REFLEX OSCILLATORS 633 



of symmetry were the same for both transits of the electron, we would do 

 well to average /3 . If the electrons got thoroughly mixed up in position 

 between their two transits, we would do well to average jS and then square 

 the average. We will present both average and r.m.s. values of /3. The 

 averaged value of ^ will be denoted as ^a , the r.m.s. value as /3s ; the value 

 on the axis will be called j3o . 



From (bl5) and (bl6) we obtain by simple averaging for the two dimen 

 sional case, 



yy 



0s = /3o 



n^i^^ + i 



(bl8) 



and for an axially symmetrical case 



/3„ - 0,2I,(yr)/iyr) (bl9) 



0. - ^oUliyr) - Iliyr)]'" (b20) 



It is convenient to rewrite these in a slightly different form, using (bl5) 

 and (bl6). 



0a = 0y (tanh yy/yy) (b21) 



sinh 2yy -\- 2yy 



0s = 0y 



] 



(b22) 



_27y(cosh 27y + 1) 

 0a = 0r2h{yr)/{yr)h[yr) (b23) 



0.. = 0\\ - l\{yr)/ll{yr)]"' (b24) 



Now consider two similar cases: two pairs of parallel semi-inhnite plates 

 with a very narrow gap between them, and two semi-infinite tubes of the 

 same diameter, on the same axis and with a very narrow gap between them 

 (.see Fig. 120). For electrons traveling very near the conducting surface, 

 V is zero save over a very short range at the gap, and the modulation coeffi- 

 cient is unity. Thus, by putting /3y = jS^ = 1, we can use expressions 

 (bl5)-(b24) directly to evaluate 0a , 0s and 0^ for the configurations de- 

 scribed. These quantities are shown in Figs. 121 and 122. 



Suppose, now, that the gap between the plates or tubes is not very small. 

 In this case, we need to know the variation of potential with distance 

 across the space d long which separates the edges of the gap in order to get 

 the modulation coefficient at the very edge of the gap, 0y or 0r . 



If the tubing or plates surrounding the gap are thick, we might reasonably 



