Rapid charge redistribution leading to core hollowing in a high-intensity ion beam
RRapid charge redistribution leading to core hollowing in a high-intensity ion beam
K. Ruisard, A. Aleksandrov ∗ Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA (Dated: August 18, 2020)Recently, the first direct measurement of a full 6D accelerator beam distribution was reported [1].That work observed a correlation between energy and transverse coordinates, for which the energydistribution becomes hollowed and double-peaked near the transverse core. In this article, a similarstructure is shown to emerge in expansion of an initially uncorrelated, high density bunched beamas the result of velocity perturbation from nonlinear space charge forces. This hollowing is obscuredwhen the 6D phase space is projected onto one- and two-dimensional axes. This phenomenon hasnot been widely recognized in accelerator systems, but parallels can be drawn to observations oflaser-ionized nanoclusters and electron sources for diffraction. While this effect provides insight intothe origin of the measured core correlation, it does not provide a complete description. A betterreproduction of the measured structure can be obtained via self-consistent simulation through theradio-frequency quadrupole.
I. INTRODUCTION
Understanding beam dynamics in the early stages ofcapture and acceleration is crucial in high-intensity ac-celerators. Space charge dynamics in low- to medium-energy transport are suspected to initiate halo formation[2, 3], causing uncontrolled losses at higher-power stages.Improved loss mitigation requires predictive capability ofaccelerator models. At present, no model has been shownto deliver accurate representation of the loss-prone beamtails and halo. This shortcoming has been attributedto incomplete information of the initial distribution [4],motivating work at the Spallation Neutron Source (SNS)Beam Test Facility (BTF) on complete characterizationof the 6D beam phase space [1].The measurements in [1] revealed an interplane corre-lation that appears as a hollowed, bimodal energy distri-bution in particle populations near the beam core. Thisfeature, which was seen to scale with beam current, wascharacterized as new and unexpected. However, as thisarticle will discuss, a similar phenomenon is broadly ob-served in a variety of charged particle systems. It willbe shown that conditions leading to core hollowing inthese systems can be present in a high-intensity acceler-ator front-end such as the BTF.Work in [1] also showed that the energy-hollowingcould be reproduced via particle-in-cell (PIC) simula-tions. This article extends that simulation work by tak-ing a closer look at the emergence of this feature. Aninitially uncorrelated compact bunch at the exit planeof the radio-frequency quadrupole (RFQ) is seeded andtransported through a model of the BTF medium-energybeam transport (MEBT). Very quickly the bunch devel-ops a transverse-energy dependence resembling observa-tions. It will be shown that the energy splitting is ac-companied by rearrangement of charge into a holloweddistribution, in which excess density “piles up” near the ∗ [email protected] edge as a result of an outwardly-propagating density per-turbation. This perturbation is launched by nonlinearspace charge in the initial distribution which causes thecore to expand more rapidly than the edge.The key to observing core hollowing is examining slicesrather than full projections of the 6D phase space. As ob-served in [1], fully projected distributions may be convex,with a smooth and monotonically decreasing density pro-file, but simultaneously the profile of a core slice may beflat-topped or hollowed. As a result, reliance on standard2D phase space visualizations may neglect significant corefeatures, particularly for high-charge bunches. The slice-emittance approach has not previously been adopted forcharacterization of ion beams. However, this approachis already established in the FEL community, as varia-tion of transverse phase space along the bunch can reducebrightness of the emitted coherent radiation [5, 6].As the demand for high current at high reliabil-ity grows, the details of internal, space charge drivencore structure will be important for accurately model-ing downstream beam dynamics and losses. One risk ofremaining blind to internal core structure is the over-simplification of beam distributions; e.g., adopting a 6DGaussian on the basis that 1D and 2D projections appearGaussian. It will be shown that the bunch produced byself-consistent tracking of particles through RFQ fieldsis distinctly non-Gaussian, containing significant inter-nal core structure in qualitative agreement with obser-vations. This article will argue that the pre-existenceof internal structure precludes the rapid hollowing fromoccuring in the MEBT.The paper is organized as follows. Section II summa-rizes the