Astrophysical $S$-factor of the direct $\alpha(d,\gamma)^6$Li capture reaction in a three-body model

TL;DR

This paper calculates the astrophysical S-factor for the (,Li) reaction using a three-body model, achieving agreement with experimental data without adjustable parameters.

Contribution

It introduces a three-body +n+p model that accurately predicts the S-factor, addressing limitations of previous potential models.

Findings

The model reproduces experimental S-factor data from LUNA.

Electric dipole transitions are forbidden at long wavelengths in isospin-zero states.

Using exact masses avoids the disappearance of E1 transitions in potential models.

Abstract

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.

Equations

$$\displaystyle{\cal M}_{\mu}^{E\lambda}=e\sum_{j=1}^{A}(\mbox{$\frac{1}{2}$}-t_{j3})r^{\prime\lambda}_{j}Y_{\lambda\mu}(\Omega^{\prime}_{j}),$$

$${\mathcal{M}}_{\mu }^{E1}=-e\sum _{j=1}^A{t}_{j3}{r}_j^{\prime }{Y}_{1\mu }({\Omega }_j^{\prime })={\mathcal{M}}_{\mu }^{E1,\mathrm{IV}}.$$

$${\mathcal{M}}_{\mu }^{E1,\mathrm{IS}}$$

$$\displaystyle\widetilde{\mathcal{M}}_{\mu}^{E1,{\rm IV}}=\mbox{$\frac{1}{2}$}erY_{1\mu}(\Omega_{r}),$$

$$Z_{\mathrm{eff}}^{(\mathrm{E}1)}\propto \left(\frac{Z_1}{A_1}-\frac{Z_2}{A_2}\right)\to Z_{\mathrm{eff}}^{(\mathrm{E}1)}\propto m_N\left(\frac{Z_1}{M_1}-\frac{Z_2}{M_2}\right)$$

$$m_N\left(\frac{Z_1}{M_1}-\frac{Z_2}{M_2}\right)\approx \frac{1}{2m_N{c}^2}\left(\frac{B(A_1,Z_1)}{A_1}-\frac{B(A_2,Z_2)}{A_2}\right).$$

Full text

radiative capture, isospin forbidden dipole transition, three-body model

I Introduction

A radiative-capture reaction is an electromagnetic transition between an initial scattering state and a final bound state. Astrophysical collision energies can be very low with respect to the Coulomb barrier and cross sections are then tiny. The dominant multipolarity is $E1$ in general. In the special case of reactions between $N=Z$ nuclei however, $E1$ transitions are « forbidden » by an isospin selection rule at the long-wavelength approximation (LWA) and $E2$ transitions become crucial. Nevertheless, $E1$ transitions are not exactly forbidden since isospin is an approximate symmetry. The analysis of the recent LUNA data ATM14 ; TAA17 for the $\alpha(d,\gamma)^{6}$Li reaction indicates that $E1$ cross sections dominate the $E2$ cross sections below about 0.1 MeV. Since $E1$ transitions vanish in many models, recent calculations use the « exact-masses » prescription to avoid their disappearance TKT16 . Here we present results of the simplest model allowing $E1$ transitions thanks to small $T=1$ components, i.e. the $\alpha+n+p$ three-body model.

II Isospin-forbidden $E1$ transitions

The electric multipole operators read at the LWA,

[TABLE]

where $A$ is the number of nucleons, \mbox{\boldmathr}^{\prime}_{j}=(r^{\prime}_{j},\Omega^{\prime}_{j}) is the coordinate of nucleon $j$ with respect to the centre of mass of the system, and $t_{j3}$ is the third component of its isospin operator \mbox{\boldmatht}_{j}. The isoscalar (IS) part of the $E1$ operator vanishes at the LWA since $\sum_{j=1}^{A}r^{\prime}_{j}Y_{1\mu}(\Omega^{\prime}_{j})=0$ and this operator becomes an isovector (IV),

[TABLE]

At the LWA, $E1$ matrix elements thus vanish between isospin-zero states. This leads to the total isospin $T$ selection rule in $N=Z$ nuclei and reactions: $T_{i}=0\rightarrow T_{f}=0$ is forbidden. But $E1$ transitions are not exactly forbidden in these $N=Z$ systems because isospin is not an exact quantum number. Small $T=1$ admixtures appear in the wave functions. The main isovector $E1$ contributions are due to $T_{f}=1$ admixtures in the final state or to $T_{i}=1$ admixtures in the initial state. Moreover, the isoscalar $E1$ operator reads beyond the LWA,

[TABLE]

up to terms that should give only a small contribution Ba12 , contrary to other expressions often used in the literature. The isoscalar $E1$ contribution to the capture involves the $T=0$ parts of the wave functions.

