Rainbows with a tilted 〈111〉 Si very thin crystal

Abstract

The angular distributions of protons channeled in a tilted (111) Si very thin crystal together with the corresponding rainbow patterns are considered. The proton energy is 10 MeV while the crystal thickness is 121.1 nm, corresponding to the reduced crystal thickness of 0.15. The tilt angle of the crystal is varied from zero up to the critical angle for channeling. The angular distributions were obtained using the numerical solution of the proton equations of motion and the computer simulation method. The rainbow lines were determined by applying the theory of crystal rainbows. The analysis has shown that the evolution of the angular distribution with the tilt angle can be fully explained by the evolution of the crystal rainbow. We have discovered that the crystal rainbow effect turns into the well-know doughnut effect in ion channeling for the larger values of the tilt angle. The interior of the doughnut is the analogue of the Alexanders dark band, which occurs in between the primary and secondary meteorological rainbows.