14 The Electron Microscope 



By the Hamiltonian analogy, the electron optical equivalent of 

 intensity is the electron current per unit cross section and unit 

 solid angle, and equation (9) is immediately applicable if we 

 replace the square of the refractive index by the potential <^, i.e., 



-(?) 



I = h(~] (10) 



We have already seen that the factor of Iq can approach a mil- 

 lion. In order to obtain an exact expression for the intensities 

 obtainable with thermionic cathodes, it is necessary to apply 

 equation (10) to every one of the electron groups with homo- 

 geneous velocity which are emitted by the cathode, and form 



the average of the quantity — . This has been first carried out 

 by D. B. Langmuir,^ and the result is 



/ = 3,700foY (11) 



where i^ is the current density at the cathode, i.e., the current 

 per unit area (not per unit area and unit solid angle), T is the 

 temperature of the cathode in degrees Kelvin, and the numerical 



factor is — '- . V is the accelerating potential, measured in 



volts. For example, with a barium oxide cathode with T = 



1000°K and i^ = 0.2 amp/cm^ we obtain, if V = 60,000 volts, 



/ = 44,400 amp/cm- steradian, which is 6.9 X 10^ times more 



0.2 

 than the maximum intensity -^- := 0.064 amp/cm- steradian at 



TT 



the cathode. 



In light optics, the intensity is a measure of the energy con- 

 centration in the beam, as the energy of a light beam does not 

 change during its propagation. In electron optics, however, the 

 energy of the electrons increases continually with acceleration, 

 therefore, in order to obtain a fair comparison, we must calculate 

 not the intensity /, but the power I.V per unit cross section and 



solid angle. This is — — =-^ — watts/cm^ steradian. 



