In this lecture, we are going to learn about the Free electron theory of metals. classification of Free Electron Theory of Metals and will discuss each type in detail. So let’s start with the basic definition of it.

**Free Electron Theory of Metals**

The study of valance electrons present in a band that controls the various properties of metals is known as the free electron theory of metals.

The electron theory of metals is classified into the following three types:

1. | Classical free electron theory or Drude-Lorentz theory |

2. | Quantum free electron theory or Sommerfeld theory |

3. | Zone theory or Band theory of solids |

we have separated covers the band theory of solids in the different lectures. You can learn by clicking the below link.

The remaining two classifications we will cover in this lecture. So let’s start one by one.

**Classical Free Electron Theory Of Metals**

This theory was developed by **Drude and Lorentz**. Even though it is a macroscopic theory it successfully explained most of the properties of metals.

In this theory, the free electrons in a metal are treated like molecules in a gas, and Maxwell-Boltzmann statistics is applied.

**The main assumptions or postulates of this theory are:**

- A metal is composed of positive metal ions fixed in the lattice.
- All the valence electrons are free to move among the ionic array. Such freely moving electrons contribute towards conduction (electrical and thermal) in metals.
- There are a large number of free electrons in a metal and they move about the whole volume like the molecules of a gas.
- The free electrons collide with the positive ions in the lattice and also among themselves. All the collisions are elastic, i.e., there is no loss of energy.
- The electrostatic force of attraction between the free electrons and the metallic ions is neglected, i.e., the total energy of the free electron is equal to its kinetic energy.
- All the free electrons in metal have a wide range of energies and velocities.
- In the absence of an electric field, the random motion of the free electrons is equally probable in all directions. So, the net current flow is zero.
- When an electric field is applied as shown in the figure below, the electrons gain a velocity called drift velocity vd and move in the opposite direction to the field, resulting in a current flow in the direction of the field.

Thus drift velocity is the average velocity acquired by an electron on applying an electric field.

**9. Relaxation Time (\tau)**

**Definition 1:**It is defined as the time required for the drift velocity to reduce to (1/e) times its initial value just when the field is switched off.

**Definition 2:**It is defined as the time taken by the free electron to reach its equilibrium position from its disturbed position just when the field is switched off.

**10. Mean Collision time (t _{c})**

- The average time between two consecutive collisions of an electron with the lattice points is called collision time.

**11. Mean free path (\lambda)**

- It is the average distance traveled by the conduction electron between successive collisions with the lattice ions.

**The Success of Classical Free Electron Theory**

The free-electron theory successfully explained:

- It verifies Ohm’s law.
- It explains the thermal conductivity and electrical conductivity of metals.
- It is used to deduce the Wiedemann-Franz law.
- It explains the optical properties of metals.

**Drawbacks of Classical Free Electron Theory**

- The theoretical value obtained for specific heat and electronic specific heat of metals based on this theory is not in agreement with the experimental value.
- The classical free electron theory is not able to explain the electrical conductivity of semiconductors and insulators.
- According to classical theory, \frac{K}{\sigma T} is constant at all temperatures. But this is not constant at low temperatures.
- The theoretical value of paramagnetic susceptibility (\chi) is greater than the experimental value; also, ferromagnetism cannot be explained.
- The phenomena such as the photoelectric effect Compton effect and black body radiation cannot be explained by this theory.

**Quantum Free Electron Theory of Metals**

To overcome the drawbacks of the classical free electron theory of metals, in 1928, by applying quantum mechanical principles ARNOLD SOMMERFIELD proposed a new theory called quantum free electron theory or Sommerfield theory.

In this theory, all the essential features of classical free electron theory are retained and the problem is treated quantum mechanically (the electron is treated as a wave) using Fermi-Dirac statistics. The electrons are assumed to obey the Paulis exclusion principle.

**The main assumption of quantum free electron theory are**:

- The energy levels of the conduction electrons are quantized.
- The distribution of electrons in the various allowed energy level occurs as per Paulis’s exclusion principle.
- The electrons are assumed to possess a wave nature.
- The free electrons are assumed to obey Fermi-Dirac statistics. Except for the above modifications, sommerfield kept the following assumptions of classical free electrons theory to be applicable in quantum free electron theory also,
- The electrons are free to move inside the metal but confined to stay within its boundaries.
- The potential energy of the electrons is uniform or constant inside the metal.
- The attraction between the electrons and the lattice ions and the repulsion between the electrons themselves are ignored.

Quantum free electrons theory provides an explanation for electrical conductivity, thermal conductivity, the specific heat capacity of metals, electronic specific heat capacity, the Compton effect, the photoelectric effect, etc.

This theory fails to make a distinction between metals, semiconductors, and insulators, Also fails to explain the positive value of the Hall coefficient and some transport properties of metals.

**Band theory of solids**

we have separated covers the band theory of solids in the different lectures. You can learn by clicking the below link.