106 The Electron Microscope 



plate is therefore in the same plane as the object. The error 

 disk may have a diameter d, which is the measure of inde- 

 terminacy of the position of an electron when it arrives at the 

 plate. By Heisenberg's principle there is a corresponding in- 

 determinacy Amz/p in its radial momentum mVry connected with 

 d by 



d.AmVj. ^ h (31) 



where h is Planck's constant. This, however, means that the 

 zone r of the lens or mirror, from which this electron came is 

 known only with an inaccuracy 



Ar = |-^ (32) 



which, however, cannot be larger than the physical aperture 2a/ 



of the lens, as we know that the electron has passed the aperture. 



By the indeterminacy contained in (31) and (32) we have 



replaced the zvave picture and we can go on considering the 



electron as a particle of which we know certain data within 



certain limits only. We can therefore equate d to the error in 



dsc, which would be produced by an electron straying by ±^Ar 



from the zone which it should pass in order to arrive at the 



r 

 geometrical image point. Writing a = "T we obtain by differen- 



. . . E 

 tiating (29) and by eliminating Trby means of (30) 



d = 



/3Cr^- 2E\ ^ AE /2Cr~\ ^ AE,_, 



(— j2- — ^lAr — 2r^= (— p-JAr — 2r^(33) 



For completeness we have added the error arising from an 

 electron passing a zone with the wrong energy, but neglect it in 

 the following. From (31), (32) and (33) we obtain Ar 



= \/-^ (34) 



Ar 



2Cr2 



