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% 0.1 BCBrown 6-Oct-2015 Initial entry.
% 0.2 BCBrown 12-Oct-2015 Complete Draft
% 0.22 BCBrown 14-Oct-2015 Small Corrections
% 0.24 BCBrown 15-Oct-2015 More Small Corrections + captions
% 1.00 BCBrown 16-Oct-2015 Ready for Beams-doc-4982
% 1.10 BCBrown 09-May-2016 Ready to correct known errors
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\title{Aperture Implications for Recycler Assuming Full Coupling and
Nominal Vertical Apertures}
\author{Bruce C. Brown \\ Accelerator Division, Main Injector Department\\
{\em Fermi National Accelerator Laboratory }
\thanks{Operated by Fermi Research Alliance
under contract with the U. S. Department of Energy}
\\ \em P.O. Box 500 \\ \em Batavia, Illinois \\
}
\begin{document}
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\maketitle
%%%\newpage
\tableofcontents
\newpage
\begin{abstract}
In order to specify the apertures needed for Recycler Collimators and
Masks, we will explore the vertical aperture requirements implied by
the nominal vertical beam pipe size and examine the lattice functions
to find the largest vertical beta. For horizontal aperture
requirements we will assume full vertical to horizontal coupling and
add to this a requirement for the momentum spread and slip stacking
offsets. To these requirements an allowance for injection errors will
be added to arrive at an aperture requirement. Using a nominal beam
emittance of 15 pi-mm-mr and the same momentum spread and slip
stacking momentum requirements, we will determine the nominal beam
size. The amplitudes required to move this beam edge to the aperture
edge will set the maximum bumps needed for collimation. Since
collimators will define apertures inside of the Recycler aperture, the
bump amplitudes for collimation will be smaller than determined here.
\bf{This is not a specification document but it will describe the basis
we will use to define collimation system requirements.}
\end{abstract}
%%\newpage
\section{Introduction}
\pgph The Fermilab Recycler is now employed as a accumulation ring for
8 GeV protons to provide high intensity beam to inject into the Main
Injector. With the available beam quality from the Booster, we have
losses for all operating scenarios but studies of slip stacking
indicate higher losses with these modes. To localize these losses, we
are designing a collimation system which is expected to absorb protons
which are outside the acceptance of the ring and the secondaries
produced when those protons interact in large steel blocks. For
designing this system we need to know the boundaries for beam which is
transmitted through the limited vertical aperture of the Recycler, and
compare that aperture to the expected beam size in order to understand
the magnitude of potential orbit bumps required to move beam into the
collimation system. We will use a nominal Recycler lattice and a
tentative collimator placement for which we will document the
important geometry.
\section{Recycler Characteristics}
\pgph The Courant-Snyder lattice parameters we will employ come from
the R90 Console Program. We employed R90 file 16 from 8 September
2015 to obtain a lattice description. The lattice included
preliminary descriptions for a number of potential primary and
secondary collimator locations and many potential locations for dipole
corrector magnets. Using the text output from R90 we loaded this
description into an Excel spreadsheet (RRApertureForBeam.xlxs) to
calculate beam properties. This `design' lattice has been compared
with measurements and found to be in reasonable agreement. We choose
not to add an allowance for the difference between `design' and
`measured' lattice to our already conservative assumptions below.
