A measure of the probability that a specific nuclear reaction will occur when two particles collide — quantified in units of barns and central to predicting fusion reaction rates.
In nuclear and particle physics, the cross-section (σ) is a quantity that characterizes the likelihood of a particular interaction between two particles. Conceptually, it can be thought of as the effective target area that one particle presents to another: a larger cross-section means a higher probability of reaction. Cross-sections are measured in barns (b), where 1 barn = 10−24 cm2 = 10−28 m2.[1]
The fusion cross-section is a strong function of the relative kinetic energy (or equivalently, temperature) of the colliding nuclei. At low energies it is vanishingly small because the Coulomb barrier suppresses tunnelling. As energy increases, the cross-section rises, typically reaching a broad peak before declining at very high energies.
For the D–T reaction, the cross-section peaks at a centre-of-mass energy of approximately 64 keV, reaching a maximum of about 5 barns — exceptionally large by nuclear standards. By contrast, the D–D cross-section peaks near 1 MeV at only about 0.1 barns, and the D–3He cross-section peaks around 200 keV at roughly 0.7 barns.[2]
In a thermal plasma, particles have a distribution of velocities described by a Maxwellian. The quantity that determines the volumetric fusion power is the reactivity, ⟨σv⟩, which is the cross-section averaged over the velocity distribution:
⟨σv⟩ = ∫ σ(v) · v · f(v) dv
where v is the relative velocity and f(v) is the Maxwellian distribution function. The fusion power density is then proportional to n1 n2⟨σv⟩, where n1 and n2 are the fuel-ion densities. For D–T, ⟨σv⟩ peaks near an ion temperature of 70 keV.[3]
Fusion cross-sections are measured experimentally using accelerator-based beam-target experiments and are among the most precisely known quantities in nuclear physics. Widely used parameterizations include the Bosch–Hale fits, which provide analytic expressions accurate to better than 1% over the energy ranges relevant to fusion reactors.[2]