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ANALYSIS OF THE TRANSVERSESCHOTTKY SIGNALS IN THE LHC
K. Lasocha∗ , D. Alves, CERN, Geneva, Switzerland
Abstract
Schottky-based diagnostics are remarkably useful tools
for the non-invasive monitoring of hadron beam and machine
characteristics such as the betatron tune and the chromaticity. In this contribution recent developments in the analysisof the transverse Schottky signals measured at the Large
Hadron Collider will be reported. A fitting-based technique,where the measured spectra are iteratively compared withtheoretical predictions, will be presented and benchmarked
with respect to the previously known methods and alternative
diagnostic.

INTRODUCTION
Schottky signals are the fluctuations of macroscopic beamcharacteristics, such as intensity or dipole moment, due tothe discrete nature of the particle ensemble. They were firstmeasured at the CERN’s Intersecting Storage Ring in theearly 1970s [1] and immediately used to provide information on momentum spread and betatron tune, as well as therate of growth of betatron amplitudes at tune values close toresonant lines. Notably, it was the observation of Schottky
signals what convinced Simon van der Meer to persue theimplementation of his eminent concept of stochastic cooling [2].
Since then, the Schottky signal analysis has become astandard diagnostic technique for both coasting and bunchedhadron beams, used among others in Tevatron [3], RHIC [4],SIS18 [5], CSR [6], ELENA [7] and LHC [8]. The basic
theory of Schottky signals of coasting beams [9] has beenenriched by studies on the Schottky spectrum deformation
due to space charge, impedance and collective effects present
in dense cooled beams [10, 11].
The case of Schottky signals of bunched beams is muchless understood. The theory describing such signals is more
complex, as it also has to take into account the particles’
synchrotron motion. Some main principles can be foundin Refs. [12, 13], as well as the effect of space-charge [14],but the knowledge on the impact of impedance, octupole
magnets or beam-beam effects on Schottky spectra is so far
very limited.
In addition, bunching introduces a finite degree of correlation in the motion of individual particles. In steady-state
conditions, thanks to the filamentation associated with thebetatron and synchrotron motion of particles, this correlation (also known as coherent motion) has a limited impact
on the observed Schottky spectrum. Nevertheless, in certain conditions coherent effects are not negligible, yet very
difficult to describe quantitatively.
∗ kacper.lasocha@cern.ch

In this contribution we shall recall the theory of the transverse Schottky spectra of bunched beams. Using experimental Schottky spectra measured during the ongoing LHC
Run3 (2022-2026), we will address the issues of how andwhen the theory can be used to derive parameters such as thebetatron tune and chromaticity. We shall also apply a fittingbased technique, proposed in [15, 16], which can be used toderive transverse beam characteristics from the spectrum,
even in the presence of coherent components.

THEORYLet us consider first the Power Spectral Density (PSD) of atransverse Schottky signal originating from a single particle.
Around every harmonic ℎ of the revolution frequency 𝜔0 ,the PSD consists of two bands, described by the following
expression
𝑃𝑇± (𝜔, 𝜏̂, Ω𝑠0 , 𝑄, 𝑄𝜉) =∞±= 𝑎 ̂ ∑ 𝐽𝑝2 (𝜒̂𝜏)], (1)𝜏,ℎ∓𝑄 ) 𝛿[𝜔−(ℎ±𝑄𝐹 )𝜔0 −𝑝Ω𝑠 (̂𝐼𝑝=−∞where 𝑄𝐼 and 𝑄𝐹 denote respectively the integer and fractional part of the betatron tune 𝑄, 𝑄𝜉 is the chromaticity, 𝜏̂is the time amplitude of synchrotron motion, Ω𝑠 (̂𝜏) is theamplitude-dependent synchrotron frequency and 𝑎 ̂ is a constant proportional to the squared amplitude of the betatronmotion at the pick-up location. The argument of the Bessel
function is given by

±
𝜒̂
𝜏𝑖 ±
𝜏 ,𝑛 = (𝑛̂
𝑖

̂𝑖
𝜔 𝑝̂
𝑄
) 𝜔0 = (𝑛𝜂 ± 𝑄𝜉) 0 ,Ω𝑠𝑖
Ω𝑠𝑖 𝑝0(2)

with 𝜂 denoting the slip factor and 𝑝̂ the amplitude of momentum oscillations around the nominal value 𝑝0 . We assume that particles perform harmonic synchrotron motion,with the synchrotron frequency given by the theory of themathematical pendulum [17]:
Ω𝑠 =

