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

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C. Lannoy1,∗ , D. Alves, K. Lasocha, N. Mounet, CERN, Geneva, Switzerland
T. Pieloni, EPFL, Lausanne, Switzerland
1
also at EPFL, Lausanne, Switzerland
Abstract

LONGITUDINAL SCHOTTKY SPECTRA

Schottky signals are used for non-invasive beam diagnostics as they contain information on various beam andmachine parameters. The instantaneous Schottky signal is,
however, only a single realisation of a random process, implicitly depending on the discrete distribution of synchrotron
and betatron amplitudes and phases among the particles. To
estimate the expected value of the Schottky power spectrum,and reveal the inner structure of the Bessel satellites described by the theory, the averaging of instantaneous Schottky spectra is required. This study describes this procedure
quantitatively by analysing the statistical properties of the
Schottky signals, including the expected value and varianceof Schottky power spectra. Furthermore, we investigate howthese quantities evolve with the number of particles in thebunch, the observed harmonic of the revolution frequency,
the distribution of synchrotron oscillation amplitudes, andthe bunch profile. The theoretical findings are comparedagainst macro-particle simulations as well as Monte Carlocomputations.

As shown in the literature [2–5], the intensity signal 𝑖(𝑡)of a bunch consisting of N particles can be written

INTRODUCTION
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,
requiring the aggregation of numerous spectra to attain themean value. This study aims at exploring the inherent variability of consecutive instantaneous Schottky spectra and at
characterising their statistical properties.
The LHC Schottky monitors measure the power spectral density (PSD) of a signal (intensity for the longitudinal
spectrum, or dipole moment for the transverse one). For arandom process 𝑋(𝑡), the PSD is defined in general by [1,
Eq. (10.9)]
1𝑃(𝜔) = lim|𝑋 (𝜔)|2 ,(1)
𝑇 →∞ 2𝑇 𝑇
where 𝑋𝑇 (𝜔) is the Fourier transform of the truncated process
𝑇𝑋𝑇 (𝜔) = ∫ 𝑋(𝑡)𝑒−𝑗𝜔𝑡 𝑑𝑡.
−𝑇

In these proceeding we will first investigate the statisticalproperties of the longitudinal spectrum, before elaboratingon the transverse one. Then we will compare our resultswith simulations and the available measurements, and finallypresent our conclusions.
∗ christophe.lannoy@cern.ch𝑁∞𝑗(𝑛𝜔0 𝑡+𝑝Ω𝑠𝑖 𝑡+𝑝𝜑𝑠𝑖 )𝑖(𝑡) = 𝑞𝑓0 ∑ ∑ 𝐽𝑝 (𝑛𝜔0 𝜏̂𝑖 ) 𝑒, (2)𝑖=1 𝑛,𝑝=−∞where 𝐽𝑝 is the Bessel function of the first kind of order𝑝, 𝑞 the charge of the particle, 𝑓0 = 𝜔0 /2𝜋 the revolution
frequency, and 𝜏
̂𝑖 , Ω𝑠𝑖 , and 𝜑𝑠𝑖 , respectively, the synchrotron
time amplitude1 , frequency, and initial phase of particle 𝑖.
The summations over 𝑛 and 𝑝 are related to the revolution
frequency harmonic and the Bessel satellite number 𝑝.
Equation (2) is a deterministic signal entirely defined bya set of 𝑁 amplitudes 𝜏
̂𝑖 and 𝑁 phases 𝜑𝑠𝑖 (the synchrotron
frequencies Ω𝑠𝑖 can be determined from the synchrotron
amplitudes as in [6, 7]). To examine the statistical propertiesof the Schottky spectra, we define the corresponding random
process 𝐼(𝑡) of the intensity signal 𝑖(𝑡)̂𝑖 ) 𝑒𝑗(𝑛𝜔0𝑡+𝑝Ω𝑠𝑖 𝑡+𝑝Φ𝑠𝑖 ) ,𝐼(𝑡) = 𝑞𝑓0 ∑ 𝐽𝑝 (𝑛𝜔0 𝑇

