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ENGINEERING PHYSICS II
CONDUCTING MATERIALS
1.7 WIEDEMANN – FRANZ LAW AND LORENTZ NUMBER
Wiedemann – Franz Law
The ratio between the thermal conductivity and electrical conductivity of a metal is directly
proportional to the absolute temperature of the metal.
K
(i.e)
σ
K
σ
𝛼 𝑇
= LT
Where, L is a constant called as Lorentz number whose value is 2.44 x 10−8WΩK-2(Quantum
mechanical value) at temperature T = 293 K.
Proof:
(i)
By Classical theory:
We know electrical conductivity (from classical theory)
ne2 τ
σ=
m
K=
K
nv2 KB τ
2
nv2 KB τ
2
ne2 τ
m
=
σ
K
=
σ
v2 KB m
… (1)
2e2
1
3
We know kinetic energy of an electron = mv 2 = 2 𝐾𝐵 T
2
Substituting this in equation (1) we can write
K
σ
KB
=
e2
3
. 2 KBT
(or)
K
3 K
= ( B )2 T
σ
2 e
K
(or)
σ
Where
= LT
L=
3
2
K
( eB )2
Substituting the value in of Boltzmann constant 𝐾𝐵 = 1.38 𝑋 10−23 𝐽𝐾 −1 and the charge of electron
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ENGINEERING PHYSICS II
CONDUCTING MATERIALS
e = 1.6021 X 10−19Joules, we get
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Lorentz number L = (
1.38 𝑋 10−23
2 1.6021 𝑋 10−19
)2
L = 1.12 x 10−8WΩK-2
It is found that the classical value of Lorentz number, is only one half of the experimental value
(i.e) 2.44 x 10−8WΩK-2.
This Discrepancy in the experimental and theoretical value of ‘L’ is the failure of classical theory. This
discrepancy can be rectified by quantum theory.
(ii)
By Quantum theory:
In quantum theory the mass of the electron (m) is replaced by the effective mass m*
ne2 τ
The electrical conductivity σ =
m
Rearranging the expression for thermal conductivity and substituting the electronic specific
heat,
the thermal conductivity can be written as
K=
K
σ
(or)
=
σ
σ
where
=
K
K
π2 nK2B τ
3
(
m∗
)T
2
π2 nKB τ
(
)T
3
m∗
ne2 τ
m
π2 KB 2
(e)T
3
= LT
L=
π2 K B
(e)
3
Substituting the values for Boltzmann constant () and the charge of the electron e we get
Lorentz number L = 2.44 x 10−8WΩK-2.
This quantum theory verifies Wiedeman-franz law and has good agreement with the experimental
value of Lorentz number.
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