Lawson Criterion Calculator

The Lawson criterion is the minimum value of the product of plasma density (n) and energy confinement time (τ_E) needed for a fusion reaction's self-heating power to balance the energy lost from the plasma, first derived by John Lawson in 1955. It sets the practical bar every magnetic confinement fusion device must clear.

nτ_E ≥ 12T / (⟨σv⟩E_ch), where T is temperature, ⟨σv⟩ is the fusion reactivity (a function of T), and E_ch is the charged-particle fusion energy (3.5 MeV for D-T, the alpha particle's share). This comes from setting fusion charged-particle heating power equal to the thermal energy loss rate 3nT/τ_E.

The exact D-T reactivity ⟨σv⟩(T) has no closed-form expression, only numerical fits to measured cross-sections. Over the practically relevant 10-20 keV range, ⟨σv⟩ ≈ 1.1×10⁻²⁴T² (T in keV, result in m³/s) is a well-known and widely cited approximation that keeps the criterion in exact closed form.

Each D-T reaction releases 17.6 MeV split between an alpha particle (3.5 MeV) and a neutron (14.1 MeV). The neutron is uncharged and escapes the magnetic field entirely, carrying its energy out of the plasma, so only the alpha's 3.5 MeV can heat the plasma and help sustain the reaction.

It is your entered nτ_E divided by the minimum required nτ_E at the given temperature, expressed as a percentage. 100% or higher means the idealized Lawson criterion is satisfied at that operating point; below 100% means losses still exceed fusion self-heating.

The fusion triple product nTτ_E is compared against one fixed round-number benchmark (roughly 3×10²¹ m⁻³·keV·s) regardless of temperature. The Lawson criterion computes an nτ_E threshold that explicitly varies with temperature through ⟨σv⟩(T), giving a more physically grounded, temperature-specific target. See the related Fusion Triple Product Calculator for the simpler comparison.

Within the 10-20 keV range covered by this calculator's ⟨σv⟩ ≈ 1.1×10⁻²⁴T² approximation, the required nτ_E actually decreases as T increases, since reactivity grows faster than the linear T term in the numerator. Outside this range the real reactivity curve flattens and eventually falls, so the required nτ_E rises again at both very low and very high temperatures, a detail this simplified model does not capture.

Energy confinement times in present-day tokamaks typically range from tens of milliseconds in small devices to a few seconds in the largest machines like JET, generally too short combined with achievable densities to fully satisfy the Lawson criterion, which is exactly why ITER is being built at a much larger scale.

No. Different fuel combinations (D-T, D-D, D-He3) have very different reactivity curves ⟨σv⟩(T) and charged-particle energy fractions, so each has its own Lawson criterion curve. D-T has by far the most favorable (lowest) Lawson criterion of any practical fusion fuel, which is why it is the near-term reactor fuel of choice.

τ_E is typically inferred from the ratio of the plasma's total stored thermal energy to the heating power needed to sustain it in steady state, τ_E = W/P_heat, a standard diagnostic computed from measured plasma parameters rather than a directly observed single quantity.

The energy balance that produces the criterion, fusion heating power equal to the thermal energy loss rate, naturally reduces to a condition on the single combined quantity nτ_E, since a lower density can always be compensated by a proportionally longer confinement time (or vice versa) without changing whether the self-heating condition is met.