Confinement criterion for a finite sized bunched beam

Summary form only given. A confinement criterion for a finite sized azimuthally symmetric relativistic bunched electron beam propagating through a perfectly conducting cylindrical pipe with a slight off-axis displacement while in the presence of either a uniform or periodic solenoidal field is studied. We compute the center-of-mass force on each off-axis beam bunch due to the induced surface charges on the pipe using a Green´s function technique. By analyzing the center-of-mass dynamics of each bunch due to magnetic focusing and its interaction with the pipe, a space-charge limit for beam confinement is derived. A comparison of the theory to high-current periodic permanent magnet (PPM) klystron experiments is established.