Longitudinal mode-coupling instabilities of proton bunches in the CERN Super Proton Synchrotron Ivan Karpov

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Longitudinal mode-coupling instabilities of proton bunches

in the CERN Super Proton Synchrotron

Ivan Karpov∗

CERN, CH-1211 Geneva, Switzerland

(Dated: October 4, 2022)

In this paper, we study single-bunch instabilities observed in the CERN Super Proton Syn-

chrotron (SPS). According to the linearized Vlasov theory, radial or azimuthal mode-coupling in-

stabilities result from a coupling of bunch-oscillation modes, which belong to either the same or

adjacent azimuthal modes, respectively. We show that both instability mechanisms exist in the SPS

by applying the Oide-Yokoya approach to compute van Kampen modes for the realistic longitudinal

impedance model of the SPS. The results agree with macroparticle simulations and are consistent

with beam measurements. In particular, we see that the uncontrolled longitudinal emittance blow-

up of single bunches observed before the recent impedance reduction campaign (2018-2021) is due

to the radial mode-coupling instability. Unexpectedly, this instability is as strong as the azimuthal

mode-coupling instability, which is possible in the SPS for other combinations of bunch length and

intensity. We also demonstrate the signiﬁcant role of rf nonlinearity and potential-well distortion

in determining these instability thresholds. Finally, we discuss the eﬀect of the recent impedance

reduction campaign on beam stability in single- and double-rf conﬁgurations.

I. INTRODUCTION

Longitudinal single-bunch instability is a possible per-

formance limitation in many synchrotrons and its mech-

anism is a subject of various studies since long time [1–

16]. The standard approach to evaluate beam stability is

based on a solution of the linearized Vlasov equation for

a small initial perturbation of a stationary distribution

function. To simplify the analysis, the modiﬁcation of

a stationary potential well by self-induced ﬁelds, called

potential-well distortion (PWD), is often neglected. The

only possible mechanism of longitudinal single-bunch in-

stability without PWD, a coupling of diﬀerent azimuthal

modes, was proposed by Sacherer [1].

Another type of instability can be caused by asym-

metry of the potential well due to PWD, resulting in

a coupling of two radial modes within one azimuthal

mode [8, 9]. An explicit condition required for this

instability to occur was found for the double-waterbag

model [10]. For an impedance model consisting of one

broad-band resonator with frequency fr=ωr/2π, the

instability thresholds computed with and without PWD

are similar for ωrσ&0.4, where σis the rms bunch

length, [8]. This result was also conﬁrmed in calcula-

tions based on the orthogonal polynomial expansion [14].

The azimuthal mode-coupling was also found in the self-

consistent analysis of electron bunches for ωrσ≈π[13].

Similar to electron bunches, the thresholds of the

single-bunch instability for proton bunches are often com-

puted neglecting bunch asymmetry due to PWD and rf

nonlinearity, as for example, in [11, 17], and thus only

azimuthal mode-coupling instability was found. To our

best knowledge, for proton bunches, so far a radial mode-

coupling instability was not observed in measurements

nor in calculations.

∗ivan.karpov@cern.ch

In the SPS, the longitudinal instability of single pro-

ton bunches occurs during the acceleration ramp. The

attempts to cure this instability by reducing the volt-

age in a single rf system and thus increasing the syn-

chrotron frequency spread for a constant longitudinal

emittance were not successful. Instead, a higher rf volt-

age was more beneﬁcial [18]. In operation, this insta-

bility is cured by the application of a higher-harmonic

(HH) rf system. Due to strong frequency dependence

of the SPS impedance [19] (Fig. 1), the observed insta-

bility was mainly studied in macroparticle simulations

using the code BLonD [20]. The latest results of simu-

lations through the ramp are consistent with measure-

ments and the agreement has been improved with the

reﬁned impedance model [21].

In the present work, the mechanism of the SPS single-

bunch instability is studied using code MELODY [22]

which is able to ﬁnd in a fully self-consistent way the

numerical solutions of the semianalytic matrix equa-

tions derived from the Vlasov equation for the full SPS

impedance model. We show that the previously observed

instability during the ramp was due to the coupling of

multiple radial modes within one azimuthal mode. For

a speciﬁc set of bunch parameters, we ﬁnd an instability

caused by the coupling of neighboring azimuthal modes

for which Landau damping is lost. The main results are

conﬁrmed by macroparticle simulations with BLonD and

are consistent with previous measurements [18].

