††thanks: This work is supported by DITANET (novel DIagnostic Techniques for
future particle Accelerators: A Marie Curie Initial Training NETwork), Project
Number ITN-2008-215080

# Interpretation of transverse tune spectra in a heavy-ion synchrotron at high
intensities

R. Singh  GSI Helmholzzentrum für Schwerionenforschung mbH, Planckstrasse 1,
64291 Darmstadt, Germany  Technische Universität Darmstadt,
Schlossgartenstr.8, 64289 Darmstadt, Germany  O. Boine-Frankenheim  GSI
Helmholzzentrum für Schwerionenforschung mbH, Planckstrasse 1, 64291
Darmstadt, Germany  Technische Universität Darmstadt, Schlossgartenstr.8,
64289 Darmstadt, Germany  O. Chorniy  P. Forck  R. Haseitl  W. Kaufmann  P.
Kowina  K. Lang  GSI Helmholzzentrum für Schwerionenforschung mbH,
Planckstrasse 1, 64291 Darmstadt, Germany  T. Weiland  Technische Universität
Darmstadt, Schlossgartenstr.8, 64289 Darmstadt, Germany

###### Abstract

Two different tune measurement systems have been installed in the GSI heavy-
ion synchrotron SIS-18. Tune spectra are obtained with high accuracy using
these fast and sensitive systems . Besides the machine tune, the spectra
contain information about the intensity dependent coherent tune shift and the
incoherent space charge tune shift. The space charge tune shift is derived
from a fit of the observed shifted positions of the synchrotron satellites to
an analytic expression for the head-tail eigenmodes with space charge.
Furthermore, the chromaticity is extracted from the measured head-tail mode
structure. The results of the measurements provide experimental evidence of
the importance of space charge effects and head-tail modes for the
interpretation of transverse beam signals at high intensity.

###### pacs:

29.20.-c,29.20.D-,41.85.-p,29.27.-a,41.75.Ak,51.75.Cn

##  I Introduction

Accurate measurements of the machine tune and of the chromaticity are of
importance for the operation of fast ramping, high intensity ion synchrotrons.
In such machines the tune spread $\delta Q_{x,y}$ at injection energy due to
space charge and chromaticity can reach values as large as 0.5. In order to
limit the incoherent particle tunes to the resonance free region the machine
tune has be controlled with a precision better than $\Delta Q\approx 10^{-3}$.
In the GSI heavy-ion synchrotron SIS-18 there are currently two betatron tune
measurement systems installed. The frequency resolution requirements of the
systems during acceleration are specified as $10^{-3}$, but they provide much
higher resolution ($10^{-4}$) on injection and extraction plateaus. The Tune,
Orbit and Position measurement system (TOPOS) is primarily a digital position
measurement system which calculates the tune from the measured position
Kowina2010 . The Baseband Q measurement system (BBQ) conceived at CERN
performs a tune measurement based on the concept of diode based bunch envelope
detection Gasior2012 . The BBQ system provides a higher measurement
sensitivity than the TOPOS system. Passive tune measurements require high
sensitivity (Schottky) pick-ups, low noise electronics and long averaging time
to achieve reasonable signal-to-noise ratio. For fast tune measurements with
the TOPOS and BBQ systems, which both use standard pick-ups, the beam has to
be excited externally in order to measure the transverse beam signals.

For low intensities the theory of transverse signals from bunched beams and
the tune measurement principles from Schottky or externally excited signals
are well known Chattopadhyay1984 ; Boussard1995 . In intense, low energy
bunches the transverse signals and the tune spectra can be modified
significantly by the transverse space charge force and by ring impedances.
Previously, the effect of space charge on head-tail modes had been the subject
of several analytical and simulation studies Blaskiewicz1998 ; Boine-
Frankenheim2009 ; Burov2009 ; Balbekov2009 ; Kornilov2010 .

Recently, the modification of transverse signals from high intensity bunches
was observed in the SIS-18 for periodically excited beams Singh2012  and for
initially kicked bunches Kornilov2012 , where the modified spectra was
explained in terms of the space charge induced head-tail mode shifts.

This contribution aims to complement the previous studies and extract the
relevant intensity parameters from tune spectra measurements using the TOPOS
and BBQ tune measurement systems. Section II presents the frequency content of
transverse bunched beam signals briefly. Sec. III presents the space charge
and image current effects on tune measurements and respective theoretical
models. Sec. IV report on the experimental conditions and compares the
characteristics of two installations as well as the various excitation
methods. Sec. V presents the experimental results in comparison with the
theoretical estimates of various high intensity effects.

##  II Transverse bunch signals

Theoretical and experimental work related to transverse Schottky signals and
beam transfer functions (BTFs) for bunched beam at low intensities can be
found in the existing literature Chattopadhyay1984 ; Sacherer1968 ;
Linnecar1981 . The transverse signal of a beam is generated by the beam’s
dipole moment

$d(t)=\sum_{j=1}^{N}x_{j}(t)I_{j}(t)$ (1)

where $x_{j}(t)$ is the horizontal/vertical offset and $I_{j}(t)$ the current
of the $j^{th}$ particle at the position of a pick-up (PU) in the ring. The
sum extends over all $N$ particles in the detector. The Schottky noise power
spectrum as a function of the frequency $f$ is defined as $S(f)=|d(f)|^{2}$
where $d(f)$ is the Fourier transformed PU signal. For a beam excited by an
external amplitude spectrum $G(f)$, the transverse response function is
defined as $r(f)=d(f)/G(f)$ Borer1979 .

If the transverse signal from a low intensity bunch is sampled with the
revolution period $T_{s}$, then the positive frequency spectrum consists of
one set of equidistant lines

$Q_{k}=Q_{0}+\Delta Q_{k},$ (2)

usually defined as baseband tune spectrum, where $Q_{0}$ is the fractional
part of the machine tune, $\Delta Q_{k}=\pm kQ_{s}$ are the synchrotron
satellites and $Q_{s}$ is the synchrotron tune.

For a single particle performing betatron and synchrotron oscillations, the
relative amplitudes of the satellites are Chattopadhyay1984

$\mid d_{k}\mid\sim\mid J_{k}(\chi/2)\mid$ (3)

where $\chi=2\xi\phi_{m}/\eta_{0}$ is the chromatic phase, $\xi$ is the
chromaticity, $\phi_{m}$ is the longitudinal oscillation amplitude of the
particle and $\eta_{0}$ the frequency slip factor. $J_{k}$ are the Bessel
functions of order $k$. In bunched beams, the relative height and width of the
lines depends on the bunch distribution. The relative height is also affected
by the characteristics of the external noise excitation or by the initial
transverse perturbation applied to the bunch. In the absence of transverse
nonlinear field components the width of each satellite with $k\neq 0$ is
determined by the synchrotron tune spread $\delta
Q_{k}\approx|k|Q_{s}\phi^{2}_{m}/16$. An example tune spectrum obtained from a
simulation code for a Gaussian bunch distribution is shown in Fig. 1 .

