A high precision method to determine fusion in the capture of $alpha$-particles by nuclei is presented. For $alpha$-capture by $^{40}{rm Ca}$ and $^{44}{rm Ca}$, such an approach gives (1) the parameters of the $alpha$--nucleus potential and (2) fusi
on probabilities. This method found new parametrization and fusion probabilities and decreased the error by $41.72$ times for $alpha + ^{40}{rm Ca}$ and $34.06$ times for $alpha + ^{44}{rm Ca}$ in a description of experimental data in comparison with existing results. We show that the sharp angular momentum cutoff proposed by Glas and Mosel is a rough approximation, Wongs formula and the Hill-Wheeler approach determine the penetrability of the barrier without a correct consideration of the barrier shape, and the WKB approach gives reduced fusion probabilities. Based on our fusion probability formula, we explain the difference between experimental cross-sections for $alpha + ^{40}{rm Ca}$ and $alpha + ^{44}{rm Ca}$, which is connected with the theory of coexistence of the spherical and deformed shapes in the ground state for nuclei near the neutron magic shell $N=20$. To provide deeper insight into the physics of nuclei with the new magic number $N=26$, the cross-section for $alpha + ^{46}{rm Ca}$ is predicted for future experimental tests. The role of nuclear deformations in calculations of the fusion probabilities is analyzed.
