# Kinetic energy of a free electron in a lattice

In summary, the task is to show that for a simple square lattice in 2-D with lattice spacing a, the kinetic energy of an electron at a corner of the first Brillouin zone (point A) is twice that of an electron at the midpoint of a side of the zone (point B). This question requires knowledge of Brillouin zones and their significance in solid physics. Understanding the concept of Brillouin zones can aid in solving this problem.

## Homework Statement

Show that for a simple square lattice (in 2-D) with the lattice spacing = a, the kinetic energy of a free electron at a corner (point A in the figure below) of the first Brillouin zone is higher than that of an electron at the midpoint of a side of the zone (point B in the figure below) by a factor of 2.
http://imageshack.com/a/img856/2000/zuci.jpg

No clue

## The Attempt at a Solution

So...like I have no clue about this...has anyone got a hint for me? I don't even know which equations to use...thanks guys...

Last edited by a moderator:
I guess this is quantum mechanics applied to solid physics?
In QM, electrons can't be totally at rest, which gives some sense to the question ...
You need first to remember everything about what a Brillouin zone is!
Then you simply need to translate "corner" and "midpoint" to answer this question.

This question contains little very physical meaning.
It is more a check that you know what a Brillouin zone is.
Therefore, I also suggest you the question yourself about why this concept (Brillouin zone) emerged and what its utility is.

## 1) What is the definition of kinetic energy of a free electron in a lattice?

The kinetic energy of a free electron in a lattice refers to the energy that an electron possesses due to its motion within a crystal lattice structure. It is a form of potential energy that is released when the electron moves from one energy state to another.

## 2) How is the kinetic energy of a free electron in a lattice related to its velocity?

The kinetic energy of a free electron in a lattice is directly proportional to its velocity. This means that as the velocity of the electron increases, so does its kinetic energy. The equation for kinetic energy is KE = 1/2 * mv^2, where m is the mass of the electron and v is its velocity.

## 3) What factors affect the kinetic energy of a free electron in a lattice?

The kinetic energy of a free electron in a lattice is affected by several factors, including the temperature of the lattice, the strength of the lattice bonds, and the mass and velocity of the electron. Additionally, any external forces acting on the electron, such as an electric field, can also affect its kinetic energy.

## 4) How does the kinetic energy of a free electron in a lattice impact its behavior?

The kinetic energy of a free electron in a lattice plays a crucial role in determining its behavior and properties. Higher kinetic energy levels can result in increased electron mobility and conductivity, while lower levels can lead to a more stable and localized electron state within the lattice.

## 5) Can the kinetic energy of a free electron in a lattice be measured experimentally?

Yes, the kinetic energy of a free electron in a lattice can be measured experimentally using various techniques such as electron spectroscopy or electron diffraction. These methods involve analyzing the energy and momentum of electrons within a lattice to determine their kinetic energy levels.

Replies
1
Views
696
Replies
1
Views
1K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
1
Views
3K