%
%****************************** Copyright Notice *******************************
% *
% Copyright (c) 2000 Universities Research Association, Inc. *
% All Rights Reserved. *
% *
%*******************************************************************************
%
%********************************************************************
%********************************************************************
%**** ****
%**** ****
%**** Fermilab Main Injector Department ****
%**** ****
%**** Document Subject: Accelerator Dsgn ****
%**** ****
%**** ****
%********************************************************************
%********************************************************************
%
%+ MI-0259
%
% Environment: LaTeX source file
%
% Related Software:
% Related Hardware:
%
% Document Cross References: MI-0162, MTF-96-0008, MI-0204, MI-0207...
%
% Modification History:
% Version Author Date Description of modification,deficiencies
% 0.1 BCBROWN 04-Jan-2000 Initial version
% 0.5 BCBROWN 10-Jan-2000 Completed first draft
% 0.55 BCBROWN 14-Jan-2000 Another
% 0.6 BCBROWN 24-Jan-2000 For Review
% 0.67 BCBROWN 25-Jan-2000 Final Draft
% 1.1 BCBROWN 28-Jan-2000 Completed
% 1.2 BCBROWN 02 Feb-2000 Improved description of orbit integration
% -- 16-Feb-2000 and end dipole (not there). Added
% Appendix on second order orbit shape.
% BCBROWN 03-Apr-2000 Include Thornton Murphy comments
% BCBROWN 05-May-2000 Give up on getting further comments
% 1.3 BCBROWN 08 May-2000 Words after equation 17 in Appen A
%-
%******************************************************************************
%******************************************************************************
%
% Use LATeX
%
\documentstyle[11pt,fermiarticle,epsf,rotate]{article}
\pagestyle{myheadings}
\def\topfraction{.99}
\def\bottomfraction{.99}
\def\textfraction{0.01}
\newcommand{\mtfdocnum}{MI-0259}
\newcommand{\mtfdocver}{1.3
}
\date{ 5/8/00
}
\markboth{\mtfhead}{\mtfhead}
\title{Transverse Placement of Recycler Gradient Magnets}
\author{Bruce C. Brown and David E. Johnson \\
Main Injector Department, Beams Division\\
{\em Fermi National Accelerator Laboratory}
\thanks{Operated by the Universities
Research Association under contract with the U. S. Department of Energy}
\\ \em P.O. Box 500 \\ \em Batavia, Illinois 60510\\
}
\begin{document}
\bibliographystyle{unsrt}
\mtfdocnv{\mtfdocnum}{\mtfdocver}
\maketitle
%\begin{center}
%\huge DRAFT -- DRAFT -- DRAFT
%\end{center}
\newpage
\tableofcontents
\newpage
\def\topfraction{.99}
\def\bottomfraction{.99}
\def\textfraction{0.01}
\begin{abstract}
To bend the 8 GeV antiprotons on the Recycler design orbit,
straight gradient magnets have been built to provide the design
integral field on a straight path through the magnet as well as on the
design orbit. This requires installation with a transverse
displacement by $d/3$ at the magnet center where $d$ is the sagitta.
The transverse alignment of the gradient magnets for the Fermilab
Recycler Ring used a slightly different set of numbers which will be
described. Bend effects of design sextupole fields and the fields
from end shims will be calculated.
\end{abstract}
\section{Introduction}
\pgph The Recycler Ring\cite{Jackson:FRRTDR} uses gradient magnets for
bending and most of the focusing in the ring. They are specified as
straight (rectangular) magnets with a uniform dipole, quadrupole and
sextupole components along the length. At the design stage, several
notes were written exploring the requirements for placing these
magnets in the ring. Norman Gelfand (MI-0200\cite{Gelfand:RCFM} and
Steve Holmes (MI-0195\cite{Holmes:ROTCFM},
MI-0196\cite{Holmes:TMMRPRCFM}) explored issues concerned with the
interaction between the specifications for straight magnets and the
curved particle orbit which will be experienced in the Recycler.
Issues of transverse placement were also discussed in
MI-0207\cite{BCBrown:ARRRGNLU}. We will review these issues and
document the alignment procedures used for Recycler Magnet
installation through the end of 1999.
\section{Mathematical Description}
\pgph We obtain an explicit polynomial description of the lattice and
magnets in the following way.
\begin{itemize}
\item Since the magnet design is for a straight hybrid permanent gradient
magnet, the normalized harmonics of the design field are uniform along
the length (neglecting end shim effects).
\begin{equation}
B_y(x) = B_0 (1 + b_2 \frac{x}{a} + b_3 (\frac{x}{a})^2)
\label{Eq:B_transv}
\end{equation}
where a is the reference (normalization) radius for the harmonic
representation of the fields.