(3)

𝑖,𝑛,𝑝

where the summation bounds have been omitted for readabil̂𝑖 and Φ𝑠 describe respectively,
ity. The random variables 𝑇𝑖the synchrotron time amplitude and initial phase for particle
𝑖. Equation (2) can be seen as one realisation of the random
process defined by Eq. (3).
In the following, Φ𝑠𝑖 is assumed to follow a uniform diŝ𝑖 is described by various probatribution on [0, 2𝜋], while 𝑇
bility density functions 𝑔(̂𝜏), such as the Rice distribution
as in [7]22𝜏̂ − 𝜏̂ − 𝑏𝑔(̂𝜏) = 2 𝑒 2𝜎2 2 𝐼0 (̂𝜏𝑏/𝜎),𝜎or a distribution corresponding to a Gaussian bunch profile
2

𝑔(̂
𝜏) =

𝜏
̂
𝜏
̂ − 2𝜎2𝑒,
𝜎2in order to assess the impact of different bunch profiles.
Applying the PSD definition from Eq. (1) to Eq. (3)yields [8]𝑗(𝑝Φ𝑠𝑖 −𝑝′ Φ𝑠 ′ )̂𝑖 )𝐽𝑝′ (𝑛𝜔0 𝑇̂𝑃(𝜔) = 𝑞2 𝑓02 ∑ 𝐽𝑝 (𝑛𝜔0 𝑇
𝑖′ )𝑒

𝑖

𝑖,𝑛,𝑝
𝑖′ ,𝑝′× 2𝜋𝛿 (𝑛𝜔0 + 𝑝Ω𝑠𝑖 − 𝜔) 𝛿𝐾 (𝑝Ω𝑠𝑖 − 𝑝′ Ω𝑠𝑖′ ) , (4)1 By time amplitude, we mean the maximum time arrival difference be-

tween a given particle and the synchronous particle

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STATISTICAL PROPERTIES OF SCHOTTKY SPECTRA

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where 𝛿 is the Dirac delta distribution and withsin(𝑥𝑇 ) ⎧
{1,
=⎨
𝑇 →∞
𝑥𝑇
{
⎩0,

𝛿𝐾 (𝑥) = lim

where the sum is taken over all combinations of non-negative
integer indices 𝑘1 through 𝑘𝑁 .
Substituting Eqs. (7) and (10) in Eq. (9), we obtain thevariance of the power for the central satellite [8]if 𝑥 = 0,otherwise.

Central Satellite
The power (i.e. the integrated PSD) of the central satellite
𝑝 = 0, at a given harmonic 𝑛 = 𝑛ℎ can be deduced from
Eq. (4) and is given by𝑁𝑁̂𝑖 )𝐽0 (𝑛ℎ 𝜔0 𝑇
̂𝑃(0) = 𝑞2 𝑓02 ∑ ∑ 𝐽0 (𝑛ℎ 𝜔0 𝑇𝑖′ )𝑖=1 𝑖′ =1
𝑁

2

(5)

= 𝑞2 𝑓02 (∑ 𝑍𝑖 ) ,𝑖=1

where we defined the random variables
̂𝑖 ),𝑍𝑖 = 𝐽0 (𝑛ℎ 𝜔0 𝑇

∀𝑖 ∈ [1, 𝑁],(6)

In the following we derive the expected power and variance of the central satellite.
Expected Power Using the linearity of the expectationand the fact that the expected value of a product is the productof the expected values for independent random variables, we
have𝑁Non-Central Satellites
For non-central satellites (i.e. 𝑝 ≠ 0), we extend thenotation introduced in Eq. (6) to different order 𝑝 of theBessel function
̂𝑖 ),𝑍𝑖,(𝑝) = 𝐽𝑝 (𝑛ℎ 𝜔0 𝑇

∀𝑖 ∈ [1, 𝑁].

