Beam optics for the compensation of third-order structure resonances Open Access

Abstract

The third-order structure resonance due to sextupole magnetic fields was compensated in the Japan Proton Accelerator Research Complex main ring synchrotron using new beam optics. In the present optics, the third-order structure resonance ν_x − 2ν_y = −21 was so strong that the working point was limited. The new optics cancels the resonance by adjusting the phase advances in the arc sections. The resonance compensation was verified by aperture survey simulations and further demonstrated by three types of experiment. First, transverse coupling measurements were performed. The coupling was observed only for the optics without the compensation. This was also verified with the tracking simulation. Second, beam losses were measured using the current and new optics. The measured beam losses were noticeably reduced at the tunes around the resonance using the new optics. The beam loss reduction was also verified with the space charge tracking simulation. Third, Fourier analyses of the transverse dipole oscillations were conducted for both optics. Fourier spectra derived from the resonances were observed, and their peaks were clearly suppressed with the new optics. This was also consistent with the results of the space charge simulations.

1. Introduction

In circular accelerators, low-order resonances limit the available tune region. In particular, third-order resonances driven by the sextupole magnetic fields exist in nearly every synchrotron. In large synchrotrons that use sextupole magnets to correct the chromaticity, the magnets can cause third-order resonances. Even in synchrotrons that do not use sextupole magnets, the sextupole field components of the bending magnets can drive third-order resonances.

Methods for suppressing third-order resonances have been studied extensively. There are a number of studies reporting the compensation of nonstructure resonances, such as Refs. [1–4]. They compensated for the third-order nonstructure resonances by applying additional sextupole magnetic fields or by changing the strengths of the sextupole magnets. Unlike nonstructure resonances, there is a small number of references for the compensation of structure resonances. In many cases, structure resonances are too strong to be compensated using corrector sextupole magnets. One example for suppressing the effect of the third-order structure resonance was proposed in Ref. [5]. They found that the emittance growth by the ν_x − 2ν_y = n resonance, known as the Walkinshaw resonance [6], can be minimized by setting the transverse emittance according to 2ε_x = ε_y.

In this study we propose a new method to compensate for the third-order structure resonance [7]. Our technique does not require additional corrector sextupole magnets and does not impose constraints on the beam distribution. The requirement for our compensating method is symmetry of the arc section of the synchrotron. Figure 1 shows a schematic of the main ring synchrotron (MR) of the Japan Proton Accelerator Research Complex (J-PARC) [8], where we demonstrated the effectiveness of the compensation method [7]. In the J-PARC MR, one arc section is composed of eight symmetric structures. Our compensation method cancels the third-order structure resonance by properly adjusting the phase advance in the arc sections. The tune can be maintained by adjusting the straight sections. The detailed theory is described in the next section.

The J-PARC MR is a proton synchrotron with three-fold symmetry of the focusing lattice delivering 30 GeV protons for neutrino and hadron experiments. At present, the average beam power is 515 kW for neutrino and 65 kW for hadron experiments. The beams are injected with a kinetic energy of 3 GeV and an intensity of 3.3 × 10¹³ protons per bunch (ppb). In total, eight bunches are injected within 130 ms and then the beams are accelerated to 30 GeV. In the present neutrino user operation conditions, the tune is set to (ν_x, ν_y) = (21.35, 21.43) [9,10]. The relevant parameters for neutrino user operation are presented in Table 1.

The beam power is planned to be upgraded to 1.3 MW for neutrino and to 100 kW for hadron experiments [4,11]. The upgrade plan assumes a hardware upgrade for faster cycling and higher intensity, and relies on lower beam losses in operation. Beam loss reduction is a general concern in high-intensity proton and ion accelerators, such as the Fermilab Booster and Main Injector [12–15], CERN PSB, PS, and SPS [16–20], AGS at BNL [21], ISIS at RAL [22], SIS18 at GSI [23], and CSNS/RCS [24]. The beam loss in the MR is strongly related to space charge effects [4,9^,–11]. Most of the beam losses can be observed in the first 300 ms, where the beam energy is below 5 GeV. In this period, the space charge effect is strongest, resulting in a large tune spread Δν_SC ≃ −0.34.¹ The particle-in-cell tracking simulation software Space Charge Tracker (SCTR) [25,26] indicates that the protons are lost because of resonance crossings [9,–,11].

