Lattice VibrationsLattice VibrationsPart IIIPart III

Solid State PhysicsSolid State Physics

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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