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THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



gap^ is A£ = 2 I Kg I where Vg is the Fourier coefficient of potential. The 

 discontinuities in energy are the Brillouin zone boundaries and form a 

 family of polyhedra in reciprocal or K space. For example, a simple square 

 lattice gives rise to a square reciprocal lattice as seen in Fig. 3. One reciprocal 

 lattice point is taken as the origin, 0, and the remainder of the lattice is 



generated by the vector g where \ g\ = -, the reciprocal of the cell constant 



of the original, direct lattice. The Brillouin zone boundaries are the per- 

 pendicular bisectors of the reciprocal lattice vectors^ and define the series 

 of zones shown in Fig. 3a. Whenever the incident electron wave vector, K^ 



AE=2|Vg| 



■^K 



|K| = |K+277-g| 



Fig. 2 — Plot of energy, E, vs. wave number \K\, along K vector showing discontinuity 

 when \K\ = \K -\- lirg] or when Bragg condition is realized. 



terminates on a Brillouin zone boundary, a diffracted wave is possible. 

 However, when the boundary conditions at the surfaces of the crystal are 

 apphed, it turns out that for a fixed total energy, jE, there are two incident 

 crystal wave vectors kI and K\ which must be considered. Consequently 

 there are also two diffracted wave vectors KI and KI with kI = kI + lirg 

 and Kg = Ko -\- Irg as shown in Fig. 3b. KI and KI are related by a beat 

 wave vector AK or K], = kI -{- AK. The net result is two waves of slightly 

 different wave length traveling nearly parallel which may undergo inter- 

 ference. This beating of the diffracted waves makes itself known by passing 

 the energy back and forth between the incident and diffracted beams. This 

 is the motivation for the name ''dynamical" theory. 



The intensity of a diffracted beam for the case when the incident wave 

 vector terminates near a Brillouin zone boundary but far from an edge or 

 corner is found to be^: 



' This treatment is the case of loose binding which is applicable for fast electrons. It 

 turns out that for the valence electrons in a crystal, this approximation is not very good 

 and that the value AE - 2\Vg\ is not correct. 



• L. Brillouin, "Wave Propagation in Periodic Structures," McGraw-Hill Book Com- 

 pany, Inc., New York (1946). 