In the J-PARC MR there are seven kinds of quadrupole electromagnets in the straight sections and four kinds in the arc sections. The same kinds of magnets have the same structures, and their magnet currents are from the same power supply. The quadrupole magnets in the arc sections are carefully set to realize an achromat lattice, and their currents are rarely changed after adjustment. The fine tuning of the working point is performed only by the quadrupole magnets in the straight sections.

Figure 2 shows the result of an aperture survey simulation near the working point in neutrino user operation. The initial actions of the particles which did not hit the aperture were investigated by tracking simulations. The strengths of the resonances can be seen. Since the MR has three-fold symmetry, the third-order structure resonances appear at ν_x = 21, ν_y = 21, ν_x − 2ν_y = −21, and ν_x + 2ν_y = 63 in Fig. 2. While the integer resonances are excited by several kinds of effects [27], the ν_x − 2ν_y = −21 and ν_x + 2ν_y = 63 resonances are induced by the sextupole magnetic fields in the sextupole and bending magnets. These resonances appear even if there are no imperfections of the magnets. Some resonances emerge when there are magnet imperfections, known as nonstructure resonances. For example, the third-order nonstructure resonances 3ν_x = 64 and ν_x + 2ν_y = 64 can be seen in Fig. 2. Although the magnet imperfections also affect the ν_x − 2ν_y = −21 and ν_x + 2ν_y = 63 resonances, their effects are almost hidden because the structured effects are much stronger.

From the next section, the compensation of third-order structure resonances is discussed. In Sect. 2, the theory of the compensation method is described. From Sect. 3, the application to the J-PARC MR is discussed. In this paper we focus on the third-order structure resonance ν_x − 2ν_y = −21. In Sect. 3, we verify the compensation by aperture survey simulations. The compensation method is demonstrated by three types of measurement: via transverse coupling measurements in Sect. 4, via beam loss measurements in Sect. 5, and via the Fourier spectra of dipole oscillations in Sect. 6. The discussion is concluded in Sect. 7.

2. Principle of compensation

We suppose that the sextupole magnets are located in the arc sections. Since the bending magnets are also located in the arc section, the resonance driving terms of the third-order structure resonances arise in the arc sections.

Figure 3 shows images of the compensation of G_{1, −2, l} in the case of M = 8. The left image shows the resonance driving terms without the compensation, assuming the present optics for neutrino user operation. The phase advance per module is Δψ_{mod, x} − 2Δψ_{mod, y} = −2π(11/16), which does not satisfy Eq. (14). As a result, each resonance driving term per module, represented by a black vector, points in a different direction. The resonance driving term in one arc section appears as the sum of those, indicated by the red vector. The case of the compensation optics is shown on the right. In this case we assume Δψ_{mod, x} − 2Δψ_{mod, y} = −2π(5/8), which corresponds to (j, M) = (−5, 8) in Eq. (14). All the black vectors have the same norm, and their directions are now isotropic. As a result, the sum of the vectors becomes zero and the resonance is compensated.

3. New optics for the J-PARC MR

As shown in Fig. 1, the number of superperiods in the MR is N = 3, and the number of modules in one superperiod is M = 8. All sextupole magnets are located in the arc sections.

Since the number of modules in one superperiod is M = 8, the horizontal phase advance of one arc section is Δψ_{arc, x} = 8Δψ_{mod, x} = 12π in the present optics. The horizontal phase advance Δψ_{arc, x} was carefully adjusted such that the arc section became achromat. The optics was verified by measurements of the betatron functions and dispersions [29,30]. Moreover, the present phase advance satisfies the condition for compensation of the resonance driving term G_{3, 0, n} given in Eq. (13). Therefore, the horizontal phase advance Δψ_{mod, x} should not be changed. On the other hand, there were no restrictions for the vertical phase advance Δψ_{mod, y}.

