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12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - W E P 0 3 4EFFECT OF LONGITUDINAL BEAM-COUPLING IMPEDANCE ON THESCHOTTKY SPECTRUM OF BUNCHED BEAMS
C. Lannoy1,∗ , D. Alves, K. Lasocha, N. Mounet, CERN, Geneva, Switzerland
T. Pieloni, EPFL, Lausanne, Switzerland
1
also at EPFL, Lausanne, Switzerland
Abstract

Synchrotron Oscillation

Schottky spectra can be strongly affected by collective
effects, in particular those arising from beam-coupling
impedance when a large number of bunch charges are involved. In such conditions, the direct interpretation of themeasured spectra becomes difficult, which prevents the extraction of beam and machine parameters in the same wayas is usually done for lower bunch charges. Since no theory
is yet directly applicable to predict the impact of impedanceon such spectra, we use here time-domain, macro-particle
simulations and apply a semi-analytical method to compute
the Schottky spectrum for various machine and beam conditions, such as the ones found at the Large Hadron Collider.
A simple longitudinal resonator-like impedance model is introduced in the simulations and its effect studied in different
configurations, allowing preliminary interpretations of theimpact of longitudinal impedance on Schottky spectra.

INTRODUCTION
Theoretical reconstructions of Schottky spectra, such as
the matrix formalism proposed in [1, 2], or the Monte Carloapproach used in [3, 4], are based on the assumption thatthe synchrotron frequency distribution is known. Undercertain conditions, one can derive an analytical relation between the amplitude of the synchrotron oscillation and its
frequency (see below) allowing these methods to reconstruct
the Schottky spectrum from the synchrotron amplitude distribution. However, this relation has to be modified when
external forces, such as the one coming from beam-coupling
impedance, affect the longitudinal dynamics.
This study will briefly present the available theory relating
the amplitude of the synchrotron oscillation to its frequency,
as well as a commonly adopted approximation. The second section will deal with the additional external forces
coming from impedance, extending the theory presented in[5] to the case of a non-linear radio frequency (RF) bucket.
Finally, we will apply the developed theory to the particularcase of a longitudinal broad-band resonator, and will benchmark it against macro-particle simulations performed with
PyHEADTAIL [4, 6, 7], in the case of a proton bunch in theLarge Hadron Collider (LHC).

For an impedance-free environment, the equation of motion for the RF phase 𝜙 1 of a given particle is [8, Eq. (9.51)]:𝑑 2𝜙+ Ω20 sin 𝜙 = 0,𝑑𝑡 2(1)

assuming that the synchronous phase 𝜙𝑠 is such that sin 𝜙𝑠 issmall enough to be neglected (i.e. no acceleration or energy
loss compensation). The nominal synchrotron frequency2
reads
̂−𝜂ℎ𝑒𝑉Ω20 = 𝜔20cos 𝜙𝑠 ,(2)2𝜋𝐸0 𝛽2where 𝑒 is the elementary charge, and where the relevant
machine parameters are: the revolution frequency 𝜔0 , thê, the rf
slippage factor 𝜂, the amplitude of the RF voltage 𝑉
harmonic number ℎ, the relativistic factor 𝛽, and the reference energy 𝐸0 . Note that by convention 𝜂 is positive abovetransition, such that 𝜂 and cos 𝜙𝑠 always have opposite sign.
Equation (1) is similar to the non-linear pendulum equation, hence the synchrotron frequency Ω𝑠 of the particle canbe written [9]̂ =Ω𝑠 (𝜙)𝜋̂
𝜙2𝒦 [sin ( 2 )]

Ω0 ,(3)̂ is the RF phase amplitude of the synchrotron oswhere 𝜙cillation and 𝒦 is the complete elliptic integral of the firstkind [10, Eq. (8.112.1)].
Equation (1) can also be approximated by replacing thesine function with its Maclaurin series expansion up to thethird order, which yields
3𝑑 2𝜙2 (𝜙 − 𝜙 ) + 𝒪(𝜙5 ) = 0.
+
Ω06𝑑𝑡 2(4)This last equation has been studied in [9], and an approximation of the oscillation frequency is given bŷ = Ω0 (1 −Ω𝑠 (𝜙)

̂2𝜙).
16(5)

Figure 1 illustrates how the exact solution from Eq. (3) compares with the approximation from Eq. (5). The amplitude
distribution corresponding to a Gaussian bunch profile ofstandard deviation 𝜎 = 31 ns is also shown in order to compare the most populated region with the zone where the1 Above transition 𝜙 has to be taken as the difference between the RF phaseof the particle and 𝜋.

