Pattern Center and Distortion Determined from Faint, Diffuse Electron Diffraction Rings from Amorphous Materials

Abstract

Diffuse rings from amorphous materials sit on a steep background resulting in a monotonically decreasing intensity over scattering vector length, frequently with no clear local maximum that could be used to determine the center of the ring. The novelty of the method reported here is that it successful processes such weak patterns. It is based on separating the angular dependence of the positions of the maxima on the azimuthal angle in the measured two-dimensional pattern for a manually preselected peak. Both pattern center and elliptical distortion are simultaneously refined from this angular dependence. Both steps are based on nonlinear least square fitting, using the Levenberg-Marquardt method. It can be successfully applied to any amorphous patterns provided they were recorded with experimental conditions that facilitate dividing them into sectors with acceptable statistics. Patterns with the center shifted to the camera corner (recording a quadrant of a ring) can also be reliably evaluated, keeping precalibrated values of the elliptical distortion fixed during the fit. Finally, the limited number of counts in any pattern is overcome by cumulating many patterns (from equivalent areas) into a single pattern. Eliminating false effects is facilitated by masking out unwanted parts of any recorded pattern from processing.

Study of the Microstructure of Amorphous Silica Nanostructures Using High-Resolution Electron Microscopy, Electron Energy Loss Spectroscopy, X-ray Powder Diffraction, and Electron Pair Distribution Function

“…From the collected ED data (samples S1–S4), the electron pair distribution function (G( r )), which provides a measure of the probability of finding two atoms separated by distance r ( Figure 8 ), was calculated using the e-PDFSuite and the Process Diffraction software [ 34 , 35 , 36 , 52 ] developed to analyze the ED patterns of amorphous and nanocrystalline materials. During the calculation of the 1D distribution from the 2D electron diffraction patterns of silica materials, the contribution of amorphous carbon support was subtracted where the distortions in the 2D diffraction patterns were corrected.…”

“…The beam stopper area and the dead pixels in the detector were masked and eliminated from the ED pattern during e-PDF calculations. The detailed procedure of the G( r ) calculation from the 1D distribution data using the e-PDFSuite and Process Diffraction software program has been described elsewhere [ 34 , 35 , 36 ].…”

“…1.5 µm, and for optimal background subtraction after recording the ED patterns from the silica particles, an additional ED pattern was recorded from the empty carbon foil under identical beam conditions. Radial averaging of the ED patterns was performed using process diffraction and the e-PDFSuite software [ 33 , 34 , 35 , 36 ]. The ED intensities were integrated radially into one-dimensional (1D) profiles and the background intensity from the carbon foil was subtracted from that of the silica particles.…”

Strain Measurement in Single Crystals by 4D-ED

“…Since the measurement is relative (to that measured in an unstrained region) any small distortion of the diffraction pattern (that might be caused by the lenses of the TEM) can only cause second-order effect in measured strain. Even this second-order error can be corrected for with one of the methods published by Lábár [ 18 , 19 , 20 ]).…”

Acquisition and evaluation procedure to improve the accuracy ofSAED

“…Since residual elliptical elongation of the diffraction patterns due to imperfect optics and alignments is always present, even in aberration corrected microscopes, its correction is inevitable (Gorelik et al, 2019 ; Mahr et al, 2019 ; Saitoh et al, 2013 ). Procedure for determination of elliptical distortion parameters was developed by Capitani et al ( 2006 ) and similar procedures are implemented in most of diffraction processing softwares (e.g., Lábár et al, 2012 ; Lábár & Das, 2017 ; Gammer et al, 2010 , Mitchell, 2008 ; Carvalho & Morales, 2012 ; Li ( 2007 ); Zou et al, 1993 ; see also https://www.iucr.org/resources ) and also for evaluation of 4D STEM data (Savitzky et al, 2021 ). Note that higher order elliptical distortion may also be present and none of the above softwares treat them.…”