Simulations of the fast transverse instability in the SPS

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN – AB DEPARTMENT CERN-AB-2005-088-RF

Simulations of the fast transverse instability in the SPS G. Rumolo∗, V. G. Vaccaro Università di Napoli “Federico II”, Napoli, Italy E. Shaposhnikova CERN Abstract In 2002 and 2003 measurements carried out at the CERN SPS on beams with low longitudinal emittance have revealed, probably for the first time in a proton machine, the crossing of the transverse mode coupling instability (TMCI) threshold far from transition. According to the present understanding, this threshold could be reached due to the increase of the microwave longitudinal instability threshold, as a result of the longitudinal impedance reduction program at the SPS in recent years. Comparing measured thresholds with those found in numerical simulations we estimate the transverse broad-band impedance parameters in 2002, and how they possibly changed in 2003 after the installation of 5 new MKE kickers. The expected variation in 2006, when 4 more MKE kickers will be installed, is extrapolated and a new threshold proposed. Simulations show the important role of the space charge which can on the one hand push the threshold to higher intensity, but on the other hand cause the emittance to blow-up well below this higher value.

Geneva, Switzerland December 15, 2005 ∗

[email protected]

1

Introduction

In the past the transverse mode coupling instability (TMCI) was a serious intensity limitation for a lepton beam in the SPS [1]. For protons this instability was expected for high intensity ppbar bunches [2] and LHC beam, but was never observed before 2002. First explanations of proton bunch stability were based on the difference in bunch length [2]. However for leptons the TMCI was observed not only at high energies during the cycle, where bunch emittances became small due to radiation damping, but also at injection (3.5 GeV/c) where beam parameters were comparable with those of a proton beam [3]. Another possible explanation is that in the past a low microwave instability threshold was leading to uncontrolled emittance blow-up of proton bunch. However microwave instability at that time was also observed for lepton bunches at injection [3]. The possible role of space charge in increasing proton beam stability was suggested in [4] and is also studied in the following. The SPS impedance reduction programme, which practically eliminated longitudinal microwave instability, was completed during shutdown 2000/2001. Measurements carried out in 2002 gave first evidence of fast transverse instability for proton bunches injected into the SPS at 26 GeV/c with small longitudinal and transverse emittances [5]. More detailed studies were performed in 2003 [6]. Direct measurement of mode coupling was not done so far, but observed dependence of losses on chromaticity points clearly in the direction of transverse instability. First analysis of measurements based on analytical estimations, code MOSES [7], and numerical simulations with HEADTAIL [8] gave estimations of transverse SPS impedance necessary to drive this instability, but also revealed a strong dependence of thresholds on chamber geometry and space charge effect [9]. In this paper numerical simulations done with HEADTAIL code [8] for the flat SPS chamber, taking into account space charge effect, are presented for different experimental conditions including mismatched beam parameters at injection.

2

Summary of the recent experimental observations

In 2002 while investigating the possibility of producing pilot and TOTEM bunches for LHC with only minor modifications to the SPS machine settings, a fast transverse instability was encountered at injection to the SPS (26 GeV/c) for intensities higher than N th = 7 × 1010 p/bunch and SPS parameters shown in Table 1 [5]. The measured tunes were 26.185/26.13. The chromaticities were corrected to 0.375/0.3. Momentum compaction factor is 1.92 × 10 −3 . Table 1: SPS parameters and observed thresholds measurement Parameter 2002 2003 set1 set2 set3 RF voltage (200 MHz) MV 2.0 0.6 0.6 1.0 Long. emittance (2σ) z eVs 0.13 0.15 0.3 0.3 Bunch length σz ns 0.7 0.5 0.9 0.8 Norm. r.m.s. emittances x,y µm 2.3/1.0?? 1.0/1.0 1.0/1.0 1.0/1.0 Chromaticities ξx,y 0.375/0.3 ∼0 ∼0 ∼0 Observed threshold Nth 1011 0.7 0.4 1.2 1.0 The bunch length was measured in the PS using a Gaussian fit to bunch profile. The longitudinal emittance was estimated in the PS from the known voltage. The injected bunch was not matched to the bucket and executed quadrupole oscillations at twice the synchrotron frequency. For a low intensity bunch the synchrotron oscillations died out after a few periods due to the bucket nonlinearities. The

