Three-dimensional description of vibration-assisted electron knock-on damage

Elastic knock-on is the main electron irradiation damage mechanism in metals including graphene. Atomic vibrations influence its cross-section, but only the out-of-plane direction has been considered so far in the literature. Here, we present a full three-dimensional theory of knock-on damage including the effect of temperature and vibrations to describe ejection into arbitrary directions. We thus establish a general quantitative description of electron irradiation effects through elastic scattering. Applying our methodology to in-plane jumps of pyridinic nitrogen atoms, we show their observed rates imply much stronger inelastic effects than in pristine graphene.