Superfocusing and zero-degree focusing in planar channeling of protons in a thin silicon crystal

Abstract

This is a theoretical study of the superfocusing and zero-degree focusing effects appearing in channeling of protons of energy of 2 MeV in the {1 1 0} channel of a thin silicon crystal. We prove that the ultimate origin of these effects is the crystal rainbow effect, which has been discovered and explored in ion transmission through axial crystal channels, nanotubes and graphene. Moreover, the effect is the origin of the shapes of the whole spatial and angular distributions of channeled protons. The incident proton velocity vector is taken to be parallel to the channel midplane. We use the Molière's approximation of the Thomas-Fermi proton-atom interaction potential and the continuum approximation. The effect of thermal vibrations of the crystal's atoms is included. We solve numerically the proton equation of motion and analyze the spatial and angular proton transmission functions. The extrema of these functions are the spatial and angular rainbow points, respectively. When the whole proton beam is taken into account, a spatial rainbow pattern, composed of the lines emerging from the superfocusing points, and an angular rainbow pattern, consisting of the lines emerging from the origin and the zero-degree focusing points, appear along the channel. We consider the region of crystal thickness comprising the first, second and third superfocusing and zero-degree focusing points. When the effect of thermal vibrations is neglected, each of these rainbow lines look like the bifurcation set of the cusp catastrophe, and when the effect is included, the line resembles the bifurcation set of the butterfly catastrophe. We demonstrate that the spatial and angular distributions of transmitted protons are fully determined by the spatial and angular rainbow lines, respectively. The superfocusing and zero-degree focusing effects weaken with the increase of the crystal thickness. This is a consequence of the anharmonicity of the continuum proton-crystal interaction potential, which makes the proton beam propagation through the channel incoherent. We also explore the influence of the effect of proton collisions with the crystal's electrons on the beam dynamics in the channel. This effect additionally contributes to the incoherence of the beam motion along the channel. Our opinion is that the peaks of the spatial and angular distributions of ions transmitted through the planar channels lying off the origin registered before were in fact the rainbow peaks.