lattice vibrations part ii solid state physics 355

Lattice VibrationsLattice VibrationsPart IIPart II

Solid State PhysicsSolid State Physics

355355

Three DimensionsThree Dimensions

For each mode in a given propagation direction,the dispersion relation yields acoustic and optical branches:

• Acoustic• Longitudinal (LA)• Transverse (TA)

• Optical• Longitudinal (LO)• Transverse (TO)

If there are p atoms in the primitive cell, there are 3p branches in the dispersion relation: 3 acoustic and 3p -3 optical.

NaCl – two atoms per primitive cell

6 branches:

1 LA

1 LO

2 TA

2 TO

CountingCounting

This enumeration follows from counting the number of degrees of freedom of the atoms. For p atoms in N primitive cells, there are pN atoms. Each atom has 3 degrees of freedom, one for each of the 3 directions x, y, and z. This gives 3Np degrees of freedom for the crystal.

q = ±/a

q.

q

Three DimensionsThree Dimensions

Al Ge

) ( trqin ueu

Quantization of Elastic WavesQuantization of Elastic WavesThe energy of an elastic mode of angular frequency is

It is quantized, in the form of phonons, similar to the quantization of light, as both are derived from a discrete harmonic oscillator model.

Elastic waves in crystals are made up of phonons. Thermal vibrations are thermally excited phonons.

En n 12

Phonon MomentumPhonon Momentum

A phonon with a wavevector q will interact with A phonon with a wavevector q will interact with particles, like neutrons, photons, electrons, as if it particles, like neutrons, photons, electrons, as if it had a momentum (the crystal momentum)had a momentum (the crystal momentum)

qp

• Be careful! Phonons do not carry momentum like photons do. They can interact with particles as if they have a momentum. For example, a neutron can hit a crystal and start a wave by transferring momentum to the lattice.

• However, this momentum is transferred to the lattice as a whole. The atoms themselves are not being translated permanently from their equilibrium positions.

• The only exception occurs when q = 0, where the whole lattice translates. This, of course, does carry momentum.

R

rr R

Proton AProton B

electron

Phonon MomentumPhonon Momentum

H2

For example, in a hydrogen molecule the internuclearvibrational coordinater1 r2 is a relative coordinate and doesn’t have linear momentum.

The center of mass coordinate ½(r1 r2 )corresponds to the uniform mode q = 0 and can have linear momentum.

r1 r2

O

Phonon MomentumPhonon Momentum

Earlier, we saw that the elastic scattering of x-rays from the lattice is governed by the rule:

Gkk

If the photon scattering is inelastic, with a creation of a phonon of wavevector q, then

Gkqk

qp

If the photon is absorbed, then

k k G q

Phonon Scattering (Normal Phonon Scattering (Normal Process)Process)

q1

q2

q3 = q1 + q2

q3 = q1 + q2 or q3 = q1 + q2 + G

Measuring PhononsMeasuring Phonons

Gkqk

scattered neutron

phonon wavevector(+ for phonon created, for phonon absorbed)

incident neutron

reciprocal lattice vector

Stokes or anti-Stokes Process

Measuring PhononsMeasuring Phonons

q

Measuring PhononsMeasuring Phonons

Measuring PhononsMeasuring Phonons

• Inelastic X-ray Spectroscopy• Raman Spectroscopy (IR, near IR, and visible light)• Microwave Ultrasonics

Other Techniques

Heat CapacityHeat Capacity

VV T

UC

Cv = yT+T3

You may remember from your study of thermal physics thatthe specific heat is the amount of energy per unit mass required to raise the temperature by one degree Celsius. Q = mcT

Thermodynamic models give us this definition:

electrons phonons

Heat CapacityHeat Capacity

Equipartition Theorem:The internal energy of a system of N particles is

Monatomic particles have only 3 translational degrees of freedom. They possess no rotational or vibrational degrees of freedom. Thus the average energy per degree of freedom is

It turns out that this is a general result.

TNkB23

TNkB21

The mean energy is spread equally over all degrees of freedom, hence the terminology – equipartition.

Heat CapacityHeat Capacity

Answer: You need to use quantum statistics to describe this properly.

Bosons and FermionsBosons: particles that can be in the same

energy state (e.g. photons, phonons)Fermions: particles that cannot be in the same

energy level (e.g. electrons)

Heat CapacityHeat Capacity

Planck DistributionPlanck Distribution

Max Planck – first to come up Max Planck – first to come up with the idea of quantum with the idea of quantum energyenergy

worked to explain blackbody worked to explain blackbody radiationradiation empty cavity at empty cavity at

temperature temperature TT, with which , with which the photons are in the photons are in equilibriumequilibrium

Planck DistributionPlanck Distribution

Einstein ModelEinstein Model

1907-Einstein developed first reasonably 1907-Einstein developed first reasonably satisfactory theory of specific heat satisfactory theory of specific heat capacity for a solidcapacity for a solid

assumed a crystal lattice structure assumed a crystal lattice structure comprising comprising NN atoms that are treated as an atoms that are treated as an assembly of 3assembly of 3NN one-dimensional one-dimensional oscillatorsoscillators

approximated all atoms vibrating at the approximated all atoms vibrating at the same frequency (unrealistic, but makes same frequency (unrealistic, but makes things easier)things easier)

Planck DistributionPlanck Distribution

number of phonons in energy level n

total number of phonons

all possible energy levels 0, 1, 2, etc.

Fraction of small as n gets large

Phononsat energy n a constant

Planck DistributionPlanck Distribution

n n nhf

/ /

/ /

0 0 0

n

n

kT n kTn

kT n kTn

N e e

N e e

Planck DistributionPlanck Distribution

average occupied

energylevel

//

0 0 0

1 1 1

1 1 1B

B

n k Tn nk T

n n n

x e ex e e

0 0

1 1

1 1n n

n n

x ex e

0

1

1n

n

xx

0 0

n n

n n

nx xx

20 0

1

1 1

n n

n n

xnx x x x

x x x x

0

n

n

nx

Einstein ModelEinstein Modelaverage

energyper oscillator

We have 3N such oscillators, so the total energy is

Einstein ModelEinstein Model

2 and

let

TkdT

dv

Tkv

B

B

Einstein ModelEinstein Model

T

How did Einstein do?

Einstein ModelEinstein Model

How did Einstein do?

K0T

Einstein ModelEinstein Model

The Einstein model failed to identically match the behavior of real solids, but it showed the way.

In real solids, the lattice can vibrate at more than one frequency at a time.

Answer: the Debye Model