Electron trajectories in a plane diode - a general result

A simple mathematical proof is given of this result: In a plane diode with hot cathode and space charge, operated under a voltage whose variation with time is arbitrary, electron trajectories never cross each other provided the current never becomes negative. But if the current is negative during a certain interval of time it is possible that electrons falling back on the cathode may cross electrons emitted from the cathode. These results also apply to a cylindrical diode, except when a magnetic field is present, in which case the behaviour is very different. The case where electrons are emitted with a finite velocity is discussed briefly.