High dimensional characterization of the longitudinal phase space formed in a radio frequency quadrupole
K. Ruisard, A. Aleksandrov, S. Cousineau, A. Shishlo, V. Tzoganis, A. Zhukov
HHigh dimensional characterization of the longitudinal phase space formed in a radiofrequency quadrupole
K. Ruisard , A. Aleksandrov, S. Cousineau, A. Shishlo, V. Tzoganis, A. Zhukov ∗ Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA (Dated: August 18, 2020)Modern accelerator front ends almost exclusively include radio-frequency quadrupoles for initialcapture and focusing of low-energy CW beams. Dynamics in the RFQ define the longitudinal bunchparameters. Simulation of the SNS RFQ with PARMTEQ seeded with a realistic LEBT distributionproduces a 2.5 MeV, 40 mA H- beam with root-mean-square emittance of 130 deg-keV. A detailedcharacterization of the longitudinal phase space is made, including a novel study of the dependence oflongitudinal emittance on transverse coordinates. This work introduces a new virtual slit techniquethat provides sub-slit resolution in an energy spectrometer as well as an approach for visualizing4D phase space data. Through simulation and measurement, the RFQ-formed bunch is confirmedto have significant internal correlated structure. The high-dimensional features are shown to be inqualitative agreement. However, the measured rms emittances are up to 30% lower than predicted,closer to the design value of 95 deg-keV.
I. INTRODUCTION
Low-level beam loss is a fact of life in high-intensityaccelerator facilities. Controlling and reducing lossesto maintain a safe accelerator environment is achievedmainly through online empirical optimization. One toolcurrently missing from the arsenal is high fidelity simula-tion capable of predicting these losses. As high-intensityaccelerators continue towards higher demands in beampower, the need for this capability becomes more accute[1]. A strong contributor for losses in a linear acceleratoris beam halo [2, 3]. As the beam distribution is both thesource and driver of halo particles, loss-level simulationaccuracy will require an equally accurate representationof the initial distribution.There are two approaches for generating a realisticfront-end initial distribution. One is a pure “end-to-end”approach, which applies self-consistent simulation of theentire beam transport system starting at or downstreamof the ion source. This may include self-consistent model-ing of the ion source/extraction electrodes, abstracting toan idealized distribution in the Low Energy Beam Trans-port (LEBT) section, or measuring the transverse phasespace of the LEBT beam. The LEBT distribution isthen propagated through the radio frequency quadrupole(RFQ). The longitudinal bunch is formed inside the RFQ,where dynamics are complicated by nonlinear focusingfrom both the vane structure and space charge. Thecomplexity of the simulation may limit the accuracy ofoutput bunch, as there is large potential for errors.Previous work at Los Alamos [4, 5], found that thebunch generated through simulation of the RFQ wasnot sufficiently accurate to model beam dynamics ina medium-energy transport line (MEBT). Particularly,transport with mismatched optics was seen to be verysensitive to the initial distribution. In this effort, cor- ∗ [email protected] recting the simulated bunch by rescaling to match ob-served rms parameters was not sufficient to reach goodagreement.Alternatively, a bunch may be generated from mea-surements in the MEBT, after the longitudinal bunch isfully formed but at an energy where detailed measure-ments are still possible. Characterization of the beam inthe MEBT circumvents the need to model the complexinternal RFQ dynamics, and arguably results in morea representative bunch. However, internal correlationsare neglected in this approach, which typically relies on2D projections or Twiss parameters (for example, see[6, 7]). Direct measurement of the 6D beam distributionhas been demonstrated [8], but for now end-to-end simu-lation remains the most accessible option for generatingfully-correlated particle coordinates.Knowledge of the fully-correlated distribution is nec-essary to accurately portray a bunched beam. As thisarticle will demonstrate, the bunch formed in the RFQhas non-trivial internal structure. The formation of thelongitudinal phase space is mediated by the space chargeforce, which couples the three planes [8–11]. Observa-tions reported here show how both the bunch shape andenergy profile to vary with distance from the high-densitycore. As core mismatch is known to excite halo growth[3, 12], it is almost certain that loss-predictive simula-tions will require knowledge of the realistic 6D structure.Given that simulation is the most readily availablesource of fully correlated bunches, one may wonder towhat extent simulation reproduces the true 6D structure.If there is discrepancy in the rms predictions, can one stilltrust the high-dimensional features? To begin addressingthis question, a detailed characterization of longitudinalphase space is compared with predictions from RFQ sim-ulation. While the primary metric is the rms emittance,it is applied to slice emittances rather than full emit-tance. By varying slice dimensionality and location, thedependence of longitudinal emittance on transverse co-ordinates is studied. This provides a method to visualizethe high-dimensional features inside the RFQ bunch. a r X i v : . [ phy s i c s . acc - ph ] A ug A. SNS Beam Test Facility
The SNS Beam Test Facility is a one-to-one replica ofthe SNS front-end, composed of 50 mA H- ion source,65 kV LEBT, 402.5 MHz RFQ and 1.3 meters of MEBTquadrupoles. In addition, the BTF is equipped with ex-tensive diagnostics enabling direct measurement of the6D phase space distribution. This phase space diagnos-tic includes two pairs of vertical/horizontal slits for isola-tion of the transverse phase space coordinates, followedby an energy spectrometer comprised of a 90 ◦ dipole andvertical slit. Finally, time-of-arrival measurement of sec-ondary electron emission from a beam-intersecting wireserves as a bunch shape monitor. The energy spectrom-eter and bunch shape monitor are used for longitudinalphase space measurements described here.Accelerator physics studies at the BTF are motivatedby the goal of demonstrating halo-predictive simulation.Ongoing efforts have followed a three-pronged approach:extensive characterization of the initial MEBT beam dis-tribution [8], deployment of high dynamic range phasespace diagnostics for halo detection, and extension of theBTF MEBT to support studies of halo evolution [13].The work described here falls in the first category, as itaddresses the applicability of RFQ simulations to high-fidelity simulation. B. Emittance Convention
With the BTF apparatus for longitudinal emittancemeasurement, a dynamic range in excess of 10 has beendemonstrated. As the rms parameters can depend heav-ily on the threshold, it is necessary to speficy the appliedthreshold. In order to standardize emittance values be-tween simulation and measurement, we adopt the follow-ing metrics for reporting emittances: • •
1% emittance, and •