Figure 1: Baseband tune spectrum from a simulation code for a low intensity
bunch.

##  III Tune spectrum for high intensity

At high beam intensities, the transverse space charge force together with the
coherent force caused by the beam pipe impedance will affect the motion of the
beam particles and also the tune spectrum. The space charge force induces an
incoherent tune shift $Q_{0}-\Delta Q_{sc}$ for a symmetric beam profile of
homogeneous density where

$\Delta
Q_{sc}=\frac{qI_{p}R}{4\pi\epsilon_{0}cE_{0}{\gamma_{0}}^{2}\beta_{0}^{3}\varepsilon_{x}}$
(4)

is the tune shift, $I_{p}$ the bunch peak current, $q$ the particle charge and
$E_{0}=\gamma_{0}mc^{2}$ the total energy. The relativistic parameters are
$\gamma_{0}$ and $\beta_{0}$, the ring radius is $R$ and the emittance of the
rms equivalent K-V distribution is $\varepsilon_{x}$. In the case of an
elliptic transverse cross-section the emittance $\varepsilon_{x}$ in Eq. 4
should be replaced by

$\frac{1}{2}\left(\varepsilon_{x}+\sqrt{\varepsilon_{x}\varepsilon_{y}\frac{Q_{0x}}{Q_{0y}}}\right)$
(5)

For the vertical plane the procedure is the same, with $x$ replaced by $y$.
The image currents and image charges induced in the beam pipe, assumed here to
be perfectly conducting, cause a purely imaginary horizontal impedance

$Z_{x}=-i\frac{Z_{0}}{2\pi(\beta_{0}\gamma_{0}b_{x})^{2}}$ (6)

and real coherent tune shift

$\Delta Q_{c}=-i\frac{qIR^{2}Z_{x}}{2Q_{x0}\beta_{0}E_{0}}$ (7)

For a round beam profile with radius $a$ and pipe radius $b$ the coherent tune
shift is smaller by $\Delta Q_{c}=\frac{a^{2}}{b^{2}}\Delta Q_{sc}$ than the
space charge tune shift. Therefore the contribution of the pipe is especially
important for thick beams $(a\sim b)$ at low or medium energies.

In the presence of incoherent space charge, represented by the tune shift
$\Delta Q_{sc}$, or pipe effects, represented by the real coherent tune shift
$\Delta Q_{c}$, the shift of the synchrotron satellites in bunches can be
reproduced rather well by Boine-Frankenheim2009

$\Delta Q_{k}=-\frac{{\Delta Q_{sc}+\Delta Q_{c}}}{2}\pm\sqrt{(\Delta
Q_{sc}-\Delta Q_{c})^{2}/4+(kQ_{s})^{2}}$ (8)

where the sign $+$ is used for $k>0$. For k=0 one obtains $\Delta
Q_{k=0}=-\Delta Q_{c}$. The above expression represents the head-tail
eigenmodes for an airbag bunch distribution in a barrier potential
Blaskiewicz1998  with the eigenfunctions

$\bar{x}(\phi)=\cos(k\pi\phi/\phi_{b})\exp(-i\chi\phi/\phi_{b})$ (9)

where $\bar{x}$ is the local transverse bunch offset,
$\chi=\xi\phi_{b}/\eta_{0}$ is the chromatic phase, $\phi_{b}$ is the full
bunch length and $\eta_{0}$ is slip factor. The head-tail mode frequencies
obtained from Eq. 8 are shown in Fig. 2 . In Ref. Blaskiewicz1998  the
analytic solution for the eigenvalues Eq. 8 is obtained from a simplified
approach, where the transverse space charge force is assumed to be constant
for all particles. This assumption is correct if there are only dipolar
oscillations. In Ref. Boine-Frankenheim2009  it is has been pointed out, that
in the presence of space charge there is an additional envelope oscillation
amplitude. For the negative-$k$ eigenmodes the envelope contribution dominates
and therefore those modes disappear from the tune spectrum. In Ref. Boine-
Frankenheim2009  Eq. 8 has been successfully compared to Schottky spectra
obtained from 3D self-consistent simulations for realistic bunch distributions
in rf buckets. Analytic and numerical solutions for Gaussian and other bunch
distribution valid for $q_{sc}\gg 1$ were presented in Burov2009 ;
Balbekov2009 .

In an rf bucket the synchrotron tune $Q_{s}$ is a function of the synchrotron
oscillation amplitude $\hat{\phi}$. For short bunches $Q_{s}$ corresponds to
the small-amplitude synchrotron tune

$Q_{s0}^{2}=\frac{qV_{0}h|\eta_{0}|}{2\pi m\gamma\beta^{2}c^{2}}$ (10)

where $V_{0}$ is the rf voltage amplitude and $h$ is the rf harmonic number.

Figure 2: Head-tail mode frequencies as a function of the space charge
parameter using Eq. 8 . The red curves represent the result obtained for
$q_{c}=q_{sc}/10$.

For head-tail modes the space charge parameter is defined as a ratio of the
space-charge tune shift ( Eq. 4 ) to the small-amplitude synchrotron tune,

$q_{sc}=\frac{\Delta Q_{sc}}{Q_{s0}}$ (11)

and the coherent intensity parameter as,

$q_{c}=\frac{\Delta Q_{c}}{Q_{s0}}$ (12)

An important parameter for head-tail bunch oscillations in long bunches is the
effective synchrotron frequency which will be different from the small-
amplitude synchrotron frequency in short bunches. For an elliptic bunch
distribution (parabolic bunch profile) with the bunch half-length
$\phi_{m}=\sqrt{5}\sigma_{l}$ (rms bunch length $\sigma_{l}$), one obtains the
approximate analytic expression for the longitudinal dipole tune boine_rf2005
,

$\frac{Q_{s1}}{Q_{s0}}=\sqrt{1-\frac{\sigma_{l}^{2}}{2}}$ (13)

Using $Q_{s1}$ instead of $Q_{s0}$ in Eq. 8 shows a much better agreement with
the simulation spectra for long bunches in rf buckets (see Ref. Kornilov2012
).

Figure 3: Simulation result for $q_{sc}=2$. Figure 4: Simulation result for
$q_{sc}=10$.

