\section{Baseline Magnetic Mitigation Scheme}

\subsection{Overview}

\subsubsection{Introduction}\label{BMMS_Introduction}

This section details how the Mice Hall Model has been used to give an overview of the stray fields likely to be observed in the MICE hall and the surrounding areas.

To start this section it was through that it would be useful to provide the reader with a set of overview and reference plots of the predicted field level throughout the entire volume of the hall model at beam height for both solenoid and flip mode at step IV (240~MeV/c) - see figures \ref{fig:Boundary_Sol_Step_IV_On} to \ref{fig:Boundary_Flip_Step_IV_Off}. Of course at this scale a lot of detail is lost; we will return to areas of detail later. 

All of these plots are plotted on a five gauss scale.  Note that where the field is $>$5~gauss the field is not plotted, so either details of structures behind the plot can be seen more clearly, or where there are no structures a white background can be observed.

All of the 2D plots are produced using Nodal interpolation with integral fields. This may be slightly less accurate than using a full integration method but given the number of elements in the hall model the full integration method will not produce 2D plots in a tractable time. As alluded to earlier Vector Fields demonstrated that the mesh on a test patch within the hall volume was accurate to $<$1\%. It is likely that that the error will increase as one moves towards the edge of the model as the mesh size increases, however as this is a larger error on a very small field value this is of no significance. 

%As will be demonstrated later, Biot Savart calculations at the location of the West Wall compare favourably with the field calculations.

%***Possibly DO a comparison plot of Model~100 with model~91. Note that model~100 was linear and not quadratic!***

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_0_Structures_On.png}
\end{center}
\caption{Bmod 5 gauss scale. Step IV 240~Mev/c Solenoid Mode. Overview of Model Volume at Beam Height from Model 91. The Model is shown with structures turned on. One can immediately see that the majority of the stray field is contained within the MICE hall.}\label{fig:Boundary_Sol_Step_IV_On}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_0_Structures_Off.png}
\end{center}
\caption{Bmod 5 gauss scale. Step IV 240~Mev/c Solenoid Mode. Overview of Model Volume at Beam Height from Model 91. The Model is shown with structures turned off.}\label{fig:Boundary_Sol_Step_IV_Off}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_94_Bmod_0_500_uT_XZplane_y_0_Structures_On.png}
\end{center}
\caption{Bmod 5 gauss scale. Step IV 240~Mev/c Flip Mode. Overview of Model Volume at Beam Height from Model 94. The Model is shown with structures turned on. One can immediately see that the majority of the stray field is contained within the MICE hall.}\label{fig:Boundary_Flip_Step_IV_On}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_94_Bmod_0_500_uT_XZplane_y_0_Structures_Off.png}
\end{center}
\caption{Bmod 5 gauss scale. Step IV 240~MeV/c Flip Mode. Overview of Model Volume at Beam Height from Model 94. The Model is shown with structures turned off.}\label{fig:Boundary_Flip_Step_IV_Off}
\end{sidewaysfigure}

\clearpage

\subsubsection{Field Outside of the MICE Hall}\label{FOMH}

The plots in this section show the maximum observable field calculated on the external faces of the MICE Hall walls/roof/floor. The images shown are for Step IV solenoid mode only as this mode of operation produces the highest external fields for step IV as there is less field cancellation in the cooling channel. No images have been shown for the East wall but this is far enough away not to be of concern.

The only area external to the MICE hall that indicates that there may be fields above 5~gauss is on the roof. As these are simulated predictions it is clear that once the magnets are installed, then field measurements would need to be taken to establish the validity of these particular simulation results. It is understood that public access to areas above 5~gauss would need to be limited; as the MICE hall roof is not considered to be public access then restricted access to this area should not be a problem. The effects of any stray fields upon equipment installed on the MICE hall roof has not been established.

%The images for flip mode are available online (NOTE I NEED TO PRODUCE THESE PLOTS!!!)

