The aim of the present work is to measure the $^{121}$Sb($alpha,gamma$)$^{125}$I, $^{121}$Sb($alpha$,n)$^{124}$I, and $^{123}$Sb($alpha$,n)$^{126}$I reaction cross sections. The $alpha$-induced reactions on natural and enriched antimony targets were investigated using the activation technique. The ($alpha$,$gamma$) cross sections of $^{121}$Sb were measured and are reported for first time. To determine the cross section of the $^{121}$Sb($alpha$,$gamma$)$^{125}$I, $^{121}$Sb($alpha$,n)$^{124}$I, and $^{123}$Sb($alpha$,n)$^{126}$I reactions, the yields of $gamma$-rays following the $beta$-decay of the reaction products were measured. For the measurement of the lowest cross sections, the characteristic X-rays were counted with a LEPS (Low Energy Photon Spectrometer) detector. The cross section of the $^{121}$Sb($alpha$,$gamma$)$^{125}$I, $^{121}$Sb($alpha$,n)$^{124}$I and $^{123}$Sb($alpha$,n)$^{126}$I reactions were measured with high precision in an energy range between 9.74 MeV to 15.48 MeV, close to the astrophysically relevant energy window. The results are compared with the predictions of statistical model calculations. The ($alpha$,n) data show that the $alpha$ widths are predicted well for these reactions. The ($alpha$,$gamma$) results are overestimated by the calculations but this is due to the applied neutron- and $gamma$ widths. Relevant for the astrophysical reaction rate is the $alpha$ width used in the calculations.While for other reactions the $alpha$ widths seem to have been overestimated and their energy dependence was not described well in the measured energy range, this is not the case for the reactions studied here. The result is consistent with the proposal that additional reaction channels, such as Coulomb excitation, may have led to the discrepancies found in other reactions.
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