The betatron functions of the present and new optics are shown in Fig. 4, and are not significantly different. This means that the physical apertures do not change dramatically in the action coordinates. The betatron functions and the dispersion function were also measured with the new optics, as described in Appendix  B.

The aperture survey simulation was also performed with the new optics to verify the compensation. The tune was changed by maintaining the phase advance in the arc sections. In other words, the tune was adjusted only by the quadrupole magnets in the straight sections. The strengths of the sextupole magnets were fixed. The resonance driving terms must not have changed during the tune scan, because the phase advance in the arc sections and the strengths of the sextupole magnet fields were fixed.

Figure 5 shows the results of the aperture survey simulation with the new arc optics. Compared with the results of the present arc optics shown in Fig. 2, it is obvious that the ν_x − 2ν_y = −21 resonance was compensated. The values used in the aperture survey simulation are listed in Table 2.

According to Eq. (13), the new arc optics compensate not only the resonance driving term G_{1, −2, n} but also G_{1, 2, n}. The corresponding resonance shown in Figs. 2 and 5 is ν_x + 2ν_y = 63. Comparing the two results, it can be verified that the ν_x + 2ν_y = 63 resonance was also compensated.

In the result of the aperture survey simulation for the new arc optics, the ν_x − 2ν_y = −21 resonance can be seen slightly. This is because it was driven by imperfections of the magnets. In other words, the ν_x − 2ν_y = −21 resonance existed as a “nonstructure” resonance.

4. Verification of the compensation by coupling measurements

The compensation for the third-order structure resonance ν_x − 2ν_y = −21 was verified not only by the aperture survey simulations but also using three types of experiment. First, the transverse coupling was measured. The ν_x − 2ν_y = −21 resonance couples horizontal and vertical motions of the beam.

To observe the transverse coupling the turn-by-turn transverse beam positions were measured. The beams were intentionally injected with dipole errors in the horizontal plane. While in the vertical plane, the beams were injected without dipole errors. The tune was set at (ν_x, ν_y) = (21.44, 21.22), where the beam was on the ν_x − 2ν_y = −21 resonance. If the beam is affected by the ν_x − 2ν_y = −21 resonance, the horizontal dipole oscillation propagates to the vertical plane.

4.1. Simulation prediction

Figure 6 shows the results of the turn-by-turn actions obtained from the simulations. The vertical action became larger in the present arc optics, although it was initially zero. This is because the transverse dipole motions were coupled, and the horizontal amplitude propagated to the vertical plane. In the new arc optics, no noticeable dipole oscillation was seen in the vertical plane. This indicates that the ν_x − 2ν_y = −21 resonance was compensated for in the new arc optics. The oscillations were damped by the effects of the chromatic tune spread and space charge.

The tune dependence of the coupling was also studied. Figure 7 shows the turn-by-turn vertical actions in the present arc optics. The different colors correspond to the different tunes. In the top figure, the vertical tunes are fixed at ν_y = 21.22. It indicates that the coupling may not be measured if the horizontal tune is shifted by 0.006. In the MR, the tune itself fluctuates (see Appendix  B), and the typical fluctuation amplitude is 0.005. It can be predicted that the turn-by-turn motions will differ shot by shot. The bottom figure shows the vertical tune dependence. In these simulations, the horizontal tunes were fixed at ν_x = 21.44. The results indicate that the vertical tune is more sensitive to the coupling than the horizontal tune. This can be attributed to the resonance proximity parameter δ ≡ ν_x − 2ν_y + 21 [28]. The vertical tune ν_y is twice as effective as the horizontal tune ν_x on δ.

4.2. Coupling measurements

The coupling was also observed in the measurements. The turn-by-turn transverse beam positions were measured by the beam position monitors (BPMs) [31–33]. In the same way as the simulations, the beams were injected with dipole errors in the horizontal plane, and the vertical dipole errors at the beam injection were carefully corrected. The tune was set at (ν_x, ν_y) = (21.44, 21.22). Low-intensity beams were used to suppress the space charge effects.

