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Recently, we have determined surface distributions of $\alpha$ clusters inthe ground state of $^{20}\mathrm{Ne}$ from $\alpha$-transfer cross sections,without investigating the properties of its excited states. In this paper weextend our comprehension of $\alpha$-cluster structures in excited states ofnuclei through reaction studies. In particular we focus on $^{16}\mathrm{O}$,for which attention has been paid to advances of structure theory andassignment regarding $4^+$-resonance states. We study the surface manifestationof the $\alpha$-cluster states in both the ground and excited states of$^{16}\mathrm{O}$ from the analysis of the $\alpha$-transfer reaction$^{12}\mathrm{C}(^6\mathrm{Li},d)^{16}\mathrm{O}$. The $\alpha$-transferreaction is described by the distorted-wave Born approximation. We test twomicroscopic wave functions as an input of reaction calculations. Then aphenomenological potential model is introduced to clarify the correspondencebetween cluster-wave functions and transfer-cross sections. Surface peaks ofthe $\alpha$-wave function of $^{16}\mathrm{O}(0^+)$ are sensitively probed bytransfer-cross sections at forward angles, while it remains unclear how wetrace the surface behavior of $^{16}\mathrm{O}(4^+)$ from the cross sections.We are able to specify that the $\alpha$-cluster structure in the $0_1^+$ and$0_2^+$ states prominently manifests itself at the radii $\sim 4$ and $\sim4.5$~fm, respectively. It is remarkable that the $4_1^+$ state has the$^{12}\mathrm{C}+\alpha$-cluster component with the surface peak at the radius$\sim 4$ or outer, whereas the $^{12}\mathrm{C}+\alpha$-cluster component inthe $4_2^+$ state is found not to be dominant. The $4_2^+$ state is difficultto be interpreted by a simple potential model assuming the$^{12}\mathrm{C}+\alpha$ configuration only.

Investigation of spatial manifestation of $\alpha$ clusters in $^{16}$O via $\alpha$-transfer reactions