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Register a new userTheoretical study of the direct $alpha+d$ $rightarrow$ $^6$Li + $gamma $ astrophysical capture process in a three-body model II. Reaction rates and primordial abundance
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The astrophysical S-factor and reaction rate of the direct capture process $alpha+d$ $rightarrow$ $^6$Li + $gamma$, as well as the abundance of the $^6$Li element are estimated in a three-body model. The initial state is factorized into the deuteron bound state and the $alpha+d$ scattering state. The final nucleus $^6$Li(1+) is described as a three-body bound state $alpha+n+p$ in the hyperspherical Lagrange-mesh method. Corrections to the asymptotics of the overlap integral in the S- and D-waves have been done for the E2 S-factor. The isospin forbidden E1 S-factor is calculated from the initial isosinglet states to the small isotriplet components of the final $^6$Li(1+) bound state. It is shown that the three-body model is able to reproduce the newest experimental data of the LUNA collaboration for the astrophysical S-factor and the reaction rates within the experimental error bars. The estimated $^6$Li/H abundance ratio of $(0.67 pm 0.01)times 10^{-14}$ is in a very good agreement with the recent measurement $(0.80 pm 0.18)times 10^{-14}$ of the LUNA collaboration.
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The astrophysical capture process $alpha+d$ $rightarrow$ $^6$Li + $gamma$ is studied in a three-body model. The initial state is factorized into the deuteron bound state and the $alpha+d$ scattering state. The final nucleus $^6$Li(1+) is described as a three-body bound state $alpha+n+p$ in the hyperspherical Lagrange-mesh method. The contribution of the E1 transition operator from the initial isosinglet states to the isotriplet components of the final state is estimated to be negligible. An estimation of the forbidden E1 transition to the isosinglet components of the final state is comparable with the corresponding results of the two-body model. However, the contribution of the E2 transition operator is found to be much smaller than the corresponding estimations of the two-body model. The three-body model perfectly matches the new experimental data of the LUNA collaboration with the spectroscopic factor 2.586 estimated from the bound-state wave functions of $^6$Li and deuteron.
A comparative analysis of the astrophysical S factor and the reaction rate for the direct $ alpha(d,gamma)^{6}{rm Li}$ capture reaction, and the primordial abundance of the $^6$Li element, resulting from two-body, three-body and combined cluster models is presented. It is shown that the two-body model, based on the exact-mass prescription, can not correctly describe the dependence of the isospin-forbidden E1 S factor on energy and does not reproduce the temperature dependence of the reaction rate from the direct LUNA data. It is demonstrated that the isospin-forbidden E1 astrophysical S factor is very sensitive to the orthogonalization procedure of Pauli-forbidden states within the three-body model. On the other hand, the E2 S factor does not depend on the orthogonalization method. This insures that the orthogonolizing pseudopotentials method yields a very good description of the LUNA collaborations low-energy direct data. At the same time, the SUSY transformation significantly underestimates the data from the LUNA collaboration. On the other hand, the energy dependence of the E1 S factor are the same in both methods. The best description of the LUNA data for the astrophysical S factor and the reaction rates is obtained within the combined E1(three-body OPP)+E2(two-body) model. It yields a value of $(0.72 pm 0.01) times 10^{-14}$ for the $^6$Li/H primordial abundance ratio, consistent with the estimation $(0.80 pm 0.18) times 10^{-14}$ of the LUNA collaboration. For the $^6{rm Li}/^7{rm Li}$ abundance ratio an estimation $(1.40pm 0.12)times 10^{-5}$ is obtained in good agreement with the Standard Model prediction.
At the long-wavelength approximation, electric dipole transitions are forbidden between isospin-zero states. In an $alpha+n+p$ model with $T = 1$ contributions, the $alpha(d,gamma)^6$Li astrophysical $S$-factor is in agreement with the experimental data of the LUNA collaboration, without adjustable parameter. The exact-masses prescription used to avoid the disappearance of $E1$ transitions in potential models is not founded at the microscopic level.
The astrophysical S-factor for the direct $ alpha(d,gamma)^{6}{rm Li}$ capture reaction is calculated in a three-body model based on the hyperspherical Lagrange-mesh method. A sensitivity of the E1 and E2 astrophysical S-factors to the orthogonalization method of Pauli forbidden states in the three-body system is studied. It is found that the method of orthogonalising pseudopotentials (OPP) yields larger isotriplet ($T=1$) components than the supersymmetric transformation (SUSY) procedure. The E1 astrophysical S-factor shows the same energy dependence in both cases, but strongly different absolute values. At the same time, the E2 S-factor does not depend on the orthogonalization procedure. As a result, the OPP method yields a very good description of the direct data of the LUNA collaboration at low energies, while the SUSY transformation strongly underestimates the LUNA data. keywords{three-body model; orthogonalization method; astrophysical S factor.
At the long-wavelength approximation, $E1$ transitions are forbidden between isospin-zero states. Hence $E1$ radiative capture is strongly hindered in reactions involving $N = Z$ nuclei but the $E1$ astrophysical $S$ factor may remain comparable to, or larger than, the $E2$ one. Theoretical expressions of the isoscalar and isovector contributions to $E1$ capture are analyzed in microscopic and three-body approaches in the context of the $alpha(d,gamma)^6$Li reaction. The lowest non-vanishing terms of the operators are derived and the dominant contributions to matrix elements are discussed. The astrophysical $S$ factor computed with some of these contributions in a three-body $alpha+n+p$ model is in agreement with the recent low-energy experimental data of the LUNA collaboration. This confirms that a correct treatment of the isovector $E1$ transitions involving small isospin-one admixtures in the wave functions should be able to provide an explanation of the data without adjustable parameter. The exact-masses prescription which is often used to avoid the disappearance of the $E1$ matrix element in potential models is not founded at the microscopic level and should not be used for such reactions. The importance of capture components from an initial $S$ scattering wave is also discussed.
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