%\begin{sidewaysfigure}[!htb]
%\begin{center}
%\hspace*{-1.5cm}
%\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_YZplane_x_-8567_Structures_On_XYZ.png}
%\end{center}
%\caption{Bmod 5 gauss scale. Step IV 240~MeV/c Solenoid Mode. North Wall Isometric view.}\label{fig:North_Wall_Sol_3D}
%\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_YZplane_x_-8567_Structures_Off.png}
\end{center}
\caption{Bmod 5 gauss scale. Step IV 240~MeV/c Solenoid Mode. North Wall Side view.}\label{fig:North_Wall_Sol}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_YZplane_x_6094_Structures_Off.png}
\end{center}
\caption{Bmod 5 gauss scale. Step IV 240~MeV/c Solenoid Mode. South Wall Side view.}\label{fig:South_Wall_Sol}
\end{sidewaysfigure}

%\begin{sidewaysfigure}[!htb]
%\begin{center}
%\hspace*{-1.5cm}
%\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_BMAX_XYplane_z_20960_Structures_On_XYZ.png}
%\end{center}
%\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. West Wall Isometric view. The predicted field is the same magnitude as that of the earth's magnetic field.}\label{fig:West_Wall_Sol_3D}
%\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_BMAX_XYplane_z_20960_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. West Wall Isometric view. The predicted field is the same magnitude as that of the earth's magnetic field.}\label{fig:West_Wall_Sol_3D}
\end{sidewaysfigure}

%ROOF

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_7953_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. External to Roof. This is the only plot where the external field goes above 5 gauss.}\label{fig:Roof_Sol_3D}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_1_mT_XZplane_y_7953_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. External to Roof. This is a 10~gauss plot of the roof. The peak areas are only just above 5 gauss.}\label{fig:Roof_Sol}
\end{sidewaysfigure}

\clearpage

\subsubsection{Mice Hall West Wall}

It was clear from earlier analysis that the compressors were not ideally located and that they required moving, this was discussed in section [REFERENCE TO JASON's SECTION ON THE WEST WALL MEZZANINE].

Early models of the MICE hall indicated that the West Wall of the MICE Hall was a relatively safe haven, this view has been confirmed as the hall models have been further improved. Furthermore, the path to the MICE Hall West Wall is relatively free of large iron structures so it was possible to do a Biot-Savart calculation to give a first order approximation of the expected field in this region; this also provided a double check on the FEA model. In context of the simplification the Biot-Savart results gave good agreement with the FEA model. As the moving of the compressors requires significant structural work to the MICE Hall this cross check is valuable as it is not possible to do an in-situ measurement of the field until after the fact.

Figure \ref{fig:Comp_Loc_WW} shows a drawing of the proposed location of the compressor locations along the West Wall.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Compressor_Locations_West_Wall.png}
\end{center}
\caption{Location of the the Sumitomo and Cryomech compressors against the West Wall for Step IV. [***Does this need a REF to Jason's Document?***] }\label{fig:Comp_Loc_WW}
\end{figure}

The Cryomech compressors have a mass of 178~kg each and the Sumitomo compressors have a mass of 96~kg each. The Cryomech compressors have a depth of 612~mm and are set back 300~mm from the West Wall.

The Sumitomo compressors have a depth of 500~mm and the centres are offset by a distance of 2000~mm from the West Wall (West Wall situated at z=19,736~mm).

Although not insignificant the overall iron density is quite low in such an extended volume and the compressor detail means that it is difficult to include this mass representatively in the hall model in a meaningful way. To create these objects and mesh them would require a fine mesh that would make the Hall model time consuming to solve. It is likely that the effect of these iron boxes on the model would be to partially shield the inner contents and locally distort the field, however given their relative distance from the bulk of the iron in the hall it is unlikely that there is enough iron in these compressors to significantly affect the magnitude of the field in this region.\footnote{Some earlier models were run with models of the compressors in location to try and ascertain the effect of placing this iron at the west end of the MICE hall. These models suffered from problems with meshing resolution and the results will not be presented. However it could be ascertained that the iron in these compressors did not have a significant gross effect upon the field in this region.} For this reason the iron mass of the compressors was excluded from the hall FEA model and it is acknowledged that this is a known but unquantified error in the results.