∗ christophe.lannoy@cern.ch

2 By nominal synchrotron frequency, we mean the limit of the synchrotron

frequency for synchrotron amplitudes approaching zero.

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12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3approximation is valid. As can be seen, the approximation̂ ≲ 0.8𝜋, which is the region whereholds for an amplitude 𝜙
the vast majority of the particles are. Noting that 𝜙 = ℎ𝜔0 𝜏,

at the current position of the particle, i.e. at a distance𝑧 = 𝛽𝑐𝜏 behind the synchronous particle. Note that thê(𝜔) of [11, Eq. (4.18)], has been replacedbunch spectrum 𝜆̂by the beam spectrum Λ(𝜔)in order to take into accountthe possible multi-turn wake (this approach is equivalent toadding the forces coming from different turns).
The beam spectrum is the Fourier transform of the beamprofile. For a single bunch in a circular accelerator, the lattercan be expressed as the convolution of the bunch profile 𝜆(𝑡)with a Dirac comb of the revolution period 𝑇0 . Using theconvolution theorem, the beam spectrum can be written̂Λ(𝜔)= ℱ{𝜆(𝑡) ∗ Ш𝑇0 (𝑡)}Figure 1: Comparison between the exact and approximate
expressions of the synchrotron frequency as a function ofthe oscillation amplitude. The distribution of amplitudes fora Gaussian bunch is also shown.
with 𝜏 the arrival time difference between a given particle
and the synchronous particle, we can also express the amplitude of the synchrotron oscillation in terms of the time
amplitude defined bŷ𝜏̂ ≡ 𝜙/ℎ𝜔
0.

(6)= ℱ{𝜆(𝑡)} ℱ{Ш𝑇0 (𝑡)}̂(𝜔)𝜔0 Ш𝜔 (𝜔),=𝜆0where ∗ stands for the convolution operation and where theDirac comb of period 𝑇0 is defined as follows∞

Ш𝑇0 (𝑡) = ∑ 𝛿(𝑡 − 𝑝𝑇0 ).
𝑝=−∞

Substituting Eq. (10) in Eq. (9) , we have𝐹𝐼𝑚𝑝 (𝑡) =THEORETICAL DESCRIPTION=In this section, we investigate the influence of impedanceon the longitudinal dynamics of a particle and, more specifically, how it can affect its oscillation frequency. A givenforce 𝐹𝑖 acting longitudinally on the particle will impact thephase dynamics with𝐹𝑖 =𝑝0
𝑑𝑝 𝑑(𝑝0 + 𝛿𝑝) 𝑑(𝛿𝑝) 𝑝0=== 𝜏̈ =𝜙,̈ (7)
𝜂𝑑𝑡
𝑑𝑡
𝑑𝑡
𝜂ℎ𝜔0

where we used the fact that 𝑝0 is constant, the definition ofthe slippage factor, and Eq. (6). The equation of motion withan additional external force is given by combining Eqs. (1)and (7), which yields
𝜙̈ + Ω20 sin 𝜙 =𝜂ℎ𝜔0𝐹 (𝑡).
𝑝0 𝑖(8)𝑁𝑒2 ∞𝑗 𝜔𝑧𝛽𝑐̂∫ 𝑍 (𝜔)Λ(𝜔)𝑒𝑑𝜔,2𝜋𝐶 −∞ ∥−𝐼𝑒 ∞̂(𝑝)𝑒𝑗𝑝𝜔0𝜏(𝑡) ,∑ 𝑍 (𝑝)𝜆𝐶 𝑝=−∞ ∥Combining Eqs. (2), (6), (8), and (11) yields
𝜙̈ + Ω20 sin 𝜙 = Ω20∞𝑝𝐼
̂(𝑝)𝑒𝑗 ℎ 𝜙 . (12)∑ 𝑍∥ (𝑝)𝜆̂ cos 𝜙𝑠 𝑝=−∞𝑉By expanding the sine and exponential functions into theirMaclaurin series∞𝑝(−1)𝑛1 𝑗𝑝 𝑛
𝜙2𝑛+1 , 𝑒𝑗 ℎ 𝜙 = ∑ ( ) 𝜙𝑛 ,(2𝑛 + 1)!
𝑛! ℎ𝑛=0𝑛=0∞

sin 𝜙 = ∑Eq. (12) can be written in the compact form∞𝜙̈ + Ω20 ∑ 𝑆𝑛 𝜙𝑛 = 0,(9)where 𝐶 stands for the circumference of the accelerator,
𝑁 for the number of particles in the bunch, 𝑍∥ (𝜔) for thêlongitudinal impedance, and Λ(𝜔)for the beam spectrum
normalised by the bunch intensity. The force has to be taken(11)

