Lawson Criterion - Inertial Confinement

Inertial Confinement

The Lawson criterion applies to inertial confinement fusion as well as to magnetic confinement fusion but is more usefully expressed in a different form. Whereas the energy confinement time in a magnetic system is very difficult to predict or even to establish empirically, in an inertial system it must be on the order of the time it takes sound waves to travel across the plasma:

Following the above derivation of the limit on n_eτ_E, we see that the product of the density and the radius must be greater than a value related to the minimum of T3/2/<σv>. This condition is traditionally expressed in terms of the mass density ρ:

ρR > 1 g/cm²

To satisfy this criterion at the density of solid D-T (0.2 g/cm³) would require an implausibly large laser pulse energy. Assuming the energy required scales with the mass of the fusion plasma (E_laser ~ ρR3 ~ ρ-2), compressing the fuel to 103 or 104 times solid density would reduce the energy required by a factor of 106 or 108, bringing it into a realistic range. With a compression by 103, the compressed density will be 200 g/cm³, and the compressed radius can be as small as 0.05 mm. The radius of the fuel before compression would be 0.5 mm. The initial pellet will be perhaps twice as large since most of the mass will be ablated during the compression.

The fusion power density is a good figure of merit to determine the optimum temperature for magnetic confinement, but for inertial confinement the fractional burn-up of the fuel is probably more useful. The burn-up should be proportional to the specific reaction rate (n²<σv>) times the confinement time (which scales as T-1/2) divided by the particle density n:

burn-up fraction ~ n²<σv> T-1/2 / n ~ (nT) (<σv>/T3/2)

Thus the optimum temperature for inertial confinement fusion is that which maximizes <σv>/T3/2, which is slightly higher than the optimum temperature for magnetic confinement.