10% emittance, representing the core of the beam.In simulation it is common to report emittances based onpercentage of enclosed particles, such as 90%, 99% emit-tances. This is distinct from the definitions here. How-ever, the term “100% emittance” is still used to indicateinclusion of all simulation particles when no threshold isapplied.In plots of the longitudinal phase space, positive phasecorresponds to positive time. The tail of the bunch,which arrives at a later time than the head, has φ tail >φ head . This phase convention matches the conventionused in RFQ simulation. FIG. 1: Illustration of high dimensional slices in 3Dspace. A slice of the cube can be made along onedimension (eg, ˜ z = 0, the green volume), two dimensions(˜ z, ˜ x = 0, blue) or all three dimensions (˜ z, ˜ x, ˜ y = 0, red).If the 1D and 2D slice volumes are projected onto the y, z plane, they will be indistiguishable, as the slice ˜ x isalong a hidden (projected) dimension. C. Dimensionality
In addition to threshold, the reported emittances willdepend on the dimensionality of the phase space used forthe calculation. Typically, emittances are calculated forthe fully projected phase space. That is, the ( φ, w ) coor-dinates of every particle regardless of location in trans-verse phase space are included. The BTF longitudinalemittance apparatus is set up to sample phase space fora slice in the transverse coordinates. This is referredto as a partially projected phase space, and the result-ing emittance a partial or slice emittance. It should beimmediately clear that many unique partial projectionsof the longitudinal phase space are possible. Slices canbe made in one or several or all of the transverse coor-dinates, and the slices can be taken at varying distancefrom the beam core. Figure 1 illustrates the slice con-cept in three dimensions. As slices are generally madeon hidden (unplotted) dimensions, it is not explicitly ap-parent whether a phase space plot represents a full orn-dimensional partial projection.Therefore, when discussing slice emittances, it is im-portant to indicate both the dimensionality (how manyslices) and the slice location. Here, f indicates the full6D phase space density f ( x, x (cid:48) , y, y (cid:48) , φ, w ). ˆ f indicatesa partial projection, where a slice is made in at leastone dimension. A tilde is used to indicate a slice of fi-nite width. Unless otherwise specified, the slice width isequal to the width determined by physical apertures inthe measurement. If coordinates do not appear as argu-ments or slices of ˆ f , the density along that coordinateis integrated. With this notation, a partially projectedlongitudinal phase space representing only the particleswithin a finite slice centered at x = 0, x (cid:48) = 0 is describedas ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) =0 . (a) Horizontal phase space of RFQinput distribution. (b) Vertical phase space of RFQinput distribution. (c) Fully projected longitudinalphase space at the RFQ exit. FIG. 2: Fully projected phase space plots for initial (2a,2b) and output (2c) distributions from Parmteq simulation.Density is plotted in linear scale. Contours on (2c) show the 10%, 1% and 0.1% threshold levels.TABLE I: rms parameters of the realistic LEBTdistribution at the entrance to the RFQ.
Quantity horizontal vertical (cid:15) [norm, mm-mrad] 0.24 0.24 α β [mm/rad] 51 49 TABLE II: rms parameters of the bunch at the RFQoutput, based on Parmteq simulation for different inputdistributions. Transmission calculated for 50 mA inputcurrent. Longitudinal emittances are 100%, unnorm.
Input distribution Realistic 4D Waterbag KVTransmission 82% 90% 88% (cid:15) z [deg-keV] 127.1 88.8 102.2 α z β z [deg/keV] 0.88 1.38 1.16 (cid:15) x , (cid:15) y [norm, mm-mrad] 0.22 0.12 0.15 The organization of this article is as follows. Sec-tion II defines the “expected distribution,” as determinedthrough RFQ simulation. After that, the measurementtechnique is introduced in Section III, including accoun-ing for the dominant error sources. The largest erroris shown to be through point spread in the phase co-ordinate. Correction of the point spread error is justi-fied through simulation and application of a virtual slitmeasurement. The high dimensional characterization oflongitudinal phase space is reported in Section IV. Theresults show the dependence of the longitudinal slice dis-tribution on RFQ amplitude and transverse coordinates.Finally, Section V summarizes the comparison betweenthe expected and measured distributions.
II. SIMULATIONA. RFQ output distribution
Original design studies for the SNS RFQ used the LosAlamos code PARMTEQ[14]. The RFQ accelerates H − from 65 kV to 2.5 MeV, achieved with vane voltage 83kV and 449 cells. Additionally, the design is constrainedto produce ≤
95 keV-deg at maximum current output.This goal was met with normalized input emittance 0 . >
90% [15–17]. Inthis paper, the PARMTEQ simulation is repeated withan input beam based on LEBT measurements.The PARMTEQ space charge calculation uses theSCHEFF module with a cylindrical geometry. Satura-tion of the PARMTEQ simulation was judged by the rmsTwiss parameters of the output bunch. 40,000 particlesand a grid spacing of 10 radial segments and 20 longitu-dinal segments was sufficient. However, for the results re-ported here up to 5,000,000 macroparticles are used. Thehigher particle number was necessary for good statisticswhen calculating rms emittances for high-dimensionalslices.The input beam is initially mono-energetic with w ≡ T − T = 0 for all particles, and initial uniform ran-dom phase. The transverse distribution is generated frommeasurements of the horizontal and vertical phase spacedistributions in the LEBT. These measurements were ac-quired in 2012 for ion source output of 50 mA. The mea-sured distribution is transformed from the measurementplane to the RFQ entrance using a matrix equations. Theresulting transverse phase space distributions are plottedin Figures 2a and 2b. The rms Twiss parameters are re-ported in Table I.As the motivation of this study is to address therole of RFQ simulations in high-fidelity modeling, themeasurement-based LEBT distribution is used to gener-ate the expected distribution, under the assumption thatthis is the most likely to resemble the actual beam pa-rameters. However, comparison to equivalent transverseFIG. 3: Dependence of calculated longitudinalemittance on width of the slice in coordinates ( x, x (cid:48) )centered at (0 , (cid:15) z = 127 . φ rms = 10 . ◦ and w rms = 12 . x, x (cid:48) )slice centered over the beam core. The narrowest slices(left-most points) have widths comparable to the mea-surement resolution, ˜ x = ± . x (cid:48) = ± . Position [m] L eff [m] (cid:82) B · dl [T] Polarity0.1306 0.061 1.12 F0.3139 0.066 -1.25 D0.5751 0.096 1.08 F0.7709 0.096 -0.61 D B. Expected distribution in the MEBT
