ELECTRON TRANSMISSION THROUGH THIN METAL SECTIONS 



869 



diffract the incident beam of wave length X through an angle 26. d is the 

 interplanar spacing. The numerical aperture of the electron lens is such that 

 in general 26 >a^^ with the result that the diffracted beams are removed 

 from the optical system and not focused in the image plane. In Fig. 1, crys- 

 tals 1 and 3 are suitably oriented to diffract and so will appear dark in the 

 final image since electrons have been removed from these regions by Bragg 

 reflection. The calculations of the intensities for this case (the Laue case) 

 were first made by Bethe.^ A similar treatment^ -^ employing the zone theory 

 of crystals can be given which yields the same final results. The procedure 

 consists in solving the Schrodinger equation for an electron moving in the 



INCIDENT BEAM 



CONJUGATE 

 FOCAL PLANE 



POLYCRYSTALLINE 

 FILM 



OBJECTIVE 

 DIAPHRAGM 



Fig. 1 — Diffraction of electrons outside objective aperture by suitably oriented crystals 

 in polycrystalline film. Crystals 1 and 3 would appear dark in final image. 



periodic potential inside the crystal and then fitting the solutions so ob- 

 tained to the plane wave solution found for the vacuum incident and dif- 

 fracted waves. The first result of the solution inside the crystal is that the 

 total energy of the electron, E, is not a continuous function of the wave 



27r 

 number K {\ K\ = — ) as it is in field-free space, but exhibits discontinuities 



A 



as illustrated in Fig. 2. These discontinuities occur whenever the Bragg or 

 Laue condition is satisfied; i.e., if g is a vector of the reciprocal lattice, 

 then discontinuities in the E ys K curve occur when \K\ = \ K -{- 2Tg\ 

 which is equivalent to the Bragg formula. The magnitude of the energy 



^H. A. Bethe, Ann. d. Physik 87, 55 (1928). This treatment was intended to explain 

 the results of Davisson and Germer which were published about a year before. Had Bethe 

 examined the behavior of the total energy in his solution of the Schrodinger equation he 

 would have discovered the band theory of crystals. 



