THE RAY-TRACING CODE ZGOUBI F. M eot CEA & IN2P3, LPSC - PowerPoint PPT Presentation

FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 THE RAY-TRACING CODE ZGOUBI F. M´ eot CEA & IN2P3, LPSC Grenoble

Contents FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 1 Introduction 3 1.1 What Zgoubi can do . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The numerical inegration method . . . . . . . . . . . . . . . . 5 2 Tracking FFAGs 8 Radial scaling triplet using ‘‘FFAG’’ , 6-D tracking simula- 2.1 tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Radial triplet represented by 3-D field maps, using ‘‘TOSCA’’ 16 2.2 ‘‘FFAG-SPI’’ , 17 → 180 MeV acceleration in RACCAM . . . 2.3 19 NuFact linear FFAG using ‘‘MULTIPOL’’ , 6-D transmission 2.4 20 2.5 EMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 EMMA using ‘‘MULTIPOLE’’ , full acceleration cycle 2.5.1 22 Closer to actual field shape : EMMA using ‘‘DIPOLES’’ 25 2.5.2 2.5.3 The idea, more precisely : . . . . . . . . . . . . . . . . 26 A new procedure for field maps, ‘‘EMMA’’ . . . . . . 2.5.4 27 Pumplet-cell I-FFAG using ‘‘DIPOLES’’ , full acceleration 2.6 cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Synchrotron radiation 31 4 Spin 35

Introduction FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 1 1.1 What Zgoubi can do Calculate trajectories of charged particles in magnetic and electric fields. • At the origin (early 1970’s) developped for design and operation of • beam lines • magnetic spectrometers • Zgoubi has so evolved that it allows today the study of • systems including complex sequences of optical elements -30 -25 -20 -15 -10 -5 0 • periodic structures -25 • and allows accounting for additional properties as -20 BEND QUAD MULT QUAD BEND MULT MULT BEND BEND QUAD QUAD -15 BEND • synchrotron radiation and its dynamical effects BEND MULT MULT BEND BEND -10 QUAD QUAD • spin tracking -5 MULT MULT 0 QUAD MULT BEND BEND • in-flight decay MULT 5 MULT BEND BEND QUAD QUAD BEND • etc... 10 BEND MULT MULT BEND QUAD MULT QUAD BEND 15 • FAQ : not space charge (not yet ?) 20

it provides numerous Monte Carlo methods FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 • object definition • stochastic SR • in-flight decay • etc. • a built-in fitting procedure including – arbitrary variables ∗ any data in the input file can be varied – a large variety of constraints, ∗ easily extendable to even more • multiturn tracking in circular accelerators including – features proper to machine parameter calculation and survey, – simulation of time-varying power supplies, ∗ any element individually (allows tune-jump, etc.) – etc.

) M 1 ( u Z 1.2 The numerical inegration method M FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 1 u (M 0 ) Y MOTION, from M 0 to M 1 M X R (M 1 ) Z ) 0 M The equation of motion ( Y R X 0 v × � d ( m� v ) = q ( � e + � b ) dt Reference • is solved using truncated Taylor expansions of � R and � u = � v/v : u ′ ( M 0 ) ∆ s 2 u ′′′′′ ( M 0 ) ∆ s 6 R ( M 1 ) ≈ � � R ( M 0 ) + � u ( M 0 ) ∆ s + � 2! + ... + � 6! (1) u ′′ ( M 0 ) ∆ s 2 u ′′′′′ ( M 0 ) ∆ s 5 � u ( M 1 ) ≈ � u ( M 0 ) + � u ( M 0 ) ∆ s + � 2! + ... + � 5! • In non-zero � E environement, rigidity at M 1 is re-computed : ( Bρ )( M 1 ) ≈ ( Bρ )( M 0 ) + ( Bρ ) ′ ( M 0 )∆ s + ... + ( Bρ ) ′′′′ ( M 0 )∆ s 4 (2) 4! • When necessary, time-of-flight is computed in a similar manner : ds ( M 0 ) ∆ s + d 2 T ds 2 ( M 0 ) ∆ s 2 + d 3 T ds 3 ( M 0 ) ∆ s 3 3! + d 4 T ds 4 ( M 0 ) ∆ s 4 T ( M 1 ) ≈ T ( M 0 ) + dT (3) 2 4!

