Difference between revisions of "Compton Scattering"

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\def\ddt{{d \over dt}}
 
\def\ddt{{d \over dt}}
\def\mean#1{\left\langle #1\right\rangle}
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\def\mean#1{\left\langle {#1}\right\rangle}
 
\def\sigot{\sigma_{12}}
 
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\def\e#1{\cdot10^{#1}}
 
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\def\hf{\frac12}
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\def\^{\hat }
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\def\.{\dot }
 
\usepackage{fullpage}
 
\usepackage{fullpage}
 
\usepackage{amsmath}
 
\usepackage{amsmath}
 
\usepackage{eufrak}
 
\usepackage{eufrak}
 
\begin{document}
 
\begin{document}
\subsection*{ Compton Scattering }
+
 
 +
\section*{ Compton Scattering}
 +
 
 +
Compton scattering is the scattering of a photon off of an electron.  If
 +
the photon {\it loses} energy, this is called {\bf Compton Scattering}.  If
 +
the photon gains energy, this is called {\bf Inverse Compton Scattering}, or
 +
``Compton Up-Scattering''.  The ``comptonization'' of an electron gas is the
 +
gain in energy of an electron gas as the result of a photon gas.\par
 +
Some examples of applications of Compton scattering are:
 +
\begin{itemize}
 +
\item  Compton exchange keeps electrons in thermal equilibrium with
 +
photons at redshifts $z\ge10^3$.
 +
\item  The spectra of AGN and xray binaries are altered by Compton
 +
Scattering (e.g. radio emission to optical wavelengths).
 +
\item  CMB photons get upscattered by galaxy cluster plasma.  This is
 +
called the Sunyaev-Zeldovich effect.
 +
\end{itemize}
 +
The classic model for Compton scattering is where a photon of energy $E=h\nu$
 +
going in the $\^n$ direction scatters off of a stationary $e^-$.  After
 +
scattering, the photon leaves with a new energy $E_1$ in a new direction
 +
$\^n_1$ (at an angle $\phi$ to its original direction $\^n$), and the $e^-$
 +
now moves in direction $\^p$ with energy $E$ and momentum $p$.  Energy
 +
conservation requires that:
 +
$$E+m_ec^2=E_1+E$$
 +
and momentum conservation requires that:
 +
$${E\over c}\^n={E_1\over c}\^n+p\^p$$
 +
Squaring these two equations and subtracting them, we find that:
 +
$$\lambda_1-\lambda={h\over m_ec}(1-\cos\phi)\leftrightarrow
 +
E_1={E\over1+{E\over m_ec^2}(1-\cos\phi)}$$
 +
Comments:
 +
\begin{itemize}
 +
\item $\lambda_1-\lambda>0$: the shift here is tiny.  The maximum possible
 +
value is $\lambda_1-\lambda={2h\over m_ec}=0.04\AA$.  The momentum
 +
tends to be shared between the electron and photon, but no so much the energy.
 +
\item  Remember this is {\it scattering}, not absorption.  Photon \# is
 +
conserved.
 +
\item  Note that for $h\nu\ll m_ec^2$, $\sigma=\sigma_T$.  For
 +
$h\nu\gg m_ec^2$, $\sigma\sim\sigma_T\left({m_ec^2\over h\nu}\right)$.  This
 +
additional term is called the Klein-Nishna correction.
 +
\item  The mean scattering angle is $\phi={\pi\over 2}$.
 +
${d\sigma_T\over d\Omega}\propto1+\cos^2\phi$.  Beware that:
 +
$${d\sigma\over d\Omega}\eval{Klein-Nishna}\ne{d\sigma_T\over d\Omega}$$
 +
\end{itemize}
 +
What we've done so far was for a stationary electron.  For a moving electron,
 +
we need to consider the dependence on the angle at which the photon
 +
is coming in with respect to the direction of velocity ($\theta$).  To make
 +
this situation similar to the one we just considered, we need to be in the
 +
frame of the electron.  In this frame, the photons has a new energy as a
 +
result of time dilation:
 +
$$E^\prime=\gamma E(1-{v\over c}\cos\theta)$$
 +
The $v\over c$ term is just the classic Doppler shift of the photon.  Now
 +
suppose that the photon (which entered at angle $\theta^\prime$ in the
 +
electron frame) rebounds at an angle $\theta_1^\prime$.  To relate this to
 +
our previous derivation, we want to find $\phi$.  So note that
 +
$\theta_1^\prime-\phi^\prime=\theta^\prime$.  Thus, in the electron's frame:
 +
$$E_1^\prime={E^\prime\over1+{E^\prime\over m_ec^2}(1-\cos\phi^\prime)}$$
 +
Transforming this back into the lab frame:
 +
$$E_1=E_1^\prime\gamma(1+{v\over c}\cos\theta_1^\prime)$$
 +
This generally follows Rybicki \& Lightman.  However, it might help to know
 +
that in R\&L, 7.7b follows from 7.8a, which follows from 7.7a.\par
 +
Comments:
 +
\begin{itemize}
 +
\item  If $E^\prime\ll m_ec^2$, then:
 +
$$\begin{aligned}E_1&\approx E^\prime\gamma(1+{v\over c}\cos\theta_1^\prime)\\
 +
&\approx\gamma^2E(1+{v\over c}\cos\theta_1^\prime)(1-{v\over c}\cos\theta)\\ \end{aligned}$$
 +
In a ``typical'' collision, $\theta\sim\theta_1^\prime\sim{\pi\over2}$, so
 +
$E_1\sim\gamma^2E$.
 +
\item  If $E^\prime\gg m_ec^2$, then:
 +
$$\begin{aligned}E_1^\prime&={E^\prime\over1+{E^\prime\over m_ec^2}
 +
(1-\cos\phi^\prime)}\approx m_ec^2\\
 +
E_1&=E_1^\prime\gamma(1+{v\over c}\cos\theta_1^\prime)\\
 +
&\approx m_ec^2\gamma(1+{v\over c}\cos\theta_1^\prime)\approx\gamma m_ec^2\\ \end{aligned}$$
 +
This final term defines the maximum rebound of the photon.
 +
\end{itemize}
 +
 
