CBETA Note 029

Online Simulation Tools for Orbit Correction in the CBETA Machine

Antonett Nunez-delPradoDepartment of Physics, University of Central Florida

Colwyn GullifordWilson Synchrotron Lab, Cornell University

(CBETA Collaboration)

Control of beam trajectory is essential in particle accelerators. The trajectory of a beam issensitive to various types of imperfections that cause it to deviate from its desired path along theaccelerator. Orbit correction is hence fundamental for the control of beam trajectory. In thiswork we aim demonstrate orbit correction using singular value decomposition techniques applied todistorted orbits generated in the CBETA-V Virtual Machine. In particular we present two concreteexamples of such correction in different sections of the CBETA Fractional Arc Test layout. Our goalis to construct the backbone of an orbit correction software toolkit to be tested in the CBETA-VVirtual Machine and eventually implemented in the completed CBETA crontrol system.

I. INTRODUCTION

A. CBETA Machine

The Cornell Brookhaven-National-Laboratory Energy-Recovery-Linac Test Accelerator (CBETA) is a particleaccelerator being built at Cornell University that willserve to test several accelerator physics concepts relevantto development of the electron and Relativistic Heavy IonCollider (eRHIC) at Brookhaven National Lab, as well asto the Energy Recovery Linac (ERL) community in gen-eral.

Linacs offer the advantage of producing beam whosequality is still roughly determined at the beam’s source,as opposed to rings, where the beam quality is deter-mined by an equilibrium distribution set by the ring’smagnets. The drawback of a conventional linac approachincludes their tendency to be wasteful due to their dis-carding of beams right after use and their creation of lowaverage beam currents. The goal of CBETA, as an en-ergy recovery linac (ERL), is to attempt to resolve bothof these issues by: after the beam goes through the mainaccelerating cryomodule, a return loop utilizing a Non-scaling Fixed Field Alternating Gradient magnet designwill direct the beam in a circular path back to the cry-omodule where it will deliver its energy back to the SRFcavities, making the energy available for new beams.

In CBETA, this process starts in the in the DC Gunsection, as seen in Fig. 1. By shining a laser onto a pho-tocathode, electrons are ejected from the material andgiven an initial acceleration up to 300 keV. These elec-trons are then molded through the buncher section intoan electron beam which then passes through the Injec-tor Cryomodule (ICM) where the electron beam gains aninitial 6MeV of energy due to the series of five two-cell su-perconducting radiofrequency (SRF) cavities. The beamis then sent through the Merger section in to the MainLinac Cryomodule (MLC), which consists of six SRF cav-ities, each of which can add up to an additional 6 MeVof energy to the beam. The beam is then passed into

the low energy Splitter line which serves to provide thenecessary matching into the first permanent magnet arcgirder of the curved return loop comprised of Fixed FieldAlternating Gradient (FFAG) magnets, shown in Fig. 2.The novelty of CBETA is that it will be the first energyrecovery accelerator to control different-energy beams inthe same structure relying on superconducting accelerat-ing cavities and FFAG magnets in the return loop. Totest the progress in construction made thus far, CBETAunderwent the Fractional Arc Test (FAT) in the springof 2018 with the purpose of testing a 42 MeV electronbeam starting from the injector through the MLC all theway to the FA section.

B. CBETA-V Virtual Machine

The CBETA-V Virtual Machine (VM) is an onlinemodel of the CBETA machine. It is a combination ofopen-source simulation packages written in Python andFortran. It is based on BMAD, a subroutine libraryfor charged-particle dynamics simulations in acceleratorsand storage rings [2], as well as TAO, an analysis sim-ulation program based on BMAD which includes toolsto solve problems such as designing lattices under spec-ified constraints, simulating errors and changes in ma-chine parameters, and simulate machine commissioning[3]. When the VM is started, it creates a copy of theCBETA EPICS control system, thereupon allowing oneto control the elements of the simulated accelerator in thesame way one would control the elements of the CBETAmachine, i.e. the same command is used to change thestrength of an element in the VM as it is to change itin the physical machine. The CBETA-V VM is of highimportance to the CBETA project because of its abilityto produce simulated physics data of real measurementscripts and to debug such scripts for use in either the vir-tual or real machine. During FAT, the virtual machineunderwent preliminary testing. Several capabilities areyet to be added to the CBETA-V VM. This paper deals

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MA3C[H,V]D0(1-4)MA3QUA0(1-4) contains

MB1C[H,V]G0(1,2)MB1QUB0(1,2) contains

Summary of combined magnets

Figure 1. Layout of the injector and linac sections of the FAT experiment. The beam begins in the gun section (A1), has initialacceleration to 6 MeV in the Injector Cryomodule (A2), and is then merged into the Main Linac Cryomodule (B1, D1) [1].