\item The orbit through a single gradient dipole is adequately
represented by a polynomial (parabola), since the deviation from a
circular orbit is small (See Appendix~\ref{Appen:Orbit}). The
``natural'' description for the design orbit thru a {\em curved}
dipole would describe $x$ as $x(s) = x_{o}$, where s is the parameter
which identifies the coordinate along the (curved) orbit. For a
uniform, straight dipole, if z is a rectilinear coordinate with zero
at the magnet center, we describe the orbit by
\begin{equation}
x = x_0 - d(\frac{2z}{L})^2.
\label{Eq:Para_nooff}
\end{equation}
$d$ is called the sagitta and is the distance from the chord
to the arc of the circle. For a magnet of length $L$ which deflects
by an angle $\theta$, the sagitta is given, in the small angle approximation, by
\begin{equation}
d = R (1- \cos (\frac{\theta}{2}) ) = \frac{R \theta^2}{ 8} = \frac{L \theta}{8}.
\label{Eq:sagitta}
\end{equation}
\end{itemize}
We note that the path length difference between the straight line used
for measurement and the circular orbit is $\delta s = R \theta - R (2
\sin \frac{\theta}{2}) \approx L \theta^2/24 = 8.14 \times 10^{-5} m$
or 18.1 ppm for the regular cell gradient magnets in the Recycler.
\subsection{Bend of Gradient Magnet}
\pgph The Recycler gradient magnets were specified by the integrated
fields which are measured by a straight harmonic probe which is
inserted along and rotated about the transverse center of the gradient
magnet. The field integral was selected to be that which would bend
an 8 GeV (kinetic energy) proton or antiproton in a circle using 301
1/3 regular cell bends (assuming that the dispersion suppressor bends
provide 2/3 of the deflection of a regular cell bend). Let us
calculate the bend which will be achieved for a particle on a
parabolic (circular) orbit. We integrate Equation~\ref{Eq:B_transv}
along the trajectory given by Equation~\ref{Eq:Para_nooff}. Let us at
first neglect the small sextupole ($b_3$) component.
\begin{equation}
\int_{-L/2}^{L/2} B d\ell = B_0 L (1 -
\frac{b_2}{a}(\frac{d}{3}-x_0))
\end{equation}
Thus, by placing the dipole at a transverse offset of $x_0=d/3$, the
integrated field on a circular orbit will be independent of the quadrupole term
and have the same value as the integral on a straight line. Restating this
description, we place the magnet center such that the circular orbit is at
a displacement of $d/3$ to cancel the first order bend effects of the gradient.
This places the design orbit at a displacement of $-2d/3$ at each end of the
magnet.
With this in mind, for the rest of this presentation, we will describe
the orbit with this term built into the description.
\begin{equation}
x = x_{offset} + x_{slope} z + d (\frac{1}{3} - (\frac{2z}{L})^2).
\label{Eq:Orbit}
\end{equation}
\section{Magnet and Alignment Details}
\pgph Gradient magnets for the recycler\cite{MT15:rrg} consist of a
pair of shaped poles, flux sources consisting of ferrite and
compensator, and an outer shell for flux return. The slot length of a
recycler gradient is set by the length of the shell but the length
over which the bending occurs is determined by the pole length. The
pole consists of steel pieces which are uniform in transverse
dimensions with a fixed length set by the design. To this pole is
added at each end a field shaping shim whose length varies across the
radial direction to create a correction to the integral field. We
designate the fixed length as $L$ in the above formulas, recognizing
that the bend effects are modified by the longitudinal distribution of
the bricks as well as the effects of the variable length end
shims\footnote{The central value (at $x=0$) of the end shim length is
also fixed.} but these variations are small and will be unimportant
for this set of calculations.
\begin{table}[b]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
&Length &dipole &\multicolumn{2}{c|}{quadrupole}&\multicolumn{2}{c|}{sextupole}\\ \hline
Magnet & L & $B_0$ & $B_2$ & $b_2$ & $B_3$ & $b_3$ \\ \hline
Name & m & $Tesla$ & $T/m$ & @$1^{\prime\prime}$ & $T/m^2$ & @$1^{\prime\prime}$ \\ \hline\hline
RGF & 4.4958 & 0.13752 & .3355 & .06197 & 0.1853 & 8.696e-04\\ \hline
RGD & 4.4958 & 0.13752 & -.3238&-.05981 & -0.3209 & -15.05e-04\\ \hline
SGF & 3.0988 & 0.13301 & .66816 & .1276 & 0.0000 & 0.0000 \\ \hline
SGD & 3.0988 & 0.13301 & -.68236 &-.1303 & 0.0000 & 0.0000 \\ \hline
\end{tabular}
\caption{A Current Set of Design Properties of Gradient Magnets for
the Recycler Ring. These are restated from the Recycler Magnet Web
page. Normalized harmonics are quoted at a reference radius of
$1^{\prime\prime}$.}
\label{Tab:MagDesign}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% The following Text was extracted from the Recycler Web %%%%%%%%%%
%%%%%%%%%% Page on Gradient Magnets %%%%%%%%%%
%% http://www\-fermi3.fnal.gov/recycler/magnets/gradient\_list.html %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%Recycler Ring Gradient List
%
%Overview
%
%The Recycler an 8 GeV kinetic energy antiproton storage ring of circumference 3.3 km built only with permanent
%magnets. The present version of the magnetic lattice is v18. In this document the magnet counts and design properties
%are listed. Note that the value of B[[pi]] for the Recycler ring is 296.5 kG-m. The phase advance per cell and betatron
%tunes for lattice version 18 are:
%
% Horizontal Phase Advance per Cell: 85.387 deg.
