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Classical free electron theory
Q 1. Write a note on Concept of electron
Concept of electron:
According to classical theory all metals are good conductors of heat and electricity as they
contain large number of free electrons. These electrons are loosely bound to their respective
atoms. While forming a solid, large numbers of atoms are joined so that boundaries of
neighbouring atoms slightly overlap on each other. Due to thermal energy these loosely
bound electrons get detached from the atoms and move randomly throughout the metal but
cannot escape from the metal, unless external voltage is applied. These electrons are called
free electrons. When external voltage is applied to the metal, electrons flow opposite to the
direction of applied electric field which constitutes an electric current and hence free
electrons are also called as conduction electrons.
The atoms of a metal become ion core after losing electrons. The boundaries of ion of an
atom are called as ion core. The arrangement of ions over the lattice of the metal is known as
lattice core. Lattice core is in a state of thermal vibration about their mean position. During
motion these electrons collide with other lattice ions.
Q 2. State the assumptions of classical free electron theory
To account for large electrical conductivity in metals Drude proposed a theory and
later it was extended by Lorentz. Hence this theory is called Drude Lorentz theory. It is
based on classical theory and it is also called as classical free electron theory.
This theory is based on following assumptions
(a) A metal is imagined to be made up 3-dimensional array of ion cores. Free electrons move
in between the ions. Such free moving electrons cause electrical conduction under an
applied field and hence referred to as conduction electrons.
(b) The free electrons are equivalent to gas molecules and they are assumed to obey the laws
of kinetic theory of gases. In the absence of applied electric field, the thermal energy
associated with each electron at temperature T is equivalent to the kinetic energy
associated with the electrons. It is related as
3
1
2
KT = mv th
2
2
Where K is the Boltzmann constant=1.38X10-23JK-1, m=mass and Vth thermal velocity of
the electron.
(c) The ionic potential due to lattice core is considered to be constant throughout the metal
and effect of repulsion between the electrons is negligible.
1

(d) The electric current in a metal due to an applied field is due to drift of electrons in a
direction opposite to the direction of the field.
Q3. Explain/Define the terms: a) Thermal velocity b) Drift velocity c) Relaxation time
c) Mean free path d) Mean collision time
(a) Thermal velocity:
A conductor consists of large number of free electrons of about
1029electrons/m3. Due to thermal energy these electrons are
moving in between the ions with a speed of 106m/s and collide
with ion cores of the conductor. After each collision velocity of
the electron becomes zero. There after start moving in random
direction. Thus in the absence of applied electric field, there is a
kind of randomness in the motion of electrons. Though the free
electrons are in motion, the net flow of current is zero or does
not give rise to any current.
“The average velocity with which the free electrons move inside the conductor due to
thermal energy is called thermal velocity”.
b) Drift velocity (vd):
When an electric field is applied, an electric field is developed inside the conductor. As a
result potential difference is developed between the ends of a conductor. The electrons start
moving opposite to the field direction and collide with ion cores. After each collision velocity
of electrons become zero; and again they gain velocity in a fresh direction but always
opposite to the direction of applied electric field. Even though randomness exists; distance
travelled as well as time of collision between the successive collisions is different.
“The average velocity with which electrons move in a conductor under the
influence of applied electric field is called drift velocity”.
The expression for drift velocity.
Consider a conductor of length „L‟ is subjected to an electric field E. In the steady state,
conduction electrons are drifted opposite to the direction of applied electric field. If „m‟ is
the mass of an electron, „vd‟ drift velocity, „τ‟ is the mean collision time, and then resistance
force „Fr‟ offered to its motion is given by
Fr =ma=
mv d
(1)
If „e‟ is the charge on the electron, „E‟ is the electric field, then force experienced by electron
due to applied electric field is F = eE
(2)
In the steady state
F = Fr
eE=
mv d
The drift velocity is given by
2
vd
eE
m

(c) Relaxation time (τr) :
In the absence of electric field, the conduction electrons move in random direction, and hence
the probability of finding an electron moving in any given direction is zero.
i.e. Vav = 0
When an external field is applied, net positive value
V1av for the average velocity of the conduction
electrons in a direction opposite to the direction of
field; which is equal to the drift velocity i.e
Vav=V1av
If the field is turned off, the average velocity Vav
starts reducing exponentially as shown in the figure
and is according to the equation.
(1)
Time counted from the instant the field is turned off,
If =
Relaxation time
equation (1) becomes
Hence relaxation time is defined as
“The time interval during which drift velocity of electrons reduces to
1
e
times
the maximum value attained by them when applied field is turned off”.
(d) Mean free path (λ):
The average distance travelled by the conduction electrons between two successive collisions
of conduction electrons under the influence of applied electric field is called mean free path.
d) Mean collision time (τ): The average time interval between two consecutive collisions of
an electron with the lattice cores in a conductor under the influence of applied electric field is
called mean collision time.
τ = λ/vth
Where „λ‟ is the mean free path, v≈vth is velocity same as combined effect of thermal & drift
velocities.
Note: But ½ mvth2=3/2 KT
Question (4): Discuss the failures of classical free electron theory.
Classical free electron theory has failed to explain specific heat, dependence of conductivity
with temperature and dependence of conductivity on electron concentration as follows
(a) Specific heat:
3

As per classical free electron theory, free electrons in a metal behave as gas molecules and
3
hence the molar specific heat of electrons at constant volume is given by Cv= R
2
Where R is universal gas constant
But experimentally molar specific heat of free electrons in a metal is given by
CV=10-4RT
This is not only less than the experimental value but also depends on temperature. Hence
classical free electron theory failed to explain specific heat.
(b) Dependence of electrical conductivity on Temperature:
According to the assumptions of classical free electron theory
√
√ ---------------------- (1)
The mean collision time „τ‟ is inversely proportional to the thermal velocity. (vth=
)
i.e.
√
From the expression
------ ------------------------- (2) From (1)
σ=
ne 2 -------------- (3)
m
Substituting for
from equation (3) gives
1
σα
-----------------------(4)
T
Therefore according to classical theory, electrical conductivity is inversely proportional to the
square root of absolute temperature. But experimentally σ is inversely proportional to the
temperature T.
i.e.
cxpt
1
T
----------- (5)
Therefore classical theory failed to explain conductivity with temperature.
(c) Dependence of electrical conductivity on electron concentration:
According to classical theory, electrical conductivity is directly proportional to electron
concentration
i.e σ =
ne 2
m
σαn
Thus as n increases conductivity should increase. But it is contrary to the observation.
Consider the data from the following table.
4

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