Tokamak Fusion Reactor: Cross-Section and the Lawson Criterion

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ITER parameters: R₀=6.2m, a=2m, B_T=5.3T, I_p=15MA.

With ITER-like parameters (R₀=6.2m, a=2m, B_T=5.3T, I_p=15MA), the aspect ratio is 3.1, the edge safety factor is about 3, and the target triple product is 3.5×10²¹ m⁻³·keV·s — sufficient for Q=10 fusion gain.

Formula

q(a) = 2π × a² × B_T / (μ₀ × R₀ × I_p) — cylindrical safety factor
nTτ_E > 3 × 10²¹ m⁻³·keV·s — Lawson criterion for D-T ignition

The Doughnut That Could Power the World

A tokamak confines plasma inside a toroidal (doughnut-shaped) vacuum vessel using powerful magnetic fields. The concept was invented in the 1950s by Soviet physicists Andrei Sakharov and Igor Tamm. Today it is the most advanced approach to controlled nuclear fusion, with ITER — a 23,000-tonne machine in southern France — aiming to demonstrate net fusion energy for the first time. The tokamak's elegant geometry balances engineering feasibility with the physics requirements of stable plasma confinement.

Magnetic Field Geometry

A tokamak uses two superimposed magnetic fields. The toroidal field, generated by large external coils, wraps around the torus the long way. The poloidal field, generated by the plasma current itself (induced by a central solenoid), wraps the short way. Together they create helical field lines that spiral around the torus. This helical twist is essential: without it, charge-dependent drifts would quickly push the plasma outward. The safety factor q measures the pitch of these helical lines and must stay above critical thresholds to avoid violent instabilities.

The Lawson Criterion

John Lawson showed in 1957 that fusion power production requires the triple product nTτ_E — plasma density times temperature times energy confinement time — to exceed a critical value. For deuterium-tritium fusion at the optimal temperature of about 15 keV, this threshold is approximately 3×10²¹ m⁻³·keV·s. Reaching this condition is the central goal of fusion research. ITER's design targets a triple product sufficient for Q=10: producing ten times more fusion power than the heating power injected into the plasma.

From ITER to Power Plants

ITER will demonstrate scientific feasibility but is not a power plant. The next step — DEMO — will be the first tokamak to generate electricity for the grid, targeted for the 2040s-2050s. Beyond tokamaks, compact high-field designs using high-temperature superconductors (like MIT's SPARC) promise smaller, cheaper reactors. Stellarators, which use complex 3D coil shapes instead of plasma current, offer inherently steady-state operation. The race to fusion energy is no longer a question of physics possibility but of engineering, materials, and investment.

FAQ

What is a tokamak?

A tokamak is a magnetic confinement device shaped like a doughnut (torus). It uses a combination of toroidal (along the ring) and poloidal (around the cross-section) magnetic fields to confine hot plasma. The name comes from the Russian acronym for 'toroidal chamber with magnetic coils'. It is currently the leading concept for controlled fusion energy.

What is the Lawson criterion?

The Lawson criterion states that for a fusion plasma to produce net energy, the triple product nTτ_E (density × temperature × energy confinement time) must exceed a critical threshold. For deuterium-tritium fusion at optimal temperature (~15 keV), this threshold is about 3×10²¹ m⁻³·keV·s.

What is the safety factor q?

The safety factor q measures how many times a magnetic field line goes around the torus the long way (toroidally) for each time it goes the short way (poloidally). When q is a rational number, the field line closes on itself and can drive instabilities. Tokamaks must maintain q > 1 everywhere and q > 2-3 at the edge for stability.

What is ITER and when will it achieve fusion?

ITER (International Thermonuclear Experimental Reactor) is a tokamak under construction in southern France. With R₀=6.2m and B_T=5.3T, it aims to produce 500 MW of fusion power from 50 MW of input heating (Q=10). First plasma is expected in the late 2030s.

Sources

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