Presentatada:DiegoQuatraroCoordinatoreDottorato:Chiar.moProf.FabioOrtolaniRelatore:Chiar.moProf.GiorgioTurchettiCo-relatore:Dr.GiovanniRumoloEsameﬁnaleanno2011 COLLECTIVEEFFECTSFORTHELHCINJECTORS:NON-ULTRARELATIVISTICAPPROACHES AlmaMaterStudiorum · Univer

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Alma Mater Studiorum · Universit`a di Bologna

FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI

DOTTORATO DI RICERCA IN FISICA

Ciclo 23

Settore scientifico-disciplinare di afferenza: FIS/07.

COLLECTIVE EFFECTS FOR THE LHC INJECTORS:

NON-ULTRARELATIVISTIC APPROACHES

Presentata da: Diego Quatraro

Coordinatore Dottorato:

Chiar.mo Prof.

Fabio Ortolani

Relatore:

Chiar.mo Prof.

Giorgio Turchetti Co-relatore:

Dr. Giovanni Rumolo

Esame finale anno 2011

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A mio padre, mia madre, mio fratello . . .

. . .

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Introduction

The upgrade of the CERN accelerator complex has been planned in order to further increase the performances of the Large Hadron Collider (LHC) in exploring high-energy frontiers. One of the main limitations to the intensity upgrade scheme is represented by collective instabilities. They are caused by the interaction of travelling charged particles, via electromagnetic fields, with the geometry and the conductivity of the beam environment and they are proportional to the beam intensity. These electromagnetic interactions are expressed in terms of wake fields (time domain) or impedances (frequency domain). High intensity beams might be driven unstable by collective phenomena and beam coupling impedances are expected to be among the main limitations to the intensity upgrade.

Impedances are usually studied assuming ultrarelativistic bunches while low and medium energy regimes have been less explored. In this thesis work, however, we present a detailed analysis of the impedance structure for low and medium energy particle accelerators. The models developed in this thesis have been applied to the Proton Synchrotron Booster (PSB) which accelerates the proton bunch from a kinetic energy of 50 MeV up to 1.4 GeV. In this energy range high intensity phenomena need to be studied with a different formalism than in high energy (ultrarelativistic) machines. The impedance of the PSB has been analysed through dedicated measurements, numerical simulations and analytical modelling, in a non-ultrarelativistic approach. Through these analysis we quantified the impedance structure of the PSB and identified the main sources of intensity-dependent instabilities.

To further understand the impedance structure of the LHC injector chain a i

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lot of efforts have been recently put into the Super Proton Synchrotron (SPS) impedance modelling. In this context, we developed a comprehensive set of numerical libraries in order to simulate collective effects taking into account the detailed description of a particle accelerator. In particular, through dedicated numerical modelling, we proved how direct space charge can mitigate wake field driven collective instabilities.

Chapter 1 presents a brief overview of the LHC and its injector chain. Chap- ter 2 presents an introduction on the main concepts related to beam dynamics, impedances and wake fields. In Chapter 3 the PSB impedance model has been analysed taking into account both the bunch parameters and the pipe geometric structure. A non-ultrarelativistic approach has been applied to analyse both resistive wall and broad band impedances. In the same chapter, we also applied symplectic high-order numerical schemes and semi-analytical electric field ap- proximations to develop a multiparticle numerical code in order to study the impact of space charge on collective effects. The code has been applied to study the SPS bunch stability at injection energy. In this regime the direct space charge effects might play a role for intensity-dependent instabilities like the Transverse Mode Coupling Instability (TMCI) which has been observed in the SPS many times. Chapter 4 presents the results obtained from experimental campaigns. With the numerical and analytical tools developed in Chapter 3, the coherent betatron oscillation data allowed us to obtain coherent space charge tune shifts, resistive wall impedance and the broad band transverse impedance estimations. The coherent tune shift is linked to the imaginary part of the impedance. In view of the intensity upgrade, the transverse instability dampers need to be strengthened. For this reason the main collective instability in the PSB has been analysed via both experimental and numerical analysis. In general, the growth rate of a collective instability is linked to the real part of the impedance. Through dedicated modelling we pointed out to the resistive wall impedance as the source of the instability. Chapter 5 is dedicated to the numerical tools which we have developed to further improve the HEADTAIL code, widely used to study collective instabilities. We developed a comprehensive set of numerical libraries in order to simulate collective effects taking into account

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Introduction iii

the detailed description of a particle accelerator. On a different note, Chapter 6 briefly describes how we obtain a fully analytical description of the high- intensity beam phase space geometry in the context of the CERN Neutrinos to Gran Sasso (CNGS) experiment.

This thesis presents several experimental, analytical and numerical tech- niques which have been developed and applied to the CERN machine complex.

The key objective has been to identify some of the factors which might limit the LHC luminosity via studying low and medium energy collective effects in the injection chain.

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List of Figures

1.1 Artistic’s view of the CERN site on surface and underground, highlighting the LHC tunnel and the four particle detectors AL- ICE, ATLAS, CMS and LHCb. . . 6 1.2 Schematics of the CERN accelerator complex. The proton accel-

erator chain starts with LINAC2 and follows the PSB, the PS, the TT2 transfer line, the SPS, the TT60/TI2 (for beam 1) and TT40/TI8 transfer line (for beam 2) to finally reach the LHC. . 10 2.1 Sketch of a proton on the ideal reference orbit. . . 14 2.2 Sketch of a proton following an orbit different from the reference

one. The coordinate system is solidal with the travelling particle and is defined by the (ˆx, ˆy, ˆz) orthonormal versors. . . 14 2.3 Sketch of a proton on an orbit different from the reference one.

The cylindrical coordinate system (ûρ, ûθ, ûy) is shown. . . 16 2.4 Phase space (ϕ, ∆E) for the synchrotron motion. The black

curves refer to γ = 4 which means before the transition crossing (γtr = 6.09). The red curves refer to γ = 8. The used parameters are listed in Tab. 2.1. The bucket is plotted in both cases. . . . 26 2.5 A leading particle 1 and a trailing particle 2 travelling in free

space with parallel velocities v. The coordinate system x, z is also shown. . . 28 2.6 A bunch of particles travelling inside the beam pipe. Shown are

the image charges on the wall generated by the bunch. . . 30 ix

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3.1 Two particles model: the particle 1 exchanges its leading position with the particle 2 every t = T_s/2 = π/ω_s. . . 40 3.2 Left: stability region in the (ω_β/ω_s, Γ) plane. The unstable region

is shown dashed. Right: frequency spectrum of the center-of- charge of the beam. The instability occurs for that value of Γ where the mode frequencies merge. . . 41 3.3 Stability region in the (ωβ/ωs, Γ) plane for the two particle beam

against the TMCI. The blue dotted regions (•) are those where the system is unstable. The red line (−) stands for the border of stability of the system at the ultrarelativistic regime. The values used for the simulations are: α1 = −0.1α² plot I), α1 = −0.5α² plot II), α1 = 0.1α2 plot III) and α1 = 0.1α2 plot IV) . . . 43 3.4 Frequency spectrum of the centre of charge of the beam versus the

bunch intensity. The instability occurs when the mode frequencies merge. For the non-ultrarelativistic one the system becomes unstable at Γ ≈ 2 for the first time: the mode frequency then splits for Γ ≈ 4, and the system return to be stable and then gets unstable again for Γ ≈ 7. . . 44 3.5 Longitudinal resistive wall impedance Eq. (3.39) for the PSB

aperture model. The impedance are plotted for four different kinetic energies: 50 MeV (−), 160 MeV (−), 1 GeV (−), 1.4 GeV (−) and γ ≫ 1 (−). . . 55 3.6 Transverse resistive wall impedance Eqs. (3.39), (3.40) for the

