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< \usepackage{booktabs}
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< \setcitestyle{comma}
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< In the testbeam setup, the module was placed in between two sets of Mimosa26 silicon detector planes that provided external high-precision tracking and the whole detector setup was slid into the PCMAG magnet at DESY.
---
> In the testbeam setup, the module was placed in between two sets of Mimosa26 silicon detector planes that provided external high precision tracking and the whole detector setup was slid into the PCMAG magnet at DESY.
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<  As a next step towards a Pixel Time Projection Chamber for a future collider experiment \cite{Ligtenberg:2021,Lupberger:2017uiv}, a module consisting of 32 GridPix chips based on the TPX3 chip was constructed.
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>  As a next step towards a Pixel Time Projection Chamber for a future collider experiment \cite{Ligtenberg:2021}, \cite{Lupberger:2017uiv}, a module consisting of 32 GridPix chips based on the TPX3 chip was constructed.
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<  Based on the experience gained with these detectors a 32-GridPix detector module - consisting of 8 quad detectors - was built. A drift box defining the electric field and gas envelope was constructed. A read out system for up to 128 chips with 4 multiplexers read out by one Speedy Pixel Detector Readout (SPIDR) board  \cite{Visser:2015bsa,Heijden:2017iqd} was designed. After a series of tests using the laser setup \cite{Hartjes:1990} and cosmics in the laboratory at Nikhef, the detector was taken to DESY for a two week testbeam campaign.
---
>  Based on the experience gained with these detectors a 32 GridPix detector module - consisting of 8 quad detectors - was built. A drift box defining the electric field and gas envelop was constructed. A read out system for up to 128 chips with 4 multiplexers read out by one Speedy Pixel Detector Readout (SPIDR) board  \cite{Visser:2015bsa} \cite{Heijden:2017iqd} was designed. After a series of tests using the laser setup \cite{Hartjes:1990} and cosmics in the laboratory at Nikhef, the detector was taken to DESY for a two week testbeam campaign.
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<  The performance of the 32-GridPix detector module was measured using these data sets.
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>  The performance of the 32 GridPix detector module was measured using these data sets.
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<   In this paper, part I of the results will be presented with the main focus on the detector spatial resolution and tracking performance. A second follow-up paper will discuss the $dE/dx$ (or $dN/dx$) and other results. 
---
>   In this paper, part I of the results will be presented with the main focus on the detector spatial resolution and tracking performance. A second follow up paper will discuss the $dE/dx$ (or $dN/dx$) and other results. 
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<  A 32-GridPix detector module was built using the quad detector module \cite{Ligtenberg:2020ofy} as a basic building block.
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>  A 32 GridPix detector module was built using the quad detector module \cite{Ligtenberg:2020ofy} as a basic building block.
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<     \caption{Schematic drawing of the 32-GridPix module detector with one example quad as viewed from the top of the quad detectors. The chips are numbered and the beam direction is shown in purple. A guard electrode of a quad detector is shown in orange. The four surrounding guard strips are shown -not to scale- in orange. Six guard wires - are shown with dashed lines (one colored red) and the pillars of the drift box are shown as circles with a P in the centre.}
---
>     \caption{Schematic drawing of the 32-GridPix module detector with one example quad as viewed from the top of the quad detectors. The chips are numbered and the beam direction is shown in purple. A guard electrode of a quad detector is shown in orange. The four surrounding guard strips are shown -not to scale- in orange. Six guard wires - are shown with dashed lines (one colored red) and the pillars of the drift box are shown a circles with a P in the centre.}
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< The data acquisition system of the quad module was adopted to allow for reading out multiple quad detectors. A multiplexer card was developed that handles four quad detectors or 16 chips and combines the TPX3 data into one data stream. For the 32-GridPix module two multiplexers are connected to a SPIDR board that controls the chips and read out process. The read out speed per chip is {160}{ Mbps} and for the multiplexer {2.56}{ Gbps}: this corresponds to a maximum rate of {21}{ MHits/s}. For each pixel the precise Time of Arrival (ToA) using a 640 MHz TDC and the time over threshold (ToT) are measured. 
---
> The data acquisition system of the quad module was adopted to allow for reading out multiple quad detectors. A multiplexer card was developed that handles four quad detectors or 16 chips and combines the TPX3 data into one data stream. For the 32 GridPix module two multiplexers are connected to a SPIDR board that controls the chips and read out process. The read out speed per chip is {160}{ Mbps} and for the multiplexer {2.56}{ Gbps} this corresponds to a maximum rate of {21}{ MHits/s}. For each pixel the precise Time of Arrival (ToA) using a 640 MHz TDC and the time over threshold (ToT) are measured. 
