%*********************************************************************
\clearpage
\newpage
\lhead{SNS Ring Design Manual}
\rhead{Accelerator Design and Beam-dynamics Issues}
\rfoot{November 1, 2002}
\section{Accelerator Design and Beam-dynamics Issues}
%*********************************************************************
%\clearpage
%\newpage
\lhead{SNS Ring Design Manual}
\rhead{HEBT layout}
\rfoot{October 10, 2002}
\subsection{Layout}
%%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
%\rhead{HEBT layout}
%\rfoot{July 2002}
\subsubsection{HEBT layout}
HEBT is about 180 meter long and carry $^-$ ion with peak average
current of 38 mA in 1 ms long pulses at the rate of 60 Hz. The HEBT
not only carry the H$^-$ ion but also optically matches linac and
ring, correct the energy jitter from the linac, increase the energy
spread of beam to avoid beam stability in the ring, clean the
transverse and longitudinal halo coming from the linac, characterize
the beam from linac, and protect ring from the fault conditions. The
general features of this line are, the magnetic filed in dipole and
quadrupoles and vacuum are low enough to control Lorentz and gas
stripping of H$^-$ for energies 800 - 1300 MeV. The ratio of aperture
to the rms beam size is kept greater than 10 through out the line
except energy corrector, spreader and
collimators. Fig.~\ref{fig:hebtlayout} shows the layout of the HEBT.
For ease of operation and stability reasons the lattice of the is
chosen is FODO which matches the linac and ring doublet-FODO lattices.
\begin{figure}[hpb]
{\parindent = 0pt
\centerline{\psfig{file=hebt_layout.eps,width=8.5in,angle=90}}
\caption{High Energy Beam Transport layout. \label{fig:hebtlayout}}
}
\end{figure}
We can consider the HEBT as having three sections: Linac-Achromat
Matching Section (LAMS), Achromat, and Achromat-Ring Matching Section
(ARMS). In addition to the bend to the ring, there is a straight beam
line used for linac beam characterization, as shown in
Fig.~\ref{fig:hebtlayout}. The first five cells (8 m/cell) after the
linac (LAMS) are used to characterize the linac beam, match beam into
the achromat, collimate beam halo. The energy corrector cavity is
located in the last half cell of the LAMS. Following this, the four
cell long achromat (14 m/cell) provides momentum selection by cleaning
up the beam energy halo at the point of maximum dispersion
($\eta_x$=6.4~m). The remaining six cells (8 m/cell) are used for
matching the beam into the accumulator ring, diagnostics. The energy
spreader cavity is located in the first cell following the achromat
(in the ARMS), where the dispersion and its derivative are zero. There
are eight horizontal and eight vertical dipole correctors placed in
strategic positions of small apertures in the line. The line has
following new features to control losses.
%*********************************************************************
%\clearpage
%\newpage
\lhead{SNS Ring Design Manual}
\rhead{Ring layout}
\rfoot{October 10, 2002}
\subsubsection{Ring layout}
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{wei_fig06.eps}
\caption{Schematic layout of the SNS accumulator ring.
The four straight sections are
designed for injection, collimation, the RF system, and extraction. \label{fig:snslayout}}
}
\end{figure}
%*********************************************************************
%\clearpage
%\newpage
\lhead{SNS Ring Design Manual}
\rhead{RTBT layout}
\rfoot{October 10, 2002}
\subsubsection{RTBT layout}
Ring to Target Beam Transfer (RTBT) line is about 150 meter long and
carry the beam from the ring extraction region to the target and
provide the desired footprint for the accelerator complex. The
general features of this line are, the line is immune to one kicker
failure and the ratio of acceptance to rms emittance is more than
20. Fig.~\ref{fig:rtbtlayout} shows the layout of RTBT.
The line has following function (a) extraction (b) beam dump (c)
protect from device failure (d) beam spreader and (e)
diagnostics. Following the extraction system beam can be dumped
straight through a 16.8$^{\circ}$ dipole magnet. After the magnet two
cell are used for the collimator system which protect line from the
dipole failure. Following another six cells of transport, the last
five quadrupoles in the line are used for final beam spreading to
produced the beam sized required at the target. The phase advance
between kickers and target is an integral multiple of $\pi$, so that
in event of a kicker failure beam will not move at the target.
\begin{figure}[hpb]
{\parindent = 0pt
\centerline{\psfig{file=rtbt_layout.eps,width=8.5in,angle=90}}
\caption{Ring to Target Beam Transport layout. \label{fig:rtbtlayout}}
}
\end{figure}
%\begin{figure}[hpb]
%{\parindent = 0pt
%\centerline{\psfig{file=revS.eps,width=7.in,angle=0}}
%\caption{Layout of the original FODO-doublet hybrid lattice in the omega configuration for the
%Spallation Neutron Source ring.}
%}
%\label{fig:hybrid}
%\end{figure}
%\begin{table}[hp]
%{\parindent = 0pt
%\newcommand{\hl}{\hline}
%\caption{Major machine parameters for the original hybrid lattice
%Spallation Neutron Source ring.}
%\vspace*{5mm}
%\centerline{
%\begin{tabular}{lll} \hl\hl
%Quantity & Value & unit \\ \hl\hl
%Circumference & 220.88 & m \\
%Average radius & 35.154 & m \\
%Injection energy & 1 & GeV \\
%Extraction energy & 1 & GeV \\
%Beam power & 2 & MW \\
%Repetition rate per ring & 60 & Hz \\
%Number of proton & 2.08 & 10$^{14}$ \\
%Ring dipole field & 0.7406 & T \\
%RF harmonic & 1, 2 & \\
%Peak RF voltage, $h=1$ & 40 & kV \\
%Peak RF voltage, $h=2$ & 20 & kV \\
%Normalized emittance & 289 & $\pi$ mm mr\\
%Unnormalized emittance (99$\%$) & 160 -- 240 & $\pi$ mm mr\\
%Betatron acceptance & 480 & $\pi$ mm mr \\
%%Additional beam clearance & 5 & mm\\
%%\svv
%Momentum acceptance (full beam) & $\pm$ 2 & $\%$ \\
%Momentum acceptance (zero amplitude) & $\pm$ 3.8 & $\%$ \\
%Magnetic rigidity, $B\rho$ & 5.6575 & Tm\\
%Bending radius, $\rho$ & 7.1301 & m \\
%Horizontal tune & 5.8 -- 6.8 & \\
%Vertical tune & 4.8 -- 5.8 & \\
%Transition energy, $\gamma_T$ & 4.95 & \\
%Horizontal natural chromaticity & $-7.5$ & \\
%Vertical natural chromaticity & $-6.3$ & \\
%Number of super-period & 4 & \\
%Arc lattice & 4 FODO cells & \\
%Arc cell length & 8 & m \\
%Straight section lattice & 2 doublets & \\
%Straight section drift length & 9.04, 2$\times$5.45 & m \\
%\hl\hl
%\end{tabular}
%}
%}
%%\vspace*{-2cm}
%%\label{tab:main}
%%\vspace*{-1.5in}
%\end{table}
%
%*********************************************************************
\clearpage
\newpage
\lhead{SNS Ring Design Manual}
\rhead{Lattice}
\rfoot{October 10, 2002}
\subsection{Lattice}
%*********************************************************************
%\clearpage
%\newpage
\lhead{SNS Ring Design Manual}
\rhead{Optics}
\rfoot{October 10, 2002}
\subsubsection{HEBT optics}
\vspace*{1mm}
{\noindent\em 2.2.1.1 Introduction}
\vspace*{1mm}
The SNS High Energy Beam Transport line (HEBT) connects the 1 GeV
linac to an accumulator ring. The HEBT not only matches the beam into
the accumulator, but also determines the beam quality at injection.
HEBT also capable to transport 1.3 GeV H minus to the accumulator ring
\cite{rapariahebt}.
\vspace*{1mm}
{\noindent\em 2.2.1.2 Design Requirements}
\vspace*{1mm}
Table~\ref{tab:hebtopt} gives the required Courant-Snyder parameters
at the entrance (middle of the first HEBT quadrupole) and exit
(injection stripping foil) of the HEBT. In addition, a major
requirement of all parts of this accelerator is minimization of
uncontrolled beam losses to allow hands-on maintenance. This is
achieved by maintaining adequate tolerances on elements, appropriately
located collimators, and sufficient beam diagnostics.
\begin{table}[thb]
\caption{Courant-Snyder parameters at the entrance and exit of the HEBT for a 1.4 MW beam.
\label{tab:hebtopt}}
\vspace{2mm}
\centering{
\begin{tabular}{l l l l} \hline
Quantity & Entrance & Exit & Units \\
& (end of linac) & (stripping foil) & \\ \hline
$\alpha_x$ & 0.00 & 0.053 & \\
$\beta_x$ & 2.377 & 10.44 & $\pi$ mm/mrad \\
$\epsilon_x$ & 1.26 & 1.40 & mm mrad (5 rms, unnorm.) \\
$\alpha_y$ & 0.00 & 0.045 & \\
$\beta_y$ & 13.543 & 12.12 & mm/mrad \\
$\epsilon_y$ & 1.26 & 1.40 & $\pi$ mm mrad (5 rms, unnorm.) \\
$\alpha_z$ & 0.0005 & 0.14 & \\
$\beta_y$ & 0.005 & 0.05 & deg/keV \\
$\epsilon_z$ & 1500 & 1500 & $\pi$ keV deg (5 rms) \\ \hline
\end{tabular}
}
\end{table}
\vspace*{1mm}
{\noindent\em 2.2.1.3 Description of the HEBT Line}
\vspace*{1mm}
The HEBT has following functions: (a) matching of the beam from the
linac into the transport line, (b) momentum correction, (c) momentum
spread, (d) matching bean into the accumulator ring, (e)
characterization of the beam out of the linac and before injection,
and (f) halo and momentum tails cleanup. We have managed to decouple
the first four of these functions, and can consider the HEBT as having
three sections: Linac-Achromat Matching Section (LAMS), Achromat, and
Achromat-Ring Matching Section (ARMS). In addition to the
90$^{\circ}$ bend to the ring, there is a 0$^{\circ}$ beamline used
for linac beam characterization.
The first four and half cells (8.0 m/cell) after the linac (LAMS) are
used to characterize the linac beam, match beam into the achromat,
collimate beam halo. Following this section, the four cell long
achromat (14 m/cell) bends the beam 90$^{\circ}$ and provides momentum
selection by cleaning up the beam energy halo at the point of maximum
dispersion ($\eta$=6.4 m). The energy spreader cavity is located in
the first cell following the achromat (in the ARMS), where the
dispersion and its derivative are zero. The remaining six cells (8
m/cell) are used for matching the beam into the accumulator ring,
diagnostics. There are 16 small dipole magnets for steering of the
beam in the quadrupole focusing-plane located strategically to align
the beam in the narrow apertures. To reduce the probability of
uncontrolled beam losses, HEBT is equipped with five sets of beam halo
scrapers and three beam absorbers. The collimators are the minimum
apertures in the line, chosen to be 10 times the rms beam size. The
maximum magnetic field in dipoles and quadrupoles is kept less than
2.1 kG, to keep H$^-$ stripping losses below 0.1 nA/m. The alignment
tolerances required to keep the beam losses low (Raparia, 97) are
given in Table~\ref{tab:hebterr}.
\begin{table}[thb]
\caption{Expected misalignment for the HEBT magnets. \label{tab:hebterr}}
\vspace{2mm}
\centering{
\begin{tabular}{l l} \hline
Translation (x and y) & $\pm$0.1 mm \\
Pitch and yaw & $\pm$1 mrad \\
Rotation & $\pm$0.5 deg \\ \hline
\end{tabular}
}
\end{table}
\vspace*{1mm}
{\noindent\em 2.2.1.4 Linac to Achromat Matching Section}
\vspace*{1mm}
The linac has a FDOO lattice with a phase advance of about
90$^{\circ}$/cell, and the achromat has a FODO lattice with
90$^{\circ}$/cell phase advance. To remove any linac beam halo, there
are four movable (two in each plane) and two fixed collimators located
in the 3rd through 7th half-cells in this section (details of the
configuration are discussed in the section on halo scraping). The
space between quadrupoles in the first cell of the HEBT is occupied by
beam diagnostics, as discussed in a later section.
