Accelerator physics needs advanced modeling and simulation techniques, in particular for beam stability studies. A deeper understanding of the effects of magnetic fields nonlinearities will greatly help in the improvement of future colliders design and performance. This paper presents a study of quadrupole tracking using realistic field maps and measured or simulated longitudinal harmonics. The main goal is to describe the effect of the longitudinal dependence of high order non-homogeneity of the field in the case of the HL-LHC inner triplet

Bonaventura, Luca

Dalena, Barbara

De Maria, Riccardo

Molson, James

Pugnat, Thomas

Simona, Abele

International audienceAccelerator physics needs advanced modeling and simulation techniques, in particular for beam stability studies. A deeper understanding of the effects of magnetic fields nonlinearities will greatly help in the improvement of future colliders design and performance. This paper presents a study of quadrupole tracking using realistic field maps and measured or simulated longitudinal harmonics. The main goal is to describe the effect of the longitudinal dependence of high order non-homogeneity of the field in the case of the HL-LHC inner triplet

de Maria, Riccardo

HAL-CEA

Accurate and Efficient Tracking in Electromagnetic Quadrupoles

Accelerator physics needs advanced modeling and simulation
techniques, in particular for beam stability studies.
A deeper understanding of the effects of magnetic fields
nonlinearities will greatly help in the improvement of future
colliders design and performance. This paper presents a
study of quadrupole tracking using realistic field maps and
measured or simulated longitudinal harmonics. The main
goal is to describe the effect of the longitudinal dependence
of high order non-homogeneity of the field in the case of the
HL-LHC inner triplet

