[email protected] Numerical Simulations of Multipole Errors and Fringe Fields in the 1.5-TeV Muon Collider Lattice Using MADX and COSY codes Valery Kapin* & Yuri Alexahin (FNAL APC) APC seminar, FNAL, Batavia, 8-Sep-2011 *G.S. at APC, on leave from Moscow Engineering Physics Institute (State Univ.) The Object of the study Muon collider (MC) lattices with IR~1cm are featured by large -functions and beam sizes at IR (interection region) of SC-magnets S.C. magnets provide essential systematic multipolar errors, while special constructions against muon decay products increase them Fringe fields in quadrupoles become important due to large and varying beam sizes in q-magnets at IR These features require to use adequate simulation tools MADX code is considered as an appropriate candidate The aim of the study is to test and adapt MADX for MC simulations

Modified MADX tracking can import magnet maps from COSY code This work discuss and demonstrate abilities of MADX to be all-in-one code for MC simulations Background for MADX usage MAD is a code for modeling the beam dynamics in accelerators. MAD-X is the successor of MAD-8 (frozen in 2002). MAD-X offers most of the MAD-8 functionality. The most important addition is the interface to PTC, the Polymorphic Tracking Code by E. Forest (KEK, Japan).

PTC guarantees a proper description for thick elements (arbitrary exactness with various symplectic integrators) MAD-X has a modular organization => Team: custodian (F.Schmidt) + Module Keepers PTC-TRACK module is developed by V.Kapin (ITEP) & F.Schmidt (CERN) in frames of ITEP-CERN collaboration 2004-2006 Also high-order kicks are incorporated into to thick-magnets via MADX PTC module by VK & FS in 2005. MAD-X Home Page: http://mad.home.cern.ch/mad/ Subjects Refs. on status of M.C. lattice design Needs for simulations of multipole errors and fringe fields

MADX modules: abilities for simulations of multipoles errors and fringe fields COSY: export of magnet maps DA vs. fringe fields DA vs. sextupolar errors Chromaticity vs. multipole errors USER'S GUIDE Table of contents EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH TFS File Format TOUSCHEK

MAD-X Copyright Statement Conventions Command and Statement Format Control Statements o PTC Set-up Parameters Physical Elements and Markers o Overview of MAD-X Tracking Modules

Sequences Using aperture in MAD-X Conversion to Sixtrack Input Format Twiss Module o Thin-Lens Tracking Module (THINTRACK) MAKETHIN Conversion to Thin Lens o Thick-Lens Tracking module (PTC_TRACK) Dynap Module o

Emit Module o Ripken Optics Parameters (PTC_TWISS) Error Assignment Module IBS module Matching Module o Non-Linear Machine Parameters (PTC_NORMAL)

Orbit Correction Module o PTC Auxiliary Commands PLOT o Known Differences to Other Programs o Keyword and Subject Index o References SODD

Survey, geometric SXF file input and output Line Tracking Module (ptc_track_line) M.C. lattice design and study: refs. & team 1. Yu.Alexahin, M.C. Lattice Design Status, M.C. workshop, Telluride CO, July, 2011 (Beams-doc-3895-v1) 2. Y.I.Alexahin, E.Gianfelice-Wendt, V.Kapin, Chromaticity Correction for M.C. Optics, PAC11, New York, 2011 3. Y.I.Alexahin, E.Gianfelice-Wendt, V.Netepenko, Conceptual Design of the M.C. Ring Lattice, IPAC10, Kyoto, 2010

4. Y.I.Alexahin, E.GianFelice-Wendt, V.V.Kashikin, N.V.Mokhov, A.V.Zlobin, V.Alexakhin, M.C. Interaction Region Design, ibid. 5. A.V.Zlobin, Y.I.Alexahin, V.V.Kashikin, N.V.Mokhov, Magnet design for M.C. ring & IR, ibid 6. A.Netepenko, M.C. Lattice Design, Beams-doc-3579-v1, 2010 7. Y.I.Alexahin, E.Gianfelice-Wendt, V.Netepenko, New M.C. lattice design, Beams-doc-3477-v1, 2009 MC lattice and magnet requirements [1] zoom Beam energy (TeV)

