A NEW PARADIGM FOR MODELING, SIMULATIONS AND ANALYSISOF INTENSE BEAMS

E. Nissen, B. Erdelyi, Department of Physics, Northern Illinois University, Dekalb, IL 60115, USA

Abstract

Currently when the effects of space charge on a beamline are calculated the problem is solved using a particle incell method to advance a large number of macroparticles. Ifquantities such as space charge induced tune shifts are de-sired it is difficult to determine which of the many variablesthat make up the beam is the cause. The new method pre-sented here adds the effects of space charge to a nonlineartransfer map, this allows us to use normal form methods todirectly measure quantities like the tune. This was done us-ing the code COSY Infinity which makes use of differentialalgebras, which allow the direct calculation of how the tunedepends on the beam current. The method involves findingthe high order statistical moments of the particles, deter-mining the distribution function, and finally the potential.In order to advance the particles as accurately as possiblea fast multipole method algorithm is used. In this talk wepresent the new methods and how they allow us to followthe time evolution of an intense beam and extract its non-linear dynamics. We will also discuss how these methodscan improve the design and operation of current and futurehigh intensity facilities.

INTRODUCTION

The purpose of this study is to create a method wherebythe effects of space charge in a particle beam are includedin the transfer map of the machine the beam is passingthrough. Currently the transfer map governs the motion ofsingle particles in the machine, which is useful for steering,bare tunes, dynamic apertures, and many other quantities ofinterest governing the motion of single particles. Creatingthis new space charge added map will allow for the analysisof the effects of space charge on quantities that are directlyextracted from the map using normal form methods such astunes and chromaticities.

The calculation of the space charge effect involves firstcreating a distribution of particles which will serve as prox-ies for the beam. These are used to calculate the dis-tribution function throughout the beam pipe, the distribu-tion function is then integrated with an appropriate Green’sfunction to determine the potential. The potential is usedto find the electric fields, which are used to create an elec-tric field map which is applied to the map of the elementusing Strang splitting. We will begin with an overview ofthe process before examining some results.

SOFTWARE ENVIRONMENT

The software being used is COSY Infinity 9.0 [1]; thispackage uses differential algebras to perform exact numer-ical differentiation as well as to create Taylor models of theelements in question. Differential algebras work by creat-ing vectors with their elementary mathematical operationsredefined in such a way that they retain the derivatives ofeach quantity as they move through an algorithm. This al-lows for not only non-linear Taylor maps, but for coordi-nate transforms to normal form coordinates that retain anyvariable dependances that the original map had. These Tay-lor models allow for high order transfer maps, as well asnon-linear normal form transformations. The easy inclu-sion of non-linearity in the transfer maps, which allows out-side calculation using differential algebras makes the taskof adding space charge to the transfer map significantly lesscumbersome than a traditional code.

DISTRIBUTION CALCULATION

In order to create a map of the effects of space chargein a region, the distribution must be calculated within thatregion. We use a set of discrete test particles as proxies forthe distribution which can be used to calculate the Taylorseries.

The particles are formed into a Taylor series using theirstatistical moments. If two distributions have the same mo-ments, mathematically they are identical [2]. The momentsare calculated by,

Mnm =

Nparticles∑

i=1

xni y

mi . (1)

If we assume that the distribution is a Taylor series of theform, ρ(x, y) =

∑i

∑j

Cijxiyj , then the moments can be

connected to the coefficients with the equation,

Mnm =∑

i

∑

j

∫ xr

−xr

∫ yr

−yr

Cijxn+iym+jdxdy, (2)

where xr and yr are the x and y boundarys. This is triv-ially integrated, forming a matrix equation. This matrix isinverted using truncated single value decomposition, whichgives the proper values for the Taylor series coefficients.This is the same way that the coefficients are found forthree dimensions.

WEO2C06 Proceedings of HB2010, Morschach, Switzerland

534 Computational Challenges in High-Intensity Linacs, Rings incl. FFAGs, Cyclotrons

POTENTIAL INTEGRATION

Now that the distribution function has been determined,the potential must be found. This is accomplished byintegrating the distribution function multiplied with theappropriate Green’s function over the region of interest.The Green’s function takes the form of either 1

|r−r′| orln(|r − r′|) for 3D and 2D distributions respectively. Themethods involved for both 2D and 3D are similar, but 2Dwill be shown in this paper. The integral I which computesthe potential at point (x0, y0) due to the distribution func-tion ρ(x, y) in the domain D ∈ [a, b]× [c, d] is,

I =

d∫

c

b∫

a

ρ(x, y) ln(√(x− x0)2 + (y − y0)2)dxdy. (3)

If our point of interest (x0, y0) is outside of D then stan-dard numerical integration can be performed. However, ifthe point is within the distribution there will be a singu-larity at that point where standard numerical integrationbreaks down. This singularity can be removed using aDuffy transformation[3], which is explained below.

