Compute the Q factor, Lawson triple product and IPB98 energy confinement time of a D-T fusion tokamak (D+T → He-4 + n + 17.6 MeV) in real time. The defaults reproduce the ITER design point (R=6.2 m, a=2.0 m, B=5.3 T, T=15 keV), giving Q≈10 and clearing the ignition condition — so you can build intuition for the design trade-offs that govern ITER, SPARC and the DEMO commercial reactors.
Parameters
Electron density n_e
×10²⁰ m⁻³
Central electron density. ITER targets about 1.0×10²⁰ m⁻³
Plasma temperature T
keV
Ion temperature. 1 keV ≈ 1.16×10⁷ K, so 15 keV ≈ 170 million K
Energy confinement time τ_E
s
Time over which the thermal energy decays by 1/e. The figure of merit H-mode chases
Major radius R
m
Centre of the torus to the plasma centre. ITER = 6.2 m
Minor radius a
m
Cross-section radius of the plasma. ITER = 2.0 m, DIII-D = 0.67 m
Toroidal field B_T
T
ITER = 5.3 T (Nb₃Sn), SPARC = 12 T (ReBCO high-Tc superconductors)
Auxiliary heating P_aux
MW
Total NBI + ECRH + ICRH input power. ITER plans 50 MW
Results
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Reactivity <σv> (m³/s)
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Plasma volume (m³)
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Fusion power P_fus (MW)
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Q value
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Lawson triple product (m⁻³·keV·s)
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IPB98 τ_E (s)
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Tokamak cross-section — D-shape plasma with alpha and neutron emission
D-shaped plasma confined inside the toroidal field produced by the superconducting coils. Yellow: alpha particles (He-4, 3.5 MeV). Blue: neutrons (14.1 MeV). The right gauge shows the current Q value.
D-T reactivity <σv> vs temperature T
Lawson plane — triple product and machine operating points
Fusion power P_fus is the product of reactivity <σv>, the fuel-density product, the energy per reaction E_DT = 17.6 MeV and the plasma volume V. Q is the gain over external heating, and the Lawson triple product sets the ignition condition.
Left: toroidal volume (circular-section approximation). Right: H-mode confinement scaling IPB98(y,2). Plasma current and size dominate the gain, and auxiliary heating actually degrades τ_E.
Nuclear Fusion Tokamak Q-factor — Lawson criterion and ITER design
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Everyone talks about "fusion energy" in the news. What exactly is this Q factor that keeps showing up?
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In plain words it is a gain: "for every unit of heating you pump in, how many units of fusion energy come out?" Q = P_fus / P_aux. Q = 1 is breakeven, Q = 5 is the floor for ITER, Q = 10 is the ITER high-gain goal, and Q → ∞ is "ignition", where the plasma burns by itself. A DEMO commercial reactor aims for Q = 20-40. NIF made headlines in late 2022 by reaching target gain Q > 1 with laser inertial fusion — but that was Q at the fuel capsule. Once you include the whole facility, NIF is still Q ≈ 0.01.
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OK… and with the default sliders I see Q = 12.86. That is in the ITER ballpark, right?
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Exactly — the defaults track ITER (R = 6.2 m, a = 2.0 m, B = 5.3 T, n = 1×10²⁰, T = 15 keV, P_aux = 50 MW). Plasma volume V = 2π²Ra² ≈ 490 m³, reactivity <σv> ≈ 1.86×10⁻²² m³/s, fusion power ≈ 644 MW, alpha self-heating about 130 MW, so Q ≈ 12.9. The real ITER ends up limited by alpha losses and operates closer to Q = 10 with 500 MW out, but the design intent is exactly this point.
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The "Lawson triple product" below shows 4.5×10²¹. That is the value you have to exceed to ignite, right?
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Yes — Lawson's criterion. Once n·T·τ_E ≥ about 3×10²¹ m⁻³·keV·s, the alpha particles from D-T fusion deliver more heat to the plasma than is lost, and the plasma burns on its own energy. The default is 4.5×10²¹, comfortably above. Try dropping the temperature to 5 keV: <σv> collapses exponentially and Q falls below 1. Push T too high and the pressure limit p = nT bites instead, so the sweet spot lands around 13-20 keV.
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So what really sets τ_E in practice? The slider lets me pick anything, but a real device cannot — what fixes it?
