Module 4 - Thermal Radiation
I.
History of Blackbody Radiation
A.
What is a blackbody?
A blackbody is an object that absorbs all radiation that is incident upon it.
When radiation falls upon an object, some of the radiation may be absorbed,
reflected, or transmitted. Most objects that appear black are poor reflectors of
optical radiation but they are not blackbodies since they still reflect some
radiation.
A good approximation of a blackbody is a cavity. In the cube below, the radiation
entering the cavity is continually reflected around the cavity. Thus, the cavity acts
like a minnow trap. It is easy for the light to enter but not exit. This is why a
cavity always looks black at room temperature regardless of the color of the
inside of the cavity.
Front View
B.
Side View of Cavity
Emission Spectra
When a body is at thermal equilibrium, it must re-emit all the radiation that it
absorbs. Blackbodies are interesting because the color of their emission depends
only on their temperature and not upon their shape or composition. Thus, their
emission spectrum can be used as a pyrometer (temperature gauge). The
processing of steel became of great importance during the industrial revolution of
the late 1800's. Thus, blackbody pyrometers were extremely useful tools for
determining the temperature of steel ovens and other processing equipment.
Although light bulb filaments and many other regular objects are not true black
bodies, their temperature can be approximated from their emission spectra by
assuming that the object is a blackbody. Thus, blacksmith's used the color of the
heated steel to determine the processing of steel when making buggy springs,
horse shoes etc.
C.
Experimental Emission Spectra
Blackbody Emission
Energy Density
Higher T
0
0
Frequency
We can see from the graph that the maximum emission frequency shifts to higher
frequency as the temperature is raised.
We can also see that the total energy emitted (area under curve) increases at
increasing temperature.
D.
Stephan-Boltzman Equation
The power density emitted by a blackbody is proportional to the 4th power
of its temperature.
P σ T4
A
σ  5.670x108 W K/m 2
E.
Wien's Displacement Law
The peak of the blackbody emission spectra is inversely proportional to the
temperature.
λ max  2.898 mm  K
T
F.
Ultra-violet Catastrophe
The theory of blackbody radiation would appear to be a straight forward
application of classical electromagnitism and classical physics. Although classical
physics produced a workable theory at low frequency (long wavelengths), it failed
at high frequencies (short wavelengths). Since the theory predicted an infinite
amount of energy radiated at short wavelengths, it was called the "ultra-violet
catastrophe."
II.
Density of States For An E&M Wave In A Blackbody Radiator
Since the emission spectrum of a blackbody radiator is independent of the shape
of the blackbody, we are free to choose the simplest shape possible in deriving the
density of state function. Thus, we choose a cube with sides of length L.
L
Blackbody (3-D Cube )
L
1-D E&M Wave Boundary
(analogous to waves on string)
A.
Wave Equation
We start by writing the wave equation for a wave in free space. We developed this
equation using Maxwell's Equations in PHYS2424.

 1 2 E
2 E  2
0
c  t2
where
2
2
2
 2   î   ĵ   k̂
x 2 y 2 z 2
This equation is a separable differential equation and can be solved by separation
of variables. This is a standard math technique that you either may have already
studied or will learn later in your course work. For now, we are only concerned
with the fact that the electric field can be written as the product of three spatial
functions (X,Y,Z) and a time function as shown below:
E  E o Xx Yy Zz e iω t
B.
Using Boundary Conditions
From PHYS2424, we know that the electric field inside a perfect conductor in
electrostatic equilibrium is ZERO!
Thus, we have six boundary conditions that must be imposed upon our standing
wave solution. These six conditions are due to the six metal conductors that form
the faces of our cube.
1)
X(0) = 0
2)
X(L) = 0
3)
Y(0) = 0
4)
Y(L) = 0
5)
Z(0) = 0
6)
Z(L) = 0
By looking at the figure of the standing wave on a string, you will notice that
conditions 1,3, and 5 are satisfied by a sine function. Thus, we have that
X(x) = Sin ( kx x)
Y(y) = Sin ( ky y)
Z(z) = Sin (kz z)
We also see that only certain wave numbers (k's) will satisfy the boundary
conditions 2,4, and 6. The argument of the sine function must change by an
integer number of  radian as the spatial variable changes by L. This corresponds
to the conditions that
k x L  jx π where jx is either 0,1, 2, 3,....
k y L  jy π where jy is either 0,1, 2, 3,....
k z L  jz π where jz is either 0,1, 2, 3,....
Thus, we have that the electric field in one direction can be written as
 jx
E  E o sin