% Vertical Phase Advance per Cell: 79.22 deg.
% Horizontal Tune: 24.4254
% Vertical Tune: 24.4188
% Gamma-T: 19.968047
%
%
%
%There are 4 different kinds of gradient magnets in the Recycler ring lattice. Their magnet type specifications are:
%
% 1.RGF: Focussing gradient magnets in the normal arcs
% 2.RGD: Defocussing gradient magnets in the normal arcs
% 3.SGF: Focussing gradient magnets in the dispersion suppression cells
% 4.SGD: Defocussing gradient magnets in the dispersion suppression cells
%
%
%
%RGF
%
% Pole Length 177 in. 4.4958 m
% Central Field 1.3752 kG
% Angular Deflection 20.9 mrad
% Magnet Count 108
% BL 6.182432402 kG-m
% k1 0.01131616 1/m2
% B' 3.3552531 kG/m
% B'L 15.084547 kG
% b1 619.7358305 units
% k2 0.01250234 1/m3
% B" 3.706956663 kG/m2
% B"L 16.66573576 kG/m
% b2 8.695660047 units
%
%
%RGD
%
% Pole Length 177 in. 4.4958 m
% Central Field 1.3752 kG
% Angular Deflection 20.9 mrad
% Magnet Count 108
% BL 6.182432402 kG-m
% k1 -0.01092083 1/m2
% B' -3.2380373 kG/m
% B'L -14.55757 kG
% b1 -598.085362 units
% k2 -0.0216429 1/m3
% B" -6.417142099 kG/m2
% B"L -28.85018745 kG/m
% b2 -15.05312612 units
%
%
%SGF
%
% Pole Length 122 in. 3.0988 m
% Central Field 1.3301 kG
% Angular Deflection 13.9 mrad
% Magnet Count 64
% BL 4.121621601 kG-m
% k1 0.022534721 1/m2
% B' 6.681568 kG/m
% B'L 20.704843 kG
% b1 1275.961405 units
% k2 0.0 1/m3
% B" 0.0 kG/m2
% B"L 0.0 kG/m
% b2 0.0 units
%
%
%SGD
%
% Pole Length 122 in. 3.0988 m
% Central Field 1.3301 kG
% Angular Deflection 13.9 mrad
% Magnet Count 64
% BL 4.121621601 kG-m
% k1 -0.023013743 1/m2
% B' -6.8235985 kG/m
% B'L -21.14497 kG
% b1 -1303.084596 units
% k2 0.0 1/m3
% B" 0.0 kG/m2
% B"L 0.0 kG/m
% b2 0.0 units
%
%
%
%Entered by Gerry Jackson gpj@fnal.gov on 2/16/98
%
%
Let us at this point document the design features of the Recycler.
The magnet design was specified on 2/16/98 in\\
http://www\-fermi3.fnal.gov/recycler/magnets/gradient\_list.html and
this information was then entered into the MTF database where it was
accessed for magnet measurement. The values stored there are reported
in the database report at\\
http://www-ap.fnal.gov/MagnetData/PAGE/series\_report\_main.html. At
this point we will document the features which these specifications
imply. For a proton (antiproton) kinetic energy of 8 GeV, the
corresponding momentum is 8.88889 GeV/c. The magnetic rigidity is
$B\rho = 29.6501$ T-m. The integrated bend field is $\int B d\ell = 2
\pi B\rho = 186.297$ T-m. The regular cell bends (RGF and RGD) each provide
1/(301 1/3) of this or 0.618243 T-m. The dispersion suppressor bends
(SGF and SGD) provide 2/3 of that or 0.412162 T-m. Precisely these
numbers appear in the Recycler specification and the MTF Database.
The pole length is specified as $L= 177^{\prime\prime}$ for the regular
cell magnets and $L= 122^{\prime\prime}$ for the dispersion suppressor cells.
