.. ----------------------------------------------------------------------- SCPN Fusion Core -- Plasma Physics Primer Copyright 1998-2026 Miroslav Sotek. All rights reserved. License: GNU AGPL v3 | Commercial licensing available ----------------------------------------------------------------------- ============================== 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 :math:`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: .. list-table:: :header-rows: 1 :widths: 25 40 35 * - Parameter - Definition - Fusion-relevant value * - Temperature :math:`T` - Kinetic energy per particle, usually in keV (1 keV = 11.6 MK) - 10--20 keV (100--200 million K) * - Density :math:`n` - Number of particles per unit volume - :math:`10^{20}` m\ :sup:`-3` * - Debye length :math:`\lambda_D` - :math:`\sqrt{\varepsilon_0 T / (n e^2)}`; the distance over which charge is screened - :math:`\sim 10^{-4}` m * - Plasma frequency :math:`\omega_p` - :math:`\sqrt{n e^2 / (\varepsilon_0 m_e)}`; fastest electrostatic response - :math:`\sim 10^{11}` rad/s A plasma behaves collectively when the number of particles in a Debye sphere :math:`N_D = \frac{4}{3}\pi n \lambda_D^3 \gg 1`, so that the mean-field approximation is valid. For fusion plasmas, :math:`N_D \sim 10^8`. Why Fusion? ----------- Nuclear fusion combines light nuclei into heavier products, releasing energy from the mass deficit (:math:`E = \Delta m \, c^2`). The most accessible reaction is deuterium-tritium (D-T): .. math:: \text{D} + \text{T} \;\longrightarrow\; {}^4\text{He}\,(3.5\;\text{MeV}) + n\,(14.1\;\text{MeV}) This reaction has the largest cross-section :math:`\langle\sigma v\rangle` at accessible temperatures (:math:`\sim 10`--:math:`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**: .. math:: :label: lawson n \, T \, \tau_E \;\geq\; 3 \times 10^{21} \;\text{m}^{-3}\,\text{keV}\,\text{s} where: - :math:`n` is the plasma density - :math:`T` is the temperature - :math:`\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: 1. **High temperature** (:math:`T \sim 10`--:math:`20` keV) to maximise :math:`\langle\sigma v\rangle` 2. **High density** (:math:`n \sim 10^{20}` m\ :sup:`-3`) 3. **Good confinement** (:math:`\tau_E \sim 1`--:math:`10` s) Magnetic Confinement --------------------- Charged particles gyrate around magnetic field lines with a radius :math:`\rho_L = m v_\perp / (q B)` (the Larmor radius). For a 10 keV deuterium ion in a 5 T field, :math:`\rho_L \approx 4` mm -- much smaller than the plasma size (:math:`\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: 1. **Toroidal field** :math:`B_\phi` -- produced by external coils wrapped around the torus. Typically 2--13 T. 2. **Poloidal field** :math:`B_\theta` -- produced by the plasma current :math:`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 :math:`\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 ----------------- .. math:: R &= R_0 + r \cos\theta \\ Z &= r \sin\theta where: - :math:`R_0` is the major radius (distance from the machine axis to the plasma centre) - :math:`a` is the minor radius (half-width of the plasma cross-section) - :math:`r` is the radial coordinate from the magnetic axis - :math:`\theta` is the poloidal angle - :math:`\phi` is the toroidal angle (around the torus) Key dimensionless parameters: .. list-table:: :header-rows: 1 :widths: 20 35 25 20 * - Symbol - Name - Definition - Typical value * - :math:`A = R_0/a` - Aspect ratio - Major / minor radius - 2.5--4.0 * - :math:`\varepsilon = a/R_0` - Inverse aspect ratio - Minor / major radius - 0.25--0.4 * - :math:`\kappa` - Elongation - Vertical / horizontal half-widths - 1.6--2.0 * - :math:`\delta` - Triangularity - Horizontal shift of extremal points - 0.2--0.5 * - :math:`q` - Safety factor - :math:`\frac{r B_\phi}{R_0 B_\theta}` - 1 (axis) to 3--5 (edge) The safety factor :math:`q` measures the pitch of the helical field lines: a field line completes :math:`q` toroidal transits for every poloidal transit. Rational values :math:`q = m/n` (where field lines close on themselves) are sites of MHD instabilities. The :math:`q = 1` surface is where sawteeth occur; :math:`q = 2` and :math:`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 :math:`\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 :math:`\psi` where the separatrix crosses itself. .. admonition:: What's Next? Now that you understand what a plasma is, why we want fusion, and how a tokamak confines it, proceed to :doc:`fusion_engineering_101` to learn about the physics models (equilibrium, transport, stability, control) that SCPN-Fusion-Core implements.