Signatures of $\alpha$-clustering in $^{12}$C and $^{13}$C

TL;DR

This paper investigates the cluster structures of carbon isotopes $^{12}$C and $^{13}$C using the cluster shell model, deriving relations for form factors and transition probabilities, and showing experimental data align with specific symmetric triangular configurations.

Contribution

It introduces simple relations for form factors and transition probabilities within the cluster shell model and demonstrates their applicability to experimental data for $^{12}$C and $^{13}$C.

Findings

Experimental data for $^{12}$C and $^{13}$C fit well with triangular cluster structures.

Derived relations accurately describe ratios of form factors and transition probabilities.

Triangular configurations exhibit ${ m D}_{3h}$ and ${ m D'}_{3h}$ symmetry, respectively.

Abstract

We study the cluster structure of $^{12}$C and $^{13}$C in the framework of the cluster shell model. Simple relations are derived for ratios of longitudinal form factors as well as transition probabilities. It is shown that the available experimental data for $^{12}$C and $^{13}$C can be well described by a triangular structure with ${\cal D}_{3h}$ and ${\cal D}'_{3h}$ symmetry, respectively.