Thermal averaging

Reaction rates 〈σv〉 and lifetimes

The reactions that we have discussed so far all have cross sections σ(E) that depend strongly on center-of-mass energy E. This dependence may be because of a repulsive Coulomb barrier, so that σ(E) = E-1 exp(-2πη)S(E) for an ‘astrophysical S-factor’ S(E) that is relatively less variable with energy. It may be because neutrons have a σ(E)∝ 1/v behavior for projectile-target relative velocity v near zero. Or itmaybe because of resonances giving sharply peaked cross sections, like Г/[(E - ER)2 + Г2/4] for a single resonance centered at ER with a full width at half maximum of Г.

In a stellar plasma of a mixture of two or more nuclear species, there will be a considerable range of relative energies E (or, velocities v) because of the statistical distribution of thermal energy among all the particles in the plasma. The actual rate of nuclear reactions will therefore require an averaging of the cross sections σ(E) over the thermal distribution of relative energies. We will therefore define an average reaction rate as the number of reactions per second per unit volume, and find an expression for this in terms of σ(E) and the distribution φ(v) of the relative velocities of the interacting particles.