Difference between revisions of "Thomson Scattering"

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[[Radiative Processes in Astrophysics|Course Home]]
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===Videos===
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* [https://youtu.be/B7FgipOtPN4 Thomson Scattering (Abdurrahman, UC Berkeley)]
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===Reference Materials===
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* [https://en.wikipedia.org/wiki/Thomson_scattering Thomson Scattering (Wikipedia)]
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* [http://farside.ph.utexas.edu/teaching/em/lectures/node96.html Thomson Scattering (UTexas)]
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===Need to Review?===
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* [[Larmor Formula]]
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* [[Electromagnetic Plane Waves]]
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===Related Topics===
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* [[Compton Scattering]] for the high energy limit.
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* [[Opacity]]
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\documentclass[11pt]{article}
 
\documentclass[11pt]{article}
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\usepackage{eufrak}
 
\usepackage{eufrak}
 
\begin{document}
 
\begin{document}
\section*{ Coulomb Focusing}
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\section*{Basics}
  
Electric fields exert a force on charged particles causing them to accelerate. As seen in the derivation of the Larmor formula, accelerated charged particles emit radiation. So overall, incoming radiation might be off-scattered by charged particles. Here, we will consider incoming radiation being off-scattered by an electron, the so called Thomson scattering. Note that this requires a low energy, low intensity $\vec{E}$-field and a low energy electron. Other limits will be discussed later in the course, see [[Compton Scattering]].
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[[File:thomson.png|thumb|400px|center|Thomson Scattering.]]
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Electric fields exert a force on charged particles causing them to accelerate. As seen in the derivation of the [[Larmor Formula]], accelerated charged particles emit radiation. So overall, incoming radiation might be off-scattered by charged particles. Here, we will consider incoming radiation being off-scattered by an electron, the so called Thomson scattering. Note that this requires a low energy, low intensity $\vec{E}$-field and a low energy electron. Other limits will be discussed later in the course, see [[Compton Scattering]].
  
 
An incoming electromagnetic wave with an angular frequency $\omega_0$ passes an electron of mass $m_e$. Model this wave in its easiest form by
 
An incoming electromagnetic wave with an angular frequency $\omega_0$ passes an electron of mass $m_e$. Model this wave in its easiest form by
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$$m\ddot{z}=eE_0 \sin(\omega_0 t).$$
 
$$m\ddot{z}=eE_0 \sin(\omega_0 t).$$
  
The root-mean-square average acceleration from the equation above is
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The root-mean-square average acceleration, which will be important for determining the radiated power is
  
 
$$a_\text{rms}=\sqrt{\langle a^2 \rangle} = \frac{eE_0}{2m_e}.$$
 
$$a_\text{rms}=\sqrt{\langle a^2 \rangle} = \frac{eE_0}{2m_e}.$$
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where $\langle S\rangle=\frac{dP}{d\sigma}$ is the flux passing the cross-section of the electron, which is related to the amplitude of the incoming field via the time-averaged Poynting flux.
 
where $\langle S\rangle=\frac{dP}{d\sigma}$ is the flux passing the cross-section of the electron, which is related to the amplitude of the incoming field via the time-averaged Poynting flux.
  
Integrating the above equation and recalling the Larmor formula for $P$, we obtain:
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Integrating the above equation and recalling the [[Larmor Formula]] for $P$, we obtain:
  
 
\begin{align}
 
\begin{align}
\sigma &= \frac{8\pi}{cE_0^2} P \\
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\sigma_T &= \frac{8\pi}{cE_0^2} P \\
 
&=\frac{8\pi}{cE_0^2}\frac{e^2 a_\text{rms}^2}{3c^3} \\
 
&=\frac{8\pi}{cE_0^2}\frac{e^2 a_\text{rms}^2}{3c^3} \\
 
&= \frac{8\pi}{3} \frac{e^4}{m_e^2c^4}=\frac{8\pi}{3} r_0^2,
 
&= \frac{8\pi}{3} \frac{e^4}{m_e^2c^4}=\frac{8\pi}{3} r_0^2,
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$$\sigma_T = \frac{8\pi}{3} r_0^2 = 6.65 \cdot 10^{-25} \text{cm}^2$$
 
$$\sigma_T = \frac{8\pi}{3} r_0^2 = 6.65 \cdot 10^{-25} \text{cm}^2$$
  
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\section*{Examples}
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<center>[[File:BAO.png|thumb|300px|center|The Power spectrum of BAO over angular scales.]]</center>
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Thomson scattering is important on very different scales in Astronomy. For example:
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\begin{itemize}
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    \item At high temperatures, Thomson scattering can dominate the photon diffusion in a star and thus determines the energy flux making its way out of the star.
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    \item At the epoch of recombination ($z \approx 1100$), Thomson scattering caused the high opacity of the plasma. The effects of Thomson scattering are, for example, visible in the diffusion damping of Baryon Acoustic Oscillations. For high l-moments ($l\geq 1000$), Thomson scattering set the length over which photons could diffuse. This directly determines the damping of small scales (high moments) as inhomogeneities can be leveled out by photons.
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\end{itemize}
  
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Latest revision as of 10:25, 20 August 2021

Course Home

Videos[edit]

Reference Materials[edit]

Need to Review?[edit]

Related Topics[edit]

Basics

Thomson Scattering.

Electric fields exert a force on charged particles causing them to accelerate. As seen in the derivation of the Larmor Formula, accelerated charged particles emit radiation. So overall, incoming radiation might be off-scattered by charged particles. Here, we will consider incoming radiation being off-scattered by an electron, the so called Thomson scattering. Note that this requires a low energy, low intensity -field and a low energy electron. Other limits will be discussed later in the course, see Compton Scattering.

An incoming electromagnetic wave with an angular frequency passes an electron of mass . Model this wave in its easiest form by

The electron is accelerated by this external electric field, so that its motion is described by

The root-mean-square average acceleration, which will be important for determining the radiated power is

In order to oscillate with the same frequency as the incoming radiation, the photon energy needs to be significantly smaller than the rest energy of the electron. This statement is equivalent saying that the wavelength of the incoming wave needs to be larger than the Compton wavelength , a characteristic scale at which quantum mechanics becomes important. At the same time, we assumed the electron to be at low velocity and the amplitude of the incoming wave to be small, otherwise a relativistic description is needed.

We now try to determine the cross-section of this interaction. A trick to be used here is to expand the differential power scattered into a solid angle with .

where is the flux passing the cross-section of the electron, which is related to the amplitude of the incoming field via the time-averaged Poynting flux.

Integrating the above equation and recalling the Larmor Formula for , we obtain:

where is the classical electron radius, which can be easily re-derived by requiring that the potential Coulomb energy makes up all of the electron’s rest energy by bringing in charge within a radius that suffices this condition.

Concluding, the cross-section is off from a pure, classical cross-section for this interaction by a factor of .

Examples

The Power spectrum of BAO over angular scales.

Thomson scattering is important on very different scales in Astronomy. For example:

  • At high temperatures, Thomson scattering can dominate the photon diffusion in a star and thus determines the energy flux making its way out of the star.
  • At the epoch of recombination (), Thomson scattering caused the high opacity of the plasma. The effects of Thomson scattering are, for example, visible in the diffusion damping of Baryon Acoustic Oscillations. For high l-moments (), Thomson scattering set the length over which photons could diffuse. This directly determines the damping of small scales (high moments) as inhomogeneities can be leveled out by photons.