T. Pugnat

B. Dalena

SIMONA, ABELE

L. Bonaventura

R. DeMaria

J. Molson

Archivio istituzionale della ricerca - Politecnico di Milano

ACCURATE AND EFFICIENT TRACKING IN ELECTROMAGNETICQUADRUPOLEST. Pugnat∗, B. Dalena, CEA, Irfu, DACM, Université Paris-Saclay, F-91191, Gif-sur-Yvette, FranceA. Simona and L. Bonaventura, MOX, Politecnico di Milano, Milano, ItalyR. De Maria, J. Molson, CERN, Geneva, SwitzerlandAbstractAccelerator physics needs advanced modeling and sim-ulation techniques, in particular for beam stability studies.A deeper understanding of the effects of magnetic fieldsnonlinearities will greatly help in the improvement of futurecolliders design and performance. This paper presents astudy of quadrupole tracking using realistic field maps andmeasured or simulated longitudinal harmonics. The maingoal is to describe the effect of the longitudinal dependenceof high order non-homogeneity of the field in the case of theHL-LHC inner triplet.INTRODUCTIONIn Ref. [1], a method to evaluate the non-linear fringe fieldeffect on the long term beam dynamics has been presented.Following Ref. [2], the magnetic field map or the fieldharmonics are used to compute first a representation of thevector potential, which enters in the expression of the Hamil-tonian, after the non-linear transfer map of the quadrupoleis derived using Lie algebra techniques for tracking simula-tions.This paper presents further studies and improvements ofthe same procedure. First, a specific gauge for the represen-tation of the vector potential is employed in order to reducecomputational cost [3]. Then, a summary of the comparisonwith other integrators than Lie of second order performedin [3] is given. The implementation of the method in Six-Track and further improvement to the code presented in [1]are described. Finally, tracking results are discussed.METHODThe motion of a charged particle in a magnetic field is de-scribed in terms of its position and its canonical momentumby the Hamilton equations. Ref. [4] have shown that therelativistic Hamiltonian can be expressed as an 8 D equiva-lent Hamiltonian written as follows (see also pages 5 to 8 inRef. [3]):K = pz − δ −az +(px−ax )22(1+δ)+(py−ay )22(1+δ)= K1 +K2 +K3 +K4(1)where p is the normalised momentum vector, δ is the mo-mentum deviation, P0 is the nominal momentum vector anda = qA/P0 is the normalised vector potential.As shown in Eq. (1), the equivalent paraxial Hamiltonianis a sum of 4 parts that are used to generate transfer mapfor the tracking of charged particles using the Lie Algebra,∗ thomas.pugnat@cea.fras explained in Ref. [4], [5] and [6]. It can also be shownthat between the formulation of SixTrack (Ref. [7]) and theone used here, the longitudinal deviation for the position(respectively σ and l) and momentum (respectively pσ andδ) are connect by ∆σ = L f + ∆lββ0 and δ + 1 = ββ0pσ +β/β0 where L f is the length of integration.Generalized GradientUp to now, all the studies on beam dynamics used hard-edge approximation in order to consider the effect of themagnet. This paper explores the effect of longitudinal de-pendency of magnetic field on beam stability’s observable,such as Dynamic Aperture and detuning with amplitude. Inparticular, the effect of the generalized gradient C[n]m (see Eq.(20) at the page 15 in Ref. [1]) can be cancelled in the HardEdge approximation.The generalized gradients dump the high longitudinal fre-quency components of the harmonics. Then, using moregeneralized gradients derivatives means considering higherfrequency components of the harmonics and a better recon-struction of the magnetic field longitudinally. Only consider-ing the integrated strength (equivalent to∫C[0]m dz) is a fastand simple approach, but it generates an approximation ofthe magnetic field effect in the fringe field’s regions.Vector Potential RepresentationIn Ref. [1], the vector potential is reconstructed startingfrom the field harmonics using the AF gauge to derive theexpression of the vector potential components (see Eq. (22)in Ref. [3]). In Ref. [3], a new expression has been derivedusing a horizontal free gauge HFC (see Eq. (28) in Ref.[3]). Table 1 summarizes the evaluation’s cost in termsof number of operations needed to reconstruct the vectorpotential monomials A(x, y, z) =∑i, j ai, j(z)xiyj , whereai, j(z) are coefficients that only depend on z and ND is thenumber of derivatives.Table 1: Vector Potential Evaluation’s CostND=2 ND=16Normal Skew Normal SkewAF 80 68 352 330HFC 64 52 251 225HFC/AF 0.80 0.76 0.72 0.68Comparison of IntegratorsThe Hamilton equations that describe the motion of thecharged particle in the quadrupole system have been solvedProceedings of IPAC2018, Vancouver, BC, Canada - Pre-Release Snapshot 06-May-2018 12:00 UTCTHPAK0040ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.05 Beam Dynamics and EM FieldsD02 Non-linear Single Particle Dynamicsusing different symplectic and non symplectic integrators:mid-point, 4th order and 6th order Gauss, 4th order non-symplectic Runge-Kutta and Lie maps of equivalent order(including the one presented in Ref. [1]), as discussed inRef. [3]. A comparison of the different integrators is reportedin Ref. [3], in terms of speed and accuracy, using analyticas well as more realistic fields, generated with the ROXIEcode [8] for the inner triplet of HL-LHC.The main conclusions of the study are:• For a dz greater than 40mm, the longitudinal informa-tion is greatly deteriorated (Fig. 1).