Lattice Vibrations Part IV
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Transcript of Lattice Vibrations Part IV
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Lattice VibrationsLattice VibrationsPart IVPart IV
Solid State PhysicsSolid State Physics
355355
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Thermal ExpansionThermal Expansion
32)( AxCxxU
• 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.
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Thermal ExpansionThermal ExpansionThe average displacement is determined from...
dxe
dxxex
TBkxU
TBkxU
0
/)(
0
/)(
After some manipulation...
TkC
Ax B24
3 If A is zero, there is no
thermal expansion.
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Thermal ExpansionThermal Expansion
Dilatometer
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Thermal ExpansionThermal Expansion
TL
L
0
alumina (Al2O3)
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Thermal ExpansionThermal Expansion
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Negative Thermal ExpansionNegative 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.
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Negative Thermal ExpansionNegative Thermal Expansion
Many tetrahedrally bonded materials show negative thermal expansion at lowtemperatures; for example, the thermal expansion of ice Ih becomes negative below80 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
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Thermal ConductivityThermal 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.
L
TA
dt
dQ
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Thermal ConductivityThermal ConductivityMaterial Thermal Conductivity
(W/m-K)
CC
CuAgCuAg
AgAg
CuCu
AuAu
AlAl
brassbrass
PlPl
quartzquartz
glassglass
waterwater
woolwool
polystyrenepolystyrene
aerogelaerogel
1000-26001000-2600
>430>430
430430
390390
320320
236236
111111
7070
88
11
0.60.6
0.050.05
0.030.03
0.0000170.000017
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Thermal ConductivityThermal 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.
• From the kinetic theory of gases...
cv31
mean free path
average particle velocity
specific heat capacity per unit volume
TH TC
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Thermal ConductivityThermal 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
• The amount of energy carried by each phonon is then,
• The number of phonons passing through a unit area per unit time is the
vdx
dTd
dx
dTT
Cvdx
dTTCE
vn flux phonon
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Thermal ConductivityThermal Conductivity
• The net flux of energy is then,
dx
dT
dx
dTcv
dx
dTCvn
dx
dTCvn
dx
dTCvvn
TCvn j
x
xx
x
31
231
2
2312
2222
vv
vvvv
x
zyx
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Thermal DiffusivityThermal Diffusivity
VC v
31
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Laser Flash Diffusivity
222 /
10
)1(21 dtn
n
neV
V
mm t
d 2
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Thermal DiffusivityThermal Diffusivity
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Thermal ConductivityThermal Conductivity
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Thermal ConductivityThermal Conductivity
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Phonon ScatteringPhonon Scattering
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Phonon ScatteringPhonon 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.
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Phonon ScatteringPhonon Scattering
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1q
2q 3q
213
213 qqq
1q 2q
3q
321
321 qqq
Phonon Scattering: N Phonon Scattering: N ProcessesProcesses
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Phonon Scattering: U Phonon Scattering: U ProcessesProcesses
G
1q
2q
3q
213
321 Gqqq
zone. louinfirst Bril theinit keep toGby reduced
bemust wavevectorresultant the then,a
q q If 21
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.
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Phonon ScatteringPhonon Scattering