The BTF measurements are made with respect to aplane 1.36 meters downstream of the RFQ. The refer-ence point is the location of the first vertical slit used inthe phase space measurement. After this point, at least98% of the beam is intercepted. In the remaining 2%“beamlet,” there should be no contribution from spacecharge on the beam evolution, and downstream mea-surements can easily be mapped to this plane via matrixequations. For the purposes of comparison, it is consider-ably more straighforward to propagate the self-consistent6D Parmteq distribution to the measurement plane thanback-propagate the measured phase space. With this inmind, the expected distribution is defined as the outputfrom Parmteq simulation seeded with the initial mea-sured LEBT distribution at the plane of the first slit inthe emittance apparatus.Modeling of the MEBT is done with the particle-in-cell code PyORBIT [18]. Between the RFQ and the firstslit, the MEBT contains four quadrupoles. A hard-edgedmodel was used, with parameters listed in Table III. Astepsize of 1 cm is used for the space charge calculation.The expected distribution at the measurement plane isplotted in Figure 4. For the fully projected phase space, slice (cid:15) z [deg-keV] rms φ [deg] rms w [keV]none (full) 122 5.4 22.8˜ y = 0 117 5.1 22.9˜ y, ˜ x = 0 135 5.2 25.8˜ y, ˜ x, ˜ x (cid:48) = 0 128 4.9 26.4˜ y, ˜ x, ˜ x (cid:48) , ˜ y = 0 144 4.9 29.8 TABLE IV: Dependence of rms quantities of expecteddistribution on dimensionality of phase space slice. Allslices are centered over the beam core. rms values arecalculated with 1% threshold applied. For these results,the RFQ simulation is seeded with 5,000,000 particlesto obtain good statistics in high-dimensional slices. Theslice widths are twice as large as in measurement for thesame reason.the 100% rms emittance is (cid:15) z = 131 deg-keV and the rmswidths are 5 . ◦ and 24 keV. The phase width is reportedfor the “shear-corrected” frame, where the linear phasecorrelation has been subtracted. For comparison, theuncorrected phase width at this location is 43 ◦ . Thebunch has significantly different aspect ratio than at theRFQ exit (10 . ◦ and 12 . ∼ × and thephase width to decrease by ∼ × .Table IV compares the 1%-thresholded emittance andrms widths for slices of different dimensionality in the ex-pected distribution. The rms emittance does not have astrong dependence on the dimensionality of the slice. Inthe context of measurement, the variation in emittancevalues is comparable to the uncertainty in measurement,which will be shown to be around 10 − f ( w ) = (cid:82) dφ ˆ f ( φ, w ) | (cid:126)x on the dimensionality of aslice in transverse dimensions. This illustrates the in-crease of rms energy width reported in Table IV as wellas the presence of very non-Gaussian internal structure.This structure strongly resembles with the initial obser-vation of high-dimensional correlations reported in [8]. III. APPARATUS
The apparatus for longitudinal emittance measure-ment at the SNS BTF is a combination of an energyspectrometer and bunch shape monitor. This apparatus FIG. 5: Dependence of partially projected energyprofile ˆ f ( w ) on dimensionality of slice.employs a dipole-slit system to isolate a narrow band ofenergy, followed by a bunch shape monitor to measure thephase distribution. The apparatus was first described in[8], where it was utilized in measurements of the full 6Ddistribution.Figure 6 shows the transverse optics used during mea-surements. Energy selection is made after a 90 ◦ dipole,located around s = 3 meters in Figure 6. Upstream ofthe dipole, two vertical slits select thin slices in x and x (cid:48) . This ensures a very narrow beam enters the dipole,such that the horizontal spread at dipole exit is createdmainly by the beam energy spread. A third vertical slitdownstream of the dipole blocks all but a thin slice inthe energy distribution. All slits are the same width.The width of the third slit is measured to be 0 . ± . µ m-wide horizon-tal wire that intersects the beam. This wire emits sec-ondary electrons, which are collected and focused onto amicrochannel plate. Between the wire and plate, an rf de-flecting field streaks the beam so that vertical position atthe plate corresponds to time-of-arrival. The microchan-nel plate amplifies the electron signal which is then im-aged via a phosphor screen and camera. The signal at theBSM camera is the partial phase distribution ˆ f ( φ ) | ˜ x, ˜ x (cid:48) , ˜ y , representing the fraction of beam selected by the verti-cal slits and BSM wire. Thanks to the sensitivity of theBSM screen and the high bit depth of the BSM camera,a signal-to-noise ratio of 10 . was achieved. The BSMconcept is explained in more detail in [13, 20, 21].By varying the energy selected by the third slit, thephase space ˆ f ( φ, w ) can be reconstructed. The centralslice emittance ˆ f ( φ, w ), which includes the peak densityvalue, is shown in Figure 7. In this measurement, phasespace is sampled with stepsize ∆ w = 0 . φ − w correlation is subtracted.FIG. 6: Optics view of longitudinal emittance measurement, showing the location and effect of the three verticalslits and bunch shape monitor (BSM). Position s is measured from the exit face of the RFQ. After Slit f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y ∼ . A threshold isapplied at 6 × − of the peak density. A. Dimensionality of measurement
Due to selection by upstream slits and the BSM wire,the measured ˆ f ( φ, w ) phase space is a partial projectionbased on only a fraction of the total phase space volume.As shown above in Figure 3, for the expected distributionthe core emittance could be as much as 10% lower thanthe full rms emittance. The partial projection measuredwith BTF longitudinal emittance apparatus is:ˆ f ( φ, w ) = (cid:90) dy f ( x, x (cid:48) , y , y , φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y where ˜ x = x ± ∆ x , ˜ x (cid:48) = x (cid:48) ± ∆ x (cid:48) , ˜ y = y , ± ∆ y .Notice that the vertical coordinates are in a frame y , y rather than the standard y, y (cid:48) . At the BSM loca-tion, the vertical slice that is selected is ˜ y = y wire ± ∆ y .However, the horizontal coordinates are referenced to thelocation of the first vertical slit, which is upstream of theBSM wire by 2.2 meters, four quadrupoles and one 90 ◦ dipole. The slice made by the BSM wire is rotated in the y, y (cid:48) phase space at the reference plane. FIG. 8: Vertical phase space at the first slit, showingthe BSM wire bisecting the beam core. The intensityscale is logarithmic.Figure 8 shows the vertical phase space of the beamat the first slit location, measured using a slit-scan ap-proach. The shadow of the BSM wire is visible.Because the longitudinal emittance apparatus imagesa 3-fold slice in the six-dimensional phase space, a five-dimensional scan is required to measure the emittance ofthe “full” beam. However, a five-dimensional scan withhigh dynamic range and reasonable resolution would havea very long duration. The measurements have an effectiverepetition rate of approximately 2.5 Hz, which is half thebeam repetition rate. At each point in the camera image f ( φ ) is averaged for 20 shots in order to improve dynamicrange. Sampling the phase space on a grid of size 14 x14 x 14 x 40 in ( y , x, x (cid:48) , w ) would take an estimated9 days of continuous measurement. Instead, this papertakes the approach of conducting four-dimensional scans,iterating over coordinates ( x, x (cid:48) , w ). A 4D scan requiresapproximately 16 hours. The dependence on the fifthcoordinate, y , is explored by repeating the 4D scan overa range of BSM wire positions.TABLE V: Values for rms point-spread function and1 σ errorbars Quantity w [keV] φ [ ◦ ]uncertainty 0 . . . . FIG. 9: Illustration of virtual slit concept. Two phaseprofiles (thin curves) are measured with the BSMcamera for two dipole magnet settings separated by 0.05A. The profiles plotted are obtained with a much widerslit (1 mm) than the standard 0.2 mm slit used foremittance measurements, and the phase profile nearlyfills the camera frame. The heavy black line is thedifferential profile, which recovers two narrower profilescorresponding with the the two edges of the wide profile.