Figures \ref{fig:West_17000_Off} and \ref{fig:West_19000_Off} show the predicted field at z=17,000~mm and z=19,000~mm for Model 91

%\begin{sidewaysfigure}[!htb]
%\begin{center}
%\hspace*{-1.5cm}
%\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_100_uT_XYplane_z_17000_Structures_On_XYZ.png}
%\end{center}
%\caption{Bmod 1 gauss scale. Step IV 240~MeV/c Solenoid Mode. The predicted field at the West end of the MICE Hall is realtively low.}\label{fig:West_17000_XYZ}
%\end{sidewaysfigure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Model_91_Bmod_0_100_uT_XYplane_z_17000_Structures_Off.png}
\end{center}
\caption{Bmod Field Profile towards the West end of the MICE Hall at z=17,000~mm, 1 gauss Scale. Even a couple of metres within the hall the field level is not much above the earth's magnetic field. The hot areas are where there are ferrous objects.}\label{fig:West_17000_Off}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Model_91_Bmod_0_100_uT_XYplane_z_19000_Structures_Off.png}
\end{center}
\caption{This is on the same scale as the previous plot but the plot is taken 2~m further away from the magnets, z=19,000~mm, close to the West Wall. (736~mm away from the wall itself.)}\label{fig:West_19000_Off}
\end{figure}

%*******************************************************

%DO we wish to include?\\
%FEA plot of field  in front of Cryomech Compressors?\\
%FEA Plot in front of the Sumitomo compressors?

%*******************************************************

\clearpage

\subsubsection{ISIS Control Rooms}

The ISIS control rooms are three rooms that form part of the South Side Buildings (SSB); a nomenclature used for the model as they are on the south side of the MICE hall. These buildings, the operations room, main control room and diagnostics room are located adjacent to the MICE Hall South wall but on its upper level and are partly shielded from the MICE magnets by the South Shield Wall. The ISIS control rooms contain a significant amount of diagnostic and control equipment that is used for the operation of the ISIS accelerator. No iron mass is assumed to exist in the ISIS control rooms, although in reality there is likely to be some structural steelwork and racks.

Figure \ref{fig:ISIS_CR_Overview} shows the relative location of these rooms with respect to the the MICE hall in the MICE hall model.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Overview_ISIS_004.png}
\end{center}
\caption{This shows the relative location of the ISIS control rooms to the MICE hall. The South Shield wall partially covers the diagnostics room.}\label{fig:ISIS_CR_Overview}
\end{figure}

Figures \ref{fig:South_Side_ISIS_CR} through to \ref{fig:Ceiling_ISIS_CR} show the field predicted on the outer boundaries of these rooms for step IV solenoid mode 240~MeV/c.  

Figure \ref{fig:ISIS_CR_Vector} shows the predicted vector field that passes at  a height of y=3000~mm, this is ${\approx}$1~m above the height of the floor in the ISIS control rooms. Be aware that there will be a component out of the plane that will be difficult to distinguish in this figure.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Model_91_B_VECTORS_0_500_uT_XZplane_y_3000_Structures_On.png}
\end{center}
\caption{This shows the predicted direction of the magnetic field as it passes through the ISIS control rooms at ${\approx}$1~m above the height of the ISIS control room floor.}\label{fig:ISIS_CR_Vector}
\end{figure}

The reader should be aware that there is some concern that the Shield walls may not be quite as effective in reality as they are calcualted to be in the MICE Hall Models, although quantifying this is difficult. This may have an impact upon the field seen in these areas. Further information on this issue is described in section \ref{Shield_Wall_Issues}.

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_YZplane_x_6100_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. South Side of the ISIS control rooms.}\label{fig:South_Side_ISIS_CR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XYplane_z_4280_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. East side of the ISIS control rooms.}\label{fig:East_Side_ISIS_CR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_2080_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. Floor of the the ISIS control rooms.}\label{fig:Floor_ISIS_CR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_5050_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. Ceiling of the the ISIS control rooms.}\label{fig:Ceiling_ISIS_CR}
\end{sidewaysfigure}

\clearpage

\subsubsection{MICE Local Control Room (MLCR)}

The MLCR is currently split into two sections. The section of the control room closest to the MICE hall contains much of the controls equipment and DAQ including the PPS interlock system. The section further away serves as the Control Room where the MICE staff operate the experiment from.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{MLCR_002.png}
\end{center}
\caption{Location of the MLCR with respect to the MICE magnets}\label{fig:MLCR_Loc}
\end{figure}

Figure \ref{fig:MLCR_Loc} shows the location of the MLCR with respect to the MICE magnets.