\vspace*{1mm}
{\noindent\em 2.2.1.5 Momentum Selection (Achromat)}
\vspace*{1mm}
A 90$^{\circ}$ achromatic bend starts at the 5th cell of the HEBT
line, and finishes in four cells, containing eight 11.25$^{\circ}$
dipoles. The total phase advance in the achromat is 360$^{\circ}$. A
beam energy-halo scraper is located at the middle cell of the
achromat, where the dispersion is maximum (6.4~meter). The first
dipole of the achromat is a switching magnet to provide beam to the
0$^{\circ}$ linac dump.
\vspace*{1mm}
{\noindent\em 2.2.1.6 Momentum Correction and Momentum Spread}
\vspace*{1mm}
The H minus beam from the linac suffers from the energy (2.0~MeV)and
pahse (3.5$^{\circ}$) jitter due to the phase and amplitude errors of
0.5$^{\circ}$ and 0.5\% respectively. To correct this energy jitter
there is corrector cavity in the HEBT which operate at the same
frequency as linac namely 805 MHz and phase lock with last cavity of
the linac such that the particle with design energy (synchronous) sees
the $-$90$^{\circ}$ (zero voltage) hence does not change the
energy. The particle having more energy than the design energy arrives
earlier than the synchronous particle and sees the negative voltage
hence lose the excess energy. The particle having less energy then the
synchronous particle arrive later than the synchronous and sees the
positive voltage and gain the required energy. The energy gain (lose)
depend on the time difference (phase difference) between the off
energy particles and the synchronous particle and the cavity voltage.
There is a limit to this correction scheme, which is if the phase
difference is more than 90$^{\circ}$ than the particle does not gain
(lose) the correct amount of energy. Ideally we want maximum phase
difference of less than 60$^{\circ}$. The phase slip per MeV for a
distance $L$ is given by
\begin{equation}
\Delta\phi_L \equiv \frac{\gamma}{\gamma(\gamma+1)}\frac{\Delta T}{T}\frac{L}{\beta c}2\pi f
\end{equation}
where $\beta$ and $\gamma$ are the relativistic parameters, $c$ is the
speed of light, $f$ is the RF frequency and $T$ and $\Delta T$ are the
design energy and energy difference. Fig.~\ref{fig:hebtphase} shows
the phase slip per MeV as function of the distance.
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{hebt_optics1.eps}
\caption{Phase slip per MeV as function of distance for three different energies. \label{fig:hebtphase}}
}
\end{figure}
The required voltage is given by $V_0=\Delta E/\sin f_{slip}$. The
energy jitter after the corrector cavity 0.2 MeV was achieved which is
within the requirement of the ring Momentum correction is accomplished
with a 2.6 meter long, 16 cell, 805 MHz rf cavity, operating at a
gradient ($E_0T$) of 3.6 MV/m. This cavity is similar to the last
cavity of linac. The cavity is located in the first half-cell before
the achromat (100 m from the linac).
SNS accumulator ring also need $\pm$4 MeV energy spread for stability
reason. This is achieved by placing another RF cavity after the
achromat which frequency is about 100 kHz different than the linac
frequency. The linac bunches see different phase as they arrive at
this cavity and gain/loss different energies depending on the cavity
voltage, hence creating required energy spread without creating energy
tails. Results from beam tracking, integrated for the entire
injection period, are shown on Fig.~\ref{fig:hebtedist}. The width of
the energy spread is controlled by RF amplitude of the
cavity. Simulations using a debuncher cavity are shown for
comparison. Since the spreader cavity only translates the energy,
there is no energy tail in this scheme while the other scheme produces
a very long energy tail, which can cause the protons to spill over
into the extraction gap.
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{hebt_optics2.eps}
\caption{Time integrated energy distribution using constant amplitude energy
spreader cavity(black) and debuncher cavity (red). \label{fig:hebtedist}}
}
\end{figure}
\vspace*{1mm}
{\noindent\em 2.2.1.6 Ring Matching Section}
\vspace*{1mm}
After the achromat, two cells are provided for the diagnostics. At
the end of the achromat this line is parallel to the ring straight
section, but offset by 10~m, allowing one to have the required ``dog
leg'' for injection into the ring. These bends are necessary to allow
the dispersion and its derivative to be zero at the injection stripper
foil. The dispersion has a minimum and maximum of similar amplitude
but opposite sign through the ``dog leg''. This section has enough
``knobs'' (quadrupoles) to match six variables (four amplitude
functions and two dispersion functions). There is no vertical bend and
no vertical dispersion. The locations of the dipoles are determined
by the injection scheme Fig.~\ref{fig:hebtpara} shows the amplitude
function ($\beta_x$, $\beta_y$) and the dispersion function ($\eta_x$)
along the HEBT.
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{hebt_optics3.eps}
\caption{The lattice function $\beta_x$, $\beta_y$, and $\eta_x$ along the HEBT.
\label{fig:hebtpara}}
}
\end{figure}
\subsubsection{Ring optics}
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{wei_fig09.eps}
\caption{SNS ring lattice superperiod of FODO/doublet
structure. The lattice periodicity is 4.
Along the indicated beam line are dipoles (centered square boxes), focusing quadrupoles
(upper bars), and defocusing quadrupoles (lower bars). \label{fig:snslattice}}
}
\end{figure}
The ring optics file is included in Section~\ref{sec:ringoptics}.
\subsubsection{RTBT optics}
\vspace*{1mm}
{\noindent\em 2.2.3.1 Introduction}
\vspace*{1mm}
Ring to Target Beam Transfer (RTBT) line is about 150 meter long and
carry the beam from the ring extraction region to the target and
provide the desired footprint for the accelerator complex. The
general features of this line are, the line is immune to one kicker
failure and the ratio of acceptance to rms emittance is more than 20.
\vspace*{1mm}
{\noindent\em 2.2.3.2 Design Requirements}
\vspace*{1mm}
The beam requirements at the target are given in
Table~\ref{tab:targetreq}. Table~\ref{tab:rtbtopt} gives the
Courant-Snyder parameters of the beam at the beginning of the
extraction kicker magnet and at the target.
\begin{table}[thb]
\caption{Beam requirements at the target. \label{tab:targetreq}}
\vspace{2mm}
\centering{
\begin{tabular}{l l} \hline
Beam width & 200 mm \\
Beam height & 70 mm \\
Time-average beam current density, over beam footprint & $\leq$ 0.147 A/m$^2$ \\
Beam power within target and outside nominal spot & < 10\% \\
Peak time-average beam current density, over 1 cm2 & $\leq$ 0.25 A/m$^2$ \\ \hline
\end{tabular}
}
\end{table}
\begin{table}[thb]
\caption{Courant-Snyder parameters at the ring extraction magnet and at the target for a 1.4 MW beam.
\label{tab:rtbtopt}}
\vspace{2mm}
\centering{
\begin{tabular}{l l l l} \hline
Quantity & Output of Ring & Target & Units \\ \hline
$\alpha_x$ & -0.3180 & 0.0 & \\
$\beta_x$ & 11.468 & 83.0 & $\pi$ mm/mrad \\
$\epsilon_x$ & 120 & 120 & mm mrad (5 rms, unnorm.) \\
$\alpha_y$ & -0.2729 & 0.0 & \\
$\beta_y$ & 13.002 & 10.2 & mm/mrad \\
$\epsilon_y$ & 120 & 120 & $\pi$ mm mrad (5 rms, unnorm.) \\
\hline
\end{tabular}
}
\end{table}
\vspace*{1mm}
{\noindent\em 2.2.3.3 Design Description and Functions of the RTBT Line}
\vspace*{1mm}
The RTBT uses a FODO lattice up to the beam spreading section. The
line has following elements: (a) extraction, (b) beam dump, (c) halo
collimation, (d) beam spreader, and (e) diagnostics. The first four
functions have essentially been decoupled in the RTBT. The extraction
system starts in the ring with a kicker magnet and continues through
four cells in the RTBT. Following the extraction system, the beam can
be dumped straight through a 16.8$^{\circ}$ dipole magnet. After this
16.8$^{\circ}$ bend, two cells are used for the halo collimation.
Following another 6 cells of transport, the last five quadrupoles in
the line are used for final beam spreading to produce the beam size
required at the target. Every quadrupole in the RTBT is followed by a
small dipole corrector magnet for steering of the beam in the
quadrupole focusing plane. To reduce the probability of uncontrolled
beam losses and define the beam size precisely on the target, RTBT is
equipped with four transverse beam halo scrapers and several types of
diagnostic devices. To keep uncontrolled beam losses low, a study of
the required alignment tolerances \cite{rapariahebtpac}, has led to
the requirements given in Table 3.
\begin{table}[thb]
\caption{Expected misalignment for the RTBT magnets. \label{tab:rtbterr}}
\vspace{2mm}
\centering{
\begin{tabular}{l l} \hline
Translation (x and y) & $\pm$0.1 mm \\
Pitch and yaw & $\pm$1 mrad \\
Rotation & $\pm$0.5 deg \\ \hline
\end{tabular}
}
\end{table}
Fig.~\ref{fig:rtbtphase} shows the amplitude functions ($\beta_x$, $\beta_y$) and
the dispersion function ($\eta$)
along the RTBT. This line is designed such that it can accommodate the beam
current required for the upgrade to 2 MW.
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{rtbt_optics1.eps}
\caption{The lattice functions along the RTBT.
The phase advance between kickers and target is an integral multiple of $\pi$,
so that in event of a kicker
failure beam will not move at the target.Fig.~\ref{fig:codmisskicker} shows the closed orbit for
different kickers failure. \label{fig:rtbtphase}}
}
\end{figure}
\begin{figure}
\centering{
\epsfysize=3.5in
\epsfbox{rtbt_optics2.eps}
\caption{Close orbit due to kickers failure. \label{fig:codmisskicker}}
}
\end{figure}
\vspace*{1mm}
{\noindent\em 2.2.3.4 Extraction}
\vspace*{1mm}
The extraction of the beam is done in a single turn with full aperture
at a pulse repetition frequency of 60 Hz. The extraction system
consists of two sets (seven each) kicker magnets and a Lambertson
magnet septum and a dipole magnet. The Lambertson septum magnet will
receive the vertically kicked down beam and will provide large
deflection angle (16.8$^{\circ}$) to enable ejection horizontally from
the accumulator ring. A dipole, which is 540$^{\circ}$ phase advance
away from the Lamberton magnet, bends the beam horizontally in the
same direction by 16.8$^{\circ}$, making the extraction system
achromatic. The Lambertson magnet is rotated 2.55$^{\circ}$ anti-clock
wise to neutralize the vertical kick from the kickers and making the
beam about 9 inches difference between the ring and RTBT beam height.
\vspace*{1mm}
{\noindent\em 2.2.3.5 Beam spreader}
\vspace*{1mm}
The beam spreader consists of five radiation hard quadrupoles near the
end of the RTBT. These five 36 cm diameter aperture quadrupoles
provide the desired beam size at the target. Due to thermal
considerations of the target, the beam current density on the target
must remain below 0.25 A/m$^2$. This requirement results in a
non-Gaussian beam distribution in space (with un-normalized rms
emittance of 24 $\pi$ mm mrad). The required current density
distribution is achieved using painting scheme described elsewhere
\cite{beebewangpaint}.
%*********************************************************************
\clearpage
\newpage
\lhead{SNS Ring Design Manual}
\rhead{Ring optics tuning}
\rfoot{November 5, 2002}
\subsubsection{Ring optics tuning}
Detailed studies of the SNS ring lattice optics have been conducted using
the MAD Computer Code \cite{MAD}. The horizontal and vertical tuning
ranges for matched solutions with well-behaved lattice functions were
determined and the power supply margin verified. Using stability criteria,
potential working points have been identified. For these working points,
the behavior of the lattice functions was examined with increasingly complex
assumptions, including chromatic effects and sextupoles, lattice perturbations
due to injection chicane and to dynamic bumps for painting, and magnet field
and alignment errors. The results have been used to guide beam dynamics
calculations to determine injection painting schemes, dynamic apertures, orbit
and resonance correction schemes, collimator settings, and to resolve other
design issues.