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<     \begin{tabular}{l r} 
<         \toprule
---
>     \begin{tabular}{l r} %\toprule
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<         \bottomrule
---
> %        \bottomrule
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<  Telescope tracks were required to have hits in at least 5 out of the 6 planes and a total $\chi^2$ of better than 25 per degree of freedom.  The uncertainties on the telescope track prediction in the middle of the GridPix detector module are dominated by multiple scattering.
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>  Telescope tracks were selected  were required to have hits in at least 5 out of the 6 planes and a total $\chi^2$ of better than 25 per degree of freedom.  The uncertainties on the telescope track prediction in the middle of the GridPix detector module are dominated by multiple scattering.
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<  GridPix hits outside an acceptance window of 30 mm wide in $x$ and 15 mm wide in $z$, corresponding to the size of the quartz window, were not used in the track finding and reconstruction.
---
>  GridPix hits outside an acceptance window of 30 mm wide in $x$ and 15 mm wide in $z$ were not used in the track finding and reconstruction.
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<  A TPC track was required to have at least 100 hits in each multiplexer. At least 25\% of the total number of hits should be on track and the $\chi^2$ per degree of freedom had to be less than 3 in $xy$ and $zy$. All track parameters were expressed at a plane in the middle of the TPC module. 
---
>  A TPC track was required to have a least 100 hits in each multiplexer. At least 25\% of the total number of hits should be on track and the $\chi^2$ per degree of freedom had to be less than 3 in $xy$ and $zy$. All track parameters were expressed at a plane in the middle of the TPC module. 
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<  the track selections are summarised in Table \ref{tab:selections}.
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>  the track selections are summarised in table \ref{tab:selections}.
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<     \begin{tabular}{c} 
<        \toprule
<         Track/Event Selection \\ \midrule
---
>     \begin{tabular}{c} %\toprule
>         Track/Event Selection \\ %\midrule
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<        \bottomrule
---
> %        \bottomrule
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<  The individual TPX3 chips were iteratively aligned fitting a shift in $x$ ($z$) and two slopes $dx(z)/dx(y)$.
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>  The individual TPX3 chips were iteratively aligned fitting a shift in $x$ ($z~drift$) and two slopes $dx(z~drift)/d row(column)$.
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<     \caption{An event display for run 6916 without $B$ field, with in total 1293 TPC hits (black dots) in the (top) precision plane ($x,y$) and (botom) drift plane ($z~drift,y$). The fitted TPC track (red line) with 1130 hits on track and the telescope track (blue line) with 5 Mimosa26 planes (blue hits) on track are shown. In green the off track Mimosa26 hits are shown.}
---
>     \caption{An event display for run 6916 without $B$ field, with in total 1293 TPC hits (black dots) in the precision plane ($x,y$) and drift plane ($z~drift,y$). The fitted TPC track (red line) with 1130 hits on track and the telescope track (blue line) with 5 Mimosa26 planes (blue hits) on track are shown. In green the off track Mimosa26 hits are shown.}
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< The track residual in $xy$ is defined as the closest distance - defined as the 2D $xy$ projection of the 3D distance - between the hit at the center of the pixel and the track. The residual in $z$ is calculated at this point of closest approach.  
<  The single-electron hit resolutions in $xy$ and $z$ will be extracted from the track residuals. 
<  In order to study the single-electron resolution for the data with and without magnetic field, additional selections on the telescope and TPC tracks were applied. Due to the trigger time jitter of 25 ns (corresponding to $~$1.5 mm drift), the prediction of the telescope track in $z$ must be used as the reference for $z$. Therefore the $z$ hits of the TPC track were fitted to correct for the common time shift and the - biased - $z$ residuals were calculated with respect to the fitted TPC track. In the $xy$ plane the residuals of TPC hits with respect to the telescope track were used to extract the single-electron resolution in $xy$.
---
> The track residual in $xy$ is the closest point of the the hit at the center of the pixel to track in the $xy$ plane. The residual in $z$ is calculated at this point of closest approach.  
>  The single electron hit resolutions in $xy$ and $z$ will be extracted from the track residuals. 
>  In order to study the single electron resolution for the data with and without magnetic field, additional selections on the telescope and TPC tracks were applied. Firstly, due to the trigger time jitter of 25 ns (corresponding to $~$1.5 mm drift), the prediction of the telescope track in $z$ must be used as the reference for $z$. Secondly, the $z$ hits of the TPC track were fitted to correct for the common time shift and the $z$ residuals were calculated with respect to the fitted TPC track. In the $xy$ plane the residuals of TPC hits with respect to the telescope track were used to extract the single electron resolution in $xy$.
