Time-dependent current-voltage relation in electron guns

We studied mathematical models for degradation behaviors of current and voltage in electron guns especially using oxide cathodes. We found that the current and the voltage, $I(t)$ and $V(t)$⁠, follow the stretched exponential decay in oxide cathodes. On this basis, we derived a general expression for the time-dependent current–voltage relation as $I(t)=p(t)V(t)δ(t)$⁠, where $δ(t)$ is a time-dependent exponent and the perveance, $p(t)$⁠, is a function of $δ(t)$⁠. The exponent $δ(t)$ indicates the deviation of the classical Child–Langmuir relation $(I=pV3∕2)$⁠. This deviation is attributed to the gradual change of the electron gun geometry over time.