Particle Orbit Theory¶

\[ \begin{align}\begin{aligned} \label{eq:lorentzeq} m_j \dv[2]{\vec{r}_j}{t} = q_j \qty[ \vec{E}(\vec{r},t) + \dot{\vec{r}_j} \times \vec{B}(\vec{r}, t) ]\\Whenever charge distributions or current densities are signifigant a\end{aligned}\end{align} \]

statistical or fluid approach will be required.

Producing a Plasma¶

There are a number of approaches to producing a plasma in the lab:

1. photoionisation—for this we need photons with sufficient energy to remove electrons from the neutral species, e.g. 13.6 eV for Hydrogen, 24.6 eV for Helium, and 15.6 eV for molecular nitrogen. All of these energies lie within the ultraviolet region of the electromagnetic spectrum. The existence of long-lived, metastable states can help with ionisation processes (He has two, at 19.8 eV (with a half-life of 700s), and 20.61 eV; \({\rm H}_{2}\) at 11.75 eV (\(10^{-3} {\rm s}\)), and H at 0.52 eV (1 month).)

2. electron impact—for this we accelerate any free electrons (for example, the seed electrons) in an electric field until they reach the ionising threshold. This process makes use of the Townsend process, where \(N\) electrons move along the \(x\)-axis in the presence of a uniform electric field, \(E\); then \(\delta N\) electrons are produced by electron-impact ionisation in a distance \(\dif x\), according to

   \[ \begin{align}\begin{aligned} \label{eq:townsend} \delta N = \alpha_{{\rm T}} N \dif{x}\\with :math:`\alpha_{{\rm T}}` being the ionisation co-efficient.\end{aligned}\end{align} \]

   Thus,

   \[\label{eq:townsendint} N(x) = N_0 \exp(\alpha_{\rm T} x)\]

Basic Physics of Plasmas¶

Cold Magnetised Plasma Model¶

\[ \begin{align}\begin{aligned} \label{eq:momentumelectron} m \vec{\dot{v}} &= q(\dot{\vec{v}} \times \vec{B})\\and assume that :math:`\vec{B}` lies in the\end{aligned}\end{align} \]

\(\vec{\hat{z}}\)-direction, then

\[ \begin{align}\begin{aligned}\vec{B} = \vec{\hat{z}} B_0\\so\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\dot{\vec{v}} = \frac{q}{m} \left[ \vec{v} \times ( \hat{\vec{z}} B_0) \right]\\so in components,\end{aligned}\end{align} \]

\[\dot{v_x} = \frac{q}{m} \left[ \dot{v_y}B_0 - \dot{v}_z 0\right] = \frac{q B_0}{m} v_y\]

\[\dot{v_y} = \frac{q}{m} \left[ \dot{v_z} 0 - \dot{v}_x B_0\right] = - \frac{q B_0}{m} v_x\]

\[ \begin{align}\begin{aligned}\dot{v_z} = \frac{q}{m} 0 = {\rm const}\\so, defining the cyclotron frequency\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:cyclotron} \omega_{\rm c} = \frac{|q|B}{m}\\and for electrons,\end{aligned}\end{align} \]

\(\nu_{\rm c} = 28\ \giga \hertz\ \tesla^{-1}\), hence

\[ \begin{align}\begin{aligned}\ddot{v}_{x} = \omega_{\rm c} \dot{v_y} = \omega_{\rm c} [-\omega_{\rm c} v_x]\\that is\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\ddot{v}_x + \omega_{\rm c}^2 v_x = 0\\which is plane perpendicular to the charged particle (e.g. an\end{aligned}\end{align} \]

electron), which will thus undergo circular motion. THis motion is, however, uniform along the field. The net effect is that the particle will describe a helix.

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} n_{\rm s} & = n_0_{\rm s}+ n_1(\vec{x},t) \\ \vec{v_{\rm s}} & = \vec{v_{0,{\rm s}}} + \vec{v_1}(\vec{x},t)\end{aligned}\end{split}\\Now, under perturbation, we can linearlise the most important\end{aligned}\end{align} \]

equations:

Recall Maxwell’s equations

(\(\vec{v}(E)\)), then, \(\vec{J} = \vec{J}(\vec{v}) = \vec{J}(\vec{E})\), with \(\vec{J} = \vec{\sigma} \vec{E}\), then if we have a conductivity law we can move to a dielectric description.