PSB aperture model. The impedance are plotted for the three different kinetic energies at the flat top: 50 MeV (−), 160 MeV (−), 1 GeV (−), 1.4 GeV (−) and γ ≫ 1 (−). . . 56 3.7 Longitudinal wake function per unit length Wk(z) Eq. (3.42) for

the PSB aperture model. The wake functions are plotted for the four different kinetic energies at: 50 MeV (−), 160 MeV (−), 1 GeV (−), 1.4 GeV (−) and γ ≫ 1 (−). . . 58

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Index xi

3.8 Longitudinal resistive wall wake fields for the PSB. The impedance are plotted for the three different kinetic energies at the flat top:

50 MeV (−), 160 MeV (−), 1 GeV (−) and 1.4 GeV (−). The found scaling laws show a good agreement. . . 59 3.9 Transverse wake functions W⊥(z) Eq. (3.40) for the circular PSB

beam pipe. The wake functions are plotted for the three different kinetic energies at: 50 MeV (−), 160 MeV (−), 1 GeV (−) and 1.4 GeV (−). The dashed vertical lines stand for the range of validity between which the approximation Eq. (3.40) holds true. 59 3.10 Semi-analytical transverse wake functions W⊥(z) for the round

PSB beam pipe. E = 1.4 GeV. . . 60 3.11 A Elliptical structure of the PSB beam pipe. B Yokoya’s factors

for the horizontal and vertical, both dipolar and quadrupolar, component of the wake field forces. C Horizontal and vertical components of the wake field forces, coming from an elliptical beam pipe structure such as the PSB one. . . 60 3.12 Transverse wake field functions for the elliptic PSB beam pipe,

at four different energies. . . 61 3.13 Comparison between longitudinal (left column) and transverse

(right column) ultrarelativistic wake function Eq. (3.44) (−) and the the proposed non-ultrarelativistic one Eq. (3.49) (−) for three different energies. . . 64 3.14 The Γ curve for calculating the Fourier transform. . . 65 3.15 Comparison between the imaginary (left column) and real (right

column) part of the longitudinal ultrarelativistic broad band impedance using Eq. (3.45) (−) and the proposed non-ultrarelativistic one Eq. (3.52) (−) for three different PSB energies. . . 66 3.16 Numerical solution of the Haissinski equation using Eq. (3.49)

as wake function. We used three different bunch populations N:

N = 1 · 10¹³ (−), N = 2 · 10¹³ (−) and N = 3 · 10¹³ (−). . . 68

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3.17 Numerical solution of the Haissinski equation using Eq. (3.44)− and Eq. (3.49)−as a wake function. The numerical solution has been compared against the numerical data acquired•at the PSB for a bunch with N = 4.9 · 10¹²(left) and N = 6.9 · 10¹²(right) particles at 1.4 GeV kinetic energy. . . 69 3.18 The j − th particle (•) , located in the k − th slice of the bunch,

feels the wake field force W₁ excited by the l − th slice. The force is proportional to the number of particles Nl and the centroid position ¯xl of the l − th slice. . . 74 3.19 Mode shifting for resistive wall and no space charge. . . 76 3.20 Mode shifting for resistive wall and space charge. . . 77 3.21 TMCI intensity threshold as a function of the horizontal beam

size. . . 78 3.22 Mode shifting for broad band wake fields without space charge. 79 3.23 Mode shifting for broad band wake fields with space charge. . . 80

4.1 Illustration showing the electric forces from the images of the beam, off centred vertically by ˆy1, acting on the witness particle at location y1 inside the beam between two infinite horizontal conducting parallel plates separated vertically by distance 2h. . 86 4.2 Illustration showing the magnetic forces from the images of the

beam, off centred vertically by ˆy₁, acting on the witness particle at location y1 inside the beam between two infinite horizontal conducting parallel plates separated vertically by distance 2h.

The normal component of the non penetrating magnetic fields vanish at the plates. The beam of image currents flowing into or out of the paper are labelled “in” and “out”. . . 88

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Index xiii

4.3 Illustration showing the magnetic forces from the images of the beam, off centred vertically by ˆy₁, acting on the witness particle at location y1 inside the beam between two infinite horizontal conducting parallel plates separated vertically by distance 2g.

The parallel components of the penetrating magnetic fields vanish at the pole faces. Here, the beam and all the image currents flow into the paper. . . 89 4.4 Programmed tune for three different energy cycles at PSB. The

dashed lines show the point of the magnetic cycle at which we took the bunch length and tune measurements. . . 93 4.5 Measured bunch length (4σ) obtained from the longitudinal beam

profile at Ring4. Similar values were found in Ring2. . . 93 4.6 Tune shift measurements taken at 160 MeV (β ≃ 0.519) for Ring2

(left) and Ring4 (right). . . 94 4.7 Tune shift measurements taken at 1 GeV (β ≃ 0.875) for Ring2

(left) and Ring4 (right). . . 94 4.8 Tune shift measurements taken at 1.4 GeV (β ≃ 0.916) for Ring4

only. . . 95 4.9 An example of the horizontal tune shift measurement (right) at

160 MeV for Ring2. We have subtracted the calculated con- tribute of the indirect space charge forces: Q^Z_x = Q_x− Q^s.cx (b_t). . 95 4.10 Typical pattern of the losses observed at 378 ms (left) and 478

ms (right) the injection. The kinetic energies are ≃131 MeV and

≃330 MeV respectively. . . 100 4.11 Device used at the PSB in order to obtain the sum signal Σ and

the difference ∆. They are two different kind of measurements in order to obtain the longitudinal density of the bunch (Σ) and its

“internal” structure (∆). Picture from H. Schmickler, CAS 2007. 101 4.12 Typical pattern from the pick-up signal. Both the pictures refer

to the horizontal plane. The horizontal axis refers to the time of the pick up acquisition. During the time of the acquisition 500 data signal have been collected. . . 102

