  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) fusion 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, Wong's 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.