Determination of the accessible region of tune space involves a number of
degrees of freedom and some constraints. Constraints include the necessity
of matched solutions, beta functions within aperture-dictated limits, magnet
field strengths within specifications, and achromatic arcs. Degrees of
freedom include the phase advances in the straight sections and the vertical
phase advance in the arcs. The horizontal phase advance in the arcs is
restricted to 2$\pi$ by the achromaticity constraint. The accessible region
of tune space for the SNS ring lattice is $6 < \nu_x < 7$ and $4 < \nu_y < 7$.
Figure \ref{fig:LattFuncMax} shows the behavior of the maxima of the lattice
functions $\beta_x$, $\beta_y$, $D_x$ and of the doublet focusing quadrupole
strength as a function of $\nu_y$ for different values of $\nu_x$.
Considerations of these and other parameters aided in our definition of the
accessible tune space.
\begin{figure}[!ht]
\begin{minipage}[l]{3.0in}
\centerline{\epsfig{figure=Fig1a.eps,height=2.0in,width=2.5in,angle=-0,clip=}}
\end{minipage}
\begin{minipage}[l]{3.0in}
\centerline{\epsfig{figure=Fig1b.eps,height=2.0in,width=2.5in,angle=-0,clip=}}
\end{minipage}
\begin{minipage}[l]{3.0in}
\centerline{\epsfig{figure=Fig1c.eps,height=2.0in,width=2.5in,angle=-0,clip=}}
\end{minipage}
\begin{minipage}[l]{3.0in}
\centerline{\epsfig{figure=Fig1d.eps,height=2.0in,width=2.5in,angle=-0,clip=}}
\end{minipage}
\caption{Parameters found in tune space survey. \label{fig:LattFuncMax}}
\end{figure}
For tunes above 7, some magnet strengths exceed specifications and beta
functions exceed limits. For tunes below the specified ranges, there is
insufficient focusing for solutions to be obtained. Even within the specified
range there are inaccessible gaps due to structure resonances. These occur
around $\nu_x, \nu_y = 6$ which is a second order structure resonance and also
as $\nu_y$ approaches 4, which is a first order structure resonance. As a tune
approaches one of these structure resonance values, the corresponding beta
functions become excessive, and sufficiently close to the resonance no
solutions exist. Particularly severe is the case of $\nu_x$ approaching 6.
Because of the achromatic constraint, the horizontal phase advance is fixed in
the arcs, and an excessive change in the rate of phase advance between arc and
straight sections causes $\beta_x$ to become too large at the injection foil by
the time $\nu_x = 6.15$. Because the rate of vertical phase advance can be
kept more uniform throughout the ring, the effect when $\nu_y$ is in the
vicinity of the vertical resonant regions is more localized. However, we
follow the practice of keeping both $\nu_x$ and $\nu_y$ well removed from
these integer values.
Selection of potential operating points from this large region in tune space
becomes simpler with the aid of a resonance analysis. To start, we consider
the most dangerous resonances: first and second order resonances, third order
structure resonances, and fourth order structure sum resonances. In horizontal
tune space, there is a second order resonance at $\nu_x = 6.5$ and a third
order structure resonance at $\nu_x = 6.67$. Given the tune spread of the
intense SNS beam ($\Delta \nu_x = 0.2$), avoidance of these resonances requires
choosing $6.15 < \nu_x < 6.5$. For the same reason, we must also choose
$\nu_y < 6.5$. Analysis of the vertical tune space reveals only two regions
with working space between resonances at this level: $4 < \nu_y < 4.5$ and
$6 < \nu_y < 6.5$. Of these, the stronger vertical focusing yields better
behaved vertical beta functions in the second region, so we concentrate on
operating points in the region $6.15 < \nu_x, \nu_y < 6.5$. Figure
\ref{fig:TuneSpace} shows this region, including all resonance lines to fourth
order and two potential operating points. Of these, the point at
$(\nu_x, \nu_y) = (6.4, 6.3)$ is further removed from the second order
structure resonance at 6.0, but nearby third and fourth order sum resonances
will require correction to prevent beam loss. The point at
$(\nu_x, \nu_y) = (6.23, 6.2)$ avoids the higher order resonances, but at high
intensities the tune spread may excite the structure resonance at 6.0. Both of
these operating points have been subjected to full beam dynamic studies
\cite{Fedotovpac01, Papa, Wei}.
\begin{figure}[!ht]
\centerline{\epsfig{figure=Fig2.eps,height=4.0in,width=4.0in,angle=-0,clip=}}
\caption{Potential operating points in tune space. All resonance lines
through fourth order are shown. \label{fig:TuneSpace}}
\end{figure}
Because these two operating points are reasonably close in tune space, their
lattice functions are quite similar. The horizontal and vertical beta
functions both peak at the doublets in the straight sections, at approximately
$\beta_x = 27m$ and $\beta_y = 15m$, respectively; and the dispersion in the
arcs peaks at about $D_x = 3.7m$. At the injection foil, however, the beta
values are $\beta_x = 7.2m$ and $\beta_y = 11.0m$ for the
$(\nu_x, \nu_y) = (6.4, 6.3)$ operating point and $\beta_x = 10.4m$ and
$\beta_y = 12.2m$ for the $(\nu_x, \nu_y) = (6.23, 6.2)$ operating point.
Thus, the injection painting scheme must be optimized independently for each
operating point. The horizontal and vertical chromaticities are about
$\xi_x = -8$ and $\xi_y = -7$, respectively. Consequently, for
$\Delta p/p_0 = \pm 0.01$, which corresponds to $\Delta E = 15MeV$ for $1 GeV$
protons, the closed orbit deviation reaches $37mm$ in the arcs and about
$1-2mm$ in the straight sections, and the horizontal and vertical tunes are
shifted by about $\Delta \nu_x = -0.08$ and $\Delta \nu_x = -0.07$,
respectively.
The injection chicane consists of four static horizontal dipole magnets in one
of the $12.5m$ straight section drifts and eight dynamic bump magnets (four
horizontal, four vertical) arrayed symmetrically in the corresponding $6.85m$
drifts. When the static injection dipoles are included in the ring lattice,
the horizontal closed orbit is bumped about $10cm$ in the injection region.
This bump leads to a local blip, also of $10cm$, in the dispersion function and
to beta-beating of about 5\% in $\beta_y$ at the doublets. When the dynamic
bump magnets are activated, the beta-beating is not strongly affected, but they
do generate a residual dispersion function around the ring. In the case of a
large bump ($40mm$ in x and $46mm$ in y - sufficient to direct the closed orbit
through the foil), the residual horizontal dispersion peaks at $D_x = 25cm$
in the straight section doublets, and there is a vertical dispersion function
of up to $D_y = 15cm$ distributed around the ring. For
$\Delta p/p_0 = \pm 0.01$ the horizontal closed orbit deviation peaks at about
$6mm$ and the vertical closed orbit deviation reaches about $1.5mm$ in the
straight sections.
Among the corrector magnets are four sets of chromatic sextupoles in the arc
sections. With these sextupoles activated and set to eliminate the first order
chromaticities and reduce the variation of the beta functions with energy, the
main effect is the reduction of the chromaticities to near zero. The tunes
remain essentially constant ($\Delta \nu < 0.01$) for particles with
$\Delta p/p_0 = \pm 0.01$. The effect of eliminating the variation of the beta
functions with energy is to make the ring maximum beta function values at
$\Delta p/p_0 = 0$ minima when plotted versus energy. In spite of eliminating
the first order energy variation, the maximum beta function values at
$\Delta p/p_0 = \pm 0.01$ typically exceed those at $\Delta p/p_0 = 0$ by a few
percent, and for one case, the $(\nu_x, \nu_y) = (6.23, 6.20)$ operating point,
the increase in maximum $\beta_x$ is 20\%.
The effects of magnet errors have also been studied. One example involves the
large arc quadrupoles, which are located at the maxima in the dispersion
function and which accordingly have $26cm$, rather than $21cm$, bore radii.
Introducing random magnetic field errors, using a Gaussian distribution of
0.4\% width and truncated at 1\% maximum error, into these magnets results in
beta-beating of about 4\% in $\beta_x$ and in an increase in the maximum of the
horizontal disperion function $D_x$ from $4.00m$ to $4.35m$ for the
$(\nu_x, \nu_y) = (6.23, 6.20)$ operating point. Because these are focusing
quadrupoles, the vertical lattice functions are little affected by the errors.
To avoid these errors, the large arc quadrupoles will be placed on their own
separate power supply.
Another possible source of error would be a dynamic mismatch of the injection
kicker strengths. We considered an extreme worst case situation of 1\% errors
in the kicker field strengths at the large kick size of $40mm$ in x and $46mm$
in y. The errors were assigned systematically to maximize their effect. The
result was a mild closed orbit distortion of $1.6mm$ distributed around the
ring. Thus, the control of the beam painting afforded by the kickers will be
good.
The results of these optical ring lattice studies have been and are continuing
to be used to guide beam dynamic investigations such as determination of
injection painting schemes, dynamic aperture and loss calculations, determining
collimator settings, and other design issues.
%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
%\rhead{Acceptance}
%\rfoot{July 2002}
\subsubsection{Acceptance}
%*********************************************************************
\clearpage
\newpage
\lhead{SNS Ring Design Manual}
\rhead{Beam Loss}
\rfoot{November 8, 2002}
%\subsection{Beam Loss Mechanisms}
\subsection{Beam Loss}
\label{sec:beamlosses}
In this section, we will calculate the total loss budget expected under
nominal conditions and assuming no cleaning systems in both the
accumulator ring and transfer lines.
Cleanig systems described in section~\ref{sec:collimation} are then
introduced to concentrate these losses in special areas where restricted
acces and special safety procedures will be put in place. The design
and layout of the cleaning sections is based on this loss budget and the
collimators are required to whitstand the heat load and deformations
produced by the losses.
After the various cleaning systems are introduced, a large percentage
of these losses will in this way become ``controlled losses''
and only a small fraction will attain the machine becoming what we
call ``uncontrolled'' loss. These uncontrolled losses are
mainly due to the inefficiency of the cleaning systems or to loss
mechanism itself. The estimations of the prompt and remanent
radiation levels and the calculaions of the required shielding are
based in the distribution and intensity of uncontrolled losses. This
will be presented after the introduction description of the cleaning
systems in section \ref{sec:collimation}.
\subsubsection{HEBT beam-loss budget}
\label{sec:loss_HEBT}
Along the HEBT, the sources of uncontrolled loss are scattering with
the residual gas and magnetic H$^-$ stripping.
At 1 GeV the cross sections of H$^-$ stripping for Nitrogen and Hydrogen are
\[\sigma_N=9.14\cdot 10^{-19}~cm^2\]
\[\sigma_H=1.30\cdot 10^{-19}~cm^2.\]
With a vacuum of $5\cdot 10^{-8}$, the fractional stripping losses account for
$\approx 2.8 \cdot 10^{-5}$ along 170 meters or $1.6\cdot 10^{-7}$ per
meter.
The scattering produces a flux of neutraly charged halo particles that
are homogeneously distributed along the line and inside the magnetic
fields. Most of then will thus become uncontrolled loss.
These losses take place continuously during normal operation. We may
expect though variations if the vacuum pressure or the composition of
the residual gas changes unexpectedly.
For magnetic stripping losses, magnet strength is chosen so that the
magnetic stripping is at the $10^{-8}$per meter level much lower than
the vacuum stripping loss.
Steady losses happenning in the HEBT are mostly linked to tails coming
from the LINAC.