0.75 IR beta-function * (cm) ~1 Geom. r.m.s. Emitt. (nm) 3.5 Dist. (m) IP -> 1st Q 6m B in arcs (T) 8-10 Q length (m) ~2 D length <6

Needs for simulations of multipole errors Cross-section and good field region of IR dipole coil of open mid-plane design allowing the passage of the decay electrons Geometrical harmonics [4,5] Rref 40mm b1 104 b3 -5.875 b5 -18.320 b7

-17.105 b9 -4.609 The standard multipole field expansion: B y ( x, y ) iBx ( x, y ) x iy Bref 10 4 (bn ian ) n 1 rref n 1 Relatively large values of geometrical harmonics in comparison with the traditional cos() design

With MAD notation the circular multipoles are written as B y iBx ikmskew x iy B m 0 m! m=0, 1, 2, for dipole, quadr., sext., resp. k norm m m for L 6m, the MAD kicks With the relation knl knnorm 1 L :

m n 1, and at Bref 8T, m knl B 2500 T m : n 1! b k norm 3.2 10 7 2 -1.41.10-2 4 -3.30.10+2 6 -5.77.10+6

8 -5.44.10+10 n 1 rrefn 1 n Needs for simulations of fringe fields in Qs Discussion and comprehensive list of reference about fringe fields in review paper*: general tendency to believe that contributions from opposite ends of Qs cancel each other. It is true for most rings. Q-FF are neglected in codes, e.g. MAD8, etc. FF effects may be important: in small rings with a large emittances (~10m.rad) and short magnets; in colliders for magnets in IR, where the -function variation is big. * Y.Papaphilippou, J.Wei, R.Talman, Deflections in magnet fringe fileds, PR-E67, 2003 QFF simulations: approaches Old studies for QFF formulae: K.Steffen (1965 - book), G.E. Lee-Whiting (1970)

D.L. Smith (1970) H.Wollnik (1970, 1972), P.Krejcik (1987), S.Koscielniak (2007), Lee-Whitinga Symplectic maps for FF: 1) E.Forest (1988 book -> PTC) 2) M.Berz & H.Wollnik (1990) -> M.Berz & Co. (1990 -> a) G.E. Lee-Whiting, Third Order Aberations of a Magnetic Quadrupole Lens, NIM-83, pp.232-244 (1970) present) Symplectic maps for QFF E.Forest, Beam Dynamics: a new attitude and framebook, 1998. PTC lib. by E.Forest implements the above Lee-Whitings HE-limit formulae for QFF . M.Berz, K.Makino, B.Erdelyi, Magnet Fringe Fields,

.., PAC2001. Details on COSYs QFF comes in next slides MADX modules: abilities for simulations of multipoles errors and fringe fields MADX consist of two type of modules: a) traditional modules with the most of MAD-8 functionality; b) PTC modules as an interface to PTC library by E.Forest Relevant traditional modules Twiss module (tunes, chromaticity, Wx,y, etc) THINTRACK via MAKETHIN (tracking in KICK-DRIFT lattice) SODD (Second Order Detuning and Distortion) as extension by FS Multipole errrors in thick magnet via splitting; Fringe fields in quadrupoles are absent (FFQ is not an octupole!) PTC modules: PTC Set-up Parameters BE1: RBEND, l=0.2, ANGLE=0.1, PTC_TWISS (Ripken Optics, w/o Wx,y)

knl:={0,0,-1.4E-02,0,-3.3E+02}; PTC-TRACK (Thick-Lens Tracking) PTC_NORMAL (Non-Linear Machine Parameters) Multipole errors in thick magnet are incorporated (available to all modules) LW-HE FF in Qs can be switched ON (by MADX source code modification) FFQ extension via export of Quad. map from COSY INFINITY COSY INFINITY 9.0 Beam Physics Manual: Maps can be printed to ext. file with command PM; Command FR invokes fringe field computations Maps for general elements from measurements are also possible (!); Modification of MADX PTC-TRACK done by VK allows to import externally generated map and use in particle tracking (ELEMENT_BY_ELEMENT): STEPS: testing and understanding COSY maps for Qs; coordinate transformation COSY-> MADX; subroutines for reading and calculating maps; testing runs) Particle tracking with PTC_TRACK Phase space trajectories at x0=0 y0=6; 12; 18 24m Phase space trajectories at x0=6m

