Download Blackbody radiation and ultra-violet catastrophe

Dualisme Cahaya Sebagai
Gelombang dan Partikel
Wave Properties
Light intensity = 𝐼 ∝ 𝐸 2
Particle Properties
Light intensity: 𝐼 = 𝑁ℎ𝑓
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Quantum Theory of Light
Under a constant frequency, the smallest
energy unit of light is quantized.
For one photon
ℎ𝑐
𝐸 = ℎ𝑓 =
𝜆
For N photon
𝐸 = 𝑁ℎ𝑓
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Spectrum of electromagnetic radiation
Free electrons in resonance
Transition of bound electrons in atoms
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Nucleus
1.3.2 Photoelectric effect
http://lectureonline.cl.msu.edu/~mmp/kap28/PhotoEffect/photo.htm
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Photoelectric effect applet
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Observations
• Monochromatic light is incident to one of the
electrodes made by a particular metal. The
induced current called photocurrent is
collected.
• If V is fixed, there exists a threshold frequency
o, below which there is no photocurrent.
• Different electrode materials have threshold
frequency o
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Observations
• The photocurrent is constant above the threshold
frequency under a constant illumination, regardless of
the frequency.
• The photocurrent is proportional with the intensity.
• The energy of electron is proportional with the
frequency.
• For a particular electrode and frequency of light, a
stopping voltage Vs exists. No photocurrent can be
collected regardless of the intensity of light.
• There is no measurable tie lag between the illumination
of light and the release of photoelectrons. (10-9s)
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Graphical presentations

o
Energy of photoelectrons is
proportional to the frequency;
Existence of threshold frequency
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Photocurrent is proportional to light
intensity
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Different frequency of light has different
stopping voltage
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Stopping voltage varies with
(1)electrode material for the same
frequency
(2) increases with frequency of light for
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the same electrode
Einstein’s interpretation
Under a constant frequency, the smallest
energy unit of light is quantized.
E  h
One photon can at most
release one electron,
regardless of the photon
energy
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Work Function, W– the minimum energy for
electron to escape from the metal surface
h  K  W
where K is the kinetic energy of the photoelectron
Photocurrent I p  nq
where  is the quantum efficiency, n number of
photons striking to the electrode per second, q
electron charge
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The number of photons striking to the electrode is
related to the light intensity IL by
A
n  IL
h
Where A is the area of the electrode exposed to
light.
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Threshold frequency
When K = 0, there is no photoelectrons. The
frequency reaches the threshold frequency, i.e.
h o  W
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Stopping voltage
The reverse biasing voltage: where the photoelectrode
is positively biased and the other negatively biased.
When the biasing
voltage increases
from zero, the
photocurrent
decreases. At V  VS,
IP = 0
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Stopping voltage
The physical meaning is that the potential
difference barrier is decelerating the electron,
and finally consumes all its kinetic energy. Thus,
the stopping voltage gives us a tool to determine
the kinetic energy,
K  qVS
v0
K
1 2
mv
2
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mgh qVS
Stopping voltage against frequency
qVS  K  h  W
h
W
VS   
q
q
Stopping
Voltage
VS
Slope = h /q
o
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
Example:
Cahaya ultraviolet dengan panjang gelombang 350 nm dan intensitas 1 W/m2
mengenai permukaan elektroda yang “work function” nya 2.2 eV.
a. Hitung energi kinetik elektron yang dilepas oleh elektroda tersebut.
b. Jika 1% dari photon yang datang diubah menjadi fotoelektron, berapa jumlah
elektron yang dilepas per detik jika luas permukaan 1 cm2
Diketahui konstanta Planck =6.626x10−34 Js = 4.136x10−15 eVs
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Failure of classical theory
• In 1902, Philipp Eduard Anton von Lenard
observed that the energy of the emitted
electrons increased with the frequency of the
light. This was at odds with James Clerk
Maxwell's wave theory of light, which
predicted that the energy would be
proportional to the intensity of the radiation.
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• Within the experimental accuracy, (about 10-9
second), there is no time delay for the emission
of photoelectrons. In terms of wave theory, the
energy is uniformly distributed across its
wavefront. A period of time is required to
accumulate enough energy for the release of
electrons.
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Inconsistency of Classical wave theory and
experiment
I~
8hc
1

