R-matrix theory is a rigorous framework used in nuclear physics to describe scattering states resulting from the interactions of particles or a system of particles, including nucleons, nuclei, electrons, atoms, or molecules. While this theory was developed and used to describe reactions with two particles in both entrance and exit channels, as well as three-particle elastic scattering, the framework to compute the elements of the scattering matrix connecting two-particle with three-particle reaction channels is lacking. We explored the feasibility of generalizing the R-matrix approach to the treatment of reactions with three particles in the final state. The form of the Bloch operator required to compute the elements of the scattering process connecting two-particle channels to three-particle channels was identified following the methodology originally adopted for the simpler case of two-particle reactions. The formalism was then implemented within the context of the no-core shell model with continuum, a state-of-the-art ab initio approach for the description of low-energy reactions of light nuclei. It was found that the generalization of the R-matrix framework is relatively simple for transfer reactions such as the 3H+3H 4He+n+n fusion reaction, where the final three-particle state does not result from the breakup of one of the nuclei in the initial state. In such cases, the matrix representation of the generalized Bloch operator assumes a rather simple block-diagonal form, with the two blocks given by the standard Bloch operators for two-particle and three-particle scattering, respectively. We also provided an explanation for a recently observed energy dependence for the neutron spectrum from the 3H+3H 4He+n+n reaction at center-of-mass energies in the range of 16–50 keV and supported the compilation of a review paper on ab initio techniques for the description of light and unbound nuclei.
Nuclear reactions leading to the emission of three or more particles, such as the (n,2n) reaction on 9Be used to multiply neutrons in reactor blankets and the fusion reactions 3H+3H 4He+n+n (required to interpret the inertial confinement fusion neutron spectrum at Lawrence Livermore National Laboratory's National Ignition Facility) and 3H+3H 4He+p+p (to determine the flux of solar neutrinos detected on Earth [Adelberger et al. 2011]) are extremely difficult to characterize. A complete experimental determination requires difficult coincidence measurements of the outgoing energies and angles of all but one of the reaction products. Even in the rare cases in which such experiments are feasible, the range of energies and/or angles covered is usually limited. Therefore, a robust theoretical description is critical to provide all reaction data required by applications. Unfortunately, due to the complexity of the three-particle scattering problem and a historical lack of computing power, evaluations for reactions emitting multiple fragments are usually based on very simple models and lack credible predictive power. This can lead to large uncertainties in evaluated cross sections and angular distributions, with important repercussions for applications. As an example, the 9Be(n,2n) cross section is one of the largest sources of uncertainty in the criticality simulations of reactors (Papadimitriou 2000). An accurate description of light-nuclei reactions starting (i.e., the entrance channel) with the collision of two particles, but ending (i.e., the exit channel) with the emission of three particles could be obtained with the state-of-the art ab initio (literally, from first principles) approach to light-nucleus reactions developed at Livermore (Navrátil et al. 2016), which is based on the microscopic R-matrix theory (Lane and Thomas 1958, Descouvemont and Baye 2010). While this theory has been developed and successfully applied to describe reactions with two particles (Kumar et al. 2016, Calci et al. 2016, Hupin et al. 2013) in both entrance and exit channels, as well as three-particle elastic scattering (Romero-Redondo et al. 2014 and 2016, Quaglioni et al. 2013), the framework to compute the elements of the scattering matrix connecting two-particle with three-particle reaction channels is still missing. The goal of this project was to investigate what it would take to develop a generalized microscopic R-matrix or equivalent theory to address these missing matrix elements and whether it would be feasible to implement such a generalized framework within our in-house ab initio approach to light-nucleus reactions.
In R-matrix theory, the matchup of internal and external solutions can be conveniently and elegantly implemented with the help of the Bloch operator, a singular operator defined at the boundary between the two solutions (or regions). The form of the Bloch operator is known when either only two-particle or three-particle channels are present, but it has not been derived for the more general case in which entrance and exit channels contain a different number of particles. The first objective of this project was to identify the mathematical expression for such a generalized Bloch operator and rigorously prove its correctness through a formal derivation. This objective has been fully met.
Originally, a second research objective of the project was to test the generalized R-matrix formalism by applying it to the description of the n+d n+n+p reaction process, for which accurate benchmark calculations are available. This test turned out to be more difficult than expected. The n+d n+n+p process is part of a subcategory of reactions terminating in a three-body channel in which the final state results from the breakup of one of the nuclei in the initial state. The derivations conducted as part of this project showed that the Bloch operator for this type of reaction may contain non-trivial couplings between two-particle and three-particle basis states, which had not been anticipated and went beyond the scope of the project. This achievement of this objective was determined to be unlikely within the project duration and was left to a future investigation.
An opportunity emerged during the course of the project to provide theoretical support in the interpretation of measurements of the neutron spectrum from the 3H+3H 4He+n+n reaction in inertial confinement fusion experiments at the Laboratory for Laser Energetics OMEGA Laser Facility (Rochester, NY). Therefore, part of the project was dedicated to carrying out ab initio calculations of 3H+3H scattering and researching previous microscopic calculations for this process, with the goal of explaining the underlying physics behind an unexpected dependence on the center-of-mass energy of the measured spectrum in the range of 16–50 keV (Johnson et al. 2018).
Following the seminal work of Bloch (1957) as well as the that of Robson (1969), the mathematical expression for the desired Bloch operator was formally derived by assuming that the wave function in the internal region is represented in terms of an expansion over two-particle as well as three-particle basis states. The derivation was then repeated within the more complex model space of the no-core shell model with the continuum (NCSMC) approach, where continuous ‘microscopic-cluster’ states made of pairs and triplets of nuclei in relative motion with respect to each other are further combined with static many-body wave functions of the aggregate system to arrive at a fully integrated description of the structure and dynamics of the reacting nuclei in the internal region. The final result was the mathematical expression of the generalized Bloch operator for the NCSMC approach. It was found that the generalization of the microscopic R-matrix framework is relatively simple for transfer reactions such as the 3H+3H 4He+n+n fusion reaction, where the final three-particle state does not result from the breakup of one of the nuclei in the initial state. In such cases, the matrix representation of the generalized Bloch operator assumes a rather simple block-diagonal form, with the two blocks given by the standard Bloch operators for two-particle and three-particle scattering, respectively. On the other hand, processes such as the n+d n+n+p reaction, where the final state is given by the breakup of one of the nuclei in the initial state, in principle contain non-trivial couplings between two-particle and three-particle basis states. The applicability of the generalized Bloch operator derived in this project to this type of reactions is not clear and requires further investigations.
This research supports the Laboratory's stockpile stewardship mission and inertial fusion science and technology focus by enabling the future development of an essential capability needed to accurately model light-nucleus reactions with three-particle final states. Further, it helps maintain continued leadership in the Laboratory's core competencies of nuclear, chemical, and isotopic science and technology, and high-performance computing, simulations, and data science.
Support for continued research on the development of the NCSMC approach for the description of light-nucleus reactions with three-particle final states will be provided starting in fiscal year 2019 by the SciDAC-4 "NUCLEI" collaboration, jointly funded by the DOE’s Office of Nuclear Physics and the NNSA.
Adelberger, E. G., et al. 2011. "Solar Fusion Cross Sections. II. The pp Chain and CNO Cycles." Review of Modern Physics 83(1). doi: 10.1103/RevModPhys.83.195.