Difference between revisions of "Einstein Coefficients"

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Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms.  There are three coefficients:
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Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms.   
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Say we have 2 energy levels with an energy difference of $\Delta E=h\nu_0$.  Einstein coefficients govern the
 +
interaction of radiation with these discrete energy
 +
levels. There are three coefficients:
  
 
\begin{figure}\centering
 
\begin{figure}\centering
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\includegraphics[width=2in]{a21.png}
 
\includegraphics[width=2in]{a21.png}
 
\includegraphics[width=2in]{b21.png}
 
\includegraphics[width=2in]{b21.png}
\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 $B_{21}$.}
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\caption{Left: Photon absorption is described by $B_{12}$.  Center: Spontaneous photon emission is described by $\ato$.  Right: Stimulated photon emission is described by $B_{21}$.}
 
\end{figure}
 
\end{figure}
  
These coefficients govern the interaction of radiation with discrete energy
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\subsection{Spontaneous Emission, $\ato$}
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
 
now.
 
  
There are 3 coefficients:\par
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$\ato$ governs decay from energy state 2 to 1.  It is the transition  
\begin{itemize}
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probability per unit time for an atom, and has units of $s^{-1}$More specifically, the probability an atom
\item  $\ato$ governs decay from 2 to 1, and is the transition  
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undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate
probability per
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$\ato$. So $\ato^{-1}$ is the mean lifetime of the excited state.   As an example,
unit time.  The probability of spontaneous
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$H_\alpha$ (n=3 to n=2 in hydrogen) has an Einstein A coefficeint of $\ato\approx 10^9 s^{-1}$.
de-excitation and release of photon is Poisson-distributed with mean rate
 
$\ato$. So $\ato^{-1}$ is the mean lifetime of the excited state. e.g.
 
For $H_\alpha$ (n=3 to n=2): $\ato\approx 10^9 s^{-1}$.
 
  
\item  $\bot$ governs absorptions causing transitions $1\to2$.
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\subsection{Spontaneous Absorption, $\bot$}
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$\bot$ governs absorptions causing transitions $1\to2$.
 
The transition probability per unit time is $\bot J_\nu$, where $\bot$ is
 
The transition probability per unit time is $\bot J_\nu$, where $\bot$ is
 
the probability constant, and $J_\nu$ is:
 
the probability constant, and $J_\nu$ is:
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constant which makes $J_\nu$ the mean intensity, instead of the total intensity.
 
constant which makes $J_\nu$ the mean intensity, instead of the total intensity.
 
However, we have to remember that there are uncertainties in the energy-level
 
However, we have to remember that there are uncertainties in the energy-level
separations.  $\phi(\nu)\equiv$ is called the line profile function.  It  
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separations.  $\phi(\nu)$ is called the line profile function.  It  
 
describes
 
describes
 
some (maybe gaussian) distribution of absorption around $\nu_0$ (the
 
some (maybe gaussian) distribution of absorption around $\nu_0$ (the
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$$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$
 
$$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$
  
\item  $\bto$ governs stimulated emission.  In this example, we are in energy
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\subsection{Stimulated Emission, $\bto$}
 +
 
 +
$\bto$ 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
 
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 $\bto\Jbar$.
 
emission of 2 photons.  The transition per unit time is $\bto\Jbar$.
\end{itemize}
 
  
 
\section{ Einstein Relations among coefficients}
 
\section{ Einstein Relations among coefficients}

Revision as of 12:24, 18 September 2014

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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 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:

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 .