384 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



slope is therefore produced as is common to all thin lens approximations. 

 The diverging effect of electric field lines which originate on charges 

 which have passed the anode plane is then normally accounted for by 

 the universal beam spread curve/" In our attempt to evaluate the lens 

 effect more accurately, we will still depend upon using the universal 

 beam spread curve in the region following the lens and on treating the ; 

 equivalent anode lens as thin. Consequently our improved accuracy 

 must come from a mathematical treatment which allows the electric 

 field lines originating in the cathode-anode region to leave the beam grad- 

 ually, rather than a treatment where all of these flux lines leave the beam , 

 at the anode plane. In practice the measured perveances, P(= I/V^'^), 

 of active guns of the type considered here have averaged within 1 or 2 

 per cent of those predicted for corresponding gridded Pierce guns. There- 

 fore the total space charge between cathode and anode is much the 

 same with and without the use of a grid, even though the charge dis- 

 tribution is not the same in the two cases. The total flux which must 

 leave our beam is therefore the same as that which will leave the cor- , 

 responding idealized beam and we may write 



yp = I EndA = TT/VFidea/ (5) 



w^here En is the electric field normal to the edge of the beam, ra = rdfa/fc) 

 is the beam radius at the anode lens, and Videai is the magnitude of the 

 field at the corresponding gridded Pierce gun anode. 



To find the appropriate thin lens focal length we will now find the 

 total integrated transverse impulse which would be given to an elec- 

 tron which follows a straight-line path on both sides of the lens (see Fig. 

 2), and we will equate this impulse to wAw where An is the transverse 

 velocity given to the electron as it passes through the equivalent thin 

 lens. In this connection we will restrict our attention to paraxial elec- 

 trons and evaluate the transverse electric fields from (4) and from the 

 tank plot outlined in Section B, respectively. The total transverse im- 

 pulse experienced by an electron can be written 



f Fn dt = e [ —dl (()) 



J Path J Path U 



where u is the velocity along the path and Fn is the force normal to the 

 path. 



We will usually find that the correction to (1) is less than about 20 

 per cent. It will therefore be worthwhile to put (6) in a form which in 

 effect allows us to calculate deviaiions from Fu as given by (1) instead 