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} -i\omega v_x &= \frac{q}{m} E_x - \frac{\omega_c}{i \omega} \qty[ \frac{q}{m}E_y - \omega_cv_x] \\ -i \omega v_x - \frac{\omega_c^2}{i \omega} v_x &= \frac{q}{m} E_x - \frac{q \omega_c}{i \omega m}E_y \\ &= \frac{q}{m}\qty[E_x - \frac{\omega_c}{i \omega} E_y] \\ \qty(1 - \frac{\omega_c^2}{\omega^2}) v_x &= - \frac{q}{i \omega m} \qty[E_x - \frac{\omega_c}{i \omega}E_y] \\ \text{since } \qty(1- \frac{\omega_c^2}{c^2}) v_y &= - \frac{q}{i \omega m} \qty[ E_x + \frac{\omega_c}{i \omega} E_y] \\ (1 - \frac{\omega_c^2}{\omega^2} \vec{v} &= M \cdot \vec{E} \\ \vec{v} &= \frac{1}{\qty(1 - \frac{\omega_{c}^2}{\omega^2})} \cdot M \cdot \vec{E}\end{aligned}\end{split}\\So, we know that we can get to this expression for\end{aligned}\end{align} \]

\(\vec{J}\)—how does this help? We now bring Maxwell’s equations into the mixture.

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \nabla \times \vec{E} &= \pdv{\vec{B}}{t} \\ \nabla \times ( \nabla \times \vec{E}) &= - \nabla \times \qty( \pdv{\vec{B}}{t}) \\ &= - \pdv{t} \qty(\nabla \times \vec{B}) \\ &= - \pdv{t} \qty[\mu_0 \vec{J} + \frac{1}{c^2} \pdv{\vec{E}{t}}] \\ &= - \pdv{t} \qty[\mu_0 \vec{\sigma} \cdot \vec{E} + \frac{1}{c^2} \pdv{\vec{E}}{t}]\end{aligned}\end{split}\\We are interested in waves, where solutions are proportional to\end{aligned}\end{align} \]

\(e^{i \vec{k}\cdot \vec{r} - i \omega t}\)

\[ \begin{align}\begin{aligned} \begin{aligned} \nabla \times (\nabla \times \vec{E}) &= - \vec{k} \times (\vec{k} \times \vec{E})\end{aligned}\\RHS:\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \begin{aligned} i \omega \qty[ \mu_0 \vec{\sigma} \cdot \vec{E} - \frac{i \omega}{c^2} \vec{E}]\end{aligned}\\So, the full equation is\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\vec{k} \times (\vec{k} \times \vec{E}) + \frac{\omega^2}{c^2} K \cdot \vec{E} &= 0\\with\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}K = I + \frac{i \sigma}{\epsilon_0 \omega}\\being the dielectric tensor.\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\vec{n}= \frac{\vec{k}c}{\omega}\\(a refractive index with “directional complications”),\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\label{eq:genrefind} \vec{n} \times ( \vec{n} \times \vec{E} ) + K \cdot \vec{E} = 0\\To help to understand the significance of equation ([eq:genrefind]),\end{aligned}\end{align} \]

let’s choose a geometry—let’s put \(\vec{B}_0 = \vec{\hat{z}} B_0\), and let’s take a wave in the \(\vec{\hat{x}}-\vec{\hat{z}}\)-plane (with one component parallel to \(\vec{B}_0\), and one perpendicular),

\[ \begin{align}\begin{aligned} \begin{aligned} \vec{n} &= \hat{x} n \sin(\theta) + \hat{z} n \cos(\theta)\end{aligned}\\Then expand the vector cross-product\end{aligned}\end{align} \]

\(\vec{n} \times (\vec{n} \times \vec{E})\) for this choice of geometry.

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \begin{pmatrix} S-n^2 \cos^2\theta & - iD & n^2 \cos(\theta) \sin(\theta) \\ iD & S-n^2 & 0 \\ n^2 \cos\theta \sin\theta & 0 & P- n^2 \sin^2 \theta \end{pmatrix} \begin{pmatrix} E_x \\ E_y \\ E_z \end{pmatrix} = 0\end{aligned}\end{split}\\Where we have written\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \label{eq:dielectricten} K= \begin{pmatrix} S & -iD & 0 \\ iD & S & 0 \\ 0 & 0 & P \end{pmatrix}\end{split}\\Where\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} S &= \frac{1}{2}(R+L) \\ D &= \frac{1}{2}(R-L)\end{aligned}\end{split}\\and\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} R &= 1 - \sum_S \frac{\omega^2_{\rm P_s}}{\omega^2} \qty( \frac{\omega}{\omega+\epsilon_0 \omega_{\rm c_s}}) \\ &= 1 - \frac{\omega_{\rm P}^2}{(\omega+\omega_{\rm C_+})(\omega-\omega_{\rm c^-})} \\ L &= 1 - \sum_S \frac{\omega_{\rm P_s}^2}{\omega^2} \qty(\frac{\omega}{\omega-\epsilon_{\rm s} \omega_{\rm c_s}}) \\ &= 1 - \frac{\omega_{\rm p}^2}{(\omega-\omega_{\rm c_+})(\omega+\omega_{\rm c_-})} \\ \epsilon_{\rm s} &= \begin{cases} +1 & \text{ for positive ion } \\ -1 & \text{ for negative electron} \end{cases} \\ \omega_{\rm P}^2 &= \omega_{\rm P_+}^2 + \omega_{\rm P_{-}}^2 \\ P &= 1 - \frac{\omega_{\rm P}}{\omega^2}\end{aligned}\end{split}\\For a non-trivial electric field the determinant of the matrix must\end{aligned}\end{align} \]

