The $\alpha$ condensate states of atomic nuclei ${}^{12}C$, ${}^{16}O$ and ${}^{20}Ne$ in an analytical solvable model

TL;DR

This paper investigates alpha condensate states in certain nuclei using an analytical model, comparing calculated energies and radii with experimental and other theoretical results, revealing relationships between radius and energy.

Contribution

It introduces an analytical solvable model to study alpha condensate states, providing insights into energy ratios and radii that align or differ from existing models and experiments.

Findings

Calculated energy ratio of Hoyle states matches experimental data.

Root-mean-square radii are around 9fm, differing from THSR wave function results.

Radius decreases as the alpha condensate energy increases.

Abstract

The $\alpha$ condensation in the ${}^{12}C$, ${}^{16}O$ and ${}^{20}Ne$ nuclei is investigated within an analytical solvable model. It is found that the calculated ratio of the ground state energies of the Hoyle state of ${}^{12}C$ and the Hoyle-like state of ${}^{16}O$ is consistent with that of the experimental values. Along this clue, the ground state energy of ${}^{20}Ne$ is obtained to be 1MeV approximately, which is far less than the experimental value of 3MeV. Additionally, the root-mean-square radii of these nuclei are also calculated, and all of them lies around 9fm, which is different from the result calculated with the Tohsaki-Horiuchi-Schuck-Ropke(THSR) wave function. Since the root-mean-square radius is relevant to the ground state energy of the $\alpha$ condensate nucleus, the root-mean-square radii of ${}^{16}O$ and ${}^{20}Ne$ are also calculated with the ground state energies used in the THRS wave function. As a result, the root-mean-square radii of ${}^{16}O$ and ${}^{20}Ne$ reduced to 5fm, and it is similar to the result obtained with the THRS wave function. The calculation result manifests that the root-mean-square radius of $\alpha$ condensate nuclei decreases with the energy increasing.