Plasma Physics Primer¶

This chapter introduces the physics of plasmas and magnetic confinement fusion from first principles. No prior knowledge of plasma physics is assumed; familiarity with classical electromagnetism at an undergraduate level is sufficient.

What Is a Plasma?¶

A plasma is a quasi-neutral gas of charged and neutral particles that exhibits collective behaviour. When a gas is heated to temperatures above roughly \(10^4\) K, atoms ionise: electrons separate from nuclei, producing a soup of free electrons and ions. This is the fourth state of matter – distinct from solids, liquids, and gases because the long-range Coulomb force between charged particles gives rise to collective phenomena (waves, instabilities, shielding) that have no counterpart in neutral gases.

Over 99% of visible matter in the universe is plasma: stars, the solar wind, interstellar medium, and lightning. On Earth, plasmas must be created and sustained artificially.

Key plasma parameters:

Parameter	Definition	Fusion-relevant value
Temperature \(T\)	Kinetic energy per particle, usually in keV (1 keV = 11.6 MK)	10–20 keV (100–200 million K)
Density \(n\)	Number of particles per unit volume	\(10^{20}\) m^-3
Debye length \(\lambda_D\)	\(\sqrt{\varepsilon_0 T / (n e^2)}\); the distance over which charge is screened	\(\sim 10^{-4}\) m
Plasma frequency \(\omega_p\)	\(\sqrt{n e^2 / (\varepsilon_0 m_e)}\); fastest electrostatic response	\(\sim 10^{11}\) rad/s

A plasma behaves collectively when the number of particles in a Debye sphere \(N_D = \frac{4}{3}\pi n \lambda_D^3 \gg 1\), so that the mean-field approximation is valid. For fusion plasmas, \(N_D \sim 10^8\).

Why Fusion?¶

Nuclear fusion combines light nuclei into heavier products, releasing energy from the mass deficit (\(E = \Delta m \, c^2\)). The most accessible reaction is deuterium-tritium (D-T):

\[\text{D} + \text{T} \;\longrightarrow\; {}^4\text{He}\,(3.5\;\text{MeV}) + n\,(14.1\;\text{MeV})\]

This reaction has the largest cross-section \(\langle\sigma v\rangle\) at accessible temperatures (\(\sim 10\)–\(20\) keV) and releases 17.6 MeV per event. The alpha particle (3.5 MeV) deposits its energy in the plasma, sustaining the burn; the neutron escapes and is captured in a lithium blanket to breed fresh tritium and extract heat.

Fusion fuel is abundant: deuterium from seawater (1 in 6700 hydrogen atoms), tritium bred from lithium. A 1 GW fusion plant would consume roughly 250 kg of fuel per year. There is no long-lived radioactive waste and no chain-reaction risk.

The Lawson Criterion¶

A self-sustaining fusion plasma requires that the alpha heating exceeds all power losses. Lawson (1957) showed that this imposes a minimum on the triple product:

(1)¶\[n \, T \, \tau_E \;\geq\; 3 \times 10^{21} \;\text{m}^{-3}\,\text{keV}\,\text{s}\]

where:

\(n\) is the plasma density
\(T\) is the temperature
\(\tau_E\) is the energy confinement time (the e-folding time for stored energy loss)

The triple product is the single most important figure of merit for any confinement scheme. Achieving it requires simultaneously:

High temperature (\(T \sim 10\)–\(20\) keV) to maximise \(\langle\sigma v\rangle\)
High density (\(n \sim 10^{20}\) m^-3)
Good confinement (\(\tau_E \sim 1\)–\(10\) s)

Magnetic Confinement¶

Charged particles gyrate around magnetic field lines with a radius \(\rho_L = m v_\perp / (q B)\) (the Larmor radius). For a 10 keV deuterium ion in a 5 T field, \(\rho_L \approx 4\) mm – much smaller than the plasma size (\(\sim 1\) m). This means magnetic fields can confine charged particles perpendicular to the field.

The problem is parallel transport: particles stream freely along field lines. To prevent end losses, the field lines must close on themselves. This is achieved by bending the magnetic field into a torus.

The Tokamak¶

A tokamak (from the Russian acronym for “toroidal chamber with magnetic coils”) is the most successful magnetic confinement device. It uses two magnetic field components:

Toroidal field \(B_\phi\) – produced by external coils wrapped around the torus. Typically 2–13 T.
Poloidal field \(B_\theta\) – produced by the plasma current \(I_p\) flowing in the toroidal direction. This current is driven inductively by a central solenoid (transformer action) or non-inductively by neutral beam injection (NBI) or radiofrequency waves (ECCD, LHCD).

The combination \(\mathbf{B} = B_\phi \hat\phi + B_\theta \hat\theta\) produces helical field lines that wrap around nested flux surfaces (topological tori). Particles confined to flux surfaces undergo only slow cross-field transport (diffusion), while parallel losses are eliminated by the closed topology.

Tokamak Geometry¶

\[\begin{split}R &= R_0 + r \cos\theta \\ Z &= r \sin\theta\end{split}\]

where:

\(R_0\) is the major radius (distance from the machine axis to the plasma centre)
\(a\) is the minor radius (half-width of the plasma cross-section)
\(r\) is the radial coordinate from the magnetic axis
\(\theta\) is the poloidal angle
\(\phi\) is the toroidal angle (around the torus)

Key dimensionless parameters:

Symbol	Name	Definition	Typical value
\(A = R_0/a\)	Aspect ratio	Major / minor radius	2.5–4.0
\(\varepsilon = a/R_0\)	Inverse aspect ratio	Minor / major radius	0.25–0.4
\(\kappa\)	Elongation	Vertical / horizontal half-widths	1.6–2.0
\(\delta\)	Triangularity	Horizontal shift of extremal points	0.2–0.5
\(q\)	Safety factor	\(\frac{r B_\phi}{R_0 B_\theta}\)	1 (axis) to 3–5 (edge)

The safety factor \(q\) measures the pitch of the helical field lines: a field line completes \(q\) toroidal transits for every poloidal transit. Rational values \(q = m/n\) (where field lines close on themselves) are sites of MHD instabilities. The \(q = 1\) surface is where sawteeth occur; \(q = 2\) and \(q = 3/2\) surfaces host neoclassical tearing modes (NTMs).

The Magnetic Axis and Separatrix¶

The magnetic axis is the closed field line at the centre of the nested flux surfaces (the O-point of the poloidal flux function \(\psi\)). The separatrix (or last closed flux surface, LCFS) is the outermost closed flux surface; beyond it, field lines intersect material surfaces (the divertor). The X-point is a saddle point of \(\psi\) where the separatrix crosses itself.

What’s Next?

Now that you understand what a plasma is, why we want fusion, and how a tokamak confines it, proceed to Fusion Engineering 101 to learn about the physics models (equilibrium, transport, stability, control) that SCPN-Fusion-Core implements.