# Spill ripple mitigation by bunched beam extraction with high frequency
synchrotron motion

S. Sorge<EMAIL_ADDRESS>P. Forck  R. Singh  GSI Darmstadt, Planckstraße 1,
64291 Darmstadt, Germany

(March 7, 2022)

###### Abstract

Slow extraction of bunched beams is widely used for mitigating the formation
of micro structures due to ripples of the accelerator magnets power supplies
on the extracted beam which is also referred to as spill. That helps, in
particular, experiments with slow detectors or with low extraction rates.
Furthermore, this technique is widely used for avoiding long gaps in the spill
measured in ionisation chambers which in turn would trigger interlocks.

On the other hand, the bunches create spill structures on time scales defined
by the rf frequency. In addition, macroscopic spill structures of duration up
to some hundred milliseconds can be created by the synchrotron motion for a
sufficiently large synchrotron tune.

The aim of this report is to study the influence of the synchrotron tune
determined by amplitude and harmonic number of the rf voltage which
additionally depends on the transverse emittance and the longitudinal bunch
area of the circulating beam as well as the distribution of the particles’
transit times.

##  I Introduction

Slow extraction from synchrotrons and storage rings is widely used for
delivering hadron beams to many fixed target experiments as well as for the
irradiation of tumours in hadron cancer therapy. In most cases, a third
integer lattice resonance is excited with sextupoles which results in the
formation of a triangular area in the phase space plane used for the
extraction with stable betatron motion inside, where usually the particles of
the whole beam fit in at the beginning of the extraction. The extraction is
executed by subsequent feeding the resonance which means that the particles
are successively pushed out of that stable phase space area such that their
betatron motion becomes unstable, they leave the beam, and become extracted by
reaching the extraction septum. The extracted beam is also referred to as
spill. Typical extraction time intervals reach from some hundred milliseconds
up to hours. Several resonance feeding techniques are presently in operation
such as tune sweep, betatron core driven, rf knock-out extraction, where the
first two mechanisms are based on the reduction of the size of the stable
phase space area, whereas the betatron amplitudes of the particles are slowly
increased beyond the edge of the stable phase space area in the last method.
The choice of the resonance feeding mechanism depends on the parameters of the
beams required by the users such as beam energy and intensity, spill duration,
or safety requirements like the ability to suddenly interrupt the extraction
if the spill is utilised for the irradiation of patients in medical
facilities.

A frequent user requirement to the spill is a low level of undesired temporal
structures. Ripples and noise on the current supplied to the accelerator
quadrupole magnets could be identified as major source of fluctuations of the
machine tune which translate in a ripple of the size of the stable area in the
phase space plane and, hence, in structures on the particle flow out of the
stable phase space area. These spill micro structures have a characteristic
time from $\sim 10\ {\rm ms}$ down to $\sim 10\ {\rm\mu s}$ which is too short
to be efficiently mitigated by a feedback system [1, 2] and significantly
longer than the revolution time.

The particles need after leaving the stable phase space area a certain time
$T_{\rm tr}$ for the transit to the extraction septum. These transit times are
not uniform and follow a distribution instead. The width of the distribution
defines a transit time spread $\Delta T_{\rm tr}$. Thereby, temporal
structures formed by particles which left the stable phase space area at the
same instant are smeared out. That mechanism can be utilised for reduction,
especially, of spill micro structures while applying an extraction technique
which does not rely on longitudinal dynamics as for example slow extraction of
unbunched beam by tune sweep. On the other hand, spill micro structures can be
mitigated with help of longitudinal dynamics, where the basic mechanism
consists in the change of the momenta and, thus, the chromatic tunes of
particles towards the resonance tune by rf fields which results in a
transition across the edge of the stable phase space area much faster than
that caused by the resonance feeding. Thereby, the duration of stay of
particles near the separatrix is reduced and the level of structures imprinted
on the particle flux out of the stable phase space area is reduced. The
particle momenta can be quickly changed by applying rf fields of defined
harmonic number resulting in synchrotron motion during slow extraction of
bunched beams or channelling between empty rf buckets, or by driving
longitudinal diffusion by rf noise with a band limited power spectrum during
stochastic extraction. The aim of the present report is to study mitigation
and also formation of of spill structures during slow extraction of bunched
ion beams out of the present heavy ion synchrotron SIS18 of the GSI facility
within a simultation study for synchrotron motion significantly faster than
usually applied in present operation. Hence, the work is restricted to slow
extraction by tune sweep performed with two fast ramped quadrupoles which is
the standard slow extraction technique of SIS18. That topic is motivated by
the plan to install a special high frequency rf cavity in order to provide rf
fields with a harmonic number increased by an order of magnitude which is
required by an experiment collaboration with the goal to take advantage of
spill micro structure reduction provided by bunched beam extraction and to
avoid spill structures at typical present rf frequencies. That requirement
implies a strongly increased synchrotron tune.

