114 The Electron Microscope 



tial of the cathode *'C" must be positive against the magnetic 

 casing by a suitable amount. By these two measures the electrons 

 are trapped in a space, the approximate outlines of which are 

 shown in the second figure 41 in dotted lines. They cannot 

 escape anywhere, except back to the cathode, and as the filament 

 is very thin, this will happen very rarely, in general only after 

 the electron has traversed the accessible space many times. 

 Therefore, according to this rather over-simplified picture, the 

 negative space charge could be maintained in a stationary con- 

 dition with no input at all. 



The actual distribution of space charge within the accessible 

 space presents a very difficult problem. The static magnetron is 

 usually explained by Hull's simple theory, which has been only 

 slightly modified by Pidduck ^^ and Brillouin.''' According to 

 these theories, the distribution is the same as if the electrons 

 were all rotating on coaxial circles, held in equilibrium by the 

 radial electric field and by the inward directed force exerted by 

 the magnetic field; the law of the electric field is given by the 

 condition that this equilibrium should be possible at any radius. 

 This gives a quadratic increase of the potential with the radius, 

 and by Poisson's equation a constant space charge density 



-^==8^^ (40) 



Numerically this is 2.33 X lO^^H^ electrostatic units, if H is 

 substituted in gauss. At H := 2,000 gauss, this gives 93.2 

 elst.units/cm^, or 1.94 XlO^^ electrons/cm^, which is of the de- 

 sired order of magnitude. But in fact, the effect would be zero, 

 as the concentrating effect of the magnetic field would exactly 

 counterbalance the dispersing effect of the space charge. This 

 can be immediately understood from the way in which Hull's 

 equation was derived. It is also shown explicitly by the path 

 equation of paraxial electrons which is conveniently written in 

 a form due to Scherzer : 



R" = 



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