Morphological Analysis of the Rainbow Patterns Created by Point Defects of Graphene

Abstract

We have investigated the transmission of the 5 keV proton beam through a graphene sheet containing monovacancy, adatom, and Stone–Wales defects. The proton–graphene interaction potential was constructed using the Doyle–Turner's proton–carbon interaction potential. The closed form of the scattering law was obtained using the momentum approximation. Angular distributions of the transmitted protons were analyzed using the morphological method based on the inspection of the rainbow patterns in the impact parameter and scattering angle planes generated by the rainbow scattering. We have demonstrated that rainbows in the impact parameter plane are attracted and repelled by the nearest saddles and maxima of the reduced proton–graphene interaction potential. This explains why the rainbow pattern is so sensitive to the redistribution of the potential extrema caused by defects. Each defect type produces its distinctive rainbow pattern that dominantly determines the shape of the angular distribution. The ridge maxima of the angular distributions were investigated and related to the spectrum of the Jacobian matrix of the map generated by the scattering law. In the end, it has been shown how observed rainbow patterns could be used to determine the unknown defect densities of the complicated sample containing a combination of the different defect types.

Describing the scattering of keV protons through graphene

“…To capture the behavior described previously in the literature [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ], we consider the transmission of protons with a kinetic energy of 5 keV through single-layer graphene. These particles propagate along the z -direction with velocity v z until they interact with the 2D membrane where they are scattered, as shown in Figure 1 .…”

“…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].…”