  Recently, we have determined surface distributions of $\alpha$ clusters in
the ground state of $^{20}\mathrm{Ne}$ from $\alpha$-transfer cross sections,
without investigating the properties of its excited states. In this paper we
extend our comprehension of $\alpha$-cluster structures in excited states of
nuclei through reaction studies. In particular we focus on $^{16}\mathrm{O}$,
for which attention has been paid to advances of structure theory and
assignment regarding $4^+$-resonance states. We study the surface manifestation
of 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 $\alpha$-transfer
reaction is described by the distorted-wave Born approximation. We test two
microscopic wave functions as an input of reaction calculations. Then a
phenomenological potential model is introduced to clarify the correspondence
between cluster-wave functions and transfer-cross sections. Surface peaks of
the $\alpha$-wave function of $^{16}\mathrm{O}(0^+)$ are sensitively probed by
transfer-cross sections at forward angles, while it remains unclear how we
trace 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 $\sim
4.5$~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 in
the $4_2^+$ state is found not to be dominant. The $4_2^+$ state is difficult
to be interpreted by a simple potential model assuming the
$^{12}\mathrm{C}+\alpha$ configuration only.

Nuclear Theory