Kinetic stability theory of space-charge-limited flow

The effect of a spread in energy of the emitted beam on the stability of the space-charge-limited diode is investigated. It is found that for small thermal spreads the so-called emittron or transit time instability persists and is correctly described by cold fluid equations. Thermal effects become important when ωT∼ε−1/4T, where ω is the mode frequency, T is a typical particle transit time, and εT is the ratio of the thermal energy of the beam to potential energy associated with the applied voltage. For modes with frequencies above this value, phase mixing of the particles causes a dramatic reduction in growth rate, such that other damping mechanisms will be able to stabilize the high-frequency modes. The analysis presented here applies specifically to transverse electromagnetic (TEM) modes. However, the results can be expected to qualitatively describe other modes as well.