Generally, the results of the study show that a gain in spill quality by
increasing the synchrotron tune and according reduction of the duration of
stay of the particles near the oscillating edge of the stable phase space area
can be achieved only up to an optimum of the synchrotron tune, whereas further
rise reduces the spill quality. Reasons are that the transit time distribution
during bunched beam slow extraction has a reduced but still significant
influence on the spill quality and, in addition, is itself modified by the
synchrotron motion. Furthermore, the creation of macroscopic spill structures
is found for a sufficiently large synchrotron tune. The conditions resulting
in such phenomena are analysed and possible ways of mitigation are proposed.

##  II Conditions for slow extraction from SIS18 assumed in the study

The present study is focused on the present GSI heavy ion synchrotron SIS18,
but the results are applicable to other machines like SIS100 or hadron therapy
synchrotrons.

As in most synchrotrons, slow extraction from SIS18 is based on the excitation
of a one-dimensional third integer resonance with sextupoles resulting in the
formation of a triangular area of stable betatron motion in the phase space
plane used for the extraction which is the horizontal phase plane in SIS18.
The stable phase space area formation can be described by the Kobayashi theory
[3]. The size of the stable phase space area is given by its corners which are
unstable fixed points (UFP) of betatron motion. Written in normalised
coordinates of the phase space vector $\vec{X}=(X,X^{{}^{\prime}})$ with

$X\equiv\frac{x}{\sqrt{\beta_{x}}}\ \mbox{and}\
X^{{}^{\prime}}\equiv\sqrt{\beta_{x}}x^{{}^{\prime}}+\frac{\alpha_{x}}{\sqrt{\beta_{x}}}x,$
(1)

the UFPs have all the same absolute value [4]

$\left|\vec{X}_{\rm UFP}\right|=8\pi\left|\frac{Q_{\rm r}-Q_{\rm p}}{S_{\rm
v}}\right|,$ (2)

where $Q_{\rm r}$ and $Q_{\rm p}$ are resonance tune and on axis tune of the
particles. The latter can be modified by chromaticity $\xi$ and relative
momentum deviation $\delta$ of the particles to

$Q_{\rm p}=Q_{\rm m}+\xi\delta.$ (3)

$S_{\rm v}$ in Equation (2) is the strength of a virtual sextupole and given
by

$S_{\rm v}\cdot{\rm e}^{3{\rm i}\psi_{\rm v}}=\sum\limits_{n}S_{n}{\rm
e}^{3{\rm i}\psi_{n}}$ (4)

with the normalised sextupole strength of the $n$th sextupole

$S_{n}=\frac{1}{2}\beta_{x,n}^{3/2}(k_{2}L)_{n}$ (5)

and the betatron phase advance $\psi_{n}$ between the location of the $n$th
sextupole and the considered location which is usually the entrance of the
electro-static septum. The $\psi_{n}$ have to be determined for the third
integer tune of the resonance $Q_{\rm r}$. The resulting area of the stable
phase space area is

$A_{\rm stable}=3\frac{1}{2}\ |\vec{X}_{\rm UFP}|^{2}\ \sin
120^{o}=\frac{\sqrt{27}}{4}\ |\vec{X}_{\rm UFP}|^{2}.$ (6)

The corresponding emittance is

$\epsilon_{\rm stable}=\frac{A_{\rm stable}}{\pi}.$ (7)

The dependence of $A_{\rm stable}$ on the tune can be used for extracting the
beam by moving the machine tune $Q_{\rm m}$ slowly across the resonance tune
by changing the focusing strengths of quadrupoles such that it shrinks and the
particle become successively unstable according to their transverse emittance
and momentum dependent tune such that they leave the synchrotron. This
technique referred to as quadrupole driven extraction is the present standard
technique for slow extraction from SIS18. Hence, this manuscript is focused on
the usage of this method.

A very comprehensive series of machine experiments was performed with beams of
Ar18+ ion at the beam energy $E=500\ {\rm MeV/u}$. Typical particle numbers
were between $10^{5}$ and $10^{6}$ ions per cycle which is lower than in user
experiments. The purpose was to use a scintillation counter for measuring the
spill which has a limited count rate. But it allows for experiments with a
high time resolution which was actually chosen to $10\ {\rm\mu s}$. In
addition, it provides the opportunity to perform particle tracking simulations
with particle numbers similar to those of the measurements. These experiments
delivered the most important results the particle tracking results achieved in
this study are compared with. For that reason, we apply conditions in the
simulation of the present study similar to those in these experiments.