B. Accounting for point-spread increase tomeasured phase
As noted in Section II B, space charge defocusingcauses a narrowing in phase downstream of the RFQ.This brings the phase width close to the phase resolutionof the measurement. The phase resolution is not limitedby the resolution of the BSM, but by the point spreadfunction originating from the finite slit widths.Point spread is a systematic, asymmetric error thatacts to inflate the measured rms values. While the to-tal point-spread is the combination of the three verticalslits, BSM wire and internal BSM electron focusing, thedominant contribution is the width of the third (energy)slit, which affects measurement of both phase and en-ergy. Table V summarizies the rms point spread widths,as well as systematic uncertainty originating primarilyfrom uncertainty in calibration curves. As seen in TableV, the rms phase point-spread is much larger than theuncertainty, and comparable to the expected rms width5 . ◦ . The origin and calculation of errors are discussedin more detail in Appendix A.The majority of the 3 . ◦ point-spread is due to thelarge φ − w correlation at the BSM plane. For the se- FIG. 10: Comparison of analytic error estimate withactual errors for simulated and measured emittancemeasurements. The lines connect the raw rms values ofthe reconstructed phase space (higher values) and thevalue after applying correction, including 1 σ errorbars.The open circles show the “true” emittance and phasewidth, as determined from the virtual slit measurementand simulated emittance reconstruction. The calculatedcorrection for the phase space shown in Figure 7 isincluded, for which the “true” values are not known.The separation between emittance values at the samephase width is due to the difference in energy widths,which vary between 20 keV and 26 keV.lected energy slice, the measured phase profile will bewider than the shear-corrected profile that the appara-tus is intended to measure. In comparison, the effect ofthe energy point-spread is negligible, as the estimated0 . (cid:10) φ (cid:11) = (cid:10) φ meas. (cid:11) − (cid:68) φ p.s.f. (cid:69) .However, both the expected distribution described inSection II B and the point-spread function have signif-icantly non-Gaussian features. Therefore, in this analy-sis correction to the rms values is estimated on the basisof simulated and measured recovery of the “true” phasewidth, which suggest a much smaller correction than es-timated through propagation of Gaussian errors.In simulation, the point spread error is esimated bypropagating the expected distribution through a PyOR-BIT model of the longitudinal phase space apparatus,including all three slits as shown in Figure 6. Details ofthe approach are included in Appendix B. In measure-ment, it is possible to obtain sub-slit phase resolutionthrough application of a novel virtual slit method.The virtual slit method requires collecting two phaseprofiles separated by a differential step in energy slit po- (a) Measured slice emittance compared to fully-projectedsimulation emittance. Both calculations apply 1% threshold. (b) Measured vs. simulated transmitted current. As theRFQ has degraded transmission, and the ion source wasproducing less than 50 mA at the time of measurement, thesimulated current is rescaled to saturate at the same outputcurrent. FIG. 11: Dependence of transmission and emittance on RFQ vane voltage. Results are compared with PARMTEQsimulation of the realistic LEBT distribution as well as an rms-equivalent waterbag, both at 50 mA input.sition and subtracting one from the other. The differ-ence waveform includes peak and an anti-peak alignedwith the leading and trailing edges of the phase profile,as illustrated in Figure 9. The difference profiles cor-respond to the phase profile of a beamlet selected by avirtual slit of width equal to the step size. The tech-nique is analagous to the use of scrapers in beam profilemeasurements, in which transmission is measured as afunction of scraper position and differentiated to recoverthe spatial profile. As the technique doubles data collec-tion time and reduces dynamic range, it is not applied tothe measurements reported in Section IV. More detailson the implementation of the virtual slit technique arediscussed in Appendix C.Comparison of the “true” to “measured” rms values inboth simulated reconstruction and virtual slit measure-ment allows determination of an appropriate correctionfactor. In this case, a multiplicative correction to therms phase, energy and emittance reduces the systematic,point-spread error to well within the uncertainty interval.As expected, the point-spread function has a relativelysmall effect on the near-flat-topped energy distribution:the “true” rms energy width was roughly 95% the raw“measured” width in both simulation and experiment.The correction to phase width is larger, as expected. Ad-ditionally, the required correction has a threshold depen-dence; as more tails are included in the rms calculation,the relative point-spread error is smaller. At 1% thresh-old, a correction factor of 87% minimizes the residualerror in simulation and experiment. At 10% threshold,the corrected value decreases to 83% of the raw width.There is not enough dynamic range in simulation (lim-ited by particle count) and measurement (limited by vir-tual slit method) to recommend a correction at the 0.1% threshold. For the analysis here, the same 87% correctionfactor is applied to the 0.1% threshold values.Figure 10 illustrates the magnitude of the rms phaseand emittance correction against the “true” error. Theone-sigma uncertainty interval is plotted as well; theuncertainty on emittance is estimated by Gaussian er-ror propagation of Table V values under the assump-tion (cid:15) z ≈ ∆ φ ∆ w . (This is valid in the upright, shear-corrected frame as apparent from rms values in TableIV.) Applying the estimated corrections to the measuredslice emittance shown in Figure 7, the rms values are:rms (cid:15) φ = 126 − ±
14 deg-keV,rms φ = 5 . ◦ − . ◦ ± . ◦ andrms w = 22 . − . ± . IV. MEASUREMENTS
Measurements of the longitudinal emittance, which asdescribed above represents a high-dimensional slice in thetransverse phase spaces, are repeated many times to mapthe dependence on several parameters. First, emittanceis measured over a range of RFQ voltages, which is afree parameter that may be set to obtain minimum out-put emittance. Second, a four-dimensional scan is usedto map dependence of the emittance on the transversedimensions. This is then integrated to reconstruct thelower dimensional partial projection ˆ f ( φ, w ) | ˜ y . Finally,the 4D scan is repeated for several BSM wire locations, tomeasure dependence on the coordinate ˜ y . Measurements (a) Minimally-processed data from the 4D scan of ˆ f ( x, x (cid:48) , φ, I ) | ˜ y . Each frame shows a partial projection ˆ f ( φ, I ) | ˜ x , ˜ x (cid:48) , withvertical axis I and horizontal axis φ . The axis limits are held fixed for all subplots, but the color scale is not. Each sub-framecorresponds to a different location in x, x (cid:48) . Color is signal strength in logarithmic scale. Data has been cleaned of spurioussignals and averaged.(b) Image of the 4D scan along the x, x (cid:48) axes, which arereferenced to the location of the first vertical slit. The color ofeach point is the integrated signal in the φ, w dimensions.Points with no signal above 10 − . threshold are not filled.The intensity scale is logarithmic. (c) Longitudinal phase space integrated over horizontalcoordinates, (cid:82) dxdx (cid:48) ˆ f ( φ, w ) | ˜ y . Contour levels are shown forthe three threshold cuts at 0.1%, 1% and 10% FIG. 12: Data from a 4D scan with BSM wire positioned for peak signal strength near the core of the beam.0of the longitudinal emittance are done at nominally 20-25mA average current out of the RFQ.
