Spontaneous Emission, Stimulated Emission, and Absorption
- Draine Ch. 6: Spontaneous Emission, Stimulated Emission, and Absorption
- Ryden and Pogge Ch. 2.1: Radiative Transfer
- Ryden and Pogge Ch. 2.2: Absorbers and Emitters
- Lecture notes on Radiative Transfer in Astrophysics by C.P. Dullemond.
- Radiative Processes by Rybicki and Lightman
General rules for absorption and emission of radiation by absorbers with quantized energy levels. Can be atoms, ions, molecules, dust grains, or anything that has (quantized) energy levels.
Absorption of photons
Some absorber is in a lower level, there is radiation present that has
Rate of this reaction
There is some
Emission of photons
Spontaneous emission
Random process (independent of radiation field), and occurs with a probability per unit time
Stimulated emission
Occurs if photons of identical
- frequency
- polarization
- direction are present in a radiation field.
Rate of emission
From state
Einstein coefficients are not independent
Strength of stimulated emission (
Radiation fields
(Review from Rybicki and Lightman, Ch 1)
It has dimensions
- 3D space
- direction
- frequency
Intensity itself is not a vector quantity but it is a scalar field that is function of direction. Draine and Rybicki and Lightman write the angular direction vector as
If we have a defined reference frame, we would probably write
Flux
Once you’ve defined

How to calculate flux from specific intensity. Credit: Rybicki and Lightman Ch. 1
In astronomy settings,
Note that
What is true is
You’ll sometimes also see a spectral energy distribution plotted
I find the question of whether we’re referring to
: specific intensity (Jy/ster) : spectral flux density (Jy) : Bolometric flux .
to be endlessly confusing in conversation, because it’s very common to colloquially use “flux” or “brightness” to mean a range of quantities, even though they have specific definitions in many settings. I find the clearest thing is to state the variable
Energy density and photon occupation number
The “mean” (directionally averaged) intensity is
and the mean radiation density is
For a thermal, blackbody spectrum, we have
In equilibrium, the absorbers must have levels populated according to
The following doesn’t depend on thermal equilibrium
We can also write down the photon occupation number
which we can average over all directions to get
These photon occupation numbers make it easy to rewrite the transition rates.
From
Stimulated emission is not important when
From
Absorption cross section
Because we’re talking about photons, instead of a velocity-dependent cross section, we will write a frequency-dependent cross section,
Photon density [1/cm^3] per unit frequency
Like in Ch 2, we would rate a rate as
For the
We can relate this back to the Einstein B coefficient for absorption to find
Oscillator strength
You can also write these relationships with something called the oscillator strength,
Intrinsic line profile
A Lorentzian is a good description of the intrinsic line profile (need quantum mechanical calculation to get it exact).
The intrinsic width of the absorption line reflects the uncertainty in the energies of levels
Intrinsic widths of lines
We’ll talk more about absorption line profiles, Doppler broadening, Voigt profiles, etc… in a few lectures.