Statistical Properties of Schottky Spectra
CHRISTOPHE LANNOY1,2, DIOGO ALVES1, KACPER LASOCHA1,NICOLAS MOUNET1, TATIANA PIELONI2
1 CERN, Geneva, Switzerland
2 EPFL, Lausanne, Switzerland

I. Introduction
Schottky signals are used for non-invasive beam diagnostics as they contain information on various beam andmachine parameters. The Large Hadron Collider (LHC) Schottky system provides a single spectrum every second,
based on the signal acquired over approximately the last 16 000 revolutions. Notably, consecutive spectra
exhibit significant dissimilarity, necessitating the aggregation of numerous spectra to attain the mean value. This
study explores the variability inherent in consecutive instantaneous Schottky spectra and analyses
quantitatively their statistical properties, including the expected value and variance of Schottky power spectra.
Furthermore, we investigate how these quantities evolve with the number of particles in the bunch, theobserved harmonic of the revolution frequency, the distribution of synchrotron oscillation amplitudes, and thebunch profile. The theoretical findings are compared against macro-particle simulations as well as Monte Carlocomputations.

II. Theoretical description

Longitudinal Schottky spectrum (i.e. the PSD of 𝐼 𝑡 )With 𝛿 the Dirac delta andThe longitudinal Schottky spectrum is the power spectral density(PSD) of the beam current.
The intensity signal 𝑖 (𝑡) of a bunch consisting of N particles is [1] :𝒑 = 𝟎Central Satellite
Deterministic

Signalentirely defined by a setof 𝑁 amplitudes 𝜏𝑖 and 𝑁
phases 𝜑𝑠,𝑖 .

With:• 𝐽𝑝 : Bessel function of order 𝑝.
• 𝑞 : charge of the particle.
• 𝑓0 = 𝜔0/2𝜋 : revolution frequency.
• 𝜏ෝ𝑖 : synchrotron time amplitude.
• Ω𝑠,𝑖 : synchrotron frequency.
• 𝜑𝑠,𝑖 : synchrotron initial phase.

𝒑 ≠𝟎Non-Central Satellites

withwithThis equation is formally identical to the squared distance of a
1D random walk of 𝑁 steps of variable length 𝑍𝑖 .

This equation is formally identical to the squared distance of a 2D random walk of 𝑁 steps of variable
length 𝑍𝑖,(p) and direction 𝑝Φ𝑠,𝑖 .

Expected power and variance are given by:

Expected power and variance are given by:

CV of the longitudinal satellites around harmonic 𝑛ℎ = 427 725 for a Rice amplitude distribution(dashed lines, 𝜎 = 0.31 ns and 𝑏 = 1.306) and a Gaussian bunch profile (solid lines, 𝜎 = 0.31 ns).

Random Process defined by random variables:𝑇෡𝑖 : synchrotron time amplitude following a given pdf 𝑔 𝜏Ƹ .
Φ𝑠𝑖 : synchrotron initial phase following a uniform distribution on [0,2𝜋].
The PSD of a random process 𝑋(𝑡) isdefined by:With 𝛼, 𝛽, 𝛾 and 𝛿, the first four moments of 𝑍𝑖 .

For typical LHC conditions, the coefficientsof variation (CV) converge to:•where 𝑋𝑇 (𝜔) is the Fourier transformof the truncated process:𝟐 for the central satellite.

• 1 for non-central satellites.

Transverse Schottky spectrum

Harmonic scan of the expected powers of the transverse Bessel satellites for a Gaussian bunch profile (𝜎 = 0.31 ns). The solid lines
correspond to a zero chromaticity 𝑄𝜉 = 0 while the dashed lines correspond to 𝑄𝜉 = 20.

The transverse Schottky spectrum is the power spectral density (PSD) of thedipole moment of the beam.
A similar development can be conducted for the transverse spectra. Thefigures on the right present how the expected power of transverse satellites
vary with the harmonic number and the chromaticity.
• As the harmonic number increases, the power of lower ordersatellites decreases, while the opposite occurs for higher ordersatellites.
• The effect of chromaticity can be observed, resulting in a higher
and narrower upper band compared to the lower band.

III. Simulations• The theoretical equations derived in this study are compared against macro-particle [2-4] and Monte-Carlo [4, 5]
simulations of a realistic LHC ion fill.
• Both simulation techniques show a good agreement with the theoretical predictions and a clear convergence to aCV of 𝟐 and 1 can be observed.
Macro-particle and Monte-Carlo simulations of a Schottky spectrum. The average spectrum and CV are based on 1000
random draws of the bunch for the macro-particle simulation and 5000 random draws for the Monte-Carlo simulation.

IV. Measurements• The previously developed theory is studying the mean and variance over different random draws of the bunches(phases and amplitudes of synchrotron and betatron motions).
• In the real machine, there is a unique instance of the bunch.
• However, the relative initial phase differences are drifting over time due the non-linearities of the machine.
• After a given period, this unpredictable drift is equivalent to drawing new random phases, making the averaging oversuccessive instantaneous spectra equivalent to the averaging over different random draws of the bunch.
• The same conclusion can not be drawn for the longitudinal central satellite, since the synchrotron phases do not appear
in their power.
• The CV for the transverse sidebands and the non-central (|𝑝| > 2) longitudinal satellites agree well with the theory.
• The CV for 𝒑 = ±𝟏 and 𝒑 = ±𝟐 longitudinal satellites deviate from the predictions, which suggest that thesesatellites are subject to effect beyond the adopted theory of Schottky spectra and should not be used for diagnostic.
Experimental LHC Schottky spectrum taking during fill 8412 from ’2022-11-18 17:44:00’ to ’2022-11-18
18:17:00’. Upper plots: average of 1980 instantaneous Schottky spectra. Lower plots: CV of the PSD.

Frequencies on all the plots have been shifted from the LHC Schottky harmonic, ℎ = 427725, to the first harmonic.

V. Conclusion• The aim of this study was to quantify the statistical properties of Schottky spectra and reveal the dependencies onparticle count, harmonic number, and oscillation amplitude distribution.
• Our analysis demonstrated the convergence of the coefficient of variation to specific, non-zero values, for different
Bessel satellites.
• Our findings were validated through simulations and data from the LHC Schottky monitors and while transverse andnon-central longitudinal satellites exhibit consistent convergence in the experimental data, the behaviour of the central
longitudinal satellites still requires further investigation.

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