Cross sections of $\alpha$-induced reactions slightly below doubly-magic $^{40}$Ca from the statistical model

New experimental data for the $^{38}$Ar($\alpha$,n)$^{41}$Ca and $^{38}$Ar($\alpha$,p)$^{41}$K reactions were used to find a best-fit set of parameters for statistical model calculations. The very good agreement between the experimental data and the best-fit calculation confirms the applicability of the statistical model for nuclei in the vicinity of the doubly-magic $^{40}$Ca despite of their relatively low level densities. The present study investigates the sensitivities and finds that the $\alpha$-nucleus and the nucleon-nucleus potentials are the most important ingredients for the calculation of ($\alpha$,n) and ($\alpha$,p) reactions in the statistical model. Furthermore, the width fluctuation correction plays an essential role in the peculiar case of $^{38}$Ar. The best-fit parameters from $^{38}$Ar are applied to the mirror nucleus $^{38}$Ca and the neighboring $^{36}$Ar and $^{40}$Ar nuclei. For $^{38}$Ca this results in an astrophysical reaction rate of the $^{38}$Ca($\alpha$,p)$^{41}$Sc reaction which has a flatter temperature dependence compared to all previous calculations. For $^{40}$Ar a better reproduction of $^{40}$Ar($\alpha$,p)$^{43}$K data from literature is obtained. The disagreement between calculation and an early experimental data point for the $^{36}$Ar($\alpha$,p)$^{39}$K reaction persists.