Perturbation matrix: free electron model on a square lattice

In summary, the conversation discusses the nearly free electron model in a 2D lattice, specifically a divalent 2D metal with a square lattice and one atom per primitive lattice cell. The periodic potential has two Fourier components, V10 and V11, corresponding to G = (1,0) and (1,1) with both being negative and mod(V10) > mod(V11). The secular equation is written down and an expression for electron energies at k = (pi /a, 0) is obtained. The potential between the two final states is expected to be zero, as they are separated by a reciprocal lattice vector (0,1).
  • #1
Fek
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Homework Statement


Nearly free electron model in a 2D lattice. Consider a divalent 2D metal with a square lattice and one atom per primitive lattice cell. The periodic potential has two Fourier components V10 and V11, corresponding to G = (1,0) and (1,1). Both are negative and mod(V10) > mod(V11).

Write down the secular equation and obtain an expression for the electron energies at k = (pi /a, 0).

Homework Equations

The Attempt at a Solution


Please see attached file (question also attached (part (i). I believe this is wrong but I cannot see what the expectation of the potential between the two final states can be other than zero (as they are separated by a reciprocal lattice vector (0,1).
 

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  • #3
All sorted thank you.
 

FAQ: Perturbation matrix: free electron model on a square lattice

1. What is a perturbation matrix?

A perturbation matrix is a mathematical representation of the changes in energy levels of a system due to the introduction of a perturbation, or disturbance. In the context of the free electron model on a square lattice, the perturbation matrix describes the changes in energy levels of the electrons when placed in a periodic potential.

2. How is the perturbation matrix calculated?

The perturbation matrix is calculated by solving the Schrödinger equation for the free electron model on a square lattice in the presence of a periodic potential. This involves using mathematical techniques such as perturbation theory and matrix algebra to determine the changes in energy levels due to the perturbation.

3. What is the significance of the free electron model on a square lattice?

The free electron model on a square lattice is a simplified theoretical model used to understand the behavior of electrons in a crystalline solid. It assumes that the electrons are not affected by the atoms in the lattice and can move freely within the crystal. This model has been successful in explaining many properties of metals, such as electrical conductivity and thermal conductivity.

4. How does the square lattice affect the behavior of electrons?

The square lattice creates a periodic potential for the electrons, causing them to form energy bands. In the free electron model, the electrons can move through the lattice with different energies, but the periodic potential affects their movement and can lead to the formation of energy bands with gaps in between them.

5. What are the applications of the perturbation matrix in physics?

The perturbation matrix has many applications in physics, particularly in the study of quantum mechanics and solid state physics. It is used to understand the effects of perturbations in various systems, such as atoms, molecules, and crystals. It is also used to calculate properties of materials, such as electrical and thermal conductivity, and to design new materials with desired properties.

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