5. Light-matter interactions:
Blackbody radiation
The electromagnetic spectrum
Sources of light
Boltzmann's Law
Blackbody radiation
The cosmic microwave background
The electromagnetic spectrum
All of these are electromagnetic waves. The amazing diversity is a result of the fact that
the interaction of radiation with matter depends on the frequency of the wave.
The boundaries between regions are a bit arbitrary…
Sources of light
Accelerating charges emit light
Linearly accelerating charge
http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html
Vibrating electrons or nuclei of an atom or molecule
Synchrotron radiation—light emitted by charged particles
whose motion is deflected by a DC magnetic field
Bremsstrahlung ("Braking radiation")—light emitted when
charged particles collide with other charged particles
The vast majority of light in the universe
comes from molecular motions.
Core (tightly bound) electrons vibrate in their
motion around nuclei
 ~ 1018 - 1020 cycles per second.
Valence (not so tightly bound) electrons
vibrate in their motion around nuclei
 ~ 1014 - 1017 cycles per second.
Nuclei in molecules vibrate with respect to
each other
 ~ 1011 - 1013 cycles per second.
Molecules rotate
 ~ 109 - 1010 cycles per second.
One interesting source of x-rays
Sticky tape emits x-rays when you unpeel it.
Scotch tape generates
enough x-rays to take
an image of the bones
in your finger.
This phenomenon, known as
‘mechanoluminescence,’ was
discovered in 2008.
For more information and a video describing this:
http://www.nature.com/nature/videoarchive/x-rays/
Three types of molecule-radiation interactions
key idea: conservation of energy
Before
After
Absorption
•Promotes molecule to a higher energy state
•Decreases the number of photons
Spontaneous Emission
•Molecule drops from a high energy state to a lower state
•Increases the number of photons
Stimulated Emission
•Molecule drops from a high energy state to a lower
state
•The presence of one photon stimulates the
emission of a second one
•This process has no analog in classical physics - it
can only be understood with quantum mechanics!
Unexcited
molecule
Excited
molecule
Atoms and molecules have discrete energy
levels and can interact with light only by making
a transition from one level to another.
A typical molecule’s energy levels:
2nd excited
electronic state
Energy
1st excited
electronic state
Lowest vibrational and
rotational level of this
electronic “manifold”
Excited vibrational and
rotational level
Transition
Ground
electronic state
Computing the energies of these
levels is usually difficult, except
for very simple atoms or
molecules (e.g., H2).
Different atoms emit light at different
frequencies.
Wavelength
Frequency (energy)
the “Balmer series”
Johann Balmer
(1825 - 1898)
This is the emission spectrum from
hydrogen. It is simple because
hydrogen has only one electron.
high-pressure
mercury lamp
Bigger atoms have much more complex spectra, because they are more complex
objects. Molecules are even more complex.
Ludwig Boltzmann computed the
relative populations of energy levels
In the absence of collisions,
molecules tend to remain
in the lowest energy state
available.
Low Temperature
Collisions can knock a molecule into a higher-energy state.
The higher the temperature,
the more this happens.
High Temperature
In equilibrium at a temperature T, the ratio of the populations of any
two states (call them number 1 and number 2) is given by:
N 2 exp   E2 / k BT 

 exp  E / k BT 
N1 exp   E1 / k BT 
Boltzmann population factor
Boltzmann didn’t know quantum mechanics. But
this result works equally well for quantum or
classical systems.
A higher temperature
A lower temperature
Ludwig Boltzmann
(1844 - 1906)
energy
E3
E3
eE/kThigh
E2
E1
E2
eE/kTlow
E1
population
population
Blackbody radiation
Blackbody radiation is the radiation emitted from a hot body.
It's anything but black!
It results from a balance between the absorption and emission
of light by the atoms which comprise the object, at a given
temperature.
Imagine a box, the
inside of which is
completely black.
Consider the
radiation emitted
from the small hole.
Inside: a gas of
photons at some
temperature T
The classical physics approach
to the blackbody question
Lord Rayleigh figured out how to count the
electromagnetic modes inside a box.
Basically, the result follows from requiring
that the electric field must be zero at the
internal surfaces of the box.
