BEAM DYNAMICS IN NS-FFAG EMMA WITH DYNAMICAL MAPS∗

Y. Giboudot† , R. Nilavalan, Brunel University, Uxbridge, UKR. Edgecock, STFC Rutherford Appleton Laboratory, Didcot,UK

A. Wolski, University of Liverpool, Liverpool, UK

Abstract

The Non-Scaling Fixed Field Alternating Gradient ac-celerator EMMA has a compact linear lattice, in which theeffects of magnet fringe fields need to be modelled care-fully. A numerical magnetic field map can be generatedfrom magnet measurements or magnet design software. Wehave developed a technique that produces from the numer-ical field map, a dynamical map for a particle travelling ina full EMMA cell, for a given reference energy, withoutacceleration. Since the beam dynamics change with en-ergy, a set of maps have been produced with various refer-ence energies between 10 MeV and 20 MeV. For each ref-erence energy, the simulated tune and time of flight havebeen compared with results in Zgoubi - tracking directlythrough numerical field map. The range of validity of asingle map has been investigated by tracking particles withlarge energy deviation: the results can be used to imple-ment a model of acceleration based on dynamical maps.

INTRODUCTION

In tracking studies, the behaviour of a single particle canbe defined by six dynamical variables: the horizontal (ver-tical) positionX (Y ) and momemtumPX (PY ), the lon-gitudinal position with respect to a reference particleZ,and the energy deviationδ. Tracking a particle through asequence of magnetic elements in a beamline consists ofcomputing the values of these variables at different loca-tions or steps. If the numerical values of the magnetic fieldcomponents are known on grid points throughout the mag-netic element, then the equations of motion for the dynam-ical variables may be integrated numerically for given ini-tial conditions, to find the values of the variables at the exitof the magnetic element. However, this method requirestracking through the beamline each time one wants a char-acterization of the beam behaviour. When tracking manyparticles through many steps, the process can be highly de-manding in terms of computing time and memory.

In EMMA [1], a highly compact doublet cell is achievedusing short quadrupole magnets. A large aperture require-ment then leads to potentially significant fringe fields. Ac-curate simulations of the beam dynamics in EMMA requirea dense description of the magnetic field, and numerous in-tegration steps. Solving Maxwell’s equations in an EMMAcell (by a Finite Element code, OPERA [2]) we have gen-erated a 3D magnetic field map that can be used for numer-ical tracking in EMMA with PyZgoubi [3, 4, 5]. In most

∗Work supported by the Engineering and Physical Sciences ResearchCouncil, UK.

† yoel.giboudot@stfc.ac.uk

cases, PyZgoubi routines are fast and reliable. However, analternative approach based on dynamical maps could pro-vide some benefits, particularly where speed is important;for example, when tracking many particles through manycells. Dynamical maps also provide the possibility of read-ing significant quantities (such as tunes and chromaticities)directly from the map, giving an insight into the dynamicsthat is not provided directly by purely numerical methods.

To generate a dynamical map, one propagates a variablethrough the cell as a function instead of a numerical value.The magnetic field must be expressed in analytical form:an appropriate form can be obtained from a numerical fieldmap by fitting an appropriate three-dimensional mode ex-pansion [6]. Then, we use a symplectic integrator imple-mented in the differential algebra (DA) code COSY [7], topropagate a vector of six power series (one series for eachof the six dynamical variables) through the field.

DYNAMICAL MAP DESCRIPTION

The DA integration routine outputs the dynamical map inexplicit form as shown in Table 1. The first column givesthe name of each coefficient following the TRANSPORTcode nomenclature. The final six columns indicate, as ex-ponents for the six dynamical variables, the term in the mapto which the coefficient in second column refers. Thus,each variable is expressed as a power series in the valuesof the dynamical variables at the entrance of the cell. Forinstance the expression for the horizontal position is:

X1 = 1.0344X0 + 0.2683PX0 − 0.0103 δ0 +

4.6261X20+ 1.7204X0PX0 · · ·+ 0.0096 δ

2

0.

Therefore, once the dynamical map has been obtained,tracking particles in the EMMA cell simply involves calcu-lating the ouput values for a given set of input values. Thepower series is truncated at a certain order (in the exam-ple in Table 1, at 2nd order). Even though the integration

Table 1: Selected Terms from the 2nd Order DynamicalMap for the X Variable, for one EMMA Cell at 15 MeVReference Energy.

