This document discusses the electron theory of metals according to classical free electron theory. It describes the classical free electron theory proposed by Drude and Lorentz in 1900, which treats electrons in metals as a free electron gas obeying classical mechanics. The theory postulates that electrons move freely within the metal lattice and collide elastically with positive ions. When an electric field is applied, electrons drift through the metal. The document derives equations for drift velocity, current density, conductivity, resistivity, and mobility based on this classical free electron model. It also briefly mentions quantum free electron theory and band theory as later improvements over the classical model.

1. 1 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
ENGINERING PHYSICSENGINERING PHYSICSENGINERING PHYSICSENGINERING PHYSICS
Mr. Gouri Kumar Sahu
Senior Lecturer in Physics
C.U. T. M.

2. 2 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
3.1. Electron Theory of metals
The electron theory of metals explain the following concepts
Structural, electrical and thermal properties of materials.
Elasticity, cohesive force and binding in solids.
Behaviour of conductors, semi conductors, insulators etc.
So far three electron theories have been proposed.
1. Classical Free electron theory:
• It is a macroscopic theory.
• Proposed by Drude and Loretz in 1900.
• It explains the free electrons in lattice
• It obeys the laws of classical mechanics.

3. 3 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
Electron Theory of metals
2. Quantum Free electron theory:
It is a microscopic theory.
Proposed by Sommerfield in 1928.
It explains that the electrons move in a constant potential.
It obeys the Quantum laws.
3. Brillouin Zone theory or Band theory:
Proposed by Bloch in 1928.
It explains that the electrons move in a periodic potential.
It also explains the mechanism of semi-conductivity, based on bands and
hence called band theory.

4. 4 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
3.2.1. Classical Free electron theory
(Drude-Lorentz Theory)
3.2.1. Classical Free electron theory
(Drude-Lorentz Theory)
Postulates of Classical free electron theory:
1. All the atoms are composed of atoms. Each atom have central nucleus
around which there are revolving electrons.
2. The electrons are free to move in all possible directions about the
whole volume of metals.
3. In the absence of an electric field the electrons move in random
directions making collisions from time to time with positive ions which are
fixed in the lattice or other free electrons. All the collisions are elastic i.e.; no
loss of energy.
4. When an external field is applied the free electrons are slowly drifting
towards the positive potential.
5. Since the electrons are assumed to be a perfect gas they obey classical
kinetic theory of gasses.
6. Classical free electrons in the metal obey Maxwell-Boltzmann
statistics.

5. 5 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
3.2 Drude-Lorentz Theory 1
Consider a conductor subjected to an electric field E in the x-
direction. The force on the electron due to the electric field = -eE.
From Newton’s law,
െ݁‫ܧ‬௫ ൌ ݉ܽ௫
‫,ݎ݋‬ ܽ௫ ൌ
߲ ൏ ‫ݒ‬௫ ൐
߲‫ݐ‬
ൌ െ
݁‫ܧ‬௫
݉
ሾ2.1ሿ
‫݁ݎ݄݁ݓ‬ ൏ ‫ݒ‬௫ ൐ ݅‫ݏ‬ ‫݄݁ݐ‬ ܽ‫݃ܽݎ݁ݒ‬ ݀‫ݐ݂݅ݎ‬ ‫ݕݐ݅ܿ݋݈݁ݒ‬
Integrating equation ሾ2.1ሿ
൏ ‫ݒ‬௫ ൐ ൌ െ
݁‫ܧ‬௫
݉
‫ݐ‬ ൅ ‫ܥ‬
ܵ݅݊ܿ݁ ൏ ‫ݒ‬௫ ൐ ൌ 0, ܽ‫ݐ‬ ‫ݐ‬ ൌ 0 ‫݁ݓ‬ ݃݁‫ݐ‬ ‫ܥ‬ ൌ ܱ.
ܵ‫,݋‬ ൏ ‫ݒ‬௫ ൐ ൌ െ
݁‫ܧ‬௫
݉
‫ݐ‬ ሾ2.2ሿ

6. 6 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
3.2 Drude-Lorentz Theory 2
ܵ‫,݋‬ ൏ ‫ݒ‬௫ ൐ monotonically proportional with time (not true).
Retarding force is provided scattering of electrons by the vibrating atoms
(phonon) and the imperfection in the lattice.
The retarding force is directly proportional to the drift velocity.
So,
డழ௩ೣவ
డ௧ ௦௖௔௧௧.
ൌ െ
ழ௩ೣவ
ఛ
ሾ2.3ሿ
ଵ
ఛ
has the unit of sec-1. So, ߬ is called relaxation time.
In steady state,
డழ௩ೣவ
డ௧ ௙௜௘௟ௗ
ൌ െ
డழ௩ೣவ
డ௧ ௦௖௔௧௧.
Using eq[2.1] and [2.3], we have
െ
݁‫ܧ‬௫
݉
ൌ
൏ ‫ݒ‬௫ ൐
߬
‫,ݎ݋‬ ൏ ‫ݒ‬௫ ൐ൌ െ
݁߬
݉
‫ܧ‬௫ ሾ2.4ሿ

7. 7 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
We know that, Current density ݆௫ ൌ െ݊݁ ൏ ‫ݒ‬௫ ൐ ሾ2.5ሿ
Where n is the concentration of electrons in the metal.
From eq[2.4] and [2.5],
݆௫ ൌ
݊݁ଶ
߬
݉
‫ܧ‬௫ ሾ2.6ሿ
3.2 Drude-Lorentz Theory 3
SESSION-3

8. 8 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
3.2.3 Ohm’s law in terms of E and J
From Ohm’s law
V=Potential across the conductor, I= Current through the conductor
R=Resistance
=conductivity of the conductor
Let L=length
d=width of conductor
A=area of cross section
Now
And
So, Ohm’s law can be written as
Or,
Equation [2.7] is the ohm’s law

9. 9 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
SESSION-3
3.2.4 Electrical Conductivity
Comparing eqs[2.6] and [2.7], we get
࣌ ൌ
௡௘మఛ
௠
ሾ2.8ሿ
The electrical conductivity is directly proportional to the
concentration of free electrons and the relaxation time.
Electrical Resistivity: ࣋ ൌ
૚
࣌
ൌ
࢓
௡௘మఛ
[2.9]
Mobility: It is defined as the average drift velocity per unit applied electric
field.
So,
ߤ௘ ൌ
‫ݒ‬
‫ܧ‬
ൌ
݁߬
݉
െ ሾ2.10ሿ
Hence
ߪ ൌ ݊݁ߤ௘. ሾ2.11ሿ

10. 10 ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics.
END OF SESSION -3
SESSION-3
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