# lecture+13+mak +phonon2

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• 7/30/2019 Lecture+13+MAK +Phonon2

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When a material is heated, its temperature rises. The

rise in temperature is a measure of the change in thetotal internal energy of all atoms or molecules in the

materials. The total internal energy has contributions

mainly from (i) lattice vibration (ii) kinetic energy of free

electrons. Different materials require different amountsof heat energy to raise their temperature by one unit.

The specific heat of a material is defined as the

amount of thermal energy required to raise the

temperature of one kilogram of the substances throughone degree Celcius or 1 K

Specific Heat of Solids

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Classical Theory of

Heat Capacity

Postulate an atom to be a sphere that is held by two

springs at its site. The thermal energy that the harmonicoscillator can absorb is proportional to the absolute

temperature of the environment. Therefore the average

energy of the oscillator is then

TkB

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In 3 D, an atom in cubic crystal

responds to 3 directions. Thus,

each atom represents three

oscillators, each of whichabsorbs the thermal energy =

kBT. Therefore the average

energy per atom in 3-D is

TkB3The total internal energy per mole is TkN B03

The molar heat capacity isB

V

VkN

T

C0

3

Limitation: temperature independent and independent of

material.

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Upon cooling to low temperature, scientists found that this law was no

longer valid.

It also wasnt true for some materials, like diamond.

Also, it should be pointed out that the shape of the curves look

different for different materials

Can we use what we know about phonons to calculate the heat

capacity?

Some of our heat capacity goes to the electrons, and other sources,but in most materials the lattice vibrations absorb most of the energy

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This arises because the solid is treated as an assembly

of independent oscillators

the oscillations of a given ion affect those of its

neighbors. These in turn influence their neighbors and

so on.

In addition, if the internal energy of a solid resides

primarily in the ions, their amplitudes of oscillation must

be expected to vary with temperature.

Thus, a more realistic description of heat capacity must

take these factors in account.

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Quantum Mechanical

Considerations

the phonon

Einstein utilized a quantum

mechanics approach and

incorporated Plank's hypothesisof discrete vibrational

frequencies.

• 7/30/2019 Lecture+13+MAK +Phonon2

An object at any temperature emits

Stefans Law describes the total power

The spectrum of the radiation depends onthe temperature and properties of the

object

• 7/30/2019 Lecture+13+MAK +Phonon2

Experimental data fordistribution of energy inblackbody radiation

As the temperatureincreases, the total amountof energy increases Shown by the area under the

curve

As the temperatureincreases, the peak of thedistribution shifts to shorterwavelengths

• 7/30/2019 Lecture+13+MAK +Phonon2

Wiens Displacement Law

The wavelength of the peak of the

blackbody distribution was found to

follow Weins Displacement Law max T = 0.2898 x 10

-2m K

maxis the wavelength at which the curves

peak T is the absolute temperature of the object

• 7/30/2019 Lecture+13+MAK +Phonon2

The Ultraviolet Catastrophe

Classical theory did not matchthe experimental data

At long wavelengths, the matchis good

At short wavelengths, classicaltheory predicted infinite energy

At short wavelengths,

experiment showed no energy This contradiction is called the

ultraviolet catastrophe

• 7/30/2019 Lecture+13+MAK +Phonon2

Plancks Resolution

Planck hypothesized that the blackbodyradiation was produced by resonators

Resonators were submicroscopic chargedoscillators

The resonators could only have discreteenergies E

n= n h

n is called the quantum number

is the frequency of vibration

h is Plancks constant, 6.626 x 10-34 J s

Key point is quantized energy states

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Max Planck

1858 1947

Introduced a

quantum of action,h

Awarded Nobel

Prize in 1918 for

discovering the

quantized nature of

energy

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Quantum MechanicalConsiderations

the phonon

Einstein postulates that a solid of N atoms can be

considered as being 3N simple harmonic oscillators with

3N quantum states.

The quantum oscillators vibrate independently with the

same frequency =/2.

Each oscillator has only discrete energy given by

En=n

If n=0, it means that the oscillator is not oscillating, n=1,

it means that the oscillator is oscillating with thefundamental frequency. If n=2, the oscillator is oscillating

with twice the fundamental frequency and so on.

The Einstein Model

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2

1nn

Make a transition to Q.M.

Represents equally spaced

energy levels

Energy, E

Energy levels of atoms

vibrating at a single

frequency

Energy of harmonic oscillator

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The average number of phonons, Nph, at a given

temperature vibrating with frequency was found by

Bose and Einstein to obey a special type of statistics:

1

1

Tk

N

B

phonon

exp

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The average energy of an isolated oscillator is then the

average number of phonons times the energy of aphonon:

1

Tk

N

B

phononosc

exp

The thermal energy of a solid can now be calculated by

taking into account that a mole of a substance contains

3N0 oscillators. Therefore, the thermal energy per mole

1

3 0

Tk

N

B

osc

expIf we assume the same frequency

for all the 3N0 oscillators

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Finally, the molar heat capacity is

2

2

0

1

3

Tk

Tk

TkkN

TC

B

B

B

B

V

V

exp

exp

Einstein temperature T is defined by

B

E

EEBE

kk

Rewriting the molar heat capacity in terms of the

Einsteins temperature

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2

2

0

1

3

T

T

T

kNC

E

E

EBV

exp

exp

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For large temperatures, using the

approximation ex 1 + xyielded CV

3N0kBTin agreement with the classical

case or the Dulong-Petit value.

For low temperature such that T

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The Debye Model

Take into account that the atoms interact with each other

and therefore vibrate interdependently. When interaction

occurs between atoms, many more frequencies are

thought to exist. The thermal energy per mole can then

be obtained by modifying the Einstein equation byreplacing the 3N0oscillators of a single frequency with the

number of modes in a frequency interval, d, and by

summing up over all allowed frequencies. The total

energy of vibration for the solid is then

dDosc )(

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The density of states or more accurately thedensity of vibrational modes, D() is defined

so that D() dis the number of modes whose

frequencies lie in the interval and +d. For

continuous medium the density of modes is

3

2

22

3

s

v

VD

)( where vs is the velocity of

sound. where vs is the velocity

of sound.

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D

d

Tkv

V

B

s

0

3

32

12

3

exp

The integration is performed between = 0and a cutoff

frequency, called the Debye frequ