REFLEX OSCILLATORS 639 



Here the grid is assumed to lie in the y, z, plane. This gives a mesh of wires 

 about squares a on a side, the wires bulging at the intersections. We take 

 r to be the wire radius midway between intersections. 



We see that /3o will be the same in this case as in the case of a parallel wire 

 grid. Thus the added wires, which intercept electrons, haven't helped us 

 as far as this part of the expression goes. 



As a further appro.ximation, an averaging will be carried out as if the 

 apertures had axial symmetry. Averaging will be carried out to a radius 

 giving a circle of area a\ The steps will not be indicated. 



Further a factor analogous to / will be worked out. Again, the steps 

 will not be indicated. The results are 



^a = gisin {yd/2)/{'yd/2)\Gy{ya) (b41) 



Ci-ya) = 2h{ya/V^)/(ya/\/^)Go{ya) (b42) 



g = 1 + (.365 a/d) (log.o (a/Tr) - .69) (b43) 



The quantity 6'i(7,a) is plotted in Fig. 125 for comparison with the parallel 

 wire case. It should be emphasized that these expressions assume r « a, 

 and that Gi(ya) is really only an estimate based on a doubtful approximation. 

 The indications are, however, that the only beneficial affect of going from a 

 parallel wire grid to a mesh with the same wire spacing lies in a small de- 

 crease in 5F (a small increase in the mu of the grid), while by doubling the 

 number of wires in the parallel wire grid, a can be halved, both raising mu 

 and increasing G(ya,n). 



APPENDIX III 



Approximate Treatment of Bunching 



We assume that the conditions are as shown in Fig. 126 where the elec- 

 tron energy on first entering the gap is specified by the potential Vo . Across 

 the gap there exists a radio frequency voltage, V sin co/. The ratio 

 of the energy gained by the electron in crossing the gap to the energy 

 which it would gain if the transit time across the gap were zero is 

 called the modulation coefficient and is denoted by a factor, /3. We assume 

 that the modulation coefficient is the same for all electrons. We also neg- 

 lect the effects of space charge throughout. After leaving the gap the 



F/e + Vo 



electrons enter an electrostatic retarding field of strength Eq 



I 



2' This analysis follows the method given In- Webster. J. Ann. Phys. 10, Julv 1939, nn 

 501-508. ' - 'M 



