Lecture+16+MAK +Heat+Cap+Electron+Gas

7/30/2019 Lecture+16+MAK +Heat+Cap+Electron+Gas

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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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Source: Chris Wiebe


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Source: Chris Wiebe


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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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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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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


7/12PHY 3201 FIZIK KEADAAN PEPEJAL

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)[()()()(



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



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,



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



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

Lecture+16+MAK +Heat+Cap+Electron+Gas

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