Geometrical Electron Optics 5 



Therefore, the time of transit between two points is 



IS 



n ds 

 cJ 



and this must be a minimum. Comparing the two law^s, we see 

 that the dynamical trajectory and the light ray become identical 

 ij the refractive index of the medium is assumed proportional to 

 the electron velocity. The proportionality constant is obviously 

 of no importance. 



At first sight it might appear that the analogy is very im- 

 perfect, as the velocity belongs to the electron whereas the 

 refractive index is a property of the medium, that is to say, it 

 must be a function of the position only. But this difference 

 vanishes in the cases most important in electron optics, in which 

 only one cathode is considered, which emits electrons with 

 initial velocities negligible as compared to the velocities which 

 they acquire in the field. In this case, the refractive index is 

 proportional to the square root of the potential (f>, measured 

 from the cathode as zero level, and this is a function of the 

 position only. 



The analogy extends also to cases in which the initial veloci- 

 ties cannot be neglected. Let (f>o be the initial energy of an 

 electron emitted by the cathode, measured in potential units, 

 e.g., in volts. When this electron comes to a point of potential </>, 



its velocity measured in the same units will be V<^ + ^o- The 

 refractive index will not be the same for all electrons, but this 

 also has its optical counterpart in the chromatic dispersion of 

 ordinary media. We talk, therefore, also in electron optics of 

 chromatic effects, meaning by this the effects of initial velocities, 

 or of varying cathode potentials. 



In the case of a thermionic cathode with absolute tempera- 

 ture T, the average energy of an electron leaving the cathode 

 will be 2kT ergs,'^ where k is Boltzmann's constant, 1.37 X 10"^^ 



* The average energy of electrons in an electron cloud of temperature 



^ . 3 T 



T is 2kT or 7 720 volts. But more fast electrons will leave the cathode 



than slow^ ones, therefore, the average energy of the emitted electrons 



4 

 will be larger by a factor t . 