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4.13 Measured horizontal (left) and vertical (right) chromaticities. . 103 4.14 Longitudinal bunch length measurements at 378 ms. Left: the ac-

quired bunch lengths as a function of the bunch intensity. Right:

an example of the bunch profile acquired for a bunch intensity of 3 · 10¹² protons. . . 104 4.15 Bunch profile spectrum for the PSB bunch with a parabolic dis-

tribution. We used the data Tab. 4.3 and the beam parameters at time 378 ms. . . 106 4.16 Growth rate of the PSB instability as a function of the mode

number n. . . 107 4.17 Pick-up signal at 378 ms. Increasing the bunch intensity we ob-

served that the number of nodes passed from 3 to 2. . . 108 4.18 Growth time of the first (100 ms after the injection) PSB insta-

bility as a function of the beam intensity. . . 108 4.19 Envelope curve of the ∆ signal of the beam. We have analysed a

set of 90 pick up traces separated by 21 revolution periods each. 109 4.20 ∆ signal from the HEADATIL simulation. The growth rate time

has been found to be τ ≈ 10ms. . . 109 4.21 Structure of the resonances in the PSB. The tunes have been

scanned in the horizontal and the vertical plane. The colour indicates the percentage of beam losses in logarithmic scale. . . 110 5.1 Arrangement of the several elements along the ring. sj is the

starting point for the transport through the Mj matrix. . . 115 5.2 Plots of the sampled β function for the SPS lattice. The first plot

(left) represents the β_x sampled through all its range of variation (option 0). The second is (right) for the random choice of the βx

(option 2). For these plots we have chosen 50 interaction points. 120 5.3 Example of the sampling along the machine. The several type of

elements are shown. The command to generate that lattice was hdtl sps.dax 2 500 ec MBB wf MKV ob BPV . . . 123

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Index xv

5.4 Horizontal and vertical components, both dipolar and quadrupolar components, of one SPS extraction kicker (MKE.61634). . . 124 5.5 Simulated bunch mode spectrum vs. bunch population Nb using

multi-kick approach (white) and the single-kick approach (red).

Vertical plane (left) and horizontal (right). The size and of the either the red or the white dots depends on the spectral amplitude. . . 125 5.6 Growth rates comparison between the former version (one inte-

grated kick ) of the code − and the new one (tabulated kicker wakes) −. . . 126 5.7 Bottom: Slope of the vertical phase advance between consecutive

BPMs determined from a linear fit as a function of the intensity.

Top: Source localization from phase advance slope using linear response matrix. The top bar plot represents the location of the impedance sources used in the simulation. . . 129

6.1 (Left) Iteration of the map Eq. (6.2), with N = 2 and ω = 2π · 0.21. (Right) Magnification of the neighbourhoods of an unstable fixed point. . . 135 6.2 (Left) Iteration of the map Eq. (6.2), with N = 2 and ω = 2π ·

0.20. (Right) Flow generated by the interpolating Hamiltonian calculated up to the 6-th order. . . 139 6.3 Phase portraits for a polynomial map and different values of the

tune ω = 2π/4 + ǫ. We pass from a phase space with no islands (top left) to a phase space with 4 islands (bottom right). . . 144 6.4 Evolution of the initial distribution during the resonance crossing

process. The detuning term has been suppressed and thus the resonance is unstable. . . 145 6.5 Tune variation used for the splitting presented in Fig. 6.4. The

tune curve is of type 2 − 2. . . 146

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6.6 (Left) Fraction of untrapped particles as a function of the initial emittance. N indicates the number of turns involved in the capture process. The curve 2 − 2 has been used. (Right) Fit- ting of the amplitude of the exponential function, used to fit the simulation results of the trapping process. . . 147 6.7 Fraction of particles remaining in the beam core as a function of

δ. The fitted parabola is also shown. The numerical simulations are performed with σ = 0.08 for the initial distribution. . . 149

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List of Tables

1.1 LHC beam parameters relevant for the peak luminosity. IP1, IP2, IP5 and IP8 refer to ATLAS, ALICE, CMS and LHCb detectors respectively. . . 8 2.1 Parameters used to solve the equation of motion for the PS. . . 26 3.1 Parameters used to calculate the resistive wall impedance Eq. (3.39). 54 3.2 Parameters used to solve Eq. (3.53) for the PSB bunch at 1.4 GeV. 68 3.3 SPS parameters used for the simulations. . . 75 4.1 Total measured tune shift in the vertical plane, as well as the

slope of the fitted lines. . . 96 4.2 Calculated space charge tune shift. . . 97 4.3 Parameters used to calculate the contribution of the resistive

wall to the effective impedance. See the next section for the measurements of the chromaticity. . . 98 4.4 Resistive Wall characteristic frequencies ˆω coming from the lead-

ing term of Eq. (4.18). The effective resistive wall impedance as well as the tune shifts are calculated. . . 98 4.5 The effective broad band impedance as well as the respective

tune shifts are calculated subtracting the space charge and the resistive wall contribution from the experimental observations. . 99 4.6 PSB beam parameters for Ring4 at time 378 ms. . . 101 4.7 Head-tail modes for ξ = −1, for a parabolic distribution Eq.(4.24).

We used the parameters of the PSB. . . 105 xvii

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5.1 Outline of the arguments command line for hdtl. argvj stands for the j−th argument to type on the shell. The number of the arguments depends on how many elements we want to use. . . 122 5.2 Parameters used to simulate the SPS TMCI. . . 125 6.1 Fit parameters for the scaling laws determined with the numerical

simulations. . . 148

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Index 1

List of acronyms

CERN European Organization For Nuclear Research PSB Proton Synchrotron Booster (injector)

PS Proton Synchrotron (injector) SPS Super Proton Synchrotron (injector) LHC Large Hadron Collider

LEP Large Electron Positron Collider CNGS Cern Neutrinos to Gran Sasso

TMCI Transverse Mode Coupling Instability MTE Multi-Turn Extraction

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Chapter 1 The CERN Large Hadron

Collider and its Injector Chain

Founded in the 1954, the European Organization for Nuclear Research (CERN) is a multinational laboratory for particle physics located at the French-Swiss border near Geneva. As of September 2009, CERN has 20 European member states and more than 40 other states from all over the world are involved in CERN programmes. With around 2500 employees and 8000 visiting scientists representing 580 universities, 85 nationalities and half of the wold’s particle physicists, CERN is the largest particle physics laboratory in the world [1].

1.1 LHC

The accelerator complex at CERN is a succession of machines with increasing energy The Large Hadron Collider (LHC) accelerates each particle beam up to the record energy of 7 TeV. This energy has never been reached before in a lab and the goal of the LHC is to let bunches of protons, travelling in opposite direction, collide. A collider has a big advantage over other kinds of accelerator where a beam collides with stationary target. When two beams collide, the energy of the collision is the sum of the energies of the two beams. In addition to accelerating protons, the accelerator complex also accelerates lead ions produce from a highly purified lead samples heated to high temperatures.