A. Dependence on RFQ amplitude
The longitudinal phase space is determined by theRFQ parameters. Particularly, the RFQ vane voltagemay be tuned to produce the optimal (minimal) outputemittance. For each voltage amplitude, the slice emit-tance ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y is measured for a fixed ˜ x, ˜ x (cid:48) , ˜ y slice.The slice center in each dimension is chosen to be nearthe peak density.Figure 11 shows the result of varying RFQ amplitudeon longitudinal slice emittance as well as the transmit-ted current, measured on a Faraday cup positioned af-ter two 90 ◦ dipoles. The raw emittance values are cor-rected and uncertainty assigned according to the correc-tion described in Section III B. Simulated values of thefull emittance at the RFQ exit are included for compari-son, for both the realistic LEBT and rms-equivalent wa-terbag initial distributions. For these simulations, 40,000macroparticles are used.While the simulated vane voltage is applied exactly, thetest-stand voltage is not measured. Instead, the constantof proportionality between setpoint and vane voltage ischosen for maximum overlap with the simulated curves.Relative vane voltage is with respect to design value of 83kV. The minimum measured emittance occurs at relativeamplitude 0.96, corresponding to simulation amplitude80 kV. The RFQ amplitude is set to relative amplitude0.96 for the all measured results reported here. The sim-ulated voltage is 83 kV unless otherwise indicated.The measured 1% emittance is significantly lower thanexpected for most voltages, particularly near the setpointwith lowest emittance. The predictions of waterbag-seeded distributions agree well with measurement for rel-ative voltages near and below 1. However, the increasein measured emittance at high RFQ voltage is not re-produced in either simulation. There is agreement in thesharp emittance increase at low voltages, that coincideswith formation of a low-energy tail. B. Integrated 4D emittance
As described above, the technique for measuring lon-gitudinal emittance requires making three slices of the6D distribution in the transverse dimensions. Therefore,the measured longitudinal phase space represents a three-way slice in phase space, ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y . In order to re-construct the integrated emittance ˆ f ( φ, w ) | ˜ y , a 4D scanover variables ( x, x (cid:48) , φ, w ) is performed.The 4D scan is programmed as a nested loop of thethree actuators that select the three dimensions ( x, x (cid:48) , w ).These are the first two vertical slits and the dipole cur-rent: ( x , x , I ). The slice ˜ y was chosen to give thepeak signal strength at the BSM, which corresponds to FIG. 13: Dependence of longitudinal emittance onposition of BSM wire. The rms emittance is calculatedfor three threshold levels for both the 1D sliceˆ f ( φ, w ) | ˜ y (shown with errorbars) as well as 3D sliceˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y (square points without errorbars) at fixedlocation ˜ x, ˜ x (cid:48) ∼ y ∝ BSM wire.the BSM wire bisecting the core of the beam. Figure12b illustrates the resolution of the 4D scan in transversephase space by plotting the partial projection ˆ f ( x, x (cid:48) ) | ˜ y .The minimally-processed 4D scan data is shown in Fig-ure 12. Each sub-plot in Figure 12a is the phase spaceˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y for a point in x, x (cid:48) , y space, correspondingwith the scatter points in Figure 12b.Figure 12c shows the same data integrated over x and x (cid:48) to construct the 1D partial projection, ˆ f ( φ, w ) | ˜ y . Inaddition to integration, significant processing of the datahas been done, including thresholding, correcting for vari-ation in microchannel plate response and slow drifts inphase and RFQ output current. The output current overthe 15.3 hour scan duration was on average 20 . ± . − ±
12 deg-keV. This can be compared to the emit-tance for the central frame only, 126 − ±
14 deg-keV.As expected from realistic simulations (Figure 3, TableIV), the emittance of a 3D core slice is very close to theemittance of the lower-dimensional 1D slice.
C. Dependence on vertical slice
The integrated 4D emitttance shown in Figure 12 isstill a partial projection, due to the intersection of theBSM wire with the vertical phase space. Dependenceon the BSM wire location is measured by repeating the4D scan procedure at different wire positions. Figure 13shows the resulting rms slice emittances for ˆ f ( φ, w ) | ˜ y and ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y versus wire position. BSM wire posi-tion is reported in terms of distance from beam centerat the plane of the wire. The center is determined to bethe BSM wire position with the highest recorded signalintensity, with a precision ± .
25 mm.There is a clear trend of lower emittances in the 1D y slice near the core compared to edge slices. For the0.1% and 1% emittances, the emittance of the high-1TABLE VI: Comparison of simulated (expected values)and measured emittances for partially projected phasespace ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y =0 . Expected distribution valueshave slice width twice that of physical slit and wirewidth, for improved particle statistics. Comparison isnot made for 0 .
1% threshold due to low number ofparticles in 3D slice.
Threshold Quantity Measured Expected0.1% (cid:15) z [deg.-keV] 147 − ±
13 –1% (cid:15) z [deg.-keV] 126 − ±
13 13110% (cid:15) z [deg.-keV] 81 − ±
11 961% rms φ [deg.] 5 . − . ± . w [keV] 22 . − . ± . TABLE VII: Comparison of simulated (expectedvalues) and measured emittances for partially projectedphase space ˆ f ( φ, w ) | ˜ y =0 . The slice applied to thesimulated (expected) distribution is comparable to BSMwire width. Threshold Quantity Measured Expected0.1% (cid:15) z [deg.-keV] 133 − ±
12 1221% (cid:15) z [deg.-keV] 119 − ±
12 11410% (cid:15) z [deg.-keV] 86 − ±
11 861% rms φ [deg.] 5 . − . ± . w [keV] 21 . − . ± . dimensional slice ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) , ˜ y is within error-bars of thesingle-slice ˆ f ( φ, w ) | ˜ y emittance, reinforcing the observa-tion that the core slice emittance has low dependenceon slice dimensionality. This breaks down at the 10%threshold, where the high-dimensional 3D slice emittanceis noticeably lower than the 1D slice and has a flat de-pendence on transverse position. V. COMPARSON OF MEASUREMENT TOSIMULATION
The purpose of this study was to evaluate the degree ofconfidence that can be placed on predictions from “end-to-end” acccelerator models that include RFQ dynam-ics. The method is to compare a measured beam distri-bution with output from RFQ simulations, using bothrms values and internal structure as metrics. In general,the measured rms emittance was 20-30% lower than ex-pected, a discrepancy that exceeds the 1 σ uncertainty.This was illustrated in the previous sections in the com-parison of RFQ voltage dependence (Figure 11). In thiscase, RFQ simulation seeded with a “less realistic” wa-terbag distribution was a better predictor of the outputemittance. More detailed comparison to the expecteddistribution, shown in Tables VI and VII, shows thatthis discrepancy persists at all threshold levels. This is FIG. 14: Dependence of measured (blue curve, withuncertainty interval) and simulated slice emittances(black curve) for slice ˜ x, ˜ x (cid:48) , ˜ y = 0. The correction topoint-spread error is applied to measured values.visualized further in Figure 14.Although the rms emittances were not reproduced,there was qualitative agreement of the internal, high-dimensional bunch structure. In measurement, the emit-tance of a core slice in y was shown to be lower thanthe edge slices by ∼
30 deg-keV (shown in Figure 13).In analysis of the 4D scan data, the same convex de-pendence can be seen along coordinate x (cid:48) , but not x .Along x , the emittance monotonically decreases with dis-tance from core. The same general behavior is seen inthe expected bunch, as plotted in Figure 15. However,despite similar transverse rms parameters between simu-lation and experiment, the scale of this feature does notagree; while the width of the simulated feature is ∼ σ ,in measurement it is closer to 0 . σ , where σ is the rmswidth in transverse coordinate.As previously discussed, the rms emittance has a weakdependence on dimensionality. In addition, the shapeof the high dimensional profiles are in qualitative agree-ment. Figure 16 compares the measured 1D and 3D par-tial energy projections against simulated distributions.The general shape is reproduced, particularly for the 3Dslice profile which is lop-sided with a peak on the low en-ergy side. Fine-tuning of the energy width can be doneby adjusting the current in simulation, as the initial head-tail deceleration is driven by space charge. However thiseffect cannot explain the emittance discrepancy, as thisprocess does not lead to significant emittance growth.Finally, one prominent feature not recreated in simu-lation is the tail trailing the main bunch. This feature isvery visible, for example in Figures 7 and 12c. The tail isincluded in the 1% and 0.1% emittance calculations, butexcluded when a 10% threshold is applied. This can beseen in the emittance curve in Figure 14, where a slightknee is visible just under the 10% threshold level. Theamplitude of the tail diminishes with RFQ amplitude; itis possible this is an artifact of non-optimal RFQ voltagethat may vanish completely at a higher setpoint.2FIG. 15: Dependence of 1% rms emittance on position of slices in the transverse coordinates, compared againstsimulation of the expected distribution (dotted lines) The left-most plot shows emittances for a 1D slice ˜ y . Themeasured points are the same as shown in Figure 13. The remaining two plots show emittance for the 3D slice˜ x, ˜ x (cid:48) , ˜ y . In the middle, the center of slice ˜ x (cid:48) is varied while keeping ˜ x, ˜ y = 0. On the right, the center of slice ˜ x isvaried for ˜ x (cid:48) , ˜ y = 0. Distance from core is normalized to rms beam width, to account for difference in simulated andmeasured transverse beam size.FIG. 16: Comparison of partially projected energyprofile for measurement (solid lines) and simulation(dashed lines) for 1D (left plot) and 3D (right plot)slices. Here, (cid:126)y ≈ (cid:126)y (cid:48) is approximated. VI. DISCUSSION
The question driving this research is: what is the beststrategy for defining an initial distribution, particularlywhen high accuracy for loss-level predictions is desired?As seen, the output bunch from the RFQ includes sig-nificant internal structure. The end-to-end simulationapproach provides a high degree of information, both interms of resolution and interplane correlations. However,the complexity of RFQ dynamics means that small errorsmay result in large discrepancy, as seen here with therms emittance. A measurement-based approach avoidsthis drawback, but with the challenge of typically lowerresolution and dimensionality.In this study, the simulated longitudinal rms emittancewas sensitive to the initial LEBT distribution, to a degreethat exceeded the measurement uncertainty. Interest-ingly, the measured emittance was nearest the predictionsfrom an idealized 4D waterbag. The “most realistic”PARMTEQ simulation, based on measured LEBT dis-tribution, predicted rms emittances 20-30% larger thanmeasured. This is consistent with errors typically seen inRFQ benchmarking (eg, [22, 23]), and is an improvement on the 80% discrepancy found with independent measure-ments in the SNS MEBT [24], but is still unsatisfactory.Despite the discrepancy, the simulation qualitatively re-produced the observed high-dimensional structure.The failure to achieve rms-level accuracy is likely dueto errors in the simulation parameters, which may be am-plified by nonlinear effects in the RFQ. The most likelysource of error is the LEBT distribution, which was cre-ated from quite an old measurement. A better under-standing of this distribution, particularly the variabilityduring operation and between source changes, may allowfor better agreement.Another candidate is the simulated beam current,which operationally was significantly lower than the de-sign value. The realistic LEBT distribution was mea-sured at 50 mA, and this was used as the input current forPARMTEQ simulations. However, at the time of mea-surements, the LEBT current was measured to be near40 mA. In addition, the transmission of the BTF RFQis significantly lower than the design value. During thesemeasurements it was operating around 60%, compared to ≥
82% seen in simulation. Through combination of theseeffects, the measured current in the MEBT is about 50%less than the current of the expected bunch. While thisdiscrepancy may account for some differences in the corestructure, the simulated rms emittance was not seen todepend strongly on the input LEBT current. In MEBTsimulations, the energy-phase aspect ratio increases withspace charge but the rms emittance has a flat depen-dence.Finally, as mentioned the RFQ vane voltage is not pre-cisely known, and likely contributes a systematic error tothe simulated value. As discussed in the previous sec-tion, a lower-than-optimal operating voltage can accountfor the presence of tails in the measurement. It is alsopossible operating at a higher RFQ voltage may also re-sult in better agreement in rms emittance, as the mea-sured voltage dependence shows an increasing trend athigher voltages. This will be addressed in future studies,through implementation a non-intrusive bremsstrahlungvoltage diagnostic [25].3Another limitation of this study was the phase resolu-tion, which is too low to accurately measure the beamphase profile. The system was originally designed onthe basis of the output from RFQ simulations, which asshown in Section II A has nearly equal aspect ratio. How-ever, due to space charge defocusing in the first meter ofthe MEBT, the longitudinal bunch rapidly elongates inphase space. At the plane of the BSM the phase profileis considerably narrower than at the RFQ exit. The ex-periment resolution can be improved by either decreasingthe width of the energy slit or reducing the linear φ − w correlation at the energy slit location via installation ofa rebunching cavity upstream of the BSM.As beam halo mitigation becomes a more pressing mat-ter for high-intensity accelerators, the demand for pre-dictive accelerator models will grow. As such, the ques-tion of generating realistic and representative distribu-tions needs to progress beyond rms equivalence and 2Dcharacterization. For planned studies in halo evolutionat the SNS BTF, agreement with dynamic range ≥ issought. The bunch produced via PARMTEQ simulationdoes not benchmark at the 10%-0.1% threshold level, andtherefore is not trusted to deliver good halo predictions.Ongoing work is focused on generating an initial MEBTbunch on the basis of direct 6D measurement.Full 6D characterization is still an impractical solutionfor wide application. As such, RFQ simulations will con-tinue to be a powerful tool for generating fully-correlateddistributions. From previous efforts, rescaling bunch co-ordinates to match measured rms widths may not beenough for predicting downstream evolution [5]. Look-ing forwards, the need for improved simulation accuracywill require more sophisticated strategies for generatingdistributions. This will entail reconciling simulated 6Dcoordinates with both low- and high-dimensional mea-surements, and extending the metrics for achieving agree-ment beyond rms equivalency. A. Acknowledgments
The authors acknowledge the contributions of Bran-don Cathey, who not only authored the initial highdimensional beam study [8] but also the data collec-tion software used for the 4D characterization reportedhere. The authors are also grateful for the assistanceof SNS operations, whose personnel monitored datacollection during long study times. This manuscripthas been authored by UT-Battelle, LLC under Con-tract No. DE-AC05-00OR22725 with the U.S. Depart-ment of Energy. This research used resources at theSpallation Neutron Source, a DOE Office of ScienceUser Facility operated by the Oak Ridge National Lab-oratory. The United States Government retains andthe publisher, by accepting the article for publication,acknowledges that the United States Government re-tains a non-exclusive, paid-up, irrevocable, world-widelicense to publish or reproduce the published form of this manuscript, or allow others to do so, for UnitedStates Government purposes. The Department of En-ergy will provide public access to these results of fed-erally sponsored research in accordance with the DOEPublic Access Plan(http://energy.gov/downloads/doe-public-access-plan).
