Clustering effects in the ${}^6\mathrm{Li}(p,{}^3\mathrm{He}){}^4\mathrm{He}$ reaction at astrophysical energies

Background: The understanding of nuclear reactions between light nuclei at energies below the Coulomb barrier is important for several astrophysical processes, but their study poses experimental and theoretical challenges. At sufficiently low energies, the electrons surrounding the interacting ions affect the scattering process. Moreover, the clustered structure of some of these nuclei may play a relevant role on the reaction observables.

Purpose: In this article, we focus on a theoretical investigation of the role of clustered configurations of ${}^6\mathrm{Li}$ in reactions of astrophysical interest.

Methods: The ${}^6\mathrm{Li}$($p{,}^3\text{He}$)${}^4\mathrm{He}$ reaction cross section is described considering both the direct transfer of a deuteron as a single pointlike particle in the distorted-wave Born approximation (DWBA), and the transfer of a neutron and a proton in second-order DWBA. A number of two- and three-cluster structure models for ${}^6\mathrm{Li}$ are compared.

Results: Within the two-cluster structure model, we explore the impact of the deformed components in the ${}^6\mathrm{Li}$ wave function on the reaction of interest. Without explicit adjusting of the calculation inputs to the transfer channel, we obtain reaction cross sections in good agreement with the results of more microscopic models. Within the three-cluster structure model, we gauge the degree of $\alpha \text{−}d$ clustering and explicitly probe its role in specific features of the transfer cross section; however, cross-section absolute values are overestimated. We compare the energy trend of the astrophysical $S$ factor deduced in each case.

Conclusions: Clustered ${}^6\mathrm{Li}$ configurations lead in general to a significant enhancement of the astrophysical factor in the energy region under study. This effect only originates from clustering, whereas static deformations of the ground-state configuration play a negligible role at very low energies.

Figure 1

Top panel: Experimental ${}^6\mathrm{Li}+p\to {}^3\mathrm{He}+\alpha$ astrophysical factors as a function of the collision energy, in particular data below $1\mathrm{MeV}$ from Ref. [21] (black circles), inverse-kinematics data from Ref. [24] (orange full squares), data from Ref. [28] including insulator-target data first published in Ref. [23] (red upward triangles), and Trojan Horse method data from Ref. [5] (green open diamonds). Bottom panel: Green open diamonds are the same as in the top panel. All other points represent the predicted bare-nucleus astrophysical factor obtained from data in the top panel using Eq. (2) with $U=182\mathrm{eV}$. Each line corresponds to Eq. (4) with a given value of $R_n$, as per the legend, and $C$ adjusted so that the line agrees with experimental data at $350\mathrm{keV}$ (see text).

Figure 2

Points are experimental ${}^6\mathrm{Li}+p$ elastic-scattering phase shifts, collected in Ref. [49] from earlier experiments (see references therein). Lines in corresponding colors are the prediction of the optical potential discussed in Sec. 4a. In the figure legend, $L$ is the ${}^6\mathrm{Li}\text{−}p$ relative orbital angular momentum modulus quantum number, $S$ the modulus quantum number for the sum of ${}^6\mathrm{Li}$ and $p$ spins, $J$ the modulus quantum number for the sum of $\vec{L}$ and $\vec{S}$, and $\pi$ the parity of the state.

Figure 3

${}^6\mathrm{Li}$($p{,}^3\text{He}$)${}^4\mathrm{He}$ astrophysical $S$ factor. Points are a subset of the rescaled data in the bottom panel of Fig. 1 (only some data points are shown for readability). The magenta dotted line is the resonating-group-method calculation in Ref. [14]. The other lines are the present-work deuteron-transfer post-form DWBA calculation. The brown dashed line was computed including only a relative orbital angular momentum $\ell =0$ ($l$ in the figure legend) for the ${}^4\mathrm{He}\text{−}d$ wave function. The turquoise dot-dashed and blue solid lines were computed including also a $d$-wave component (constructed as reported in Sec. 4a) with relative norms of about $0.8\%$ and $6.6\%$ respectively.

