\section{Per-calorimeter and per-bunch analysis}
\label{sec:per_calobunch}

This section will detail the fast rotation analysis performed per-calorimeter and per-bunch.

\subsection{Per-calorimeter analysis}

The results presented in Sec.~\ref{sec:nominal} correspond to all the 24 calorimeters combined, i.e., the results correspond to the azimuthal averaging around the ring.
It is important to perform the analysis per-calorimeter to ensure the results are consistent all around the ring. Any significant difference would have to be
understood and, if needed, included in the anomalous spin precession analysis. The per-calorimeter fast rotation analysis is almost identical to the one performed
on all the calorimeters combined. The only difference being that the definition of the background is fixed to the ones from the nominal analysis. Not fixing the
background definition would introduce a systematic bias in the per-calorimeter results, thus making less straightforward the comparisons of the results.
The \tz\ parameter is optimized for each calorimeter and is expected to jump by a $24^{ \text{th}}$ of the cyclotron revolution period between two consecutive calorimeters.
Figure~\ref{fig:t0_vs_caloNum} shows the optimized \tz\ values as a function of calorimeter number. 
The statistical uncertainty on each data point (see Sec.~\ref{sec:stat_percalo}) is about \mbox{0.055 ns} and thus too small to be seen. 
The fit to the data point returns:

\begin{equation}
t_0 = 6.218(2) \pm 121.85(2)~\text{ns.}
\end{equation}

The number in () is the uncertainty associated with the last digit, i.e., $6.218(2)$ means \mbox{$6.218 \pm 0.002$} and $121.85(2)$ means $121.85 \pm 0.02$.
The magic cyclotron period of 149.14 ns corresponds to a time shift of 6.214 ns between two consecutive calorimeters. 
From the nominal measured cyclotron frequency  of 6,698.9 kHz, using all the calorimeters and bunches, the measured cyclotron period period is 149.28 and thus 
the time shift is 6.220 ns. The time shift obtained from the per-calorimeter results is statistically 1 standard deviation away from the expected results of 6.220 ns.

\begin{figure}[bt]
\centering
\includegraphics[width=0.82\textwidth]{fig/t0_vs_caloNum}
\caption{Optimized \tz\ value as a function of calorimeter number. The black line is a linear fit to the data point. 
The fit results gives the \tz\ variation as a function of calorimeter number: $t_0 = 6.218(2) \cdot \text{Calo}~\# + 121.85(2)$ ns.}
\label{fig:t0_vs_caloNum}
\end{figure}


\begin{figure}[bt]
\centering
\includegraphics[width=0.82\textwidth]{fig/radial_allCalos}
\caption{Radial distributions for all the 24 calorimeters overlaid.}
\label{fig:rad_allcalos}
\end{figure}

Figure~\ref{fig:rad_allcalos} shows the radial distributions for all the 24 calorimeters overlaid. The by-eye agreement is satisfying.
Fig.~\ref{fig:results_percalo} shows $x_e$, $\sigma$ and $C_E$ as a function of calorimeter number. The error bars are the statistical uncertainty.
The results per-calorimeter are statistically in reasonable agreement, the largest deviation being of the order of 4 standard deviations\footnote{The systematic
uncertainty is expected to be larger than the statistical uncertainty but is not estimated per-calorimeter.}.
Assuming that each calorimeter has the same statistical uncertainty, the averaging of them all yields:

\newpage
~
\newpage

\begin{align*}
x_e & = 6.39 \text{ mm,}\\
\sigma & = 9.18 \text{ mm,}\\
C_E & = -521 \text{ ppb,}\\
\end{align*}

which are in good agreement with the results presented previously for all the calorimeters combined. The standard errors\footnote{The standard error is the
uncertainty on the average value using the spread of the results.} of the per-calorimeter results are:

\begin{align*}
\delta x_e & = 0.020 \text{ mm,}\\
\delta \sigma & =  0.008 \text{ mm,}\\
\delta C_E & =  1.38 \text{ ppb,}\\
\end{align*}

which for $x_e$ and $C_E$ are twice the statistical uncertainty estimated when performing the analysis on all the calorimeters combined. This indicates some non-statistical
effect in the per-calorimeter analysis\footnote{In the presence of statistical effect only: the results and the associated uncertainties should be the same if performing the
analysis on the combined inputs or combining the outputs of the individual analysis.}.
Table~\ref{tab:results_percalo} summarizes all the numbers.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_caloNum}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_caloNum}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_caloNum}}
\caption{Analysis results of the fast rotation analysis per-calorimeter: (a) $x_e$, (b) $\sigma$, and (c) $C_E$.
The error bars show the statistical uncertainty (see Sec.~\ref{sec:stat_percalo}).}
\label{fig:results_percalo}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{lcccc} \hline\hline
& $t_0$ [ns] & $x_{e}$ [mm] & $\sigma$ [mm] & $C_{E}$ [ppb]\\ \hline
calo \#1  & 128.175 & 6.204 & 9.157 &-509.1 \\
calo \#2  & 134.464 & 6.373 & 9.160 &-518.2 \\
calo \#3  & 140.589 & 6.421 & 9.208 &-524.5 \\
calo \#4  & 146.702 & 6.473 & 9.151 &-522.9 \\
calo \#5  & 153.016 & 6.438 & 9.226 &-526.7 \\
calo \#6  & 159.081 & 6.512 & 9.185 &-527.6 \\
calo \#7  & 165.304 & 6.494 & 9.183 &-526.5 \\
calo \#8  & 171.639 & 6.437 & 9.164 &-522.0 \\
calo \#9  & 177.681 & 6.521 & 9.176 &-527.4 \\
calo \#10 & 183.908 & 6.466 & 9.193 &-525.6 \\
calo \#11 & 190.187 & 6.355 & 9.131 &-515.0 \\
calo \#12 & 196.271 & 6.341 & 9.160 &-516.6 \\
calo \#13 & 202.586 & 6.231 & 9.187 &-512.9 \\
calo \#14 & 208.893 & 6.379 & 9.158 &-518.4 \\
calo \#15 & 214.970 & 6.486 & 9.187 &-526.3 \\
calo \#16 & 221.399 & 6.367 & 9.140 &-516.3 \\
calo \#17 & 227.490 & 6.436 & 9.181 &-523.2 \\
calo \#18 & 233.882 & 6.226 & 9.069 &-503.6 \\
calo \#19 & 240.012 & 6.222 & 9.208 &-513.9 \\
calo \#20 & 246.288 & 6.313 & 9.198 &-518.0 \\
calo \#21 & 252.487 & 6.465 & 9.191 &-525.5 \\
calo \#22 & 258.589 & 6.491 & 9.267 &-532.8 \\
calo \#23 & 264.938 & 6.334 & 9.241 &-522.4 \\
calo \#24 & 271.180 & 6.267 & 9.217 &-517.0 \\
\hline
\hline
\end{tabular}
\caption{Per-calorimeter results of the fast rotation analysis.}
\label{tab:results_percalo}
\end{center}
\end{table}

