Minor changes

\usepackage[version=3]{mhchem}

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\usepackage[protrusion=true,expansion=true,final]{microtype}

		The peak positions are at $\mathrm{4.57nm^{-1}}$, $\mathrm{5.4nm^{-1}}$, $\mathrm{7.61nm^{-1}}$, $\mathrm{8.65nm^{-1}}$ and $\mathrm{11.4nm^{-1}}$. These are the peaks that will be used for the first round of corrections. To correct the aberrations, we scale the peaks with the radial peak values. The scaled peaks are shown in \autoref{fig:scaled_peaks}. This is performed by finding the nearest radial peak value (from \autoref{fig:bragg_peaks}) to the peak value and dividing the peak value by the radial peak value. As we observe in \autoref{fig:scaled_peaks}\blu{a}, the different diffraction peaks are marked by different colors, with each color corresponding to one of the peak positions in \autoref{fig:bragg_peaks}, with the scaled peaks shown in \autoref{fig:scaled_peaks}\blu{b}, where each of the diffraction peak locations scaled to the nearest peak position. As we observe in \autoref{fig:scaled_peaks}\blu{b}, the peaks are now more consistent with respect to the angle, and thus it can be understood that the aberration function is a function of radial distance.

		But there is still a residual waviness in the data in \autoref{fig:scaled_peaks}\blu{b}, which we have to correct. One way to express the aberrations is through \autoref{eq:ps_aberrations}.

		But there is still a residual waviness in the data in \autoref{fig:scaled_peaks}\blu{b}, which we have to correct. If we express these aberrations purely as a function of overlapping sine waves, as expressed by \autoref{eq:angular_aberrations}, where $a_i$ represents the amplitude of the $\mathrm{i^{th}}$ sine wave, and $\theta_i$ represents the phase offset of the $\mathrm{i^{th}}$ sine wave.

		\begin{equation}\label{eq:angular_aberrations}

			\mathcal{A}_{\theta} = \sum_{i=1}^{n}a_i \sin \left(\theta + \theta_i\right)

		\end{equation}

		Plotting the effects of these sine waves in \autoref{fig:scaled_peaks_aberrations}, we observe that the increasing the order sine waves capture the waviness more and more accurately, with a fifth order corrector being remarkably accurate in capturing the features. However, it is to be noted that there is no radial correction for aberrations here and this correction is purely angular. Yet we have noticed how scaling the peaks by the nearest Bragg peak height (as shown in \autoref{fig:scaled_peaks}) remarkably reduces the aberrations -- thus there is a radially increasing component to the aberrations too.

		\begin{figure}[h]

			\centering

			\includegraphics[width=0.6\textwidth]{GeneratedFigures/Scaled_Peaks_Aberrations.pdf}

			\caption{\label{fig:scaled_peaks_aberrations}\textbf{Scaled Polar Peaks (in purple) with overlaid aberration order (in green)}}

		\end{figure}

		To express the radial nature of the aberrations we thus modify our equation slightly in \autoref{eq:ang_rad_aberrations}, the amplitude is modified from $a_i$ to $r \times ^{a_i}$/$_{r_i}$, which means the amplitude of the sine wave is now radially dependent.

		\begin{equation}\label{eq:ang_rad_aberrations}

			\mathcal{A}_{\left(\theta, r\right)} = r \times \sum_{i=1}^{n}\frac{a_i}{r_i} \sin \left(\theta + \theta_i\right)

		\end{equation}

		One way to express the aberrations is through \autoref{eq:ps_aberrations}.

		\begin{equation}\label{eq:ps_aberrations}

			r_i^a = r_i^0 \times \left[\Pi_{i=1}^{n}\left(1 + a_i\sin\left(\frac{\theta_i}{n}\right)\right)\right]

		\end{equation}