The Duffy transformation is performed by cutting D intofour rectangles, which all share a vertex at (x0, y0), as seenin Fig. 1. This has the effect of splitting the one integralI into four integrals Iac, Iad, Ibc, and Ibd. These are allintegrals of the same form, so we will focus on Iac. Thenext part of the transform is to scale the sides so that it isan integral over a unit square,

Figure 1: The integration region is subdivided into foursmaller regions with their corners on the expansion point.This moves the singularity from the center to a corner.

u1 =x− x0

a− x0; u2 =

y − y0c− y0

, (4)

dxdy = (a− x0)(c− y0)du2du1. (5)

To speed things up we will use λ1 = (a− x0) and λ2 =(c − y0). As is shown in Fig. 2, the square is then cut intotriangles which will be separately integrated.

Iac =

∫ 1

0

∫ u1

0

λ1λ2ρ(λ1u1 + x0, λ2u2 + y0)×

Figure 2: This shows how the integration region is furthersubdivided into triangles.

× ln(√λ21u

21 + λ2

2u22)du1du2

+

∫ u2

0

∫ 1

0

λ1λ2ρ(λ1u1 + x0, λ2u2 + y0)×

× ln(√λ21u

21 + λ2

2u22)du1du2. (6)

The final step is to convert the triangles to unit squares withthe transformations,

u1 = w1; u2 = w1w2, (7)

and,u1 = w1w2; u2 = w2, (8)

which gives the integral in the following form,

Iac =

∫ 1

0

∫ 1

0

λ1λ2ρ(λ1w1 + x0, λ2w1w2 + y0)× (9)

×(w1 ln(w1) + w1 ln(√λ21 + λ2

2w22))dw1dw2

+

∫ 1

0

∫ 1

0

λ1λ2ρ(λ1w1w2 + x0, λ2w2 + y0)×

×(w2 ln(w2) + w2 ln(√λ22 + λ2

1w21))dw1dw2. (10)

Notice that there is now no singularity in the integrand. The3D version involves cubes and pyramids instead of squaresand triangles, but the transformation and the result are sim-ilarly effective.

ELECTRIC FIELD KICK

Once the potentials have been calculated, COSY’s abil-ity to find exact numerical derivatives comes into play, andis used to find the electric fields for the region in question,which are then added into the map using a technique knownas Strang splitting [4]. This is a method of finding a solu-tion to a differential equation that can be thought of as acombination of two differential equations with known so-lutions,

d�z

ds= �g1(�z, s) =⇒ �f1(s), (11)

d�z

ds= �g2(�z, s) =⇒ �f2(s), (12)

Proceedings of HB2010, Morschach, Switzerland WEO2C06

Computational Challenges in High-Intensity Linacs, Rings incl. FFAGs, Cyclotrons 535

Which can be solved in the autonomous case as,

d�z

ds= �g1(�z)+ �g2(�z)=⇒ �f1(

s

2)◦ �f2(s)◦ �f1(s

2)+O(s3). (13)

The space charge kicks are calculated over a region smallenough to allow us to assume an autonomous system. SinceCOSY already contains the maps for the single particle el-ements, the kick is combined with the pre-existing maps.

PARTICLE ADVANCEMENT

In cases where the distribution is symmetric and easilymodeled using a Taylor series of the desired order then thekickmap that is generated can be used to advance the testparticles. In some cases the distribution can be more com-plex than the Taylor series is capable of advancing. Sincequantities such as the space charge induced tune shift affectthe reference particle the information is still useful, but amore stable method for advancing the particles is still re-quired. This has been accomplished using a single levelinstance of the fast multipole method [5].

The method works by subdividing the region of inter-est into smaller squares. It then calculates the multipolemoments of each square up to a desired order. Then a Tay-lor series for each square is calculated from the multipoleexpansions of the squares that do not share an edge withthe one in question. Finally for each point the potential iscalculated using the Taylor series and the individual inter-actions of each of the particles in the neighboring squares.