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transverse emittance values were measured in the SPS, but could be 30% less than indicated due to a problem with wire-scanners discovered later [5]. A few different sets of measurements of the fast transverse instability done in 2003, also at 26 GeV/c, [6, 9] are available for analysis, see Table 1. Chromaticities ξ x,y were corrected to zero in all cases. The transverse emittance values come from measurements in the PS and do not take into account a blow-up during transfer from the PS to the SPS. The threshold was found to be Nth = (3 − 4) × 1010 for the smallest longitudinal emittance, (2003/set1). Bunches with larger longitudinal emittance are more stable, therefore the instability threshold grew to Nth = 12 × 1010 for the parameters of (2003/set2). The parameters that showed the crossing of the instability threshold for N th = 1011 are given in (2003/set3). The 2003/set1 and 2003/set2 have bunches well matched to their buckets, whereas in 2003/set3 the bunch was mismatched at injection. For the 2003 measurements the transverse impedance is expected to be larger with respect to the previous year because of the 5 MKE kickers installed in the SPS during 2002/2003 shutdown. The vertical impedance of one MKE kicker can be approximated by a broad-band resonator centered at 2 GHz with peak value 0.6 MOhm/m [10]. The contribution of five kickers to the imaginary part of the vertical impedance is then ∼ 3 MOhm/m. The comparison of simulations with measurements can give an estimation of the SPS transverse impedance. Our ultimate question will then be: is this impedance compatible with a threshold above the nominal LHC intensity (1.2 × 10 11 ) for a bunch with σz = 1 ns, z = 0.32 − 0.35 eVs, voltages of 0.75 MV and 2 MV for zero (corrected) chromaticity? Estimations done with code MOSES (round chamber geometry, no space charge) suggest that this threshold could be very close to the nominal LHC intensity [9]. Simulations need to take into account the flat chamber geometry in the SPS as well as the space charge effect, which might not be negligible at injection (26 GeV/c).

3

HEADTAIL simulations

All the simulations presented below have been carried out using the HEADTAIL code [8]. The effect of a broad-band impedance has been considered on the single bunch: the bunch is subdivided into N slices and each slice feels the sum of the wakes from the preceding slices. The wake is constructed from the transverse impedance Zt by applying the corresponding dipolar and quadrupolar kicks in both horizontal and vertical planes, weighed by the appropriate Yokoya coefficients (depending on the aspect ratio of the cross section of the SPS vacuum chamber) [8]. HEADTAIL has been already used for the study of the TMCI threshold in Ref. [9]. For a matched bunch in a round chamber, excluding space charge effects, the results were found to be in excellent agreement with the ones obtained with MOSES code [7] based on numerical solution of the matrix eigenvalue problem. Results reported in [9] also show that flat chamber and space charge can significantly change the TMCI threshold (differences by factors between 2 and 5). To reproduce the experimental observations presented above we always used a flat chamber geometry (as the SPS chamber is indeed flat and the corresponding behaviour of the transverse impedance has been confirmed by all tune shift and growth/decay rate measurements carried out in recent years [11, 12, 13]). A bunch was longitudinally matched or not according to the set of measurements. We also ran two sets of simulations to analyze cases with and without space charge. 3.1 Neglecting space charge The set of measurements in 2002 was simulated using the parameters of Table 1 and a broad-band impedance with Zt = 10 MΩ/m or 15 MΩ/m, fr = 1.3 GHz and Q = 1. The thresholds found in absence of space charge are reported in Table 2.

