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,  
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* [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)]
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* [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  
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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}
 
Derivation identical to Rybicki's.  We should memorize these.\par
 
  
 
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.\par
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now.
 +
 
 
There are 3 coefficients:\par
 
There are 3 coefficients:\par
 
\begin{itemize}
 
\begin{itemize}
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\end{itemize}
 
\end{itemize}
  
\subsection*{ Einstein Relations among coefficients}
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\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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$${\ato\over\bto}={2h\nu^3\over c^2}$$
 
$${\ato\over\bto}={2h\nu^3\over c^2}$$
  
\subsection*{ Rewriting $j_\nu, \alpha_\nu $ in terms of Einstein coeffs}
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\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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$$\boxed{\alpha_\nu={(n_1\bot-n_2\bto)\phi(\nu)h\nu\over4\pi}}$$
 
$$\boxed{\alpha_\nu={(n_1\bot-n_2\bto)\phi(\nu)h\nu\over4\pi}}$$
  
\subsection*{ Estimating Cross-Sections }
+
\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

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Einstein Coefficients

Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms. 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 .

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 .