Instabilities in a Neutralized Electron Stream in a Finite-Length Drift Tube

Abstract

Instabilities in one and two drifting‐electron streams neutralized by nondrifting ions and bounded transversely and axially are presented in mathematical and physical detail. Solutions for growth rates are obtained from the dispersion relation for a cylindrical stream of cold electrons and cold ions plus the boundary conditions at the gridded ends of the drift region. The new results in addition to the critical current (or critical perveance=190×10⁻⁶) given by Pierce in 1944 are: (i) less approximate inclusion of finite drift length, (ii) ac motion of ions, (iii) detailed physical description, (iv) growth rates over a range of parameters. The results show instability for immobile ions with the critical perveance value agreeing with that given by Pierce, but show marginal stability for mobile ions, contrary to Pierce and to the experiments of Atkinson. The difference between mobile and immobile ion solutions is ascribed to the ability of ions to neutralize (or not) excess electron charge in the vicinity of the output grid. The relation to stream‐ion wave amplification is discussed. The relation of the solutions presented here to those for a cylindrical stream which is not neutralized but is thin enough to have only small velocity variation radially is also discussed.