Thin-lens tracking through the Combined-Function Magnets in the PS - PowerPoint PPT Presentation

Thin-lens tracking through the Combined-Function Magnets in the PS Malte Titze April 27, 2016

Introduction Symplecticity Results & Conclusion References Outline 1 Introduction 2 Symplecticity 3 Results & Conclusion M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 2 / 15

Introduction Symplecticity Results & Conclusion References Introduction • Topic : Accelerator physics; Space Charge (SC) effects (= interaction between charged particles). • LHC Injector Upgrade (LIU): Deliver twice higher brightness beams compared to today’s nominal values. SC effects play an increasing role [Bartosik et al. (2016)]. • SC effects are larger at low energies (time dilation @ high energies) M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 3 / 15

Introduction Symplecticity Results & Conclusion References Current Situation • Usually the tracking of the individual particles and the simulation of their interaction goes in an alternating fashion (i.e. tracking, interaction, tracking, interaction, ... ). • In particular, there are so-called ’Analytical’ or ’Frozen’ SC models (using analytical solutions of the Poisson equation) and ’Self Consistent’ PIC Codes (solving the Poisson equations on a grid; PIC = Particle In Cell). • The Frozen SC model we currently use at CERN is based on the widely used tracking program MAD-X. M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 4 / 15

Introduction Symplecticity Results & Conclusion References Current Situation p. 2 • Recall: Simulate SC effects at CERN in with the Frozen SC model requies in particular tracking with MAD-X through the entire complex of preaccelerators, especially the PSB, PS and the SPS. • The PS is somewhat special in that it includes 100 so-called Combined-Function Magnets (CFMs) which consist of the superposition of bending magnets with multipoles of higher order. Figure 1: CFM in the PS, courtesy by Schoerling (2014) M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 5 / 15

Introduction Symplecticity Results & Conclusion References Purpose of these magnets? • Bending magnets and multipoles are used to keep particles on track. • Like marbles in a larger tracking pipe, the particles generally perform oscillations around the reference trajectory. Figure 2: Marble in a pipe with offset • Tune: phase advance of these oscillations during one revolution in the ring. • Chromaticity: dependency of tune w.r.t. the longitudinal momentum relative to the reference particle. M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 6 / 15

Introduction Symplecticity Results & Conclusion References Purpose of these Magnets? p. 2 • It turns out that many aspects of beam physics are similar to optics. In particular, one can describe most focusing / defocussing elements as thin lenses. • A thin lens description usually leads to fast tracking algorithms. • For ’thick’ elements like bending magnets, one is looking for a corresponding decomposition into several thin slices . Figure 3: Schematic view of the bending situation on the dark side of the moon ... • CFM = all in one = save space for other instruments. However that makes a thin-lens description not easy. The answer requires a small excursion. M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 7 / 15

Introduction Symplecticity Results & Conclusion References Excursion: Symplecticity Hamilton Equations with z := ( q , p ) tr and � � 0 1 z = J ( d H ( z , s )) tr ˙ (1) J = . − 1 0 2 n × 2 n If z 0 is a solution to (1) and M is a differentiable map on phase space, leaving H invariant, then z 1 = M ( z 0 ) is another solution if = J ( d H ( z 1 , s )) tr = J ( d H ( z 0 , s )( M ′ ( z 0 )) − 1 ) tr ! M ′ ( z 0 ) ˙ z 0 = ˙ z 1 = J [( M ′ ( z 0 )) − 1 ] tr ( d H ( z 0 , s )) tr = J [( M ′ ( z 0 )) tr ] − 1 ( − J ) ˙ z 0 , Symplecticity Condition consequently 1 : M ′ ( z 0 ) tr JM ′ ( z 0 ) = J 1 with a grain of salt ... M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 8 / 15

Introduction Symplecticity Results & Conclusion References Symplectic vs. non-symplectic integrators Simulating “ z i = M ( z i − 1 )” 3 3 3 10 2 2 2 1 1 5 1 0 0 0 0 1 1 1 2 2 5 2 3 3 3 15 10 5 0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 (d) gauss-legendre 2 (a) explicit euler (b) implicit euler (c) midpoint Figure 4: Arnold cats under the effect of various Runge-Kutta solvers for the harmonic oscillator with same step sizes. Message: Numerical errors introduce artificial effects, like blow up of phase space in single particle tracking, which leads to wrong predictions over long periods. 2 Symplectic; for a proof see Hairer et al. (2006). M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 9 / 15