John William Strutt,
3rd Baron Rayleigh
(1842 - 1919)
# modes ~ 2
And the energy
per mode is kBT,
(this is called the
“equipartition
theorem”, also
due to Boltzmann)
The Rayleigh-Jeans Law
Rayleigh-Jeans law (circa 1900):
energy density of a radiation field u() = 82kT/c3
Note: the units of this expression are correct. Strictly
speaking, u() is an energy density per unit bandwidth, such
that the integral  u   d gives an answer with units of
energy density (energy per unit volume).
Total energy radiated from a black body:
 u   d  
energy density u()
uh-oh… the "ultraviolet catastrophe"
keeps going
up forever!
frequency 
Rayleigh (and others) knew this was wrong. This
failure of equipartition led Max Planck to propose
(1900) that the energy emitted by the atoms in the
solid must be quantized in multiples of h. This
revolutionary idea troubled Planck deeply because
he had no explanation for why it should be true…
Einstein A and B coefficients
In 1916, Einstein considered the various transition rates between
molecular states (say, 1 and 2) involving light of energy density u:
Absorption rate = B12N1 u
Spontaneous emission rate = A N2
Stimulated emission rate = B21 N2 u
In equilibrium, the rate of upward transitions equals the rate of
downward transitions:
B12 N1 u = A N2 + B21 N2 u
using Boltzmann’s
result
Rearranging:
(B12 u ) / (A + B21 u ) = N2 / N1 = exp[–E/kBT ]
Einstein A and B coefficients
We can solve this expression for the energy density of the field:
(B12 u ) / (A + B21 u ) = exp[E/kBT ]
Rearrange to find: u 
A B21
B12
exp  E / k BT   1
B21
For blackbody radiation, we would expect that u() should be infinite
in the limit that T = . But, as T  , exp[E/kBT ]  1
So: B12 = B21  B
We can eliminate*
the ratio A/B to find: u   
 Coeff. up = coeff. down!
8 hv 3 c3 
exp  hv / kBT   1
(we have used E=h)
* We require that this expression has to be the same as the Rayleigh-Jeans law in the limit of very low frequencies.
energy density u()
This solves the ultraviolet catastrophe.
Rayleigh-Jeans: u() ~ 2
3
Planck-Einstein: u   ~
exp  h / k BT   1
Max Planck
1858-1947
frequency 
At low frequencies, the two
results are basically equivalent.
At high frequencies,
Einstein’s result goes
back down to zero, so the
integral of u() is finite.
Albert Einstein
1879-1955
(photo taken in 1916)
Blackbody emission
The higher the temperature, the more the emission
and the shorter the average wavelength.
Wien’s Law: the peak wavelength
is proportional to 1/T.
"Blue hot" is hotter
than "red hot."
Notice: for an object near
room temperature, the peak
of the blackbody spectrum is
around  = 10 microns – the
so-called ‘thermal infrared’
region.
Blackbody emission from hot objects
Molten glass and heated
metal are hot enough to shift
the blackbody spectrum into
the visible range, so that
they glow: “red hot”.
Even hotter
metal is
“white hot”.
Cooler objects emit
light at longer
wavelengths: infrared
The sun is a black body
Irradiance of the sun vs. wavelength
When measured at sea level, one sees absorption
lines due to molecules in the atmosphere.
Cosmic microwave background
As an object cools slowly, the radiation it emits retains the blackbody
spectrum. But the peak shifts to lower and lower frequencies.
This is true of all objects. Even the universe.
The 2.7° cosmic microwave
background is blackbody
radiation left over from
the Big Bang!
Arno Penzias and Bob Wilson
Nobel prize, 1978
Cosmic microwave background
Measurements of the microwave
background tell us about the
conditions in the early formation
of the universe.
Theory and observation
agree so well that you
cannot distinguish them
on this plot!
The stunningly good fit (and the
amazing level of isotropy) are
considered to be convincing proof
of the Big Bang description of the
birth of the universe.
The latest results:
Planck satellite
(European Space
Agency)
The temperature of the blackbody is not the
same everywhere in the sky. The small
variations reflect the structure of the universe
at the time that the radiation was emitted.
Tiny fluctuations in the
temperature of the blackbody
spectrum at different points in
the sky (hundreds of
microKelvins).
Planck
data from 2013