Figures \ref{fig:North_Side_MLCR} through to \ref{fig:Ceiling_MLCR} show the predicted field strength on the boundaries of the volume for Step IV of the experiment. These field maps were produced assuming no iron in the MLCR which is only an approximation as the racks are magnetic and so will draw some flux. The effect of this missing iron is unknown but given that the racks will represent small localised volumes of iron they are unlikely to significantly alter the field map, only localised effects would be expected. As anticipated the area closest to the hall will experience the largest air field, although typically for step IV the field doesn't appear to be much greater than ${\approx}$2.5 gauss.

Figure \ref{fig:MLCR_Vector} shows a vector field map through the plane of the MLCR at x=-1000~mm. Both the field gradient through the MLCR volume and the direction of the imposed field can be inferred from this. 

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Model_91_B_VECTORS_0_500_uT_XZplane_y_-1000_Structures_On.png}
\end{center}
\caption{Vector Field map through the MLCR and surrounding buildings at a height of ${\approx}$1.3~m below the ceiling of the MLCR. Note that the yellow walls are the outline of the ISIS control room walls as seen from above.}\label{fig:MLCR_Vector}
\end{figure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_YZplane_x_6090_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. MLCR North Side.}\label{fig:North_Side_MLCR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_BMAX_YZplane_x_16300_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. MLCR South Side. Note the different scale on this plot!}\label{fig:South_Side_MLCR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XYplane_z_5670_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. MLCR East Side.}\label{fig:East_Side_MLCR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XYplane_z_9125_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. MLCR West Side.}\label{fig:West_Side_MLCR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_-2630_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. MLCR Floor.}\label{fig:Floor_MLCR}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_500_uT_XZplane_y_360_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. MLCR Ceiling.}\label{fig:Ceiling_MLCR}
\end{sidewaysfigure}

\clearpage

\subsubsection{Rack Room 2 (RR2)}

Rack Room 2 is located in the South East Corner of the South Side Buildings and is indicated by the volume enclosed by magneta coloured wall in the ealier figure \ref{fig:MLCR_Loc}. The shape of the room as indicated in the model is only an approximation of its true shape, which is quite complex, however the volume shown by the FEA model encompasses the whole of RR2. The approximation was made to ease the modelling and the subsequent meshing, as the walls are non-magnetic and so are not a critical component of the simulation.

This room has been provided as overflow for MICE, and will be utilised to contain many of the racks that can no longer be placed in the MICE hall due to the stray field. Further details on this were presented in section [REFERENCE TO JASON'S SECTION]

RR2 is far enough away from the beamline that the additional fields imposed from Step IV running are negligible. Figure \ref{fig:RR2_Field} shows a cross section of the field through this region at a height of y=-1000~mm below beam height level (All of the buildings on the south side of the MICE hall are slightly lower in elevation than the MICE hall itself.) and gives some indication of the predicted field strength in this region from the MICE magnets. The direction of the field can be seen in the earlier figure \ref{fig:MLCR_Vector}. As the predicted field strength is of the order of ${\approx}{\leq}$0.5 gauss there are therefore no concerns with putting equipment into this room for MICE for Step IV.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Model_91_Bmod_0_100_uT_XZplane_y_-1000_Structures_On.png}
\end{center}
\caption{Bmod at y=-1000~mm. The fields through RR2 are negligble for Step IV. Note the scale on this plot is 1 gauss.}\label{fig:RR2_Field}
\end{figure}

\clearpage

\subsubsection{Behind the North Shield Wall (NSW)}

The North Shield Wall is of interest because there are a number of electrical systems that sit behind this wall. Beyond Step IV the RF tubes, amplifiers and their associated electronics will be placed here.
The North shield wall also partially shields the sub-station transformer, although this will be dealt with seperately in section \ref{sub_station}.

The MICE fibre trackers sit inside the bore of the largest superconducting magnets; however the associated DAQ electronics are constrained to be within 10~m of the tracker due to the length of the leads. The length of these leads means that it is possible to place the tracker racks behind the North Shield Wall. ***LINK TO CRAIG'S PHOTO*** This will only work if several other key components of the tracker system can be successfully shielded against the magnetic fields from MICE which are particularly large in this area due to their proximity to the magnets. The section [*** INSERT LINK TO KIRIL'S WORK ***] shows the work done to understand the local sheilding requirements that would be necessary if global sheilding was not to be employed.