D.
π x   jy π y   jz π z  ω t
sin
sin
e
L   L   L 

Wave Number Vector - k
The individual wave numbers (kx ky, kz) found in the previous section are
components of the three dimensional wave vector as shown below:
kz
kx
ky
The magnitude of this vector, k, is called the wave number and is connected to
the wave length, , by the equation that we developed in PHYS2424:
k  k x 2  k y 2  k z 2  2π
λ
We can now convert this to a connection between j and  by
k kx  ky  kz 
2
2
2
2
π
 
 
L
π
 
 
L
2
j
2
2
 2
2  j 2 

j
 j
y
z 
 x
 2π 
  
 λ 
 2π 
  
 λ 
2
2
j  2L .
λ
Thus, only certain wavelengths exist in the cavity as determined by the magnitude
of the j-vector. We will study this material again when we deal with wave guides
in the junior E&M class.
E.
Density of States Function
We now wish to determine the number of available states with wavelengths such
that j is between some value j to j+dj. A spherical shell of radius j and thickness dj
has a volume of
V  4 π j2 dj
j
However, all components of j must be positive so we are restricted to only 1/8th
of the sphere's volume.
1
π j2 dj
2


V   4 π j dj 
2
8
We must also account for the fact that an E&M wave can have two different
polarization states. Thus, the total number of states with wave numbers between j
and j + dj is
Gjdj  π j2 dj
We now convert this into the number of states in the cavity between the
frequency  and  + d using the relationship c  λν . Substituting the
relationship into our previous results, we have that
j 2L ν
c
dj 2L dν
c
G ν  dν 
2

 4πL

 c2

Gν  dν 
 
 2  2 L 
dν
ν 
  c 

2

8π ν

 c3



 dν


L3
We now divide by the volume of the cube to obtain the density of state function:
gν  dν 
2

8π ν

 c3






dν
DENSITY OF STATES
This is the number of allowed energy states per unit volume of the cavity that
emit radiation between the frequency  and  + d. Everyone used this result in
their work even Max Planck.
III.
Calculating Blackbody Energy Spectra
To calculate the energy per volume emitted by the blackbody, you multiply the
average energy emitted per state by the number of states per volume. Thus, we
have that
u νdν   ε  gvdv
Thus, the problem with the classical result had to either reside in the calculation
of the density of states using Maxwell's electromagnetic theory or the average
energy calculation using classical mechanics (Maxwell-Boltzman distribution /
equipartition theorem).
IV.
Classical Physics (Rayleigh - Jeans Theory)
A.
The atoms of the blackbody radiator are considered to be classical harmonic
oscillators whose energy is given by
ε  1 k x2
2
Noting that a classical harmonic oscillator has two degrees of freedom and using
the equipartition theorem (from Maxwell-Boltzman Statistics), we have the
average energy of the oscillator states as


 ε   2  1 k T   k T
2

Since the oscillators are in thermal equilibrium with their environment, this is also
the average energy of the radiation they emit.
B.
Rayleigh - Jeans Radiation Formula
u ν  dν   ε  gν  dν 
2


8π ν kT 

 dν
3


c


The formula can also be written in terms of wavelength by noting that
ν c
λ
dν  c dλ .
λ2
Substituting this into our Rayleigh - Jeans formula, we obtain
u λ dλ 
2


 8 π c k T   c


 λ 2 c 3   λ 2



dλ  8 π k T dλ

λ4

Thus, the classical theory suggested that energy density was proportional to the
square of the frequency or inversely to the fourth power of the wavelength.
Theory fits the experimental data at low energy but predicts infinite energy as
   or equivalently   0.
C.
Max Planck's Solution
Planck was a former student of Kirchhoff 's and had developed a research
program on applying the Second Law of Thermodynamics to problems at a time
when the law was not widely applied. Planck's application of thermodynamics
convinced him that the average energy calculation was incorrect.
Planck decided that the energy of the harmonic oscillator states shoul be
represented by
ε  n h ν where n is an integer 0, 1, 2, ....
Planck's arguments for this statement are beyond the scope of this course. The
interested student can find a discussion in The Quantum Physicists and an
Introduction to Their Physics by William H. Cooper.
We now calculate the average energy for an energy state using this relationship
between energy and frequency.

