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BEAM HALO IN HIGH-INTENSITY
HADRON LINACS
vorgelegt von
Diplom-Ingenieur
Frank Gerigk
von der Fakult¨at IV - Elektrotechnik und Informatik
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
- Dr.-Ing. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. R. Orglmeister
Berichter: Prof. Dr. H. Henke
Berichter: Prof. Dr. I. Hofmann
Tag der wissenschaftlichen Aussprache: 21. Dezember 2006
2006 Berlin
D 83
ii
Abstract
English:
This document aims to cover the most relevant mechanisms for the development of beam halo in high-
intensity hadron linacs. The introduction will outline the various applications of high-intensity linacs
and it will explain why, in the case of the CERN Superconducting Proton Linac (SPL) study a linac was
chosen to provide a high-power beam, rather than a different kind of accelerator. The basic equations,
needed for the understanding of halo development will be derived and employed to study the effects of
initial and distributed mismatch on high-current beams. The basic concepts of the particle-core model,
envelope modes, parametric resonances, the free-energy approach, and the idea of core-core resonances
will be introduced and extended to study beams in realistic linac lattices. The approach taken is to study
the behavior of beams not only in simplified theoretical focusing structures but to highlight the beam
dynamics in realistic accelerators. All effects which are described and derived with simplified analytic
models, are tested in realistic lattices and are thus related to observable effects in linear accelerators. This
approach involves the use of high-performance particle tracking codes, which are needed to simulate the
behavior of the outermost particles in distributions of up to 100 million macro particles. In the end a set
of design rules will be established and their impact on the design of a typical high-intensity machine, the
CERN SPL, will be shown. The examples given in this document refer to two different design evolutions
of the SPL study: the first conceptual design report (SPL I) and the second conceptual design report (SPL
II).
Deutsch:
Das Ziel dieser Arbeit ist, die relevantesten Mechanismen der Haloentwicklung f¨ur Teilchenstrahlen
in Hochintensit¨atslinearbeschleunigern zu behandeln. In der Einleitung werden die vielf¨altigen An-
wendungen dieser Linearbeschleuniger (kurz: Linac) vorgestellt. Es wird weiterhin erkl¨art warum im
Falle der CERN Studie zur Konstruktion eines supraleitenden Protonenlinacs (SPL) ein Linac gew¨ahlt
wurde um einen hochintensiven Protonenstrahl zu liefern, anstatt eines anderen Beschleunigertyps. An-
schließend werden die grundlegenden Gleichungen abgeleitet, welche zum Verst¨andnis der Haloentwick-
lung ben¨otigt werden. Diese Gleichungen werden dann benutzt um den Einfluss von anf¨anglicher und
statistisch verteilter Strahlfehlanpassung auf hochintensive Teilchenstrahlen zu untersuchen. Grundle-
gende Konzepte wie: das Teilchen-Kern Modell (particle-core model), Enveloppenmoden, parametrische
Resonanzen, der “freie Energie” Ansatz und die Idee der Kern-Kern Resonanzen werden eingef¨uhrt
und erweitert um Teilchenstrahlen in realistischen Fokussierungskan¨alen zu studieren. Eine Grundidee
dieser Arbeit ist, das Strahlverhalten nicht nur in vereinfachten theoretischen Fokussierungsstrukturen
zu beschreiben, sondern die Strahldynamik in realistischen Beschleunigern zu untersuchen. Alle Ef-
fekte welche mit vereinfachten analytischen Modellen abgeleiten werden, werden so mit beobachtbaren
Effekten in Linearbeschleunigern in Zusammenhang gebracht. Dieser Ansatz bringt es mit sich, dass
leistungsf¨ahige Simulationsprogramme benutzt werden um die Trajektorien der ¨außersten Randteilchen
einer Verteilung zu verfolgen, welche aus bis zu 100 Millionen Makropartikeln besteht. Gegen Ende der
Arbeit wird eine Reihe von Regeln aufgestellt und es wird aufgezeigt, welchen Einfluss diese Regeln
auf das Design eines typischen Linacs f¨ur hochintensive Teilchenstrahlen (den CERN SPL) hat. Die
Beispiele in dieser Arbeit beziehen sich auf zwei Entwicklungsstadien dieses Linearbeschleunigers: den
ersten konzeptionellen Designreport (SPL I) und den zweiten revidierten Report (SPL II).
ii
1 Introduction 2
1.1 High-power hadron linacs: machine types and their applications . . . . . . . . . . . . . 2
1.1.1 H−injection and beam chopping . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Beam loss in linacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 High-intensity linacs versus other accelerator types . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Requirements for a CERN-based proton driver . . . . . . . . . . . . . . . . . . 6
1.2.2 High-power cyclotrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Rapid cycling synchrotrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Accelerator choice for a CERN-based proton driver . . . . . . . . . . . . . . . . 11
1.3 High-power linac studies and the goal of this thesis . . . . . . . . . . . . . . . . . . . . 13
2 Basic equations 15
2.1 3D envelope equations with space-charge . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The principle of smooth approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Quadrupole and RF focusing terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Smooth approximation for a FODO channel with RF cavities . . . . . . . . . . . . . . . 21
3 Multi-particle simulations with the IMPACT code 23
3.1 The IMPACT code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Using IMPACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Initial mismatch 28
4.1 Space-charge and beam stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 The particle-core model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Initial mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 3D envelope eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 The “free energy” limit for r.m.s. emittance growth . . . . . . . . . . . . . . . . . . . . 37
4.5 Particle distributions for simulations in 6D phase space . . . . . . . . . . . . . . . . . . 41
4.5.1 KV, waterbag, and Gaussian distributions . . . . . . . . . . . . . . . . . . . . . 41
4.5.2 6D distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.3 Distributions and emittance definitions . . . . . . . . . . . . . . . . . . . . . . 43
4.6 Mismatch for realistic linac beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6.1 Particle redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6.2 Maximum halo extent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6.3 Beam collimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
iii
5 Distributed mismatch 53
5.1 Particle-core model for statistical gradient errors . . . . . . . . . . . . . . . . . . . . . . 54
5.1.1 Average effects and evidence for a resonant process . . . . . . . . . . . . . . . . 56
5.1.2 Halo development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Limitations of and conclusions from the particle-core model . . . . . . . . . . . . . . . 59
5.3 3D particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4 Conclusions on statistical gradient errors . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Core-core resonances 65
6.1 Application of stability charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2 Core-core resonances & beam halo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Practical linac design 70
7.1 general rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.2 Low-energy beam chopper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.3 The SPL project at CERN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3.2 Layout and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A R.m.s. envelope equations and the smooth approximation 82
A.1 Space-charge force term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.2 Thin lens approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A.3 Quadrupole focusing in the smooth approximation . . . . . . . . . . . . . . . . . . . . . 84
A.4 RF focusing in the smooth approximation . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.4.1 Longitudinal focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.4.2 Transverse defocusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B Accelerating structures 88
C Derivation of envelope eigenmodes 90
D Stability charts 92
iv
Symbols
s.........................................................................longitudinalposition
x,y,z......................................................s-dependent single particle positions
a...............................................................transverser.m.s.beamenvelope
ax,y .......................................................transverser.m.s.beamenvelopeinx,y
b..............................................................longitudinalr.m.s.beamenvelope
ˆa,ˆ
b.... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .matched r.m.s. envelopes in the smooth approximation
κ(s).......................................................focusingforcedependingonposition
Fsc .....................................................................space-chargeforceterm
I......................................................................................current
K3..................................................................3Dspace-chargeparameter
K1..................................................................1Dspace-chargeparameter
σ0........................................................zero-currentphaseadvanceperperiod
σ..........................................................full-currentphaseadvanceperperiod
kQ................................................................quadrupolefocusingconstant
k0.. . zero-current phase advance per unit length (wave number for the transverse betatron oscillations)
k......................................................full-currentphaseadvanceperunitlength
αi,βi,γi,(i=x,y,z)..........................................................Twissparameters
εi,(i=x,y,z)..................................................................r.m.s.emittance
εt,l .......................................................transverse,longitudinalr.m.s.emittance
q......................................................................................charge
G...........................................................................magneticgradient
l.......................................................................................length
lQ...........................................................................quadrupolelength
lC................................................................................cavitylength
L..............................................................lengthbetweentwoquadrupoles
Lp....................................................................lengthoffocusingperiod
m...................................................................................restmass
c................................................................................speedoflight
v.....................................................................................velocity
β.....................................................................normalisedvelocity(v/c)
γ........................................................................relativisticmassfactor
E0.......................................................................electricfieldgradient
T............................................................................transittimefactor
φ.................................................................................phaseangle
λ..............................................................................RFwavelength
q...............................................................................electriccharge
qn(r)..............................................................radialelectricchargedensity
ε0.....................................................................permittivityoffreespace
µ0...................................................................permeabilityoffreespace
P......................................................................................power
W.....................................................................................energy
A........................................................................................area
v
ABBREVIATIONS
AD Antiproton Decelerator at CERN
ADS Accelerator Driven Systems
AGS Alternating Gradient Synchrotron at BNL
APT Accelerator Production of Tritium
BNL Brookhaven National Laboratory
BooNE Booster Neutrino Experiment at FNAL
CCDTL Coupled Cavity DTL
CCL Coupled Cavity Linac
CERN Centre Europ´eenne pour la Recherche Nucl´eaire
CNGS Cern Neutrinos to Gran Sasso
CONCERT COmbined Neutron Centre for European Research and Technology
CW Continuous Wave
DOE Department of Energy (USA)
DTL Drift Tube Linac
ESS European Spallation Source
EURISOL European Isotope Separation On-Line
FFAG Fixed Field Alternating Gradient
FNAL Fermi National Accelerator Laboratory
IFMIF International Fusion Materials Irradiation Facility
IMPACT Integrated Map and Particle ACcelerator Tracking Code
ISOLDE Isotope Separation On-Line
JPARC Japan Proton Accelerator Complex
KOMAC KOrea Multi-purpose Accelerator Complex
linac linear accelerator
LAMPF Los Alamos Meson Physics Facility
LANL Los Alamos National Laboratory
LANSCE Los Alamos Neutron Science Center
LBNL Lawrence Berkeley National Laboratory
LEBT Low Energy Beam Transport
LEIR Low Energy Ion Ring at CERN
LEP Large Electron Proton Collider at CERN
LHC Large Hadron Collider at CERN
MiniBooNE first stage of the BooNE experiment
Nb/Cu Niobium on Copper
NuMI Neutrinos at the Main Injector at FNAL
PARMILA Phase And Radial Motion In Linear Accelerators
PS Proton Synchrotron at CERN
PSB Proton Synchrotron Booster at CERN
PSI Paul Scherrer Institute
PSR Proton Storage Ring at LANL
RAL Rutherford Appleton Laboratory
RCS Rapid Cycling Synchrotron
RF Radio Frequency
1
RFQ RF Quadrupole
r.m.s. root mean square
SC SuperConducting
SDTL Separated DTL
SPL (I/II) Superconducting Proton Linac (I - design report 2000 [1], II - design report 2006 [2] )
SPS Super Proton Synchrotron at CERN
SNS Spallation Neutron Source
TRASCO TRAsmutazione SCOrie (design of an Accelerator Driven System for Nuclear Waste Transmutation)
WNR Weapons Neutron Research Facility at LANL
XADS eXperimental Accelerator Driven System
2
1. Introduction
1.1 High-power hadron linacs: machine types and their
applications
High-power LINear ACcelerators (linacs) are under study for various applications since the early 70’s.
The first machine to actually achieve beam power in the megawatt range was the linac of the Los Alamos
Meson Physics Facility (LAMPF), delivering its first 800MeV beam at low duty cycle in 1972. After the
construction of the proton storage ring (PSR) the neutron scattering community becomes the main user
of the facility which in 1983 produces an average beam power of 1MW using a pulsed beam. In 1993
the Department of Energy (DOE) drops the support for the LAMPF facility and the Los Alamos Neutron
Science Center (LANSCE) is created. In 2005 the LANSCE machine remains the most powerful linear
proton accelerator worldwide (Fig. 1.1).
Figure 1.1: The Los Alamos Neutron Science Center
Today LANSCE provides two different types of proton beams onto targets: a) a long-pulse proton beam
(≈1ms) with an average current of 1mA for the Weapons Neutron Research Facility (WNR), and b) a
short-pulse proton beam (≈0.1µs) with an average current of up to 125µA for the Lujan Center which
is produced by injecting an H−beam into the PSR where the beam is accumulated and compressed. The
long-pulse beam is used either directly for irradiation experiments or to produce high-energy neutrons
which complement those delivered by the Lujan Center, where “cold neutrons” are produced which allow
CHAPTER 1. INTRODUCTION 3
precise time-of-flight measurements used for general purpose material sciences. The Lujan Center is a
national user facility for generally “non-military” basic and applied research in the fields of material
science, engineering, condensed matter physics, polymer science, structural biology, chemistry, earth
sciences, and neutron-nuclear-science research. It falls into the same science category as the ISIS facility
at the Rutherford Appleton Laboratory (RAL) in the UK which, with an average beam power on target
of 160 kW, is until today the most powerful pulsed spallation neutron source. In recent years neutron
spallation sources became one of the most popular applications for high-intensity linacs. However, so
far only the Spallation Neutron Source (SNS) project at Oakridge, Tennessee [3] was constructed and
recently (spring 06) delivered the first beam on target. Similar projects are planned in Europe, India,
China, and Korea.
1.1.1 H−injection and beam chopping
From the “accelerator point of view” long-pulse and short-pulse proton drivers have an important dif-
ference. Long-pulse proton drivers dump the beam from a linear accelerator directly onto the target,
while short-pulse proton drivers need additional accumlator rings to compress the length of the linac
pulses from the order of milliseconds to the order of microseconds. The size of the accumulator ring
then determines the length of the compressed linac pulse as depicted in Fig. 1.2.
Figure 1.2: Compression of linac pulses with accumulator rings
In order to achieve pulse compression factors of up to 1000 (as in the case of SNS),linac bunches must be
accumulated over a large number of injection turns (e.g. 1000). If one uses classical multi-turn injection
with protons then the position of the circulating beam is always slightly shifted to make space for the
injection of the incoming beam. With this technique the emittance or phase space area (i.e. the planes
x/x0,y/y0, and z/z0, respectively; see section 2.1) of the circulating beam increases with each turn
approximately by the emittance of the injected beam. This principle is usually applied for up to a few
tens of injection turns because one quickly reaches the acceptance limit of the circular machine which is
basically given by the physical beam pipe aperture. This bottleneck can be removed by using H−charge-
exchange injection which allows to inject particles over a large number of turns without increasing the
emittance of the circulating beam. As shown in Fig. 1.3 the incoming H−beam and the circulating beam
are deflected by the same dipole onto a common trajectory which makes it possible to inject repeatedly
into the same phase space area. The stripping foil removes two electrons from the H−beam and the
particles then continue on the same trajectory.
The maximum beam intensity that can be accumulated in a ring via H−injection is mainly limited
by: a) the space-charge limit in the ring which is defined by the injection energy and the number of
accumulated particles in the ring, b) heating of the stripper foil, c) repetition rate of the accumulation
process, d) the existence of high duty-cycle, high-current H−sources for the linac, e) electron cloud and
other instabilities in the rings.
To minimise losses when the H−beam is injected into a circular accelerator, one can employ low-energy
beam choppers (see Section 7.2). These devices usually operate in an energy range of 2 to 3MeV and are
4 1.1. HIGH-POWER HADRON LINACS: MACHINE TYPES AND THEIR APPLICATIONS
D1 D4
linac beam
circulating
beam
H
H
+
−stripping Foil
D2 D3
Figure 1.3: H−multi-turn injection via charge exchange
located between Radio Frequency Quadrupole (RFQ) and Drift Tube Linac (DTL). The principle works
as follows: when a continuous bunch train is injected into a ring RF system all bunches in the transition
area between two ring RF buckets are either partially or completely lost as illustrated in Fig. 1.4. The
effect is especially pronounced for non-accelerating RF buckets (as used in accumulator/compressor
rings), where it can yield up to 30% of beam loss at injection. In case of synchrotrons the RF buckets are
not fixed with respect to the location of the bunches and can thus still “collect” some of the particles in the
transition areas. The purpose of low-energy beam choppers is to create gaps in the otherwise continuous
bunch train to allow the transition from one ring RF bucket to the next without losing linac bunches.
Depending on the acceptable losses the chopping ratio should be between 25 - 40% (see Fig. 1.4).

!
"$#&%$(')(*+
-,./01
Figure 1.4: Low energy beam chopping
While Los Alamos was able to achieve one megawatt of beam power with its proton beam, only 80kW
could be produced with H−which are injected into the PSR via charge-exchange injection. Many of
the high-power hadron linacs which are under study or under construction today foresee H−injection
together with accumulator and compressor rings to produce short pulses of protons in the µs range con-
taining short intense bunches down to the ns range. The two main applications for which this time
structure is needed are short-pulse spallation targets [e.g. Spallation Neutron Source (SNS [3]), Euro-
pean Spallation Source (ESS study [4])] and neutrino factories [5] or super-beam facilities [6] (here only
an accumulator ring is needed). Applications that use a direct proton beam on target can be partitioned
into four main categories: a) long-pulse spallation targets, e.g: the WNR at LANSCE, the International
Fusion Materials Irradiation Facility (IFMIF, under study), ESS, b) Accelerator Driven Systems (ADS,
under study) for the transmutation of nuclear waste from conventional nuclear reactors or energy produc-
tion with energy amplifiers (under study), c) radioactive ion beam facilities, e.g. On-Line Isotope Mass
Separator (ISOLDE [7], in production), European Isotope Separation On-Line (EURISOL [8], under
study), and d) the Accelerator Production of Tritium, e.g. APT [9], abandoned study, COmbined Neu-
tron Centre for European Research and Technology (CONCERT [10], abandoned study). Table 1.2 lists
the main characteristics of these linac-based projects.
CHAPTER 1. INTRODUCTION 5
1.1.2 Beam loss in linacs
The main concern in all high-power linac projects aiming at beams in the megawatt range is the main-
tainability of the machine. Beam loss leads to activation of accelerator components and may require long
“cool-down” periods before hands-on repairs become possible. The currently accepted limit for beam
loss that still allows hands-on maintenance is 1W/m and stems from the experience at LANSCE (see
Fig. 1.5)1. Even though this limit was already achieved by an accelerator that was conceived in the 70’s
one has to keep in mind that it took over 10 years to reach full beam power (with protons) under the con-
dition of assuring hands-on maintenance for large parts of the machine. Until today only 80kW of H−
can be accelerated at LANSCE and then compressed by the PSR. This means that low-loss acceleration
in high-power proton linacs remains a “hot topic” and especially so for accelerators using H−beams
which suffer from the additional difficulties of: a) ionisation losses throughout the accelerator (e.g. via
rest gas in the beam pipe, magnetic fields, or blackbody radiation of the beam pipe), b) generally worse
beam quality out of H−sources when compared to proton sources, c) deterioration of beam quality by the
use of low-energy beam choppers, and d) injection into subsequent accumulator and compressor rings.
0.1
1
10
100
0 500 1000 1500 2000
average loss current [nA]
proton energy [MeV]
1 W/m
measured LANSCE losses
estimated SNS losses
Figure 1.5: General beam loss budget for 1W/m, measured losses at LANSCE, and estimated losses for
SNS [11], [12]
With envisaged beam powers of up to 10MW the loss limit of 1W/m implies the need to control beam
losses to a level of 10−6−10−7/m in order to avoid performance limitations due to machine activation.
This poses not only a challenge to theoretical predictions of beam losses but also to the simulation
tools, used to cross-check and quantify theoretical predictions. This challenge led to the development of
multi-particle codes like IMPACT [13] which use parallelised space-charge routines. With these codes it
became possible to routinely simulate 106particles and to perform detailed halo studies with up to 108
particles within reasonable time scales. On the other hand the imperative for low-loss operation in many
linac projects (APT, SNS, JPARC [14], CONCERT, TRASCO [15], ESS, SPL, etc; see also Section 1.3)
led to a significant increase in the understanding of beam halo formation which is nowadays considered
to be the dominant loss mechanism in high-intensity hadron linacs.
1the number 1W/m assumes that the majority of the beam losses takes place inside of quadrupoles mainly consisting of
iron, which provides a natural shielding for the radiation from the activated beam pipe.
6 1.2. HIGH-INTENSITY LINACS VERSUS OTHER ACCELERATOR TYPES
Various mechanisms can contribute to the development of a low-density particle halo which surrounds
the core and eventually results in beam loss. The most prominent are: a) parametric 2:1 resonances
between the oscillations of a mismatched beam core and the movements of single particles, b) envelope-
lattice resonances which occur between the beam envelopes and the elements of a periodic focusing
structure, c) intra-beam scattering, and d) coherent space-charge coupling resonances. The envelope-
lattice resonances can yield rapid r.m.s. emittance growth and halo development but are easily avoided
by keeping the zero-current phase advance per period in all three planes below 90◦. Intra-beam scattering
may be of importance in the Low-Energy Beam Transport (LEBT) section, where the ionisation of gas
in the beam pipe is used to compensate the high space-charge forces at low energy. At higher energies,
where the vacuum in the beam pipes is kept at much lower values than in the LEBT, this process is
considered to be of minor importance (see e.g. [16], [17]). It will be shown in Chapter 6 that space-
charge coupling resonances [18] are responsible for emittance exchange between the transverse and the
longitudinal plane of the beam but that they do not (by themselves) contribute to the development of beam
halo. This leaves the parametric resonances as the most important halo mechanism in high-intensity
linacs.
1.2 High-intensity linacs versus other accelerator types
High-intensity linacs are not the only type of accelerator that is suited to produce high beam power.
Promising candidates are cyclotrons and Rapid Cycling Synchrotrons (RCS). Another machine type that
has been studied in recent years is the Fixed Field Alternating Gradient (FFAG) accelerator. So far,
however, only small scale prototypes have been built.
The type of accelerator that is chosen for a particular high-power application depends to a large degree on
the required beam pulse structure. It was already mentioned that short pulse operation (with pulses in the
range of µs) necessitates the use of an accumulator ring. If the single bunches out of the accumulator ring
need to be very short (ns range) then an additional compressor ring is needed to reduce the bunch length.
A second decisive criterion for or against a certain accelerator type is the required output energy. For low-
energy (≤500 MeV) high-power applications, cyclotrons appear as the most economical solution, while
for high-energy (≥10 GeV) high-power applications, RCSs would clearly be the cheapest solution. In
the following we compare the characteristics of a high-power linac with cyclotrons and RCSs and we
review the arguments for a linac-based proton driver in case of the SPL project at CERN.
1.2.1 Requirements for a CERN-based proton driver
The most suitable machine type for a high-power proton driver at CERN needs to be matched to the
requirements of potential high-power users at CERN. Presently, the most likely high-power proton
applications to be located at CERN in the future are EURISOL and/or various types of neutrino facilities
which are detailed in the following (compare also [2]). Apart from the high-power applications the SPL
would also be beneficial for direct injection into the CERN Proton Synchrotron (PS), by-passing the PS
booster (PSB), which is presently used to produce proton beams of 1.3GeV. An increase of the injection
energy together with the smaller emittances which can be obtained from a linac would result in higher
brightness beams out of the PS which in turn would improve the beam quality out of the subsequent Super
Proton Synchrotron (SPS) and the Large Hadron Collider (LHC) which is presently under construction
at CERN. Figure 1.6 shows a diagram of the accelerator chain at CERN.
CHAPTER 1. INTRODUCTION 7
Figure 1.6: Accelerator chain at CERN
8 1.2. HIGH-INTENSITY LINACS VERSUS OTHER ACCELERATOR TYPES
EURISOL
EURISOL is a study for a next generation radioactive ion beam facility which produces a wide range
of exotic ions for users in the areas of nuclear physics, nuclear astrophysics, and material sciences.