##  III Spill simulations and characterisation of spill micro structures

Spill structures can be investigated by simulating the process of slow
extraction with particle tracking. We use the MAD-X code for the simulations.
Gaussian distribution functions truncated at $2\sigma$ are applied to set the
initial coordinates of the test particles. $10^{5}$ particles were tracked in
each simulation for $5\cdot 10^{5}$ revolutions after finishing the bunching
process. The $\sigma$ values of corresponding Gaussian distribution functions
of the transverse particle coordinates are defined by the rms beam emittances
$\epsilon_{x,{\rm rms}}=3.9\ {\rm mm\ mrad}$ and $\epsilon_{y,{\rm rms}}=1.3\
{\rm mm\ mrad}$. The resulting maximum beam emittances are $\epsilon_{x,{\rm
max}}=15.5\ {\rm mm\ mrad}$ and $\epsilon_{y,{\rm max}}=5.2\ {\rm mm\ mrad}$,
respectively. They arise from the beam width given by the machine acceptance
during injection and subsequent adiabatic shrinkage during acceleration. In
order to study the influence of the horizontal beam width, transverse beam
emittances reduced by the factor $0.2$ are applied. The maximum momentum
deviations $\delta_{\rm m}$ reach from $\delta_{\rm m}=5\cdot 10^{-4}$ up to
$\delta_{\rm m}=10^{-3}$, where the actual value depends on the rf voltage
defined by amplitude $V_{\rm rf}$ and harmonic number $h$.

The time resolution for measuring spills is given by the measurement rate of
the spill measurement system, where $f_{\rm sample}=100\ {\rm kHz}$ in SIS18.
The resulting upper limit to the frequency range of the spill micro structures
is then $f_{\rm up}=f_{\rm sample}/2=50\ {\rm kHz}$.

The quantity used for characterising the spill quality defined by the micro
structure level in this study is the time dependent duty factor [5]

$F(t)=\frac{\langle N\rangle^{2}(t)}{\langle N^{2}\rangle(t)}=\frac{N_{\rm
av}^{2}(t)}{N_{\rm av}^{2}(t)+\sigma_{N}^{2}(t)}$ (8)

is applied, where $\langle x\rangle$ denotes the average of the variable $x$
given in time intervals according to the measurement resolution $t_{\rm
meas}=1/f_{\rm meas}$ in time intervals $t_{\rm rec}$ which define the average
length. The latter defines the recording resolution. The actual size of the
measurement and recording time intervals used for this study were $t_{\rm
meas}=10\ {\rm\mu s}$ and $t_{\rm rec}=10\ {\rm ms}$. A good spill quality
given by a low spill structure level results in large value of the time
dependent duty factor.

The upper limit to $F(t)$ under realistic beam conditions and for $t_{\rm
rec}\gg t_{\rm meas}$ is reached in the Poisson limit which denotes stochastic
distribution of extracted particles in the time intervals. For Poisson’s
distribution function to find $N$ particles in a measurement time interval,

$f_{\rm P}={\rm e}^{-\lambda}\frac{\lambda^{N}}{N!}$ (9)

one finds $N_{\rm av}^{2}+\sigma_{N}^{2}=(N^{2})_{\rm av}=N_{\rm
av}\left(N_{\rm av}+1\right)$ such that

$F_{\rm P}(t)=\frac{N_{\rm av}(t)}{N_{\rm av}(t)+1}.$ (10)

It can be useful to reduce the time dependent duty factor to the weighted
factor defined in [6] by

$\displaystyle F_{\rm av}$ $\displaystyle=$ $\displaystyle\frac{\int{\rm d}t\
\dot{N}(t)F(t)}{\int{\rm d}t\ \dot{N}(t)}$ (11) $\displaystyle=$
$\displaystyle\frac{\sum\limits_{k}N_{k}F_{k}}{\sum\limits_{k}N_{k}}$

which is, basically, the time average of the weight function $\dot{N}(t)$
which is the extraction rate at instant $t$ denoting the spill. On the other
hand, spill and time dependent duty factor are recorded with the time
resolution given by $t_{\rm rec}$ such that, approximately,
$N_{k}=\dot{N}(t_{k})\cdot t_{\rm rec}$ with $t_{k}=k\cdot t_{\rm rec}$ and
the integrals represented by the sums in the second row of Equation (11).

##  IV Mitigation of spill micro structures with bunching

The spread of the particles’ transit times $\Delta T_{\rm tr}$ arising from
the width of the transit time distribution results in a finite interval of
arrival times of particles which became unstable at the same instant. Hence,
spill micro structures imprinted by oscillations of the stable phase space
area’s size due to a tune ripple are smeared out. This mechanism could be
identified as a major spill smoothing mechanism for slow extraction of
coasting beams [7, 6].

In contrast to the aforementioned mechanism, spill smoothing for bunched beam
slow extraction is mainly based on the periodic change of the particles’ tune
due to synchrotron motion which yields a transition from stable to unstable
betatron motion significantly faster than caused by the tune sweep. Therefore,
the particles will stay much shorter near the oscillating edge of the stable
phase space area such that resulting structures imprinted on the spill are
strongly reduced. Following this reasoning, one would expect that the spill
quality defining duty factor always grows for an increasing synchrotron tune
$Q_{\rm s}$ or, at least, saturates because the time the particles are near
the separatrix can not be reduced below zero. However, it is found by
performing particle tracking simulations of the extraction that the weighted
duty factor has a maximum at a certain $Q_{\rm s}$ and is decreased if $Q_{\rm
s}$ is further increased which one can see in Figure 1. Our analysis reveals
that particles are re-captured into the stable phase space area after becoming
unstable if the synchrotron motion is too fast such that they can not become
extracted. As a result, the distribution of the transit times of extracted
particles is altered. That suggests that the influence of the transit time
distribution is only reduced but not irrelevant.

