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Lawson Criterion (Triple Product)

\[n\,T\,\tau_E \;\gtrsim\; 3\times10^{21}\ \text{keV}\cdot\text{s}\cdot\text{m}^{-3}\]

Variables

Symbol Meaning Units
\(n\) fuel ion density m⁻³
\(T\) ion temperature keV
\(\tau_E\) energy confinement time s

Assumptions

  • Deuterium–tritium fuel in a 50:50 mix
  • Temperature in the 10–20 keV window, where \(\langle\sigma v\rangle \propto T^2\)
  • Ignition (alpha self-heating alone balances losses), not merely breakeven
  • Losses characterised by a single \(\tau_E\); bremsstrahlung folded into the threshold

Derivation sketch

Alpha heating per unit volume \(P_\alpha = \tfrac14 n^2\langle\sigma v\rangle E_\alpha\) against losses \(P_{\rm loss} = 3nT/\tau_E\). Setting \(P_\alpha \ge P_{\rm loss}\) gives

\[n\tau_E \ \ge\ \frac{12\,T}{\langle\sigma v\rangle E_\alpha}\]

Because \(\langle\sigma v\rangle \propto T^2\) near the operating point, the right-hand side goes as \(1/T\); multiplying both sides by \(T\) leaves a nearly temperature-independent threshold. That is why the triple product, rather than \(n\tau_E\), is the quoted figure of merit.

Notes

The same threshold is met by wildly different machines: magnetic confinement at \(n \sim 10^{20}\) m⁻³ with \(\tau_E \sim\) seconds, inertial confinement at \(n \sim 10^{30}\) m⁻³ with \(\tau_E \sim 10^{-10}\) s. Ten orders of magnitude traded each way.