Download Heat Capacity 16

16
1
16.1. Classical (Atomistic) Theory of Heat Capacity
Average energy of one-dimensional harmonic oscillator
E  kBT
Average energy per atom ( three-dimensional harmonic
oscillator )
E  3kBT
It was shown that average kinetic energy of a particle is
expressed as
Ekin
2
3
 k BT
2
(from previous chapter)
Average potential energy of vibrating atom has the same average magnitude
as the kinetic energy.
So the total energy of vibrating atom is expressed as Etot=Epot.+Ekin.
Namely, the total internal energy per mole
E  3N0 kBT
Finally, the molar heat capacity is
 E 
Cv  
  3N 0 k B
 T v
Inserting numerical values for N0 and kB
Cv  25 J/mol  K  5.98 cal/mol  K
Simple model can readily explain the experimentally observed heat capacity except
that the calculated heat capacity turned out to be temperature independent - need
QM explanation.
3
16.2. Quantum Mechanical Considerations-The Phonon
16.2.1 Einstein Model
The energy of the i th energy level of a harmonic oscillator


1
 i   i   hv
2

h = Planck's constant of action
v = frequency of harmonic oscillator
In 1907, Einstein postulated that the energies of the above-mentioned classical
oscillators should be quantized, i.e., he postulated that only certain vibrational modes
should be allowed – these lattice vibration quanta were called phonons.
Phonon waves propagate through the crystal with the speed of sound.
They are elastic waves vibrating in a longitudinal and/or in a transverse mode.
4
The allowed energies of a single oscillator,
En  n 
The average number of phonons at a given temperature is given as
1
N ph 
exp(

k BT
(Bose-Einstein statistics)
) 1
Eosc   N ph 
exp(
Boson: any particle with an integral number of spins.
Photon and phonon are boson particles


k BT
E  3 N 0 Eosc  3N 0
exp(
) 1


k BT
The average energy of an isolated oscillator.
) 1

exp(
)
2


k BT
E

Cv  ( )v  3 N 0 k B 
(Einstein model)


T
 k BT  (exp
 1) 2
k BT
5
Heat capacity at a constant volume
 U  
hv / kT
2 hv
hv / kT
Cv  

3
nhv
(
e

1)
e

kT 2
 T  v
2
e hv / kT
 hv 
 3nk 
 hv / kT
 1) 2
 kT  (e
For large temperatures, e x  1  x applies, which yields
Cv  3nk - follows classical Dulong-Petit value
At low temperatures, Einstein theory predicts an exponential reduction
which is conradictory with the experimental results which show by T 3
Einstein temperature is
hv
E 
k
So,
e E / T
 E 
cv  3R    E / T
 1) 2
 T  (e
6
2
16.2.2 Debye Model
We now refine the Einstein model by taking into account that the atoms in a crystal
interact with each other oscillators are thought to vibrate interdependently.
 Einstein model considered only one frequency of vibration D.
When interactions between the atoms occur, many more frequencies are thought to
exist, which range from about the Einstein frequency down to the acoustical modes of
oscillation.
Debye modified the Einstein equation by replacing the 3N0 oscillators of a single
frequency with the number of modes in a frequency interval, d, and by summing
up over all allowed frequencies.
Applying periodic boundary conditions over N3 primitive cells within a cube of side
length L,

  vs k (velocity of sound is constant: = s )
exp[i ( K x x  K y y  K z z )]
k
 exp[i ( K x ( x  L)  K y ( y  L)  K z ( z  L)]
whence,
L 3 4 3
V3
N ( )
k  2 3
2 3
6 vs
dN
V2
2
4
D( ) 

K x , K y , K z  0; 
; 
; ...
d

2 2 vs 3
L
L
Therfore, there is one allowed value of K per unit volume of
3
V
 L 



2  8 3
7
16.2.2 Debye Model
Total energy of vibration for the solid
Eosc = the average energy of one oscillator
E   Eosc D( )d 
E
3V
2 
2
3
s

e
 / kT
1


 1)
kT
3V  2
D( )  2 3 (in 3-D)
2  s
(exp
3
D
0
=
d
Heat capacity at a constant volume
E
3V 2
Cv  ( ) v  2 3 2
T
2  s kT

D
0
 4e
(e
 / kT
 / kT
 1)
2
d
Or indicating with Debye temperature  D
Cvph
8
T 
 9kN 0  
 D 
3

 D /T
0
x 4e x
d
x
2
(e  1)
12 4
T
Cv 
nk ( )3
5
D
x

kT

hv
,
kT
D 
D
k

hvD
k
D  debye frequency
16.3. Electronic Contribution to The Heat
Capacity
Thermal energy at given temperature
Ekin 
3
3
kBTdN  k BTN ( EF )kBT
2
2
(consider electrons as non-interacting particles)
The Heat capacity of the electrons
 E 
2
Cvel  

3
k
BTN ( EF )

 T v
Population density at E
For E < EF
3N *  electrons 
N ( EF ) 


2 EF 
J

N *  number of eletrons which have an energy equal to or smaller than EF
*
9 N *kB2T
 E 
2 3N
C 

  3kBT
2 EF 2 EF
 T  v
el
v
F
V 2m 3 2 1 2
( )  E (6.8)
2 2 2
*
2
2
2 N
3
EF  (3
) 
(6.11)
V
2m
N (E) 
J
 
K
So far, we assumed that the thermally excited electrons behave like a classical gas.
9
In reality, the excited electrons must obey the Pauli principle. So,
C 
el
v
 2 N *kB2T
2
=
EF
2
2
N *kB
T
TF
EF  k BTF
If we assume a monovalent metal in which we can reasonably assume
one free electron per atom, N* can be equated to the number of
atoms per mole.
C 
el
v
 2 N0 kB2T
2
EF
=
2
2
N 0 kB
T
TF
 J 


 K  mol 
Below the Debye temperature, the heat capacity of metals is
sum of electron and phonon contributions.
Cvtot  Cvel  Cvph   T   T 3
Cvtot

   T 2
T
  3k B2 N ( EF )
 1 
  9kn  
 D 
10
3

 D /T
0
x 4e x
d
x
2
(e  1)
Thermal effective mass
mth*  (obs.)

m0  (calc.)
11