The facility would provide radioactive ion beams with intensities which are on average three orders of
magnitude larger than current ISOL installations. Another recently discovered capability of EURISOL
is the so-called beta-beam facility, which would make EURISOL an ideal partner user of a proton driver
that delivers beam to a “super-beam” target (more details on beta-beams and super-beams in the follwing
section). The “green field” EURISOL study foresees a continuous (CW) beam with 5MW beam power
at 1–2 GeV. The continuous operation as well as the target energy are chosen to minimise the impact
of thermal stresses in the target. The proton beam will then be shared between one high-power target
(max. 5MW) and up to three low-power targets (<100 kW each). In terms of target stability and
lifetime all targets would benefit from continuous beam operation which means that a pulsed beam from
the SPL is certainly not the optimum choice. However, in order to mitigate the effects of thermal shocks
and shock-waves created by a pulsed beam, the targets can be heated. Presently the minimum pulse
length which is still considered suitable for the operation of EURISOL type targets is estimated to be
in the range of ms. First studies on using pulsed beams (at energies above 1–2GeV) from the SPL for
EURISOL targets have recently started and it seems likely that a satisfactory operation can be achieved
with pulsed operation [19] even if the lifetime of the targets will be reduced.
Neutrino facilities
In recent years neutrino physics has emerged as a new potential user of high-power accelerators and
its supporters argue that the physics case is independent of the explorations at high-energy colliders
[20, 21]. For neutrino facilities at CERN one has to distinguish three types of possible facilities which
can be characterised as follows (see also [2]):
i) super-beam: 4MW proton beam + accumulator ring + pion production target,
ii) beta-beam: ≈200 kW proton beam + ISOL type target + (existing) CERN Proton Synchrotron
(PS) + (existing) CERN Super Proton Synchrotron (SPS) + decay ring,
iii) neutrino factory: 4MW proton beam + accumulator and compressor ring + pion production target
+ muon cooling and capture channel + muon acceleration + muon decay ring.
All scenarios aim to study the topic of neutrino oscillations, which is currently being discussed as one
of the major candidates for future high-energy physics studies at CERN and/or elsewhere in the world.
The super-beam and neutrino factory schemes are based on a high-power (4MW) proton beam hitting
a pion-producing target. The pions decay within a few tens of metres into muons which then decay
into neutrinos. In case of the neutrino factory the muons are captured and accelerated to energies of
up to 20–50 GeV. Due to their short mean lifetime (≈2.2ms at rest) the muons must be accelerated
quickly to relativistic speeds. The decay then takes place in a dedicated decay ring with long straight
sections pointing to two detectors: one “near-detector” at a distance of several hundred kilometres and
one “far-detector” at a distance of several thousand kilometres.
While the neutrino factory scheme offers the ultimate potential for neutrino physics, a combination of
the super-beam and beta-beam scheme in the same energy range appears as a possible viable alterna-
tive. The beta-beam scheme would make use of the CERN infrastructure (PS and SPS) and it would use
CHAPTER 1. INTRODUCTION 9
the same target technology needed for EURISOL. The optimum energy for a CERN-based beta-beam
plus super-beam facility is estimated to be around 3.5GeV [22]. Simulations for pion production tar-
gets indicate an optimum energy range for the proton beam of 5–10GeV, promising up to 50% higher
transmission through an assumed muon front-end than at energies of 1–2GeV [23, 24]. However, since
these simulations rely on pion production models which are not yet experimentally verified it would be
premature at this point in time to establish an optimum proton driver energy.
All three neutrino schemes can be based on a pulsed linac with a time structure which is dictated by the
installations after the target. In the most challenging scheme, the neutrino factory, very short pulses in the
order of µs are needed in combination with ultra short bunches in the ns range. The restriction in pulse
length comes from the fact that in the final stage of all neutrino factory schemes, a muon beam decays
in a dedicated ring into neutrinos. Assuming a circumference of ≈2km for such a ring, the length of
the muon pulse that can be injected within one turn is ≈6µs. The short bunches are preferred in order
to reduce the energy spread of the particles that are produced by the MW proton beam impinging on
the pion production target. A small energy spread facilitates the task of capturing the pion beam (which
quickly decays into a muon beam) and to accelerate the muons to energies between 20 and 50GeV. The
decay time of muons at 50GeV is in the order of a few ms which imposes a timing restriction on the
repetition rate with which one can re-fill the muon decay ring. This means that the repetition rate should
be somewhere between 10 and 100Hz. Considering different repetition rates while assuming the same
average beam power one finds that the lowest repetition rate is limited by the space-charge forces of
the accumulated beam in the accumulator/compressor rings. The highest repetition rate is limited by
an increased power consumption of the accelerator components between the pion target and the muon
decay ring. (This comes from the fact that for every beam pulse one has to add a certain time needed to
fill all accelerating cavities with energy. For high repetition rates with relatively low currents the power
consumption of the whole facility would be dominated by the power needed to fill the cavities before
the arrival of the actual beam pulse.) The time structure of µs pulses and ns bunches can be achieved by
combining an H−linac with an accumulator ring (see Section 1.1.1) and a compressor ring [25].
The synergy
Forthe time being, a pulsed beam at 3.5 GeVseems a viable compromise between the needs of EURISOL
and possible neutrino facilities. At this energy both users can be supplied with high-power beams in a
time-sharing operation mode. Another advantage of 3.5GeV is that one can profit from an increased
injection energy into the CERN PS, lowering the space-charge tune shift at injection and thus removing
the present intensity limitation. While EURISOL(including beta-beams) could be operated with a proton
front-end it is mandatory to foresee H−operation to drive a super-beam facility or a neutrino factory.
The high-power long-pulse operation at or above 1GeV needed for EURISOL can only be obtained
from a linac and not from any other type of high-power accelerator as we will show in the following.
Furthermore, a linac offers the unique opportunity to share the cost of the proton driver between two
high-power users, both of which will certainly not operate for 12 months a year (due to maintenance,
target exchanges, etc).
1.2.2 High-power cyclotrons
Average beam power in the MW range has already been achieved with cyclotrons, e.g. with the separated
sector cyclotron at the Paul Scherrer Institute in Switzerland PSI [26]. A cyclotron makes use of the
magnetic force on moving charges (F=q[v×B]) to bend particles into a circular path. The magnetic
10 1.2. HIGH-INTENSITY LINACS VERSUS OTHER ACCELERATOR TYPES
field is supplied by large dipole magnets above and below the acceleration path (see Fig. 1.7). Between
the two “dees” of magnetic field region an oscillating electric field accelerates the particles. After each
acceleration step the radius of the circular path is increased due to the higher particle energy until finally
the beam is extracted.
Cyclotrons are generally used in Continuous Wave (CW) operation with currents up to the order of mA.
They are uneconomical for H−acceleration above energies of 50 to 100 MeV because due to the high
magnetic fields the extra electron of the H−particles will be stripped. This means that very large magnets
with low fields would be necessary to avoid H−stripping. A general limitation for all particle species is
the maximum energy which depends on the strength of the magnetic field and the maximum radius of the
machine. So far proton energies in the order of ≈500MeV have been realised with chains of cyclotrons
but it seems unlikely that future machines will reach significantly higher values. For applications that
need long pulses or CW beams of protons at low or medium energies (≤500 MeV), cyclotrons are a
viable alternative to linacs, especially because of the compact size and because the RF power is used
much more efficiently.
extracted beam
RF field
injection magnetic field
S S
N N
Figure 1.7: Cyclotron principle: (left) top view, (right) side view
1.2.3 Rapid cycling synchrotrons
The second type of circular accelerator which has already produced high-power beams is the rapid cy-
cling synchrotron (RCS). In a synchrotron the beam is accelerated by one or more RF cavities, which
can be matched in frequency to the revolution frequency of the particles. Contrary to the cyclotron the
beam is kept on a constant-radius orbit by increasing the magnetic field in the dipoles, used to bend the
beam, in synchronism with the increasing beam energy. Hence the name synchrotron. Within typically
several thousand turns the beam is accelerated to its full energy and can be extracted. For an RCS the
whole acceleration cycle only takes a few ms, which means one can have many cycles per second and
CHAPTER 1. INTRODUCTION 11
thus obtain a high average beam power. A review of existing high-power synchrotrons [27] in Table 1.1
shows that so far beam powers close to the MW range have only been achieved in high energy machines
with low cycling rates. At lower beam energy the neutron spallation source ISIS remains since many
years the RCS with the highest beam power (0.16MW).
Table 1.1: Beam parameters of existing and proposed proton synchrotrons
machine rep. rate [Hz] energy [GeV] power [MW]
RAL ISIS 50 0.8 0.16
BNL AGS 0.5 24 0.13
Fermilab MiniBooNE 7.5 8 0.05
Fermilab NuMI 0.5 120 0.3
CERN CNGS 0.17 400 0.5
Keeping the uncontrolled losses below the 1W/m limit is one of the major design challenges for RCS
accelerators. For example, a ring with 200 m circumference handling a 1MW beam can accept a total
fractional beam loss of ≈10−4at its top energy (or 10−3at injection energy assuming a typical energy
multiplication factor of ≈10 in a synchrotron). In existing machines losses of around 10% or more
occur mostly at injection. They are related to injection losses due to limited longitudinal acceptance (as
explained in Section 1.1.1), a high space-charge tune shift during the accumulation phase, premature H−
stripping and injection foil scattering, magnet field errors and misalignments, various beam instabilities
(e.g. head-tail instability, coupled bunch instability, negative mass and microwave instability, electron
clouds, see also [28]), and accidental beam losses due to malfunctioning of single elements in the ac-
celerator chain. The employment of low-energy beam choppers should help to drastically reduce the
injection losses but with rising beam power requirements the space-charge forces at injection will rise
and limit the current density in the ring. The losses during acceleration have to be controlled by dedicated
beam collimation sections which scrape off the outermost particles before they get lost on the beam pipe.
One of the main cost drivers in an RCS are the main power converters for the fast cycling magnets,
which become more expensive with rising repetition rates. Slower repetition rates, on the other hand,
mean that the beam intensity at injection has to be very high in order to produce beams in the MW range.
Although an RCS is generally considered to be more economic than a linac the beam dynamics and the
technology becomes very challenging if beam energies of only a few GeV in combination with high-
intensity beams are required. In the energy range of a few GeV existing machines, like ISIS are still an
order of magnitude short of reaching MW beam power.
1.2.4 Accelerator choice for a CERN-based proton driver
We have already seen that the choice of a particular type of accelerator for a certain high-power proton
application does not only depend on the beam power. The possible energy range as well as the needed
time structure have to be taken into account along with possibly existing parts of an infrastructure that
can be reused for the new facility.
The initial reason to choose a linear accelerator as proton driver for future high-intensity applications at
CERNwas triggered by the idea to re-use decommissioned RF equipment from the Large Electron Proton
(LEP) Collider at CERN (compare also Section 7.3). The first idea was to put the super conducting (SC)
LEP RF cavities in a straight line and to add a low-energy front-end. During subsequent design revisions
12 1.2. HIGH-INTENSITY LINACS VERSUS OTHER ACCELERATOR TYPES
the actual percentage of recuperated material was drastically reduced and as a consequence also other
accelerator types were considered as alternatives. The cyclotron idea was discarded because of its limited
beam energy potential (especially for the acceleration of H−) and the difficulties with high beam currents
in such a machine. A rapid cycling synchrotron would be the most viable alternative, assuming that the
stability problems for high-intensity beams at injection can be solved and that the uncontrolled losses
can be kept below 1W/m. Furthermore it would be extremely challenging to produce a MW beam at
relatively low energies of only a few GeV which would be suitable for EURISOL.
In comparison to a synchrotron a linac offers a simpler beam dynamics which can be simulated in great
detail with existing codes. In synchrotrons it is still difficult to simulate a complete acceleration cycle,
and it is virtually impossible to make hundreds of simulations with different sets of statistical errors in
order to test the machine lattice. This makes it difficult to judge wether new RCS designs can fulfil the
need for low losses, which is required to ensure hands-on-maintenance. Another advantage of linacs is
that there is a certain freedom to adapt the time structure of the pulses to the needs of various users. It is
possible to deliver longer pulses with lower average currents while maintaining the same average beam
power. In the case of the SPL at CERN, pulses in the ms range for EURISOL are extracted directly from
the linac, while the more demanding time structure for neutrino applications is produced with a separate
accumulator and compressor ring. This means that two completely different user requirements can be
fulfilled with one proton front-end. A linac can deliver beam to several users within the same pulse
simply by using a low-energy beam chopper to produce a gap (≈0.1ms) in the bunch train, long enough
to switch the high-power beam between different beam lines. This is important for instance in the case
of EURISOL targets in order to avoid thermal stresses caused by the cooling of the target between beam
pulses. Another option which still has to be verified experimentally is to share the beam within each
pulse by using partial laser stripping at the linac output to convert a fraction of the beam from H−to H0.
The beams can then be separated in a bending magnet and diverted to the respective users.
In the context of CERN a linac-based proton driver like the SPL offers the possibility for a staged ap-
proach towards neutrino physics and nuclear physics (EURISOL):
i) In the first stage the SPL would deliver a low-power beam (e.g. 0.5MW) to a EURISOL / beta-
beam facility. The targets for this facility basically exist or can be derived from the ISOLDE
facility at CERN. In this stage it is not even necessary to have a high-duty H−ion source but one
can use existing proton sources in the front-end. Gradually one can then increase the beam power
and start the testing of high-power targets first for EURISOL and then for a super-beam facility.
ii) Once the full beam power is reached (which will take several years) and once there is enough
confidence in the short-pulse (µs) super-beam target technology, one can add an accumulator ring
and use an H−beam for charge-exchange injection into the ring. At this point one can supply
the EURISOL low- and high-power targets, along with the low-power beta-beam targets plus the
high-power super-beam target. At the same time one can start the prototyping and testing of the
short-pulse neutrino factory target which must be able to withstand ns bunches of extremely high
instantaneous power.
iii) The addition of a compressor ring now completes the installation to a complete neutrino factory
proton driver and by this time it should also be clear if the energy of 3.5GeV is sufficiently high.
If necessary one could now still increase the final linac energy (and possibly also the beam power)
to meet the required values. In parallel one should now also have enough confidence in neutrino
factory targets and the necessary key technologies for the construction of a complete neutrino
factory.
CHAPTER 1. INTRODUCTION 13
During all stages and even during the ramp-up in beam power at the very beginning, can the SPL deliver
beam to the users including the regular CERN physics program which includes a luminosity upgrade
for the LHC. Upgrades in terms of beam energy or beam power are achieved simply by adding more
accelerating cavities at the end of the linac or by lengthening the beam pulses. This feature is important
for the CERN scenario since it is not yet clear if 3.5GeV will be sufficient for a neutrino factory. In the
energy range of a few GeV and for beam powers of several MW (or higher) a linac offers more flexibility
than circular accelerators. At this point in time FFAGs are not yet considered as a viable alternative, even
though the technology has the potential to produce high-power beams. Future R&D work will have to
show if FFAGs can indeed fulfil this potential.
1.3 High-power linac studies and the goal of this thesis
It was mentioned in the previous sections that the understanding of beam dynamics and halo develop-
ment was boosted by the large number of high-power linac studies which have been undertaken in the
last two decades (see Table 1.2). In order to simplify the problem, most of the analytical studies were
performed using simplified accelerator lattices and beam geometries. One usually neglects acceleration
and in the first stage one often uses two-dimensional beams, assuming un-bunched long cylinders of
charged particles. To further simplify the analytic treatment all periodic transverse focusing forces are
often averaged (smoothed) over the focusing periods. Without these simplifications it is in many cases
impossible to study beam characteristics and one has to accept a drastic change of the original lattice.
In the following chapter we will outline the analytic treatment of 3D beams with space-charge, which is
used in most studies concerned with halo development. We will stress, which simplifications are made
in order to derive the commonly used formulas.
The analytic findings of the last decades were usually verified with numerical multi-particle simulations.
However, in most of these simulations the same simplifications as in the analytic treatment were used,
meaning that 2D beams in continuous focusing lattices with artificially created particle distribution were
used to test the analytic theories. In many cases the studies stopped at this point and the theories were
not tested on realistic accelerator lattices including periodic focussing, acceleration, and transitions be-
tween different focussing lattices. This thesis was motivated by a different, maybe naive but entirely
practical approach: during the design process of the SPL I [1] the beam dynamics simulations exhibited
certain features which could not easily be related to well-known phenomena. In subsequent discussions
with I. Hofmann, K. Bongardt and other colleagues, the relevant theory was revisited and amended if
necessary. The first feature was the observation of emittance exchange between the transverse and lon-
gitudinal planes for certain settings of the phase advance. This phenomenon could be identified as a
coherent space-charge resonance as described in [18]. The corresponding theory and the simulation
results of this study are reported in Chapter 6.
After a number of tests with mismatched beams, the mismatch modes were used to systematically study
halo development in the SPL (see Chapter 4). In an advanced stage of the design, when the SPL was
simulated with statistical errors, features similar to the ones found for the mismatch modes were observed
and were then related to the same theory (the particle-core model) that describes the basic effects of initial
mismatch (see Chapter 5).
The work presented in this thesis is entirely based on phenomena that can be observed for realistic 3D
beams in realistic linac lattices. These phenomena are then related to simplified analytic expressions
(where possible), which will be derived in the following.
14 1.3. HIGH-POWER LINAC STUDIES AND THE GOAL OF THIS THESIS
Table 1.2: High-power, high-energy linac studies and projects worldwide
machine particle energy beam power rep. rate peak current
species [GeV] [MW] [Hz] [mA]
existing:
LANSCE (USA) p/H−0.8 1/0.08 16.5/9.5 100/20
SNS* (USA) H−1 1.4 60 38
under construction
JPARC†H−0.6 0.23 25 30
proposal
SPL II (CERN, Switzerland) H−3.5 5 50 64
ESS‡(Europe) H−1.33 10 50 114
CONCERT‡(Europe) H−, p 1.33 25 50 114
KOMAC (Korea) p 1 20 CW 20
TRASCO‡(Italy) p 1 30 CW 30
EURISOL (Europe) p 1 5 CW 5
APT‡(USA) p 1 100 CW 100
FNAL (USA) H−, p 8 2 10 25
XADS (Europe) p 0.6 6 CW 10
IFMIF D+0.04 2x5 CW 2x125
* quoted are the goals, †if the final upgrade is approved, ‡suspended study
CHAPTER 2. BASIC EQUATIONS 15
2. Basic equations
Charged particle beams have velocity components which are perpendicular to the direction of propa-
gation. These components arise for instance in the particle source where particles are extracted from
a plasma with random thermal motions. After extraction space-charge forces will push particles away
from the axis and without transverse focusing system the beam will soon start to diverge and be lost on
the beam pipe. The acceleration of particles to energies beyond a few MeV is usually accomplished by
radio frequency (RF) electric fields which partition the longitudinal axis in accelerating and decelerating
areas, changing in time with the frequency of the RF system. For successful acceleration the particles
must be confined longitudinally to the accelerating areas (RF buckets), meaning that the beam must be
bunched in the longitudinal plane. Since the particles inside a bunch have different longitudinal mo-
menta, a longitudinal focusing system must be put into place to complement the transverse focusing of
the beam. In modern accelerators one often uses a periodic focusing lattice of alternating quadrupoles for
transverse beam confinement, while longitudinal confinement is achieved by using off-crest acceleration,
which ensures that faster particles are decelerated and that slower particles are accelerated with respect
to the central (synchronous) particle.
In order to calculate the beam evolution and the stability properties of the complete ensemble of particles,
it is useful to work with equations describing the r.m.s. beam envelopes. For effects that go beyond the
regular oscillations of the beam one often uses the smoothed form of these equations, where one averages
over the regular beam oscillations to study superimposed oscillations, e.g. due to mismatch.
In later chapters we will often use expressions that come from the smoothed approximation of Sacherers
envelope equations [29]. Therefore this chapter will provide a derivation of these equations together with
a set of useful formulae which are tailored for later employment.
2.1 3D envelope equations with space-charge
The smoothed envelope equations average over the rapid flutter of the beam envelopes which are caused
by the focusing and defocusing forces, and provide relations for averaged r.m.s. beam sizes. The three
main forces acting on the beam shall be considered, which are either external (quadrupoles and RF
gaps), or internal (space-charge force) caused by the charged particles of the beam. For this purpose
one assumes a uniform particle distribution in the first place, and can then, as Sacherer showed in [29],
generalise the equations for various distributions without major changes.
Before deriving the actual envelope equations one needs to define some basic properties of a generic
particle distribution. The variables used for the following derivations are:
•sis the independent variable denoting the longitudinal position on the beam axis,
•x,y, and zare the s-dependent single particle coordinates within a bunch at position s, which refer
to the particle positions relative to the bunch center,
•ax,ay, and bare the s-dependent r.m.s. beam envelopes,
•rx,ry, and rzare the semi-axes of an ellipsoidal bunch of particles, and
16 2.1. 3D ENVELOPE EQUATIONS WITH SPACE-CHARGE
•ˆaand ˆ
bare the matched r.m.s. beam envelopes with space-charge which are in a simplified case
(the smooth approximation) assumed constant over one focusing period, ˆa0and ˆ
b0are used for the
zero-current case.
Assuming linear focusing forces all particle trajectories lie on ellipses in the the x/x0plane which is
called trace-space and which represents atwo-dimensional projection of the six-dimensional beam. Often
one also refers to the x/x0plane as the unnormalized phase space projection (the x/(px/mc)plane
would be the normalized phase space projection). In the following we will simply refer to “phase space”,
regardless of the normalization of the vertical axis (see Fig. 2.1). For a matched beam the iso-density
contours in phase space are concentric and geometrically similar to the trajectory ellipses. This statement
remains valid even for mismatched beams as long as the focusing system remains linear. It is thus
convenient to describe the properties of a linac beam with parameters that relate to r.m.s. ellipses in all
three phase space planes (x/x0,y/y0,z/z0). For this purpose one can use the Courant-Snyder (or Twiss)
parameters (α,β,γ) of Eq. (2.1) to describe the contour of a generic ellipse as shown in Fig. 2.1, which
is centred at the origin and rotated by a certain angle.
slope
PSfrag replacements
xmax =√εxβx
x0
max =√εxγx
slope =−
αx
βx
x
x0
Figure 2.1: Generic r.m.s. ellipse for particle distributions in the x/x0phase space
γxx2+ 2αxxx0+βxx02=εx(2.1)
The Twiss parameters in each plane αi,βi, and γiare linked by
γiβi−α2
i= 1 i=x,y,z (2.2)
and the area of the ellipse in each plane is given by Ai=πεiwhich introduces the definition of the
un-normalised r.m.s. emittance εiof an arbitrary particle distribution in phase space. When designing a
linear accelerator one usually refers to the normalised emittance which remains constant during accel-
eration (provided that there is no emittance growth). It is calculated by multiplying the un-normalised
emittance with the relativistic factors (βγ). In the following, however, we will continue to use the un-
normalised r.m.s. emittance which relates to the physical properties of the beam.
CHAPTER 2. BASIC EQUATIONS 17
Averaging over the particle distribution one can express the r.m.s. quantities of the beam which are only
dependent on the longitudinal position s.
a2
x=x2=βxεxa02
x=x02=γxεxxx0=−αxεxεx=qx2x02−xx02(2.3)
The same definitions apply in the y-plane for ay,εyand in the z-plane for b,εz. Using Eq. (2.3) one can
derive the envelope equations starting from the transverse equation of motion for a single particle:
x00 +κ(s)x= 0 transverse equation of motion,
Hill’s equation (2.4)
Hill’s equation is a second-order linear differential equation with κ(s)denoting the linear external focus-
ing forces. In hadron linacs, one is usually interested in beam transport from low to high energy, which
means one has to consider the space-charge force term Fsc, describing the forces between the charged
particles which is most pronounced at low energy:
x00 +κ(s)x−Fsc(s) = 0 transverse equation of motion
with space-charge (2.5)
We note that in Eq. (2.5) the space-charge forces (in the term Fsc) acting on the single particle are also
dependent on the transverse position of the particle with respect to the bunch centre. If we want to make
the transition from the single particle equation to an envelope equation describing the r.m.s. quantities
of all particles we will therefore, at some stage, need to average over Fsc in order to obtain a purely
s-dependent quantity.