\subsection{Vertical Beam Size}
\pgph We assume that the principal limitation is in the vertical
acceptance due to the standard beam pipe height. This will manifest
itself at the largest $\beta_y$ around the ring. In this lattice file
we find $\beta_{peak} = 58.6$ m. For the maximum aperture we use
$y_{pipe} = 20$ mm. The internal beam pipe height at the center is a
bit more than 22 mm but there are many welds at high $\beta_y$ so we
assume that this should cause us to assume a smaller maximum available
size. The vertical boundaries will then be
\begin{eqnarray}
\label{Eq:YBoundaries}
y_{max}(s)= \frac{\sqrt{\beta(s)} }{\sqrt{\beta_{peak}}} y_{pipe} \\
y_{min}(s)= -\frac{\sqrt{\beta(s)}}{\sqrt{\beta_{peak}}} y_{pipe}
\end{eqnarray}
Using typical Booster 95\% normalized beam emittance of $\epsilon =
15$ pi-mm-mr we calculate typical RMS vertical beam sizes from
\begin{equation}
\sigma_y(s) = \sqrt{\frac{\epsilon \beta_y(s)}{6 \; \beta_{rel}\gamma_{rel}}}
\label{Eq:Ysigma}
\end{equation}
where $\beta_{rel}$ and $\gamma_{rel}$ describe the proton
relativistic motion. From this we find a three sigma boundary
\begin{eqnarray}
y_{up}(s) = 3 \sigma_y(s)\\
y_{down}(s) = -3 \sigma_y(s)
\label{Eq:ysize}
\end{eqnarray}
%\begin{figure}[bthp]
\begin{figure}[t]
\centering
\includegraphics [height=4.5in]{RRApertureForBeam_Vert.eps}
\caption{Vertical beam properties in the region from RR611 to RR620
are shown. The orange curves mark the upper and lower vertical
boundaries of beam acceptance. Beams which can circulate through
the vertical aperture limit at the maximum beta will fit inside of
these boundaries after correcting injection offsets. The blue
curves describe the beam envelope for 15 pi-mm-mr beam. Red with
green triangles mark positions under consideration for collimation.
The horizontal axis (STATION) is the distance from the MI10
injection marker (in meters).}
\label{Fig:VertBeamSize}
\end{figure}
For the region from RR611 to RR620 where the collimators and
collimator bumps are planned, we show the typical beam size and
boundaries of accepted beam in Figure~\ref{Fig:VertBeamSize}.
Locations under consideration for collimation are identified with
green triangles with red outlines. These include a primary identified
in the spreadsheet as PCOLL613B and Secondary Collimators identified
as SCOLL613D and SCOLL616 (for initial phase of collimation - 2016)
and potential additional Secondary collimators (Alt Sec - potential
second phase installation) at SCOLL614 and SCOLL619U.
\begin{figure}[bthp]
%\begin{figure}[b]
\centering
\includegraphics [height=4.5in]{RRApertureForBeam_Horiz.eps}
\caption{Horizontal beam properties in the region from RR611 to RR620
are shown. Assuming full coupling, beam which circulates through the
vertical aperture limit at the maximum vertical beta will fit within
the orange curves after correcting injection offsets. Blue curves
show the beam size for 15 pi-mm-mr beam with the momentum width
allowed by slip stacking acceptance. The green curves show the same
beam which has been displaced for slip stack slipping. The x offset
curve shows the displacement of the beam center for the slipping
beam (momentum orbit for slipping beam). We see that the edge of
the on momentum beam is just beyond the center (offset beam
position) of the slipping beam. We are using the convention that x
positive is radially out (not the convention used in R90).}
\label{Fig:HorizBeamSize}
\end{figure}
\subsection{Horizontal Beam Size}
\pgph For each injection from the Booster (at 15 Hz), the injected
beam will circulate for about 6000 turns before the next injection.
We assume (round beam) that some combination of linear (skew
quadrupole) and non-linear coupling will mix all the beam vertical and
horizontal motion and especially so for the beam at the boundaries.
This assumption implies that we calculate the horizontal betatron beam
size with the same parameters as for the vertical beam size.
To this we must add a contribution for momentum effects. Two
contributions are of interest. The longitudinal admittance for 15 Hz
slip stacking contributes an RMS momentum spread of 3.5 MeV/c for
injected beam which can be accepted. The 20 Hz Booster operation may
allow a larger admittance which we will take as 4/3 larger or 4.667
MeV/c. For each of the slip stacked beams we will calculate a
horizontal beam sigma which is the RMS of the betatron RMS and
and momentum RMS. For an 8 GeV or 8.889 GeV/c beam these imply
$\frac{\sigma_p}{p}$ of 0.000394 (15 Hz) or 0.000525 (20 Hz).