The approximation used in this note will assume that the specified bend is
achieved on a circular arc of chord length $L$. Known corrections due to
longitudinal offsets and Mean Squared Length differences are second order
for this calculation and have a small effect.
\subsection{Offset Desired}
\begin{table}[b]
\centering
\begin{tabular}{|l|r|r|c|l|}
\hline
Series & RGF / RGD & SGF / SGD & Units &Comments\\ \hline
Offset & 0.024475012& 0.010793394 & feet &For Installation 1999 \\ \hline
Offset &7.46 & 3.29 & mm &For Installation 1999 \\ \hline
$2d/3$ &7.81 & 3.59 & mm & displacement for parabola\\ \hline
$x_{offset}$&-0.35 & -0.30 & mm & For Equation~\ref{Eq:Orbit}\\ \hline
Corr Offset & 7.87 7.76 & 3.61 3.56 & mm & Inc. Quartic Terms\\ \hline
$x_{offset}$&-0.41 -0.30&-0.32 -0.27 & mm & w/ Quartic for Eq~\ref{Eq:Orbit}\\ \hline
%Series RGx SGx
%feet 0.024475012 0.010793394
%inches 0.293700144 0.12952073
%mm 7.459983658 3.289826552
%
%L 4.4958 3.0988
%Theta 0.020851279 0.013900852
%d 0.011717897 0.005384495
%
%d/2 5.858948666 2.692247598
%d/3 3.905965778 1.794831732
%2d/3 7.811931555 3.589663464
%
% -0.3519479 -0.29983691
\end{tabular}
\caption{The radial displacement of the design ends of the poletip in
the accelerator reference frame (LTCS). The first row gives the
prescription from MI-0196 translated to survey feet which was
specified to the alignment group. The second row gives this in mm. A
displacement from a parabolic orbit of $2d/3$ eliminates the first
order bend change due to the gradient. $x_{offset}$ is defined above
to specify the radial displacement from the desired design orbit. The
displacement shown on the line labeled Corr Offset is the correct
shift including the first order orbit shape changes (orbit in gradient
%field). See Appendix~\ref{Appen:Orbit}. The specification used for
field). See Appendix~A. The specification used for
alignment was based on MI-0196 which corresponded to a shorter magnet
design. }
\label{Tab:Offsetspec}
\end{table}
\pgph If we used the prescription above to set the magnets, the ends
($z=\pm L/2$) would be displaced so that in the magnet's reference
frame (as we are using for these calculations), the design orbit
enters at $-2d/3$ (inside). In the tunnel reference frame, the orbit
is fixed and we displace RGF or RGD magnets (outside) by 7.81 mm and
SGF or SGD magnets by 3.59 mm. As shown above, this will eliminate
bending effects due to the gradient. In Table~\ref{Tab:Offsetspec} we
show these values and the similar numbers assuming the orbit
calculated for a gradient (see Appendix~\ref{Appen:Orbit}). Since the
installation used values from MI-0196, we calculate the orbit offset
we expect in the magnet frame. Corrections for the effects of
sextupole body field and other effects will be shown below.
\subsection{Prescription for Magnet Placement}
The procedure for installation of the gradient magnets in the
tunnel\cite{IAA99_Recycler} involved several steps. The lattice
program MAD was used with version RR19 of the Recycler lattice
description to produce a survey output file in terms of the Main
Injector LTCS (Local Tunnel Coordinate System) metric site coordinate
system. These coordinates specified the design central orbit
trajectory and were given at the magnetic pole tips of the gradient
magnets and quads, and at the center of the bpm assembly. The design
orbit had a circumference of 3319.418828 meters. These site
coordinates were converted into survey feet (39.37/12) and transmitted
to the Survey and Alignment group to be transferred to the ceiling of
the Main Injector enclosure and used for initial placement of the
gradient magnets. These pole-tip marks on the ceiling specified the
design longitudinal bend center of all the gradient magnets.
The magnet fabrication determined the relation between the iron poles
and the fiducial cups (survey plugs) which were built into 4 locations
at each end of each magnet. For radial positioning, adequate
precision was obtained by fabrication. For the vertical position, the
pole shape was sufficiently accurate but the tolerance buildup
including the vertical depth of the fiducial cup required measurement.
A procedure was implemented and documented in the magnet fabrication
traveler to measure (redundantly) the vertical offset between the back
face of the pole and the reference face of the fiducial cup. James
Volk supervised this work and prepared a summary for all the magnets
which was then passed to the survey and alignment group. The results
were histogrammed, revealing that the corrections were typically less
than 0.25 mm (0.01$^{\prime\prime}$).
