54 BELL SYSTEM TECHNICAL JOURNAL 



For the second group of curves (Type B) a certain fraction of the 

 current denoted by Z (and so indicated on the curves) will be trans- 

 mitted while the fraction 1 — Z will be reflected toward the first 

 plane. These potential curves have zero slopes at the reflection planes 

 so that they correspond to solutions of Child's equation on both sides 

 of a so-called virtual cathode existing at the reflection plane. For 

 each value of Z from to 1 a possible potential distribution is obtained. 



It will be observed that the portions of the solid curves to the left 

 of the zero points are drawn lighter than the rest. These portions 

 correspond to potential distributions resulting from conditions with 

 reflected current and so while applying as extensions of the heavy 

 curves to the right they cannot be entered directly with values of <p and 

 a. Should values used to enter the figure fall in this region the ab- 

 sence of a potential zero is indicated and the correct distribution is 

 obtained by entering Fig. 3. Before leaving Fig, 2 the existence should 

 be noted of a small domain lying between the curves marked h and c 

 for which two different B solutions are possible for the same injected 

 current, spacing, and potential, one corresponding to larger values of Z 

 than the other. The significance of this region will be apparent when 

 the current voltage relationships are considered. All solutions of the 

 A and B types are represented on Fig. 2. 



Solutions of the C and D types are to be found on Fig. 3 except for a 

 small overlap region which for clearness is shown on Fig. 4. Values of 

 if and a which cannot be entered on Fig. 2 as well as values common to 

 both B and C types are to be found on these figures. The additional 

 overlap solutions of the C type are shown in Fig. 4. These are ex- 

 tensions of the potential distribution curves of Fig. 3 after they reach 

 the right-hand boundary curve where they turn inward as shown and 

 overlap the other solutions. 



For convenience each curve of the C type is labeled by the value of 

 its minimum potential. The parameters of the D curves do not have 

 this simple physical significance. However, for all cases the value of 

 the curve parameter is simply related to the electric field at the first 

 plane by the equations: 



type C ^ = - t VI - ^n.in}i'. (6) 



Vl - a"\ (7) 



Off _ ^ 



d(T 3 



^=±tvrT^. (8) 



type D 



type A ^ ^ _ ;x>^ Vr+^'/^. (9) 



