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Role of non-coplanarity in nuclear reactions using the Wong formula based on the proximity potential

we assessed Wong's formula for its angular momentum $\ell$-summation and "barrier modification" effects at sub-barrier energies in the dominant fusion-evaporation and capture (equivalently, quasi-fission) reaction cross-sections. For use of the multipole deformations (up to $β_4$) and (in-plane, $Φ$=0$^0$) orientations-dependent proximity potential in fusion-evaporation cross-sections of $^{58}$Ni+$^{58}$Ni, $^{64}$Ni+$^{64}$Ni and $^{100}$Mo, known for fusion hindrance phenomenon in coupled-channels calculations, and the capture cross-sections of $^{48}$Ca+$^{238}$U, $^{244}$Pu and $^{248}$Cm reactions, forming superheavy nuclei, though the simple $\ell$=0 barrier-based Wong formula is found inadequate, its extended version, the $\ell$-summed Wong expression fits very well the above noted capture cross-sections at all center-of-mass energies $E_{c.m.}$'s, but require (additional) modifications of the barriers to fit the fusion-evaporation cross-sections in the Ni-based reactions at below-barrier energies. Some barrier modification effects are shown \cite{kumar09} to be already present in Wong expression due to its inbuilt $\ell$-dependence via $\ell$-summation.