III Three-body model of $\alpha(d,\gamma)^{6}$Li reaction

The present wave functions BT18 are adapted from the $\alpha+n+p$ model of Ref. TKT16 . The $J_{f}=1^{+}$ final bound state is described in hyperspherical coordinates and the initial scattering states are described in Jacobi coordinates. Three-body effective $E1$ and $E2$ operators are constructed which assume that the $\alpha$ particle or cluster is in its $0^{+}$ ground state. For example, the isovector part of the effective three-body $E1$ operator reads at the LWA,

[TABLE]

where $r$ is the Jacobi coordinate between $n$ and $p$. The expressions of the isoscalar part of the $E1$ operator beyond the LWA and of the $E2$ operator can be found in Ref. BT18 .

The three-body states contain $S=0$ and 1 components. Because of the isospin zero of the $\alpha$ particle and the antisymmetry of the $n+p$ subsystem with orbital momentum $l$, the components with $l+S$ odd correspond to $T=0$ and those with $l+S$ even to $T=1$. The initial scattering state is described by the product of a frozen deuteron wave function ($l_{i}=0$, $S_{i}=1$) and $\alpha+d$ $L$ partial scattering waves. Hence, it is purely $T_{i}=0$. The $J_{f}=1^{+}$ final bound state contains a small $T_{f}=1$ component (about 0.5 %). The $E1$ transitions start from $L_{i}=1$ and the $E2$ transitions from $L_{i}=0$ and 2.

This model requires an asymptotic correction to the $E2$ matrix elements. Indeed the overlap integrals $I_{L}(R)$ of the deuteron and $\alpha+n+p$ final wave functions decrease too fast beyond 10 fm as shown for $L=0$ by Fig. 1. This is corrected by matching at 7.75 fm the overlap integrals with the exact Whittaker asymptotic function multiplied by realistic asymptotic normalization coefficients.

Total $E1+E2$ astrophysical $S$ factors calculated in the three-body model with the $E2$ correction are compared in Fig. 2 with experimental data. The isoscalar $E1$ capture contribution is small and can be neglected in first approximation. The isovector $E1$ contribution dominates below about 0.1 MeV. Model A and B correspond to different $\alpha+N$ potentials (see Ref. BT18 for details).

IV Comment on the exact-masses prescription

To obtain non-vanishing $E1$ transitions in the two-body or potential model, experimental masses are used in the effective charge of $N=Z$ nuclei,

[TABLE]

where $m_{N}$ is the nucleon mass and $Z_{1,2}$, $A_{1,2}$ and $M_{1,2}$ are the charges, mass numbers and experimental masses of the colliding nuclei, respectively. This exact-masses prescription is unfounded.

(i) $E1$ transitions would remain exactly forbidden in the $d(d,\gamma)^{4}$He reaction, in contradiction with ab initio calculations.

(ii) Using the mass expression M=Am_{N}+(N-Z)\mbox{\frac{1}{2}}(m_{n}-m_{p})-B(A,Z)/c^{2}, effective charges would depend on the binding energies $B(A_{1,2},Z_{1,2})$,

[TABLE]

Binding energies per nucleon $B(A,Z)/A$ mostly depend on the main $T=0$ components of the wave functions and not on the small $T=1$ components physically responsible for the non vanishing of “forbidden” $E1$ transitions.

(iii) $E1$ matrix elements would be unphysically sensitive to the long $T_{f}=0$ $\alpha+d$ tail of the 6Li wave function.

V Conclusion

Isovector $E1$ transitions with $T=1$ admixtures in the final state and $E2$ transitions explain the order of magnitude of the LUNA data ATM14 ; TAA17 , without any adjustable parameter. Isoscalar $E1$ transitions beyond the LWA are negligible for the $\alpha(d,\gamma)^{6}$Li reaction. The exact-masses prescription is not founded and should not be trusted for reactions between $N=Z$ nuclei, such as $\alpha(d,\gamma)^{6}$Li and 12C$(\alpha,\gamma)^{16}$O. A three-body model with $T=1$ admixtures in both initial and final states should be developed. Microscopic six-body and ab initio calculations are difficult but possible and necessary for a deeper understanding of this reaction.