Potential absence of observed $\pi^2$ linear-chain structures in $^{14}$O via $^{10}$C($\alpha,\alpha$) resonant scattering

Background: The preference for light nuclear systems to coagulate into $\alpha$-particle clusters has been well-studied. The possibility of a linear chain configuration of $\alpha$-particles would allow for a new way to study this phenomenon. Purpose: A rotational band of states in $^{14}$C has been claimed showing a $\pi^2$ linear chain structure. The mirror system, $^{14}$O, has been studied here to examine how this linear chain structure is affected by replacing the valence neutrons with protons. Method: A beam of $^{10}$C was incident into a chamber filled with He:CO$_2$ gas with the tracks recorded inside the TexAT Time Projection Chamber and the recoil $\alpha$-particles detected by a silicon detector array to measure the $^{10}\mathrm{C}(\alpha,\alpha)$ cross section. Results: The experimental cross section was compared with previous studies and fit using R-Matrix theory with the previously-observed $^{14}$O states being transformed to the $^{14}$C using mirror symmetry. The measured cross section does not replicate the claimed states, with the predicted cross section exceeding that observed at several energies and angles. Conclusion: A series of possibilities are highlighted with the most likely being that the originally-seen $^{14}$C states did not constitute a $\pi^2$ rotational band with a potentially incorrect spin assignment due to the limitations of the angular correlation method with non-zero spin particles. The work highlights the difficulties in measuring broad resonances corresponding to a linear chain state in a high level density.