1. Sources of Light and and Blackbody Radiation
Prof. Rick Trebino
Georgia Tech
Sources of light: gases, liquids, and solids
Boltzmann's Law and blackbody radiation

2. Where does light come from?
We’ve seen that Maxwell’s Equations (i.e., the wave equation) describe the propagation of light.
But where does light come from in the first place?
Some matter must emit the light. It does so through the matter’s polarization:
Note that matter’s polarization is analogous to the polarization of light. Indeed, it will cause the emission of light with the same polarization direction.
where N is the number density of charged particles, q is the charge of each particle, and is the position of the charge. Here, we’ve assumed that each charge is identical and has identical motion.

3. Polarized and unpolarized media
Unpolarized medium
Polarized medium
On the right, the displacements of the charges are correlated, so it is polarized at any given time (and its polarization is oscillating).

4. Maxwell's Equations in a Medium
The induced polarization, , contains the effect of the medium and is included in Maxwell’s Equations:
This extra term also adds to the wave equation, which is known as the Inhomogeneous Wave Equation:
The polarization is the source term and tells us what light will be emitted.
Notice that the induced polarization, and hence , gets differentiated twice. But is just the charge acceleration!
So it’s accelerating charges that emit light!

5. Sources of light Accelerating charges emit light!
Linearly accelerating charge
Synchrotron radiation— light emitted by charged particles deflected by a magnetic field
Bremsstrahlung (Braking radiation)— light emitted when charged particles collide with other charged particles

6. But the vast majority of light in the universe comes from molecular vibrations emitting light.
Electrons vibrate in their motion around nuclei.
High frequency: ~ cycles per second.
Nuclei in molecules vibrate with respect to each other.
Intermediate frequency: ~ cycles per second.
Nuclei in molecules rotate.
Low frequency: ~ cycles per second.

7. Water’s vibrations
Movies from

8. Atomic and molecular vibrations correspond to excited energy levels in quantum mechanics.
Energy levels are everything in quantum mechanics.
Excited level
DE = hn
Energy
Ground level
The atom is vibrating at frequency, n.
The atom is at least partially in an excited state.

9. Excited atoms emit photons spontaneously.
When an atom in an excited state falls to a lower energy level, it emits a photon of light.
Excited level
Energy
Ground level
Molecules typically remain excited for no longer than a few nanoseconds. This is often also called fluorescence or, when it takes longer, phosphorescence.

10. Different atoms emit light at different widely separated frequencies.
Each colored emission line corresponds to a difference between two energy levels.
These are emission spectra from gases of hot atoms.
Frequency (energy)
Atoms have relatively simple energy level systems (and hence simple spectra).

11. Collisions broaden the frequency range of light emission.
A collision abruptly changes the phase of the sine-wave light emission. So atomic emissions can have a broader spectrum.
Quantum-mechanically speaking, the levels shift during the collision.
Collision
Electric field
time
New frequencies in the emission
Gases at atmospheric pressure have emission widths of ~ 1 GHz.
Solids and liquids emit much broader ranges of frequencies (~ 1013 Hz!).

12. Molecules have many energy levels.
A typical molecule’s energy levels:
E = Eelectonic + Evibrational + Erotational
2nd excited
electronic state
Lowest vibrational and rotational level of this electronic “manifold”
Energy
1st excited
electronic state
Excited vibrational and rotational level
Transition
There are many other complications, such as spin-orbit coupling, nuclear spin, etc., which split levels.
Ground
electronic state
As a result, molecules generally have very complex spectra.

13. Atoms and molecules can also absorb photons, making a transition from a lower level to a more excited one.
Excited level
This is, of course, absorption.
Energy
Ground level
Image from
Absorption lines in an otherwise continuous light spectrum due to a cold atomic gas in front of a hot source.

14. Decay from an excited state can occur in many steps.
Infra-red
Energy
Ultraviolet
Visible
Microwave
The light that’s eventually re-emitted after absorption may occur at other colors.

