Clustering and $\alpha$-capture reaction rate from ab initio symmetry-adapted descriptions of ${}^{20}\mathrm{Ne}$

We introduce a new framework for studying clustering and for calculating $\alpha$ partial widths using ab initio wave functions. We demonstrate the formalism for ${}^{20}\mathrm{Ne}$, by calculating the overlap between the ${}^{16}\mathrm{O}+\alpha$ cluster configuration and states in ${}^{20}\mathrm{Ne}$ computed in the $abinitio$ symmetry-adapted no-core shell model. We present spectroscopic amplitudes and spectroscopic factors, and compare those to no-core symplectic shell-model results in larger model spaces, to gain insight into the underlying physics that drives $\alpha$ clustering. Specifically, we report on the $\alpha$ partial width of the lowest $1^{-}$ resonance in ${}^{20}\mathrm{Ne}$, which is found to be in good agreement with experiment. We also present first no-core shell-model estimates for asymptotic normalization coefficients for the ground state, as well as for the first excited $4^{+}$ state in ${}^{20}\mathrm{Ne}$ that lies in a close proximity to the $\alpha {+}^{16}\mathrm{O}$ threshold. This outcome highlights the importance of correlations for developing cluster structures and for describing $\alpha$ widths. The widths can then be used to calculate $\alpha$-capture reaction rates for narrow resonances of interest to astrophysics. We explore the reaction rate for the $\alpha$-capture reaction ${}^{16}\mathrm{O}{(\alpha ,\gamma )}^{20}\mathrm{Ne}$ at astrophysically relevant temperatures and determine its impact on simulated x-ray burst abundances.

Figure 1

Illustration of the coordinates used in the formalism: ${\vec{r}}_i$ are the particle coordinates in the laboratory frame; $\overrightarrow{R^{\prime }}$ and $\overrightarrow{R^{\prime \prime }}$ are the center-of-mass coordinates of the individual clusters, $\vec{R}$ is the center-of-mass coordinate of the $A$-body system; ${\vec{\zeta }}_i$ are the relative coordinates with respect to $\vec{R}$. The relative separation of the clusters is ${\vec{r}}_{(A-a),a}$.

Figure 2

Wave function of the $l=0\alpha {+}^{16}\mathrm{O}$ for the ${}^{20}\mathrm{Ne}$ ground state ($\hslash \mathrm{\Omega }=15$ MeV), calculated in the $abinitio{N}_{\max }=8$ SA-NCSM using an $\mathrm{Sp}(3,\mathbb{R})$ basis [red, labeled as “SA-NCSM/$\mathrm{Sp}(3,\mathbb{R})$”] and an $\mathrm{SU}\left(3\right)$ basis [gray, labeled as “SA-NCSM/$\mathrm{SU}\left(3\right)$”], and in the $N_{\max }=22$ NCSpM (dotted black). Results are reported for a single symplectic irrep, $\sigma =48.5\left(80\right)$ (see text for details). The interior and exterior contributions to the wave function are indicated with solid and dashed lines, respectively. (Inset) The same “SA-NCSM/$\mathrm{Sp}(3,\mathbb{R})$” wave function but compared to its spectroscopic amplitude $r{u}_{\nu Il}^{J^{\pi }}$, prior to matching to the Whittaker function and normalizing to one (dotted red).

