lattice vibrations part iii solid state physics 355
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Lattice VibrationsLattice VibrationsPart IIIPart III
Solid State PhysicsSolid State Physics
355355
Back to Dispersion CurvesBack to Dispersion Curves
We know we can measure the phonon dispersion We know we can measure the phonon dispersion curves -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.
q p
pqq p
pq nUU ,,
PlanckDistributionsum over all
wavevectorssum over allpolarizations
Density of StatesDensity of States
,, /
,
/
1
( )1
B
B
q pq p k T
q p q p
q p
k Tp
U ne
D de
number of modes
unit frequency
D()
Density of States: One Density of States: One DimensionDimension
tpqipqs esqautsqauu ,
0,0 )sin()sin(
determined by the dispersion relation
If the ends are fixed, what modes, or wavelengths, are allowed?
Density of States: One Density of States: One DimensionDimension
# of # of wavelengtwavelengt
hshs
wavelengtwavelengthh
wavevectorwavevector
0.50.5 22LL //LL
11 LL 22//LL
1.51.5 22LL/3/3 33//LL
22 LL/2/2 44//LL
2
q
L
Nq
)1(max
aN
L2
1
2min
Density of States: One Density of States: One DimensionDimension
To calculate the density of states, use
number of modes
unit frequency
D()
There is one mode per interval q = / L with allowed values...
L
N
LLLq
)1(,...,
3,
2,
So, the number of modes per unit range of q is L / .
Density of States: One Density of States: One DimensionDimension
( )
/
g
dND d d
ddN dq
ddq d
L dqd
dL d
d dq
L d
v
To generalize this, go back to the definition...the number of modes is the product of the density of states and the frequency unit.
There is one mode for each mobile atom.
Density of States: One Density of States: One DimensionDimension
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 ConditionsPeriodic Boundary Conditions• No fixed atoms – just require that u(na) = u(na + L).• This is the periodic condition.• The solution for the displacements is
• The allowed q values are then,
( )0 sin( ) i nqa t
su u nqa e
LN
LLq
2,...,
4,
2,0
Density of States: 3 Density of States: 3 DimensionsDimensions
• Let’s say we have a cube with sides of length L.
• Apply the periodic boundary condition for N3 primitive cells:
))()()(()( LzzqLyyqLxxqizzqyyqxxqiee
LN
LLq zyx
2,...,
4,
2,0,,
Density of States: 3 Density of States: 3 DimensionsDimensions
There is one allowed value of q per volume (2/L3) in q space or
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
3
3
82 VL
33 3
4
8q
VN
2
2
( )
2
dND
d
Vq dq
d
qz
qy
qx
Debye Model for Heat Debye Model for Heat CapacityCapacity
,, /
,
/
1
( )1
B
B
q pq p k T
q p q p
q p
k Tp
U ne
D de
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.
Debye Model for Heat Debye Model for Heat CapacityCapacity
2
2
2
2
2
2 2 3
( )2
2
1
2 2
dN Vq dqD
d d
V d
v d v
V V
v v v
Debye Model for Heat Debye Model for Heat CapacityCapacity
D
TBk
p
D
TBk
dev
V
dev
VU
0 /32
2
0 /32
2
123
12
qD
32
333
3
6
v
3
4
2
3
4
2
vV
N
Lq
LN
D
DD
Debye Model for Heat Debye Model for Heat CapacityCapacity
kTxlet
Debye Model for Heat Debye Model for Heat CapacityCapacity
Debye Temperature is related to1. The stiffness of the bonds between atoms2. The velocity of sound in a material, v3. The density of the material, because we can
write the Debye Temperature as:
Debye Model for Heat Debye Model for Heat CapacityCapacity
How did Debye do??
Debye Model for Heat Debye Model for Heat CapacityCapacity
Debye Model for Heat Debye Model for Heat CapacityCapacity
Debye Model for Heat Debye Model for Heat CapacityCapacity
Debye Model for Heat Debye Model for Heat CapacityCapacity
• 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.
Debye Model for Heat Debye Model for Heat CapacityCapacity
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
The temperature in gives a rough demarcation between quantum mechanical regime and the classical regime for the lattice.
DBTk
DDBk
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 Measuring Specific Heat CapacityCapacityDifferential 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 Measuring Specific Heat CapacityCapacity
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