Realistic analytical approach for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> decay and clustering

We approximate the realistic $\ensuremath{\alpha}$-core nuclear interaction between the inner turning point and Coulomb barrier, derived within the double folding procedure, by a parabolic dependence. It turns out that the corresponding harmonic oscillator frequency is concentrated in a narrow interval around $\ensuremath{\hbar}{\ensuremath{\omega}}_{1}\ensuremath{\approx}$ 9 MeV for all analyzed transitions from even-even and odd-mass $\ensuremath{\alpha}$ emitters. The penetrability through the nuclear barrier has an exponential dependence on the ratio between the fragmentation potential and this harmonic oscillator frequency. On the other hand, the Coulomb penetrability has the standard dependence on the Coulomb parameter. Our analysis revealed that the reduced width, extracted by using the nuclear plus Coulomb penetrability factors, exponentially decreases versus the number of $\ensuremath{\alpha}$ clusters along all analyzed $\ensuremath{\alpha}$ lines, except one region where it increases by approaching the neutron magic number ${N}_{\mathrm{mag}}=126$. In that case, the number of valence proton pairs is much smaller than the number of valence neutron pairs. The reduced width exponentially depends on the magic neutron-proton asymmetry above doubly magic nuclei. These dependencies allow us to propose a systematics for reduced widths in terms of the quartet number and magic asymmetry evaluated above the closest doubly magic nuclei.