\begin{equation}
\sigma_x(s) = \sqrt{\frac{\epsilon \beta_x(s)}{6 \; \beta_{rel}\gamma_{rel}}
+ (1000 \; \eta \frac{\sigma_p}{p})^2}
\label{Eq:Xsigma}
\end{equation}
The factor of 1000 is because we express beam sizes in mm. We also
comment here that $\eta$ is negative in this lattice. Before
calculating the positions of the beam edges, we will calculate the
displacement of the beam caused by the momentum offset for slip
stacking. This is determined by the time available for slipping and
is $dp =24.5$ MeV/c ($dp/p = 0.276$\%) for 15 Hz Slip Stacking or $dp
=32.66$ MeV/c ($dp/p = 0.367$\%) for 20 Hz Slip Stacking.
\begin{equation}
x_{\textrm{ssoffset}} = 1000 \; \eta \; \frac{dp}{p}
\label{Eq:xoffset}
\end{equation}
The 3 sigma edges for the beam at momentum center are given by
\begin{eqnarray}
x_{in}(s) = 3 \sigma_x(s) \\
x_{out}(s) = -3 \sigma_x(s)
\label{Eq:xedge}
\end{eqnarray}
For the beam which is decelerated for slipping, the edges are
\begin{eqnarray}
x_{\textrm{off-in}}(s) = x_{\textrm{ssoffset}} +3 \sigma_x(s)\\
x_{\textrm{off-out}}(s) = x_{\textrm{ssoffset}} -3 \sigma_x(s)
\label{Eq:xoffedge}
\end{eqnarray}
For the maximum horizontal beam size we will be more conservative by
adding the momentum width, momentum offset, and betatron sizes
linearly. We seek to know the size we need to leave when we build new
devices. Slip stacking may employ beam on momentum center along with
decelerated beam (Spring 2015 option) or on center with accelerated
beam (possible future option) or we may inject off center and displace
the slipping beam from that orbit. We will never use the aperture we
describe with this calculation but it will allow full flexibility for
future decisions.
\begin{eqnarray}
\label{Eq:XBoundaries}
x_{max}(s)= \frac{\sqrt{\beta(s)} }{\sqrt{\beta_{peak}}} y_{pipe}
- 1000 \;\eta \; (dp + \sigma_p) \\
x_{min}(s)= -\frac{\sqrt{\beta(s)} }{\sqrt{\beta_{peak}}} y_{pipe}
+ 1000 \;\eta \; (dp + \sigma_p)
\end{eqnarray}
The horizontal beam features are illustrated in
Figure~\ref{Fig:HorizBeamSize}. Again we show typical beam size and
boundaries of accepted beam but the beam boundaries are shown both for
on momentum beam and beam decelerated for slipping. Also shown as (x
offset) is the beam center (momentum orbit) for the decelerated beam
\section{Boundaries at Proposed Collimator Locations}
\pgph With the above formulas implemented in the spreadsheet
RRApertureForBeam.xlxs, we can find the expected beam sizes and
maximum beam sizes for the proposed collimator locations. will
provide the spreadsheet with momentum parameters for 20 Hz slip
stacking but for 15 Hz values, the user can substitute in the two
cells as desired.
\subsection{Vertical Boundaries}
\pgph For the vertical boundaries, we can put everything in
Table~\ref{Table:VertBeamBndries}. The position of the largest
vertical displacement we can require at each location is y min or y
max. We see that this is a bit smaller than the 20 mm size we assume
for the beam pipe inside aperture. We suggest that an allowance for
injection steering errors of up to 2 mm should be added to this.
We can consider whether the appropriate vertical size of a collimator
should be uniform or whether smaller collimation apertures
corresponding to y max as small at 9.37 mm might permit a reduced
external activation.
