IMAGE FORMATION MECHANISM 



about the specimen except the phase. 

 Through the use of proper optical equipment, 

 the image can be reconstructed from such a 

 hologram in a process called "microscopy by- 

 reconstructed wave fronts." Although prac- 

 tical difficulties until now made it impossible 

 to reach high resolutions by this method, its 

 existence amply shows the importance of dif- 

 fraction in the image forming process. 



At extremely high resolution, crystal lat- 

 tices and other periodic structures can be ob- 

 served provided that the aperture of the sys- 

 tem allows the zero order spectrum to pass 

 together with at least one of the first-order 

 diffraction spectra. This is in agreement with 

 the Abbe theory of image formation in light 

 optics. Moire patterns are due to the com- 

 bination of a doubly diffracted beam, orig- 

 inating on overlapping crystals, with the in- 

 cident beam. 



The intensity at any point of the image is 

 governed by the number of electrons which 

 have been scattered within the solid angle 

 formed by the aperture of the objective lens. 

 This number can be WTitten as 



for the elastically scattered electrons is given 

 by 



where 



/ = loe- 



N<rx 



(5) 



Z is the atomic number of the element con- 

 stituting the specimen, / is the so-called 

 form factor, ?? and cp are the angles defining 

 the solid angle, mo is the rest mass of the 

 electron, e is its charge, and h again Planck's 

 constant. 



For the description of the inelastic process, 

 Lenz adds an inelastic scattering function, S, 

 to the right side of equation (5) and obtains 



^ = ^ (Z - /)^ + S (6) 



in this equation, {5'") is modified so that 



27r 



where /o is the nmnber of electrons in the 

 primary beam before interacting with the 

 specimen, x is the thickness of the specimen, 

 A^ is the number of scattering atoms per unit 

 volume, and a is the scattering cross section. 

 The scattering cross section is essentially 

 made up of two components 



/-©' 



(6') 



and the inelastic scattering function is 

 S = Z -•'- for ^ « 1 



(6") 



<r = ffe + ai 



(4) 



where o-g is the elastic scattering cross sec- 

 tion, i.e., the scattering cross section for all 

 electrons which have not suffered any change 

 of energy in the scattering act; o-j is the in- 

 elastic scattering cross section i.e., the cross 

 section for those electrons which suffered an 

 energy loss in the scattering act. 



There have been a number of attempts to 

 calculate these cross sections. As an example, 

 we follow the treatment of Lenz as given in 

 a recent paper. This calculated cross section 



provided Wentzel's approximations are cor- 

 rect. In equation (6'), E represents the 

 energy of the incident electron and AE the 

 energy lost in the scattering act. In earlier 

 calculations for this last quantity, half of the 

 average ionization energy was assumed. In 

 some cases, it may be more justified to substi- 

 tute, instead of the ionization energy, the 

 measured characteristic energy losses in the 

 specimen. 



Attention should be called to the fact 

 that calculations of cross sections at best are 

 approximations. Accurate wave functions 

 are generally unavailable and, in their ab- 

 sence, all calculations have to be taken with 



161 



