THERMIONIC ELECTRON EMISSION 433 



Usually thermionic experiments are not performed with plane 

 parallel cathodes and anodes but with a small cylindrical cathode con- 

 centric with a cylindrical anode. In the cylindrical case, the normal 

 or radial component of velocity is not the only one which determines 

 whether the electron will reach the anode. Schottky ^* derived the 

 following formula for this case on the assumption that the emitted 

 electrons leave the filament with a velocity distribution given by 

 Maxwell's law (equation (7)) for a temperature T. As we have seen 

 above both the classical and the Fermi-Dirac theory predict this dis- 

 tribution for the electrons which escape from the filament. This 

 formula replaces equation (41). 



i = i^i{2/-ir\ 



iVre/kTjexp. (- Vre/kT) 



Xco 

 exp. { — x-)dx 



(VrClkT) 



(46) 



It is assumed that the diameter of the cathode is small compared to 

 the diameter of the anode, and that the current is not limited by space 

 charge. Table I gives values of logio (io/i) for values of VrC/kT 

 taken from an article by Germer.^" 



TABLE I 

 Values of logio /u/i for Various Values of Vre/kT (Germer) -" 



Figure 6 shows various ideal plots of log i versus V for cathodes of 

 clean tungsten and thoriated tungsten. It is assumed that the anode is 

 clean tungsten. Curves 1, 2 and 3 are for a clean tungsten cathode at 

 temperatures of 1400, 1550 and 1700° K., respectively. Curves 4 and 

 5 are for a thoriated tungsten cathode at 1400° K. activated to such 

 an extent that the work function is 4.03 and 3.53 volts, respectively. 

 The dashed lines indicate the currents for a plane cathode and parallel 

 anode. 



Curves 1, 4 and 5 illustrate an important theorem which follows 

 from the analysis on contact potential given in connection with Figs. 

 3, 4 and 5. The theorem is: The current collected by an anode is 

 independent of the work function of the cathode provided that the 



