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BELL SYSTEM TECHNICAL JOURNAL 



Using (5.20), (5.21), (5.22) in connection with (5.19), we obtain 



= 11.16 {J,\<?/v,yi' 



(5.23) 



We have in (5.23) an expression for the gain parameter C in case longi- 

 tudinal fields only are present and in case there is a uniform current density 

 /o wherever there is a longitudinal field. 



In a number of cases, as in case of a large-diameter helix, or of a resonator 

 with large apertures, the stored energy due to the transverse field is about 

 equal to that due to the longitudinal field and C will be 2~^i^ times as great 

 as the value of C given by (5.23). Thus, the value of C given by (5.23), or 

 even 2~^'^ times this, represents an unattainable ideal. It is nevertheless 

 of interest in indicating how limiting behavior depends on various parame- 

 ters. For instance, we see that if the wavelength Xo is made shorter, a higher 

 current density must be used if C is not to be lowered; for a constant C 

 the current density must be such as to give a constant current through a 

 square a wavelength on a side. 



In the table below, some values of C have been computed from (5.23) 

 for various wavelengths and current densities. The broad-band condition 

 of equal phase and group velocities has been assumed, and the voltage has 

 been taken as 1,000 volts. 



\ 



\ 



WavelengthX Amp/cm^ 

 Cm \ 



For larger voltages, C will be smaller. C can of course be made larger by 

 making the group velocity smaller than the phase velocity. 



Of course, if the electron stream does not pass through some portions of 

 the field, C will be smaller than given by (5.23). C will also be less if there 

 are "harmonic" field components which do not vary in the z direction as 

 exp(yco2/t)). 



5.3b Narroiv Beams 



Usually, no attemj)t is made to iill tlie entire field with electron flow even 

 though this is necessary in getting a large value of C for a given current 

 density. Instead a narrow electron beam is shot through a region of high 