Similarly, we also define the following moments of 𝑍𝑖,(𝑝)2 ),𝛽(𝑝) = 𝐸(𝑍𝑖,(𝑝)

(12)

4 ).
𝛿(𝑝) = 𝐸(𝑍𝑖,(𝑝)̂𝑖 )𝐽𝑝 (𝑛ℎ 𝜔0 𝑇
̂𝑃(𝑝) = 𝑞2 𝑓02 ∑ 𝐽𝑝 (𝑛ℎ 𝜔0 𝑇𝑖′ )𝑒

𝑖=1 𝑗=1𝑗≠𝑖= 𝑁𝛽 + 𝑁(𝑁 − 1)𝛼2 ,(7)where we defined the notations(8)

𝛽 = 𝐸(𝑍𝑖2 ),for the first and second moments of 𝑍𝑖 . These are independent of 𝑖 since all the particles have the same synchrotron
amplitude distribution. Note that if 𝛼 ≈ 0, the dominantterm of the expected value of the power for the central satellite, scales as 𝑁 rather than 𝑁 2 .
Variance The variance can be expressed as
22 ) − (𝐸(𝑃 )) ,𝑉 𝑎𝑟 (𝑃(0) ) = 𝐸 (𝑃(0)
(0)

𝑖

𝑖,𝑖′

𝑖=1

(9)

where the second term is the square of the average power
calculated in Eq. (7). From the multinomial theorem, thefirst term can be expanded using
4

𝑁
𝑁
4!
𝐸 ⎡⎢(∑ 𝑍𝑖 ) ⎤⎥ =∑𝐸 [∏ 𝑍𝑖𝑘𝑖 ] ,𝑘 !𝑘 !...𝑘𝑁 ! 𝑖=1⎣ 𝑖=1⎦ 𝑘1+𝑘2+...+𝑘𝑁=4 1 2
𝑘1 ,𝑘2 ,...,𝑘𝑁 ≥0

where 𝛾 and 𝛿 represent the third and fourth moments of 𝑍𝑖 .
It is worth noting that Eq. (5) is formally identical to
the squared distance of a one-dimensional random walk of
𝑁 steps of variable length 𝑍𝑖 . By reducing Eq. (5) to theparticular case of unit steps, we have 𝛼 = 𝛾 = 0, 𝛽 = 𝛿 = 1,
and we find the same expression for the mean and varianceas in the one-dimensional random walk of unit steps [9].

𝑗𝑝(Φ𝑠𝑖 −Φ𝑠 ′ )𝑁= ∑ 𝐸(𝑍𝑖2 ) + ∑ ∑ 𝐸(𝑍𝑖 )𝐸(𝑍𝑗 )𝛼 = 𝐸(𝑍𝑖 ),+ (4𝑁 3 − 16𝑁 2 + 12𝑁)𝛽𝛼2 + (−4𝑁 3 + 10𝑁 2 − 6𝑁)𝛼4 ,(11)

The power contained in a given satellite 𝑝, can be derived
from Eq. (4) and is given by [8]𝑁 𝑁
𝐸(𝑃(0) )
⎛
⎜
⎟
=
𝐸
∑
∑ 𝑍𝑖 𝑍𝑗 ⎞𝑞2 𝑓02⎝𝑖=1 𝑗=1⎠
𝑁

𝑉 𝑎𝑟 (𝑃(0) )= 𝑁𝛿 + (4𝑁 2 − 4𝑁)𝛾𝛼 + (2𝑁 2 − 3𝑁)𝛽2
𝑞4 𝑓04

(10)

𝑗𝑝Φ𝑠𝑖

= 𝑞2 𝑓02 ∣∑ 𝑍𝑖,(𝑝) 𝑒

2

∣ .

