Understanding Free Electron Kinetic Energy on a Square Lattice

Click For Summary
The discussion focuses on calculating the kinetic energy of a free electron on a square lattice, specifically comparing the energy at the corner of the first Brillouin zone to that at the midpoint of a side face, with the expectation that the former is twice as high. Participants clarify that the positions refer to k-vectors in k-space, emphasizing the relevance of the lattice structure to the energy spectrum. The tight binding model is suggested as a necessary framework for understanding the electron behavior in this context. The geometry of the lattice is crucial for determining the shape of the Brillouin zone and the resulting energy levels. Overall, the conversation aims to clarify the relationship between lattice structure and electron kinetic energy in solid-state physics.
aaaa202
Messages
1,144
Reaction score
2

Homework Statement


Show for a simple square lattice that the kinetic energy of a free electron is higher at the corner of the first zone than at the midpoint a side face by a factor of 2.


Homework Equations


Simple geometry.


The Attempt at a Solution


I think I know how to solve, but I am not completely sure I understand the question. The position of the electron refers to the position of its k-vector in k-space right? Well this is not real space, and I can't really see how this has anything to do with the lattice being square in real space.
 
Physics news on Phys.org
I suspect the hidden assumption here is that you should use a tight binding model with nearest neighbor hopping on the lattice given. So in your case the square lattice is specifying the shape of the Brillouin zone and telling you the energy spectrum.

Does this help?
 

Similar threads

Some questions about reciprocal lattice vectors
  • · Replies 3 ·
Replies
3
Views
2K
Kinetic energy of a free electron in a lattice
  • · Replies 1 ·
Replies
1
Views
4K
Finding Energy in FCC Lattice Using Weak Potential Method
  • · Replies 1 ·
Replies
1
Views
1K
Perturbation matrix: free electron model on a square lattice
  • · Replies 1 ·
Replies
1
Views
2K
Energy gaps for quasi-free electrons in a 2D lattice
  • · Replies 1 ·
Replies
1
Views
1K
The kinetic energy of the electrons outside the FBZ
  • · Replies 1 ·
Replies
1
Views
2K
Solving for A in a Simple Cubic Lattice
  • · Replies 4 ·
Replies
4
Views
2K
Solved: Kittel Problem 7.1 About Kinetic Energy in Square Lattice
  • · Replies 1 ·
Replies
1
Views
5K
Figuring out Bravais lattice from primitive basis vectors
  • · Replies 9 ·
Replies
9
Views
4K
How Is the Polyacetylene Chain Structured in a 1D Lattice Model?
  • · Replies 12 ·
Replies
12
Views
2K