306 BELL SYSTEM TECHNICAL JOURNAL 



spaced obstacles (the atoms in a crystal lattice, for instance) diffraction 

 patterns are obtained, similar to those which may be obtained with waves 

 of X-rays or light. It appears that electrons get around sufficiently small 

 objects just as sound waves get around telephone poles, automobiles, and 

 even houses, and if the objects are sufficiently small their effect on the 

 electron flow will either be absent or will consist of a few ripples which are 

 meaningless in disclosing the shape or size of the object. 



The electron wave-length, which varies inversely as the momentum of the 

 electron, may be simply expressed in terms of the energy V in electron volts. 

 A simple non-relativistic expression which is only 5% in error at 100,000 

 volts (a high voltage for electron microscopes), is* 



X - Vl50/F X 10~' cm (1) 



Thus for 30,000-volt electrons the wave-length is 7 X 10"^° cm or about 

 1.4 X 10~' times the diameter of a hair and 1.2 X 10"'' times the length 

 of a wave of yellow light. 



In terms of this wave-length X and the half angle of the cone of rays 

 accepted by the objective, a, we can express the distance d between point 

 objects which can just be distinguished in an electron microscope. This 

 distance is 



d = .61X/sin a (2) 



For small values of a 



la = 1/f (3) 



where / is the well known photographic / number, the ratio of the focal 

 length to the lens diameter. We see that, just as with cameras, the smaller 

 the/ number the better. In electron microscopes a small/ enables us to 

 distinguish smaller objects. 



Aberrations 



Just as in cameras, the limitation to the / number is imposed by lens 

 aberrations. But in electron lenses the aberrations are much more severe. 

 Why is this so? Because with electron lenses we have less freedom of design 

 than with optical lenses. 



Consider an electric lens. The quantity analogous to the index of 

 refraction for light is the square root of the potential with respect to the 

 cathode. Now suppose that with a light lens we know the index of re- 

 fraction at every point along the axis. Suppose, for instance, that the in- 

 dex of refraction is 1 everywhere along the axis except for a space L long 



* The relativistic expression is * 



X = (ViSO/F/Vi -}- .98 X 10-6 F) X 10-8 cm 



