SOMMERFELD MODEL Maya yadav ppt

2. RAI SAHAB BHAWAR SINGH COLLEGE
Page 2

3. Page 3

4. Kit Building Class Lesson 3
Page 4

5. Kit Building Class Lesson 3
Page 5

6. SOMMERFELD’S FREE ELECTRON
MODEL
According to the sommerfeld each
free electron inside the metal
experience an electrostatic attractive
force due to all the positive ions and
an electrostatic repulsion force due
to the other electrons.

7. SOMMERFELD’S FREE ELECTRON MODEL
 Thus each free electrons inside metal is in
the attractive force electric field due to
positive ions and a repulsion electric field
due to remaining all electrons.
 The force of repulsion due to the mutual
interaction of electron can be assumed to
negligible and the attractive field due to the
positive ions can be considered to be uniform
every where inside the crystal.

8. SOMMERFELD’S FREE ELECTRON MODEL
 Thus each free electron inside the metal is in
an attractive potential field. since the crystal
structure of solid is periodic that is why in a
solid is crystal each positive ions is at a
definite distance form each other, therefore
this potential field inside the metal must also
be periodic.
 But for the sake of convenience sommerfeld.

9. FREE ELECTRON MODEL
+ + + + +
+ + + + +
+ + + + +
+ + + + +
+ + + + +
Schematic model of metallic
crystal, such as Na, Li, K, etc.
The equilibrium positions of
the atomic cores are
positioned on the crystal lattice
and surrounded by a sea of
conduction electrons.
For Na, the conduction
electrons are from the 3s
valence electrons of the free
atoms. The atomic cores
contain 10 electrons in the
configuration: 1s22s2p6.

10. Sommerfeld Constant
• The Sommerfeld constant is proportional to the density of states at
the Fermi energy, since
• Now, we look at this and say, “Obviously!” – because only the
electrons very close to the Fermi energy can absorb energy.
)(
3
1
F
2
B
2
 gk
The Sommerfeld constant is
also related to another,
rather important, concept in
Solid State Physics –
effective mass.
3/2
2
22
F
2
F 3
22







V
N
mm
k



11. Effective Mass
m~
effective~ m
Electrons interact with
• periodic lattice potential
• phonons
• other electrons
+ + + + +
+ + + + +
+ + + + +
+ + + + +
+ + + + +

12. Sommerfeld Constant
mm /e
observed
 2
KmolmJ/ 
calculated
 2
KmolmJ/ 
Observed values
come from the linear
heat capacity
measurements.
Calculated values are
determined using the
conduction electron
density and from
assuming me = m,

13. 13

kkF
A couple of much-used graphs relating to the
Sommerfeld model:
a) The free electron
dispersion
Probability
of state
occupation
1
0 , k
F or
kF
b) The T = 0 state occupation
function.

14. Recap: Free Electron Model
Some successes:
1. electrical conductivity
2. heat capacity
3. thermal conductivity
Some failures:
1. physical differences between conductors, insulators,
semiconductors, semi-metals
2. positive Hall coefficients – positive charge carriers
??

15. Quantum Sommerfeld’s gas: do wave mechanics and then think in an
‘equivalent particle’ picture
Random velocity dominantly
quantum (due to Pauli principle):
m
V
N
mkv FF /3/
3/1
2






 









F
B
Bel
Tk
nkc
2
2
Small effective number of particles moving very fast, due to special
quantum mechanical constraints.