Progress in the LHC Schottky Spectra Analysis
64th SY-BI Students and R&D meeting
K. Łasocha, D. Alves, T. Levens, O. Marqversen (BI-IQ)
C. Lannoy, N. Mounet (ABP-CEI)
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Presentation outline
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Synchrotron motion
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While circulating along the machine the particles, due to the acceleration obtained in RF cavities, particles undergo longitudinal synchrotron motion.
The turn-by-turn longitudinal position of a particle can be written as:
As a consequence, particle’s momentum also oscillates.
Oscillation amplitude
Synchrotron frequency
Synchrotron phase
Revolution period
Betatron motion
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As the result of focusing, mostly affected by quadrupole magnets, particles also undergo transverse betatron motion.
The turn-by-turn position of a particle in given location can be written as:
Oscillation amplitude
Betatron tune
Betatron phase
Tune & Chromaticity
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To preserve beam stability, betatron tunes should avoid values such that:
nQH + mQV = k,
where n,m,k are integers.
Tune & Chromaticity
Courtesy: R. Steinhagen
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To preserve beam stability, betatron tunes should avoid values such that:
nQH + mQV = k,
where n,m,k are integers.
Coll
INJ
Tune & Chromaticity
Courtesy: R. Jones
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To preserve beam stability, betatron tunes should avoid values such that:
nQH + mQV = k,
where n,m,k are integers.
Coll
INJ
Tune & Chromaticity
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Betatron tune is related to the particle momentum via chromaticity Qξ (another notation: Q’):
In the LHC dp/p ≈ 3×10-4, and natural, uncorrected Qξ ≈ -140, so Δq ≈ 0.04.
Tune & Chromaticity
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Betatron tune is related to the particle momentum via chromaticity Qξ (another notation: Q’):
In the LHC dp/p ≈ 3×10-4, and natural, uncorrected Qξ ≈ -140, so Δq ≈ 0.04.
Tune & Chromaticity
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Betatron tune is related to the particle momentum via chromaticity Qξ (another notation: Q’):
In the LHC dp/p ≈ 3×10-4, and natural, uncorrected Qξ ≈ -140, so Δq ≈ 0.04.
In addition, certain values of chromaticity have to be avoided to prevent the onrise of so called head-tail instabilities.
In the LHC, H-T instabilities arise for Q’ below or equal to 0.
Tune & Chromaticity
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Betatron tune is related to the particle momentum via chromaticity Qξ (another notation: Q’):
In the LHC dp/p ≈ 3×10-4, and natural, uncorrected Qξ ≈ -140, so Δq ≈ 0.04.
Chromaticity can be corrected
using sextupole magnets.
In the LHC target value of chromaticity Q’ is in order of 5-20.
How to know that we are on target?
Tune & Chromaticity: measurements
The direct (and invasive) way of measuring Q and Q’ is based on kicking/modulating the beam:
For machine protection, this can be done only on a probe beam, before the injection of physics beam.
The non-invasive alternatives are:
LHC design report put a requirement of controlling Q’ within 1 unit at INJ, and within 3 units at Flattop.
This can be verified only since the last year’s upgrade of Schottky system.
Courtesy: M. Krupa
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LHC Schottky Monitor
Instrument designed to observe transverse oscillations of a selected bunch at frequencies dominated by incoherent particle motion.
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M. Betz, O.R. Jones, T. Lefevre, M. Wendt, Nuclear Instruments and Methods in Physics Research Section A, vol. 874, 2017
LHC Schottky Monitor
Transverse sidebands
Longitudinal (central) band
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Transverse beam spectrum around 427725th harmonic of the revolution frequency (≈ 4.81 GHz ± 7.5 kHz) with a sub-hertz resolution.