1.8 meters

MD1DIP01MS1DPB01

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FAT Splitter (S1) and 1st Arc Girder (FA) Layout

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Figure 2. Layout of the post linac section of the FAT experiment. Beam exits the linac and enters the S1 splitter line at thebottom right and proceeds to the fractional FA arc (top right) [1].

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with orbit correction of beam trajectories based on thethe current FAT lattice of CBETA.

The CBETA elements simulated in the virtual machinethat will be commonly referred to afterward in this paperinclude: Beam Position Monitors (BPMs), which are pri-marily used to measure the horizontal and vertical beampositions; quadrupole magnets, which prevent beam lossby trying to focus the beam; dipole magnets, which areelectromagnets whose main task is to create the magneticguide field that defines the reference orbit [4]; correctors,which are physically equivalent to dipoles except smallerand hence do not modify the reference orbit but serve tocorrect imperfections in the beam’s trajector; and lastlySRF cavities, which create a longitudinal field inside thecryomodule, and depending whether this field is parallelor anti-parallel to the beam longitudinal trajectory, theywill accelerate or decelerate the beam respectively [5].

II. ORBIT CORRECTION

It is well known in the field of accelerator physics thatone of the most indispensable operations in a particle ac-celerator is orbit correction. Typically the ideal path fora beam in an accelerator is to travel through the longi-tudinal axis, i.e. the center of the pipe. In general errorsin magnet production and placement cause the beam or-bit to deviate from this ideal. Thus, an orbit correctioncapability is of primary importance to ensure adequatecontrol of the beam orbit

A. Response Matrix

The (linear) relationship between changes in currentto corrector magnets and the changes in orbit positiondownstream caused by them can be represented by thecorrector-to-BPM mapping called the response matrix.A system with M BPMs and N correctors generates anM ×N dimensional response matrix

RM×N =

r11 r12 r13 . . . r1Nr21 r22 r2Nr31 r3N...

...rM1 rMN

.

Each element rij in the response matrix represents theorbit shift diagnosed by the i-th BPM caused by the aunit current change in the j-th corrector magnet. Hence,all elements in the i-th row preceding the j-th column inthat same row will be zero if the current given to j-thBPM was the only adjustment in the system. An orbitresponse matrix can either be calculated via simulationor measured.

B. Orbit Correction Theory

Given the importance of orbit correction, numerousorbit correction algorithms have been proposed over theyears. One that is widely used is the technique of singularvalue decomposition (SVD). Suppose the desired orbit isrepresented by rd, the position of a non-ideal orbit in theaccelerator (as diagnosed by the BPMs) represented by r,and the change in current settings of the electromagnetsthat will shift the beam position from the uncorrectedtrajectory to the desired one by ∆I. The equation toattain the desired trajectory is thus

rd = r + R · ∆I, (1)

where R is the response matrix obtained from speci-fied correctors and diagnostics. This equation howeverassumes previous knowledge of the change in correctorcurrents and finds the resulting change in beam position.Nevertheless, the way in which the practical problem isformulated is the reverse of that: knowing the desiredorbit trajectory, we intend to find the necessary ∆I thatwill generate the best orbit shift that will bring rc as closeas possible to rd, i.e. that will minimize the differencebetween r and rd. If we define ∆r to be that difference,

∆r = rd − r, (2)

then orbit correction techniques essentially aim to invertthe matrix equation

∆r = R · ∆I. (3)

Were the number of correctors M the same as the num-ber of diagnostics N , thus producing a square responsematrix R, the next step would entail finding the inverseof R and the procedure would be nearly spotless (assum-ing R is nondegenerate). This, however, is usually notthe case. Most of the times we deal with non-square re-sponse matrices, and these do not have an inverse, butrather a pseudo-inverse, which will produce the identitymatrix when multiplied with R depending on the orderof their multiplication.

