Nuclear Reaction Rates in a Plasma

The problem of determining the effects of the surrounding plasma on nuclear reaction rates in stars is formulated ab initio, using the techniques of quantum statistical mechanics. Subject to the condition that the nuclear reactions ensue only at very close approach of the fusing ions and the condition that the reaction be slow, the authors derive a result that expresses the complete effects of Coulomb barrier penetration and of the influence of the surrounding plasma in terms of matrix elements of well defined operators. The corrections do not separate into the product of initial state and final state effects. When the energy release in the reaction is much greater than thermal energies, the corrections reduce, as expected, to evaluation of the equilibrium probability of one ion being very near to the position of another ion. We address the calculation of this probability in an approach that is based on perturbation theory in the couplings of the plasma particles to the two fusing particles, with the Coulomb force between the fusing particles treated nonperturbatively and interactions among the plasma particles treated in the one-loop approximation. We recapture standard screening effects, find a correction term that depends on the quantum mechanical nature of the plasma, and put an upper bound on the magnitude of the further correction terms for the case of a weakly coupled plasma. We find that possible “dynamical screening” effects that have been discussed in the literature are absent. The form of our results suggests that an approach that relies on numerical calculations of the correlation functions in a classical Coulomb gas, followed by construction of an effective two body potential and a quantum barrier penetration calculation, will miss physics that is as important as the physics that it includes. 2