# 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 $\Delta E=h\nu _{0}$ . Einstein coefficients govern the interaction of radiation with these discrete energy levels. There are three coefficients:   Left: Photon absorption is described by $B_{12}$ . Center: Spontaneous photon emission is described by ${A_{21}}$ . Right: Stimulated photon emission is described by $B_{21}$ .

### 1 Spontaneous Emission, ${A_{21}}$ ${A_{21}}$ governs decay from energy state 2 to 1. It is the transition probability per unit time for an atom, and has units of $s^{-1}$ . More specifically, the probability an atom undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate ${A_{21}}$ . So ${A_{21}}^{-1}$ is the mean lifetime of the excited state. As an example, $H_{\alpha }$ (n=3 to n=2 in hydrogen) has an Einstein A coefficeint of ${A_{21}}\approx 10^{9}s^{-1}$ .

### 2 Spontaneous Absorption, ${B_{12}}$ ${B_{12}}$ governs absorptions causing transitions $1\to 2$ . The transition probability per unit time is ${B_{12}}J_{\nu }$ , where ${B_{12}}$ is the probability constant, and $J_{\nu }$ is:

$J_{\nu }\equiv {\int {I_{\nu }d\Omega } \over 4\pi }\,\!$ It depends on $I_{\nu }$ (the intensity), but it does not depend on direction, so we integrate over all angles. The $4\pi$ is a normalization constant which makes $J_{\nu }$ the mean intensity, instead of the total intensity. However, we have to remember that there are uncertainties in the energy-level separations. $\phi (\nu )$ is called the line profile function. It describes the relative absorption probability around $\nu _{0}$ (the absorption frequency), and is subject to the requirement that: $\int _{0}^{\infty }{\phi (\nu )d\nu }=1$ . We can approximate the width of $\phi (\nu )$ as an effective width $\Delta \nu$ . $\Delta \nu$ is affected by many factors:

• ${A_{21}}$ (the natural, uncertainty-based broadening of at atom in isolation),
• $\nu _{0}v_{\rm {therm}}/c$ (Dopper broadening from thermal motion), and

• n_collσ_collv_rel$(collisionalbroadening,a.k.a.pressurebroadening).$ 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:

$R_{ex}^{-1}={B_{12}}\int _{0}^{\infty }{J_{\nu }\phi (\nu )d\nu }\approx {B_{12}}{\bar {J}}\,\!$ ### 3 Stimulated Emission, B_21

B_21$governsstimulatedemission.Inthisexample,weareinenergystate2,andanincomingphotoncausesatransitiontoenergylevel1andtheemissionof2photons.Thetransitionperunittimeis$ B_21J$.$ ## 1 Einstein Relations among coefficients

Assume we have many atoms with 2 energy states, and n_1$isthe\#densityinstate1,dittofor$ n_2$\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_21$.Usingthat$ n_2n_1=g_2g_1e^-hν_0 kT$\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_ν\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 J$\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$intermsofEinsteincoeffs$ In a small volume dV\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_ν$intermsoftheEinsteincoefficients.Theexcitationprobabilitypertimeis$ n_1B_12J$,andtheenergylostincrossingthesmallvolume$ ∝n_1B_12I_νdΩ4πφ(ν)dν$(itistheprobabilitypertimepervolumeofgoing$ 12$byabsorbing$ I_ν$fromaconeofsolidangle$ dΩ$andfrequencyrange$ [ν,ν+dν]\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 ds$\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:

$\alpha _{\nu }={n_{1}{B_{12}}\phi (\nu ) \over 4\pi }h\nu =n_{1}\sigma _{12}\,\!$ Thus, the cross-section of an atom for absorption of a photon is:

$\sigma _{12}={{B_{12}}\phi (\nu )h\nu \over 4\pi }\,\!$ To estimate B_12$\displaystyle , we use the fact that, ignoring g’s,$ B_12∼B_21$,and$ A_21B_21=2hν^3c^2$.Thenusingtheapproximationthatthat$ φ(ν)∼$\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_21$,so$ σ_12∼λ^28π$.$ 