PEDERELIASSON Emittancepreservationandluminositytuninginfuturelinearcolliders 409 DigitalComprehensiveSummariesofUppsalaDissertationsfromtheFacultyofScienceandTechnology

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(1)Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 409. Emittance preservation and luminosity tuning in future linear colliders PEDER ELIASSON. ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2008. ISSN 1651-6214 ISBN 978-91-554-7130-9 urn:nbn:se:uu:diva-8576.

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(215) List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I. II. III. IV. V. VI. VII. VIII. Luminosity Tuning Bumps in the CLIC Main Linac P. Eliasson and D. Schulte EUROTeV-Report-2005-006, PAC 2005 Proceedings p. 1141 Dispersion Free Steering and Emittance Tuning Bumps in the ILC Linac P. Eliasson and D. Schulte EUROTeV-Report-2005-021 Luminosity Tuning at the Interaction Point P. Eliasson, M. Korostelev, D. Schulte, R. Tomás, and F. Zimmermann EUROTeV-Report-2006-039, EPAC 2006 Proceedings p. 774 Design of main linac emittance tuning bumps for the Compact Linear Collider and the International Linear Collider P. Eliasson and D. Schulte Phys. Rev. ST Accel. Beams 11, 011002 (2008) Dynamic imperfections and optimized feedback design in the Compact Linear Collider main linac P. Eliasson CLIC-Note-738, submitted to Phys. Rev. ST Accel. Beams (2008) Performance of CLIC emittance tuning bumps in a dynamic environment P. Eliasson and D. Schulte EUROTeV-Report-2008-013 Study of an ILC Main Linac that Follows the Earth Curvature P. Eliasson, A. Latina, D. Schulte, F. Poirier, and N. Walker EUROTeV-Report-2006-040, EPAC 2006 Proceedings p. 786 Implications of a Curved Tunnel for the Main Linac of CLIC A. Latina, P. Eliasson, and D. Schulte EUROTeV-Report-2006-031, EPAC 2006 Proceedings p. 864. Reprints were made with permission from the publishers..

(216) Other papers to which I contributed: i. ii. iii. iv. v. vi. vii. viii. Recent Improvements of PLACET A. Latina, H. Burkhardt, P. Eliasson, L. Neukermans, J. Resta López, G. Rumolo, D. Schulte, and R. Tomás EUROTeV-Report-2006-030, EPAC 2006 Proceedings p. 2251 Recent Improvements of PLACET A. Latina, P. Eliasson, L. Neukermans, G. Rumolo, and D. Schulte ICAP 2006 Proceedings p. 188 First Design of a Post-Collision Line for CLIC at 3 TeV T. Ekelöf, P. Eliasson, A. Ferrari, and V. Ziemann EUROTeV-Report-2006-023, EPAC 2006 Proceedings p. 765 A Study of Failure Modes in the ILC Main Linac P. Eliasson, E. Elsen, D. Krücker, A. Latina, F. Poirier, D. Schulte, N. Walker, and G. Xia EUROTeV-Report-2006-040, EPAC 2006 Proceedings p. 789 Benchmarking/Crosschecking DFS in the ILC Main Linac J. Smith, P. Eliasson, A. Latina, D. Schulte, F. Poirier, N. Walker, P. Lebrun, K. Ranjan, K. Kubo, and P. Tenenbaum SLAC-TN-06-035 (2007) Comparison of tracking codes for the International Linear Collider J. Smith, P. Eliasson, A. Latina, D. Schulte, F. Poirier, N. Walker, P. Lebrun, K. Ranjan, K. Kubo, and P. Tenenbaum PAC 2007 Proceedings p. 3020 Bunch compressor for beam-based alignment A. Latina, D. Schulte, and P. Eliasson EUROTeV-Report-2007-066, PAC 2007 Proceedings p. 2844 Dynamic effects during beam-based alignment D. Schulte, A. Latina, and P. Eliasson EUROTeV-Report-2007-068, PAC 2007 Proceedings p. 2847.

