Undergrad Attractive Kronig-Penney Potential Dispersion Relation Confirmation

[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.

 

[Charles Link]

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.

 

[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?

 

[Charles Link]

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?

I am also on a learning curve with material such as this. Many years ago, I had a couple of graduate level solid state physics courses, but I don't have very much expertise in that area.