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Analysing the collisions/interactions of hadron particles at very high energy, as the one reached at the LHC, is a key point in order to improve our knowledge of the fundamental building blocks of the matter [2]. In particular, particle physicists expects to encounter new phenomena reaching in this range of energy.

Physicists hope that new phenomena could validate or invalidate our present knowledge of the matter which is mainly based on the Standard Model [3].

The Standard model has been tested by various experiments and it has proven particularly successful in anticipating the existence of previously undiscovered particles. However, our current understanding of the Universe in incomplete since many unsolved questions remain unsolved

• the Standard Model does not explain the origin of the mass, nor why some particles are very heavy while other have no mass at all. The answer to that might be given by the so-called Higgs mechanism. According to this theory, the whole of space is filled with a “Higgs field”, and by interacting with this field, the particles acquire their masses. Whether a particles as a big mass or not depends on how intensely they interact with the Higgs field: particles that have feeble interactions are light while those that interact intensely are heavy. The Higgs field has at least one new particle associated with it, the Higgs Boson. If such a particle exists, experiments at the LHC should be able to detect it.

• the Standard Model does not offer a unified description of all the fundamental forces, as it remains difficult to construct a theory of gravity similar to those for the other forces. Supersymmetry - a theory that hypothesises the existence of more massive partners of the standard particles we know - could facilitate the unification of fundamental forces. If supersymmetry should turn out to be right, then the lightest supersymmetric particles should be found at the LHC.

• cosmological and astrophysical observation have shown that all of the visible matter accounts for only 4% of the Universe. We still miss for particles or phenomena responsible for dark matter (23%) and dark energy (73%). The existence of the dark matter came in 1933, when astronomical

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1.1 LHC 5

observation and calculations of gravitational effects revealed that there must be more “stuff” present in the Universe. Dark energy is a form of energy that appears to be associated with the vacuum in space and it is homogeneously distributed throughout the Universe and time. In other words, its effect is not diluted as the Universe expands.

• the LHC will also help us to investigate the mystery of antimatter and why we observe our Universe as only made of matter when antimatter must have been produced in the same amounts.

• in addition to the studies of proton-proton collisions, heavy-ion collisions at the LHC will provide a window onto the state of matter that would have existed in the early Universe, called quark-gluon-plasma.

LHC project milestones

Due to its impressive size and number of worldwide collaboration, the time scale of the LHC spans over more than 30 years. The idea of the LHC began in the early 1980s. Although CERN’s Large Electron Positron Collider (LEP), which ran from 1989 to 2000, was not built yet, scientists already were looking further into the future of particle physics. They imagined re-using the 27 kilometre LEP ring for an even more powerful hadron machine. In 1984, a symposium organised in Lausanne, Switzerland, became the official starting point for work on the LHC. Working groups were set up to consider various aspects of the physics that could be studied with a proton collider. Consequently, the LHC became a priority for CERN. Since then several meeting and working groups were organised. Meanwhile, research and developments efforts were put in elab- orating models and prototype for the high field superconducting magnets, the most challenging components of the LHC project. These meetings and studies led to the first approval of the project by the CERN Council in December 1994.

The four main experiments at the LHC are all run by international collab- orations. Each experiment is distinct and characterised by its particle detec- tor. Two large experiments, ATLAS and CMS, are based on general-purpose detectors to analyse the myriad of particles produces by the collisions in the accelerator. They are designed to investigate the largest range of physics pos-

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sible. In order to confirm any new discoveries made the two detectors have to both confirm independently. Two medium-size experiments, ALICE and LHCb, have specialised detectors for analysing the LHC collisions in relation to specific phenomena. In Fig. 1.1 we can observe the LHC site together with its main detectors.

Figure 1.1: Artistic’s view of the CERN site on surface and underground, highlighting the LHC tunnel and the four particle detectors ALICE, ATLAS, CMS and LHCb.

A dedicated Design Report was published in 2004 [4] with all the technical information about the LHC complex. The last superconducting magnet has been put in place in May 2007 and in November same year the interconnections of all the arcs of the LHC have been competed. In August 2008 the 27 kilo- metres of the LHC have attained the -271^◦C needed for the experiment and in September, for few days, particles have circulated in the LHC for the first time.

1.2 Brief Technical Overview

As we have already seen, the LHC aims to discover the Higgs particle and to study rare events with centre of mass collision energies of up to 14 TeV. The number of events per second generated in the LHC collision is given by

N_event = Lσ_event (1.1)

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1.2 Brief Technical Overview 7

where σevent is the cross section of the event under study and L is the machine luminosity. The machine luminosity depends only on the beam parameters and can be written for a Gaussian beam distribution as

L = N_b²nbfrevγr

4πǫnβ^∗ F, (1.2)

where N_b is the number of particles per bunch, n_b the number of bunches per beam, f_rev the revolution frequency, γ_r the relativistic gamma factor, ǫ_n the normalised transverse emittance of the beam (in other words the transverse size of the beam σ), β^∗ the beta function (other parameter related to the transverse size of the beam: size beam σ ≈√

βǫ) at the Interaction Point (IP) and F the geometric luminosity reduction factor due to the crossing angle of the colliding beams at the IP

F = 1/

s

1 + θcσz

2σ^∗

2

, (1.3)

where θ_c is the full crossing angle at the IP, σ_zthe RMS bunch length and σ^∗ the transverse RMS beam size at the IP. The exploration of rare events in the LHC collisions therefore requires both high beam energies and high beam intensities together with small beam sizes. The LHC has tow high luminosity experiments, ATLAS and CMS, aiming at a peak luminosity of L = 10³⁴cm⁻²s⁻¹ in the LHC proton operation. Tab.1.1 shows the main parameters required to reach the peak luminosity in the LHC proton operation. In addition the LHC has one dedicated ion experiment ALICE aiming at a peak luminosity of L = 10²⁷cm⁻²s⁻¹ for nominal Pb-Pb ion operation. The high beam intensities exclude the use of anti-proton beams and one common vacuum and magnet system for both circulating beams (as it is done in TEVATRON) and implies the use of two proton beams. To collide tow beams of equally charged particles requires opposite magnet dipole fields in both beams. The LHC is therefore designed as a proton-proton collider with separate magnet fields and vacuum chamber in the main arcs and with common sections only at the insertion regions where the experimental detectors are located. The tow beams share an approximately 130 m long common beam pipe along the interaction regions. Together with the large number of bunches (2808 for each proton beam), and a nominal bunch

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Injection Collision

Proton energy GeV 450 7000

Relativistic gamma 479.6 7461

Number of bunches 2808

Longitudinal emittance (4σ) [eVs] 1.0 2.5

Transverse normalized emittance [µm rad] 3.5 3.75

Circulating beam current [A] 0.584

Stored energy per beam [MJ] 23.3 362

RMS bunch length cm 11.24 7.55

RMS beam size at IP1 and IP5 µm 375.2 16.7

RMS beam size at IP2 and IP8 µm 279.6 70.9

Geometric factor F - 0.836

Table 1.1: LHC beam parameters relevant for the peak luminosity. IP1, IP2, IP5 and IP8 refer to ATLAS, ALICE, CMS and LHCb detectors respectively.

