The reaction 10^{10}B(p,α)7(p,\alpha)^{7}Be produces the 53-day radioisotope 7^{7}Be and thereby poisons the aneutronic fusion process 11^{11}B(p,2α)4(p,2\alpha)^{4}He planned for boron-proton fusion energy; the same low-energy cross section enters stellar boron-destruction networks. Below roughly 1 MeV the cross section is governed by near-threshold states of the compound nucleus 11^{11}C — the subthreshold 7/21+7/2^+_1, the threshold-aligned 5/22+5/2^+_2, and the 5/23+5/2^+_3 resonance — whose energies, widths, and channel decompositions are known only from indirect, model-dependent measurements. A shell-model-embedded-in-the-continuum (SMEC) study (Okołowicz, Płoszajczak, and Nazarewicz, 2023) predicted a significant continuum-coupling-induced enhancement of the low-energy cross section driven mainly by the 5/22+5/2^+_2 state, but the prediction is unvalidated: accelerator data at the required sub-MeV energies are unavailable or mutually inconsistent, and the theoretical sensitivity to the shell-model monopole terms, the continuum-coupling strength V0V_0, and the restricted model space has not been quantified. The open problem is to determine the low-energy bare-nucleus cross section (astrophysical S-factor) and the near-threshold 11^{11}C state properties well enough — by direct or indirect measurement, including laser-driven plasma environments, and/or by systematic theory — to confirm, refute, or quantitatively refine the predicted enhancement.

Background

Boron-proton fusion is pursued as an aneutronic energy source: the reaction 11^{11}B(p,2α)4(p,2\alpha)^{4}He — proton capture on boron-11 with breakup into three α\alpha particles — releases 8.7 MeV without neutron activation or long-lived radioactive products. Natural boron, however, contains about 19% 10^{10}B, which undergoes 10^{10}B(p,α)7(p,\alpha)^{7}Be and generates 7^{7}Be, a long-lived electron-capture radioisotope that poisons the reactor environment. The same reaction destroys boron in pre-main-sequence stars and other stellar sites. At the low energies of interest — from the near-threshold region around 10 keV up to sub-MeV energies — the charged-particle cross section is Coulomb-suppressed to values unmeasurable by ordinary laboratory techniques, and it is conventionally discussed through the astrophysical S-factor, obtained by factoring the Coulomb-barrier penetration out of the cross section: σ(E)=S(E) E−1exp⁡(−2πη)\sigma(E)=S(E)\,E^{-1}\exp(-2\pi\eta), with η\eta the Sommerfeld parameter, the standard dimensionless Coulomb-suppression measure η=Z1Z2e2/(ℏv)\eta = Z_1 Z_2 e^2/(\hbar v) for relative velocity vv of the collision partners.

The reaction proceeds through the compound nucleus 11^{11}C, whose low-energy behavior is controlled by states close to the proton separation threshold Sp≈8.690S_p\approx 8.690 MeV (levels are labeled by spin and parity JπJ^\pi, with a subscript giving the running index among levels of the same JπJ^\pi): the subthreshold 7/21+7/2^+_1 level, the threshold-aligned 5/22+5/2^+_2 resonance at Ex≈8.70E_x\approx 8.70 MeV (corresponding to a center-of-mass energy near 10 keV), and the nearby 5/23+5/2^+_3 resonance. Their properties are known mainly from indirect measurements: the Trojan Horse Method (THM) extracts the bare-nucleus low-energy cross section from a chosen three-body reaction — here 10^{10}B(d,nα)7(d,n\alpha)^{7}Be, with the deuteron acting as the 'Trojan horse' carrying a virtual proton — but carries large model-dependent uncertainties, while phenomenological R-matrix analyses (which parametrize the cross section by resonance energies and reduced widths) extrapolate inconsistently from higher-energy data. Direct measurements extending to low energies (M. Wiescher et al., 2017) proved inconsistent with R-matrix expectations, and the most recent direct dataset (B. Vande Kolk et al., 2022) begins at 0.8 MeV, above the region of interest; Trojan-horse measurements (C. Spitaleri et al., 2017; A. Cvetinović et al., 2018) cover 5 keV–2.2 MeV but rest on model-dependent normalization.

