132 The Electron Microscope 



If an electron is scattered by a hydrogen atom, we know with 



a probabiHty amounting almost to certainty that it will be de- 



7 

 fleeted by an angle 6^^ less than about — == radian. For 10,000 ev 



electrons this is about 4° and the uncertainty 26^ is 8°. The 

 corresponding uncertainty in transversal momentum is 2mv6m, 

 and with this is associated, by Heisenberg's principle an uncer- 

 tainty dn in the position from which the electron appears to 

 originate. This is given by 



dH.2mvC = h (55) 



With the de Broglie wavelength 



mv 



this can be written 



dH = 



26 



m 



which is the same as Abbe's relation, with the diflference that 

 dm. is not the aperture of the objective, but the half-angle of 

 diffraction at the object. This explains the difficulty which we 

 have met in the case of beams not filling the aperture. The 

 diffraction error is no error at all, but a measure of the size of 

 the object. This becomes even clearer if we substitute the values 

 of A and ^m 



12.2 



^^=2^;:=^- -0.87 A (56) 



This is the apparent diameter of the hydrogen atom. It is a 

 little less than the Bohr diameter of the older quantum theory, 

 the diameter of the nearest orbit on which the electron was sup- 



