Download P86 1,2,3,5

Chapter 3 Crystal vibration and
thermal properties
§3.1 vibration of a monatomic linear chain
1. The model
Consider the elastic vibration of a 1D lattice (lattice
constant a) with one atom (mass m) in the primitive cell.
2. The vibration of a monatomic linear chain
(1) The total force on the n-th atom:
f    xn 1  xn 1  2 xn 
(2) The equation of motion:
mxn   xn1  xn1  2 xn 
(3) Tentative solution:
xn  Ae i qna t 

qa
sin
(4) Dispersion relation:   2
m
2
(5) Lattice wave: when considering only the nearest
neighbors
and
harmonic
approximation,
the
propagation of a monatomic linear chain can be
expressed as a harmonic plane wave with wavelength
1


2
v

q.
q and velocity
3. The properties of lattice wave
Comparison: elastic wave in continuum medium
xn  Aei qx  t 
4. Periodic boundary conditions (Born-Karman model)
xn  xn N
→
2
q
l
Na
2
§3.2 Two atoms per primitive basis
1. The model
Complex lattice with two different atoms; the
periodicity is 2a
 type p：m
 type q：M （M > m）
2. The vibration of a diatomic linear chain
(1) the total force on the 2n-th atom of type p and the
(2n+1)-th atom of type q:
f p  f left  f right   x2n 1  x2n 1  2 x2n 
f q  f left  f right   x2n 2  x2n  2 x2n 1 
(2) equation of motion
type p: mx2n    x2n 1  x2n 1  2 x2n 
type q: M x2n 1    x2n 2  x2n  2 x2n 1 
(3) Tentative solution
x2n  Ae i q 2na t 
x2n 1  Bei q 2n 1a t 
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(4) dispersion relation
2
2  m 2 A  2 cos qa B  0
 2  cos qa  A  2   M 2 B  0
 

 m  M   m  M  2mM cosq2a  
mM 
2
2
1 2
2   q 
 :  2
   q 
2
(5) conclusion：for a diatomic linear chain, there exist
two independent lattice waves for the same q, and each
has its own dispersion relation. (In general, for 1D
complex lattice with n atoms per primitive cell,
n branches of the dispersion relation: one
acoustical and n  1 optical)
there are
4


3. Properties of acoustical and optical branches
(1) angular frequency 
acoustical
optical branches
branch
min  0
q0
q

2a
min 
2
M
max 
max 
(2) vibration of neighbor atoms


acoustical branch
optical branches
B
m 2  2 

A
2  cos qa
A 2   M 2

B 2  cos qa
4. Periodic boundary conditions
the n-th and (n+N)-atom is identical:
x2n1  x2( n N )1 →
5
eiNq2a  1
2

2
m
Consider a crystal with N primitive cells, each cell has n
atoms:
 the number of frequency branches＝n * dimension
 the number of wave vector q＝the number of
primitive cells
 the number of frequency＝degree of freedom (nN *
dimension）
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§3.3 vibration of 3D complex lattice
1. The model
  
the primitive vectors: a1 , a2 , a3
number of primitive cells along three directions:
N1 , N 2 , N 3
total number of primitive cells: N  N1  N 2  N 3
n atoms per cell with mass m1 , m2 ,, mn




lattice vector of the l-th cell: Rl  l1a1  l2 a2  l3a3
the equilibrium position of the s-th atom in the l-th cell
 l   
R   Rl   s
 s
the displacement in the  direction of the s-th atom in
l
the l-th cell: u  
 s
2. The vibration of 3D complex lattice
(1) the force of the s-th atom in the l-th cell
l
U
 ll '   l ' 
f    
   D '  u '  
l
 s
 ss '   s' 
l ' ' s '
u  
 s
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(2) equation of motion
l
 ll '   l ' 
ms u      D '  u '  
 s
 ss '   s' 
l ' ' s '
lattice wave solution
 
l 
1
i q  Rl t 
u   
A ( s)e
ms
s
(3) dispersion relation

q
D '     2  ' ss'  0 3n  3n dynamic matrix
 ss ' 

q
1
 ll '  iq( Rl  Rl ' )
where D '    
D '  e



ss
'
ms ms ' Rl  Rl '
 
 ss ' 

