Thermo & Stat Mech - Spring 2006 Class 22 1 Thermodynamics and Statistical Mechanics Fermi-Dirac...
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Thermo & Stat Mech - Spring 2006 Cla 1 Thermodynamics and Statistical Mechanics Fermi-Dirac Statistics
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Transcript of Thermo & Stat Mech - Spring 2006 Class 22 1 Thermodynamics and Statistical Mechanics Fermi-Dirac...
Thermo & Stat Mech - Spring 2006 Class 22
1
Thermodynamics and Statistical Mechanics
Fermi-Dirac Statistics
Thermo & Stat Mech - Spring 2006 Class 22
2
Free Electrons in Metals
A model based on Fermi-Dirac statistics explains why metals do not have a much larger heat capacity than insulators. At one time it was expected that the “free electron gas” would provide a large addition to the lattice heat capacity, but it does not.
Thermo & Stat Mech - Spring 2006 Class 22
3
Fermi Function
stateper Number
1
1)(
1
1
kT
kTj
jj
e
f
eg
Nf
j
Thermo & Stat Mech - Spring 2006 Class 22
4
Properties of Fermi Function
0)(,For
1)(0,For
)(1,For
1
1)(
21
fe
fe
fe
e
f
kT
kT
kT
kT
Thermo & Stat Mech - Spring 2006 Class 22
5
Properties of Fermi Function
0)(,For
1)(0,For
0 asConsider
1
1)(
fee
fee
T
e
f
kT
kT
kT
Thermo & Stat Mech - Spring 2006 Class 22
6
Properties of Fermi Function
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2 0.25
E(eV)
FD(T1)
FD(T2)
FD(T3)
= 0.1 eVT1 = 1KT2 = 200KT3 = 400K
Thermo & Stat Mech - Spring 2006 Class 22
7
Free Electron Gas
dmV
dg
dmV
dg
2/3
22
2/3
22
2
2)(
electronsfor 2
2
4)(
17) (Lec Particlesfor States ofDensity
Thermo & Stat Mech - Spring 2006 Class 22
8
Energy of Electrons
1
2
2
2
21
1)()(
)(
2/32/3
22
2/3
22)(
kT
kT
e
dmVdU
dmV
e
dgfdU
Thermo & Stat Mech - Spring 2006 Class 22
9
Fermi Energy
2/3
2/3
22
3/22
2
2/32/3
22
0
2/12/3
220
2
32
23
2
3
22
2
2
2)(
FF
F
NmV
V
N
m
mVN
dmV
dgNFF
F Energy of highest energy state occupied at T = 0 K
Thermo & Stat Mech - Spring 2006 Class 22
10
Electrons’ Energy at T = 0 K
F
FF
F
NU
NU
dN
dgUFF
5
3
5
2
2
3
2
3)(
0
2/52/30
0
2/32/300
Thermo & Stat Mech - Spring 2006 Class 22
11
Electrons’ Energy at T > 0 K
2)(5
3
5
3
kTN
NU
kTkT
NNU
FF
FF
Thermo & Stat Mech - Spring 2006 Class 22
12
Heat Capacity of Electrons
FV
FFV
V
FF
kTNkC
kTNkTk
N
T
UC
kTN
NU
2 :resultExact
22
)(5
3
2
2
2
Thermo & Stat Mech - Spring 2006 Class 22
13
Heat Capacity of Metals
At normal temperatures, the heat capacity due to the electrons is much less than the heat capacity of the lattice, since kT << F.
At low temperatures, the lattice heat capacity gets very small, and the contribution of the electrons becomes significant.
Thermo & Stat Mech - Spring 2006 Class 22
14
At Low Temperatures
3
42
223
234
5
12
2
25
12
D
metal
Fmetal
Felectrons
DLattice
NkBBTA
T
C
NkABTATC
kTNkC
TNkC
Thermo & Stat Mech - Spring 2006 Class 22
15
Better Value for Electron’s Energy
0
)()( dgfU
To calculate the energy of the electrons correctly, it is necessary to carry out the integral below. In order to do that, a value is needed for the chemical potential, .
