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A standard way of defining the Einstein coefficients is as follows:

  • The Einstein coefficient $A_{21}$ is defined such that the number of atoms that transition from level 2 to level 1 via spontaneous emission per unit time, per unit volume is $A_{21} N_2$.
  • The Einstein coefficient $B_{21}$ is defined such that the number of atoms that transition from level 2 to level 1 via stimulated emission per unit time per unit volume due to a photon in the frequency range $[\nu, \nu+d\nu]$ is $B_{21}N_2\phi(\nu)\rho(\nu)d\nu$ where $\phi(\nu)$ is the line profile.
  • The Einstein coefficient $B_{12}$ is defined such that the number of atoms that transition from level 1 to level 2 via absorption per unit time per unit volume due to a photon in the frequency range $[\nu, \nu+d\nu]$ is $B_{21}N_2\phi(\nu)\rho(\nu)d\nu$ where $\phi(\nu)$ is the line profile.

(Note they may also be defined in terms of $J_\nu=\frac{c}{4\pi} \rho(\nu)$ instead of $\rho(\nu)$).

My problem with these definitions is the presences of the $\phi(\nu)$ term. For these to hold we must assume that the system is predominately homogeneously broadened else we could not stick the line profile in like this. My question is therefore; for a system which is inhomogeneously broadened do we /can we use the Einstein coefficients? If so how do we define them and if not what is done instead?

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