LHC optics measurement & correction procedures M. Aiba, R. Calaga, A. Morita, R. Tomás & G. Vanbavinckhove Thanks to: I. Agapov, M. Bai, A. Franchi, M. Giovannozzi, V. Kain, G. Kruk, J. Netzel, S. Redaelli, F. Schmidt, J. Wenninger and F. Zimmermann Extended LTC - March 2008 Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.1/37
Contents • Calibration independent measurements: + Phase measurements: - from FT or SVD of turn-by-turn BPM data - from Closed Orbit Distortion + Betas from phases + Normalized Dispersion + Coupling from FT of turn-by-turn BPM data • Correction + Response matrix inversion + Simulations + RHIC tests and future SPS tests • Controls application Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.2/37
Turn-by-turn BPM data Fake LHC BPM data (pilot bunch) 4 3mm kick 3 2 Decoherence due to ∆ Q 1 x [mm] 0 σ bpm =0.2mm -1 -2 -3 -4 0 100 200 300 400 500 600 700 800 900 1000 Turn number Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.3/37
FFT of BPM data Fake LHC BPM FFT 300 Qx 250 Amplitude [arb. units] 200 150 100 Qs, 2Qs, ... -2Qx 50 Qy 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Frequency [tune units] ∆ φ x between BPM1 and BPM2 = φ bpm 2 Qx - φ bpm 1 Qx A bpmN � � Coupling inferred from the amplitude of Qy Qy Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.4/37
FFT versus SUSSIX 3mm oscillation + 0.2mm Gaussian error (no decoherence) 3 SUSSIX FFT 2.5 Phase error [deg] 2 ∝ N -1/2 1.5 1 0.5 0 100 200 300 400 500 600 700 800 900 1000 Number of turns In presence of noise SUSSIX reduces the phase error by a factor 2-3 showing same scaling with N. Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.5/37
Closed Orbit Distortion The x CO change at location s produced by a corrector c is given by: � ∆ x CO ( s ) ∝ β s cos( | φ s − φ c | − Qπ ) A collection of orbits using different correctors allows to fit β s and φ s at all BPMs yielding: • calibration dependent β s • calibration independent φ s A. Morita, PRSTAB 10 , 072801 operational in KEK-B Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.6/37
LHC COD performance simulation 0.20 10.00 σ /A (worst) 5.94 σ /A (typ.) 4.35 σ /A (best) wo calibration error 0.15 Maximum error of ∆φ w calibration error = 2 ◦ needs σ σ max A =0.3% φ 0.10 0.05 0.00 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 BPM resolution Maximum Closed-Orbit Excitation: σ BPM /A Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.7/37
Betas from phases in the LHC arcs Betas from phases ( σ φ =0.25 o ) 5 Max. β -measurement error [%] 4 3 2 cot φ 12 − cot φ 13 β 1 = m 11 /m 12 − N 11 /N 12 1 Horizontal Vertical 0 0 5 10 15 20 25 30 Model rms ∆β/β [%] Model unknowns and BPM noise contribute to β error Using β 1 , β 2 , β 3 improves the error, works in IRs. Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.8/37
� to errors in LHC � D/ √ β x Robustness of 1.02 〈 D/ β 1/2 〉 err / 〈 D/ β 1/2 〉 ideal 〈 D 〉 err / 〈 D 〉 ideal 〈 β〉 err / 〈β〉 ideal 1.015 Ratio (err/model) 1.01 1.005 1 0.995 0 0.02 0.04 0.06 0.08 0.1 0.12 Horizontal rms beta-beating Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.9/37
Normalized dispersion � A bpmN β bpmN = c bpmN c global x Qx Let X bpmN be the radial steering= c bpmN D bpmN δ , X bpmN D bpmN D bpmN δ = c , A bpmN = global c global � � β bpmN β bpmN x x Finally averaging X bpmN /A bpmN over all BPMs: � D bpmN � X bpmN � � = c , ← frommodel global A bpmN � β bpmN x Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.10/37
Simulation of D/ √ β x measurement σ BPM =0.2mm, kick=2 σ , N=512turns, dp/p=0.15% Max. D β -1/2 measurement error [m 1/2 ] 0.04 0.03 0.02 0.01 0 0 5 10 15 20 Model rms ∆β/β [%] Effectively a model independent measurement! Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.11/37
Dispersion from D/ √ β and β σ BPM =0.2mm, kick=2 σ , N=128turns, dp/p=0.1% 3 2 1 Dispersion [m] 0 -1 -2 Simulation -3 MAD -4 0 2 4 6 8 10 12 14 16 Longitudinal location [km] Very good measurement of D from D/ √ β and β Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.12/37