For Gaussian bunches with a bunching factor $B_{f}=0.3$ ($B_{f}=I_{0}/I_{p}$,
$I_{0}$ is the dc current), the transverse tune spectra obtained from PATRIC
simulations Boine-Frankenheim2009  for different space charge factors and thin
beams ($q_{c}=0$) are shown in Figures 1, 3 and 4. The dotted vertical lines
indicate the positions of the head-tail tune shifts obtained from Eq. 8 with
$Q_{s}=Q_{s1}$. For the low-$k$ satellites there is a good agreement between
Eq. 8 and the simulation results. Lines with $k>2$ can only barely be
identified in the simulation spectra. The positions of the satellites for
$k=0,1,2$ together with the predicted head-tail tune shifts from Eq. 8 are
shown in Fig. 5 . The error bars indicate the obtained widths of the peaks in
the tune spectra.

Figure 5: Head-tail tune shifts and their width obtained from the simulations
for $q_{c}=0$. The error bars indicate the widths of the lines.

It is important to notice that the simulations for moderate space charge
parameters ($q_{sc}\lesssim 10$) require a 2.5D self-consistent space charge
solver. The theoretical studies rely on the solution of the Möhl-Sch nauer
equation Moehl1995 , which assumes a constant space charge tune shift for all
transverse particle amplitudes. In contrast to the self-consistent results,
PATRIC simulation studies using the Möhl-Sch nauer equation gave tune spectra
with pronounced, thin satellites also for large $k$. In order to account for
the intrinsic damping of head-tail modes Burov2009 ; Balbekov2009 ;
Kornilov2010 , which is the main cause of the peak widths obtained from the
simulations, a self-consistent treatment is required.

For thick beams (here $q_{c}=0.15q_{sc}$, which corresponds to the conditions
at injection in the SIS-18) the positions of the synchrotron satellites
obtained from the simulation are indicated in Fig. 6 . Again, the error bars
indicate the widths of the peaks. From the plot we notice an increase in the
spacing between the $k=0,1,2$ satellites, relative to the analytic expression.
Also the peak width for $k=1,2$ does not shrink with increasing $q_{sc}$.

Figure 6: Head-tail tune shifts and their width obtained from the simulations
for $q_{c}=0.15q_{sc}$. The error bars indicate the widths of the peaks.

The tune spectra obtained for $q_{sc}=3$, $q_{sc}=5$ and $q_{sc}=10$ are shown
in Fig. 7 , Fig. 8 and Fig. 9 . One can observe that for thick beams (here
$a\approx 0.4b$) the $k=1$ peak remains very broad up to $q_{sc}=10$.

Figure 7: Tune spectrum obtained from the simulation for $q_{sc}=3$ and
$q_{c}=0.15q_{sc}$. Figure 8: Tune spectrum obtained from the simulation for
$q_{sc}=5$ and $q_{c}=0.15q_{sc}$. Figure 9: Tune spectrum obtained from the
simulation for $q_{sc}=10$ and $q_{c}=0.15q_{sc}$.

This observation is consistent with a very simplified picture for the upper
$q_{sc}$ threshold for the intrinsic Landau damping of head-tail modes. For a
Gaussian bunch profile, the maximum incoherent tune shift, including the
modulation due to the synchrotron oscillation is

$\Delta Q_{\max}=-\Delta Q_{sc}+kQ_{s}$ (14)

where $\Delta Q_{sc}$ is determined by Eq. 4. The minimum space charge tune
shift is (see Refs. Balbekov2009 ; Kornilov2010 )

$\Delta Q_{\min}=-\alpha q_{sc}Q_{s}+kQ_{s}$ (15)

where $\alpha$ is determined from the average of the space charge tune shift
along a synchrotron oscillation with the amplitude $\hat{\phi}=\phi_{m}$. For
a parabolic bunch we obtain $\alpha=0.5$. For a Gaussian bunch and
$\hat{\phi}=3\sigma_{l}$, we obtain $\alpha=0.287$. Each band of the
incoherent transverse spectrum has a lower boundary determined by the maximum
tune shift $\Delta Q_{\max}$ and an upper boundary determined by $\Delta
Q_{\min}$. Landau damping, in its very approximate treatment, requires an
overlap of the coherent peak with the incoherent band. The head-tail tune for
low $q_{sc}$ can be approximated as

$\Delta Q_{k}=-\frac{1}{2}\left(\Delta Q_{sc}+\Delta Q_{c}\right)+kQ_{s}$ (16)

The distance between the coherent peak and the upper boundary of the
incoherent band for fixed $k$ is

$\delta Q_{k}=\left(\frac{1}{2}-\alpha\right)\Delta Q_{sc}+\frac{1}{2}\Delta
Q_{c}$ (17)

For large $q_{sc}$ the head-tail modes with positive $k$ converge towards
$Q_{k}=-\Delta Q_{c}/2$. For a given $k$ the mode is still inside the
incoherent band if

$k\gtrsim\alpha q_{sc}-\frac{1}{2}q_{c}$ (18)

holds. In order to illustrate the above analysis, the incoherent band for
$k=2$ is shown in Fig. 10 (shaded area). For $q_{c}=0$ the coherent head-tail
mode frequency crosses the upper boundary of the band at $q_{sc}\approx 4.5$.
For $q_{c}=0.15q_{sc}$ the $k=2$ head-tail mode remains inside the incoherent
band until $q_{sc}\approx 12$. For the $k=1$ modes the above analysis leads to
thresholds of $q_{sc}\approx 2$ (thin beams) and $q_{sc}\approx 6$ for
$q_{c}=0.15q_{sc}$.

Figure 10: Head-tail mode frequencies as a function of the space charge
parameter. The grey shaded area indicates the incoherent band for a Gaussian
bunch and $k=2$. The red curves represent the head-tail mode frequencies
obtained for $q_{c}=0.15q_{sc}$.

##  IV Measurement setup for transverse bunch signals

In this section, a brief description of the transverse beam excitation
mechanisms as well as the two different tune measurement systems, TOPOS and
BBQ, in the SIS-18 is given. Further the experimental set-up, typical beam
parameters and uncertainty analysis of the measured beam parameters is
discussed.

###  IV.1 Transverse Beam Excitation

The electronics used for beam excitation consist of a signal generator
connected to two $25$ W amplifiers which feed power to $50\Omega$ terminated
stripline exciters as shown in Fig. 11 . Excitation types such as band limited
noise and frequency sweep are utilized at various power levels to induce
coherent oscillations.

####  IV.1.1 Band limited noise:

Band limited noise is a traditionally used beam excitation system for slow
extraction in the SIS-18. The RF signal is mixed with Direct Digital Synthesis
(DDS) generated fractional tune frequency, resulting in RF harmonics and their
respective tune sidebands. This signal is further modulated by a pseudo-random
sequence resulting in a finite band around the tune frequency. The width of
this band is controlled by the frequency of the pseudo-random sequence.
Typical bandwidth of band limited exciter is $\approx 5\%$ of the tune
frequency. There are two main advantages of this system; first it is an easily
tunable excitation source available during the whole acceleration ramp and
second, the band limited nature of this noise results in an efficient
excitation of the beam in comparison to white noise excitation. The main
drawback is the difficulty in correlating the resultant tune spectrum with the
excitation signal.