Figure 8 shows the results of the turn-by-turn beam positions using the present arc optics. It is clear that the horizontal and vertical dipole oscillations were coupled. On the other hand, Fig. 9 shows the results using the new arc optics. The amplitude of the vertical dipole oscillation was maintained at approximately 1 mm. No clear coupling was seen.

In these experiments, as predicted, the dipole oscillations varied shot by shot. Even when using the present arc optics, the couplings were not seen in some results. This was because the dipole oscillations were very sensitive to the tune, as indicated by the simulations. The shot-by-shot differences can be explained by the fluctuations in the tunes.

To cope with the shot-by-shot differences, the dipole oscillations were measured many times under the same conditions. We judged whether the coupling occurred by checking that the amplitude of the vertical position was greater than 3 mm. Since the betatron function at the BPM used was β_y = 26.5 m in the new arc optics, the 3 mm amplitude corresponds to the action of |$2J_y = 0.34 \, \mu$|m.

Table 4 shows the results of the coupling measurements. We defined the “coupled event ratio” as the ratio of the number of coupled events to the total number of events. In the new arc optics, coupling was not observed in any measurements. It is clear that the ν_x − 2ν_y = −21 resonance was compensated in the new arc optics.

Figure 10 shows the actions 2J_x, 2J_y as a function of the turn in the present arc optics. The data shown in Fig. 10 are the same as in Fig. 8. It is clear that the horizontal and vertical actions were coupled with each other.

Ideally, the value J_c = 2J_x + J_y is constant, but this equation neglects the damping effects of the dipole oscillations. As can be seen in Figs. 8 and 10, the dipole oscillations were damped in the real measurements. This indicates that the value J_c was also damped. However, since the speed of the damping was not very fast, J_c can be considered constant over a short time scale. Figure 11 shows the relationship between the transverse actions. The plots are colored according to the number of turns. The motions of turn numbers 160–309, shown as blue circles, were on the line |$2J_x + J_y = 1.7 \, \mu$|m. Similarly, the motions of turn numbers 310–424, shown as sky blue circles, were on the line |$2J_x + J_y = 1.2 \, \mu$|m. The motions of turn numbers 425–494, shown as green circles, were on the line |$2J_x + J_y = 0.75 \, \mu$|m. Thus, the J_c = 2J_x + J_y value was constant over a short time range. This reveals that the source of the coupling is the ν_x − 2ν_y = −21 resonance.

5. Verification of compensation by beam loss measurements

The second experiment for the verification of the compensation for the third-order structure resonance ν_x − 2ν_y = −21 involved beam loss measurements. If the ν_x − 2ν_y = −21 resonance is compensated, the beam loss due to the resonance is reduced. We compared the beam losses of the present and new arc optics. The tune was scanned across the ν_x − 2ν_y = −21 resonance to confirm that the resonance caused the beam loss.

Figure 12 shows the major resonances and the scanned range, which was ν_y = 21.20–21.29. The horizontal tune was fixed at ν_x = 21.44. Other resonances can also cause beam losses. A significant resonance was the third-order nonstructure resonance 3ν_x = 64, because it could affect many particles considering the space charge tune shift. To suppress its effect, the two nonstructure resonances 3ν_x = 64 and ν_x + 2ν_y = 64 were compensated by using the trim coils of the sextupole magnets [4,9]. The reason for fixing the horizontal tune is to consider the residual nonstructure resonance. Even if a residual nonstructure resonance 3ν_x = 64 arises, its effects do not change during the scan.

5.1. Experiments

Table 5 lists the conditions used in the experiments. The collimator aperture was set to be small to observe the beam losses clearly. The position of the collimator, determined based on the betatron functions at the vertical tune ν_y = 21.24, was fixed during the tune scan. The setting values were calculated by using the Strategic Accelerator Design (SAD) [34] computer program. The betatron functions at the position of the collimator calculated by SAD were (β_x, β_y) = (20.79, 10.06) m for the present arc optics and (β_x, β_y) = (23.30, 8.08) m for the new arc optics. The maximum beta deviations resulting from the change of the optics during the tune scan were |$(|\Delta \beta _x|/\beta _x, |\Delta \beta _y|/\beta _y) = (0.8\%, 2.4\%)$| for the present arc optics and |$(|\Delta \beta _x|/\beta _x, |\Delta \beta _y|/\beta _y) = (0.4\%, 1.1\%)$| for the new arc optics.

