Absence of saturation for finite injected currents in axially symmetric cavity diode

The Child-Langmuir law is investigated numerically using a fully electromagnetic particle-in-cell code for a closed axially symmetric diode. It is found that the average current transmitted to the anode increases with the injected current even after the formation of virtual cathode in both the non-relativistic and relativistic cases. The increase is found to be a power law. In other words, the time averaged fraction $f$ of electrons reaching the anode varies with the input current as, $f\sim J_{\rm IN}^{-\beta}$ where $\beta < 1$. In contrast, for an infinite parallel plate diode, $f \sim J_{\rm IN}^{-1}$. The possibility of asymptotic saturation is also discussed.