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PHY 3201 FIZIK KEADAAN PEPEJAL
Heat Capacity of the
Electron Gas
The heat capacity, Cis the amount of heat dQ, which is
needs to be transferred to a substance on order to raise
its temperature by a certain temperature interval.
V
VT
UC
The specific heat capacity is the heat capacity per unit
mass
m
Cc
Heat Capacity,C
The heat capacity at constant volume is defined as
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PHY 3201 FIZIK KEADAAN PEPEJAL
Source: Chris Wiebe
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PHY 3201 FIZIK KEADAAN PEPEJAL
Source: Chris Wiebe
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PHY 3201 FIZIK KEADAAN PEPEJAL
Electronic Heat
Capacity
The increase UU(T)-U(0) in the total energy of a
system of N electrons when heated from 0 to T is
F
DdfDdU
00
)()()(
where f(,T) is the Fermi distribution function and D() is
the density of states.
11
TkTf
B/exp),(
The Fermi-Dirac Distribution Function is given as
The function f(, T) gives the probability that the energy
level , is occupied by an electron at Temperature T
At 0 K, the upper limit is F
and the FD function is 1
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PHY 3201 FIZIK KEADAAN PEPEJAL
We multiply the identity
F
DdfDdN
00
)()()(
By F to obtain
F
F
F
DdfDd FF
00
)()()()(
0)()()()()( 00
F
F
F
DdfDdfDd FFF
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PHY 3201 FIZIK KEADAAN PEPEJAL
F
DdfDdU
00
)()()(
F
F
DfdfDdU FF
0
)()](1)[()()()(
F
F
F
DdfDdfDdU
00
)()()()()(
F
F
F
DdfDdfDdU
00
)()()()()(
))()()()()((
00
F
F
F
DdfDdfDd FFF
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F
F
DfdfDdU FF
0)()](1)[()()()(
The energy needed to take electrons from F to
the orbital of energy > F
Energy needed to bring the electrons
to F from orbital below F
dT
DfdfDdd
dT
dUC
F
F
FF
el
0
)()](1)[()()()(
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F
F
DdTdfd
dTdfDd
dTdUC
FFel
0
)()()()()()(
0
)(),(
)(
D
dT
TdfdC Fel
Since the electrons at F are of importance, we make
D() = D(F)
0
),()()(dT
TdfdDC FFel
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If we ignore the variation of the chemical potential with
temperature and assume that = F, which is goodapproximation at room temperature and below. Then
22
1
Tk
Tk
TkdT
Tdf
BF
BF
B
F
/)exp(
/)exp(),(
Tk x
x
B
BF
Tk
Tk
B
FFel
BFB
F
B
F
dx
e
e
Tk
TkxDd
e
e
Tk
DC/)(
)(
)()(
)(
22
32
0
22
11
Therefore,
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Taking into account that F >>kBT, we can put the low
integration limit to minus infinity and obtain
TkDdx
e
exTkDC BF
x
x
BFel
22
2
22
31)()(
Using the density of states for a free electron gas
F
F
ND
2
3)(
we finally obtain
F
BelT
TNkC
2
2 where we defined the Fermi
temperature TF= EF/kB
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The total heat capacity, taking into account the electron
and the lattice contribution, equals to:
3ATTC For temperatures below both the Debye temperature and Fermi Temperature, TF
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Metal expt FEG expt/FEFG =m*/m
Li 1.63 0.749 2.18
Na 1.38 1.094 1.26
K 2.08 1.668 1.25
Cu 0.695 0.505 1.38
Ag 0.646 0.645 1.00
Au 0.729 0.642 1.14
Al 1.35 0.912 1.48
Results for simple metals (in units mJ/mol K)
show that the FEFG values are in reasonable
agreement with experiment, but are always
too high:
The departure from unity involves three
separate effects:
2. Interaction of conduction electrons withphonons (phonons can distort the
lattice that the electrons see .electron-
phonon scattering is common in most
materials)
3. Interaction of the conduction electronswith themselves (these are negative
charges, which should repel one
another!)
1. Interaction of conduction electrons with
the periodic potential of the lattice