• In a general manner, the truncated Taylor series FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 � = � R ( M 1 ) R ( M 0 ) + � u ( M 0 ) ∆ s + ... u ′ ( M 0 ) ∆ s + ... u ( M 1 ) � = � u ( M 0 ) + � (4) ( Bρ )( M 1 ) = ( Bρ )( M 0 ) + ( Bρ ) ′ ( M 0 )∆ s + ... = T ( M 0 ) + dT T ( M 1 ) ds ( M 0 ) ∆ s + ... require computation of the derivatives u ( n ) = d n � u / ds n � ( Bρ ) ( n ) = d n ( Bρ ) / ds n d n ( T ) / ds n

Integration in magnetic fields FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 • Let’s introduce simplified notations : � u = � v u ′ = d� u b � � v, ds = v dt, � ds, m� v = mv� u = q Bρ � u B = (5) Bρ v × � d ( m� v ) = q ( � e + � b ) dt (with � e = 0 ) then writes u ′ = � u × � � B u ( n ) = d n � u ′ = � u / ds n needed in the Taylor expansions : � u × � This yields the � B u ′ × � u × � u ′′ B ′ � = � B + � u ′′ × � u ′ × � B ′ + � u × � u ′′′ B ′′ � = � B + 2 � (6) u ′′′ × � u ′′ × � B ′ + 3 � u ′ × � B ′′ + � u × � u ′′′′ B ′′′ � = � B + 3 � u ′′′′′ = � u ′′′′ × � u ′′′ × � B ′ + 6 � u ′′ × � B ′′ + 4 � u ′ × � B ′′′ + � u × � B ′′′′ � B + 4 � B ( n ) = d n � � B / ds n . where

Tracking FFAGs FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 2 • Accelerator R&D domains concerned : • NuFact – scaling lattice – linear non-scaling lattice • EMMA • Medical – scaling lattice – linear non-scaling lattice – quasi-linear non-scaling lattice • p-driver – scaling lattice – linear non-scaling lattice – quasi-linear non-scaling lattice – pumplet lattice • etc. • In all cases : 2-D and 3-D field map based ray-tracing

• Optics : FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 • Scaling FFAG simulations need special elements, like - [sector, spiral,...] dipoles with “arbitrary” axial (including fringe field effects) and radial de- pendence of magnetic field B zi ( r, θ ) = B z 0 ,i F i ( r, θ ) R i ( r ) (“ FFAG ” procedure) : - Scaling FFAG, NC magnets R i ( r ) = ( r/R 0 ,i ) K i (“ DIPOLES ” procedure) : - Scaling FFAG, SC magnets R i ( r ) = b 0 i + b 1 i ( r − R 0 ,i ) /R 0 ,i + b 2 i ( r − R 0 ,i ) 2 /R 2 0 ,i + ... - accounting for possible variable gap g = g 0 ( R 0 /r ) K , and overlapping of fringe fields • Linear FFAGs are built from quadrupole fields, the “ MULTIPOLE ” procedure in Zgoubi is employed to simulate that. These FFAGs did not necessitate any particular developement. • The isochronous type of FFAG lattice (“pumplet”) is simulated combining “ MULTIPOLE ” and “ DIPOLES’’ procedures. 5 5 -2.25 -2.5 4 4 -2.75 3 3 -3 2 -3.25 1 2 -3.5 -0.15 -0.1 -0.05 0.05 0.1 0.15 1 -3.75 -1 -2 -0.15 -0.1 -0.05 0.05 0.1 0.15 -0.15 -0.1 -0.05 0.05 0.1 0.15

• RF : not a real issue up to now : FFAG09 Wrkshp, FermiLab, 21-25 Sept. 2009 • Regular point transforms seems to answer FAQs • The question of TOF in terms of a reference particle, or absolute TOF computation sometimes arise - well managed by the ray-tracing methods up to now • This has been confirmed experimentally : acceleration is achieved successfully in a number of different RF regimes (detailed simulations published, see FFAG workshop series) : – RF swing in radial S-FFAG lattice (cf. KEK 150 MeV) – RF swing in spiral S-FFAG lattice (cf. RACCAM) – RF swing in linear NS-FFAG lattice (cf. PAMELA by T. Yokoi et al.) – fixed, low RF frequency, stationary bucket mode, in S-FFAG (cf. PRISM ; muon accelera- tors in NuFact) – fixed, high RF frequency, serpentine mode, in NS-FFAG (cf. EMMA ; muon accelerators in NuFact) – fixed, high RF frequency, quasi-isochronous mode, in semi-NS-FFAG (“Pumplet” lattice) • More realistic, pill-box or other type of cavities will be introduced for NuFact R&D, • as well as 4-D electromagnetic field maps