Post on 14-Apr-2018
7/30/2019 Lecture+13+MAK +Phonon2
1/24
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
7/30/2019 Lecture+13+MAK +Phonon2
2/24
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
7/30/2019 Lecture+13+MAK +Phonon2
3/24
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.
7/30/2019 Lecture+13+MAK +Phonon2
4/24
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Upon cooling to low temperature, scientists found that this law was no
longer valid.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
7/30/2019 Lecture+13+MAK +Phonon2
5/24
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.
7/30/2019 Lecture+13+MAK +Phonon2
6/24
PHY 3201 FIZIK KEADAAN PEPEJAL
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
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
7/30/2019 Lecture+13+MAK +Phonon2
8/24PHY 3201 FIZIK KEADAAN PEPEJAL
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
7/30/2019 Lecture+13+MAK +Phonon2
9/24PHY 3201 FIZIK KEADAAN PEPEJAL
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
10/24PHY 3201 FIZIK KEADAAN PEPEJAL
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
11/24PHY 3201 FIZIK KEADAAN PEPEJAL
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
7/30/2019 Lecture+13+MAK +Phonon2
12/24
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
13/24
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
7/30/2019 Lecture+13+MAK +Phonon2
14/24
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
7/30/2019 Lecture+13+MAK +Phonon2
15/24
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
7/30/2019 Lecture+13+MAK +Phonon2
16/24
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
7/30/2019 Lecture+13+MAK +Phonon2
17/24
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
7/30/2019 Lecture+13+MAK +Phonon2
18/24
PHY 3201 FIZIK KEADAAN PEPEJAL
2
2
0
1
3
T
T
T
kNC
E
E
EBV
exp
exp
7/30/2019 Lecture+13+MAK +Phonon2
19/24
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
7/30/2019 Lecture+13+MAK +Phonon2
20/24
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 )(
7/30/2019 Lecture+13+MAK +Phonon2
21/24
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.
7/30/2019 Lecture+13+MAK +Phonon2
22/24
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 frequency,
Dwhich isdetermined by postulating that the total number of modes
must be equal to the number of degrees of freedom.
The molar heat capacity, CV is then
D
d
Tk
Tk
Tkv
VC
B
B
Bs
V
0
2
4
232
2
12
3
exp
expor
7/30/2019 Lecture+13+MAK +Phonon2
23/24
PHY 3201 FIZIK KEADAAN PEPEJAL
T
x
x
B
phD
V
dxe
exTNkC
/
0 2
43
0
19
7/30/2019 Lecture+13+MAK +Phonon2
24/24
PHY 3201 FIZIK KEADAAN PEPEJAL
T
x
x
B
phD
V
dxe
exTNkC
/
0 2
43
0
19
BB kTkx Dand
is called the Debye temperature
At very low temperatures, the heat capacity is found fromthe assumption that /T , when
323
00
3
2341511
TNkC
e
xdx
e
xdx BVx
x
x
D
Therefore, at very low temperatures, the T3 approximation
is quite good model for the acoustic modes.