PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 12, 054203 (2009)Experimental investigation of random noise-induced beam degradationin high-intensity accelerators using a linear Paul trapMoses ChungAccelerator Physics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USAErik P. Gilson, Ronald C. Davidson, Philip C. Efthimion, and Richard MajeskiPlasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA(Received 3 February 2009; published 26 May 2009)A random noise-induced beam degradation that could affect intense beam transport over longpropagation distances has been experimentally investigated by making use of the transverse beamdynamics equivalence between an alternating-gradient focusing system and a linear Paul trap system.For the present study, machine imperfections in the quadrupole focusing lattice are considered, which areemulated by adding small random noise on the voltage waveform of the quadrupole electrodes in the Paultrap. It is observed that externally driven noise continuously increases the rms radius, transverseemittance, and nonthermal tail of the trapped charge bunch almost linearly with the duration of thenoise. The combined effects of collective modes and colored noise are also investigated and comparedwith numerical simulations.DOI: 10.1103/PhysRevSTAB.12.054203 PACS numbers: 52.59.Sa, 29.27. a, 41.85.Ja, 52.27.JtI. INTRODUCTIONHigh-intensity accelerators have a wide range of applicationsranging from basic scientific research in high energyand nuclear physics, to applications such as heavy ionfusion, ion-beam-driven high energy density physics, tritiumproduction, nuclear waste transmutation, and spallationneutron sources for material and biological research[1–3]. One critical but unavoidable problem in highintensityaccelerators is the presence of undesired randomnoise due to machine imperfections, and its influence onthe long-time-scale beam dynamics [4,5]. The machineimperfections may include quadrupole magnet and rf cavityalignment errors, quadrupole focusing gradient errors,rf field amplitude and phase errors, to mention a fewexamples [6]. Usually, random noise in the machine componentsacts as a continuous supply of free energy to thebeam, which results in irregular mismatch oscillations ofthe beam envelope [5], enhanced halo formation [4,7], andemittance growth [8], particularly over long propagationdistances. Consequently, the associated beam degradationdefines the practical and/or economic tolerances in themachine design and operation [3,9]. In transforming randomnoise effects into emittance growth, the action of thenonlinear space-charge force plays a critical role [8].Hence, it is increasingly important to understand the effectsof random noise on long-distance beam propagationwith moderate space-charge forces. From various multiparticlesimulations with both space-charge and randomnoise effects, considerable progress has been made indeveloping an improved understanding of the randomnoise-induced beam degradation [5,6,8,10–12]. However,experimental verification of these effects has been **somewhat**limited due to the lack of dedicated experimentalfacilities that allow the study of long-time-scalephenomena.The Paul trap simulator experiment (PTSX), which is alinear Paul trap [13] that can experimentally simulate thenonlinear transverse dynamics of intense beam propagationover large equivalent distances through an alternatinggradient(AG) transport lattice [14,15], provides a compactand flexible laboratory facility for the experimental investigationof random noise effects. The idea of using a linearPaul trap confining a pure ion plasma to study intense beampropagation was proposed by Davidson et al. [14] and byOkamoto and Tanaka [16]. The physics equivalence betweenthe AG system and the linear Paul trap system,including the self-field forces, is described in detail inRef. [14]. The PTSX device consists of three collinearcylinders with radius r w ¼ 0:1 m, each divided into four90 azimuthal sectors (see Fig. 1 in Sec. II for details). Thepure ion plasma is confined transversely in the central 2 m-long cylinder by oscillating voltages. The static voltages onthe two 0.4 m-long end cylinders confine the ions axially. Abrief description of the PTSX device is given in Sec. II ofthis paper. The amplitude of the oscillating voltage waveformapplied to the central electrodes in the PTSX devicecorresponds to the quadrupole focusing gradient in an AGlattice system. Hence, by slightly modifying the voltageamplitude V 0 max in every half-focusing period T=2, we cansimulate the effect of randomly distributed quadrupolefocusing gradient errors in the actual transport channel.As mentioned earlier, in intense beams, the action of thenonlinear space-charge force plays a critical role in transformingrandom noise effects into emittance growth [8].The relative importance of space-charge effects can be1098-4402=09=12(5)=054203(11) 054203-1 Ó 2009 The American Physical Society