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Table 2: Simulated TMCI thresholds: space charge not included Nth /1011 measured simulated for Zt year set 10 MΩ/m 15 MΩ/m 20 MΩ/m 2002 0.7 0.7 0.5 2003 set1 0.4 0.6 0.4 2003 set2 1.2 1.2 0.8 2003 set3 1.0 1.0 0.8

The transition from the stable to the unstable regime at intensity ∼ 6 × 10 10 is illustrated in Fig. 1 for the case Zt = 10 MΩ/m. Fig. 2 shows that the bunch was mismatched in this case and executed quadrupole oscillations, which were eventually damped because of the bucket nonlinearity. The longitudinal SPS impedance (ImZ/n 5 Ω), which usually leads to undamped quadrupole oscillations for intensities higher than (3 − 4) × 10 10 , was not included in these simulations. 30

N=5 x 1010 N=7 x 1010

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Figure 1: Vertical centroid evolution below and above threshold for TMCI. Simulations were done for the 2002 set of measurements. The blue curve has an offset to show two traces on one plot distinctly.

The three sets of measurements in 2003 (2003/set1), (2003/set2) and (2003/set3) were simulated using the parameters of Table 2 and a broad-band impedance with Z t = 15 or 20 MΩ/m, fr = 1.3 GHz and Q = 1. In the absence of space charge, the thresholds are summarized in Table 2. 3.2 Including space charge In this section we discuss the results of simulations including space charge effect (transverse). The thresholds in Table 3 are found from simulation scans over the bunch intensities for a broad-band impedance with fr = 1.3 GHz, Q = 1 and Zt = 10, 15 or 20 MΩ/m. The set of measurements in 2002 was simulated using the parameters of Table 1 and results are shown in Table 3. The main difference with the case without space charge is that, even below the instability threshold, 4

0.65 0.6 0.55

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Figure 2: Oscillation of the bunch r.m.s. length due to initial mismatch of the bunch to the bucket (simulations for set 2002)

Table 3: Simulated TMCI thresholds with space charge included Nth /1011 measured simulated for Zt year set 10 MΩ/m 15 MΩ/m 20 MΩ/m 2002 0.7 > 1.9 1.5 1.2 2003 set1 0.4 1.0 0.7 2003 set2 1.2 2.0 1.2 2003 set3 1.0 1.4 1.0

the combined effect of space charge and broad band impedance causes a quick emittance blow-up in both transverse planes, which is more than a factor two for intensities above 5 × 10 10 (see Figs. 3, 4). One could think that the factor three increase in the TMCI threshold is due to the very small transverse emittances used in the simulation, which give initial maximum Laslett tune shifts ∆Q x,y in the 5 4.5 4

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Figure 3: Horizontal (left) and vertical (right) r.m.s. beam size evolution for different values of bunch current and an impedance Zt = 15 MΩ/m. Parameters from the 2002 measurements were used and space charge is included.

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Figure 4: Horizontal (left) and vertical (right) r.m.s. beam size evolution for different values of bunch current and an impedance Zt = 20 MΩ/m. Parameters from the 2002 measurements were used and space charge is included.

range 0.1–0.35 for the bunch currents considered in our scans. If that were true, the TMCI thresholds should approach the values found in the previous section (no space charge) as the transverse emittances increase. However, simulations have shown that for about doubled r.m.s. size values (emittances increased by a factor 4) the TMCI threshold hardly changes. What happens is that for the higher emittance values the blow-up below threshold becomes less significant and only occurs as we approach the TMCI threshold very closely (see Fig. 5). Fig. 6 shows that, when starting from a beam with doubled transverse sizes below the TMCI threshold, the absence of fast blow up in the horizontal plane causes space charge to still affect the bunch even more than in the original case with initial half sizes (because the Laslett tune shift scales like 1/[σx,y (σx + σy )]). More than double values of initial emittances are needed to effectively make space charge weaker and therefore lower the TMCI threshold toward the ideal value when space charge is not taken into account. Space charge gets also suppressed at higher bunch energies, since the Laslett tune shift scales like 1/γ 3 (but the overall effect of an energy change is far more complex because the transverse emittances also change with energy, and the TMCI threshold has a dependence on the slip factor η). Besides, horizontal and vertical planes become coupled through space charge so that the TMCI appears also in x-plane, Fig. 7, while it was seen only in the vertical plane in the absence of space charge (which is correct, since the flat chamber cancels the tune shift in the horizontal plane and therefore makes mode coupling impossible). Three sets of measurements in 2003 (2003/set1) were simulated using the parameters of Table 1 and a broad-band impedance with fr = 1.3 GHz, Q = 1 and Zt = 15, 20 MΩ/m. The thresholds with space charge are shown in Table 3. For the 2003/set1 of measurements there is again a strong emittance blow-up below threshold induced by the combined effect of impedance and space charge, which becomes evident for bunch currents above 4 × 1010 , as shown in Fig. 8. For the 2003/set2, within the current scans we have considered here (0.9-2.0)×10 11 ), the emittance blow-up is always significant and the Laslett tune shift can amount to 0.35 in both planes. For intensities used in simulations of the third set of measurements in 2003 (2003/set3), (0.51.5)×1011 , the emittance blow-up is always significant for Z t = 20 MΩ/m and it becomes also large above (6 − 7) × 1010 for Zt = 15 MΩ/m. The Laslett tune shift can reach 0.3 in both planes.

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Figure 5: Horizontal (left) and vertical (right) r.m.s. beam size evolution for different values of bunch current and an impedance Zt = 15 MΩ/m. Parameters from the 2002 measurements were used and space charge is included. Initial bunch transverse r.m.s. sizes are σx,y = 2.5/3.6 mm.

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Figure 6: Horizontal (left) and vertical (right) r.m.s. beam size evolution for a bunch current of 1.3 × 1011 and an impedance Zt = 15 MΩ/m. Parameters from the 2002 measurements were used and space charge is included. The two cases are shown with initial bunch transverse r.m.s. sizes σx,y = 1.2/1.8 mm and σx,y = 2.5/3.6 mm.

4

Extrapolation of the transverse impedance value

Simulations without space charge would clearly allow us to extrapolate a transverse impedance value in 2002 and define its increment in 2003 after installation of 5 MKE kickers. An impedance value of Zt = 10 MΩ/m seems plausible in 2002, whereas a value of Z t between 15 and 20 MΩ/m seems to fit the observations from 2003. One could then conclude that from 2002 to 2003 the impedance grew by an amount 5 MΩ/m. which is coherent with recent estimations of the transverse impedance introduced by these ferrite elements [10]. Space charge changes the picture. If the measurements were really done with low transverse emittances as given in Table 1, then Table 3 shows that the intensity thresholds for TMCI are significantly higher than those given in Table 2, and somewhat higher values of transverse impedance are to be inferred. The 2002 measurements would require an impedance above 20 MΩ/m to be justified. The 2003 sets of measurements also seem to be compatible with transverse impedance around 20 MΩ/m, except 2003/set1 which requires a value of Zt higher than 20 MΩ/m. From the simulations including space charge the effect of the 5 MKE kickers does not appear clearly. On the other hand, the experimental observations gave evidence of beam loss above the indicated

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Figure 7: Horizontal centroid motion for bunch currents of 1.3 − 1.5 × 1011 and an impedance Zt = 15 MΩ/m. Parameters from the 2002 measurements were used and space charge is included. The two cases are shown with initial bunch transverse r.m.s. sizes σx,y = 1.2/1.8 mm (left) and σx,y = 2.5/3.6 mm (right). The blue curves have an offset to show two traces on one plot distinctly.

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Figure 8: Horizontal (left) and vertical (right) r.m.s. beam size evolution for two different values of bunch current and impedance (as labeled). Parameters from the 2003/set1 measurements were used and space charge is included.

thresholds, which could be due to TMCI induced dipole motion or due to the emittance blow-up even below the TMCI threshold caused by the interplay of impedance and space charge. If the latter were the case, the values of Zt would be smaller and closer to those extrapolated from the simulations not including space charge; simulations show in fact strong emittance blow-up in a strong space charge regime (i.e. for very low transverse emittances) at about the same intensities where TMCI would set in if space charge was absent. The exact reason for emittance blow-up in space charge regime should clarify the answer. However, simulations with different chromaticity values have shown that, even if the horizontal emittance blow up is reduced by chromaticity, the larger effect in the vertical plane (where, moreover, the aperture is smaller) is not cured. Since in the measurements a decrease of losses was observed for higher chromaticities [6], the simulated behavior with chromaticity seems to point to TMCI rather than to the emittance blow-up below threshold as responsible for the observed losses. Certainly, more precise information on the transverse beam sizes during the measurements would be the key for a deeper experimental understanding and would greatly help for better modeling.