C. Singular Value Decomposition

One of the most common approaches to producing a‘pseudo-inverse’ for the response matrix R is through thewell-known matrix decomposition method called Singu-lar Value Decomposition (SVD). Using SVD, one can de-compose an M × N matrix R into a product of threematrices:

R = U · S · VT (4)

where U and V are both orthonormal matrices of sizesM × M and N × N , respectively, and U has columnsthat are left singular vectors and V has rows that are

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right singular vectors. S is a diagonal matrix of the samesize as R, i.e. M ×N , whose diagonal elements are non-negatives values called singular values, usually producedin descending order, and have the form

Sij = swδij , (5)

where 1 ≤ w < min(M,N). The singular values cor-respond to the coupling efficiency between the first wdiagnostics and w correctors.

To ‘invert’ R, as decomposed in Eq. (4), we noticethat by definition of orthonormal matrix, the inverses ofU and VT are UT and V, respectively. The ‘inverse’ ofS, is a N ×M diagonal matrix whose elements are

(S−1)ij = twδij , (6)

where, again, 1 ≤ w < min(M,N) and

tw =

0 sn ≤ τ1

snotherwise.

(7)

The tolerance τ refers to the threshold for which all sin-gular values in S below it are zeroed out so that whendealing with S−1, the inverse of these singular values donot result in diverging values [6].

The expression for R−1 is thus

R−1 = V · S−1 · UT, (8)

which allows for us to invert Eq. (4) to obtain the desiredchange in corrector strengths that will take the currentbeam position as close as possible to the desired position

R−1 · ∆r = ∆I. (9)

It is very important to note a couple of points. Oneis that when applying orbit correction using SVD thereis a free parameter that must be adjusted for, namelythe tolerance. The tolerance must be chosen carefully:a threshold that is too small will result in unsuitablyhigh corrector strengths, whereas one that is too highwill diminish the utility of a higher number of diagnos-tics and result in a mediocre solution. The second im-portant point is to emphasize that the orbit correctionprocedure is iterative. Since the response matrix is onlya linear approximation (accurate within a bounded offsetdisplacement from the design orbit) of a response fromthe actual system, repeating the orbit correction proce-dure a number of times will indeed push the non-idealorbit to converge to the desired corrected trajectory [4].

III. SIMULATIONS

We tested orbit correction using SVD using sev-eral combinations of BPMs and correctors/dipoles, withquadrupole offsets and scaling errors included. We alsopresent a slight modification of orbit correction theorythat will aid in finding the periodic orbit in the curvedFA section of the FAT layout.

1. Correction in Injector and Merger section withQuadrupole Offsets and Errors

This example deals with the A3 Quad Telescopesection, which is part of the Injector section, and theB1 Merger sections of the FAT layout (Refer to Fig. 1).We chose to apply orbit correction in the injector linebecause there had been no previous orbit correctionattempts in this section and was thus seen as a naturaltest case. Nevertheless, since the method outlined inthis paper is a general procedure, if it works in thissection, it should also apply to the other sections of theaccelerator.

Procedure - The simulation starts by producingthe response matrix, which is determined from theset of diagnostics and correctors that will play a partin the orbit correction process. For this example, allavailable BPMs from the A1 Gun section to the S1Splitter section were used as the set of diagnostics andthe set of correctors consisted of all those in the A3and B1 sections. A ‘perfect’ response matrix was thusobtained with all nominal settings, that is one that didnot take into account errors on any of the virtual opticalelements, and a tolerance of 0.4345 was manually chosenfor use in forming the psuedo-inverse via SVD. To createrandomized ‘bad’ orbits, the same set of correctors fromthe A3 and B1 sections were randomly set with currentsranging in the range of ±50 mA. Next, the quadrupolesin the A3 section were given a scaling error of 5%RMS, and transverse offsets of 100 µm. A total of 200randomized ‘bad’ orbits were tested.

Results - Fig. 3 depicts the orbit correction of 200 ran-domized orbits produced from random current changesgiven to the A3 and B1 correctors and random offsetsand scaling errors to the quadrupole magnets. The fourplots show results where the vertical axes represent thechange in position in the corresponding transverse di-rections from the design values as a function of positionalong the accelerator. The lines in all four plots repre-sent the orbits as simulated by TAO. The location of theBPMs are represented by the horizontal coordinates ofthe red dots, and their vertical coordinates represent theposition readings of the beams. The left top and bottomplots depict the original uncorrected orbits. We can seethat initial section of the orbits in the left plots are ze-roed out since they correspond to the A1 and A2 sectionsof the lattice and the randomization of the orbits beginsin A3. The right top and bottom plots depict the ‘cor-rected’ orbits in the corresponding transverse directionsafter iterating the SVD orbit correction process 10 times.