spacing of 25 ns, the long common beam pipe implies 34 parasitic collision points for each experimental insertion region (for four experimental IR’s this implies a total of 136 unwanted collision points). Dedicated crossing angle orbit bumps separate the two LHC beams left and right from the central interaction point in order to avoid collisions at these parasitic collision points. These is not enough room for two separate rings of magnets in the LEP tunnel. Therefore the LHC uses twin bore magnets which consists of two sets of coils and beam channels within the same mechanical structure and cryostat. The peak beam energy in a storage ring depends on he integrated dipole field along the storage ring circumference. Aiming at peak energies of up to 7 TeV inside the existing LEP tunnel implies a peak dipole field (for steering the particles) of 8.33 T and the use of superconducting magnet technology. In fact reaching such a high magnetic field strength requires very large current densities in the coil, which was made possible by using Niobium,-Titanium (NbTi) alloys cooled down to a temperature of 1.9 K (-271^◦C) using Helium. At this temperature, the NbTi

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1.3 The LHC injectors 9

alloy drops its resistance close to zero, thus becoming superconducting and al- lowing a critical density current of 1000 to 2000 A/mm² though the cable.

The LHC is not a perfect circle (as all the synchrotrons). It is made of eight arcs and eight “insertions”. The arcs contain the dipole “bending” magnets, with 154 in each arc. An insertion consists of a long straight section plus two (one at each end) transition regions. A sector is defined as the part of the machine between two insertion points. The eight sectors are the working units of the LHC. In an accelerator like LHC, particles circulate in a vacuum tube (beam pipe) and are manipulated using electromagnetic devices: dipole magnets keep the particles in their nearly circular orbits, quadrupole magnets focus the beam, and accelerated cavities are electromagnetic resonators that accelerate particles and then keep them at a constant energy by compensating for energy losses.

The LHC uses eight cavities per beam, each delivering 2 MV (an accelerating field of 5MV/m) at 400 MHz. Vacuum in the LHC is ensured by a complex systems which keep the beam vacuum pressure constant at around 10⁻¹³ atm because of the necessity to avoid collisions with gas molecules.

1.3 The LHC injectors

The accelerator complex at CERN is a succession of machines with increas- ingly higher energies. In Fig. 1.2 we can observe the CERN accelerator complex.

A chain of smaller linear accelerator, synchrotrons and transfer lines is used to accelerate the beam from the proton source and provide the 450 GeV/c proton beam to inject into the LHC. Each machine injects the beam into the next one, which takes over to bring the beam to an even higher energy and so on.

In order to be injected into the LHC, the path of each proton that reaches the LHC is the following

a) LINAC2 (source and linear accelerator)

The protons are produced by stripping electrons from hydrogen atoms.

The protons are then focused and preaccelerated to 750 KeV in an RF Quadrupole and finally accelerated in the 30m drift tube linear accelerator to reach 50 MeV of kinetic energy at the exit of the LINAC2.

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Figure 1.2: Schematics of the CERN accelerator complex. The proton accelerator chain starts with LINAC2 and follows the PSB, the PS, the TT2 transfer line, the SPS, the TT60/TI2 (for beam 1) and TT40/TI8 transfer line (for beam 2) to finally reach the LHC.

b) Proton Synchrotron Booster (PSB)

The protons are injected into the PSB and accelerated to a kinetic energy of 1.4 GeV. The PSB consists of 4 25 m radius rings on top of each other.

c) Proton Synchrotron (PS)

The proton beam is then fed to the PS which is a 100m radius machine.

Here the protons are accelerated up to a momentum of 26 GeV/c. During the accelerating process lots of beam manipulation is going on in order to achieve the nominal bunch parameters for the LHC beams.

d) Super Proton Synchrotron (SPS)

Protons are then sent to the SPS where they are accelerated up to a momentum of 450 GeV/c. It is in here that the clockwise and anticlockwise beams separate. The transfer line T T 60/T I2 sees the clockwise beam 1 passing by before reaching the LHC. On the other side T T 40/T I8 injects beam 2 circulating anticlockwise in the LHC.

The importance of the injector chain is crucial. As we have already seen the nominal luminosity depends on the bunch parameters. These parameters need a sophisticate “pre-processing” which is mainly carried out in the injection chain.

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1.4 Plans for the Upgrade of the LHC Complex 11

Any changes in the LHC beam needs to be studied from the beginning already in the injection chain.

1.4 Plans for the Upgrade of the LHC Complex

Conditional on the achievements of the nominal beam parameters, two possible scenarios for the LHC upgrades are possible [5]. The first one is a further increase in the beam energy. An increase in energy is a natural upgrade path to pursue further the search for very massive new particles and to explore extremely small distances. This scenario requires higher field magnets. Research already aims towards this direction of reaching a dipolar field of 20T in order to steer the particles in the dipoles.

The second possible scenario for the LHC upgrade relies on the possibility of a luminosity upgrade. What is desirable is an increase of the luminosity to 10³⁵cm⁻²s⁻¹, which means one order of magnitude higher than the nominal one. Higher LHC luminosity will benefit a large number of Standard Model measurements that we already know will be possible. They include the Higgs physics, electroweak measurements, supersymmetric particles and many more like searches for new physics. The main knobs to play with in order to achieve the desired luminosity upgrade are the following [6]

• upgrade of the current injectors.

• reduce the crossing angle at the IPs.

• a smaller beam size. This goal might be achieved with either improving the focusing of the beam at the IPS or improving the injector chain in delivering high-intensity and small size bunches to the LHC.

• increase in the beam intensity. All the current upgrade schemes are aiming at increasing he intensity per bunch from the nominal 1.15 · 10¹⁵ protons per bunch to the value of 1.7 · 10¹⁵or 4.9 · 10¹⁵. This is another reason for understanding the effects which might limit the beam intensity upgrade.

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The interaction of the charged particles, via electromagnetic fields, with conducting boundaries of the beam pipe can result in collective beam instabilities.

Generally speaking the collective effects are a function of the vacuum system geometry and its surface properties. They are usually proportional to the beam currents and can therefore limit the maximum attainable beam intensities in the injector chain. In the upgrade scheme the PSB is meant to keep on running.

It is therefore particular important to understand its limitations in term of collective effects which we have briefly described before. This thesis work is one contribution to the studies for the beam intensity upgrade of the LHC chain.

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Chapter 2 Linear optics and collective effects in synchrotrons

In this chapter we briefly describe the linear motion of a charged particle in a synchrotron machine. We will derive the equation of motion for the single particle longitudinal and transverse dynamics.

In addition we will define the wake field forces and their effects on the collective motion of the bunch.