Coefficient Order ExponentsR11 1.0344 1 1 0 0 0 0 0R12 0.2683 1 0 1 0 0 0 0R16 -0.0103 1 0 0 0 0 0 1T111 4.6261 2 2 0 0 0 0 0T112 1.7204 2 1 1 0 0 0 0

......

......

T166 0.0096 2 0 0 0 0 0 2

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routine is symplectic, the truncation results in a symplecticerror. A symplectic transformation satisfiesJT ·S ·J = S,whereS is a block diagonal matrix constructed from the‘unit’ 2×2 antisymmetric matrix, andJ is the Jacobian ofthe transformation. A symplectic error in a map may besignificant if the map is applied iteratively many times, orif small non-symplectic physical effects are being investi-gated. If the map is computed using a symplectic integrator(as is the case for the results presented here), then the sym-plectic error depends on the order of truncation, rather thanintegration step size.

The symplectic integrator that we use requires the parax-ial approximation. This requires some care, since in anFFAG, beam excursions can be large (of the order of a fewcm). However, when one computes a dynamical map, it isnecessary to make a choice of reference trajectory. Sinceenergy and transverse position are correlated, a sensiblechoice is to look for closed orbits for various reference en-ergies over the full energy range (in EMMA, from 10 MeVto 20 MeV), and use these closed orbits as reference tra-jectories. For small energy deviations, particle trajectoriesshould then remain close to the reference trajectory, and theparaxial approximation should be valid.

In practice, we do not use exactly the closed orbit asthe reference trajectory at a given reference energy. Forsimplicity in the integration, we use instead a straight linestarting (and ending) at the middle of a long drift, wherethe field is close to zero; the position of the straight lineis chosen to minimize the excursion of the closed orbitwith respect to this reference trajectory. Since there are42 cells in total in EMMA, concatenating the dynamicalmap around the straight reference trajectory with a rotationthrough 2π/42 (about a vertical axis) produces a map forone periodic section of the EMMA lattice.

MAPS WITH VARIOUS REFERENCEENERGIES

We first compare the results of the dynamical map withthe results of numerical tracking in Zgoubi. Eleven dynam-ical maps were calculated around the closed orbits for ref-erence energies from 10 MeV to 20 MeV, in steps of 1 MeV.The tunes (phase advances per cell) can be obtained fromthe eigenvaluesλ of the linear part of a given map:

λ = e±2πiν ,

whereν is the tune.The zeroth-order term in the map for the fifth variable

(longitudinal coordinate,Z) represents the difference inpath length of the closed orbit with respect to the referencetrajectory.

The comparisons of these features with numerical track-ing through the magnetic field map with PyZgoubi are plot-ted in Figs. 1 and 2. The two codes show good agree-ment for the horizontal tune and path length, although aslight discrepancy occurs in the path length for high en-ergy. This might be due to a truncation in the relativistic

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Figure 1: Horizontal (left) and vertical (right) tune versuskinetic energy using multiple dynamical maps.

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Figure 2: Path length versus kinetic energy from dynamicalmap (blue) and PyZgoubi (red).

conversion from time to flight to path length and has to bestudied in more detail. A hard-edge model of the magnetsin Zgoubi is also shown (in green) on the plot, and indicatesthe impact of the fringe field on the vertical tune: a discrep-ancy of about 10% is found at 10 MeV. Betatron motionmay be studied in more detail by applying the dynamicalmaps to particles with some initial transverse offset withrespect to the reference trajectory. Figure 3 shows the hor-izontal phase space for reference energies from 10 MeV to20 MeV, constructed by applying the appropriate dynami-cal map iteratively to particles with 1 mm initial transverseoffset (with respect to the reference trajectory). We noticethat there is some non-physical growth in the amplitudeover time for 10 MeV and 11 MeV: this is a consequenceof the truncation of the dynamical map to 2nd order. Theeffect disappears if terms up to 4th order are retained.

MAPS WITH ENERGY DEVIATION

The number of dynamical maps required to model thedynamics over the full energy range in EMMA will de-pend on the range of validity of each map with respectto variations in the energy deviationδ. Figure 4 shows

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Figure 3: Betatron motion with multiple dynamical maps.

