318 BELL SYSTEM TECHNICAL JOURNAL 



has overcome the divergmg effect of the space charge, we still have a space 

 charge limitation of the beam current. C. J. Calbick has calculated the 

 value of this limiting current. ^^ If the beam completely fills a conducting 

 tube at a potential V with respect to the cathode, the limiting beam current 

 is independent of the diameter of the beam and is 



/ = 29.3 X 10~°F'^^ (20) 



If the beam diameter is less than that of the conducting tube, the limiting 

 current is lower. 



But perhaps we can completely overcome the effects of space charge. 

 Suppose we put a very little gas in the discharge space. Then positive ions 

 will be formed. Any tendency of the electronic space charge to lower the 

 potential and slow up the electrons will trap positive ions in the potential 

 minimum and so raise the potential. Thus the gas enables us to get rid 

 of the the slowing up effect of the space charge as well as its diverging 

 effect. 



Before we congratulate ourselves unduly, it might be well to make sure 

 about the stability of an electron beam in which the electronic space charge 

 is neutralized by heavy positive ions. Langmuir and Tonks, in their work 

 on plasma oscillations, introduced a concept, extended later by Hahn and 

 Ramo, which enables us to investigate this problem. The concept is that 

 of space charge waves. It is found that in a cloud of electrons whose net 

 space charge is neutralized by heavy, relatively immobile positive ions, 

 small disturbances of the electron charge density produce a linear restoring 

 force; and this, together with the mass of the electrons, makes possible a 

 type of space charge wave which may be compared roughly with sound 

 waves, although much of the detailed behavior of space charge waves is 

 quite different from that of sound waves. We may express a disturbance in 

 an electron beam in terms of these space charge waves and then examine the 

 subsequent history of the disturbance as a function of time. This has been 

 done'^ and the perhaps surprising result is that even when the electronic 

 space charge is neutralized by hea\y positive ions, the flow tends to collapse 

 if the current is raised above a limiting value 



/ = 190 X 10""]'''' (21) 



It is true that this current is 6.5 times the limiting current in the absence of 

 ions, but it is a limit nevertheless. 



If this limit in the presence of ions seems unnatural, perhaps we should 

 recall a mechanical analogy. Consider a vertical long column subjected to 

 a load F. If we subject it to a sidewise force aF proportional to F, as shown 

 in Fig. 11a, the behavior on increasing F will be a gradual deformation 

 (analogous to the space charge lowering of potential in the absence of ions) 



