Positron lifetime measurements are reported for a linear polyethylene and several ethylene-1-hexene copolymers synthesized with metallocene catalysts. The lifetimes, their intensity, the relaxation time, and the relative free volume fraction obtained are correlated with comonomer content, crystallinity, location of P relaxation, and some mechanical parameters, such as elastic modulus and microhardness. A structural discontinuity is found with comonomer molar fraction that evidences the existence of two different regimes: the first one including the homopolymer and copolymers with contents up to around 5-7 mol % and the second regime for higher compositions. Possible reasons for this behavior are discussed. In addition, a linear relationship is found between the location of the P relaxation, which is associated with large-scale cooperative motions in the amorphous environments, and the free volume fraction: the higher the relaxation temperature is, the lower free volume value is obtained.