A measurement of the scale of the error of the orbitcorrection procedure in the injector and merger sectionsis shown in Fig. 4. Our metric of the error is the RMS

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(a) Horizontal orbit offsets of randomized uncorrected orbits. (b) Horizontal orbit offsets of corrected orbits after 10 iterations.

(c) Vertical orbit offsets of randomized uncorrected orbits. (d) Vertical orbit offsets of corrected orbits after 10 iterations.

Figure 3. Orbit correction of 200 random trajectories simulated by CBETA-V with quadrupole offsets and quadrupole errorsusing A3 Injector and B1 Merger correctors. The two left plots show the transverse offset of the randomized beam orbit. Thetwo right plots show the beam orbit after ten orbit correction iterations. The transverse offsets are the change in position inthe corresponding transverse direction from the design value in the VM. The red dots represent the position of the diagnosticsalong the accelerator pipe as well as the diagnostics’ readings.

value of the BPM residuals, given by

σRes =

√√√√ ∑i=x,y

(ri − rdesign,i)2

N. (10)

The results suggest that the RMS values for each of the200 tests converge after about 5 iterations and the orbitcorrection errors are at the micron and sub-micron scales.

0 2 4 6 8 10

Iteration

10 -6

10 -4

10 -2

10 0

10 2

σR

es (

mm

)

Figure 4. Measurement correction error for the 200 simulatedorbits in the simulation presented in Fig. 3 as a function ofthe number of iterations of orbit correction. The error metricused measures the offset of the diagnostic readings of an orbitfrom the design value, and it is calculated with Eq. (10).

2. Finding the Periodic Orbit in the FA section

In this example we show that by slightly modifying theorbit correction approach previously outlined, it is pos-sible to find the periodic orbit in the FA 1st Arc Girdersection of the FAT layout. The particularity that comeswith applying orbit correction to this first part of thereturn loop of the final CBETA machine is that, unlikeall the other sections previous to it, it is a curved trajec-tory with a repeating collection of magnets called cells.For this section it turns out it is possible to find a spe-cific input trajectory into the the arc such that the beamfollows a trajectory that will repeat itself in every cell,and hence making the diagnostics in each repeated cellread the same position. The desired beam position canbe represented by the constant vector

C =

CC...C

= C

11...1

. (11)

In simulation, we know what this orbit position C forthe nominal 42 MeV beam energy. In contrast, whenusing the real CBETA machine we allow for a deviationfrom the design value of C. To try and simulate thisscenario with the virtual machine, we aim to set up a sit-uation where we do not know the value of this periodictrajectory C. Because the periodic orbit is a function ofenergy, a straightforward way of arriving at such situa-tion is to simply change the energy of the beam. This canbe achieved by changing the cavity voltages in the mainaccelerating cryomodule. After applying the change in

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(a) Horizontal orbit offset of randomized uncorrected orbit. (b) Horizontal orbit offset of corrected orbit after 10 iterations.

Figure 5. Finding of the periodic orbit in the FA section for an orbit with beam energy of 52 MeV using the last two dipolesin the S1 splitter section as correctors. The left plot shows the transverse offset of the randomized beam orbit. The two rightplot shows the beam orbit after ten orbit correction iterations. The red dots represent the position of the diagnostics along theaccelerator pipe as well as the diagnostics’ readings.

beam energy (which occurrs at the MLC), it is necessaryto scale the dipoles in the successive S1 Splitter to bringthe beam back to the center of the pipe. However, scalingthe dipoles for the beam to arrive at the FA section doesnot guarantee that it will go into the FA section with theright position and angle to get onto the periodic orbit.

It is important to emphasize that this problem isdifferent from the one tackled in the first simulation.Previously, we knew what our desired orbit was (the

Figure 6. Correction error of the simulated orbit with a beamenergy of 52 MeV presented in Fig. 5 as a function of thenumber of iterations of orbit correction. The error metricused measures the deviation of the diagnostics’ readings fromone another, and it is calculated with Eq. (14)

nominal beam path), but in the FA section we donot. Hence, in this scenario not only do we need tofind a vector of current changes that will steer ontothe periodic trajectory, we must also identify whatthis periodic orbit is. We can make use of the factthat our desired orbit takes a special form, namely,that it is constant, and thus be able to solve this problem.

