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We developed a general method for evaluating the energy spectrum evolution ofrelativistic charged particles that have undergone small quantum losses, suchas the ionization losses when the electrons pass through matter and theradiation losses in the periodic fields. These processes are characterized by asmall magnitude of the recoil quantum as compared with the particle's initialenergy. The `detector' function for arbitrary recoil spectrum is derived inaddition to the straggling function. These functions are determined by theaverage number of scattering events undergone by the particle over the lengthof the detector (ionization losses) or the periodic field and the specificspectrum of the recoil. Moments for both the straggling function and thedetector function are derived. The minimum average number of recoils, abovewhich both functions depend only upon the mean energy of recoil quantum, isestimated. The average of this number is about 10 recoil events.

Kinetics of relativistic electrons undergoing small recoils in periodic structures and matter