The rms emittance of the beam coming from the linac is expected to be
small ($\epsilon_{rms}$~=~0.5~$\pi$mm$\cdot$mrad normalised) but
large tails in the distribution may contain a fraction of the beam
larger than $10^{-4}$. Fig.~\ref{fig:losses} shows the beam fraction
contained in the tails of a gaussian beam depending on the beam rms
emittance. A factor of two in the beam emittance coming from the linac
yields an increase of three orders of magnitude in the halo
population.
These losses are suceptible of being removed by collimators placed on
the path of the beam before the achromat. This collimation system
is described in section \ref{sec:collimation}.
\begin{figure}[htbp]
\begin{center}
\centering
\includegraphics*[width=0.7\columnwidth]{hebt_losses.eps}
\caption{Fraction of the beam exceeding a fixed aperture in the
HEBT line. The aperture is chosen to match 13$\pi$mm$\cdot$mrad,
half of the HEBT acceptance and equal to the collimator
acceptance.}
\label{fig:losses}
\end{center}
\end{figure}
Longitudinal loses may ocurr if the energy jitter or the energy spread
of the beam are larger than expected. No quantification has been done
of this effect as it will depend in a large measure of operating
conditions. A cleaning device has been foreseen to catch this
longitudinal tails before the achromat.
\subsubsection{Ring beam-loss budget}
\label{sec:loss_ring}
During the process of injection and accumulation, space-charge and
resonance crossing are expected to make the protons diffuse outwards
frm the beam center. Following tracking studies and an optimization of
the painting schemes, a fraction of $2.0\cdot 10^{-3}$ of the beam is
assumed to be in the tails of the beam...
Special attention is required at the injection as extraction sections
\paragraph{Injection area}
The main source of loss in the injection section is the nuclear
scattering of the beam in the Carbon foil. Besides, we also need to
consider the magnetic stripping of the H$^-$ beam in the second dipole of
the injection chicane INJB2 where the H$^-$ beam traverses an area
close to the magnet coil and of high megnetic field.
The losses produced at the injection foil are dominated by nuclear
scattering. Once the energy and foil thickness have been defined, the
loss is determined by the size of the incoming beam and
the painting scheme.
The maximum number of foil crossings has been estimated by
simulation. The average number of foil
crossings per proton is $\approx$ 7 in nominal conditions. Yet, if the
beam emittance increases or deviates from a Gaussian distribution, this
number increases up to 12 crossings per proton. For a carbon foil of
300~$\mu$g/cm$^2$, the fractional loss at the foil due to nuclear
scattering will be $\approx 3.7\cdot 10^{-5}$ under nominal conditions
and up to $\approx 6.3\cdot 10^{-5}$ for an exceptional large beam.
The fractional loss per meter due to magnetic stripping of H$^-$ is
given by the formula
\[\frac{N_{loss}}{NL}=\frac{B}{K_1}e^{-\frac{K_2}{\beta\gamma c B}}\]
\[K_1=8\cdot 10^{-6};K_2=4.3\cdot 10^9\]V/m
where B is the magnetic field in Tesla and $\beta$ and $\gamma$ are the
relativistic parameters.
We assume a magnetic field of 0.3 Tesla. For this magnetic field,
$1.3\cdot 10^{-7}$ of the beam will be lost along the effective
magnetic length of the dipole ($\approx$ 1 meter) at 1 GeV. For a
momentum of 1.3 GeV, the magnet should be replaced with a longer
version and the magnetic field reduced to 0.25 to keep the same level
of stripping losses.
\subsubsection{RTBT beam-loss budget}
\label{sec:loss_RTBT}
The beam reaching the RTBT line is assumed to be well inside tha
aperture thanks to careful painting and collimation in the ring.
No steady losses are expected from halo development or partial
cleaning.
In addition to that, the optics of the RTBT has been calculated such
that a failure of one of the fourteen extraction kickers will produce an
orbit deviation along the RTBT line but no beam hits the vacuum
pipe and the beam impacts the target at the nominal location. In the
event of a failure of two kickers, approximately 10\% of the beam
would be lost in the transfer line. Figure \ref{fig:codmisskicker} shows the
trayectory of the beam along the RTBT when any one of the fourteen
kickers fail to work. Also maximum deviation of the closed orbit along
the line in the case f two kicker failures is indicatedas an envelope
by the black line. Form this plot is it clear the losses would be localized
inside a range of 10 to 20 meters. In the rare case
of more than two kicker failures, the whole pulse would be lost.
The average beam losses will be given by the probability of kicker
failure. Nevertheless, we have to draw attention to the fact
that prompt losses would be high and very localized. The RTBT
collimators are essential to prevent transient losses and protect the
target vessel from the extraction misfunction. The design of the
collimators has been done to resist two whole consecutive pulses after
which the machine should be stopped and the kickers fixed.
One should include in this section the losses in the target window due to
nuclear scattering.
%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
%\rhead{Controlled beam loss}
%\rfoot{July 2002}
%\subsubsection{Controlled beam loss}
%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
%\rhead{Uncontrolled beam loss}
%\rfoot{July 2002}
%\subsubsection{Uncontrolled beam loss}
%*********************************************************************
\clearpage
\newpage
\lhead{SNS Ring Design Manual}
\rhead{Single-particle Effects}
\rfoot{November 4, 2002}
\subsection{Single-particle Effects}
Typical values for the tune
spread due to different mechanisms in the SNS ring are given
in Table~\ref{tab:tunespread}.
\begin{table}[hpt]
\center
\caption{Tune spread produced by various mechanisms
on a 2 MW beam with transverse emittance of 480 $\pi$~mm~mrad and
momentum spread of $\pm1\%$.}
\label{tab:tunespread}
\vskip 5pt
{\newcommand{\hl}{\hline}
\begin{tabular}{ll} \hl
Mechanism & Full tune spread \\ \hl
Space charge & 0.15-0.2 (2 MW beam) \\
Chromaticity & $\pm$0.08 ($1\%~ \Delta~ p/p$)\\
Kinematic nonlinearity ($480 \pi$) & 0.001 \\
Fringe field ($480 \pi$) & 0.025 \\
Uncompensated ring magnet error ($480 \pi$) & $\pm$0.02 \\
Compensated ring magnet error ($480 \pi$) & $\pm$0.002 \\
Fixed injection chicane & 0.004 \\
Injection painting bump & 0.001 \\
\hl
\end{tabular}
}
\end{table}
%*********************************************************************
%\clearpage
%\newpage
\lhead{SNS Ring Design Manual}
\rhead{Kinematic nonlinearity}
\rfoot{July 2002}
\subsubsection{Kinematic nonlinearity}
Note that, even in the absence of any field, the motion of a
relativistic particle in free space is a non-linear function of the
canonical momentum ${\bf p}$. The ``kinematic non-linearity'' refers
to these high-order terms in the expansion of the classical
relativistic Hamiltonian which contain only the transverse momenta,
$p_x$ and $p_y$. This non-linearity is negligible in high-energy
colliders (\emph{e.g.} RHIC, LHC), where $p_{x,y}\ll p_z$ but it
becomes noticeable in low-energy high-intensity rings.
In fact, a measure of the impact of this non-linearity is given by the
first-order tune-shift. By keeping all the kinematic terms in the
expansion of the Hamiltonian, one can obtain a general expression for the
first-order tune-shift they induce~\cite{fringe}.
%~\cite{yannis}:
%\begin{equation} \label{eq:kinem}
% \delta Q_{x,y} =
% \frac{1}{2\pi} \sum_{k=2}^\infty \frac{(2k-3)!!}{2^k(2k)!!}
% \times\sum_{\lambda=0}^k \,\lambda\binom{2\lambda}{\lambda}
% \binom{k}{\lambda} \binom{2(k-\lambda)}{k-\lambda}
% J_{x,y}^{\lambda-1} J_{y,x}^{k-\lambda} G_{x,y}
%\end{equation}
%where $G_{x,y}=\oint_\mathrm{ring} \gamma_{x,y}^\lambda
%\gamma^{k-\lambda}_{y,x} ds$
The first, usually
dominant, term in the series gives an octupole-like tune-shift,
\emph{i.e.} linear in the actions. For a high-intensity rings, where the
emittance is large and the gamma functions in the straight sections
exceed unity, the kinematic terms give a non-negligible tune-shift. For
the SNS accumulator ring it is about $10^{-3}$ at 480~$\pi$~mm~mrad.
%*********************************************************************
\subsubsection{Sextupole Effects}
The most common magnet non-linearity encountered in small rings
arises from high-field sextupoles introduced for chromaticity control.
The SNS ring contains twenty chromatic sextupoles,
placed in the arcs in high $\beta$ and dispersion areas~\cite{tsoupas}.
Their non-linear effect has been quantified and
found small. Sextupole-like contributions also come from
the leading-order fringe-field effect of the thirty-two arc
dipoles. These sextupole effects, also small, can easily be corrected by
the eight dedicated sextupole correctors.
%\clearpage
%\newpage
\lhead{SNS Ring Design Manual}
\rhead{Magnet imperfection and nonlinearities}
%\rfoot{July 2002}
\subsubsection{Magnet imperfection and nonlinearities}
In a magnet with normal quadrupole symmetry the first allowed
multipole error is the normal dodecapole, $b_6$. In the absence of
pole-tip shaping, this error can be exceedingly large: for the SNS
21~cm quadrupole~\cite{fringe}, an {OPERA-3d}
\cite{opera} simulation (with un-shaped ends) gives a dodecapole
component of about 120 (in units of $10^{-4}$, normalized with respect
to the main, quadrupole, component).
%\begin{figure}[t]
% \centering
% \rotatebox{0}{\scalebox{0.35}{\includegraphics*{MOP6A013.eps}}}
% \vskip -10 pt
% \caption{Dodecapole component in an SNS 21~cm quadrupole with un-shaped ends.
% The reference radius is 10~cm, and the origin, $z=0$, is at the magnet's center.}
% \label{evoldodec}
% \vskip -10 pt
%\end{figure}
Because the dodecapole error is quite localized, its effect can be computed
using a thin-element approximation. Applying first-order perturbation theory,
one finds the tune-spread induced by dodecapole errors is given by
\begin{subequations} \label{eq:dodtune}
\begin{equation}
\begin{pmatrix}\delta\nu_x \\ \delta\nu_y \end{pmatrix}
= \sum_i \frac{b_{6i} Q_i}{8\pi B\!\rho} {\cal D}_i
\begin{pmatrix}J_x^2 \\ J_x J_y \\ J_y^2 \end{pmatrix},
\end{equation}
where ${\cal D}_i$ denotes the $3\times2$ matrix
\begin{equation}
\begin{pmatrix}
\beta_{xi}^3 & -6\beta^2_{xi}\beta_{yi} & 3\beta_{xi}\beta^2_{yi} \\
-3\beta^2_{xi}\beta_{yi} & 6\beta_{xi}\beta^2_{yi} & -\beta^3_{yi} \\
\end{pmatrix}.
\end{equation}
\end{subequations}
Here the index $i$ runs over all dodecapole kicks in the ring,
\emph{i.e.} over the entrances and exits of all quadrupoles. Note
that this effect depends linearly on the error strength, but
quadratically on the amplitude. A comparison of this analytic result
with \textsc{MaryLie} tracking data has shown a striking
agreement~\cite{fringe}. In addition, it was made put in evidence that
the very large uncorrected dodecapole error gives a tune-spread
roughly twice that caused by the quadrupole fringe fields.
\begin{figure}[t]
\centering
\rotatebox{0}{\scalebox{0.35}{\includegraphics*{MOP6A014.eps}}}
\vskip -10 pt
\caption{Tune footprints of the SNS ring with a
dodecapole error in the quadrupoles of $b_6=60$units; results are from
tracking data (blue) or the analytic result \eqref{eq:dodtune} (red).}
\label{dodec}
\vskip -10 pt
\end{figure}
By shaping the ends of the quadrupoles, one can reduce the $b_6$ error
to 1 unit or less~\cite{Steffen}. This constitutes \emph{local}
compensation. One might also correct the $b_6$ error by adding a
small negative dodecapole component through the body of the magnet.
