Transcript Document

ME 381R Lecture 6: Lattice Specific Heat
Dr. Li Shi
Department of Mechanical Engineering
The University of Texas at Austin
Austin, TX 78712
www.me.utexas.edu/~lishi
[email protected]
•Reading: 1-3-2 in Tien et al
•Reference: Ch5 in Kittel
Density of Phonon States in 1D
A linear chain of N=10 atoms
with two ends jointed
a
Only N wavevectors (K) are allowed (one per mobile atom):
K=
-8p/L
-6p/L -4p/L -2p/L
0 2p/L
4p/L
6p/L 8p/L p/a=Np/L
Only 1 K state lies within a DK interval of 2p/L
# of states between K and K + dK is: (L/2p)dK
# of K-vibrational modes between w and w+dw :
D(w)dw 
D(w): density of states
L dw
2p dw / dK
2
Density of States in 3D
2p 4p
Np
K x , K y , K z  0;
;
;...;
L
L
L
N3: # of atoms
Kz
Ky
Kx
2p/L
D(w ) 
VK 2
2p
2
1
;
dw / dK
V  L3
3
Lattice Specific Heat
El  
p


1

  n w K , p  2 w K , p
K
p: polarization(LA,TA, LO, TO)
K: wave vector
1
4pK 2 dK

El     n wK , p  wK , p
2
p

2p L3


Dispersion Relation: K  g w 
Energy Density:
D(w ) 
VK 2
2p 2
1
dw / dK
1



l     n w  wDw dw / V
2
p

Lattice Specific Heat:
d l
Cl 

dT
p

d n
dT
wDw dw
4
Debye Approximation:
w  vs K
2
2
Debye Density of Dw   V g w  dg  V w
States
2p 2 dw
2p 2vs3
Number of Atoms:
N
Debye cut-off Wave Vector
Debye Cut-off Freq.
4
Frequency, w
Debye Model
w  vs K
3
p
K
D
3
2p L
3

K D  6p 
2

1
3
Wave vector, K p/a
0
Debye Temperature [K]
w D  vs K D

w D vs 6p 
Debye Temperature  D 

kB
kB
2

1
3
C(dimnd)
Si
Ge
B
Al
1860
625
360
1250
394
Ga
NaF
NaCl
NaBr
NaI
240
492
321
224
164
5
Lattice Specific Heat
Energy Density
l  
p
Specific Heat
wD

0
D


T 3
1  w
x dx  4
 9k B



n
w

d
w

 3  x T

2  2p 2vs3
  D 0 e  1


w
x
k BT
3
3  D T
 T  

Cl  9k B 

 D 


0

e x x 4 dx 
2
e x 1 


10 7
C  3 kB  4.7 10 6 J m3 K
When T << D,
l  T 4 , Cl  T 3
3
Specific Heat, C (J/m -K)
10 6
Diam ond
10 5
10 4
10
3
10
2
10 1 1
10
C  T3
Classical
Regime
Quantum
Regime
 D  1860 K
10 2
10 3
Te m pe r atur e , T (K)
10 4
6