relevant observations at the SNS BTF. SectionIII describes a mechanism for space charge driven hollow-ing observed in mainly non-accelerator charged particlesystems. Then, Section IV examines particle-in-cell sim-ulation of an initial Gaussian distribution in a model ofthe BTF MEBT. A beam with excessively high chargedensity is used to visualize this mechanism. Section Vcompares this simple model to output from RFQ simu-lations. Finally, Section VI addresses the implications to a r X i v : . [ phy s i c s . acc - ph ] A ug FIG. 1: Measured dependence of energy distribution ondimensionality of a core slice at 21 mA. Thelongitudinal phase space is plotted in a shear-correctedplane, where ˆ φ = φ − (cid:68) dφdw (cid:69) w .simulation and measurement of high-intensity bunchedbeams. II. OBSERVATIONS AT THE SNS BEAM TESTFACILITY
In prior work [1], direct measurement of the 6D distri-bution was made downstream of the RFQ exit, where asecond-order correlation between the energy distributionand all transverse coordinates ( x, x (cid:48) , y, y (cid:48) ) was observed.For transverse coordinates near the beam core, the en-ergy distribution is hollowed. For coordinates near thetransverse edge and for the fully projected energy distri-bution, the energy profile is convex and single-peaked.The importance of studying slices rather than full pro-jections was a key insight adopted during these measure-ments. In order to discuss internal structure of the 6Ddistribution, it is necessary to distinguish between fullprojections , where information from all particles is pro-jected into a 1D or 2D profile, and partial projections, which represent only a fraction of particles selected byslices in the hidden (non-plotted) dimensions. In thenomenclature adopted here, slice:x,x’ indicates selectionof slices in the x and x’ coordinates. Unless otherwisenoted, slice:x,x’ bisects the beam core, selecting particlesnear x=0 and x’=0. The width of the slice is determinedby the physical width of slit apertures used in the mea-surement: ∆ x ∼ . x (cid:48) ∼ . III. SELF-FIELD-DRIVEN CORE HOLLOWINGIN CHARGED PARTICLE BUNCHES
Although not widely seen in accelerator beams, thisphenomenon has been observed across diverse fields. Inthe field of laser-plasma interactions, laser ionization of agaseous nanocluster results in Coulumb explosion (CE).A uniformly filled sphere is an equilibrium distributionin this expansion; however, even perturbative nonunifor-mity is shown to result in the formation of density ringson the outer edge of the expanding bunch as long as thedensity decreases from core to outer edge [7–9]. The ini-tial non-uniformity results in core ions with higher out-ward velocity than edge ions. This velocity eventuallyreaches a crossover point, where inner particle trajecto-ries catch up to outer particles and the radial velocityprofile becomes multi-valued. Density begins to pile upagainst a critical spherical surface and eventually formsa shock front at the expanding edge.This effect is also seen in the field of ultrafast electrondiffraction (UED), where laser-driven photoemission cre-ates a pancake-shaped bunch ( r >> z ) at the cathodesurface. In this system, a transverse ring-like densityperturbation appears at the expanding edge, resulting ina hollowed transverse profile [10, 11]. As in the case withthe BTF ion beam, the hollow xy profile is only visiblefor a slice through the longitudinal core, while the fully-projected profile is uniform. This study in [10] points tothis as one reason this effect was previously undetected.A second reason was the space charge dependence, asthe ring only forms at electron densities relevant to next-generation UED sources.As far as this author is aware, the longitudinal hol-lowing of a bunched ion beam has not been previouslyFIG. 3: Snapshots of phase space evolution in 1D longitudinal simulation at even intervals of 2.5 meter. A line-outof the spatial profile is also shown.observed. This may be explained not only by the need toexamine high-dimensional phase space slices, but also bythe uniqueness of conditions leading to this phenomenon:a compact, high-density expanding bunch. The transi-tion from RFQ to the MEBT provides such conditions.A similar scenario can be found in injectors for brightelectron beams. Indeed, similar behavior has been notedin modeling of rf photoinjectors [12], where rapid trans-verse emittance growth was seen to correlate with longi-tudinal position and was accompanied by a density spikeat the outer edge. Another parallel can be drawn tothe application of laser-wakefield accelerators (LWFA)as electron injectors to colliders or FELs [13, 14], wherevery small emittance leads to significant space charge ef-fect during bunch expansion between plasma boundaryand RF structure. Modeling of this system places thesame emphasis on knowing the full 6D phase space dis-tribution, and reveals a similar “deepening” of a bimodalenergy distribution with increased space charge [15].In two University of Maryland experiments with low-energy, high density unbunched electron beams, thetransverse profiles developed visible density rings [16, 17].This study concludes that “the perturbation is the re-sult of a strong beam-edge imbalance produced by anaperture in an expanding beam” [16]. However, trans-verse space charge “rings” may also be indistinguishablydriven by external nonlinearity. Additional structure in[16] is shown to originate from velocity perturbation dueto field distortion at the cathode grid.The systems discussed here share a mechanism forspace charge driven redistribution. All cases share thecharacteristics of non-negligible space charge forces, non-uniformity in initial charge distribution and a freely ex-panding bunch. In the following section it will be shownthat this same phenomenon is observed in simulationsof the BTF MEBT and generates interplane correlationssimilar to observations (Figures 1 and 2). In the BTFMEBT, the beam initially starts as a tightly focusedbunch at the RFQ exit. Within the first meter, the beamexpands in all dimensions. Quadrupole focusing limitstransverse size, but longitudinally the bunch expandsfreely. Therefore, it is consistent with this mechanismthat hollowing is observed mainly along the longitudinaldimension. IV. HOLLOWING OF AN INITIALLYGAUSSIAN ACCELERATOR BEAM
Previous work [1] demonstrated that the two-peakedenergy profile can be recreated through self-consistentparticle-in-cell simulation. The same simulation case isrevisited here in greater detail. In addition, a 1D longi-tudinal solver is used for illustration. Both simulationsuse the PyORBIT code [18]. Simulations are done fora 2.5 MeV H − bunch with 1 . × ions per bunch.This is equivalent to 100 mA averaged over an RF cy-cle, significantly higher than the nominal operating cur-rent of 40 mA. Increasing the space charge drives a morerapid charge redistribution with a deeper and more vis-ible structure. The case at nominal SNS parameters isaddressed in Section V.First, the 1D longitudinal model is useful for illus-trating the effect without high-dimensional slicing. Aninitial bunch of 200,000 macroparticles is seeded with around and uniform transverse distribution of radius 1.3mm (this is rms equivalent to the profile at the RFQexit). The transverse profile is held constant in the sim-ulation as if subject to uniform transverse focusing. Theinitial longitudinal has zero emittance and the spatialdistribution is Gaussian with σ z = 16 . × longer than the expected length at the RFQ exit. In thismodel, which takes an impedance-based approach to thespace charge calculation, a perfectly-conducting cylindri-cal pipe boundary is assumed at 2 × the beam radius.Figure 3 shows the longitudinal evolution in phasespace, with the initial distribution shown in frame 0. Inframe 1, a nonlinear velocity kick from the space chargefields is visible and the profile has broadened. Due tothe initial space charge kick, particles with the highestenergies are found midway between the core and edge.By frame 2, the profile is more uniform. Frame 3 cap-tures a crossover point, where inner particles have caughtup to outer particles. At this point the energy distri-bution w ( z ) becomes multi-valued and the lower-energytails start to fold under the expanding core. In the spa-tial profile, excess density has collected near the edgesand two prominent peaks have formed. Between frames3 and 4, the bunch expands with the two peaks “lockedin” at the outer edge. (a) Evolution of core energy slice (b) Evolution of core phase slice (c) Evolution of rms and peak energy(d) Evolution of rms and peak phase FIG. 4: Evolution of the phase and energy profiles of a 10% core slice. Figures (a) and (b) show the stacked phaseand energy profiles for the first 0.5 meters of evolution in the MEBT. The gaps indicate the quadrupole locations.Colored lines highlight the profile at 4 cm, 8 cm and 22 cm. Figures (c) and (d) compare the growth rate of the rmswidth versus the location of the distribution peak. When the distribution is single-peaked it is centered at z, w = 0.Once two peaks are formed they propagate outwards and quickly exceed the rms width. Although only one peak istracked in (c) and (d), the splitting occurs when the peak deviates from 0.TABLE I: Parameters for quadrupoles in first 1.3meters of BTF MEBT s[m] (cid:82) B · dz [T] L eff [ cm ] polarityQuad 1 0.131 1.149 6.1 +Quad 2 0.314 1.357 6.6 − Quad 3 0.575 1.08 9.6 +Quad 4 0.771 0.61 9.6 − TABLE II: rms parameters for 100 mA Gaussian initialdistribution. Emittances are un-normalized. x y z α -2.0 2.0 0 β [mm/mrad] 0.2 0.2 1.0 (1.3 deg/keV) (cid:15) [mm-mrad] 2.2 2.2 1.5 (50 deg-keV) The same signatures can be seen in a full 3D simu-lation of a Gaussian beam when core slicing is applied.The same case is