Effect of quadrupole focusing-field fluctuation on the transverse stability of intense hadron beams in storage rings

... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A systematic experimental study is performed to clarify the parameter dependence of the noiseinduced beam instability previously demonstrated by a Princeton group [M. Chung et al., Phys. Rev. Lett. 102, 145003 (2009)]. Because of the weakness of the driving force, the instability develops very slowly, which substantially limits the application of conventional experimental and numerical techniques. In the present study, a novel tabletop apparatus called “S-POD” (Simulator of Particle Orbit Dynamics) is employed to explore the long-term collective behavior of intense hadron beams. S-POD provides a many-body Coulomb system physically equivalent to a relativistic charged-particle beam and thus enables us to conduct various beam-dynamics experiments without the use of large-scale machines. It is reconfirmed that random noise on the linear beam-focusing potential can be a source of slow beam quality degradation. Experimental observations are explained well by a simple perturbation theory that predicts the existence of a series of dangerous noise frequency bands overlooked in the previous study. Those additional instability bands newly identified with S-POD are more important practically because the driving noise frequencies can be very low. The dependence of the instability on the noise level, operating tune, and beam intensity is examined and found consistent with theoretical predictions. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Introduction
Recent progress in accelerator technologies has made it feasible to provide extremely intense and/or high-power hadron beams for diverse purposes [1,2].The operating condition of an advanced hadron ring is carefully chosen to avoid possible collective instabilities that deteriorate the beam quality [3].Ideally, the beam can stay in the machine for many turns with no major particle losses if the operating point is set sufficiently away from all dangerous resonance bands on the tune diagram.In reality, however, the machine operating condition is not perfect but various noise sources exist in lattice elements.For instance, the electric currents exciting quadrupole electromagnets along the beam orbit have random ripples.The linear focusing fields acting on the stored beam fluctuate more or less about the ideal design strength all the time.Since high-performance hadron rings of the next generation are supposed to store a huge number of protons or heavy ions for a very long period, extra attention must be paid even to such a weak source of beam disturbance.
The field fluctuation in quadrupole magnets has been carefully minimized in modern accelerators, so the resultant instability, if any, causes no significant effect within a short timescale.The noise-induced effect will be an important issue only in the case where long-term beam stability is PTEP 2018, 023G01 K. Ito et al. strongly required.An experimental study of this type of imperfection effect is generally quite troublesome because the effect is so feeble and, furthermore, any real large-scale accelerators possess uncontrollable noise sources.The use of a multi-particle tracking code seems an alternative approach to the problem, but long-term simulations become unreasonably time-consuming once interparticle Coulomb interactions are taken into account.To overcome these difficulties in conventional methods, we here employ the novel experimental apparatus "S-POD" (Simulator of Particle Orbit Dynamics) developed at Hiroshima University [4,5].The main components of S-POD include a compact "linear Paul trap" (LPT), many AC and DC power supplies, vacuum pumps, plasma diagnostic devices, a Doppler laser-cooling system, and a personal computer that controls the whole experimental process.As briefly outlined in the next section, S-POD provides a non-neutral plasma physically equivalent to a charged-particle beam in an alternating-gradient (AG) focusing potential [6].In contrast to conventional beam-dynamics experiments relying on large-scale machines, the fundamental parameters of interest to us are highly flexible in S-POD.It is even possible to vary the strength and frequency spectrum of the focusing noise over a wide range.Because of the compactness and simple nature of the system, we are not bothered by unwanted noise sources that complicate output signals.
A similar ion trap system was constructed at Princeton Plasma Physics Laboratory and applied to some beam physics issues [7].Gilson and his co-workers used a different type of LPT to investigate the beam quality degradation induced by random noise [8][9][10].They concluded that quadrupole noise gives rise to continuous emittance growth due to the development of a non-thermal tail around the beam core.This intriguing result was linked to a theoretical expectation that random lattice imperfections can expedite the halo formation process [11][12][13].The noise level assumed by the Princeton group was around 1% of the design focusing strength, much greater than the typical noise expected in recent high-performance machines.Consequently, the instability grows only within a few thousand AG focusing periods, which enables them to carry out particle-in-cell (PIC) simulations for comparison with experimental observations.In the present paper, we consider a noise range at least one order of magnitude lower than the Princeton experiment.The instability is, therefore, much weaker and exhibits real long-term development for which reliable PIC simulations are no longer possible.We follow the plasma behavior over millions of AG periods, changing the operating point and beam intensity.A detailed mathematical description of the instability is also given to predict the noise frequencies that demand particular attention.We experimentally verify the existence of additional instability bands overlooked in the previous study by the Princeton group.

