Spontaneous Emission, Stimulated Emission, and Absorption

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  • 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 hν=EuEl the absorber can absorb a photon and undergo an upward transition Xl+hνXu.

Rate of this reaction

There is some n(Xl). The rate per volume at which absorbers absorb photons (lu) is proportional to the density of photons of the appropriate energy and the number density of absorbers

(dnudt)lupopulatelevelu=(dnldt)depopulatelevell

=nlBluuν

uν is the radiation energy density per unit frequency.

Blu is a proportionality constant called the Einstein B coefficient for the transition lu.

Emission of photons

Spontaneous emission

XuXl+hν

Random process (independent of radiation field), and occurs with a probability per unit time Aul called the Einstein A coefficient.

Stimulated emission

Xu+hνXl+2hν

Occurs if photons of identical

  • frequency
  • polarization
  • direction are present in a radiation field.

Rate of emission

From state ul

(dnldt)=(dnudt)=nu(Aul+Buluν)

Bul is the Einstein B coefficient for the downward transition.

Einstein coefficients are not independent

Bul=c38πhν3Aul

Blu=guglBul=guglc38πhν3Aul

Strength of stimulated emission (Bul) and absorption (Blu) are both determined by Aul (spontaneous emission) and the ratio of the degeneracies gugl.

Radiation fields

(Review from Rybicki and Lightman, Ch 1)

Iν is the specific intensity of radiation, you can think of it as the energy carried along by an infinitesimal “bundle” of rays.

It has dimensions ergss1cm2ster1Hz1 I like calling Iν the “specific intensity,” and that seems to be common usage in astronomy. In non-astronomy settings, this is called “spectral intensity.” If it is integrated over all frequency, it’s sometimes called the radiant intensity I.

Iν can be a little mind-bending to think about… it can be a function of

  • 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 Ω and the solid angle surrounding that vector as dΩ.

If we have a defined reference frame, we would probably write Ω as a vector in spherical coordinates and define the components along ϕ^ and θ^ and let dΩ=dϕsinθdθ.

Bν is exactly the same type of variable, we just use it instead of Iν when we’re referring to blackbody radiation, specifically of the form Bν=2hν3c21ehν/kT1 it also has units of ergss1cm2ster1Hz1

Flux

Once you’ve defined Iν, then it’s relatively easy to calculate quantities like energy density, flux, momentum, etc, as integrals of the specific intensity field.

How to calculate flux from specific intensity. Credit: Rybicki and Lightman Ch. 1

How to calculate flux from specific intensity. Credit: Rybicki and Lightman Ch. 1

Fν=IνcosθdΩ (intensity passing through some differential area dA, lowered by the effective angle).

Fν has units of ergss1cm2Hz1 (i.e., angular dependence has been integrated out). I like using the unit of Jansky, which is 1Jy=1023ergss1cm2Hz1

In astronomy settings, Fν is called the spectral flux density. In non-astronomy settings, this is called spectral irradiance. Sometimes you will see Fλ, which has units ergss1cm2cm1 (or per Å, depending on your wavelength range).

Note that Fν and Fλ are flux densities or distribution function, which means that FνFλ (their units are different)!

What is true is Fνdν=Fλdλ because we’re comparing two quantities with units ergss1cm2

You’ll sometimes also see a spectral energy distribution plotted Fνν=Fλλ this comes about because ν=cλ dλdν=cν2=λν and the minus sign goes away because fluxes are defined per positive unit of frequency or wavelength.

I find the question of whether we’re referring to

  • Iν: specific intensity (Jy/ster)
  • Fν: spectral flux density (Jy)
  • F: Bolometric flux ergss1cm2.

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 Iν or Fν (or Fλ) and the units.

Energy density and photon occupation number

The “mean” (directionally averaged) intensity is Jν=14πIνdΩ=14πIνsinθdθdϕ. it has a different meaning than flux. In an isotropic radiation field, the net flux will be 0, whereas the mean intensity will have some positive value.

and the mean radiation density is uν=4πcJν. these quantities are not a function of direction, since we’ve averaged over it.

For a thermal, blackbody spectrum, we have uν=4πcBν(T)

In equilibrium, the absorbers must have levels populated according to nunl=gugle(ElEu)/kT

The following doesn’t depend on thermal equilibrium

We can also write down the photon occupation number nγ, i.e., a dimensionless quantity tracking how many photons exist “per given solid angle”

nγ=c22hν3Iν

which we can average over all directions to get nbar,γ=c38πhν3uν (a truly dimensionless quantity).

These photon occupation numbers make it easy to rewrite the transition rates.

From ul (dnldt)=nuAul(1+nbar,γ)

Stimulated emission is not important when nbar,γ1.

From lu (dnudt)=nlguglAulnbar,γ

Absorption cross section

Because we’re talking about photons, instead of a velocity-dependent cross section, we will write a frequency-dependent cross section, σ(ν). The velocity will be the speed of light c.

Photon density [1/cm^3] per unit frequency uνhν

Like in Ch 2, we would rate a rate as nAnBσv we’ll do the same thing with nl, uνhν, and σ(ν).

For the lu transition, (dnudt)=nldνσlu(ν)cuνhν.

We can relate this back to the Einstein B coefficient for absorption to find Blu=chνdνσlu(ν) relate this back to the Aul coefficient, dνσlu(ν)=guglc28πνlu2Aul. Then solve for σlu(ν) σlu(ν)=guglc28πνlu2Aulϕν where ϕν is a normalized line profile ϕνdν=1.

Oscillator strength

You can also write these relationships with something called the oscillator strength, flu. Aul=8π2e2νlu2mec3glguflu

Intrinsic line profile

A Lorentzian is a good description of the intrinsic line profile (need quantum mechanical calculation to get it exact).

ϕ(ν)=4γul16π2(ννul)2+γul2

The intrinsic width of the absorption line reflects the uncertainty in the energies of levels l,u due to the finite lifetimes against transitions to other levels.

Intrinsic widths of lines hν, which means x-ray transitions can have considerably larger line widths than radio transitions (assuming the medium is stationary).

We’ll talk more about absorption line profiles, Doppler broadening, Voigt profiles, etc… in a few lectures.