Fast electrons in solids lose energy primarily to plasmons which subsequently decay into an electron‐hole pair. The electron and hole may create additional electron‐hole pairs if they have sufficient energy. The average energy to create an electron‐hole pair,Ex, is thus given essentially by ℏ &ohgr;p/n, where ℏ &ohgr;pis the plasmon energy andnis the number of pairs which are created by the plasmon and its decay products. The values ofnand henceExdepend on the ratio ℏ&ohgr;p/Eg, whereEgis the usual band gap in semiconductors and insulators. We have found that for a simple model of direct and indirect materials, sharp thresholds exist for various values ofn. For direct‐gap materials,n=1 for 1< ℏ&ohgr;p/Eg<4,n=3 for 4< ℏ&ohgr;p/Eg<12, etc. Similar relations for indirect‐gap materials have also been obtained. Consideration of phonon losses, effective masses, and problems withkconservation indicate that the changes innvs ℏ&ohgr;p/Egshould be smoothed for real materials. but not eliminated. Our theoretical values forEx/Egallow values lower than previous treatments and give good agreement with known experimental values. Some problems associated with determining the theoretical value of &ohgr;pfor materials withdelectrons are discussed. Some experiments for testing the theory are suggested. The importance ofExin determining the efficiency of phosphors is also discussed.