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Signal perspective:
Schottky Spectrum: emittance
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Schottky Spectrum: tune
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Schottky Spectrum: chromaticity
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Schottky Spectrum: chromaticity
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Schottky Spectrum: chromaticity
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Schottky Spectrum: synchrotron frequency
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Fs
Synchrotron frequency determines the distance between the satellites
Theory
Vague understanding on of how Schottky spectra are affected by:
Diagnostics
Challenges:
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Schottky spectra analysis: status for 2023
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What we expected
What we saw
Theory of Schottky spectra
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Simulation techniques: matrix formalism
Details in: K. Lasocha’s PhD Thesis, K. Lasocha and Diogo Alves, Phys. Rev. Accel. Beams 23, 062803
Matrix formalism provides a fast way to calculate Schottky spectra when the theory is known. In short, it can be described by the following diagram:
Since recently matrix formalism is implemented as Acc-Py package SchMatrix:
Schottky matrix, determined by the machine parameters
Synchrotron amplitude distribution or longitudinal bunch profile
Schottky spectrum
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Simulation techniques: macroparticle simulations
Details in: C. Lannoy et al. JINST 19 P03017, C. Lannoy’s PhD Thesis
If the theory is not yet known, or is to be benchmarked, the Schottky spectra can be calculated from one of the simulation packages, as PyHeadtail and XSuite.
C. Lannoy developed an effective procedure that calculates Schottky spectra from turn-by-turn position of macroparticles.
Such simulations are significantly slower than Matrix Formalism, but can include a myriad of effects not predicted in the Schottky theory.
The Schottky Monitor module is planned to be included in XSuite package soon.
Courtesy: C. Lannoy
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Impact of octupoles on Schottky spectra
Details in: C. Lannoy et al., IPAC2024 WEPG32
Octupole magnets provide additional tune spread to stabilise LHC beam:
Transverse actions, proportional to the betatron oscillation amplitude
Detuning coefficients, dependent on beam optics and octupole currents
Expansion under assumption of uniform octupole currents
Modification of the Schottky spectrum is defined:
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Impact of octupoles on Schottky spectra
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∗
convolution
=
∗
convolution
Impact of octupoles on emittance estimate
Theory:
No impact, the power of the spectrum is invariant on octupole current.
Experiment:
Power of spectrum over the theoretical predictions for low octupole current
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MD 9263
Impact of octupoles on tune estimate
Theory:
Center of sidebands shifted, and coincides with the average tune, weighted by Jx. For Gaussian bunches:
Qx, Schottky = Qx,0 + 2 αx εx + αxy εy
Experiment:
Effect of octupoles clearly visible and agree with the theoretical predictions.
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MD 9263
Impact of octupoles on chromaticity estimate
Theory:
The width of sidebands increases, changing the relation between the width and chromaticity. For Gaussian bunches:
σ± = σ±,0 + 2 αx2 εx2 + αxy2 εy2
This correction is however negligible for the LHC.
Experiment:
Changes in octupole current does not affect the chromaticity estimate.
Exception: Ioct = 0.
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MD 9263
Surrounded by vacuum pipe, beam interacts with the components creating EM field, that may affect trailing particles.
Integrated over the whole beam spectrum, the effect is called beam coupling impedance. It can significantly affect the beam dynamics, and deform the Schottky spectrum.
The effect of the impedance can be seen in Schottky Spectra by scanning bunches of different intensities.
Courtesy: C. Lannoy
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Longitudinal impedance
C. Lannoy developed the theory for Schottky spectra that includes the longitudinal impedance.
Dependent on the impedance model, the synchrotron frequency is changed as a function of the synchrotron amplitude.
The theory was implemented in the matrix formalism and benchmarked wrt to the macroparticle simulations, assuming the broadband resonator impedance model.
Courtesy: C. Lannoy
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Nominal frequency shift
Longitudinal impedance
Courtesy: C. Lannoy
Comparing the theory and simulations to the real measurements had two goals:
The longitudinal model is assumed to follow the broadband resonator model with parameters:
The data collected during dedicated MD data matched the model, but with Im(Z||)/n = 135 mΩ.
Investigations ongoing on the sources of the discrepancy.
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General RF system
Longitudinal motion has a dominant impact on the shape of the Schottky spectrum.
We have good models which address the dependence of:
Nevertheless:
What can we say about Schottky spectrum without taking any assumptions on RF system?
