the free electrons are non-interacting and obey?
Answers
Answer:
In simple terms, the reason that free electron theory works is that the (non-interacting) electrons in a metal form a fermi-surface. If in the absence of interactions the fermi surface is the ground state of the system. If you add coulomb interaction this will enable electrons to interact.
The electron theory of metals aims to explain the structure and properties of solids
through their electronic structure. The electron theory is applicable to all solids i.e., both
metals and non metals. It explains the electrical, thermal and magnetic properties of
solids etc. The theory has been developed in three main stages.
The classical free electron theory
Drude and Lorentz proposed this theory in 1900. According to this theory, the
metals containing the free electrons obey the laws of classical mechanics.
The quantum free electron theory
Somerfield developed this theory in 1928. According to this theory the free
electrons obey quantum laws. According to this theory the free electrons are moving in a
constant potential.
The zone theory
Bloch stated this theory in 1928. According to this theory, the free electrons move
in a periodic field provided by the lattice. According to this theory the free electrons are
moving in a constant potential.
2. Definitions and relations
Drift velocity
It is defined as the average velocity acquired by the free electrons of a metal in a
particular direction by the application of electric field.
Relaxation time
It is defined as the time taken by the free electrons to reach its equilibrium position
from the disturbed position in the presence of electric field.
Collision time(τ )
It is defined as the average time taken by the free electrons between two successive
collisions.
Current density j)(
It is defined as the magnitude of current passing through unit area.
A
I
J = or ∫
I = J • ds
Expression for Relaxation time ( )
r
τ
When the metal is subjected to an external electric field, the electrons move opposite to
the applied field. After removal of electric field the drift velocity decays exponentially as
r
t
d o
v v e
τ
−
=
Where o
v is the initial velocity of an electron, before application of electric field and r
τ is
the relaxation time. Unit –III Free Electron Theory Engineering Physics
Dr. P.Sreenivasula Reddy M.Sc, (PhD) Website: www.engineeringphysics.weebly.com
Page 2
If r
t = τ then
−1
v = v e d o or
e
v
v
o
d =
Thus the relaxation time may be stated as the time taken for the drift velocity to decay
to
e
1 of its original initial value.
Mean free path λ)(
Free electrons in a metal are continuously moving in all directions and with various
speeds. They frequently collide with one another. Therefore, they move in straight line
with constant speeds between two successive collisions. The distance traveled by the
electron between two successive collisions is called as free path and their mean is called
the mean free path.
The average distance traveled by the electron between two successive collisions is
called mean free path.
Or
The mean free path is the average distance traveled by an electron between two
successive collisions with other free electrons.
λ = c τ
Where c is the mean square velocity of electron.
m
K T
c
3 B =
Expression for Mean collision time:-
The average time taken by the electrons between two consecutive collisions of
electron with the lattice points is called mean collision time.
If ‘ v ’ be the total velocity of electron i.e. thermal and drift velocity, then the mean collision
time is given by
v
λ
τ =
d th v = v + v
If d th v << v then th v = v
Expression for drift velocity
When electric field is applied on an electric charge ‘ e ’, then it moves in opposite
direction to the field with a velocity d
v . This velocity is known as drift velocity.
The Lorentz force acting on the electron is FL −= eE
The resistance force acting on the electron is Fr = r