Free electron theory of metals - Engineering Physics

Introduction

Free Electron means unbound electron of the atom of any metal which can move freely. Many properties of metal-like electrical conductivity, magnetic susceptibility, Thermal conductivity, Mechanical strength, optoelectronic properties, etc. depend on the free electrons.

Free electron theory are of two types-

1. Classical or Drude-Lorentz Theory

2. Quantum or Sommerfield Theory

This theory was first proposed by Lorentz and Drude with the help of classical laws of physics. The classical theory of free electron explains the properties like ohm’s law, metallic luster, etc. but it could not explain heat capacity and the paramagnetic susceptibility of the conduction electrons. The Quantum theory removed the limitations of classical theory by using statistics developed by Sommerfield and succeeded in explaining many experimental facts.

Drude-Lorentz or Classical Theory

In 1900, P. Drude suggested that the metal crystals consist of positive ion cores with their valence electrons free to move among these positive metal ions. These valence electrons are also known as conduction electrons since they are responsible for the conduction of electricity in metals. The Coulomb’s force of attraction between these positive ions and negatively charged electrons do not permit these free electrons to leave the metal surface. The potential field due to these ion's cores is supposed to be uniform and hence potential energy of electrons may be taken as constants (which is taken as zero for convenience). Hence we have to deal only with the kinetic energy of the electrons. The mutual repulsion among the electrons is neglected.

It has been suggested that the free electrons I  the metal behave like these of atoms or molecules in a perfect gas. Hence, these electrons are sometimes known as the free-electron cloud. Fermi gas is constituted by electrons which are charged particles while the atoms or molecules which constitute ordinary gas are neutral The concentration of electrons in Fermi gas is large 10²⁹per m³ as compared with the concentration of atoms or molecules of ordinary gas 10²⁵ per m³. That’s why Drude-Lorentz classical theory also known as Free Electron Gas Model.

In 1909, Lorentz suggested that this free electron gas behaves like perfect gas obeys Maxwell-Boltzmann statistics. Hence, the classical theory is also known as the Drude-Lorentz theory.

Success of Free Electron Theory

The free-electron theory successfully explained

1. Electrical conductivity

2. Ohm’s Law

3. Thermal conductivity

4. Weidmann-Franz law

5. Complete opacity of metals and their high luster

1. Electrical conductivity

It is defined as the amount of electricity that flows in unit time per unit area of cross-section of the conductor per unit potential gradient.

According to free-electron theory, in a solid, the electron moves freely. If E is the applied electric field, then the acceleration of an electron having charge is given by  

 ....(1.1)

Integrating eq. (1.1), we get 

At t=0, 

, so C=0

Hence

If λ is the mean free path of electrons, then the relaxation time τ between two successive collisions is given by

 ...(1.2)

So average velocity between two successive collisions

=

Putting the values of τ from eq. (1.2), we get

 ...(1.3)

Since 

(T is the absolute temperature and K is Boltzmann constant)

Putting this value of (mv) in eq. (1.3), we get 

 ...(1.4)

If n is the number density of electron in the conductor, then-current density i is given by 

Putting eq. (1.4) in above eq., we get 

 ...(1.5)

If q charge is flowing through a conductor of cross-section area A in time t, then

Or, 

 ...(1.6)

For the unit area of cross-section

Using eq. (1.5), we get 

 ...(1.7)

This expression show that conductivities of different materials depending on the number of free electron, T, and A.

2. Ohm’s law

From eq. (1.6) we have

 ...(1.8)

This is microscopic form of ohm’s law.

3. Thermal conductivity

There is no transfer of energy, if there is no temperature difference between two points in a system. So to discuss the thermal conductivity of metals, we suppose that a temperature gradient exists across the system instead of the voltage gradient, hence the transport of thermal energy takes place due to this gradient.

Suppose A and B be the two ends of the system in the form of a metallic rod and end A is at a higher temperature than B, then the thermal conductivity from A to B takes place by electrons. In a collision, the electron near A loose their kinetic energy while the electrons near B gain energy.

The amount of heat Q passing through a cross-section of the rod per unit area per second is given by 

 ...(1.9)

Where λ is mean free path, v is the velocity of electrons and n is the number density of free electron.

From Kinetic theory of gas

Hence 

Or 

 ...(1.10)

This value of K is verified experimentally and the theory of the free electron is found to be successful to explain thermal conductivity.

4. Wiedemann-Franz relation

In 1853, Weidmann and Franz discovered that all good electrical conductors are also good thermal conductors and the ratio of thermal conductivity to the electrical conductivity at any temperature is constant for all metals.

i.e., 

using eqns. (1.7) and (1.10), we get

This is Weidmann-Franz relation.

5. Lustre and opacity of metals

When electromagnetic radiations fall on metal, it produces forced oscillations in the free electrons having the same velocity as that of electromagnetic radiations. Thus the energy of incident radiations is absorbed by free electrons and the metal appears opaque. The excited electron on returning to its initial state emits a photon having the same energy as is absorbed initially. This energy is given out in the form of visible light in all directions, but only the light rays directed towards the metal surface can get through. Hence, the metal appears to reflect virtually all the light that is incident on it, giving it the characteristic metallic luster.

Failure of the theory

1. It fails to explain the heat capacity of materials. According to this theory 

But experimentally, 

2. It does not explain the paramagnetic susceptibility of the conduction electrons, Experimentally χ is independent of T, but by this theory  

3. It does not explain why some crystals are metallic.

4. It is unable to explain why metals prefer certain structures.

5. It does not explain the temperature variation of electrical resistivity because this theory predicts the variation of resistivity as  which actually is linear.

6. it is unable to differentiate insulator, semiconductor and conductor.  

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