Origin of symmetry energy in finite nuclei and density dependence of nuclear matter symmetry energy from measured $\alpha$-decay energies

Based on the Skyrme energy density functional, the spatial distribution of the symmetry energy of a finite nucleus is derived in order to examine whether the symmetry energy of a finite nucleus originates from its interior or from its surface. It is found that the surface part of a heavy nucleus contributes dominantly to its symmetry energy compared to its inner part. The symmetry energy coefficient $a_{\text{sym}}({A})$ is then directly extracted and the ratio of the surface symmetry coefficient to the volume symmetry coefficient $\kappa$ is estimated. Meanwhile, with the help of experimental alpha decay energies, a macroscopic method is developed to determine the symmetry energy coefficient of heavy nuclei. The resultant $a_{\text{sym}}({A})$ is used to analyze the density dependence of the symmetry energy coefficient of nuclear matter around the saturation density, and furthermore, the neutron skin thickness of $^{208}\text{Pb}$ is deduced which is consistent with the pygmy dipole resonance analysis. In addition, it is shown that the ratio $\kappa$ obtained from the macroscopic method is in agreement with that from the Skyrme energy density functional. Thus the two completely different approaches may validate each other to achieve more compelling results.