𝜋
Ω𝑠0 ,
ℎ 𝜔 𝜏
̂
2𝒦 [sin ( 𝑅𝐹 0 )]2

(3)where Ω𝑠0 is the nominal, limit synchrotron frequency,ℎ𝑅𝐹 𝜔0 𝜏̂ is the the RF phase amplitude of synchrotron oscillations and 𝒦 ([0, 1]) → [𝜋/2, ∞] denotes the complete
elliptic integral of the first kind [18, p. 590].
In the absence of coherent components which, in the caseof the transverse Schottky spectrum is equivalent to having
a uniform distribution of both betatron and synchrotron oscillation phases, the total multiparticle PSD is given simply

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𝑁𝑃𝑇± (𝜔, Ω𝑠0 , 𝑄, 𝑄𝜉) = ∑ 𝑃𝑇± (𝜔, 𝜏̂𝑖 , Ω𝑠0 , 𝑄, 𝑄𝜉).
𝑖=1

(4)

PSD [arb. units]

As we can see, given a set of parameters Ω𝑠0 , 𝑄 and 𝑄𝜉,the Schottky spectrum can be fully determined by the distribution of synchrotron amplitudes 𝜏
̂𝑖 within a given bunch.
In [15] we also showed that, in the absence of coherent
motion, one can also use the longitudinal bunch profile or
the distribution of synchrotron frequencies to determine the
distribution of synchrotron amplitudes.
An example theoretical transverse Schottky spectrum isshown in Fig. 1. Two sidebands, located around the frequencies (ℎ ± 𝑄𝐹 )𝜔0 , consist of a series of Bessel satellites. The
distance between consecutive satellites is given by the synchrotron frequency. Individual satellites have a finite width,
which reflects the spread in synchrotron frequencies among
the particles. For non-zero chromaticity, the upper (right)
and lower (left) sidebands are different, what is a result of adifferent argument of the Bessel functions, Eq. (2).
One has to keep in mind, that the observed Schottky signal
is an intrinsically random process and in each time instance
one can observe only its single realization [19]. Meanwhile
the theory presented here describes the expected value, i.e.
the ensemble average, of the spectrum.

−0.40

−0.35

−0.30

−0.25

0.25

0.30

0.35

0.40

Frequency - fr ev [fr ev ]Figure 1: Theoretical transverse Schottky spectrum.
Assuming a discrete macroscopic distribution of synchrotron amplitudes 𝑔(̂𝜏𝑖 ) for an apriori chosen grid of synchrotron amplitudes 𝜏
̂1 , … , 𝜏̂𝑚 , the DFT (Discrete Fourier
Transform) of the total transverse Schottky signal at frequencies 𝜔1 , … , 𝜔𝑛 can be expressed in a form of a matrix
𝑇 ,±equation (Eq. (5)), where single particle 𝑃𝐷𝐹𝑇
s can be calculated directly from Eq. (1), taking into the account the
sampling frequency, period and the windowing function.
The vector 𝒜 denotes the discrete synchrotron amplitude
distribution, while 𝒮𝐷𝐹𝑇 is the DFT of the total Schottky
signal. The number of considered amplitudes 𝜏
̂1 , … , 𝜏̂𝑛has to be chosen as a trade-off between the time-complexityand discretisation error. In our studies we have used 50
amplitudes uniformly distributed over the whole RF bucket.