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4.1 Stability of the LHC-type bunch in the SPS The parameters in Table 4 were used to simulate the case of the LHC-type bunch in the SPS. Simulations were done for three values of the ring transverse impedance Zt (15, 20 and 23 MΩ/m) extrapolated from the 2002 and 2003 measurements. We considered regimes with and without space charge, which can simulate the cases at low and high energy and/or different transverse emittances. The 1/γ 3 scaling law for the Laslett tune shift means that the effect of space charge is reduced by a factor 2 × 10−4 after acceleration to the top energy and can be completely neglected. Table 4: Nominal LHC beam parameters in the SPS Parameter Momentum GeV/c 26 450 11 Nominal intensity 10 1.2 1.1 Ultimate intensity 10 11 1.9 1.7 Longitudinal emittance (2σ) z eVs 0.35 0.6 Bunch length (1σ) σz ns 1.05 0.4 RF voltage V MV 2.0 7.0 Norm. r.m.s. emittances x,y µm 3.0/3.0 < 3.5/3.5 Chromaticities ξx,y corrected corrected

with space charge, V=0.7 MV with space charge, V=2 MV w/o space charge, V=0.7 MV w/o space charge, V=2 MV

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Figure 9: TMCI thresholds for different impedance and capturing voltage values. Simulations were carried out with and without space charge. The lines 1.1 × 1011 and 1.9 × 1011 highlight the nominal and ultimate values of the bunch current in the SPS at injection energy. Thresholds are shown here for zero chromaticity in both planes.

Simulations of the LHC-type bunch in the SPS at 26 GeV/c have delivered the results shown in Fig. 9 in terms of expected TMCI thresholds. Without the stabilization introduced by space charge, the LHC-type bunch in the SPS would be unstable for impedance with Z t > 15 MΩ/m, when captured in a 0.7 MV voltage. In a 2 MV voltage bucket, too, thresholds for the TMCI are rather low, and impedances with Zt > 17 MΩ/m are not tolerable. Space charge has a stabilizing effect and raises the TMCI threshold. Figure 9 shows that the threshold lies above the nominal intensity in most of the simulated cases, except Z t = 23 MΩ/m and a capture 9

voltage of 0.7 MV. However taking into account that the extrapolation discussed in the previous subsection suggests impedance values above 20 MΩ/m, thresholds in space charge regime also seem quite low. Furthermore, the emittance blow-up below the TMCI threshold, which appears to be worse for lower initial emittances (see Figs. 10 and 11), can set in for values below the nominal intensity and would not be tolerable for the required beam quality. At top energy, simulations show that space charge does not significantly influence the beam dynamics (as was expected) and the bunch (with longitudinal emittance artificially blown up to 0.6 eVs, see Table 4) keeps stable for impedance values up to 23 MΩ/m and intensities even higher than the ultimate intensity. A scan in longitudinal emittances from 0.2 to 0.6 eVs shows that the TMCI threshold for the LHC bunch in the SPS (bunch always matched to its bucket and Z t = 23 MΩ/m) reduces to 1.5 × 1011 p/bunch when z = 0.2 eVs, but is 1.9 × 1011 p/bunch when z = 0.3 eVs. 8 7 6

σx0=1.8mm, N=0.5 x 1011 σx0=1.8mm, N=0.7 x 1011 σx0=1.8mm, N=0.9 x 1011 11 σx0=1.8mm, N=1.1 x 10 σx0=1.8mm, N=1.3 x 1011

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Figure 10: Horizontal (left) and vertical (right) r.m.s. beam size evolution for a current scan below TMCI threshold, impedance Zt = 15 MΩ/m. Parameters of the LHC-type bunch were used and space charge is included.

