The irradiation hardening of polyethylene (PE) crystals is explained in terms of the intersection and interaction between cross-links and dislocations. The elastic energies and forces of interaction between cross-links and the dislocations responsible for the various plastic deformation modes are calculated using a force dipole model of a cross-link and the strain field of the dislocation. The elastic energies of interaction are in all cases less than 0.7 eV (1.12×10−19 J) and they are greater for edge than for screw dislocations. The hardening which arises from the direct intersections is calculated using a Morse potential model for the cross-link strength, and it is found that these interactions involve energies of the order of 3.6 eV (5.76×10−19 J). From these results it is concluded that at 0 K both types of interaction produce similar hardening. However, since the elastic interaction energies are small, the hardness of cross-linked PE crystals at moderate temperatures is due solely to direct intersections of cross-links and dislocations. The strongest interactions take place between cross-links and those dislocations which produce chain-axis slip and this explains why this mode of deformation is readily suppressed by irradiation. The forces of interaction between cross-links and twin dislocations are not negligible, but since their interaction energies are of the order of 0.1 eV (0.16×10−19 J), twinning deformation, at moderate temperatures, should not be affected by irradiation. By combining all the possible deformation modes, the relationshipτ c=4χ1/2, is derived for the increase in yield stress,τ c (GPa), in terms of the atomic concentration of cross-links,χ, provided that these are uniformly distributed in the crystal.