In our discussion of recombination, we related the emission of Lyman photons to their absorption. This

In our discussion of recombination, we related the emission of Lyman α photons to their absorption. This involves some important ideas in the theories of radiation and thermodynamics.

(a) Consider a population of two-state atoms. Let the number of atoms in the lower state be N₁ and in the upper state N₂. The probability per unit time of an upperstate atom changing to a lower state and releasing a photon of energy hν equal to the energy difference of the states is denoted by A. We expect that the rate of upward, 1→2 transitions is proportional to the occupation number η_γ of the photons with frequency ν. Call this rate K_uη_γ. By requiring that the atoms should be able to remain in Boltzmann equilibrium with the Planckian radiation field of the same temperature, show that there must also be downward, stimulated emission at a rate per state-2 atom of Kdηγ , and show that K_u = K_d = A.

(b) The absorption cross section σ for an atom at rest can be written in the Lorentz or Breit-Wigner form as:

Either make a classical model of an atom as an electron oscillator with natural frequency ν₀, or use time-dependent perturbation theory in quantum mechanics to justify the form of this formula.
(c) Identify the frequency probability function P_ν introduced in Sec. 28.4.4, and plot the natural line profile for an emission line.

(d) Atoms also have thermal motions, which Doppler shift the photon frequencies. Modify the line profile by numerically convolving the natural profile with a 1- dimensional Gaussian velocity distribution and replot it, drawing attention to its behavior when A is much more than the thermal Doppler shift.

(e) We have restricted our attention to a two-state system. It is usually the case that we are dealing with energy levels containing several distinct states, and the formalism we have described has to be modified to include the degeneracies g_i of these levels. Make the necessary corrections and recover the formulas used in the text.

(f) A second complication is polarization. Discuss how to include this.

Transcribed Image Text:

6= ΠΑ2 (ν – vo)2 + A2