The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of energy generated by fusion reactions within fusion fuel to the rate of energy losses from the plasma environment. When the rate of energy production exceeds the rate of energy loss, the fusion system can produce net energy. If enough of this energy is retained within the plasma to sustain further reactions, the plasma reaches ignition.^[1]

The concept was first developed by John D. Lawson in a classified 1955 report at the Atomic Energy Research Establishment in Harwell, United Kingdom.^[2] The work was later declassified and formally published in 1957.^[1]

As originally formulated, the Lawson criterion defines a minimum value for the product of plasma density and energy confinement time required to obtain net fusion power. Later refinements introduced the more useful fusion triple product, which combines plasma density, temperature, and confinement time into a single performance measure.^[1]

On 8 August 2021, researchers at the National Ignition Facility at Lawrence Livermore National Laboratory announced an inertial confinement fusion experiment that exceeded the Lawson criterion for ignition conditions.^[3]^[4]

Lawson criterion and triple-product performance of major magnetic confinement fusion experiments.

Energy balance

The Lawson criterion is based on the energy balance of a fusion plasma. A fusion reactor can only produce net power when the fusion heating exceeds all energy losses.

Net power = Efficiency × (Fusion − Radiation loss − Conduction loss)

Fusion is the energy generated by nuclear fusion reactions.
Radiation loss is the energy lost through electromagnetic radiation such as X-rays.
Conduction loss is the energy carried away by escaping plasma particles.
Efficiency measures how effectively the reactor converts fusion energy into useful power.

Lawson estimated the fusion power using a thermalized plasma obeying a Maxwell–Boltzmann distribution.^[5]

The volumetric fusion reaction rate is approximately:

$Fusion rate = n_{A} n_{B} ⟨ σ v ⟩ E$

where:

$n_{A}$ and $n_{B}$ are the number densities of the fusion fuels,
$σ$ is the fusion cross section,
$v$ is the relative particle velocity,
$E$ is the fusion energy released per reaction.

Lawson also estimated radiation losses using the bremsstrahlung relation:

P_B = 1.4 \times 10^{-34} N^2 T^{1/2} \frac{\mathrm{W}}{\mathrm{cm}^3}

where $N$ is the plasma number density and $T$ is the plasma temperature.^[1]

By balancing fusion power against radiation losses, Lawson estimated minimum ignition temperatures of approximately:

30 million K (2.6 keV) for the deuterium–tritium reaction,
150 million K (12.9 keV) for the deuterium–deuterium reaction.^[1]^[6]

Confinement time

The energy confinement time $τ_{E}$ measures how long a plasma retains its energy before losses dominate.

\tau_E = \frac{W}{P_{\mathrm{loss}}}

where:

$W$ is the plasma energy density,
$P_{l o s s}$ is the power loss density.

For a steady-state fusion reactor, fusion heating must at least balance energy losses:

f E_{\rm ch} \ge P_{\rm loss}

For a 50–50 deuterium–tritium plasma, the Lawson criterion becomes:

n\tau_E \ge \frac{12T}{E_{\rm ch}\langle\sigma v\rangle}

For the D–T reaction, the minimum required value is approximately:

n\tau_E \ge 1.5 \times 10^{20}\ \frac{\mathrm{s}}{\mathrm{m}^3}

The minimum occurs near a plasma temperature of approximately 26 keV.^[1]

Triple product

A more useful performance parameter is the fusion triple product:

nT\tau_E

This combines plasma density, temperature, and confinement time. For the D–T reaction, the minimum required triple product is approximately:

nT\tau_E \ge 3 \times 10^{21}\ \mathrm{keV\ s\ m^{-3}}

The optimum temperature for this condition is near 14 keV.^[7]

Major fusion experiments such as JT-60, TFTR, JET, and ITER are often evaluated using the triple-product diagram shown above.^[8]

Inertial confinement

The Lawson criterion also applies to magnetic confinement fusion and inertial confinement fusion. In inertial confinement systems, confinement time is determined by the expansion time of the compressed fuel pellet.

For inertial confinement fusion, the criterion is often written as:

\rho R \ge 1\ \mathrm{g/cm^2}

where:

$ρ$ is the fuel density,
$R$ is the fuel radius.

Achieving this condition generally requires extremely high compression ratios produced by powerful laser systems.^[9]

Non-thermal systems

The Lawson criterion was originally derived for thermalized plasmas, but some fusion concepts use non-thermal particle distributions. Examples include the fusor, polywell, and migma concepts.

In such systems, energy losses from radiation and particle conduction remain major obstacles to net energy production.^[10]