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Background: For nuclei heavier than $^{208}$Pb $\alpha$ decay is a dominatingdecay mode, and in the search of new superheavy elements one often observeschains of $\alpha$ decays. Purpose: Explore and test microscopic descriptionsof $\alpha$ decay based on theories with effective nuclear interactions.Methods: The nuclear ground states are calculated with theHartree-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 sphericaleven-even $\alpha$ emitting nuclei shows that the model consistently predictstoo small formation amplitudes while relative values are in good agreement withexperiment. Conclusions: The method was found to be numerically practical evenwith a large basis size. The comparison of formation amplitudes suggests thatthe pairing type correlations included in the HFB approach cannot producesufficient $\alpha$-particle clustering.

$\alpha$-decay calculations of heavy nuclei using an effective Skyrme interaction