• Lie methods profit more from the change of gauge thanthe other methods.• Lie methods are faster with respect to other symplecticmethods. The explicit, non-symplectic Runge-Kuttamethod is the fastest.• All the methods display the same low accuracy for stepsize bigger than 4 cm for the realistic field considered.−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04−4−3−2−101234·10−9yinynon-linearities∆z = 2 cm∆z = 4 cm∆z = 8 cmReferenceFigure 1: Information losses on the position with dz.IMPLEMENTATION IN SIXTRACKA 2nd-order Lie map (Lie2, described in Ref. [1]), hasbeen integrated in SixTrack (Ref. [7]). Historically, it waschosen for its speed, symplecticity and, as a 2nd order track-ingmethod like the thin lens used by SixTrack, real measuredharmonic (consisting on the mean value by sections dz) canbe used without interpolations. Other maps are also con-sidered to increase the accuracy as shown in Ref. [3] butrequire more accurate measurement. Here, the code hasbeen implemented as a 4 D Tracking.As input, it needs a config file and files (A-files) containingthe vector potential’s coefficients discretised by section ofsame length, dz. Figure 2 shows the implementation inSixTrack with the aim to do not change the internal structureof the main code. In Fig. 2, it is assumed that QSixTrack isthe normal SixTrack routine for one full quadrupole and foreach A-file (in or out), I is a 2nd order Lie integrator, D is adrift of length L f − Lq (with the integration length L f andthe equivalent magnetic length Lq =∫B2(z)dz/B2,center inthe config file) and Q−1 = (∏Qi)−1 is the Anti-Quad withQi a quadrupole’s thin matrix with length dz and strengthKi = 2C[0]2(zi) for the section zi in the A-file (see Eq. (20)in Ref [3]).in out z QSixTrack Din-1 Iout Qout-1 Din Dout-1 Dout Lq,in Lf,out 1) 2) 3) 4) 5) 9) 8) 7) 6) Figure 2: Implementation in SixTrack.The routines that compute A(x, y) and its derivatives, areimproved using specific tables (Matrix Market format, Ref.[9]) for the coefficient of A, which reduces the computationtime by almost 50%.TRACKING RESULTS USING HL-LHCThe Dynamic Aperture (DA) is computed using the HLL-HCV1.0 optics, simulating the particles’ motion over 104revolutions. A set of initial conditions distributed on a polargrid is considered in such a way as to have 30 particles (dif-ferent initial conditions) for each interval of 2 sigma from0 to 28. Five phase space angles and 60 different machines(also called seeds), according to dipole field errors, are con-sidered. The initial momentum offset δ is set to 2.7e-4. Theinner triplet is modelled using 8 different vector potentialfiles with only the natural harmonics of the quadrupole (2-6-10-14), according to the connector side or the non-connectorside, and to the polarity. 7 8 9 10 11 12 13 14 15 16 17 10  20  30  40  50  60  70  80DA [σ]angle (degrees)Lie2 B2-6-10-14 ND=6Lie2 B6-10-14 ND=6Lie2 B2-6-10-14 ND=0Lie2 B6-10-14 ND=0sixtrackFigure 3: Dynamic aperture comparison for δ = 2.7e − 4.As shown in Fig. 3, the DA computed using B6-10-14is on average systematically better than the one using themultipoles model of SixTrack but compatible in the hori-zontal plan considering the minimum and maximum valuesProceedings of IPAC2018, Vancouver, BC, Canada - Pre-Release Snapshot 06-May-2018 12:00 UTC05 Beam Dynamics and EM FieldsD02 Non-linear Single Particle DynamicsTHPAK0041ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.(error bars). If B2 is added in the tracking, the DA is al-most the same for the horizontal plane, but the discrepancyin the vertical plane increases, showing the impact of thelongitudinal dependency of the field, among other effects,like interference between B2 and the others multipoles orincertitudes of the method (interface procedure for example).Fig. 3 also show the influence of ND on the DA. One wouldexpect the DA to converge to the SixTrack value as ND tendto 0. But in the case of B6-10-14, the DA doesn’t have majorchange and when adding B2, the convergence can be seen,but if the DA is the same for the x-plane, the discrepancyreduce in the y-plane but still significative. The origin ofthis difference has yet to be explained quantitatively.0.2 0.25 0.3 0.35 0.401020304050Six. multi. Dlt0|fft(hx)|0.2 0.25 0.3 0.35 0.401020304050|fft(hy)|0.2 0.25 0.3 0.35 0.401020304050Lie2 C6:14 Dlt00.2 0.25 0.3 0.35 0.401020304050Part.9Part.1Ref0.2 0.25 0.3 0.35 0.401020304050QxLie2 C2:14 Dlt00.2 0.25 0.3 0.35 0.401020304050QyFigure 4: Tunes for 1000 revolutions for δ = 0.By extracting the positions and momenta of the particlesat one point on the ring (here, at ip3), the fractional partof the tunes can be computed using the DFT. Figures 4and 5 shows the tunes computed over 1000 revolutions forparticles of different δ with a horizontal amplitude equal to0.1993mm (Part.1) and 0.4599mm (Part.9) and a ratio of0.19281 between horizontal and vertical phase space.0.2 0.25 0.3 0.35 0.401020304050Six. multi.