Kronig-Penney Model Homework Solution

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Homework Statement

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.

Homework Equations

$$cos(kd)=cos(k_{1}d)+P\frac{sin(k_{1}d)}{k_{1}d}$$

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
$$-1=cos(k_{1}d)+P\frac{sin(k_{1}d)}{k_{1}d}$$
and then
$$1+P\frac{sin(k_{1}d)}{k_{1}d}=cos(k_{1}d)$$

but after that I'm stuck and I'm not sure what kind of expression I'm supposed to find for the band gap.

 

[klile82]

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).