the use of truncated taylor maps in dynamic aperture studies

Particle Accelerators, 1993, Vol. 41, pp 117-132 Reprints available directly from the publisher Photocopying permitted by license only

@1993 Gordon & Breach Science Publishers, S.A. Printed in the United States of America.

THE USE OF TRUNCATED TAYLOR MAPS IN DYNAMIC APERTURE STUDIES R. KLEISS 1 , F. SCHMIDT2 , F. ZIMMERMANN3 ITH Division, CERN, Geneva, Switzerland 2SL Division, CERN, Geneva, Switzerland 3DESY, Hamburg, Germany (Received 9 October 1992,. in final form 9 December 1992)

For dynamic aperture studies in large proton storage rings, the use of truncated Taylor maps is a possible alternative to element-by-element tracking. Since such maps are inherently non-symplectic, additional symplectification procedures are called for. Results are presented of a detailed comparison of two such symplectification schemes, kick factorization and dynamic rescaling. The usefulness of map tracking was studied for HERA and the LHC and the considerable differences found could be related to the strength of high-order nonlinear field errors. KEY WORDS: Truncated Taylor Maps

INTRODUCTION For large proton storage rings, simulations of particle motion over 107 - 108 turns are required to provide a reliable measure of the dynamic aperture. Conventional elementby-element tracking (which we shall denote by DI tracking in what follows) is limited to about 106 turns even with today' s supercomputers I. To overcome this limitation alternative methods have to be found. An appealing approach is to represent the accelerator by a high order truncated Taylor map, and to use that map for tracking 2 instead. * Such maps can easily be extracted from standard tracking codes, making use of automatic differentiation techniques such as the differential algebra package of Berz4 . This approach has the drawback, however, that truncated Taylor maps do not, in general, conserve the phase-space volume, and the particle motion is consequently not symplectic. In a previous papers we have studied the feasibility of Taylor map tracking for the LHC, and found that, due to the lack of symplecticity, such tracking (hereafter referred to as DA tracking) is of limited use. Some procedure to symplectify the motion is necessary if DA tracking is to be useful for long-term stability studies of the LHC. In this paper, we consider the possibility of 'symplectifying' the DA tracking by either one of two methods, kick factorization and dynamic rescaling. We made a detailed comparison between DI and DA tracking (with and without symplectification) for two machines, HERA and the LHC. As we shall see, our study reveals a large difference in * see als0 3 for a rather complete review on many aspects concerning maps

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TABLE 1: Comparison of some machine parameters for HERA and LHC.

Accelerator Parameters HERA Parameter 157T mm mrad Normalized Emittance (20-) Unnormalized Emittance (20-) 0.47T mm mrad 40 GeV Injection Energy 76 m Maximal Beta in the Arc 95 Number of FODO Cells 75 mm Coil Diameter in sc. Magnets 7 . 10- 4 Half Bucket Height (~p)/p

LHC 157T mm mrad 0.037T mm mrad 450 GeV 167 m 224 50mm 1.3.10- 3

the applicability of map tracking for the two accelerators, which, we conjecture, can be directly related to the size of the high-order nonlinear field errors.

2 2.1

MAP TRACKING Comparison o/parameters/or LHC and HERA

To set the stage of the argument, we present in Tab. 1 some of the relevant parameters of our machine models. The number of FODO cells in the LHC is about twice that in HERA: as we shall see, this has the effect of increasing the importance of higherorder nonlinear field effects arising from cross-terms between lower-order ones. Another important point is that the coil radiu~ in the LHC is smaller than in HERA (see Tab. 1), leading to larger multipole errors. In Tab. 2 we give the multipole errors used in our studies. The skew and normal multipole coefficients an and bn are defined by the equation 00

"(b n+lan .)(X+lZ. )n-I/ r (n-I) . B z+l'B x == B oL.....,. o

(1)

n=1

All multipole values are specified for a reference radius ro == 25 mm in terms of 10- 4 relative to the main dipole field B o. In case of HERA, the measured multipole errors for each individual magnet are used in the simulation. Quoted in the table is the mean value over all magnets. In HERA the average part of the 10-pole and of the 12-pole error is corrected for by correction coils, both in the real machine and in the simulation. Sextupole correctors are adjusted to provide about zero chromaticity. A large systematic 20-pole component in the quadrupole magnets remains uncompensated. Its contribution will therefore be evident in the Taylor map. For the LHC, there are sextupole, octupole and decapole correctors, which are adjusted such as to minimize high order chromaticity and detuning. No field errors in the quadrupoles have been taken into account here. Intuitively, one would expect already on the basis of these numbers that high-order effects may be larger (and, consequently, map tracking less reliable) in the LHC than in HERA.

TRUNCATED TAYLOR MAPS

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TABLE 2: Systematic multipole errors for HERA and the LHC superconducting magnets, as used in our simulation studies.

Systematic Multipole Errors Dipole Quadrupole Magnets Magnets HERA LHC HERA Machine Reference 25 25 25 Radius (mm) Normal Components -32.0 -20.9 0.02 b3 0.08 0.8 0.05 b4 12.0 18.0 0.02 bs -0.1 -2.4 b6 -2.0 34.0 0.02 b7 0.14 -0.17 bs 0.44 52.0 0.30 bg -0.12 3.04 b lO Skew 0.05 Components a3 -0.36 0.83 -0.01 a4 0.21 -0.02 as -0.02 -0.69 a6 0.01 -0.19 a7 -0.56 0.236 as -0.46 0.10 ag -0.25 0.26 alO 2.2 Error of the average amplitude A xy per turn and taylor map coefficients

We first discuss the performance of DA tracking without any attempt at symplectification. Our construction of the DA map for the LHC has been described extensively in our previous papers; the HERA map was produced in the same way, where of course the actual parameter values rather than estimates have been used. To start we like to define how we do our normalization, what amplitudes we use and how we calculate amplitude errors. As a normalization the Courant-Snyder transformation T is applied to our initial DA map M to yield the linearly normalized DA map N.

M==ToNoT- I .

(2)

Considering only one degree of freedom this transformation reads

0) (~ f?t)' T- == (~ ~v7J I

T=

(3)

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where (n, (3) are the twiss parameters. Of course we use the transformation generalized to six dimensions considering the coupling between all three planes of motion, so that the map N is linearly normalized and completely decoupled. This procedure transforms linear six-dimensional motion into circles in each motion plane and removes the machine dependent beta values, so that different accelerators may be compared. After this normalization we denote the phase space coordinates of the particles in the transverse part by x, X I, y, and y , and in the longitudinal part by z, Z I where the primes indicate the derivative with respect to the path length. As stated above the coordinates are now independent of the {3 functions. We have evaluated various amplitudes Ax

== Vx 2 +x '2 ,

Axy =

JA; +A;,

Ay

==

Vy2+y'2,

Axyz = JA~ +A;,

Az

==

Vz 2 +zl2, (4)

all defined in units of Vmm x mrad. In our dynamic aperture studies we vary the two initial transverse amplitudes Ax, A y keeping their ratio constant, usually equal to one, while in the longitudinal plane the relative momentum deviation is kept fixed at 0.35 x 10- 3 and 1.0 x 10- 3 for HERA and the LHC respectively (compare Tab. 1). As we are mainly interested in the transverse plane we only quote the amplitude Axy in the following. In the error analysis we average over a number of turns in order to separate the global, long-term evolution from short-term fluctuations. To assess the reliability of tracking with Taylor maps, a first point of interest is the error of the average amplitude (A xy ) per tum, as derived from the difference between OA and 01 tracking over a few thousand turns. For start amplitudes Axy well below thedynamic aperture, this quantity appears to be well-defined, i.e. nonlinear deviations in the average only start to be evident for significantly higher tum numbers. This is shown in Fig. 1a. We have plotted the relative amplitude error (Axy(D~.~~,)(DI)) per tum for an 11th-order linearly normalized map for the LHC, and two maps (of order 7 and 11) for HERA, as a function of the normalized initial amplitude A xy • This relative amplitude error per turn depends on the location of the particle in phase space and has therefore to be averaged over many turns. Evidently, with increasing initial amplitude A xy the error increases due to the finite order of the OA maps. However, the HERA OA results follow 01 tracking much more closely than the LHC map: even a 7th -order map for HERA performs better than an 11th-order map for the LHC. To get a qualitative understanding of these results, we have· studied the order dependence of the coefficients of the two 11th order maps. In order to compare the two maps M of the two accelerators we had to construct (see eq. 2) the linearly normalized maps N in both cases, so as to suppress the dependence on the different beta functions. These maps are then polynomials in the normalized coordinates x, x', y,.y', z and Zl. Hence the nth order Taylor map coefficients are quoted in units of (mm x mrad) I ;n . In Fig. 1b, the average magnitude of these map coefficients is given as a function of the term order for the two lattices. The average (absolute) value of the coefficients is only one of the many possible measures, but since (for high orders) the number of distinct terms with

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