We can use Eq. (1) in a way that the desired orbit isthe constant vector C,

C = r + R · ∆I. (12)

We can rewrite this as

R · ∆I − C = −r. (13)

In matrix form, this equation is represented byr11 r12 r13 . . . r1Nr21 r22 r2Nr31 r3N...

...rM1 rMN

·

∆I1∆I2

...∆IN

−

CCC...C

= −

r1r2r3...rM

.

As it is, this system seems to be unsolvable, given thatthere are two unknowns, ∆I and C. However, it is pos-sible to write this problem as only a tiny difference fromthe problem we have dealt with before. By expandingthe response matrix R such that it contains an addedcolumn of a series of ones, and by multiplying this with anew vector identical to ∆I except for an added last entry

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of value −C, we can obtain the same vector −r,r11 r12 r13 . . . r1N 1r21 r22 r2N 1r31 r3N 1...

......

rM1 rMN 1

·

∆I1∆I2

...∆IN−C

= −

r1r2r3...rM

.

This step allows us to put all unknown values in onevector, and thus ending up with a matrix equation wherewe solve for the extended version of the ∆I vector,which includes the constant position to be read from thediagnostics. From here on, the steps of matrix inversionthrough singular value decomposition are the same.

Procedure - As described in this subsection, in thissimulation we alter our previous approach of producinga randomized orbit. Instead of randomly setting correc-tors or dipoles to a random value, we increase the energyof the beam from the nominal value of 42 MeV to 52MeV by applying a 10 MeV increase in the last cavity ofthe MLC (RD1CAV01). We set out to avoid beam lossin the S1 splitter section by autoscaling the dipoles in S1accordingly from their design values. Once the beam ar-rives at the FA section entrance, it is imperative to ensurethat it enters the section at the correct position and an-gle. To solve for these two variables, only two dipoles areneeded for orbit correction. Thus, the last two dipolesin S1 are used to produce the response matrix for thissimulation, along with the set of all diagnostics in FA,and the tolerance for the response matrix was manuallyset to 0. Orbit correction using SVD as presented on thispage is applied 10 consecutive times.

We repeated this same procedure for 10 sampleenergies ranging from the minimum energy needed toget a periodic orbit in FA, 39 MeV, to the maximumoutput energy from the MLC, 59 MeV.

Results - Fig. 5 shows our finding of the periodic orbitfor an arbitrary energy beam value of 52 MeV. It is worthmentioning that the design value for the beam positionthroughout the length of the accelerator is 0, except inthe FA section, where we know in advance that the designbeam position is at about 15mm for the nominal beamenergy of 42 MeV. The two plots in this figure displayresults where the vertical axes represent the difference inBPM position from the periodic orbit at 42 MeV as afunction of position along the accelerator. The left plotshows the transverse offset of the uncorrected producedorbit at 52 MeV and the right one shows the transverseoffsets after 10 correction iterations. In these plots thebeam offset is zero until it encounters the two dipoles atthe very end of the splitter section. The uncorrected or-bit goes through the FA section in a seemingly randomfashion, as displayed by the random BPM readings rep-resented by the red dots in the left plot. In the rightplot, we can see that after 10 iterations the BPM diag-nostics in FA read nearly the same value, suggesting that

the entering beam orbit has matched the periodic orbittrajectory at 52 MeV.

Measurement of the error scale of our finding of theperiodic orbit corresponding to the 52 MeV energy beamis shown in Fig. 6. In this case our error metric makesuse of the difference between FA BPM diagnostics:

σRes =

√√√√∑i,j

(xi − xj)2

2N. (14)

This gives a sense of how much, on average, the BPMreadings deviate from one another; consequently givinginsight into how perfect the periodic orbit can be found.Fig. 5 shows us that the uncorrected orbit achieves pe-riodicity to some level of accuracy in each of the firstthree iterations, and it seems to stop improving there-after, suggesting that this uncorrected orbit will matchinto the corresponding periodic orbit by the third itera-tion.

Having tested our periodic orbit-finding technique witha random energy supported by CBETA, we proceed toshow the results for this same procedure for 10 energiesbetween 39 MeV to 59 MeV. In Fig. 7, both plots havebeen zoomed into the section of the orbits that gets mod-ified by altering the energy of the beam. The differentlines represent the simulated orbits produced from the 10different energies. The left plot depicts the uncorrectedorbits as somewhat chaotic in the curved FA section,whereas the right plot shows the corrected orbits afteronly 3 iterations to be rather periodic. The BPM read-ings in both plots align well with the simulated orbits,furthering agreeing with the realization of periodicity.