vanish, giving a relationship between \(\omega\) and \(k\) so th dispertion relation

\[ \begin{align}\begin{aligned} \label{eq:dispertionrelation} An^4 - Bn^2 + C = 0\\with\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} A &= S \sin[2](\theta) + P \cos[2](\theta) \\ B &= RL \sin[2](\theta) + PS (1 + \cos[2](\theta) \\ C &= PRL\end{aligned}\end{split}\\*N.B. There are two spherical cases, :math:`\theta=0`, and\end{aligned}\end{align} \]

:math:`theta= frac{pi}{2}`*.

\[ \begin{align}\begin{aligned}\begin{split} \begin{pmatrix} S-n^2 & -iD & 0 \\ iD & S-n^2 & 0 \\ 0 & 0 & P \end{pmatrix} \begin{pmatrix} E_x \\ E_y \\ E_z \end{pmatrix} = 0\end{split}\\i.e. :math:`(S-n^2)^2 - D^2 = 0` if :math:`E_x, E_y \neq 0` or\end{aligned}\end{align} \]

:math:` P=0` if \(E_z\neq 0\).

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} n^2 = R & \frac{iE_x}{E_y} = - \frac{S-n^2}{D} = - \frac{S-R}{D} = 1 & \text{right circ. pol.} \\ n^2 = L & \frac{i E_x}{E_y} = - \frac{S-n^2}{D} = - \frac{S-L}{D} = -1 & \text{left circ. pol.}\end{aligned}\end{split}\\*N.B. Recall that\end{aligned}\end{align} \]

\(S = \frac{1}{2} (R+L) \therefore S-R = \half{}(-R+L) = -D\) etc.*

\[ \begin{align}\begin{aligned} \label{eq:cutoff} \omega^2 \mp (\omega_{\rm re}-\omega_{\rm c_i}) \omega - (\omega_p^2 + \omega_{c_i}\omega_{c_e})=0\\i.e. at\end{aligned}\end{align} \]

\(\omega \sim \mp \half \omega_{\rm c_e}+ \qty(\omega_{\rm p}^2 + \frac{1}{4}\omega_{c_{e}}^2)^{\frac{1}{2}}\). At the low-frequency limit, if \(\omega \ll \omega_{\rm c_i}\) (the lowest natural frequency is \(\omega_{\rm c_i}\), then \(R \sim L \sim 1+ \frac{c^2}{c_A^2}\), where \(C_A^2 = \frac{B_0^2}{\mu_0\rho_0}\) is the Alfven speed. The refractive index is \(n^2 = 1+\frac{c^2}{c_A^2}\), i.e. \(\omega^2 = \frac{k^2 c^2}{1 + \frac{c^2}{c_A^2}} \approx k^2c_A^2\) which descripe non-dispersive Alfven waves (c.f. Magnetohydrodynamics later).

For \(\theta = \frac{\pi}{2}\), waves propagating perpendicular to \(B_0\), the dispertion relation becomes

Characteristics of these modes are best seen in a matrix system.

1. For \(E_x = E_y = 0\), and \(E_z \neq 0\), we have \(n^2=P\), i.e. we have the same dispertion relation as in the unmagnetic case. Note, \(\vec{k}\cdot \vec{E} = 0\), so we have transverse electormagnetic waves which are independent of \(B_0\), and \(E\) is parallel to \(B_0\), so all motion is aligned with \(B_0\), and therefore

   \[ \begin{align}\begin{aligned}n^2 = P \qquad \text{(Ordinary mode, O-mode)}\\and the cutoff is at the plasma frequency, with no resonance.\end{aligned}\end{align} \]

2. Now suppose we consider \(E_z=0\), \(E_x, E_y \neq 0\), then \(n^2 = \frac{RL}{S}\) in solution,

   \[ \begin{align}\begin{aligned}\frac{iE_x}{E_y} = - \frac{S-n^2}{D} = - \frac{D}{S}\\(first row gives :math:`SE_x - iD E_y = 0`). Hence the wave is\end{aligned}\end{align} \]

   partly longitudinal, and partly transverse, since both \(E_x, E_y \neq 0\), and thus \(\vec{k} \cdot \vec{E} \neq 0\). The fact that \(E\) is perpendicular to \(B_0\) means that same for the gyration about the magnetic field which is generated, so wave properties depend on \(B_0\),

