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:

F = qE; Mₑ a = qE, where F = ma and Mₑ is the mass of an electron; a = qE/mₑ → Eqn 1

v - u = at, where u = 0 as overall movement cancel out; V_f bar = α τ, where V_f bar is the average velocity (final); V_f bar = a λ/(⟨v⟩) → Eqn 2

, 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)

V_d = V_f bar; V_d = qE/m (λ/(⟨v⟩))

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