Resonant Nuclear Fusion at Second Order

We study the possibility of low energy nuclear fusion assisted by a low energy resonance. We use a simple potential model and replace repulsive Coulomb barrier with a step potential barrier. This is convenient since it allows us to perform analytic calculations and allows better control on approximations. The fusion process involves a proton and another nucleus. We first consider a process in which this transition takes place by emission of a single photon at first order in perturbation theory. At this order we find a very large cross section over a very narrow range of initial momenta, corresponding to the resonance width. The cross section rapidly drops to very small values as we move away from resonance. In any experimental situation, the rate for this process would be negligible since it is practically impossible to have a significant number of initial state particles at precisely the resonant energy. We next consider another process, which involves emission of two photons and gets dominant contribution at second order in perturbation theory. In this case the initial state energy is taken to be larger than the resonance energy. We find a significant cross section, provided the initial energy is larger than the resonant energy. The exponential suppression factors cancel out in this calculation. The experimental signature for this process is two photon emission within very narrow range of energies. One of the photon would be emitted at low energy of order of the initial state energy and the second would correspond to the nuclear energy.