In order to derive the envelope equations one starts by averaging over the particle positions and the
second moments of the distribution:
ax=qx2a0
x=xx0
ax(2.6)
a00
x=xx00
ax+x02
ax−xx02
a3
x=xx00
ax+x2x02
a3
x−xx02
a3
x(2.7)
Using the definition of the r.m.s. quatities in Eq. (2.3) one can simplify Eq. (2.7) to
a00
x=xx00
ax+ε2
x
a3
x(2.8)
Differentiating xx0and using the equation of motion (2.5) yields
xx00=x02+xx00 =x02−x2κ(s) + xFsc(s)(2.9)
or
xx00 =−x2κ(s) + xFsc(s)(2.10)
Replacing the xx00 term in Eq. (2.8) with Eq. (2.10) one arrives at the r.m.s. envelope equation which is
similar to the equation of motion (2.5) but now contains an additional emittance term. After repeating
18 2.1. 3D ENVELOPE EQUATIONS WITH SPACE-CHARGE
the same steps in the yand zplane one obtains:
a00
x+axκx(s)−xFsc,x(s)
ax−ε2
x
a3
x= 0
a00
y+ayκy(s)−yFsc,y(s)
ay−ε2
y
a3
y= 0
b00 +bκz(s)−zFsc,z(s)
b−ε2
z
b3= 0
r.m.s. envelope equations with
space-charge (2.11)
These equations were first derived in 1959 by Kapchinskiy and Vladimirskiy [30] for a continuous (un-
bunched) beam and in this context they are often referred to as the KV-envelope equations.
The term κ(s)in the envelope equation takes on different forms depending on the type of focusing lattice.
Generally one considers periodic focusing channels with period length LPso that κ(s) = κ(s+LP).
The emittance term in Eq. (2.11) is negative and acts as a defocusing force on the r.m.s. beam size.
Onecan simplify the envelope equations (2.11) by replacing the averaged space-charge terms inEq. (2.11)
using the results of Sacherer [29] and Lapostolle [31]. Sacherer showed that these terms depend very lit-
tle on the actual particle distribution (uniform, Gaussian, hollow, parabolic), making the r.m.s. envelope
equations a widely applicable tool to model the r.m.s. beam quantities.
In the next step we replace the space-charge terms by expressions relating to the particle mass and charge,
their velocity and the aspect ratio of the bunches.
Lapostolle defines the electric potential for an ellipsoidal bunch in free space assuming similar dimen-
sions for the semi-axes (rx,ry,rz) of the bunch. The electric field components can then be written as
[32]:
Ex,sc(s) = 3Iλ[1 −f(s)]x
4πε0c(rx+ry)rzrx≈3Iλ[1 −f(s)]x
8πε0crxryrz
Ey,sc(s) = 3Iλ[1 −f(s)]y
4πε0c(rx+ry)rzry≈3Iλ[1 −f(s)]y
8πε0crxryrz(2.12)
Ez,sc(s) = 3Iλf(s)z
4πε0crxryrz
where Iis the averaged current over one RF period, λis the (free space) RF wavelength, cis the speed
of light and ε0is the permittivity of free space. The ellipsoidal form factor f(s)depends on the aspect
ratio of the bunch (which changes in a periodic focusing channel). Even though it will not be needed in
the following we cite here Lapostolle’s approximation [31] for f(s)
f(s)≈√axay
3γb (2.13)
In a three-dimensional uniform ellipsoid one can relate the semi-axes of the bunch to the r.m.s. value
with (see [31]):
ri=√5ai(2.14)
The space-charge force terms can be replaced by their field components (see Appendix A.1)
Fi,sc(s) = qEi,sc(s)
β2γ3mc2(2.15)
CHAPTER 2. BASIC EQUATIONS 19
and one can express the space-charge terms in Eq. (2.11) as:
xFx,sc(s)
ax=K3[1 −f(s)]
2ayb
yFy,sc(s)
ay=K3[1 −f(s)]
2axb
zFz,sc(s)
b=K3f(s)
axay(2.16)
with K3being a 3-D space-charge parameter defined as:
K3=3qIλ
20√5πε0mc3β2γ3(2.17)
Using these expressions we can now write the envelope equations as:
a00
x+axκx(s)−K3[1 −f(s)]
2ayb−ε2
x
a3
x= 0
a00
y+ayκy(s)−K3[1 −f(s)]
2axb−ε2
y
a3
y= 0
b00 +bκz(s)−K3f(s)
axay−ε2
z
b3= 0
3D envelope equations with
space-charge (2.18)
In the following sections we will introduce some approximations to replace κ(s)by averaged wave-
numbers kfor a periodic quadrupole channel with RF cavities.
2.2 The principle of smooth approximation
The smooth approximation is often used to simplify the envelope equations for practical usage. The
idea is to average the focusing forces over one full focusing period of length Lp. For this purpose the
s-dependent focusing term κ(s)is replaced by the smoothed (averaged over one period) wave numbers
kwhich are related to the phase advance per period σby
κx(s)−→ k2
x,0 =σx,0
LP2
κy(s)−→ k2
y,0 =σy,0
LP2
κz(s)−→ k2
z,0 =σz,0
LP2
smooth approximation w/o
space-charge (2.19)
When considering beams with space-charge the smooth approximation can be used in the following way:
κx(s)−K3[1 −f(s)]
2axayb−→ k2
x=σx
LP2
κy(s)−K3[1 −f(s)]
2axayb−→ k2
y=σy
LP2
κz(s)−K3f(s)
axayb−→ k2
z=σz
LP2
smooth approximation with
space-charge (2.20)
20 2.3. QUADRUPOLE AND RF FOCUSING TERMS
The last step represents a strong simplification since it completely removes the dependence of the space-
charge terms on the beam envelopes. From practical experience one finds that for weak or moderate
space-charge the equations are still reasonably precise, while for very strong space-charge forces this
approximation becomes inadequate.
In the next step one can relate the transverse and longitudinal wave numbers to actual properties of the
focusing periods such as magnetic gradients, electric fields, period length, etc. For the transverse plane
this can be done by comparing the transfer matrix of the thin lens approximation with a general periodic
solution in matrix form (see Appendix A.2).
In the presence of RF cavities the situation becomes more complex since they not only provide accel-
eration and longitudinal focusing but also transverse defocusing. The transverse effects of RF cavities
are again evaluated using the transfer matrix method as in the transverse plane. For the longitudinal ef-
fects one simplifies the longitudinal equation of motion without space-charge to specify the longitudinal
wave-number kl,0 (see Appendix A.4).
2.3 Quadrupole and RF focusing terms
Using the smooth approximation together with the thin lens approximations for quadrupoles and RF
cavities one can replace the s-dependent focusing κ(s)in the envelope equations (2.11).
In the longitudinal plane the longitudinal zero-current wave number kl,0 can be derived as (see Ap-
pendix A.4)
κz,0(s)→k2
l,0 =2πqE0Tsin(−φs)
mc2λβ3γ3
smooth approximation for
longitudinal focusing (2.21)
For the quadrupole focusing forces one has to specify the type of focusing lattice. Here we will use a
standard FODOlattice, consisting of focusing quadrupole, drift, defocusing quadrupole, and drift without
acceleration to obtain (compare Appendix A.3):
κx,Q(s) = κy,Q(s)−→ k2
Q=qGlQ
2mcβγ 2smooth approximation for
transverse focusing in FODO
lattices w/o RF
(2.22)
Adding the transverse defocusing from RF cavities located in the drifts, the transverse focusing constants
become (compare Appendix A.4)
κx(s) = κy(s)−→ k2
t,0
≈qGlQ
2mcβγ 2
−πqE0Tsin(−φs)
mc2λβ3γ3
=k2
Q−k2
l,0
2
smooth approximation for
transverse focusing in FODO
lattices with RF
(2.23)
CHAPTER 2. BASIC EQUATIONS 21
2.4 Smooth approximation for a FODO channel with RF
cavities
Starting with the 3D envelope equations with space-charge (same as Eq. 2.18)
a00
x+axκx(s)−K3[1 −f(s)]
2ayb−ε2
x
a3
x= 0
a00
y+ayκy(s)−K3[1 −f(s)]
2axb−ε2
y
a3
y= 0
b00 +bκz(s)−K3f(s)
axay−ε2
z
b3= 0
3D envelope equations with
space-charge (2.24)
one can use the smooth approximation [Eq. (2.20)] to remove the s-dependence of the focusing forces.
a00
x+axk2
t−ε2
x
a3
x
= 0
a00
y+ayk2
t−ε2
y
a3
y
= 0
b00 +bk2
l−ε2
z
b3= 0
smooth approximation for the
3D envelope equations with
space-charge in a FODO
lattice with RF
(2.25)
The focusing constants are defined as [see Eq. (2.20)]
k2
t=k2
t,0 −K3[1 −f(s)]
2axayb=σt
LP2
k2
l=k2
l,0 −K3f(s)
2axayb=σl
LP2
smooth approximation with
space-charge (2.26)
As stated in the previous section the zero-current constants of a FODO quadrupole lattice are given by
k2
l,0 =2πqE0Tsin(−φs)
mc2λβ3γ3k2
t,0 =qGlQ
2mcβγ 2
−k2
l,0
2(2.27)
Tocalculate the matched r.m.s.beam envelopes one simply sets the second derivations to zero and obtains
ˆax,y =rεt
kt=sεtLP
σt
ˆ
b=rεl
kl=sεlLP
σl
smooth approximation for the
matched full-current
r.m.s. beam envelopes in a
FODO lattice with RF
(2.28)
In the zero-current case the space-charge terms vanish and the solution is found as
22 2.4. SMOOTH APPROXIMATION FOR A FODO CHANNEL WITH RF CAVITIES
ˆax,y,I=0 =sεt
kt,0 =sεtLP
σt,0
ˆ
bI=0 =sεl
kl,0 =sεlLP
σl,0
smooth approximation for the
matched zero-current
r.m.s. beam envelopes in a
FODO lattice with RF
(2.29)
We note that if the beam is matched Eq. (2.25) will describe a beam with constant beam envelopes in
all three planes. Only in the case of mismatch will Eq. (2.25) describe a beam with changing r.m.s. en-
velopes. One can argue that this is a drastic simplification of the actual beam transport where a periodic
lattice keeps the beam oscillating. On the other hand, having basically removed the lattice oscillations
from the equations, one can now use them to determine the characteristics of mismatch oscillations in a
relatively simple way. In the next chapter multi-particle simulations will be used to study the effects of
mismatch and there we will plot the ratio of mismatched over matched r.m.s. beam envelopes to compare
the results with the predictions from a simplified analytical model, the basis of which was derived in this
chapter.
CHAPTER 3. MULTI-PARTICLE SIMULATIONS WITH THE IMPACT CODE 23
3. Multi-particle simulations with the
IMPACT code
In the first stage of the SPL design, the standard beam dynamics code PARMILA [33] was used to estab-
lish a first layout [34, 35]. PARMILA is used since the 1960s for the design of linacs and was still used
as the major design code for the SNS project [3]. PARMILA combines the creation of the accelerator
layout with particle tracking and thus makes it difficult to isolate specific lattice characteristics. Further-
more, when the SPL study started, PARMILA only offered a 2D space-charge routine which assumes
rotational symmetry of the beam. Another limitation of PARMILA was that only 105particles could be
used. While this number is perfectly adequate to calculate the r.m.s. quantities of a beam it is not enough
to study detailed halo dynamics (see next section). In searching for a more precise particle tracking tool,
subsequent simulations were done with IMPACT which is described in the following. IMPACT, how-
ever, was developed as a “verification code” rather than a “design code” which means that it lacks all
features that make it easy for the user to build a linac with several acceleration sections. In order to use
IMPACT for design improvements and simulations of the complete SPL linac, a number of tools had to
be developed which are described in Section 3.2.
3.1 The IMPACT code
The IMPACT [13] code was originally developed at the Los Alamos National Laboratory (LANL) by
R. Ryne and J. Qiang and is now largely maintained by J. Qiang at the Lawrence Berkeley National
Laboratory (LBNL). IMPACT stands for Integrated Map and Particle ACcelerator Tracking Code. The
design goal was to provide the accelerator community with a fast and reliable multi-particle code, capable
of using the resources of modern parallelised supercomputers. With up to several hundred processors
working in parallel it becomes possible to use large (up to 256x256x256 in routine runs) space-charge
grids together with a high number of particles (up 108in routine runs) within reasonable time frames of
typically a few hours.
One of the basic ideas in the code development was to use symplectic split-operator methods which
allow to include space-charge effects in single-particle beam dynamics codes. The space-charge forces
are then computed using time-efficient parallelised Particle In Cell (PIC) techniques which were largely
developed by the plasma physics community. The single-particle Hamiltonian can be written as:
H=Hext +Hsc (3.1)
where Hext refers to the external focusing forces of quadrupoles, RF cavities, etc. and Hsc stands for
the space-charge forces of the beam. The Hamiltonians for standard beam line elements can be derived
from standard magnetic optics and are listed in [13]. To calculate the space-charge potential, the charge
is deposited on the grid and the potential is obtained by convolving the charge density with a Green’s
function.
24 3.1. THE IMPACT CODE
Once the Hamiltonians are known, the mapping Mext and Msc can be computed, which corresponds to
Hext and Hsc. For 2nd order accuracy one can then use the following algorithm to advance the particles
M(τ) = Mext(τ/2)Msc(τ)Mext(τ/2).(3.2)
In the standard version of IMPACT τequals the longitudinal coordinate z. The symplectic split operator
method can easily be generalised to higher order but then needs more space-charge calculations per full
step. Since the space-charge calculation dominates the execution time, this option should only be used
for highly space-charge dominated beams.
A standard integration step thus involves: 1) transport of a numerical particle distribution through a half
step based on Mext, 2) solution of Poisson’s equation based on the particle positions and performance
of a space-charge kick based on Msc which only affects the particle moments, and 3) transport through
the 2nd half step. For intense space-charge this sequence can be used repeatedly on successive sections
of one beam-line element. For weak space-charge forces, one can achieve good accuracy by including
several beam line elements in one half step. The split operator method thus de-couples the rapid variation
of external focusing forces from the slowly varying space-charge fields and allows to adapt the step size
for each calculation separately.
Another feature of the IMPACT code is the use of on-axis RF field maps for all accelerating cells which
allow a precise calculation of longitudinal focusing and acceleration as well as a good approximation of
the transverse defocusing.
The justification for using large numbers of particles is illustrated in Figs. 3.1 to 3.3. A high number of
particles provides a high resolution in the outermost areas of phase space, where halo formation takes
place and where particles are lost on the beam pipe. It is quite obvious that a resolution of 10−6, which
is required for linacs with an output power in the MW range, can only be achieved with particle numbers
of at least 107or even above. In Fig. 3.1 the transverse particle density at the end of a linac is shown for
simulations with various particle numbers.
0.1
1
10
100
1000
10000
0123456789
particles per unit length
r.m.s. radii
10e5 particles
10e6 particles
10e7 particles
Figure 3.1: Radial particle density (in arbitrary units) at the end of a linac simulated with different particle
numbers, IMPACT multi-particle simulation
One can see that the simulation with 105particles shows a double-Gaussian profile, which resembles
the superposition of two Gaussian distributions. Using 107particles shows that a double-Gaussian does
not correctly describe the tails in the distribution but that the outermost particles rather follow a third
distribution type superimposed on the double-Gaussian.
CHAPTER 3. MULTI-PARTICLE SIMULATIONS WITH THE IMPACT CODE 25
Figure 3.2 shows an interesting example for the resolution that can be obtained with high numbers of
particles [36]. The graph shows the halo distribution in longitudinal phase space for a mismatched
beam. The distribution was calculated using 25 million particles on a 2563space-charge grid. Out of
this distribution 105particles were chosen randomly for the plot. In the core region (blue) the density
changes from 1 to 1/10 at the edge of the blue area. In the subsequent green, yellow, and red areas the
density changes from 1/10 to 1/102, to 1/103, and to 1/104. The figure also illustrates the need for
suitable post-processing of the simulation results which can significantly facilitate the interpretation of
the results.
Figure 3.2: Post-processed data showing the longitudinal halo development for a mismatched beam, IM-
PACT multi-particle simulation, source: R.D. Ryne
For the evaluation of statistical errors one rather uses many simulations with fewer particles in order
to estimate the average particle loss due to errors. Also here one can profit from the short simulation
times of IMPACT by using for instance 100 simulations with 105particles rather than one simulation
with 107particles. As an example Fig. 3.3 shows the losses per beam line element in the drift tube linac
of the ESS. The DTL is simulated with two different beam pipe over aperture ratios assuming a 1%
randomly distributed quadrupole gradient error. Using a high number of error runs one can make very
precise predictions on beam loss, assuming that the input distribution of the simulated lattice is known
in sufficient detail. As demonstrated in Fig. 3.3 one can for instance make predictions on possible “hot
spots” in a linac design.
Finally, Table 3.1 shows some typical simulation times for Linac4 and the SPL using a Linux PC cluster
at Rutherford and an IBM supercomputer at NERSC. Linac4 has a length of 67m and consists of 270
beam line elements while the SPL has a length of 580m and consists of 460 elements.
26 3.2. USING IMPACT
Table 3.1: Simulation time for Linac4 and SPL on a linux PC cluster†or on the NERSC IBM SP RS/6000
supercomputer*
machine space-charge grid Nparticles Nprocessors cpu time
linac4†3235·1042 25min
linac4†6431064 220min
SPL†12831078 1130 min
SPL* 64310664 40min
SPL* 643107128 70min
SPL* 128310764 430min
SPL* 1283107128 285min
Q2 Q9 Q10 Q12 Q15 Q17 Q18 Q21 Q24 Q26 Q30 Q32 Q36 Q39 Q41 Q44 Q46 Q51 Q53 Q3 Q6 Q8 Q10 Q11 Q14 Q16 Q17 Q19 Q20 Q22 Q23
1.0E−7
1.0E−6
1.0E−5
1.0E−4
1.0E−3
element
average loss
1.0% small aperture
1.0% large aperture
Q2 Q9 Q10 Q12 Q15 Q17 Q18 Q21 Q24 Q26 Q30 Q32 Q36 Q39 Q41 Q44 Q46 Q51 Q53 Q3 Q6 Q8 Q10 Q11 Q14 Q16 Q17 Q19 Q20 Q22 Q23
Figure 3.3: Average transverse losses in the ESS DTL for two different beam pipe over aperture ratios
(4.5/6.5) assuming a 1% statistical quadrupole gradient error [37], IMPACT multi-particle
simulation
3.2 Using IMPACT
In order to use IMPACT the input files have to be prepared by means of several “helper codes”. Addi-
tionally it may be necessary to fine-tune the lattice that has been prepared in a first step with an envelope
code like TRACE3D [38]. This is due to the fact that a beam that is matched with an envelope code
is not necessarily matched for the use with a multi-particle code, especially if high space-charge forces
are present. Different methods of transporting the beam through a lattice will results in slightly different
matched beams and thus one may have to re-match the initial beam (and the lattice around structure tran-
sitions) when using IMPACT. For the simulations in this thesis a number of small scripts were written
CHAPTER 3. MULTI-PARTICLE SIMULATIONS WITH THE IMPACT CODE 27
to simplify the lattice set-up and the evaluation of results from IMPACT simulations. The idea was to
keep the IMPACT code package (IMPACT, FIX3D, THETA) unchanged and to use scripts to start and
evaluate the codes of the IMPACT package. This approach has the advantage that in case of changes in
the IMPACT codes (bug fixes, implementation of new features), the scripts can be easily adapted to new
formats of the input and output files.
The code asks for an element-by-element lattice description and does not use any graphical user inter-
faces. One of the interesting features of IMPACT is for example the exact integration over the on-axis
electric fields in accelerating cavities, while most of the standard codes simplify this integration to a
drift-kick-drift approach. In order to perform this integration correctly, however, the code needs as input
the RF phase at the point when the synchronous particle enters the cavity (and the on-axis field map),
rather than just the synchronous phase and field amplitude in the centre of the cavity. The first approach
(by R.D. Ryne) to calculate this phase value was to use the fitting capabilities of the beam dynamics
code MARYLIE [39]. Later on these preparatory calculations were simplified by J. Qiang who wrote the
“THETA” code which calculates the correct phase settings for the different types of accelerating cavities
and which uses the same lattice description format as IMPACT.
To improve the calculation of matched beams for IMPACT, the 3D envelope matching code FIX3D [40]
was written (R.D. Ryne). It uses the same cavity field integration as IMPACT and thus produces a much
better beam matching than TRACE3D. FIX3D also calculates the zero-current and full-current phase
advance for the matched lattice period. This capability was then used by a script, written to calculate the
phase advance evolution for a complete linac and to change the magnet settings in the lattice in order
achieve a desired phase advance profile. This approach was used for the simulations in Section 6.1 where
a certain ratio of longitudinal to transverse full-current phase advance is needed.
Together with R.D. Ryne a transition matching system was developed using again the fitting capabilities
of MARYLIE and the FIX3D code. With this system the beam was propagated with FIX3D up to a
certain lattice point before the transition. Then the matched beam was calculated for the first period
after the transition and compared with the previously calculated beam that ended before the transition.
Using the fitting routines of MARYLIE,the lattice before the transitions was changed until the difference
between matched and propagated beam became minimal. With this approach IMPACT could be used to
simulate complete linacs consisting of several different sections [41].
To run and evaluate a large number of simulations needed for the statistical studies presented in Chap-
ter 5, several (python and bash) scripts were written and combined with fortran codes, to automatise this
process and to perform error studies with only a few small input files. The input files specify the type of
errors (quadrupole gradients, RF phases, etc.), their amplitude, their distribution type (uniform, Gaussian
with or without cut-off at a certain multiple of the r.m.s value, parabolic), whether the errors occur in
groups (e.g. all RF phases of one RF tank) or should be different for each element, and the number of
simulations. A new error set is then created for each simulation and written into a new directory, from
where the job is submitted to a linux cluster. After the completion of all runs (which is also monitored by
a script) a small Fortran code is used to read the simulation output from each directory and to evaluate
the results in terms of r.m.s. emittance growth, maximum envelope deviations, particle loss, energy and
phase jitter, etc.
28
4. Initial mismatch
4.1 Space-charge and beam stability
In the design of linacs using hadron beams for high-intensity applications one has to take into account
the effect of space-charge forces and all related beam instabilities. Some of these instabilities are known
since many years and are well documented in text books on accelerator physics (e.g. [32]). One of
the classic rules in hadron linac design is derived from perturbation studies of the envelope equations
(e.g. [42]) which show that envelope instabilities (in beams with space-charge) occur for zero-current
phase advance (σ0) values above 90◦. As a consequence, lattices for high-intensity beams are generally
designed to operate below 90◦in all three planes. Below this classic threshold beam halo and emittance
exchange are considered to be two of the most detrimental effects with respect to beam quality. The latter
is triggered by coherent core-core resonances and will be treated in in detail in Chapter 6.
The development of a diffused beam halo is mainly triggered by single particle resonances, which can
occur between single particles and either the periodicity of the lattice or the envelope oscillations of
an r.m.s. mismatched beam core. For σ0<90◦, however, only the particle-envelope (particle-core)
resonances are of real importance.
For both resonance types (core-core and particle-core) space-charge plays an important role. In case
of the single particle resonances space-charge provides a large spread of single particle tunes and so
increases the probability for single particles to enter into a resonance with the core. For core-core reso-
nances space-charge increases the width of the stop bands, which finally, for very intense beams, yield a
“sea of instability” in which no stable working point can be found for non-equipartitioned beams.
In the following sections and chapters the theoretical models for both resonances are discussed and
tested with 3D tracking simulations of realistic linac lattices (3D in real space or 6D in phase space). The
particle-core model is used to explain the basic mechanism for particle-core resonances. Then the three
eigenfrequencies for a 3D beam envelope are approximated and used to excite mismatch oscillations
for 3D tracking studies. The particle-core model is then extended to study statistical gradient errors
(Chapter 5) and to show that even these random errors yield beam halo via particle-core resonances.
Finally we consider the effect of core-core resonances and we discuss their occurrence in realistic designs
(Chapter 6).
