Banach fixed-point between SEM image and EBSD diffraction pattern from a cylindrically symmetric rotating crystal

TL;DR

This paper demonstrates a novel mathematical relationship between SEM images and EBSD Kikuchi patterns in a cylindrically symmetric rotating crystal, illustrating a fixed-point theorem application in electron diffraction analysis.

Contribution

It reveals a new fixed-point relationship between SEM images and EBSD Kikuchi patterns in a specific crystal symmetry, bridging microscopy and mathematical fixed-point theory.

Findings

Identified a fixed-point relationship between SEM images and EBSD Kikuchi patterns.

Observed Kikuchi patterns from a cylindrically symmetric rotating crystal.

Established a mathematical connection explained by Banach fixed-point theorem.

Abstract

The Kikuchi bands arise from Bragg diffraction of incoherent electrons scattered within a crystalline specimen and can be observed in both the transmission and reflection modes of scanning electron microscopy (SEM). Converging, rocking, or grazing incidence beams must be used to generate divergent electron sources to obtain the Kikuchi pattern. This paper report the observation of Kikuchi pattern from SEM images of an exceptional rotating crystal with continuous rotation in the local crystal direction and satisfying cylindrical symmetry, named a cylindrically symmetric rotating crystal. SEM images of cylindrically symmetric rotating crystals reflect the interactions between electrons and the sample in both the real- and momentum-space. Furthermore, we identify an unexpected mathematical relationship between the electron backscattered diffraction (EBSD) Kikuchi pattern matrix map and the SEM image of the present sample which can be rationalized as a concrete example of the Banach fixed-point theorems in the field of EBSD technique.