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< where  $\sigma_\text{track}$ is the uncertainty from the track prediction,  $d_\text{pixel}$ is the pixel pitch size, $z_0$ is the position of the grid, and $D_T$ is the transverse diffusion coefficient. The last two terms correspond to the single-electron detector resolution (squared). The resolution at zero drift distance $d_\text{pixel}/\sqrt{12}$ was fixed to {15.9}{ $\rm \mu$m} and $\sigma_\text{track}$ to {30}{ $\rm \mu$m} for {$B$ = 0 T} and  {42}{ $\rm \mu$m} for {$B$ = 1 T} data. The uncertainty on the track prediction was measured and is larger than the Mimosa plane resolution because of multiple scattering in the sensors and in the entrance and exit windows.
---
> where  $\sigma_\text{track}$ is the uncertainty from the track prediction,  $d_\text{pixel}$ is the pixel pitch size, $z_0$ is the position of the grid, and $D_T$ is the transverse diffusion coefficient. The last two terms correspond to the single electron detector resolution (squared). The resolution at zero drift distance $d_\text{pixel}/\sqrt{12}$ was fixed to {15.9}{ $\rm \mu$m} and $\sigma_\text{track}$ to {30}{ $\rm \mu$m} for {$B$ = 0 T} and  {42}{ $\rm \mu$m} for {$B$ = 1 T} data. The uncertainty on the track prediction was measured and is larger than the Mimosa plane resolution because of multiple scattering in the sensors and in the entrance and exit windows.
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<     \caption{Measured spread on the residuals in the pixel plane (black points) for (left) $B$ = 0 T and (right) $B$ = 1 T, fitted with equation \eqref{eq:sigmax} (blue line).}
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>     \caption{Measured spread on the residuals in the pixel plane (black points) fitted with equation \eqref{eq:sigmax} (blue line).}
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< The spread on the residual in $z$ of the ionisation electrons $\sigma_z$ is given by:
---
> The spread on the residuals in $z$ of the ionisation electrons $\sigma_z$ is given by:
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<     \sigma_z^2= \sigma_\text{track}^2 + \sigma_{z0}^2  + D_L^2(z_{\rm drift} -z_0),
---
>     \sigma_z^2= \sigma_\text{track}^2 + \sigma_{z0}^2+ + D_L^2(z_{\rm drift} -z_0),
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< The last two terms in the equation correspond to the single-electron detector resolution (squared).
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> The last two terms in the equation correspond to the single electron detector resolution (squared).
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< The systematic uncertainty on $\sigma_\text{track}$ is estimated to be {25}{ $\rm \mu$m}.  
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> The systematic uncertainty on $sigma_\text{track}$ is estimated to be {25}{ $\rm \mu$m}.  
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<     \caption{Measured spread on the residuals in the drift plane for hits with a ToT above {0.60}{ $\rm \mu$s} for (left) $B$ = 0 T and (right) $B$ = 1 T. The data are fitted with the expression of equation \eqref{eq:sigmaz}.}
---
>     \caption{Measured spread on the residuals in the drift plane for hits with a ToT above {0.60}{ $\rm \mu$s}. The data are fitted with the expression of equation \eqref{eq:sigmaz}.}
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< It is important to measure possible deformations in the pixel ($xy$) and drift ($z$) plane to quantify the tracking precision. For the construction of a large Pixel TPC, deformations in the pixel plane should be controlled to better than typically {20}{ $\rm \mu$m} because these affect the momentum resolution. 
< The mean residuals in the pixel and drift planes are shown in Figure \ref{fig:deformationsB0} for the {$B$ = 0 T} data set using a large set of runs to cover the whole module. The residuals were calculated with respect to the telescope track prediction. Because of limited statistics, groups of {16$\times$16}{} pixels were combined into one bin.  
---
> It is important to measure possible deformations in the pixel ($xy$) and drift ($z$) plane to quantify the tracking precision. For the construction of a large Pixel TPC, deformations in the pixel plane deformation should be controlled to better than typically {20}{ $\rm \mu$m} because these affect the momentum resolution. 
> The mean residuals in the pixel and drift planes are shown in Figure \ref{fig:deformationsB0} for the {$B$ = 0 T} data set using a large set of runs to cover the whole module. The residuals were calculated with respect to the telescope track prediction. Because of limited statistics, bins were grouped into {8$\times$16}{} pixels.  
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< Deformations are present at the four corners of the drift box (chips 1, 10, 19 and 24) and close to the right upper corner edge (chip 16) of the drift box. These come from inhomogeneities in the drift field near the supporting pillars, where the field wires are too close to the chip to provide a constant electric field. 
