Plasma Frequency

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<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>