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The gamma-radiation emitted when positrons are annihilated in a material has a peak whose line width depends upon the Doppler broadening. We have developed an isotope-based, variable-energy positron beam system at the National Institute of Materials and Chemical Research (NIMC) which allows us to use this Doppler broadening technique for depth-selective analysis of damage and defects in near-surface regions. In this paper an outline of our positron beam system which uses 22Na is given. Its use is described in investigating ion-irradiated amorphous poly(aryl-ether-ether-ketone) (PEEK) and amorphous silicon. The damaged region of Au+ bombarded PEEK was found to extend to a greater depth than the average implantation depth for the same material. This is in contrast to the case of O+ bombardment. Different widths of annihilation lines observed for as-received and heat-treated samples of amorphous silicon were discussed in relation to structural changes due to annealing.