Search for novel $\alpha$-cluster structure in light unstable nuclei

The formation of $\alpha$-cluster structure in nuclei is an example of an emergent ordering that appears under certain conditions. It is still an open question of what the exact conditions are for $\alpha$ cluster-formation. In order understand these conditions, we have a program to search for $\alpha$-cluster states in light unstable nuclei. To access these unstable nuclei we use in-flight radioactive beams that are produced with TwinSol, a pair of superconducting solenoids. The Prototype Active-Target Time-Projection Chamber was used to measure differential cross sections with high precision. Recent experiments for confirming a novel type of cluster structure, the linear-chain structure, in $^{14}$C and the search for cluster structure in $^{14}$O will be presented. The presence or absence of cluster structures in $^{14}$C and $^{14}$O will help elucidate the role of extra nucleons on the formation of cluster structures.