# 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. Say we have 2 energy levels with an energy difference of . Einstein coefficients govern the interaction of radiation with these discrete energy levels. There are three coefficients:

*Left: Photon absorption is described by . Center: Spontaneous photon emission is described by . Right: Stimulated photon emission is described by .*

### 1 Spontaneous Emission,

governs decay from energy state 2 to 1. It is the transition probability per unit time for an atom, and has units of . More specifically, the probability an atom undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate . So is the mean lifetime of the excited state. As an example, (n=3 to n=2 in hydrogen) has an Einstein A coefficeint of .

### 2 Spontaneous Absorption,

governs absorptions causing transitions . The transition probability per unit time is , where is the probability constant, and is:

It depends on (the intensity), but it does not depend on direction, so we integrate over all angles. The is a normalization constant which makes the mean intensity, instead of the total intensity. However, we have to remember that there are uncertainties in the energy-level separations. is called the line profile function. It describes the relative absorption probability around (the absorption frequency), and is subject to the requirement that: . We can approximate the width of as an effective width . **Failed to parse (unknown function "\n"): {\displaystyle \Delta \n }**
is affected by many factors:

- (the natural, uncertainty-based broadening of at atom in isolation),