Experimental method
The many-body dynamical system created in the LPT of S-POD obeys the Hamiltonian where (x, y) are the transverse spatial coordinates of a confined ion that has mass M 0 and charge state q, (p x , p y ) are the dimensionless canonical momenta conjugate to (x, y), the independent variable is τ = ct with t and c being time and the speed of light respectively, φ is the collective Coulomb potential, and K(τ ) denotes the linear focusing function proportional to the radio frequency (rf) voltages ±V (τ ) on the quadrupole electrodes of the LPT.Since the fluctuation of the linear focusing potential has no direct effect on the synchrotron motion, we have ignored the longitudinal degree of freedom for simplicity.Equation intense hadron beam propagating through an AG transport channel (see Ref. [14] and references therein), except for the detail of constant coefficients.A major advantage in S-POD experiments is the high flexibility of the waveform of K(τ ).In a real accelerator, the lattice design determines the focusing function, which means that its waveform is no longer controllable after the machine construction.In S-POD, an arbitrary waveform is available for K(τ ) because it is determined not by the mechanical structure of the LPT but by the time variation pattern of the electrode voltage V (τ ).We can easily emulate a variety of AG focusing lattices simply by changing the rf waveform from the power source.Among a wide range of choices for the waveform of V (τ ), we here adopt the simplest sinusoidal focusing model where the geometry of an AG cell is approximated by a sine curve; namely, V (τ ) = V 0 sin(2πf 0 τ/c).where f 0 is the rf frequency fixed at 1 MHz.It has been verified experimentally and theoretically that this simplified model can reproduce well the collective beam behavior in the most standard AG configuration called "FODO" [15].The FODO lattice consists of two quadrupoles of opposite polarities, one of which is for beam focusing (F) and the other for defocusing (D).The rf amplitude V 0 is ideally a constant parameter that determines the bare tune of the betatron motion or, in other words, the bare phase advance of the betatron oscillation per AG period.Since the sinusoidal waveform is symmetric, the horizontal and vertical bare tunes (ν 0x , ν 0y ) take an identical value, i.e., ν 0x = ν 0y ≡ ν 0 .The same model was employed in the Princeton experiment mentioned above.
We apply random noise to V (τ ) in order to replicate the common situation where excitation currents to quadrupole magnets fluctuate about a certain central value.In the previous experiment by the Princeton group [8][9][10] the ideal sinusoidal amplitude V 0 was replaced by V 0 [1 + δ(τ )], where δ(τ ) stands for random fluctuation that satisfies |δ(τ )| 1.The perturbing term is thus the product of the noise amplitude V 0 δ(τ ) and the sinusoidal function sin(2π f 0 τ/c).In addition, δ(τ ) was defined as a series of square pulses that have the same width 1/2f 0 but randomly varying heights.Presumably, they tried to imitate the gradient error originating from mechanical imperfections of quadrupole focusing magnets.In the following study, we simply add a random perturbation V (τ ) to the quadrupole electrodes instead of using complex square pulses; namely, V (τ ) = V 0 sin(2πf 0 τ/c) + V (τ ).This model approximately reproduces the effect of random ripples in quadrupole excitation currents from the power source rather than the effect of mechanical errors in the magnets themselves.Strictly speaking, the latter type of machine imperfection is inconsistent with our original assumption, i.e., the complete randomness of the error field over a long timescale, because the linear focusing potential (including the mechanical error) becomes periodic every turn around the ring.
A schematic view of the LPT is shown in Fig. 1.It consists of three quadrupole sections plus two End Caps, all of which are electrically isolated from each other.The axial plasma confinement is achieved by adding DC bias voltages to the central Gate and End Cap sections.A large number of 40 Ar + ions generated by the electron bombardment process are accumulated in a relatively long quadrupole section between the Gate and one of the two End Caps.The electrode rods in this plasma confinement region are 50 mm long, while the aperture is 5 mm in radius.According to profile measurement with a phosphor screen, the transverse plasma extent is typically a few mm. 1 The ion cloud in the LPT is, therefore, like a long bunch.The initial plasma density is controllable by changing the neutral Ar gas pressure and/or the electron beam current from an e-gun.Note that the species of ions is of no essential importance because the ion's mass and charge state are nothing but  scaling parameters in an S-POD experiment.At high density, the bare tunes (ν 0x , ν 0y ) are depressed to (ν x , ν y ) by the Coulomb potential φ.The root-mean-squared (rms) average of the space-chargeinduced tune shift can be written as ν = (1 − η)ν 0 , with η being the rms tune depression.The lowest value of η in the S-POD system is currently limited to around 0.8 for technical reasons,2 but this number already exceeds the range of beam density achievable in typical high-intensity circular accelerators.The random noise from an arbitrary function generator is introduced only to the section for ion accumulation.After a certain period of plasma confinement, the DC potential barrier at the Gate section is dropped to extract the ions toward the Faraday-cup detector.