  
 
\end{document}
 
\end{document}
 
<\latex>
 
<\latex>

Revision as of 20:05, 18 November 2013

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Reference Material

<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle {#1}\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\hf{\frac12} \def\^{\hat } \def\.{\dot } \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document}

\section*{ Compton Scattering}

Compton scattering is the scattering of a photon off of an electron. If the photon {\it loses} energy, this is called {\bf Compton Scattering}. If the photon gains energy, this is called {\bf Inverse Compton Scattering}, or ``Compton Up-Scattering. The ``comptonization of an electron gas is the gain in energy of an electron gas as the result of a photon gas.\par Some examples of applications of Compton scattering are: \begin{itemize} \item Compton exchange keeps electrons in thermal equilibrium with photons at redshifts $z\ge10^3$. \item The spectra of AGN and xray binaries are altered by Compton Scattering (e.g. radio emission to optical wavelengths). \item CMB photons get upscattered by galaxy cluster plasma. This is called the Sunyaev-Zeldovich effect. \end{itemize} The classic model for Compton scattering is where a photon of energy $E=h\nu$ going in the $\^n$ direction scatters off of a stationary $e^-$. After scattering, the photon leaves with a new energy $E_1$ in a new direction $\^n_1$ (at an angle $\phi$ to its original direction $\^n$), and the $e^-$ now moves in direction $\^p$ with energy $E$ and momentum $p$. Energy conservation requires that: $$E+m_ec^2=E_1+E$$ and momentum conservation requires that: $${E\over c}\^n={E_1\over c}\^n+p\^p$$ Squaring these two equations and subtracting them, we find that: $$\lambda_1-\lambda={h\over m_ec}(1-\cos\phi)\leftrightarrow E_1={E\over1+{E\over m_ec^2}(1-\cos\phi)}$$ Comments: \begin{itemize} \item $\lambda_1-\lambda>0$: the shift here is tiny. The maximum possible value is $\lambda_1-\lambda={2h\over m_ec}=0.04\AA$. The momentum tends to be shared between the electron and photon, but no so much the energy. \item Remember this is {\it scattering}, not absorption. Photon \# is conserved. \item Note that for $h\nu\ll m_ec^2$, $\sigma=\sigma_T$. For $h\nu\gg m_ec^2$, $\sigma\sim\sigma_T\left({m_ec^2\over h\nu}\right)$. This additional term is called the Klein-Nishna correction. \item The mean scattering angle is $\phi={\pi\over 2}$. ${d\sigma_T\over d\Omega}\propto1+\cos^2\phi$. Beware that: $${d\sigma\over d\Omega}\eval{Klein-Nishna}\ne{d\sigma_T\over d\Omega}$$ \end{itemize} What we've done so far was for a stationary electron. For a moving electron, we need to consider the dependence on the angle at which the photon is coming in with respect to the direction of velocity ($\theta$). To make this situation similar to the one we just considered, we need to be in the frame of the electron. In this frame, the photons has a new energy as a result of time dilation: $$E^\prime=\gamma E(1-{v\over c}\cos\theta)$$ The $v\over c$ term is just the classic Doppler shift of the photon. Now suppose that the photon (which entered at angle $\theta^\prime$ in the electron frame) rebounds at an angle $\theta_1^\prime$. To relate this to our previous derivation, we want to find $\phi$. So note that $\theta_1^\prime-\phi^\prime=\theta^\prime$. Thus, in the electron's frame: $$E_1^\prime={E^\prime\over1+{E^\prime\over m_ec^2}(1-\cos\phi^\prime)}$$ Transforming this back into the lab frame: $$E_1=E_1^\prime\gamma(1+{v\over c}\cos\theta_1^\prime)$$ This generally follows Rybicki \& Lightman. However, it might help to know that in R\&L, 7.7b follows from 7.8a, which follows from 7.7a.\par Comments: \begin{itemize} \item If $E^\prime\ll m_ec^2$, then: $$\begin{aligned}E_1&\approx E^\prime\gamma(1+{v\over c}\cos\theta_1^\prime)\\ &\approx\gamma^2E(1+{v\over c}\cos\theta_1^\prime)(1-{v\over c}\cos\theta)\\ \end{aligned}$$ In a ``typical collision, $\theta\sim\theta_1^\prime\sim{\pi\over2}$, so $E_1\sim\gamma^2E$. \item If $E^\prime\gg m_ec^2$, then: $$\begin{aligned}E_1^\prime&={E^\prime\over1+{E^\prime\over m_ec^2} (1-\cos\phi^\prime)}\approx m_ec^2\\ E_1&=E_1^\prime\gamma(1+{v\over c}\cos\theta_1^\prime)\\ &\approx m_ec^2\gamma(1+{v\over c}\cos\theta_1^\prime)\approx\gamma m_ec^2\\ \end{aligned}$$ This final term defines the maximum rebound of the photon. \end{itemize}


\end{document} <\latex>