Einstein Coefficients

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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:

B12.png A21.png B21.png
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 . is affected by many factors:

  • (the natural, uncertainty-based broadening of at atom in isolation),
  • (Dopper broadening from thermal motion), and
  • (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.

Suffice to say, the transition probability per unit time associated with spontaneous absorption is:

3 Stimulated Emission,

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:

and

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 .