The source study (J. Okołowicz, M. Płoszajczak, and W. Nazarewicz, 2023) approached this system with the shell model embedded in the continuum (SMEC), an extension of the nuclear shell model in which discrete shell-model configurations are coupled to open particle-decay channels; near thresholds this coupling can 'collectivize' a state, aligning it with a decay channel and strongly modifying its structure. Using the WBP interaction (a Warburton–Brown shell-model Hamiltonian for the pp and sdsd shells, employed in the restricted (psd)(psd) model space) with monopole terms adjusted to reproduce the mirror nucleus 11^{11}B, and a Wigner–Bartlett contact force of strength V0V_0 as the continuum-coupling interaction, they found that the 5/22+5/2^+_2 state is aligned with the [10[^{10}B(3+)⊗p(s1/2)]5/2+(3^+)\otimes p(s_{1/2})]_{5/2^+} channel, carries a large s1/2s_{1/2} spectroscopic factor (the fraction of its wave function consisting of 10^{10}B in the 3+3^+ state plus an s1/2s_{1/2} proton), and acquires a large continuum-coupling correlation energy. They concluded that the low-energy 10^{10}B(p,α)7(p,\alpha)^{7}Be cross section should be significantly enhanced at energies accessible to laser-driven hot plasma facilities such as NIF (the National Ignition Facility laser) and OMEGA EP (the extended-performance petawatt laser at the Laboratory for Laser Energetics), energies that accelerator-based measurements cannot reach. Their affordable calculation variants — different monopole shifts and two values of V0V_0 — changed the level ordering, correlation-energy behavior, and spectroscopic factors, so the prediction carries unquantified model uncertainty, and no measurement has yet tested it. The agenda of measuring threshold-region reactions such as 10^{10}B(p,α)7(p,\alpha)^{7}Be in plasma environments was set out in M. Wiescher, R. J. deBoer, and J. Görres (2022).

Problem Statement

Determine the low-energy bare-nucleus 10^{10}B(p,α)7(p,\alpha)^{7}Be cross section — equivalently the astrophysical S-factor S(E)S(E) — over the near-threshold region from the ∼10\sim 10 keV resonance through sub-MeV energies, and establish the properties of the near-threshold 11^{11}C states that control it: the excitation energies, proton and α\alpha partial widths, and channel decompositions (spectroscopic factors) of the subthreshold 7/21+7/2^+_1, threshold-aligned 5/22+5/2^+_2, and 5/23+5/2^+_3 states. The specific objective posed by the source study is to validate, refute, or refine the predicted continuum-coupling-induced enhancement of the low-energy cross section driven by the 5/22+5/2^+_2 and 7/21+7/2^+_1 states, either (i) by a direct or indirect experimental determination of the sub-MeV cross section — including measurements in laser-driven hot plasma environments of the NIF/OMEGA-EP type (large laser facilities such as the National Ignition Facility and the OMEGA EP laser), or improved accelerator-based techniques — and/or (ii) by a systematic theoretical study quantifying the dependence on the WBP monopole adjustments, the continuum-coupling strength V0V_0, and enlarged model spaces beyond the restricted (psd)(psd) space (the shell-model space spanned by the pp and sdsd shells). The problem is solved when S(E)S(E) and the near-threshold resonance parameters are constrained, with stated uncertainties, to the point where the predicted near-threshold enhancement of the low-energy 10^{10}B(p,α)7(p,\alpha)^{7}Be reaction rate is quantitatively confirmed or excluded.

