Difference between revisions of "Plasma Frequency"

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(Created page with '=== Topical Videos === * [http://www.youtube.com/watch?v=xYZfXV0PFkc&feature=youtu.be Plasma Effects (Polin)]')
 
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=== Topical Videos ===
 
=== Topical Videos ===
 
* [http://www.youtube.com/watch?v=xYZfXV0PFkc&feature=youtu.be Plasma Effects (Polin)]
 
* [http://www.youtube.com/watch?v=xYZfXV0PFkc&feature=youtu.be Plasma Effects (Polin)]
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 +
=== Reference Material ===
 +
 +
<latex>
 +
\documentclass[11pt]{article}
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\def\sigot{\sigma_{12}}
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\def\sigto{\sigma_{21}}
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\def\wz{\omega_0}
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\def\dce{\vec\tr\times\vec E}
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\def\dcb{\vec\tr\times\vec B}
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\def\ato{{A_{21}}}
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\def\bto{{B_{21}}}
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\def\bot{{B_{12}}}
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\def\inv#1{\frac1{#1}}
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\def\hf{\frac12}
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\def\bfield{{\vec B}}
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\def\eval#1{\big|_{#1}}
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\def\tr{\nabla}
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\def\ef{\vec E}
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\usepackage{fullpage}
 +
\usepackage{amsmath}
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\usepackage{eufrak}
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\begin{document}
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 +
\section{Plasma Frequency}
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\subsection{ Damped Simple Harmonic Oscillator}
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 +
We can model $e^-$ and their atoms as a lattice of springs connecting $e^-$
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to fixed points.  Say that we send a plane wave at these springs that
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looks like:
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\def\eikrwt{e^{i(\vec k\vec r-wt)}}
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$$\ef=E_0\eikrwt$$
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We'll say that the displacement of $e^-\ll{2\pi\over k}$.  The equation of
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motion for a single electron is then:
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$$\ddot x+\gamma\dot x+\wz^2x={eE_0\eikrwt\over m_e}$$
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where $\gamma$ is our dampening factor.  Note that $\bfield$ is absent
 +
here; we're neglecting it because it is small for
 +
reasonable energies. It turns out this
 +
equation has the steady-state solution:
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$$x={-e\over m_e}{\left((\wz^2-\omega^2)+i\omega\gamma\right)\over
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\left((\wz^2-\omega^2)^2+(\omega\gamma)^2\right)}E_0\eikrwt$$
 +
The limiting cases of this equation explain many phenomena.  For example:
 +
$\gamma=0$ (loss-less propagation).\par
 +
In this case, our solution looks like:
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$$x={-e\over m_e}{E_0\eikrwt\over(\wz^2-\omega^2)}$$
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From this we can use the dispersion relation to relate $\omega$ (frequency) and
 +
$k$ (phase).
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\subsection*{ The Dispersion Relation}
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 +
Let's solve Maxwell's equations.  Recall that $\vec J$ is the current density:
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$$\begin{aligned}\vec J&=n\cdot e\cdot\dot x\\
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&={ne(-i\omega)(-eE_0\eikrwt)\over m_e(\wz^2-\omega^2)}=\sigma\ef\\ \end{aligned}$$
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where $\sigma\equiv conductivity={-ne^2i\omega\over m_e(\wz^2-\omega^2)}$, and
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n is the \# density of electrons.
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So on to Maxwell's equations:
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$$\begin{aligned}\dce&=-{1\over c}{\partial\bfield\over\partial t}\\
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ikE&={i\omega B\over c}\\ \end{aligned}$$
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$$\begin{aligned}\dcb&={4\pi\vec J\over c}+{1\over c}{\partial\ef\over\partial t}\\
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-ikB&=\left({4\pi\sigma\over c}-{i\omega\over c}\right)E\\ \end{aligned}$$
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Combining these two equations we get:
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$$\boxed{\left({ck\over\omega}\right)^2=1+{4\pi ne^2\over
 +
m_e(\wz^2-\omega^2)}}$$
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This is our dispersion relation.  The plasma frequency is defined as:
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$$\omega_p^2\equiv {4\pi ne^2\over m_e}$$
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and the index of refraction is:
 +
$$\eta\equiv{ck\over\omega}$$
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Rewritten in these terms, our dispersion relation in plasma is:
 +
$$\boxed{\eta^2=1-{\omega_p^2\over\omega^2}}$$
 +
 +
 +
 +
\end{document}
 +
<\latex>

Revision as of 14:19, 15 October 2014

Topical Videos

Reference Material

<latex> \documentclass[11pt]{article} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\wz{\omega_0} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\inv#1{\frac1{#1}} \def\hf{\frac12} \def\bfieldTemplate:\vec B \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\ef{\vec E} \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document}

\section{Plasma Frequency}

\subsection{ Damped Simple Harmonic Oscillator}

We can model $e^-$ and their atoms as a lattice of springs connecting $e^-$ to fixed points. Say that we send a plane wave at these springs that looks like: \def\eikrwt{e^{i(\vec k\vec r-wt)}} $$\ef=E_0\eikrwt$$ We'll say that the displacement of $e^-\ll{2\pi\over k}$. The equation of motion for a single electron is then: $$\ddot x+\gamma\dot x+\wz^2x={eE_0\eikrwt\over m_e}$$ where $\gamma$ is our dampening factor. Note that $\bfield$ is absent here; we're neglecting it because it is small for reasonable energies. It turns out this equation has the steady-state solution: $$x={-e\over m_e}{\left((\wz^2-\omega^2)+i\omega\gamma\right)\over \left((\wz^2-\omega^2)^2+(\omega\gamma)^2\right)}E_0\eikrwt$$ The limiting cases of this equation explain many phenomena. For example: $\gamma=0$ (loss-less propagation).\par In this case, our solution looks like: $$x={-e\over m_e}{E_0\eikrwt\over(\wz^2-\omega^2)}$$ From this we can use the dispersion relation to relate $\omega$ (frequency) and $k$ (phase).

\subsection*{ The Dispersion Relation}

Let's solve Maxwell's equations. Recall that $\vec J$ is the current density: $$\begin{aligned}\vec J&=n\cdot e\cdot\dot x\\ &={ne(-i\omega)(-eE_0\eikrwt)\over m_e(\wz^2-\omega^2)}=\sigma\ef\\ \end{aligned}$$ where $\sigma\equiv conductivity={-ne^2i\omega\over m_e(\wz^2-\omega^2)}$, and n is the \# density of electrons. So on to Maxwell's equations: $$\begin{aligned}\dce&=-{1\over c}{\partial\bfield\over\partial t}\\ ikE&={i\omega B\over c}\\ \end{aligned}$$ $$\begin{aligned}\dcb&={4\pi\vec J\over c}+{1\over c}{\partial\ef\over\partial t}\\ -ikB&=\left({4\pi\sigma\over c}-{i\omega\over c}\right)E\\ \end{aligned}$$ Combining these two equations we get: $$\boxed{\left({ck\over\omega}\right)^2=1+{4\pi ne^2\over m_e(\wz^2-\omega^2)}}$$ This is our dispersion relation. The plasma frequency is defined as: $$\omega_p^2\equiv {4\pi ne^2\over m_e}$$ and the index of refraction is: $$\eta\equiv{ck\over\omega}$$ Rewritten in these terms, our dispersion relation in plasma is: $$\boxed{\eta^2=1-{\omega_p^2\over\omega^2}}$$


\end{document} <\latex>