Difference between revisions of "Einstein Coefficients"

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Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms.   
 
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 $\Delta E=h\nu_0$.  Einstein coefficients govern the  
+
Suppose we have an atom with 2 energy levels with an energy difference of $\Delta E=h\nu_0$.  Einstein coefficients describe the  
interaction of radiation with these discrete energy
+
transition rates caused by the interaction of radiation with these discrete energy
 
levels. There are three coefficients:
 
levels. There are three coefficients:
  
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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 $\ato$.  Right: Stimulated photon emission is described by $B_{21}$.}
+
\caption{Left: Photon absorption rates are described by $B_{12}$.  Center: Spontaneous photon emission rates are described by $\ato$.  Right: Stimulated photon emission rates are described by $B_{21}$.}
 
\end{figure}
 
\end{figure}
  
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undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate
 
undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate
 
$\ato$. So $\ato^{-1}$ is the mean lifetime of the excited state.  As an example,
 
$\ato$. So $\ato^{-1}$ is the mean lifetime of the excited state.  As an example,
$H_\alpha$ (n=3 to n=2 in hydrogen) has an Einstein A coefficeint of $\ato\approx 10^9 s^{-1}$.
+
$H_\alpha$ ($3\to2$ transition in hydrogen) has an Einstein A coefficient of $\ato\approx 10^9 s^{-1}$.
 +
 
 +
If $n_2$ describes the number density of atoms in the upper energy state, then the transition rate per volume is given by:
 +
\begin{equation}
 +
r_{\rm spon}=n_2\ato
 +
\end{equation}
  
 
\subsection{Spontaneous Absorption, $\bot$}
 
\subsection{Spontaneous Absorption, $\bot$}
  
$\bot$ governs absorptions causing transitions $1\to2$.
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$\bot$ governs photon absorption that causes a transition from the lower to upper energy state ($1\to2$). In contrast
The transition probability per unit time is $\bot J_\nu$, where $\bot$ is
+
to the $\ato$ case, absorption requires the presence of photons, so translating $\bot$ to an excitation rate requires
the probability constant, and $J_\nu$ is:
+
some knowledge of the background radiation field.
$$J_\nu \equiv {\int{I_\nu d\Omega} \over 4\pi}$$  
+
 
It depends on $I_\nu$ (the intensity), but it does not depend on  
+
To describe the background radiation field, we define the spherically averaged specific intensity:
direction, so we integrate over all angles.  The $4\pi$ is a normalization
+
\begin{equation}
constant which makes $J_\nu$ the mean intensity, instead of the total intensity.
+
J_\nu \equiv \frac1{4\pi}\int{I_\nu d\Omega}
 +
\end{equation}
 +
We use $J_\nu$ instead of $I_\nu$ (the intensity) because atomic absorption does not depend on direction.
 
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.  $\phi(\nu)$ is called the line profile function.  It  
+
separations, which means that atoms absorb photons that are not perfectly tuned to
 +
the energy difference between electronic statesTo incorporate this, we use the line profile function, $\phi(\nu)$.  It  
 
describes
 
describes
 
the relative absorption probability around $\nu_0$ (the
 
the relative absorption probability around $\nu_0$ (the
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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.
 
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:
+
Using the line profile function, we get the transition probability per unit time associated with spontaneous absorption:
$$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$
+
\begin{equation}
 +
t_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar
 +
\end{equation}
  
 
\subsection{Stimulated Emission, $\bto$}
 
\subsection{Stimulated Emission, $\bto$}

Revision as of 13:28, 18 September 2014

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Reference Material

Einstein Coefficients

Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms. Suppose we have an atom with 2 energy levels with an energy difference of . Einstein coefficients describe the transition rates caused by the interaction of radiation with these discrete energy levels. There are three coefficients:

B12.png A21.png B21.png
Left: Photon absorption rates are described by . Center: Spontaneous photon emission rates are described by . Right: Stimulated photon emission rates are 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, ( transition in hydrogen) has an Einstein A coefficient of .

If describes the number density of atoms in the upper energy state, then the transition rate per volume is given by:

2 Spontaneous Absorption,

governs photon absorption that causes a transition from the lower to upper energy state (). In contrast to the case, absorption requires the presence of photons, so translating to an excitation rate requires some knowledge of the background radiation field.

To describe the background radiation field, we define the spherically averaged specific intensity:

We use instead of (the intensity) because atomic absorption does not depend on direction. However, we have to remember that there are uncertainties in the energy-level separations, which means that atoms absorb photons that are not perfectly tuned to the energy difference between electronic states. To incorporate this, we use 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.

Using the line profile function, we get the transition probability per unit time associated with spontaneous absorption:

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 .