References

  1. J. Okołowicz, M. Płoszajczak, and W. Nazarewicz, 'Near-threshold resonances in 11C and the 10B(p,α)7Be aneutronic reaction cross section', Phys. Rev. C 107, L021305 (2023). DOI: 10.1103/PhysRevC.107.L021305; arXiv:2211.02555; https://arxiv.org/abs/2211.02555
  2. J. Okołowicz, M. Płoszajczak, and W. Nazarewicz, 'Erratum: Near-threshold resonances in 11C and the 10B(p,α)7Be aneutronic reaction cross section [Phys. Rev. C 107, L021305 (2023)]', Phys. Rev. C 109, 059902 (2024). DOI: 10.1103/PhysRevC.109.059902; https://doi.org/10.1103/PhysRevC.109.059902
  3. A. M. Mukhamedzhanov, 'Proton 0.01 MeV resonance width and low-energy S-factor of p+10B fusion', Phys. Rev. C 108, 054603 (2023). DOI: 10.1103/PhysRevC.108.054603; arXiv:2307.08963; https://arxiv.org/abs/2307.08963
  4. M. R. D. Rodrigues, A. Bonasera, M. Scisciò, J. A. Pérez-Hernández, M. Ehret, F. Filippi, P. L. Andreoli, M. Huault, H. Larreur, D. Singappuli, D. Molloy, D. Raffestin, M. Alonzo, G. G. Rapisarda, D. Lattuada, G. L. Guardo, C. Verona, Fe. Consoli, G. Petringa, A. McNamee, M. La Cognata, S. Palmerini, T. Carriere, M. Cipriani, G. Di Giorgio, G. Cristofari, R. De Angelis, G. A. P. Cirrone, D. Margarone, L. Giuffrida, D. Batani, P. Nicolai, K. Batani, R. Lera, L. Volpe, D. Giulietti, S. Agarwal, M. Krupka, S. Singh, and Fa. Consoli, 'Radioisotope production using lasers: From basic science to applications', Matter Radiat. Extremes 9, 037203 (2024). DOI: 10.1063/5.0196909; arXiv:2312.09145; https://doi.org/10.1063/5.0196909
  5. M. Wiescher, R. J. deBoer, and J. Görres, 'Threshold effects in the 10B(p,α)7Be, 12C(p,γ)13N and 14N(p,γ)15O reactions', Front. Phys. 10, 1009489 (2022). DOI: 10.3389/fphy.2022.1009489; https://doi.org/10.3389/fphy.2022.1009489
  6. M. Wiescher, R. J. deBoer, J. Görres, and R. E. Azuma, 'Low energy measurements of the 10B(p,α)7Be reaction', Phys. Rev. C 95, 044617 (2017). DOI: 10.1103/PhysRevC.95.044617; https://doi.org/10.1103/PhysRevC.95.044617
  7. C. Spitaleri, S. M. R. Puglia, M. La Cognata, L. Lamia, S. Cherubini, A. Cvetinović, G. D'Agata, M. Gulino, G. L. Guardo, I. Indelicato, R. G. Pizzone, G. G. Rapisarda, S. Romano, M. L. Sergi, R. Spartà, S. Tudisco, A. Tumino, M. Gimenez Del Santo, N. Carlin, M. G. Munhoz, F. A. Souza, A. Szanto de Toledo, A. Mukhamedzhanov, C. Broggini, A. Caciolli, R. Depalo, R. Menegazzo, V. Rigato, I. Lombardo, and D. Dell'Aquila, 'Measurement of the 10B(p,α0)7Be cross section from 5 keV to 1.5 MeV in a single experiment using the Trojan horse method', Phys. Rev. C 95, 035801 (2017). DOI: 10.1103/PhysRevC.95.035801; https://doi.org/10.1103/PhysRevC.95.035801
  8. A. Cvetinović, C. Spitaleri, R. Spartá, G. G. Rapisarda, S. M. R. Puglia, M. La Cognata, S. Cherubini, G. L. Guardo, M. Gulino, L. Lamia, R. G. Pizzone, S. Romano, M. L. Sergi, and A. Tumino, 'Trojan horse measurement of the 10B(p,α0)7Be cross section in the energy range from 3 keV to 2.2 MeV', Phys. Rev. C 97, 065801 (2018). DOI: 10.1103/PhysRevC.97.065801; https://doi.org/10.1103/PhysRevC.97.065801
  9. B. Vande Kolk, K. T. Macon, R. J. deBoer, T. Anderson, A. Boeltzig, K. Brandenburg, C. R. Brune, Y. Chen, A. M. Clark, T. Danley, B. Frentz, R. Giri, J. Görres, M. Hall, S. L. Henderson, E. Holmbeck, K. B. Howard, D. Jacobs, J. Lai, Q. Liu, J. Long, K. Manukyan, T. Massey, M. Moran, L. Morales, D. Odell, P. O'Malley, S. N. Paneru, A. Richard, D. Schneider, M. Skulski, N. Sensharma, C. Seymour, G. Seymour, D. Soltesz, S. Strauss, A. Voinov, L. Wüstrich, and M. Wiescher, 'Investigation of the 10B(p,α)7Be reaction from 0.8 to 2.0 MeV', Phys. Rev. C 105, 055802 (2022). DOI: 10.1103/PhysRevC.105.055802; https://doi.org/10.1103/PhysRevC.105.055802
  10. M. Wiescher, C. A. Bertulani, C. R. Brune, R. J. deBoer, A. Diaz-Torres, L. R. Gasques, K. Langanke, P. Navrátil, W. Nazarewicz, J. Okołowicz, D. R. Phillips, M. Płoszajczak, S. Quaglioni, and A. Tumino, 'Quantum physics of stars', Rev. Mod. Phys. 97, 025003 (2025). DOI: 10.1103/RevModPhys.97.025003; https://doi.org/10.1103/RevModPhys.97.025003