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H. Damerau, RF manipulations, RF CAS 2023
PSB/SPS like RF bucket
LHC like RF bucket
Higher order chromaticity
The linear dependence of tune on the momentum is only an approximation.
In the LHC non-linear chromaticities were measured up to the fifth order. It is important to know if these high-order terms affect our spectra, and how.
One could develop theory for Schottky spectra with higher order terms included (e.g. K. Lasocha et al. IPAC’24 WEPG30), but
what can we say about Schottky spectrum without taking any assumptions on highest included chromaticity term?
Courtesy: M. Le Garrec,
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Beam 1 Horizontal
Main conclusions:
Using a realistic longitudinal motion (X-Suite simulation) and high-order chromaticities measured in the LHC, the developed theory estimates the required chromaticity corrections below 0.1.
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Example Schottky spectrum for 2 harmonic RF system
Statistical properties of spectra
Schottky spectra are inherently random, and require averaging if their shape is to be determined.
The power of each bin follows exponential distribution in time:
This fact can be used to filter out the noise from the signal in the spectrum, as for the bins containing Schottky signal, we expect:
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Number of samples
Coupling between the planes
If the motion is coupled between the planes, we see a copy of vertical signal in the horizontal signal, and vice versa.
The coupled components can be detected by calculating Pearson correlation coefficients of the time signal between the respective bins from both planes.
This can serve as an additional filter, or be used to correct the spectra and determine the coupling coefficients.
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During MD 14343 we tried to measure previously introduced coupling. The results were not conclusive.
LHC Schottky Diagnostic system
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Online Schottky analysis pipeline (2024)
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NXCALS
44.9 kHz
1 Hz
0.1 Hz
Accpit
Expert GUI
Curve fitting
Previous attempts to measure the Q’ from the Schottky spectra were based on fitting the spectra.
The chromaticity is however given by:
so fitting the spectrum puts too big focus on centres, and too little on tails.
The adopted procedure assumes that the original spectrum is bell-shaped (q-Gaussian or super-Gaussian) and fits the variance with 5 parameters:
(Frequency - tune)2 ⋅ Spectrum
Spectrum
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Typical performance (physics fill)
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Fill 10074
Q’ measurements overview (proton Flattop)
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Median, IQR and histograms for 90 “good” fills that made it to the STABLE between 26.07 and 10.10
B2 hardware swap
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Based on 53 fills
Based on 37 fills
What influences Schottky estimate
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Hardware imprints
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Hardware imprints
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Hardware imprints
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Hardware imprints
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Fill 10226
Hardware imprints
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Fill 10226
LOs power is monitored continuously since last YETS, with a dedicated UCAP converter to report fast variations
Relative transverse emittance measurements
The emittance is mostly determined by the noisy spectrum centers.
As the result, we cannot rely on the fit of the filtered spectrum.
For clean spectra (Pb82+), plausible emittance estimates can be obtained even from instantaneous 1-second spectra.
During the MD 14343 the Schottky-based emittances were compared to WS. Despite good correlation, the absolute values deviated:
Study requires more work, but is interesting, as Schottky can measure the ion emittance at injection.
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MD 14343
ML Schottky analysis
SchMatrix package makes it possible to efficiently generate a big number of artificial Schottky spectra (in this moment we have 2.7 TB of artificial Schottky spectra).
These can serve as a dataset for machine-learning studies, aiming at:
The pioneering work was done by M. Bradicic. During his Summer Student project he trained an autoencoder model obtaining good results in predicting chromaticity and betatron tune.
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LHC Fill 9996, STABLE
2025 analysis code upgrade
Original analysis code was a bricolage of various corrections and improvements.
This year’s upgrade had a few goals:
The analysis is now split into 8 distinct UCAP converters. Each converter comes with tests, including
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Schottky Analysis
Stats
Schottky Preprocess
Schottky Analysis Correlation
Schottky Analysis
Tune Fast
Schottky Analysis Emittance Fast
Schottky Analysis
Tune
Schottky Analysis Emittance
Schottky Analysis
Chroma
FESA
Conclusion
Last three years witnessed a significant progress in the Schottky signal analysis.
Theory:
Diagnostics:
Promising studies have been started on:
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