In Fig.~\ref{compdodec} we compare the tune-spreads \eqref{eq:dodtune}
after local and body compensation. In this example, the compensation
works well in both cases, with local compensation being slightly
better. But, in fact, it is essential to use local compensation:
because the tune-spreads \eqref{eq:dodtune} depend cubically on the
$\beta$ functions, the results of body compensation will be very
sensitive to the ring optics.
\begin{figure}[t]
\centering
\rotatebox{0}{\scalebox{0.60}{\includegraphics*{MOP6A015.eps}}}
\vskip -10 pt
\caption{Comparison of tune-shift plots using body (red) and local
(blue) compensation of the dodecapole component in the SNS ring
quadrupoles~\cite{fringe}.}
\label{compdodec}
\vskip -10 pt
\end{figure}
%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
\rhead{Magnet fringe fields}
%\rfoot{July 2002}
\subsubsection{Magnet fringe fields}
The relative impact of a longitudinal fringe field on a particle's
transverse momentum is proportional to the ratio of transverse
emittance to magnetic length~\cite{scal}. Hence, the effect of
quadrupole fringe-fields is usually small in low-emittance, low
aspect-ratio machines ({\em e.g.} RHIC, LHC) but is very important for
high-emittance, high aspect-ratio machines such as the SNS. For a
quadrupole one can evaluate the fringe-field contribution in the limit
of zero fringe length. The corresponding Hamiltonian for a single
fringe (to leading order) is \cite{ForMilut,Forest}
\begin{equation}
H_{f} = \frac{\pm Q}{12B\!\rho(1\!+\!\frac{\delta p}{p})}
(y^3 p_y - x^3 p_x + 3 x^2 y p_y - 3 y^2 x p_x),
\label{eq:hamfringe}
\end{equation}
where $Q_i$ is the quad strength, and the $+$ and $-$ signs are used
at, respectively, the entrance and exit of the magnet. It follows, as
Lee-Whiting showed many years ago~\cite{Lee}, that a quadrupole
fringe-field induces an octupole-like transverse kick. Using
\textsc{MaryLie} \cite{marylie}, one can build quadrupole maps that
include fringe fields based on either~\eqref{eq:hamfringe} or an exact
representation~\cite{Venturini}. We created tune footprints by
applying Laskar's frequency analysis~\cite{Laskar} to 1200 turns of
\textsc{MaryLie} tracking data. Particles were launched in different
directions out to $1000\pi$~mm~mrad, and the only non-linearities
included were those caused by thick elements and magnet fringe fields.
Figure~\ref{fringe} shows that quadrupole fringe fields have an
important impact on the dynamics of the SNS ring, giving tune spreads
of about (0.04,0.03) at $1000\pi$~mm~mrad, roughly one-third the
space-charge tune spread~\cite{Alexei2}. In addition,
Fig.~\ref{fringe} shows that the hard-edge model slightly
overestimates the fringe-field effect and therefore represents a
conservative estimate.
\begin{figure}[t]
\centering
\rotatebox{0}{\scalebox{0.35}{\includegraphics*{MOP6A011.eps}}}
\vskip -10 pt
\caption{Tune footprints of the SNS ring, based on
realistic (blue) and hard-edge (red) quadrupole fringe fields.}
\label{fringe}
\vskip -10 pt
\end{figure}
In our case the tune spread can be accurately represented by the results
of first-order perturbation theory \cite{ForMilut,Forest}:
\begin{equation}
\begin{pmatrix}\delta\nu_x \\ \delta\nu_y \end{pmatrix} =
\begin{pmatrix} a_{hh} & a_{hv} \\ a_{hv} & a_{vv} \end{pmatrix}
\begin{pmatrix} 2 J_x \\ 2 J_y \end{pmatrix},
\label{eq:ffield}
\end{equation}
where the normalized anharmonicities are given by
\begin{equation}
\begin{split}
a_{hh} & = \frac{-1}{16\pi B\!\rho}\sum_i \pm Q_i\beta_{xi}\alpha_{xi},\\
a_{hv} & = \frac{ 1}{16\pi B\!\rho}\sum_i \pm Q_i
(\beta_{xi}\alpha_{yi}-\beta_{yi}\alpha_{xi}),\\
a_{vv} & = \frac{ 1}{16\pi B\!\rho}\sum_i \pm Q_i\beta_{yi}\alpha_{yi}.\\
\end{split}
\label{eq:anharm}
\end{equation}
Here the index $i$ runs over the entrances and exits of all
quadrupoles in the ring, and the $+$ and $-$ signs are as in
\eqref{eq:hamfringe}. Note that the entrance and exit fringe fields
do \emph{not} cancel one another: even if the $\beta$ functions are
equal at the entrance and exit, the $\alpha$ functions usually change
sign, leading to an additive effect. For the SNS lattice we find
$(a_{hh},a_{hv},a_{vv})\approx(49,22,42)~{m^{-1}}$, and these values
closely match (apart from the obvious resonance) the results shown in
Fig.~\ref{fringe}.
%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
%\rhead{Chromatic effects}
%\rfoot{July 2002}
%\subsubsection{Chromatic effects}
%*********************************************************************
%\clearpage
%\newpage
%\lhead{SNS Ring Design Manual}
\rhead{Resonances and dynamic aperture}
%\rfoot{July 2002}
\subsubsection{Resonances analysis and frequency maps}
%We choose a standard SNS ring lattice matched for the four working points in
%study. The fringe fields in the ends of the quadrupoles are modeled as
%thin kicks using 5th order maps issued by
%MaryLie~5~\cite{marylie}. Systematic and random magnet errors at the
%required level of 10$^{-4}$ of the main magnet field are also
%included~\cite{magnet}. The chromaticity sextupoles are switched off
%in order to study the effect of the chromatic tune-shift and then
%tuned to give 0 chromaticity. In all the simulations, the RF voltage
%is set to 0, for simplicity. Finally, space-charge forces are not
%included, as we would like at a first stage to study the single
%particle resonance effects to the SNS beam. A total of 1500 particles
%are uniformly distributed on the phase-space with zero initial
%momentum and different initial transverse coordinates, up to
%480$\pi$~mm~mrad (the transverse physical aperture) and tracked with
%the FTPOT module of the UAL environment~\cite{UAL}. They are given 9
%different $\delta p/p$ values, from +2\% to -2\% corresponding to the
%beam momentum aperture. The tracking is conducted for 500 turns to
%provide a good precision in the tune determination.
%\begin{figure}[htb]
%\centering
%\includegraphics*[width=70mm]{WOPB0111.eps}
%\caption{Frequency maps for the working point (6.4,6.3), for 9
%different $\delta p/p$, from +2\% (left bottom corner) to -2\% (right
%upper corner)}
%\label{fig:tunediag1}
%\vspace{-15pt}
%\end{figure}
%
%Frequency maps for the w.p. (6.4,6.3) and natural chromaticity are
%displayed in Fig.~\ref{fig:tunediag1}, where different colors
%correspond to different $\delta p/p$. The triangular shape of the maps
%reflects the octupole-like nature of the leading order quadrupole
%fringe fields~\cite{fringe}, which apparently dominate the
%single-particle non-linear dynamics. The first evident observation is
%that the beam is crossing a multitude of excited resonant lines, which
%appear as distortions of the frequency map, due to the huge chromatic
%tune-shift of more than 0.3, for $\pm$2\% . This can be clearly
%viewed in Fig.~\ref{fig:tunediag2}, where we zoom on one of the
%frequency maps of Fig.~\ref{fig:tunediag1}. For $dp/p$=-1\%, the
%chromatic tune-shift moves the beam very close to the half-integer
%resonance which is excited by quadrupole errors and quadrupole fringe
%fields, as well. Its width is quite big and can be measured on the
%frequency map to be around 0.004. In addition, the structural 5th
%order resonance $Q_x+4Q_y=32$ is excited. This effect can be mainly
%attributed to first order off-momentum ``feed-downs'' from the
%dominant dodecapole error in the edges of the quadrupoles and a small
%contribution from the systematic decapole error on the dipoles. A
%complementary feature of the frequency map is the diffusion map
%(Fig.\ref{fig:tunediag2}). One may compute the tune difference along a
%trajectory for two consecutive time-spans. The difference of the two
%values is a good indication of the particle diffusion. Then, a plot of
%the real space can be produced, mapping each initial condition with
%the diffusion coefficient, using a different color code. In this
%particular case, the grey dots correspond to stable particles, the
%green to weakly unstable and the blue/purple/brown to very unstable,
%most of them excited due to the half-integer resonance (outer stripe
%of initial conditions) and a few of them due to the 5th order normal
%decapole resonance. The frequency - diffusion maps for the
%w.p. (6.23,5.24) and on-momentum particles are given in
%Fig.~\ref{fig:tunediag3}. In this case, the phase space is quite
%perturbed, as the w.p. is very close to the crossing of major 4th
%order resonances which are mostly excited by the quadrupole fringe
%fields. This can be also seen in the diffusion map, where the unstable
%area occupies more than half of the initial conditions space.
%
%Finally, the frequency - diffusion map for the w.p. (6.3,5.8) and
%$\delta p/p=1\%$ is shown in Fig.~\ref{fig:tunediag4}. This map puts
%in evidence the major limitation of this w.p., i.e. the excitation of
%the structural coupling resonance $(1,1)$, due to quadrupole fringe
%fields and/or skew quadrupole errors. This resonance in combination
%with the space-charge can enhance particle loss to a level as high as
%10\%~\cite{Fedotov01}.
%\begin{figure}[t]
%\centering
%\includegraphics*[width=40mm]{WOPB0112a.eps}
%\includegraphics*[width=40mm]{WOPB0112b.eps}
%%\includegraphics*[width=40mm]{tunediag_6.23_5.24_ord5.eps}
%%\includegraphics*[width=40mm]{6.23_5.24_diff_0.eps}\\
%%\includegraphics*[width=40mm]{tunediag_6.3_5.8_p_10.eps}
%%\includegraphics*[width=40mm]{6.3_5.8_diff_p10.eps}
%\caption{Frequency (left) and diffusion map (right), for the working
%point (6.4,6.3) and $\delta p/p = -1\%$ . The different colors on the
%diffusion map represent different tune differences in logarithmic
%scale.}
%\label{fig:tunediag2}
%\end{figure}
%\begin{figure}[htb]
%\centering
%\includegraphics*[width=40mm]{WOPB0113a.eps}
%\includegraphics*[width=40mm]{WOPB0113b.eps}
%\caption{Frequency (left) and diffusion map (right) for the
%working point (6.23,5.24) and $\delta p/p = 0$ .}
%\label{fig:tunediag3}
%\end{figure}
%\begin{figure}[htb]
%\centering
%\includegraphics*[width=40mm]{WOPB0114a.eps}
%\includegraphics*[width=40mm]{WOPB0114b.eps}
%\caption{Frequency (left) and diffusion map (right), for the
%working point (6.3,5.8) and $\delta p/p = 1\%$ .}
%\label{fig:tunediag4}
%\end{figure}
\begin{figure}[htb]
\centering
\includegraphics*[width=85mm]{WOPB0115a.eps}\\
\includegraphics*[width=85mm]{WOPB0115b.eps}\\
\includegraphics*[width=85mm]{WOPB0115c.eps}
\caption{Tune diffusion coefficients for all working points versus the
different momentum spreads, for natural chromaticity (top), for zero
chromaticity (center) and for natural chromaticity and only positive
momentum spread (bottom). The dashed lines are the average of the
coefficient over all momentum spreads.}
\label{fig:tunecomp}
\vspace{-15pt}
\end{figure}
We used frequency maps to study the impact of different resonances to
the single-particle dynamics of the SNS~\cite{frmap}. In particular,
the efficiency of different working points was studied regarding
single particle dynamics considerations.