METHOD EFFECTIVENESS

Now that the method of adding space charge to the maphas been derived we need to determine what the optimumconditions are for its use. Since a series of test particles arebeing used it becomes important to determine how manyparticles are needed to achieve convergence. If we look atthe moments of the distribution when compared to the idealmoments in Fig. 3 we see that there is good convergence atthe 106 particle mark.

Since the Taylor models are given at a particular orderas are the moments being used it becomes necessary to de-termine which orders in the Taylor series or the momentorder are ideal. The differences in the potentials over awide range of the space are averaged to determine how ac-curate the potential as calculated using the moment methodis with respect to a point by point coulomb solver. As canbe seen in Fig. 4, the most accurate area is in the regionat 17th order in the moment method with the Taylor seriesorder at or above 17 For a uniform distribution.

METHOD IN PRACTICE

A number of tests must be done to see how well thisnew method works in comparison with systems that canbe solved analytically or numerically with high accuracy.First, we compare the expansion of the beam due to spacecharge as it moves through a drift. Assuming a uniform

0 1 � 10 6 2 � 10 6 3 � 10 6 4 � 10 6 5 � 10 6Particle Number

0.330

0.332

0.334

0.336

0.338Moment Value

Statistical Moments �0,2�

1 � 10 6 2 � 10 6 3 � 10 6 4 � 10 6 5 � 10 6Particle Number

�0.004

�0.002

0.002

0.004Moment Value

Statistical Moments �4,3�

0 1 � 10 6 2 � 10 6 3 � 10 6 4 � 10 6 5 � 10 6Particle Number

0.074

0.076

0.078

0.080

0.082Moment Value

Statistical Moments �0,12�

1 � 10 6 2 � 10 6 3 � 10 6 4 � 10 6 5 � 10 6Particle Number

�0.004

�0.002

0.002

0.004Moment Value

Statistical Moments �5,11�

Figure 3: This are the moments calculated for a uniformsquare using an increasing number of test particles, Theblue points indicate the moment as calculated, while thepurple line indicates the ideal moment.

beam with a diameter of 1 cm with 1 A of current over adrift of 20 cm with an energy of 100 KeV, the relation,

rmr0

= 1 + 5.87× 10−5 I

(γ2 − 1)32

(z

r0)2, (14)

predicts a quadratic increase in radius of 33% [6]. We cal-culate the increase from the map method in two separateways; first by placing test particles at the edges of the be-ginning distribution, the other is by examining the linearmap element that compares the initial to the final size ofthe beam. Projections of the initial and final distributionare shown in Fig. 5 while a comparison of the methods and

WEO2C06 Proceedings of HB2010, Morschach, Switzerland

536 Computational Challenges in High-Intensity Linacs, Rings incl. FFAGs, Cyclotrons

5 10 15 20 25

5

10

15

20

25

Integration Order

Mom

entO

rder

Accuracy for Various Orders and Moments

Figure 4: This shows the potential of a uniform circular dis-tribution for different integration and moment orders. Thiswas done by creating a grid of the orders. The darker thearea the higher the accuracy. The contours show the regionbetween 0 and .002.

sizes are shown in Table 1. A profile of the increase isshown in Fig. 6 which shows the quadratic character of thesize growth.

�0.006 �0.004 �0.002 0.002 0.004 0.006

�0.006

�0.004

�0.002

0.002

0.004

0.006

Figure 5: (color) A comparison of the initial and final hori-zontal and vertical test particle positions in the 20cm drift.Purple indicates the initial points, blue the final.

The next set of experiments involves the addition ofquadrupoles to the system, we use a drift quad drift quaddrift FODO cell system to study how the tune of the sys-tem changes with increasing space charge. Both the num-ber of particles and the initial distribution are the same,the only change is in the peak current. In Fig. 7 there is

0.20

0.20E-01

0.20E-01

X-motion

Figure 6: (color) A profile view of the increase in beamsize through a drift.

Table 1: Table of different methods for finding the increasein size from beginning to end in the described problem.

Method GrowthEdge Point x 35.27 %Edge Point y 35.30 %

Map Element x 31.21 %Map Element y 31.34 %

a distinct change in the tune as the current changes, but thequadrupole currents remain constant.

