A mathematical model which describes the high-perveance instabilities of long drifting electron beams

It has been recognized for more than twenty years that space-charge effects set a definite limit on the stable electron beam current that can be injected into a given conducting enclosure at a particular energy, If this value of ‘limiting current’ or ‘critical perveance’ is exceeded, the beam becomes unstable. Haeff [1], Pierce [2], Smith and Hartman [3], and Fay et al [4], many years ago derived the expressions predicting these critical perveance magnitudes for the most likely enclosure and beam configurations. More recently, the actual dynamic behaviour in this range of operation beyond critical perveance has come under investigation by Birdsall and Bridges [5], Buneman [6], Lomax [7], Lauer and Twombly [8], and others.