(217) My contributions to the papers. Papers I-II were both written almost exclusively by me, and I also carried out the simulations treated in the papers. The co-author assisted with helpful advice. In Paper III, the work was shared equally between me and the second author. I performed the luminosity tuning simulations and documented this part of the work, and the second author constructed the Final Focus knobs and wrote that part. In Papers IV-VI, the simulations and writing were carried out exclusively by me. The co-author of Papers IV and VI provided useful feedback. In Paper VII, we used a new curved ILC main linac lattice that I had designed. In addition, I performed the simulations presented, except the ones related to dynamic imperfections. My contribution to Paper VIII was the study of correction methods in a curved CLIC main linac. I also co-authored Papers i-viii, but with smaller contributions..

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(219) Contents. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Physics at future linear colliders . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Physics beyond the Standard Model . . . . . . . . . . . . . . . . . 1.2 Linear colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Why linear colliders? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Acceleration principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 General layout of a linear collider . . . . . . . . . . . . . . . . . . 1.2.4 The International Linear Collider (ILC) . . . . . . . . . . . . . . 1.2.5 The Compact Linear Collider (CLIC) . . . . . . . . . . . . . . . 2 Beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Beam optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 The beta function and the phase space ellipse . . . . . . . . . . 2.1.6 Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Wakefield effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Beam-beam interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Collision parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Emittance and luminosity preservation . . . . . . . . . . . . . . . . . . . . . . 3.1 Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Beam-based alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 One-to-one steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Dispersion Free Steering . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Alternative beam-based alignment methods . . . . . . . . . . . 3.2.4 RF alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Tuning bumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Emittance tuning bumps . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Beam-laser luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Luminosity tuning bumps . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Collision parameter tuning . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Trajectory feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 10 10 11 11 12 13 16 17 21 21 21 21 23 24 24 26 26 27 27 27 28 31 31 33 33 34 35 36 36 36 37 39 39 39 40.

(220) 3.4.2 Emittance-optimised feedback . . . . . . . . . . . . . . . . . . . . . 4 Summary of papers I-VIII and obtained results . . . . . . . . . . . . . . . . 4.1 Paper I: Luminosity Tuning Bumps in the CLIC Main Linac . . 4.2 Paper II: Dispersion Free Steering and Emittance Tuning Bumps in the ILC Linac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Paper III: Luminosity Tuning at the Interaction Point . . . . . . . . 4.4 Paper IV: Design of main linac emittance tuning bumps for the Compact Linear Collider and the International Linear Collider . 4.5 Paper V: Dynamic imperfections and optimized feedback design in the Compact Linear Collider main linac . . . . . . . . . . . . 4.6 Paper VI: Performance of CLIC emittance tuning bumps in a dynamic environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Paper VII: Study of an ILC Main Linac that Follows the Earth Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Paper VIII: Implications of a Curved Tunnel for the Main Linac of CLIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary in Swedish: Optimering av emittans och luminositet vid framtida linjäracceleratorer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Framtida acceleratorer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Emittans och luminositet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Optimering av emittans och luminositet . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 43 43 44 45 46 47 49 50 50. 53 54 54 55 57 59.

