On a Strong Increase in the Amplitude of the Wave Function of a Massive Nonrelativistic Particle Incident on a Crystal (One-Dimensional Approximation)

Abstract—

An increase in the amplitude of the wave function of a massive nonrelativistic particle incident on a one-dimensional crystal is studied. Similarly to the case of light propagation in a one-dimensional periodic medium, a perturbation theory is constructed for small deviations in the energy of the incident particle from the boundary of the band gap. Formulas are derived for the wave function of the particle in the crystal and the reflection and transmission coefficients. The general features and differences between the initial equations, the obtained characteristics for the massive particle and the properties of the light field are analyzed in detail. It is found that the main properties of the wave function have the same features as the properties of the light field. Numerical estimates are given for the increase in the amplitude of the wave function of the particle inside the one-dimensional periodic medium with a period that is equal to the lattice constant of palladium. It is shown that, as the energy of the incident particle decreases, the amplitude of the wave function increases significantly, which correlates with the experimentally observed increase in the yield of D–D reactions for low-energy particles compared with the values obtained by extrapolating data from the high-energy region.