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Analysis of $M1$ capture in the $α(d,γ)^6$Li reaction

Ergash M. Tursunov, Daniel Baye

TL;DR

This work analyzes the $M1$ component of the radiative capture $\alpha(d,\gamma)^6\mathrm{Li}$ by introducing an effective $M1$ operator that is exactly equivalent to the long-wavelength form. In a three-body $p+n+\alpha$ framework, the operator's isoscalar part links to the total spin and is suppressed for $S$-wave entrance due to state orthogonality, while isovector contributions depend on small isospin mixing. The calculated $M1$ $S$-factor is negligibly small across astrophysical energies, reinforcing the dominance of $E2$ and aligning with several non-ab initio studies, though it leaves unresolved the large $M1$ signal found in some ab initio works. The paper argues that applying the effective operator to other models could clarify discrepancies and guide future ab initio investigations.

Abstract

An effective operator is exactly equivalent to the long-wavelength form of the $M1$ operator in transition matrix elements. It allows us to analytically and numerically analyze the $M1$ contribution to the $α(d,γ)^6$Li reaction. Isoscalar $M1$ transitions from an initial $S$ wave are shown to be forbidden in radiative capture reactions when distortion is neglected in the initial state. A calculation in a three-body model with proton, neutron, and a structureless $α$ interacting through effective forces leads to a negligible $M1$ $S$-factor at small energies. The dominant $M1$ contribution comes from transitions from an initial $S$ wave to isospin 1 components of the $^6$Li ground state. It is suggested that using this effective $M1$ operator in other models should clarify the origin of large discrepancies between $M1$ $S$-factors appearing in the literature.

Analysis of $M1$ capture in the $α(d,γ)^6$Li reaction

TL;DR

This work analyzes the component of the radiative capture by introducing an effective operator that is exactly equivalent to the long-wavelength form. In a three-body framework, the operator's isoscalar part links to the total spin and is suppressed for -wave entrance due to state orthogonality, while isovector contributions depend on small isospin mixing. The calculated -factor is negligibly small across astrophysical energies, reinforcing the dominance of and aligning with several non-ab initio studies, though it leaves unresolved the large signal found in some ab initio works. The paper argues that applying the effective operator to other models could clarify discrepancies and guide future ab initio investigations.

Abstract

An effective operator is exactly equivalent to the long-wavelength form of the operator in transition matrix elements. It allows us to analytically and numerically analyze the contribution to the Li reaction. Isoscalar transitions from an initial wave are shown to be forbidden in radiative capture reactions when distortion is neglected in the initial state. A calculation in a three-body model with proton, neutron, and a structureless interacting through effective forces leads to a negligible -factor at small energies. The dominant contribution comes from transitions from an initial wave to isospin 1 components of the Li ground state. It is suggested that using this effective operator in other models should clarify the origin of large discrepancies between -factors appearing in the literature.
Paper Structure (12 sections, 32 equations, 3 figures)

This paper contains 12 sections, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Effective $M1$ transition operator
  3. Microscopic $M1$ operator
  4. Transition matrix elements
  5. Three-body model of $M1$ transitions
  6. Three-body $M1$ operator
  7. Three-body wave functions
  8. Three-body $M1$ matrix elements
  9. Results
  10. Discussion
  11. Conclusion
  12. Reduced matrix elements in the three-body model

Figures (3)

  • Figure 1: Size of $T = 0$ and $T = 1$ isospin components displayed as $M1$$S$-factors.
  • Figure 2: Partial $M1$$S$-factors from the $^3S_1$, $^3D_1$, and $^3D_2$ initial partial waves.
  • Figure 3: $E1$, $E2$, and $M1$$S$-factors of the $\alpha(d,\gamma)^6$Li reaction and their sum (full line). The $E1$ and $E2$ contributions are obtained as in Refs. BT18 and TTK18, respectively, compared with experimental data of Robertson et alRDW81, Kiener et alKGR91, Mohr et alMKW94, Anders et al (LUNA14) ATM14, and Trezzi et al (LUNA17) TAA17.