# Attractive Kronig-Penney Potential Dispersion Relation Confirmation

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• essil
In summary, the dispersion relation for the attractive Kronig-Penney potential can be found on Wikipedia. However, the calculation of the determinant, as well as literature discussing the attractive variant of the potential, is scarce. The book "Solid State Physics" by Ashcroft and Mermin may provide helpful insights, as they discuss the Kronig-Penney model for a potential barrier. The transmission coefficient for the potential well can also be found in other sources, but reproducing the dispersion relation may be challenging. The role of ##\delta## in this problem is unclear, and it is uncertain if the equation ##\frac{cos(Ka+\delta)}{|t|}=cos(ka)## can be used.
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.

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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.

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?

essil said:
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.

## 1. What is the Kronig-Penney potential and its significance in condensed matter physics?

The Kronig-Penney potential is a mathematical model used to describe the periodic potential experienced by a particle in a crystal lattice. It is an important concept in condensed matter physics as it helps explain the behavior of electrons in solid materials and their electronic band structure.

## 2. How does the dispersion relation for the Kronig-Penney potential confirm the existence of energy bands?

The dispersion relation is a mathematical relationship between the energy and momentum of a particle. In the Kronig-Penney potential, it shows that the energy bands are formed due to the periodicity of the potential, resulting in allowed energy ranges for particles with certain momenta. This confirms the existence of energy bands in a crystal lattice.

## 3. Can the Kronig-Penney potential dispersion relation be applied to other types of potentials?

Yes, the Kronig-Penney potential dispersion relation is a general concept and can be applied to other potentials as well. However, the specific form of the dispersion relation may vary depending on the type of potential being studied.

## 4. How does the Kronig-Penney potential dispersion relation explain the behavior of electrons in a crystal lattice?

The dispersion relation for the Kronig-Penney potential shows that the allowed energy ranges for electrons in a crystal lattice are determined by the periodicity of the potential. This explains the formation of energy bands and the behavior of electrons in a crystal lattice, such as their ability to conduct electricity and their response to external electric fields.

## 5. Are there any limitations to the Kronig-Penney potential dispersion relation?

One limitation of the Kronig-Penney potential dispersion relation is that it assumes an idealized, perfectly periodic crystal lattice. In reality, most crystals have some degree of imperfections and disorder, which can affect the behavior of electrons and may not be accurately described by the dispersion relation.

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