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Homework Help: Kronig-Penney model

  1. Mar 6, 2008 #1
    1. The problem statement, all variables and given/known data

    My homework has to do with the Kronig-Penney model for an electron moving in a 1-D periodic lattice. I already figured out part A, which asked for me to show that E(k) approached the energy of a free electron for electrons with high energies in the lattice.

    Part B is asking: Find an expression for the lowest possible energy of an electron. Why isn't this zero?

    Part C is asking : find an expression for the band gap at k = pi/d.

    2. Relevant equations

    [tex]cos(kd)=cos(k_{1}d)+P\frac{sin(k_{1}d)}{k_{1}d}[/tex]

    3. The attempt at a solution

    I'm having a lot of trouble with the implicit nature of this equation in this problem. For part B, I know that cos(kd) has to be between +1 and -1, but at lower values of E, the right hand side of the equation is greater than 1, resulting in a band. That's why there is some lowest possible energy above zero. I'm just stuck on showing this numerically.

    For Part C, I got
    [tex]-1=cos(k_{1}d)+P\frac{sin(k_{1}d)}{k_{1}d}[/tex]
    and then
    [tex]1+P\frac{sin(k_{1}d)}{k_{1}d}=cos(k_{1}d)[/tex]

    but after that I'm stuck and I'm not sure what kind of expression I'm supposed to find for the band gap.
     
  2. jcsd
  3. Mar 6, 2008 #2
    I'm not sure what's hiding in your k1's and P's (is k the same as k1?), but there's a really good treatment of this in McKelvey's Solid State Physics (section 8.3 in my version).
     
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