From <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> and cluster emission:

This study presents a unified theoretical analysis of $\ensuremath{\alpha}$, simultaneous $2\ensuremath{\alpha}$, and heavier cluster emissions from a series of actinium (Ac) and thorium (Th) isotopes in the mass ranges $^{207--229}\mathrm{Ac}$ and $^{210--230}\mathrm{Th}$, within the framework of preformed cluster model (PCM). The emphasis is focused on identifying and characterizing the rare yet physically significant, $2\ensuremath{\alpha}$ decay mode, treated as the simultaneous emission of two $\ensuremath{\alpha}$ particles from the parent nucleus. The study aims to understand the isotopic evolution of decay modes, particularly the transition from dominant $\ensuremath{\alpha}$ decay to the emergence of simultaneous $2\ensuremath{\alpha}$ and the heavier cluster emissions. The model incorporates detailed computations of $Q$ values, barrier penetrabilities, and cluster preformation probabilities to systematically assess the viability of various decay channels. Through this approach, we aim to understand the structural and dynamical conditions that prefer the simultaneous $2\ensuremath{\alpha}$ emission and identify specific isotopes where such processes become increasingly probable. The even-odd staggering observed in reference to the calculated decay half-lives and preformation probabilities reflects the underlying nuclear structure effects and pairing correlations, offering additional insight into decay dynamics. Moreover, the Geiger-Nuttall plots constructed for both $\ensuremath{\alpha}$ and $2\ensuremath{\alpha}$ modes confirm linear systematics, validating the quantum tunneling nature of these emissions. The present work not only extends the applicability of PCM to rare decay modes but also provides a theoretical basis for future experimental validations related to rare decay mechanisms of radioactive nuclei.