Proceedings of IPAC’10, Kyoto, Japan TUPEC058

05 Beam Dynamics and Electromagnetic Fields

D06 Code Developments and Simulation Techniques 1857

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Figure 4: Betatron tunes computed with a set of dynamicalmaps at different reference energies (blue dots), and with asingle dynamical map with different energy deviations (redcrosses).

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Figure 5: Betatron motion at different energies, simulatedwith dynamical map up 4th order (left), and 9th order(right).

the tune as a function of energy obtained in two differ-ent ways: first, from different dynamical maps computedfor different reference energies (blue dots); and second,from a single dynamical map at a single reference energy(15 MeV), but with different values for the energy devia-tion δ (red crosses). The good agreement between the twomethods suggests that it may be possible to use a singlemap to describe the transverse dynamics with good accu-racy, even for large energy deviations. Figure 5 shows thehorizontal phase space for different energies, generated us-ing maps with different reference energies (dots), and withfixed 15 MeV reference energy but different energy devia-tions (crosses). We see a strong effect from the symplecticerror with a map truncated at 4th order; the effect of theerror is greatly reduced by retaining terms in the map upto 9th order. For modelling acceleration in EMMA, it isimportant that the dynamical maps also describe accuratelythe variation in path length with energy. Figure 6 shows thelength of the closed orbit computed in two different ways:first, from a set of dynamical maps at different referenceenergies (blue dots); and second, from a single dynamicalmap (15 MeV reference energy) with different values forthe energy deviation (red crosses). We see there are largediscrepancies (more than 10%) at large energy deviationswhen the maps are truncated at 2nd order; however, thereis good agreement for the maps truncated at 4th order.

SUMMARY AND NEXT STEPS

The large range of transverse positions in an FFAG canbe modelled using multiple dynamical maps with differ-

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Figure 6: Closed orbit path length computed with a set ofdynamical maps at different reference energies (blue dots),and with a single dynamical map with different energy de-viations (red crosses). Left: dynamical map up to 2nd or-der. Right: dynamical map up to 4th order.

ent reference trajectories. The phase advance per cell andthe path length computed using a dynamical map showgood agreement with the results obtained using a numer-ical tracking code, Zgoubi. Accurate description of the be-tatron motion requires the dynamical maps to be computedto at least the 4th order, at small energy deviation. If theenergy deviation is large (for example, to cover the full en-ergy range in EMMA from 10 MeV to 20 MeV in a singledynamical map), then a map up to 9th order may be re-quired to model the horizontal motion.

Acceleration may be included in the dynamics by mak-ing an appropriate adjustment to the energy deviation at theend of each cell (representing the effect of an RF cavity inthe cell). The results presented here suggest that it may bepossible to achieve a reasonable description of the dynam-ics using a fixed reference energy at the mid point of theenergy range. However, if a very accurate description isrequired, then it may be necessary to change to a differentreference energy at one or more points during the acceler-ation.

REFERENCES

[1] J.S. Berg, “The EMMA main ring lattice,” Nuclear Instru-ments and Methods in Physics Research A 596 (2008) 276-284.

[2] Vector Field OPERA 3D, http://www.vectorfield.com/.

[3] F. Méot, “The ray-tracing code Zgoubi,” Nuclear Instru-ments and Methods in Physics Research A 427 (1999) 353-356; F. Méot, S. Valero, “Zgoubi users’ guide,” CEA DSMDAPNIA/SEA-97-13 (1997).

[4] Y. Giboudot ,F. Méot, “Optical matching of EMMA cell pa-rameters using field map sets,” Procs. PAC 09, Vancouver.

[5] S. Tygier, D. Kelliher, “PyZgoubi interface to Zgoubi,”http://sourceforge.net/projects/pyzgoubi/.

[6] Y. Giboudot, A. Khan, R. Edgecock, A. Wolski, “Particletracking studies using dynamical map created from finite ele-ment solution of the EMMA cell,” Procs. PAC 09, Vancouver.

[7] COSY Infinity, http://bt.pa.msu.edu/index_cosy.htm.

TUPEC058 Proceedings of IPAC’10, Kyoto, Japan

1858

05 Beam Dynamics and Electromagnetic Fields

D06 Code Developments and Simulation Techniques