\clearpage

\subsection{Per-bunch analysis}

Figure~\ref{fig:frbunches} in App.~\ref{app:frbunches} shows that the 8 bunches in the accelerator cycle have different longitudinal profiles.
Given the length of the incoming pulse of about 200 ns and the time inhomogenous kick provided by the three kickers inside the ring, 
one can expect different stored radial distributions for each bunch. It is therefore interesting to look at the fast rotation results
for each bunch individually\footnote{The anomalous spin precession frequency is nominally performed combining all the bunches together.}
combining all the calorimeters together.
Similarly to the per-calorimeter analysis, the definition of the background is fixed to the ones from the nominal analysis. Not fixing the
background definition would introduce a systematic bias in the per-bunch results, thus making less straightforward the comparisons of the results.
The \tz\ parameter is optimized for each bunch and is expected to be randomly distributed because of the differences in the beam profiles (the time centroid
of each bunch depends on its profile).
Figure~\ref{fig:results_perbunch}(a) shows the optimized \tz\ values as a function of bunch number.
The statistical uncertainty on each data point (see Sec.~\ref{sec:stat_perbunch}) is about \mbox{0.03 ns} and thus too small to be seen.
Assuming each bunch has the same statistical uncertainty, the averaging of the \tz\ values yields:

\begin{align*}
t_0=127.75~\text{ns},
\end{align*}

which is close to the optimized \tz\ value when all the calorimeters and bunches are combined together: $t_0= 128.091$ ns.
Figure~\ref{fig:rad_allbunches} shows the radial distributions for all the 8 bunches overlaid.

\begin{figure}[bt]
\centering
\includegraphics[width=0.82\textwidth]{fig/radial_allBunches}
\caption{Radial distributions for all the 8 bunches overlaid.}
\label{fig:rad_allbunches}
\end{figure}

Fig.~\ref{fig:results_perbunch}(b, c, d) show $x_e$, $\sigma$, and $C_E$ as a function of bunch number. The error bars are the statistical uncertainty.
Assuming that each bunch has the same statistical uncertainty, the averaging of them all yields:

\begin{align*}
x_e & = 6.34 \text{ mm,}\\
\sigma & = 9.15 \text{ mm,}\\
C_E & = -516 \text{ ppb,}\\
\end{align*}

which are in good agreement with the results presented previously for all the calorimeters and bunches combined.
These results indicate that the per-bunch and per-calorimeter information can be linearly combined together before or after performing the fast rotation analysis.
Table~\ref{tab:results_perbunch} summarizes all the numbers.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/t0_vs_bunchNum}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/xe_vs_bunchNum}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/width_vs_bunchNum}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/ce_vs_bunchNum}}
\caption{Analysis results of the fast rotation analysis per-bunch: (a) \tz\, (b) $x_e$, (c) $\sigma$, and (d) $C_E$.
The error bars show the statistical uncertainty (see Sec.~\ref{sec:stat_perbunch}).}
\label{fig:results_perbunch}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{lcccc} \hline\hline
& $t_0$ [ns] & $x_{e}$ [mm] & $\sigma$ [mm] & $C_{E}$ [ppb]\\ \hline
bunch \#0 & 132.315 & 6.439 & 9.202 & -524.9 \\
bunch \#1 & 120.581 & 5.887 & 9.089 & -488.0 \\
bunch \#2 & 125.740 & 6.126 & 9.175 & -506.4 \\
bunch \#3 & 137.372 & 6.746 & 9.116 & -535.3 \\
bunch \#4 & 124.868 & 6.282 & 9.158 & -513.2 \\
bunch \#5 & 131.306 & 6.291 & 9.215 & -518.0 \\
bunch \#6 & 111.938 & 6.300 & 9.101 & -509.9 \\
bunch \#7 & 137.839 & 6.656 & 9.170 & -534.4 \\
\hline
\hline
\end{tabular}
\caption{Per-bunch results of the fast rotation analysis.}
\label{tab:results_perbunch}
\end{center}
\end{table}

\clearpage