####  IV.1.2 Frequency sweep:

Frequency sweep (chirp/harmonic excitation) using a network analyzer for BTF
measurements is an established method primarily for beam stability analysis
Boussard1995 . However, using this method for tune measurements during
acceleration is not trivial, and thus the method is not suitable for tune
measurements during the whole ramp cycle. Nevertheless, this method offers
advantages compared to the previous excitation method for careful
interpretation of tune spectrum in storage mode, e.g., injection plateau or
extraction flat top. Thus frequency sweep is used during measurements at
injection plateau to compare and understand the dependence of tune spectra on
the type of excitation.

###  IV.2 TOPOS

Following the beam excitation, the signals from each of the 12 shoe-box type
BPMs Forck2004  at SIS-18 pass through a high dynamic range (90 dB) and
broadband (100 MHz) amplifier chain from the synchrotron tunnel to the
electronics room, where the signals are digitized using fast 14 bit ADCs at
125 MSa/s. Bunch-by-bunch position is calculated from these signals using
FPGAs in real time and displayed in the control room. The spatial resolution
is $\approx$0.5 mm in bunch-by-bunch mode. Further details can be found in
Kowina2010 . Hence, TOPOS is a versatile system which provides accurate bunch-
by-bunch position and longitudinal beam profile. This information is analyzed
to extract non trivial parameters like betatron tune, synchrotron tune, beam
intensity evolution etc.

Figure 11: Scheme of the excitation system and the Tune, Orbit and Position
(TOPOS) measurement system.

###  IV.3 BBQ

The BBQ system is a fully analog system and its front end is divided into two
distinct parts; a diode based peak detector and an analog signal processing
chain consisting of input differential amplifier and a variable gain filter
chain of 1 MHz bandwidth. The simple schematic of BBQ system configuration at
SIS-18 is shown in Fig. 12 and the detailed principle of operation can be
found in Ref. Gasior2012 .

Figure 12: BBQ: Baseband Q measurement system. Diode detectors (a) and signal
chain (b).

###  IV.4 Comparison of TOPOS and BBQ

The sensitivity of BBQ has been measured to be $\approx$10-15 dB higher than
that of TOPOS under the present configuration. The main reason for the
difference is the relative bandwidth of the two systems and their tune
detection principles. In BBQ, the tune signal is obtained using analog
electronics (diode based peak detectors and differential amplifier)
immediately after the BPM plates, while in TOPOS position calculation is done
after passing the whole bunch signal through a wide bandwidth amplifier chain.
Even though the bunches are integrated to calculate position in TOPOS which
serves as a low pass filter, the net signal-to-noise ratio is still below BBQ.
Operations similar to BBQ could also be performed digitally in TOPOS to obtain
higher sensitivity but would require higher computation and development costs.
TOPOS can provide individual tune spectra of any of the four bunches in the
machine, while BBQ system provides ”averaged” tune spectra of all the bunches.
Both the systems have been benchmarked against each other. Tune spectra shown
in Sec. V are mostly from the BBQ system while TOPOS is primarily used for
time domain analysis, nevertheless this will be pointed out when necessary.

###  IV.5 Beam parameters during the measurements

Experiments were carried out using N7+ and U73+ ion beams at the SIS-18
injection energy of 11.4 MeV/u. The data were taken during 600 ms long
plateaus. At injection energy space charge effects are usually strongest. Four
bunches are formed from the initially coasting beam during adiabatic RF
capture. The experiment was repeated for different injection currents. At each
intensity level several measurements were performed with different types and
levels of beam excitation in both planes. Tune measurements were done
simultaneously using the TOPOS and BBQ systems. The beam current and the
transverse beam profile are measured using the beam current transformer
Reeg2001  and the ionization profile monitor (IPM) Giacomini2004
respectively. An examplary transverse beam profile is shown in Fig. 13 . The
dipole synchrotron tune $(Q_{s1})$ is deduced using the residual longitudinal
dipole fluctuations of the bunches. $Q_{s1}$ has been used as an effective
synchrotron tune for all experimental results and will be referred as $Q_{s}$
from hereon. The momentum spread is obtained from longitudinal Schottky
measurements Caspers2008 . Important beam parameters during the experiment are
given in Tab. 1 and Tab. 2 . It is important to note that all the parameters
required for analytical determination of $q_{sc},q_{c}$ are recorded during
the experiments.

Table 1: Beam parameters during the $N^{7+}$ experiment Beam/Machine parameters | Symbols | Values
---|---|---
Atomic mass | $A$ | 238
Charge state | $q$ | 73
Kinetic energy | $E_{kin}$ | 11.4 MeV/u (measured
Number of particles | $N_{p}$ |  $1,5,12\cdot 10^{8}$ (measured)
Tune | $Q_{x},Q_{y}$ | 4.31, 3.27 (set value)
Chromaticity | $\xi_{x},\xi_{y}$ | -0.94, -1.85 (set value)
Transverse emittance | $\epsilon_{x},\epsilon_{y}(2\sigma)$ |  $45,22$ mm-mrad (measured)
Slip factor | $\eta$ | 0.94
Bunching factor | $B_{f}$ | 0.4 (measured)
Synchrotron tune | $Q_{s0},Q_{s1}$ | 0.007,0.0065 (measured)
Momentum spread | $\frac{\Delta p}{p}$ | 0.001 (measured)
Lattice parameter |  $\beta_{x,trip}$, $\beta_{y,trip}$ | 5.49, 7.76
Table 2: Beam parameters during the $N^{7+}$ experiment Beam/Machine Parameters | Symbols | Values
---|---|---
Atomic mass | $A$ | 14
Charge state | $q$ | 7
Kinetic energy | $E_{kin}$ | 11.56 MeV/u (measured)
Number of particles | $N_{p}$ |  $3,6,11,15\cdot 10^{9}$ (measured)
Tune | $Q_{x},Q_{y}$ | 4.16, 3.27 (set value)
Chromaticity | $\xi_{x},\xi_{y}$ | -0.94, -1.85 (set value)
Transverse emittance | $\epsilon_{x},\epsilon_{y}(2\sigma)$ |  $33,12$ mm-mrad (measured)
Slip factor | $\eta$ | 0.94
Bunching factor | $B_{f}$ | 0.37 (measured)
Synchrotron tune | $Q_{s0},Q_{s1}$ | 0.006,0.0057 (measured)
Momentum spread | $\frac{\Delta p}{p}$ | 0.0015 (measured)
Lattice parameter |  $\beta_{x,trip}$, $\beta_{y,trip}$ | 5.49, 7.76

From Tab. 1 and Tab. 2 one can estimate that in the measurements the head-tail
space charge and image current parameters were in the range $q_{sc}\lesssim
10$ and $q_{c}\lesssim 0.2q_{sc}$ for the horizontal and vertical planes.

