852 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



where </> is the spread measured in electron volts, and V is the voltage in 

 the stream. Then (84) may be written in the form 



8V 

 The last term is small compared to rj — and may be neglected, and it fol- 



ox 



lows that the idealized theory is an approximation for physical electron 



streams when 



it being understood that the inequality holds for the gradients of the d-c. 

 components, and also for the gradients of the a-c. components of <l> and V. 

 This requirement is satisfied over a wide range of conditions, and the ideal- 

 ized equations are applicable in a corresponding manner.* 



It is thought that these considerations explain why the single-velocity 

 theory of electron streams is so successful in explaining the characteristics 

 of electronic tubes. ^' ^ 



Conclusion 



It is believed that the preceding pages serve to unify our theories of elec- 

 tron streams in some such manner as follows: 



(1) They develop the general solution for a single velocity stream, and 

 this solution contains all particular solutions. 



(2) The small signal theory is considered in detail as a special case of the 

 general solution, and the a-c. stream is shown to be the sum of two com- 

 ponents: a wave traveling with a finite velocity plus an oscillation that is in 

 phase over the entire diode space. 



(3) The wave expression gives identically the same results as previous 

 calculations based on the motions of electrons as individual particles. 



(4) The idealized stream is shown to be an approximation for multi- 

 velocity streams over a wide range of conditions, and this correspondence 

 explains why the single velocity theory is so successful in describing the 

 characteristics of electronic tubes. 



Acknowledgments 



The writer wishes to thank F. B. Llewellyn and L. C. Peterson for their 

 assistance in the study of electron streams, and to thank R. K. Potter for 

 his encouragement in writing the article at this time. 



* It should be noted that this requirement is not satisfied by a velocity-modulated 

 stream of small current density in a long, field-free drift space. 



