Analysis of the Transverse Schottky Signals in the LHC

Kacper Łasocha, Diogo Alves, CERN Beam Instrumentation Group14.09.2023, Saskatoon, IBIC'23

Schottky signals•

Fluctuations of the macroscopic beam signal due to discrete motion (synchrotron or betatron) ofindividual particles within the bunch

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Schottky signals•

Fluctuations of the macroscopic beam signal due to discrete motion (synchrotron or betatron) ofindividual particles within the bunch

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Schottky signals•

Fluctuations of the macroscopic beam signal due to discrete motion (synchrotron or betatron) ofindividual particles within the bunch

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Schottky signals•

Fluctuations of the macroscopic beam signal due to discrete motion (synchrotron or betatron) ofindividual particles within the bunch

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Schottky signals•

Fluctuations of the macroscopic beam signal due to discrete motion (synchrotron or betatron) ofindividual particles within the bunch•

Most pronounce for long, low intensity, transversly large bunches

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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Transverse Schottky signals
Lower sideband

fs: synchrotron frequency

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

Upper sideband

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Transverse Schottky signals
2 Q frev

Betatron tune

Chromaticity

Mirrored difference method,minimize the cost function:: RMS width of upper/lower sideband
In certain conditions +/- signs flip, see
K. Lasocha and D. Alves, Phys. Rev. Accel. Beams 25, 062801

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Transverse Schottky signals: assumptions
Synchrotron and betatron motion

Uniform distribution of phases;no "coherent" components

Synchrotron motion – harmonic, with amplitudedependent frequency:

Uniform distribution of pjand ph implies
PSD proportional to the number of particles N.

The theory predicts only the expected,ensemble averaged spectrum. Time averaging
required to have a correspondence.

Otherwise the power can be proportional to N2:

f

Sufficiently long time averaging

Analyzed LHC spectra are averaged for 100 s.

Betatron motion – harmonic, with frequency
modulated wrt to the momentum:

See C. Lannoy et al., WEP035, this conference.

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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Typical LHC beam parameters:

Schottky signals in LHC
Nparticles(per bunch)••

One system for two particle species: protons and Pb82+ ions,one device per beam and plane
Pair of slotted waveguides, probing beam field at 4.81 GHz,filtering and downmixing signal to 11.2 kHz•

Gating system enables observation of single bunches•

The only instrument measuring the LHC chromaticity inthe non-invasive way

Details on the LHC Schottky system in
M. Betz et al., NIM, vol. 874, pp 113-126, 2017

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

p+

Pb82+

1011

108

Bunch length (4σ)

1-1.4 ns

Normalized
transverse emittance

1.5-2.5 μm

Energy Inj/Flattop(per nucleon)

0.45 - 6.8
TeV

0.18 - 2.6
TeV

4.81 GHz = 427725 frev

23 September 2023

11Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly usedK. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•

Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly usedK. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

14

Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•

Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly usedK. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

15

Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly used

III. Transient effects:•Tune shifts, RF modulation, energy ramp•

Theory cannot be directly used to averaged spectra, but
spectograms are easy to readK. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

16

Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly used

III. Transient effects:•Tune shifts, RF modulation, energy ramp•

Theory cannot be directly used to averaged spectra, but
spectograms are easy to readK. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

17

Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly used

III. Transient effects:•Tune shifts, RF modulation, energy ramp•

Theory cannot be directly used to averaged spectra, but
spectograms are easy to readK. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

18

Schottky spectra examples
I.

Steady beam conditions:•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly used

III. Transient effects:•Tune shifts, RF modulation, energy ramp•

Theory cannot be directly used to averaged spectra, butspectograms are easy to readIV. Beyond current theory effects:•Octupole magnets, betatron coupling, impedance, all intrument
problems•

Theory to analyse such spectra is still to be developed

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

19

Schottky spectra examples
I.

Steady beam conditions:

Analysis techniques derived•

Mostly at flattop energy of ion fills, shorter periods at flatbottom•

Easy to analyse: just use the theory

II. Local distortions:•Caused by residual coherence, beam parameter mesurements,
beam interaction with the surrounding•

Theory cannot be directly used

III. Transient effects:

Under development•

Tune shifts, RF modulation, energy ramp•

Theory cannot be directly used to averaged spectra, butspectograms are easy to readIV. Beyond current theory effects:•Octupole magnets, betatron coupling, impedance, all intrument
problems•

Theory to analyse such spectra is still to be developed

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Signal analysis in stable conditions
Example: 8 hour long ion collisions in Nov 2022

Betatron tune

Chromaticity

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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Signal analysis in presence of local distortions
Example: Early proton fill in March 2023

Chromaticity

Offset of over 4 units...
K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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Matrix formalism for Schottky spectra
Mathematically, Schottky spectra are given as afunction of:•Synchrotron amplitude distribution - 2 parameters•

Nominal synchrotron frequency - 1 parameter•

Betatron tune - 1 parameter•

Chromaticity - 1 parameter

For given parameters the spectrum can be calculated with a simple
matrix transform.

Details in: K. Lasocha and D. Alves, Phys. Rev. Accel. Beams 23, 062803
K. Lasocha and D. Alves, Phys. Rev. Accel. Beams 25, 062801
K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Matrix formalism for Schottky spectra
Mathematically, Schottky spectra are given as afunction of:•Synchrotron amplitude distribution - 2 parameters•Nominal synchrotron frequency - 1 parameter•Betatron tune - 1 parameter•Chromaticity - 1 parameter

For given parameters the spectrum can be calculated with a simple
matrix transform.

Use case 1: fast Schottky spectra simulationfd

Use case 2 (spectra fitting): given an experimentallymeasured spectrum, true parameters would minimize
the cost function:

Minimizing routines iteratively simulate Schottky
spectra and compare them with the measurement.
K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

Details in: K. Lasocha and D. Alves, Phys. Rev. Accel. Beams 23, 062803
K. Lasocha and D. Alves, Phys. Rev. Accel. Beams 25, 062801
23 September 2023

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Matrix formalism: excluding frequency bins
Contrary to previous methods, fitting procedure allow to exclude the spectral regions with undesired components.

K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

23 September 2023

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Signal analysis in presence of local distortions
Example: Early proton fill in March 2023

Betatron tune
Only "valid" frequencies taken into sum

Chromaticity
Nominal synchrotron tune calculated independently,
Cost function minimization using Differential Evolution algorithm.
K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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LHC Schottky online signal analysis pipeline
K. Lasocha, D. Alves MOCO03, IBIC'19

C. Lannoy et al., WEP035, IBIC'23

Obtain a Schottky spectrum

Get the distribution ofsynchrotron amplitudes fromlongitudinal bunch profile

Determine "valid" frequencies:
1) Coefficient of variation
2) Correlation between planes

Determine chromaticity byfitting the transversespectrum

Calculate betatron tuneusing MD algorithm

Calculate nominal fs based on
RF voltage and longitudinal
Schottky spectrum
K. Lasocha, D. Alves MOCO03, IBIC'19

Implementation in the final stage of development, planned be in use in the end of 2023
K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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Conclusions
Summary of the talk:•Theory of transverse Schottky spectra reviewed and succesfully applied to stable beams,•Local spectral distortions mitigation technique proposed and demonstrated on proton spectra,•

Automation of the analysis will be tested in the coming days.

Possible next steps:•Handling of transient effects: 1D ---> 2D analysis; image recognition techniques?

•Expanding the theory of Schottky spectra: impedance, octupoles, ...

See C. Lannoy et al., WEP034, IBIC'23

Thanks for your attention!
K. Łasocha | Analysis of the Transverse Schottky Signals in the LHC

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