Morphological study of the rainbow scattering of protons by graphene

Abstract

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.

Describing the scattering of keV protons through graphene

“…In that model, the charge of the C nucleus is spread over a volume , replacing the Coulombic singularity at r = 0 by a Gaussian. Its inflection point produces a second outer rainbow, which was predicted to be sensitive to temperature and the orientation of the membrane [ Ćosić et al (2018) ; Ćosić et al (2019) ; Ćosić et al (2021) ; Hadžijojić et al (2021) ]. However, it also entails that protons are transmitted during head-on collisions.…”

“…In this context transmission of protons at 5 keV through graphene has been proposed as an analytic tool [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ; Ćosić et al (2021) ]. Describing the interaction of protons with the membrane via a thermally averaged potential, two rainbow features are predicted: an “inner” and an “outer” one.…”

“…The outer one stems from a maximum in the deflection function close to a carbon atom, leading to a signal at around 170 mrad [ Ćosić et al (2018) ]. The authors argue that in combination these give very detailed insights into the internal temperature [ Ćosić et al (2019) ] and the interaction potential [ Ćosić et al (2021) ], the defect concentration [ Hadžijojić et al (2021) ] as well as the membrane’s orientation and inclination [ Ćosić et al (2018) ; Ćosić et al (2021) ]. However, there are several conceptual issues with the underlying theoretical description.…”

“…Additionally, we account for the membrane’s corrugation as well as the experimental resolution and quantify their impact on the pattern at the detector. Based on this, we evaluate the predictions from the literature [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ; Ćosić et al (2021) ] and test whether classical scattering through membranes can be used as an analytic tool. In our analysis we find that the predicted outer rainbow is an artifact from an improper description of the scattering process.…”

The Morphological Analysis of the Collagen Fiber Straightness in the Healthy Uninvolved Human Colon Mucosa Away From the Cancer

“…A similar approach proved its usefulness in the scattering theory where the existence of the rainbow effect forces regular, abrupt metamorphoses (or in this case catastrophic changes [12]) of the angular distribution. The shape analysis was at the core of the newly developed morphological method [13], which was being used for the determination of the proton-graphene interaction potential and characterization of graphene samples [13,14].…”