2.1 Coordinate system and Hamiltonian in a circular accelerator

The first step in writing down the equation of motion, is to define the coordinate system. In a particle accelerator the ideal trajectory any particle should follow is called reference/designed orbit. In the special case of a synchrotron the reference orbit is given by a succession of straight lines and arcs of circumference. In the following we will assume the reference orbit to be a perfect circumference of radius ρ, whereas in reality we should have a radius ρ as function of the position s of the particle along the reference orbit. Fig. 2.1 shows the reference orbit.

Three main references have been extremely useful in explaining the follow- 13

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x y

s ρ Designed orbit

θ

θ0 = 0

Proton on the designed orbit

θ = θ(t)

Figure 2.1: Sketch of a proton on the ideal reference orbit.

ing concepts of transverse dynamics [7, 8, 9]. The particle travelling on the reference orbit will be called reference particle. The reference particle is also assumed to have the velocity v = βc, which will be assumed to be also the speed of the bunch. Each proton of the bunch does not exactly follow the reference orbit, but it will rather follow the orbit (−) reported in Fig. 2.2.

ˆ x ˆ y

s y

ˆ z Proton trajectory

ρ Designed orbit

θ

θ₀ = 0

Proton

θ = θ(t)

Figure 2.2: Sketch of a proton following an orbit different from the reference one.

The coordinate system is solidal with the travelling particle and is defined by the (ˆx, ˆy, ˆz) orthonormal versors.

The horizontal, vertical and longitudinal positions (x, y, z) are function of the time t and represent the position of a particle respect to the position of the reference particle. We will now derive the equation of the linear motion of a proton. Under the influence of an electrical E and magnetic B field, a charged

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2.1 Coordinate system and Hamiltonian in a circular accelerator 15

particle experiences the following force

F = e(E + v × B), (2.1)

being e the charge of the particle and v its velocity. It is now useful notice that

kvk =q

v²_x+ v²_y + v²_z ≈ |v^z| (2.2) and will will also assume that Bs≈ 0. This consideration allows us to decou- ple the longitudinal and transverse motion, provider the so called chromaticity is zero. The forces will then be











Fx = e(Ex+ vsBy) Fy = e(Ey− v^sBx) F_s = eE_s

(2.3)

In order to control (get the particles closer to the reference orbit, guiding them, etc. ) the particle orbit magnetic elements are widely used. The forces in the transverse plane then read







Fx = evsBy

F_y = −evsB_x. (2.4)

The relativistic equations of motion read

d

dt(m₀γv) = ev × B ⇒











˙vx = evsBy

m₀γ

˙vy = −evsBx

m₀γ

(2.5)

We now refer to the cylindrical coordinate system (ûρ, ûθ, ûy) as reported in Fig. 2.3.

If we denote with r, u and a the position, the velocity and the acceleration of the particle in the transverse plane we have

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x x y y

s

Magnetic field By

Bx

Bs= 0

ˆ u_y

ˆ uθ

ˆ u_r ρ

Designed orbit

θ

θ0 = 0

Proton on the designed orbit θ = θ(t)

Figure 2.3: Sketch of a proton on an orbit different from the reference one. The cylindrical coordinate system (û_ρ, û_θ, û_y) is shown.











r = rˆu_r+ uˆu_y

u = ˙rûr+ r ˙θûθ+ ˙yûy

a =

r − r ˙θ¨ ² ˆ u_r+

2 ˙r ˙θ + r ¨θ ˆ

u_θ+ ¨yˆu_y

(2.6)

Being vs = r ˙θ we have for the transverse component of Eq. (2.5)





 m0γ

r −¨ v_s²

r

= evsBy

m0γ ¨y = −ev^sBx

(2.7)

The equations of motion for the reference particle are







p0 = −eBy⁰ρ

B_x⁰ = 0 (2.8)

where we have defined p0 = γm0v0 as the designed momentum and B_x⁰/B_y⁰ are the horizontal/vertical magnetic fields needed in order to keep the reference particle travelling on the reference orbit.

Letting ρ = x + y we have from Eq. (2.7)







m0γ(¨x − v_s²

x + ρ) = evsBy

m0γ ¨y = −ev^sBx

(2.9)

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We now assume that x/ρ ≪ 1 1

x + ρ = 1 ρ

1 − x

ρ

+ O (x/ρ)²

(2.10) and also write the polynomial expansion for the magnetic field components











By = B⁰_y+ x d

dxBy + O(x²) Bx= y d

dyBx

(2.11)

being B_x⁰ = 0. We then obtain the following equations











¨

x + x v_s²

ρ² − evs

m0γ d dxBy

= evsBy

m0γ +v_s² ρ

¨

y + y evs

m0γ d

dyB_x = 0.

(2.12)

It is customary in the accelerator literature to replace the independent coordinate t with s = βct. This change in the derivative (ds = βcdt) leads to











x^′′+ x 1

ρ² − e m0γvs

d dxBy

= eBy

m0γvs

+ 1 ρ y^′′+ y e

m0γvs

d

dyBx = 0.

(2.13)

In absence of electrical currents and fields the Maxwell equation ∇ × B = 0 holds true in the vacuum region inside the magnet. This equation, together with the assumption of B_s = 0, gives as d/dxB_y = d/dyB_x. If we name with g the quantity d/dxB_y = g and we assume the momentum of the particle follows the relation p ≈ p^s = γm0vs we obtain











x^′′+ x 1 ρ² − eg

p

= eB_y⁰ p + 1

ρ y^′′+ yeg

p = 0.

(2.14)

Making the assumption that the proton has only a slightly different momentum from the reference particle, p/p0 ≈ 1 + ∆p/p⁰ and ∆p/p0 ≪ 1, we can write

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1 p ≈ 1

p0 − ∆p

p²₀ (2.15)

and so, finally, we get the result







x^′′+ x 1 ρ² − k

= 1 p

∆p p0

y^′′+ yk = 0.

(2.16)

where we defined the quadrupole strength k = eg/p0 and used the relation eB_y⁰ = −1/ρ. The quantity ρ and k might also be functions of the position of the particle along the machine: ρ = ρ(s), k = k(s). So we have







x^′′+ xKh(s) = 1 p

∆p p0

y^′′+ yKv(s) = 0.

(2.17)

where we have cast Kh(s) = 1/ρ²(s) − k(s) and K^v(s) = k(s). In any case, being L the circumference length, those functions are anyway periodic with period L: ρ(s) = ρ(s + L), k(s) = k(s + L). It is worth stressing the fact that in the above derivation we did not assume any coupling between the horizontal and vertical plane.

We will now show the solution of Eq. (2.17) for a particle with the momentum equal to the one of the reference particle: ∆p = 0. We will now focus on the vertical plane only, but the following treatment also applies unchanged to the horizontal plane.

