Role of a universal function of the nuclear proximity potential: A systematic study of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> decay of heavy and superheavy nuclei and <mml:math xmlns:mml="http://www.w

The idea of the universal function is a fundamental advantage of the proximity potential model. In recent years, several formulas for this function have been given in the framework of the original proximity potential of 1977 (Prox 77). We present a systematic study of the role of the universal function of the proximity potential model in an $\ensuremath{\alpha}$-decay process for 250 ground-state-to-ground-state transitions using the WKB approximation. In order to realize this goal, five universal functions proposed in the proximity models gp 77, Guo 2013, Ngo 80, Zhang 2013, and Prox 2010 have been incorporated into the formalism of Prox 77. The obtained results show that the radial behavior of universal function affects the penetration probability of the $\ensuremath{\alpha}$ particle. The theoretical $\ensuremath{\alpha}$-decay half-lives are calculated and compared with the corresponding experimental data. It is determined that the Prox 77 with Guo 2013 universal function provides the best fit to the available data (with standard deviation $\ensuremath{\sigma}=0.49$). The Geiger-Nuttall plots for various isotopic groups have been investigated using this modified form of the proximity potential model. The role of the different universal functions in the $\ensuremath{\alpha}$-decay half-lives of superheavy nuclei with $Z=104\text{--}118$ are also studied. It is shown that the experimental half-lives in the superheavy mass region are described well using the Prox 77 with Zhang 2013 universal function. In this paper, the validity of the original and modified forms of the proximity 77 potential is also examined for complete fusion reactions between $\ensuremath{\alpha}$ particles and ten different target nuclei. Our results show that the measured $\ensuremath{\alpha}$-induced fusion reaction cross sections can be well reproduced using the Prox 77 with Zhang 2013 universal function for the reactions involving light and medium nuclei, whereas the Prox 77 with Guo 2013 universal function model demonstrates a reliable agreement with the experimental data at sub-barrier energies for the heaviest systems.