 n h ν  e
ε  n 0


 n h ν 
 kT 

 e
h ν 
kT 
n


n 0
The bottom summation can be found in a math handbook and is given by

h ν 
k T 
n
 e


n0
1

h ν 

k T

1 e

We can use the following Calculus trick to help evaluate the top summation
d e n x
  n e n x
dx
Where we will define x to be
x  hν
kT

 n h ν  e
h ν 
kT 
n


n0

 n h ν  e
n0
h ν 
kT 
n



h ν  n e
h ν 
kT 
n


n 0

n x 
   n x 
  h ν  de
 h ν d  e
dx
dx 
n0

n  0






 n h ν  e
h ν 
kT 
n


n0

 n h ν  e
h ν 
kT 
d  1 
h ν
dx  1 e  x 
n


n0

 n h ν  e
h ν
h ν 
kT 
n


n0




2

  e - x 
 x   

1 e

1
hν



kT
-
 h νe



 hν  

 kT 
 1 e





2
We now substitute our results back into average energy equation and obtain

ε 
 n h ν  e
h ν 
kT 
n


n0

 e
n 0


 n h ν 
 kT 
hν

 
 kT 


hν e
hν



kT
hν


1 e
e
hν




kT
1
Planck now multiplied the average energy by the classical density of states to
obtain his radiation formula of






2 


hν
8π ν 

 dν
u ν  dν   ε  gν  dν  

3


 c

hν



 


 e  k T  1 


uν  dν 
8 π h ν3
  h ν 
  
c3  e k T 




1


dν
We see that the exponential in the denominator prevents the ultra-violet
catastrophe by causing the intensity to decrease exponentially at high
frequencies.
We also can show that Planck's result give the Rayleigh - Jean's result for low
frequency by expanding the exponential function as follows
e
uν  dν 
hν