A direct-current current transformer (DCCT) [31,32,35,36] was used to obtain the beam losses. The beam current can be measured by the DCCT in the MR with an accuracy of 1% [35] and a relative deviation of 0.1% [36]. The time resolution is about 100 μs and is short enough compared to the circulating time of 130 ms.

The experiments were conducted with relatively high-intensity beams to observe clear responses of beam losses. The low-intensity beams were not lost because their emittances were small. In the MR, the transverse emittance at the beam injection was nearly linear with respect to the beam intensity [11]. The beam intensity was scanned and set to 2.3 × 10¹³ ppb because the beam loss with the present arc optics was about 10% which was larger than the DCCT accuracy. The chromaticity setting was the same as for neutrino user operation. The distortions of the closed orbit were less than 1 mm at every point. The injection beta matching was performed by scanning the quadrupole magnets in the beam transport line.

Figure 13 shows the results of the beam loss ratios, defined as the ratio of the decrease of the beam current in 130 ms measured by the DCCT. The tunes were measured by the tune meter [31,32] using weak-intensity beams (2.5 × 10¹² ppb). In the results for the present arc optics, the beam loss reached its peak at ν_y = 21.24–21.25. For the new arc optics, the peak was suppressed compared to the present arc optics. These show that the ν_x − 2ν_y = −21 resonance was successfully weakened.

Although the on-resonance condition was ν_y = 21.22, the peak was at ν_y = 21.24–21.25. This was due to the space charge detuning. Since the horizontal and vertical emittances are nearly the same in the MR, the peak of the loss was shifted above the resonance.

Ideally, the peak due to the ν_x − 2ν_y = −21 resonance is flattened in the new arc optics. The small peak in the new arc optics was derived from imperfections of the magnets. This indicates that the structure resonance was compensated, but the ν_x − 2ν_y = −21 resonance remained as a nonstructure resonance. This can be compensated by using the trim coils of the sextupole magnets, if needed.

5.2. Simulations

Resonance compensation using the beam loss scan was also performed using SCTR simulations. In these simulations, the particles were lost when their actions became larger than the aperture. The conditions were set to be the same as those in the experiments; they are listed in Table 6.

In the experiments, there were systematic shifts due to the distortions of the closed orbits at the collimator. Using the BPMs near the collimator, the distortions were estimated as (Δx, Δy) = (0.5, 0.1) mm for the present arc optics, and (Δx, Δy) = (0.7, 0.8) mm for the new arc optics. The distortion decreased the effective aperture size. The effective aperture was estimated as (H,V) = (38.6π, 39.6π) mm mrad for the present arc optics, and (H,V) = (38.2π, 36.5π) mm mrad for the new arc optics. The distortions of the closed orbits were slightly different shot by shot. In addition, the collimator position had certain errors. The error in the distance between the beam and the collimator was assumed to be 0.5 mm. This corresponds to aperture errors of (H,V) = (2.7π, 4.0π) mm mrad for the present arc optics, and (H,V) = (2.6π, 4.2π) mm mrad for the new arc optics.

The beam was tracked across 24 200 turns, which corresponded to the circulating time of the beam (130 ms) in the experiments. The initial transverse emittance was the typical value for a beam with an intensity of 2.3 × 10¹³ ppb. The initial longitudinal distribution, the bunching factor, and the momentum spread were determined using the data from the wall current monitor [37].

Figure 14 shows the results of the simulations. The error bars show the systematic errors due to the displacement of the collimator. Comparing the two kinds of optics, the peak due to the ν_x − 2ν_y = −21 resonance was suppressed in the new arc optics. This shows that the resonance was weakened, and is consistent with the experiments. In both optics, the peaks were above the on-resonance condition ν_y = 21.22. This is also consistent with the experimental results.