For designing bumps, we see that the allowed displacement (y max - y
up) also varies by x2 and while that does go with the Beta at the
collimator, one needs to design specific bumps to see what magnet
strength is needed.
\begin{table}[tbhp]
\begin{center}
\caption{Vertical Lattice and Beam Boundaries and Maximum Bump Amplitude}
\begin{tabular}{|l|lccc|cccc|c|}
\hline
Name & Station & phase & Beta & Alpha & y dn & y up & y min & y max & y max - y up \\ \hline
& m & Rad/2$\pi $& m & & mm & mm & mm & mm & mm \\ \hline
PCOLL613B & 2853 & 20.93 & 50.746 & 0.04275 & -10.978 &10.978 & -18.612 & 18.612 & 7.633 \\ \hline
SCOLL613D & 2861 & 20.96 & 31.782 & 1.7997 & -8.688 &8.688 & -14.729 & 14.729 & 6.041 \\ \hline
SCOLL614 & 2878.1 & 21.12 & 25.12 & -1.65082 & -7.724 &7.724 & -13.095 & 13.095 & 5.371 \\ \hline
SCOLL616 & 2906.7 & 21.34 & 12.849 & -0.9175 & -5.524 &5.524 & -9.3652 & 9.3652 & 3.841 \\ \hline
SCOLL619U & 2939.8 & 21.66 & 57.391 & -0.04248 & -11.675 &11.675 & -19.793 & 19.793 & 8.118 \\ \hline
\end{tabular}
\label{Table:VertBeamBndries}
\end{center}
\end{table}
\subsection{Horizontal Boundaries}
\pgph We need to display the horizontal information in more tables to
keep them of manageable size. For finite beam momentum spread, the
dispersion ($\eta$ or Eta) increases the beam size. But more
importantly, we will be working with slip stacking momentum offsets of
one of the beams. We consider that in one direction and exhibit the
required displacements. The vertical collimation will impact both
beams. With the preferred two secondary collimator solution, we will
seek to collimate the on momentum beam. Only by adding at least one
of the alternative collimators will it be possible to provide
horizontal scraping for the offset (slipping) beam. Perhaps the
collimator vertical boundaries will be sufficient to control beam
halo. In Table~\ref{Table:HorizLattice20Hz} we show the lattice
parameters with the RMS beam size and the slip stacking displacement.
\begin{table}[tbhp]
\begin{center}
\caption{Horizontal Lattice, beam size, and offset for 20 Hz Slip Stacking}
\begin{tabular}{|l|cccccc|c|c|}
\hline
Name & Station & phase & Beta & Alpha & Eta & Etap & sigma & x offset \\ \hline
& m &Rad/2$\pi$& m & & m & & mm & mm \\ \hline
PCOLL613B & 2853 & 21.797 & 11.514 & -0.004 & -1.131 & 0.0006 & 1.841 & -4.154 \\ \hline
SCOLL613D & 2861 & 21.885 & 24.034 & -1.630 & -1.478 &-0.07923 & 2.635 & -5.431 \\ \hline
SCOLL614 & 2878.1 & 21.949 & 34.967 & 2.0269 & -1.667 & 0.08288 & 3.161 & -6.124 \\ \hline
SCOLL616 & 2906.7 & 22.164 & 22.163 & 2.1675 & -1.246 & 0.12183 & 2.505 & -4.579 \\ \hline
SCOLL619U & 2939.8 & 22.530 & 12.372 & 0.0459 & 0.003 &-0.00354 & 1.807 & 0.011 \\ \hline
\end{tabular}
\label{Table:HorizLattice20Hz}
\end{center}
\end{table}
Table~\ref{Table:HorizBoundaries20Hz} shows the boundaries for 15
pi-mm-mr beams for both the beam displaced for slipping and the beam
on momentum center (as injected). Also shown is the maximum beam
extent due to betatron beam size, momentum beam size and momentum
displacement for slipping (20 Hz slip stacking) added linearly.