Theoretical expectation
High-precision PIC simulations are too time-consuming in the timescale corresponding to beam transport over millions of AG periods.Developing an analytic theory is also extremely difficult because of the complexity of the basic equations describing the collective beam motion.It would, however, certainly be informative if any theoretical expectations are available for comparison with experimental data.We here slightly modify the one-dimensional (1D) Vlasov theory constructed in Ref. [17] and derive approximate formulas concerning the beam stability.The mathematical description helps to understand the basic mechanism and parameter dependence of the instability even if it is based on a 1D model.
We are allowed to treat the horizontal and vertical degrees of freedom separately, provided that the Coulomb coupling between the two transverse directions does not play an important role.Following the mathematical formalism in Ref. [17], we eventually reach the differential equation whose solution predicts the condition of beam stability under the influence of a linear imperfection potential.The definitions of the symbols in Eq. ( 2) have been summarized in Appendix A. In our PTEP 2018, 023G01 K. Ito et al.
noise experiment with S-POD, the perturbation amplitude δu 2 can be expressed as where r 0 denotes the minimum distance from the LPT axis to the surfaces of the quadrupole electrode rods.r 0 = 5 mm in the LPT used for this experiment.Since the unperturbed betatron function is periodic, we can expand β 2 x on the right-hand side of Eq. ( 2) into a Fourier series as where a n and ϕ n are constant parameters dependent on the harmonic number n.If the ring consists of N sp lattice superperiods, the Fourier coefficients a n become particularly large for every N sp harmonic numbers, i.e., n = 0, N sp , 2N sp , . . .Other harmonics generated by lattice imperfections are not exactly zero but strongly suppressed after careful beam optics and orbit correction.
The noise function δu 2 has nothing to do with the lattice design here.Although δu 2 is the superposition of oscillatory components over a wide frequency range, we pick just one component for the sake of simplicity, writing δu 2 ∝ cos(κθ) where κ is a non-integral constant.The κ parameter is related to the noise frequency f noise as κ = f noise /f rev , where f rev denotes the revolution frequency of the beam around the ring.The product of β 2 x and δu 2 on the right-hand side of Eq. ( 2) yields the oscillatory terms proportional to cos(n ± κ)θ, etc.The driven harmonic oscillation of G 2 , therefore, becomes resonantly unstable under the condition 2 = |n ± κ|. ( Considering the weakness of the driving force, this type of instability is most likely negligible within a short timescale, but over a very long timescale it may give rise to noticeable beam loss.The resonance condition of Eq. ( 5) together with Eq. (A7) in Appendix A tells us to avoid the quadrupole field fluctuation of the frequency ranges near The frequencies with n = 0, N sp , 2N sp , . . .should be especially dangerous.
The simple sinusoidal focusing model adopted for the present experiment suffices to validate the instability criterion in Eq. ( 6).The unit AG structure of this model corresponds to a single sinusoidal period, so N sp = 1 and the bare tune ν 0 per lattice period is limited within a narrow range ν 0 < 0.5.Since f rev = f 0 = 1 MHz in the current S-POD setup, we expect weak instability of an ion plasma when the linear perturbing potential V (τ ) contains the noise components of the frequencies In our model, the modulation of the plasma envelope is nearly sinusoidal.The first two terms with n = 0 and 1 in Eq. ( 4) are then much more important than others, leading to the following resonance  conditions: The instability should be most severe at the first frequency (corresponding to n = 0) because a 0 > a 1 .