In order to compare the performance of each working point, we used the tune
diffusion indicator, which is computed by the average of the tune
differences used for the construction of the diffusion maps and
normalized by the initial emittances, for all the integrated
orbits~\cite{Laskar}. This indicator is correlated with other global
chaos indicators, as the resonances driving terms norm and the dynamic
aperture~\cite{Yannis}. We plot the value of the tune diffusion
coefficient versus the $\delta p/p$ for all w.p., on the top of
Fig.~\ref{fig:tunecomp}. The pick values on the diffusion indicators,
for all w.p., correspond to areas of the phase space that are
perturbed due to 4th order resonances, showing once more the
destructive effect of quadrupole fringe fields. The doted lines on the
plots represent the average values of the diffusion indicators for all
tracked momentum spreads. It is clear that (6.23,6.20) is the best
w.p. choice, followed by (6.4,6.3). Their performance can be further
improved by using the available multi-pole correctors~\cite{Yannis2},
for correcting the normal and skew 3rd order resonances, in the case
of (6.4,6.3), and the 4th order normal resonances in the case of
(6.23,6.20). The other two w.p. have the disadvantage of crossing
major structural coupling and 4th order resonances, which are very
difficult to correct.
In the center of Fig.~\ref{fig:tunecomp}, we plot the tune diffusion
coefficient when using the 4 families of chromaticity
sextupoles~\cite{tsoupas} in order to set the chromaticity to 0. In
that case, the chromatic tune-shift is completely canceled and all the
particles with different $\delta p/p$ are located in the same area of
the tune diagram. This is reflected in the very weak dependence of the
tune diffusion coefficient with respect to the momentum
spread. Furthermore, the values of the average diffusion coefficient
are almost equal to the ones for $\delta p/p=0$, when the chromatic
sextupoles are switched-off. This proves the very small non-linear
effect introduced by the chromaticity sextupoles. Finally, note that,
in this case, (6.3,5.8) seems to be the best
w.p. choice. Nevertheless, this is not a fair comparison, as the
space-charge force depresses the tunes. A better picture of the
w.p. performance can be estimated by the average tune diffusion
coefficient of all particles with positive momentum spread and
non-zero chromaticity (at the bottom of Fig.~\ref{fig:tunecomp}). The
results are pretty much the same as in the case of natural
chromaticity.
\subsubsection{Dynamic aperture}
\begin{figure}[htb]
\mbox{
\epsfig{file=da_nosext.eps,width=0.5\columnwidth}
\hskip.1cm
\epsfig{file=da_sext.eps,width=0.5\columnwidth}}
\vskip .1cm
\caption{Dynamic aperture for the working point (6.3,5.8), without
(left) and with (right) sextupoles.}
\label{fig:da}
\end{figure}
In Figs.~\ref{fig:da}, we plot the maximum survival amplitude (in
terms of total emittance) of particles launched in 5 different initial
ratios of the transverse emittances, with three different momentum
spreads ($\delta p/p=0,\pm0.2$). The momentum spread of $\pm 0.2$ is
indeed higher than the actual RF bucket size of $\pm
0.7$. Nevertheless, it corresponds to the momentum acceptance of the
ring and halo particles can reach this level before they are
``cleaned'' by the Beam-In-Gap kicker. By Figs.~\ref{fig:da}, one may
observe the unacceptable reduction in the dynamic aperture of the SNS
ring below the physical aperture of 180~$\pi$~mm~mrad for a momentum
spreads of -0.2 (green curve on the left). This is attributed to the
fact that the chromaticity pushes the particles' vertical tune towards
the very dangerous integer resonance, at $Q_y=6$ and the particles get
rapidly lost. A less pronounced reduction of the dynamic aperture can
be attributed to the half-integer resonance at $2Q_y=11$ for particles
with momentum spread of 0.2 (red curve on the left). Finally, the on
momentum particles have very similar dynamic aperture (blue curves).
%*********************************************************************
\clearpage
\newpage
\lhead{SNS Ring Design Manual}
\rhead{Multi-particle Effects}
\rfoot{November 1, 2002}
\subsection{Multi-particle Effects}
One of the primary tasks in the design of the SNS ring is to control
collective effects. Dominant collective effects include space charge,
impedance driven instabilities and electron cloud. Transverse
painting is used to alleviate space charge force. The variety of
space-charge induced halo growth mechanisms and space-charge limit for the SNS ring
were explored
both analytically and numerically \cite{Holmes99}-\cite{Fedotov02}.
Estimates of the coupling impedance
and its effect on beam stability are performed through the design
stage of the project. The key impedance contributions are identified and
benchmeasured with the instability thresholds being estimated \cite{Zhang99}-
\cite{Davino02}. The cures
to prevent instabilities are implemented and feedback system is being evaluated
\cite{Wei02}, \cite{Danilov02asac}.
\subsubsection{Space-charge effects}
In the longitudinal direction, space charge contributes a defocusing
force below transition energy.
The corresponding potential is
\begin{equation}
V_{sc}=-I Z_{sc}; ~ Z_{sc}=-j\frac{nZ_0g_0}{2\beta \gamma^2};
~g_0=1+2\ln \frac{b}{a},
\label{LongSC}
\end{equation}
where $I$ is the peak current, $n=\omega/\omega_0$ is the frequency
harmonic, $Z_0=377 \Omega$, $a$ and $b$ are the average radii of the
beam and vacuum chamber, respectively.
The bunch spread may cause particle
leakage from the RF bucket requiring of enhanced RF field focusing.
The SNS ring will have a dual harmonic RF system with peak amplitudes
of 40 kV for harmonic $h=1$ and 20 kV for $h=2$ which brings the bunch
leakage to a negligible level.
In the transverse direction, the space-charge tune shift sets
space-charge limit due to the excessive beam loss
associated with the low order machine resonances. The maximum
incoherent tune shift for the particles near the center of beam
distribution can be estimated by
\begin{equation}
\Delta \nu_{inc} = -\frac{ N_0 r_0 R_0}{ \pi \nu_{0} \beta^2 \gamma}
\bigg[
\frac{\gamma^{-2} - \eta_e}{2\sigma_{x,y} (\sigma_x+\sigma_y)}\frac{F_{sc}}{B_f}
+\Big (
\beta^2+\frac{\gamma^{-2}-\eta_e}{B_f} \Big ) \frac{\epsilon_1}{b^2} +
\kappa \beta^2\frac{\epsilon_2}{g^2}
\bigg ],
\label{inctuneshift}
\end{equation}
where $r_0$ is the classical radius of protons, $R_0$ is the
average radius of ring circumference, $\nu_{0}$ is the zero-current
betatron tune, $\sigma_{x,y}$ is the rms beam size,
$b$ is the radius of vacuum chamber, $g$ is the distance to magnetic
poles from the beam pipe center,
$B_f$ is the
bunching factor, $\eta_e$ is the neutralization factor, $\kappa$
is a factor showing portion of ring circumference covered with the
magnetic poles and $F_{sc}$ is a form factor depending on beam
distribution changing from $F_{sc}=1/2$ for the uniform density
beam to $F_{sc}=1$ for the Gaussian beam. The coherent tune shift
of a dipole transverse oscillation of a beam with penetrating
magnetic fields is given by
\begin{equation}
\Delta \nu^{p}_{coh} = -\frac{ N_0 r_0 R_0}{ \pi \nu_{0} \beta^2
\gamma} \bigg[ \Big(\frac{\gamma^{-2} - \eta_e}{B_f}+\beta^2 \Big
) \frac{\xi_1}{b^2}+ \kappa \beta^2\frac{\xi_2}{g^2} \bigg ],
\label{cohpen}
\end{equation}
and, with non-penetrating magnetic fields, by
\begin{equation}
\Delta \nu^{non-p}_{coh} = -\frac{ N_0 r_0 R_0}{ \pi \nu_{0}
\beta^2 \gamma} \bigg[ \Big(\frac{\gamma^{-2} - \eta_e}{B_f} \Big
) \frac{\xi_1}{b^2}+ \beta^2\frac{\epsilon_1}{b^2} +\kappa
\beta^2\frac{\epsilon_2}{g^2} \bigg ]. \label{cohnpen}
\end{equation}
The Laslett image coefficients $\epsilon_{1,2}$, $\xi_{1,2}$ depend
on the geometry of beam pipe, with the subscripts $1,2$ referring
to the electric and magnetic problems, respectively.
Other high-order coherent beam modes are also depressed by a direct space-charge
force similar to a single particle depression (which is different
from the first order dipole mode not effected by a direct
space-charge force with the dipole coherent
tune shift $\Delta \nu_{coh}$ in Eqs.~\ref{cohpen}-\ref{cohnpen}
caused by the image
effects). The accurate treatment of beam response to machine
resonances should include collective beam behaviour. As a result,
the space-charge limit associated with the crossing of
the half-integer resonance may be significantly altered when the
quadrupole oscillation of beam envelope is taken into account
\cite{Fedotov02}.
For the SNS ring, space-charge tune shifts for a realistic
distribution based on multi-turn injection painting were
calculated using the simulation program UAL/ORBIT \cite{Malitsky02}.
The transverse
painting and dual harmonic rf system are used to control the
direct incoherent space-charge tune shift with the goal not to
exceed $\Delta \nu =0.2$. The choice of the working tune is then based on
consideration of the space-charge induced resonances and the avoidance
of dangerous nonlinear resonances excited by magnet imperfections
in the presence of the space-charge induced tune spread.
\begin{figure}[ht]
\center
\mbox{\epsfig{file=t3int623620.eps,width=0.65\columnwidth}}
\vskip .2cm
\caption{Tune spread at the end 1060-turn injection
due to space charge and $\Delta p/p=0.6 \%$ for three
final intensities 1)$N=1 \cdot 10^{13}$ (red),
2) $N=1 \cdot 10^{14}$ (pink),
3) $N=2 \cdot 10^{14}$ (green).
}
\label{t3int623620}
\end{figure}
For the base line working point, the tune spread for three beam
intensities is shown in Fig.~\ref{t3int623620}. Figure
~\ref{t3dp6463} shows combined tune spread of a 2MW beam due to the
space charge and
$\Delta p/p$. The variety of the
space-charge effects and associated beam halo were explored and
the space-charge limit for the base line working point was found
to be
around $N=2\cdot 10^{14}$ protons.
\begin{figure}[ht]
\center
\mbox{\epsfig{file=t3dp6463.eps,width=0.65\columnwidth}}
\vskip .2cm
\caption{Tune spread of a 2 MW beam for 1)space-charge alone
(green), 2) space charge and $\Delta p/p=0.6 \%$ (red),
3) space charge and $\Delta p/p=1.0 \%$ (pink).
}
\label{t3dp6463}
\end{figure}
The most effective approach to resolve space-charge problems is to
rise the injection energy. Other ways of space-charge compensation
can be considered but they should be carefully evaluated. For example,
space-charge compensation with inductive inserts can lead to
a microwave instability if ferrite material have significant losses.
\subsubsection{Impedance budget}
The longitudinal coupling impedance is defined as
\begin{equation}
Z_{\parallel}(\omega)= \frac{\int E_z exp(jkz)dz}{I},
\label{inst1}
\end{equation}
assuming a harmonic excitation current of amplitude $I(\omega)$
which excites a harmonic field with complex amplitude
$E_z(\omega)$. The transverse coupling impedance is defined as the
integral of the deflecting fields over one turn normalised by the
dipole moment of the excitation beam current
\begin{equation}
Z_{\bot}(\omega)=j
\frac{\int \Big [ E_r+\beta c B_{\theta} \Big ] exp(j\omega z/v)dz}{ I \Delta y},
\label{inst2}
\end{equation}
where $\Delta y$ is the horizontal or vertical offset of the beam from the axis.
At low frequency the impedance is dominated by the skin effect of
the vacuum chamber; at medium and higher frequencies the impedance
behaves as that of a broad band resonator - thus the word
``broadband'' impedance. At certain high frequencies there will be
strong local resonances, for example, from cavities.