General Equation of Motion(13)

𝑛=0with the coefficients 𝑆𝑛 defined by⎧{−𝑍𝑛𝑆𝑛 = ⎨ 𝑗 𝑛−1{
⎩ 𝑛! − 𝑍𝑛⃗⃗⃗ ⃗⃗ + 𝛽𝑐⃗⃗⃗ ⃗⃗ (𝑡, 𝑧 = 𝛽𝑐𝜏(𝑡))⃗⃗⃗ ⃗ ⃗ × 𝐵]
𝐹𝐼𝑚𝑝 (𝑡) = 𝑒 [𝐸∥
=−𝜔𝑧∞
−𝐼𝑒 ∞
̂(𝜔)𝑒𝑗 𝛽𝑐 𝛿(𝜔 − 𝑝𝜔0 )𝑑𝜔∑ ∫ 𝑍∥ (𝜔)𝜆𝐶 𝑝=−∞ −∞where we used the compact notation 𝑍∥ (𝑝) ≡ 𝑍∥ (𝑝𝜔0 ),̂(𝑝) ≡ 𝜆̂(𝑝𝜔0 ), and the average current 𝐼 ≡ 𝑁𝑒 𝜔0 .
𝜆2𝜋

Impedance ForceIn this study, we will focus on the force arising fromthe longitudinal beam-coupling impedance, which is givenby [11, Eq. (4.18)](10): 𝑛 even,: 𝑛 odd,(14)and𝑛∞𝐼̂(𝑝) 1 ( 𝑗𝑝 )∑ 𝑍∥ (𝑝)𝜆𝑛! ℎ̂ cos 𝜙𝑠 𝑝=−∞𝑉∞
⎧𝑅𝑒[𝑍∥ (𝑝)] : 𝑛 even,𝐼𝑗 𝑛̂(𝑝)𝑝𝑛 × {=∑𝜆⎨𝑛̂ cos 𝜙𝑠 𝑛!ℎ 𝑝=−∞{𝑉⎩𝑗 𝐼𝑚[𝑍∥ (𝑝)]: 𝑛 odd.

𝑍𝑛 =WEP034
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Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

IBIC2023, Saskatoon, CanadaJACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - W E P 0 3 4

̂(𝑝), 𝑅𝑒 [𝑍∥ (𝑝)],In the last equality, we used the fact that 𝜆and 𝐼𝑚 [𝑍∥ (𝑝)] are respectively even, even, and odd functions.
The idea behind the expansion performed above is that,for small oscillation amplitudes, only the first order terms
can be kept. The shift of the nominal synchrotron frequency
is given by 𝑆1 while the non-linear terms will be responsible
for an amplitude dependent synchrotron frequency shift.

Broad-Band Resonator
We will now apply Eq. (13) to the particular case of abroad-band resonator described by the following function
𝑍∥𝐵𝐵 (𝜔) =𝑅∥𝜔

1 − 𝑗𝑄 ( 𝜔𝑟 − 𝜔𝜔𝑟 ),where 𝑅∥ is the shunt impedance, 𝜔𝑟 the angular cut-off
frequency, and 𝑄 the quality factor. The values we will
use for these parameters are shown in Table 1 and correspond to a first estimate of the broad-band part of the LHC
impedance [12, p. 71].
The even terms in Eq. (13) are responsible for the synchronous phase shift and we assume that, in the particularcase of a broad-band resonator, their contribution can beneglected. Expanding Eq. (13) up to the third order gives𝜙̈ + Ω20 (𝑆1 𝜙 + 𝑆3 𝜙3 ) + 𝒪(𝜙5 ) = 0,(15)which is similar to Eq. (4) where the factors 1 and − 16 havebeen generalised to the arbitrary coefficients 𝑆1 and 𝑆3 . In
this case, the approximate synchrotron frequency is givenbŷ = Ω0 √𝑆1 (1 + 3𝑆3 𝜙
̂2 ) ,
Ω𝑠 (𝜙)(16)
8𝑆1

as can be seen by substituting the function 𝜙(𝑡) = √𝐴Φ(𝑡)(with 𝐴 an arbitrary number) in Eq. (4) and identifying thecoefficients.