|fft(hx)|0.2 0.25 0.3 0.35 0.401020304050|fft(hy)|0.2 0.25 0.3 0.35 0.401020304050Lie2 C6:140.2 0.25 0.3 0.35 0.401020304050Part.9Part.1Ref0.2 0.25 0.3 0.35 0.401020304050QxLie2 C2:140.2 0.25 0.3 0.35 0.401020304050QyFigure 5: Tunes for 1000 revolutions for δ = 2.7e − 4.As shown in Fig. 4, the tunes measured are very closeto the reference ones (Qx = 62.31 and Qy = 60.32) with aprecision of 10−3. Showing that the linear part introduced byour method called Lie2 is well compensated by the combi-nation of anti-drift, anti-quadrupoles and drifts described inthe previous section. The amplitude of the frequency distri-butions is well preserved in our method (Lie2) with respectto nominal Sixtrack ones. The tune spread in Fig. 5 causedby the δ-oscillation (periodicity ≈ 600 revolutions) doesnot change, while the amplitude seems affected. Statisticalstudies and full 6D Tracking need to be done to confirm ornot the phenomena.0 0.01 0.02 0.03 0.04 0.05 0.060.2750.280.2850.290.2950.30.305Jx [um]QxDelta=0 1000turns0 0.005 0.01 0.015 0.020.320.3220.3240.3260.3280.330.3320.334Jy [um]QysixtrackLie2 C6:14Lie2 C2:14Figure 6: Tune vs action (J) (1000 revolutions with seed 1).Finally, Fig. 6 shows the evolution of the tunes with theaction variables (J), using positions andmomenta of particleswith the same phase space angle. The tunes are computedon the first 1000 revolutions with δ = 0. The small tune shiftbetween the multipole and our method is probably caused bysome small difference on the field error used for the tracking.The detuning observed appear to be the same whatever themethod used.CONCLUSIONWe report the first results of the impact of the longitudinaldescription of the HL-LHC inner triplet field harmonics onthe long term tracking. A method, called Lie2, is describedand implemented into SixTrack. DA computed with thetwo methods are pretty similar in the horizontal plane andslightly different in the vertical one. First order detuningwithamplitude is also very consistent between the two methods,proving the robustness of the SixTrack present model. Thefull 6 D Tracking is still to be investigated to have a morequantitative comparison between themodels. Further studieson the uncertainties introduced by the methods and on thesecond order detuning with amplitude are planned as well.ACKNOWLEDGEMENTThe authors would like to thanks S. Izquierdo-Bermudezand E. Todesco for providing the field harmonics computedwith ROXIE of the inner triplet prototype for the HL-LHCproject, K.N. Sjobak for the help and comment during theimplementation in SixTrack and A. Chancé for useful discus-sions and for the first implementation of the Horner routines.Proceedings of IPAC2018, Vancouver, BC, Canada - Pre-Release Snapshot 06-May-2018 12:00 UTCTHPAK0042ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.05 Beam Dynamics and EM FieldsD02 Non-linear Single Particle DynamicsREFERENCES[1] B. Dalena et al., "Fringe Field Modeling for the High Lumi-nosity LHC Large Aperture quadrupole", in Proc. IPAC’14,Dresden, Germany, June 2014, paper TUPRO002, pp. 993-996.[2] M. Venturini, A. J. Dragt„ "Accurate computation of transfermaps from magnetic field data", Nucl. Instr. Meth., vol. 427,pp. 387-392, May 1999.[3] A. Simona, L. Bonaventura, T. Pugnat, B. Dalena, High ordertime integrators for the simulation of charged particle motionin magnetic quadrupoles, arXiv:1802.08157, submitted forpublication.[4] Y. K. Wu, E. Forest and D. S. Robin, "Explicit symplecticintegrator for s-dependent static magnetic field", Phys. Rev. E,vol. 68, pp. 046502, Oct. 2003.[5] A. J. Dragt, Lie Methods for Nonlinear Dynamics with Appli-cations to Accelerator Physics, University of Maryland, MD,USA, 1997.[6] H. Yoshida, "Construction of higher order symplectic integra-tors", Phys. Lett. A, vol. 150, no. 5, pp. 262-268, 1990.[7] SixTrack, http://sixtrack.web.cern.ch/SixTrack[8] ROXIE, https://espace.cern.ch/roxie[9] Matrix Market Exchange Formats, https://math.nist.gov/MatrixMarket/formats.htmlProceedings of IPAC2018, Vancouver, BC, Canada - Pre-Release Snapshot 06-May-2018 12:00 UTC05 Beam Dynamics and EM FieldsD02 Non-linear Single Particle DynamicsTHPAK0043ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.

ACCURATE AND EFFICIENT TRACKING IN ELECTROMAGNETIC QUADRUPOLES

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Accurate and Efficient Tracking in Electromagnetic Quadrupoles

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