The impedance
budget of the SNS ring is being constantly evaluated with the key
contributors benchmeasured. The low-frequency contribution is
shown in Table~\ref{tab:implow}, for frequencies below 10MHz. Table
~\ref{tab:imp50} shows the impedance contribution at an arbitrary chosen
frequency of 50MHz.
Some formulas and methods which were used for impedance budget estimates
are summarized below:
\paragraph{Space charge}
The longitudinal space-charge impedance is calculated using
\begin{equation}
Z_{\parallel}=-j\frac{Z_0}{2\beta \gamma^2}\frac{\omega}{\omega_0}
\bigg ( 1 + 2\ln \frac{b}{a} \bigg ).
\label{sc_imp_long}
\end{equation}
The transverse space-charge impedance is given by
\begin{equation}
Z_{\bot}=-j\frac{RZ_0}{ \beta^2 \gamma^2}
\bigg ( \frac{1}{a^2}-\frac{1}{b^2} \bigg ).
\label{sc_imp_tr}
\end{equation}
\paragraph{Extraction kicker}
A prototype of the SNS extraction kicker was measured \cite{Davino02}.
The impedance for all 14 kickers was then obtained using
the scaling formula \cite{Davino022}.
\paragraph{RF cavity}
The RF cavity impedance was measured \cite{Hahn02}. The contribution from
the high-order modes to the longitudinal impedance is negligible.
For the transverse impedance,
a resonant mode at 17.6 MHz was measureed when the cavity was in the first
harmonic configuration. The second harmonic configuration did not give
that mode. This resonant impedance was strongly decreased by placing
four 40$\Omega$ carborundum rods (glow-bars) on top and bottom of each gap
\cite{Davino023}.
\paragraph{Injection foil assembly}
The impedance of the foil assembly was calculated with
the MAFIA code and is expected to be negligible.
A possible resonant impedance at 170 MHz can be damped
with a lossy material. This
impedance will be measured when the device will be built.
\paragraph{Resistive wall}
Assuming a smooth cylindrical beam pipe of radius $b$, the
longitudinal and transverse impedances are
\begin{equation}
Z_{\parallel}=\bigg (
sgn(\omega)+j \bigg )
\frac{\beta Z_0 \delta_s}{2b}\frac{\omega}{\omega_0},
\label{rw_imp_lon}
\end{equation}
\begin{equation}
Z_{\bot}=\bigg (
sgn(\omega)+j \bigg )
\frac{R Z_0 \delta_s}{b^3},
\label{rw_imp_tr}
\end{equation}
where $\delta_s$ is the skin depth at the frequency $\omega$.
\paragraph{BPM, Beam-in-Gap kicker and Tune kikers}
All these devices are based on the dual plane striplines.
For one strip plate
\begin{equation}
Z_{\parallel}=Z_c \bigg (
\frac{\phi_0}{2 \pi} \bigg )^2
\bigg ( \sin^2 \frac{\omega l}{c}+
j \sin \frac{\omega l}{c} \cos \frac{\omega l}{c} \bigg ),
\label{bpm_imp_lon}
\end{equation}
where $Z_c$ is the characteristic impedance of the stripline.
Each stripline has a length $l$ and subtends an angle $\phi_0$
to the beam pipe axis.
For a pair of striplines
\begin{equation}
Z_{\bot}=\frac{c}{b^2}\bigg (\frac{4}{\phi_0}\bigg )^2
\sin^2 \frac{\phi_0}{2} \bigg ( \frac{Z_{\parallel,2}}{ \omega} \bigg ),
\label{bpm_imp_tr}
\end{equation}
where $Z_{\parallel,2}$ is the longitudinal coupling
impedance for a pair of striplines.
\paragraph{Broadband impedance}
The broadband impedance is caused by the bellows, steps,
vacuum ports, valves and collimators. The impedance of
these components is estimated using the low-frequency
approximation formulas for the longitudinal impedance
\cite{Zhang99}. The transverse impedance is then obtained
using an approximate relation
\begin{equation}
Z_{\bot}\approx
\frac{2R}{\beta b^2} \bigg (
\frac{Z_{\parallel}}{n} \bigg ).
\label{tr_lon_imp}
\end{equation}
The impedance of the collimators was estimated using calculations
with the MAFIA code, which was found to be in a good agreement
with the estimates given by the simple analytic formulas \cite{Kurennoy00}.
\begin{table}[hpt]
\center
\caption{SNS impedance budget below 10MHz}
\label{tab:implow}
{\newcommand{\hl}{\hline}
\begin{tabular}{ccc}
\hl
& $\mathbf{Z_{\parallel}/n}$ $[\Omega]$ & $\mathbf{Z_{\bot}}$
$[k\Omega/m]$ \\ \hl
~&~& \\
Space charge & -j196 & j(-5.8+0.45)$\times 10^3$ \\
~&~& \\
Extraction kicker & 0.6n+j50 & 33+j125 \\
(25 $\Omega$ termination) & & (measured)\\
~&~& \\
RF cavity & per cavity: & total: \\
(measured) & $0.6~ (f=7.49MHz,~Q=88)$ & 18 at 17.6MHz\\
~ & $0.2~ (f=11.37MHz,~Q=59)$ & (damped by glow-bar)\\
~ & $0.1~ (f=35MHz,~Q=1)$ & \\
~ & $0.9~ (f=87MHz,~Q=20)$ & \\
~&~&\\
Injection foil assembly & j0.05 &
j4.5 \\
(MAFIA calculation) & & \\
~&~&\\
Resistive wall & (j+1)0.71, at $\omega_0$ &
(1+j)8.5, at $\omega_0$ \\
~&~& \\
Broadband & & \\
~&~& \\
BPM & j4.0 & j18.0 \\
~&~& \\
BIG and TK& j1.1 & j7 \\
~&~& \\
Bellows & j1.3 & j11 \\
~&~& \\
Steps & j1.9 & j16 \\
~&~& \\
Ports & j0.49 & j4.4 \\
~&~& \\
Valves & j0.15 & j1.4 \\
~&~& \\
Collimator & j0.22 & j2.0 \\
~&~& \\
Total BB & j9 & j60\\
~&~& \\ \hl
\end{tabular}
}
\end{table}
\begin{table}[hpt]
\center
\caption{SNS impedance budget at 50MHz}
\label{tab:imp50}
{\newcommand{\hl}{\hline}
\begin{tabular}{ccc}
\hl
& $\mathbf{Z_{\parallel}/n}$ $[\Omega]$ & $\mathbf{Z_{\bot}}$
$[k\Omega/m]$ \\ \hl
~&~& \\
Space charge & -j196 & j(-5.8+0.45)$\times 10^3$ \\
~&~& \\
Extraction kicker & 19.4+j12 & 12.5+j65 \\
(25 $\Omega$ termination, measured) & & \\
~&~& \\
RF cavity & per cavity: & total: \\
(measured) & see before & 0 (at 17.6MHz)\\
~&~&\\
Injection foil assembly & j0.05 &
j4.5 \\
(MAFIA calculation) & & \\
~&~&\\
BPM & 2.0+j3.5 & 9+j16\\
~&~& \\
BeamInGap and TuneKicker& 0.7+j0 & 5.0+j0 \\
~&~& \\
Broadband & & \\
~&~& \\
Bellows & j1.3 & j11 \\
~&~& \\
Steps & j1.9 & j16 \\
~&~& \\
Ports & j0.49 & j4.4 \\
~&~& \\
Valves & j0.15 & j1.4 \\
~&~& \\
Collimator & j0.22 & j2.0 \\
~&~& \\
Total BB & j4.1 & j35\\
~&~& \\ \hl
\end{tabular}
}
\end{table}
\subsubsection{Impedance minimization}
Impedance budget estimates and its minimization is an important
measure to prevent collective instabilities.
The imaginary part of the impedance leads to the frequency tune
shift while the real part directly contributes to the instability
growth rates. The largest contribution to impedance budget in
high-intensity rings is typically due to the space-charge, as can
be seen in the Tables. However, space-charge contribution is purely
imaginary and thus should not be directly used in the stability
estimates. On the other hand, large space-charge impedance should not be
forgotten, since, in combination with the real part of the
impedances coming from other sources, it can strongly influence
beam stability as discussed in the following sections.
As can be seen from the impedance budget table, the largest
contribution to the real part of the impedance is from the
extraction kickers. The presented measured values for the
extraction kickers contribution is a result of significant
minimization efforts with the largest reduction achieved by
low termination impedance (25 $\Omega$) of the pulse forming
network (PFN) circuits \cite{Davino02}.
Except for the injection kicker and some beam diagnostics
sections, stainless steel vacuum chamber is used. Ceramic
vacuum pipes are used to avoid eddy-current heating corresponding
to an injection bump risetime of about 200 $\mu s$. Their coupling
impedance is reduced by an internal 1 $\mu m$ copper-film coating
(much thinner than the skin depth of 0.9 $mm$).
A second, 0.1 $\mu m$ thin TiN coating is applied to supress the
secondary yeild of electron generation.
The steps in vacuum pipe are designed with the tapering length at least
three time longer than the step height to avoid sharp discontinuities
and resonances. Vacuum ports and valves are shielded. The bellows are not
shielded to avoid mechanical complications.
\subsubsection{Measurements of coupling impedance}
%Subsection written by H.Hahn
%The Spallation Neutron Source is a high intensity machine and its
%performance depends to a large degree on reducing the coupling impedances,
%both longitudinal and transverse. The transverse impedance of kicker magnets
%represent in typical collider accelerator or storage rings the largest
%single contributing element. Thus, achieving the design performance of the
%SNS Accumulator Ring is expected to depend largely on sufficiently reducing
%the extraction kicker impedance. Of particular concern is the impedance at
%the low frequency end with focus on frequencies below 100 MHz. Minimizing
%the kicker impedance is obviously one of the major design objectives.
%Exhaustive impedance reduction studies involving a resistive termination
%have resulted in a satisfactory solution.
%Obtaining values of the coupling impedances is essential for performing beam
%dynamic studies.
Although good estimates for simple accelerator components
can be obtained from theoretical formulas, it is mandatory to determine the
major impedance contributions in bench measurements. The validity of these
measurements in which the beam is simulated by a single wire for
longitudinal or a twin-wire for transverse impedance measurements has been
documented by theoretical studies and experience at various machines.
The longitudinal coupling impedance of a component is conveniently measured
by inserting a wire in the center of the beam pipe to form a coaxial
transmission line. The forward scattering coefficient $s_{21} $is measured
both for the device under test and a reference tube of the same length. The
ratio,$S_{21} = s_{21}^{DUT} / s_{21}^{REF} $, yields the coupling impedance
by an appropriate expression, either the conventional hp-formula or the
log-formula \cite{hahn2000}.
In the typical situation, the characteristic impedance,$R_c $, of the line
is different from the standard impedance of the network analyzer, $R_0 $.
Consequently a matching network, such as an impedance transformer, needs to
be inserted. Often, for simplicity's sake, a resistive matching is applied.
On the input side, forward and backward matching is achieved with a series
and parallel resister,
\[
R_p = G_p^{ - 1} = \frac{R_0 }{\sqrt {1 - \eta } } \sim R_0 \left( {1 +
\frac{1}{2}\eta } \right)
\]
\[
R_{in} = R_c \frac{\eta - \left( {1 - \sqrt {1 - \eta } } \right)}{1 - \sqrt
{1 - \eta } } \sim R_c \left( {1 - \frac{1}{\mbox{2 }}\eta } \right)
\]
\noindent
with $\eta = R_0 / R_c $. Furthermore, on the output side, forward matching
is achieved with a series resistor
\[
R_{out} = R_c \left( {1 - \eta } \right)
\]
As example for the SNS rf cavity measurement, the characteristic impedance
of the 1.25 mm $\emptyset $ wire in the $\sim $15 cm beam tube is
$R_c = \frac{Z_0 }{2\pi }\ln \frac{r_o }{r_i } \approx 288\Omega $(vs. 265
$\Omega $ measured)
\noindent
requiring the matching resistors, $R_p \approx $54 $\Omega $, $R_{in}
\approx $263 $\Omega $, and $R_{out} \approx $213 $\Omega $.
