Abstract
The trajectory differential equation governing the motion of relativistic electrons is derived in terms of the scalar and vector potentials of the system using the principle of least action. The conservation of energy and momentum are used to develop the paraxial-ray differential equation describing the beam radius of a laminar-flow relativistic electron beam. The focusing of the electron beam in drift and accelerating regions has been examined and the conditions for perfect balancing and nonspreading of a laminar-flow drifting beam are also derived. It is shown that the equilibrium condition for Brillouin flow is that 2ωL2−ωp2[1 − (uz/c)2]=01 where ωL is the Larmor precession frequency and ωp the electron-plasma frequency. uz and c denote, respectively, the axial beam velocity and the speed of light in vacuum. The variation of the normalized ripple amplitude and the scallop wavelength of a drifting beam is discussed. The profile of a beam accelerated in a uniform longitudinal electrostatic field is also illustrated.
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