The paper is organized as follows. In Sec. II, we brieﬂy

discuss the main deﬁnitions and semianalytical methods

to evaluate single-bunch instabilities. Two possible insta-

bility mechanisms in the SPS and a comparison of cal-

culations with measurements are presented in Sec. III.

We consider diﬀerent instability mitigation measures in

Sec. IV and, ﬁnally, present the main conclusions.

arXiv:2210.00080v1 [physics.acc-ph] 30 Sep 2022

10−1

100

101

102

103

ReZk/k (Ω)

1.25 1.50

100

101

0123456

Frequency (GHz)

−100

−50

100

ImZk/k (Ω)

Before LS2

After LS2

23456

−1

FIG. 1. SPS impedance model separated by real (top) and

imaginary (bottom) parts before and after the impedance

reduction campaign during the 2nd Long Shutdown (LS2),

which ended in March 2021.

II. MAIN EQUATIONS AND DEFINITIONS

The longitudinal motion of a particle in a synchrotron

can be described in terms of its energy and phase devi-

ations, ∆Eand φ, relative to the synchronous particle

with the energy E0. For beam stability analysis it is con-

venient to introduce another set of variables, the energy

Eand the phase ψof synchrotron oscillations

E=˙

φ2

2ω2

+Ut(φ),(1)

ψ= sgn(η∆E)ωs(E)

√2ωs0Zφ

φmax (E)

dφ0

pE − Ut(φ0).(2)

Here η= 1/γ2

tr −1/γ2is the phase slip factor, γis the

relativistic Lorentz factor, γtr is the Lorentz factor at

transition energy, fs0=ωs0/2πis the frequency of small-

amplitude synchrotron oscillations in a bare single-rf sys-

tem, ωs(E) is the synchrotron frequency as a function

of E, and φmax(E) is the maximum phase of the parti-

cle with synchrotron oscillation energy E=Ut[φmax(E)].

The total potential includes contributions from both the

rf system and the beam-induced ﬁelds

Ut(φ) = 1

V1cos φs0Zφ

∆φs

[Vrf (φ0) + Vind(φ0)−δE0/q]dφ0.

(3)

where V1is the rf voltage amplitude of the main rf

system, δE0is the energy gain per turn of the syn-

chronous particle with charge qexcluding intensity ef-

fects, ∆φsis the synchronous phase shift due to inten-

sity eﬀects that satisﬁes the relation δE0/q =V1sin φs0=

Vrf (∆φs) + Vind(∆φs), and φs0is the synchronous phase

in a bare single-rf system. Below we consider a double-rf

system with a total voltage of

Vrf (φ) = V1[sin (φ+φs0) + rsin (nφ +nφs0+ Φn)] ,

(4)

where Φnis the relative phase oﬀset between the main

and the HH rf systems with harmonic numbers hand nh,

respectively, and Vn=rV1is the voltage amplitude of the

HH rf system. For particular values of Φnone can de-

ﬁne two distinct regimes: bunch-shortening mode (BSM)

when both rf systems are in phase at the bunch center

and bunch-length mode (BLM) when both rf systems are

in counter-phase at the bunch center. In the SPS oper-

ation, they are chosen such that the contribution of the

HH rf system is zero at φ= 0 and it does not contribute

to a shift of the synchronous phase. Thus, Φn=π−nφs0

for BSM and Φn=−nφs0for BLM.

In general, the total potential Utdepends on the par-

ticle distribution function, impedance model, and bunch

intensity. It can be calculated using an iterative proce-

dure [12, 23]. In this work, we consider a particle distri-

bution of the binomial family

F(E) = 1

2πωs0AN1−E

Emax µ

,(5)

with the normalization constant

AN=ωs0ZEmax

ωs(E)1−E

Emax µ

.(6)

For µ→ ∞, the bunch has a Gaussian line density and

the corresponding bunch length τ4σis typically deﬁned

as four times the rms bunch length σ, i.e. τ4σ= 4σ.

The bunch length τ4σis related to the full-width at half-

maximum (FWHM) bunch length τFWHM as

τ4σ=τFWHMp2/ln 2.(7)

In practice, SPS proton bunches are far from being Gaus-

sian, and the best ﬁt to the measured bunch proﬁles is for

µ≈1.5. This value will be assumed in the present work

for all calculations and simulations. For easy compari-

son with measurements, we use Eq. (7) as a deﬁnition of

the bunch length. We also deﬁne the total longitudinal

emittance in units of eVs as

=I∆E(φ)

hω0

dφ =s−V1cos φs0qβ2E0

πηω2

0h3N,(8)

with a dimensionless emittance

N= 2 Zφmax

φmin p(Emax −Ut(φ)) dφ, (9)

where ω0= 2πf0is the revolution frequency, φmin and

φmax are the minimum and maximum phases of the par-

ticle with the energy of synchrotron oscillation Emax.