Figure 3

Spectroscopic amplitudes for the $l=1\alpha {+}^{16}\mathrm{O}$ at energy $E=1.06$ MeV, calculated from the lowest $1^{-}$ state in ${}^{20}\mathrm{Ne}$. Results are reported for a single symplectic irrep, $\sigma =49.5\left(90\right)$ (see text for details). (a) The ${}^{20}\mathrm{Ne}$ wave functions are calculated using the $abinitio$ SA-NCSM [with $\mathrm{SU}\left(3\right)$ basis (gray) and $\mathrm{Sp}(3,\mathbb{R})$ basis (red)] with $N_{\max }=5$ and $\hslash \mathrm{\Omega }=15$ MeV. Interior and exterior (real part) components are shown with solid and dashed lines, respectively. For the $\mathrm{Sp}(3,\mathbb{R})$ case, we show the spectroscopic amplitude without matching to the Coulomb-Hankel functions (dotted red), and the asymptotic oscillatory behavior of the real part of the wave functions (inset). (b) The composite ${}^{20}\mathrm{Ne}$ wave functions are calculated using the SA-NCSM/$\mathrm{SU}\left(3\right)$ with $N_{\max }=9$ [$\hslash \mathrm{\Omega }=13\mathrm{MeV}$ (dash-dotted blue) and $15\mathrm{MeV}$ (solid gray)], $N_{\max }=7$ [$\hslash \mathrm{\Omega }=15\mathrm{MeV}$ (small-dash green) and $17\mathrm{MeV}$ (large-dash red)], and NCSpM with $N_{\max }=23,\hslash \mathrm{\Omega }=15\mathrm{MeV}$ (dotted black).

Figure 4

$\alpha$ partial width ${\mathrm{\Gamma }}_{\alpha }$ as a function of the channel radius $r_c$ for the $1^{-}$ resonance at 1.06 MeV (relative to the ${}^{16}\mathrm{O}+\alpha$ threshold) in ${}^{20}\mathrm{Ne}$. The ${}^{20}\mathrm{Ne}$ wave functions are calculated using the $abinitio$ SA-NCSM [with $\mathrm{SU}\left(3\right)$ basis] (blue) and NCSpM (gray $\times$) for increasing $N_{\max }$ model spaces and with (a) $\hslash \mathrm{\Omega }=13$ MeV, (b) $\hslash \mathrm{\Omega }=15$ MeV, and (c) $\hslash \mathrm{\Omega }=17$ MeV. The extrapolation (black) is determined with Shanks transformation on the $N_{\max }=5,7,$ and 9 data.

Figure 5

ANC ($C_0$) for the ${}^{20}\mathrm{Ne}$ ground statein the $l=0{}^{16}\mathrm{O}+\alpha$ channel as a function of the channel radius $r_c$. The ${}^{20}\mathrm{Ne}$ ground state is calculated using the $abinitio$ SA-NCSM [with $\mathrm{SU}\left(3\right)$ basis] (blue) and NCSpM (gray $\times$) for increasing $N_{\max }$ model spaces and with (a) $\hslash \mathrm{\Omega }=13$ MeV, (b) $\hslash \mathrm{\Omega }=15$ MeV, and (c) $\hslash \mathrm{\Omega }=17$ MeV. The extrapolations (black) are determined with the Shanks transformation on the $N_{\max }=6,8,$ and 10 data.

Figure 6

${}^{16}\mathrm{O}{(\alpha ,\gamma )}^{20}\mathrm{Ne}$ reaction rate (in ${\mathrm{cm}}^3{\mathrm{mol}}^{-1}{\mathrm{s}}^{-1}$) through the 1.06-MeV $1^{-}$ resonance in ${}^{20}\mathrm{Ne}$ determined with Eq. (28) as a function of the temperature (in GK). The reaction rate determined with the extrapolated $\alpha$ partial width ${\mathrm{\Gamma }}_{\alpha }=10\left(3\right)$ eV derived from $abinitio$ SA-NCSM wave functions [red, labeled as “SA-NCSM /$\mathrm{SU}\left(3\right)$”] is compared to the database reaction rate (blue). The error in the database rate is given by the thickness in the curve.

Figure 7

The difference between the initial mass fractions of the neutron star and the mass fractions 24 h after the burst begins, based on the MESA XRB simulation that uses the database reaction rate (blue circles) and the reaction rate derived from $abinitio$ SA-NCSM wave functions, shown in Fig. 6 (red $\times$). All isotopes in the network with mass differences greater than ${10}^{-10}$ are shown, and we label some isotopes of interest. The inset shows a detailed look at the abundance pattern for isotopes of H, He, Mn, and Fe.