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Background: Recent theoretical and experimental researches usingproton-induced $\alpha$-knockout reactions provide direct manifestation of$\alpha$-cluster formation in nuclei. In recent and future experiments,$\alpha$-knockout data are available for neutron-rich beryllium isotopes. In$^{12}$Be , rich phenomena are induced by the formation of $\alpha$-clusterssurrounded by neutrons, for instance, breaking of the neutron magic number$N=8$.  Purpose: Our objective is to provide direct probing of the $\alpha$-clusterformation in the $^{12}$Be target through associating the structure informationobtained by a microscopic theory with the experimental observables of$\alpha$-knockout reactions.  Method: We formulate a new wave function of theTohsaki-Horiuchi-Schuck-R{\"o}pke (THSR) type for the structure calculation of${}^{12}$Be nucleus and integrate it with the distorted wave impulseapproximation framework for the $\alpha$-knockout reaction calculation of$^{12}$Be$(p,p\alpha)^{8}$He.  Results: We reproduce the low-lying spectrum of the $^{12}$Be nucleus usingthe THSR wave function and discuss the cluster structure of the ground state.Based on the microscopic wave function, the optical potentials and$\alpha$-cluster wave function are determined and utilized in the calculationof ${}^{12}$Be($p,p\alpha$)${}^{8}$He reaction at 250 MeV. The possibility ofprobing the clustering state of $^{12}$Be through this reaction is demonstratedby analysis of the triple differential cross sections that are sensitivelydependent on the $\alpha$-cluster amplitude at the nuclear surface.  Conclusions: This study provides a feasible approach to validate directly thetheoretical predictions of clustering features in the $^{12}$Be nucleus throughthe $\alpha$-knockout reaction.

Direct probing of the cluster structure in ${}^{12}$Be via $\alpha$-knockout reaction