The longitudinal offset (z\_cent\_off) of the bend center with respect
to the physical pole tip length was measured for each gradient
magnet\cite{TD98015v1.2}\cite{TD98013v1.1}. During the initial
placement of the gradient magnets in the tunnel, this longitudinal
offset was applied to displace the magnet along the tangent to the
design orbit at the bend center\cite{Gattuso:RMAO}. The initial
transverse placement of the gradient magnets placed the magnet
centerline on the design orbit trajectory without taking into account
any sagitta correction.
The MAD survey output file was further processed to provide final
alignment specifications. The transverse offsets (sagitta correction)
applied to the Recycler magnets used the prescription described in
note MI-0196\cite{Holmes:TMMRPRCFM}. Table~\ref{Tab:Offsetspec}
describes the transverse displacements (offset) used for the RGF and
RGD magnets (regular arc cells) and the SGF and SGD magnets
(dispersion suppressor cells). Individual gradient magnet
longitudinal offsets (z\_cent\_off) were taken from the z-scan
measurements. The program read the MAD survey output file.
Transverse offsets were applied to shift the magnet centerline at the
bend center perpendicular to the tangent of the reference orbit.
Since the stands provided no longitudinal adjustment, up to 8 mm of
error in longitudinal placement was permitted. Since the offsets were
specified to the laser tracker with respect to the local tangent, the
horizontal and vertical setting tolerance of 0.25 mm were not
compromised. The longitudinal offset was applied parallel to the
tangent. Coordinates for the pole tip corners were calculated. A
revised file was produced of site coordinates (X,Y,Z) at each pole-tip
which represented the shifted location of the center of the magnet
steel. These new coordinates were then transferred to the survey and
alignment group where Babatunde Oshinowo used these revised site
coordinates, the design locations of the four fiducial cups at each
end, and the measured vertical offsets to calculate the site
coordinates of the fiducial cups. These were installed into the laser
tracker\footnote{The recycler alignment used an interferometric laser
tracker system. Model SMX Tracker4000 was employed with associated
Insight software. Its properties as well as the specifications and
procedures for recycler alignment are documented by O'Sheg
Oshinowo\cite{IAA99_Recycler}.} software for final survey.
\section{Bend Effects for Magnets as Installed}
\pgph We have described the Recycler gradient magnet fabrication and
installation along with a description of the desired offset for an
ideal gradient magnet. Let us now calculate some effects due to
higher order fields, end shim corrections and non-ideal magnet
placement.
\subsection{Bend of Gradient Magnet with Sextupole}
If we include the design sextupole term and integrate the field
(Equation~\ref{Eq:B_transv}) along the circular orbit
(Equation~\ref{Eq:Orbit}), we find
\begin{equation}
\int_{-L/2}^{L/2} B d\ell = B_0 L (1 + b_2\frac{x_{offset}}{a} +
b_3[\frac{4d^2}{45a^2}+ \frac{x_{offset}^2}{a^2}+\frac{L^2 x_{slope}^2}{12a^2}]).
\label{Eq:Bdl_full}
\end{equation}
If we substitute for the sagitta with Equation~\ref{Eq:sagitta}, we have
\begin{equation}
\int_{-L/2}^{L/2} B d\ell = B_0 L (1 + b_2\frac{x_{offset}}{a} +
b_3[\frac{L^2\theta^2}{720a^2}+ \frac{x_{offset}^2}{a^2}+\frac{L^2 x_{slope}^2}{12a^2}]).
\label{Eq:Bdl_alt}
\end{equation}
We can solve Equation~\ref{Eq:Bdl_full} for the offset for which the
integral is $B_0 L$. We find
\begin{equation}
x_{offset} = \frac{- a b_2 \pm \sqrt{(a b_2)^2 - (16/45) b_3^2
d^2}}{2 b_3} \approx -\frac{4}{45} \frac{ b_3 d^2}{a b_2}
\label{Eq:b2b3offset}
\end{equation}
for the case where $x_{slope}=0$. Evaluating this for the RGF
gradient magnet we require only a displacement of $x_{offset} =
-6.74\times 10^{-6}$ m to have the same integral on the circular
orbit as on the central straight path. Alternatively, $(\int B
d\ell/ B_0 L)-1 = 1.645 \times 10^{-5}$ on the an orbit with
$x_{offset} = 0 $.
\subsection{Bend Correction with End Shims}
\pgph The error fields in the Recycler are corrected using shims which
are attached at the ends of the poles. Note that the dipole field is
adjusted (changing the permanent magnet material which drives the
flux) after modifying the end shims so that no dipole field need be
ascribed to the end field. (Therefore there is not a term $(1 +
\cdots)$ in Equation~\ref{Eq:B_l_ends}.) The installation
displacement by z\_cent\_off allows one to achieve the desired bend
center. The error correction applied by the end shims was selected to
create the design field on the straight measurement path. Let us
calculate its effect on the bend field seen by particles on the
expected circular orbit.
