Short Topical Videos
Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms. There are three coefficients:
Left: Photon absorption is described by . Center: Spontaneous photon emission is described by . Right: Stimulated photon emission is described by .
These coefficients govern the interaction of radiation with discrete energy levels. Say we have 2 energy levels with a difference . There is some uncertainty associated with , but we’ll say it’s small for now.
There are 3 coefficients:
- governs decay from 2 to 1, and is the transition probability per unit time. The probability of spontaneous de-excitation and release of photon is Poisson-distributed with mean rate . So is the mean lifetime of the excited state. e.g. For (n=3 to n=2): .
- 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 some (maybe gaussian) distribution of absorption around (the absorption frequency), and is subject to the requirement that:
Say that is the width of the distribution around . is affected by many factors: (the natural, uncertainty-based broadening of at atom in isolation), (the thermal, Doppler-based broadening), and (collisional broadening, a.k.a. pressure broadening). So really, the transition probability per unit time is:
- 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 .
1 Einstein Relations among coefficients
Assume we have many atoms with 2 energy states, and is the # density in state 1, ditto for . Assume we are in thermal, steady-state equilibrium, so:
This is because as many atoms need to be going from energy state 1 to 2 as visa versa. A second relation is: . Using that :
In thermal equilibrium :
Combining this with earlier, we get:
2 Rewriting in terms of Einstein coeffs
In a small volume :
We can express in terms of the Einstein coefficients. The excitation probability per time is , and the energy lost in crossing the small volume (it is the probability per time per volume of going by absorbing from a cone of solid angle and frequency range ). Thus, the energy is given by:
Recognizing that :
Correcting for stimulated emission, we get:
3 Estimating Cross-Sections
The absorption coefficient, written in terms of Einstein constants is:
Thus, the cross-section of an atom for absorption of a photon is:
To estimate , we use the fact that, ignoring g’s, , and . Then using the approximation that that , we get:
In a single atom, , so .