MOSES CHUNG et al. Phys. Rev. ST Accel. Beams 12, 054203 (2009)FIG. 1. Schematic of the PTSX device showing: (a) side viewof the quadrupole electrodes, cesium ion source, and chargecollector, and (b) end view of the central electrode set.described either in terms of the effective tune depression= 0 [17], or the normalized intensity parameter ^s ¼! 2 pð0Þ=2! 2 q [18], where ! 2 pðrÞ ¼nðrÞq 2 =m 0 is the plasmafrequency squared, and ! q is the average smooth-focusingfrequency of the beam particles’ transverse oscillations inthe applied focusing field. Here, 0 is the permittivity offree space, nðrÞ is the radial ion density profile, r is theradial distance from the beam axis, and q and m are the ioncharge and mass, respectively. For example, the newlycommissioned Spallation Neutron Source (SNS) is designedto be operated at = 0 * 0:6 in the linac section(total length 331 m), and = 0 * 0:9 in the accumulatorring (circumference 248 m) [19,20], which correspondto ^s & 0:9 and ^s & 0:35, respectively. In the linac section,space-charge effects play a more important role for thenoise effect to be significant. In the accumulator ring, onthe other hand, the long propagation distances associatedwith the beam lifetime can have a larger impact on noiseinducedbeam degradation. Since the PTSX device coversthe operating range of 0 < ^s

EXPERIMENTAL INVESTIGATION OF RANDOM NOISE- ... Phys. Rev. ST Accel. Beams 12, 054203 (2009)when an oscillating voltage V 0 ðtÞ is applied with alternatingpolarity on adjacent segments, the resulting electricfield becomes an oscillating quadrupole field near the trapaxis. This quadrupole electric field exerts a ponderomotiveforce that confines the pure ion plasma radially. To trap theplasma axially, the two end electrodes are biased to aconstant positive voltage þ ^V, after the charge bunch isinjected into the central section.The cesium ion source is located on the trap axis near thecenter of one of the short electrode sets so that ion injectionis not affected by the fringe fields [24]. The amount ofcharge injected can be controlled easily by adjusting thevoltages on the emitter surface (V s ), acceleration grid (V a ),and deceleration grid (V d ) of the ion source. A DC powersupply keeps the source temperature around 1000 C sothat cesium is ionized through contact ionization on theemitter surface. The charge collector is mounted on a linearmotion feedthrough at the other end of the short electrodeset, and moves in the transverse direction along a null ofthe applied potential in order to minimize the perturbationof the quadrupole potential configuration during the profilemeasurement [25]. Using the sensitive electrometer withLABVIEW interface, the total axially integrated ion chargeQðrÞ through the collector plate centered at radius r can bemeasured accurately to as low as the 1 fC range, which isadequate to detect the formation of halo particles. Theradial ion density profile nðrÞ is related to QðrÞ by nðrÞ QðrÞ=qr 2 cL p . Here, r c is the size of the circular collectingplate and L p is the plasma length. The effect of finite widthof the collector plate has been corrected during the dataprocessing stage, and the resultant error is only a fewpercent.The PTSX device manipulates the charge bunch usingan inject-trap-dump-rest cycle, and the one-componentplasmas created in the trap are highly reproducible [15].A sinusoidal waveform V 0 ðtÞ ¼V 0 max sinð2f 0 tÞ is appliedby an arbitrary function generator with a 20 MHzclock rate. Random noise is excited through anotherLABVIEW interface which samples a uniformly distributedrandom number in the range jj max , and adjustsV 0 max to V 0 max ð1 þ Þ in every half-focusing period1=2f 0 . Application of the noise signal begins after theinjected ion bunch becomes sufficiently stabilized (typicallyin about 12 ms), and ends before the dumping stagebegins. In the dumping stage, most of the trapped ions arecollected within 2 ms without significant changes in theradial density profile. Since the collected charge is necessarilyaveraged over many focusing periods during thedumping process, the value of the rms radius calculatedfrom the measured radial profile can be interpreted as therms radius of the beam in the smooth-focusing approximation[18].In order to minimize the effects of neutral collisions onthe plasma behavior, the base pressure of PTSX is keptbelow 5 10 9 Torr after a week-long bake at 200 C.When the ion source is on, the operating pressure risesup to 10 8 10 7 Torr. Even in this case, the characteristicion-neutral collision time is in * 2 sec , and thetrapped plasma is collisionless to very good approximation.The vacuum phase advance used in this experiment is v ¼ 52 , which is considerably below the envelope instabilitylimit [18]. When v < 72 , it is well establishedthat the smooth-focusing model, in which the averagebeam cross section is assumed to be circular, is a goodapproximate description of a beam with elliptical crosssection in an AG lattice [18]. Note also that recent axialcurrent measurements [26] show that the axial nonuniformityin the initial beam is smoothed out in about 6 ms.III. SMOOTH-FOCUSING THEORETICAL MODELFor simplicity in the theoretical analysis presented inthis section, we assume that the average effects of thequadrupole focusing field are described by an equivalentsmooth-focusing force [18]. In this smooth-focusing approximation,the evolution of the rms radius R b ðtÞ of acharge bunch with line density N is effectively describedbyd 2 R bþ ! 