Download - Lattice Vibrations Part IV

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.


Dilatometer


TL

L

0

alumina (Al2O3)

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


• 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

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.

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.