Figure 1: Weighted duty factors according to Equation (11) of spills obtained
in simulations of the slow extraction of Ar18+ beams from SIS18 at the beam
energy $E=500\ {\rm MeV/u}$ as function of the synchrotron tune according to
the the rf voltages $V_{\rm rf}=(0,\ 1.2,\ 6.2,\ 12,\ 70,\ 200)\ {\rm kV}$,
where $V_{\rm rf}=12\ {\rm kV}$ is the maximum which can be reached in SIS18.
The rf voltage at the beginning of the simulations is switched off and slowly
turned on during the first 50000 turns. The initial maximum momentum width is
$\delta_{\rm m}=5\cdot 10^{-4}$ and is increased during the bunching process.
Hence, also longitudinal dynamics with low rf voltages with particles captured
and others not captured in the rf bucket are well described.

We apply a model based on the analytic expression for the transit time from
Equation (4.13) in [8]

$T_{\rm
tr}\approx\frac{1}{\sqrt{3}\varepsilon}\ln\left|\frac{1}{1-\Lambda_{0}^{2}}\frac{n}{n+3}\frac{3}{\lambda_{0}}\right|,$
(12)

in order to demonstrate the effect of the synchrotron motion on the transit
time distribution. The variables in this equation are
$\varepsilon=6\pi|Q_{x,{\rm r}}-Q_{x}(\delta)|$, $\lambda_{0}\equiv-\Delta
X_{\rm start}/h$ and $\Lambda_{0}\equiv-\Delta X_{\rm
start}^{{}^{\prime}}/(2\sqrt{3}h)$ with the the coordinates of the start of a
particles transition in horizontal phase space $\Delta X_{\rm start},\Delta
X_{\rm start}^{{}^{\prime}}$, and $n\equiv-\Delta X_{\rm ESS}/h$. One should
note that $\Delta X_{\rm start}$, $\Delta X_{\rm start}^{{}^{\prime}}$, and
the normalised horizontal coordinate of the electro-static septum $\Delta
X_{\rm ESS}$ are defined relative to the point $(\Delta X,\Delta
X^{{}^{\prime}})=(0,0)$ at the upper left corner of the triangular stable
phase space area shown in Figure 2 and, hence, negative. The transit time in
this formula is dimensionless and given in units of three subsequent
revolutions. By applying the analytic formula, the effect can be directly
depicted. That is an advantage compared to the use of simulation data, where
it is difficult to determine the instant when a particle becomes unstable and
its transit from the stable phase area to the extraction septum begins.

Figure 2: Schematic representation of the stable phase area in normalised
coordinates $X,X^{{}^{\prime}}$ defined relative to he centre of the triangle.
The starting coordinates $\Delta X_{\rm start}$, $\Delta X_{\rm
start}^{{}^{\prime}}$ relevant for the time of a particle’s transit from the
stable phase area to the electro-static septum are defined relative to the
upper left corner of the triangle.

From Equation (12), it follows that the transit time of a particle is the
shorter the larger $\lambda_{0}$ is, where a large value of $\lambda_{0}$
corresponds to a large distance of a particles’ starting point $\Delta X_{\rm
start}$ from the separatrix. The maximum distance $\Delta X_{\rm start}$
during a synchrotron period is reach at the cusp of synchrotron motion, where
the momentum dependent tune

$Q(\delta)=Q_{\rm m}+\Delta Q(\delta)\ \mbox{with}\ \Delta
Q(\delta)=\xi\delta$ (13)

is closest to the resonance tune $Q_{\rm r}$. On the other hand, the chromatic
tune of particle will move away from the resonance after it has passed the
cusp and after a certain time it will re-enter the stable phase space area if
it has not reached the extraction septum, yet. This limiting time $t_{\rm
lim}$ can approximately be defined by

$t_{\rm lim}=T_{\rm s}\frac{\psi_{\rm s}}{2\pi},$ (14)

where $\psi_{\rm s}$ is the synchrotron phase angle shown in Figure 3. The
ellipse in this figure represents the trajectory in longitudinal phase space
with the largest synchrotron amplitude which corresponds to the maximum
$\psi_{\rm s}$. $\psi_{\rm s}$ will be the smaller the less the cusp of the
trajectory goes beyond the stability frontier at the instant $t+T_{\rm s}$. In
addition, particles with a short $t_{\rm lim}$ will start their transit from
only a little outside the stable phase space area at $t+T_{\rm s}$. Hence,
$\Delta X_{\rm start}$ and $\lambda_{0}$ of such particles in Equation (12) is
small and will result in a long transit time. In other words, longer transit
times $T_{\rm tr}$ will be suppressed by shorter limiting times $t_{\rm lim}$.

