ELECTRON MICROSCOPY 



a. b. 



Fig. 1. Methods of achieving selected diffrac- 

 tion images: (a) using normal circular aperture 

 off-centre, (b) using annular aperture. 



objective aperture diaphragm, simply by 

 positioning the aperture off-center so that 

 it accepts the desired diffracted beams. This 

 method is generally used when the specimen 

 consists of a few isolated single crystals, giv- 

 ing rise to a diffraction pattern consisting of 

 a number of isolated spots. The second 

 method is generally used with an object 

 containing a large number of minute crys- 

 tals the diffraction pattern of which consists 

 of concentric rings. An annular aperture 

 concentric with the optical axis of the micro- 

 scope is adjusted along the optical axis in 

 the space between the object and the back 

 focal plane of the objective lens to accept 

 the desired cone of electrons corresponding 

 to a particular ring of the diffraction pat- 

 tern. 



Resolution and Optimum Operating 

 Conditions. For simplicity consider the 

 object to be a disc-shaped crystal of diam- 

 eter a, illuminated by a coherent electron 

 beam normal to the plane of the crystal. 

 Assume the crystal to be so orientated that 

 it gives rise to a Bragg reflection. This will 

 be of angular aperture 2X/a, where X is the 

 wavelength. If the lenses are perfectly cor- 

 rected so that the resolution is limited only 

 by diffraction then the resolution d is given 

 by the well-known formula 



d = 0.6X/N.A. 



In the electron microscope where apertures 

 are small the semi-angular aperture is very 

 nearly equal to the numerical aperture so 

 that 



d = 0.6X/X/a = O.Ga, 



i.e., three-fifths of the crystal diameter. 



Decreasing the coherence of the illumina- 

 tion until the distance I in the object over 

 which the illumination is coherent becomes 

 smaller than the crystal diameter makes the 

 angular aperture of the diffracted beam 

 2\/l, and the resolution d = O.Ql. 



The radius of the circle of confusion in 

 the Gaussian plane due to spherical aberra- 

 tion alone is 





{dhH)n 



where C« is the spherical aberration constant 

 and dhki the interplanar spacing of the crys- 

 tal planes responsible for the Bragg reflec- 

 tion (1). Owing to the marked spherical 

 aberration of electron lenses the Gaussian 

 plane is not the plane of best focus for se- 

 lected diffraction images. In the plane of 

 best focus the radius of the circle of confusion 

 due to spherical aberration is given approxi- 

 mately by 



r = Cs(\/iy, 



for small Bragg angles. 



The combined effects of diffraction and 

 spherical aberration result in a resolution 



d = V(0.602 + iC.(X//)'p. 



Best resolution is obtained when rf is a mini- 

 mum, i.e., when 



A typical value of Cs would be lO^A and 

 for X, 5 X 10~2A, giving an optimum value 

 for / of 14A. In most microscopes this value 

 of / is approached at critical illumination. 

 In practice best resolution is usually attained 

 very close to critical illumination, where the 

 image deteriorates. This is probably due to 

 the presence of astigmatism in the objective. 



252 



