# Plasma Frequency

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## 1 Plasma Frequency

### 1.1 Damped Simple Harmonic Oscillator

We can model ${\displaystyle e^{-}}$ and their atoms as a lattice of springs connecting ${\displaystyle e^{-}}$ to fixed points. Say that we send a plane wave at these springs that looks like:

${\displaystyle {\vec {E}}=E_{0}e^{i({\vec {k}}{\vec {r}}-wt)}\,\!}$

We’ll say that the displacement of ${\displaystyle e^{-}\ll {2\pi \over k}}$. The equation of motion for a single electron is then:

${\displaystyle {\ddot {x}}+\gamma {\dot {x}}+\omega _{0}^{2}x={eE_{0}e^{i({\vec {k}}{\vec {r}}-wt)} \over m_{e}}\,\!}$

where ${\displaystyle \gamma }$ is our dampening factor. Note that ${\displaystyle {\vec {B}}}$ is absent here; we’re neglecting it because it is small for reasonable energies. It turns out this equation has the steady-state solution:

${\displaystyle x={-e \over m_{e}}{\left((\omega _{0}^{2}-\omega ^{2})+i\omega \gamma \right) \over \left((\omega _{0}^{2}-\omega ^{2})^{2}+(\omega \gamma )^{2}\right)}E_{0}e^{i({\vec {k}}{\vec {r}}-wt)}\,\!}$

The limiting cases of this equation explain many phenomena. For example: ${\displaystyle \gamma =0}$ (loss-less propagation).

In this case, our solution looks like:

${\displaystyle x={-e \over m_{e}}{E_{0}e^{i({\vec {k}}{\vec {r}}-wt)} \over (\omega _{0}^{2}-\omega ^{2})}\,\!}$

From this we can use the dispersion relation to relate ${\displaystyle \omega }$ (frequency) and ${\displaystyle k}$ (phase).

### The Dispersion Relation

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

{\displaystyle {\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 ${\displaystyle \sigma \equiv conductivity={-ne^{2}i\omega \over m_{e}(\omega _{0}^{2}-\omega ^{2})}}$, and n is the # density of electrons. So on to Maxwell’s equations:

{\displaystyle {\begin{aligned}{\vec {\nabla }}\times {\vec {E}}&=-{1 \over c}{\partial {\vec {B}} \over \partial t}\\ikE&={i\omega B \over c}\\\end{aligned}}\,\!}
{\displaystyle {\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:

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

This is our dispersion relation. The plasma frequency is defined as:

${\displaystyle \omega _{p}^{2}\equiv {4\pi ne^{2} \over m_{e}}\,\!}$

and the index of refraction is:

${\displaystyle \eta \equiv {ck \over \omega }\,\!}$

Rewritten in these terms, our dispersion relation in plasma is:

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