The LHC Schottky Monitor consists of four pairs of approximately 1 m waveguides, one for each of the 2 beams
and the 2 planes, sensitive to the beam field within a bandof approximately 400 MHz around 4.8 GHz. An electronic
gating system allows one to isolate the signal coming from asingle bunch. The coupled signal is then sequentially filteredand down-mixed, so that the final digitized Schottky signal
consists of a narrow (approximately 14 kHz wide) band,
originally located around the 427725𝑡ℎ harmonic of the revolution frequency, but mixed down to the 1𝑠𝑡 harmonic.
The sampling rate, locked to the RF frequency to avoid
discrepancies during the energy ramp, is kept equal to 4×𝑓𝑟𝑒𝑣 .
The spectra are calculated every second using the last
216 = 65536 samples, which results in a frequency resolution of approximately 0.69 Hz, and the expected value ofthe spectrum is estimated by performing a moving averageof the last 100 spectra on a bin-by-bin basis.
The typical Schottky spectrum measured by the LHC
Schottky monitor, in the absence of any coherent effects,
consists of three bands: two transverse bands described by
Eq. (4) plus the central longitudinal band. Although bydesign the strong longitudinal signal should not be seen by
the monitor, imperfections of the common mode rejection
make this signal visible and available for the diagnostic
purposes, as discussed in Ref. [15].
The particle species accelerated in the LHC are either
protons or 𝑃𝑏82+ ions. For the purpose of Schottky signals
analysis, ion beams are significantly more favorable. Ionbunches consist of a lower number of particles, but each having a higher individual charge, they are therefore less likely
to produce coherent effects in the spectrum. In addition,
the higher transverse emittance of ion bunches increases
the measured dipole moment, and therefore intensifies thetransverse Schottky signal.
In Fig. 2 one can observe spectrograms and averaged
spectra during typical stages of the ion beam operation. For
ilustrative purposes only the upper transverse sideband isshown.
Figure 2 a) presents an unperturbed spectrum during
steady conditions. Such a spectrum reliably reflect the theory
discussed in the previous section, and can be easily analyzed.
Spectra in Fig. 2 b) and c) were measured under conditions
which introduced a local deformation of the averaged spectrum. The majority of the spectrum is however not affected,
and can be matched with the theoretical description. The
spectrogram shown in Fig. 2 d) is an example of a varietyof transient effects, including different types of beam beam
parameters change. Inevitably under these non-stationary
conditions, and while the spectrogram is understandable,
information is lost in the averaging process and the time averaged spectrum becomes very complex to analyse. Finally,
Fig. 2 e) shows the spectrogram and average Schottky spectrum measured while the current in the octupole magnets
is being ramped up. In this case individual Bessel satellites smear into wider bands (and eventually disappear for
TH2C0307 Machine Parameter Measurements

463

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by the sum of individual PSDs

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𝑇 ,±𝑃𝑇 ,± (𝜔1 , 𝜏
̂1 , Ω𝑠0 , 𝑄, 𝑄𝜉) ⋯ 𝑃𝐷𝐹𝑇
(𝜔1 , 𝜏̂𝑛 , Ω𝑠0 , 𝑄, 𝑄𝜉)𝑃𝑇 ,± (𝜔1 )̃ 𝜏1 )⎡ 𝐷𝐹𝑇⎤ ⎡𝑔(̂
⎤⎤ ⎡ 𝐷𝐹𝑇
𝑇 ,±
𝑇 ,±
𝑇 ,±
̃ 𝜏2 )⎥ ⎢ 𝑃𝐷𝐹𝑇
̂1 , Ω𝑠0 , 𝑄, 𝑄𝜉) ⋯ 𝑃𝐷𝐹𝑇 (𝜔2 , 𝜏̂𝑛 , Ω𝑠0 , 𝑄, 𝑄𝜉) ⎥ ⎢𝑔(̂⎢ 𝑃𝐷𝐹𝑇 (𝜔2 , 𝜏(𝜔2 ) ⎥⎢⎥⋅ ⎢ . ⎥ = ⎢
⎥,
..
..
..
⎥
⎢
⎥ ⎢ .. ⎥ ⎢
.
.
⋱
.
⎥ ⎢
⎥
⎢
⎥ ⎢
𝑇 ,±
𝑇 ,±
𝑇 ,±
𝑔(̂
̃
𝜏
)
𝑃
(𝜔
)
⎣
⎦
𝑃
(𝜔
,
𝜏
̂
,
Ω
,
𝑄,
𝑄𝜉)
⋯
𝑃
(𝜔
,
𝜏
̂
,
Ω
,
𝑄,
𝑄𝜉)
⎣
𝑛
⎣
⎦
𝑚
𝑚
1
𝑠
𝑚
𝑛
𝑠
⏟
⏟
⏟
⏟
⏟
𝐷𝐹𝑇
𝐷𝐹𝑇
𝐷𝐹𝑇
⏟⏟⏟⏟⏟
⏟⏟⎦
0
0
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ℳ(Ω𝑠0 ,𝑄,𝑄𝜉))Time [s]50

