Einstein Coefficients

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Einstein Coefficients

Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms. Say we have 2 energy levels with an energy difference of . Einstein coefficients govern the interaction of radiation with these discrete energy levels. There are three coefficients:

B12.png A21.png B21.png
Left: Photon absorption is described by . Center: Spontaneous photon emission is described by . Right: Stimulated photon emission is described by .

1 Spontaneous Emission,

governs decay from energy state 2 to 1. It is the transition probability per unit time for an atom, and has units of . More specifically, the probability an atom undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate . So is the mean lifetime of the excited state. As an example, (n=3 to n=2 in hydrogen) has an Einstein A coefficeint of .

2 Spontaneous Absorption,

governs absorptions causing transitions . The transition probability per unit time is , where is the probability constant, and is:

It depends on (the intensity), but it does not depend on direction, so we integrate over all angles. The is a normalization constant which makes the mean intensity, instead of the total intensity. However, we have to remember that there are uncertainties in the energy-level separations. is called the line profile function. It describes the relative absorption probability around (the absorption frequency), and is subject to the requirement that: . We can approximate the width of as an effective width . is affected by many factors:

  • (the natural, uncertainty-based broadening of at atom in isolation),

(Dopper broadening from thermal motion), and

  • n_collσ_collv_rel

Line profile functions are of special interest for studying line emission/absorption, and we have more discussion in a separate section on line profile functions.

Suffice to say, the transition probability per unit time associated with spontaneous absorption is:

3 Stimulated Emission, B_21


1 Einstein Relations among coefficients

Assume we have many atoms with 2 energy states, and n_1n_2Failed to parse (syntax error): {\displaystyle . Assume we are in thermal, steady-state equilibrium, so: \[ n_1{B_{12}}{\bar J}=n_2{A_{21}}+n_2{B_{21}}{\bar J} \] This is because as many atoms need to be going from energy state 1 to 2 as visa versa. A second relation is: } J= n_2A_21n_1B_12-n_2B_21n_2n_1=g_2g_1e^-hν_0 kTFailed to parse (syntax error): {\displaystyle : \[ {\bar J}={{{A_{21}}\over {B_{21}}}\over {g_1{B_{12}}\over g_2{B_{21}}}e^{{-h\nu _0 \over kT}}-1} \] In thermal equilibrium } J_ν=B_νFailed to parse (syntax error): {\displaystyle : \[ \begin{align} {\bar J}& \equiv \int _0^\infty {J_\nu \phi (\nu )d\nu }\\ & =\int _0^\infty {B_\nu \phi (\nu )d\nu }\\ & \approx B_\nu (\nu _0)\\ & ={2h\nu _0^3\over c^2(e^{{-h\nu _0 \over kT}}-1)}\\ \end{align} \] Combining this with } JFailed to parse (syntax error): {\displaystyle earlier, we get: \[ {g_1{B_{12}}=g_2{B_{21}}} \] \centerline{and} \[ {{A_{21}}\over {B_{21}}}={2h\nu ^3\over c^2} \] }

2 Rewriting j_ν, α_ν

In a small volume dVFailed to parse (syntax error): {\displaystyle : \[ \begin{align} j_\nu & \equiv {dE\over dt\, dV\, d\nu \, d\Omega }\\ & ={h\nu _0{A_{21}}n_2\phi (\nu )\over 4\pi }\\ \end{align} \] We can express } α_νn_1B_12J∝n_1B_12I_νdΩ4πφ(ν)dν12I_νdΩ[ν,ν+dν]Failed to parse (syntax error): {\displaystyle ). Thus, the energy is given by: \[ \begin{align} E& =n_1{B_{12}}{I_\nu d\Omega \over 4\pi }\phi (\nu )d\nu h\nu dt\, dV\\ & =\alpha _\nu I_\nu ds\, dt\, d\Omega \, dA\, d\nu \\ \end{align} \] Recognizing that } dV=dA dsFailed to parse (syntax error): {\displaystyle : \[ \alpha _\nu ={n_1{B_{12}}\phi (\nu )\over 4\pi }h\nu \] Correcting for stimulated emission, we get: \[ {\alpha _\nu ={(n_1{B_{12}}-n_2{B_{21}})\phi (\nu )h\nu \over 4\pi }} \] }

3 Estimating Cross-Sections

The absorption coefficient, written in terms of Einstein constants is:

Thus, the cross-section of an atom for absorption of a photon is:

To estimate B_12Failed to parse (syntax error): {\displaystyle , we use the fact that, ignoring g’s, } B_12∼B_21A_21B_21=2hν^3c^2φ(ν)∼Failed to parse (syntax error): {\displaystyle , we get: \[ \sigma _{12}\sim {{A_{21}}\over \left({2h\nu ^3\over c^2}\right)} \frac1{\Delta \nu } \] \[ {\sigma _{12}\sim {\lambda ^2\over 8\pi }{{A_{21}}\over \Delta \nu }} \] In a single atom, } Δν∼A_21σ_12∼λ^28π