1. Substitute Lecturer: Jason Readle Thurs, Sept 17th, 2009
Lecture #8: Blackbody Radiation, Einstein Coefficients, and Homogeneous Broadening
Substitute Lecturer: Jason Readle
Thurs, Sept 17th, 2009

2. Topic #1: Blackbody Radiation

3. What is a Blackbody? Ideal blackbody: Perfect absorber
Appears black when cold!
Emits a temperature-dependent light spectrum

4. Blackbody Energy Density
The photon energy density for a blackbody radiator in the ν → ν + dν spectral interval is

5. Blackbody Intensity The intensity emitted by a blackbody surface is
(Units are or J/s-cm2 or W/cm2)

6. Blackbody Peak Wavelength
The peak wavelength for emission by a blackbody is
where 1 Å = 10–8 cm

7. Example – The Sun
Peak emission from the sun is near 570 nm and so it appears yellow
What is the temperature of this blackbody?
Calculate the emission intensity in a 10 nm region centered at 570 nm.
Tk = 5260 K

8. Example – The Sun
Also
570 nm → 17,544 cm–1
kT (300 K) eV

9. Example – The Sun or
Dn = 9.23 · 1012 s–1
= 9.23 THz

10. Example – The Sun

11. Since hν = 2.18 eV = 3.49 · 10–19 J → ρ(ν) d ν / hν = 1.58 · 1010
Example – The Sun
Since hν = 2.18 eV
= 3.49 · 10–19 J
→ ρ(ν) d ν / hν = 1.58 · 1010

12. Intensity = Photon Density · c
Example – The Sun
Remember,
Intensity = Photon Density · c or
 = 4.7 · 1020 photons-cm–2-s–1
= 164 W-cm–2

13. Example – The Sun

14. Topic #2: Einstein Coefficients

15. Absorption
Spontaneous event in which an atom or molecule absorbs a photon from an incident optical field
The asborption of the photon causes the atom or molecule to transition to an excited state

16. Spontaneous Emission
Statistical process (random phase) – emission by an isolated atom or molecule
Emission into 4π steradians

17. Stimulated Emission Same phase as “stimulating” optical field
Same polarization
Same direction of propagation

18. Putting it all together…
Assume that we have a two state system in equilibrium with a blackbody radiation field.

19. Einstein Coefficients
For two energy levels 1 (lower) and 2 (upper) we have
A21 (s-1), spontaneous emission coefficient
B21 (sr·m2·J-1·s-1), stimulated emission coefficient
B12 (sr·m2·J-1·s-1), absorption coefficient
Bij is the coefficient for stimulated emission or absorption between states i and j

20. Two Level System In The Steady State…
The time rate of change of N2 is given by:
Remember, ρ(ν) has units of J-cm–3-Hz–1

21. Solving for Relative State Populations
Solving for N2/N1:

22. Solving for Relative State Populations
But… we already know that, for a blackbody,

23. Einstein Coefficients
In order for these two expressions for ρ(ν) to be equal, Einstein said:
and

24. Example – Blackbody Source
Suppose that we have an ensemble of atoms in State 2 (upper state). The lifetime of State 2 is
This ensemble is placed 10 cm from a spherical blackbody having a “color temperature” of 5000 K and having a diameter of 6 cm
What is the rate of stimulated emission?

25. Example – Blackbody Source

26. Example – Blackbody Source
hν = 3.2 eV
 l = nm
n = 7.7 · 1014 s–1

27. Example – Blackbody Source
Blackbody emission at the surface of the emitter is

28. Example – Blackbody Source
Assuming dν = Δν = 100 MHz,
At the ensemble, the photon flux from the 5000 K blackbody is:
0(ν)dν = 3.7 · 10–5 J-cm–2-s–1
 7.2 · 1013 photons-cm–2-s–1
at nm
= 6.48 · 1012 photons-cm–2-s–1

29. Example – Blackbody Source
And or
ρ(ν)dν = 3.46 · 10–17 J-cm–3

30. Example – Blackbody Source
The stimulated emission coefficient B21 is
= 3.5 · 1024 cm3-J–1-s–2

31. Example – Blackbody Source
Finally, the stimulated emission rate is given by
= – 3.5 · 1024 cm3-J–1-s–2

32. This is negligible compared to the spontaneous emission rate of
To reiterate…
This is negligible compared to the spontaneous emission rate of
A21 = 106 s–1 !

