Resonances and thermonuclear reaction rates for charged particle collisions

Abstract

We derive an approximate analytic expression for thermonuclear reaction rate for charged particles for the case that the cross section contains a single narrow or broad resonance described by a Breit-Wigner shape. An asymptotic series is obtained by making a Taylor series expansion which is exact at both the effective energy of the Gamow window and at the resonance energy. The resulting expression is uniformly valid as the effective energy and resonance energy coalesce or separate and as the resonance width is varied from narrow to broad. This is not true of conventional analytic approximations. We compare our uniform approximation with the exact reaction rate numerically and find it to be accurate over a large parameter range. We use our expressions to calculate the reaction rate for $^{12}\mathrm{C}(p,\ensuremath{\gamma})^{13}\mathrm{N}$.