The solution of Eq. (2.17) (Hill’s equation) is given by











y(s) = q

ǫβy(s) cos (ψy(s) + ϕy(0)) y^′(s) = −

p(ǫy)

pβy(s)[αy(s) cos(ψy(s) + ϕy(0)) + sin(ψy(s) + ϕy(0))]

(2.18)

where we have defined











ψy(s) = Z s

t

dt 1 βy(t) αy(s) = −β^′_y(s)

2

(2.19)

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The quantities ǫy and ϕy(0) are constant given by initial conditions. The motion is thus given by a cosine-like orbit with amplitude given by p

ǫ_yβ_y(s) and frequency ψy(s). These are of course not constant functions along the machine length. The functions ψ, α, β are usually called Twiss parameters. We will encounter them later on in the Chapter 5.

Being L = 2πR the circumference of the machine and R its radius, the number of betatron oscillation per turn, defined as the tune Qy, is given by

Qy = 1 2π

Z s+L s

ds 1

βy(s) (2.20)

It can be easily checked that the following relation

ǫy =

y²(s) + βy(s)y^′(s) + αy(s)ys(s)

/βy(s) (2.21) holds. Defining the quantity γy(s) = (1+α²_y(s))/βy(s) we can write Eq. (2.21) in the following more compact way

ǫy = γyy²+ 2αyyy^′+ βyy^′2 (2.22) The above Eq. (2.22) tells us that no matters the shape, the orientation of the ellipse in the phase space, its area remains constant. A bunch of protons travelling in the machine has an initial distribution in the phase space for s = 0.

During the acceleration process the emittance is no longer preserved but it is shrinking. The quantity which stays constant is instead the normalised emittance defined by ǫN = γβǫ.

If the travelling particle has a different momentum respect to p0 of the reference particle, then it will experience a different focalization. In this case the particle orbit will be the betatron oscillation Eq. (2.18) plus the particular solution of the Hill equation. This particular solution normalised by the momentum offset ∆p/p₀ is defined as the dispersion function D(s). From Eq. (2.14) keeping the second order in ∆p, we have

k(s) = eg

p = eg(s) p₀

1 − ∆p p₀

= k0(s) + ∆k(s). (2.23)

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This different focusing strength causes a change in the number of turns of the particle while travelling along the machine. We can thus introduce a ∆Q with respect to the tune of a particle on-momentum (∆p = 0). This shift will be very important in our studies of the collective instabilities. This shift ∆Q is directly related to the chromaticity ξ

∆Q Q0

= ξ∆p p0

. (2.24)

2.2 Longitudinal Synchrotron Motion

In order to accelerate (in module) a charged particle we need an electric field with longitudinal component. Radio Frequency (RF) fields are widely used in order to accelerate charged particles.

Two main references [10], [8] have been extremely useful in preparing the exposition of the following longitudinal dynamics related concepts.

The synchrotron motion of the particle along the machine and inside the bunch plays a crucial role in the stability studies for the collective effects. In the following we will assume only one RF cavity working in the ring.

Longitudinal electric field in an RF cavity

In the gap of an ideal RF cavity the longitudinal component E_s(t) of the electric field might be written as

Es(t) = V

g sin (φrf(t) + ϕ) (2.25)

where ϕ is the phase of the particle passing by the cavity, ωrf is the RF voltage angular frequency, and V is the cavity voltage amplitude, which is assumed to be constant throughout the cavity.

The RF cavity is synchronized with the motion of the reference particle. The RF “kicks” the particle every time it enters the cavity. The reference proton is designed to always arrive at the RF frequency with the same phase ϕ = ϕ0

respect to the RF phase. The phase of the RF should be such that accelerates the reference particle which has a revolution frequency of ν₀ = ω₀/2π = T₀⁻¹ =

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2.2 Longitudinal Synchrotron Motion 21

2πR/βc.

The RF angular frequency ω_rf is so chosen as an integer multiple h of the designed angular frequency ω0

ωrf = hω0. (2.26)

The harmonic number h indicates the maximum number of bunches that can be accelerated in the machine.

2.2.1 Energy gain in a RF cavity

We now estimate the amount of energy ∆E each particle of the bunch gains for each time it passes through the RF accelerating cavity.

The magnetic field is not able to accelerate (change the quantity kvk) a charge q. So an electric field Es(s) is applied along the whole length g of the RF cavity

∆E = q Z g/2

−g/2

dsEs(s) = q Z tout

tin

dtEs(t)βc (2.27) where tin = −g/2βc is the time the particle enters the cavity and t^out = g/2βc is the time the time the particle leaves the cavity.

The quantity ˙β is small compared to β so we can write

∆E = qβcV g

Z tout

tin

dt sin (hω0t + ϕ0) = e ˆV sin(ϕ0) (2.28) where we have defined ˆV = T V and T = sin(x)/x x being x = hω0g/2βc.

In the absence of other sources of energy change, the energy rate dE/dt is related to the energy gained per turn

dE dt = ω

2ω∆E = ω

2πe ˆV sin φ, (2.29)

where ω is the angular revolution frequency of each proton and φ is the phase each particle has while approaching the cavity.

2.2.2 Closed orbit length change with momentum

As we said each particle of the bunch has a different longitudinal momentum.

So each particle has a different revolution frequency.

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Being L the length of the closed orbit, which differs from the reference orbit L₀, we can define the momentum compaction factor α_cp which links the momentum offset ∆p/p0 of the bunch to the closed orbit change ∆L/L0

αcp = ∆L/L0

∆p/p₀ . (2.30)

The quantity L0 is obviously given by the length of the on momentum particle L0 =

I ds =

I

ρ(s)dθ. (2.31)

From the definition of the dispersion D(s) each proton with a momentum deviation ∆p follows the trajectory xD = D(s)∆p/p0 whit is characterised by a radius R(s) = ρ(s)+xD(s). The orbit LD of a general particle can be calculated by

LD = I

dsD = I

(ρ(s) + xD(s)) dθ = I

1 + xD(s) ρ(s) ds

= L0+ ∆L. (2.32) We than observe that

∆L L0

= αcp

∆p p0

, with αcp= 1 L0

I

dsD(s)

ρ(s). (2.33)

The coefficient αcpis in general positive which means that a positive momentum deviation increases the length of the closed orbit trajectory.

2.2.3 Revolution frequency change with momentum and transition energy

We now introduce some concepts which will be useful while studying the dynamics of the PSB and the SPS. The collective motion of the particles in the bunch will be influenced by the following quantities. In particular because of the coupling the wake force and the chromaticity induce between the longitudinal and the transverse plane.

The phase slip η is here define as

η = − ∆ω/ω0

∆p/p₀

. (2.34)

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Differentiating the logarithm of the relation ω = βc/R gives dω

ω = dβ β − dR

R = 1 γ²

dp p − dR

R , (2.35)

where we used the relation dβ/β = (1 − β²)dp/p = γ⁻²dp/p.

