# Einstein Coefficients

Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms. Suppose we have an atom with 2 energy levels with an energy difference of $\Delta E=h\nu _{0}$ . Einstein coefficients describe the transition rates caused by the interaction of radiation with these discrete energy levels. There are three coefficients:   Left: Photon absorption rates are described by $B_{12}$ . Center: Spontaneous photon emission rates are described by ${A_{21}}$ . Right: Stimulated photon emission rates are 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, $H_{\alpha }$ ($3\to 2$ transition in hydrogen) has an Einstein A coefficient of ${A_{21}}\approx 10^{9}s^{-1}$ .

If $n_{2}$ describes the number density of atoms in the upper energy state, then the transition rate per volume is given by:

$r_{\rm {spon}}=n_{2}{A_{21}}\,\!$ ### 2 Spontaneous Absorption, ${B_{12}}$ ${B_{12}}$ governs photon absorption that causes a transition from the lower to upper energy state ($1\to 2$ ). In contrast to the ${A_{21}}$ case, absorption requires the presence of photons, so translating ${B_{12}}$ to an excitation rate requires some knowledge of the background radiation field.

To describe the background radiation field, we define the spherically averaged specific intensity:

$J_{\nu }\equiv {\frac {1}{4\pi }}\int {I_{\nu }d\Omega }\,\!$ We use $J_{\nu }$ instead of $I_{\nu }$ (the intensity) because atomic absorption does not depend on direction. However, we have to remember that there are uncertainties in the energy-level separations, which means that atoms absorb photons that are not perfectly tuned to the energy difference between electronic states. To incorporate this, we use the line profile function, $\phi (\nu )$ . 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),
• $\nu _{0}v_{\rm {therm}}/c$ (Doppler broadening from thermal motion), and
• $n_{coll}\sigma _{coll}v_{rel}$ (collisional broadening, a.k.a. pressure broadening).

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.

Using the line profile function, we get the transition probability per unit time associated with spontaneous absorption:

$t_{ex}^{-1}={B_{12}}\int _{0}^{\infty }{J_{\nu }\phi (\nu )d\nu }\approx {B_{12}}{\bar {J}}\,\!$ ### 3 Stimulated Emission, ${B_{21}}$ ${B_{21}}$ governs stimulated emission. In this example, we are in energy state 2, and an incoming photon causes a transition to energy level 1 and the emission of 2 photons. The transition per unit time is ${B_{21}}{\bar {J}}$ .

## 1 Einstein Relations among coefficients

Assume we have many atoms with 2 energy states, and $n_{1}$ is the # density in state 1, ditto for $n_{2}$ . 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: ${\bar {J}}={n_{2}{A_{21}} \over n_{1}{B_{12}}-n_{2}{B_{21}}}$ . Using that ${n_{2} \over n_{1}}={g_{2} \over g_{1}}e^{-h\nu _{0} \over kT}$ :

${\bar {J}}={{{A_{21}} \over {B_{21}}} \over {g_{1}{B_{12}} \over g_{2}{B_{21}}}e^{h\nu _{0} \over kT}-1}\,\!$ In thermal equilibrium $J_{\nu }=B_{\nu }$ :

{\begin{aligned}{\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{aligned}}\,\! Combining this with ${\bar {J}}$ earlier, we get:

${g_{1}{B_{12}}=g_{2}{B_{21}}}\,\!$ and

${{A_{21}} \over {B_{21}}}={2h\nu ^{3} \over c^{2}}\,\!$ ## 2 Rewriting $j_{\nu },\alpha _{\nu }$ in terms of Einstein coeffs

In a small volume $dV$ :

{\begin{aligned}j_{\nu }&\equiv {dE \over dt\,dV\,d\nu \,d\Omega }\\&={h\nu _{0}{A_{21}}n_{2}\phi (\nu ) \over 4\pi }\\\end{aligned}}\,\! We can express $\alpha _{\nu }$ in terms of the Einstein coefficients. The excitation probability per time is $n_{1}B_{12}{\bar {J}}$ , and the energy lost in crossing the small volume $\propto n_{1}{B_{12}}{I_{\nu }d\Omega \over 4\pi }\phi (\nu )d\nu$ (it is the probability per time per volume of going $1\to 2$ by absorbing $I_{\nu }$ from a cone of solid angle $d\Omega$ and frequency range $[\nu ,\nu +d\nu ]$ ). Thus, the energy is given by:

{\begin{aligned}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{aligned}}\,\! Recognizing that $dV=dA\,ds$ :

$\alpha _{\nu }={n_{1}{B_{12}}\phi (\nu ) \over 4\pi }h\nu \,\!$ 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 ${B_{12}}$ , we use the fact that, ignoring g’s, ${B_{12}}\sim {B_{21}}$ , and ${{A_{21}} \over {B_{21}}}={2h\nu ^{3} \over c^{2}}$ . Then using the approximation that that $\phi (\nu )\sim {\frac {1}{\Delta \nu }}$ , we get:

$\sigma _{12}\sim {{A_{21}} \over \left({2h\nu ^{3} \over c^{2}}\right)}{\frac {h\nu }{4\pi \Delta \nu }}\,\!$ ${\sigma _{12}\sim {\lambda ^{2} \over 8\pi }{{A_{21}} \over \Delta \nu }}\,\!$ In a single atom, $\Delta \nu \sim {A_{21}}$ , so $\sigma _{12}\sim {\lambda ^{2} \over 8\pi }$ .