(221) 1. Introduction. During the past 50 years, an impressive development of particle colliders, with ever increasing luminosities and centre-of-mass energies, has resulted in a number of fundamental discoveries in particle physics. In recent decades, experiments at hadron colliders such as SPS and Tevatron, at e+ e− colliders such as LEP and SLC, and at the electron-proton collider HERA have explored the energy range up to several 100 GeV and established beyond doubt the validity of the Standard Model at these energies [1]. In order to extend the exploration into the TeV regime, new more powerful colliders have to be constructed. The Large Hadron Collider (LHC) installed at CERN in the old LEP tunnel is scheduled to start up in May 2008 [2] and will give a first glimpse of TeV physics. In addition, two major linear colliders are planned for the postLHC era: the Compact Linear Collider (CLIC) and the International Linear Collider (ILC). Both are intended to provide precision measurements of phenomena discovered at the LHC. They will also have the potential to discover new physics. A number of important accelerators and their main properties are listed in Table 1.1. Table 1.1: Examples of important accelerators. The first four are no longer in operation, and the Tevatron will be operational only for a few more years. The LHC is due to start up in May 2008, while ILC and CLIC are still only in the planning stage. The listed maximum energies, taken from [3–5], are all approximate. Similarly, the stated luminosities are only estimates.. Accelerated particles Machine, site. (max. energy [GeV]). SLC, SLAC. e+ (50) + e− (50). LEP, CERN. e+ (100). +. e− (100). Luminosity [1033. cm−2 s−1 ]. Accelerator type. 0.003. Linear. 0.1. Circular. HERA, DESY. e− (30). + p(820). 0.075. Circular. SPS, CERN. p(450) + p¯ (450). 0.006. Circular. Tevatron, Fermilab. p(1000) + p¯ (1000). 0.04. Circular. LHC, CERN. p(7000) + p(7000). 10. Circular. ILC, (not chosen). e+ (500). 20. Linear. 60. Linear. CLIC, (not chosen). +. e− (500). e+ (2500) + e− (2500). 9.

(222) 1.1. Physics at future linear colliders. Even though the Standard Model is a very successful theory, in perfect agreement with the results from accelerator experiments, it has its limitations. Apart from incorporating many arbitrary parameters, it cannot explain gravity, does not fully account for the behaviour of neutrinos, and cannot explain the dark matter observed in cosmos [6]. The main motivation for building new highenergy colliders is indeed the search for physics beyond the Standard Model. The LHC is a proton-proton collider with a centre-of-mass energy of 14 TeV. Because of the composite structure of hadrons, not all of this energy is available for the creation of new particles. As a consequence, the LHC centre-of-mass energy is sufficient to explore the energy range only up to approximately 1 TeV. While hadron collisions produce large backgrounds, much cleaner collision are obtained with leptons and antileptons, and all of the centre-of-mass energy is available to produce final state particles. The production of new particles is in this case “democratic” in the sense that the different species of elementary particles are produced in equal amounts within the given energy range. While hadron-hadron colliders offer excellent conditions for discovering new physics, a lepton-antilepton collider is better suited for precision studies of already discovered particles. In this way, the LHC and CLIC/ILC complement each other. This kind of complementarity was previously successfully exploited at the SPS (discovery of the W ± and Z 0 bosons) and the LEP (precision measurements of the properties of these bosons). Because of their precision, CLIC and ILC may also discover completely new phenomena. In addition, in the multi-TeV CLIC collider, new particles may be produced that are to heavy to create at the LHC.. 1.1.1. Physics beyond the Standard Model. Questions that may be answered by experiments at new high-energy colliders include those related to the mechanism behind electroweak symmetry breaking and mass generation. In the Standard Model, this symmetry breaking is explained by the Higgs mechanism [1, 6, 7]. If a Higgs boson exists, it will most likely be discovered at the LHC. While the mass of the Higgs boson could be measured by the ATLAS and CMS experiments at the LHC, it would be difficult to measure its spin and couplings. A TeV lepton-antilepton collider would offer the possibility to accurately measure these properties and consequently establish the precise nature of the Higgs boson. New physics may also be much more complex. According to supersymmetry, there exist several Higgs bosons and a spectrum of so-called supersymmetric particles (sparticles). These sparticles have the same interactions as the Standard Model particles but are bosons instead of fermions and vice versa. If signals of sparticles are seen at the LHC, TeV lepton-antilepton colliders could later be used to precisely determine their properties (eg. mass,. 10.