Coupling (local) (For global coupling correction see: R. Jones et al, CERN-AB-2005-083 BDI0) Let A H Qy be the amplitude of the vertical tune in the horizontal plane, hence � A H A V | f 1001 | = 1 Qy Qx A H A V 2 Qx Qy • Calibration independent but BPM-tilt dependent • Close to the resonance: ∆ Q min ≈ 4∆ | f 1001 | PRSTAB 8 , 034001; PRSTAB 10 , 064003 Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.13/37
Coupling: measurement around LHC 0.105 MADX 0.1 Simulation 0.095 0.09 0.085 |f 1001 | 0.08 0.075 0.07 0.065 0.06 0.055 0 5 10 15 20 25 Longitudinal location [km] Random 2mrad BPM tilts, 400 turns, 4mm kick → Lo- cal coupling is measurable. Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.14/37
Optics correction • Using the model response matrix R , the change in the quadrupole circuits ∆ � k to achieve correction is given by: � D x � � ∆ � � k = − R − 1 ∆ √ β x φ, , Q x , Q y This equation applies the same for correction on 1 beam only or on 2 beams simultaneously. • Similarly for local coupling correction: � � ∆ � ℜ ( � f 1001 ) , ℑ ( � f 1001 ) , ℜ ( � f 1010 ) , ℑ ( � k s = − R − 1 f 1010 ) s Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.15/37
Correction: Simulation in 2006 6 Before Before 0.6 After After Peak ∆ D x / √β x [x10 -2 ] 4 Peak ∆β y / β y 0.4 Peak Vs. RMS 2 0.2 Peak β -Beat Dispersion 0 0 0 0.1 0.2 0.3 0.4 0 1 2 RMS ∆ D x / √β x [x10 -2 ] Peak ∆β x / β x 2 0.3 Before After Tune Shift: ∆ Q 1 RMS ∆β y / β y ∆ Q y [x10 -2 ] 0.2 0 0.1 -1 RMS β -Beat 0 -2 0 0.05 0.1 0.15 0.2 -2 -1 0 1 2 ∆ Q x [x10 -2 ] RMS ∆β x / β x • Errors: 80% measured & as installed, 20% extrapolated • Additional 2mm random sext. misalignments + 5 units random B2 Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.16/37
Correction: present status, preliminary 0.6 10 Before Before After After 8 Peak ∆ D x / √β x [x10 -2 ] LHC Beam 1, 60 Seeds 0.4 Peak ∆β y / β y 6 (Mutlipole Errs upto B10/A10) 3 RMS Spec 4 0.2 Peak Vs. RMS 2 Peak β -Beat Dispersion 0 0 0 0.2 0.4 0 1 2 3 4 5 RMS ∆ D x / √β x [x10 -2 ] Peak ∆β x / β x 0.3 3 Before After 2 RMS ∆β y / √β y 0.2 1 ∆ Q y , ∆ Q’ y , ∆ y 0 0.1 -1 Tune Shift [x 10 -2 ] -2 Chrom Shift [x 10] RMS β -Beat Orbit Shift [cm] 0 -3 0 0.1 0.2 0.3 -3 -2 -1 0 1 2 3 RMS ∆β x / √β x ∆ Q x , ∆ Q’ x , ∆ x • Errors: 100% measured & as installed • No additional misalignments added, no orbit correction Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.17/37
Correction: observations Correction is achieved for most of the seeds if: • σ φ < 1 ◦ • failing BPMs < 10% • σ D √ β ≈ 0 . 01 m 1 / 2 (see support slides for details) → Might be tight for LHC commissioning... Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.18/37
Coupling correction Using all the skew quadrupole correctors: 0.12 Uncorrected Corrected 0.1 0.08 ∆ Q min ≈ 0 . 01 |f 1001 | 0.06 0.04 ∆ Q min ≈ 0 . 001 0.02 0 0 5 10 15 20 25 30 Longitudinal location [km] → Not perfect due to the particular distribution of errors/correctors. Best local correction is realignment Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.19/37
RHIC exp.: 6 random quads used 1.2 Ideal Model Opp Polarity BPMs Baseline 1 Horizontal Quad Trimmed (∆φ) [Q Units] 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 1.2 Ideal Model Baseline 1 Vertical Quad Trimmed (∆φ) [Q Units] 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitudinal Position [km] Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.20/37
RHIC exp.: MAD reconstruction, ∆ φ 6 Horizontal Reconstructed MADX δ(∆φ) [x10 -2 , Q Units] 4 Meas 2 0 -2 -4 Effect of 6 Trim Quads -6 0 0.5 1 1.5 2 2.5 3 3.5 4 6 δ(∆φ) [x10 -2 , Q Units] Vertical 4 2 0 -2 -4 -6 0 0.5 1 1.5 2 2.5 3 3.5 4 Longitudinal Position [km] Rogelio Tom´ as Garc´ ıa LHC optics measurement & correction procedures – p.21/37
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