(13)

𝑖

̂𝑖 and Φ𝑠 are independent random
By assuming that 𝑇𝑖variables, one can show that the expected value and variance2
for non-central satellites are respectively given by [8]𝐸(𝑃(𝑝) ) = 𝑞2 𝑓02 𝑁𝛽(𝑝) ,𝑉 𝑎𝑟(𝑃(𝑝) ) = 𝑞4 𝑓04 (𝑁𝛿(𝑝) + (𝑁 2 − 2𝑁)𝛽2(𝑝) ) .
Note that Eq. (13) is formally identical to the squared distance of a two-dimensional random walk of 𝑁 steps of variable length 𝑍𝑖,(𝑝) and directions 𝑝Φ𝑠𝑖 . By reducing Eq. (13)to the particular case of unit steps, we have 𝛽(𝑝) = 𝛿(𝑝) = 1,
and we find the same expression for the mean and varianceas in the two-dimensional random walk of unit steps [10].
Figure 1 illustrates the coefficient of variation (CV), defined for both central (𝑝 = 0) and non-central satellites as
𝐶𝑉(𝑝) =√𝑉 𝑎𝑟(𝑃(𝑝) ).
𝐸(𝑃(𝑝) )As can be seen, for typical LHC conditions, CV is √2 forthe central satellite and 1 for non-central satellites.
2 The variance was derived under the assumption of a linear RF bucket.

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2
1

2

Proton bunch 1.5 × 1011
1.8 × 108
0 Ion bunch
2
6
101014 1018 1022101010Number of particles N

1026

Figure 1: CV of the longitudinal satellites around the harmonic 𝑛ℎ = 427 725 for a Rice amplitude distribution(dashed lines, 𝜎𝑟 = 0.22 ns and 𝑏𝑟 = 1.306) and a Gaussian bunch profile (solid lines, 𝜎𝑔 = 0.31 ns).

TRANSVERSE SCHOTTKY SPECTRA
The transverse Schottky spectrum is given by the power
spectral density of the dipole moment signal. By assumingthat chromaticity 𝑄𝜉 is the only source of betatron frequency
spread, the corresponding random process of the dipole
moment can be written [2, 5]
𝑁

0.05

̂𝑖 𝑞𝑓0𝑋
𝐽𝑝 (𝜒±𝑇̂ ,𝑛∓𝑄 )2𝑖𝐼𝑛,𝑝=−∞(±) 𝑖=1
× 𝑒𝑗(𝑡[(𝑛±𝑄𝐹 )𝜔0 +𝑝Ω𝑠𝑖 ]±Φ𝛽𝑖 +𝑝Φ𝑠𝑖 ),where the nominal tune is decomposed into its integer andfractional part 𝑄 = 𝑄𝐼 + 𝑄𝐹 and with
𝜒±𝑇̂ ,𝑛 = (𝑛 ±
𝑖

𝑄𝜉
̂𝑖 .
) 𝜔0 𝑇𝜂̂𝑖 , Φ𝛽In the above equations, 𝜂 is the slippage factor, and 𝑋
𝑖are respectively, the random variable of the amplitude, andthe initial phase of the betatron motion of particle 𝑖. The summation over ± corresponds to the upper and lower transverse
sidebands of the Schottky spectrum.
The PSD of the dipole moment can then be computedwith Eq. (1), and the mean and variance of the power for agiven satellite 𝑝 around the Schottky harmonic 𝑛 = 𝑛ℎ canbe derived, leading to the following expressions [8]:𝑁𝑞2 𝑓02 𝑥̂2 ±𝛽(𝑝) ,(14)
4
𝑞4 𝑓04±
𝑉 𝑎𝑟 (𝑃(𝑝))=(𝑁̂
𝑥4 𝛿±(𝑝) + (𝑁 2 − 2𝑁)̂𝑥2 2 (𝛽±(𝑝) )2 ) ,16±
𝐸(𝑃(𝑝))=

̂𝑖 , andwith 𝑥̂2 and 𝑥̂4 , the second and fourth moments of 𝑋
where we extended the notations introduced previously with
𝛽±(𝑝) = 𝐸 [𝐽𝑝2 (𝜒±𝑇̂ ,𝑛 ∓𝑄 )] , 𝛿(𝑝)± = 𝐸 [𝐽𝑝4 (𝜒±𝑇̂ ,𝑛 ∓𝑄 )] .
𝑖