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There is an empirical scaling fitted to dozens of tokamaks called IPB98(y,2): τ_E ∝ I_p^0.93 · B^0.15 · n^0.41 · R^1.39 · a^0.58 · P^(-0.69). Notice the 1.39 on R — size dominates everything, which is exactly why ITER is so huge (R = 6.2 m). Auxiliary power actually hurts τ_E (exponent -0.69, known as "power degradation"), so brute-forcing more heat is not free. The right-hand "IPB98 τ_E" stat in this tool is that prediction; with default sliders it lands at about 1.21 s, while we entered τ_E = 3 s on the left — i.e. we are assuming a good H-mode confinement enhancement.
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Are there fusion approaches other than tokamaks?
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Broadly two families: magnetic confinement and inertial confinement. Magnetic includes tokamaks (ITER, JT-60SA, DIII-D, EAST, KSTAR) and stellarators (Wendelstein 7-X). Tokamaks twist the magnetic field with a plasma current; stellarators bake the twist into external coils. Inertial includes NIF (lasers) and Z-pinches, which crush tiny D-T capsules. Recent private efforts like SPARC (B = 12 T ReBCO HTS) and Helion (FRC) target compact, faster paths. ITER will start D-T around the mid-2030s, with DEMO around 2050 as the first electricity-producing prototype.
Frequently Asked Questions
Q is the dimensionless ratio of the fusion power produced in the plasma to the auxiliary heating power injected, Q = P_fus / P_aux. Q = 1 is breakeven, Q = 5 is the minimum ITER target, Q = 10 is the ITER high-gain goal, and Q → ∞ is "ignition" where the plasma sustains itself with no external heat. DEMO commercial reactors aim for Q = 20-40. JET reached Q ≈ 0.12 in 1991, Q ≈ 0.65 in 1997, and NIF inertial-confinement laser fusion first achieved target gain Q > 1 in December 2022.
The Lawson criterion is the condition under which the self-heating from fusion (alpha particles in D-T) compensates the energy losses from the plasma. For D-T fusion, n·T·τ_E ≥ about 3×10²¹ m⁻³·keV·s is the ignition condition. The triple product collapses three design knobs — density, temperature and confinement — into one figure of merit. ITER targets about 4×10²¹ m⁻³·keV·s. The default values in this simulator give 4.5×10²¹ and clear the ignition threshold.
The D-T reactivity <σv> peaks near 70 keV (about 800 million K), but practical tokamaks only reach 10-25 keV. More importantly, the pressure-limited figure of merit <σv>/T² is maximised between 13 and 20 keV, where you get the most fusion power per unit of magnetic-field pressure. ITER therefore runs at 10-20 keV and this tool defaults to 15 keV.
IPB98(y,2) is the H-mode energy confinement scaling law regressed in 1998 from a multi-device tokamak database for ITER design. It has the form τ_E ∝ I_p^0.93 · B_T^0.15 · n^0.41 · R^1.39 · a^0.58 · P_loss^(-0.69). Confinement grows strongly with plasma current and size, and degrades with auxiliary heating power. At the ITER design point IPB98 predicts τ_E ≈ 3.7 s. This tool assumes I_p = 15 MA so you can compare the IPB98 estimate against the τ_E slider value.
Real-World Applications
ITER (International Thermonuclear Experimental Reactor): The largest tokamak in the world, under construction at Cadarache in France with R=6.2 m, a=2.0 m, B_T=5.3 T, I_p=15 MA and Nb₃Sn superconducting coils. The design goal is Q=10 and 500 MW of fusion power sustained for at least 400 s — exactly the operating point this tool's defaults try to reproduce. Funded by the EU, Japan, US, Russia, China, South Korea and India. First plasma was originally 2025, now pushed to the mid-2030s.
JT-60SA (Japan): A superconducting tokamak in Naka used as ITER's pre-operation complement. R=2.96 m, B_T=2.25 T, I_p=5.5 MA. Achieved first plasma in 2023 and is producing H-mode and non-inductive current-drive data ahead of ITER. Enter R=3.0 m, a=1.1 m, B=2.3 T into this tool to see a JT-60SA-style operating point.
SPARC and ARC (MIT / Commonwealth Fusion Systems): Compact high-field tokamaks built around ReBCO high-temperature superconductors. SPARC targets B_T=12 T at 1/40 of ITER's volume to reach Q>2. Enter R=1.85 m, a=0.57 m, B=12 T to feel the SPARC design sensitivity. Successor ARC aims to demonstrate electricity in the 2030s.
