The role of the Wigner function in charged-particle beam transport

Abstract

The role of the Wigner function in the dynamics of charged particle beams in high-energy accelerating machines is discussed. This is done within the quantum-like de- scription of the thermal wave model (TWM). A brief review of the numerical experiments showing satisfactory agreement between TWM and the particle tracking simulations is presented. A simple analysis in phase space in terms of the Wigner quasidistribution, showing that TWM is capable of reproducing the beam dynamics in the presence of the space charge effects, is put froward. The branch of electron optics describing the transport of non-laminar paraxial high-energy charged-particle beams involves the dynamics of an extremely large number of charged particles that is mainly dominated by the electromagnetic interactions, while the effects of the temperature cannot be neglected. The behavior of such a system is collective and affected by the thermal spreading among the particles (thermal regime). This discipline has been strongly developed in particle accelerator physics (1). A peculiar aspect of the electron optics of non-laminar beams is the mixing of the particle tra- jectories, i.e., mixing among the electron rays (1). In paraxial beams, this effect causes a slight random deviation of the electron rays slopes (with respect to the propagation direction). In vacuo, the electron ray slopes are Boltzmann-distributed, hence the instantaneous transverse momentum spread (normalized to the longitudinal relativistic particle momentum) is σp ∼ vT /c � 1( vT and c being the transverse thermal velocity and the speed of light, respectively). However, it is a matter of fact, both theoretically predicted and experimentally observed, that during the beam motion the instantaneous spread of the electron rays positions in the transverse plane, say σ, is related to σp by the following inequality: