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

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.


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 δQ 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 ∆Q ≈ 10 −3 . In the GSI heavy-ion synchrotron  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 [1]. The Baseband Q measurement system (BBQ) conceived at CERN performs a tune measurement based on the concept of diode based bunch envelope detection [2].
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 [3,4]. 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 [5][6][7][8][9].
Recently, the modification of transverse signals from high intensity bunches was observed in the SIS-18 for periodically excited beams [10] and for initially kicked bunches [11], 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. Section III presents the space charge and image current effects on tune measurements and respective theoretical models. Section IV report on the experimental conditions and compares the characteristics of two installations as well as the various excitation methods. Section 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 [3,13,14]. The transverse signal of a beam is generated by the beam's dipole moment  [15].
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 usually defined as baseband tune spectrum, where Q 0 is the fractional part of the machine tune, ∆Q k = ±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 [3] | d k |∼| J k (χ/2) | 3 where χ = 2ξφ m /η 0 is the chromatic phase, ξ is the chromaticity, φ m is the longitudinal oscillation amplitude of the particle and η 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 = 0 is determined by the synchrotron tune spread δQ k ≈ |k|Q s φ 2 m /16. An example tune spectrum obtained from a simulation code for a Gaussian bunch distribution is shown in Fig. 1 .

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 − ∆Q sc for a symmetric beam profile of homogeneous density where is the tune shift, I p the bunch peak current, q the particle charge and E 0 = γ 0 mc 2 the total energy.
The relativistic parameters are γ 0 and β 0 , the ring radius is R and the emittance of the rms equivalent K-V distribution is ε x . In the case of an elliptic transverse cross-section the emittance ε x in Eq. 4 should be replaced by 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 and real coherent tune shift For a round beam profile with radius a and pipe radius b the coherent tune shift is smaller by ∆Q c = a 2 b 2 ∆Q sc than the space charge tune shift. Therefore the contribution of the pipe is especially important for thick beams (a ∼ b) at low or medium energies.
In the presence of incoherent space charge, represented by the tune shift ∆Q sc , or pipe effects, represented by the real coherent tune shift ∆Q c , the shift of the synchrotron satellites in bunches can be reproduced rather well by [6] where the sign + is used for k > 0. For k=0 one obtains ∆Q k=0 = −∆Q c . The above expression represents the head-tail eigenmodes for an airbag bunch distribution in a barrier potential [5] with the eigenfunctionsx wherex is the local transverse bunch offset, χ = ξφ b /η 0 is the chromatic phase, φ b is the full bunch length and η 0 is slip factor. The head-tail mode frequencies obtained from Eq. 8 are shown in Fig. 2 . In Ref. [5] 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. [6] 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. [6] 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 1 were presented in [7,8].
In an rf bucket the synchrotron tune Q s is a function of the synchrotron oscillation amplitudeφ.
For short bunches Q s corresponds to the small-amplitude synchrotron tune where V 0 is the rf voltage amplitude and h is the rf harmonic number.
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, and the coherent intensity parameter as, 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 φ m = √ 5σ l (rms bunch length σ l ), one obtains the approximate analytic expression for the longitudinal dipole tune [17], 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. [11]).
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 [6] for different space charge factors  It is important to notice that the simulations for moderate space charge parameters (q sc 10) require a 2.5D self-consistent space charge solver. The theoretical studies rely on the solution of the Möhl-Schnauer equation [12], 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-Schnauer equation gave tune spectra with pronounced, thin satellites also for large k. In order to account for the intrinsic damping of head-tail modes [7][8][9], 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 .
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 ≈ 0.4b) the k = 1 peak remains very broad up to q sc = 10.
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 where ∆Q sc is determined by Eq. 4. The minimum space charge tune shift is (see Refs. [8,9]) where α is determined from the average of the space charge tune shift along a synchrotron oscillation with the amplitudeφ = φ m . For a parabolic bunch we obtain α = 0.5. For a Gaussian bunch peak with the incoherent band. The head-tail tune for low q sc can be approximated as The distance between the coherent peak and the upper boundary of the incoherent band for fixed k is For large q sc the head-tail modes with positive k converge towards Q k = −∆Q c /2. For a given k the mode is still inside the incoherent band if 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 ≈ 4.5. For q c = 0.15q sc the k = 2 head-tail mode remains inside the incoherent band until q sc ≈ 12. For the k = 1 modes the above analysis leads to thresholds of q sc ≈ 2 (thin beams) and q sc ≈ 6 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.

A. Transverse Beam Excitation
The electronics used for beam excitation consist of a signal generator connected to two 25 W amplifiers which feed power to 50Ω 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.

