# 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 ${\displaystyle \Delta \nu }$. $\displaystyle \Delta \n$ is affected by many factors:

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