Exploring sensitivity of charge-exchange ($p, n$) reactions to the neutron density distribution

$Background:$ The determination of the nuclear neutron properties suffers from uncontrolled uncertainties, which attracted considerable attention recently, such as in the context of the PREX experiment. $Purpose:$ Our aim is to analyze the sensitivity of charge-exchange ($p, n$) reactions to the neutron density distribution $\rho_{n}$ and constrain the neutron characteristics in the nuclear structure models. $Method:$ By combing the folding and the mean-field models, the nucleon-nucleus ($NA$) potential can be obtained from the nuclear density distribution. Further, the ($p, p$) and ($p, n$) cross sections for $^{48}$Ca and $^{208}$Pb are calculated following the distorted-wave Born approximation (DWBA) method. $Results:$ Compared with the ($p, p$) cross section, the effects of $\rho_{n}$ variation on the ($p, n$) cross section are significant, which is due to the impact of isovector properties. Based on the global folding model analyses of data, it is found that $^{48}$Ca and $^{208}$Pb have relatively large neutron skin thickness $\Delta R_{n p}$. $Conclusions:$ Results illustrate that the charge-exchange ($p, n$) reaction is a sensitive probe of $\rho_{n}$. The results in this paper can offer useful guides for future experiments of neutron characteristics.