  Background: For nuclei heavier than $^{208}$Pb $\alpha$ decay is a dominating
decay mode, and in the search of new superheavy elements one often observes
chains of $\alpha$ decays. Purpose: Explore and test microscopic descriptions
of $\alpha$ decay based on theories with effective nuclear interactions.
Methods: The nuclear ground states are calculated with the
Hartree-Fock-Bogoliubov (HFB) method using the Skyrme interaction. Microscopic
$\alpha$-decay formation amplitudes are calculated from the HFB wave functions,
and the $R$-matrix formalism is utilized to obtain decay probabilities.
Results: Using a large harmonic-oscillator basis we obtain converged
$\alpha$-decay widths. A comparison with experiment including all spherical
even-even $\alpha$ emitting nuclei shows that the model consistently predicts
too small formation amplitudes while relative values are in good agreement with
experiment. Conclusions: The method was found to be numerically practical even
with a large basis size. The comparison of formation amplitudes suggests that
the pairing type correlations included in the HFB approach cannot produce
sufficient $\alpha$-particle clustering.
