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Based on the Skyrme energy density functional, the spatial distribution ofthe symmetry energy of a finite nucleus is derived in order to examine whetherthe symmetry energy of a finite nucleus originates from its interior or fromits surface. It is found that the surface part of a heavy nucleus contributesdominantly to its symmetry energy compared to its inner part. The symmetryenergy coefficient $a_{\text{sym}}({A})$ is then directly extracted and theratio of the surface symmetry coefficient to the volume symmetry coefficient$\kappa$ is estimated. Meanwhile, with the help of experimental alpha decayenergies, a macroscopic method is developed to determine the symmetry energycoefficient of heavy nuclei. The resultant $a_{\text{sym}}({A})$ is used toanalyze the density dependence of the symmetry energy coefficient of nuclearmatter around the saturation density, and furthermore, the neutron skinthickness of $^{208}\text{Pb}$ is deduced which is consistent with the pygmydipole resonance analysis. In addition, it is shown that the ratio $\kappa$obtained from the macroscopic method is in agreement with that from the Skyrmeenergy density functional. Thus the two completely different approaches mayvalidate each other to achieve more compelling results.

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