\textbf{h,} Calculated projected unit cell volume from individual strain patterns.}
\end{figure*}
\subsection{\label{ssec:electron_diffraction}Generating Combined Electron Diffraction Patterns}
One of the most materials science techniques 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.
Theoretically calculated electron diffraction for the Platinum-Cobalt alloy, with an electron accelerating voltage of 200kV.
\textbf{b,} The normalized probability of locating an electron diffraction spot for BOL nanoparticles (in maroon), and EOL nanoparticles (in dark green).
Theoretically calculated electron diffraction for the Platinum-Cobalt alloy, at an electron accelerating voltage of 200kV - similar to experimental conditions.
\textbf{b,} The normalized probability of locating an electron diffraction spot for BOL nanoparticles (in brick-red), and EOL nanoparticles (in purple).
\textbf{c,} The probability of locating electron diffraction spots for BOL nanoparticles, as a function of the \textit{d-spacing} and the distance from the edge of the nanoparticle.
\textbf{d,} The probability of locating electron diffraction spots for EOL nanoparticles, as a function of the \textit{d-spacing} and the distance from the edge of the nanoparticle.}
\end{figure*}
@@ -170,12 +173,12 @@
The \href{https://doi.org/10.17188/1189002}{platinum}, \href{https://doi.org/10.17188/1186809}{cobalt} and \href{https://doi.org/10.17188/1313278}{platinum-cobalt alloy} crystal structure CIF files were obtained from the Materials Project \cite{materials_project}.
\section*{Author Contributions}
D.M., D.A.C. and M.J.Z. designed the study. C.W. and J.S. grew the nanoparticles.
D.M., M.J.Z. and D.A.C. designed the study.
D.M. performed the 4D-STEM strain quantification, developed the analysis algorithms, performed the data analysis, wrote the Python codes, created the figures and wrote the manuscript, with contributions from other authors.
C.W. and J.S. grew the nanoparticles.
M.J.Z. performed the 4D-STEM experiments.
D.M. developed the 4D-STEM strain directed the data analysis, developed the quantification algorithms, analyzed the 4D-STEM data and wrote the manuscript, with contributions from other authors.
H.Y. and D.A.C. developed the particle picking algorithms.
J.D.H. developed the iterative post-specimen aberration corrector and wrote the Python codes for the same.
D.A.C. and M.J.Z. supervised the project.
H.Y. and D.A.C., along with D.M. developed the particle picking algorithms.
J.D.H. and D.M. developed the iterative post-specimen aberration corrector and wrote the Python codes for the same.
All authors discussed the results and commented on the manuscript.
\section*{Competing Interests}
@@ -317,9 +320,23 @@
\end{figure}
Fitting
\section{Generating corrected strain maps}
To generate the corrected strain maps, after the aberration correction, we also need to get corrected median spots.
This is because our strain is calculated with respected to the median diffraction position in each nanoparticle.
Thus, first we apply our aberration correction parameters to the median positions.
Following that, we compare these positions with the peaks in the theoretical diffraction data (\autoref{fig:medianspots}\blu{a}), and observe that the pattern has multiple peaks.
We apply a peak finding algorithm, and choose the most prominent peaks, as shown by the blue dots in \autoref{fig:medianspots}\blu{a}.
These are our now our reference median positions for their radial distances.
The median positions that we obtain from the data are in Cartesian co-ordinates, which we first convert to polar co-ordinates, and replace the radial value of the polar co-ordinate with the nearest theoretical peak.
We then convert the polar co-ordinates back to Cartesian co-ordinates, and use these corrected median positions to calculate the strain maps.
The advantage of this approach is that now our median positions are \textit{consistent} across all the datasets, and thus we can compare disparate nanoparticles' strains together. The corrected and the uncorrected median positions for the BOL and EOL nanoparticles are shown in \autoref{fig:medianspots}\blu{b} and \autoref{fig:medianspots}\blu{c} respectively.
While, calculating these corrected median spots, we also update our calibration parameters. The original microscope generated calibration parameters for the reciprocal space was $\mathrm{0.128nm^{-1}/pixel}$. We modified our calibration values such that the ratio between the corrected and uncorrected median spots is as close to unity as possible. This gave a new calibration parameter of $\mathrm{0.12777nm^{-1}/pixel}$, which is approximately 0.2\% lower than the original value. We also compared the ratios of the corrected median positions for both the BOL and EOL data. Interestingly, the BOL median data is smaller than the new common median positions by 0.71\%, while the EOL data is larger by 0.43\%. 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.
\caption{\label{fig:probabilities}\textbf{Effects of corrections. a,} Uncorrected radial probability density (red). \textbf{b,} Corrected radial probability density (green).}