DEMO and commercial reactors: EU-DEMO, J-DEMO and CFETR (China) target electricity around 2050 with Q=20-40, 1-2 GW output, steady-state operation, and tritium self-breeding (Li(n,α)T blanket reactions giving TBR > 1.0). To maximise Q in this tool, grow the minor radius (volume ∝ a²) and push the field (pressure budget ∝ B²) — the most effective design moves.
Common Misconceptions and Pitfalls
The biggest misconception is the press coverage of "NIF achieved fusion power in 2022". What Lawrence Livermore's National Ignition Facility actually achieved was "more fusion energy out of the D-T fuel capsule than the laser energy that hit it" (target gain Q > 1, namely 3.15 MJ out vs 2.05 MJ in). The total laser plant consumes about 300 MJ to fire that shot, so the facility-wide Q is still ≈ 0.01. For a working power plant you need facility-Q > 10, which no experiment has achieved. Always check which Q the article is quoting — plasma Q, target Q or wall-plug Q.
Second, the simplification that "a bigger toroidal field is always better". Raising B_T relaxes the beta limit (β = 2μ₀p/B², a stability cap on plasma pressure), so the allowed pressure p = nT scales as B² and fusion power scales as p²V ∝ B⁴. But doubling the field quadruples the magnetic forces and exponentially raises the structural stress on the coil supports. Nb₃Sn taps out at about 13 T in practice; going higher needs ReBCO HTS (good for 20-25 T) at much higher cost and tougher quench protection. Push the slider to B=12 T and you will see that IPB98 τ_E barely budges (exponent 0.15) while the pressure margin grows dramatically.
Finally, "ignition is not commercialisation". Even with a Lawson-satisfying ignited plasma, a commercial fusion reactor still needs to solve: (1) tritium breeding ratio TBR > 1.0 in lithium blankets (tritium half-life 12.3 years means you cannot stockpile it), (2) first-wall damage from 14 MeV neutrons (replacing components below 100 dpa) and low-activation materials (vanadium alloys, ODS steels), (3) divertor heat exhaust at 10-20 MW/m², (4) ELM suppression and disruption mitigation, (5) economics below a few cents per kWh. Even a setting that gives Q → ∞ in this tool ignores those constraints, which is why neutron sources like IFMIF-DONES are being built in parallel.
How to Use
Enter plasma density in m⁻³ (typical range: 1×10¹⁹ to 3×10²⁰ for ITER-scale tokamaks)
Input plasma temperature in keV (D-T fusion optimized near 20–25 keV; SPARC target ~12 keV)
Specify energy confinement time τ_E in seconds (empirical IPB98 scaling predicts 0.3–1.2 s for 10 MW input)
Set major radius R in meters (ITER = 6.2 m; SPARC = 1.85 m; JT-60SA = 3.0 m)
Review Q factor output: Q > 1 indicates net fusion gain; Q = 10 is breakeven in commercial reactors
Worked Example
ITER baseline scenario: n_e = 1.0×10²⁰ m⁻³, T_plasma = 21.3 keV, τ_E = 0.31 s (from IPB98 correlation with P_aux = 50 MW), R = 6.2 m. Plasma volume ≈ 840 m³. Reactivity <σv> ≈ 1.1×10⁻²² m³/s at 21.3 keV. Fusion power P_fus ≈ 10.0 MW. Lawson triple product n·T·τ ≈ 1.1×10²¹ m⁻³·keV·s. Q ≈ 10 for 50 MW input (target for ITER operations).
Practical Notes
Confinement time scales weakly with density and strongly with temperature (IPB98L-mode: τ_E ∝ n⁰·⁴¹ T⁰·⁸⁴ P⁻⁰·⁶⁹); increase confinement via triangularity, aspect ratio, or magnetic field
Q-factor sensitivity: density variations ±10% shift Q by ±20%; temperature changes drive σv exponentially (T²⁰ dependence near 20 keV)
Lawson criterion (n·T·τ > 10²¹ m⁻³·keV·s) mandates simultaneous optimization; trading lower T for higher n·τ products reduces alpha heating self-sufficiency
SPARC (12 keV, n = 2×10²⁰ m⁻³) achieves Q ≈ 2–3 with compact geometry; NIF and laser fusion use different burn physics (ignition vs. confinement scaling)