Plasma lifetime in the presence of white noise
Figure 2 shows the results of plasma lifetime measurements with and without white noise.A multifunction generator (WF1974, NF Corporation) was used as the source of the artificial noise applied to the quadrupole rods.The bare tune has been fixed at ν 0 = 0.145, where no intrinsic resonance is supposed to exist in the absence of noise.The effective noise strength has been defined by V ≡ f , where f is the frequency resolution.We set f = 20 Hz, considering the typical width of a resonance stop band.V0 is the effective amplitude of the sinusoidal focusing field given by V0 = V 0 / √ 2. The open circles (•) in Fig. 2(a) correspond to the ideal case with no random noise, while the other three filled symbols ( , , ) show the perturbed cases where weak white noise is added.It is evident that the ion plasma in the LPT decays faster as the noise is strengthened.
Although the ion number N ion appears to decrease linearly in Fig. 2(a), we have realized that the decay speed is changed in an early stage before t = 2 s; after that, the decay becomes slower.The time evolution of N ion can actually be fitted very well with a two-component exponential function, where T S and T L are, respectively, the short-term and long-term decay constants.N S and N L are other fitting constants that satisfy N S < N L in general.We have noticed, from our past experiences, that T L depends on the Ar gas pressure and on the initial number of ions in the LPT.This suggest that interparticle (Ar + -Ar + and Ar + -Ar) collisions should be the main cause of the long-term decay after  2 s. 3 On the other hand, T S is insensitive to the initial ion number.A possible cause of the relatively fast plasma decay in the early stage is an initial mismatch of the ion distribution to the external focusing potential.Panel (b) in Fig. 2 summarizes the fitting results obtained from the S-POD data in panel (a).The long-term plasma lifetime T L without white noise exceeds 10 s, corresponding to beam transport over 10 7 AG focusing cells.T L becomes shorter exponentially as the noise is enhanced.T S is also shortened with increasing V , while the V dependence is not so strong as T L .

