Lawson Criterion

John D. Lawson, a British physicist, published his seminal paper in 1957 (originally an internal UKAEA report in 1955). He derived the criterion by balancing the fusion power output against the energy losses from a hypothetical fusion device. At the time, the focus was on achieving break-even conditions, and he simplified the plasma losses to convective and radiative terms. His original paper didn't consider the temperature dependence fully (he used a fixed temperature of 10 keV), but it laid the foundation for all subsequent fusion efficiency metrics. The criterion was instrumental in justifying funding for large-scale fusion experiments by setting a clear, measurable target.

The value 10²⁰ m⁻³·s for D-T arises from balancing the fusion power density (n²⟨σv⟩E/4) with the power loss density (n kT / τ_E). Solving for nτ_E yields (nτ_E) > (12 k T) / (⟨σv⟩ E) for ignition (ignoring radiation). At T ≈ 10-15 keV, ⟨σv⟩ ≈ 10⁻²² m³/s, E = 17.6 MeV = 2.82×10⁻¹² J, k = 1.38×10⁻²³ J/K. Plugging in: nτ > 12×1.38×10⁻²³× (10×10³×1.6×10⁻¹⁹) / (10⁻²² × 2.82×10⁻¹²) ≈ 1.05×10²⁰ s/m³. The original Lawson paper used a slightly different temperature and assumed 50% energy recovery, arriving at 10²⁰. This threshold assumes a thermal Maxwellian plasma, no radiation losses, and equal ion-electron temperatures.

For D-D fusion, the reactivity ⟨σv⟩ is about 100 times lower than D-T at the same temperature (peak ~10⁻²³ vs 10⁻²² m³/s). Also, the energy per reaction is lower (3.27 or 4.03 MeV vs 17.6 MeV). Substituting into the power balance, the required nτ for D-D is roughly (⟨σv⟩·E)_DT / (⟨σv⟩·E)_DD times higher, about a factor of (10⁻²²×17.6)/(10⁻²³×3.6) ≈ 50-100. Thus, the Lawson criterion for D-D is approximately nτ > 10²² m⁻³·s, which is 100 times more stringent. This makes D-D uncompetitive for near-term reactors; however, it's attractive for advanced fuels due to reduced tritium handling.

The simple Lawson criterion neglects radiative losses (bremsstrahlung, line radiation, synchrotron). When these are included, the power balance becomes: P_fusion = P_loss_thermal + P_radiation. Bremsstrahlung power scales as n² T^0.5, while fusion power scales as n² ⟨σv⟩. For T > 10 keV, bremsstrahlung is significant. Including this raises the required nτ by about 20-30%. For a realistic reactor with impurities, the threshold can be 1.5–2 times higher, around 2×10²⁰ m⁻³·s. Also, the criterion assumes 100% α-heating efficiency; in practice, only ~70-80% of α energy is deposited in the plasma, raising the required triple product further.

Ignition means the plasma is self-sustaining (no external heating needed), requiring nτ > 10²⁰ at T≈10 keV. Break-even (Q=1) means P_fusion = P_input, which requires a lower nτ—about 3-5×10¹⁹ m⁻³·s—because some external heating can compensate for the losses. However, the term 'break-even' is often used loosely: in the 1990s, the scientific break-even was considered Q=1, but true commercial break-even (Q_eng>1) requires Q>10, which corresponds to nτ > 5-7×10¹⁹ for ITER-like parameters. The confusion arises because many popular articles conflate ignition, Q=1, and net electricity.

The Lawson criterion sets a physics target for nτ, but meeting it at high density and temperature produces a specific fusion power density. The neutron wall loading (MW/m²) is proportional to n²·T·R (where R is the plasma radius). To satisfy the Lawson criterion, one can choose a low n (large τ) or high n (small τ). However, neutron wall loading is limited by materials to ~2-5 MW/m² for steady-state operation. If n is too high, the wall loading exceeds this limit, shortening the first-wall lifetime. Thus, the reactor design must find a sweet spot: achieve the required nτ without exceeding the wall loading constraint. This often leads to the choice of large, moderate-density reactors (like ITER) over compact high-density designs.