Figure 13: Normalized transverse beam profile in vertical plane at
$N^{7+}=15\cdot 10^{8}$ ions. The dotted lines shows the normal distribution
for the rms width obtained by evaluation of beam profile around its centre.

##  V Experimental Observations

Tune spectra measurements at different currents are presented and interpreted
in comparison with the predictions of Sec. III . The effect of different
excitation types and power on the transverse tune spectra is studied.
Transverse impedances in both planes are obtained from the coherent tune
shifts. Incoherent tune shifts are obtained from the relative frequency shift
of head-tail modes in accordance with Eq. 8 . Chromaticity is measured using
head-tail eigenmodes and the obtained relative height of the observed peaks
for different chromaticities is analyzed.

###  V.1 Modification of the tune spectrum with intensity

Figure 14: Horizontal tune spectra for $U^{73+}$ ions and beam parameters
given in Tab. 1 (see text). The dashed lines indicate the head-tail tune
shifts from Eq. 8 .

Fig. 14 shows the horizontal tune spectra obtained with the BBQ system using
band-width limited noise at different intensities. Fig. 14 (a) shows the
horizontal tune spectrum at low intensity. Here the $k=1,0,-1$ peaks are
almost equidistant, which is expected for low intensity bunches. The space
charge parameter obtained using the beam parameters and Eq. 4 is
$q_{sc}\approx 0.15$. The vertical lines indicate the positions of the
synchrotron satellites obtained from Eq. 8 (with $Q_{s}=Q_{s1}$). Fig. 14 (b)
shows the tune spectrum at moderate intensity ($q_{sc}\approx 0.7$). The
$k=2,-2$ peaks can both still be identified. Fig. 14 (c) shows the tune
spectra at larger intensity ($q_{sc}\approx 1.7$). An additional peak appears
between the $k=0$ and $k=-1$ peaks which can be attributed to the mixing
product of diode detectors (since at this intensity $30-40V$ acts across the
diodes pushing it into the non-linear regime). The $k=0,1,2$ peaks can be
identified very well, whereas the amplitudes of the lines for negative $k$
already start to decrease (see Sec. III ). In the horizontal plane the effect
of the pipe impedance and the corresponding coherent tune shift can usually be
neglected because of the larger pipe diameter.

Figure 15: Vertical tune spectra for $N^{7+}$ ions and beam parameters given
in Tab. 2 . The dashed lines indicate the head-tail tune shifts from Eq. 8 .

Fig. 15 shows the vertical tune spectrum obtained by the BBQ system with band
limited noise excitation for $N^{7+}$ beams with $q_{sc}$ values larger than
$2$. Here the negative modes $(k<0)$ could not be resolved anymore. In the
vertical plane the coherent tune shift is larger due to the smaller SIS-18
beam pipe diameter ($q_{c}\approx q_{sc}/10$). The shift of the $k=0$ peak due
to the effect of the pipe impedance is clearly visible in Fig. 15 .

In the measurements the width of the peaks is determined by the cumulative
effect of non-linear synchrotron motion, non-linearities of the optical
elements, closed orbit distortion, tune fluctuation during the measurement
interval as well as due to the intrinsic Landau damping ( Sec. III ). From the
comparison to the simulations we conclude that the intrinsic Landau damping is
an important contribution to the width of the $k=1,2$ peaks.

###  V.2 Determination of coherent and incoherent tune shifts

The coherent tune shift $\Delta Q_{c}$ can be obtained by measuring shift of
the $k=0$ line as a function of the peak bunch current as shown in Fig. 16 .
The transverse impedance is obtained by a linear least square error fit of the
measured shifts in both planes to Eq. 6 and Eq. 7 . The impedance values are
obtained in the horizontal and vertical planes at injection energy are $0.23$
M$\Omega/m^{2}$ and $1.8$ M$\Omega/m^{2}$ respectively, which agrees very well
with the expected values for the average beam pipe radii of the SIS-18.

Figure 16: Coherent tune shift obtained from the measurement for the
horizontal and for the vertical planes as a function of the peak beam current.
The dotted lines correspond to a linear least square error fit. The error bars
in the horizontal plane are due to uncertainties in the current measurements.
In the vertical plane errors result from the width of $k=0$ mode. The FFT
resolution is $\approx 5\cdot 10^{-4}$ and is always kept higher than the mode
width. Figure 17: The measured positions of the peaks in the horizontal tune
spectra for different $U^{73+}$ beam intensities together with the analytical
curves from Eq. 8 using the space charge tune shift estimated from the beam
parameters in Tab. 1 . The dotted line corresponds to the incoherent tune
shift. The error bars for the vertical plane correspond to the width of
measured modes (see text). In the horizontal plane the error bars are
estimated by the propagation of uncertainties (see text). Figure 18: This plot
shows the predicted shifts from analytical Eq. 8 and measured head-tail mode
frequencies are overlaid in vertical plane for $U^{73+}$ ion beam at various
current levels ( Tab. 1 ). Figure 19: This plot shows the predicted shifts
from analytical Eq. 8 and measured head-tail mode frequencies are overlaid in
horizontal plane for $N^{7+}$ ion beam at various current levels ( Tab. 2 ).
Figure 20: This plot shows the predicted shifts from analytical Eq. 8 and
measured head-tail mode frequencies are overlaid in vertical plane for
$N^{7+}$ ion beam at various current levels ( Tab. 2 ).

Figures 17, 18, 19 and 20 show the measured positions of the peaks in the tune
spectra for different intensities. In comparison the analytical curves (solid
lines) obtained from Eq. 8 for the head-tail tune shifts are plotted using
$q_{sc}$ estimated from the beam parameters in Tab. 1 and Tab. 2 for each
intensity. The error bars in the vertical plane ($\delta q_{k}=\frac{\Delta
Q_{k}}{Q_{s}}$) correspond to the $3$ dB width of the measured peak due to
accumulation of various effects (see subsection V.1 ). In the horizontal
plane, error bars ($\delta q_{sc}$) are estimated by propagation of parameter
uncertainties mentioned in Measurement uncertainties .

In this subsection, we introduce another space charge parameter $q_{sc,m}$
which is the measured space charge parameter using the following method. It is
not to be confused with $q_{sc}$ which is predicted for a given set of beam
parameters by Eq. 4 . The incoherent space charge tune shift can be determined
directly from the tune spectra by measuring the separation between the $k=0$
and $k=1$ peaks, i.e. $(q_{k01}=\frac{\Delta Q_{k01}}{Q_{s}}$) and fitting it
with the parameter $q_{sc}$ in the predictions from Eq. 8 . The value of
$q_{sc}$ for the best fit is denoted as $q_{sc,m}$.

