Ab initio Calculations of the Isotopic Dependence of Nuclear Clustering

Nuclear clustering describes the appearance of structures resembling smaller nuclei such as alpha particles (${}^4\text{He}$ nuclei) within the interior of a larger nucleus. In this Letter, we present lattice Monte Carlo calculations based on chiral effective field theory for the ground states of helium, beryllium, carbon, and oxygen isotopes. By computing model-independent measures that probe three- and four-nucleon correlations at short distances, we determine the shape of the alpha clusters and the entanglement of nucleons comprising each alpha cluster with the outside medium. We also introduce a new computational approach called the pinhole algorithm, which solves a long-standing deficiency of auxiliary-field Monte Carlo simulations in computing density correlations relative to the center of mass. We use the pinhole algorithm to determine the proton and neutron density distributions and the geometry of cluster correlations in ${}^{12}\mathrm{C}$, ${}^{14}\mathrm{C}$, and ${}^{16}\mathrm{C}$. The structural similarities among the carbon isotopes suggest that ${}^{14}\mathrm{C}$ and ${}^{16}\mathrm{C}$ have excitations analogous to the well-known Hoyle state resonance in ${}^{12}\mathrm{C}$.

Figure 1

In panel (a) we show the ground state energies versus number of nucleons $A$ for the hydrogen, helium, beryllium, carbon, and oxygen isotopes. The errors are 1 standard deviation error bars associated with the stochastic errors and the extrapolation to an infinite number of time steps. In panel (b) we show ${\rho }_3/{\rho }_{3,\alpha }$ and ${\rho }_4/{\rho }_{4,\alpha }$ for the neutron-rich helium, beryllium, carbon, and oxygen isotopes. The error bars denote 1 standard deviation errors associated with the stochastic errors and the extrapolation to an infinite number of time steps. For comparison we show also the number of alpha clusters, $N_{\alpha }$.

Figure 2

Plots of the proton and neutron densities for the ground states of ${}^{12}\mathrm{C}$, ${}^{14}\mathrm{C}$, and ${}^{16}\mathrm{C}$ vs radial distance. We show data for $L_t=7$, 9, 11, 13, 15 time steps. We show ${}^{12}\mathrm{C}$ in panel (a), ${}^{14}\mathrm{C}$ in panel (b), and ${}^{16}\mathrm{C}$ in panel (c). The errors are 1 standard deviation error bars associated with the stochastic errors. For comparison we show the experimentally observed proton densities for ${}^{12}\mathrm{C}$ and ${}^{14}\mathrm{C}$ [31].

Figure 3

The two red spheres with arrows indicate the first two spin-up protons, and the line connecting them is the longest side of the triangle. We show the third spin-up proton probability distribution in ${}^{12}\mathrm{C}$ in panel (a), ${}^{14}\mathrm{C}$ in panel (b), and ${}^{16}\mathrm{C}$ in panel (c). The results are computed at $L_t=15$ time steps. In panel (d) we show the third spin-up proton probability distribution for a simple Gaussian lattice model of the distribution of the spin-up protons.