When extended across the entire field of view, this quantification thus tracks the lattice parameter variations, aka strain, with the precision that cannot be beaten by conventional STEM imaging.
Indeed, 4D-STEM has been used for picometer precision strain quantification in two-dimensional crystals, semiconductor heterojunctions, and catalyst nanoparticles \cite{nbed_strain1, nbed_strain2, 4dstem_nanoparticles, yimo_strain, holo_strain, strain_graphene}.
Most notably, it has been demonstrated that the errors in 4D-STEM measurement are significantly lower than the errors from even drift-corrected annular dark field (ADF) STEM imaging, even when looking at the same particle\cite{4dstem_nanoparticles}.
In addition to strain, analysis of diffraction disks from 4D--STEM datasets has proven to be a powerful method for quantifying the twist angles and interlayer distances between twisted bilayer two-dimensional Moir\'e systems\cite{twisted_graphene, michael_twisted}.
In addition to strain, analysis of diffraction disks from 4D-STEM datasets has proven to be a powerful method for quantifying the twist angles and interlayer distances between twisted bilayer two-dimensional Moir\'e systems\cite{twisted_graphene, michael_twisted}.
However, the second issue for electron microscopy-based lattice quantification is present in both conventional STEM and 4D-STEM imaging.
This is because lattice parameter quantification can be performed when the region is imaged is oriented along a low-index crystallographic zone axis, with the problem being absent only for two-dimensional crystals \cite{4dstem_nanoparticles, yimo_strain, holo_strain,disk_registration, nbed_strain2}.
Thus, almost every electron microscopy-based lattice parameter variation studies report results from a single particle.
Important questions about the statistics of surface strain, the effect of particle size on surface strain have thus remained unanswered.
Two recent works have proposed methods to break this logjam -- strain measurements from multiple zone axes, and using cepstrum functions to process CBED datasets, and both of them have shown promising results\cite{elliot_strain,strain_tensor}.
Two recent works have proposed methods to break this logjam --- strain measurements from multiple zone axes, and using cepstrum functions to process CBED datasets, and both of them have shown promising results\cite{elliot_strain,strain_tensor}.
Yet another recent work has demonstrated that using unsupervised approaches, complicated microstructures can be reconstructed and identified from each other from 4D-STEM datasets\cite{many_4dstem}.
A very recent preprint demonstrated that 4D-STEM, among all other techniques can break this ``statistical logjam'' - the inability to quantify lattice parameter variations with high spatial precision from multiple particles - and can be used to quantify the lattice parameter variations from multiple particles\cite{elliot_strain_many}.
Precession electron diffraction has also been applied to 4D-STEM datasets to perform orientation mapping\cite{4dstem_precession}.
@@ -241,13 +241,11 @@
In this work, we pursue a slightly different workflow, that is also computationally less intensive. We first collect all the observed diffraction spots and then correct for the post-specimen projector lens aberrations (detailed in \autoref{ssec:aberration_correction}).
Once corrected, we use theoretically calculated powder diffraction peaks from simulations of PtCo alloys (detailed in \autoref{ssec:correcting_to_known}) to assign zone axes to the observed diffraction spots.
This allows us to calculate the strain in each nanoparticle with respect to a common reference, and thus compare the strain across multiple nanoparticles, and also observe the contribution of individual zone axes to strain evolution from the particle surface to the particle core.
\subsection{\label{ssec:aberration_correction}Correction of lens aberrations}
\subsection{\label{ssec:correcting_to_known}Reference Correction to known diffraction patterns}
\subsection{\label{ssec:aberration_correction}Common Referencing to Quantify Strains Across Particles}
\caption{\label{fig:montage}\textbf{Volumetric Strain across all nanoparticles. a,}
The theoretically calculated diffraction pattern for PtCo alloy, with the peak positions and the corresponding Miller indices in red.
\textbf{b,} Original median spots for \textit{beginning-of-life (BOL)} nanoparticles in red, with the corrected spots shown in blue.
\textbf{c,} NOriginal median spots for \textit{end-of-life (EOL)} nanoparticles in red, with the corrected spots shown in blue.}
@@ -272,6 +270,14 @@
However, this is in inverse space - which means the BOL particles are slightly more strained than the EOL nanoparticles, something that is borne out by our other observations too.
\subsection{\label{ssec:montage_strain}Volumetric Strain across all nanoparticles}
\caption{\label{fig:diff_correction}\textbf{Correcting median spots to known PtCo pattern. a,}
The theoretically calculated diffraction pattern for PtCo alloy, with the peak positions and the corresponding Miller indices in red.
\textbf{b,} Original median spots for \textit{beginning-of-life (BOL)} nanoparticles in red, with the corrected spots shown in blue.
\textbf{c,} NOriginal median spots for \textit{end-of-life (EOL)} nanoparticles in red, with the corrected spots shown in blue.}
\end{figure*}
\subsection{\label{ssec:electron_diffraction}Generating Combined Electron Diffraction Patterns}
One of the most widely used approaches in materials science for crystal structure determination is powder diffraction. In powder diffraction, a polycrystalline sample is irradiated commonly with X-rays, and since the sample is polycrystalline, all possible allowed Bragg planes diffract. The resulting pattern, also called the Debye-Scherrer pattern, gives a series of peaks corresponding to the allowed Bragg planes. The peak positions are then used to calculate the lattice parameters of the sample.
@@ -496,4 +502,10 @@
Uncorrected diffraction spot positions in polar-coordinates of all the nanoparticles. \textbf{b,} Radially corrected diffraction spot positions in polar-coordinates of all the nanoparticles. This correction is done purely for the angle, and not for the radial distance. \textbf{c,} Bragg-vector corrections of the nanoparticles of all the nanoparticles, where red corresponds to the uncorrected positions, and blue corresponds to the angular corrected positions.}
\caption{\label{fig:montage_correction}\textbf{Effect of Correction on Strain. a,} Montage of nanoparticles with the uncorrected volume strain maps. \textbf{b,} Montage of nanoparticles' volume strains with the projector lens aberrations corrected. \textbf{c,} Montage of nanoparticle strains' volume strains with both projector lens aberrations corrected and the median spots corrected to the theoretical diffraction peaks.}
