# Kronig-Penney Model Homework Solution

• lightfalcon
In summary, the conversation revolved around solving a problem involving the Kronig-Penney model for an electron moving in a 1-D periodic lattice. Part A was already solved, which required showing that the energy of electrons with high energies in the lattice approached that of a free electron. Part B asked for an expression for the lowest possible energy of an electron and for an explanation of why it is not zero. The discussion also touched on the implicit nature of the equation and the difficulty of showing this numerically. Part C involved finding an expression for the band gap at a specific value of k. The conversation ended with a mention of a helpful resource for understanding this problem.
lightfalcon

## 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.

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).

## 1. What is the Kronig-Penney Model?

The Kronig-Penney Model is a theoretical model used in solid state physics to describe the behavior of electrons in a periodic potential. It was developed by physicists Ralph Kronig and Walter Penney in the 1930s.

## 2. What is the purpose of the Kronig-Penney Model?

The Kronig-Penney Model is used to study the energy levels and band structure of electrons in a periodic potential. It helps to explain various properties of crystalline materials, such as electrical conductivity and optical properties.

## 3. How does the Kronig-Penney Model work?

The Kronig-Penney Model uses the Schrödinger equation to describe the motion of electrons in a periodic potential. It considers the potential energy of the electrons and the kinetic energy of their wave-like behavior to determine the allowed energy levels and band structure.

## 4. What are some applications of the Kronig-Penney Model?

The Kronig-Penney Model has been used to study the properties of various materials, including semiconductors, metals, and insulators. It has also been applied in the development of electronic and photonic devices, such as transistors and solar cells.

## 5. Are there any limitations to the Kronig-Penney Model?

While the Kronig-Penney Model is a useful tool for understanding the behavior of electrons in periodic potentials, it has some limitations. It assumes a perfect crystal lattice and does not take into account factors such as impurities and defects, which can affect the material's properties. Additionally, it does not consider the effects of electron-electron interactions.

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