Fig. 8 depicts the same metric represented by Eq. (14)plotted versus the number of iterations. Because in thissimulation no offsets or errors were added to any opti-cal elements, the orbit correction procedure is completein under 10 iterations. This occurs because orbit correc-tion has indeed found the exact solution, i.e. a perfectlyperiodic orbit for a corresponding energy, and hence theall BPM diagnostics in FA read the same value. Conse-quently, the residual function at that iteration is zero. Inthe log plot scale depicted in Fig. 8, this would becomenegative infinity, and is thus not depicted in the plot.

In Fig. 9 we plot the periodic orbit Shift, measured by

∆x = x(E) − x(Edesign), (15)

where x(E) is the orbit position diagnosed by the BPMsin FA (all of which read the same, or nearly the samevalue, per input beam energy) and x(Edesign) representsorbit position at the nominal beam energy of 42 MeV.We can see that the shift from the nominal positionincreases as you increase the input beam energy.

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(a) Horizontal orbit offset of randomized ‘uncorrected’ orbit. (b) Horizontal orbit offset of ‘corrected’ orbit after 3 iterations.

Figure 7. Finding of the Periodic Orbits in the FA section for 10 orbits with different beam energies ranging from 39 MeV to59 MeV using the last two dipoles in the S1 splitter section as correctors.

Figure 8. Correction error of the 10 simulated orbits withenergies ranging from 39 MeV to 52 MeV, as a function of thenumber of iterations of orbit correction.

Figure 9. Periodic orbit shift from the nominal position of anorbit with beam energy of 42 MeV in FA section, as a functionof beam energy. The orbit shift is calculated with Eq. (15)

IV. CONCLUSION

Two cases were tested with the CBETA-V virtual ma-chine using the current FAT lattice, and presented twoimportant ones in this paper. The first one demonstratedorbit correction of two hundred randomized orbits us-ing SVD along the A3 line in the Injector and the B1Merger sections when taking quadrupole magnets offsetsand scaling errors into account. After ten iterations, theerror scale of the corrections was shown to be at the mi-cron and sub-micron level. In the second case, a slightlymodified version of the orbit correction procedure withSVD was used to find both the necessary current changesthat will match the input beam to a periodic orbit, andthe periodic orbit itself. This was tested at ten differentenergies and it was shown that, after ten iterations, theerror for this simulation was also at the micron and submicron scale.

Thus far, this orbit correction toolkit has been suc-cessfully tested to work based on the FAT lattice imple-mented in the virtual machine. Work is currently beingdone to scale up this software in order to test it with thefull 1-pass CBETA lattice. Once this is completed andall procedures are in place for quickly switching betweenlattices, the scaled-up version of this software shall beimplemented in the future completed CBETA machine.

V. ACKNOWLEDGEMENTS

I would like to express my most sincere thanks to mymentor Colwyn Gulliford for his enormous help and guid-ance throughout this project as well as for his patienceand words of encouragement. I would also like to expressmy gratitude to Adam Bartnik for his insights, wonderfulcollaboration, and delectable explaining style. Thanks to

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both Colwyn and Adam, whose collaborative spirit werethe main ingredients in creating such a pleasurable workenvironment. I would also like to thank Ivan Bazarovfor helping organize the CLASSE REU program whichhas given me an amazing opportunity to grow as a re-

searcher. Lastly, I would like to thank the National Sci-ence Foundation for making all this possible by fundingthis program with the award PHY-1757811 for the REUSite Accelerator Physics and Synchrotron Radiation Sci-ence at Cornell University.

[1] C. Gulliford and A. a. Bartnik, “Beam commissioning re-sults from the cbeta fractional arc test,”.

[2] D. Sagan, “Introduction and tutorial to bmad and tao,”(2017).

[3] D. Sagan and J. C. Smith, in Particle Accelerator Confer-ence, 2005. PAC 2005. Proceedings of the (IEEE, 2005)

pp. 4159–4161.[4] A. Zupanc, .[5] V. Ziemann, “Accelerator physics and technology,”

(2008).[6] Y. Chung, G. Decker, and K. Evans, in Particle Accel-

erator Conference, 1993., Proceedings of the 1993 (IEEE,1993) pp. 2263–2265.