   \[ \begin{align}\begin{aligned}n^2 = \frac{RL}{S} \qquad \text{(Extraordinary mode, X-mode)}\\the cutoff :math:`R=0` or :math:`L=0`, and resonance at :math:`S=0`.\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} R & = 1 - \frac{\omega_{\rm p}^2}{(\omega+\omega_{\rm c_+})(\omega-\omega_{c_-})} \\ L & = 1 - \frac{\omega_{\rm p}}{(\omega-\omega_{\rm c_+})(\omega+\omega_{\rm c_-})}\end{aligned}\end{split}\\since :math:`\omega \ll \omega_{c_i}`\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} R &\approx 1 - \frac{\omega_p^2}{\omega_{c_i}(-\omega_{c_e})} \\ &= 1 + \frac{\frac{ne^2}{\epsilon_0 m_e} + \frac{n e^2}{\epsilon_0 m_i} } {\frac{eB_0}{m_i} \frac{e B_0}{m_e}} \\ &= 1 + \frac{n(m_i+m_e)}{\epsilon_0 B_0^2} \\ &\approx 1 + \frac{\mu_0 \rho_0 c^2}{B_0^2} \\ &= 1 + \frac{c^2}{c_A^2} \approx \frac{c^2}{c_A^2}\end{aligned}\end{split}\\For :math:`\theta = 0`, :math:`\omega \ll \omega_{c_i}`, which are\end{aligned}\end{align} \]

both circular polarisations for transverse Alfven waves. Then,

\[ \begin{align}\begin{aligned}n^2 = \frac{kL}{S} \therefore \omega^2 = k^2 c_A^2\\and\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}S = \half (R+L)\\but :math:`n^2 = P`, so there is a cutoff; no low frequency is\end{aligned}\end{align} \]

possible.

\[ \begin{align}\begin{aligned}\begin{split} \begin{pmatrix} S & -iD & 0 \\ iD & S-n^2 & 0 \\ 0 & 0 & P-n^2 \end{pmatrix}\end{split}\\So\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} S E_x - iD E_y & = 0 \\ \frac{i E_x}{E_y} &= - \frac{D}{S}\end{aligned}\end{split}\\So what are the implications for the Alfven wave? Since :math:`R`,\end{aligned}\end{align} \]

\(L\) are approximately equal, for \(\omega \ll \omega_c\),

\[ \begin{align}\begin{aligned}\qty|\frac{E_x}{E_y}| \ll 1\\and so :math:`|E_y| \gg |E_x|`. From the equations of motion,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \dot{v}_x &= - \frac{e}{m} E_x - \omega_c v_y \\ \dot{v}_y &= - \frac{e}{m} E_y - \omega_c v_x\end{aligned}\end{split}\\we can differentiate the system with respect to :math:`t` to show\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\qty| \frac{v_x}{v_y} | \approx \frac{\omega_c}{\omega} \quad \text{when}\quad \qty| \frac{E_x}{E_y} | \approx 0\\We know that the O-mode cuts off at :math:`\omega=\omega_{\rm p}`.\end{aligned}\end{align} \]

For the X-mode, \(n^2= \frac{RL}{S}\), the two cutoffs occur at \(R=0\) or \(L=0\). This is the same as the circuarly polarised waves as before, but we have two cutoff frequencies for the same mode. Resonance occurs at \(S=0\), and again, this occurs at two places, an upper and a lower hybrid frequency:

\[\begin{split}\begin{aligned} \omega_u^2 &= \omega_p^2 + \omega_{c_e}^2 \\ \omega_l^2 &= \omega_p^2 \frac{\omega_{c_i}\omega_{c_e}}{\omega_p^2+\omega_{c_e}^2}\end{aligned}\end{split}\]

Ideal Magnetohydrodynamic Plasmas¶

Here we consider the behaviour of plasmas at long wavelengths, and low frequencies; in this limit we can retrieve classical thermodynamic relations. Starting with the model equations,

Notably, Ampere’s law contains no mention of displacement current, and there is no allowance for charge separation. An ideal MHD plasma exhibits perfect conductivity, so

at the normal modes, just as with the cold plasma case. Take a perturbation,

Then we assume \(\vec{u}_0 = 0\) (stationary equilibrium), and \(\vec{J}_0 = \frac{\nabla \times \vec{B}}{\mu_0} = 0\).

Then linearise,

then, assuming all perturbed quantities are proportional to \(\exp[ i(\vec{k} \cdot \vec{r} - \omega t)]\), so

\(\nabla \cdot \vec{B} = 0\),

obtain everything in terms of \(u_1\). \(p_1\) can be eliminated using (IV), \(\rho_1\) using (I), and \(B_1\) using (III).

This process yields

Then,

the minimum cyclotron frequency for ions and plasma frequency, and the wavelength is much greater than the Debye length of the Larmour Radius. Consider the component of \(\vec{u}_1\) perpendicular to the direction of motion, \(\hat{z}\). To find this direction take the cross-product of the whole equation ([eq:mhdincoordis]), with \(\hat{z}\). Now take \(\hat{z} \cdot \hat{b} = \cos(\theta)\).

