382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



■i 



trary point in our gun in terms of the well known solution for space 

 charge limited flow between two concentric spheres, Vl , and a potential 

 distribution, Vb , which does not depend on space charge and can there- 

 fore be obtained in the electrolytic tank. Once the potential distribution 

 is found, electron trajectories may be calculated, and an equivalent lens 

 sj^stem found. Equation (4) is used in this way in Part C as one basis for 

 estimating a correction to the Davisson equation. (It will be noted that i 

 (4) predicts a small but finite negative field at the cathode. This is be- 

 cause the space charge density associated with Fsc is slightly greater 

 near the cathode than that associated with F(sc)' , and it is this latter 

 space charge which will make the field zero at the cathode under real 

 space charge limited operation. Equation (4), as applied in Part C of this 

 section, is used to give the voltage as a function of position at all points 

 except near the cathode where the voltage curves are extended smoothly 

 to make the field at the cathode vanish.) 



B. Use of a False Cathode in Treating the Anode Lens Problem 



Before evaluating the lens effect by use of (4), it will be useful to de- 

 velop another approach which is a little simpler. The evaluation of the 

 lens effect predicted by both methods will then be pursued in Part C 

 where the separate results are compared. 



In Part A we noted that no serious error is made in neglecting the dif- 

 ference between the two space charge configurations considered there 

 because these differences were mainly in the very low space charge 

 region near and beyond A2 . It similarly follows that we can, with only 1 

 a small decrease in accuracy, ignore the space charge in the region near 

 and beyond A2 so long as we properly account for the effect of the high 

 space charge regions closer to the cathode. To place the foregoing obser- 

 vations on a more quantitative basis, we may graph the Langmuir po- 

 tential (for space charge limited flow between concentric spheres) versus 

 the distance from cathode toward anode, and then superpose a plot of 

 the potential from LaPlace's equation (concentric spheres; no space 

 charge) which will have the same value and slope at the anode. The La- 

 Place curve will depart significantly from the Langmuir in the region of 

 the cathode, but will adequately represent it farther out." Our experi- 

 ence has shown that the representation is "adequate" until the difference 

 between the two potentials exceeds about 2 per cent of the anode voltage. 

 Then, since space charge is not important in the region near the anode 

 for the case of a gridded Pierce gun, corresponding to space charge 

 limited flow between concentric spheres, it can be expected to be similarly 

 unimportant for cases where the grid is replaced by an aperture. Let us 



I 