2 K 2qRdt 2 b2R b 4R 3 ¼ 0; (1)bwhere ! q is the smooth-focusing frequency in theabsence of noise, K ¼ 2Nq 2 =4 0 m is the effective (dimensional)self-field perveance, and ¼ 2R b ½h _x 2 þ_y 2 i f ðdR b =dtÞ 2 Š 1=2 is the average transverse emittancedefined in the beam frame [18]. Here, h i f denotes thestatistical average over the particle distribution function inthe smooth-focusing approximation. Note that the effectiveself-field perveance K and the average transverse emittance have the dimensions of ðvelocityÞ 2 and (length velocity), respectively. For a sinusoidal voltage waveformV 0 ðtÞ ¼V 0 max sinð2f 0 tÞ used in the PTSX, the smoothfocusingfrequency occurring in Eq. (1) is given approximatelyby [18]! q ¼ 8qV 0 max 1mr 2 pwf 0 2 ffiffiffi : (2)2 When there are random errors in voltage amplitudes V 0 maxin every half-focusing period T=2 ¼ 1=2f 0 , the trappedplasma encounters a time series in the smooth-focusingfrequencies given by ! q ð1 þ 1 Þ; ...;! q ð1 þ i Þ;...;! q ð1 þ N1=2Þ [8]. Here, i is a random number whichbelongs to a uniform distribution in the range max i max , and i and N 1=2 are the index and the totalnumber of half periods of the noise duration, respectively.Hence, an error sample is composed of N 1=2 statisticallyindependent random numbers, representing white noise inthe present analysis.054203-3

MOSES CHUNG et al. Phys. Rev. ST Accel. Beams 12, 054203 (2009)An example of the numerical solution of Eq. (1), havingan error sample with max ¼ 1%, shows that irregularoscillations of the beam envelopes are excited in a mannersimilar to beam mismatch [5] and grow continuously toconsiderable amplitudes, depending on the duration of thenoise [Fig. 2(a)]. Since the envelope equation itself cannotself-consistently describe the emittance growth resultingfrom the envelope oscillations, the center of the oscillationsremains nearly the same as the initial equilibriumradius R b0 . To describe the time evolution of the transverseemittance self-consistently, we make use of particle-in-cell(PIC) simulations. The result of WARP 2D PIC simulations[27] with the same error sample used in the envelopeequation demonstrates that the oscillation amplitudes aresaturated to some extent, and that the oscillation centerincreases linearly with the noise duration [Fig. 2(b)], implyingthe conversion of free energy available from theenvelope oscillations into emittance growth [28]. Eventhough there are as many negative energy kicks as positivekicks on the average, the simulation results suggest that therandom noise has a cumulative effect [5], which ends upproducing an overall increase in transverse energy (temperature)of the beam particles.Since the error limit max in the focusing force istypically only a few percent, any possible changes due tothe random noise in the global quantities such as the linedensity N, the effective transverse temperature T ? , and theaverage transverse emittance , will take place quiteslowly. In this case, we can still assume that the beam isin a quasiequilibrium state, which means that the averagefocusing force balances both the thermal pressure force ofthe plasma and the space-charge force over a slow timescale. Therefore, the global force balance equation can beexpressed approximately as [18]m! 2 qR 2 b ¼ 2 T ? þ Nq24 0; (3)and the evolution of the average transverse emittance canbe approximated byðtÞ ¼2R b! 2 qR 2 bNq 2 1=2:(4)4 0 mIndeed, Eqs. (3) and (4) are equivalent in form to theenvelope equation (1), provided we set d 2 R b =dt 2 ¼ 0 inEq. (1) (equilibrium case). Equation (4) allows us to estimatethe emittance growth due to the random noise, simplyby using the values of N and R b measured in the experiments.Note that errors in calculating the emittance usingEq. (4) result primarily from the instrumental uncertaintiesat large radii (see error bars in Fig. 6). However, when thereexists a significant population of low-density halo particlesbelow the detection limit ( & 1fC) of the charge collector,then the emittance calculated from Eq. (4) necessarilyunderestimates the actual mean transversepemittance.Particles far away from the beam core ( >ffiffiffi2 Rb ) are ofcourse weighted more heavily in calculating the emittancein the simulations [12,27].FIG. 2. Evolution of the rms radius R b ðtÞ calculated (a) fromthe envelope equation with constant emittance, and (b) from 2DWARP simulations with emittance growth. Note that the sameerror sample with max ¼ 1% is used for (a) and (b).IV. EXPERIMENT RESULTSThe initial plasma is obtained 12 ms after the ion injectionis completed. Figure 3 shows that the measured radialcharge profile QðrÞ is approximately a Gaussian functionof r (a straight line in the log versus r 2 plot as low as to the1 fC range) with a temperature comparable to the thermaltemperature of the cesium ion source ( 1000 C), indicatingthe validity of the assumption of thermal-equilibrium-likebeam.To characterize the statistical properties of the beamresponse to the random noise, we make use of the onaxischarge ^Q ¼ Qðr ¼ 0Þ, which is the most easily andaccurately measurable quantity in the present experimentalsetup. Figure 4 shows the time history of the statisticalaverage (h ^Qi), and the standard deviation (square root of054203-4

EXPERIMENTAL INVESTIGATION OF RANDOM NOISE- ... Phys. Rev. ST Accel. Beams 12, 054203 (2009)FIG. 3. (Color) Measured radial profiles of trapped plasma withoutapplied noise. The charge bunch is maintained in the quasiequilibriumstate for about 38 ms with a slight increase in theeffective transverse temperature [calculated from Eq. (3)]. Astraight line in the log of QðrÞ versus r 2 plot indicates that theradial profile is a Gaussian function of r.FIG. 4. Time history of (a) the statistical average, and (b) thestandard deviation of the on-axis charge computed over anensemble of 20 random error samples.the variance 2^Q ¼h^Q 2 i h^Qi 2 ) of the on-axis chargecomputed over an ensemble of 20 random error samples.The average of the on-axis charge decays almost linearlywith the amplitude and the duration of the noise up to25 ms. After 25 ms, the decay rate becomes **somewhat**rapid, which is likely related to the production of haloparticles [see also Figs. 5(a) and 7] and the resultantenhanced particle loss. It should be noted that both beamexpansion due to emittance growth and halo productiondue to parametric resonance are responsible for the on-axischarge decay observed in this experiment. As illustrated inFig. 2, emittance growth leads to beam expansion, which isnecessarily accompanied by core depletion when the linedensity N is fixed. In addition, a particle-core model [5]predicts that noise-induced mismatch oscillations can alsopopulate low-density halo particles at large radii, particularlyafter long propagation distances (trapping time in thisexperiment). Detailed numerical studies (similar toRef. [5]) may help to understand more clearly the role ofsuch mechanisms on core depletion. Because the initialbeam state is not a perfect equilibrium, and there is intrinsicnoise (either physics originated, or device originated) inthe PTSX device, such as two-stream interactions, collisionswith residual gas, drift in the ion source conditions,jitter in the voltage waveform, or mechanical vibrations ofthe vacuum pumps, the on-axis charge for the case with noapplied noise also decreases slightly during the 30 ms oftrapping. The fact that this process has a nearly constantstandard deviation suggests that instrumental uncertaintiesare dominant over small-amplitude intrinsic noise.Because of the relatively small number of error samples,it is not clear if the beam response is a random-walk-likediffusion process ( ^Q / t1=2 )[29]. However, the generaltendency is that the standard deviation of the on-axischarge increases with time, which strongly suggests thatthe fluctuations in the on-axis charge measurements originatefrom the applied noise rather than from instrumentaluncertainties (i.e., ^Q const) or statistical fluctuations(i.e., ^Q /h^Qi 1=2 )[30].When the beam is in a quasiequilibrium state, the onaxischarge (or equivalently ^s) can be a single parameterthat effectively characterizes the variation of the equilibriumdensity profile from one error sample to another [17].Hence, except for the case where there is a significantdeparture from the equilibrium state (such as the formationof a broad halo), we can infer the statistical characteristicsof the radial density profile from those of the on-axischarge. For example, when the normalized intensity parameter^s is moderately low (^s 0:2 in this study), theequilibrium density profile of the beam is nearly Gaussian[18]. In this case, we can relate N and R b to the on-axisdensity ^n ¼ nðr ¼ 0Þ by N ^nR 2 b. Since, N is constantunless beam particles are lost to the wall, we can furtherconsider R b to be a function of the parameter ^n according054203-5

