Measuring sextupole and octupole resonance driving terms using

Measuring sextupole and octupole resonance driving terms using
turn-by-turn BPM data Andrea Franchi on behalf of the ASD Beam Dynamics Group Low Emittance Rings 2014 Workshop Frascati, 17th -19th September 2014

the physics behind the analysis from TbT BPM data to magnet strengths
Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

the physics behind the analysis
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Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX I->e X->t L->h I->e E->a R->s M->I Z->m V->r + some extrapolations reveal Edgar Allan Poe’s novel “The Gold-Bug” Hereupon Legrand arose, with a grave and stately air, and brought me the beetle from a glass case in which it was enclosed. It was a beautiful scarabaeus, and, at that time, unknown to naturalists—of course a great prize in a scientific point of view. There were two round black spots near one extremity of the back, and a long one near the other. The scales were exceedingly hard and glossy, with all the appearance of burnished gold. The weight of the insect was very remarkable, and, taking all things into consideration, I could hardly blame Jupiter for his opinion respecting it. Andrea Franchi Resonance Driving Terms

Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

human voice Storage Ring’s voice Andrea Franchi Resonance Driving Terms

language spectrum: Storage Ring’s transverse spectra Andrea Franchi Resonance Driving Terms

from TbT BPM data to magnet strengths
X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM Andrea Franchi Resonance Driving Terms

X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM from TbT BPM data Andrea Franchi Resonance Driving Terms

X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM from linear lattice model from TbT BPM data Andrea Franchi Resonance Driving Terms

X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM
from TbT BPM data to magnet strengths X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM from linear lattice model Courant-Snyder TbT data for FFT from TbT BPM data Andrea Franchi Resonance Driving Terms

|V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0|
from TbT BPM data to magnet strengths |H(1,0)|=½(2Ix)1/2 |V(0,1)|=½(2Iy)1/2 |H(-2,0)|=(2Ix) |FNS3| |H(0,-2)|=(2Iy) |FNS2| |V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM Courant-Snyder TbT data for FFT s-dep. s-dep. s-indep. kick-dep. kick-indep. Andrea Franchi Resonance Driving Terms

|V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0|
from TbT BPM data to magnet strengths |FNS3| ≈ M3(β,φ) × K2(sext) |FNS2| ≈ M2(β,φ) × K2(sext) |FNS1| ≈ M1(β,φ) × K2(sext) |FNS0| ≈ M0(β,φ) × K2(sext) from linear lattice model from magnet calibration |H(1,0)|=½(2Ix)1/2 |V(0,1)|=½(2Iy)1/2 |H(-2,0)|=(2Ix) |FNS3| |H(0,-2)|=(2Iy) |FNS2| |V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| from the model from measurement Andrea Franchi Resonance Driving Terms

Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Andrea Franchi Resonance Driving Terms

Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Large kicker strength to increase FFT resolution Vs. second & higher-order contribution to RDTs (checks are possible, simulations and data analysis) unwanted modulation induced by 2nd order effects of sextupoles Andrea Franchi Resonance Driving Terms

Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Large kicker strength to increase FFT resolution Vs. second & higher-order contribution to RDTs (checks are possible, simulations and data analysis) linear model (β,φ) needs to be robust otherwise model & measured RDTs get corrupted (check BPM phase advance and invariance of tune line amplitude) Andrea Franchi Resonance Driving Terms

Magnet calibration Vs cycling at different currents
Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Large kicker strength to increase FFT resolution Vs. second & higher-order contribution to RDTs (checks are possible, simulations and data analysis) linear model (β,φ) needs to be robust otherwise model & measured RDTs get corrupted (check BPM phase advance and invariance of tune line amplitude) Magnet calibration Vs cycling at different currents Andrea Franchi Resonance Driving Terms

results for sextupole magnets
Andrea Franchi Resonance Driving Terms

more than 50% of the RDT modulation stems from focusing errors (beta beating)!! model with perfect sextupoles Andrea Franchi Resonance Driving Terms

model with perfect sextupoles Andrea Franchi Resonance Driving Terms

model with fitted sextupole errors Andrea Franchi Resonance Driving Terms

fit of 224 sextupole errors S4C15 calibration S4C S4C29-30 calibration of sextupole correctors Andrea Franchi Resonance Driving Terms

fit of 224 sextupole “effective rotations” (better call them “skew-sextupole fields”) Andrea Franchi Resonance Driving Terms

beam-based nonlinear magnet calibration

correction of resonance driving terms & lifetime
sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: Andrea Franchi Resonance Driving Terms

sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: inj. septum Andrea Franchi Resonance Driving Terms

sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: hypothesis: the more “matched” the RDTs, the larger the dynamic aperture, and (hopefully) the longer the lifetime Andrea Franchi Resonance Driving Terms

sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: hypothesis: the more “matched” the RDTs, the larger the dynamic aperture, and (hopefully) the longer the lifetime filling pattern P= 6.03 GeV/c Ey Ex=4nm lifetime correctors OFF manual correction RDT correction multibunch (1 mA/bunch) low chroma 7 pm 16.2 h 24.2 h 22.4 h 16 bunch (~6 mA/bunch) large chroma 5 pm 2.5 h 3.2 h 2.0 h Andrea Franchi Resonance Driving Terms

RDTs are a powerful tool to check linear and nonlinear lattice models
Conclusion the harmonic analysis of turn-by-turn BPM data provides a direct measurement of the resonant driving terms (RDTs, i.e. the β-functions of the nonlinear lattice) RDTs are a powerful tool to check linear and nonlinear lattice models RDTs are robust observables for beam-based calibration of nonlinear magnets Correcting RDTs (alone) seems not really effective in improving beam lifetime, specially for Touschek- dominated beams: off-momentum RDTs needed? Andrea Franchi Resonance Driving Terms

Measuring sextupole and octupole resonance driving terms using

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Measuring sextupole and octupole resonance driving terms using

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