The intensity dependence of the beam losses were also simulated to confirm that the shift of the peak tune was due to the space charge effects. Figure 15 shows the results of the beam losses with three kinds of beam intensities using the present arc optics. The peak was approximately at ν_y = 21.25 for the 2.3 × 10¹³ ppb beam, at ν_y = 21.24 for the 1.2 × 10¹³ ppb beam, and at ν_y = 21.23 for the 0.6 × 10¹³ ppb beam. The peak became close to the on-resonance condition ν_y = 21.22 as the beam intensity became weak. This supports that the shift of the peak was due to the space charge effects.

6. Verification of compensation by frequency analyses

The third set of experiments for the verification of the resonance compensation were Fourier analyses of the turn-by-turn positions of the beams. Fourier analysis is often used to obtain the tune of the optics. If the beam is affected by a resonance, the spectral line driven by the resonance can also be observed [38,39]. We compared the results of Fourier analyses for the present arc optics and for the new arc optics.

6.1. Simulation approach

First, the spectra of the resonances were analyzed using SCTR simulations. The tune was set at (ν_x, ν_y) = (21.44, 21.24) to observe the spectrum of the ν_x − 2ν_y = −21 resonance. At this tune, the spectra of the ν_x − 2ν_y = −21 resonance appear at (f_x, f_y) = (0.48, 0.20) in the units of the tune. The on-resonance condition was avoided because the spectra of the ν_x − 2ν_y = −21 resonance and the spectra of the tune would appear at the same frequencies.

The beams were injected with dipole errors in both the horizontal and vertical planes. The turn-by-turn transverse beam positions were acquired. Fourier amplitudes were calculated using numerical analysis of fundamental frequency (NAFF) [40] with Hanning windows. The number of turns used for the analyses was 512. The simulations were performed with both the present and new arc optics. For each, two kinds of simulation were conducted: one excluding the imperfections of the magnets, and another including these effects. The spectra appearing in the simulations without errors correspond to the structure resonances. On the other hand, all possible spectra appeared in the simulations with errors. The conditions used in the simulations are listed in Table 8.

The images on the left of Fig. 16 show the results of the simulations using the present arc optics. In both the horizontal and vertical planes, the largest peaks corresponded to the tunes f_x = 0.44 in the horizontal plane and f_y = 0.24 in the vertical plane. The second largest peak in the horizontal plane appeared at f_x = 0.48. This matched the frequency where the ν_x − 2ν_y = −21 resonance should appear. In the vertical plane, the second largest peak appeared at f_y = 0.20. This also matched the frequency of the spectrum of the ν_x − 2ν_y = −21 resonance.

As can be seen in Table 7, the spectrum at f_x = 0.48 in the horizontal plane can be induced by the resonances ν_x + 2ν_y = n (jkml = 1020) and ν_x − 2ν_y = n (jkml = 1002). A candidate for the former is ν_x + 2ν_y = 63. However, it is too far from the working point. Thus, the peak at f_x = 0.48 was excited by the ν_x − 2ν_y = −21 resonance. The vertical peak f_y = 0.20 can be excited by the resonances ν_x = n (jkml = 1011) and ν_x − 2ν_y = n (jkml = 0120). A candidate for the former is ν_x = 21, but this resonance is too far as well. Consequently, it can be verified that the peak at f_y = 0.20 was excited by the ν_x − 2ν_y = −21 resonance.

The images on the right of Fig. 16 show the results of the simulations using the new arc optics. Comparing the results with those of the present arc optics, it is obvious that the peaks at f_x = 0.48 in the horizontal plane and f_y = 0.20 in the vertical plane disappeared in the simulations without errors. This clearly shows that the ν_x − 2ν_y = −21 resonance was compensated in the new arc optics. In the results including errors, there were small peaks at f_x = 0.48 and f_y = 0.20 even in the new arc optics. This was because the ν_x − 2ν_y = −21 resonance was excited by the effects of magnet imperfections. The ν_x − 2ν_y = −21 resonance affected the beam as a nonstructure resonance. However, the peak height was still smaller than in the present arc optics. This is the difference between the structure and nonstructure resonance.