$\displaystyle q_{sc,m}=\frac{1-q_{k01}^{2}}{\gamma q_{k01}}\hskip
142.26378pt0\leq q_{k01}\leq 1$ (19)

Eq. 19 is obtained by rearranging Eq. 8 for $k=0,1$ while
$\gamma=\frac{q_{sc}-q_{c}}{q_{sc}}$. The linearized absolute error on
measured $q_{sc,m}$ ($\delta q_{sc,m}$) is given by

$\displaystyle\delta q_{sc,m}=\frac{-(1+q_{k01}^{2})}{\gamma
q_{k01}^{2}}\cdot\delta q_{k01}$ (20)

$\delta q_{k01}$ is given by either the width of the $k=0,1$ lines or by the
frequency resolution of the system. In a typical tune spectra measurement
using data from 4000 turns $\delta q_{k01}\approx 0.04$. The absolute error is
a non-linear function of $q_{sc}$ in accordance to the Eq. 19 . It is possible
to define the upper limit of $q_{sc}$ where this method is still adequate
based on the system resolution and Eq. 19 . If we define a criterion that,
$q_{k01}\gtrsim\delta q_{k01}$ to resolve the head-tail modes. This gives the
limit to be $q_{sc}\lesssim 8$ where the measurement error is still within the
defined criterion.

Fig. 21 shows a plot of the predicted space charge tune shifts ($q_{sc}$)
versus the ones measured from the tune spectra using the above procedure
($q_{sc,m}$). For $q_{sc}\lesssim 3.5$ the space charge tune shifts measured
from the tune spectra are systematically lower by a factor $=0.74$ than the
predicted shifts. It is shown by the dotted line in Fig. 21 which is obtained
by total least squares fit of the measured data points. For larger $q_{sc}$
the factor decreases to $\approx 0.4$. Thus the method for measuring the
incoherent tune shift based on head-tail tune shifts is found to be
satisfactory only in the range $q_{sc}\lesssim 3.5$. A possible explanation is
the effect of the pipe impedance. Similar observations are made by the results
of self-consistent simulations in Sec. III , where for $q_{sc}\gtrsim 2$ the
separation of the $k=0$ and $k=1$ peaks observed is underestimated by Eq. 8
(see Fig. 6 ).

Figure 21: Combining the results from the Figures 17, 18, 19 and 20, a plot of
predicted $q_{sc}$ using Eq. 4 against measured $q_{sc}$ using the distance
between modes $k=0$ and $k=1$ is obtained.

###  V.3 Effect of excitation parameters on tune spectrum

Figure 22 presents the tune spectra obtained from BBQ system at various
excitation power levels of band limited noise. The beam is excited with
$0.25,1.0$ and $2.25$ mW/Hz power spectral density on a bandwidth of $10$ KHz.
Signal-to-noise ratio (SNR) increases with excitation power whereas the
spectral position of various modes is independent of excitation power.

Figure 22: Tune spectra at $5\cdot 10^{8}U^{73+}$ particles with beam excited
at three different excitation amplitudes. The beam is excited with $0.25,1.0$
and $2.25$ mW/Hz power (blue, red and black respectively). The dotted lines
mark the relative positions of the head-tail modes. The frequencies of various
head-tail modes are unaffected.

Beam excitation using two other excitation types i.e. frequency sweep and
white noise is also performed to study the effect of excitation type on the
tune spectra. Figure 23 shows the tune spectra under same beam conditions for
different types of beam excitation obtained from the BBQ system. The
frequencies of various modes in the tune spectra are independent of the type
of excitation. The signal-to-noise ratio (SNR) is optimum for band limited
noise due to long averaging time compared to ”one shot” spectra from sweep
excitation.

###  V.4 Time domain identification of head tail modes

Figure 24 shows the 2-D contour plot for frequency sweep excitation in
vertical plane obtained from the TOPOS system, where various head-tail modes
are individually excited as the excitation frequency crosses them. Frequency
sweep excitation allows resolving the transverse center of mass along the
bunch for various modes which helps in identifying each head-tail mode in time
domain. This serves as a direct cross-check for the spectral information and
leaves no ambiguity in identification of the order (k) of the modes. Figure 25
shows the corresponding transverse center of mass along the bunch for $k=0,1$
and $2$ at the excited time instances. This method works only with sweep
excitation and requires high signal-to-noise ratio in the time domain, which
amounts to higher beam current or high excitation power.

Figure 23: Horizontal tune spectra over time with $U^{73+}$ at $5\cdot 10^{8}$
particles with band limited noise (RF Noise), frequency sweep (chirp noise)
and white noise (top to bottom). The dotted lines mark the relative positions
of the head-tail modes. Figure 24: Vertical tune spectra with frequency sweep
over time at $N^{7+}$ with $1.5\cdot 10^{9}$ particles. The various modes get
individually excited as the sweep frequency (marked by the arrow) coincides
with the mode frequency. The $k=0,1,2$ modes are marked. Symmetric sidebands
due to coherent synchrotron oscillations are visible around the sweep
frequency spaced with $Q_{s}$ and are clearly distinguished from the head-tail
modes. Figure 25:  Transverse center of mass along the bunch for $k=0,1$ and
$2$ modes corresponding to Figure 24 . Frequency sweep is used to excite the
beam which enables to resolve the distinct head-tail modes temporally. The rms
bunch length ($2\sigma_{rms}$) is indicated.

###  V.5 Measurement of the chromaticity

As highlighted in the previous section, the frequency sweep allows to resolve
the different head-tail modes both spectrally and temporally. This procedure
can be used for the precise determination of the chromaticity by fitting the
analytical expression for the head-tail eigenfunction Eq. 9 to the measured
bunch offset, with the chromaticity ($\xi$) as the fit parameter as shown in
Fig. 26 for $k=0,1$ and $2$. The measured chromaticity is independent of the
order of head-tail eigenfunction used to estimate it.

The fitting method is shown in Eq. 21 ; the head-tail eigenfunction from Eq. 9
is multiplied with the beam charge profile $\hat{S(t)}$ and corrected for the
beam offset $\Delta{x}$ at the BPM where the signal is measured.

$\displaystyle
F(\phi,\xi,A)=\Delta{x}\cdot\hat{S(t)}\cdot(1+A\cdot\bar{x}(\phi))$ (21)
$\displaystyle E(\xi,A)=(M(\phi)-F(\phi,\xi))^{2}$

The fit error $E(\xi,A)$ is reduced as a function of independent variables;
chromaticity $\xi$ and head-tail mode amplitude $A$. The fit error gives the
goodness of the fit. It is used to determine the error bars on measured
chromaticity. This method has been utilized for the determination of
chromaticity at SIS-18 as shown in Fig. 27 . The set and the measured
chromaticity can be fitted by linear least squares to obtain the form
$\xi_{s,y}=1.187\xi_{m,y}+0.804$ as shown by red dashed line in Fig. 27 . Fig.
27 also shows a coherent tune shift due to change in sextupole strength which
is used to adjust the chromaticity. This is due to uncorrected orbit
distortions during these measurements. These chromaticity measurements agree
with the previous chromaticity measurements using conventional methods
Paret2009 .