At sufficiently low frequencies, the component is considered as a lumped
element and the forward scattering coefficient follows as
\[
s_{21} = \frac{2R_0 }{R_0 + \left( {R_{in} + R_c + Z_\parallel }
\right)\left( {1 + G_p R_0 } \right)}
\]
The expression for the longitudinal coupling impedance is obtained from the
ratio of the scattering coefficients,
\[
Z_\parallel = \left( {R_c + R_{in} + \frac{R_0 }{1 + G_p R_0 }}
\right)\left( {\frac{1}{S_{21} } - 1} \right) = 2R_c \left( {\frac{1}{S_{21}
} - 1} \right)
\]
\noindent
in full analogy to the well known hp-formula.
The numerical results for the coupling impedance are preferably presented as
real and imaginary part, with the real part of direct interest to the beam
stability analysis,
\[
R_\parallel = 2R_c \left( {\frac{\Re (S_{21} )}{\parallel S_{21} \parallel
^2} - 1} \right)
\]
\noindent
where $\parallel S_{21} \parallel $is the magnitude of the scattering
coefficient ratio. Typically, this ratio is stored in the network analyzer
as data/memory. By using the conversion from scattering to impedance format,
the instrument can produce directly the real and imaginary part of the
coupling impedance.
Alternatively, the measurements can be interpreted using the log formula,
\[
Z_\parallel = - 2R_c \log _e S_{21}
\]
This expression is derived as approximation for the case $Z_\parallel \ll
R_c $and is not suited for the wire measurement of the cavity shunt
impedance. However, in contrast to the hp-formula, which is derived for a
lumped, localized structure, the log formula sums correctly the small values
of separate impedances by taking into account the phase shift between
different locations. Apart from the fundamental resonance, the real part of
the coupling impedance is given by
\[
R_\parallel = - 2R_c \log _e \parallel S_{21} \parallel
\]
\noindent
with $\parallel S_{21} \parallel $ being downloaded in text format from the
instrument.
The transverse coupling impedance of kickers can be measured on the bench by
using the standard method in which a twin-wire ``Lecher'' line, simulating
the beam, is inserted into the ``Device Under Test''. The forward
transmission coefficients $S_{21}^{DUT} $ of the kicker is compared with the
$S_{21}^{REF} $ obtained in a reference tube of at least equal length and is
interpreted according to the HP-formula for lumped units,
\[
Z^{DUT} \approx 2Z_L \left( {1 - S_{21}^{DUT} / S_{21}^{REF} } \right) /
\left( {S_{21}^{DUT} / S_{21}^{REF} } \right),
\]
\noindent
or alternatively the log-formula,
\[
Z^{DUT} = - 2Z_L \log \left( {S_{21}^{DUT} / S_{21}^{REF} } \right)
\]
\noindent
with $Z_L $ the characteristic impedance of the line. Finally, one obtains
the transverse impedance as \cite{walling}
\[
Z_ \bot = \frac{c}{\omega }\frac{Z^{DUT}}{\Delta ^2},
\]
\noindent
with $\Delta $ being the spacing of the two wires. This relation requires
the knowledge of essentially three quantities: the measured impedance of the
device under test, the effective wire spacing and the characteristic
impedance. In the typical case of the wire diameter much smaller than the
wire spacing, the effective $\Delta $ is given by the center-to-center
distance of the wires. For wires of diameter d, with center to center
spacing $D$, the effective spacing is known to be
\[
\Delta = D\sqrt {1 - \left( {d / D} \right)^2} .
\]
In all other practical cases, it is necessary to make ad-hoc measurements of
the effective spacing, either by a comparison with a line of known
properties or by a direct measurement of the mutual inductance. [ D. Davino
and H.Hahn, to be published in Phys.Rev. S.T. Accel. Beams]
The twin-wire measurements of the SNS extraction kicker were performed by
using a line, home-made from 5 $\times $ 7.5 mm rectangular tubes, shown in
Fig.~\ref{fig:hahn1}. The center-to center spacing is 45.6 mm, which in good
approximation can be taken as $\Delta $ for the interpretation of the
measurements based on the HP-formula. The other value required in the
formula, the characteristic impedance of the line, was measured with a
communication network analyzer, Tektronix CSA803, to be 260 $\Omega $.
Matching of the line characteristic impedance to the 50 $\Omega $ cables of
the network analyzer, Agilent 8753ES, is achieved by means of 300 $\Omega $
transformers (North Hills 0501BB) with a center-tapped secondary winding,
serving as 180\r{ } hybrid. The transformer covers the frequencies from 30
kHz up to 100 MHz. The network analyzer was set for a logarithmic frequency
range from 100 kHz to 100 MHz, with 1601 points, and a 100 Hz bandwidth. In
view of the various measuring errors, a 10-20{\%} uncertainty in the quoted
transverse impedances is expected.
%\begin{center}
%\begin{figure}[htbp]
%\centerline{\includegraphics[width=6.75in,height=3.81in]{hahn1.eps}}
%\label{fig1}
%\end{figure}
%
%\end{center}
\begin{figure}[hpb]
{\parindent = 0pt
%\centerline{\psfig{file=fig_2_5_4_cable.eps,width=3.in,angle=0}}
\centerline{\psfig{file=hahn1.eps,width=3.in,angle=0}}
\caption{The homemade cable for the twin-wire measurements. The dimensions
are: $a$= 7.5 mm, $b$=5 mm and $d$=45.6 mm. \label{fig:hahn1}}
}
\end{figure}
A direct relation exists between the input impedance, $Z_{in} $, measured at
the bus bar port and the external source contribution to the transverse
impedance seen by the beam, and measured by the twin-wire method. Ignoring
the smaller ferrite contribution for the sake of a simple approximation, a
coupling impedance estimate follows by scaling the input impedance according
to
\[
Z_ \bot ^{in} = \frac{c}{\omega h^2}Z_{in}
\]
\noindent
where $h$ is the aperture in kick direction. This points to the possibility of
getting a reasonable idea of the coupling impedance by measuring$Z_{in} $,
especially for kickers mounted in accelerators, provided that an access to
the bus-bar is available. The possibility of obtaining the coupling
impedance of an installed magnet by combining external impedance
measurements at the kicker terminals with an analytical formula is obviously
tempting. However, it must be emphasized that only the coupled flux
contribution is accessible and the coupling impedance perpendicular to the
kick direction is not seen. The comparison of results for the SNS extraction
kicker demonstrated that an estimation of the real part is possible, whereas
the imaginary part can differ.
\subsubsection{Longitudinal instabilities}
Disregarding the effect of space charge, the Boussard-Keil-Schnell
criterion for longitudinal beam stability is
\begin{equation}
\mid Z_{\parallel}/n \mid \leq
F_{\parallel}\frac{\mid \eta \mid B_f E_s}{e\beta^2 I_0}
\bigg ( \frac{\Delta E}{E_s} \bigg )^2_{FWHM},
\label{longstab}
\end{equation}
where $I_0$ is the average bunch current, $\eta$ is the slip
factor, and $F_{\parallel}$ is the distribution dependent form factor.
One can see that large values of energy or momentum spread can
help to damp the instability. Such a damping mechanism is known
as Landau damping.
The largest resistive contribution to $Z_{\parallel}$ in the SNS
ring come from the extraction kickers and high-order modes of the
RF cavities. The space-charge effect plays a stabilizing role in the longitudinal
beam stability, significantly raising the instability threshold.
For the present impedance budget, the instability threshold
with space charge is above $N=3 \cdot 10^{14}$ \cite{Woody01}.
\subsubsection{Transverse instabilities}
The transverse stability condition is given by
\begin{equation}
\mid Z_{\bot x,y}/n \mid \leq
F_{\bot}\frac{4 B_f E_s}{e\beta \beta_{x,y} I_0}
\bigg ( \frac{\Delta E}{E_s} \bigg ) \mid
(n-\nu_{x,y,0})\eta +\xi_{x,y} \mid ,
\label{transtab}
\end{equation}
where $F_{\bot}$ is a form factor which depends
on the transverse beam distribution, $n$ is an arbitrary
integer and $\xi_{x,y}$ is the chromaticity. Instability
occurs only for slow waves with $n > \nu_{x,y}$.
The space charge plays destibilizing role for the transverse
microwave instability. Minimization of the tranverse impedance
significantly increased the intstability threshold. With the present
impedance budget the growth rate for the intensity $N=2 \cdot 10^{14}$ is expected
to be only about 1$ms^{-1}$ which is negligible \cite{Fedotov022},
\cite{Fedotov02asac}.
An additional
Landau damping can be introduced by means of the chromatic sextupoles.
For the working point below the integer, such as $(\nu_x,\nu_y)=(6.3,5.8)$
one can have a resistive wall instability (at 200KHz) with the growth rate of
5$ms^{-1}$. This instability can be damped by introducing Landau
damping with the chromatic sextupoles.
\subsubsection{Electron-cloud effects}
Beam-induced multipacting is believed to be the leading source
of sustained electron production. Depending on beam parameters,
one of the two multipacting models usually apply: multi-bunch
passage multipacting or single-bunch trailing-edge multipacting.
The electron-cloud buildup is sensitive to the intensity, spacing and
length of proton bunches, and on the secondary electron yield
(SEY) of the beam pipe surface.
The elecrton cloud can in turn lead to e-p instability.
This effect was extensively studied for the SNS ring \cite{Danilov012}-\cite{Wei022}.
For the SNS baseline intensity of $N=1.5 \cdot 10^{14}$, the electron
line density threshold is about 5$nC/m$ \cite{Blaskiewicz02asac}.
Also, the first harmonic voltage of the SNS is designed to have
40kV which is estimates to be sufficient to stabilize the beam
for the high-intensity operation of the SNS.
\subsubsection{Cures and Feedback}
\paragraph{Landau damping with sextupoles}
The design of the SNS ring has chromatic sextupoles which
allows to introduce a relatively large frequency spread.
For the resistive wall instability at 0.2 MHz the chromaticity
provides strong Landau damping, with $\xi=-7$ completely damping
the instability.
For the instability associated with the extraction kicker impedance
at frequencies 2-20 MHz the chromaticity has only partial effect,
with $\xi=-7$ leading to a $20 \%$ thershold increase in the presence
of space charge.
For the elecrton cloud instability at frequencies 100-200 MHz,
the chromaticity has little effect on the thresholds.
\paragraph{Effect of octupoles}
The purpose of correction octupoles it to correct imperfection
machines resonances. With a maximum coil current in correctors
not to exceed 17A the introduced tune spread was found to be unsufficient
to introduce effective Landau damping.
\paragraph{Effect of $b/a$}
The spread of coherent tunes along the bunch length due to longitudinal
current distribution plays effective role in trasnverse beam stability. Increasing
the $b/a$ ratio decreases such detuning thus making beam more unstable.
\paragraph{E-cloud cures}
All internal surfaces, including ferrite are coated with
TiN to suppress electron multipacting. The 100 $nm$
thickness is expected to sustain proton-halo
bombardment without eddy-current heating.
Injection stripped electrons are guided to the collectors
with a low backscattering yield.
A voltage up to $\pm$1 kV can be applied to 42 BPMs to clear electrons.
In addition, dedicated electrodes will be placed in the injection
region. Also, solenoids can be wound in straight sections to reduce
multipacting.
\paragraph{Feedback system}
A wide-band feedback system can be implemented to damp
the instabilities. Three frequency ranges are of interest:
1. 0.2-0.8 MHz (depending on working point) with the growth
rates of about 5$ms^{-1}$ due to resistive wall impedance,
2. 2-20 MHz with the growth rates of about 1-3 $ms^{-1}$
due to the extraction kicker impedance,
3. 100-200 MHz with growth rates of about 50 $ms^{-1}$ due to the
electron cloud instability.
Only for the electron cloud instability this results in a significant
power requirements.
The feedback system will be developed using the three tune measurement
kickers each of 50 cm length to cover the range of frequncies up to
200 MHz \cite{Danilov02asac}.