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  • 7/30/2019 Lecture+13+MAK +Phonon2

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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    Quantum Mechanical

    Considerations

    the phonon

    Einstein utilized a quantum

    mechanics approach and

    incorporated Plank's hypothesisof discrete vibrational

    frequencies.

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    7/24PHY 3201 FIZIK KEADAAN PEPEJAL

    Blackbody Radiation

    An object at any temperature emits

    electromagnetic radiation

    Sometimes called thermal radiation

    Stefans Law describes the total power

    radiated

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

    object

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    Blackbody Radiation Graph

    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

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    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

    emitting the radiation

  • 7/30/2019 Lecture+13+MAK +Phonon2

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    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

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    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    Max Planck

    1858 1947

    Introduced a

    quantum of action,h

    Awarded Nobel

    Prize in 1918 for

    discovering the

    quantized nature of

    energy

  • 7/30/2019 Lecture+13+MAK +Phonon2

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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    2

    2

    0

    1

    3

    T

    T

    T

    kNC

    E

    E

    EBV

    exp

    exp

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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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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    PHY 3201 FIZIK KEADAAN PEPEJAL

    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