4.2 The particle-core model
The particle-core model is a simple means to study how single particles are influenced by oscillations of
the beam core. A bunch of particles is treated as a “blob” of charge with distinct boundaries, which then
interacts with a single particle crossing the core.
CHAPTER 4. INITIAL MISMATCH 29
4.2.1 Introduction
The particle-core model described here follows the approach taken in [43] to derive maximum amplitudes
for parametric beam halo. To study the principle of halo development it is sufficient to use the simplest
description of the beam. An azimuthally symmetric continuous beam in a uniform focusing channel
is described using the smooth approximation of the envelope equations. As outlined in Chapter 2 one
averages over the external forces of a periodic channel and simplifies the space-charge calculation by
assuming a uniform spatial distribution of the particles within the core. Using the derivations presented
in Chapter 2 one can write the one-dimensional transverse equation of motion for the core-radius rcas:
r00
c+k2
0rc−ε2
r3
c−K1
rc= 0 (4.1)
where k0represents the transverse focusing forces as well as the zero-current phase advance per unit
length. K1is a 1D space-charge constant containing mass, axial velocity, charge, and the number of core
particles. Together with k0it defines the depressed phase advance per unit length (k2=k2
0−K1
r2
0). In
this case εis the
total
un-normalised emittance (as opposed to the un-normalised
r.m.s.
emittance used
in Chapter 2). For a matched beam the core radius remains constant (⇒r00
c= 0) and becomes:
r2
c=r2
0=ε
k0hu+p1 + u2iwith u=K1
2εk0
(4.2)
For a single particle, moving transversely in the field of the core, the transverse equation of motion can
be written as:
x00 +k2
0x−Fsc = 0 (4.3)
where Fsc contains the space-charge forces acting on the single particle at the transverse position x. In
contrast to the core radius rcwhich is by definition symmetric in (x, −x)and (y, −y), the single particle
particle will assume positive and negative values in the transverse plane, which is why the nomination
xwas chosen. Assuming the core as an infinitely long cylinder of uniform charge, particles experience
linear forces inside the core and non-linear forces when leaving the core area:
Fsc =(K1x/r2
c:|x|< rc
K1/x :|x| ≥ rc(4.4)
Outside the core, according to Gauss’ law, the space-charge forces acting on the particles no longer
depend on the actual size of the core. By using particles with different initial transverse positions in the
range of 0< x < nrc, the particle-core model is used to study the circumstances under which particles
can be driven to large-amplitude oscillations. In all of the following simulations the initial transverse
moments of the single particles are set to zero.
Compared to the equations derived in Chapter 2 we do not use r.m.s. quantities for calculations with the
particle-core model. This is simply due to the fact that in Eq. (4.4) we need to establish when the particle
is inside or outside the beam core, rather than when it is inside or outside the r.m.s. core radius. The
variables used in this section are:
30 4.2. THE PARTICLE-CORE MODEL
•rc=xmax =ymax (100%) core radius or core envelope,
•r0(100%) matched core radius,
•xtransverse single particle position,
•µmismatch parameter: µ=rc/r0,
•τtune depression: τ=k/k0,
•εtotal (100%) emittance.
4.2.2 Initial mismatch
In the matched case the core radius as well as the amplitudes of the single particle oscillations remain
constant. If the core is initially mismatched with µ=rc/r06= 1, its radius starts to oscillate around its
equilibrium value r0. In the simple 1D model an initial envelope mismatch corresponds to the excitation
of the breathing mode (fast mode), which is the only beam eigenmode for an azimuthally symmetric,
continuous beam. Single particles traversing the core can obtain a net energy gain if the core radius is
different for a particle when entering and leaving the core. As an example we integrate the equations of
motion [Eq. (4.1), (4.3), and (4.4)] for a tune depression of τ=k/k0= 0.7, a core mismatch of µ= 0.6,
and an initial single particle amplitude of x(0) = 0.9r0. The parameters of the transport channel that
is used in this section are listed in Table 4.1. For the integration the standard fourth-order Runge-Kutta
algorithm with constant step size is used. The resulting core and single particle oscillations are shown in
Fig. 4.1.
1
-3
-2
-1
0
1
2
3
0 5 10 15 20 25 30 35 40
PSfrag replacements
zk0/2π
rc/r0x/r0
particle
core
Figure 4.1: Particle and core envelopes for initial mismatch, µ= 0.6,x(0) = 0.9r0,τ= 0.7
If the oscillation frequency of the single particle has a 1:2 parametric ratio with the oscillation frequency
of the core, the particle gains energy and increases its oscillation amplitude until the resonant condition
is no longer fulfilled. As a result one finds a certain maximum particle amplitude as well as so-called
fixed-points in phase space around which halo particles start to conglomerate (compare stroboscopic
CHAPTER 4. INITIAL MISMATCH 31
Table 4.1: Parameters for the simulated transport channels
tune depression (τ)0.5 - 0.9
energy (E)11.4 MeV
focusing period (l)0.333 m
zero-current phase advance per period (lk0)38.5 deg
plots in Fig. 4.2). The maximum halo extent is determined by the amount of initial mismatch, while
the time constant for the development of beam halo is influenced by the tune depression. Wangler [43]
also found that the position of the fixed-points is fairly insensitive to changes in tune depression or
mismatch amplitude. In [44], however, it was shown that the fixed-point core distance is dependent
on the equipartitioning ratio (εtkt/εlkl). To visualise the fixed-points as well as the maximum halo
amplitudes one can use stroboscopic plots showing the phase space position of the single particle once
per core oscillation period. For this purpose we integrate Eqs. (4.1) and (4.3) and plot in Fig. 4.2 (a)
and 4.2 (b) the transverse momentum and position of the particle every time the core oscillation reaches
its minimum, which coincides with the occurrence of the maximum/minimum single particle amplitude
(compare Fig. 4.1). To obtain a coherent resonance pattern a large number of focusing periods is needed
(4000 in this example). Each dotted line represents a single particle with a certain initial amplitude and
zero initial transverse momentum.
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
PSfrag replacements
x0/(r0k0)
x/r0
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
PSfrag replacements
x0/(r0k0)
x/r0
Figure 4.2: (a) Stroboscopic plot for µ= 0.95,
0.2≤x(0)/r0≤3.5,τ= 0.8
(b) Stroboscopic plot for µ= 0.6,
0.2≤x(0)/r0≤3.5,τ= 0.8
In the case of weak initial mismatch [e.g. 5%, Fig. 4.2 (a)] particles that start within the core (|x|/r0≤1)
remain basically undisturbed in their trajectory. They either need a non-zero initial momentum or an
initial amplitude larger than the core radius to gain energy by means of a parametric resonance. In
contrast, for strong initial mismatch [e.g. 40%, Fig. 4.2 (b)], the resonant regions surrounding the fixed-
points become very large and the core area with undisturbed trajectories shrinks considerably. In both
cases, for strong and weak mismatch particles with very large initial amplitudes (e.g. |x|/r0≈3), remain
as unperturbed in their orbits as the inner core particles. This can be explained by the fact that in order
to gain or to loose energy, the single particles must be in resonance (1:2 parametric ratio) with the core
32 4.3. 3D ENVELOPE EIGENMODES
oscillations. Due to the space-charge forces single particles with small amplitudes (within the core)
have different oscillation frequencies than particles with larger amplitudes. Only the particles that can
gain enough energy via a parametric resonance with the core will change their trajectories and will be
“caught” by the resonance. The effect becomes stronger for high space-charge forces, providing larger
“kicks” to the single particles. It becomes also stronger for larger amplitudes of the core oscillations. In
that case the oscillation energy, stored in the core oscillations is large, meaning that the single particles
can receive large “kicks” to change their orbits.
This classic 1D particle-core model predicts maximum halo amplitudes of ≈3r0. In terms of multiples
of r.m.s. beam envelopes (in 3D) this corresponds to ≈7a. It is worth noting that particles which are
initially outside the core and get into resonance with the core oscillations have the same maximum halo
amplitudes as particles starting inside the core. Studies with a more sophisticated 3D particle-core model
[45] show that the maximum halo extent due to initial mismatch can amount to >5r0if the longitudinal
to transverse focusing ratio is larger than one.
4.3 3D envelope eigenmodes
In the previous section it was shown that initial mismatch yields the excitation of regular core oscillations
with a fixed frequency. For a 3D bunched beam one can expect to find three such eigenmodes, which
are likely to have different oscillation frequencies. Since the 3D envelope equations are coupled, one
can also expect that each mode has different oscillation amplitudes in all three planes. Using the 3D
envelope equations [Eq. (2.18)] and the smooth approximation for a FODO channel with RF, one can
derive and analyse the eigenmodes of a 3D bunched beam. In the following the approach of Bongardt &
Pabst (presented in [46] and [47]) is used to derive an approximation for the three eigenfrequencies of a
mismatched beam. Similar, more exhaustive derivations are given in [48] and [49]. The derived formulae
will then be used to excite the eigenmodes and to systematically study the effects of initial mismatch with
a particle tracking code.
Starting point are the 3D envelope equations [see Eq. 2.18] which were derived in Chapter 2:
a00
x+axκx(s)−K3[1 −f(s)]
2ayb−ε2
x
a3
x= 0
a00
y+ayκy(s)−K3[1 −f(s)]
2axb−ε2
y
a3
y= 0
b00 +bκz(s)−K3f(s)
axay−ε2
z
b3= 0
3D envelope equations with
space-charge (4.5)
For these equations one can define a general matched s-dependent solution which has the same period-
icity as the focusing system
˜ax,y(s+LP) = ˜ax,y(s)
˜
b(s+LP) = ˜
b(s)matched solution (4.6)
The mismatched solution will oscillate around the matched one with a periodicity different from the
length LPof the focusing periods (compare Fig. 4.1). Starting point of the derivation is the perturbation
of the s-dependent matched beam envelopes:
CHAPTER 4. INITIAL MISMATCH 33
ax,y(s) = ˜ax,y(s) + ∆ax,y(s)
b(s) = ˜
b(s) + ∆b(s)mismatched solution (4.7)
The perturbed solution is then re-inserted into the envelope equations (4.5). For the sake of simplicity
we will assume equal transverse emittances (εx=εy=εt). Further simplification is achieved by using
the smooth approximation for the zero-current focusing terms [κx,y(s)→k2
t,0,κz(s)→k2
l,0, compare
(2.19)] and by ignoring the changing of the bunch shape along the transport channel [f(s)→f(s)]. We
also use the smooth approximation to replace the s-dependent periodic solutions ˜ax,y(s)and ˜
b(s)by the
matched (constant) beam sizes ˆax= ˆay= ˆaand ˆ
b. After some further manipulation (see Appendix C)
one can derive three eigenmodes which are usually referred to as follows:
quadrupolar mode:
σenv,Q = 2 ·σt(4.8)
with the eigensolution:
∆ax(s)
ˆa=Am·cos σenv,Q ·s
LP+φ
∆ay(s)
ˆa=−Am·cos σenv,Q ·s
LP+φ
∆b(s)
ˆ
b= 0
(4.9)
high-frequency mode or breathing mode or fast mode:
σenv,H =σ2
t,0 +σ2
t+1
2σ2
l,0 +3
2σ2
l+(4.10)
sσ2
t,0 +σ2
t−1
2σ2
l,0 −3
2σ2
l2
+ 2 ·σ2
t,0 −σ2
t·σ2
l,0 −σ2
l
with the eigensolution:
∆ax,y(s)
ˆa=Am·cos σenv,H ·s
LP+φ
∆b(s)
ˆ
b=Am
gH·cos σenv,H ·s
LP+φ
gH>0
(4.11)
34 4.3. 3D ENVELOPE EIGENMODES
low-frequency mode or slow mode:
σenv,L =σ2
t,0 +σ2
t+1
2σ2
l,0 +3
2σ2
l−(4.12)
sσ2
t,0 +σ2
t−1
2σ2
l,0 −3
2σ2
l2
+ 2 ·σ2
t,0 −σ2
t·σ2
l,0 −σ2
l
with the eigensolution:
∆ax,y(s)
ˆa=Am·cos σenv,L ·s
LP+φ
∆b(s)
ˆ
b=Am
gL·cos σenv,L ·s
LP+φ
gL<0
(4.13)
The form factors gH,gLare defined as
gH,L =σ2
t,0 −σ2
t
σ2
env,H,L −2·σ2
t,0 +σ2
t(4.14)
Amis the mismatch amplitude, and ˆaandˆ
bare the matched beam envelopes in the smooth approximation.
The nomenclature of the modes becomes clear when considering the characteristics of each mode:
- The quadrupolar mode consists of envelope oscillations of the transverse beam envelopes around
their matched equilibrium with 180◦phase difference between the planes. The longitudinal plane
is unaffected (example in Fig. 4.3).
- The fast mode “breathes” in all three planes with the same phase but with different amplitudes
in the transverse and longitudinal planes. From the eigenvalues one can see that its oscillation
frequency is always higher than that of the slow and quadrupolar mode (example in Fig. 4.4).
- The slow mode also has different amplitudes in the transverse and longitudinal planes. The longi-
tudinal and transverse envelopes oscillate with a phase difference of 180◦(example in Fig. 4.5).
Since the three eigenmodes were derived with some strong simplifications, the solutions are only ap-
plicable within certain limits. For high space-charge forces or rapidly changing particle velocities, the
derived excitation becomes less precise. Its quality can be judged by the smoothness of the ratio of
mismatched over matched r.m.s. beam envelopes. An improvement can be achieved by measuring the
envelope tunes in the simulation output and recalculating the mismatch excitation with these tunes. An
even better excitation can be obtained by numerical computation of the envelope tunes.
Excitation of mismatch eigenmodes
Going back from the smoothed r.m.s. beam envelopes to the s-dependent envelopes (ˆa→˜ax,y(s)) one
can excite the eigenmodes using the eigensolutions [Eqs. (4.9), (4.11), (4.13)] and the form factors
CHAPTER 4. INITIAL MISMATCH 35
[Eq. (4.14)]. The matched Twiss parameters at the beginning of a focusing period (s= 0) are ˜αx,y,z(0)
and ˜
βx,y,z(0). The goal is to modify αand βsuch that the mode envelope maximum is located at s= 0.
The modified αand βvalues can then be used in a simulation code to study the mismatched beam
behaviour.
For the excitation one considers the eigensolutions and their derivations
ax,y(s) = ˜ax,y(s)·1±Amcos σenv s
LP+φ
d
dsax,y(s) = d
ds˜ax,y(s)·[1 ±Amcos σenv s
LP+φ]∓(4.15)
˜ax,y(s)Amσenv sin σenv s
LP+φ
with Ambeing the amplitude of the mismatch oscillation. The maximum mismatch envelope value at
s= 0 is obtained by setting φ= 0:
ax,y(0) = ˜ax,y(0) ·[1 ±Am]d
dsax,y(0) = d
ds˜ax,y(0) ·[1 ±Am]
(4.16)
Obviously the envelope value and the momentum have to be changed by the same factor, and since
α=−aa0
εβ=a2
ε(4.17)
the Twiss parameters αand βalso have to be changed by the same factor, which in case of the quadrupole
mode means:
αm,x(0) = (1 + Am)2˜αx(0)
βm,x(0) = (1 + Am)2˜
βx(0)
αm,z(0) = ˜αz(0)
αm,y = (1 −Am)2˜αy(0)
βm,y = (1 −Am)2˜
βy(0)
βm,z =˜
βz(0)
(4.18)
Figure 4.3 shows an example for the excitation of the quadrupole mode using the above formulae. The
30% envelope mismatch is excited at the beginning of two sections of the SPL I lattice (120 - 380MeV)
which has FODO quadrupole focusing and using a 6D waterbag distribution1. The oscillations in the x
and yplanes are of opposite phase and have the same amplitude. The longitudinal plane remains almost
unperturbed (matched) and only small oscillations can be observed. Even though the excitation is not
perfect, meaning that the oscillation maxima are not perfectly constant, the oscillations carry on almost
unchanged. The lengthening of the oscillation period is due to the changing phase advance per metre,
which decreases with increasing energy.
The same principle can be used to derive the excitation for the fast and slow mode and Fig. 4.4 and 4.5
show the resulting mismatch oscillations using the same lattice and using again a maximum envelope
mismatch of 30%.
One can observe that also in case of the fast and slow mode, the oscillation characteristics fulfil the
theoretical predictions. For both modes the transverse oscillations are in phase and of the same amplitude.
1The 6D waterbag distribution assumes a uniform distribution in all 6 phase space dimensions (compare Section 4.5).
36 4.3. 3D ENVELOPE EIGENMODES
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 50 100 150 200
length [m]
PSfrag replacements
1 + ∆ax/ax1 + ∆ay/ay1 + ∆b/b
Figure 4.3: 30% quadrupole mode excitation with an initial 6D waterbag distribution in two sections
of the SPL I SC lattice (120 - 380 MeV), plotted is the ratio of mismatched over matched
r.m.s. beam envelopes, IMPACT multi-particle simulation
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 50 100 150 200
length [m]
PSfrag replacements
1 + ∆ax/ax1 + ∆ay/ay1 + ∆b/b
Figure 4.4: 30% fast mode excitation with an initial 6D waterbag distribution in two sections of the SPLI
SC lattice (120 - 380 MeV), plotted is the ratio of mismatched over matched r.m.s. beam
envelopes, IMPACT multi-particle simulation
The longitudinal oscillations have a different amplitude and in case of the slow mode also a different
phase than in the transverse plane. Also here one can observe that the excitation is not perfect, especially
in case of the fast mode. Despite the imperfection, the oscillations carry on basically undamped over the
length of the lattice.
CHAPTER 4. INITIAL MISMATCH 37
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 50 100 150 200
length [m]
PSfrag replacements
1 + ∆ax/ax1 + ∆ay/ay1 + ∆b/b
Figure 4.5: 30% slow mode excitation with an initial 6D waterbag distribution in two sections of
the SPL I SC lattice (120 - 380MeV), plotted is the ratio of mismatched over matched
r.m.s. beam envelopes, IMPACT multi-particle simulation
4.4 The “free energy” limit for r.m.s. emittance growth
So far we have established that initial mismatch creates mismatch oscillations of the beam core and that
these oscillations, via parametric resonances, are responsible for the development of beam halo and prob-
ably a certain r.m.s. emittance growth. The envelope equations assume a constant r.m.s. emittance and
are thus not suited to describe the process of growing emittances. However, an estimation for r.m.s. emit-
tance can be obtained from the “free energy” approach, which was introduced by Reiser [50]. Reiser uses
a procedure that avoids the problem of frozen emittances in the envelope equations by stating that sta-
tionary (matched) and non-stationary (mismatched) beams have different average energies per particle.
Borrowing an expression from thermodynamics he states that a matched beam is in its thermodynamic
equilibrium which depends on the parameters of the beam, the focusing lattice and on the associated
average particle energy Wi(which can be calculated out of the lattice and beam parameters). If the beam
is mismatched, the associated particle energy changes to Wf=Wi+ ∆Wand the beam is no longer in
its equilibrium. In this context the “excess energy” ∆Wis called the “free energy” of the beam. He then
assumes that the beam relaxes (or thermalises) into a new stationary distribution that corresponds to the
increased average energy per particle Wf. By calculating the initial and final average particle energies
(Wiand Wf) and by associating them to a beam described by the one-dimensional envelope equation
(wave number k0, emittance ε, space-charge constant K1, beam radius rc) it becomes possible to calcu-
late the final emittance, circumventing the problem of constant emittances by excluding the transitional
process from the calculation.
The thermalisation process can be triggered by non-linear space-charge forces, instabilities, or collisions,
and causes emittance growth, increased beam radii, and possibly beam halo as the beam approaches a new
stationary state at the higher average energy per particle. Reiser quotes three examples for non-stationary
initial beams, which are: a) mismatch in the density profile, b) mismatch in the initial r.m.s. envelope,
and c) off-centring.
38 4.4. THE “FREE ENERGY” LIMIT FOR R.M.S. EMITTANCE GROWTH
In the following we will describe the three main steps of Reisers derivation to describe the emittance
increase due to initial envelope mismatch. The complete calculation can be found in [42] or [50]. For the
derivation Reiser uses an azimuthally symmetric beam in a uniform focusing channel using the same one-
dimensional smoothed envelope equation [Eq. (4.1)] that was already used for the particle-core model in
Section 4.2.
r00
c+k2
0rc−ε2
r3
c−K1
rc= 0 (4.19)
If the initial beam is stationary the core radius is constant (rc=r0), the 2nd derivations vanish, and one
can write Eq. (4.19) as
k2
0r0−ε2
r3
0−K1
r0
= 0 or k2r0−ε2
r3
0
= 0 or ε=kr2
0(4.20)
where the relation
k2=k2
0−K1
r2
0
(4.21)
was employed.
For the subsequent derivation the following set of variables will be used:
•k0zero-current wave number,
•rpbeam pipe radius,
•r0general stationary beam radius,
•ri,ki,εibeam radius, full-current wave number, and emittance of an initially stationary beam,
•rmbeam radius of an initially mismatched beam,
•rf,kf,εffinal beam radius, final full-current wave number, and final emittance after thermalisation.
Step I: average particle energy for a stationary beam
In the first step the average particle energy of a stationary beam is defined as the sum of its transverse
kinetic energy Wk, its average potential energy due to the external focusing forces Wp, and the energy
associated to its self fields Ws. For the latter it is necessary to introduce the radius rpof the beam pipe.
Summing up the three contributions Reiser derives the total energy per particle in a stationary beam as
W=βγmc2
4r0
0
2+k2
0r2
0+1
2K11 + 4 ln rp
r0 (4.22)
Using Eq. (4.20) and (4.21) together with r0
0=k0r2
0one can write
W=βγmc2
4(k2+k2
0)r2
0+1
2(k2
0−k2)r2
01 + 4 ln rp
r0 (4.23)
CHAPTER 4. INITIAL MISMATCH 39
The energy equation relating the final energy (index f), the initial energy (index i), and the “free energy”
(Wf=Wi+ ∆W) can then be written as
βγmc2
4(k2
f+k2
0)r2
f+1
2(k2
0−k2
f)r2
f1 + 4 ln rp
rf=
βγmc2
4(k2
i+k2
0)r2
i+1
2(k2
0−k2
i)r2
i1 + 4 ln rp
ri+ ∆W
(4.24)
where the “free energy” ∆Wcan be introduced in the following convenient form
∆W=1
2βγmc2k2
0r2
ih(4.25)
The parameter his the “free energy” parameter which can be calculated for each of the three afore
mentioned cases (distribution mismatch, envelope mismatch, off-centring).
Step II: for the emittance increase
From Eq. (4.21) we get
k2
i=k2
0−K1
r2
iand k2
f=k2
0−K1
r2
f⇒k2
f=k2
0−ri
rf2
(k2
0−k2
i)(4.26)
and together with Eq. (4.25) substituted into Eq. (4.24) one finds an expression that relates the change in
beam radii to the “free energy” parameter h
h=rf
ri2
−1− 1−k2
i
k2
0!ln rf
ri(4.27)
Using Eq. (4.20) one can write the emittance difference between the final and the initial beam as
∆ε2=ε2
f−ε2
i=k2
fr4
f−k2
ir4
i(4.28)
With Eq. (4.26) and ε2
i=k2
ir4
ione can transform Eq. (4.28) into an expression for the final emittance
εf
εi=rf
riv
u
u
t1 + k2
0
k2
i"rf
ri2
−1#(4.29)
Assuming that the mismatch is not too big (rf−riri) one can simplify Eq. (4.27) to
rf
ri≈1 + h
1 + (ki/k0)2(4.30)
and get a first order approximation of the maximum emittance growth due to initial envelope mismatch.