*** CRAIG CAN WE HAVE A COPY OF A DRAWING THAT SHOWS WHERE THE RACKS COULD BE PLACED? (and can this be included here?)***

Before showing these plots it is once again worth pointing out that the MICE hall model assumes that the shield walls are dual skinned contiguous sheets of magnetic AISI~1010 steel. In reality these walls are not contiguous but are constructed from plates built onto an I-beam frame and unfortunately there are signficant gaps between where these plates join onto the beams. These gaps lead to flux leakage, which means that the performance of the shield wall may be worse than that indicated in these models. This issue is looked at in more detail in section \ref{Shield_Wall_Issues}. It is worth commenting that without the advantage of any measurements to compare with these models, the magnitude of this effect remains an uncertainty.

Figures \ref{fig:NSW_y0} and \ref{fig:NSW_x-6000} give some indication of the expected field strength behind the NSW. Note that both of these plots are on a 10 gauss scale.

*** CRAIG IS THERE A CONCLUSION THAT WE WISH TO DRAW FROM THESE PLOTS? FEEL FREE TO INSERT YOUR OPINION ***

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_1_mT_XZplane_y_0_Structures_On.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. 10 gauss Scale. This is a plot at beam height. The area behind the NSW that sits between the Linac wall, marked with a black box, is noticeably warmer than the area behind the west end of the NSW.}\label{fig:NSW_y0}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_91_Bmod_0_1_mT_YZplane_x_-6000_Structures_Off.png}
\end{center}
\caption{Bmod. Step IV 240~MeV/c Solenoid Mode. 10 gauss Scale. This is a plot at x=-6000~mm. See figure~\ref{fig:NSW_y0} to reference the plane. Once again it is clear that the area between the Linac wall and the NSW is seeing higher fields than behind the rest of the NSW}\label{fig:NSW_x-6000}
\end{sidewaysfigure}

\clearpage

\subsubsection{Quads Q7-Q9}

Without the addition of a return yoke the Quads Q7-Q9 will see a significant magnetic field. In the hall model the quads have been modelled as a solid lump of iron, which clearly is an inaccurate overassumption, but on the scale of the hall model this simplification was necessary.

Figures \ref{fig:Q9_Surface} through to \ref{fig:Q9_y0} show the effect that the quads are having on the field produced by the MICE magnets. There are a few points worth noting.  

Firstly the downstream face of Q9 sees signficant magnetisation, however without a more realistic representation of the ferrous mass of a quad it is difficult to know whether this is a signficant real issue or not.

Secondly in the hall model the line of quads act as a low reluctance path to the flux providing a path to the DSA and the fridge plates on the South Shield Wall. This path to the fridge is unintended and has the effect of shifting the field further upstream. Once again the sensitivity of this to a real quadrupole ferrous mass is unknown.

In conclusion if a return yoke was not to be employed this is an area that would require further refinement and study.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Quads_0_5T_001.png}
\end{center}
\caption{Bmod. 0.5T scale. Step IV 240 Mev/c Solenoid Mode. This shows the induced magnetisation in the ferrous components close to the magnets. The Quads, Q7 to Q9 left to right, have each been modelled as a solid ferrous lump, so the amount of iron has been overestimated. It is unclear what effect a more realistic model would have but it is clear that Q9 will see a large field from the cooling channel.}\label{fig:Q9_Surface}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Quads_100mT_001.png}
\end{center}
\caption{Bmod. 100mT Scale. Step IV 240 Mev/c Solenoid Mode . This plot shows how the quads pull the field from the magnets and how there is a hot spot around the end of the fridge plates; these are the plates that are part of the SSW and are perpendicular to the second quad, Q8, in the figure.}\label{fig:Q9_y0}
\end{figure}

%\clearpage

\subsection{Sub Models}

In order to study some features in further details, some sub-models were built. This section details these submodels and the results obtained from them.

\subsubsection{Trench}

*** I have added this subsection as I think Mike Courthold may have done some work on this and might wish to add something. If not then remove. *** Note I haven't yet asked him about this!

\subsubsection{Shield Wall Model}\label{Shield_Wall_Issues}

Both the north and south shield wall are constructed from plates of AISI~1010 steel bolted onto a steel frame, however in the MICE hall model the shield wall is modelled as a contiguous sheet of AISI~1010 steel with no supporting framework.

In principle this should be a reasonable approximation, however it has been noted that the plates from which the real steel wall are constructed have significant gaps between the plates. Given that the structural steel behind the plates (I beams) has poorer magnetic properties than the face plates and that additionally the I beam it is much thinner in cross section there was some concern that the real wall will not perform quite as well as the modelled one. Will there be significant field leakage behind these gaps thereby increasing the effective field behind the shield wall? 

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{Shield_Wall_Gaps_NSW.png}
\end{center}
\caption{Photograph of the north shield wall. The gaps between the plates can be clearly observed.}\label{fig:Gaps_NSW}
\end{figure}

In order to understand this issue a separate high resolution model of the shield wall was built incorporating these gaps to see if this effect was significant. The vertical I beams were included in the model, however the horizontal I beams were not included as this would have made the meshing difficult. As the return flux will be dominated by the plates in the horizontal direction the horizontal I beams are less important. The meshing of the model was very fine to capture the detail in the gap and to get an accurate representation of what is happening in the gap region. The model included the Step IV magnets at 240~MeV/c Solenoid mode.

The First section of wall that was modelled was composed of two by three panels. This model demonstrated that gap leakage would be observed but the model was of insufficient size to make suitable comparisons with the model of the shield walls in the MICE hall model. For brevity these results are not included here but are available online.

Subsequently a second model with a larger wall composed of three by seven panels was modelled. This wall is approximately half the size of one of the real shield walls. A 7~mm panel gap was used in all of the models.
Figure \ref{shield_wall_model_01} illustrates how the shield wall looks with respect to the Step IV magnets. The I beams can be seen jutting out from the between the two layers of plates. Figures \ref{shield_wall_plot_01} and \ref{shield_wall_plot_03} shows the magnitude of the field behind the 7${\times}$3 shield wall model at a distance of 25~mm and 250~mm respectively behind the shield wall. There are clear peaks behind the location of the I beams where the continuity of the plates are broken. In this particular model the plates are standing off from the I beams by 1.5~mm which is to simulate the fact that the join between the plates and the I beams is not too good. The 1.5~mm is excessive and so represents a worse case scenario but some of the two by three panel models had perfect contact between the plates and the I-beams and an increase in the magnitude of the field behind the shield wall was still predicted, although the magnitude of the effect was much subdued. The models are sensitive to the reluctance of this magnetic connection between the plates and the I-beams and it is difficult to estimate what the reluctance of this connection is. This remains a large uncertainty in the modelling.

Finally plots \ref{shield_wall_plot_02} and \ref{shield_wall_plot_04} shows a comparison with the predicted field behind the South shield wall in model 91. There is a small discrepancy in the location of these two shield walls (${\approx}$3\%) but the difference in the predicted field due to the gaps inbetween the plates is striking.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{shield_wall_model_01.png}
\end{center}
\caption{This shows the 7${\times}$3 shield wall model. The model is symmetrical in x and y to speed up the solve time. Consequently there is a symmetrical shield wall at the same distance on the -x axis that is not shown in this figure.}\label{shield_wall_model_01}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{shield_wall_plot_01.png}
\end{center}
\caption{This shows the magnitude of the field behind the 7${\times}$3 shield wall model at a distance of 25~mm behind the shield wall. There is  a 1.5~mm offset between the plates and the I-beams in this model} \label{shield_wall_plot_01}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{shield_wall_plot_03.png}
\end{center}
\caption{This shows the magnitude of the field behind the 7${\times}$3 shield wall model at a distance of 250~mm behind the shield wall. There is  a 1.5~mm offset between the plates and the I-beams in this model} \label{shield_wall_plot_03}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{shield_wall_plot_02.png}
\end{center}
\caption{This is a comparison from plot (figure \ref{shield_wall_plot_01}) compared with the predicted field 25~mm behind the South Shield Wall from model 91. The South shield wall is slightly further away from the magnets than the standalone shield wall model so the south shield wall data is taken slightly further away from the axis (3752~mm vs 3636~mm). Despite this small discrepancy in distance to the shield wall (${\approx}$3\%) the difference in the predicted field behind the wall is striking.}\label{shield_wall_plot_02}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{shield_wall_plot_04.png}
\end{center}
\caption{This is an identical comparison from plot (figure \ref{shield_wall_plot_03}) compared with the predicted field 250~mm behind the South Shield Wall from model 91. As previously, the South shield wall is slightly further away from the magnets than the standalone shield wall model so the south shield wall data is taken slightly further away from the axis (3966~mm vs 3861~mm).}\label{shield_wall_plot_04}
\end{figure}

***
NOTE: I have included the results that we have but ideally I would like to re-run the shield wall model at the NSW distance with no gap, a 1~mm gap and no gap with a  contiguous wall for comparison with model~91. As this is a computationally expensive model I will try to run this model over the summer holidays. If not then we will be stuck with what we have.