It is also possible to determine the relative response amplitude of each head-
tail mode to the beam excitation both in time and in frequency domain with
TOPOS. Fig. 28 shows the tune spectrum obtained with sweep excitation for
different chromaticity values. The beam parameters are kept the same
($N^{7+},14\cdot 10^{9},q_{sc}\approx 10$). The spectral position and and
relative amplitude of each head-tail mode peak are confirmed using the time
domain information (see Figure 25 ). In Fig. 29 the single particle response
amplitudes for different $k$ ( Equation 3 ) are plotted as a function of the
chromaticity. The measured relative amplitudes are indicated by the colored
symbols. The comparison indicates that the simple single particle result (
Equation 3 ) describes quite well the dependence of the relative height of the
peaks obtained from the TOPOS measurement.

Figure 26:  Analytical curves from Eq. 9 (dotted) are fitted to the local
transverse offset for the $k=0$ and $k=1$ modes with the head-tail phase shift
as the fit parameter. This method has been used for the precise determination
of the chromaticity. Figure 27: Chromaticity measurement using the method
shown in Figure 25 over the possible range of operation at SIS-18. The black
dashed line marks the measured natural vertical chromaticity of SIS-18. The
error bars for the chromaticity measurement are determined mainly by the
electronic noise in the amplifier chain of the TOPOS system. Figure 28:
Relative signal amplitudes of various head-tail modes are plotted with respect
to the measured chromaticity. The centroid of the spectrum moves toward higher
order modes with increase in chromaticity. Figure 29: Relative signal
amplitudes of various head-tail modes are plotted with respect to the
coefficient of Bessel’s function in Equation 3 $(\chi_{f}=\chi/2)$. All the
modes are relative to k=1 mode, which has a significant amplitude during the
whole measurement range.

##  VI Application to tune measurements in SIS-18

In this section we will discuss the application of our results to tune
measurements in the SIS-18 and in the projected SIS-100, as part of the FAIR
project at GSI FAIR2010 . As shown in the previous section the relative
amplitudes of the synchrotron satellites in the tune spectra are primarily a
function of chromaticity and possibly the excitation mechanism. In order to
determine the coherent tune with high precision the position of the $k=0$ mode
has to be measured. Depending on the machine settings, if the relative height
of the $k=0$ peak with respect to the other modes is small, then the $k=0$
mode may not be visible at all. To estimate the bare tune frequency in this
case, the information of space charge parameter, coherent tune shift and
chromaticity are all simultaneously required with good precision.

Another important point is the tune measurement during acceleration. The space
charge parameter for $1\cdot 10^{10}Ar^{18+}$ stored ions in the SIS-18 from
injection to extraction reduces only by $\approx 20\%$ as shown in Fig. 30 .
The dynamic shift of head-tail modes during acceleration is shown in Fig. 31
obtained from TOPOS system under same conditions. The asymmetry of $k=1,-1$
modes around $k=0$ mode can only be understood in view of the space charge
effects predicted by Eq. 8 . Thus, a correct estimate of this parameter plays
an important role in understanding the tune spectra not only during dedicated
experiments on injection plateau, but also during regular operations.

The measurement time required to resolve the various head-tail modes ($\Delta
Q_{k}$) is a complex function of $Q_{s}$, $q_{sc}$, beam intensity and
excitation power. To give some typical numbers for SIS-18; on a measurement
time of 600 ms on the injection plateau, if one spectrum is obtained in
$\approx 20$ ms ($\approx 4000$ turns), an improvement of factor $\approx 6$
in SNR by averaging 30 spectra. Following the calculations in subsection V.2 ,
$q_{sc}\lesssim 8$ can be resolved under typical injection operations.
However, the constraints on measurement time are much higher during
acceleration, where the tune/revolution frequency increases due to
acceleration. This allows the measurement of a single spectrum typically only
over $500-1000$ turns (depends on ramp rate as well). There are no averaging
possibilities since the tune is moving during acceleration due to dynamic
changes in machine settings as seen in Fig. 31 . In addition, the synchrotron
tune reduces with acceleration making it practically very difficult to resolve
the fine structure of the head-tail modes for $q_{sc}\gtrsim 2$.

Figure 30: Change in synchrotron tune and space charge tune shift on
acceleration from injection to extraction for $Ar^{18+}$ with $1\cdot 10^{10}$
particles. The estimated space charge parameter reduces by $\approx 20\%$ in
this typical case from injection at 11.4 MeV/u to 300 MeV/u at extraction.
Figure 31: The movement of head-tail modes during acceleration. The head-tail
modes are getting closer since the synchrotron tune reduces with acceleration,
but the asymmetry of the $k=1$ and $k=-1$ modes around $k=0$ is maintained
throughout the ramp which depends primarily on the space charge parameter
$(q_{sc})$.

##  VII Conclusion

Two complimentary tune measurement systems, TOPOS and BBQ, installed in the
SIS-18 are presented. Analytical as well as simulation models predict a
characteristic modification of the tune spectra due to space charge and image
current effects in intense bunches. The position of the synchrotron satellites
corresponds to the head-tail tune shifts and depends on the incoherent and the
coherent tune shifts. The modification of the tune spectra for different bunch
intensities has been observed in the SIS-18 at injection energy, using the
TOPOS and BBQ systems. From the measured spectra the coherent and incoherent
tune shift for bunched beams in SIS-18 at injection energies were obtained
experimentally using the analytic expression for the head-tail tune shifts.
Head-tail modes were individually excited and identified in time domain and
correlated with the spectral information. A novel method for determination of
chromaticity based on gated excitation of individual head tail mode is shown.
The dependence of the relative amplitudes of various head-tail modes on
chromaticity is also studied. The systems were compared against each other as
well as with different kinds of excitation mechanisms and their respective
powers. These measurements give a clear interpretation of tune spectra at all
stages during acceleration under typical operating conditions. The
understanding to tune spectra provides an important input to new developments
related to planned transverse feedback systems for SIS-18 and SIS-100. The
measurement systems also open new possibilities for detailed beam
investigations as demonstrated in this contribution.

## Acknowledgement

We thank Marek Gasior of CERN BI Group for the help in installation of BBQ
system and many helpful discussions on the subject. GSI operations team is
also acknowledged for setting up the machine. We also thank Klaus-Peter Ningel
from the GSI-RF group who helped setting up the amplitude ramp for the
adiabatic bunching.