\(\hat{z} \cdot \vec{u}_1 = 0\), and we still have \((\omega^2 - k^2 \cos[2](\theta))(\hat{z} \times \vec{u}_1) = 0\). Clearly a non-trivial solution, governed by the dispertion relation is possible: a wave which satisfies

frequency–for \(\theta=0\) this is just like a cold plasma solution, but at \(\theta=\frac{\pi}{2}\) there is no transverse wave solution, but the cold plasma has the low frequency limit of the X-mode, the compressed Alfvén mode.

The general solution for MHD waves is that we have a dispertion reltion,

• Alfven

  \[\omega^2 = k^2 c_A^2 \cos[2](\theta)\]

• Fast (+) and Slow (-) Magnetosonic

  \[2 \frac{\omega^2}{k^2} = c_s^2 + c_A^2 \pm \sqrt{( c_s^2 + C_A^2)^2 - 4k^2 c_s^2 c_A^2 \cos[2](\theta)}\]

For the fast MS mode, \(\frac{B^2}{2 \mu_0}\) magnetic pressure enhanes the thermodynamic pressure, \(p\), by varying in phase with it. Fr the slow MS mode the magnetic presure and thermodynamic pressure oppose one another. Magnetic pressure plays a powerful conceptual role in MHD.

Recall the equilibrium; to see how significant the magnetic pressure can be,

\(\vec{B}_0\) and \(p_0\), however, we can generalise this, since

\(\vec{B}_0 \cdot \nabla \vec{B}_0 = 0\), for equilibrium,

Plasma tends to avoid the strongest field regions in equilibrium. This is a possible mechanism for confinement. How could this go wrong? Suppose we have a cylinder of plasma, which is carrying current. It has an azimuthal \(\vec{B}_0\)

(2,0) ellipse (.2 and .5);

(-2,-.5) rectangle (2, .5); (-2,0) ellipse (.2 and .51);

in -1,-.5,…, 1.5 (, -.5) ..controls (+.2, -.3) and (+.2, .3) .. (,.5); (-1.8, -.5) ..controls (-1.8+.2, -.3) and (-1.8+.2, .3) .. (-1.8,.5) node [midway, right] \(\vec{B}_0\);

(0,.9) – (2,.9) node [midway, fill=white] \(\vec{J}\);

If we bend the cylinder, but wish to maintain \(\vec{J}_0\),

The magnetic pressure is stronger on the inside of the bend than the outside. Thus the plasma is push upwards, wosening the distortion. This gives a kink instability.

Plasmas with Collision¶

Consider a binary collision of two plasma particles, where the Coulomb force between the particles is

ion.

(0,0) circle (0.5) node [midway, white] +; (-4,1) circle (0.2) node [white] -;

(-3.7,1) – (0,1) arc (90:40:1) – ++(130:-2);

(0,1) – (4,1) (1,0)–(4,0);

(3,1) – (3,0) node [right, midway] \(b = r \sin(\theta)\);

Consider the change of \(v_{\perp}\), with a massive ion, \(m_{\rm ion} \gg m_{\rm e}\). The change of perpendicular momentum from the equation of motion can be written

\(90^{\circ}\)), the change of \(m_{\rm e} v_{\perp}\) is of the order of \(mv\) itself, so,

The interaction between two particles can be described using the interaction cross-section, \(\sigma\), where

with \(b\) being the area of the disc. We simply assume the interaction is happening for impact parameters

For this interaction the cross-section is

\(n_{\rm i} \approx n_{\rm e}\), and \(q_{\rm i} = q_{\rm e} = e\) is

plasma, and a more rigorous estimate indicates

\(\Lambda \sim n_{\rm e} \lambda^3_{\rm De}\). \(\log(\Lambda)\) is the Coulomb logarithm, and is normally assumed to be constant, with a value in the range \(\log(\Lambda) \in [10,30]\).

Kinetic Theory of Plasmas¶

There are some phenomena which neither MHD nor single-particle descriptions of a plasma can describe. For these situations we need to consider the velocity distribution, \(f(\vec{v})\) of the plasma. In fluid theory the independent variables are functions of \(\vec{r}\) and \(t\) only, which is because the velocity distribution is taken to be Maxwellian everywhere, so can be uniquely speciied by the temperature, \(T\), and the number density, \(n(\vec{r}, t)\),

If \(f\) is correctly normalised it describes the probbaility of finding a particle in the range \(\vec{r} \in (\vec{r}, \vec{r}+\dd{\vec{r}})\) and \(\vec{v} \in (\vec{v}, \vec{v}+\dd{\vec{v}})\). \(f\) is a function in seven variables, and, if it is a Maxwellian distribution, it has the form,

Ignoring collisions, and assuming the plasma to be a closed system, with no sources or sinks of particles, the function will obey the Liouville theorem, so

for the time derivative along a trajectory in \((\vec{r}, \vec{v})\)-phase space, so

electromagnetic the equation takes the form

of Maxwell’s equations, as \(\vec{E}\) and \(\vec{B}\) are the average values of the electric and magnetic fields from particles in the plasma.

