Electron Streams in a Diode* 



By FRANK GRAY 



A general solution of the electron stream equations is developed for a parallel 

 plane diode, under the assumption that the electron velocity is single valued. This 

 solution contains all particular solutions. It serves to unify the wave theory and 

 the particle theory of electron flow, and it is an approximation for multi- velocity 

 streams over a wide range of conditions. 



Introduction 



THE theory of an electron stream flowing in a diode has received much 

 attention ;^~^^ because the tetrodes, pentodes and other modern tubes 

 are cascade arrangements of individual diodes. The theory of the diode is 

 the foundation for considering the circuit characteristics and the noise 

 characteristics of these tubes. In earlier days when communication channels 

 were confined to relatively low frequencies, an electron could traverse 

 a diode in a short period of time compared to an oscillation of any electrical 

 signal, and the theory could be developed rather simply from the known 

 d-c equations. But in these days of high and ultra-high frequencies, the 

 situation is quite different. A signal voltage may oscillate several times 

 while an electron is traversing a diode, and the electron stream flows in 

 bunches or waves. The present article is primarily concerned with this more 

 compUcated type of flow. It is confined to the case of parallel plane electrodes, 

 and developed under the usual assumption that the electron velocity is a 

 single valued function of space and time. It is shown to be an approximate 

 solution for physical electron streams over a wide range of conditions. 



Particular solutions for an electron stream under small signal conditions 

 are given in various published articles. These theories approach the subject 

 in two different manners. In one approach attention is confined to the 

 motions of electrons as individual particles,^"^ and the other approach may 

 best be described as a wave theory of electron streams. But the two lines 

 of approach have not hitherto given identical results, and the disagreement 

 can probably be attributed to neglected factors in the wave theory. 



The present article considers electron streams without regard to any other 

 than a mathematical approach to the subject. The differential equations 

 are linear in the derivatives, and they should therefore have a general 

 solution that contains all particular solutions. The theory seeks and obtains 



* The paper was presented at a meeting of the American Physical Society in Columbus, 

 Ohio in 1945. 



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