MOSES CHUNG et al. Phys. Rev. ST Accel. Beams 12, 054203 (2009)samples by using the statistical properties of the on-axisdensity. By measuring a single radial profile for a givennoise amplitude and duration with a certain error samplethat gives ^n h^ni, and calculating the corresponding rmsradius using R 2 b ¼ð1=NÞ R r w0nðrÞ2r 3 dr, we can effectivelyestimate hR b i. Otherwise, it would be necessary tomeasure nðrÞ for every error sample to calculate theensemble-averaged quantities, which requires 360 independentradial profile measurements to scan the entireparameter range presented in this study. This processwould be very demanding for the present diagnostic setup,and is vulnerable to any drift in the experimental conditions(mostly from the ion source). In the present study, theprocesses of choosing an appropriate error sample andmeasuring the corresponding radial profile have been performedin 5 ms intervals (300 focusing periods) with threedifferent noise amplitudes, which require only 18 independentradial profile measurements. In Figs. 5 and 7, theFIG. 5. (Color) Evolution of (a) the measured radial profile, and(b) the corresponding rms radius under the influence of 1%uniform white noise.pto R b ð ^nÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiN=^n . From Fig. 4, we can infer that thestatistical fluctuations in ^n around its ensemble averagevalue h ^ni are small, i.e., ^n ¼h^niþ ^n with h ^ni ¼0 andj ^n=h ^nij 1. Therefore, for a given noise amplitude andduration, we obtain the following simple expressions forthe ensemble average of the rms radius hR b i and its variance 2 R b:sffiffiffiffiffiffiffiffiffiffiNhR b i ; (5)h ^ni 2 R b N 4 2^nh ^ni 3 : (6)The error propagation formula [31] has been applied inobtaining Eq. (6). Although these approximate expressionsare not explicitly used, they provide an indication that ifthere is a correlation between the on-axis density ^n and therms radius R b (or equivalently emittance), then we caninfer an rms radius averaged over the ensemble of errorFIG. 6. The emittance growth is estimated from (a) radialprofile measurements, and (b) WARP 2D PIC simulations. Theemittance is calculated from Eq. (4), and is normalized to itsinitial value i . For the WARP simulations, 20 random errorsamples are used to calculate the ensemble-averaged emittance.The dotted lines indicate the background emittance growth in theabsence of the applied noise.054203-6

EXPERIMENTAL INVESTIGATION OF RANDOM NOISE- ... Phys. Rev. ST Accel. Beams 12, 054203 (2009)1 þ t, where const. Hence, using Eq. (4), we wouldexpect the emittance to increase according to = i ðR b =R b0 Þ 2 ð1 þ tÞ 2 1 þ 2t, for small growth rate. If the growth rate becomes sufficiently large, the emittanceincreases approximately parabolically with time. Forthe case where max ¼ 1:5%, the experimentally determinedemittance is **somewhat** underestimated after a noiseduration of 15 ms. This is most likely due to the formationof a significant halo population under the detection limit( 1fC) of the charge collector, which is too low to bemeasured in the experiment, but contributes considerablyin the simulations. The formation of a significant halopopulation is apparent in Fig. 7(b). On the other hand,for the case with max ¼ 0:5%, the experimentally estimatedemittance has a slightly larger value than the simulationresults, which is likely due to the intrinsic noisepresent in the PTSX device. Since the correlation betweenthe intrinsic and applied noise, and its effect on the resultantemittance growth are **somewhat** unclear, the backgroundemittance growth level is plotted separately,rather than with the standard procedure of subtraction.For the WARP 2D PIC simulation presented in Fig. 6(b),40 000 macroparticles are used, which is comparable to thenumber of macroparticles adopted for other noise simulations[8]. The time step for the PIC simulation is t ¼0:05=f 0 2=! p .FIG. 7. (Color) Measured radial profiles with different noiseamplitudes and duration. Initially, the trapped plasma is in athermal equilibrium state, for which the radial density profile is astraight line in the log versus r 2 plot.measured radial charge profiles are shown in 10 ms intervalsrather than 5 ms intervals without error bars to indicatethe evolution of the low-density tail more clearly. Thetypical scale of the error bars for these plots is similar tothat of Fig. 3.The typical evolution of the measured radial profilesunder the influence of uniform white noise (with max ¼1%) is shown in Fig. 5(a). It is clear that low-density tailsare developing in the radial charge profiles, and the correspondingrms radius R b grows almost linearly with thenoise duration [see Fig. 5(b)]. Note that there is goodagreement between the rms radius obtained in the WARP2D simulations presented in Fig. 2(b) and the experimentalresults presented in Fig. 5(b). The evolution of the averagetransverse emittance given in Eq. (4) is estimated from theradial profile measurements for a given noise amplitudeand duration, using a total of 18 independent radialprofiles. Consistent with the WARP 2D simulations, weobserve a continuous emittance growth which is approximatelylinear with the duration of the noise (Fig. 6). Wenote from Figs. 2(b) and 