5
 exp( hc / kT )  1
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8kT
4
1.3.1 What is blackbody?
• Under thermal equilibrium, an object of a finite
temperature emit radiation (supposed to be EM
wave).
• It is found that an object that absorbs more also
emits more radiation.
• A perfect absorber is an object with black surface
that must of the incident energy is absorbed. It is
also expected to be a perfect radiator.
• Consequently, a blackbody is a perfect radiation.
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Phenomena for a blackbody
• Temperature-dependence emission spectrum.
• Reciprocal relation between wavelength at the peak
intensity max and ambient temperature T
• Inconsistency between observed spectrum and the
prediction from classical theory (Reyleigh and Jeans)
• A model suggested by Planck (quanta of light energy)
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Temperature dependence of emission
spectrum
http://webphysics.davidson.edu/alumni/MiLee/java/bb_mjl.htm
http://surendranath.tripod.com/AppletsJ2.html
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Classic: Rayleigh – Jeans Formula
𝐸 = 𝑘𝐵 𝑇
Total Energy in the cavity with frequency interval between v and v+dv
8𝜋𝑘𝐵 𝑇 2
𝑢 𝑣 𝑑𝑣 = 𝐸 𝐺 𝑣 𝑑𝑣 =
𝑣 𝑑𝑣
𝑐3
Planck Radiation
ℎ𝑣
𝐸=
ℎ𝑣
exp
−1
𝑘𝐵 𝑇
8𝜋ℎ
𝑣3
𝑢 𝑣 𝑑𝑣 = 𝐸 𝐺 𝑣 𝑑𝑣 = 3
𝑑𝑣
𝑐 exp ℎ𝑣 − 1
𝑘𝐵 𝑇
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Inconsistency of Classical wave theory and
experiment
I~
8hc
1

5
 exp( hc / kT )  1
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8kT
4
Blackbody radiation – Stefan’s Law
An object having a surface temperature T will emit radiation power P which is
proportional to the surface area A of the object and to the fourth power of the
temperature.
P  AT
4
 is the Setfan’s constant and is equal to 5.67 x 10-8 Wm-2K-4(Jm-2s-1K-4)
 is the emissivity of the object and is
0   1
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Stefan ‘s law
The total power (area
below a constant
temperature curve P)
P  AT
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4
Wein displacement Law
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-Hitung panjang gelombang yang dipancarkan tubuh anda
- Berapa energi yang dipancarkan oleh tubuh anda?
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Blackbody radiation and ultra-violet catastrophe
Blackbody radiation and ultra-violet catastrophe
From Maxwell E.M. theory we know that a dipole oscillating with frequency  will on average emits energy r()
r() = const. 2
E 
E  is the average energy of the oscillating dipole.
where
Rayleigh - Jean's Law
Maxwell-Boltzman distribution gives the energy state number density N(E) at energy E as
 total number of possible energy state   N ( E )dE
N ( E )  e  E / kT

0
and
Total energy
r()

  EN ( E )dE
Blackbody
radiation
0
Hence the average energy:
 E 
Rayleigh-Jeans
Totale.nergy
Total.number.of .energy .states
UV
catastrophe
as we do the integration


 EN ( E )dE
0

 N ( E )dE
0



0
Ee  E / kT dE


0
e  E / kT

 kT
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Ultra-violet catastrophe
This result is in fact well known from kinetic theory where the energy of
vibrational degree of freedom
 kT
The energy of rotational / bending degree of freedom
 12 kT
So now we have
r()  Const. 2T
Rayleigh-Jeans Law describes the beginning part of the blackbody radiation correctly at
low frequency (long wavelength) but it obviously wrong at high . As  increases r()
increases without limit, i.e. At   , as we do the integration  , r(), the
0
radiate energy goes to infinity!
This is called UV catastrophe.
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Planck's derivation of blackbody radiation


0
0
 EN ( E )dE   Ee
 E 
 N ( E )dE
e
Rayleigh - Jean's Law:

0
 E / kT

dE
 E / kT
 kT
0
This integration assume continuous distribution of energy, i.e. the oscillator can take up any value.
Plank made the hypothesis that the oscillator will only take up discrete energies
0, E0, 2E0, 3E0, .....etc.
the average energy is obtained from a summation instead of integration.

 E 
 nE e
n 0
 nE0 / kT
0  E0 e  E0 / kT  2 E0 e 2 E0 / kT  3E0 e 3E0 / kT  .....

1  e  E0 / kT  e 2 E0 / kT  e 3E0 / kT  .....
0

e
 nE0 / kT
n 0
Let
x  e  E0 / kT
(1  x )1
1  2 x  3x 2  ..... 
  E   E0 x 

2
 1  x  x  ..... 

E0 x
(1  x )

E0
1
(  1)
x
E 
(1  x )2
E0
e
E0 / kT
1
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Blackbody radiation
http://surendranath.tripod.com/Applets.html
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Conclusion
• Rayleigh – Jeans ‘s derivation is only valid at
long wavelengths (low frequencies) and fails
at short wavelength.
• Planck made an assumption: energy emitted
from the radiator at a frequency , E = nh,
where n and integer, h the Planck constant
(energy is discretized)
• Implication: classical theory predicts energy is
continuous
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