Effect of colored noise
We made use of controlled "colored" noise to check if all noise components over the whole frequency range contribute equally to the observed lifetime shortening.The Princeton group previously performed a similar experiment and found the existence of a resonance stop band corresponding to the first condition in Eq. ( 8) [10].In our S-POD experiment, we numerically design the colored noise that has the central frequency of f c and bandwidth f noise .The numerical noise data is transmitted to the multifunction generator to produce the perturbation voltage V .An example of the colored noise from WF1974 is displayed in Fig. 3.The noise components except for the narrow range When the LPT operating point is adjusted to ν 0 = 0.145, we expect the linear resonant instability around the noise frequencies noise [kHz] ≈ 290, 710, and 1290 according to the theoretical prediction in Eq. ( 8) if tune shift ν is not too large.The time evolution of ion number in the LPT measured at three different values of f c is plotted in Fig. 4. The noise level V / V0 is more than one order of magnitude lower compared with the case of white noise experiments in Fig. 2.This is simply due to the limited bandwidth; f noise has been set at 100 kHz in Fig. 4. The colored noise of V / V0 = 80 ppm here corresponds to white noise of V / V0 = 0.13% in terms of spectrum PTEP 2018, 023G01 K. Ito et al.The number of ions surviving in the LPT after 1 s is plotted as a function of the central frequency f c of colored noise with the bandwidth f noise = 50 kHz.The noise strength is set at V / V0 = 88 ppm in the frequency range below 1 MHz.V / V0 has been doubled above 1 MHz because the third instability in this range is much weaker than the other two.The initial ion number is fixed at about 10 6 in all measurements.density.At f c = 50 kHz, sufficiently distanced from all of the predicted instability bands, the plasma lifetime is independent of V / V0 .T L is longer than 10 s, no matter whether the colored noise is on.In contrast, T L has been obviously shortened at f c = 300 kHz and 700 kHz with increasing V / V0 .The instability at f c = 300 kHz is stronger than that at f c = 700 kHz, which is consistent with the theoretical expectation.We have confirmed that the noise-dependent ion losses also occur near f noise = 1290 kHz, and this third instability is weaker than the other two.
Figure 5 shows the number of ions remaining in the LPT after 1 s.The dangerous noise frequency regions are now apparent.The upper panel corresponds to the case where ν 0 = 0.145.The three instability bands are observed in the vicinity of the noise frequencies evaluated from Eq. ( 8).The operating tune has been changed from 0.145 to 0.190 in the lower panel.The instability bands are then moved to the positions close to the resonant frequencies f noise [kHz] ≈ 380, 620, and 1380  predicted by Eq. ( 8).In both cases (a) and (b), we have used noise twice as strong in the frequency range above 1 MHz to enhance ion losses at the weak third stop band.

Ion density dependence
The experimental results given above are convincing evidence that the collective mechanism described in Sect. 3 can seriously affect the long-term stability of an intense hadron beam.An important remaining task is to corroborate the intensity dependence of the instability.The general resonance condition of Eq. ( 6) includes the rms tune shift ν that could be a significant size in advanced high-intensity rings.In order to clarify the effect of ν on the instability bands, we conducted ion-loss measurements as in Fig. 5, changing the initial plasma density.Four different cases have been considered in Fig. 6.The number of ions stored in the LPT at the beginning is gradually increased from the top panel (a) to the lower.The initial plasma density is the highest in the bottom panel (d).We recognize that the theory in Sect. 3 can explain the experimental observation in Fig. 6.Since the operating point is chosen at ν 0 = 0.145 in this example, the two instability bands in the frequency range below 1 MHz appear around f noise [MHz] ≈ 0.29 − 2C 2 ν and 0.71 + 2C 2 ν according to Eq. ( 8), and the former instability is stronger than the latter.These resonance conditions indicate that, as the initial ion number is increased, the first severe band shifts to the lower frequency side while the second relatively weak band moves the other way.The average of the two resonant frequencies is always equal to 0.5 MHz, which means that the movements of the two bands take place symmetrically with respect to f c = 0.5 MHz.The S-POD data in Fig. 6 have verified all these expectations.The most severe ion losses have occurred at f c ≈ 0.235 MHz when N 0 ≈ 1 × 10 7 .Assuming that this frequency corresponds to the central position of the first instability band, the magnitude of the coherent band shift is roughly 19% of the bare tune.If we adopt Sacherer's theoretical prediction for the C m factor, i.e., C 2 = 3/4 [18], the tune depression is estimated to be 0.75.