In ICF, the confinement time is the disassembly time of the compressed fuel pellet (~ns to ps), while the density is extremely high (~10³¹ m⁻³, i.e., solid density). The Lawson criterion nτ is massively exceeded (nτ ~ 10²⁵ s/m³), but the plasma temperature is transient. The 'Lawson criterion' for ICF is often expressed in terms of the areal density ρR (where ρ is mass density and R is radius), and the ignition condition is ρR > 0.3 g/cm² for D-T. This is analogous to nτ because ρR ∝ n·R, and τ ∝ R/v_sound, so nτ ∝ ρR. Thus, it's a generalized form, but the numerical threshold differs due to the different physics of compression and burn. The NIF achieved ignition with ρR ~ 0.4 g/cm², corresponding to an effective nτ several orders of magnitude above the magnetic confinement value, but the energy accounting is fundamentally different.

The product nτ represents the total number of confinement intervals per particle—essentially the integrated density over the confinement time. For a given reaction rate, the fusion yield is proportional to the number of collisions per particle during its lifetime. Since fusion reactions require two particles to meet, the number of reactions per unit volume is proportional to n² times the collision rate. The loss term is proportional to n/τ. Balancing these gives nτ as the critical parameter. Physically, nτ is the 'exposure' of the plasma: how many reactive encounters a typical particle experiences before it leaves the plasma. A high nτ means each particle has many opportunities to fuse before being lost.

The Lawson criterion assumes a thermal plasma with a Maxwellian distribution. In practice, auxiliary heating produces a non-Maxwellian fast ion tail, which enhances the fusion reaction rate above the thermal rate (since the reactivity ⟨σv⟩ increases for high-energy ions). This means the effective fusion power for a given n and T is higher than predicted by the thermal criterion. This can reduce the required nτ for a given Q by 10-20%. However, fast ions also have shorter confinement times and can drive instabilities (Alfvén eigenmodes), which may worsen thermal confinement. For accurate projections, fusion codes use the fast-ion distribution function, not the simple Maxwellian, to calculate the effective reactivity.

The traditional Lawson criterion assumes a steady-state plasma with constant n and T. In actual experiments, the plasma evolves over time (ramp-up, flat-top, ramp-down). During the transient phase, the fusion rate is lower, and the heat losses are not balanced. The 'time-dependent Lawson criterion' requires integrating the power balance over time, which is more complex and leads to a condition such that the total fusion energy produced exceeds the total input energy over the entire pulse. For long-pulse or steady-state devices like ITER, the steady-state criterion is sufficient. However, for short-pulse devices (e.g., some stellarators with limited pulse length), achieving ignition requires reaching the steady-state threshold quickly, which is more difficult because the temperature and confinement time are evolving.

The original Lawson criterion, nτ > 10²⁰, implicitly assumes a fixed optimal temperature (~10 keV) for D-T. However, since the reaction rate ⟨σv⟩ is strongly temperature-dependent, the criterion is more accurately stated as a function of T: nτ > f(T), where f(T) = 12 k T / (⟨σv⟩ E) (plus radiation terms). The function f(T) has a minimum around T=10-15 keV. The triple product n·T·τ_E was introduced to separate the variables and create a single, device-independent figure of merit. The ignition condition is then n T τ > 3×10²¹ m⁻³·keV·s. This triple product is not a different criterion; it's the same Lawson criterion with the temperature explicitly included and evaluated at the optimal T. It is preferred because it's independent of assumptions about the specific T.

Real plasmas have radial profiles; the core is hotter and denser than the edge. The simple criterion uses volume-averaged n and T, which underestimates the fusion yield because the fusion power scales as n²⟨σv⟩, which is non-linear. A peaked profile (high core n, T) gives a higher fusion power than a flat profile with the same average n and T. To account for this, physicists use the volume-integrated fusion power and volume-integrated loss. The effective nτ required is lower if the profile is peaked because the reaction rate is enhanced in the core. Advanced scenarios with high core peaking can achieve ignition with a lower global nτ than the simple criterion. The Lawson criterion should be applied to the core values for accuracy, but in practice, it's used as a global figure of merit for comparing machines.