(It would also be nice to re-run these models at a distance that is comparable with SSW given that current model has a small offset error in it but this is less important.)

***

\subsubsection*{Conclusions}

The shield wall models have indicated that for the case where the shield walls are not contiguous that any gaps in the shield wall do have an effect upon the field that is observed behind the shield wall. The increase and the localisation of this increase in the field are dependent upon a number of assumptions that are implicit within the model. It would be difficult to further quantify or justify these assumptions without recourse to some magnetic measurements. As we would be unable to obtain such measurement until the MICE magnets are in-situ and turned on this leaves us in an uncomfortable situation that we have a known problem that is difficult to quantify.  

This does leave several questions unanswered.

Is there a global increase in the field behind the shield wall due to the gaps (as the shield wall is less effective than anticipated) and how far does this extend?
	
How far do the local incursions extend?
 
Do these incursions affect any equipment? If we don't build a return yoke the is the shield wall considered to be a safe haven for the tracker racks?

Do these incursions affect any walkways that were assumed to be safe?

It was already anticipated that a return yoke would be needed for Step V/VI of the experiment so on this assumption any magnetic leaks from the shield wall are less likely to be of concern for the RF (which wouldn't be used until any return yoke was in-situ). However even with reduced external fields do these leaks create local non homogenous areas in the field that would have an adverse effect on the RF tubes? Would MICE be better off without the shield walls at all in this area? This is something that will need separate consideration once a decision has been made. 

\subsubsection{Sub-Station Sub Model}\label{sub_station}

%*** Ideally I'd like to rerun the models with more iron mass in them but this is unlikely to happen before this report can be published. ***

The MICE hall contains a modern substation that provides power for the experiment. This substation sits in the North West corner of the MICE hall and is partially shielded by the North Shield Wall. As the substation is relatively new there is a significant amount of control electronics, including programmable logic controllers, embedded into the substation's front panels.  There is some concern that these electronics could be adversely affected by stray magnetic fields.

Figure \ref{sub_station_photo} shows a photograph of the MICE substation. The base MICE hall model did not contain a model of the substation but the substation represents a signficant mass of iron, estimated at ${\approx}$5300~kg. In order to ascertain whether the substation would have an impact upon the field distribution in the MICE hall 5300~kg of mild steel in a 14~mm shell that was an approximate cuboidal representation of the substation geometry and location was added to model~91 and run as model~113. The results from this model through the plane y=0 can be seen in figure \ref{model_113_y0_on}. Other figures can be viewed on-line. Compared with model~91, there is minimal change to the field distribution in the area that the substation is located and it appears that the maximum air field that the substation will see is approximately 2~gauss, although for the most part the field is much lower.  One can also ascertain that a significant field component runs South to North, although there is an East to West component at the West end of the substation. An equivalent model for Step~VI of the experiment has not been run.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=13cm,keepaspectratio]{sub_station_01.jpg}
\end{center}
\caption{This photograph of part of the substation in the MICE hall clearly shows that a signficant amount of electronics has been mounted in the front panels.}\label{sub_station_photo}
\end{figure}

To understand the effect of an external field on electronic equipment mounted on the surface of the substation panels several submodels were run where the substation was modelled as a series of racks with a 2~mm thick mild steel skin. The first set of submodels contained a single window in each rack that measured 450~mm high by 300~mm wide. The second set contained four windows in each rack, with each window measuring 100~mm by 100~mm on 150~mm width centres and 200~mm height centres. It should be noted that as only the outer panels were modelled these sub-models only give a total mass of 1000~kg but as we are primarily interested in the field level in the  windows of these 2~mm thick panels and that by and large the internal iron in these racks is some distance from these windows, this was considered to be sufficient to give a first estimate of the effect.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=10cm,keepaspectratio]{Sub_Station_02_Model_14_Bmod_101.png}
\end{center}
\caption{Plot of the steel around the second window from the West end in the substation model. 5 gauss external field running from South to North (from +x). 750 gauss scale.}\label{sub_station_window_1_750}
\end{figure}