εf
εi=v
u
u
t 1 + 2k2
0
k2
i
h!(4.31)
40 4.4. THE “FREE ENERGY” LIMIT FOR R.M.S. EMITTANCE GROWTH
Step III: the “free energy” parameter hfor radial mismatch
A stationary (matched) beam in the smooth approximation can be represented by a circle in the (x/x0)
trace space. During the transport of the beam the circle rotates clockwise but the values for riand r0
i
remain constant as shown in Fig. 4.6. If the beam is radially mismatched it has initially still the same
emittance but an elliptic shape in trace space, denoted by rmand r0
min Fig. 4.6. Due to the clockwise
rotation of the ellipse the beam radius rmnow oscillates between the depicted position of rmand rmax.
x
x’
rm
ri
rf
rmax
r0
f
r0
m
r0
i
Figure 4.6: Upright trace-space ellipses for the initial mismatched beam (rm,r0
m), the initial stationary
beam (ri,r0
i), and the final stationary beam (rf,r0
f)
Using Eq. (4.22) we can calculate the energy difference ∆Wbetween the initial mismatched beam
(subscript m) and the initial matched beam (subscript i). Replacing K1with Eq. (4.26) then yields
∆W=βγmc2
4r02
m−r02
i+k2
0(r2
m−r2
i) + 2r2
i(k2
0−k2
i) ln ri
rm(4.32)
Initially the emittances of the matched and mismatched beam are equal so that rmr0
m=rir0i=εi(valid
for upright emittances as shown in Fig. 4.6). Using
r0i=kiriand r0
m=rir0
i
rm=kir2
i
rm(4.33)
and comparing Eq. (4.32) with Eq. (4.25) one can now express the “free energy” parameter hfor a
radially mismatched beam as
CHAPTER 4. INITIAL MISMATCH 41
h=k2
i
2k2
0 r2
i
r2
m−1!−1
2 1−r2
m
r2
i!+ 1−k2
i
k2
0!ln ri
rm(4.34)
where ki/k0is the initial tune depression and rm/riis the initial envelope mismatch.
In anisotropic 3D beams thermalisation not only occurs within one plane but can also occur between the
planes (e.g. between the longitudinal and transverse planes) if certain resonance conditions are met (see
Chapter 6). Generalising the “free energy” approach Hofmann suggested in [51] to relate the averaged
r.m.s. emittance growth to the averaged mismatch in all three planes. Using Eq. (4.31) this yields an
expression stating the theoretical upper limit for r.m.s. emittance growth in a 3D beam for an arbitrary
initial mismatch assuming full thermalisation:
1
3 ∆εx,f
εx,i +∆εy,f
εy,i +∆εz,f
εz,i !=1
3X
n=x,y,z v
u
u
t1 + 2k2
n,0
k2
n,i
hn(4.35)
The formulae assume a complete thermalisation of the free energy into r.m.s. emittance growth, while in
practice one should expect slightly lower values.
4.5 Particle distributions for simulations in 6D phase space
Multi-particle simulations of realistic accelerating structures with space-charge suffer from the fact that
there are no analytic solutions for stationary (self-consistent) particle distributions in 6D phase space.
For simulations of very long bunches (or un-bunchd beams) one usually simplifies the problem to 4D
phase space (x/y/x0/y0) where stationary solutions can be found for constant focusing lattices. In the
following we will briefly describe the characteristics of the three most commonly used 4D distributions
and then look at the 6D case which is needed for realistic simulations of high-intensity linacs.
4.5.1 KV, waterbag, and Gaussian distributions
A distribution often used for theoretical studies is the Kapchinskij-Vladimirskij distribution (KV) [30]
which is the only self-consistent distribution where the external and internal forces (focusing system
and space-charge) are linear and where the emittances are preserved. The KV distribtion is charac-
terised by the fact that all particles have the same constant transvere energy, resulting in a distribu-
tion which covers the surface of 4-dimensional hyperellipsoid in 4D phase space. All 2D projections
(x/x0, y/y0, x/y, x0/y0, x/y0, x0/y) of this distribution are uniformly filled ellipses for beams with or
without space-charge. In continuous focusing systems where the external nonlinearities are negligible
and where the bunches are very long with respect to their transverse dimensions, the KV distribution
is a useful tool to study space-charge related effects. However, it is clear that the KV distribution is
not very realistic since all real beams will have a certain finite energy spread. Apart from that realistic
accelerators hardly resemble the idealized beam transport systems (constant focusing, no acceleration)
which are used in theoretical beam studies. It was also found that the KV distribution shows instabilities
[52] which are not found in simulations with more realistic beams or in experiments. Another serious
limitation is that there is no 6D solution for a KV-type beam which basically excludes its use for realistic
linac simulations.
For these reasons one often uses the so-called waterbag distribution, where the particles are not mono-
energetic but where particle energies are uniformly distributed between 0 and a certain maximum. Here
42 4.5. PARTICLE DISTRIBUTIONS FOR SIMULATIONS IN 6D PHASE SPACE
Table 4.2: Definition and properties of 4D particle distributions often used in multi-particle simulations
distribution definition: f(r4)particle density in real space
r2
4=x2+x02+y2+y02r2=x2+y2
Kapchinskij-Vladimirskij (KV)
(r2
c=x2
max +y2
max)1
2π2r3
cδ(r4−rc)1
πr2
c
waterbag (WB)
(r2
c=x2
max +y2
max)2
π2r4
cfor 0< r4< rc2
πr2
c1−r2
r2
c
Gauss
(a2=x2)1
4π2a2e−r2
4
2a21
2πa2e−r2
2a2
the 4D phase space is not just populated on the surface of a hyperellipsoid but it is represented by
uniformly filled hyperspheres meaning that it is uniform in 4D phase space (x/y/x0/y0). In the zero-
current case the 2D projections are again of elliptical shape with a density function in real space that
depends linearly on the space-charge forces. For very high space-charge the 2D phase space projections
(x/x0, y/y0) become more and more square and the real space density becomes basically uniform (see
Struckmeier [53] for detailed derivations and plots). However, even if the waterbag distribution is more
realistic than the KV distribution it is still somewhat artificial since it imposes sharp beam boundaries
within which all particles are confined. In realistic beams there will always be a “fuzzy” beam boundary
and a certain amount of halo particles which are not adequately described with a waterbag distribution.
For a less sharp beam boundary one can use a Gaussian distribution which is defined by a Gaussian
energy profile. For vanishing space-charge this distribution has a Gaussian density profile in real space
while for high space-charge forces the density profile becomes uniform. Also this distribution is not
completely realistic since it assumes infinite exponential tails which do not occur in realistic beams. In
computer simulations with untruncated Gaussian distributions, the largest single particle amplitudes are
basically related to the number of macroparticles generated. To obtain a better defined beam boundary
one can cut off the Gaussian tails at a certain multiple of the r.m.s. radius.
In 4D simulations most computer codes generate the above mentioned particle distributions for the zero-
current case which is summarized in Table 4.2 (from [54, 42]). One should note that for all three dis-
tribution types one can find a stationary solution for beams with space-charge in a continuous focusing
channel. However, since most codes only generate the zero-current distribution (as listed in Table 4.2)
the simulated beams are usually not stationary (apart from the KV case) and are subject to a certain
redistribution of particles.
4.5.2 6D distributions
We have seen that even for a simplified 4D beam in a continuous focusing lattice it is a challenging task to
find a realistic particle distribution that is stationary. In case of a 6D bunched beam in a realistic periodic
focusing lattice including acceleration this task becomes impossible and one usually has to accept a
certain initial emittance growth due the initial redistribution of particles. At very low energy, where
CHAPTER 4. INITIAL MISMATCH 43
beams are usually space-charge dominated, one can show [42] and experimentally observe that beams
develop a uniform density profile that is well approximated by a 6D waterbag distribution (which is
uniform in 6D phase space [x/y/z/x0/y0/z0]). At higher energy where the space-charge forces become
very small beams usually develop a Gaussian profile, meaning that the equilibrium distributions in linacs
will always be somewhere between a waterbag and a Gaussian profile. As we will show in the following
Gaussian distributions generally yield more emittance growth than waterbag distributions meaning that
one should use both distribution types to define upper and lower boundaries for the expected beam
performance in a linac.
When studying initial mismatch in realistic lattices, the effects of radial mismatch are usually much more
pronounced than the effects due to mismatch in the density profile which is the case for all multi-particle
simulations presented here. If needed one can completely seperate the effects by tracking a beam over
a certain number of equal periods without acceleration in order to provide enough time for the beam to
find its equilibrium distribution. This distribution can then be used to specifically study effects due to
radial (or other) mismatch without any influence of distribution mismatch.
For end-to-end simulations of a complete linac one is less interested in seperating different effects that
yield emittance growth. In this case the focus should be on simulating a linac as realistically as possible
and it is then sensible to use a simulated (or measured) output distribution of the RFQ which usually
contains a certain amount of halo particles. The input distribution for the RFQ simulation itself is usually
of the 4D waterbag type which becomes bunched and which develops some (usually Gaussian) tails
during the passage of the RFQ.
4.5.3 Distributions and emittance definitions
The emittance definition used in this thesis is the one for the r.m.s. emittance introduced by Sacherer
in [29]. Lapostolle suggested in [55] to use an emittance definition corresponding to four times the
r.m.s. emittance which corresponds to the total emittance of a uniform continuous beam. Weiss has
calculated [56] the ratio between the total and the r.m.s. emittance for beams which have a clearly defined
border and which are uniformly filled in n-dimensional projections as
εtotal
ε=n+ 2 (4.36)
where εis the r.m.s. emittance. The resulting ratios for waterbag and KV beams are listed in Table 4.3.
In a 4D Gaussian beam which is truncated at ptimes the r.m.s. radius the ratio of total to r.m.s. emittance
is approximately given by p2(assuming that p > 4).
The ratio of total to r.m.s. emittance is also a measure for the tails in a distribution and is thus important
for the characterisation of halo development in a linac. However, using this ratio in multi-particle simu-
Table 4.3: Ratio of total to r.m.s. emittances for distributions with well defined boundaries which are
uniform in all n-dimensional hyperellipsoidal projections
distribution n εtotal/εrms uniform in
KV 2 4 (x/x0, y/y0, x/y, x0/y0, x/y0, x0/y)
4D waterbag 4 6 (x/y/x0/y0)
6D waterbag 6 8 (x/y/z/x0/y0/z0)
44 4.6. MISMATCH FOR REALISTIC LINAC BEAMS
lations is not very useful since it is determined by one single particle which happens to have the largest
single particle emittance. For better statistics one often looks at fractional emittances which contain a
certain percentage of all particles as for instance 95% or 99%. But also the fractional emittances do not
offer a clear judgment on halo development since there is no straightforward algorithm to calculate the
smallest emittance value which contains a certain percentage of particles. The usual (numerical) proce-
dure to determine fractional emittances is to calculate the size and orientation of the r.m.s. ellipse [using
Eq. (2.1) and (2.3)] and then to gradually increase this area until the desired percentage of particles is in-
cluded, thereby maintaining the aspect ratio and orientation of the r.m.s. ellipse. Since halo development
generally yields a fragmented S-shaped distribution in the 2D projections (x/x0, y/y0, z/z0) [as shown
in Fig. 3.2)] this procedure becomes questionable and can only give an indication on the process of halo
development. Another problem is that the tails have a different angular velocity than the core particles
yielding oscillating values of the fractional emittances (an example of these oscillations can be appreci-
ated in Figs. 6.6 and 6.7 where the 99%, 99.9%, and 99.99% fractional emittances during an emittance
exchange are plotted). For this reason we recommend a halo characterisation that is based on the particle
density versus beam radius (e.g. Fig. 4.13) or even more precise on the fraction of particles which is
found outside multiples of the r.m.s. emittance (e.g. Fig. 5.10). As long as this curve is taken always at
the same lattice position it provides a useful tool to compare the effect of different mismatch types on
the development of beam halo.
4.6 Mismatch for realistic linac beams
In this section more realistic lattices and beams shall be considered, including acceleration and changing
focusing elements. As an example the whole superconducting linac section (120 - 2200MeV, simulation
current: 40mA, average tune depression: ≈0.7) of the SPLI study [1] is used to show in more detail the
effect of the three eigenmodes on the beam. Furthermore an inhomogeneous 6D Gaussian beam density
profile is used instead of the 6D waterbag distribution of the previous section. The lattice is simulated
with the 3D multi-particle code IMPACT (see Chapter 3). The resulting mismatch oscillations for the
three eigenmodes are plotted in Figs. 4.7 to 4.9.2
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 100 200 300 400 500 600
length [m]
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
1 + ∆b/b
Figure 4.7: 30% quadrupolar mismatch in the SC section of SPLI, IMPACT multi-particle simulation
2The ratio of mismatched over matched r.m.s. envelopes is used to visualise the oscillations.
CHAPTER 4. INITIAL MISMATCH 45
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 100 200 300 400 500 600
length [m]
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
1 + ∆b/b
Figure 4.8: 30% fast mode mismatch in the SC section of the SPLI, IMPACT multi-particle simulation
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 100 200 300 400 500 600
length [m]
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
1 + ∆b/b
Figure 4.9: 30% slow mode mismatch in the SC section of the SPLI, IMPACT multi-particle simulation
One can observe that the phase and amplitude characteristics of all modes are basically the same as for
the simulations with a 6D waterbag beam in the previous section. Although the tunes, and therefore also
the oscillation frequencies, of the mismatch modes change considerably along the linac, and even though
3 lattice transitions are crossed, the oscillations remain remarkably stable. However, in contrast to the
examples in Figs. 4.3 to 4.5, the oscillations now appear to be damped. This effect can be explained by
stating that the “free energy” that has been introduced into the system via mismatch is transformed into
beam halo and emittance growth. The emittance growth is visible in plots 4.7 to 4.9 through the shift of
the oscillation average (1 at the beginning, and slightly higher at the end the simulations). At the end
of the transformation the r.m.s. core has reached a new equilibrium and the oscillations basically stop.
Figure 4.10 shows the r.m.s. emittance growth for the three modes and the predicted maxima from the
free energy approach (Eq. 4.35).
In agreement with the transient behaviour in Figs. 4.7 to 4.9 the emittance growth for the quadrupolar
mode has the shortest rise time, followed by the slow mode and the fast mode. None of the three modes
surpasses the predicted maximum r.m.s. emittance growth from the free energy approach and only the
slow mode does not reach its predicted maximum value. One can see that the agreement for the fast
mode and quadrupolar mode are remarkably exact, meaning that the free energy limits seem to provide a
useful estimate for maximum r.m.s. emittance growth. The simulations also suggest that a typical high-
46 4.6. MISMATCH FOR REALISTIC LINAC BEAMS
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
0 100 200 300 400 500 600
average emittance growth
length [m]
quadrupolar mode maximum
fast mode
slow mode
slow mode maximum
fast mode maximum
quadrupolar mode
Figure 4.10: R.m.s. emittance growth for the 3 eigenmodes and predicted maxima from the free energy
approach, IMPACT multi-particle simulation
power linac is long enough for these maxima to be reached. Furthermore, one can conclude that if these
maxima are surpassed there must be more sources of mismatch than only the initial envelope mismatch
which was introduced here on purpose.
The difference in transformation speed between the example in Fig. 4.3 and those in Figs. 4.7 to 4.9 can
be explained as follows: in the first example the simulations with the 6D waterbag distribution result in
a very long rise time, while in the second example the Gaussian distribution provides enough “noise” to
quickly transform the mismatch oscillations into emittance growth. Reiser [42] suggests that the non-
uniformity of the beam is responsible for the quicker transformation and Hofmann [51] points out that
the Gaussian tails may be important for the transformation speed.
To highlight the difference Fig. 4.11 shows the transverse mismatch oscillations due to fast mode exci-
tation in a periodic transport channel without acceleration for a 6D waterbag and a 6D Gaussian beam.
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 10 20 30 40 50 60 70
dr/r0
length [m]
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 10 20 30 40 50 60 70
dr/r0
length [m]
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
Figure 4.11: Mismatched over matched transverse r.m.s. envelopes for 6D waterbag (left) and Gauss
(right), 200 focusing periods, fast mode excitation, IMPACT multi-particle simulation
CHAPTER 4. INITIAL MISMATCH 47
One can see that the transverse r.m.s.core-oscillations for a Gaussian beam with 30% fast mode excitation
are rapidly damped from 30% to ≈12% where they remain almost constant. At the same time the
waterbag beam loses only a fraction of its oscillation energy with the result of much smaller emittance
growth (7% versus 27% for 6D Gauss) and halo development in a transport channel of equal length.
Beams in realistic end-to-end simulations for high-intensity linacs usually develop particle distributions
which are somewhere in between a 6D waterbag and a Gaussian distribution.
4.6.1 Particle redistribution
The effect of parametric resonances on the beam distribution can be illustrated by the redistribution of
particles in a mismatched bunch. Since the three beam eigenmodes have three different frequencies,
one can expect that each eigenmode affects particles with different oscillation frequencies (tunes). In
order to plot a clear redistribution pattern a transport channel with 50 focusing periods (Gaussian input
distribution) is simulated, exciting separately the three beam eigenmodes [Fig. 4.12 (a)].
We start with the quadrupolar mode, which oscillates only in the transverse plane. Its eigenfrequency
is by definition [see Eq. (4.8)] twice as high as that of the transverse depressed tune. One can therefore
expect a strong resonance with the core particles which also have the lowest tunes in the bunch. Looking
at Fig. 4.12 (a) one can observe that indeed particles are removed from the core area and deposited at
a higher radius. For the slow and fast modes the situation is more complicated since they both oscillate
in all three planes. Nevertheless, it seems that one can obtain a consistent picture: the eigenfrequency
of the slow mode is lower than for the quadrupolar mode, therefore the coupling with the core particles
can only be very weak and consequently only few particles are removed from the core. Finally the fast
mode seems to affect particles with higher tunes. The intuitive conclusion is that high-frequency core
oscillations yield a large fixed point - core distance and thereby large halo amplitudes, whereas particles
with lower tunes (from the core centre) are affected by the low-frequency core oscillations and end up
around fixed points with a smaller fixed point - core distance. However, this pattern and the order of
the fixed points should not be generalised since it is not only influenced by the mode frequencies but
also by beam-lattice resonances and inherent redistribution patterns of the initial distribution (waterbag,
Gauss, etc.). The only conclusion we can draw at this stage is that each eigenmode will generally trigger
different redistribution patterns with different fixed point - core distances. It is also very likely that the
fast mode will always yield the largest halo amplitudes, which makes it very suitable for halo studies.
For small mismatch amplitudes a general mismatch excitation yields a superposition of modes as shown
in Fig. 4.12 (b). As an example we use a “+ + +” and “+−+” mismatch which stands for +30%
envelope mismatch in
x
,
y
, and
z
, or +30% in
x
and
z
and -30% in
y
, respectively.
Comparing Figs. 4.12 (a) and (b) we find that the redistribution for the “+−+” excitation is dominated
by the fixed point of the quadrupolar mode at ≈2rc,r.m.s.. This is consistent with theory since “+−” in
x
and
y
corresponds to the excitation of the quadrupolar mode. For the “+++” mismatch the quadrupolar
mode is not excited at all and one obtains a mixture of the low and fast mode redistribution patterns. In
this case the two fixed points are clearly visible, even if they are slightly shifted by the superposition.
We note that the partitioning of modes that are excited by a general mismatch is dependent on the emit-
tances and tunes of the machine and can be completely different from the one in Fig. 4.12. When studying
the effects of mismatch in a particular linac it is therefore advisable either to excite all three eigenmodes
separately or to use at least a few different sets of general mismatches.
48 4.6. MISMATCH FOR REALISTIC LINAC BEAMS
r.m.s. radii
artibrary units
−600
−400
−200
0
200
400
2 3 4 5
quadrupolar mode
low frequency mode
high frequency mode
600
10
r.m.s. radii
arbitrary units
−600
−400
−200
0
200
400
600
1 2 3 4 5
+−+ mismatch
+++ mismatch
Figure 4.12: (a) Particle redistribution when excit-
ing the three eigenmodes in a transport
channel with 50 periods. Plotted is the
difference in transverse particle density
between the matched and mismatched
distribution at the end of the channel,
IMPACT multi-particle simulation.
(b) Particle redistribution when exciting a gen-
eral 30% envelope mismatch in a transport chan-
nel with 50 periods. Plotted is the difference
in transverse particle density between the mis-
matched and the matched distribution at the end
of the channel, IMPACT multi-particle simula-
tion.
4.6.2 Maximum halo extent
The previous section confirms the prediction from the simple 1D particle-core model that halo particles
gather around fixed points and that the distance of these fixed points from the core is given by the oscil-
lation frequency of the mismatched core and its 2:1 parametric ratio with the single-particle oscillations.
Beyond the orbits around these fixed points the particles cannot gain more energy from the core oscil-
lations, simply because the 2:1 resonance moves out of phase. Although it was found that fixed point -
core distances can theoretically go to infinity [57] one observes that maximum halo amplitudes for the
actual particles are limited to a certain threshold. Hofmann suggests in [58] that even if the oscillations
are still “in phase” for increasing fixed point - core distances, there is a decrease in the space-charge cou-
pling force, without which there can be no energy transfer from the core oscillations to the single-particle
orbits.
With a simple test one can verify the idea of a maximum halo extent: using again the superconduct-
ing section of SPLI, we excite a “+++” mismatch for a Gaussian input beam. Figure 4.13 shows the
transverse distribution at the end of the linac for different amplitudes of initial mismatch.
Even in simulations with very large numbers of particles (here: 10 million) one finds that the redistribu-
tion pattern as well as the maximum halo amplitude remain basically the same for the different mismatch
amplitudes. Only the particle density towards the maximum amplitude increases with increasing mis-
match. In this example the outermost particles extend to a maximum value of ≈8times the r.m.s. enve-
lope (8a). However, due to imperfections in the lattice or transitions between sections, maximum values
of up to ≈12aare found in simulations. As an example we show in Fig. 4.14 the maximum transverse
halo extent for a strongly mismatched (40%) beam in a normal conducting linac (3 - 120MeV), with two
lattice transitions (at 18m and 41m).
CHAPTER 4. INITIAL MISMATCH 49
0.1
1
10
100
1000
10000
0123456789
particles per unit length
r.m.s. radii
40% mismatch
30% mismatch
20% mismatch
matched
Figure 4.13: Transverse particle density (in arbitrary units) at the end of SPLI (120 – 2200 MeV, 40mA
simulation current, Gaussian input beam, 107particles) when exciting a “+++” mismatch
with different amplitudes, IMPACT multi-particle simulation
2
3
4
5
6
7
8
9
10
11
12
0 10 20 30 40 50 60 70
100%(mismatched)/rms(matched)
length [m]
Figure 4.14: Transverse halo extent in multiples of the r.m.s. beam envelope for a normal conducting
linac (3 - 120MeV, 40mA simulation current, 40% “+++” mismatch, Gaussian input beam),
IMPACT multi-particle simulation
4.6.3 Beam collimation
An important aspect in the beam halo discussion is the question of the rise time for emittance growth
or beam halo to develop. As discussed in the previous section, halo amplitudes between 8 and 12aare
observed in simulations. This means that a simplistic approach to avoiding transverse beam loss would
be to use apertures larger than 12 times the r.m.s. beam envelope. In normal conducting RF accelerating
structures, however, the size of the aperture is directly linked to the power requirements of the cavities,
and large bore radii unavoidably yield a poor RF efficiency. If the focusing quadrupoles are separated
from the RF, as for instance in Coupled Cavity Drift Tube Linacs (CCDTL), Separated DTLs (SDTL), or
Coupled Cavity Linacs (CCL), one can use quadrupoles with large bore radii without decreasing the RF
efficiency3. Since the beam envelopes reach their highest values in the quadrupoles, enlarging their bore
radius is an effective means to reduce beam loss. In a classical Alvarez DTL, where the quadrupoles are
3examples of the structures can be found in Appendix B
50 4.6. MISMATCH FOR REALISTIC LINAC BEAMS
housed in the drift tubes, this trick cannot be applied and the apertures are typically in a range of 5 to 7a,
depending on the average beam power in the structure. Here, the only solution to reduce beam loss is the
use of scrapers before entering the DTL. In that case the rise time for halo formation really becomes the
crucial parameter for loss prediction in the machine. As an example Fig. 4.15 shows the evolution of the
(longitudinally) most unstable particle that was found in the simulation of an ideal transport channel (the
transverse behaviour can be assumed identical).
