Effect of Longitudinal Beam-coupling Impedanceon the Schottky Spectrum of Bunched Beams
CHRISTOPHE LANNOY1,2, DIOGO ALVES1, KACPER LASOCHA1,NICOLAS MOUNET1, TATIANA PIELONI2
1 CERN, Geneva, Switzerland
2 EPFL, Lausanne, Switzerland

I. Introduction
Schottky spectra can be strongly affected by collective effects, inparticular those arising from beam-coupling impedance when large
bunch charges are involved. In such conditions, the direct interpretationof the measured spectra becomes difficult, which prevents theextraction of beam and machine parameters in the same way as is
usually done for lower bunch charges.

Theoretical reconstructions of Schottky spectra, such as the matrix formalism[1] or the Monte Carlo approach [2, 3] are based on the following expressionof the beam current [4] and assume that the synchrotron frequency
distribution is known.

II. Theoretical description
Synchrotron oscillation without external forces
The equation of motion
for the RF phase 𝜙 of agiven particle is:With 𝛀𝟎 the nominal synchrotron frequency(i.e. the limit synchrotron frequency for෡ approaching zero).
synchrotron amplitude 𝝓When the particles are moving freely in the potential well of the radio
frequency (RF) bucket, an analytical relation between the amplitude of
the synchrotron oscillation and its frequency can be used, allowing thesemethods to reconstruct the Schottky spectrum from the synchrotron
amplitude distribution. However, this relation has to be modified whenbeam-coupling impedance affects the longitudinal dynamics.

Equation of motion with impedance
Additional external forces, such as the one comingfrom beam-coupling impedance, will influence thelongitudinal dynamics of the particle [6].
➔ With additional forces, the previous equation ofmotion becomes:

Beam-coupling impedance
Beam spectrum:−

Exact oscillation frequency−(non-harmonic motion)(K: complete elliptic integral of the first kind)With the notations:•: Fourier transform.
• 𝜂 : slippage factor.
• 𝑝0 : reference momentum.
• ℎ : RF harmonic number.
• 𝜆 𝑡 : bunch profile.
•: Dirac comb of period 𝑇0 .
• 𝑇0 = 2 𝜋/𝜔0 : revolution period.
• 𝐶 : accelerator circumference.
• 𝑍∥ 𝜔 : longitudinal impedance.
• 𝑁 : number of particle in the bunch.
• 𝑒 : elementary charge.
• 𝐼 = 𝑁𝑒/𝑇0 : bunch current.
• 𝜏 : time arrival difference between
a given particle and thesynchronous particle.

This equation is identical to the one of the non-linear physical pendulum
and solutions and approximations of this equation exist in the literature [5].

Third order approximation

Approx. osc. frequency
Expanding the sine and exponential function with their Maclaurin series.
The idea is that, for small oscillation amplitudes, only the first order terms can bekept, while for larger amplitudes, higher order terms can be taken into account.

Comparison between the exact and approximate expressions of thesynchrotron frequency as a function of the oscillation amplitude.

(1)

With the coefficients:

General equation of motion
with impedanceLongitudinal broad-band resonator impedance• The developed theory will be applied to the particular case of a longitudinal broad-band resonator.
• The even terms in Eq. (1) are responsible for the synchronous phase shift and it can be shown that,in the particular case of a broad-band resonator, their contribution can be neglected.

Broad-band resonator:

Expanding Eq. (1) up to the third order gives:

With:• 𝑅∥ : shunt impedance.
• 𝜔𝑟 : cut-off frequency.
• 𝑄 : quality factor.

Approx. osc. frequency(2)III. Simulations• The simulation is conducted with PyHEADTAIL [7] and aims to reproduce the typicalconditions of an LHC proton fill at injection.
• The value of the parameters chosen for the broad-band resonator correspond to asignificant part of the impedance in the LHC that can be modelled as a broad-bandresonator.
• The following effects of the broad-band resonator can be observed:

Comparison of the macro-particle simulation (blue) against theadapted matrix formalism, with Eq. (2) including impedanceterms 𝑍𝑛 up to the first (red) and third (green) order.

Comparison of Eq. (2) including impedance terms 𝑍𝑛 up to
the first (red) and third (green) order, against macroparticle simulation (black dots).
Nominal synchrotron
frequency shift• Shift of the nominal synchrotron frequency. All the satellites converge toward
the central one. This shift is due to the term 𝑆1 in Eq. (2) and the new nominal
synchrotron frequency is Ω0 𝑆1 .
• The broad-band resonator will reduce the nominal synchrotron frequency for amachine operating above transition.
• Amplitude dependent synchrotron frequency shift due to the higher orderterms 𝑆2𝑛+1, 𝑛 ≥ 1.
Simulated longitudinal Schottky spectra with (blue) and without (orange)
an LHC-like broad-band resonator.

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

• The macro-particle simulation is compared against thetheoretical matrix formalism, where the relation between
synchrotron amplitudes and frequencies has been
generalized with Eq. (2) to include impedance effects.
• The nominal synchrotron frequency shift is well
reproduced by the 1st order theory, while the shift for nonzero amplitude particles, requires higher order terms (𝑍𝑛).
• With the third order impedance term, the theory is ingood agreement agreement with the simulation, as it also
includes the amplitude dependent synchrotron frequency
shift.

V. Conclusion• The aim of this study was to explore the effects of impedance on the Schottky spectrum.
• The longitudinal equation of motion was generalized to include the forces coming from any impedance 𝑍(𝜔), and thecase of a broad-band resonator was studied in more detail.
• The relation between synchrotron amplitude and frequency was generalized to include the effect of a broad-bandresonator, allowing existing theoretical reconstruction methods of Schottky spectra to include impedance effects.
• The developed theory was shown to be in good agreement with macro-particle simulations, by correctly reproducing
the amplitude dependent synchrotron tune shift, and the sub-structure of the Schottky spectrum satellites.

• The validity of Eq. (2) can also be probed by෠ from the macroextracting the relation Ω𝑠 ( 𝜙)particle simulation.
• On the above figure, each black dot corresponds tothe synchrotron amplitude and frequency of a given
macro-particle.
• The nominal synchrotron frequency shift is well
reproduced by the first order term 𝑍1• The shift for larger amplitude particles is correctwith the third order theory, up to a certain
amplitude 𝜙෠ ∼ 1.25 rad.
• In order to extend the region where Eq. (2) is valid,
one would need to take into account the fifth orderterm 𝑆5, in the equation of motion.
• However, this is not crucial since the majority of theparticles are well described, as can be seen fromthe amplitude distribution.

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