(223) spin and couplings) and thereby confirm that they are indeed supersymmetric “partners” of the Standard Model particles. In [1] it is shown that for certain sparticles, detailed measurements will only be possible at the multi-TeV collider CLIC. For some sparticles, the discovery potential is also shown to be better at the future linear colliders than at the LHC. For the exploration of the supersymmetric spectrum, full complementarity with the LHC will probably require lepton-antilepton collisions with a centre-of-mass energy of at least 2 TeV [6]. Other problems that may be addressed are the unification of forces, for which supersymmetry offers an explanation, and the origin of quark flavors. Since one of the candidates for cold dark matter is a light neutral sparticle, the discovery of supersymmetry could explain the origin of dark matter. Possibly experiments could also lead to the discovery of extra space-time dimensions at small scale.. 1.2. Linear colliders. As initially mentioned, two major linear colliders, CLIC and ILC, are currently being planned for the post-LHC era. This section will explain why a circular TeV lepton-antilepton collider is not easily realised. Furthermore, the principle properties of a linear collider will be outlined, as well as the design of CLIC and ILC.. 1.2.1. Why linear colliders?. Charged particles undergoing acceleration are known to emit electromagnetic radiation [8]. The radiation emitted during longitudinal acceleration can be shown to be completely negligible for all practical purposes. When a particle trajectory is bent, and the acceleration is perpendicular to the direction of motion, the situation is, however, quite different. In a circular accelerator, the energy loss per revolution caused by the so-called synchrotron radiation is ΔE ∼. β 3γ 4 , ρ. (1.1). where ρ is the bending radius. For the extremely relativistic particles of highenergy colliders, β ≈ 1. For a given energy E = γmc2 , Eq. (1.1) may be reformulated as E4 ΔE ∼ . (1.2) ρm4 The energy loss per revolution is thus inversely proportional to the fourth power of the particle mass m. Consequently, the energy losses due to synchrotron radiation are particularly severe for light particles like electrons and 11.

(224) positrons, while the heavier protons are less affected. For the extreme energies of the LHC, the effect will, however, be noticeable even for protons. For electrons and positrons, the large synchrotron radiation losses limit the centre-of-mass energy to approximately 200 GeV for a circular machine with a circumference of 27 km. The record energy of 209 GeV was obtained at LEP in 2000. Considering the collider length and synchrotron radiation power losses, it can be shown that the cost of a collider scales linearly with energy for a linear collider and quadratically with energy for a circular collider. The most cost-efficient way of accelerating electrons/positrons to the TeV regime is, therefore, by using a linear accelerator. In a circular accelerator, particles gain energy by passing through the same accelerating structures turn after turn, as they circle around the accelerator a very large number of times. In a linear accelerator, on the other hand, a particle only passes each accelerating structure once. To reach TeV energies, the accelerator has to be very long, or the particles have to be accelerated using very strong electromagnetic fields, i.e. using a very high accelerating gradient. Each ILC linear accelerator (linac) is foreseen to be 11.3 km long for a centre-of-mass energy of 0.5 TeV. The CLIC main linacs, each 21 km long according to the most recent design, will accelerate electrons/positrons to a centre-of-mass energy of 3 TeV due to its very high gradient [9]. It may, therefore, be considered more “compact”, as its name indicates.. 1.2.2. Acceleration principle. Modern linear colliders are based on an acceleration technique suggested in 1925 by Ising [8]. While the very first accelerators used electrostatic fields to accelerate particles, Ising proposed to use a high-frequency voltage source to produce an alternating field. This could solve the problem of electrical discharges frequently occurring in electrostatic colliders, strongly limiting their performance. A few years later, Wideröe carried out the first successful test of an accelerator based on this principle, see Fig. 1.1. Instead of the simple drift tube technique used in the Wideröe linear accelerator, modern accelerators, in general, increase the beam energy using more elaborate accelerating structures in the form of iris-loaded cylindrical waveguides, see Fig. 1.2. A properly designed structure will guide a travelling wave with a phase velocity equal to that of the accelerated beam (v ≈ c). The wave can, obviously, only accelerate particles on its positive half-period, and, as a consequence, the accelerated beam has to be bunched. By matching the phase of the travelling wave to the moving bunched beam, the beam can “ride” on the wave and will be continuously accelerated in the waveguide. For stability reasons, the phase is often adjusted such that the bunches are accelerated slightly off-crest, where the field has a positive slope. This offers the possibility of increasing acceleration if needed. More importantly, it introduces an energy spread in each bunch such that the tail of the bunch gets a lower energy 12.