0 20 40 60 80 100 120 140 160
period
−40
−20
0
20
40
single particle phase (deg)
Figure 4.15: Evolution of the (longitudinally) most unstable particle in a transport channel of 160 periods
with fast mode excitation and an initial 6D waterbag distribution. Plotted is one point per
period. The dashed line indicates the maximum initial mismatched phase extent and the
solid line indicates twice that value. Example from Bongardt & Pabst.
One can see that the most unstable particle after 160 periods is
not
the one that initially has the largest
phase value. This means that even if halo scraping is employed at the low-energy stages of a linac,
particles can still be expelled to large amplitude orbits. On the other hand the rise time in this particular
example amounts to ≈60 focusing periods, which is already longer than many linacs.
Judging from the theory and the simulation results it is vital in every high-power linac design to avoid
all sources of mismatch. These include not only initial mismatch but also mismatch between different
sections. Once the beam is mismatched the only way to damp the core oscillations and to return to an
equilibrium is the development of beam halo. This means that beam collimation and halo scraping can
only help to control the effects of mismatch (beam loss) but they can certainly not correct the mismatch
itself, nor will they prevent further halo formation.
This behaviour is illustrated in the following by placing collimators, long enough to remove halo particles
from both planes of the xand yphase space, at 4 different positions in a FODO channel (Fig. 4.16). At
all 4 positions particles beyond 3.2 r.m.s. amplitudes are removed from the beam.
From Fig. 4.17 one can appreciate that in all 4 cases the amount of beam loss in the scrapers is approx-
imately the same, while the overall r.m.s. emittance increase and the maximum halo amplitude depend
on the position of the collimators. Obviously, the scraping is most effective at position 4, where the
core oscillations are already damped and where the beam has settled into a new equilibrium. From this
position onward only little additional emittance growth takes place. Considering a realistic linac with
several sources of mismatch (e.g. lattice transitions, statistical errors) this means that several collimators
should be foreseen at positions sufficiently far away from the actual mismatch sources.
CHAPTER 4. INITIAL MISMATCH 51
0.7
0.8
0.9
1
1.1
1.2
1.3
0 10 20 30 40 50 60 70
(1+∆x)/x
length [m]
x−plane
scraper 2
scraper 1
scraper 3
scraper 4
Figure 4.16: Mismatch oscillations due to a fast mode excitation with 30% initial amplitude mismatch
in a FODO channel using a Gaussian distribution and showing 4 possible scraper positions
(shown is the unscraped beam), IMPACT multi-particle simulation
1e−05
1e−04
1e−03
1e−02
1e−01
1
1 2 3 4 5 6 7 8
fraction of particles
r.m.s. radii
nominal (w/o mismatch)
scraper1
scraper2
scraper3
scraper4
no scraper
loss:3.9%,∆ε=10%
loss:3.8%,∆ε=0.5%
loss:3.7%,∆ε=2%
loss:3.5%,∆ε=5%
loss:0%,∆ε=26%
Figure 4.17: Final particle distribution and r.m.s. emittance growth for 4 scraper positions in a FODO
channel, 30% fast mode excitation, Gaussian distribution, IMPACT multi-particle simula-
tion
52 4.6. MISMATCH FOR REALISTIC LINAC BEAMS
For linear accelerators which send their beam directly onto a target longitudinal halo is usually of minor
interest. If, however, the linac beam enters a subsequent ring system for beam storage or further acceler-
ation, the longitudinal behaviour becomes more important. Especially in accumulator and/or compressor
rings with no acceleration, the longitudinal bunch shape has to be well confined to fit into the RF bucket
of the ring system. In this case the bulk of the losses is triggered by the linac RF jitter, the uncertainty
of the energy and phase level at the end of the linac which is caused by the finite tolerances of the RF
system. Losses due to halo formation in the longitudinal plane will only have a minor contribution.
CHAPTER 5. DISTRIBUTED MISMATCH 53
5. Distributed mismatch
Another contribution to the development of beam halo stems from statistical gradient errors which can
be regarded as a multitude of individual mismatch sources. The particle-core model suggests that the
main condition for halo development is the parametric 2:1 ratio between the core and single-particle
oscillations. The 3D simulations (Fig. 4.7, 4.8, and 4.9) of the SPLI lattice show that once the core os-
cillations are excited they can remain remarkably stable throughout a complete linac even though several
(matched) lattice transitions are crossed and even though the focusing constants change considerably. At
some point the oscillations become damped by the transformation of the “free energy” (introduced via
mismatch) into emittance growth and it is important to realize that this is the only process by which the
oscillation amplitudes are reduced.
In 3D error simulations for Linac4 [59], the normal conducting front-end of the SPL, it was found that
similar core oscillations can be excited by statistically distributed quadrupole gradient errors and that
also there the resulting core oscillations remain remarkably stable throughout various lattice changes.
Figure 5.1 shows an example of these core oscillations (triggered by statistical errors on the quadrupole
gradients) for the Linac4 lattice, comprising a 3MeV chopper line, followed by 3 DTL tanks and 37
CCDTL tanks.
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
1 + ∆b/b
PSfrag replacements
1 + ∆ax/ax
1 + ∆ay/ay
1 + ∆b/b
1 + ∆ax/ax
1 + ∆ay/ay
1 + ∆b/b
Figure 5.1: (a) Worst case envelope deviations
for 1% (total) quadrupole gradient er-
rors in the Linac4 front-end, IMPACT
multi-particle simulation
(b) Worst case envelope deviations for Linac4 up
to 120 MeV, IMPACT multi-particle simulation
One can see that for statistical gradient errors of only 1%, the worst case envelope deviations increase
the regular beam amplitudes by up to 40% corresponding to a 40% initial mismatch!
In the following, the 1D particle-core model from Section 4.2 is extended to study if it can predict regular
core oscillations caused by statistical gradient errors and if these core oscillations can be a source of halo
formation. Furthermore systematic 3D tracking studies will be used to quantify the effects for a realistic
lattice (see also [60]).
54 5.1. PARTICLE-CORE MODEL FOR STATISTICAL GRADIENT ERRORS
5.1 Particle-core model for statistical gradient errors
Starting point is again the 1D smoothed envelope equation [see Eq. (4.1)] and the same set of variables
and conventions introduced in section 4.2. In every focusing period a small error is added to the focusing
constant k0, simulating the effect of statistically distributed gradient errors in a transport channel.
r00
c+ (k0+ ∆)2rc−ε2
r3
c−K1
rc= 0 (5.1)
The movements of the single particle in the space-charge field of the core are then given by [compare
Eq. (4.3)]
x00 + (k0+ ∆)2x−Fsc = 0 (5.2)
with the space the space-charge force term (of a continuous round beam) being defined as
Fsc =(K1x/r2
c:|x|< rc
K1/x :|x| ≥ rc(5.3)
The errors are applied using a Gaussian error distribution with a cut-off at twice the r.m.s. value. Fig-
ure 5.2 shows the result of integrating the core and single particle equations [Eq. (5.1) and Eq. (5.2)]
assuming a randomly chosen error set with 1% (r.m.s.) focusing error. The initial single particle ampli-
tude in this example is |x(0)|/r0= 1.2with r0being the matched core radius for ∆ = 0. It is evident
that once ∆6= 0, the beam will be mismatched in this period. With ∆changing from period to period
the beam is subject to a different small mismatch in every period.
-1.5
-1
-0.5
0
0.5
1
1.5
0 5 10 15 20 25 30 35 40 45
PSfrag replacements
rc/r0x/r0
zk0/2π
core
particle
Figure 5.2: Particle oscillations & core envelope in case of a matched beam with 1% (r.m.s.) statistical
focusing gradient errors (tune depression τ= 0.8,|x(0)|/r0= 1.2)
After a certain number of periods the core starts oscillating in a similar manner as for initial mismatch.
Due to the irregular excitation of the core, its oscillation amplitude is now subject to a slowly changing
random modulation. The oscillation frequency, however, remains almost constant at the frequency of
CHAPTER 5. DISTRIBUTED MISMATCH 55
the eigenmode and only changes within a few per cent around this reference value. Only for very small
oscillations, as in the first periods of Fig. 5.2, does the oscillation frequency change by up to ≈20%.
Even though the errors are statistically distributed, the core oscillations can reach considerable am-
plitudes, a phenomenon that can be observed in realistic linac lattices as in Fig. 5.1. Since the core-
oscillations are irregular as opposed to those resulting from initial mismatch (e.g. Fig. 4.1), it is unlikely
for single particles to enter a stable parametric resonance for more than a few oscillation periods. Never-
theless, there are sections in the lattice, when the core oscillations maintain an almost constant amplitude
(e.g. periods 25-40 in Fig. 5.2) and during these it seems that the single particle enters a similar oscil-
lation pattern as for the initial mismatch in Fig. 4.1: the maximum amplitude of the single particle is
rising and falling with a more or less sinusoidal modulation. This pattern suggests that even a few os-
cillation periods which are more or less ‘in phase’ with the core oscillations suffice to transfer energy
from the core to the single particle trajectories. Comparing once more with the case of initial mismatch
in Fig. 4.1 this explanation seems very likely, since also there the energy transfer practically starts from
the first few core oscillations onwards, implying that the mechanism to transfer energy is the same as for
initial mismatch. In the case of initial mismatch the core immediately oscillates with a large amplitude
yielding an increase of single particle amplitudes by a factor of 2-3 within a few periods. For statistical
gradient errors, however, the particle-core model suggests that several hundred periods may be necessary
to achieve core oscillation amplitudes that are large and stable enough to trigger a significant increase in
single particle amplitudes.
Plotting the maximum and minimum values of core and single particle amplitudes along the channel
[using again Eq. (5.1) and Eq. (5.2)] for different initial single particle amplitudes (Fig. 5.3) confirms
that the single particle envelopes are oscillating in a manner similar to initial mismatch. It also shows
that particles from within the core boundaries are only weakly affected, whereas particles starting outside
of the core boundaries show a clear increase in their oscillation amplitudes.
-6
-4
-2
0
2
4
6
0 100 200 300 400 500
PSfrag replacements
rc/r0x/r0
zk0/2π
single particles
core
Figure 5.3: Maximum particle/core amplitude values in case of a matched beam with 1% r.m.s. statistical
focusing gradient errors (τ= 0.8,0.2≤ |x(0)|/r0≤3.5)
56 5.1. PARTICLE-CORE MODEL FOR STATISTICAL GRADIENT ERRORS
5.1.1 Average effects and evidence for a resonant process
In order to show which particles experience on average the largest amplitude growth, Figs. 5.4 (a), (b),
and 5.5 show the growth factors for single particle amplitudes as a function of their initial values. Each
of the plotted points represents the average of maximum amplitude values found in 1000 simulations
[using Eq. (5.1) and Eq. (5.2)] with different error sets. The curves connecting the points are interpolated
with cubic splines.
PSfrag replacements
PSfrag replacements
Figure 5.4: (a) Maximum amplitude growth for
single particles over 100 zero-current
betatron periods versus their initial am-
plitude and tune depression τ. Each dot
represents the averaged results of 1000
runs with different error sets with 1%
r.m.s. error.
(b) Maximum amplitude growth for single par-
ticles over 100 zero-current betatron periods as
a function of their initial amplitude and of the
r.m.s. error amplitude. Each dot represents the
averaged results of 1000 runs with different er-
ror sets for a tune depression of τ= 0.8.
Figure 5.4 (a) shows that for an emittance dominated beam (0.71 < τ < 1.0) the maximum growth
factors (or halo amplitudes) clearly depend on the tune depression of the beam, while they remain basi-
cally constant for space-charge dominated beams (0.0<τ<0.71). The same observation was made by
Wangler [43] who studied halo development due to initial mismatch using the particle core model.
Figure 5.4 (b) shows, not surprisingly, that the maximum oscillation amplitudes depend on the amplitude
of the statistical variation that is applied to the focusing forces, and finally, Fig. 5.5 explores how the
maximum amplitude growth depends on the length of the simulated transport channel.
In all three cases (Figs. 5.4 (a), (b), and 5.5) the maximum amplitude growth is found for particles with
initial amplitudes around 1.5r0, suggesting that these particles have the highest probability of entering a
parametric resonance with the core oscillation.
In order to show that the underlying mechanism is indeed based on a resonance, Fig. 5.6 shows the wave
numbers (tunes per metre) for the core (kcore) and the single particle (2kparticle) oscillations, assuming one
randomly chosen error set with different initial amplitudes for the single particles. The data is smoothed
by approximating the raw data with bezier curves of n’th order (n - number of data points) which connect
the end points. For simplicity we use the bezier smoothing which is available in gnuplot [61] and which
reduces the amplitudes of the rapidly changing wave numbers while making it easier to distinguish the
trends in the raw data.
CHAPTER 5. DISTRIBUTED MISMATCH 57
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
0.5 1 1.5 2 2.5 3 3.5
100 periods
200 periods
400 periods
600 periods
800 periods
1000 periods
PSfrag replacements
multiples of r0
average growth
Figure 5.5: Maximum amplitude growth for single particles as a function of their initial amplitude and
the length of the transport channel (in zero-current betatron periods). Each dot represents the
averaged results of 1000 runs with different error sets for a tune depression of τ= 0.8and
an r.m.s. error amplitude of 1%.
1.1
1.9
1.5
1.3
1.2
core
2.7
2.1
2.5 2.3
1.7
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
0 50 100 150 200
PSfrag replacements
kcore
zk0/2π
kcore,2kparticle
Figure 5.6: Wave numbers (arbitrary units) for core and single particle oscillations. Single particle wave
numbers are multiplied by 2 and depict particles starting with different initial amplitudes (1.1
- 2.7 times the matched core radius), τ= 0.8, r.m.s. error amplitude: 1%.
After 20 - 30 zero-current betatron periods the core wave number assumes a relatively stable value
of ≈3.7(arbitrary units). A few periods later the wave numbers of particles, that start with initial
amplitudes between 1.3 and 2.1 times the matched core radius, begin to oscillate around an average
that is half as big as the core wave number, meaning that they fulfil the condition for a parametric 1:2
resonance with the core. It is interesting to note that particles whose initial wave numbers are very close
to 0.5 times the core wave number start very quickly to resonate with the core, while it takes more time
until particles with larger or smaller wave numbers are affected. Particles that start with initial amplitudes
of |x(0)| ≤ 1.2r0or |x(0)| ≥ 2.3r0have either too small or too high a wave number to enter a resonance
within the length of the calculated transport channel.
58 5.1. PARTICLE-CORE MODEL FOR STATISTICAL GRADIENT ERRORS
One can expect that the range of particles likely to interact with the core oscillations will be larger in
a more realistic 3D beam with altogether 3 different core-eigenfrequencies. Furthermore the maximum
amplitude values and their associated time constants will certainly be different for more realistic beams
with non-uniform distributions and non-linear space-charge forces within the core area.
Even though one could imagine that for statistical gradient errors, the particles experience on average as
much negative energy kicks as positive kicks, Fig. 5.5 clearly shows that statistical gradient errors have
a cumulative effect. This can be understood by interpreting these errors as a continuous supply of ‘free
energy’ which can only increase the transverse beam temperature (or energy) but never yield a decrease
(compare [50, 62]). In the case of initial mismatch the beam obtains one large energy kick at the begin-
ning and the particles as well as the core perform regular oscillations around their equilibrium. Without
further disturbance these oscillations are maintained indefinitely in the particle-core model, while a con-
tinuous supply of ‘energy’ via statistical gradient errors yields ever increasing particle amplitudes. We
note at this point that the core oscillations of a realistic 3D beam, caused by initial mismatch, are eventu-
ally damped by the energy transfer from the core to the single particle oscillations. This transformation
from mismatch to beam halo and r.m.s. emittance growth has been found to develop much more rapidly
for Gaussian beams than for the idealised 6D waterbag beams (compare Fig. 4.11 and also [51]).
5.1.2 Halo development
While the average effects give a good indication of the general particle behaviour due to statistical gra-
dient errors, they provide little information about the development of beam halo and its maximum extent
in phase space. To illustrate the formation of halo driven by statistical gradient errors we use the same
kind of stroboscopic plot that showed the resonance pattern for initial mismatch (e.g. Fig. 4.2). Fig-
ure 5.7 depicts the pattern of one particle for one particular error set, which has a larger than average
amplitude growth. One can see that initially the particle maintains a phase space trajectory close to the
core boundaries. At some point the particle receives a kick that, within a few periods, rapidly increases
its amplitude from ≈2r0to ≈3.5r0, where it remains for another ≈150 zero-current betatron periods.
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
PSfrag replacements
x/r0
x0/r0k0
Figure 5.7: Stroboscopic plot for single particle with: |x(0)|/r0= 1.2, 1% r.m.s. error, 1000 zero-current
betatron periods
CHAPTER 5. DISTRIBUTED MISMATCH 59
Later on another kick expels the particle even further to amplitudes >4r0. Contrary to initial mismatch,
particles which are subject to statistical gradient errors seem to increase their amplitudes further and
further, provided the length of transport channel is long enough.
To assess more generally the potential for halo development we evaluate in Fig. 5.8 the probability for
particles to reach large amplitudes as a function of their initial amplitude and the length of the transport
channel.
PSfrag replacements
200 periods
PSfrag replacements
Figure 5.8: Probability for single particles to reach large amplitudes; 200 and 800 zero-current betatron
periods. The black bars represent one out of a total of 1000 simulations.
Two features in Fig. 5.8 are worth noting: a) particles that start at 1.2<|x(0)|/r0<2.1show an almost
equal probability of reaching certain large amplitudes, which can be explained by the enhanced amplitude
growth that was observed earlier for particles withinitial amplitudes around ≈1.5r0. b) Particles starting
from within the core area (x(0)/r0<1) show extremely low probability to actually transgress the core
boundaries and reach larger amplitudes. As soon as the initial particle amplitudes are slightly larger than
the matched core radius (x(0)/r0>1) the probability for amplitude growth increases considerably.
5.2 Limitations of and conclusions from the particle-core
model
A continuous, azimuthally symmetric, un-bunched beam in a constant focusing channel is a very crude
approximation for a realistic 3D linac beam propagating through a periodic, accelerating lattice which
is subject to the influence of non-linear forces. However, as a proof-of-principle, the model shows that
halo development can be triggered by statistical gradient errors and that the process of expelling single
particles to large amplitudes is based on a parametric resonance. It also suggests that in a space-charge
dominated beam the maximum halo amplitudes are independent of the tune depression. Furthermore it
shows that the maximum halo extent depends on the error amplitudes and on the length of the transport
channel. Contrary to initial mismatch there seems to be no boundary for the extent of the halo produced,
suggesting that particles will inevitably get lost even if one uses accelerating lattices that allow large
bore radii (e.g. superconducting cavities). The model cannot predict r.m.s. emittance growth and only
allows very limited conclusions about the time constants for halo development along with the associated
maximum amplitudes. While the time constants for single particles to become halo particles might seem
very long, the situation certainly changes when considering the large number of particles in a realistic
beam. There, even a tiny fraction of particles (10−5−10−7)acquiring large amplitudes, may cause
losses that can impose limits on the maximum beam power for the operation of the machine.
60 5.3. 3D PARTICLE TRACKING
5.3 3D particle tracking
To study the effects of statistical gradient errors on a realistic 3D beam we use a periodic focusing
channel without acceleration and with the same basic properties as used for the particle-core model (see
Table 4.1). For each case considered in the following, 500 randomly created error sets (Gaussian error
distribution with cut-off at twice the r.m.s. value) are simulated with the 3D tracking code IMPACT [13],
using 105particles.
The particle-core model predicts (see Fig. 5.4) that the maximum amplitude growth for single particles
only depends on the tune depression if the beam is in the emittance dominated regime (0.7<τ<1). To
verify this finding for a realistic 3D beam, IMPACT is used to calculate the r.m.s. emittance growth and
the maximum beam amplitudes for a Gaussian particle distribution. The first plot in Fig. 5.9 shows the
r.m.s. emittance growth versus tune depression assuming 0.5% and 1.0% r.m.s. errors for the quadrupoles
of the periodic focusing channel.
0
10
20
30
40
50
60
70
0.4 0.5 0.6 0.7 0.8 0.9
rms emittance growth
tune depression
scaling law
1% rms error
0.5% rms error
Figure 5.9: R.m.s. emittance growth for 0.5% and 1% r.m.s. quadrupole errors versus tune depression,
scaling law: ∼(k/k0)−3/2(200 focusing periods, 6D Gauss, 500 simulations per data point,
IMPACT multi-particle simulation)
One can see that the emittance growth rises with stronger tune depression and also with increasing errors
on the quadrupoles. The increase in r.m.s. emittance growth in the simulated region (0.4< τ < 0.9)
scales approximately with: (k/k0)−3/2and, in contrast to the predictions from the particle-core model,
no difference in the growth pattern can be seen for emittance dominated or space-charge dominated
beams. In order to judge if the maximum halo amplitude is also sensitive to changes in tune depression
for space-charge dominated beams Fig. 5.10 can be used. It shows the average fraction of particles
exceeding a certain multiple of the (average) transverse r.m.s.
input
emittance (εt= (εx+εy)/2).
Again, in contrast to the prediction from the particle-core model, the maximum halo amplitude increases
with stronger tune depression in the emittance-dominated
and
space-charge dominated regime. However,
in agreement with the particle-core model, the halo amplitudes increase with increasing amplitudes of
the r.m.s. quadrupole errors.
CHAPTER 5. DISTRIBUTED MISMATCH 61
10-5
10-4
10-3
10-2
10-1
1
0 20 40 60 80 100
fraction of particles
rms (input) emittances: εrms,t
k/k0=0.9
k/k0=0.8
k/k0=0.7
k/k0=0.6
k/k0=0.5
k/k0=0.4
10-5
10-4
10-3
10-2
10-1
1
0 20 40 60 80 100
fraction of particles
rms (input) emittances: εrms,t
k/k0=0.9
k/k0=0.8
k/k0=0.7
k/k0=0.6
k/k0=0.5
k/k0=0.4
Figure 5.10: Averaged fraction of particles exceeding multiples of the r.m.s. input emittance (εr.m.s.,t)
for transport channels with different tune depressions, left: 0.5% r.m.s. quadrupole error,
right: 1.0% r.m.s. quadrupole error (200 focusing periods, 6D Gauss, 500 simulations per
curve, IMPACT multi-particle simulation)
The next comparison with the particle-core model is shown in Fig. 5.11 which characterises the output
distribution for transport channels of different lengths.
1
0 20 40 60 80 100
PSfrag replacements
200 focusing periods
150 focusing periods
100 focusing periods
50 focusing periods
rms (input) emittances: nεrms,t
fraction of particles
100
10−1
10−2
10−3
10−4
10−5
Figure 5.11: Averaged fraction of particles exceeding multiples of the r.m.s. input emittance (εr.m.s.,t)
for transport channels of different length (100 focusing periods correspond to ≈10.7 zero-
current betatron periods), 6D Gauss with 1% (r.m.s.) quadrupole gradient errors, 500 simu-
lations, normalised to transverse
input
emittance, IMPACT multi-particle simulation
The fraction of particles exceeding the
input
distribution increases approximately linearly with the length
of the transport channel which coincides with the results from the particle-core model in Fig. 5.5. In terms
of the ‘free energy approach’ one can argue that statistical gradient errors represent a continuous feed of
‘free energy’ into the system, which is transformed into r.m.s. emittance growth and beam halo.
As predicted by the particle-core simulations, the maximum halo amplitudes in the case of statistical
gradient errors can reach significantly higher values than in the case of initial mismatch. The probability,
however, of reaching halo amplitudes in excess of 10 times the r.m.s. envelope amplitudes, seems very
62 5.3. 3D PARTICLE TRACKING
low. Nevertheless, the simulations suggest that the effects of statistical gradient errors can be seen in
linear accelerators, which have a high number of focusing elements in their low-energy sections. Below
150MeV one finds approximately between 5 and 10 zero-current betatron periods in a typical normal
conducting linac (≈200 quadrupoles plus 300 RF gaps, not counting the RFQ), which, depending on
the lattice characteristics, may already be long enough to yield significant losses due to statistical errors.
Using for instance only 5 zero-current betatron periods with a 1% r.m.s. gradient error (Fig. 5.11), one
already finds a fraction of 10−5of the particles beyond 50 r.m.s. input emittances which corresponds to ≈
7 r.m.s. envelope amplitudes (7a, an aperture limitation that often used on normal conducting accelerating
structures).