5(b) that the rms radius R bincreases approximately linearly with time as R b =R b0 V. EFFECTS OF COLORED NOISESo far, we have used uniform white noise to modelrandom errors in the quadrupole focusing gradient. In anactual accelerator system, a power supply usually drivesseveral magnets in series which are mounted on a commonmechanical structure. Thus, any ripples in the driven currentsor ground vibration can comprise a colored noise inthe quadrupole focusing gradient with a finite autocorrelationtime (e.g., several focusing periods) [32]. In recentpapers [4,7], Bohn and Sideris pointed out that the presenceof colored noise can boost a small number of particlesto much larger amplitudes than inferred from a parametricresonance alone [33], by continually kicking halo particlesback into the right phase with the core envelope oscillation.Hence, in this section, we describe a preliminary experimentalstudy that investigates the possible synergistic effectbetween colored noise and the collective modesindicated by Bohn and Sideris.To modify the voltage amplitude with colored noise inevery half-focusing period T=2, we use a numerical algorithmgenerating Gaussian colored noise, based on theintegration of the Langevin equation according to [29] iþ1 ¼ i e T=ð2 acÞ þ w i max ð1 e T= ac Þ 1=2 : (7)Here, w i is a Gaussian random number generated anew ateach step i with zero mean and unit variance, ac is theautocorrelation time which measures the memory of ran-054203-7

MOSES CHUNG et al. Phys. Rev. ST Accel. Beams 12, 054203 (2009)the quadrupole mode frequency ! Q are expressed as [36]! B 2! q112K2R 2 b0 !2 q 1=2;(8)! Q 2! q134K2R 2 b0 !2 q 1=2;(9)FIG. 8. (Color) Dependence of the on-axis charge signal on(a) noise amplitude, and (b) noise duration for white ( ac ¼ 0)and colored noise ( ac > 0). The error bars indicate the standarddeviation of the on-axis charge calculated over 20 error samples.domness, and max is the amplitude of the desired colorednoise. To excite collective modes, we perturb the initialquiescent plasma (^s 0:2 and ! q ¼ 52:2 10 3 s 1 forthis case) by instantaneously increasing the voltage amplitudeby 1.5 times, and switching back to the original valueafter one focusing period T [34]. In this way, it is expectedthat a mixed mode with breathing mode period 2=! B 3:70T and quadrupole mode period 2=! Q 3:65T isexcited [35]. Here, the breathing mode frequency ! B andwhere R b0 is the equilibrium rms beam radius. Indeed, thebreathing mode frequency is identical to the frequency ofsmall-amplitude oscillations in the envelope equation (1).Note also that the two frequencies are both equal to 2! q inthe absence of space-charge effect (K ! 0Þ.For a matched beam, where there is no mismatch oscillation,scans of the noise amplitude and duration in Fig. 8demonstrate that white noise is more detrimental thancolored noise. Indeed, when the autocorrelation time ofcolored noise is greater than 2:5T, there is no significantchange in the on-axis signal even for large amplitude andlong duration of the colored noise. This can be understoodbecause the energy kicks introduced by the white noisetransfer maximum external energy into the system bycompressing the beam most abruptly. As reported previouslyin Ref. [37], a gradual change in the focusing fieldstrength tends to compress (or expand) the beam with lessemittance growth.FIG. 9. WARP 2D PIC simulation results for the beam responses to (a) colored noise with ac ¼ 5T and max ¼ 1%, (b) beammismatch introduced by the incorrect (1.5 times stronger in this case) focusing field strength for one FODO lattice period, and (c) thecombination of colored noise and beam mismatch. The mean radius ½aðtÞbðtÞŠ 1=2 (top), the on-axis density nð0Þ (middle), and the meantransverse emittance ½ x y Š 1=2 (bottom) are normalized to their initial values, respectively, and perturbation starts at t ¼ 0ms. For afair comparison, the same colored noise sample has been applied to cases (a) and (c).054203-8

EXPERIMENTAL INVESTIGATION OF RANDOM NOISE- ... Phys. Rev. ST Accel. Beams 12, 054203 (2009)The simulation results in Fig. 9(a) also indicate that thecolored noise ( ac ¼ 5T for this case) perturbs the initialmatched beam only slightly and leaves the average on-axisdensity nearly unchanged, as expected from the experimentalresults in Fig. 8. Here, we have chosen the autocorrelationtime to be ac ¼ 5T so that the colored noiseitself will not affect the initial beam too much (when ac >2:5T), and have a time scale comparable to the collectivemodes (for instance, 2=! B 3:70T). From the beammismatch case presented in Fig. 9(b), we observe theexcitation of collective modes, resulting in a 200% emittancegrowth. The envelope oscillations last for about 5 msand eventually damp away. The most interesting case is thebeam response in the presence of both the collective modesand the colored noise [Fig. 9(c)]. Even though the colorednoise itself cannot excite significant envelope oscillations,when combined with the collective modes, it gives rise tocontinuous emittance growth and an increase in the meanradius with much higher oscillation amplitudes. Thesesimulation results can also be interpreted as indicatingFIG. 