Appendix A: Sources of error
FIG. 17: f ( φ, w ) phase space immediately afterselection at the energy slit. The solid white line plotsthe projected phase distribution f ( φ ). The dashed lineshows the partial phase distribution ˆ f ( φ ) | w =0 (alongthe thin horizontal dashed line). This distribution isgenerated through PyORBIT simulation.There are many potential sources of error. These canbe separated into four categories by origin, ordered byeffect on the measurement:1. Resolution in energy and phase, determined bythe physical width of the three slits as well as thephase resolution of the BSM.2.
Calibration errors , which are applied in the cal-culation of energy and phase coordinates. This un-certainty is determined by the variance of a linearleast squares fit of the calibration data.3.
Model geometry , including uncertainty in pathlength and strength of magnetic elements, used inthe calculation of energy. This includes uncer-tainty in machine readbacks such as slit positionand dipole current.4.
Machine variation , encompassing both slowdrifts and jitter.The largest source of error is due to the resolution of themeasurement. The finite slit width create point-spreadin both the energy and phase dimensions that result insystematic over-estimation of rms parameters. While thetotal point-spread is the combination of the three verticalslits, BSM wire and internal BSM electron focusing, the4FIG. 18: Output at beam current monitor during collection of results shown above. Red dashed line indicatesaverage over scan of 20.48 mA. Pauses are due to routine lock-outs of BTF facility.largest term is the width of the third (energy) slit. Theenergy point-spread is relatively small: 0.6 keV comparedto the rms width 23 keV of the expected distribution.However, for the phase coordinate the relative erroris much larger. The main contribution to phase spreadcomes from two sources: the finite width of the energyslice and the electron optics in the BSM. The point-spread of the BSM can be directly measured by disablingthe BSM RF deflector and recording the image of theBSM wire. The measured rms width of the internal BSMpoint-spread is 0 . ◦ .The energy spread contributes to phase spread at theBSM through time-of-flight. For a collection of particleswith rms energy spread 0.6 keV originating at the samephase in the plane of the energy slit, the phase spread atthe BSM will be 0 . ◦ . However, there is an additional,larger point spread effect due to the fact that at the en-ergy slit the bunch is already highly correlated. There-fore, the projected phase width is significantly wider thanthe phase width of a monoenergetic slice. This effect is il-lustrated in Figure 17, which is generated through PyOR-BIT simulation of the expected distribution to the energyslit. The 100% rms phase width is 5 . ◦ , but through pro-jection of the correlated phase space, the apparent widthincreases to 6 . ◦ . Assuming Gaussian phase distributionand point spread function, the rms width of the point-spread from correlation is estimated to be 3 . ◦ . Addingthe three sources of phase spread in quadrature, the to-tal rms point-spread is 3 . ◦ , roughly half the expectedwidth.Calibration errors make the next largest contribu-tion and dominate the calculation of systematic un-certainty. This includes calibration of dipole strength( − . ± .
006 mm/A relative to motion of the beamat the energy slit) and conversion of BSM camera pixelsto arrival phase (0 . ± .
008 degrees/pixel). These un-certainties grow linearly with distance from the centralphase and nominal dipole current. At the rms width ofthe expected distribution w = 23 keV, δw = ± . φ = 5 . ◦ , δφ = ± . ◦ .Finally, uncertainty in the model geometry used to cal-culate beam energy has a negligible effect on calculatederrors. Variations in the BTF beam and measured signalalso have a negligible contribution. The effect of jitter(item 5) is reduced through averaging, and the overall statistical uncertainty is low. Slow variations, includ-ing drifts in phase, RFQ output current (Figure 18) andBSM micro-channel plate response, are corrected beforeemittance is calculated.Assuming that the calibration errors are independent,they can be summed in quadrature to estimate the uncer-tainty in the measured emittance. The same can be donefor the systematic error of the point spread function. Inthe approximation (cid:15) z ≈ ∆ φ ∆ w , the error propagates as (cid:18) δ(cid:15) z (cid:104) (cid:15) z (cid:105) (cid:19) ∼ (cid:18) δφ (cid:104) φ (cid:105) (cid:19) + (cid:18) δw (cid:104) w (cid:105) (cid:19) (A1) Appendix B: Simulated estimate of point spreaderror
The analytic error estimate is based on the assumptionthat the ˆ f ( φ, w ) distribution and point spread functionsare Gaussian, which is very much not true. In orderto more carefully estimate the systematic error due tofinite slit widths, the longitudinal emittance measure-ment was reproduced with PyORBIT simulation. Theexpected distribution from the RFQ is tracked to the lo-cation of the BSM wire, with slit apertures applied asin measurement. The first two vertical slits are centeredat ˜ x = 0 ± . x (cid:48) = 0 ± . w = 0 .
25 keV. Just as in measure-ment, the emittance ˆ f ( φ, w ) is reconstructed by combin-ing these phase distributions. The selection of verticalphase space at the BSM wire is not included, as shouldhave a negligible effect on the point spread error.The expected distribution, generated through RFQsimulation with 5,000,000 particles, is resampled backto 5,000,000 to after the first two slit aperatures. Thisnumber is chosen to maintain good particle statistics inphase space slices. For a slit positioned at the densitypeak, only about 2% of particles pass through. The re-sampling results in slight artifical growth of the longitu-dinal emittance at each slit (about 1% at each). This issmaller than the point spread effect.The reconstructed distribution is compared to the lon-gitudinal distribution in the plane of the first vertical slit5 (a) Simulated phase space at the reference plane(location of first vertical slit). (b) Reconstruction of phase space fromsimulation of emittance measurement with finiteslit widths. FIG. 19: Comparison between ˆ f ( φ, w ) | ˜ x, ˜ x (cid:48) at the first slit and the reconstructed phase space after simulation of thephase space measurement. The effect of the slit-width point spread on the phase width is apparent. 1% and 10%contour lines are drawn.FIG. 20: Comparison of rms phase width at 1%threshold. Solid orange points are phase widthmeasured with virtual slit technique. Open blue pointsare result of “typical” measurement. The solid blue lineis the same measurement with 83% correction factorapplied.in Figure 19. No significant space-charge influenced evo-lution of the longitudinal emittance is expected betweenthis location and the BSM. The broadening of the distri-bution due to finite slit width is apparent, particularly inthe phase width. Appendix C: Sub-slit resolution with virtual slit
As established, the largest error in the emittance mea-surement is due to the point spread associated with thefinite width of the energy slit. However, it is possible toobtain a higher resolution that overcomes the physicallimitations of the existing apparatus, without the needto manufacture and install narrower slits. This is done TABLE VIII: Comparison of virtual slit reconstructedemittance with “typical” measurement at 1% threshold.