In order to demonstrate the effect, the transit time distribution will be
determined in two steps for conditions of an Ar18+ beam at the beam energy
$E=500\ {\rm MeV/u}$. The two rf voltages $V_{\rm rf}=6.2\ {\rm kV}$ and
$V_{\rm rf}=12.0\ {\rm kV}$ are applied which are the largest used in the
measurements. Further simplifying assumptions are:

  1. 1.

the distribution of the momentum deviations $\delta$ is symmetric with respect
to the sign of $\delta$ and the maximum for all cases is $\delta_{\rm
m}=0.0005$,

  2. 2.

the synchrotron tune does not depend on the synchrotron amplitude and is given
by [9]

$Q_{\rm s}=\sqrt{\frac{ZehV_{\rm rf}|\eta\cos\phi_{\rm
s}|}{2\pi\beta^{2}E_{\rm tot}}}$ (15)

with the charge of the ions $Ze$, harmonic number and amplitude of the rf
voltage $h,V_{\rm rf}$, the phase slip factor $\eta$, the phase of the
synchronous particle $\phi_{\rm s}$, the relativistic factor $\beta=v/c$, and
the total energy of an ion $E_{\rm tot}=Am_{\rm u}c^{2}\gamma$ with the number
of nucleons in the ion’s nucleus, $m_{\rm u}c^{2}=931.494\ {\rm MeV}$ and
$\gamma=1/\sqrt{1-\beta^{2}}$.

  3. 3.

the particle density does not depend on betatron and synchrotron coordinates.

In the first step, the transit time distribution is determined according to
the initial particles $\Delta X_{\rm start},\Delta X_{\rm
start}^{{}^{\prime}}$ as well as the momentum dependent difference between
particle and resonance tunes but neglecting the limitation of the transit
times due to the synchrotron motion. The resulting transit time distributions
are shown in Figure 5. They appear rather independent from the synchrotron
tune.

In the second step, the limitation of the transit times due to the synchrotron
motion is taken into account. It turns out that, after a single synchrotron
period, none of the particles is extracted for all considered rf voltages. For
that reason, the machine tune is further moved towards the resonance in time
steps of the synchrotron period. In addition, we assume that the betatron
amplitude of the particles has not changed during the short stay outside the
stable phase space area such that the horizontal start coordinates relative to
the centre of the stable phase area, $X_{\rm start},X_{\rm
start}^{{}^{\prime}}$ are not changed. The distance of the starting point from
the separatrix, $\Delta X_{\rm start}$ and, hence, $\lambda_{0}$, of all
particles will increase because of the shrinking stable phase area denoted by
a decreasing $h(t)$ in the transit time formula above. That will result in a
decrease of the transit times $T_{\rm tr}$ such that they will become shorter
than the limiting times $t_{\rm lim}$. The resulting in the transit time
distributions are those of the particles which reach the extraction septum.
They are shown in Figure 6.

Figure 3: Range of longitudinal coordinates of particles with certain
horizontal emittance related to the size of the stable phase space area
characterised by $h(t)$ at instant $t$ and shrunk within a synchrotron period
$T_{\rm s}$ by $\Delta h(T_{\rm s})$.

Figure 4: Schematic plot of the stable phase space area at instant $t$ and
shrunk within a synchrotron period by moving the machine tune $Q_{x,{\rm m}}$
towards the resonance tune $Q_{x,{\rm r}}$.

Figure 5: Distributions of the transit times according to Equation (12)
without taking into account the limitation arising from the re-capture of
particles into the stable phase area due to synchrotron motion for the same
conditions as those for Figure 1.

Figure 6: Distributions of the transit times as determined in Figure 5, where
the limitation is taken into account which arises from the re-capture of
particles into the stable phase area due to synchrotron motion. Hence, the
distributions in this figure represent the real transit times of the particles
which reach the extraction septum.

The corresponding averaged transit times and rms spreads are

$\displaystyle V_{\rm rf}=6.2\ {\rm kV}$ $\displaystyle:$ $\displaystyle
T_{\rm tr,av}=162\ {\rm rev.},\ \Delta T_{\rm tr,rms}=44\ {\rm rev.}$
$\displaystyle V_{\rm rf}=12.0\ {\rm kV}$ $\displaystyle:$ $\displaystyle
T_{\rm tr,av}=116\ {\rm rev.},\ \Delta T_{\rm tr,rms}=33\ {\rm rev.}$

The results suggest that the decrease of the transit time spread for the rf
voltage increased from $6.2\ {\rm kV}$ to $12\ {\rm kV}$ which results in the
decrease of the duty factor has a stronger impact than the decrease of the
stay of the particles near the oscillating edge of the horizontal stable phase
space which would result in a reduced spill micro structure level.

One should note that the differences of the mean and rms values of the transit
times found for the considered rf voltages are rather underestimated because
the increase of the synchrotron period for increased synchrotron amplitude is
neglected which rather occurs for lower voltages.