10. (Color) Radial profiles are either (a) measured from theexperiments, or (b) obtained from the PIC simulations. Threedifferent external perturbation scenarios are considered: noperturbation (red curves); instantaneous mismatch only (greencurves); and both instantaneous mismatch and colored noise with ac ¼ 5T and max ¼ 1% (blue curves).enhanced halo formation. One possible underlying mechanismfor enhanced halo formation has been proposed byBohn and Sideris [4,7]. By extending the particle-coremodel [33,38], they showed that the combination of colorednoise and core envelope oscillations (breathing modesin their case) can eject particles to a much larger degreethan would be achieved in the absence of noise.To experimentally explore the synergistic effect betweencollective modes and colored noise, and the resultant enhancedhalo formation predicted in the simulations, wemeasured the radial charge profiles corresponding to cases(b) and (c) of Fig. 9. In the case where there is neitherinduced mismatch nor applied noise, the initial matchedbeam remains nearly in a quasiequilibrium state (withR b ¼ 0:916 cm) even after 20 ms of trapping [red curvein Fig. 10(a)]. On the other hand, when the beam isinstantaneously mismatched in the same manner as in theprevious simulation,pa large nonthermal ion tail is measuredat r>ffiffiffi2 Rb 1:29 cm [green curve in Fig. 10(a)].More interestingly, in the experiment with both instantaneousmismatch and applied colored noise, we observe thedevelopment of a small bump around ð3 4ÞR b [bluecurve and arrow in Fig. 10(a)]. The error bars are determinedprimarily from the offset errors ( 1fC) in constructingthe radial profiles. Although the accuracy is notadequate to cover the range below & 1fC, it still givesenough precision to resolve the small bump illustrated inFig. 10(a). Therefore, the experimentally observed bumpmight be attributed to the effect of the enhanced haloformation expected for the case of combined perturbations.To check the size and the location of the bump, we alsoperform WARP 2D PIC simulations for similar experimentalparameters. Despite the difference in their absolutevalues, the relative sizes and locations of the bumps illustratevery good agreement between the experimental resultsand the simulations [see Fig. 10(b)]. The discrepancyin the absolute value is likely related to 3D end effectspresent in the PTSX device, which could enhance the radialparticle loss to the wall for particles with large radialexcursion [26].VI. SUMMARY AND DISCUSSIONIn this study, we have presented experimental verificationof the random noise-induced beam degradation theoreticallyexpected in high-intensity accelerators. This waspossible because the PTSX device is a compact experimentalsetup with flexible control over the external focusingfields that can simulate the nonlinear transversedynamics of an intense beam propagating through an actualAG focusing system. By adding a small random rippleon top of the applied voltage waveform, we demonstratedthat externally driven noise continuously increases the rmsradius, transverse emittance, and nonthermal tail of thetrapped charge bunch depending on the amplitude andduration of the noise. In particular, we have observed the054203-9

MOSES CHUNG et al. Phys. Rev. ST Accel. Beams 12, 054203 (2009)combined effects of collective modes and colored noise,which are consistent with theoretical predictions and numericalsimulations.The amplitude of the noise ( max 1%) used to emulatemachine imperfection effects in this study may seem**somewhat** larger than the actual tolerance limits adopted indaily accelerator operation. This is a practical compromiseto overcome the difficulty of measuring small changes incharge signals ( 1fC) in the PTSX device (particularlyfor the tail distribution or halo particles in the off-axisregions). Nonetheless, in modern high-intensity accelerators,loss of only a few particles per meter can causeradioactivation that would preclude routine hands-onmaintenance [4]. Therefore, it is highly relevant to verifythe validity of numerical tools and to test the physicsmodels for beam loss in experiments with parameterseven **somewhat** beyond the actual tolerance limits. Forthe present study, the combined effect of collectivemodes and colored noise has been investigated with afixed autocorrelation time ac ¼ 5T. By measuring radialprofiles with several different autocorrelation times, e.g.,2= ac ! q , ! B , ! Q , ð! Q þ ! B Þ=2, ð! Q ! B Þ=2; ...,it may be possible to develop a more detailed understandingof noise-enhanced halo formation (for example, howthe location of the bump in the radial profile