PSD [arb units]

100

100

0
0.15

0.20

0.25

0.30

0.35

0.40

0.35

0.40

PSD. [arb units]Time [seconds](a) Stable conditions
0

50

100

100

0
0.15

0.20

0.25

0.30

Time [seconds]
PSD. [arb units]

100

100

0
0.20

0.25

0.30

0.35

Time [seconds]
PSD. [arb units]

𝑖=𝑁

(6)

0.40

(c) Collimator movement

𝑄𝜉 = −𝜂 (𝑛

0

50

100

100

0
0.20

0.25

0.30

0.35

0.40

0.35

0.40

(d) Tune shift
Time [seconds]

Of the various parameters that can be estimated from transverse Schottky spectra, some of the most important include
the fractional betatron tune 𝑄𝐹 and the chromaticity 𝑄𝜉.
Examining Eq. (1), one sees that the transverse sidebands
are symmetric with respect to the frequencies (ℎ ± 𝑄𝐹 )𝜔0 .
The frequency bin corresponding to the tune value can thenbe determined as the one which minimizes the following
expression:

If a higher tune resolution is desired, values in between thefrequency bins can be interpolated.
In order to estimate chromaticity, it can be proven [16] that,for LHC conditions and RMS widths of the lower and uppertransverse sidebands equal to Δ𝑓− and Δ𝑓+ respectively, thechromacity is given by

50

0

50

100

100

0
0.15

ANALYSIS

𝑖=1

0

0.15

𝒮𝐷𝐹𝑇

𝐶𝑀𝐷 (𝑘) = ∑ ∣𝑃𝑇± (𝜔𝑘−𝑖 ) − 𝑃𝑇± (𝜔𝑘+𝑖 )∣ .

(b) Coupling measurement

0.15

𝒜

(5)

high enough octupole strength), corresponding to the tune
spread induced by the octupoles. The authors of this contribution are not aware of any theoretical result explaining
how transverse beam characteristics can be derived in suchcases.

0

PSD. [arb units]

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0.20

0.25

0.30

Frequency - fr ev [fr ev ](e) Octupole ramp

Figure 2: Examples of upper sidebands of Schottky spectrograms and averaged spectra of ion beam. All, apart from (d),measured at the injection energy.

Δ𝑓− − Δ𝑓+
− 𝑄𝐼 ) .
Δ𝑓− + Δ𝑓+(7)

Application of Eqs. (6) and (7) requires stationary beam
conditions and negligible coherent effects. In the case of
the LHC, this is mostly achievable during Pb82+ ion runs.
In Fig. 3 one can see the tune and chromaticity at flattop
during LHC fill 8413, estimated using these formulas. The
effect of octupole magnets is neglected here, what couldn’t
be done at injection energy due to the lower magnetic rigidity
and higher transverse size of the beam. The resolution of
the measured chromaticity is below the 3 units, the value
required at flattop energy by the LHC design report [20].
During less favorable conditions, when spurious components are present in the spectrum as in Fig. 2 b) and c), adifferent approach has to be taken. A key observation is thatEq. (5) remains valid after removing an arbitrary number ofrows from both matrix ℳ and vector 𝒮𝐷𝐹𝑇 . If an unwanted
component is present at frequency bins 𝜔𝑖 , … , 𝜔𝑗 , these canbe simply excluded from Eq. (5) forming new matrix ℳ̃
̃and vector 𝒮𝐷𝐹𝑇
. Then, the values of 𝑄 and 𝑄𝜉 can beobtained by minimizing the cost function:
̃ 𝑠 , 𝑄, 𝑄𝜉)⋅𝒜− 𝒮𝑒𝑥𝑝
̃ |2 , (8)𝐶𝑇 (Ω𝑠0 , 𝒜, 𝑄, 𝑄𝜉) = |ℳ(Ω0

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14

0.8

0.3102100.31006

0.309820.3096−2

LHC Fill 8505, Beam 2 Vertical, 29.03.2023, 22:07
Original
Fitted

0.6

0.4

0.2

00
01:

0

0
02:

0

0
03:

00
04:

05:

00

06:

00

0
07:

0

0
08:

0

Time [HH:MM of 19.11.2022]

Figure 3: 𝑄 and 𝑄𝜉, estimated in stable beam conditions
using Eqs. (6) and (7). Two tune outliers marked with dots.