##  V Spill nano structures

The improvement of the spill quality on microscopic time scales due to
bunching occurs at the price of the creation of spill structures on time
scales defined by the rf frequency

$f_{\rm rf}=f_{\rm rev}\cdot h$ (16)

with the revolution time $f_{\rm rev}$ and the harmonic number of the rf
fields $h$. The rf frequency of the present SIS18 rf cavity reaches up to
$f_{\rm rf}=6\ {\rm MHz}$ [10]. According to a typical revolution time $t_{\rm
rev}\in[0.8,1.0]\ {\rm\mu s}$ during extraction, the rf harmonic number is
usually $h=4$ or $h=5$ and the total duration of an rf bucket is $t_{\rm
bucket}\in[200,250]\ {\rm ns}$. Hence, these structure are referred to as
spill nano structures.

The length of the bunches in the extracted beam is found to be shorter than
that in the circulating beam [11]. The reason is that only particles are
extracted which are near the cusp of synchrotron motion such that their
chromatic tune reaches the phase space range of unstable betatron motion which
is, basically, the area in the longitudinal trajectory marked by the red
ellipse in Figure 3 and the black solid lines which mark the frontier between
stable and unstable motion at beginning and ending of a synchrotron period. In
addition, most particles will start with their transit towards the extraction
channel after they have passed the cusp of synchrotron motion because the
transit time reaches its minimum there which is pointed out in Section IV.

An example for resulting profiles of circulating and extracted bunches as
function of the arrival time compared to that of the synchronous particle
obtained from simulation data are shown in Figure 7. The length given by the
full length at half maximum intensity of the circulating bunch under the
chosen conditions is about $t_{\rm b,circ}=84\ {\rm ns}$, whereas that of the
extracted bunch integrated over the whole extraction process is $t_{\rm
b,ex}=40\ {\rm ns}$. The full bucket length is $t_{\rm bucket}=190\ {\rm ns}$
such that the duration of the void between two bunches is $t_{\rm
void,circ}=106\ {\rm ns}$ for the circulating beam and even $t_{\rm
void,ex}=150\ {\rm ns}$ for the extracted beam.

Figure 7: Intensity profiles as function of the time difference to the
synchronous particle of a circulating bunch and the resulting extracted
bunches averaged over the time of the whole extraction process obtained in a
simulation for conditions used in Figure 1 with $h=5$ and $V_{\rm rf}=6.2\
{\rm kV}$.

In addition, one should keep in mind that the extracted bunch in Figure 7 is
averaged over the whole extraction such that it is longer than an
instantaneous bunch. As a result, the intensity of the beam arriving at the
target will have a strong variability. That limits the maximum intensity and,
hence, the event rate of some experiments because the integration time of the
detector is comparable or shorter than the duration of a bunch. E. g. the
HADES detector has an integration time of $t_{\rm int}=140\ {\rm ns}$. For
that reason, the number of particles detected in an integration time will be
given by the maximum intensity which can lead to a pile up in case of a too
large averaged intensity or it will be zero when a void is detected which
means waste of detection capacity [13, 12]. A possible solution is to increase
the rf frequency to at least $f_{\rm rf}\geq 40\ {\rm MHz}$. Actually, the
installation of an rf cavity with a frequency $f_{\rm rf}=80\ {\rm MHz}$ is
planned such that the duration of an rf bucket would shrink to $t_{\rm
bucket}=12.5\ {\rm ns}$ and always five or six bunches arrive within an
integration time at the detector [14]. The result would be a much lower
fluctuation level of particles detected in subsequent detector integration
times. The generation of bunches with such a high rf frequency results in fast
synchrotron motion because $Q_{\rm s}\propto\sqrt{h}$.

##  VI Macroscopic spill structures arising from fast synchrotron motion

It is shown in Section IV that there is an optimum of the synchrotron tune
above which the level of spill micro structures is again increased for further
increased $Q_{\rm s}$. An additional limitation to the improvement of the
spill quality found in our studies is that synchrotron motion with
sufficiently large synchrotron tune can result in the formation of macroscopic
spill structures of duration $\sim 0.1\ {\rm s}$ or longer. Such structures
could be found for conditions of present SIS18 operation in measurements as
well as in simulations, see Figures 8 and 9. These figures show spills of the
extraction Ar18+ beams at beam energy $E=500\ {\rm MeV/u}$ bunched with an rf
voltage of amplitude $V_{\rm rf}=12\ {\rm kV}$ and harmonic number $h=5$.

Figure 8: Spill of an Ar18+ beam measured during slow extraction from SIS18 at
the beam energy $E=500\ {\rm MeV/u}$ bunched with the rf voltage of amplitude
$V_{\rm rf}=12\ {\rm kV}$ and harmonic number $h=5$.

