Direct evidence of the $alpha$-cluster manifestation in bound states has not been obtained yet, although a number of experimental studies were carried out to extract the information of the clustering. In particular in conventional analyses of $alpha$-transfer reactions, there exist a few significant problems on reaction models, which are insufficient to qualitatively discuss the cluster structure. We aim to verify the development of the $alpha$-cluster structure from observables. As the first application, we plan to extract the spatial information of the cluster structure of the $^{20}$Ne nucleus in its ground state through the cross section of the $alpha$-transfer reaction $^{16}$O($^6$Li,~$d$)$^{20}$Ne. For the analysis of the transfer reaction, we work with the coupled-channel Born approximation (CCBA) approach, in which the breakup effect of $^6$Li is explicitly taken into account by means of the continuum-discretized coupled-channel method based on the three-body $alpha + d + {}^{16}$O model. The two methods are adopted to calculate the overlap function between $^{20}$Ne and $alpha + {}^{16}$O; one is the microscopic cluster model (MCM) with the generator coordinate method, and the other is the phenomenological two-body potential model (PM). We show that the CCBA calculation with the MCM wave function gives a significant improvement of the theoretical result on the angular distribution of the transfer cross section, which is consistent with the experimental data. Employing the PM, it is discussed which region of the cluster wave function is probed on the transfer cross section. It is found that the surface region of the cluster wave function is sensitive to the cross section. The present work is situated as the first step in obtaining important information to systematically investigate the cluster structure.
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