# Difference between revisions of "Einstein Coefficients"

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* [http://en.wikipedia.org/wiki/Einstein_coefficients Einstein Coefficients (Wikipedia)] | * [http://en.wikipedia.org/wiki/Einstein_coefficients Einstein Coefficients (Wikipedia)] | ||

* [http://www.physics.byu.edu/faculty/christensen/Physics%20428/FTI/The%20Einstein%20Coefficients.htm Einstein Coefficients (Christensen, BYU)] | * [http://www.physics.byu.edu/faculty/christensen/Physics%20428/FTI/The%20Einstein%20Coefficients.htm Einstein Coefficients (Christensen, BYU)] | ||

− | * [http://www.doylegroup.harvard.edu/wiki/images/f/f4/Hilborn.pdf Einstein coefficients, cross sections, f values, dipole moments, | + | * [http://www.doylegroup.harvard.edu/wiki/images/f/f4/Hilborn.pdf Einstein coefficients, cross sections, f values, dipole moments, and all that (Hilborn, Amherst)] |

− | and all that (Hilborn, Amherst)] | + | * [http://www-star.st-and.ac.uk/~kw25/teaching/nebulae/lecture07_detbalance.pdf Relations between the Einstein coefficients (Wood, U. St. Andrews)] |

− | * [http://www-star.st-and.ac.uk/~kw25/teaching/nebulae/lecture07_detbalance.pdf Relations between the Einstein | + | |

− | coefficients (Wood, U. St. Andrews)] | ||

<latex> | <latex> | ||

\documentclass[11pt]{article} | \documentclass[11pt]{article} | ||

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\caption{Left: Photon absorption is described by $B_{12}$. Center: Spontaneous photon emission is described by $A_{21}$. Right: Stimulated photon emission is described by $B_{21}$.} | \caption{Left: Photon absorption is described by $B_{12}$. Center: Spontaneous photon emission is described by $A_{21}$. Right: Stimulated photon emission is described by $B_{21}$.} | ||

\end{figure} | \end{figure} | ||

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These coefficients govern the interaction of radiation with discrete energy | These coefficients govern the interaction of radiation with discrete energy | ||

levels. Say we have 2 energy levels with a difference $\Delta E=h\nu_0$. | levels. Say we have 2 energy levels with a difference $\Delta E=h\nu_0$. | ||

There is some uncertainty associated with $\nu$, but we'll say it's small for | There is some uncertainty associated with $\nu$, but we'll say it's small for | ||

− | now. | + | now. |

+ | |||

There are 3 coefficients:\par | There are 3 coefficients:\par | ||

\begin{itemize} | \begin{itemize} | ||

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\end{itemize} | \end{itemize} | ||

− | \ | + | \section{ Einstein Relations among coefficients} |

Assume we have many atoms with 2 energy states, and $n_1$ is the \# density | Assume we have many atoms with 2 energy states, and $n_1$ is the \# density | ||

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$${\ato\over\bto}={2h\nu^3\over c^2}$$ | $${\ato\over\bto}={2h\nu^3\over c^2}$$ | ||

− | \ | + | \section{ Rewriting $j_\nu, \alpha_\nu $ in terms of Einstein coeffs} |

In a small volume $dV$: | In a small volume $dV$: | ||

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$$\boxed{\alpha_\nu={(n_1\bot-n_2\bto)\phi(\nu)h\nu\over4\pi}}$$ | $$\boxed{\alpha_\nu={(n_1\bot-n_2\bto)\phi(\nu)h\nu\over4\pi}}$$ | ||

− | \ | + | \section{ Estimating Cross-Sections } |

The absorption coefficient, written in terms of Einstein constants is: | The absorption coefficient, written in terms of Einstein constants is: |

## Revision as of 12:02, 18 September 2014

### Short Topical Videos

### 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. 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:

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 .