# 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 ${\displaystyle \Delta E=h\nu _{0}}$. Einstein coefficients govern the interaction of radiation with these discrete energy levels. There are three coefficients:

Left: Photon absorption is described by ${\displaystyle B_{12}}$. Center: Spontaneous photon emission is described by ${\displaystyle {A_{21}}}$. Right: Stimulated photon emission is described by ${\displaystyle B_{21}}$.

### 1 Spontaneous Emission, ${\displaystyle {A_{21}}}$

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

### 2 Spontaneous Absorption, ${\displaystyle {B_{12}}}$

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

${\displaystyle J_{\nu }\equiv {\int {I_{\nu }d\Omega } \over 4\pi }\,\!}$

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

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

(Dopper broadening from thermal motion), and

• n_collσ_collv_rel${\displaystyle (collisionalbroadening,a.k.a.pressurebroadening).}$

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:

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

### 3 Stimulated Emission, B_21

B_21${\displaystyle governsstimulatedemission.Inthisexample,weareinenergystate2,andanincomingphotoncausesatransitiontoenergylevel1andtheemissionof2photons.Thetransitionperunittimeis}$B_21J${\displaystyle .}$

## 1 Einstein Relations among coefficients

Assume we have many atoms with 2 energy states, and n_1${\displaystyle isthe\#densityinstate1,dittofor}$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${\displaystyle .Usingthat}$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${\displaystyle intermsofEinsteincoeffs}$

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${\displaystyle intermsoftheEinsteincoefficients.Theexcitationprobabilitypertimeis}$n_1B_12J${\displaystyle ,andtheenergylostincrossingthesmallvolume}$∝n_1B_12I_νdΩ4πφ(ν)dν${\displaystyle (itistheprobabilitypertimepervolumeofgoing}$12${\displaystyle byabsorbing}$I_ν${\displaystyle fromaconeofsolidangle}$dΩ${\displaystyle 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:

${\displaystyle \alpha _{\nu }={n_{1}{B_{12}}\phi (\nu ) \over 4\pi }h\nu =n_{1}\sigma _{12}\,\!}$

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

${\displaystyle \sigma _{12}={{B_{12}}\phi (\nu )h\nu \over 4\pi }\,\!}$

To estimate B_12$\displaystyle , we use the fact that, ignoring g’s,$ B_12∼B_21${\displaystyle ,and}$A_21B_21=2hν^3c^2${\displaystyle .Thenusingtheapproximationthatthat}$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${\displaystyle ,so}$σ_12∼λ^28π${\displaystyle .}$