Theory of the Electron Microscope 149 



Therefore, the second term is either not calculated at all, or not 

 given, and it is just this term which is of the greatest importance 

 in microscopy. 



It has been stated before and will be proved now that for the 

 purpose of microscopic image formation not only the inelastically 

 but also the elastically scattered electrons can be considered as 

 incoherent. It is possible to decide this without going into the 

 somewhat obscure question of coherence length or length of wave 

 trains of electrons, though obviously the answer depends on this 

 to some extent. If the wave trains were of infinite length, i.e., if 

 the energy constant, E, of the electrons were defined without 

 uncertainty, any difference between E and E' would mean in- 

 coherence. But if the wave trains are extremely short, e.g., con- 

 sisting only of a few w'aves, incoherence will arise only if we can 

 expect phase shifts between electrons in the beam of the order Itt. 

 This is therefore the niinirnutn condition for incoherence, and 

 will be applied in the following. 



If an electron of mass m collides with an atom of mass M and 

 suffers a deflection by an angle 0, its energy loss will be ap- 

 proximately 



and its relative change of wavelength will be half as much, 



^ = W— (62) 



A ^ M • ^ ^ 



A phase shift of tt will be produced on the length L between 

 object and photographic plate between the original beam and a 

 deflected electron if 



AAL _ 



\2 — 2 



'- (63) 



Combining this with (62) we obtain an upper limit for the di- 

 vergence inside which the phase shift, and therefore, by the 