Quantity typical meas. virtual slit meas. uncertaintyrms (cid:15) φ [deg-keV] 120 102 13rms φ . ◦ . ◦ . ◦ rms w [keV] 23.1 22.4 0.4 by creating a virtual slit from two phase profiles sepa-rated by a differential step in slit position. The step sizemust be smaller than the physical slit width for enhancedresolution.As in the typical emittance measurements, the dipolecurrent is varied rather than actuating the energy slit.This is particularly beneficial for the virtual slit mea-surements, as the dipole current can be set with higherprecision ( ± .
005 A, equivalent to 0.005 mm response atenergy slit) than the slit actuator position ( ± .
02 mm).A virtual slit spacing of 0.05 A (0.05 mm) was found tobe sufficient in both simulation and measurement. In ap-plication of the virtual slit reconstruction on simulateddata, the recovered phase width plateaued for slit sepa-rations ≤ .
07 was within 5% of the “base truth” phasewidth. In measurement, the recovered phase width ap-pears to plateau at 0.05 A separation.Figure 20 compares the measured rms phase widths at1% threshold for a range of energies (plotted as dipolecurrent). Phase width is calculated both with and with-out application of the virtual slit technique. The recon-structed phase profiles are significantly more noisy, byton average (for all profiles measured in this data-set),the reconstructed width is equal to 83% of the measured6width without correction (shown in the figure as a solidline).Table VIII compares rms for a “typical” phase spacemeasurement with the reconstruction using the virtual slit technique. Note that there is still uncorrected point-spread from the internal BSM optics, but at rms 0 . ◦ thisleads to a much smaller error than from the energy slit. [1] S. Henderson, Accelerator and Target Technology for Ac-celerator Driven Transmutation and Energy Production ,Tech. Rep. (2010).[2] A. V. Fedotov, Mechanisms of halo formation, AIP Conf.Proc. , 3 (2003).[3] R. L. Gluckstern, Analytic Model for Halo Formationin High Current Ion Linacs, Phys. Rev. Lett. , 1247(1994).[4] C. K. Allen, K. C. D. Chan, P. L. Colestock, K. R.Crandall, R. W. Garnett, J. D. Gilpatrick, W. Lysenko,J. Qiang, J. D. Schneider, M. E. Schulze, R. L. Sheffield,H. V. Smith, and T. P. Wangler, Beam-Halo Measure-ments in High-Current Proton Beams, Phys. Rev. Lett. , 214802 (2002).[5] J. Qiang, P. L. Colestock, D. Gilpatrick, H. V. Smith,T. P. Wangler, and M. E. Schulze, Macroparticle simu-lation studies of a proton beam halo experiment, Phys.Rev. Spec. Top. - Accel. Beams , 35 (2002).[6] L. Groening, W. Barth, W. Bayer, G. Clemente, L. Dahl,P. Forck, P. Gerhard, I. Hofmann, G. Riehl, S. Yaramy-shev, D. Jeon, and D. Uriot, Benchmarking of measure-ment and simulation of transverse rms-emittance growth,Phys. Rev. Spec. Top. - Accel. Beams , 1 (2008).[7] P. K. Roy, C. E. Taylor, C. Pillai, and Y. K. Baty-gin, Comparison of profile measurements and TRANS-PORT beam envelope predictions along the 80-m LAN-SCE pRad beamline, J. Phys. Conf. Ser. , 3323(2018).[8] B. Cathey, S. Cousineau, A. Aleksandrov, and A. Zhukov,First Six Dimensional Phase Space Measurement of anAccelerator Beam, Phys. Rev. Lett. , 064804 (2018).[9] T. F. Wang and L. Smith, Transverse-Longitudinal Cou-pling in Intense Beams., Part. Accel. Print , 247(1982).[10] A. Fedotov, R. Gluckstern, S. Kurennoy, and R. Ryne,Halo formation in three-dimensional bunches with var-ious phase space distributions, Phys. Rev. Spec. Top. -Accel. Beams , 014201 (1999).[11] A. W. Chao, R. Pitthan, T. Tajima, and D. Yeremian,Space charge dynamics of bright electron beams, Phys.Rev. Spec. Top. - Accel. Beams , 19 (2003).[12] T. P. Wangler, K. R. Crandall, R. Ryne, and T. S.Wang, Particle-core model for transverse dynamics ofbeam halo, Phys. Rev. Spec. Top. - Accel. Beams ,084201 (1998).[13] Z. Zhang, S. Cousineau, A. Aleksandrov, A. Menshov,and A. Zhukov, Design and commissioning of the BeamTest Facility at the Spallation Neutron Source, Nucl. In-struments Methods Phys. Res. Sect. A Accel. Spectrom- eters, Detect. Assoc. Equip. , 162826 (2020).[14] K. R. Crandall and T. P. Wangler, PARMTEQ A beam-dynamics code fo the RFQ linear accelerator, AIP Conf.Proc. , 22 (1988).[15] S. Henderson, W. Abraham, A. Aleksandrov, C. Allen,J. Alonso, D. Anderson, D. Arenius, T. Arthur, S. As-sadi, J. Ayers, P. Bach, V. Badea, R. Battle, J. Beebe-Wang, B. Bergmann, and Others, The Spallation Neu-tron Source accelerator system design, Nucl. Instru-ments Methods Phys. Res. Sect. A Accel. Spectrometers,Detect. Assoc. Equip. , 10.1016/j.nima.2014.03.067(2014).[16] A. Ratti, R. Digennaro, R. A. Gough, M. Hoff, R. Keller,K. Kennedy, R. Macgill, J. Staples, S. Virostek, andYourd, the Design of a High Current , High Duty FactorRfq for the Sns, in Proc. EPAC 2000 (Vienna, Austria,2000) pp. 495–497.[17] A. Ratti, J. Ayers, L. Doolittle, R. Digennaro, R. A.Gough, M. Hoff, R. Keller, R. Macgill, J. Staples,R. Thomae, S. Virostek, R. Yourd, E. O. L. Berke-ley, and A. Aleksandrov, The SNS RFQ Commissioning,Linac2002 , 329 (2002).[18] A. Shishlo, S. Cousineau, J. Holmes, and T. Gorlov, Theparticle accelerator simulation code PyORBIT, ProcediaComput. Sci. , 1272 (2015).[19] K. Ruisard, Rapid charge redistribution and hollowing inthe core of a high-intensity beam, (in Prep. ) (2020).[20] A. Aleksandrov, Beam Instrumentation for High PowerHadron Beams, in Proc. 25th Part. Accel. Conf. , editedby T. Satogata, C. Petit-Jean-Genaz, and V. Schaa (JA-CoW, Pasadena, CA, 2013) pp. 380–384.[21] A. V. Feschenko, Bunch shape monitors using low energysecondary electron emission, in
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