Spin Tracking with COSY INFINITY and its Benchmarking

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Oswin Neal
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1 Spin Tracking with COSY INFINITY and its Benchmarking Marcel Rosenthal for the JEDI collaboration

2 Outline Methods Simulation toolbox New extension: Transfer maps for time-varying fields Application I : RF-B Solenoid Driven Oscillations Induced RF field spin resonance Benchmarking of measurement, analytical estimation and tracking Application II: RF-E B Wien filter Driven Oscillations EDM method based on RF fields Benchmarking of tracking and analytical estimation Systematic limitations Conclusion COSY INFINITY Benchmarking 2

estimation and tracking Application II: RF-E B Wien filter Driven Oscillations EDM method based on RF fields

3 Thomas-BMT-Equation Equation of spin motion for relativistic particles in electromagnetic fields: ds dt = S Ω MDM + S Ω EDM Ω MDM = e γm Ω EDM = e m η Gγ B + Gγ + γ 1 + γ E c + β B γ γ + 1 β β E c E β c Gγ2 γ + 1 β β B μ = 2(G + 1) e 2m S G Proton Deuteron d = η e 2mc S d η e cm ~ e cm ~ COSY INFINITY Benchmarking 3

B γ γ + 1 β β E c E β c Gγ2 γ + 1 β β B μ = 2(G + 1) e 2m S G Proton 1.792847357 Deuteron -0.

4 COSY Toolbox COSY INFINITY M. Berz, K. Makino et al. Calculator: Optical functions Closed orbit Spin tune Tracker: Static maps RF maps COSY Lattice COSY Toolbox ROOT Armadillo linear algebra operations SVD, pseudo-inverse plotting storage (ROOT files/trees) COSY ring Optical functions COSY INFINITY Benchmarking 4

COSY Lattice COSY Toolbox ROOT Armadillo linear algebra operations SVD,

5 Transfer Maps Solutions for equations of motion to arbritary order: M(z 0 ), A(z 0 ) Relate phase space and spin coordinates before and after element Static fields: Same transfer map for all particles. time-independent What about time-varying fields? COSY INFINITY Benchmarking 5

element Static fields: Same transfer map for all particles.

6 Transfer Maps for RF fields Radiofrequency fields: Split element into N maps covering the 360 phase interval of the time-varying field (currently N = 36). Map 5 Map 15 Map COSY INFINITY Benchmarking 6

interval of the time-varying field (currently N = 36).

Induce spin resonance by RF-B solenoid and measure characteristica of

7 Experimental Setup for Studies Beam setup: Polarized deuterons, 970 MeV/c Electron cooled and bunched Optimized Spin Coherence Time Idea: Induce spin resonance by RF-B solenoid and measure characteristica of vertical polarization oscillations COSY INFINITY Benchmarking 7

8 Measured time distribution Bunch center: τ = 0 Time marking system allows for determination of event time with respect to RF cavity Extraction of measured time distribution (τ-distribution) of particles in the bunch is possible COSY INFINITY Benchmarking 8

Extraction of measured time distribution (τ-distribution) of

9 Probability Deconvolved amplitude distribution deconvolution assuming harm. oscillator ρ(τ ) Bunch center: τ = 0 averaging τ in tracking τ in ns Initial longitudinal amplitude distribution required for analytical estimations and for tracking simulations. Assuming solution for an harmonic oscillator τ τ cos(2πν sync n + φ sync ), the deconvolution results in the longitudinal synchrotron amplitude τ -distribution COSY INFINITY Benchmarking 9

required for analytical estimations and for tracking simulations.

10 Benchmarking Results Radiofrequency field: B sol = B sol cos 2πν sol n + φ sol turned on at ~11 mio. turns. resonance condition: ν sol = Gγ + K, K Z preliminary K = 1 preliminary K = 1 preliminary K = COSY INFINITY Benchmarking 10

resonance condition: ν sol = Gγ + K, K Z preliminary K = 1

11 Benchmarking Results Radiofrequency field: B sol = B sol cos 2πν sol n + φ sol turned on at ~11 mio. turns. resonance condition: ν sol = Gγ + K, K Z preliminary K = 1 preliminary K = 1 preliminary K = 2 Analytical estimation: P y n = ρ τ S y n, τ dτ COSY INFINITY Benchmarking 11

resonance condition: ν sol = Gγ + K, K Z preliminary K = 1 preliminary K

12 RF Wien filter induced spin resonance Idea: RF-E B Wien filter with B e y on resonance induces buildup of vertical spin component for non-vanishing EDM. Minimized impact on beam, but interaction on spin Analytical estimation for closed orbit and S z 0 = 1: ds y α 0 dn 2 n y 2 n z sin φ WF + n y n x cos φ WF + fast osc. terms α 0 = 1 + G γ q p B L WF B WF n = B WF cos(2πν WF n + φ WF ) E WF n = βc B WF (n) n x, n y, n z : spin closed orbit of static RF Wien filter location η = 10 5 BL WF = 0.06 T mm COSY INFINITY Benchmarking 12

Minimized impact on beam, but interaction on spin Analytical estimation for closed orbit and S z 0 = 1: ds y α 0 dn 2 n y 2 n z sin φ WF

13 Buildup for different EDM magnitudes Buildup scales linearly with EDM magnitude and depends on initial RF Wien filter phase η = 10 5 BL WF = 0.06 T mm Good agreement between analytical estimates and tracking results COSY INFINITY Benchmarking 13

14 Buildup for Misalignments Introduce misalignments of the 56 lattice quadrupoles Gaussian distributed with σ mis = 0.1 mm BL WF = 0.06 T mm ds y dn α 0 2 n y 2 n z sin φ WF + n y n x cos φ WF + fast osc. terms Pure EDM: n x 0, misalignments: n x 0 and n z COSY INFINITY Benchmarking 14

06 T mm ds y dn α 0 2 n y 2 n z sin φ WF + n y n x cos φ WF + fast osc.

15 Connection to Orbit RMS Main contribution from additional radial magnetic fields Besides spin motion, also beam motion is affected Gaussian distributed quadrupole displacements ( μ = 0, σ = σ mis ) BL WF = 0.06 T mm Vertical buildup ΔS y per turn for RMS of 1 mm similar to η = 10 4 (d = e cm) COSY INFINITY Benchmarking 15

displacements ( μ = 0, σ = σ mis ) BL WF = 0.

16 Summary Method: Calculation of maps for time-varying elements implemented into COSY INFINITY extension Application I: RF-B Solenoid Successfully benchmarked with analytical estimates and measured data for RF-B solenoid induced resonance Dependence of oscillation damping on solenoid frequency has been reproduced Application II: RF-E B Wien filter Tracking results for EDM related buildup match with analytical calculations. Gaussian distributed quadrupole displacements which introduce an vertical orbit RMS of 1 mm lead to a buildup similar to an EDM of d e cm Questions? THPF031 ([email protected]) COSY INFINITY Benchmarking 16

II: RF-E B Wien filter Tracking results for EDM related buildup match with analytical calculations.

Results: Low current (2 10 12 ) Worst case: 800 MHz, 12 50 GeV, 4 turns Energy oscillation amplitude 154 MeV, where

Results: Low current (2 10 12 ) Worst case: 800 MHz, 12 50 GeV, 4 turns Energy oscillation amplitude 154 MeV, where Status Focus has shifted to a neutrino factory Two comprehensive designs of acceleration (liancs, arcs) Jefferson Lab, for Fermilab Study CERN (Keil et al.) Jefferson Lab study Low (2 10 12 ) charge per