58 The Electron Microscope 



disks would appear to merge into one is somewhat smaller. If 

 we assume this factor to be 0.67 and use equation (13) for the 

 de Broglie wavelength, we can write the resolution limit due to 

 diffraction alone in the simple form 



rfA = ^ A (22) 



Fluctuations AV of the driving voltage V produce in the mag- 

 netic microscope, but not in the electrostatic microscope with 

 unipotential lenses, as we have already seen, a disk of diameter 



AV 



d, = 2a/ -y- (20) 



If two, or all three of these errors are of approximately equal 

 size, it is a matter of some difficulty to estimate the resulting 

 error. In optics, Max Born has solved this difficult problem ^^ 

 by treating geometrical and diffraction errors with a unified 

 wave-theoretical method. It turns out that it is legitimate in first 

 approximation to superimpose the intensities calculated sepa- 

 rately from geometrical optics and from diffraction, though com- 

 bination terms will also arise. To simplify matters, we follow 

 a suggestion made by von Ardenne ^^ to consider the three disks 

 as bell-shaped probability distributions and to calculate the re- 

 sulting error as the geometrical sum of the three diameters, i.e., 

 as the square root of the sum of their squares. If we accept 

 this, it is evident that the three errors will have to be of the same 

 order of magnitude in the optimum, or most economical design. 



It can be seen from equations (21) and (22) that the spherical 

 aberration and the diffraction error vary in opposite directions 

 with variations of the aperture a. The sum of their squares 

 will be a minimum at an aperture which makes the diffraction 



error V^ = l-^^ times the spherical aberration. This gives for 

 the optimum aperture an equation 



, 5.8 X 10-« 



4 — ^_ /23) 