Figure 9: Spill of an Ar18+ beam obtained in a slow extraction simulation for
SIS18 conditions at the beam energy $E=500\ {\rm MeV/u}$, where $V_{\rm
rf}=12\ {\rm kV}$, $h=5$, and quadrupole ripples of two different amplitudes
are applied. The transverse beam emittances are correspond to the machine
acceptance at injection reduced by the factor $0.2$ and adiabatic shrinkage
during acceleration afterwards. Furthermore, the maximum momentum deviation in
the bunches is $\delta_{\rm m}=5\cdot 10^{-4}$. Both, the transverse
emittances and the maximum momentum deviation are lower than during usual
SIS-18 operation. The dotted vertical lines mark the moments when the machine
tune crosses a synchro-betatron resonance. The third integer resonance
corresponding to $m=0$ in Equation (17) is crossed at $t=0.41\ {\rm s}$.

Crossing synchro-betatron coupling resonances could be identified in tracking
simulations as a likely mechanism. These resonances are defined by the
condition [15]

$kQ_{x}+lQ_{y}+mQ_{s}=n,$ (17)

where $k,l,m,n$ are integer numbers, where $k=3$, $l=0$, and $n=3\cdot
Q_{x,{\rm r}}=13$. In simulations, the exact time behaviour of $Q_{x}$ is
known and, hence, it can be detected when such a resonance is crossed. In
doing so, a clear correlation between the instants when $Q_{x}$ is fulfilling
the resonance condition and the appearance of the structures could be found
which one can see in Figure 9. That figure makes also visible that quadrupole
ripples inhibit the formation of macroscopic structures, where the spill
obtained with the quadrupole ripple of amplitude zero resembles more the
measured spill in Figure 8.

The macroscopic structures are found to depend on the rf voltage, the
horizontal beam emittance, and the momentum width, where the macroscopic spill
structures become more visible for a higher rf voltage because the distance
between the resulting synchro-betatron resonances is larger such that they can
be better resolved. In addition, the structures shown in Figure 9 are found
for bunches with a maximum momentum deviation $\delta_{\rm max}=5\cdot
10^{-4}$ and a horizontal beam emittance $\varepsilon_{x}=3.1\ {\rm mm\
mrad}$. The latter is achieved by filling the total machine acceptance at
injection to $20\ \%$ and subsequent adiabatic shrinkage during acceleration.
Both values are lower than during regular SIS18 operation. The structures are
diminished or even vanish if the horizontal emittance or the momentum width is
increased. It turns out that the horizontal emittance has a stronger influence
than the momentum width which is demonstrated by repeating the simulations
with increased momentum width and horizontal emittance, respectively. The
resulting spills are shown in Figure 10. They suggest that macroscopic spill
structures should not play an important role in present SIS18 operation.

Figure 10:  Spills obtained for Ar18+ beam at beam energy $E=500\ {\rm MeV/u}$
bunched with the rf voltage of harmonic number $h=5$ and amplitude $V_{\rm
rf}=12\ {\rm kV}$ with varied horizontal emittance and momentum width. The
black curve corresponds to the solid black curve in Figure 9, where the
initial horizontal tune is changed from $Q_{x,{\rm ini}}=4.329$ in the figure
above to $Q_{x,{\rm ini}}=4.327$ in this figure in order to enable the
application of increased horizontal emittance and momentum width.

The situation will be different if the harmonic number of the rf field $h$ is
increased by a factor of about $20$ which will result in a strongly increased
synchrotron tune. That is demonstrated in the following by examining the
example of an Ar18+ beam extracted at the same energy as in the example above
but the harmonic number increased to $h=100$. In the first step, the rf
voltage amplitude to $V_{\rm rf}=70\ {\rm kV}$ is assumed which is planned to
be the maximum of the high frequency cavity foreseen to be installed in SIS18
[14]. The resulting spill obtained by particle tracking is shown in Figure 11.

Figure 11: Spill of the extraction of an Ar18+ beam at beam energy $E=500\
{\rm MeV/u}$ bunched with rf fields of the voltage $V_{\rm rf}=70\ {\rm kV}$
and the harmonic number $h=100$ obtained by particle tracking. The horizontal
tune is moved from $Q_{x,{\rm i}}=4.326$ to $Q_{x,{\rm f}}=4.334$. Two
synchrotron-betatron resonances are crossed which are marked by the dotted
vertical lines.

The horizontal tune in this example is shifted from $Q_{x,{\rm i}}=4.326$ to
$Q_{x,{\rm f}}=4.334$. The synchrotron tune is $Q_{\rm s}=0.015$ such that the
gap between two synchro-betatron resonances is a little lower than three times
the horizontal tune interval crossed during the extraction, $\Delta
Q_{x}=0.024$. Two resonance are crossed which are marked by the dotted lines
in Figure 11, where the horizontal third integer resonance is that crossed at
$t=0.48\ {\rm s}$. In addition, this figure shows that the extraction rate has
its maximum near the resonance crossed first at $t=0.18\ {\rm s}$. The reason
is that the number of particles in the beam when reaching the third integer
resonance is already strongly reduced.