SIMULATIONThe simulation aims at reproducing the typical conditionsof an LHC proton fill at injection. The method used toreconstruct the Schottky spectrum from the macro-particle
simulation is presented in [4] and the simulation parameters
are summarised in Table 1.
Table 1: PyHEADTAIL Simulation Parameters
Intensity
Energy per proton
Slippage factor
RF harmonic
RF voltage
LHC circumference
Bunch length (RMS)
Broad-band resonator
Shunt impedanceCut-off frequency
Quality factor

1.5 × 1011 protons per bunch450 GeV
3.436 × 10−435640
4 MV
26.659 km
𝜎 = 0.31 ns𝑅∥ = 31.1 kΩ𝜔𝑟 = 2𝜋 × 5 GHz
𝑄=1Figure 2 presents an overall view (upper plot) of the simulated longitudinal Schottky spectrum together with a detailed view of two regions (lower plots). It can be observedthat the broad-band resonator induces a shift of the nominal
synchrotron frequency (all the satellites moving toward thecentral one). This shift is due to the term 𝑆1 and the newnominal synchrotron frequency is Ω0 √𝑆1 . The broad-band
resonator will reduce the nominal synchrotron frequency for

Figure 2: Simulated longitudinal Schottky spectra with (blue) and without (orange) an LHC-like broad-band resonator.
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07 Machine Parameter Measurements

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - W E P 0 3 4a machine operating above transition, as can be seen fromEq. (14) (the opposite happens below transition). One canalso observe in Fig. 2 an amplitude-dependent synchrotron
frequency shift, as the alterations in the satellite shapes can
not be explained by a simple linear transformation involving
shifting and scaling of the original form. This effect is dueto the higher order terms 𝑆2𝑛+1 , 𝑛 ≥ 1 in Eq. (13).
Figure 3 compares the macro-particle simulation withthe theoretical matrix formalism of Ref. [1], where the relation between synchrotron amplitudes and frequencies has
been adapted, replacing Eq. (5) by Eq. (16) to take into
account the effect of impedance. The dashed red line corresponds to the matrix formalism where only the first order term of the impedance contribution has been kept (i.e.
𝑍1 ≠ 0 and 𝑍3 = 0, which corresponds to 𝑆1 = 1 − 𝑍1 and𝑆3 = −1/6 in Eq. (16)). As can be seen, the nominal synchrotron frequency shift is well reproduced by the theory,
while the shift for non-zero amplitude particles, requires
higher order terms (𝑍𝑛 ).

One can also probe the validity of Eq. (16) by directlŷ from the macro-particle simextracting the relation Ω𝑠 (𝜙)ulation. This was done in Fig. 4 where each black dot corresponds to the synchrotron amplitude and frequency of agiven macro-particle. As before, one can observe that thenominal synchrotron frequency shift is well reproduced by
the first order term 𝑍1 , while the shift for larger amplitudeparticles is correct with the third order theory, up to a certain̂ ∼ 1.25 rad. In order to extend the region whereamplitude 𝜙Eq. (16) is valid, one would need to take into account thefifth order term 𝑆5 , in the equation of motion. However,this is not crucial since the majority of the particles are welldescribed, as can be seen from the amplitude distribution.

Figure 4: Comparison of Eq. (16) including impedanceterms 𝑍𝑛 up to the first (red) and third (green) order, against
macro-particle simulation (black dots).

CONCLUSION

Figure 3: Comparison of the macro-particle simulation(blue) against the adapted matrix formalism, with Eq. (16)including impedance terms 𝑍𝑛 up to the first (red) and third(green) order. The A-B labels correspond to the shaded regions of Fig. 2.

The dashed green line also includes the third order term of
the impedance contribution (i.e. 𝑍1 ≠ 0 and 𝑍3 ≠ 0, whichcorresponds to 𝑆1 = 1 − 𝑍1 and 𝑆3 = −1/6 − 𝑍3 ). With thethird order impedance term, the theory is in good agreementwith the simulation.

The aim of this work was to explore the effects of longitudinal impedance on the Schottky spectrum. We first derived
a longitudinal equation of motion including the forces coming from any impedance 𝑍(𝜔). The particular case of abroad-band resonator was then studied in more detail andthe relation between synchrotron amplitude and frequency
was generalised to this situation, 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, bycorrectly reproducing the amplitude-dependent synchrotron
tune shift.

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12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3IBIC2023, Saskatoon, Canada
JACoW Publishing
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