strong non-linear coupling All w/o FFQ and multipole errors DA vs. fringe fields Tracking results using modified PTC-TRACK y0 (m) y0 (m) x0 (m) x0 (m) 1000 turns DA for 1.5TeV lattice in units of initial coordinates at IP (x=y=0): without (left) and with quadrupole fringe fields: center - embedded in MADX PTC hard-edge approximation, right - maps produced by COSY. Only vertical motion suffers due to y_max>> x_max PTC underestimates the effect DA vs mutipole errors: detuning coefficients Detuning coefficients (tune shifts vs. amplitude) are helpful for DA optimization:

An increase in detuning coefficients leds to to DA reduction Two procedures for det. coeff. calculations are used: a) PTC_NORMAL b) TBT analysis of PTC_TRACK data Correction sextupole error by sextupole of CORR1 via ptc_normal and TBTof PTC-TRACK PTC_NORMAL reproduce Parabolic dependence on Sext. Strength k2nl Opt.K2nl ~-0.02 TBT of PTC_TRACK Reproduce noisy Parabolic dependences Opt.K2nl ~-0.01 DA vs mutipole errors: optimization via scanning k2nl DA in the plane of initial particle coordinates:. left - no multipole errors, center sextupole error added, right - sextupole corrector placed at the 1st y maximum. Effect of the sextupole error can also be compensated with octupole (Netepenko) Sextupole error affects both x- and y-motion

Optimistic results for CORR2 Opt.K2nl ~-0.018 All multipole errors of IR Dipoles Effect of multipole components on DA in 1.5TeV case: decapole is most detrimental Head-on attempts to maximize DA (at all multipole errors) by simple scanning of every multipole k*nl of CORRs are not succesfull. => Needs for nonlinear corrector arrangements for multipole error correction Chromaticity corrections with TWISS module (traditional MADX) Lattice preparations Magnet conversion (slicing) for traditional MADX: a) BE1: rbend; b) BE1: sbend; c) BE1: sbend splitted onto 5 segmens; d) 5 sextupoles k2l are on in BE1. Chromaticity corrections with TWISS module Montague chromatic functions: a) up to c) no differences; d) 5 sextupoles k2l are ON in BE1. Matching constraints study: Matching results are the

same for both Wy and DQ2 Chromaticity corrections with TWISS module Tunes vs deltap: a) BE1: rbend; d) 5 sextupoles k2l are ON in BE1; e) after correction of liner chromaticity with MADX matching commands. Non-linear chromaticity corrections with TWISS module Final plot with Twiss All multipoles in BE1 are on It is helpful to use ploting software which provides polinomial fitting coefficients Non-linear chromaticity corrections: TWISS -> PTC_TWISS Transfer to PTC_TWISS: Small tuning by linear chrom. correctors

S1& S2-4: Plots for the range of pp=0.6% Qx,yQx,y[-0.018;+0.004] [-0.022;+0.004] Conclusions Muon collider (MC) lattice requires simulations with adequate treatments of systematic multipolar errors and Fringe fields in quadrupoles MADX code with relevant extensions and modifications can be an appropriate candidate to be all-in-one code for MC simulations Combination MADX with COSY as a map provider may cover most of types of quadrupole fringe fields formulations MADX modules TWISS, PTC_TWISS and PTC_NORMAL can be used as guiding tools at a design stage, while tracking with PTC_TRACK can provide more reliable results Some automatizations for DA calculations and TBT analysis of PTC_TRACK data are desirable. It can be done by an inclusion into source code both algorithms used in MADX-input-scripts and external codes like SUSSIX by FS.