakup at near-barrier energies', Phys. Rev. C 93 (2016) 024605, DOI 10.1103/PhysRevC.93.024605, https://doi.org/10.1103/PhysRevC.93.024605
  12. A. Diaz-Torres, 'PLATYPUS: A code for reaction dynamics of weakly-bound nuclei at near-barrier energies within a classical dynamical model', Comput. Phys. Commun. 182 (2011) 1100-1104, DOI 10.1016/j.cpc.2010.12.053, https://doi.org/10.1016/j.cpc.2010.12.053
  13. M. Gómez-Ramos, A. Deltuva, A. M. Moro, 'Benchmarking Faddeev and transfer-to-the-continuum calculations for (p,pN) reactions', Phys. Rev. C 102 (2020) 064613, DOI 10.1103/PhysRevC.102.064613, https://doi.org/10.1103/PhysRevC.102.064613
  14. Rishabh Kumar, M. Maiti, T. N. Nag, S. Sodaye, 'Exploring various features of the reaction mechanism involved in the collision of 7Li on Cu', Phys. Rev. C 104 (2021) 064606, DOI 10.1103/PhysRevC.104.064606, https://doi.org/10.1103/PhysRevC.104.064606