Einstein Coefficients

Short Topical Videos

• LASER: Stimulated emission: part 1, part 2 (Nainani, Stanford)
• Einstein Coefficients Part 1, Part 2, Part 3 (kridnix, Bucknell)

Reference Material

• Einstein Coefficients (Wikipedia)
• Einstein Coefficients (Christensen, BYU)
• Einstein coefficients, cross sections, f values, dipole moments, and all that (Hilborn, Amherst)
• Relations between the Einstein coefficients (Wood, U. St. Andrews)

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),
• $\nu _{0}v_{\rm {therm}}/c$ (Dopper 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 $\sigma _{12}\sim {\lambda ^{2} \over 8\pi }$.