In synchrotrons or storage rings for space-charge dominated beams, where the bunches are transported
through a large number of lattice periods, statistical gradient errors may well account for the development
of a substantial parametric beam halo that has to be scraped by dedicated beam collimation systems. In
order to show that statistical errors not only increase the r.m.s. emittance but do in fact produce a low-
density beam halo we plot in Fig. 5.12 the averaged fraction of particles exceeding certain multiples of
the transverse r.m.s.
output
emittance for three different r.m.s. error amplitudes. By normalising each
curve by its output r.m.s. emittance one basically removes the contribution of the r.m.s. emittance growth
from the plots. The plot also compares the halo development with respect to an initial mismatch, using a
30% fast mode excitation at the beginning of the simulation.
0 20 40 60 80 100
PSfrag replacements
nominal output (∆ε= 7.3%)
0.5% rms error (∆ε= 16.6%)
1.0% rms error (∆ε= 45.3%)
1.5% rms error (∆ε= 99.1%)
30% fast mode (∆ε= 27%)
rms (output) emittances: nεrms,t
fraction of particles
100
10−1
10−2
10−3
10−4
10−5
Figure 5.12: Averaged fraction of particles exceeding multiples or the respective r.m.s. output emittance
εr.m.s.,t for statistical gradient errors. (500 simulations per curve, 6D Gauss, 200 focusing
periods, IMPACT multi-particle simulation, ∆ε→r.m.s. emittance growth.)
In all cases one can observe a clear halo development. However, compared to the fast mode initial
mismatch we find that, on average, the effects of statistical errors are much less dramatic. For the fast
mode excitation one can observe a certain ‘hump’ in the output distribution which is likely to be a result
of the redistribution pattern of the mismatched beam core (compare particle redistribution due to initial
mismatch in Fig. 4.12a). For statistical gradient errors the resonant conditions are changing very rapidly
and thus the output distributions become very smooth.
It was stated earlier (page 46, Fig. 4.11) that in case of initial mismatch beams with Gaussian distributions
are much quicker to develop beam halo than beams with waterbag distributions. For statistical errors we
find that there is not such a distinct difference between the two distributions. Figure 5.13 shows the
averaged fraction of particles exceeding a certain multiple of the r.m.s.
output
emittance for simulations
with initial waterbag and Gaussian distributions using initial mismatch and/or statistical errors.
CHAPTER 5. DISTRIBUTED MISMATCH 63
1
0 20 40 60 80 100
PSfrag replacements
G, 30% fast mode (∆ε= 27%)
WB, 30% fast mode (∆ε= 7%)
WB, 1% rms error (∆ε= 39%)
G, 1% rms error (∆ε= 45%)
rms (output) emittances: nεrms,t
fraction of particles
100
10−1
10−2
10−3
10−4
10−5
Figure 5.13: Averaged fraction of particles exceeding multiples of the r.m.s. output emittance εr.m.s.,t
for statistical error runs. (500 simulations, 6D Gauss and waterbag, 200 focusing periods,
IMPACT multi-particle simulation, ∆ε→r.m.s. emittance growth)
While in the case of initial mismatch the difference between the Gaussian and the waterbag beam (27%
r.m.s. emittance growth versus 7%) is clearly visible, only a very small difference can be observed in
case of statistical errors (45% versus 39%). This is indeed quite surprising since the particle-core model
seems to suggest that those particles mainly affected by statistical errors are initially outside the core in
which case the Gaussian distribution should be affected much stronger than the waterbag distribution.
5.4 Conclusions on statistical gradient errors
Using a simple particle-core model one can show that statistical gradient errors can excite oscillations
of the beam core which can then, via parametric resonances, transfer energy to single particles. 3D
simulations for a periodic transport channel show that this mechanism not only results in r.m.s. emittance
growth but that it also initiates the development of a low-density beam halo surrounding the core. The
process scales more or less linearly with the error amplitude (see Fig. 5.12) and the length of the transport
channel (see Fig. 5.11) but seems to be almost insensitive to the type of particle distribution that is chosen
in 3D simulations (Fig. 5.13). The particle-core model predicts that for emittance dominated beams the
maximum amplitudes depend on the level of tune depression and reach their maximum when entering
the space-charge dominated regime (τ < 0.7) where no further dependency on the tune depression is
observed. More realistic 3D simulations, however, show that also in the space-charge dominated regime
the r.m.s. emittance growth as well as the maximum halo amplitudes rise with stronger tune depression.
For most simulations a 1% r.m.s. gradient error was applied, which may seem high in view of the spec-
ifications for new linac projects like the Spallation Neutron Source (SNS) (0.14% r.m.s. quadrupole
gradient error [63]). Nevertheless, most of todays operating accelerators do not fulfil such tight margins
and one should keep in mind that quadrupole gradient errors are only one error type that has to be added
to a whole range of error sources. For high-power linacs, which are planned to operate with a beam
power of 10MW or more (e.g. linacs as fusion drivers or for waste transmutation), even much smaller
error margins may not be sufficient to prevent the development of unwanted and harmful beam halo. As
a rule of thumb one finds that the effect of a 1% r.m.s. quadrupole gradient error in terms of emittance
growth and halo development, is comparable to the effect of a 20% initial fast mode excitation (assum-
64 5.4. CONCLUSIONS ON STATISTICAL GRADIENT ERRORS
ing a transport channel of ≈15 zero-current betatron periods and a space-charge dominated beam). In
circular accelerators, statistical gradient errors provide a continuous source for halo development as long
as the beam is in a regime with relevant space-charge forces. Here, they may at least partly explain the
characteristic ‘halo shoulders’ that can be observed in synchrotrons and that have to be controlled by
beam collimation systems.
Forthe time being the author is not aware of any significant studies on the subject of halo development for
off-centred beams, which are caused by the misalignment of lattice elements. The resulting oscillations
of the beam core will be introduced in the next chapter as 1st order eigenmodes of the beam. Since in
this case the whole beam including the outermost particles oscillates around the nominal beam axis there
is no obvious possibility for exciting a resonance between the particles and the core. It is therefore likely
that any potential halo development will only occur as a 2nd order effect, and will certainly be much
smaller than halo caused by statistical gradient errors.
CHAPTER 6. CORE-CORE RESONANCES 65
6. Core-core resonances
For a long time it was believed that equipartitioning is a necessary feature of high-current linacs to avoid
emittance exchange due to space-charge effects. A beam is called equipartitioned in all three planes if
the following equation holds
σxεx=σyεy=σzεzequipartitioning condition (6.1)
where σis the phase advance (or tune) per focusing period and εthe r.m.s. emittance. The expres-
sions in Eq. (6.1) are usually understood as energies (Wi=σiεi) in the different planes and the idea is
that anisotropic (or non-equipartitioned) beams may exchange a certain amount of “energy” so that the
beam becomes equipartitioned. This effect was first identified as a coherent space-charge instability in
[64] where a number of coherent eigenmodes were derived analytically for anisotropic two-dimensional
beams. Jameson stressed the importance of these instabilities [65] and continued to recommend their
avoidance in the design of high-intensity linacs [66] The main difference with respect to the eigenmode
analysis for bunched beams in Section 4.3 is the requirement to incorporate non-elliptical and changing
bunch shapes as well as changing emittances. Therefore the analytical treatment is based on an integra-
tion of the Vlasov equation and is limited to the evaluation of a KV distribution in two dimensions (see
[18]). Until now no three-dimensional treatment has been found. Details of the 2D derivation are found
in [18] and [67]. The analysis provides a set of 2D eigenmodes which can be characterised as in Fig. 6.1.
4th order
envelope, skew
2nd order:
1st order:
dispacement 3d order
even
odd
Figure 6.1: Characterisation of core-core eigenmodes
Since the analytic treatment could not be taken any further, computer simulations were employed to
evaluate the effect of the identified eigenmodes on the performance of simulated multi-particle beams
[68, 67]. It was found that ideal KV beams are affected by oscillating and non-oscillating modes, while
waterbag beams are only sensitive to non-oscillatory (purely growing) instabilities. In both cases the
r.m.s. emittances are only affected if they have different starting values in the different planes. In this
context one needs to explain the terms “oscillatory” and “non-oscillatory”: an eigenmode can be char-
acterised by the eigenfrequency ωwith which a beam distribution changes in time (e−jωt). If the real
part of the eigenmode frequency is non-zero (Re(ω)6= 0) one speaks of an oscillatory mode, while non-
oscillatory modes are characterised by Re(ω) = 0. Strictly speaking the non-oscillatory modes should
not be labelled as eigenmodes but rather as coherent space-charge instabilities.
66 6.1. APPLICATION OF STABILITY CHARTS
From Fig. 6.1 one can see that the even modes are symmetric with respect to the horizontal axis, while
the odd modes have no such symmetry. Interpreting the two planes as
x
and
y
, the odd symmetry
corresponds to a lack of rotational symmetry around the longitudinal axis. It should be noted that this
type of mode cannot be found with
r
-
z
simulation codes which assume azimuthal beam symmetry.
First-order modes represent the trivial case of displacement from the beam axis, and the oscillation
frequencies are just the betatron frequencies without space-charge in each direction. The second-order
modes correspond to the beam eigenmodes that were derived in the Section 4.3. However, since this
is a 2D analysis we would find only the two eigenmodes for a DC beam (quadrupolar and breathing
mode). The real novelty of the Vlasov analysis are the third- and forth-order modes and, looking at
the oscillation shape, it becomes immediately clear why these modes are found only if the mathematical
tools allow non-elliptical bunch shapes. The areas where these coherent space-charge instabilities (due to
the third and fourth-order modes) affect the beam can then be visualised in “stability charts” (see Fig. 6.2
or Appendix D, [67]).
Up to this stage the theory was developed using 2D beams in idealised constant focusing structures with-
out acceleration. Even though the theory was developed already in 1979/80 it was never systematically
applied for the design of high-intensity linacs, and still in 1995 Jameson [69] writes with respect to the
coherent space-charge instablities: “Contemporary RF linac design,... has evolved empirically to usually
avoid the unstable regions.” A common approach was to simply choose an equipartitioned design, which
puts severe constraints on the design of the machine. In the following section we report the results of a
systematic study of emittance exchange for the superconducting part of the SPLI project that was initally
performed without knowledge of the stability charts. In subsequent studies the results were compared
with the predictions of the charts and first published in [70].
6.1 Application of stability charts
We now make the transition from the simplified 2D theory to a realistic linac structure with acceleration
using 3D bunched beams. Although the mathematical model is derived for anisotropy between the two
transverse planes of a DC beam, Hofmann suggested in [18] to apply the results for anisotropy between
the transverse and the longitudinal plane. This idea was tested and verified with 3D multi-particle simu-
lations using a simple constant-focusing lattice (as opposed to a periodic focusing lattice which is found
in real accelerators) without acceleration [71]. This approach was used for the superconducting part of
the SPLI project [70]. Figure 6.2 shows the stability chart for the SPLI emittance ratio of εl/εt= 2.
The shaded areas of the chart indicate where emittance exchange between the longitudinal and the trans-
verse plane is to be expected (the degree of shading indicates the speed of the process). For strong tune
depression (kx/kx0 ≤0.4) one obtains emittance exchange for all tune ratios, while for moderate values
stable beam operation seems possible in certain areas. The dashed line indicates the condition for an
equipartitioned beam [see Eq. (6.1)].
The application of the charts for a 3D beam with anisotropy between the longitudinal and transverse
plane is demonstrated with the following example: in two linac sections of SPLI (120 - 383MeV, 40mA
simulation current, waterbag distribution), the quadrupole gradients are modified to create three different
focusing lattices that fall into different areas of the stability chart in Fig. 6.2. The original SPLI and the
“case 1” lattice should both be stable, while “case 2” is clearly located in an unstable area. The result of
the simulations is plotted in Figs. 6.3 to 6.5.
In case of the SPLI and case 1 there is no significant change in the longitudinal/transverse r.m.s. emit-
CHAPTER 6. CORE-CORE RESONANCES 67
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
kx/kox
0 0.25 0.5 0.75 1 1.25 1.5 1.75
kz/kx
0,03
0,13
0,23
0,33
0,43
0,53
0,63
0,73
4-3 betatron periods
<2 betatron periods
equipartition
SPL
case 2 case 1
Figure 6.2: Stability chart for the SPLI emittance ratio of εl/εt= 2, source: Ingo Hofmann
tance values, while for case 2 one can observe a substantial emittance exchange from the longitudinal
to the transverse plane. We note that in this case there is one “hot” plane, the longitudinal one, which
is feeding the two “cold” transverse planes. For this reason there seems to be much more longitudinal
decrease than transverse increase. In an actual linac one should try to avoid a design where the transverse
emittance is higher than the longitudinal one, because in case of an exchange two “hot” planes would
feed one “cold” plane and a high longitudinal emittance increase would be the consequence.
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0 20 40 60 80 100 120 140 160 180
[pi-mm-mrad]
length [m]
longitudinal
transverse
Figure 6.3: R.m.s. emittance evolution for the SPLI lattice, IMPACT multi-particle simulation
68 6.1. APPLICATION OF STABILITY CHARTS
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0 20 40 60 80 100 120 140 160 180 200
[pi-mm-mrad]
length [m]
longitudinal
transverse
Figure 6.4: R.m.s. emittance evolution for case 1, IMPACT multi-particle simulation
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0 20 40 60 80 100 120 140 160 180 200
[pi-mm-mrad]
length [m]
longitudinal
transverse
Figure 6.5: R.m.s. emittance evolution for case 2, IMPACT multi-particle simulation
In recent years the stability charts have been successfully applied in various high-intensity linac projects
and should be regarded as a new tool in the design of linac lattices. Looking at the various stability
charts for different emittance ratios in Appendix D, one can see that small emittance ratios close to 1.0
yield large stable areas in the charts and vice versa. Nevertheless, equipartitioning does not appear to
be an obligatory design feature as was often assumed in the past. Even though emittance ratios close
to 1.0 provide larger stable areas, the demand for equipartitioning puts severe restrictions on the lattice
design which seems no longer justified. The emittance exchange itself takes place only if we have a
combination of beam anisotropy plus a certain tune ratio plus a minimum tune depression. Up to now
the understanding is that the most harmful resonance in the charts is the fourth-order even mode, which
is always located around a tune ratio of 1.0.
CHAPTER 6. CORE-CORE RESONANCES 69
6.2 Core-core resonances & beam halo
The obvious question to ask is if core-core resonances contribute to the development of beam halo and a
simple answer can be given by looking at the fractional emittances (99%, 99.9%, etc.) for case 2 of the
previous section.
In Fig. 6.6 we see that the outermost particles seem to be unaffected by the decreasing longitudinal
r.m.s. emittance, while the rising transverse r.m.s. emittance in Fig. 6.7 yields a slight increase in higher
fractional emittances. Altogether the ratio between the 99.99% emittances and the r.m.s. values increases
by ≈25%, which can hardly be referred to as halo development. This behaviour confirms the model,
which describes this kind of instability as a space-charge driven core-core resonance, which is an alto-
gether different process than the particle-core resonances previously discussed.
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
0 20 40 60 80 100 120 140 160 180 200
r.m.s.
99%
99.9%
99.99%
length [m]
Figure 6.6: Evolution of fractional longitudinal emittances for case 2. All emittances are normalised to
an initial value of 1.0, IMPACT multi-particle simulation.
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0 20 40 60 80 100 120 140 160 180 200
r.m.s.
99%
99.9%
99.99%
length [m]
Figure 6.7: Evolution of fractional transverse emittances for case 2. All emittances are normalised to an
initial value of 1.0, IMPACT multi-particle simulation.
70
7. Practical linac design
7.1 general rules
An important choice for each linac lattice concerns the evolution of the transverse and longitudinal phase
advance. High electric fields at low energy yield strong longitudinal focusing forces [see Eq. (2.27)]
and the limitation of keeping the longitudinal phase advance <90◦can impose severe constraints on
the maximum electric fields to be used in the first stages of acceleration. If beam loss has to be kept as
low as possible then this relation enforces the use of short focusing periods and clashes with the idea
of using superconducting cavities at low energy for the following reasons: the use of SC technology is
most efficient in case of long drift spaces between quadrupoles, so that there is enough space to fit a
cryostat with preferably several accelerating gaps. Long cryostats with several gaps, however often push
the phase advance too high, so that the electric field has to be kept lower than technologically feasible
and thus one of the advantages of SC technology is lost. For this reason the most common low-energy
(2 – 50MeV) structure for high-intensity linacs is still the classic Alvarez Drift Tube Linac (DTL). New
approaches for SC linacs use transverse focusing elements within the cryostat and thus try to overcome
the space limitations.
The longitudinal focusing of the accelerating cavities decreases with increasing beam energy (for the
same accelerating gradient) and if the linac aims to accelerate beyond a few hundred MeV then the
longitudinal phase advance has to be limited even further at low energy: we have seen that the stability
charts (see Appendix D) offer more than one stable area for linac operation. The two largest areas are
either close to equipartitioning with kl< ktor on the opposite side of the 4th order instability (which is
located around kt≈kl) at kl> kt. If the final linac energy is above a few hundred MeV and if one wants
to avoid crossing the unstable areas then the area close to equipartitioning is most suitable for accelerating
the beam and the appropriate tune ratio must be maintained throughout the machine. This means that in
order to stay in the stable area we have to impose a certain maximum tune ratio which depends on the
space-charge tune depression (e.g. kl/kt<0.8). Restricting the maximum transverse zero-current phase
advance to 85◦per period and assuming that the zero-current tune ratio approximately equals the full-
current ratio we obtain a limit of ≈65 −70◦for the maximum longitudinal zero-current phase advance
per period. Using these limits one can determine the maximum energy gain per period that respects this
limit which can be translated into a certain number of accelerating gaps or a maximum number (nmax) of
βλ per period1. With Eq. (2.21) one obtains a simple relation to determine nmax:
nmax ≤65◦π
180◦smc2βγ3
2πλ∆Wptan φs(7.1)
where ∆Wpdenotes the energy gain per focusing period. Depending on the actual tune depression one
can replace the 65◦in Eq. (7.1) with an appropriate value. The choice of focusing periods for SPLI
which was chosen [72, 73] according to Eq. (7.1) is plotted in Fig. 7.1. For SPLII the same rule was
applied to define the maximum electric gradients and the maximum number of accelerating cells per
period [2].
1in a DTL (0-mode structure) one βλ corresponds to one accelerating gap, while in a CCL (π/2-mode structure) one βλ
corresponds to two accelerating gaps.
CHAPTER 7. PRACTICAL LINAC DESIGN 71
0.15 0.2 0.25 0.3 0.35 0.4 0.45 β
6
8
10
12
n[ βλ]
Figure 7.1: Dashed lines →maximum number of βλ per focusing period to keep σl,0 <65◦, solid
lines →actual number of βλ per period between 3 and 120MeV. Lattice structures from
left to right: DTL, CCDTL with 3 gaps per cavity, CCDTL with 4 gaps per cavity.
In low-energy machines (e.g. <200 MeV) resonance crossing can be avoided and it can be considered
to operate the complete linac in a regime where kl> kt. In this case SC cavities would be preferable in
order to keep the longitudinal phase advance high throughout the machine.
Another design goal is to minimise the mismatch created by transitions from one accelerating structure
to the next. Different accelerating structures have different RF efficiencies depending on the particle
velocity and the RF frequency used. As one can see from Fig. 7.1 structure transitions often entail an
increase in the number of accelerating gaps per focusing period yielding an increase in the “real estate”
electric gradient. The most suitable way to minimise transition mismatch is to keep the phase advance
per metre smooth across the transition. This measure equals the matched beam size on both sides (see
Eq. [2.28]) even for beam currents different from the design current. Thus the transition becomes less
sensitive to changes in the nominal beam conditions and the additionally created mismatch is minimised.
7.2 Low-energy beam chopper
In many high-intensity linac designs low-energy beam choppers are used to minimise losses at injection
of the linac beam into the RF buckets of a subsequent ring system (see Section 1.1). To minimise the
deflecting voltage of the chopper plates while keeping the space-charge forces at an acceptable level,
these devices are usually located at energies of 2-3 MeV. Since the lattice of these transport lines has to
provide enough space to house 1–2 deflecting plates of typically 0.5m length, they demand a consider-
able change in the length of the focusing period. Figure 7.3 shows the impact of such a transition on the
phase advance per metre (in case of the ESS chopper line which is shown in Fig. 7.2) [37] and Fig. 7.4
shows the corresponding change in transverse and longitudinal beam size.
72 7.2. LOW-ENERGY BEAM CHOPPER
3.13 m
chopper A1 chopper B2 gap gapgapgapgapgap
F D F D F D F D
chopper A2
F D F
chopper B1
D F
Figure 7.2: Layout of the ESS chopper line
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14 16
phase advance [deg/m]
length [m]
x-plane
y-plane
z-plane
Figure 7.3: Full current phase advance per metre along chopper line and two DTL tanks in the ESS,
IMPACT multi-particle simulation
-3
-2
-1
0
1
2
3
0 2 4 6 8 10 12 14 16
r.m.s. radius [mm]
length [m]
x-plane
y-plane
0 2 4 6 8 10 12 14 16
length [m]
phase width
energy width
0
10
20
30
40
50
60
70
80
r.m.s. phasewidth [deg]
r.m.s. energy width [keV]
Figure 7.4: left: r.m.s. transverse beam envelopes along chopper line and two DTL tanks in the ESS;
right: total r.m.s. phase and energy width, IMPACT multi-particle simulation
To accommodate the chopper line, the phase advance per metre has to be changed drastically, and, as a
consequence the otherwise smooth focusing is severely disturbed making the matching into and out of
CHAPTER 7. PRACTICAL LINAC DESIGN 73
the chopper line not only difficult but also very sensitive to errors.
For successful operation the deflecting field between the chopper plates has to rise between two succes-
sive bunches in order to avoid partly chopped bunches. For this purpose the chopper amplifiers have to
provide rise times in the nanosecond range for electric fields of 0.5 – 3kV (depending on the design of
the line). Two main approaches can be taken to achieve clean chopping: a) Minimising the voltage on the
chopper plates eases the task of the amplifier but implies the use of long chopper lines, so that the small
deflection angle is sufficient to deviate the beam. In this case it is mandatory to optically enhance the
deflection by using a deflecting quadrupole between the chopper plates and the dump which collects the
deflected beam. b) Minimising the length of the chopper line reduces the emittance growth but increases
the demands for high voltage at fast rise times from the amplifier. Both solutions are currently under
study at RAL [74] and CERN [75] with the goal of building two test stands to investigate the technical
feasibility of each solution. The RAL approach is a revised version of the ESS chopper line and aims at
providing a front-end for future upgrades of the ISIS injector linac [76].
7.3 The SPL project at CERN
7.3.1 Introduction
After the shut-down of the Large Electron Proton (LEP) Collider at CERN a large number of supercon-
ducting RF cavities (including the cryogenic infrastructure) became available. It was before the actual
shut-down that the idea was conceived to recuperate these RF cavities for a superconducting proton linac
(SPL) at CERN [77, 78]. The first conceptual design report [1] focused on re-using the obsolete cavi-
ties, which were produced with the CERN-specific technique of sputtering a thin layer of niobium onto
welded copper cavities (Nb/Cu). The main application of this linac was a CERN-based neutrino factory
which dictated the timing scheme for the linac. Soon after this design report it became clear that the
use of newly developed medium-βNb/Cu cavities would be more efficient than the recuperation of the
old LEP cavities [79]. In 2005 the progress in superconducting bulk niobium cavities led to abandoning
the old Nb/Cu technology and to a further revision of the design, making the linac 40% shorter while
increasing the energy by 60% to 3.5GeV. The revised design of the SPL is referred to as SPLII and is
published in a 2nd conceptual design report [2].