If there are collisions in the plasma, \(\dv{f}{t}\neq 0\), and, using the collision integral,

to include sources or sinks of particles by adding terms on the right hand side.

The collision term can sometimes be approximated as

is the collision time. This is the Krook collision term.

Plasma waves and Landau Damping¶

As an illustration of the use of the Vlasov equation, we consider the electron plasma oscillations in a uniform plasma with no applied magnetic or electric fields. Consider a first order perturbation,

Poisson’s equation (\(\epsilon_0 \nabla \vec{E} = \rho\)),

waves in the plasma are plane waves in the \(x\)-direction. Then,

\(v_z\) can be carried out, and

\(v_x = \frac{\omega}{k}\). Landau suggested (1946) letting \(\omega \to \omega + i o\) for a small \(o\), which makes the integral

This can be written symbolically as

First, concentrate on the real part, where the integral can be computed by integrating by parts,

velocities), and so we can expand \((v - \frac{\omega}{k})^{-2}\),

And now, using the expansion for the integral,

Then we can write

\(\omega_{\rm pe}^2\) in the second term, so

frac{k_{rm B} T_{rm e}}{m} frac{k^2}{omega_{rm pe}}`). Then, the group velocity is

Finally, the imaginary part. For simplicity ignore the thermal correction, so \(\omega(k) \approx \omega_{\rm pe}\), then

distribution for \(\tilde{f}\),

frac{omega_{rm pe}}{k}`,

described as “probably the most astounding result of plasma physics.”

The damping is not reandomisation by collisions, but a resonant (i.e.phase velocity of waves is the same as the velocity of interacting particles) \(v=\frac{\omega}{k}\) transfer of energy from waves to particles. It can be reversed if \(\pdv*{\tilde{f}_0}{v}>0\).

Plasma Radiation¶

in 0,30,…,360 (B) (0,0)+(:7); (0,0) +(:7) – +(:10); (0,0) circle (7);

(0,0) circle (5); in 0,30,…,360 (2,0) – (:5) coordinate (A);

(0,0) node [fill=accent-blue, circle] 1; (2,0) node [fill=accent-green, circle] 2;

in 0,30,…,360 (0,0)+(:5) – +(:7);

(0,.6) – (2,.6) node [above, midway] \(\Delta v \ t\);

(0,-.6) – (-5,-.6) node [fill=white, midway] \(r = ct\); (-170:5) – (-170:7) node [below, yshift=-.1cm, midway] \(r^{\prime} = c \Delta t\);

The radial component has the usual form,

while the \(\vec{E}_{\phi}\) component is given by the number of radial field-lines per unit area in the direction of \(\vec{\phi}\). From geometry arguments,

\(\frac{\Delta v}{\Delta t} = \ddot{r}\), and substituting \(\vec{E}_r\),

strength varying with \(\frac{1}{r}\), in contrast to the radial field which varies with \(\frac{1}{r}\). This “kink” is an outward moving pulse of electromagnetic radiation. The power per unit area per second is given by the Poynting vector,

\(\dd{\Omega} = 2 \pi \sin(\phi) \dd{\phi}\),

radiated power from an accelerated charge, and can be applied to our specific case of a particle gyrating in an electric field.

Faraday Rotation in a Cold Plasma¶

\[n^2 = R \qquad n^2 = L \quad L \neq R\]

thus different polarisations have different phase speeds—we can exploit this as a remote diagnostic of the plasma conditions.

\[n^2 = \frac{k^2 c^2}{\omega^2}\]

Thus it is possible to convert between \(n\) and \(\frac{\omega}{k}\). A superposition of the circuarly polarised waves produces an evolution along the propogation ray of the net \(\vec{E}\) polarisation direction.

