Difference between revisions of "Einstein Coefficients"

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separations.  $\phi(\nu)$ is called the line profile function.  It  
 
separations.  $\phi(\nu)$ is called the line profile function.  It  
 
describes
 
describes
some (maybe gaussian) distribution of absorption around $\nu_0$ (the
+
the relative absorption probability around $\nu_0$ (the
 
absorption frequency), and is subject to the requirement that:
 
absorption frequency), and is subject to the requirement that:
$$\int_0^\infty{\phi(\nu)d\nu}=1$$
+
$\int_0^\infty{\phi(\nu)d\nu}=1$.  We can approximate the width of $\phi(\nu)$ as an effective width $\Delta\nu$.
Say that $\Delta\nu$ is the width
+
$\Delta\n $ is affected by many factors:
of the distribution around $\nu_0$. $\Delta \nu $ is affected by many factors:
+
\begin{itemize}
$\ato$ (the natural, uncertainty-based broadening of at atom in isolation),
+
\item $\ato$ (the natural, uncertainty-based broadening of at atom in isolation),
$\nu_0{V_T\over c}$ (the thermal, Doppler-based broadening), and  
+
\item $\nu_0\frac{v_{\rm therm}{c}$ (the thermal, Doppler-based broadening), and  
$n_{coll}\sigma_{coll}v_{rel}$
+
\item $n_{coll}\sigma_{coll}v_{rel}$
(collisional broadening, a.k.a. pressure broadening).
+
(collisional broadening, a.k.a. pressure broadening).  
So really, the transition probability per unit time is:
+
\end{itemize}
 +
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:
 
$$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$
 
$$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$
  

Revision as of 12:30, 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 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 . Failed to parse (unknown function "\n"): {\displaystyle \Delta \n } is affected by many factors:

  • (the natural, uncertainty-based broadening of at atom in isolation),