33. Example – Laser Source
Let us suppose that we have the same conditions as before, EXCEPT a laser photo-excites the two level system:
Let Δνlaser = 108 s–1 (100 MHz, as before).

34. Example – Laser Source If the power emitted by the laser is 1 W, then
Power flux, P
= W-cm–2
Since hν = 3.2 eV
= 5.1 · 10–19 J
→ P = 2.5 · 1020 photons-cm–2-s–1

35. Example – Laser Source
= 4.24 · 10–17 J-cm–3-Hz–1
= 83.3 photons-cm–3-Hz–1

36. Example – Laser Source 3.5 · 1024 cm3-J–1-s–2 · 4.24 · 10–17 J-cm–3-s

37. Example – Laser Source
Remember, in the case of the blackbody optical source:
What made the difference?

38. Source Comparison
Total power radiated by 5000 K blackbody with R = 0.5 cm is 11.1 kW

39. Key Points
Moral: Despite its lower power, the laser delivers considerably more power into the 1 → 2 atomic transition.
Point #2: To put the maximum intensity of the blackbody at nm requires T  7500 K!
Point #3: Effective use of a blackbody requires a process having a broad absorption width

40. Ex. Photodissociation
C3F7I + hν → I*

41. Bandwidth In the examples, bandwidth Δν is very important
Δν is the spectral interval over which the atom (or molecule) and the optical field interact.

42. Topic #3: Homogeneous Line Broadening

43. Semi-Classical Conclusion
This diagram:
suggests that the atom absorbs only (exactly) at

44. The Shocking Truth!

45. Line Broadening
The fact that atoms absorb over a spectral range is due to Line Broadening
We introduce the “lineshape” or “lineshape function” g(ν)

46. Lineshape Function
g(ν) dν is the probability that the atom will emit (or absorb) a photon in the ν → ν + dν frequency interval.
g(ν) is a probability distribution
and Δν / ν0 << 1

47. Types of Line Broadening
There are two general classification of line broadening:
Homogenous — all atoms behave the same way (i.e., each effectively has the same g(ν).
Inhomogeneous — each atom or molecule has a different g(ν) due to its environment.

48. Homogeneous Broadening
In the homogenous case, we observe a Lorentzian Lineshape
where ν0 ≡ line center

49. Homogeneous Broadening
Δν = FWHM
Bottom line: Homogeneous → Lorentzian

50. Sources of Homogeneous Broadening
Natural Broadening — any state with a finite lifetime τ sp (τsp ≠ ∞) must have a spread in energy:
Collisional Broadening — phase randomizing collisions

51. Heisenberg’s Uncertainty Principle
Natural Broadening
ΔE Δt ≥ 
Heisenberg’s Uncertainty Principle

52. Natural Broadening
In the case of an atomic system:

53. Natural Broadening
In general

54. Example: Sodium (Na)
(Both arrows indicate “resonance” transitions)

55. ~ 2 · 10–8! Example: Sodium (Na) = 9.9 · 106 s–1 ≈ 10 MHz
Radiative lifetime of the 3p 2P3/2 state is 16 ns
= 9.9 · 106 s–1 ≈ 10 MHz
ν0 = 5.1 · 1014 Hz 
~ 2 · 10–8!

56. Example: Mercury (Hg)

57. Example: Mercury (Hg)
Remember:
In general,

58. Collisional Broadening
An atom that radiates a photon can be described as a classical oscillator with a particular phase

59. Collisional Broadening
Suppose now that we have collisions between atom A (the radiator) and a second atom, B…

60. Collisional Broadening
Such collisions alter the phase of the oscillator.
(Arrows indicate points at which oscillator suffers collision)

61. Collisional Broadening
Result? Broadening of Transition!
The rate of phase randomizing collisions is:
where:
kC (cm3 – s–1) is known as the rate constant of collisional quenching (deactivation of the excited atom)
NC (cm-3) is the number density of colliding atoms

62. Collisional Broadening

63. Total Homogenous Broadening
Is calculated by summing the rates of the various homogeneous broadening processes:

64. Example – KrF Laser KrF laser (λ = 248.4 nm) τsp = 5 ns
kC = 2 · 10–10 cm3-s–1
1 atmosphere ≡ 2.45 · 1019 cm–3

65. Example – KrF Laser
Δνtotal = 31.8 MHz +
· P(atm)

66. Example – KrF Laser
Δνtotal = MHz GHz · P
spontaneous collisions
Note that these terms are equal for P = 0.02 atm!

67. Next Time
Inhomogeneous broadening
Threshold gain