(0,-4.4) – (7,-4.4) node [midway, below] \(z\);

(0,0) circle (2); (0,-2) – (0,2) node [midway, right, black] \(\vec{E}\); (2,4) node [text width=3cm, left, text ragged] At \(z=0\) \(R\) and \(L\) are synchronised, producing a vertical polarisation.; (-1,-3) circle (.8) node \(R\); (1,-3) circle (.8) node \(L\); (-1,-2.2) circle (0.1); (1,-2.2) circle (0.1); (-1,-2) arc (90:135:1); (1,-2) arc (270:225:-1);

(0,4) node [text width=3cm, text ragged] As \(z\) increases, \(R\) and \(L\) the faster rotation of the \(L\) mode causes the polarisation to become diagonal.; (0,0) circle (2); (0,-2) – (0,2) node [midway, right, black] \(\vec{E}\);

(-1,-3) circle (.8) node \(R\); (1,-3) circle (.8) node \(L\); (-1,-2.2) circle (0.1); (1,-2.2) circle (0.1); (-1,-2) arc (90:135:1); (1,-2) arc (270:225:-1);

Suppose the polarisation vector \(\vec{E}\) shifts by an angle \(\dd{\theta}\) as a result of the varying phase interference of the superposed \(R\) and \(L\) modes. Then,

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} R & = 1 - \frac{\omega_{\rm p}^2}{\omega(\omega-\omega_{\rm c_e})} \\ L & = 1 - \frac{\omega_{\rm p}^2}{\omega(\omega+\omega_{\rm c_e})}\end{aligned}\end{split}\\with :math:`\omega_{\rm c_+} \to 0`, as :math:`m_+ \to \infty`. Thus\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \eval{n}_{\rm RCP} &\approx 1 - \frac{\omega_{\rm p}^2}{2 \omega (\omega-\omega_{\rm c_e})} \\ \eval{n}_{\rm LCP} &\approx 1 - \frac{\omega_{\rm p}^2}{2 \omega(\omega+\omega_{\rm c_e})}\end{aligned}\end{split}\\Then\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} (n_R - n_L) &= \frac{\omega_{\rm p}^2}{2 \omega} \qty[ \frac{1}{\omega - \omega_{\rm c_e}} - \frac{1}{\omega(\omega+\omega_{\rm c_e})}]\\ &= \frac{\omega_{\rm p}^2}{2 \omega} \frac{\omega + \omega_{\rm c_e}-(\omega-\omega_{\rm c_e})}{\omega^2 - \omega_{\rm c_e}^2} \\ &= \frac{\omega_{\rm p}^2 \omega_{\rm c_e}}{\omega(\omega^2-\omega_{\rm c_e}^2)} \\ &\approx \frac{\omega_{\rm p}^2 \omega_{\rm c_e}}{\omega^3} \\\end{aligned}\end{split}\\Hence,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \dd{\theta} &= \half \qty| (k_R - k_L)| \dd{z} \\ & \approx \half \int_0^2 \frac{\omega_{\rm p}^2 \omega_{\rm c_e} \dd{z}}{c \omega^2} \\ &= \half \int_0^2 \frac{ \frac{ne^2}{m_{\rm e}^2} \frac{eB}{m_{\rm e}}}{ c \omega^2} \dd{z}\\ &= \frac{e^2}{2 \epsilon_0 m_{\rm e}^2 c \omega^2} \int_0^2 nB \dd{z}\end{aligned}\end{split}\\We could allow :math:`n` and :math:`B` to vary slowly, i.e. with the\end{aligned}\end{align} \]

gradient-scale length of \(n,B\) much greater than the wavelength of the circular poalrisation modes, and still have a Faraday rotation result which is reasonably slow.

We can generalise this result, with

field direction and the line-of-sight direction.

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} S &= \half \qty( 1 + \frac{\omega_{\rm p}^2}{(\omega + \omega_{\rm ci})(\omega_{\rm ei}-\omega)} + 1 - \frac{\omega_{\rm p}^2}{(\omega - \omega_{\rm ei})(\omega+\omega_{\rm ce})}) \\ &= \half \qty( 2+ \omega_{\rm p}^2 \frac{(\omega-\omega_{\rm ci})(\omega+\omega_{\rm ce}) - (\omega+\omega_{\rm ci})(\omega_{\rm ce}-\omega)}{(\omega^2-\omega_{\rm ci}^2)(\omega_{\rm ce}^2 - \omega^2)}) \\ &= \half \qty( 2+ \omega_{\rm p}^2 \frac{\omega \qty(\omega+\omega_{\rm ce} - (\omega_{\rm ce}-\omega))}{\omega^2(\omega_{\rm ce}^2 - \omega^2)})\\ &= \half \qty( 2+ \omega_{\rm p}^2 \frac{2 \omega^2}{\omega^2 \qty(\omega_{\rm ce}^2 - \omega^2)} )\\ &= \half \qty( 2+ \omega_{\rm p}^2 \frac{2 \omega^2}{\omega^2 \qty(\omega^2_{\rm ce} - \omega^2)}) \\ &= 1 + \frac{\omega_{\rm p}^2}{\qty( \omega_{\rm ce}^2 - \omega^2)}\end{aligned}\end{split}\\Now introducing :math:`\Gamma_{\rm p} := \frac{\omega_{\rm p}^2}{\omega^2_{\rm ce} - \omega^2}`, finally,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} S &= \half (R+L) = 1+r_{\rm p}\\ D &= \half (R-L) = r_{\rm p} \frac{\omega_{\rm ce}}{\omega}\\ P &= 1 - \frac{\omega_{\rm p}^2}{\omega^2} \approx S\end{aligned}\end{split}\\In the limit :math:`\omega \gg \omega_{\rm ce}`. The general cold\end{aligned}\end{align} \]

plasma dispertion relation is equation ([eq:dispertionrelation]), which can be solved for \(n^2\),

