Lattice Vibrations
Part IV
Solid State Physics
355
Thermal Expansion
•
Anharmonic effects can be important for physical
properties.
•
As you heat up the solid, internal energy of the
lattice increases as kBT; and the lattice expands.
U ( x )  Cx 2  Ax 3
Thermal Expansion
The average displacement is determined from...


 x  0 
U ( x ) / k BT
e
dx
0
xe U ( x ) / k BT dx
After some manipulation...
 x 
3A
k T
2 B
4C
If A is zero, there is no
thermal expansion.
Thermal Expansion
Dilatometer
Thermal Expansion
L
 T
L0
alumina (Al2O3)
Thermal Expansion
Negative Thermal Expansion
Zirconium tungstate
exhibits “negative
thermal expansion”
from 0.3 K up to at
least 1400 K.
The structure of ZrW2O8
consists of a framework of
ZrO6 octahedra and WO4
tetrahedra linked at
corners, but with one of the
corners of the WO4
tetrahedra remaining
unlinked.
Negative Thermal Expansion
Many tetrahedrally bonded
materials show negative
thermal expansion at low
temperatures; for example, the
thermal expansion of ice Ih
becomes negative below
80 K. The dynamics of ice, even
in its natural hexagonal form,
are still a puzzle despite many
decades of work. The
combination of the rotational
disorder and the complexity of
the inter-molecular forces make
modeling the system difficult.
H2O, Si, Ge, ZnSe, GaP, GaAs
Thermal Conductivity
What is heat?
Heat is the spontaneous
flow of energy from an
object at a higher
temperature to an object at
a lower temperature.
dQ
T
 A
dt
L

Thermal Conductivity
Material
Thermal Conductivity (W/m-K)
C
CuAg
Ag
Cu
Au
Al
brass
Pl
quartz
glass
water
wool
polystyrene
aerogel
1000-2600
>430
430
390
320
236
111
70
8
1
0.6
0.05
0.03
0.000017
Thermal Conductivity
• Thermal conduction is a diffusion process and proceeds
via the random movement of electrons and phonons.
• These particles carry energy from one part of the solid,
where the internal energy is higher toward a region
where the internal energy is lower. T
H
• From the kinetic theory of gases...

1 cv
3
average particle velocity
mean free path
specific heat capacity per unit volume
TC
Thermal Conductivity
• As a phonon moves a distance d, it will reduce the
temperature by T as it carries energy away.
• This change in temperature is
dT
dT
T 
d
v
dx
dx
• The amount of energy carried by each phonon is then,
dT
E  CT 
Cv
dx
• The number of phonons passing through a unit area per
unit time is the
phonon flux  n v
Thermal Conductivity
• The net flux of energy is then,
j   n  v x  CT
dT
  n  v x  v x C
dx
dT
2
 n  v x   C
dx
dT
2
1
  3 n  v  C
dx
dT
dT
1
  3 v c

dx
dx
 v x2    v 2y    v z2  v 2 
  v x2  13  v 2 
Thermal Diffusivity
   CV
  13 v
Laser Flash Diffusivity

V
n  n 2 2t / d 2
 1  2 (1) e
V0
n 1
d2
  m
tm
Thermal Diffusivity
Thermal Conductivity
Thermal Conductivity
Phonon Scattering
Phonon Scattering
• Phonon scattering with other
phonons is the result of
anharmonic effects.
• If the forces between atoms
were purely harmonic, there
would be no mechanism for
collisions between different
phonons; and the mean free
path would be limited solely
by geometrical influences
such as boundaries and
imperfections.
Phonon Scattering
Phonon Scattering: N Processes

q2

q1

q1

q3

q2

q3
  
q 3  q1  q 2
3  1  2
  
q1  q 2  q 3
1  2  3
Phonon Scattering: U Processes

G

q1

q3

q2

  
q1  q 2  q 3  G
3  1  2
 

If q1  q 2  , then the resultant wavevector must be
a
reduced by G to keep it in the first Bril louin zone.
The physical result of all this is that a phonon comes along and
“experiences” a different local “stiffness” due to the strain caused by
another phonon.
Phonon Scattering