ℎ

𝐼

𝑖

ℎ

𝐼

In contrast to the longitudinal spectrum, these expressions
are also valid for the central transverse satellites due to the additional betatron phase appearing in the dipole moment. The

0.03
0.02

Lower bandp=0
p=5
p = 10p = 20p = 30p = 50

LHC Schottky harm.
nh = 427 725

0.01

0.05

∞

𝐷(𝑡) = ∑ ∑ ∑

0.04

0.00

E(P(p)+ ) [arb. units]

CV(p)

3

CVs behave similarly to those of the longitudinal satellites,
approaching 1 as 𝑁 becomes sufficiently large (including thecentral satellites).
Figure 2 illustrates how the power of different Bessel
satellites, as given by Eq. (14), are impacted by the harmonic
number and the chromaticity on each sideband. It can beobserved that as the harmonic number increases, the powerof lower order satellites decreases, while the opposite occurs
for higher order satellites. The effect of chromaticity canalso be observed, resulting in a higher and narrower upperband compared to the lower band.

E(P(p)) [arb. units]

p=0
p=1
p=5
p = 10p = 20p = 30p = 504100000 200000 300000 400000 500000

Upper band

0.04
0.03
0.02
0.01
0.00

100000 200000 300000 400000 500000
Harmonic number : n

Figure 2: Harmonic scan of the expected powers of the transverse Bessel satellites for a Gaussian bunch profile (𝜎𝑔 =0.31 ns). The solid lines correspond to a zero chromaticity
𝑄𝜉 = 0 while the dashed lines correspond to 𝑄𝜉 = 20.

SIMULATIONS AND MEASUREMENTS
The theoretical equations derived in this study are compared against macro-particle and Monte-Carlo simulationsof a realistic LHC ion fill in Fig. 3. The macro-particle simulations has been performed using PyHEADTAIL [11, 12]
and the method used to retrieve the Schottky spectra fromthe tracking simulations is presented in [13]. The procedureof the Monte-Carlo simulation is described in [13, 14]. Bothsimulation techniques show a good agreement with the theoretical predictions and a clear convergence to a CV of √2
and 1 can be observed.
It should be noted that the previously developed theory
is studying the mean and variance over different randomdraws of the bunches (phases and amplitudes of synchrotron
and betatron motions), while in the real machine, there isa unique instance of the bunch. However, due to various
non-linearities present in reality, the relative initial phasesof synchrotron and betatron motions are drifting with time.
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Lower band

Macro-particle CV

0

1.5
1.0
0.5
0.0

Central band

200

Macro-particle
Monte-Carlo

2

100

1

Monte-Carlo CV

PSD [arb. units]

2

1.5
1.0
0.5
0.0

7.5

8.0

7.5

7.5

8.0

1

0 10.5

8.5

1.5
1.0
0.5
0.0

8.5

1.5
1.0
0.5
0.0

8.0
8.5
Frequency [kHz]Upper band

1e 5

11.0

11.5

12.0

2

10.5

11.0

11.5

12.0

2

10.5

11.0
11.5
Frequency [kHz]

12.0

0
1.5
1.0
0.5
0.0
1.5
1.0
0.5
0.0

14.0

14.5

15.0

14.0

14.5

15.0

14.0
14.5
15.0
Frequency [kHz]Figure 3: 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.

PSD [arb. units]

After a given period, this unpredictable drift is equivalent
to drawing new random phases, making the averaging oversuccessive instantaneous Schottky spectra equivalent to theaveraging 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
Eq. (5). Figure 4 presents an experimental measurementfrom the LHC Schottky monitors. The CV for the transverse
sidebands and the non-central (|𝑝| > 2) longitudinal satellites agree with the theory. No conclusion can be drawn
for the central (𝑝 = 0) longitudinal satellite for the reason
given above. The CVs for 𝑝 = ±1 and ±2 longitudinal satellites deviate from the predictions, which suggests that thesesatellites are subject to effects beyond the adopted theory of
Schottky spectra and should not be used for diagnostics.

