Transcript Slide 1

Lattice Vibrations
Part III
Solid State Physics
355
Back to Dispersion Curves


We know we can measure the phonon dispersion curves
- the dependence of the phonon frequencies upon the
wavevector q.
To calculate the heat capacity, we begin by summing
over all the energies of all the possible phonon modes,
multiplied by the Planck Distribution.
U
U q, p   n  q, p
q
sum over all
wavevectors
p
q
p
Planck
Distribution
sum over all
polarizations
Density of States
U    n  q , p  
q
p
   D( )
p
q
q , p
e
 / k BT
1
p
q , p
e
 / k BT
d
number of modes
D()
unit frequency
1
Density of States: One Dimension
us  u0 sin(sqa  q, p t )  u0 sin(sqa) e
iq, pt
determined by the dispersion relation
If the ends are fixed, what modes, or wavelengths, are allowed?
Density of States: One Dimension
# of
wavelengths
2
wavelength wavevector q  
0.5
2L
/L
1
L
2/L
1.5
2L/3
3/L
2
L/2
4/L
2L
 2a
N 1
( N  1)
qmax 
L
min 
Density of States: One Dimension
To calculate the density of states, use
number of modes
D()
unit frequency
There is one mode per interval  q =  / L with allowed values...
( N  1) 
 2 3
q   ,
, ,...,

L
L L L

So, the number of modes per unit range of q is L / .
Density of States: One Dimension
There is one mode for each
mobile atom.
To generalize this, go back to the
definition...the number of modes
is the product of the density of
states and the frequency unit.
dN
D( )d  
d
d
dN dq

d
dq d 
L dq

d
 d
L d

 d / dq
L d

 vg
Density of States: One Dimension
monatomic lattice
diatomic lattice
• Knowing the dispersion curve we can calculate the group
velocity, d/dq.
• Near the zone boundaries, the group velocity goes to
zero and the density of states goes to infinity. This is
called a singularity.
Periodic Boundary Conditions
• No fixed atoms – just require that u(na) = u(na + L).
• This is the periodic condition.
• The solution for the displacements is
i ( nqa t )
us  u0 sin(nqa) e
• The allowed q values are then,
2 
 2 4
q  0,
,
,..., N

L
L
L

Density of States: 3 Dimensions
• Let’s say we have a cube with
sides of length L.
• Apply the periodic boundary
condition for N3 primitive cells:
i ( qx xq y y qz z )
e
i ( qx ( x L)q y ( y  L)qz ( z  L))
e
2 
 2 4
qx, y,z  0,
,
,..., N

L
L
L

Density of States: 3 Dimensions
qz
There is one allowed value of q per
volume (2/L3) in q space or
3
V
 L 



 2 
8 3
allowed values of q per unit volume of
q space, for each polarization, and for
each branch.
The total number of modes for each
polarization with wavevector less than
q is
 V 4
N   3  q 3
 8  3
qy
qx
dN
D( ) 
d
Vq 2 dq
 2
Debye Model for Heat Capacity
U    n  q , p  
q
p
   D( )
p
q
q , p
e
 / k BT
1
p
q , p
e
 / k BT
d
Debye Approximation:
For small values of q, there is a linear
relationship =vq, where v is the speed of
sound.
...true for lowest energies, long
wavelengths
This will allow us to calculate the density of
states.
1
Debye Model for Heat Capacity
2
dN Vq dq
D ( ) 
 2
d  2 d 
V   d  


 
2 
2  v  d   v 
2
V   1
V

  
2 
2 3
2  v   v  2 v
2
Debye Model for Heat Capacity
3
3
 L  4
 L  4
3
N 
  qD  
 
 2  3
 2  3
 
 D
 v 


3
N 3
 D  6
v
V
2
U
 D  V 2 
 0
p
 D  V 2 
0
3



d
2
3


/
k
T
 2 v  e
B 1





d
2
3


/
k
T
 2 v  e
B 1


qD
Debye Model for Heat Capacity

let x 
kT
Debye Model for Heat Capacity
Debye Temperature is related to
1. The stiffness of the bonds between atoms
2. The velocity of sound in a material, v
3. The density of the material, because we can
write the Debye Temperature as:
Debye Model for Heat Capacity
How did Debye do??
Debye Model for Heat Capacity
Debye Model for Heat Capacity
Debye Model for Heat Capacity
Debye Model for Heat Capacity
Debye Model for Heat Capacity
•
Einstein's oscillator treatment of specific heat gave
qualitative agreement with experiment and gave the
correct high temperature limit (the Law of Dulong
and Petit).
•
The quantitative fit to experiment was improved by
Debye's recognition that there was a maximum
number of modes of vibration in a solid.
•
He pictured the vibrations as standing wave modes
in the crystal, similar to the electromagnetic modes
in a cavity which successfully explained blackbody
radiation.
ωD represents the maximum frequency of a normal
mode in this model.
ωD is the energy level spacing of the oscillator of
maximum frequency (or the maximum energy of a
phonon).
It is to be expected that the quantum nature of the
system will continue to be evident as long as
k BT  D
The temperature in k B D  D gives a rough
demarcation between quantum mechanical regime
and the classical regime for the lattice.
Typical Debye frequency:
(a) Typical speed of sound in a solid ~ 5×103 m/s. A
simple cubic lattice, with side a = 0.3 nm, gives
ωD ≈ 5×1013 rad/s.
(b) We could assume that kmax ≈ /a, and use the linear
approximation to get
ωD ≈ vsound kmax ≈ 5×1013 rad/s.
A typical Debye temperature:
θD ≈ 450 K
Most elemental solids have θD somewhat below this.
Measuring Specific Heat Capacity
Differential scanning calorimetry (DSC) is a
relatively fast and reliable method for
measuring the enthalpy and heat capacity for
a wide range of materials. The temperature
differential between an empty pan and the
pan containing the sample is monitored while
the furnace follows a fixed rate of
temperature increase/decrease. The sample
results are then compared with a known
material undergoing the same temperature
program.
Measuring Specific Heat Capacity