3n solutions:    j (q ) （ j  1,2,3n ） 
dispersion relation
 acoustical branches: 3
 optical branches: 3n  3
3. Periodic boundary conditions
 N1 


l
l

N
2
u    u 
 N 
 s
 s 3


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4. The density of states (DOS):  ( )
DOS：the number of states per unit frequency range
Vc
 j   
2 3 
      j w 
ds
 q 
j
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§3.4 Quantization of lattice vibration:
phonon
1. Harmonic oscillator
(1) Normal coordinate
Q mx
   2Q  0
(2) Hamiltonian canonical equation Q
1
(3) Quantization E n  ( n  )
2
2. Lattice vibration
kinetic energy T 
1
mi ui 2

2 i
1
 2U
ui u k
potential energy U  
2 i ,k ui uk 0
(1) normal coordinates
mi ui   aij Q j
j
T
1
Q j 2

2 j
U
1
 j 2Q j 2

2 j
(2) canonical equation
(3) quantization
 j   j 2Q j  0
Q
1
2
 j  ( n j  ) j
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3. Normal coordinates of monatomic linear chain
general solution： un   Aq e
i ( qna q t )
q
(1) Normal coordinates
Let Q ( q)  Aq Nm e
iq t
, then un 
T
1
Q ( q)Q * ( q)

2 q
U
1
q 2Q ( q)Q * ( q)

2 q
1
Q(q )eiqna

Nm q
( q)  q 2Q ( q)  0
(2) Canonical equation Q
1
(3) Quantization  q  ( n q  )q
2
4. Phonon
(1) The energy of a lattice vibration is quantized. The
quantum of energy ( q ) is called a phonon in
analogy with the photon of the electromagnetic wave.
Note phonon is not real particle!
The crystal vibrations can be described by a
collection of various phonons. Two phonons do not
interact within the harmonic theory. When the
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anharmonic terms are included, the phonons are
interacting.
(2) Phonon is Boson
a) obey BE distribution
b) number of phonons is not conserved

(3) Quasi momentum of phonon: q
(4) Phonon is non-local particle
4. Inelastic scattering by phonons: measure the
dispersion relations
Lattice vibration can induce inelastic scattering of
neutron: phonon creation or adsorption
conservation of energy:
 2k 2  2k '2

  q
2M
2M
conservation of momentum:
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k  k ' q  K h
§3.5 Heat capacity of solid
1. Definition and experimental facts
 E 
(1) definition CV  


T
 V
(2) experimental facts
a) at high temperature CV  3Nk B
b) at low temperature CV  T  bT 3
2. The problems of classic theory in heat capacity
 can not explain the low temperature experimental
facts
 why the contribution from electrons is very small
3. Quantum theory of lattice heat capacity
(1) average total energy of lattice vibration
E
max

0

 ( )d
exp( / k B T )  1
 E 
CV  
 

T

V
max

0
(
 2 exp( / k B T )
)
 ( )d
k B T [exp( / k B T )  1]2
the key is to find DOS:  ( )
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(2) Einstein model
all
atoms
have
the
same
frequency:
 ()  3N (  E )
(3) Debye model
In the Debye approximation the velocity of lattice
wave is taken as constant for each polarization type, as it
would be for a classical elastic continuum.
 ( ) 
3V
2 2 v p
2

3
30
25
25
:10 rad/s
30
20
15
15
10
10
5
5
13
20
0

K
M
2D graphene

14
0
DOS
thermal capacity
2000
CV:J/kg*K
1500
1000
2D graphene
tube(3,3)
tube(4,2)
tube(5,0)
tube(5,0)+Li
500
0
0
600
1200
temperature:K
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§3.6 Anharmonic effects
1. Phonon interactions
Harmonic approximation: independent harmonic
oscillator, no phonon-phonon interactions
Anharmonic terms: harmonic oscillator is not
independent, energy exchange between phonons
→ thermal equilibrium, thermal conductivity
Because of the anharmonic effects, two phonons
interact with each other to produce a third phonon.
“phonon collision”
conservation of energy： 1  2  3




conservation of momentum： q1  q2  q3  K h

N process： K h  0

U process： K h  0
2. Equation of state and thermal expansion
dU
E
（Grüneisen approximation）

dV
V
   cV / B0
p
3. Thermal conductivity
1
  cV v p l
3
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Homework
P86 1,2,3,5
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