We describe the end contributions as if they occured at the points
$\pm L/2 - z_{off}$ (where $z_{off}$ = z\_cent\_off) with
normalization using an effective length, $L_{e}$. $L_{e}$ is
sufficiently small compared to the betatron wavelength that we can
apply the correction as if it were at a point. For $z = +L/2 -
z_{off}$,
\begin{equation}
B = B_0(b_{2l} \frac{L}{L_e}\frac{x}{a} + b_{3l}\frac{L}{L_e}(\frac{x}{a})^2)
\label{Eq:B_l_ends}
\end{equation}
For $z = -L/2 - z_{off}$,
\begin{equation}
B = B_0(b_{2o} \frac{L}{L_e}\frac{x}{a} + b_{3o}\frac{L}{L_e}(\frac{x}{a})^2)
\label{Eq:B_o_ends}
\end{equation}
were we have normalized the harmonics to the field integrated over the
magnet length $L$. The subscripts of `l' (LEAD or LABEL end) and `o'
(OTHER end) designate the positions of the end shims whose field is
being described. $L_e$ is the effective length of the end but is used
here only for a normalizing factor. We multiply the end field by
$L_e$ to get the integrated bend and evaluated the field at the orbit position
corresponding to $z=-L/2$. For the `OTHER' end we obtain
\begin{eqnarray}
\lefteqn{\int_{OTHER} B d\ell = B_0 L [}& &\\ \nonumber
& &b_{2o} (L \theta(\frac{1}{12 a}+\frac{z_{off}}{2aL}+\frac{z_{off}^2}{2aL^2})+(\frac{-x_{offset}}{a}+\frac{x_{slope}(L+2 z_{off})}{2a}))\\ \nonumber
& &+ \frac{b_{3o}}{144 a^2L^2}(L^2\theta+6L^2 x_{slope}+6\theta z_{off}^2
-12 L x_{offset}+(\theta+2x_{slope})6Lz_{off})^2]
\label{Eq:Bint_o_ends}
\end{eqnarray}
A similar result would can be obtained for the `LEAD' end. Typically,
$z_{off} < 0.05$ m. This makes all terms involving $z_{off}$ small
compared to the other terms in the Equation~\ref{Eq:Bint_o_ends}.
Let us evaluate it after setting $z_{off}=0.$ We will employ the
$L$ and $\theta$ appropriate for the regular cell magnets.
\begin{eqnarray}
\label{Eq:Bint_o_ends_num}
\lefteqn{\int_{OTHER} B d\ell = B_0 L [}& &\\ \nonumber
& &b_{2o} (-0.307556 + 39.3701 x_{offset} - 88.5 x_{slope}\\ \nonumber
& &+ b_{3o}(0.09459-24.217 x_{offset}+1550.003 x_{offset}^2\\ \nonumber
& &+54.437 x_{slope}- 6968.5 x_{offset} x_{slope}+7832.25 x_{slope}^2)
\end{eqnarray}
\subsection{Magnitudes for Installation and Closed Orbit Effects}
\pgph For most strong focusing rings, the longitudinal distribution of
fields is simple enough that there is no need to separately consider
the effects of installation offsets from the offsets created by orbit
distortions. We are in a position to determine if this is the case
for the Recycler Gradient magnets. The end shims which have been
created typically provide $<5 \times 10^{-4}$ quadrupole and sextupole
correction when normalized to the integrated magnet strength as shown
here. We can examine Equation~\ref{Eq:Bint_o_ends_num} to see what to
expect for various errors. Placing these corrections at the ends
of the pole (longitudinal effect alone) gives a coefficient of
$b_{2o}$ of -0.307556 and a coefficient of $b_{3o}$ of 0.09459.
Assuming the specified alignment tolerance of 0.25 mm (probably the
installation achieved about 0.15 mm) we conclude that
$x_{slope}<55\;\mu$Radian. Including the design specification
difference of 0.35 mm and adding linearly an installation error
of 0.15 mm we expect $x_{offset}<0.5$ mm. On the other hand the
aperture of the Recycler Ring will accommodate orbit errors $>$10 mm
and corresponding angles of $\sim$400$\mu$Radian. With these larger
errors, we find a coefficient of $b_{2o}$ of -0.737 and a a
coefficient of $b_{3o}$ of 0.543 for $x_{offset}=-10$ mm while the
other sign for $x_{offset}$ gives small coefficients. Although
systematic changes in bending of $\sim 2\times 10^{-4}$ or so are not
negligible, they do not justify further effort at this point. Note
that although the installation effects can add systematically, the
orbit dependent closed orbit effects will mostly cancel when averaged
over the ring.