The values that are used in Fig. 7 are directly calculatedfrom the map of the system, which gives direct numbersfor the x and y tunes, these values can now be used witha fitting algorithm to solve for a desired set of fractionaltunes with space charge included. The results shown inFig. 8 show that the fitting algorithm was able to frequentlydetermine the quadrupole currents necessary to bring thetune of the system back to the bare tune and counteract theeffects of space charge. This new ability to fit a system todesired parameters with space charge as an included quan-tity provides a deeper understanding of how elements canbe adjusted to get desired effects both during the design ofnew machines and during the evaluation of extant ones.

Finally, another use combines the ability to calculatespace charge maps with COSY’s ability to create exact nu-merical derivatives. This involves how the chromaticity ofthe system is determined by the effects of space charge onthe beam. Using the tunes at a different operating point wecan see in Fig. 9 that both the tunes and the chromatici-ties become nonlinear with increasing peak current. This isan excellent example of a system moving from being emit-tance dominated to being space charge dominated.

CONCLUSIONS

We developed a method for including the effects of spacecharge in the transfer map of a system. This method nowallows us to calculate how space charge can effect quan-tities such as tunes and chromaticities using normal formmethods. This was accomplished by converting a set oftest particles into a distribution function, which could be

Proceedings of HB2010, Morschach, Switzerland WEO2C06

Computational Challenges in High-Intensity Linacs, Rings incl. FFAGs, Cyclotrons 537

0.01 0.02 0.03 0.04 0.05Current �A�

0.115

0.120

Fractional X Tune

0.01 0.02 0.03 0.04 0.05Current �A�

0.42

0.43

0.44

Fractional Y Tune

Figure 7: (color) The X and Y fractional tunes as calculatedthrough a system with increasing peak beam current.

0.001 0.002 0.003 0.004Current �A�

1.01

1.02

1.03

1.04

Quad Current

0.001 0.002 0.003 0.004Current �A�

0.20

0.25

0.30

0.35

0.40

0.45

Fractional Tune

Figure 8: (color) The X and Y normalized quadrupole cur-rents are shown in the upper diagram with their resultanttunes in the lower diagram.

integrated with a Green’s function after a series of coordi-nate transforms. The resultant potential could then be usedto find the electric field and thus apply that to the motion ofthe particles. This method is a self consistant computationof space charge effects in a transfer map.

The map that we have gained from this method has al-lowed us to look deeply at what is happening to the dynam-ics of the beam at the map level. We can calculate how theeffects of space charge can alter the tune of the map as wellas the chromaticity. Since this method will give a direct nu-merical value for these quantities in the presence of spacecharge they can be used to calculate an objective function

0.005 0.010 0.015 0.020Current �A�

0.12

0.14

0.16

0.18

0.20

0.22

0.24

Fractional X Tune

0.005 0.010 0.015 0.020Current �A�

�0.185

�0.180

�0.175

�0.170

�0.165

�0.160

Chromaticity

Figure 9: (color) The first graph is a tune that initially startsout as a 90◦ phase advance cell, but is subjected to an in-creased amount of space charge. Below is the chromaticity.

in a fitting algorithm, thus allowing a user to determine di-rectly how to counteract the actions of space-charge in anaccelerator. Further work also involves using a fast multi-pole method to help with the tracking of particles that fol-low a distribution that is not easily modeled using a Taylorseries.

ACKNOWLEDGEMENTS

This work was supported by the DOE under contract DE-FG02-08ER41532 with Northern Illinois University.

REFERENCES

[1] M. Berz, “Advances in Imaging and Electron Physics vol108”, Academic Press, London, 1999

[2] M. G. Bulmer, “Principles of Statistics”, MIT Press, Cam-bridge MA, 1967

[3] M. G. Duffy, “Quadrature Over a Pyramid or Cube of In-tegrands with a Singularity at a Vertex”, Siam J. NUMER.ANAL., Vol. 19, No. 6, Dec. 1982

[4] E. Hairer et al., “Geometric Numerical Integration: StructurePreserving Algorithms for Ordinary Differential Equations”,Springer-Verlag, New York, 2002

[5] J. Carrier et. al.,“A Fast Adaptive Multipole Algorithm forParticle Simulations”, Siam J SCI. STAT. COMPU., Vol. 9,No. 4, Jul. 1988

[6] M. Reiser, “Theory and Design of Charged Particle Beams”,Wiley-VCH, Weinheim, 2008

WEO2C06 Proceedings of HB2010, Morschach, Switzerland

538 Computational Challenges in High-Intensity Linacs, Rings incl. FFAGs, Cyclotrons