0.0
−0.40
0.0

−0.35 0.2 −0.30

In addition, it is also possible to estimate Ω𝑠0 and 𝒜,
apriori, either from the longitudinal part of the spectrum asshown in [15], or by measuring using other instruments. At
the LHC, the synchrotron amplitude distributions can be calculated from measured longitudinal bunch profiles [15, 21],
while the nominal synchrotron frequency is possible to derive either from the peak detected Schottky system [22] or
calculated based on the knowledge of the voltage in the RF
cavities. The cost function given by Eq. (8) can be minimized using optimization routines, such as L-BFGS-B [23]or differential evolution algorithms [24]. Both approaches
have been successfully used in Refs. [15, 16]. This approachhas been used on transverse proton spectra acquired shortly
after the restart of the LHC in 2023 (fill 8505), at injectionenergy. As can be seen in Fig. 4, the central satellites are
significantly enhanced, which suggests the presence of residual coherence in the particles’ synchro-betatron motion. To
eliminate their impact on the analysis, the central frequency
bins have been excluded. The betatron tune was calculatedusing Eq. (6), modified as explained previously, while thenominal synchrotron frequency was determined based on the
RF voltage and corrected using the Schottky spectra in an
independent analysis of the longitudinal part of the spectrum.
The minimization of Eq. (8), using the differential evolution
algorithm from SciPy library, required finding only three
distinct parameters, under the assumption that 𝒜 follows
the Rice distribution [25]. The obtained fit, that is ℳ̃ ⋅ 𝒜
for optimal parameters, is compared with the experimental
spectrum in Fig. 4, while estimated values of the betatrontune and chromaticity are presented in Fig. 5. As a reference for comparison, both the chromaticity estimated with
the invasive RF-modulation technique [26] is shown as thedashed black line, as well as the chromaticity calculatedusing Eq. (7).

0.25
0.6

0.30

0.8 0.35

0.40
1.0Figure 4: Measured and fitted transverse sidebands. The
central part of each sideband was excluded from the analysis.
LHC Fill 8505, Beam 2 Vertical

0.310618
Fractional tune
Chromaticity
Eq. (7)0.3104Fractional Tune

where 𝒮𝑒𝑥𝑝 is the experimentally measured Schottky spectrum and | ⋅ | denotes the standard Euclidean norm. Similarly,
the betatron tune can also be obtained by modifying Eq. (6)in a way where only frequency bins which do not containcoherent components are included in the sum, and the results
is then scaled by the number of elements in the sum.

−0.25
0.4

Frequency - fr ev [fr ev ]140.3102100.31006

0.3098Chromaticity

00

00:

2
Reference Chromaticity

0.3096

58
21:

59

21:

00
22:

01

22:

02
22:

03

22:

04
22:

05
22:

06
22:

07
22:

−2

Time [HH:MM of 29.03.2023]

Figure 5: 𝑄 and 𝑄𝜉, estimated in the presence of coherent
components, using modifed Eq. (6) and spectra fitting.

CONCLUSION
In this contribution we presented results and discussed
some of the challenges associated with the analysis of the
Schottky spectra at the LHC. We have demonstrated how a
new fitting technique can help to overcome problems with theanalysis of spectra containing strong localized components.
The online implementation of the Schottky spectrum analysis is in the final stage of development and is planned to
be validated during the LHC ion campaing during the fallof 2023. In parallel, studies are ongoing in order to better
understand the spectra, for example including the effect ofoctupole magnets and beam coupling impedance [27].
The future steps towards further understanding of Schottky spectra and improvement of the beam diagnostic procedures should focus on the developing the theory of Schottky
signals which includes the effect of octupole magnets. Verybeneficial would also be to investigate the possibilities ofusing image recognition techniques to analyze 2D Schottky
spectrograms.

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[2] S. van der Meer, “Stochastic damping of betatron oscillationsin the ISR”, CERN, Geneva, Switzerland, Rep. CERN-ISR-

TH2C0307 Machine Parameter Measurements

465

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Fractional Tune

1.0

PSD [arb units]

Fractional tune
Chromaticity

0.310418

Chromaticity

LHC Fill 8413, Beam 2 Horizontal

0.311 0.311

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PO-72-31, 1972.
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TH2C03466

07 Machine Parameter Measurements