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 ≈ 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.

Frequency sweep:
Frequency sweep (chirp/harmonic excitation) using a network analyzer for BTF measurements is an established method primarily for beam stability analysis [4]. 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.
Following the beam excitation, the signals from each of the 12 shoe-box type BPMs [18] 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 ≈0.5 mm in bunch-by-bunch mode. Further details can be found in [1]. 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.

C. 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. [2].

D. Comparison of TOPOS and BBQ
The sensitivity of BBQ has been measured to be ≈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. The beam current and the transverse beam profile are measured using the beam current transformer [19] and the ionization profile monitor (IPM) [20] 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 [21]. Important beam parameters during the experiment are given in Tab. I and Tab. II . It is important to note that all the parameters required for analytical determination of q sc , q c are recorded during the experiments.
From Tab. I and Tab. II one can estimate that in the measurements the head-tail space charge and image current parameters were in the range q sc 10 and q c 0.2q sc for the horizontal and vertical planes.   Figure 14 shows the horizontal tune spectra obtained with the BBQ system using band-width limited noise at different intensities. Figure 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 ≈ 0.15.
The vertical lines indicate the positions of the synchrotron satellites obtained from Eq. 8 (with Q s = Q s1 ). Figure 14 (b) shows the tune spectrum at moderate intensity (q sc ≈ 0.7). The k = 2, −2 peaks can both still be identified. Figure 14 (c) shows the tune spectra at larger intensity (q sc ≈ 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 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 ≈ q sc /10). The shift of the k = 0 peak due to the effect of   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 = ∆Q k01 Qs ) 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 .
Equation 19 is obtained by rearranging Eq. 8 for k = 0, 1 while γ = qsc−qc qsc . The linearized absolute error on measured q sc,m (δq sc,m ) is given by δq 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 δq k01 ≈ 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 δq k01 to resolve the head-tail modes. This gives the limit to be q sc 8 where the measurement error is still within the defined criterion. Figure 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 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 ≈ 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 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 2 the separation of the k = 0 and k = 1 peaks observed is underestimated by Eq. 8 (see Fig. 6 ).
C. Effect of excitation parameters on tune spectrum against measured q sc using the distance between modes k = 0 and k = 1 is obtained.
whereas the spectral position of various modes is independent of excitation power. 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. 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. Frequency sweep is used to excite the beam which enables to resolve the distinct head-tail modes temporally.
The rms bunch length (2σ rms ) is indicated.
The fitting method is shown in Eq. 21 ; the head-tail eigenfunction from Eq. 9 is multiplied with the beam charge profileŜ (t) and corrected for the beam offset ∆x at the BPM where the signal is measured.
The fit error E(ξ, A) is reduced as a function of independent variables; chromaticity ξ 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 ξ s,y = 1.187ξ m,y + 0.804 as shown by red dashed line in Fig. 27 . Figure 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 [22].
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. Figure 28 shows the tune spectrum obtained with sweep excitation for different chromaticity values. The beam parameters are kept the same (N 7+ , 14 · 10 9 , q sc ≈ 10). The spectral position and and relative amplitude of each head-tail mode peak are confirmed using the time domain information (see Fig. 25 ). In

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 [24]. 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 The measurement time required to resolve the various head-tail modes (∆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 ≈ 20 ms (≈ 4000 turns), an improvement of factor ≈ 6 in SNR by averaging 30 spectra. Following the calculations in Sec. V B , q sc 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 2.

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. Figure 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 .

Calculation of bunching factor
where N ≈ 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 ∆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 [23].
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. [23], the relative random uncertainty (std. deviation) in beam profile width(σ x ) measurements is given by Eq. A2 .
where ∆x = 2.1mm is the wire spacing and δN i = 1 2 8 is the ADC resolution of the IPM and B is defined as σ x /∆x. If the error bars are derived from j measurements, the measured profile is given by σ av,x = σ x j , δσ av,x = σ 2 x j − σ x 2 j + δσ x 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 ≈ 5% using Eq. A2 and Eq. A3 . The relative systematic error (bias) in transverse beam width measurements is < 1% [23] 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 ≈ 5% based on 5-8 measurements at the same intensity settings for each measurement point. Bunch length and bunching factor vary by ≈ 2 − 3% due to long term beam losses only under high intensity beam conditions. The maximum relative bias in the lattice parameter β is assumed to be ≈ 5% at the IPM location. Taking all the relative errors, uncertainty propagation using familiar Eq. A4 gives relative error for estimated incoherent tune shifts ≈ 12%.
δε av,x ε av,x = 4( δσ av,x σ av,x ) 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.