The Lawson Criterion

The condition on density and confinement time under which fusion self-heating balances plasma energy losses.

The original idea

In 1955 John Lawson asked what it takes for a fusion plasma to produce more energy than it loses. Balancing fusion power against radiation and transport losses, he derived a threshold on the product of density n and energy confinement time tau_E, at a given temperature.

n tau_E >= 12 T / ( <sigma v> E_ch )  (for ignition by charged fusion products)

Here is the temperature-averaged fusion reactivity and E_ch is the energy carried by charged products (the alpha particle in D-T, since the neutron escapes). The right-hand side has a minimum near 25 to 30 keV for D-T, which is why that temperature range is targeted.

Break-even versus ignition

Lawson's original criterion is a break-even balance; the ignition criterion is stricter, requiring alpha heating alone to sustain the temperature against all losses with no external heating. Both are usefully summarized by the triple product.

How it is evaluated numerically

• Compute from tabulated reactivity fits (for example the Bosch-Hale parameterization)
• Estimate tau_E from confinement scaling laws or transport simulation
• Combine with density limits (Greenwald) and beta limits (Troyon) to define the operating window

In practice the Lawson balance is embedded in a full power-balance calculation that tracks alpha heating, auxiliary heating, bremsstrahlung, and conducted losses self-consistently.

Reading it honestly

The criterion is a necessary condition, not a guarantee of net electrical output, which also depends on capture efficiency and plant systems. Kronos machines are design and simulation studies; no hardware net-gain is claimed before first-of-a-kind operation. The Hyperion breeder targets a plasma gain Q of 3.424 in its design point, an energy-multiplying but not yet ignited regime.