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

Abstract

A: We developed a general method for evaluating the energy spectrum evolution of relativistic charged particles that have undergone small quantum losses, such as the ionization losses when the electrons pass through matter and the radiation losses in the periodic fields. These processes are characterized by a small magnitude of the recoil quantum as compared with the particle's initial energy. The 'detector' function for arbitrary recoil spectrum is derived in addition to the straggling function. These functions are determined by the average number of scattering events undergone by the particle over the length of the detector (ionization losses) or the periodic field and the specific spectrum of the recoil. Moments for both the straggling function and the detector function are derived. The minimum average number of recoils, above which both functions depend only upon the mean energy of recoil quantum, is estimated. The average of this number is about 10 recoil events. K: Beam dynamics, beam-line instrumentation, detector modelling and simulations I. A X P : 1711.09571

Statistics of relativistic electrons radiating in periodic fields

“…with γ being the dimensionless particle energy (Lorentz factor). A solution to (1) in a form of the characteristic function (Fourier transform of the distribution density), [1,2] is:…”

“…where w(ω) is the recoil's spectrum, and f 0 is the initial spectrum. The parameter x > 0 is the ensemble average number of the recoils undergone by an electron since entering the driving force, [1,2]. The equation (3) may be generalized and simplified due to the model's assumption of the independence of recoils on the electron's energy, as proposed in [4].…”

“…In a number of processes involving beams of highenergy electrons, such as radiation in the periodic structures, ionization losses in matter, etc., the energy degradation of an incident electron is in the form of the small portions (recoil quanta), which spectrum is almost independent of the electron's energy. In our previous papers [1,2], we have considered the evolution of the spectrum of such electron beam. It was shown, that the spectrum is determined by the parameters of a single recoil and the average number of the recoils.…”

“…In this paper, we use the reduced energy units: ǫ for the energy in the straggling spectrum, ω for the energy of the spectrum of the recoil quantum, both are dimensionless, normalized to the energy unit, e.g., to the charged particle rest energy, [2]. We use a convention for the Fourier transform in the form of:…”

“…A solution to (1) in a form of the characteristic function (Fourier transform of the distribution density), [1,2] is:…”