Thermo & Stat Mech - Spring 2006 Class 22
16
Approximate Method
0
0
')'()( Then,
0)0( and ,)(
)( where
)()( :integral heConsider t
dHh
hd
dhH
dHfI
Thermo & Stat Mech - Spring 2006 Class 22
17
Approximate Method
0
00
0
)()(
so ,0)0(0)(
)()()()(
)()(
dd
dfhI
hf
dd
dfhhfI
dd
dhfI
Thermo & Stat Mech - Spring 2006 Class 22
18
Graph of
Fermi-Dirac
0
20
40
60
80
100
120
140
160
0 0.05 0.1 0.15 0.2 0.25
E (eV)
-df/
d
d
df
Thermo & Stat Mech - Spring 2006 Class 22
19
Expansion
...)(')()("
)(')()(')(')(
...)(')()("
)(')()(')(')(
...)(")()(')()()(
221
0
221
00
221
dfh
dfhdfhI
dfh
dfhdfhI
hhhh
Thermo & Stat Mech - Spring 2006 Class 22
20
Evaluate each term
function oddan 0)(')()('
)()(
)()()()(')(
0
dfh
dHh
ffhdfh
Thermo & Stat Mech - Spring 2006 Class 22
21
Evaluate each term
)(')(6
)(")(6
)1())((")(')()("
then , :Let
22
22
2
22
212
21
HkT
hkT
e
dxexkThdfh
xkT
x
x
Thermo & Stat Mech - Spring 2006 Class 22
22
Apply to Chemical Potential
2123
22
00
22
00
2
3)( where
)(')(6
)()()(
)(')(6
)()()(
F
Ng
gkTdgdgfN
HkTdHdHfI
Thermo & Stat Mech - Spring 2006 Class 22
23
Chemical Potential
2123
22
22
00
0
22
00
2
3)( where
)(')(6
)( and
)(')(6
)( )( so
)( However,
)(')(6
)()()(
F
Ng
gkTdg
gkTdgdg
dgN
gkTdgdgfN
F
F
F
Thermo & Stat Mech - Spring 2006 Class 22
24
Chemical Potential
21232
221
23
21232123
22
2123
2
1
2
3)(
6)(
2
3
2
1
2
3)('
2
1
2
3)('
)(')(6
))(()(
2
3)(
FF
F
FF
F
F
NkT
N
Ng
Ng
gkTgdg
Ng
F
Thermo & Stat Mech - Spring 2006 Class 22
25
Finally, the Chemical Potential!
2222
222
2
21232
221
23
121
)(
12
)(
12
1)(
12)(
2
1
2
3)(
6)(
2
3
FF
FF
FF
FF
F
kTkT
kTkT
NkT
N
Thermo & Stat Mech - Spring 2006 Class 22
26
Electron Energy
23
212
2
0
23
2123
22
00
4
9)(
6)(
4
9)]([
2
3)(
)]([)(6
)()()(
F
FF
NkTdgU
Ng
d
dNg
gd
dkTdgdgfU
Thermo & Stat Mech - Spring 2006 Class 22
27
Approximate again!
)()(5
3
)()()(
4
9)(
6)(
00
23
212
2
0
FFFF
F
gN
dgdgdg
NkTdgU
F
F
Thermo & Stat Mech - Spring 2006 Class 22
28
Still more!
F
FF
FFF
FF
FFF
kTNUdg
NNg
kT
gUdg
22
00
2123
22
00
)(
8)(
2
3
2
3)(
)(
12)(
)()()(
Thermo & Stat Mech - Spring 2006 Class 22
29
And more!
FF
FF
F
NkT
kTNUU
NNg
d
d
kTNUdg
gd
dkTdgU
4
9)(
6
)(
8
4
9
4
9)]([
)(
8)(
)]([)(6
)(
2222
0
23
21
22
00
22
0
Thermo & Stat Mech - Spring 2006 Class 22
30
Finally!!
FVV
F
FF
kTNk
T
UC
kTNUU
NkT
kTNUU
2
)(
4
4
9)(
6
)(
8
2
22
0
2222
0