A. Linearized Vlasov equation

When the perturbation ˜

Fto the stationary particle

distribution function F(E) grows with time t, the beam

is unstable. The initial time evolution of ˜

Fis dictated

by the linearized Vlasov equation (e.g. [24])

∂˜

∂t +dF

dt +∂˜

∂ψ

dψ

dt = 0.(10)

After expansion over the azimuthal harmonics mof syn-

chrotron motion, the solution of Eq. (10) at frequency Ω

with the eigenfunctions Cm(E,Ω) [8]

F(E, ψ, t)

=e−iΩt∞

m=1

Cm(E,Ω) cos mψ +iΩ

mωs(E)sin mψ(11)

leads to the integral equation

Ω2−m2ω2

s(E)Cm(E,Ω)

= 2iζω2

s0m2ω2

s(E)dF(E)

∞

m0=1 ZEmax

dE0

ωs(E0)

×∞

k=−∞

Zk(Ω)/k

hZ0

Imk(E)I∗

m0k(E0)Cm0(E0,Ω).(12)

Here Zk(Ω) = Z(kω0+ Ω) is the longitudinal impedance

at frequency kω0+ Ω and Z0≈377 Ω is the impedance

of free space. We also introduced the dimensionless in-

tensity parameter

ζ=qNph2ω0Z0

V1cos φs0

(13)

with Npbeing the bunch intensity. The function Imk(E)

is deﬁned as

Imk(E) = 1

πZπ

eikφ(E,ψ)/h cos mψ dψ. (14)

Detailed derivations of Eq. (12) using variables (E, ψ) can

be found in [25] and it is identical to Eq. (41) therein.

For a given impedance model, the single-bunch stability

depends on the two dimensionless parameters, ζand N,

as well as on the parameters of the rf potential: n,φs0,

r, and Φn. This fact will be applied below to understand

the mechanisms of instabilities observed in the SPS.

B. van Kampen modes

The integral equation (12) is equivalent to the equation

that describes collective modes in a plasma [26–28]. An

initial perturbation can be expressed as a superposition of

van Kampen modes, which are, in general, described by

nonregular functions. To show this, one can ﬁrst perform

the substitution in Eq. (12)

Cm(E,Ω) = r−ωs(E)dF(E)

dEmωs(E)

ω2

Cm(E,Ω),(15)

which leads to

Ω2

ω2

s0−m2ω2

s(E)

ω2

s0˜

Cm(E,Ω)

=−2iζ ∞

m0=1 ZEmax

Kmm0(E,E0)˜

Cm0,Ω(E0)dE0.

(16)

Here, the kernel Kis deﬁned as

Kmm0(E,E0,Ω) = ∞

k=−∞

Zk(Ω)/k

hZ0

×mr−ωs(E)dF(E)

dEImk(E)

×m0r−ωs(E0)dF(E0)

dE0I∗

m0k(E0).(17)

Introducing a set of orthonormal functions s(m)

ZEmax

s(m)

n(E)s(m)

n0(E)dE=δnn0,

with the Kronecker delta δij , we can decompose ˜

Cmand

Kmm0, similarly to [4]:

Cm(E,Ω) = ∞

n=0

m(Ω)s(m)

n(E),(18)

Kmm0(E,E0,Ω) = ∞

n=0

∞

n0=0

Knn0

mm0s(m)

n(E)s(m0)

n0(E0).(19)

with the coeﬃcients an

mand Knn0

mm0deﬁned as

m(Ω) = ZEmax

Cm(E,Ω) s(m)

n(E)dE,(20)

and

Knn0

mm0(Ω)

=ZEmax

0ZEmax

Kmm0(E,E0,Ω)s(m)

n(E)s(m0)

n0(E0)dEdE0,

(21)

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Longitudinalmode-couplinginstabilitiesofprotonbunchesintheCERNSuperProtonSynchrotronIvanKarpovCERN,CH-1211Geneva,Switzerland(Dated:October4,2022)Inthispaper,westudysingle-bunchinstabilitiesobservedintheCERNSuperProtonSyn-chrotron(SPS).AccordingtothelinearizedVlasovtheory,radialorazimuthalmode-coupl...

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Longitudinal mode-coupling instabilities of proton bunches in the CERN Super Proton Synchrotron Ivan Karpov

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