PHYSICAL LIMITATIONS IN ELECTRON BALLISTICS 309 



the electron stream at a spot a distance L beyond it. The electrons will 

 leave the cathode with some slight sidewise velocity components; so, elec- 

 tron paths will pass at several angles through a given point on the lens. The 

 lens will bend these paths approximately equally, and hence we can see that 

 at the point where the beam is narrowest it will still have some appreciable 

 diameter TF2. 



Now consider the beam at the lens. Suppose that through a given point 

 all the paths lie within a cone of half angle 6. Then the width W2 

 is approximately 



W2 = 2Le, (8) 



We can also see that the paths at W2 will lie within an angle approximately 



02 = W1/2L (9) 



Hence we see that approximately 



diWi = diWi (10) 



In other words, we can have a small spot through which electrons pass 

 over a wide angular range, or we can have a broad beam in which all paths 

 are nearly parallel, but we can't have a narrow spot and nearly parallel 

 rays. 



We see that the actual width of spot will depend on the thermal veloc- 

 ities, which are proportional to the square root of the cathode temperature, 

 and on the forward velocity, which is proportional to the square root of the 

 accelerating voltage. By using more involved arguments we discover 

 that for any point in an electron stream, where the beam is wide, narrow, or 

 intermediate, the current in an arbitrary direction chosen as the x direction 

 can be expressed '* 



dj = ■ jQVxe dvx dVy dv^ (11) 



■Km 



V = vl -\- vl -f vl 



when I) > \/2eV/m; (12) 



or dj = (13) 



when V < \/2eV/m (14) 



Here jo is the cathode current density, V is voltage with respect to the 

 cathode, T is the absolute temperature of the cathode in degrees Kelvin, 

 and Vx, ijy, and Vz are the three velocity components; dj is the^element 



* This expression neglects the effects of electron collisions, which may actually make 

 the current density smaller. 