Let us discuss another peak. In the horizontal results, peaks appeared at f_x = 0.24, only in the simulations with errors, for both the present and new arc optics. Since the spectra did not appear in the simulation without errors, the peak was due to the magnet imperfections. Based on Table 7, the possible sources are the resonances ν_x + ν_y = n and ν_x − ν_y = n. Considering the distance from the working point, the corresponding resonance is ν_x − ν_y = 0. The main source is the skew quadrupole magnetic field due to the rotational errors of the quadrupole magnets. Although it was not implemented in the simulations, the rotations of the BPMs can cause the spectra in the experiments. Since the BPM rotation is 0.96 ± 3.3 mrad in the MR [31], the horizontal peaks can appear in the vertical plane with a factor of 0.004 and vice versa.

According to the tables, the vertical peak due to the ν_x − ν_y = 0 resonance must appear at f_y = 0.44. In fact, clear vertical peaks can be seen at f_y = 0.44 in both the present and new arc optics, including errors. Similarly, the vertical peak f_y = 0.44 is derived from the skew quadrupole magnetic field due to the rotational errors of the quadrupole magnets.

6.2. Experiments

The resonant spectra were also observed in experiments. The beams were intentionally injected with dipole errors in both the horizontal and vertical planes. The turn-by-turn transverse beam positions were acquired by the BPMs. The data where the betatron function was high were used for the analyses to obtain clear signals. The first 256 turns were used for the analyses. Though the maximum turn number of the data was approximately 2300, the spectra obtained by 256 or 512 turns were clearer than those obtained by 128 or 1024 turns because the dipole oscillations were damped. The set tune was (ν_x, ν_y) = (21.44, 21.24), which was the same as for the simulations. The spectra of the present and new arc optics were compared. The conditions used in the experiments were the same as in the simulation, and are listed in Table 8.

The images on the left side of Fig. 17 show the measured Fourier spectra with the present arc optics. The largest peaks, shown in blue (horizontal) and red (vertical), corresponded to the tunes. In the data, the tune was |$(f_x, f_y) = (0.436\, 736, 0.237\, 540)$|⁠. The frequencies of the second highest peaks, shown in green, were |$(f_x, f_y) = (0.474\, 752, 0.199\, 253)$|⁠. This matched (2ν_y, ν_x − ν_y) with an accuracy of 0.0003. Thus, the green peaks show the effects of the ν_x − 2ν_y = −21 resonance.

The spectra derived from the differential resonance ν_x − ν_y = 0 were also observed, and are colored violet. The peak position was |$(f_x, f_y) = (0.237\, 595,0.436\, 287)$|⁠, which matched (ν_y, ν_x) with an accuracy of 0.0004.

The images on the right of Fig. 17 show the results with the new arc optics. The highest peaks, colored blue (horizontal) and red (vertical), represent the tunes. The positions of the peaks were |$(f_x, f_y) = (0.443\, 970, 0.237\, 501)$|⁠. If there had been effects of the ν_x − 2ν_y = −21 resonance, resonant spectra would appear at (f_x, f_y) = (2ν_y, ν_x − ν_y). The nearest peaks are colored green in the right images of Fig. 17. The positions of the peaks were |$(f_x, f_y) = (0.470\, 968, 0.206\, 084)$|⁠. Though the vertical peak position matched ν_x − ν_y with an accuracy of 0.0004, the horizontal peak position differed from 2ν_y by 0.004. This indicates that the horizontal peak around f_x = 0.47 might be one of the noise peaks. In fact, its height was nearly at the noise level. The true horizontal peak was so small that it was hidden by other noise. Both the horizontal and vertical peak heights for the new arc optics were smaller than those for the present arc optics. Compared to the peaks of the tunes, the heights of the peaks of the ν_x − 2ν_y = −21 resonance were |$(x,y) = (3.26\%, 10.14\%)$| for the present arc optics. Meanwhile, the peak heights of the resonance were |$(x,y) = (0.95\%, 1.35\%)$| for the new arc optics. By using the new arc optics, the peaks of the ν_x − 2ν_y = −21 resonance were reduced to |$0.95\%/3.26\% = 29\%$| for the horizontal plane and |$1.35\%/10.14\% = 13\%$| for the vertical plane. The horizontal peak for the new arc optics could be smaller because the measured peak might be noise. These results clearly show that the ν_x − 2ν_y = −21 resonance was successfully weakened.