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> Deformations are present at the four corners of the drift box (chips 1, 10, 19 and 24) and close to the upper corner edge (chip 16) of the drift box. These come from inhomogenieties in the drift field near the supporting pillars, the field wires are too close to the chip to provide a constant electric field. 
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< In order to reduce the statistical fluctuations and quantify the tracking precision, the pixels were regrouped into larger bins respecting the module geometry. After the regrouping procedure, a module plane with (4$\times$256)$\times$256 bins is obtained, as shown in Figure \ref{fig:deformationsGroupedB0} \footnote[1]{The mathematical procedure is defined as follows. The original mean residual - before rebinning - is given by  residual(i,j) where i runs horizontally and j vertically. The rebinned result for the residual(4$\times$256,256) is equal to residual(i$\%1024$,j$\%256$). The mean residual(256,4$\times$256) - discussed later in the paper - is equal to residual(i$\%256$,j).}.  
---
> In order to reduce the statistical fluctuations and quantify the tracking precision, the module was regrouped in (4$\times$256)$\times$256 pixel planes put side by side on the horizontal axis, as shown in Figure \ref{fig:deformationsGroupedB0}. E.g. the selected chips from the upper left and bottom left quad detectors are superimposed into the 0-256 ($x$) and 0-256 ($y$) plane.  
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< Due to the presence a so-called dike  - that was created in the TPX3 post-processing to protect the edges of the TPX3 chip - pixels at the edge of the chip were covered and inefficient. Therefore, the region of 5 pixels in $y$ near the edge of each chip was removed. For the drift coordinate studies, a region of 10 pixels near the edge of each chip in $x$ and $y$ was removed.      
---
> Due to the presence of the dike, pixels at the edge of the chip became covered and inefficient. Therefore, the region of 5 pixels in $y$ near the edge of the chip was removed. For the drift coordinate studies, a region of 10 pixels near the edge of the chip in $x$ and $y$ was removed.      
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< The r.m.s. of the distribution of the measured mean residual over the surface in the pixel plane is {11}{ $\rm \mu$m} and in the drift plane {15}{ $\rm \mu$m}. Similarly, regrouping the module in a plane with 256$\times$(4$\times$256) pixel bins, yielded a r.m.s. in the pixel plane of {13}{ $\rm \mu$m} and {13}{ $\rm \mu$m} in the drift coordinate. The expected statistical error - obtained by propagating the uncertainties on the residuals - in $xy$ is {4}{ $\rm \mu$m} and in $z$ {5}{ $\rm \mu$m}.
---
> The r.m.s. of the distribution of the measured mean residual over the surface in the pixel plane is {11}{ $\rm \mu$m} and in the drift plane {15}{ $\rm \mu$m}. Similarly, regrouping the module in 256$\times$(4$\times$256) pixels put them side by side on the vertical axis, yielded a r.m.s. in the pixel plane of {13}{ $\rm \mu$m} and {13}{ $\rm \mu$m} in the drift coordinate. The expected statistical error - obtained by propagating the uncertainties on the residuals - in $xy$ is {4}{ $\rm \mu$m} and in $z$ {5}{ $\rm \mu$m}.
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<     \caption{Mean residuals (color coded in mm) in the pixel (top) and drift plane (bottom) for {$B$ = 1 T} data at the expected hit position.}
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>     \caption{Mean residuals (color coded in mm) in the pixel and drift plane for {$B$ = 1 T} data at the expected hit position.}
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< In Figure \ref{fig:deformationsB1} the critical areas discussed above - around chip 11, the four corner chips and chip 16 in the right upper corner edge - can be clearly observed.  
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> In Figure \ref{fig:deformationsB1} the critical areas discussed above - around chip 11, the four corner chips and chip 16 in the upper corner edge - can be clearly observed.  
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<  Similar to the no-field deformation studies, acceptance cuts had to be applied. The region of 16 pixels in $y$ near the edge of the chips was removed. For the drift coordinate studies, in addition a region of 10 pixels in $x$ near the edge of the chip was removed.      
---
>  Similar to the no-field deformations studies, acceptance cuts had to be applied. The region of 16 pixels in $y$ near the edge of the chips was removed. For the drift coordinate studies, in addition a region of 10 pixels in $x$ near the edge of the chip was removed.      
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< The r.m.s. of the distribution of the measured mean residual over the surface in the pixel plane is {13}{ $\rm \mu$m} and in the drift plane {19}{ $\rm \mu$m}. Similarly, regrouping the module in 256$\times$(4$\times$256) pixel bins, yielded a r.m.s. in the pixel plane of {11}{ $\rm \mu$m} and {20}{ $\rm \mu$m} in the drift coordinate. The expected statistical error in $xy$ is {2}{ $\rm \mu$m} and in $z$ {3}{ $\rm \mu$m}.