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Figure 11: Horizontal (left) and vertical (right) r.m.s. beam size evolution for a current scan below TMCI threshold, impedance Zt = 15 MΩ/m. Parameters of the LHC-type bunch were used and space charge is included. The data collected in this report certainly suggest that the impedance increase in the SPS caused by the installation of 4 new MKE kickers would be not tolerable and consequently an impedance reduction program should be pursued in order to ensure the stability of the LHC bunch. In principle these thresholds can be raised through operation with high positive chromaticity in both x and y planes. Chromaticity scans with MOSES and HEADTAIL have confirmed the beneficial effect of chromaticity on the TMCI threshold [9]. High chromaticity is already used to cure instability caused by e-cloud, however, at the same time, in conjunction with high voltage, it leads to continuous beam losses on the flat bottom. 10

5

Conclusions

The observation of the TMCI threshold crossing in SPS measurements allowed us to extrapolate values for the SPS transverse impedance to be in the range (10–20) MΩ/m in 2002 and some 5 MΩ/m more in 2003, after the installation of 5 MKE kickers. These values are compatible with the impedance measurements based on tune shift with intensity and growth/decay rates reported in Refs. [11, 12, 13]. The beam transverse sizes turn out to be critical because, while they do not directly influence the transverse mode coupling dynamics, they change the role of space charge in the TMCI threshold or emittance blow-up below threshold (which is completely absent without space charge). Consequently, the precise knowledge of these values during the 2002/2003 measurements would have permitted the impedance value to be determined with less uncertainty. Applying impedance values in the range 15–23 MΩ/m (eligible values after the installation in the SPS of 4 more MKE kickers) to the LHC-type bunch with variable intensity, we found that: at injection energy, where space charge still plays a role, for impedance larger than ≈20 MΩ/m the TMCI threshold is below the nominal intensity for 0.7 MV capture voltage and below ultimate intensity (1.7 × 10 11 ) for 2 MV. In addition there could be a significant fast emittance blow-up below these thresholds, which would be intolerable for LHC or even cause beam loss. Thresholds could be increased operating the SPS with high positive chromaticity. However, the emittance blow up in the vertical plane is still relevant. The low thresholds evaluated in this report for plausible SPS impedance values in 2006 certainly show that serious measures should be taken in order to avoid TMC instability for the LHC beam. Coating of the MKE kickers can be envisaged as a possible solution [14, 15]. Ceramic tubes coated inside with a conductive layer or direct painting on the ferrite, in such a way that the coating thickness allows the kicker field to go through (frequencies of tens of MHz, for the required rise-fall times of the kickers) and the higher frequency modes to be shielded instead, are viable solutions to keep the SPS broad-band impedance within an acceptable range of values.

Acknowledgments The authors would like to thank G. Arduini, E. Benedetto, H. Burkhardt, T. Linnecar, E. Métral, and F. Zimmermann for discussion and exchange of information.

References [1] D. Brandt, J. Gareyte, “Fast instability of positron bunches in the CERN SPS”, CERN-SPS-88-17 (AMS). [2] Y. H. Chin, “Transverse mode coupling instabilities in the SPS”, CERN-SPS-86-2 (DI-MST). [3] T. Linnecar, E. Shaposhnikova, “Analysis of the transverse mode coupling instability of the leptons in the SPS”, CERN/SL/93-43 (RFS). [4] M. Blaskiewicz, “Effect of incoherent space charge forces on the fast head-tail instability”, in Proc. of PAC 1997, Vancouver, Canada. [5] G. Arduini and A. Verdier, “Poor man pilot and TOTEM bunches”, CERN AB-Note-2003-017 (MD). [6] G. Arduini, H. Burkhardt and E. Métral, “Observation of a fast single-bunch transverse instability on protons in the SPS”, CERN AB-Note-2003-093 (MD). [7] Y. H. Chin, “User’s Guide for New MOSES. Version 2.0”, CERN/LEP-TH/88-05. [8] G. Rumolo, F. Zimmermann, Phys. Rev. ST Accel. Beams 5, 121002 (2002). 11

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