IMPEDANCE PROPERTIES OF ELECTRON STREAMS 467 



minimum T'om sink and move towards the electrode of higher potential, 

 curve 4. This state of affairs, with a continuously decreasing potential 

 minimum, continues until a critical value of injected current is reached. 

 The potential distribution may now be represented by curve 5. The 

 slightest further increase in injected current causes the potential 

 minimum to sink abruptly to zero: a virtual cathode is formed. This 

 abrupt change will be referred to as a Kipp. 



With this qualitativ^e discussion of the various potential distributions 

 in mind a more detailed classification may be made. If both planes 

 are assumed to be at positive potentials we may classify the different 

 potential distributions as follows: 



1. Type B 



2. Type C 



3. Type D 



Type B corresponds to virtual cathode operation. This mode of 

 operation will be of no interest in this paper. Type C corresponds to 

 the case when a potential minimum at positive potential is present 

 between the planes and type D to the case when no such potential 

 minimum is present. For the purpose of analysis it is convenient to 

 distinguish between two types of D solution, i.e., Di and D2. This 

 comes about because the equations for potential distribution between 

 the planes may exhibit a minimum outside the planes. The Di 

 solutions correspond to the case when no such minimum exists and the 

 D2 solutions to the case when such a minimum does exist. 



Let us consider Type C distributions in some more detail. For this 



purpose attention is directed to Fig. 2. Here the ratio -jy^ is shown 

 as a function of the injected current /o with the ratio ^^^ as parameter. 



1^01 



The dotted curve represents a boundary line; for currents smaller 

 than that given by this boundary no potential minimum can exist 

 between the planes. Consider the curve 0:187. At the point a. the 

 potential minimum sets in and as more current is injected the potential 

 at the minimum decreases continuously until the point |S is reached. 

 Any further increase in current causes the potential minimurn to sink 

 abruptly to zero; thus ^ corresponds to the Kipp point. Now it is 

 seen that within a section of the curve a/? there are two possible 

 solutions, namely those corresponding to the section 1871 of the upper 

 branch and those corresponding to the lower branch 187. Let us 

 designate by Ci space charge conditions corresponding to the upper 

 branch a/S and by C2 those corresponding to the lower branch 187. 



