The thermal motion of graphene atoms was investigated using angular distributions of transmitted protons. The static proton-graphene interaction potential was constructed applying the Doyle-Turner's expression for the proton-carbon interaction potential. The effects of atom thermal motion were incorporated by averaging the static proton-graphene interaction potential over the distribution of atom displacements. The covariance matrix of graphene displacements was modeled according to the Debye theory, and calculated using Molecular Dynamics approach. Proton trajectories were used for construction of angular yields. We have found that there are lines, called rainbows, along which the angular yield is very large. Their evolution in respect to different sample orientation was examined in detail. Further we found that atom thermal motion has negligible influence on rainbows generated by protons experiencing distant collisions with the carbon atoms forming the graphene hexagon. On the other hand, rainbows generated by protons experiencing close collisions with the carbon atoms can be modeled by ellipses whose parameters are very sensitive to the structure of the covariance matrix. Numerical procedure was developed for extraction of the covariance matrix from the corresponding rainbow patterns in the general case, when atoms perform fully anisotropic and correlated motion.

We have studied metamorphoses of the angular rainbow pattern generated by classical rainbow scattering of protons by graphene. To analyze the rainbow pattern, a morphological method was developed. It focuses on the shape of the rainbow pattern rather than on the exact position of rainbow lines or the particle count. It comprises elements of the catastrophe theory, which provides a local model of the rainbow pattern and the reduced potential and an index theory of algebraic topology that allows the evolution of the rainbow pattern to be linked with bifurcations of critical points of the reduced potential. The obtained insight is summarized into five principles that allow an experimentalist to sketch a qualitatively correct rainbow pattern in the impact parameter plane and the distribution of the reduced potential critical points, just by observing the evolution of the angular rainbows. The morphological method should be applicable for the analysis of all structurally stable patterns in nature.

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