Concluding remarks
The long-term stability of high-intensity hadron beams in circular machines has been investigated theoretically and experimentally.The present study focuses on the possible beam quality deterioration caused by the fluctuation of the external linear AG focusing potential.Such noise is generally weak, but exists more or less in any storage ring due, for example, to small ripples of driving electric currents to quadrupole magnets.While this type of weak disturbance is often overlooked or not taken seriously, it could be an important issue in modern high-performance rings where intense hadron bunches are stored for a very long period.
A systematic study of the noise-induced effect is extremely difficult to execute in practice not only because a real large-scale accelerator contains various uncontrollable noise sources but also because high-precision numerical simulations including interparticle Coulomb interactions are too time-consuming.We here employed the unique non-neutral plasma trap facility "S-POD" that enables us to replicate the collective motion of a relativistic hadron bunch in a local tabletop environment.Adding well-controlled random noise to the rf potential, we have confirmed that the plasma lifetime is considerably reduced by the linear focusing field fluctuation.It has turned out that only limited frequency ranges in white noise are responsible for observed ion losses.The experimental observations can be explained nicely by a simple analytic theory that predicts collective resonance about the noise frequencies given by Eq. (6).
It is reasonable to ask, from a practical point of view, whether the instability mechanism studied here can really be a source of long-term slow ion losses in typical high-intensity rings.In particular, we should recall that the actual field in an electromagnet does not precisely respond to the ripple of the driving current.Very high-frequency noise components are probably damped in the aperture and thus have no effect on beam dynamics.This instability will become a concern particularly when the revolution frequency f rev is low.f rev is around a few hundred kHz or higher in high-intensity machines such as the Rapid Cycling Synchrotron and Main Ring at J-PARC [19], the Proton Synchrotron (PS) at CERN [20], the Spallation Neutron Source at ORNL [21], etc.The value of 2f rev ν 0 then comes into the MHz range, which suggests that the third condition in Eq. ( 6) can be discarded in practice.The first (or third) condition in Eq. ( 6) with n = 0, previously discussed by the Princeton group [10], also seems practically unimportant because the resonant noise frequency is too high. 4s an example, let us briefly look into the case of CERN PS.In the future High-Luminosity LHC (HL-LHC), intense proton bunches are supposed to stay in the PS for over a second (roughly a

( 1 )
is identical to the well-known betatron Hamiltonian of an PTEP 2018, 023G01 K. Ito et al.

Fig. 1 .
Fig. 1.Schematic of the linear Paul trap for S-POD.

Fig. 2 .
Fig. 2.Plasma lifetime measurements at ν 0 = 0.145 with different white noise powers.(a) Time evolution of the number of Ar + ions remaining in the LPT.The initial ion number is fixed at 10 6 in all measurements.We have considered four different noise strengths: V / V0 = 0 (•: no noise), 0.069% ( ), 0.14% ( ), and 0.28% ( ).The solid lines are the results of curve fitting based on Eq. (9).(b) Time constants T S and T L evaluated by fitting the four curves in (a).

Fig. 3 .
Fig. 3. Example of colored noise.The upper panel shows the time dependence of the colored noise produced by the multifunction generator WF1974.The corresponding frequency spectrum is depicted in the lower panel.In this example, f c = 250 kHz and f noise = 100 kHz.

Fig. 4 .
Fig. 4. Plasma lifetime measurements at ν 0 = 0.145 with colored noise.The colored noise with f noise = 100 kHz is centered at three different frequencies f c [kHz] = 50, 300, and 700.The noise strength V / V0 has been varied from 0 (no noise applied) to 80 ppm in each panel.

Fig. 5 .
Fig. 5. Ion-loss distributions measured at two different operating tunes: (a) ν 0 = 0.145 and (b) ν 0 = 0.190.The number of ions surviving in the LPT after 1 s is plotted as a function of the central frequency f c of colored noise with the bandwidth f noise = 50 kHz.The noise strength is set at V / V0 = 88 ppm in the frequency range below 1 MHz.V / V0 has been doubled above 1 MHz because the third instability in this range is much weaker than the other two.The initial ion number is fixed at about 10 6 in all measurements.

Fig. 6 .
Fig. 6.Dependence of the noise-induced instability bands on plasma intensity.The operating bare tune is fixed at ν 0 = 0.145.The number of ions surviving in the LPT after 1 s is plotted as a function of the central frequency f c of colored noise with the bandwidth f noise = 50 kHz.The initial ion number N 0 is increased from the top panel to the lower; namely, (a) N 0 ≈ 1 × 10 5 , (b) 1 × 10 6 , (c) 5 × 10 6 , and (d) 1 × 10 7 .The noise power has been set at V / V0 = 88 ppm in the three cases from the top.In the bottom case (d) where N 0 ≈ 1 × 10 7 , the noise power has been reduced to V / V0 = 22 ppm because ion losses are too fast at this density.