An extended database of experimental data is needed to address uncertainties of the nuclear-physics input parameters for Hauser-Feshbach calculations. Especially $alpha$+nucleus optical model potentials at low energies are not well known. The in-beam technique with an array of high-purity germanium (HPGe) detectors was successfully applied to the measurement of absolute cross sections of an ($alpha$,$gamma$) reaction on a heavy nucleus at sub-Coulomb energies. The total and partial cross-section values were measured by means of in-beam $gamma$-ray spectroscopy. Total and partial cross sections were measured at four different $alpha$-particle energies from $E_alpha = 10.5$ MeV to $E_alpha = 12$ MeV. The measured total cross-section values are in excellent agreement with previous results obtained with the activation technique, which proves the validity of the applied method. The experimental data was compared to Hauser-Feshbach calculations using the nuclear reaction code TALYS. A modified version of the semi-microscopic $alpha$+nucleus optical model potential OMP 3, as well as modified proton and $gamma$ widths, are needed in order to obtain a good agreement between experimental data and theory. It is found, that a model using a local modification of the nuclear-physics input parameters simultaneously reproduces total cross sections of the $^{112}$Sn($alpha$,$gamma$) and $^{112}$Sn($alpha$,p) reactions. The measurement of partial cross sections turns out to be very important in this case in order to apply the correct $gamma$-ray strength function in the Hauser-Feshbach calculations. The model also reproduces cross-section values of $alpha$-induced reactions on $^{106}$Cd, as well as of ($alpha$,n) reactions on $^{115,116}$Sn, hinting at a more global character of the obtained nuclear-physics input.
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