Introduction Symplecticity Results & Conclusion References ... now back to CFMs • Without going into the details, the Hamiltonian of CFMs in comoving coordinates looks as follows �� � y + e η ) 2 − p 2 x − p 2 H = p σ − (1+ K x x + K y y ) (1 + ˆ A s . (2) p 0 • The mixture of x and p x etc. makes life more complicated: Hamiltons equations (1) can be written as z = −{H , z } = − : H : z and its (symplectic) solution is therefore ˙ z = exp( − s : H :) z 0 . • (2) ⇒ H = D + K , where D is the drift term in curvilinear coordinates and K the kick term involving the A s potential. Since D and K do not commute → BCH formula necessary for a drift-kick decomposition (remember: ’Beam optics’ picture). M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 10 / 15

Introduction Symplecticity Results & Conclusion References Solution • Fortunately, there is an alternative way: One can still write down an implicit symplectic solution for even the most general Hamiltonians [Titze (2015)] K 1 p ; s f ) , � ( µ + 1)!( − ∆ s ) µ +1 ( ∂ p ( −H . + ∂ s ) µ H )( q , ¯ q = q − ¯ µ =0 K 1 p ; s f ) . � ( µ + 1)!( − ∆ s ) µ +1 ( ∂ q ( −H . + ∂ s ) µ H )( q , ¯ p = p + ¯ µ =0 • This was utilized by me in first order to obtain a suitable thin-lens tracking map and implemented in MAD-X. • Important: The vector potential of the CFMs in comoving coordinates had to be determined as well. M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 11 / 15

Introduction Symplecticity Results & Conclusion References Results Preliminary 6.075 6.965 0.001 6.970 0.002 6.975 Q x ' Q y 6.980 ' 0.003 6.985 0.004 6.990 0.005 6.995 20 40 60 80 100 120 20 40 60 80 100 120 Slice number Slice number Figure 5: Chromaticity convergence in the PS against analytic Lie-Transformation result. M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 12 / 15

Introduction Symplecticity Results & Conclusion References Results p. 2 P ( x, p x ) ( y, p y ) 0.0003 r 0.04 e 0.0002 l 0.02 0.0001 i 0.00 0.0000 m 0.0001 i 0.02 n 0.0002 0.04 a 0.0003 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 0.004 r ( x, p x ) ( y, p y ) 0.0003 y 0.04 0.0002 0.02 0.0001 0.00 0.0000 0.0001 0.02 0.0002 0.04 0.0003 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.004 0.003 0.002 0.001 0.000 0.001 0.002 0.003 0.004 Figure 6: Tracking results of the unmodified code (Top left/right) and the corrected (symplectic) code (Bottom left/right). M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 13 / 15

Introduction Symplecticity Results & Conclusion References Conclusion • We have now a correct thin-lens description of CFMs available. Moreover, we have a method at hand in which we can basically slice every finite-dimensional problem which can be described by a Hamiltonian in terms of implicit symplectic steps (For example: many body problems in astronomy, resonance driven bunch envelope equations etc.). • This model is already implemented in a test version of MAD-X and currently running first Frozen SC tracking simulations in the PS. • A future official release of MAD-X is planned with includes the corrected CFM map. Many thanks for your attention! M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 14 / 15

Introduction Symplecticity Results & Conclusion References Further Reading I H. Bartosik, A. Oeftiger, F. Schmidt, and M. Titze. Space charge studies with high intensity single bunch beams in the CERN SPS. In Proceedings of the IPAC , 2016. To be published. E. Hairer, C. Lubich, and G. Wanner. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations . Springer Series in Computational Mathematics. Springer Berlin Heidelberg, 2006. ISBN 9783540306665. D. Schoerling. Prediction of the field distribution in CERN-PS magnets. In Proceedings of the IPAC , 2014. M. Titze. Symplectic maps for general hamiltonians. CERN HSC Group Meeting, November 2015. URL https://espace.cern.ch/ be-dep/ABP/HSC/Meetings/slides_MT_02-11-15.pdf . H. Yoshida. Construction of higher order symplectic integrators. Physical Letters A , 150(5), 1990. M. Titze (CERN / HU Berlin) Gentner Day 27.04.2016 15 / 15