Figure 4

Representation of the reactants structure and the Jacobi “$T$” coordinate system. Spheres depict the particles treated as elementary in the present formalism (in Sec. 4a, the transferred $np$ system is inert). Each line represents a Jacobi “$T$” coordinate (transferred system internal motion, core-transferred motion, projectile-target motion), and $l, \ell , L$ are the symbols employed here to represent the orbital angular momentum modulus quantum number associated with each of those coordinates.

Figure 5

Contour plot for the reduced radial probability density function (pdf), in ${\mathrm{fm}}^{-2}$, for the core $+ p + n$ state of ${}^3\mathrm{He}$. The $x$ axis is the distance between the transferred $p$ and $n$, while the $y$ axis is the distance between the core and the $n\text{−}p$ center of mass. The line obeys the equation in the legend. See Sec. 4b1 for details.

Figure 6

Same as Fig. 5 but for ${}^6\mathrm{Li}$. The top-left panel represents the state yielding the best agreement with available information from three-body calculations, constructed using the amplitudes in Table 1. The state represented in the bottom-left panel was constructed with the same weights but with opposite sign for the $2s_{1/2}\times 2s_{1/2}$ component in Table 1. The states represented in the top- and bottom-right panels are constructed as those in the top- and bottom-left panels but assigning equal absolute weight to the ${\left(2s\right)}^2$ and ${\left(1p\right)}^2$ configurations. The states shown in the middle-left and -right panels do not include the configurations with transferred nucleons occupying, respectively, the ${}^6\mathrm{Li} 2s$ or $1p$ shells. See Sec. 4b1 for details.

Figure 7

The red dot-dashed line represents the ${}^6\mathrm{Li}$($p{,}^3\text{He}$)$\alpha$ astrophysical $S$ factor for the ($np$)-transfer DWBA calculation using the ${}^6\mathrm{Li}$ wave function in the top-left panel of Fig. 6. The red dashed and brown dotted lines represent the $S$ factor associated with, respectively, only the first- and second-order contributions to the total transition amplitude, namely ${\mathcal{T}}^{\left(1\right)}$ and ${\mathcal{T}}^{\left(2\right)}$ in Sec. 3b. The turquoise solid line represents the first-order-only $S$ factor, but excluding all contributions due to ${}^6\mathrm{Li}$ configurations where the $\alpha \text{−}d$ relative-motion orbital angular momentum, ${\ell }_{\alpha d}$, is greater than 0 (see text for discussion). Points are the same experimental data in Fig. 3 (not shown in the legend for brevity).

Figure 8

($np$)-transfer astrophysical $S$ factors obtained for different choices of the ${}^6\mathrm{Li}$ wave function, as per the legend (matching the labels in Table 2 and Fig. 6). The green solid line corresponds to the middle-left panel of Fig. 6, red dot-dashed and violet dashed lines [“(${2s)}^2$(3.6%)” in the legend] correspond respectively to top- and bottom-left panels of Fig. 6, orange dot-dashed and magenta dashed lines [“(${2s)}^2(50\%$)” in the legend] are associated with the top- and bottom-right panels of Fig. 6, while the black dotted line refers to the middle-right panel of Fig. 6. Experimental data are the same as in Fig. 3 (not shown in legend for brevity).

Figure 9

Same lines as in Fig. 8, each divided by a constant factor proportional to the “clustered region” norm of the associated ${}^6\mathrm{Li}$ wave function, as per Table 2 (see Sec. 4b1 for details), taking the pure ${\left(1p\right)}^2$ case as reference.

Figure 10

Points are the same rescaled data in the bottom panel of Fig. 1 (not shown in legend for brevity). Lines are $d$- and ($np$)-transfer calculations obtained for different choices of ${}^6\mathrm{Li}$ wave function, as per the legend (matching the labels in Figs. 3 and 8 and Table 2), but each was multiplied by a distinct constant to match experimental data at $350\mathrm{keV}$.