The RIBRAS measurement of 7Be+9Be^{7}\mathrm{Be}+{}^{9}\mathrm{Be} at Elab=23.1E_{\mathrm{lab}}=23.1 MeV (Umbelino et al., 2019) established that two-neutron transfer between two weakly bound, cluster-structured nuclei is dominated by the direct (simultaneous) route, but left the quantitative description of the reaction incomplete: continuum and breakup couplings of both collision partners were absorbed into a São Paulo potential whose real and imaginary parts were both renormalized by a strength factor 0.6 in the entrance partition, the spectroscopic amplitudes for the 9Be→7Be+2n^{9}\mathrm{Be}\to{}^{7}\mathrm{Be}+2n overlap lacked the explicit α+α+n\alpha+\alpha+n cluster structure, the extracted amplitudes were sensitive to the choice of optical potential, and the 0.90.9 MeV experimental resolution prevented separating the 7Be^{7}\mathrm{Be} ground-state and 429429 keV (1/2−)(1/2^-) final-state contributions. The open problem is a single, mechanism-resolved coupled calculation of this reaction — transfer (simultaneous and sequential through 8Be^{8}\mathrm{Be} resonances) treated together with projectile (α+3He\alpha+{}^{3}\mathrm{He}) and target (2α+n2\alpha+n) breakup — supported by cluster-consistent structure inputs, controlled reaction-model systematics, and state-separated cross sections. Later work has advanced the adjacent pieces (CDCC for quasielastic scattering, three-body cluster models of 9Be^{9}\mathrm{Be}, transfer-induced-breakup models on heavy targets, one- vs two-step cluster-transfer decompositions in other systems) but none performs the full coupled treatment for this system, so the problem remains open.

Background

Two-neutron transfer reactions between light nuclei are a standard probe of neutron pair correlations (pairing) in nuclear wave functions: the cross section for moving a correlated neutron pair from target to projectile depends on the spectroscopic amplitude, the overlap function between the initial and final nuclear states weighted by the pair-transfer structure. The system studied here is unusual in that both collision partners are weakly bound and cluster-structured. The projectile 7Be^{7}\mathrm{Be} (binding energy 1.5871.587 MeV with respect to α+3He\alpha+{}^{3}\mathrm{He}) has its first excited state (1/2−)(1/2^-) only 429429 keV above the ground state. The target 9Be^{9}\mathrm{Be} is a Borromean nucleus — bound as α+α+n\alpha+\alpha+n although its two-body subsystems 8Be^{8}\mathrm{Be} and 5He^{5}\mathrm{He} are unbound — with a low one-neutron separation energy (≈1.67\approx 1.67 MeV) and strong α+α+n\alpha+\alpha+n cluster character. Because the two transferred neutrons must leave 9Be^{9}\mathrm{Be} and populate 7Be^{7}\mathrm{Be}, the reaction 9Be(7Be,9Be)7Be^{9}\mathrm{Be}(^{7}\mathrm{Be},{}^{9}\mathrm{Be}){}^{7}\mathrm{Be} exchanges a neutron pair between identical partitions, and the sequential route can proceed through the resonant intermediate nucleus 8Be^{8}\mathrm{Be}, notably its 2+2^+ state at 3.033.03 MeV.

Standard tools for such analyses are: the optical model (elastic scattering described by a complex potential, here a Woods–Saxon form or the São Paulo potential, a parameter-free double-folding potential built from nuclear matter densities whose strength can be renormalized by a factor NN); DWBA (distorted-wave Born approximation, a first-order treatment of transfer between distorted waves); CRC (coupled reaction channels, which iterates the coupling between elastic and transfer partitions to all orders); and CDCC (continuum-discretized coupled channels, which couples the elastic channel to a discretized representation of a projectile's or target's breakup continuum). The experiment was performed with the 7Be^{7}\mathrm{Be} radioactive secondary beam of the RIBRAS facility (Radioactive Ion Beams in Brazil, University of São Paulo) at Elab=23.1E_{\mathrm{lab}}=23.1 MeV, near the Coulomb barrier, the energy region where couplings between weakly bound structures most strongly distort the reaction dynamics.

