Evaluation of the DC parameters and noise transport in the gun region of an injected-beam crossed-field diode

A half-Maxwellian
y
-component velocity distribution and a full-Maxwellian
z
-component velocity distribution are assumed in order to evaluate the position and depth, y
m
and V
m
, of the potential minimum as a boundary-value problem. The various dc parameters such as the voltage distribution, space-charge density, velocity and current density components and trajectories are then evaluated as an initial-value problem. The results obtained in this manner agree closely with the results obtained from the Kino gun model except that the
y
-component current density is not constant, as is usually assumed in the Kino gun model. The steady-state parameters are calculated here for both temperature-limited and space-charge-llmited conditions. The Kino gun results are shown to be essentially those for space-charge-limited operation. Even though the injection conditions under the two types of operation are identical, the formation of a potential minimum considerably changes the electron trajectories and the corresponding velocity components. The growth rate of a hybrid wave is reduced as ω
p
is decreased and/or
\omega/\omega_{c}
is increased, and the propagation constants of the two conventional space-charge waves are modified, the over-all growth rate of the slow wave being greater than that of the fast wave. For large values of ω
p
the conventional fast space-charge wave is a backward wave, although it becomes a forward wave if
\omega/\omega_{c}
is large. It is noticed that the conditions in the gun region are more favorable to the existence of low-frequency perturbations. Based upon these results several experimental observations made at various laboratories are explained qualitatively.