Using dR/R = dL/L, Eq. (2.30) and assuming each change of the reference particle to be small, we can write

η = α_cp− 1

γ². (2.36)

During the acceleration process the value of γ increases and the value γtr such that (happens for the LHC beam in the PS) η = 0 is called transition energy (of course the energy is γtrm0c²). From Eq. (2.34) we can summarise the following i) γ < γ_tr Leads to η < 0 and a momentum increase leads to an higher

revolution frequency.

ii) γ > γtr Leads to η > 0 and a momentum increased leads to a smaller revolution frequency.

iii) γ = γ_tr Leads to η = 0 and a momentum change does not change the revolution frequency up to the first significant order.

Crossing the transition energy while accelerating the beam might affect the stability of the beam, which experience a free motion. We will also observe how concerning the collective effects the synchrotron motion provide a stabilising effect.

2.2.4 Longitudinal equation of motion

The parameters of a non synchronous proton are considered to be slightly different from the ones of the synchronous proton For a proton travelling behind the reference proton by a time delay of ∆t = t−t⁰we have that the orbital angle difference is given up to the first order, by ∆θ = −ω⁰∆t. This proton arrives later to the centre of the RF cavity and therefore its phase ϕ with respect to

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the RF phase is larger than the phase ϕ0 of the reference particle by an amount

∆ϕ = hω₀∆t. Then we have that ∆ϕ = −h∆θ, and we can write

∆ω = d (∆θ)

dt = −1 h

d (∆ϕ)

dt = −1 h

dϕ

dt (2.37)

since the phase of the synchronous proton ϕ0 is assumed to change slowly with the time compared to the phase ϕ. Using the phase slip factor η =

−(∆ω/ω⁰)/(∆p/p0) we can obtain

∆p = − p0

ηω0

∆ω = p0

ηhω0

dϕ

dt. (2.38)

Using the relation E² = E₀² + p²c² and differentiating EdE = c²pdp we have dE = βcdp. Assuming β ≈ β⁰ we have ∆E = β0c∆p = ω0R0∆p.

We can write so write Eq. (2.38) this way

∆E ω0

= R0p0

ηhω0

dϕ

dt. (2.39)

We thus know the change of energy for a generic particle in the bunch. Using Eq. (2.28) and the relation ∆E = βc∆p we have

∆p = q ˆV

ωRsin ϕ. (2.40)

Assuming that the momentum increases by a small amount during one machine turn, we have

dp dt = ∆p

T = q ˆV

2πRsin ϕ, (2.41)

which, for the synchronous particle, reads 2πR0

dp0

dt = q ˆV sin ϕ. (2.42)

Subtracting Eq. (2.39) from Eq. (2.38) we have 2π

Rdp

dt − R0

dp₀ dt

= q ˆV (sin ϕ − sin ϕ0) . (2.43) If we now use Eq. (2.40), we obtain

Rdp

dt − R⁰dp0

dt = ∆Rdp0

dt + R0

d∆p

dt . (2.44)

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Using again the momentum compaction definition we have up to the first order αcpR0/p0 = ∆R/∆p ≃ dR/dp ≃ dR⁰/dp0. (2.45) Therefore we can use ∆R = (dR0/dp0)∆p and finally

Rdp dt−R0

dp₀

dt = dR₀ dp0

dp₀

dt ∆p+R₀d∆p

dt = dR₀

dt ∆p+R₀d∆p

dt = dR₀∆p dt = 1

ω0

d∆E dt . (2.46) Casting Eq. (2.42) into Eq. (2.40) we get

2π ω0

d∆E

dt = q ˆV (sin ϕ − sin ϕ⁰) . (2.47) Using Eq. (2.47) and Eq. (2.39) we have

d dt

R0p0

nhηω0

dϕ dt

− q ˆV

2π (sin ϕ − sin ϕ⁰) = 0. (2.48) which we can write as

ϕ −¨ e ˆV hηω0

2πR0p0 (sin ϕ − sin ϕ⁰) = 0. (2.49) Eq. (2.49) gives us the equation of motion for a generic particle for the longitudinal coordinate ϕ. For small phase oscillations from the synchronous one (∆ϕ ≃ 0) we have sin ϕ − sin ϕ⁰ = ∆φ cos ϕ0 + O (∆ϕ²). So Eq. (2.49) finally reads

d²∆φ

dt² − e ˆV cos ϕ0ηhω0

2πR0p0

∆ϕ = 0 (2.50)

and the synchrotron motion is stable only if η cos ϕ0 < 0. In this case the synchrotron frequency (for the linear motion) ω_s is defined as

ωs = ω0Qs = βc R

s

|η cos ϕ⁰|qV h

mγ2πβ²c² (2.51)

The PS at CERN crosses the transition having α = 0.027 and thus γ_tr ≃ 6.09.

We assume the parameters listed in Tab. 2.1

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γ = 4 γ = 8

α 0.027 0.027

η -0.036 0.011

h 8 8

V [MV] 0.2 0.2

Qs× 10⁻³ 1.6 0.6

Table 2.1: Parameters used to solve the equation of motion for the PS.

Figure 2.4: Phase space (ϕ, ∆E) for the synchrotron motion. The black curves refer to γ = 4 which means before the transition crossing (γ_tr = 6.09). The red curves refer to γ = 8. The used parameters are listed in Tab. 2.1. The bucket is plotted in both cases.

2.3 Self field of a particle: relativistic case

In this section we will introduce the wake field concept. We will assume the particles travelling with β ≃ 1. In the next chapters we will discuss the case for an arbitrary β. The lecture notes of Dr. Stupakov [11] and the “reference book”

for collective effects [12] have helped sorting out this introductory section.

Relativistic field of a charged particle moving with constant velocity

Presentatada:DiegoQuatraroCoordinatoreDottorato:Chiar.moProf.FabioOrtolaniRelatore:Chiar.moProf.GiorgioTurchettiCo-relatore:Dr.GiovanniRumoloEsameﬁnaleanno2011 COLLECTIVEEFFECTSFORTHELHCINJECTORS:NON-ULTRARELATIVISTICAPPROACHES AlmaMaterStudiorum · Univer

Presentata da: Diego Quatraro

Coordinatore Dottorato:

Chiar.mo Prof.

Fabio Ortolani

Relatore:

Chiar.mo Prof.

Giorgio Turchetti Co-relatore:

Dr. Giovanni Rumolo

Esame finale anno 2011

A mio padre, mia madre, mio fratello . . .

. . .

Introduction

Contents

List of Figures

List of Tables

List of acronyms

Chapter 1

The CERN Large Hadron

Collider and its Injector Chain

1.1 LHC

1.2 Brief Technical Overview

1.3 The LHC injectors

1.4 Plans for the Upgrade of the LHC Complex

Chapter 2

Linear optics and collective effects in synchrotrons

2.1 Coordinate system and Hamiltonian in a circular accelerator

2.2 Longitudinal Synchrotron Motion

2.2.1 Energy gain in a RF cavity

2.2.2 Closed orbit length change with momentum

2.2.3 Revolution frequency change with momentum and transition energy

2.2.4 Longitudinal equation of motion

2.3 Self field of a particle: relativistic case