The 2019 analysis combined optical-model and CRC calculations with shell-model spectroscopic amplitudes, cross-checked against the 9Be(p,t)7Be^{9}\mathrm{Be}(p,t){}^{7}\mathrm{Be} reaction. It concluded that the direct, simultaneous transfer dominates, but identified the unresolved limitations that define the present problem: the breakup and continuum channels of both nuclei were represented only grossly through an effective imaginary potential — the real and imaginary parts of the São Paulo potential in the entrance partition were both multiplied by a strength factor 0.6 (outgoing partition: NR=1.0N_R=1.0, NI=0.78N_I=0.78), the 0.6-scaled imaginary part being justified as accounting for the missing couplings to dissipative processes and to continuum states not explicitly included — rather than an explicit coupled treatment (a complete calculation including projectile breakup into two fragments and target breakup into three simultaneously with the transfer channels was judged by the authors to be extremely complicated); the spectroscopic amplitudes did not include the explicit α+α+n\alpha+\alpha+n cluster configuration of 9Be^{9}\mathrm{Be}, so the calculated overlaps may overestimate certain two-neutron components and the absolute cross sections; DWBA with Woods–Saxon distorted waves required an amplitude roughly 80%80\% larger than the shell-model value, exposing uncontrolled optical-potential and reaction-model systematics; and the experimental energy resolution (EFWHM≈0.9E_{\mathrm{FWHM}}\approx 0.9 MeV) could not separate the ground-state and 429429 keV contributions, leaving the state-resolved shares of the transfer undetermined.

Problem Statement

Construct and validate a quantitative, mechanism-resolved description of two-neutron transfer in the 7Be+9Be^{7}\mathrm{Be}+{}^{9}\mathrm{Be} system at the near-barrier energy Elab=23.1E_{\mathrm{lab}}=23.1 MeV, in which all of the following are treated within one coupled framework with quantified uncertainties: (i) the elastic, inelastic, and two-neutron transfer channels, with both the simultaneous (direct, pair-correlated) route and the sequential route through resonant intermediate states such as 8Be(2+,3.03 MeV)^{8}\mathrm{Be}(2^+,3.03\ \mathrm{MeV}), replacing the present absorption of continuum effects into the São Paulo potential whose real and imaginary parts in the entrance partition are both multiplied by a strength factor 0.6 (outgoing partition: NR=1.0N_R=1.0, NI=0.78N_I=0.78) by explicit coupling of the projectile breakup continuum (7Be→α+3He^{7}\mathrm{Be}\to\alpha+{}^{3}\mathrm{He}) and the target breakup continuum (9Be→2α+n^{9}\mathrm{Be}\to 2\alpha+n) simultaneously with the transfer channels; (ii) spectroscopic amplitudes for the 9Be→7Be+2n^{9}\mathrm{Be}\to{}^{7}\mathrm{Be}+2n overlap that incorporate the explicit α+α+n\alpha+\alpha+n cluster structure of 9Be^{9}\mathrm{Be}, so that absolute cross sections no longer rest on possibly overestimated shell-model two-neutron components; (iii) a quantification of the systematic uncertainty of extracted amplitudes and cross sections associated with the optical-potential choice and reaction-model truncation (resolving, for example, the ∼80%\sim 80\% discrepancy between DWBA-extracted and shell-model amplitudes); and (iv) a determination of the relative contributions of the 7Be^{7}\mathrm{Be} ground state and the 429429 keV (1/2−)(1/2^-) first excited state to the transfer within the coupled calculation. Completion means that the calculated elastic and transfer angular distributions reproduce the existing Elab=23.1E_{\mathrm{lab}}=23.1 MeV 7Be+9Be^{7}\mathrm{Be}+{}^{9}\mathrm{Be} data within experimental uncertainties while remaining consistent with the existing 9Be(p,t)7Be^{9}\mathrm{Be}(p,t){}^{7}\mathrm{Be} data, and that the resulting decomposition into simultaneous versus sequential transfer and into ground-state versus excited-state population is stable under variation of the coupling scheme and structure inputs, as the quantification of systematic uncertainties required by item (iii) demands.

References

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