# I Kronig-Penney model

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1. Dec 29, 2016

### essil

Hi all!

Can anyone confirm (or point me to literature) that the dispersion relation for the attractive Kronig-Penney potential is correctly given on Wikipedia (https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice):
$$cos(ka) = cos(\beta b)cos(\alpha (a-b))-\frac{\alpha ^2 + \beta ^2}{2\alpha \beta} sin(\beta b) sin(\alpha (a-b))$$
I have been unsuccessful at calculating the determinant (also given on Wikipedia) despite multiple tries, and was unable to find literature which deals with the attractive variant of the potential.

Last edited: Dec 29, 2016
2. Dec 29, 2016

Ashcroft and Mermin treat the Kronig-Penney model for a potential barrier, but not an attractive potential. I think the algebra for working to the solution is likely to be similar, so you might want to check out the book "Solid State Physics" by Ashcroft and Mermin. I found the discussion on pp.148-149 of the copy that I have.

3. Dec 29, 2016

### essil

Thank you for your reply. I have tried to modify the solution from Ashcroft and Mermin, but I am not very experienced in the field yet and I do not quite understand how the fact that my potential is attractive (negative) changes the calculations made in the book. I have used the equation

$\frac{cos(Ka+\delta)}{|t|}=cos(ka)$, where a is the period of the potential. Also I've found the transmission coefficient for the potential well here: http://www.physicspages.com/2012/08/05/finite-square-well-scattering/ (solution from Griffiths), but couldn't reproduce the dispersion relation. This might be a stupid question, but what is $\delta$ in this problem? Can I even use the equation above in this form?

4. Dec 29, 2016