We shall present result of investigation of free-standing graphene using angular patterns of transmitted non-neutralized 5-keV protons. The main purpose of this study is to show that structural stability of the rainbow scattering process (i.e. its robustness and high sensitivity) could be used as a tool for studying proton-graphene interaction.
To demonstrate sensitivity of the effect on the fine details of the interaction potential we have constructed the static proton-graphene interaction potential using: Doyle-Turner, ZBL, and Molière proton-carbon interaction potentials. The effect of the thermal vibrations was incorporated by averaging the static proton-graphene interaction potentials over the distribution of the atom thermal vibrations. Proton trajectories were obtained by numerical solution of the corresponding Newton’s equations of motion and used to construct the mapping of initial positions to corresponding scattering angles. Multiplicity and singularities (i.e. rainbow lines) of the introduced mapping were used to explain important features of calculated angular distributions, and to demonstrate significant difference in the shape of transmitted yields corresponding to the different assumed proton-carbon interaction potentials.
We have also investigated graphene thermal vibrations in detail. Covariance matrix of graphene atom displacements was modeled according the Debye theory, and calculated using Molecular Dynamic approach. The shape of transmitted angular yields and their evolution with the change of the tilt angle was explained by the corresponding evolution of introduced singularities. Rainbow lines formed by protons experiencing the close collisions with the carbon atoms were modeled by elliptical lines which parameters were find to be very sensitive to the structure of the covariance matrix. Moreover, a numerical procedure was developed which enables extraction of the covariance matrix from the corresponding rainbow patterns in the general case when carbon atoms perform fully anisotropic correlated motion.