Astrophysical reaction rate for ${}^9\mathrm{Be}$ formation within a three-body approach

The structure of the Borromean nucleus ${}^9\mathrm{Be}$ ($\alpha +\alpha +n$) is addressed within a three-body approach using the analytical transformed harmonic oscillator method. The three-body formalism provides an accurate description of the radiative capture reaction rate for the entire temperature range relevant in astrophysics. At high temperatures, results match the calculations based on two-step sequential processes. At low temperatures, where the particles have no access to intermediate two-body resonances, the three-body direct capture leads to reaction rates larger than the sequential processes. These results support the reliability of the method for systems with several charged particles.

Figure 1

The three sets of scaled Jacobi coordinates.

Figure 2

${}^9\mathrm{Be}$ in Jacobi-$T$ system.

Figure 3

Convergence of the ground-state energy of ${}^9\mathrm{Be}$ with respect to the maximum hypermomentum $K_{\max }$.

Figure 4

Convergence of the matter radius (solid line) and the charge radius (dashed line) of ${}^9\mathrm{Be}$ with respect to the maximum hypermomentum $K_{\max }$.

Figure 5

Dependence on $K_{\max }^s$ of the $1/2^{+}$ contribution to the ${}^9\mathrm{Be}$ photodissociation cross section. (See the text.)

Figure 6

Contribution of the $1/2^{+}$ (thin solid), $5/2^{+}$ (dashed), $3/2^{+}$ (dotted), $5/2^{-}$ (dot dashed), and $1/2^{-}$ (double dot dashed) states to the total photodissociation cross section (thick solid).

Figure 7

Total photodissociation from our three-body calculation (solid line) compared with the results from Refs. [49] (dashed line), [15] (dot dashed), and [14] (dotted) and experimental data of Refs. [3] (triangles) and [6] (circles).

Figure 8

Contribution of the $1/2^{+}$ (thin solid), $5/2^{+}$ (dashed), $3/2^{+}$ (dotted), $5/2^{-}$ (dot dashed), and $1/2^{-}$ (double dot dashed) states to the total reaction rate.

Figure 9

Total reaction rate from our three-body calculation (solid line) and three-body Breit-Wigner calculation [14] (dashed line) compared with sequential estimations from experimental data of Refs. [11] (squares), [3] (triangles), and [6] (circles).

Figure 10

$B\left(E1\right)$ distribution to the $1/2^{+}$ states as a function of the Poisson width parameter $w$. (See text for details.)