%\begin{figure}[!htb]
%\begin{center}
%\includegraphics[width=13cm,keepaspectratio]{Sub_Station_02_Model_14_Bmod_102.png}
%\end{center}
%\caption{Plot of the air volume inside the second window from the West end in the substation model. 25 gauss scale.}\label{sub_station_window_1_25}
%\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=10cm,keepaspectratio]{Sub_Station_02_Model_14_Bmod_103.png}
\end{center}
\caption{Plot of the air volume inside the second window from the West end in the substation model. 5 gauss external field running from South to North (from +x). 10 gauss scale.}\label{sub_station_window_1_10}
\end{figure}

An external field of 5~gauss was applied to these models, this was a deliberate overestimate of the external field based upon the air field results from model~91; some of the the submodels were completed before model~113 was solved. The submodels were run with both the field in a South to North direction and in and East to West direction for comparison. As none of the steel in the submodels is saturated then the results should to first order scale to a lower air field.

Figure \ref{sub_station_overview_01} shows a cross section through the substation model at y=0 with a 5 gauss field from +x (South to North). This is a model with a single large window in each rack. One can immediately see that the area around the windows is seeing a field of approximately 5~gauss.

Figure \ref{sub_station_window_1_750} and \ref{sub_station_window_1_10} shows a couple of view of one of the windows in the submodel. The first image shows the steel surrounding the window and it's relative magnetisation. The other plot shows the predicted field strength in the window on a 10~gauss scale. One can see that the field in the window is approximately the same as the applied external field (5~gauss) although there is some intrusion of higher field from the left and the right.

When the external field is applied East to West, a similar picture emerges except that the window plots see a slightly higher field of ${\approx}$20{\%} and there is more intrusion of the field from the steel into the window space - see figure \ref{sub_station_window_1_E_10}. A similar picture is seen for models that contain sets of four smaller windows instead of the one larger window.

%\begin{figure}[!htb]
%\begin{center}
%\includegraphics[width=13cm,keepaspectratio]{Sub_Station_01_Model_15_Bmod_101.png}
%\end{center}
%\caption{Plot of the air volume inside the second window from the West end in the substation model. 25 gauss scale.}\label{sub_station_window_1_E_25}
%\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=10cm,keepaspectratio]{Sub_Station_01_Model_15_Bmod_102.png}
\end{center}
\caption{Plot of the air volume inside the second window from the West end in the substation model. 5 gauss external field running from East to West (from +z) 10 gauss scale.}\label{sub_station_window_1_E_10}
\end{figure}

\subsubsection*{Conclusions}

The conclusion from these submodels is that for Step IV the substation does not appear to significantly alter the field profile and consequently the models predict that the substation will see a maximum field of approximately 2~gauss. Further submodelling indicates that any instrumenation mounted in the 2~mm thick steel panels will not see significant amplification of the external magnetic field due to their proximity to these steel panels and that the instruments should only see a level of magnetic field that is consistent with the magnitude of the external air field or slightly above it. A slightly higher level of magnetic field can be expected at the edges of the cutouts/windows. 

It should be bourne in mind that these conclusions are based upon the predictions from the model only and these have no grounding in any field measurements. Consequently as the models are only an approximation to both the substation geometry and mass distribution the output from these models should be viewed as a guide.

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Model_113_Bmod_0_500_uT_XZplane_y_0_Structures_On.png}
\end{center}
\caption{This is an image from the Hall model with the sub-station mass added - Model 113. Step IV Solenoid mode 240~Mev/c. The addition of the substation pulls the field slightly towards the north shield wall but change in the field distribution is minor.}\label{model_113_y0_on}
\end{sidewaysfigure}

\begin{sidewaysfigure}[!htb]
\begin{center}
\hspace*{-1.5cm}
\includegraphics[width=23.5cm,keepaspectratio]{Sub_Station_02_Model_14_Bmod_07.png}
\end{center}
\caption{Bmod. Substation model. 5~gauss external field applied from +x (South to North). Note that the model boundary is much further out than the limits of the plot. 10 gauss scale. The field in the vicinity of the windows can be clearly seen.}\label{sub_station_overview_01}
\end{sidewaysfigure}

\clearpage