Lawson criterion Totally Explained

Everything about Lawson Criterion totally explained

In nuclear fusion research, the Lawson criterion, first derived by John D. Lawson in 1955 and published in 1957, is an important general measure of a system that defines the conditions needed for a fusion reactor to reach ignition, that is, that the heating of the plasma by the products of the fusion reactions is sufficient to maintain the temperature of the plasma against all losses without external power input. As originally formulated the Lawson criterion gives a minimum required value for the product of the plasma (electron) density n_e and the "energy confinement time" τ_E. Later analyses suggested that a more useful figure of merit is the "triple product" of density, confinement time, and plasma temperature T. The triple product also has a minimum required value, and the name "Lawson criterion" often refers to this inequality.

The product $n_$

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 T^3/2/<σv>. This condition is traditionally expressed in terms of the mass density ρ:
ť 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:
ť burn-up fraction ~ n²<σv> T^-1/2 / n ~ (nT) (<σv>/T^3/2)

Thus the optimum temperature for inertial confinement fusion is that which maximizes <σv>/T^3/2, which is slightly higher than the optimum temperature for magnetic confinement.

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