Engineering physics 5(Quantum free electron theory)

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

2. 2
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
2
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
SESSION-6
6.1Quantum free electron theory
SClassical free electron theory could not explain ma ny physical properties.
SIn 1928, Sommerfelddeveloped a new theory applying quantum mechanical
concepts and Fermi-Dirac statistics to the free ele ctrons in the metal. This
theory is called quantum free electron theory.
SClassical free electron theory permits all electron s to gain energy.
SBut quantum free electron theory permits only a fra ction of electrons to gain
energy.
SIn order to determine the actual number of electron s in a given energy range
(dE), it is necessary to know the number of states (dNs) which have energy
in that range. The number of states per unit energy range is called the
density of states g(E).

3. 3
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
3
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
SESSION-6
6.1Quantum free electron theory
SAccording to Fermi-Dirac statistics, the probabilit y that a particular energy
state with energy E is occupied by an electron is g iven by,

n h
1
uydx
pmp
.
I1
9
2
ccccccccccccccccccu8,y2
where
Sis the energy in the Fermi level.
SFermi level is the highest filled energy level at 0 K. Energy corresponding to
Fermi level is known as Fermi energy.
SNow the actual number of electrons present in the e nergy range dEis
dN= f(E) g(E)dE

4. 4
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
4
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
SESSION-6
6.2
Effect 
of temperature on Fermi
-
Dirac distribution function
6.2
Effect 
of temperature on Fermi
-
Dirac distribution function
Fermi-Dirac distribution function is given by, SAt T=0K, for E> , f(E)=0 and for f(E)=1
SAt T=0K, for E= , f(E)=indeterminate
SAt T>0K, for E=EF, f(E)=1/2
SFor T>0K, some of the state below
are unoccupied and some states
above are occupied. Only those
states close to get affected, and
the states far away from remain
unaffected. The energy range over which the change take place is of the order
of .

5. 5
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
5
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
SESSION-6
6.2
Elementary 
Treatment of Quantum Free 
Electron Theory of Metals
6.2Elementary Treatment of Quantum Free 
Electron Theory of Metals
SDrift velocity is v
wh g
qp
!
"
Sconductivity D h
$
%
&
!
, "is the average time elapsed after collision
SClassical concept: current carried equally by all e lectrons,
SQuantum concept: current is carried out by very few electrons only, all
moving at high velocity ( v
S)
SIf λ is the mean free paths
Sv
Sis the speed of free electrons whose kinetic energy is equal to Fermi
energy since only electrons near Fermi level contri butes to the conductivity.
SThe average time τ between collisions is given by - h
'
(
.

6. 6
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
6
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
SESSION-6
6.2
Elementary 
Treatment of Quantum Free 
Electron Theory of Metals
6.2Elementary Treatment of Quantum Free 
Electron Theory of Metals
SThus the electric conductivity
D h
$
%
'
!(
.

5.2
SSince this free path is inversely proportional to t emperature at high
temperature.
SIt follows that, * ∝
,
1
-. / ∝ 0, in agreement with experimental conclusion.
SAn energy E
F
called (very high compared with kT= 0.025 eVat 300 k) Fermi
energy is required to make all the electrons to mov e to the unoccupied states
corresponding to a temperature T
F
called Fermi temperature.
SThe unique relation connecting the various paramete rs in quantum theory of
free electron is----
Sh
,
1
23
S
1
h Q
50
S

7. 7
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
7
ENGINEERING PHYSICS
Mr. Gouri Kumar Sahu
Sr. Lecturer in Physics
.
END OF SESSION -6
SESSION-6
THANK YOU