\section{Conclusions}
\pgph We have shown that the bend field of a straight gradient magnet
will be the same on the (nearly) circular design orbit as on a
straight path through the transverse center of the magnet, provided it
is installed with the design orbit displaced by $+d/3$ at the center.
This placement will make the bend independent of the gradient
strength. The difference between this plan and the specification used
for magnet offset during installation is documented to be about -0.35
mm for RGx and -0.30 for SGx magnets resulting in a bend field
correction of $8 \times 10^{-4}$ for RGx and $15 \times 10^{-4}$ for
SGx magnets. Since these effects are of opposite sign for F and D
magnets, the momentum orbit effects should cancel but some closed
orbit effects are expected. Table~\ref{Tab:Offsetspec} shows the
change in displacement required because the orbit is not circular.
Bend effects due to the lumped end corrections for quadrupole and
sextupole are a bit smaller. Bend corrections for design sextupole
field, for longitudinal placement to correct bend center errors and
for displacement and angle dependence of the end shim corrections have
been shown to be small for the parameters of the recycler magnets.
\section{Acknowledgments}
We would like to thank Norman Gelfand, Leo Michelotti, Shekhar
Mishra, Thornton Murphy, Babatunde Oshinowo, and James Volk for
helpful discussions, suggestions for the text and assistance in
locating references.
\appendix
\section{Orbit Through a Gradient Dipole}
\label{Appen:Orbit}
It is well known that the orbit of a charged particle moving
in a uniform dipole field is described by a circle. For the short arc
section contained within a single bending magnet of a strong focusing
accelerator, one can use the parabolic approximation to the circle
without significant error. Gradient fields must change the orbit. We
will use recursive approximations to discover the magnitude of these
effects.
To describe the orbit in the magnet frame, we begin with an expression
for a circle of radius $R$. A tangent to the circle defines the $z$
axis with the origin at the tangent point. The $x$ axis is radial
with the center located along the negative $x$ direction (negative
curvature). A rectangular gradient magnet is parallel to the $z$ axis
with its centerline at $x = 0$. A circle displaced by $x_0$ at the
center of the magnet (coordinate origin) is described by
\begin{equation}
x = x_0 + \sqrt{R^2-z^2} -R.
\label{Eq:Orbit_circle}
\end{equation}
We expand in
terms of $(z/R)^2$ which gives us an equation for $x$ {\em vs.}
$z$ similar to Equation~\ref{Eq:Para_nooff}.
\begin{equation}
x = x_0 - d(\frac{2z}{L})^2-\frac{d\theta^2}{16}(\frac{2z}{L})^4+ \cdots.
\end{equation}
At the magnet end ($z=L/2$) the $x^4$ term (correction to the
parabola) is smaller than the parabolic term by the ratio
$\theta^2/16$ so for $\theta \sim 0.02$ this correction is 25 ppm of
the sagitta or about 0.25$\mu$m for an RGF.
As an alternative to a complete solution to the orbit in a gradient dipole
field, we will observe that the deviations will be small and use recursion
to obtain a next order correction to the dipole orbit. Let us begin
by establishing an equation for the bending in a gradient field.
\begin{equation}
d\theta = -\frac{B dz}{B\rho}(1 + b_2 \frac{x}{a})
\label{Eq:grad_bend}
\end{equation}
\begin{equation}
\frac{d^2x}{dz^2} = -\frac{1}{\rho}(1 + b_2 \frac{x}{a})
\end{equation}
We obtain our recursive solution by substituting Equation~\ref{Eq:Para_nooff}
into the quadrupole ($b_2$) term in Equation~\ref{Eq:grad_bend} and integrating
twice.
\begin{equation}
\frac{dx}{dz} = -\frac{L}{2\rho}((1 + \frac{b_2 x_0}{a})\frac{2z}{L} - \frac{b_2 d}{3a}(\frac{2z}{L})^3 ).
\label{Eq:dxdz}
\end{equation}
We fix the integration constant by assuming that the angle at the
magnet center is zero. Remember that for the circle, $L/\rho =
\theta$. Examining this result at $Z=L/2$, we see that if $x_0=d/3$,
the linear and cubic terms cancel such that the bend is precisely
$\theta/2$ for half the magnet.
\begin{figure}[t]
\centering
%\epsfysize=5.5in
\epsfysize=\textwidth
\rotate{\rotate{\rotate{\epsfbox{Gradient_Orbit.eps}}}}
\caption{Orbit changes due to the gradient field in an RGF magnet.
The parabolic orbit shape due to the dipole field has been subtracted.