k T


1
hν
kT
for ν 
kT
h
2
8 π h ν3
dν  8 π k3T ν dν

c
h ν 
c 3 1
1
kT 

Planck was aware that a classical harmonic oscillator should have a continuous
energy distribution. However, he thought that the ultra-violet catastrophe might be
due to a convergence problem with the integration. Thus, he made the energy
levels discrete so that the average energy calculation involved a sum. He
intended to obtain the continuous energy distribution of the classical harmonic
oscillator by letting the parameter h approach zero in his final result. However,
letting h go to zero resulted in the ultra-violet catastrophe. Thus, h had to be left
finite. An excellent fit of experimental blackbody spectra could be obtained using
the value for h of
h 1240 eV  nm
c
This constant is called Planck's constant!!
FACT: Planck was a classical physicist and never believed that this constant had
any great physical significance. In particular, he didn't believe that the oscillator's
energy levels were actually quantized! In reality, he had used the wrong energy
statistics with the wrong energy to get the right answer! A young patent clerk
named Albert Einstein had a much different view of the constant and would make
it central to his solution of a problem that would win him the Nobel prize!
IV.
Classical Theory of Light (Review)
At the turn of the Century, all physicists considered light to be an electromagnetic
wave due to the conclusive demonstration of light interference by Thomas Young,
the theoretical work of James Clerk Maxwell and the demonstration of other
electromagnetic waves (radio waves) by Hertz.
1)
The energy of a wave is directly proportional to the square of the amplitude of
the wave. Thus, the brightness of light determines the energy of light.
2)
The energy of a wave is independent of the wavelength or frequency of the wave.
3)
The energy of the wave is not localized in space. Instead, the wave’s energy is
spread out through all of space.
V.
Photoelectric Effect
A.
In his experimental verification of Maxwell’s electromagnetic wave theory of
light, Hertz noticed and interesting phenomena. When shorter wavelength light
(ultraviolet light) was directed upon his electrodes, an electrical discharge (spark)
was more likely to occur. Hertz died in his early 40’s and never fully examined
this phenomenon called the photoelectric effect. It has profound applications in a
wide range of engineering applications ranging including light detectors as well as
being a major driving force in the radical new field of quantum mechanics. It was
for his 1905 explanation of the photoelectric effect and not Relativity that Einstein
was awarded the Nobel Prize in Physics.
B.
Experimental Setup
In the figure, a light source is directed upon a metal surface. The light provides
energy to electrons in the metal which allows many bound electrons to escape
with some kinetic energy. These electrons are collected on the top plate where
they flow through an ammeter allowing us to detect the number of electrons per
second collected (current). A grid placed in front of the top plate can be biased to
a negative electrical potential (voltage). As we raise the voltage, the current
decreases since only the most energetic electrons (highest kinetic energy) can
make it to the cathode. Eventually, we reach a voltage at which no electrons have
enough energy to make it to the top plate. This voltage is called the stopping
potential or stopping voltage and enables us to measure the maximum kinetic
energy of an electron emitted by the metal.
Light Source
V
A
+
Experimental Results:
1)
If current is detected, it will be proportional to the intensity of the light
source.
2)
Maximum kinetic energy of electrons emitted from the plate depends upon
the wavelength of the light source and is independent of the intensity of
the light.
3)
For longer wavelengths of light, no current is detected no matter how great
the light source intensity.
4)
The minimum wavelength of light at which electron emission just stops is
called the cut-off wavelength and depends only on the type of metal used
for the bottom pate (assuming no surface contamination).
5)
For light whose wavelength is below the cut-off wavelength, the current is
detected immediately no matter how dim the light source.
Results 2,3 and 5 can not be explained by the wave picture of light!!!
C.
Einstein’s Explanation
Einstein proposed that light was composed of discrete bundles of energy called
photons. The energy of each photon was directly proportional to its frequency by
Planck’s equation:
Eh ν
where h is Planck’s constant. Although Planck may have used the equation first,
Einstein’s view was much more radical. The finite value of h actually produces a
graininess in the energy of light found in nature. Light exists only in these packets
and packets will have to be either totally absorbed or not absorbed at all. The
importance of the finite value of h can not be overstated as it will be the cause of
all of the amazing results which originate from Quantum Mechanics. If this
constant had any other value, the world will live in would be dramatically
different. This special equation relating energy and frequency is often called
Einstein’s energy relation since it appears in a several different forms throughout
quantum mechanics. For example using our knowledge of waves including
ω 2π ν and c λν , we can rewrite Einstein’s energy relation in equivalent
forms as
E ω
where  
hc
λ
h
.
2π
Einstein’s other assumptions are that:
1)
Energy is conserved
2)
The electron bound in a metal can be thought of as a particle in some
average potential energy well created by all the other charges in the metal.
U(x)
0 eV
-W
In the figure above, the green electron is least tightly bound (i.e. It is in the
highest energy state) to the metal. Thus, this electron will have the largest kinetic
energy if it is emitted. The minimum energy, W, needed to remove the electron
from the metal is called the metal’s work function. When physicists or engineers
want to build electron emitters, they choose materials with low work functions
and often coat the materials surface with cesium to further reduce the materials
normal work function.
By Conservation of Energy, we know that the energy available as kinetic energy
to the electron from a photon of energy E is given by
KEU .
Using Einstein’s energy relationship for the energy of the light photon, we have
for the least tightly bound electron (our green electron),
K max  h ν - W .
When the voltage on our repulsive grid reaches the stopping potential, all of the
kinetic energy of the most energetic electron is converted to electrical potential
energy by the time it reaches our grid. Thus, we have
K max  e Vstop .
Plugging this into our previous equation gives us
e Vstop  h ν  W
h W
Vstop    ν    .
e  e 
For a graph with the stopping on the y-axis and frequency on the x-axis, this
is the equation of a straight line with slope (h/e) and y-intercept of (W/e).
Thus, the experiment provides a means of finding the work function of
materials for engineering applications as well as another means of
determining the constant h!!
VI. Duality of Light
Einstein’s explanation of the photoelectric effect suggests that light is composed
of bundles of energy called photons (light baseballs) whose energy (baseball size)
is determined by the frequency/wavelength of the light. However, the total energy
contained in an E&M wave depends also on how many photons are present!!
Bright red light consists of many low energy photons while dim blue light
consists of a few high energy photons.
The model of light waves traveling as a collection of light photons may be
pleasing to a classical physicist and explain the photoelectric effect, but it is
definitely not correct. Modern physicists have now demonstrated that there are
some problems with this view of light. It appears that light and everything else in
the world has dual properties (wave and particle). The photon always interacts
with matter as a particle, but in other experiments like Young’s interference
experiment it shows wave properties. Also, it never shows both types of
properties in the same experiment. It is as if the very act of making a
measurement creates reality. The philosophical implications of these experimental
results are both intriguing and disturbing to physicists. Einstein never accepted the
philosophical implications of modern quantum mechanics. However, there is no
doubt that the physics which originates from the simple experiments we will
discuss in the next few modules is amazingly accurate in describing how things
work in the atomic world.