Coulomb Screening Effect on the Hoyle State Energy in Thermal Plasmas

Abstract

The first excited \(J^{\pi }=0^+\) state of \(^{12}\)C, the so-called Hoyle state, plays an essential role in a triple-\(\alpha \) (\(^{4}\)He) reaction, which is a main contributor to the synthesis of \(^{12}\)C in a burning star. We investigate the Coulomb screening effects on the energy shift of the Hoyle state in a thermal plasma environment using precise three-\(\alpha \) model calculations. The Coulomb screening effect between \(\alpha \) clusters is taken into account within the Debye-Hückel approximation. To generalize our study, we utilize two standard \(\alpha \)-cluster models, which treat the Pauli principle between the \(\alpha \) particles differently. We find that the energy shift does not depend on these models and follows a simple estimation in the zero-size limit of the Hoyle state when the Coulomb screening length is as large as a value typical of such a plasma consisting of electrons and \(\alpha \) particles.