We see that there is significant overlap between the 3 sigma widths of
the displaced and centered beam. Again, we suggest adding 2 mm to
allow for injection errors.
\begin{table}[tbhp]
\begin{center}
\caption{Horizontal Boundaries for 20 Hz Slip Stacking}
\begin{tabular}{|l|c|cccccc|}
\hline
Name & Station &x off out & x off in & x out & x in & x min & x max \\ \hline
& m & mm & mm & mm & mm & mm & mm \\ \hline
PCOLL613B & 2853 & -9.679 & 1.367 & -5.524 & 5.524 & -13.613 & 13.613 \\ \hline
SCOLL613D & 2861 & -13.337 & 2.475 & -7.906 & 7.906 & -19.016 & 19.016 \\ \hline
SCOLL614 & 2878.1 & -15.608 & 3.359 & -9.484 & 9.484 & -22.449 & 22.449 \\ \hline
SCOLL616 & 2906.7 & -12.095 & 2.937 & -7.516 & 7.516 & -17.533 & 17.533 \\ \hline
SCOLL619U & 2939.8 & -5.410 & 5.432 & -5.421 & 5.421 & -9.1772 & 9.1772 \\ \hline
\end{tabular}
\label{Table:HorizBoundaries20Hz}
\end{center}
\end{table}
\begin{table}[tbhp]
\begin{center}
\caption{Horizontal Displacement Requirements}
\begin{tabular}{|l|cccc|}
\hline
&\multicolumn{2}{c}{Results for 20 Hz} & \multicolumn{2}{|c|}{Results for 15 Hz} \\ \hline
& x max - x in & -x min + x off out & x max - x off out & -x min + x off out \\ \hline
& mm & mm & mm & mm \\ \hline
PCOLL613B & 8.089 & 3.935 & 7.030 & 3.913 \\ \hline
SCOLL613D & 11.110 & 5.679 & 9.710 & 5.636 \\ \hline
SCOLL614 & 12.965 & 6.841 & 11.377 & 6.782 \\ \hline
SCOLL616 & 10.017 & 5.438 & 8.823 & 5.387 \\ \hline
SCOLL619U & 3.757 & 3.767 & 3.760 & 3.768 \\ \hline
\end{tabular}
\label{Table:HorizDisp}
\end{center}
\end{table}
\section{Observations and Conclusions}
\pgph The primary collimator should have a width of at least 2
$\times$ 13.6 mm (See Table~\ref{Table:HorizBoundaries20Hz}). Note
that we will bump the beam into this after damping and do not need to
add an allowance for injection errors. The horizontal beam position
at the primary is unconstrained. But we observe that changes in the
horizontal beam position at the first secondary (SCOLL613D) will
necessarily change the horizontal position at the primary. These
position changes fall within the limits for the primary size specified
above.
\pgph The secondary collimators for the Main Injector employed an
aperture of 2 inches by 4 inches. This is over generous. We used 2
inches by 2 inches in the MI8 collimators and this looks to be
adequate for the Recycler Collimation. The required minimum size
(with these assumptions) can be determined from (y max)-(y min) in
Table~\ref{Table:VertBeamBndries} and (x max)-(x min) in
Table~\ref{Table:HorizBoundaries20Hz} and that is in some cases much
smaller. The motion provided for the secondary collimators will
accommodate larger collimator apertures. Perhaps we can learn how
much advantage a smaller aperture provides when seeking to contain the
activation within the secondary collimators by employing MARS
calculations.
\section{Further Observations}
\pgph Version 1 of this document (October 2015) had a couple of
significant errors which required a new version. The beam size was
miscalculated in the spreadsheet by $\sqrt{\pi}$ with the result that
the plots suggested much more free aperture than was real. Errors in
Equations~\ref{Eq:Ysigma} and \ref{Eq:Xsigma} were corrected. The
specification for collimator apertures is available currently in the
RecyclerCollimation web page. It will be desirable to provide a new
document with this information. The viewpoint of this document was
employed for that work but additional allowances for uncertainties
were included, making the apertures larger.
\section{Acknowledgments}
\pgph I would like to thank Ming-Jen Yang and Phil Adamson for many
discussions and suggestions.
%\bibliography{BeamsDoc}
\bibliography{compute,accelerators,magnets,mathscieng,mtf,BeamsDoc}
\end{document}