A significant reduction of the influence of the synchro-betatron resonance
crossing occurs for reduced rf voltage which is demonstrated in the second
step. Figure 12 shows a spill obtained in a simulation, where the rf voltage
is reduced to $V_{\rm rf}=5\ {\rm kV}$ and the horizontal tune interval is
changed to $(Q_{x,{\rm i}},Q_{x,{\rm f}})=(4.329,4.334)$. All other quantities
are preserved from the previous example. Hence, the synchrotron tune is
$Q_{\rm s}=0.0041$ and three times the horizontal tune interval is
$3\cdot(Q_{x,{\rm f}}-Q_{x,{\rm i}})=0.015$ such that four resonances are
crossed, where the last is the horizontal third integer resonance. The figure
shows that the maximum extraction rate occurs between crossing the second and
the third crossed resonance. That is a sign that both these resonances are
smeared out because they are closer to each other than for $V_{\rm rf}=70\
{\rm kV}$ and because the synchrotron tune varies with the synchrotron
amplitude.

Figure 12: Spill of the extraction of an Ar18+ beam at beam energy $E=500\
{\rm MeV/u}$ bunched with rf fields of the voltage $V_{\rm rf}=5\ {\rm kV}$
and the harmonic number $h=100$ obtained by particle tracking. The horizontal
tune is moved from $Q_{x,{\rm i}}=4.329$ to $Q_{x,{\rm f}}=4.334$. The dotted
vertical lines denote the instants when synchro-betatron resonances are
crossed, the dashed vertical line at $t=0.048\ {\rm s}$ denotes the ending of
the rf voltage ramp, and the dashed line at $t=0.38\ {\rm s}$ denotes the
crossing of the third integer resonance.

However, goal of the planned installation of the high harmonic rf cavity is
the reduction of spill micro structures. Figure 13 shows the time dependent
duty factors obtained in simulations for rf fields with voltages $V_{\rm
rf}=5\ {\rm kV}$ and $V_{\rm rf}=70\ {\rm kV}$ and the harmomic number $h=100$
compared with that for extraction of unbunched beam. The duty factors found
for both rf voltages are at the beginning of the extraction time interval
higher than that obtained than that obtained for unbunched beam. The duty
factor reduction towards the end of the spill found for $V_{\rm rf}=5\ {\rm
kV}$ is, basically, caused by the reduction of the extraction rate which is
denoted by the simultaneous reduction of the Poisson duty factor and which can
be seen in Figure 12. In contrast to that, the Poisson duty factor found for
$V_{\rm rf}=70\ {\rm kV}$ remains large to the end of the spill and no
reduction of the spill at the half extraction time is visible in Figure 11.
Hence, that duty factor reduction arises from an increase of the micro spill
structure level. These results indicate that an improvement of the spill
quality on micro structure level for the rf frequency increased by an order of
magnitude is possible, where the spill micro structure mitigation is more
efficient when applying a lower rf voltage.

Figure 13: Duty factors which correspond to the spills in Figures 11 and 12
compared to that obtained with with $V_{\rm rf}=0$, i.e. without bunching. The
dashed curves represent the Poisson duty factors.

##  VII Summary

Bunched beam extraction is a known technique for mitigating the formation of
spill micro structures which arise from ripples in the quadrupole magnets of
ring accelerators. The quadrupole ripples cause tune ripples and, thereby, a
modulation of the size of the stable phase space area. The resulting
structures imprinted on the flow of particles across the edge of the stable
phase space area are reduced by synchrotron motion because it decreases the
duration of stay of particles near the oscillating separatrices.

We investigated the validity of spill micro structure improvement by bunched
beam extraction for larger synchrotron tunes. That was primarily motivated by
the plan to install an rf cavity with an rf frequency of $f_{\rm rf}=80\ {\rm
MHz}$ in the GSI heavy ion synchrotron SIS18 which is approximately 20 times
the rf frequency of the present cavity.

We found spill improvement only up to an optimum synchrotron tune and
diminution of the spill quality for increasing synchrotron tune beyond the
optimum. The latter arises from the recapture of particles due to the
synchrotron motion. That recapture reduces the transit time of the particles,
where average and spread of the transit times are more reduced for a higher
synchrotron tune. Therefore, the spill quality defined by micro structures
becomes worse at higher cavity voltages.

In addition, fast synchrotron motion is found in simulations and spill
measurement to cause the formation of macroscopic spill structures of duration
$\sim 100\ {\rm ms}$. Crossing synchro-betatron resonances during the tune
sweep could be identified as the source. The structures are better visible for
a higher synchrotron tune because the distance between the resonances is
proportional to the synchrotron tune. Consequently, the structures are closer
to each other and overlap for lower synchrotron tune such that they are
smeared out. Hence, inhibiting the formation of macroscopic structures low can
be achieved by setting the rf voltage as low as possible. Further, it was
shown that the formation of the macroscopic structures can be reduced by
increasing the momentum width or the transverse emittance of the direction
used for the extraction, where the latter has a stronger impact.

Generally, the formation of macroscopic spill structures as well as the
diminution of spill quality mark, in principle, a limitation to the
possibility to mitigate spill micro structures by increasing the synchrotron
tune in order to reduce the duration of stay of particles near the separatrix.

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