• These notes are based on notes from a previous grad student, [[ Chen - 1988 - Plasma Physics and Controlled Fusion ]], and [[ Goldston and Rutherford - 2018 - Introduction to Plasma Physics ]].
• These are not meant to replace reading one of these books or taking a plasma course.
• These are meant to be an in-depth review of Plasma I and Plasma II taught at Columbia University in preparation for the PhD Written Qualifying Exam. Topics I don’t think are relevant to this exam are not included.

0 Notes on Notation

• I will always use $T_{e}$ in $\text{eV}$, so I will leave out any factors of $k_{B}$.
• 1,2,3… I will try to use numbering to denote a step that we have taken in a proof, to make it obvious when I’m talking about taking a step and just discussion stuff.
• ${} \to$ will denote “doing math”, chugging after we have decided what to plug with numbering.
• $\Rightarrow$ will denote equations to memorize (use flashcards!!)

1 Characteristics of the Plasma State

1.1 Debye Shielding

“Potentials of the order $\frac{T_{e}}{e}$ can leak into the plasma and cause finite $\vec{E}$ to exist.” - [[ Chen - 1988 - Plasma Physics and Controlled Fusion|Chen ]] pg 8

Assume $\frac{m_{i}}{m_{e}} \rightarrow \infty$ (ion’s don’t move) For an applied potential $\Phi_{0}$, we want $\phi(x)$ and we want to find how far away from “test charge” where $\phi(x)\rightarrow 0$, i.e. where the charge is shielded.

1. Start with Poisson’s equation:

^0bdfb7

In the presence of $\phi_{0}$, electrons for a distribution function (not proven in text):

1. Integrate over $u$, using $q=e$ and $\lim_{ \phi \to 0 }n_{e}=n_{0}$ (meaning as the perturbed potential goes to 0, the electron density equals the background plasma density $n_{0}$). We get:

\(n_{e}=n_{0}\exp\left( \frac{e\phi}{T_{e}} \right)\) ^f17d56

1. Sub [[ #^f17d56 ]] and $n_{i}=n_{0}$ into [[ #^0bdfb7 ]] :

\(\epsilon_0\frac{\text{d}^{2} \phi }{\text{d}x^{2}} =en_{0}\left( \exp\left( \frac{e\phi}{T_{e}} \right)-1 \right)\)

1. Expand the exponential where $

   \frac{e\phi}{T_{e}}

   \ll 1$:

1. Keep only the linear terms and we get:

Which has solution:

\(\phi(x)=\phi_{0}\exp\left( -\frac{|x|}{\lambda_{D}} \right)\) where we have coined a new term, a characteristic length over which the perturbed potential decays, called the Debye Length:

The number of particles in a “Debye Sphere” (a sphere of radius = $\lambda_{D}$) is called the Plasma Parameter:

\(N_{D}=n \frac{4}{3} \pi \lambda_{D}^{3}\) For collective behavior to dominate over single particle movements (i.e. for a charged gas to be considered a plasma), the plasma parameter must be large: $N_{D}\gg 1$

2 Single Particle Motion

This section describes the motion of a single charged particle in fields relevant to plasmas.

2.1 Equations of motion for a charged particle in $\vec{E}=0$ and $\vec{B}=B_{0}\hat{z}$

1. First we define “perpendicular velocity” being $\perp$ to the direction of $\vec{B}$ which is the x-y plane:

^989e53

1. Then we do a force balance: \(m\frac{\text{d} v_{\perp} }{\text{d}t} =q\cdot \vec{v}_{\perp}\times \vec{B} = \hat{x}(v_{y}B_{0})-\hat{y}(v_{x}B_{0})\) $\to$ Separate into two equations:

^87821f

$\to$ Take derivative w.r.t. time:

^6a751a

$\to$ Plug [[ #^87821f ]] equations into [[ #^6a751a ]] equations:

^6c9bd6

And we’ve gotten two equations for a simple harmonic oscillator! (If you’ve forgotten, it looks like: $\ddot{v}=-\omega^{2}v$) We can now define the frequency to be: \(\Rightarrow \text{Cyclotron Frequency: }\ \ \boxed{\omega_{c}=\frac{qB}{m}}\) which is called the cyclotron frequency, or Larmor frequency.

These equations tell us that the particle moves in a circular orbit around the $\vec{B}$ lines. The charge gives the “sign” of the orbit (right handed or left handed orbit, if your thumb is in the direction of $\vec{B}$)

1. To find the radius of this circle and $v_{\perp}$ we solve the ordinary differential equations in [[ #^6c9bd6 ]], with assumed initial conditions of $v_{x}(t=0)=v_{\perp}, v_{y}(t=0)=0$:

$\to$ using [[ #^989e53 ]]:

\(v_{y}(t)=\pm v_{\perp}\sin(\omega t)\) so we get:

where $r_{L}$ is called the Larmor radius.

2.2 Drifts

So we have seen that in a magnetic field, a charged particle will move in sometimes odd ways. Now we will add an electric field perpendicular to the B field and see that the particle will “drift” linearly. With different configurations of fields, these drifts will look different, so we will go through the relevant ones.

First, I’m going to introduce the general drift equation. For an arbitrary force applied to a charged particle in a magnetic field, the particle will drift with the velocity: ! [[ #^093171 ]] We can use this to find any of the other drifts, given we remember the forces. I doubt the qual will ask to prove the drift velocities but in case, I’m going to go through each proof.

2.2.1 $E\times B$ Drift

Say we have an electric field, constant and uniform, in the $\hat{x}$ direction and a magnetic field, constant and uniform, in the $\hat{z}$ direction: \(\begin{align} \vec{E}&=E_{0} \hat{x} \\ \vec{B}&=B_{0} \hat{z} \end{align}\) We know the equations of motion, at one instant in time, is

2.2.2 Polarization Drift

2.2.3 Curvature Drift

2.2.4 Toroidal Drift

2.2.5 Generalized Drift Equation

^093171

So the drift velocity is perpendicular to both the applied force and the B field. This is a good equation to memorize.

A simple example is if gravity is applied to a charged particle with $\vec{F_{g}}=mg(-\hat{z})$:

\(v_{gravity}=\frac{mg}{qB_{0}^{2}} (-\hat{z}\times \vec{B})\) This is the gravitational drift, and is irrelevant in actual plasma physics (other forces are so much stronger) but is often used in practice problems. Some features:

• the drift is charge dependent
• the drift is, again, perpendicular to both the direction of gravity and $\hat{B}$.