26_562ln08

26_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.62 Spring 2008 Lecture summary 26 Page 1 Band Theory of Solids In the free electron theory we ignored all effects of the nuclei in the lattice, utilizing a particle-in-the-box approach sans a potential. In the band theory of solids considered here, we include a very simple potential representing the nuclei that leads to “bands” of potentially occupied states that are separated by gaps. The forces on the electrons are the regularly spaced, positively charged, essentially stationary nuclei and they are represented by delta functions. Dirac Comb Potential : The simple periodic structure depicted below reproduces many interesting aspects of the band theory of metals. It is referred to as a Dirac comb , whereas a more sophisticated model, the Kronig-Penny model, employs a comb of rectangular shapes. The actual shape is not so important for our purposes. The potential periodic repeats itself after some distance a (the lattice spacing) so that we can write the potential as V ( x ) = V ( x + a ) This is the same idea that we used previously (the approach of Born and von Kármán ) in our treatment of Debye solids and in the free electron theory of metals. Bloch’s Theorem - Felix Bloch [ Z. Physik 52 , 555 (1928)] suggested an ingenious approach to treating this problem that is today know as Bloch’s theorem. For the potential like the periodic comb above, the solution to the time independent Schrödinger equation (TISE)
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5.62 Spring 2008 Lecture summary 26 Page 2 ! ! 2
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26_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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