Free Electron Theory Of Solids

Electrons can be visualised as rattling around insde the metal, randomly changing direction when they collide with positive ions, much like a ball in a pinball machine. The average velocity and displacement of the electron along any direction over a period of time is zero because electrons are equally likely to be moving in any direction on their ‘random walk’.

Imagine that a conductor under investigation is part of an electrical circuit. Imagine closing a switch that connects a battery that applies an electric field E to the conductor in the positive x-direction. This exerts a force F = qE on each electron, which will accelerate according to Newton’s second law:

$$\begin{aligned} F \, &= qE \\ M_{e} a \, &= qE, \, \text{where} \, F = ma \, \text{and} \, M_{e} \, \text{is the mass of an electron}\\ a \, &= \frac{qE}{m_{e}} \, \rightarrow \text{Eqn 1} \end{aligned}$$

$$ \begin{aligned} v – u \, &= at, \, \text{where} \, u = 0 \, \text{as overall movement cancel out} \\ \bar{V_{f}} \, &= \alpha \tau, \, \text{where} \, \bar{V_{f}} \, \text{is the average velocity (final)} \\ \bar{V_{f}} \, &= a \frac{\lambda}{ < v >} \, \rightarrow \text{Eqn 2} \end{aligned}$$

, where
λ is the mean free path, (average distance that electrons travel between collisions)
τ is the average time interval between successive collisions,
< V > is the average speed of the electrons.

Typical Speed between collision (drift velocity) is: (by combining eqn 1 and 2)

$$\begin{aligned} V_{d} \, &= \bar{V_{f}} \\ V_{d} \, &= \frac{qE}{m} \left( \frac{\lambda}{ < v > } \right) \end{aligned}$$


Back To Solid State Physics

Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.