CV

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6 1e 12Lower band

6 1e 11

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 ofthe coefficient of variation to specific, non-zero values, fordifferent Bessel satellites. This highlights the importanceof averaging instantaneous Schottky spectra to access themean spectrum described by existing theories. Our findings
were validated through simulations and data from the LHC
Schottky monitors and while transverse and non-central longitudinal satellites exhibit consistent convergence in the experimental data, the behaviour of the central longitudinal
satellite still requires further investigation.

Central band

4

4

4

2

2

2

0

1.5
1.0
0.5
0.0

7.6

7.6

7.7

7.8

7.9

7.7 7.8 7.9Frequency [kHz]

8.0

011.0

1.5
1.0
0.5
0.0
8.0 11.0

11.1

11.1

11.2

11.3

Upper band

6 1e 1211.4

11.2 11.3 11.4
Frequency [kHz]

014.5
1.5
1.0
0.5
0.0

14.5

14.6

14.6

14.7

14.8

14.9

14.7 14.8 14.9
Frequency [kHz]Figure 4: 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.
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REFERENCES[1] S. L. Miller and D. Childers, Probability and Random Processes (Second Edition). Academic Press, 2012.
doi:10.1016/B978-0-12-386981-4.50013-8[2] D. Boussard, “Schottky noise and beam transfer function
diagnostics,” p. 42, 1986.
doi:10.5170/CERN-1987-003-V-2.416[3] S. Chattopadhyay, “Some fundamental aspects of fluctuations
and coherence in charged‐particle beams in storage rings,”
AIP Conference Proceedings, vol. 127, no. 1, pp. 467–623,
1985. doi:10.1063/1.35175[4] J. L. Laclare, “Bunched beam coherent instabilities,” 1987.
doi:10.5170/CERN-1987-003-V-1.264[5] K. Lasocha, “Non-Invasive Beam Diagnostics with Schottky
Signals and Cherenkov Diffraction Radiation,” PhD thesis,
Jagiellonian University, Kraków, Poland, 2022. https://
cds.cern.ch/record/2846538[6] D. Alves and K. Lasocha, “Kalman filter-based longitudinal
phase-space reconstruction method for hadron machines,”
Phys. Rev. Accel. Beams, vol. 24, no. 9, p. 072 801, 2021.
doi:10.1103/PhysRevAccelBeams.24.072801[7] K. Lasocha and D. Alves, “Estimation of longitudinal bunchcharacteristics in the LHC using Schottky-based diagnos-

tics,” Phys. Rev. Accel. Beams, vol. 23, no. 6, p. 062 803,
2020. doi:10.1103/PhysRevAccelBeams.23.062803[8] C. Lannoy, “Statistical properties of Schottky spectra,”
CERN note, in preparation.
[9] Random Walk–1-Dimensional. https://mathworld.
wolfram.com/RandomWalk1-Dimensional.html[10] J. M. Borwein, D. Nuyens, A. Straub, and J. Wan, “Somearithmetic properties of short random walk integrals,” The
Ramanujan Journal, vol. 26, no. 1, pp. 109–132, 2011.
doi:10.1007/s11139-011-9325-y[11] PyHEADTAIL code repository. https://github.com/
PyCOMPLETE[12] K. S. B. Li et al., “Code development for collective effects,”
in Proc. HB’16, Malmö, Sweden, 2016, pp. 362–367.
doi:10.18429/JACoW-HB2016-WEAM3X01[13] C. Lannoy, D. Alves, N. Mounet, and T. Pieloni, “LHC Schottky Spectrum from Macro-Particle Simulations,” in Proc.
IBIC’22, Kraków, Poland, 2022, pp. 308–312.
doi:10.18429/JACoW-IBIC2022-TUP34[14] M. Betz, O. R. Jones, T. Lefevre, and M. Wendt, “Bunchedbeam Schottky monitoring in the LHC,” Nucl. Instrum. Methods Phys. Res., A, vol. 874, no. 14, pp. 113–126, 2017.
doi:10.1016/j.nima.2017.08.045

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