## Appendix

### Calculation of bunching factor

Figure 32: A typical normalized longitudinal beam profile from TOPOS system at
injection. The length of the RF period is $\approx 1.2$ $\mu s$.

Fig. 32 shows a typical longitudinal profile of the bunch. If $S_{j}$ is the
amplitude at each time instance $j=1,..,N$. The bunching factor is calculated
by the Eq. A1 .

$B_{f}=\frac{\sum_{j=1}^{N}S_{j}}{max(S_{j})\cdot N}$ (A1)

where N $\approx 147$ is the number of samples in one RF period (at
injection). The TOPOS system samples the bunch at 125 MSa/s, thus the
difference between adjacent samples is 8ns.

### Measurement uncertainties

The calculation of $\Delta Q_{sc}$ from Eq. 4 has a dependence on measured
current, transverse beam profiles, longitudinal beam profiles and the twiss
parameters. The measurement uncertainty on each of these measured parameters
at GSI SIS-18 were commented in the detailed analysis in Franchetti2010 . Even
though some parameters and the associated uncertainties are correlated, any
correlations are neglected in present analysis. Uncertainties in each measured
parameter are propagated to find the error bars on the calculated and measured
incoherent tune shifts.

Reproducing from Ref. Franchetti2010 , the relative random uncertainty (std.
deviation) in beam profile width($\sigma_{x}$) measurements is given by Eq. A2
.

$\frac{\delta\sigma_{x}}{\sigma_{x}}=\frac{0.043}{B}+0.33\delta N_{i}$ (A2)

where $\Delta x=2.1$mm is the wire spacing and $\delta N_{i}=\frac{1}{2^{8}}$
is the ADC resolution of the IPM and B is defined as $\sigma_{x}/\Delta x$. If
the error bars are derived from $j$ measurements, the measured profile is
given by

$\sigma_{av,x}=\langle\sigma_{x}\rangle_{j},\delta\sigma_{av,x}=\sqrt{{\langle\sigma_{x}^{2}\rangle}_{j}-\langle\sigma_{x}\rangle^{2}_{j}+\langle{\delta}{\sigma_{x}\rangle}^{2}}$
(A3)

For each tune measurement at the given intensity and excitation power, 5-8
transverse beam profiles were measured, and the relative error is obtained
$\approx 5\%$ using Eq. A2 and Eq. A3 . The relative systematic error (bias)
in transverse beam width measurements is $<1\%$ Franchetti2010  and ignored in
this analysis.

The uncertainty in the injected current is dominated by fluctuations in the
source and the relative uncertainty is estimated to be $\approx 5\%$ based on
5-8 measurements at the same intensity settings for each measurement point.
Bunch length and bunching factor vary by $\approx 2-3\%$ due to long term beam
losses only under high intensity beam conditions. The maximum relative bias in
the lattice parameter $\beta$ is assumed to be $\approx 5\%$ at the IPM
location. Taking all the relative errors, uncertainty propagation using
familiar Measurement uncertainties gives relative error for estimated
incoherent tune shifts $\approx 12\%$.

$\displaystyle\frac{\delta\varepsilon_{av,x}}{\varepsilon_{av,x}}=\sqrt{4{(\frac{\delta\sigma_{av,x}}{\sigma_{av,x}})}^{2}+{(\frac{\delta\beta_{x}}{\beta_{x}})}^{2}}$
$\displaystyle\frac{\delta\Delta Q_{sc}}{\Delta
Q_{sc}}=\sqrt{{(\frac{\delta\varepsilon_{av,x}}{\varepsilon_{av,x}})}^{2}+{(\frac{\delta
I_{p}}{I_{p}})}^{2}}$

Tune measurements done by averaging over long intervals contribute to the
width of modes due to long term beam losses. Beam losses lead to change in
coherent tune especially in the vertical plane where the image current effects
are larger. This has been highlighted at appropriate sections in the text.

## References

  * (1) P. Kowina et al., “Digital baseband tune determination”, Proc. of BIW’10, Santa Fe, U.S.A (2010)
  * (2) M. Gasior, “High sensitivity tune measurement using direct diode detection”, Proc. of BIW’12, Virginia, U.S.A (2012)
  * (3) S. Chattopadhyay, “Some fundamental aspects of fluctuations and coherence in charged particle beams in storage rings”, Super Proton Synchrotron Division, CERN 84-11, (1984)
  * (4) D. Boussard, “Schottky Noise and Beam Transfer Function Diagnostics”, CERN Accelerator School; Fifth Advanced Accelerator Physics Course”, (1995)
  * (5) M. Blaskiewicz, Phys. Rev. ST Accel. Beams 1, 044201 (1998)
  * (6) O. Boine-Frankenheim, V. Kornilov, Phys. Rev. ST Accel. Beams, 12, 114201 (2009)
  * (7) A. Burov, Phys. Rev. ST Accel. Beams 12, 044202 (2009)
  * (8) V. Balbekov, Phys. Rev. ST Accel. Beams 12, 124402 (2009)
  * (9) V. Kornilov, O. Boine-Frankenheim, Phys. Rev. ST Accel. Beams,13, 114201 (2010)
  * (10) R. Singh et al., Proc. of BIW’12, Newport News, U.S.A (2012)
  * (11) V. Kornilov, O. Boine-Frankenheim, Phys. Rev. ST Accel. Beams,15, 114201 (2012)
  * (12) D. Möhl, Part. Accel. 50, 177 (1995)
  * (13) F. J. Sacherer, “Transverse space-charge effects in circular particle accelerators”, UCRL-18454 (1968)
  * (14) T. Linnecar and W. Scandale, Proc. of PAC 1981, p. 2147 (1981)
  * (15) J. Borer, Proc. of PAC 1979 (1979)
  * (16) K. Schindl, “Space Charge”, “Joint US-CERN-Japan-Russia School on Particle Accelerators” (1998)
  * (17) O. Boine-Frankenheim, T. Shukla, Phys. Rev. ST Accel. Beams 8, 034201 (2005)
  * (18) P. Forck, “Lecture notes on beam intrumentation” (2004)
  * (19) H. Reeg et al., Proc. of DIPAC, Grenoble, France (2001)
  * (20) T. Giacomini, Proc. of BIW’04, Knoxville, U.S.A (2004)
  * (21) F. Caspers, “Schottky Signals”,CAS 2008, Dourdon, France (2008)
  * (22) S. Paret, PhD Thesis, TU Darmstadt, Germany(2009)
  * (23) G. Franchetti et al., Phys. Rev. ST Accel. Beams 13, 114203 (2010)
  * (24) O. Boine-Frankenheim, “The FAIR accelerators: Highlights and challenges”, Proc. of IPAC 2010, Japan (2010)