## Introduction

We can categorize Free Electron Theory as follow:

- Classical free electron theory
- Quantum free electron theory

Let us start our discussion with Classical theory

**Classical Free Electron Theory**

Drude and Lorentz gave classical free electron theory of metals theory in 1900. Another name for this theory is Drude –Lorentz theory of metals.

Classical theory says that the metals having free electrons obey the laws of classical mechanics.

**Assumptions **

- Just like the molecules of a perfect gas, the valence electrons of atoms are free to move about the whole volume of the metal.
- As the free electrons move randomly and keep on colliding either with other free electrons or with positive ions fixed to the lattice. But all these collisions does not result in any kind of energy loss thus are elastic in nature.
- The momentum of free electrons vary according to the laws of classical kinetic theory of gases.
- Velocity of electrons is in accordance with classical Maxwell-Boltzman distribution of velocities.
- When metal is under the effect of an electric field, the free electrons get accelerated. However the direction is opposite to that of the direction of the applied electric field.
- The mutual repulsive forces between the electrons are not taken into account. So they can move in all the directions with all possible velocities.
- When there is no field, the value of energy associated with an electron at temperature T is kT.

**Success **

- It verifies ohm’s law.
- This explains electrical conductivity and thermal conductivity of metals.
- It derives the Widemann – Franz law. Widemann – Franz law gives the relation between electrical and thermal conductivity.

**Drawbacks **

- Theory does not explain the Compton Effect, photoelectric effect and black body radiation.
- This theory is not explaining electrical conductivity of semiconductors and insulators.
- At lower temperatures Widemann – Franz law (K/σT= constant) is not applicable.
- Ferromagnetism behaviour remains unclear. The theoretical value of paramagnetic susceptibility is greater than the experimental value.
- According to classical theory the specific heat of metals comes out 4.5R but experimental value is 3R.

**Quantum Free Electron Theory**

The failure of classical theory created the way for Quantum free electron theory. However, Sommerfeld introduced this in 1928. Arnold Sommerfeld* *overcomes many drawbacks of the classical theory by applying quantum mechanical principles in 1928.

Quantization of electrical energy levels is added here. He also used the Pauli Exclusion Principle in restricting the energy values of electron. His theory is famous as Quantum Free Electron Theory.

#### Assumption

- Free electrons available in metal are responsible for electrical conduction. Electrons have quantised energy levels. However, these are responsible for the conduction.
- Inside the metal electrons can move in a constant potential but remain confined within its boundaries and come out of this potential.
- Also, the distribution of electrons in various allowed energy levels is in accordance with Pauli Exclusion Principle.
- As electrons have wave nature so with the help of Fermi-Dirac distribution function Velocity and energy distribution can determine.
- Attractive forces between the electrons and the lattice ions as well as the repulsive forces between the electrons themselves are ignored.
- The energy is loss due to interaction of the free electron with the other free electron.

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