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} n^2 &= \frac{2 PS +(RL-PS)(\sin[2](\theta)) \pm \sqrt{(RL-PS)^2 \sin[4](\theta)} }{{2 \qty[ P+(S-P) \sin[2](\theta)]}} \\& \quad+ \frac{{4 P^2D^2 \cos[2](\theta)}}{2 \qty[ P+(S-P) \sin[2](\theta)]} \end{aligned}\end{split}\\In the limit :math:`\omega \gg \omega_{\rm ce}`, :math:`P \approx S`\end{aligned}\end{align} \]

and

\[ \begin{align}\begin{aligned}RL - PS \approx RL -S^2 = -D^2\\hence\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \begin{aligned} n_{\pm}^2 &= \frac{2 S^2 - D^2 \sin[2](\theta) \pm \sqrt{D^4 \sin[4](\theta) + 4 S^2 D^2 \cos[2](\theta)}}{2S}\end{aligned}\\where the :math:`\pm` corresponds to the right and left polarisations\end{aligned}\end{align} \]

respectively. We can further simplify for small angles of wave propogation, so \(\cos(\theta)\approx 1\) and \(v \approx 0\).

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} n_{\pm}^2 &\approx S \pm D \cos(\theta) \\ & \approx 1 + r_{\rm p} \pm r_{\rm p} \frac{\omega_{\rm ce}}{\omega} \cos(\theta) \\ & \approx \frac{\omega_{\rm ce}^2 - \omega^2 + \omega_{\rm p}^2}{\omega_{\rm ce}^2 - \omega^2} \pm \frac{\omega_{\rm p}^2}{\omega_{\rm ce}^2 -\omega^2} \frac{\omega_{\rm ce}}{\omega} \cos(\theta) \\ & \approx 1 \mp \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{\omega^3} \cos(\theta)\end{aligned}\end{split}\\For :math:`\omega^2 \gg \omega_{\rm p}^2`,\end{aligned}\end{align} \]

\(\omega^2 \gg \omega_{\rm ce}^2\). Then

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} n_+ & \approx \qty( 1 - \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{\omega^3} \cos(\theta))^{\half} \approx 1 - \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{2 \omega^3 \cos(\theta)} \\ n_- & \approx \qty( 1+ \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{\omega^3} \cos(\theta) )^{\half} \approx 1 + \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{2 \omega^3} \cos(\theta)\end{aligned}\end{split}\\And so,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\Delta n = n_- - n_+ \approx \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{\omega^3} \cos(\theta)\\Since :math:`n = \frac{kc}{\omega}`, we know\end{aligned}\end{align} \]

\(k = \frac{\omega}{c} n\), and so

\[ \begin{align}\begin{aligned} \begin{aligned} K_- - K_+ \approx \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{c \omega^2} \cos(\theta) \end{aligned}\\and so finally, the rotation angle is\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:13} \dd{\phi} = \half \frac{\omega_{\rm p}^2 \omega_{\rm ce}}{c \omega^2} \cos(\theta) \dd{z}\\At the high wave frequency limit,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\omega = k c = \frac{2 \pi c}{\lambda}\\Hence, as :math:`\omega_{\rm p}\end{aligned}\end{align} \]

= sqrt{frac{4 pi e^2 n_{rm e}}{4 pi m epsilon_0}}`, and \(\omega_{\rm c} = \frac{eB}{m}\),

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \dv{\phi}{z} &= \half \frac{e^2 n }{m \epsilon_0} \frac{eB}{m} \frac{1 \lambda^2}{c (2 \pi)^2 c^2} \cos(\theta) \\ &= \frac{e^3 B n_{\rm e} \lambda^2}{8 \pi^2 m^2 \epsilon_0 c^3} \cos(\theta)\end{aligned}\end{split}\\And then, to find the Farday angle after the radiation has travelled\end{aligned}\end{align} \]

along a path \(r\),

\[ \begin{align}\begin{aligned} \label{eq:14} \phi_{\rm F} = \frac{e^3 \lambda^2}{8 \pi^2 m^2 \epsilon_0 c^3} \int_0^r B(z) n_{\rm e}(z) \cos(\theta) \dd{z}\\Assuming a uniform number density the Faraday rotation is a measure\end{aligned}\end{align} \]

of the magnetic field along the path.

1

Air generally has around \(10^{9} {\rm m}^{-3}\) ions and electrons which are caused by background sources and friction. These can be harnassed as seeds for further ionisation.