15. Einstein showed that stimulated emission can also occur.
Before After
Spontaneous emission
Absorption
Stimulated emission

16. In what energy levels do molecules reside? Boltzmann population factors
Ni is the number density of molecules in state i (i.e., the number of molecules per cm3).
T is the temperature, and kB is Boltzmann’s constant. E3 N3 E2 N2
Energy N1 E1
Population density

17. The Maxwell-Boltzman distribution
In the absence of collisions,
molecules tend to remain
in the lowest energy state
available.
Collisions can knock a mole-
cule into a higher-energy state.
The higher the temperature,
the more this happens.
Low T
High T 3 3
Energy
Energy 2 2 1 1
Molecules
Molecules
In equilibrium, the ratio of the populations of two states is:
N2 / N1 = exp(–DE/kBT ), where DE = E2 – E1 = hn
As a result, higher-energy states are always less populated than the ground state, and absorption is stronger than stimulated emission.

18. Blackbody radiation
Blackbody radiation is emitted from a hot body. It's anything but black!
The name comes from the assumption that the body absorbs at every frequency and hence would look black at low temperature.
It results from a combination of spontaneous emission, stimulated emission, and absorption occurring in a medium at a given temperature.
It assumes that the box is filled with many different molecules that that, together, have transitions (absorptions) at every wavelength.

19. Einstein A and B coefficients
In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of intensity, I:
Spontaneous emission rate = A N2
Absorption rate = B12 N1 I
Stimulated emission rate = B21 N2 I
In equilibrium, the rate of upward transitions equals the rate of downward transitions:
at frequency n
Up = B12 N1 I = A N2 + B21 N2 I = Down
Recalling the Maxwell-
Boltzmann Distribution
Solving for N2/N1:
(B12 I ) / (A + B21 I ) = N2 / N1 = exp[–DE/kBT ]

20. Blackbody Radiation
Now solve for the intensity in: (B12 I) / (A + B21 I) = exp[-DE/kBT ]
Multiply by (A + B21 I) exp[DE/kBT] : B12 I exp[DE/kBT] = A + B21 I
Solve for I: I = A / {B12 exp[DE/kBT] – B21}
Dividing numerator and denominator by B21:
I = [A/B21] / { [B12 /B21] exp[DE/kBT] – 1 }
Now, when T ® ¥, I should also. As T ® ¥, exp[DE/kBT ] ® 1.
So: [B12 /B21] – 1 = 0
And: B12 = B21 º B ¬ Coeff up = coeff down!
And: I = [A/B] / {exp[DE/kBT ] – 1}

21. Blackbody Radiation I = [A/B] / {exp[DE/kBT ] – 1}
We can eliminate A/B based on other information and, writing in terms of the intensity per unit frequency, In :
also using DE = hn
This is the total intensity per unit frequency (that is, in a range from n to n + dn ) emitted by an arbitrary blackbody.
We considered only two levels, but our approach was general and so applies to any two levels and hence to the entire spectrum.
Notice that it’s independent of the size of the object, but, to obtain the emitted power, multiply by the surface area of the blackbody.

22. Writing the blackbody spectrum vs. wavelength
Units of In(n): intensity per unit frequency
Total intensity
Change variables from n to l:
Units of Il(l): intensity per unit wavelength

23. Blackbody emission spectrum
The higher the temperature, the more the emission and the shorter the average wavelength.
Blue hot is hotter
than red hot.
The sun’s surface is 6000 degrees K, so its blackbody spectrum peaks at ~ 500 nm—in the green. However, blackbody spectra are broad, so it contains red, yellow, and blue, too, and so looks white.

24. Wien's Law: The blackbody peak wavelength scales as 1/Temperature.
dlambda/dnu = -c/nu^2 = -lambda^2/c

25. We can tell how hot a star is by its blackbody emission spectrum.

26. The earth is a blackbody, too.

27. Cosmic microwave background
Microwave background vs. angle. Note the variations.
Peak frequency is ~ 150 GHz
The 3° cosmic microwave
background is blackbody
radiation left over from
the Big Bang!
Interestingly, blackbody radiation retains a blackbody spectrum despite the expansion the universe. It does get colder, however.
Wave-number (cm-1)

28. Color temperature
Blackbodies are so pervasive that a light spectrum is often characterized in terms of its temperature even if it’s not exactly a blackbody.
Keep in mind that blackbody spectra are broad, so they usually look white with a tint of the peak color (wavelength).
Image from the magazine Digital PhotoPro, “White Balance,”, May/June 2004

29. Emissivity
Our analysis assumed a perfectly black object. However, the object may not have transitions at all wavelengths.
The emissivity takes this into account. It is the ratio of the actual emitted spectrum and the theoretical blackbody spectrum. It depends on the medium and its density.