Extension of the equation for the annihilation lifetime of ortho-positronium at a cavity larger than 1 nm in radius

Evaluations of cavity concentrations and their size distributions in industrially important materials by positron annihilation lifetime methods require a well calibrated ortho-positronium (o-Ps) lifetime standard curve as a function of pore size. A semiempirical equation with a fitting parameter obtained by Nakanishi and Jean has been successfully utilized in various porous and polymeric materials to estimate pore sizes up to approximately 1 nm in radius. However, as experimental data accumulate recently, it has become clear that the equation no longer yields a good correlation between o-Ps lifetimes and the pore size for the pore radius larger than 1 or o-Ps lifetime longer than similar to 20 ns. Therefore, we assumed for the larger pore that o-Ps behaves more like a quantum particle, bouncing back and forth between the energy barriers as the potential well becomes large. An equation derived from this hypothesis gives excellent fitting with experimental data despite a crude approximation used in our model. More importantly. this approach has produced a simple analytical relationship between the o-Ps lifetime and the pore radius. Agreement between the estimated o-Ps wave packet size at thermal energy (similar to 1.6 nm) and the fitted parameter R-Ps (similar to 1 nm) lends credence to our hypothesis that o-Ps behaves more like a quantum particle as a pore sire becomes larger.