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## 1 Plasma Frequency

### 1.1 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:

${\vec {E}}=E_{0}e^{i({\vec {k}}{\vec {r}}-wt)}\,\!$
We’ll say that the displacement of $e^{-}\ll {2\pi \over k}$. The equation of motion for a single electron is then:

${\ddot {x}}+\gamma {\dot {x}}+\omega _{0}^{2}x={eE_{0}e^{i({\vec {k}}{\vec {r}}-wt)} \over m_{e}}\,\!$
where $\gamma$ is our dampening factor. Note that ${\vec {B}}$ is absent here; we’re neglecting it because it is small for reasonable energies. It turns out this equation has the steady-state solution:

$x={-e \over m_{e}}{\left((\omega _{0}^{2}-\omega ^{2})+i\omega \gamma \right) \over \left((\omega _{0}^{2}-\omega ^{2})^{2}+(\omega \gamma )^{2}\right)}E_{0}e^{i({\vec {k}}{\vec {r}}-wt)}\,\!$
The limiting cases of this equation explain many phenomena. For example: $\gamma =0$ (loss-less propagation).

In this case, our solution looks like:

$x={-e \over m_{e}}{E_{0}e^{i({\vec {k}}{\vec {r}}-wt)} \over (\omega _{0}^{2}-\omega ^{2})}\,\!$
From this we can use the dispersion relation to relate $\omega$ (frequency) and $k$ (phase).

### The Dispersion Relation

Let’s solve Maxwell’s equations. Recall that ${\vec {J}}$ is the current density:

${\begin{aligned}{\vec {J}}&=n\cdot e\cdot {\dot {x}}\\&={ne(-i\omega )(-eE_{0}e^{i({\vec {k}}{\vec {r}}-wt)}) \over m_{e}(\omega _{0}^{2}-\omega ^{2})}=\sigma {\vec {E}}\\\end{aligned}}\,\!$
where $\sigma \equiv conductivity={-ne^{2}i\omega \over m_{e}(\omega _{0}^{2}-\omega ^{2})}$, and n is the # density of electrons. So on to Maxwell’s equations:

${\begin{aligned}{\vec {\nabla }}\times {\vec {E}}&=-{1 \over c}{\partial {\vec {B}} \over \partial t}\\ikE&={i\omega B \over c}\\\end{aligned}}\,\!$
${\begin{aligned}{\vec {\nabla }}\times {\vec {B}}&={4\pi {\vec {J}} \over c}+{1 \over c}{\partial {\vec {E}} \over \partial t}\\-ikB&=\left({4\pi \sigma \over c}-{i\omega \over c}\right)E\\\end{aligned}}\,\!$
Combining these two equations we get:

${\left({ck \over \omega }\right)^{2}=1+{4\pi ne^{2} \over m_{e}(\omega _{0}^{2}-\omega ^{2})}}\,\!$
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:

${\eta ^{2}=1-{\omega _{p}^{2} \over \omega ^{2}}}\,\!$