7. Conclusions

In this paper, a new method to compensate for the strength of the sextupole-driven third-order structure resonance was proposed, verified by simulations, and demonstrated in beam experiments at the J-PARC MR. New optics have been developed that make use of the MR synchrotron symmetry and properly adjust the vertical phase advance in the arc section Δψ_{arc, y} to cancel the resonance.

The compensation was verified by aperture survey simulations. It was demonstrated that the third-order structure resonance ν_x − 2ν_y = −21 (and the ν_x + 2ν_y = 63 resonance) were clearly compensated.

The compensation was also verified via three measurements. First, transverse couplings were measured. The beams were set on the ν_x − 2ν_y = −21 resonance and injected with dipole errors only in the horizontal plane. The turn-by-turn beam positions were measured to observe the propagation of the dipole oscillations from the horizontal to the vertical plane. The transverse beam couplings were observed only in the present arc optics. The measurements showed that the ν_x − 2ν_y = −21 resonance was well compensated in the new arc optics. The coupling measured in the present arc optics was analyzed. It was shown that the value J_c = 2J_x + J_y was preserved in a short time range. This verified that the coupling was caused by the ν_x − 2ν_y = −21 resonance.

Second, beam loss comparison was used to verify the compensation. The tune was scanned across the ν_x − 2ν_y = −21 resonance in both optics. The beam loss was clearly reduced in the new arc optics. This showed that the ν_x − 2ν_y = −21 resonance was successfully weakened. The results were well reproduced using SCTR. The peak tune was shifted from the on-resonance condition in both experiment and simulation. Since the shift became smaller when the beam intensity became weaker in the simulation, it was confirmed that the shift was due to space charge detuning.

Third, Fourier analyses were used to verify the compensation. The tune was set near the ν_x − 2ν_y = −21 resonance, and the Fourier spectra of the resonance were measured. The clear spectra of the resonance were measured in the present arc optics. Using the new arc optics, the spectra of the resonance were significantly suppressed. This showed that the ν_x − 2ν_y = −21 resonance was well compensated.

ACKNOWLEDGEMENTS

We would like to express our deepest gratitude to the power supply group in the MR for creating the various kinds of magnet patterns. We would like to give a special thanks to K. Ohmi for the implementation of SCTR. We are grateful to the monitor group in the MR for setting up the devices, such as BPMs, DCCT, Residual Gas Ionization Profile Monitor, and intra-bunch feedback systems. We appreciate the kind cooperation of the members of the RF, 3-50BT, injection/extraction, control, and vacuum groups in the MR.

Appendix A. Tuning arc section

The most important part of the optics tuning is to gain accurate control of the phase advance in the arc section. In the MR, there are four kinds of quadrupole magnets in the arc sections which change the phase advance in the arc section.

The phase constant φ_u was obtained via Fourier analysis (NAFF). The data for the first 512 turns were used in these analyses. We defined φ_u as the Fourier phase term at the peak frequency.

The phase advance of the new optics in one arc section is (Δψ_{arc, x}, Δψ_{arc, y}) = 2π(6, 5.5). First, the phase advances were measured with the setting values. The horizontal phase advance Δψ_{arc, x} was well adjusted, but the vertical phase advance Δψ_{arc, y} had a systematic error. This indicated that the calibrations of the quadrupole magnets in the arc sections around the new settings were not perfect. Therefore, the vertical phase advance Δψ_{arc, y} was calibrated.

Appendix B. Optical measurements

Before the experiments for the verification of the compensation, the tune, the betatron functions, and the dispersions were measured to verify the beam optics. In this section the results for the new arc optics are presented.

The betatron functions were obtained using the phase advances between three adjacent BPMs [42]. The phase advances were measured by using beams with dipole errors at the beam injection. All the dipole oscillations at the BPMs were analyzed via NAFF. The first 128 turns of data were used for the analysis. The results of the betatron functions are shown in Fig. B1.

Footnotes

References