380 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Hines has suggested* that a fairly accurate description of the potential 

 distribution in such guns can be obtained by a superposition method as 

 follows: 



By the usual tank methods, find suitable beam forming electrode and 

 anode shapes for conical space charge limited flow in a diode having! 

 cathode and anode radii of curvature given by fc and f„i , respectively, 

 as shown in Fig. 1(a). Using the electrolytic tank with an insulator along 

 the line which represents the beam edge, trace out an equipotential 

 which intersects the insulator at a distance fa2 from the cathode center 

 of curvature. Let the cathode be at ground potential and let the voltage 

 on anode Ai be called B. Suppose, now, that we are interested in electron 

 trajectories in a non-gridded gun where the edge of the anode hole is a 

 distance fai from the center of curvature of the cathode. Let the voltage, 

 C, for this anode be chosen the same as the value of the equipotential 

 traced out above for the case of cathode at ground potential and A\ 

 at potential B. If we consider the space charge limited flow from a 

 cathode which is followed by the apertured anode, Ai , and the full 

 anode, Ai , at potentials C and B, respectively, it is clear that a conical 

 flow of the type which would exist between concentric spheres will re- 

 sult. The flow for such cases was treated by Langmuir,^ and the associ- 

 ated potentials are commonly called the "Langmuir potentials." 



If we operate both Ai and A2 at potential C, however, the electrons 

 will pass through the aperture in anode A2 into a nearly field-free region. . 

 If the distance, fa2 — Tai , from A2 to Ai is greater than the diameter of 

 the aperture in A2 , the flow will depend very little on the shape of Ai 

 and the electron trajectories and associated equipotentials will be of the 

 type we wish to consider except in a small region near Ai . We will shortly 

 make use of the fact that the space charge between cathode and A 2 is 

 not changed much when the voltage on Ai is changed from B to C, but 

 first we will define a set of potential functions which will be needed. 



In order to obtain the potential at arbitrary points in any axially sym- 

 metric gun when space charge is not neglected, w^e may superpose po- 

 tential solutions to 3 separate problems where, in each case, the boundary 

 condition that each electrode be an equipotential is satisfied. We will 

 follow the usual notation in using f for the distance of a general point 

 from the cathode center of curvature, and r for its radial distance from 

 the axis of symmetry. Let Vdr, r), Vh(r, >') and Vsdr, r) be the three 

 potential solutions where: (1) Vaif, r) is the solution for the case of no 

 space charge with Ai and cathode at zero potential and A 2 at potential 

 C, (2) Vb{r^ r) is the solution for the case of no space charge with A2 



* Verbal disclosure. 



