The accurate determination of the nuclear interaction potential is essential for predicting fusion cross sections and understanding the reaction mechanism, which plays an important role in the synthesis of superheavy elements. In this work, a neural network, which combines with the calculations of the fusion cross sections via the Hill-Wheeler formula, is developed to optimize the parameters of the Woods-Saxon potential by comparing the experimental values. The correlations between the parameters of Woods-Saxon potential and the reaction partners, which can be quantitatively fitted to a sigmoid-like function with the mass numbers, have been displayed manifestly for the first time. This study could promote the accurate estimation of the nucleus-nucleus interaction potential in low energy heavy-ion collisions.

Multinucleon transfer (MNT) reactions have received increasing interest in the synthesis of neutron-rich nuclei due to the distinct limitations of other reactions, particularly in the $N=126$ region, which represents the last waiting point of astrophysical nucleosynthesis. However, it is still a challenging endeavor to describe the MNT process between heavy nuclei. In this study, we develop a theoretical framework that couples the Langevin dynamics iteratively with the master equation, which is based on the HICOL model (CLIM-H). The random transfer process is achieved by solving the master equation using the Monte Carlo method, where the parametric transfer probability with a $Q$ window is employed. The isotope distributions for ${}^{58,64}\mathrm{Ni}+{}^{208}\mathrm{Pb}$, as well as the angular and isotope distributions of the recently measured ${}^{206}\mathrm{Pb}+{}^{118}\mathrm{Sn}$, could be generally reproduced based on this method. Contrary to previous theoretical predictions which show a high production cross section of $N=126$ nuclei, current calculations do not reveal appreciable cross sections. The distinguishing characteristic of this approach is its ability to generate not only the mass distribution but also the charge distribution self-consistently, which could provide references for many other studies using the multidimensional Langevin equation considering only the mass asymmetry degree of freedom.

The problem of a quantum-mechanical description of a near-barrier fusion of heavy nuclei, that occurs at strong coupling of their relative motion to surface vibrations, is analyzed. To this end, an efficient finite-element method is proposed for numerically solving coupled Schrödinger equations with boundary conditions corresponding to total absorption. The method allows us to eliminate the instabilities in the numerical solutions that appear at a large number of coupled channels in some reactions. To illustrate the validity of our approach, the results of fusion cross section of the ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ and ${}^{36}\mathrm{S}+{}^{48}\mathrm{Ca}$ reactions have been re-examined. The obtained results demonstrate a remarkable agreement with the available experimental data. It is found that experimental data can be reproduced with the use of the Woods-Saxon potential, without introducing the repulsive cores. It appears that the fusion cross sections at deep sub-barrier energies are sensitive to the potential pocket profile.

We analyze the origin of the unexpected deep subbarrier heavy-ion fusion hindrance in ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$, ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$, and ${}^{28}\mathrm{Si}+{}^{64}\mathrm{Ni}$ reactions. Our analysis is based on the improved coupled-channels description, implemented by means of the finite element method. With the aid of the Woods-Saxon potential the experimental cross sections and the $S$ factors of these reactions are remarkably well reproduced within the sudden approximation approach. We found that accounting for the nondiagonal matrix elements of the coupling matrix, traditionally neglected in the conventional coupled-channels approaches in setting the left boundary conditions inside the potential pocket, and its minimal value are crucially important for the interpretation of experimental data. We found as well a good agreement with the general trend of the experimental data for the $S$ factor of the fusion reaction ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, which has no pronounced maximum for this system.

The synthesis of superheavy elements based on fusion reactions and multinucleon transfer reactions is sensitive to the reaction energies, where the Coulomb barrier plays a crucial role as it must be overcome for the projectile and target to contact each other. The Coulomb barrier cannot be measured directly, and the synthesis of superheavy elements is sensitive to it. In this study, we systematically extract the barrier information from the experimental fusion excitation functions by the empirical cross section formula, which is based on a single-Gaussian distribution of fusion barrier heights. A total of 403 sets of experimental data are fitted, among which 243 sets have good results with ${\chi }^2/pt$ less than one. The extracted fusion barrier is a dynamical barrier, which includes the overall effect of coupled channels. Different from the prediction of the WKJ formula, the new scaling law proposed in this work is almost identical to the CW76 Coulomb barrier at the $z=170–300$ region and reproduces well the centroid barrier extracted from quasielastic scattering. The comparison to the other 14 bare potential models and exploration of the very large $z$ region is also discussed. It is also remarkable to find that the predictions of the DP2015 potential, including of the contributions of the macroscopic and shell correction terms, are highly consistent with the extracted results in this work and the CW76 potential. The numerical results demonstrated that the dynamical fusion barrier and the centroid barrier extracted from the quasielastic reaction (the reaction threshold) could be the same physical quantity for superheavy reactions. This study could provide important references for the synthesis of superheavy nuclei based on fusion reactions and very heavy multinucleon transfer reactions.

$\beta$ decay of proton-rich nuclei plays an important role in exploring isospin mixing. The $\beta$ decay of ${}^{26}\mathrm{P}$ at the proton drip line is studied using double-sided silicon strip detectors operating in conjunction with high-purity germanium detectors. The $T=2$ isobaric analog state (IAS) at 13 055 keV and two new high-lying states at 13 380 and 11 912 keV in ${}^{26}\mathrm{Si}$ are unambiguously identified through $\beta$-delayed two-proton emission ($\beta 2p$). Angular correlations of two protons emitted from ${}^{26}\mathrm{Si}$ excited states populated by ${}^{26}\mathrm{P}$ $\beta$ decay are measured, which suggests that the two protons are emitted mainly sequentially. We report the first observation of a strongly isospin-mixed doublet that deexcites mainly via two-proton decay. The isospin mixing matrix element between the ${}^{26}\mathrm{Si}$ IAS and the nearby 13 380-keV state is determined to be 130(21) keV, and this result represents the strongest mixing, highest excitation energy, and largest level spacing of a doublet ever observed in $\beta$-decay experiments.