Possibilities of Future Development 117 



The calculation shows that if the outer radius vo is large in 

 comparison with ri, the saturation value of the density at Vo 

 will be very nearly one-quarter of Hull's density pu- Inside, it 

 can be very much larger, according to the approximate law 



2r2 

 P = PH— (44) 



An electron cloud of this type can therefore act as a divergent 

 lens. This can also be immediately understood. The electrons 

 in the cloud are not in radial equilibrium, as in Hull's picture of 

 the magnetron, but they are reflected to and fro between the 

 inner and the outer boundary. The forces near the inner 

 boundary are repellent, and it is easy to show that if ri is suffi- 

 ciently small in comparison with vo, a certain inner zone will act 

 as a divergent lens. But the law of repulsion is not of the type 

 required for an electron lens. Instead of increasing linearly 

 with r, it falls oflF with increasing radius, and changes sign at a 

 certain point. This is particularly marked if we arrange the 

 cathode so that the electrons can reach the axis. In this case 

 the density at the axis becomes infinite. This, of course, is a 

 consequence of certain simplifications, notably the fact that we 

 have assumed the cathode to have zero radial extension, and 

 neglected the initial velocities. 



If extended cathodes are used, the distribution of density can 

 be modified in a way as is shown in figure 42 (d). The interac- 

 tion of electrons in the cloud is very difficult to treat quantita- 

 tively, but it can be expected that moderate interaction would 

 produce a similar distribution, even if the cathode is made with 

 as small radial extension as can be practically realized. Its first 

 effect will be a smoothing of the corners, and filling up of the 

 prohibited space with electrons, approximately as shown in the 

 figure. In order to keep the interaction sufficiently small, it may 

 be necessary to remove the electrons fairly rapidly, before they 

 have time to establish perfect equilibrium by collisions. In order 

 to avoid heat development, this will be done preferably by means 

 of collector electrodes connected with the cathode, and arranged 