changes).Moreover, future experiments with different (either verylarge or very small) space-charge contribution are expectedto further improve our basic understanding of the role ofnonlinear space-charge force in random noise-inducedbeam degradation. Improving the accuracy of the chargecollector diagnostic will also facilitate a more detailedresolution of the small changes expected in this type ofexperiment.ACKNOWLEDGMENTSThis research was supported by the U.S. Department ofEnergy. The authors would like to thank Andy Carpe forhis excellent technical support, and Mikhail Dorf for usefuldiscussions regarding the WARP simulations. The researchwas carried out at Plasma Physics Laboratory while thecorresponding author (Moses Chung) was at PrincetonUniversity.[1] R. C. Davidson and H. Qin, Physics of Intense ChargedParticle Beams in High Energy Accelerators (WorldScientific, Singapore, 2001), Chap. 1, and referencestherein.[2] M. Reiser, Theory and Design of Charged Particle Beams(Wiley-VCH, Weinheim, 2008), 2nd ed., Chap. 7.[3] T. Wangler, RF Linear Accelerators (Wiley-VCH,Weinheim, 2008), 2nd ed., Chap. 12.[4] C. L. Bohn and I. V. Sideris, Phys. Rev. Lett. 91, 264801(2003).[5] F. Gerigk, Phys. Rev. ST Accel. Beams 7, 064202 (2004).[6] J. Qiang, R. D. Ryne, B. Blind, J. H. Billen, T. Bhatia,R. W. Garnett, G. Neuschaefer, and H. Takeda, Nucl.Instrum. Methods Phys. Res., Sect. A 457, 1 (2001).[7] I. V. Sideris and C. L. Bohn, Phys. Rev. ST Accel. Beams7, 104202 (2004).[8] G. Franchetti and I. Hofmann, in Proceedings of 2002European Particle Accelerator Conference, Paris, France(EPS-IGA/CERN, Geneva, 2002), p. 1353.[9] F. Gerigk, in Beam Halo Dynamics, Diagnostics, andCollimation: 29th ICFA Advanced Beam DynamicsWorkshop on Beam Halo Dynamics, Diagnostics, andCollimation, AIP Conf. Proc. No. 693 (AmericanInstitute of Physics, New York, 2003), p. 61.[10] D. V. Gorelov and P. N. Ostroumov, in Proceedings of the1998 International Linear Accelerator Conference,Chicago (ANL, Argonne, 1998), p. 654.[11] M. Ikegami, T. Ohkawa, Y. Kondo, and A. Ueno, inProceedings of 2004 International Linear AcceleratorConference, Lubeck, Germany (DESY/GSI, Darmstadt,Germany, 2004), p. 345.[12] P. S. Yoon, W. Chou, and C. L. Bohn, in Proceedings of the2005 Particle Accelerator Conference, Knoxville,Tennessee (IEEE, Piscataway, NJ, 2005), p. 117.[13] W. Paul and H. Steinwedel, Z. Naturforsch. A 8, 448(1953).[14] R. C. Davidson, H. Qin, and G. Shvets, Phys. Plasmas 7,1020 (2000).[15] E. P. Gilson, R. C. Davidson, P. C. Efthimion, and R.Majeski, Phys. Rev. Lett. 92, 155002 (2004).[16] H. Okamoto and H. Tanaka, Nucl. Instrum. Methods Phys.Res., Sect. A 437, 178 (1999).[17] R. C. Davidson and H. Qin, Phys. Rev. ST Accel. Beams 2,114401 (1999).[18] See, for example, Chaps. 3 and 5 of Ref. [1].[19] G. Franchetti, I. Hofmann, and D. Jeon, Phys. Rev. Lett.88, 254802 (2002).[20] S. Henderson, I. Anderson, N. Holtkamp, and C.Strawbridge, SNS Parameters List (SNS-100000000-PL0001-R13) (Oak Ridge National Laboratory, OakRidge, TN, 2005).[21] D. Jeon, SNS Superconducting Linac Beam Dynamics(SNS/ORNL/AP Technical Note 12) (Oak Ridge NationalLaboratory, Oak Ridge, TN, 1999).[22] M. Chung, E. P. Gilson, M. Dorf, R. C. Davidson, P. C.Efthimion, and R. Majeski, Phys. Rev. ST Accel. Beams10, 014202 (2007).[23] A. V. Fedotov, Nucl. Instrum. Methods Phys. Res., Sect. A557, 216 (2006).[24] E. P. Gilson, R. C. Davidson, P. C. Efthimion, R. Majeski,and H. Qin, Laser Part. Beams 21, 549 (2003).[25] E. P. Gilson, M. Chung, R. C. Davidson, P. C. Efthimion,R. Majeski, and E. A. Startsev, Nucl. Instrum. MethodsPhys. Res., Sect. A 544, 171 (2005).[26] M. Chung, Ph.D. thesis, Princeton University, Princeton,NJ, 2008.[27] A. Friedman, D. P. Grote, and I. Haber, Phys. Fluids B 4,2203 (1992).[28] M. Reiser, J. Appl. Phys. 70, 1919 (1991).[29] T. Stoltzfus-Dueck and J. A. Krommes (private communication).[30] P. R. Bevington and D. K. Robinson, Data Reduction and054203-10

EXPERIMENTAL INVESTIGATION OF RANDOM NOISE- ... Phys. Rev. ST Accel. Beams 12, 054203 (2009)Error Analysis for the Physical Sciences (McGraw-Hill,New York, 2003), 3rd ed.[31] Using Eq. (3.14) of Ref. [30], we find 2 R b 2^n ð@R b=@ ^nÞ 2^n¼h ^ni 2^n ðN=4Þh ^ni 3 .[32] S. Y. Lee, Accelerator Physics (World Scientific,Singapore, 2004), Chap. 2.[33] R. L. Gluckstern, Phys. Rev. Lett. 73, 1247 (1994).[34] E. P. Gilson, M. Chung, R. C. Davidson, M. Dorf, P. C.Efthimion, and R. Majeski, Phys. Plasmas 13, 056705(2006).[35] See, for example, Chaps. 4 of Ref. [2].[36] E. P. Gilson, M. Chung, R. C. Davidson, M. Dorf, P. C.Efthimion, A. B. Godbehere, and R. Majeski, Nucl.Instrum. Methods Phys. Res., Sect. A (to be published).[37] M. Chung, E. P. Gilson, M. Dorf, R. C. Davidson, P. C.Efthimion, and R. Majeski, Phys. Rev. ST Accel. Beams10, 064202 (2007).[38] T. P. Wangler, K. R. Crandall, R. Ryne, and T. S. Wang,Phys. Rev. ST Accel. Beams 1, 084201 (1998).054203-11