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

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