---
> The r.m.s. of the distribution of the measured mean residual over the surface in the pixel plane is {13}{ $\rm \mu$m} and in the drift plane {19}{ $\rm \mu$m}. Similarly, regrouping the module in 256$\times$(4$\times$256), yielded a r.m.s. in the pixel plane of {11}{ $\rm \mu$m} and {20}{ $\rm \mu$m} in the drift coordinate. The expected statistical error in $xy$ is {2}{ $\rm \mu$m} and in $z$ {3}{ $\rm \mu$m}.
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<  In summary, the deformation studies for the $B$ = 0 and 1 T data demonstrate that the systematical uncertainties in $xy$ are smaller than {13}{ $\rm \mu$m} with and without magnetic field.
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>  In summary, the deformations studies for the $B$ = 0 and 1 T data demonstrate that the systematical uncertainties in $xy$ are smaller than {13}{ $\rm \mu$m} with and without magnetic field.
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< The mean number of hits is measured to be 124 and 89 in the {$B$ = 0 T} and 1 T data sets, respectively. The most probable values are respectively 87 and 64.    
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> The mean number of hits is measured to be 124 and 89 in the {$B$ = 0 T} and 1 T data sets respectively. The most probable values are respectively 87 and 64.    
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< The mean time over threshold (ToT) is {0.68}{ $\rm \mu$s} for the {$B$ = 0 T} and {0.86}{ $\rm \mu$s} at a {$B$ = 1 T} data. A typical ToT distribution can be found in Figure 6 of \cite{Ligtenberg:2018sjs} and Figure 5.5 of \cite{Ligtenberg:2021}. The time over threshold is related to the charge after avalanche multiplication. This means that the mean deposited charge per pixel is smaller for the 0 T data.
---
> The mean time over threshold (ToT) is {0.68}{ $\rm \mu$s} for the {$B$ = 0 T} and {0.86}{ $\rm \mu$s} at a {$B$ = 1 T} data. A typical ToT distribution can be found in Figure 5.5 of ref.\cite{Ligtenberg:2021}. The time over threshold is related to the deposited charge. This means that the deposited charge per pixel is smaller for the 0 T data.
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< The mean number of hits is in agreement with the prediction of 106 electron-ion pairs for a 5 and 6 GeV/c electron at {$B$ = 0 T}  for the T2K gas by \cite{Garfield}, crossing 236 pixels or  {12.98}{ mm} and a detector running at 85\% single-electron efficiency.
< The measured single-electron efficiency at this working point is in agreement with the efficiency vs mean time over threshold curve that was measured using a Fe source \cite{Ligtenberg:2021}.
---
> The mean number of hits is in agreement with the prediction of 106 electron-ion pairs for a 5 and 6 GeV/c electron at {$B$ = 0 T}  for the T2K gas by \cite{Garfield}, crossing 236 pixels or  {12.98}{ mm} and a detector running at 85\% single electron efficiency.
> The measured single electron efficiency at this working point is in agreement with the efficiency vs mean time over threshold curve that was measured using a Fe source \cite{Ligtenberg:2021}.
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<     \caption{Distribution of the number of TPC track hits per chip for (left) {$B$ = 0 T} and (right) $B$ = 1 T data.}
---
>     \caption{Distribution of the number of TPC track hits per chip for {$B$ = 0 T} (left) $B$ = 1 T data.}
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< %In the testbeam setup, the module was placed in between two sets of Mimosa26 silicon detector planes that provided external high-precision tracking and the whole detector setup was slid into the PCMAG magnet at DESY.
---
> %In the testbeam setup, the module was placed in between two sets of Mimosa26 silicon detector planes that provided external high precision tracking and the whole detector setup was slid into the PCMAG magnet at DESY.
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< The mean number of hits is in agreement with the predictions of \cite{Garfield} and a detector running at 85\% single-electron efficiency.
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> The mean number of hits is in agreement with the predictions of \cite{Garfield} and a detector running at 85\% single electron efficiency.
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< Not all data were analysed and users are welcome to study them using the data sets on available on the Grid \footnote[2]{For more information, please contact the corresponding author.}.
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> Not all data were analysed and users are welcome to study them using the data sets on available on the Grid.
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< A follow-up paper is in preparation on the measured $dE/dx$ or $dN/dx$ resolution and other performance topics. 
---
> A follow up paper is in preparation on the measured $dE/dx$ or $dN/dx$ resolution and other performance topics. 
