Thirteen targets with mass numbers from 58 to 238 were irradiated with the antiproton beam from the Low Energy Antiproton Ring facility at CERN leading to the formation of antiprotonic atoms of these heavy elements. The antiproton capture at the end of atomic cascade results in the production of more or less excited residual nuclei. The targets were selected with the criterion that both reaction products with one nucleon less than the proton and neutron number of the target are radioactive. The yield of these radioactive products after stopped-antiproton annihilation was determined using gamma-ray spectroscopy techniques. This yield is related to the proton and neutron density in the target nucleus at radial distance corresponding to the antiproton annihilation site. The experimental data clearly indicate the existence of a neutron rich nuclear periphery, a "neutron halo", strongly correlated with the target neutron separation energy Bn, and observed for targets with Bn < 10 MeV. For two target nuclei, 106 Cd and 144 Sm, with larger neutron binding energies a proton rich nuclear periphery was observed. Most of the experimental data are in reasonable agreement with calculations based on current antiproton-nucleus and pion-nucleus interaction potentials and on nuclear densities deduced with the help of the Hartree-Fock-Bogoliubov approach. This approach was, however, unable to account for the 106 Cd and 144 Sm results.