# Nearly free electron model: band gap

1. Feb 14, 2016

### Telemachus

I was trying to determine the bandgap in the nearly free electron model. I'm having trouble to determine the band gap bewteen the second and the third band. Its a one dimensional problem.

So, the central equation reads:

$\displaystyle \left [ \frac{\hbar}{2m} (k-G)^2-E \right ]c_{k-G}+ \sum_{K'} U_{G-K'} c_{k-K'}$

So, I want to determine the band gap which would be at the point $k=2{\pi}{a}$ in the extended diagram (the band gap between the second and the third band).

And my $K'$ I think that should be: $K'=\frac{2\pi}{a},\frac{4\pi}{a}$. When I carry on the whole calculation I get something that don't make any sense to me, because I don't find any degeneracy on the energy, and I think thats wrong.

For $G=\frac{2\pi}{a}$ I get:

$\displaystyle \left [ \frac{\hbar}{2m} (k-\frac{2\pi}{a})^2-E-U_0 \right ]c_{k-\frac{2\pi}{a}}+ U_{\frac{2\pi}{a}}c_{k-\frac{4\pi}{a}}$

And for $G=\frac{4\pi}{a}$:

$\displaystyle \left [ \frac{\hbar}{2m} (k-\frac{4\pi}{a})^2-E-U_0 \right ]c_{k-\frac{4\pi}{a}}+ U_{-\frac{2\pi}{a}}c_{k-\frac{2\pi}{a}}$

I don't know what I'm doing wrong.

Thanks in advance.

2. Feb 19, 2016

### Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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