studied as in [1], which used the codePARMILA to a propagate a bunch through the first me-ter of the BTF MEBT. The first meter includes four quadrupoles, which are modeled as hard-edged elementswith the parameters reported in Table I.A Gaussian beam with Twiss parameters shown in Ta-ble II is seeded at the plane of the RFQ exit. Theseparameters were chosen to be near the rms parametersexpected from the SNS RFQ with two notable differences.In addition to increasing the current to 100 mA, the longi-tudinal emittance is half the design value of 100 deg-keV,which further enhances the space charge effect.The simulation bunch is modeled with 500,000macroparticles. This is above the requirement for satura-tion of 100% rms parameters, and is chosen to maintaingood resolution of the phase space in core slices. ThisPyORBIT space charge model uses a 3D FFT Poissonsolver with grid size 64x64x64. The self fields are calcu-lated every millimeter.In order to visualize the emergence of core structure,evolution is tracked for a partial distribution containingonly particles near the transverse core in x and y . As thetransverse size varies during transport, the slice widthis calculated dynamically to contain 10% of particles inboth the x and y
1D projections.Figure 4 depicts the evolution of the core slice phaseand energy profiles over the first 50 cm of transport. (a) Core-slice projection at s = 4 cm. (b) Core-slice projection at s = 23 cm.(c) Full projection at s = 4 cm. (d) Full projection at s = 23 cm. FIG. 5: Evolution of the 100 mA 3D Gaussian beam. Frames (a) and (b) show the projection of the 10% core slicein x and y at s = 4 cm and s = 22 cm. Frames (c) and (d) show the corresponding full projection at both locations.In frames (a) and (b) the location of two individual macro-particles are marked by a red diamond and white star.Snapshots of the longitudinal phase space at s = 4 cmand s = 22 cm are shown in Figure 5. Within the firstfew steps, the effects of the space charge force are alreadyapparent as particles accelerate away from the core andthe energy distribution broadens. At 4 cm, splitting ofthe energy distribution w ( z ) is visible (Figure 5a, bluecurve in Figure 4b). The highest-energy particle in thisframe is indicated by the red diamond marker. Around8 cm, the energy perturbation has visibly changed thespatial profile (orange curve in Figure 4b). It appearsbroader, and shortly afterwards two distinct peaks ap-pear, framing a hollowed core. This onset is also visiblearound 8 cm in Figure 4d, where the single peak centeredat z = 0 splits into two peaks moving away from the core.The crossover point occurs around 22 cm, when theperturbed inner particles outrun edge particles. This canbe seen as the high-energy macro-particle indicated bythe red diamond overtakes the edge particle marked by awhite star. The folding-over of the tails in phase space isalso visible. At the point, the edge has reached maximumsteepness. After this point, indicated by the green curve in Figures 4a and 4b, the shape of the distribution locksin. The phase profile expands linearly, while the energywidth starts to saturate.The hollowing of the 100 mA beam is limited to thehigh current density at the core. For edge particles, thelongitudinal distribution remains single-peaked. As a re-sult, the fully-projected profiles obscure the sharp corefeatures, which is consistent with the observations in [1].In Figure 5c, the initial velocity perturbation at 4 cm isapparent as a very slight hollowing in the energy profileand barely-visible nonlinear tails in the phase space. InFigure 5d, the full projection at 23 cm has a flat-toppedenergy profile, while the spatial profile is still convex.Formation of the correlation is accompanied by in-crease in the longitudinal rms emittance. Unlike thehollowing, this can readily be seen in the full distribu-tion without taking core slices. For the 100 mA casediscussed here (Table II), the 100% rms emittance morethan doubles in the first 20 cm of transport. Figure 6compares emittance growth for different beam currentsand fixed rms parameters (using the values in Table II).FIG. 6: rms emittance evolution for initially Gaussianbeam with (cid:15) z = 50 deg-keV, average current asindicated.TABLE III: Output emittances from PARMTEQsimulation of RFQ. Emittances are un-normalized. output x y z α -2.1 1.6 0.2 β [mm/mrad] 0.19 0.14 0.6 (0.8 deg/keV) (cid:15) [mm-mrad] 3.34 3.35 3.95 (132 deg-keV) As charge density decreases, the magnitude of emittancegrowth decreases and the saturation of emittance growthoccurs on a much longer timescale. As one might expect,in core slices at lower currents the depth of the hollowingdecreases and the point at which the phase distributionsplits into two peaks occurs later. In this way, growth ofthe 100% emittance can be understood as a signature ofthe less-visible redistribution of the core density.
