lattice vibrations part iv
DESCRIPTION
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 k B T; and the lattice expands. Thermal Expansion. - PowerPoint PPT PresentationTRANSCRIPT
Lattice VibrationsLattice VibrationsPart IVPart IV
Solid State PhysicsSolid State Physics
355355
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.
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.
Thermal ExpansionThermal Expansion
Dilatometer
Thermal ExpansionThermal Expansion
TL
L
0
alumina (Al2O3)
Thermal ExpansionThermal Expansion
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.
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
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
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
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
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
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
Thermal DiffusivityThermal Diffusivity
VC v
31
Laser Flash Diffusivity
222 /
10
)1(21 dtn
n
neV
V
mm t
d 2
Thermal DiffusivityThermal Diffusivity
Thermal ConductivityThermal Conductivity
Thermal ConductivityThermal Conductivity
Phonon ScatteringPhonon Scattering
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.
Phonon ScatteringPhonon Scattering
1q
2q 3q
213
213 qqq
1q 2q
3q
321
321 qqq
Phonon Scattering: N Phonon Scattering: N ProcessesProcesses
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.
Phonon ScatteringPhonon Scattering