There are two potential high-power users of the SPL II: 1) EURISOL and 2) neutrino production facil-
ities. The beam energy of 3.5GeV was chosen as a compromise between the preferred scenarios of the
EURISOL design study ([8], 1GeV, 5MW, CW) and the requirements of a neutrino production target
(energy range 5-10 GeV, 4MW, pulsed). For SPLII a relatively short linac pulse length was chosen in
order to minimise the number of injection turns into subsequent circular machines as well as to limit the
pulse length of the H−source. This means that SPLII has to use a relatively high peak current of up
to 64mA (output current) to produce a 4-5MW beam. Table 7.1 lists the main SPLII parameters for
“EURISOL” and “neutrino” operation and also lists the design differences to the first conceptual design
report.
The most dramatic change for the beam dynamics is the increase in peak current from 18.4mA to 64mA.
The resulting higher space-charge forces make it much more difficult to control the r.m.s. emittance
growth and to limit the development of beam halo.
74 7.3. THE SPL PROJECT AT CERN
Table 7.1: Main linac parameters and changes from SPLI to SPLII
SPLI SPLII (neutrino) SPLII (EURISOL)
energy 2.2 3.5 3.5 MeV
average beam power 4 4 5 MW
length 690 430 430 m
repetition rate 75 50 50 Hz
beam pulse length 2.2 0.57 0.71 + 0.014 ms
average pulse current (after chopping) 11 40 40 mA
peak bunch current (after 3MeV) 18.4 64 40 mA
beam duty cycle (after chopping) 16.5 2.9 3.6 %
injection turns (into ISR) 660 176 –
peak RF power 32 163 163 MW
no. of 352.2MHz tetrodes (0.1MW) 79 3 3
no. of 352.2MHz klystrons (1MW) 44 14 14
no. of 704.4MHz klystrons (5MW) – 44 44
cryo temperature 4.5 2 2 K
7.3.2 Layout and design
SPLII comprises the following sections: H−ion source, Low Energy Beam Transport (LEBT), RFQ,
DTL, Coupled Cavity DTL (CCDTL) (see Fig. B.2), Side Coupled Linac (SCL) (see Fig. B.3), and a
Superconducting (SC) linac that accelerates the beam from 180MeV to 3.5GeV. Figure 7.5 shows a
basic layout of SPLII. A detailed parameter list for this layout is given in Table 7.2.
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The H−source is necessary to make use of H−charge-exchange injection into a subsequent accumula-
tor/compressor ring. After a magnetic LEBT with with 2 solenoids, the RFQ accelerates the beam from
95kV to 3MeV. Following the RFQ a beam chopper allows to create gaps in the bunch train correspond-
ing to the transitions between RF buckets of a subsequent circular machine, reducing the injection losses
to acceptable levels. A performance test of H−source, LEBT, RFQ and chopper is foreseen in a dedi-
cated 3MeV test stand at CERN [75]. The RF frequency of 352.2 MHz for the normal conducting part
CHAPTER 7. PRACTICAL LINAC DESIGN 75
Table 7.2: Linac layout of the SPL II at CERN
section energy no. of no. of peak RF no. of no. of length
range cavities cells power LEP* 704.4 MHz†
[MeV] [MW] klystrons klystrons [m]
source, LEBT – 0.095 – – – – – 3
RFQ 0.095 – 3 1 560 1.0 1 – 6
chopper line 3 3 3 0.1 – – 3.7
DTL 3 – 40 3 85 3.8 5 – 13.4
CCDTL 40 – 90 24 72 6.4 8 – 25.2
SCL 90 – 180 24 264 15.1 – 5 34.4
β= 0.65 180 – 643 42 210 18.5 – 7 86
β= 1.0643 – 3560 136 680 116.7 – 32 256
total 161.6 14 44
* 352.2 MHz, 1MW, †4-5MW
of the linac is determined by the existing LEP klystrons. From 90MeV onwards 704.4MHz klystrons
are foreseen to power the normal conducting SCL which accelerates the beam up to 180 MeV. From
there 704.4MHz superconducting elliptical multi-cell cavities raise the beam energy up to its final value
of 3.5GeV. The higher frequency reduces the size and cost of the structures and it allows to use higher
RF gradients in the SCL. On the other hand the frequency transition introduces the difficulty of match-
ing a 352.2MHz beam out of the CCDTL into the 704.4MHz buckets of the SCL. After the chopper
line a classic Alvarez DTL is used to accelerate the beam to an energy of 40MeV. Due to the demands
of high current and low losses, short focusing periods are mandatory in this energy range. Having its
maximum shunt impedance at around 20MeV, a DTL is considered the most suitable structure for this
section. After 40MeV longer focusing periods can be accepted and one can separate the quadrupoles
from the RF structure. This measure simplifies the construction and eases considerably the alignment of
the quadrupoles, which is why the CCDTL structure was chosen to cover the energy range from 40 to
90MeV. Separating the quadrupoles from the RF structure has the additional advantage that larger bore
diameters can be used in the quadrupoles without compromising the RF efficiency of the accelerating
structure. With the increasing length of the drift tubes at higher energies it becomes more efficient to
use RF structures with a gap distance of βλ/2rather than βλ as used in the DTL-type structures. At
704.4MHz the chosen SCL structure can be machined out of solid copper at a reasonable price and is
used to raise the beam energy up to 180 MeV. From this energy onwards, two families of bulk-niobium
elliptical cavities with geometrical lengths adapted to particle velocities of β= 0.65 and β= 1.0ac-
celerate the beam to its final energy of 3.5GeV. This approach reduces the amount of R&D and the
production costs compared to options that foresee more families of SC cavities, which may result in a
slightly shorter linac length.
The basic beam dynamics design of SPLII follows the design rules described in the beginning of this
chapter. The zero-current phase advance in all three planes is below 90◦per period, and the ratio between
the longitudinal and transverse full-current phase advance is kept such that it avoids unstable areas in
the stability charts. Furthermore the phase advance per metre is kept smooth across all lattice transitions
(with the exception of RFQ/chopper and chopper/DTL). Table 7.3 shows the emittance growth per section
76 7.3. THE SPL PROJECT AT CERN
of the nominal beam in all three planes assuming a Gaussian beam which is matched at each structure
transition.
Table 7.3: Emittance growth and losses in SPLII for an initially Gaussian distribution [2]
RFQ chopper DTL CCDTL SCL SC total
energy [MeV] 3 3 40 90 180 3500 3–3500
∆εx[%] 8.5 29.7 -2.4 0.7 0.75 0.3 40
∆εy[%] 10.6 1.1 20.7 1.5 8.8 0.4 49
∆εz[%] – 9 11.9 0.3 0.9 4 25
transmission [%] 99.6 91.1 99.9 100 100 100 90.7
length [%] 6 3.7 13.4 25.2 34.2 341.8 425.5
In case of the CERN SPL, the chopper line also acts as a beam collimator which removes halo particles.
Nevertheless one can observe the largest emittance growth in the chopper line and after the transition
from chopper to DTL. As explained earlier, this behaviour is triggered by the extreme change in the
focusing constants (or phase advances per metre) around the transitions into and out of the chopper line.
Extensive error simulations with TRACE WIN [80] have been used to define limits for the statistical
errors in the SPL II lattice [2]. The chosen values, which are listed in Table 7.4, ensure that no additional
beam loss is caused by the statistical errors. The additional r.m.s. emittance growth due to these errors
amounts to 10%/12%/17% in the x/y/z planes of the normal conducting part and to 42%/44%/17% in
the x/y/z planes of the superconducting section. The working assumption is that these values can be
reduced or at least maintained for the real machine with the help of an orbit correction system. In case of
larger than expected emittance growth and halo development dedicated beam collimators will be needed
to localise any beam loss.
Table 7.4: Total acceptable error amplitudes in the normal and superconducting sections of the SPL II
quadrupole gradient ±0.5±0.5%
quadrupole displacement ±0.1±0.5mm
quadrupole rotation (x,y) ±0.5±0.25 deg
quadrupole rotation (z) ±0.2±0.5deg
cavity field phase ±1.0±1.0deg
cavity field amplitude ±1.0±1.0%
Acknowledgements
I want to thank K. Bongardt and I. Hofmann for introducing me to the subject of halo development and
for many discussions on the subject. Furthermore I wish to thank H. Henke and I. Hofmann for their
encouragement to write this thesis and for accepting the role of my thesis supervisors.
BIBLIOGRAPHY 77
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82
A. R.m.s. envelope equations and the
smooth approximation
A.1 Space-charge force term
In relativistic beams the expressions for fields and momenta need to take account of the relativistic
transformations. From the relativistic momentum
p=γmvwith γ=1
q1−β2
and β=v
c(A.1)
one can derive Newton’s equation for the relativistic case
F=dp
dt=γmdv
dt+mvdγ
dt(A.2)
For longitudinal forces, parallel to the direction of beam propagation, this relation can be written as
Fk=dpk
dt=γmdvk
dt+mvk
dγ
dt(A.3)
and using dγ
dt=βγ3dβ
dtone can write the (relativistic) Newton’s equation for the longitudinal plane as
Fk=γ3mdvk
dt
Newton’s equation
(longitudinal) (A.4)
For transverse motion, perpendicular to direction of beam propagation, the 2nd term on the right hand
side of Eq. (A.2) vanishes and one obtains
F⊥=γmdv⊥
dt
Newton’s equation
(transverse) (A.5)
In the next step we can now relate Eq. (A.4) and (A.5) to the forces created by the space-charge field of
the beam. A single particle of charge qwhich is moving with the beam experiences the effects of the
Lorentz Force
F=q(E+v×B)(A.6)
which means that the radial (transverse) forces are given by
Fr=q(Er−vzBφ) = qEr1−β2=qEr
γ2(A.7)
APPENDIX A. R.M.S. ENVELOPE EQUATIONS AND THE SMOOTH APPROXIMATION 83
and the longitudinal forces are given by
Fz=qEz(A.8)
Combining Eq. (A.4) with (A.8) and Eq. (A.5) with (A.7) we find that the longitudinal and transverse
acceleration can be written identically as
dvi
dt=qEi
γ3mwith i=x,y,z (A.9)
If one now makes the transformation from time derivatives to space derivatives using
dvx
dt=d2x
dt2=β2c2d2x
ds2=β2c2x00 (A.10)
one express the space-charge force term Fsc,i in the envelope equations as
Fsc,i =qEi
β2γ3mc2with i=x,y,z (A.11)
A.2 Thin lens approximation
The thin lens model is a first step to derive the transverse focusing constants for the smooth approxima-
tion. Here one assumes constant forces along each element with transitions being treated as hard-edged.
This approach allows to treat focusing elements as magnetic lenses of zero length, hence the name
thin
lens approximation
. Starting point is the transverse equation of motion
without
space-charge:
x00 +κ(s)x= 0 equation of motion w/o
space-charge (Hills equation) (A.12)
The general solution of Eq. (A.12) can be written in matrix form:
"x
x0#=M·"x0
x0
0#transfer matrix solution to
Hills Equation (A.13)
When transporting a beam through a series of lattice elements 1, 2, 3, ..., n, the total transport matrix
Mis given by the multiplication of all single element matrices M=Mn·...·M3·M2·M1. The
matrices for focusing and drift elements have the form:
Mdrift ="1l
0 1 #Mlens ="1 0
±1
f1#(A.14)
with lbeing the length of a drift and fbeing the focal length of a focusing element with length l= 0.
Once the total transfer matrix for one focusing period is calculated, one can relate the (known) focusing
constants fof the single lattice elements to the average (unknown) focusing constants k=σ/Lpof the
whole period by comparing the transfer matrix elements with the general transverse periodic solution in
matrix form:
Mp="cos σx,y,0 +αx,y sin σx,y,0 βx,y sin σx,y,0
−γx,y sin σx,y,0 cos σx,y,0 −αx,y sin σx,y,0 #(A.15)
84 A.3. QUADRUPOLE FOCUSING IN THE SMOOTH APPROXIMATION
The matrix in Eq. (A.15) can be derived from Hills equation (A.12) assuming that the beam is transported
through a periodic lattice. The derivation can be found in various text books (e.g. [81]) and is attributed
to [82].
A.3 Quadrupole focusing in the smooth approximation
Assuming magnetic gradients of equal strength in xand y(G=∂Bx
∂y =−∂By
∂x ) a particle of charge
qwill experience opposite Lorentz forces in xand yinside the quadrupole so that one can write the
s-dependent magnetic focusing components as:
κQ,x(s) = k2
Q(s) = |qG(s)|
mcβγ κQ,y(s) = −k2
Q(s)(A.16)
The focal length of a quadrupole in the thin lens approximation is given by
1
fQ=|κ|lQ=
qGlQ
mcβγ
quadrupole thin lens
approximation (A.17)
In order to replace the s-dependent quadrupole wave number kQ(s)which is only valid inside a quadrupole
by a constant wave number kQfor a full lattice period we use the smooth approximation. Starting in the
middle of a focusing quadrupole the thin lens transfer matrix for one FODO period is:
Mp=
1 0
−1
2fQ1
·"1l
0 1 #·"1 0
1
f1#·"1l
0 1 #·
1 0
−1
2fQ1
(A.18)
=
1−l2
2f2
Q2l1 + 1
2fQ
−l
2f2
Q1−l
2fQ1−l2
2f2
Q
(A.19)
Equating (A.15) and (A.19) with LP= 2lyields an expression for the transverse phase advance per
period l
2fQ=LP
4fQ= sin σQ
2≈σQ
2(A.20)
which can be used with Eq. (A.17) and k=σ/LPto define the quadrupole wave number in the smooth
approximation. The s-dependent focusing κ(s)in the envelope equation can thus be replaced by
κQ,x(s) = κQ,y(s)−→ k2
Q=qGlQ
2mcβγ 2smooth approximation for
transverse focusing in FODO
lattices w/o RF
(A.21)
Please note the difference in power and sign between the expressions for k2
Q(s)in Eq. (A.16) describing
a beam inside a quadrupole and k2
Qin Eq. (A.21) describing a FODO quadrupole channel in the smooth
approximation.
APPENDIX A. R.M.S. ENVELOPE EQUATIONS AND THE SMOOTH APPROXIMATION 85
A.4 RF focusing in the smooth approximation
The forces of the RF system on the beam have to be divided into a longitudinally focusing part and a
transversely defocusing part. The longitudinal part is based on the phase and energy difference between
the particles of a bunch and the synchronous particle. The transverse component is based on the radial
kick of the electric fields in the gap: due to the time-dependent field rise in the gap most particles
experience a higher field in the second half of the gap than in the first half, resulting in a net defocusing
force. Effects due to different radial positions of the particles and velocity change are less important for
ion linacs.
A.4.1 Longitudinal focusing
Following [42] in the case of small acceleration rate one can write the linearised longitudinal equation of
motion as
(φ−φs)00 =2πqE0T
mc2λβ3γ3sin φs(φ−φs)(A.22)
where φsdenotes the synchronous phase and where E0Tis given by the transit time factor integral over
the accelerating gap:
E0T=
L/2
Z
−L/2
E(t= 0, s) cos ωs
βc0ds(A.23)
For small amplitude oscillations and ignoring the non-linear parts, Eq. (A.22) can be simplified to
(φ−φs)00 +k2
l,0(φ−φs) = 0 (A.24)
or, using φ−φs=−ω
βc (s−ss) = −ω
βc zone can transform (A.22) into the zframe:
z00 +k2
l,0z= 0 longitudinal equation of
motion w/o space-charge (A.25)
with the longitudinal zero-current wave number kl,0being defined as
κl,0(s)→k2
l,0=2πqE0Tsin(−φs)
mc2λβ3γ3
smooth approximation for
longitudinal focusing (A.26)
A.4.2 Transverse defocusing
The electric field in an RF gap rises during the passage of a bunch yielding a net radially defocusing
force. From F=q(E+v×B)we can write the radial Lorentz Force as:
d
dtpr=q(Er−βcBΦ)(A.27)
86 A.4. RF FOCUSING IN THE SMOOTH APPROXIMATION
The corresponding field components of an RF gap as seen by a particle of phase φare [32]:
Er=−γsE0TI12πr
γsβsλcos(φ)(A.28)
BΦ=−γsβs
cE0TI12πr
γsβsλsin(φ)(A.29)
with I0and I1being modified Bessel functions and βsand γsbeing the relativistic factors for the syn-
chronous particle. Assuming small acceleration and βs≈βone can express the radially defocusing
force as: d
dtpr=−qγE0TI12πr
γβλsin(φ)(1 −β2)(A.30)
a form that nicely shows that this force vanishes for particle velocities approaching the speed of light.
Substituting
pr=mcγβr0ds=βc dt(A.31)
and approximating the modified Bessel function by I1(z)≈z/2one can write a smoothed equation
of motion (assuming a small acceleration rate and a small velocity difference between the synchronous
particle and the particles of the bunch)
1
βγ
d
dsβγr0−k2
l,0
2r= 0 (A.32)
with k2
l,0being the longitudinal zero-current wave number defined in Eq. (2.21). Integrating Eq. (A.32)
one can then derive the thin lens approximation for an accelerating gap:
1
fg=πqE0Tsin(−φ)lg
mc2λ(βγ)3gap thin lens approximation (A.33)
Now one can replace the drifts in the FODO lattice by gaps of length lgand calculate the transfer matrix
for a FODO period with RF gaps. Starting in the centre of the focusing quadrupole one obtains
MP=
1 0
−1
2fQ1
·
1lg
1
fg1
·
1 0
1
fQ1
·
1lg
1
fg1
·
1 0
−1
2fQ1
(A.34)
=
1 + lg
fg−l2
g
2f2
Q2lg+l2
g
fQ
2
fg−lg
2f2
Q−lg
fQfg+l2
g
4f3
Q1 + lg
fg−l2
g
2f2
Q
(A.35)
Equating (A.15) and (A.35) yields an approximate expression for the transverse phase advance per period
σ2
t,0≈ lg
fQ!2
− 4lg
fg!(A.36)
which one can use to define the transverse wave number for a FODO channel including RF focusing in
the smooth approximation [using Eqs. ((A.17), (2.21), (2.22)]:
APPENDIX A. R.M.S. ENVELOPE EQUATIONS AND THE SMOOTH APPROXIMATION 87
κx(s) = κy(s)−→ k2
t,0
k2
t,0=qGlQ
2mcβγ 2
−πqE0Tsin(−φs)
mc2λβ3γ3
=k2
Q−k2
l,0
2
smooth approximation for
transverse focusing in FODO
lattices with RF
(A.37)
It is important to note that the RF defocusing term in Eq. (A.37) remains constant for different lattice
types while the quadrupole term depends on the lattice structure (e.g. FDO, or FOFODODO).
88
B. Accelerating structures
area of bearing area of bearing area of bearinglocations of post−coupler
Figure B.1: Proposed 1st DTL tank for Linac4 with external girder for the alignment of the drift tubes,
drawing from VNIIEF, Sarov
b l 3 / 2 b l
c o u p l i n g c e l l
D T L t a n k l e t
d r i f t t u b e s
q u a d r u p o l eq u a d r u p o l e
Figure B.2: Basic Coupled Cavity Drift Tube Linac (CCDTL) structure with 3 gaps per tank and
quadrupoles between the tanks
APPENDIX B. ACCELERATING STRUCTURES 89
Quadrupole
1.5 βλ
coupling cells
Figure B.3: Basic Side Coupled Linac (SCL) structure
90
C. Derivation of envelope eigenmodes
Equations (4.5) together with (4.7) are linearised using Taylor expansions around the points ∆ax= 0,
∆ay= 0,∆b= 0
ε2
t
(˜ax,y + ∆ax,y)3≈ε2
t
˜a3
x,y −3ε2
t
˜a4
x,y ·∆ax,y (C.1)
ε2
l
(˜
b+ ∆b)3≈ε2
l
˜
b3−3ε2
l
˜
b4·∆b(C.2)
K3[1 −f(s)]
(˜ax,y + ∆ax,y)(˜
b+ ∆b)≈K3[1 −f(s)] · 1
˜ax,y˜
b−∆ax,y
˜a2
x,y˜
b−∆b
˜a˜
b2!(C.3)
K3f(s)
(˜ax+ ∆ay)(˜ay+ ∆ay)≈K3f(s)· 1
˜ax˜ay−∆ax
˜a2
x˜ay−∆ay
˜ax˜a2
y!(C.4)
neglecting all non-linear terms. After this modification we obtain a system of three coupled linear differ-
ential equations of Hill’s type for the perturbation:
d2
ds2
∆ax
∆ay
∆b
+
k2
t,0+3ε2
t
˜a4
xK3[1 −f(s)]
˜a2
y˜
b
K3[1 −f(s)]
˜ay˜
b2
K3[1 −f(s)]
˜a2
x˜
b k2
t,0+3ε2
t
˜a4
y!K3[1 −f(s)]
˜ax˜
b2
K3f(s)
˜a2
x˜ay
K3f(s)
˜ax˜a2
yk2
l,0+3ε2
l
˜
b4
|{z }
M
·
∆ax
∆ay
∆b
=~
0(C.5)
Employing once more the equations of the smooth approximation (2.19) - (2.20) one can replace the
matrix elements in Mwith expressions for the phase advances per period. Furthermore one can now,
after having removed the s-dependency of the matched envelopes, assume the transverse envelopes to be
equal (˜ax,˜ay→ˆax= ˆay= ˆa,˜
b→ˆ
b), which at this point corresponds to assuming equal quadrupole
strengths in x and y or to a channel with solenoid focusing. With Eqs. (2.28) and (2.29) the previous
system of equations can be re-written as
d2
ds2
∆ax
ˆa
∆ay
ˆa
∆b
ˆ
b
+1
L2
P
σ2
t,0 + 3σ2
tσ2
t,0 −σ2
tσ2
t,0 −σ2
t
σ2
t,0 + 3σ2
tσ2
t,0 −σ2
tσ2
t,0 −σ2
t
σ2
l,0 + 3σ2
lσ2
l,0 −σ2
lσ2
l,0 −σ2
l
·
∆ax
ˆa
∆ay
ˆa
∆b
ˆ
b
=~
0(C.6)
Using the ansatz
∆ax
ˆa
∆ay
ˆa
∆b
ˆ
b
=eiσenv
LPs·
A
B
C
(C.7)
APPENDIX C. DERIVATION OF ENVELOPE EIGENMODES 91
we obtain a homogeneous system of linear equations for the envelope tunes (σenv) of the eigenmodes:
1
L2
P
σ2
t,0 + 3σ2
t−σ2
env σ2
t,0 −σ2
tσ2
t,0 −σ2
t
σ2
t,0 −σ2
tσ2
t,0 + 3σ2
t−σ2
env σ2
t,0 −σ2
t
σ2
l,0 −σ2
lσ2
l,0 −σ2
lσ2
l,0 + 3σ2
l−σ2
env
·
A
B
C
=~
0(C.8)
By setting the determinant to zero one finds three non-trivial solutions for σenv.
92
D. Stability charts
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
kx/kox
0 0.25 0.5 0.75 1 1.25 1.5 1.75
kz/kx
0,03
0,13
0,23
0,33
0,43
0,53
0,63
0,73
1 1,5
m=3/odd m=4/even
m=4/odd
m=4/even
Figure D.1: Stability chart for εl/εt= 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
kx/kox
0 0.25 0.5 0.75 1 1.25 1.5
kz/kx
0,03
0,13
0,23
0,33
0,43
0,53
0,63
0,73
>10 betatron periods
<2 betatron periods
4-3 betatron periods
-1,0
-0,5
0,0
0,5
1,0
1,5
-1,5 -1 0 0,5 1 1,5
m=3/odd m=4/even
m=4/odd
m=4/even
Figure D.2: Stability chart for εl/εt= 1.2
APPENDIX D. STABILITY CHARTS 93
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
kx/kox
0 0.25 0.5 0.75 1 1.25 1.5 1.75
kz/kx
0,03
0,13
0,23
0,33
0,43
0,53
0,63
0,73
4-3 betatron periods
<2 betatron periods
m=4/even
m=4/odd
m=4/even
Figure D.3: Stability chart for εl/εt= 2.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
kx/kox
0.25 0.5 0.75 1 1.25 1.5 1.75 2
kz/kx
0,03
0,13
0,23
0,33
0,43
0,53
0,63
0,73
m=4/even
m=4/even m=3/odd
Figure D.4: Stability chart for εl/εt= 3.0