A toy model for low-energy nuclear fusion

We study the fusion of a proton with a nucleus with the emission of two photons at low incident energy of the order of eV or smaller. We use a step model for the repulsive potential between the proton and the nuclei. We consider the reaction both in free space and inside a medium. We make a simple model for the medium by assuming a hard wall potential beyond a certain length scale. This essentially leads to discretisation of the energy spectrum which is expected inside a medium and is seen both for a crystalline lattice structure and for amorphous materials. We use second-order perturbation theory to compute the transition rate. We find that the rate in free space is very small. However, in the medium, the rate may be substantial. Hence, we conclude that nuclear fusion reactions may take place at low energies at observable rates.