The remaining parabolic shape is proportional to the $x$ displacement at
$z=0$. (In fact, it depends on the dipole field on the orbit at
$z=0$.) The quartic term partly cancels this shift.}
\label{Fig:a1_symmetry_lim}
\end{figure}
\begin{equation}
x = x_0 -\frac{L^2}{8\rho}((1 + \frac{b_2
x_0}{a})(\frac{2z}{L})^2 + \frac{L^2
db_2}{48a}(\frac{2z}{L})^4 )
\end{equation}
\begin{equation}
x = x_0 -d(1 + \frac{b_2 x_0}{a})(\frac{2z}{L})^2 + \frac{b_2 d^2}{6a}(\frac{2z}{L})^4 ,
\label{Eq:x_in_grad}
\end{equation}
where we have used the relations between sagitta, chord length and
radius: $d=L \theta/8 = L^2/8\rho$. The correction to the parabolic
term is $b_2 x_0/a$. Thus, the parabolic curvature is determined by
the dipole field at $z=0$. For the RGx magnets this correction to the
parabolic curvature is $0.154 b_2$ which we evaluate as -0.00919725
(0.9\%) for an RGD.
At the magnet end, $Z=L/2$, we wish to know what displacement the beam
will have experienced, since we wish to displace the dipole
appropriately. We find with no gradient, $-2d/3=-0.00781193$ m is the
exit point when $x_0=d/3$. Evaluating Equation~\ref{Eq:x_in_grad}
with the same $x_0$ and the parameters for and RGF (RGD) magnet we
have the exit point at $x=$-0.00786777 m (-0.00775805 m). This shift
of +56 $\mu$m (-54 $\mu$) is the result of the difference between the
change in the quadratic term ($b2 d^2/(3 a)$) for RGD magnets of
0.000107772 m and the quartic term ($b2 d^2/(6 a)$) of amplitude
0.0000538862 m. For the SGF (SGD) magnets the exit position is at
$-2d/3=$-0.00358966 m ignoring the gradient whereas it is -0.00361394
m (-0.00356487 m) when we account for the gradient.
Using these results we have the following values of $x_{offset}$:
RGF: -.0408 mm RGD:-0.298 mm SGF: -0.324 mm SGD -0.275 mm.
This difference is about twice the difference found in MI-0200.
Using Mathematica for the calculation, the process is equally simple
when including a sextupole term. The resulting equation for the orbit
contains terms up to $Z^6$ and the lower order terms have a sextupole
correction. The numerical result is -0.14 $\mu$m for the RGF and
+0.14 $\mu$m for the RGD. As expected, these corrections are too
small to consider.
\section{Other Offset Results}
In addition to the numerical integration results by Norman Gelfand
(MI-0200), Leo Michelotti\footnote{Private communication. March 2000}
has used a C++ accelerator model to obtain offsets using integration
through a magnet after dividing it into 16 parts. This was done
on different occasions with slightly different input.
\begin{table}[h]
%\vspace{1ex}
\centering
\begin{tabular}{|l|r|r|r|r|l|}
\hline
& RGF & RGD & SGF & SGD & Offset in mm.\\ \hline
MI-0259 (dipole) & 7.81 & 7.81 & 3.59 & 3.59 ¶bola \\ \hline
MI-0259 (gradient) & 7.87 & 7.76 & 3.61 & 3.56 &inc. quartic\\ \hline
Michelotti (RRV18) & 7.83263& 7.77949& 3.59519& 3.56908&16 July 1998\\ \hline
Michelotti (RRV ??) & 7.833 & 7.777 & 3.477 & 3.453 &29 July 1998\\ \hline
& & & & & \\ \hline
Holmes MI-0196 & 7.46 & 7.46 & 3.29 & 3.29 &Used for Survey \\
& & & & &thru March 2000 \\ \hline
Gelfand (MI-0200) & 7.44 & 7.39 & 3.30 & 3.28 & \\ \hline
\end{tabular}
\end{table}
%\vspace{1ex}
The orbit calculations I am making analytically for MI-0259 do not
agree in detail with the results by Norman Gelfand (MI-0200) or Leo
Michelotti (private communication). At present, we have not found the
reason for the differences. Leo Michelotti has suggested that the
number of magnet steps in his calculation may not be sufficient for 50
micron accuracy. The MI-0200 results, like the MI-0196 results are
for an earlier design of the Recycler Gradient Magnets.
Since the magnet installation has a maximum error tolerance of 0.25
mm, differences of 0.05 mm are not significant for machine operation.
We make these comparisons as a check on our methods. After examining
our results we have not found any problems with these analytic
results.
%\bibliography{compute,accelerators,magnets,mathscieng,mtf}
\bibliography{accelerators,minote,magnets,tdnote}
%* remove all but relevant *.bib file names.
\end{document}