
\sect{Boltzmann equation: Emission }

%Baruch's D05.

Equilibrium and kinetics of light and matter:
\begin{itemize}
\item[(a)] Consider atoms with fixed positions that can be either in their
ground state $a_0$, or in an excited state $a_1$, which has a
higher energy $\epsilon$. If $n_0$ and $n_1$ are the densities of
atoms in the the two levels, find the ratio $n_1/n_0$ at
temperature $T$.

\item[(b)] Consider photons $\gamma$ of frequency $\omega=\epsilon/\hbar$
and momentum $|{\bf p}|=\hbar\omega/c$, which can interact with
the atoms through the following processes:

(i) {\em Spontaneous emission}: $a_1\rightarrow a_0+\gamma$

(ii) {\em Absorption}: $a_0+\gamma \rightarrow a_1$

(iii) {\em Stimulated emission}: $a_1+\gamma \rightarrow
a_0+\gamma+\gamma$.

Assume that spontaneous emission occurs with a probability
$\sigma_1$ (per unit time and per unit (momentum)$^3$) and that
absorption and stimulated emission have constant (angle
independent) differential cross-sections of $\sigma_2$ and
$\sigma_3/4\pi$, respectively.

Show that the Boltzmann equation for the density $f({\bf r},p,t)$
of the photon gas, treating the atoms as fixed scatterers of
densities $n_0$ and $n_1$ is
\begin{equation}
\frac{\partial f({\bf r},p,t)}{\partial t}+ \frac{{\bf p}c}{|{\bf
p}|}\cdot \frac{\partial f({\bf r},p,t)}{\partial {\bf r}}=
-\sigma_2n_0cf({\bf r},p,t)+\sigma_3n_1cf({\bf r},p,t)+\sigma_1n_1
\nonumber
\end{equation}

\item[(c)] Find the equilibrium solution $f_{eq}$. Equate the result, using
(a), to that the expected value per state
$f_{eq}=\frac{1}{h^3}\frac{1}{e^{\hbar\omega/k_BT}-1}$
and deduce relations between the cross sections.

\item[(d)] Consider a situation in which light shines along the $x$ axis
on a collection of atoms whose boundary is at $x=0$ (see figure).
The incoming flux is uniform and has photons of momentum
${\bf p}=\hbar\omega {\hat x}/c$ where ${\hat x}$ is a unit vector
in the $x$ direction. Show that the solution has the form
\[Ae^{-x/a}+f_{eq}\]
and find the penetration length $a$.\\
\end{itemize}


\begin{center}
\includegraphics[scale=0.55]{D05.eps}
\end{center}


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