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\begin{document}
\title{Production of Low Cost, High Field Quality Halbach Magnets}
\author{S.J. Brooks\thanks{sbrooks@bnl.gov}, G. Mahler, J. Cintorino, A.K. Jain,\\ Brookhaven National Laboratory, Upton, Long Island, New York}
\maketitle
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\begin{abstract}
A shimming method has been developed at BNL that can improve the integrated field linearity of Halbach magnets to roughly 1 unit (1 part in $10^4$) at r=10mm. Two sets of magnets have been produced: six quadrupoles of strength 23.62T/m and six combined-function (asymmetrical) Halbach magnets of 19.12T/m with a central field of 0.377T. These were assembled using a 3D printed plastic mould inside an aluminium tube for strength. A shim holder, which is also 3D printed, is fitted within the magnet bore and holds iron wires of particular masses that cancel the multipole errors measured using a rotating coil on the unshimmed magnet. A single iteration of shimming reduces error multipoles by a factor of 4 on average. This assembly and shimming method results in a high field quality magnet at low cost, without stringent tolerance requirements or machining work. Applications of these magnets include compact FFAG beamlines such as FFAG proton therapy gantries, or any bending channel requiring a $\sim$4x momentum acceptance. The design and shimming method can also be generalised to produce custom nonlinear fields, such as those for scaling FFAGs.
\end{abstract}
\section{Requirements}
The magnets were designed as prototypes for an earlier version of the CBETA \cite{CBETA} lattice, a non-scaling FFAG arc of radius $\sim$5m transmitting 67--250MeV electrons. Table~\ref{params} shows the required fields and sizes. The CBETA magnets were going to be twice the length of these prototypes with two layers of permanent magnets (PMs) longitudinally.
% Could explain QF and BD and fact they are centered around beams
\begin{table}[hbt]
\centering
\caption{Parameters of the Two Magnet Types}
\begin{tabular}{lccc}
\toprule
\textbf{Parameter} & \textbf{QF} & \textbf{BD} & \textbf{units} \\
\midrule
Length&57.44&61.86&mm\\
Dipole $B_y(x=0)$&0&0.37679&T\\
Quadrupole $dB_y/dx$&23.624&-19.119&T/m\\
Bore radius to PMs&37.20&30.70&mm\\
...to shim holder&34.70&27.60&mm\\
Max field at PMs&0.879&0.964&T\\
Max field at r=1cm&0.236&0.568&T\\
Outer radius of PMs&62.45&59.43&mm\\
...of tubular support&76.2&76.2&mm\\
\bottomrule
\end{tabular}
\label{params}
\end{table}
% These are parameters for the prototypes including the "Hall probe sign error", this is consistent with the figures
\section{Magnet Design}
The magnets are based on a 16-segment Halbach design (Fig.~\ref{magQF}). For the combined-function magnet `BD' (Fig.~\ref{magBD}), the wedge thicknesses and magnetisation angles were optimised to give the combined field directly. This uses less PM material than nesting a Halbach dipole and quadrupole.
\begin{figure}[!htb]
\centering
\includegraphics*[width=\linewidth]{THPIK007f1}
\caption{Cross-section of the quadrupole `QF' magnet. Blue arrows show magnetisation direction of the PM blocks. The orange line graphs the mid-plane field $B_y(x,0)$, with green highlighting the good field region and red showing the beam position range in the FFAG. The grid has 1cm spacing.}
\label{magQF}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics*[width=\linewidth]{THPIK007f2}
\caption{Cross-section of the BD magnet.}
\label{magBD}
\end{figure}
\newcommand{\B}{\mathbf B}
\newcommand{\Jv}{\mathbf J}
\newcommand{\jv}{\mathbf j}
\newcommand{\M}{\mathbf M}
\newcommand{\Div}{\nabla\cdot}
\newcommand{\curl}{\nabla\times}
\newcommand{\nhat}{\hat{\mathbf n}}
\newcommand{\vectwo}[2]{\left[\begin{array}{c}#1\\#2\end{array}\right]}
The field calculation for this design starts with Maxwell's magnetostatic equations in a material:
\[ \Div\B=0 \qquad \curl\B=\mu_0(\Jv+\curl\M). \]
It is approximated that the magnetisation $\M$ does not vary with applied field, i.e. that the PM material has $\mu_r=1$. It is also assumed that each PM block has a constant $\M$ vector. This means that on a boundary with outward unit normal $\nhat$, the magnetisation is equivalent to a surface current
\[ \jv_s = -\nhat\times\M. \]
Each edge of a polygonal PM block therefore produces a sheet current, which in the 2D approximation extends infinitely in $z$. Such a sheet with current $j_s$ travelling from $(0,0)$ to $(a,0)$ produces a field
\[ \B(x,y)=\frac{\mu_0j_s}{2\pi}\vectwo
{-\arctan(x/y)+\arctan((x-a)/y)}
{\frac12(\log(x^2+y^2)-\log((x-a)^2+y^2))}, \]
which can be rotated and summed over all PM block edges.
The strength is set via $\mu_0|\M|=B_{r1}$, where $B_{r1}$ is an approximation to the true remnant field $B_r$ of the material. A value $B_{r1}