Fusion Reactivity and Reaction Rate

The velocity-averaged product of cross section and speed that sets how fast a fuel mixture fuses.

From cross section to rate

The fusion reaction rate per unit volume between two species is the product of their densities and the reactivity, the average of cross section times relative velocity over the velocity distribution:

R = n_1 n_2 <sigma v>
<sigma v> = integral of sigma(v) v f(v) dv

For identical reactants a factor of one half avoids double counting. The reactivity depends only on temperature (through the distribution), so it is tabulated as a function of T for each reaction.

Temperature dependence

Fusion requires tunneling through the Coulomb barrier, so only the fast tail of the distribution contributes, at the Gamow peak. This makes rise very steeply with temperature at first, then flatten and eventually fall. D-T peaks near 65 keV but is already large at 10 to 20 keV, which is why D-T is the easiest fuel.

Comparing fuels

• D-T: largest reactivity, peaks around 65 keV, releases 17.6 MeV (most in the neutron)
• D-D: about a hundred times smaller reactivity, two branches, produces tritium and helium-3
• D-3He: aneutronic principal reaction, requires much higher temperature than D-T

How it is used

Reactivities are computed once from validated cross-section fits (Bosch-Hale) and then used as functions of local temperature in power-balance and transport codes to get the fusion power density n^2 E_fusion / 4 (for equal densities).

Kronos fuels

The Hyperion breeder uses D-T, the highest-reactivity fuel, and targets 88.7 MW of fusion power at a plasma gain of 3.424 in its design study. The Kronos burner uses D-3He, whose lower, higher-temperature reactivity and small neutron fraction (5.44 percent) shape its very different operating point. All Kronos machines are design and simulation studies.