Geometrical Electron Optics 11 



transversal spherical aberration. In fact, the bundle has its 

 smallest cross section not at p but at m, which is called the disc 

 of minimum confusion. Its radius is one quarter of the spherical 

 aberration. When an electron lens is focused, it is always m, not 

 p, which is made to coincide with the screen or plate. 



Scherzer ^ has proved the important theorem that in electron 

 lenses, whether electrostatic, magnetic, or combined, the spherical 



Fig. 4. Spherical aberration 



aberration can never be eliminated as long as there are no space 

 charges or currents in the space traversed by electrons. All 

 lenses used to date in electron optics fall into this class. The 

 spherical aberration has always the sign as shown in figure 4, 

 i.e., the strength of the lens increases with the angle a. In optics 

 this is called an undercorrected lens. Corrected electron lenses 

 are impossible. 



Scherzer has proved also that the same fundamental difficulty 

 exists in the case of chromatic aberration which is illustrated in 

 figure 5. The faster electron (dotted line) will intersect the axis 

 always beyond the slower electron. This defect cannot be cor- 

 rected, because dispersing electron lenses cannot be realized 

 with the means employed in present-day electron optics. We 

 have proved this in the case of magnetic lenses. In electrostatic 

 lenses, it is possible to realize a slight dispersing eiifect in a part 

 of the field, but this is always more than compensated by the 

 condensing efifect. Quantitative examples will be given later. 



The spherical and the chromatic aberrations are the only lens 

 errors which exist at axial and at extra-axial points of the 

 image field. The other defects — coma, astigmatism, image curva- 

 ture, and distortion — are zero at the axis, and manifest them- 



