Possibilities of Future Development 119 



aberration introduced by the first, condensing lens of the system. 

 An approximate explanation is contained in the bottom figure 

 of figure 42. In a homogeneous electron cloud the radial electric 

 field intensity is proportional to the radius. Therefore, it acts 

 very nearly like an ordinary spherical lens. The parabolic dis- 

 tribution superimposed on it, however, has an effect approxi- 

 mately equivalent to a lens with a surface figured according to 

 a parabola of the fourth order. Such correcting plates are used 

 in various optical systems, of which the Schmidt camera is the 

 first, and the best known. 



The compensation of spherical aberration by space charges 

 can be also understood by considering Scherzer's formula for the 

 aberration of a purely magnetic lens. In our notation 



^^ = klkS [(h' + H0V (<)V 4^H^] r^^^m 



r denotes a trajectory, calculated from the paraxial equation 

 (41), which starts from the focal point with / = 1, i.e., at 45°. 

 The integration has to be carried out over the axis z from the 

 object to the image. The expression can never be zero, as the 

 integrand is a sum of positive squares. But it can be seen from 

 equation (41) that a negative space charge has the same effect 

 as an imaginary magnetic field, therefore, the integrand of (45) 

 can well become negative in space charge lenses 



Equation (45) shows that the short focal length of the sug- 

 gested lens system does not by itself ensure a reduction of the 

 spherical aberration. The factors outside the bracket are of the 

 same order of magnitude as in the usual uncorrected lenses, 

 therefore improvements can be effected only by reducing the 



factor in brackets. This is bound to be hard work, as the resolu- 



1 

 tion limit depends on (C/)^, therefore, a compensation of the 

 spherical aberration to 1 per cent would reduce the resolution 

 limit to little less than one third of the uncorrected value. But 

 it may be remembered that the optical microscope has improved 

 since Abbe's time by less than a factor 2, and yet the progress 