V. RELEVANCE TO MEASUREMENTS
One notable discrepancy between the simulation andmeasurement is that a much higher average current (100mA) is needed to drive a visible correlation than wasavailable in measurements (40 mA). In addition, theGaussian distribution expands symmetrically while theobserved profile in Figure 1 is consistently lop-sided.These discrepancies can be explained by the fact that aGaussian beam is a poor model for the 6D phase space atthe RFQ output. A bunch generated by RFQ simulationalready contains interplane correlations that also resem-ble the observed structure. This is explored in detail in[19], but repeated here for illustration.The RFQ bunch is generated by PARMTEQ [20] sim-ulation of the 402.5 MHz SNS RFQ [21, 22]. The 50 mAinput beam has uniform phase, is single-valued in energy at 65 keV and has a transverse Gaussian distribution.5,000,000 macroparticles are used to keep good particlestatistics in high-dimensional slices. In this case, the slicewidth is fixed and held equal to the slice width used in6D phase space measurements. The output bunch has41 mA average current with rms Twiss parameters asreported in Table III.Figure 7a shows the energy profiles of the fully pro-jected bunch and a 4D core slice after propagation inPyORBIT to the measurement plane (1.3 meters down-stream of the RFQ). A comparison is made to a Gaussiandistribution at the RFQ exit in Figure 7b. The initialGaussian bunch is rms-equivalent to the PARMTEQ-generated bunch. Although the fully projected profileis almost identical between the two cases, the core sliceprofiles are distinctly different. The depth and widthof the hollowing is very reduced for the 41 mA Gaussianwhen compared to the 100 mA case described previously.In comparison, the RFQ output bunch both has sharperfeatures and qualitatively resembles the measured energyprofile in Figure 1.Because of the core structure formed in the RFQ, thecharge density of the core is already lower and moreuniform than in the Gaussian bunch. As a result, nosignificant charge redistribution or emittance growth isobserved in propagation through the MEBT. As thePARMTEQ-generated bunch is understood to be morerealistic than the Gaussian bunch, it is unlikely that themechanism described here is excited in the experiment.The hollowing of the bunch occurs earlier within theRFQ, most likely during the initial capture and bunchformation when space charge is most significant.
VI. DISCUSSION
In summary, previous studies at the SNS BTF ob-served a second-order correlation between energy andtransverse coordinates. This article describes a simplemechanism by which a Gaussian bunch can rapidly de-velop a similar structure in medium energy transport.The combination of nonlinear space charge fields andbunch expansion results in hollowing of the core den-sity, an effect which has been seen in other high-densitycharged particle systems. While the hollowed core quali-tatively resembles observations, self consistent simulationof the RFQ suggests that in the laboratory this structuredevelops earlier.Although insight from this mechanism may be usefulin understanding the origin of core-hollowing within theRFQ, the dynamics may be quite different. The hol-lowing described here depends on bunch expansion: asthe space charge density drops rapidly, the nonlinearityfrom the initial distribution is imprinted as a velocityperturbation. In contrast, within the RFQ the bunch isimmersed in transverse and longitudinal focusing. Anyexpansion occurs in the presence of external fields.As this phenomenon is not expected to occur in a re- (a) 41 mA RFQ output (b) 41 mA Gaussian beam
FIG. 7: Comparison of full and slice energy profiles for simulations with two different initial conditions: (a) 41 mAbunch produced from RFQ simulation and (b) 41 mA Gaussian bunch at RFQ exit. The initial MEBT distributionsare rms equivalent.alistic MEBT bunch, effort should be made to avoid ar-tificially inducing it in simulation. This study illustrateshow a space charge driven correlation may be obscured intypical full projections and reinforces the need to createinitial distributions with realistic, fully correlated 6D dis-tributions. As demonstrated, spurious emittance growthcan be driven when a convex initial distribution is inap-propriately assumed. To avoid over-estimating the spacecharge force, extra care must be taken to include the ef-fects of upstream transverse-longitudinal coupling whengenerating high intensity simulation. The implicationsof this core structure for other space charge driven ef-fects, such as halo formation and beam loss, is not yetunderstood and will be a focus in future studies.
A. Acknowledgments