(225) Figure 1.1: In the Wideröe linear accelerator, charged particles were accelerated by the electric field between so-called drift tubes connected, with alternating polarity, to a Radio-Frequency (RF) source. Inside the drift tubes the particles were shielded from any fields. The RF phase was matched to the particles such that they always saw a field in the forward direction.. Figure 1.2: Schematic drawing of an iris-loaded waveguide. A waveguide of properly adjusted dimensions may be fed with RF power and a travelling wave with the desired phase velocity (v ≈ c) will be produced.. than the head. This is necessary for so-called BNS1 damping, which is used to prevent that wakefields, caused by the head of the bunch, kick the tail resonantly in the accelerating structures, consequently causing large emittance growth [10].. 1.2.3. General layout of a linear collider. The future linear colliders will be large and very complex machines. Ultimately their purpose is to generate high-energy beams and produce collisions with high luminosity (i.e. high interaction probability). Such collisions can only be obtained using small emittance (i.e. small transverse size and little angular divergence) beams. The colliding beams will in general be extremely small (of the order of a few nm). This poses extreme requirements on the sta1 Named. after Balakin, Novokhatsky, and Smirnov.. 13.

(226) bility and precision of the collider components and it is necessary to build the machine at a site which is very quiet, i.e. free from movements and vibrations in the ground. In order to reduce the effect of noise, the collider is preferably built in a tunnel underground. For ILC, which will use superconducting accelerating structures, a large cryogenic system will be used for cooling. A tunnel that follows the earth curvature is considered beneficial for the design of such a cryogenic system [11]. For CLIC, using normal-conducting structures, this is not an issue. However, depending on the geology of a certain site, a tunnel following the curvature of the earth may still be cheaper. A curved tunnel can, for example, be constructed at constant depth below ground. For a 50 km long straight tunnel, on the other hand, the central part would have an additional depth of 50 m. It is also of interest to know if CLIC at some point could be built in the tunnel of ILC which will, most likely, be curved. A number of subsystems guarantee that the desired collider performance is achieved. The requirements on many of these subsystems are very challenging. Starting from one end of the machine, a collider consists of beam generation and injection systems, main linac, Beam Delivery System (BDS), Interaction Point (IP), and post-collision line. Each of these systems can themselves be divided into smaller parts. Beam generation and injection systems In the first part of the collider, the bunched beam is generated. For electrons this is in general carried out using a thermionic gun. The production of positrons, instead, relies on pair-production. The techniques are quite different for CLIC and ILC, see below. Observe that the beam will not be a continuous stream of bunches. It will rather consist of pulses (or trains) of a number of bunches. These pulses will be produced with a certain repetition rate. For the normal-conducting CLIC structures the pulse length will be very short, of the order of 100 ns. In ILC, with superconducting cavities, a pulse length of 1 ms will be used. Initially, a beam has a low energy and a relatively large emittance. It is, therefore, pre-accelerated and “damped”. The damping is performed in socalled damping rings, where the particles’ transverse momentum is decreased, while the longitudinal momentum is preserved. The effect of such damping is a beam with lower emittance. Further acceleration and also bunch compression are required before injection into the main linac. During the compression, the longitudinal bunch length is strongly reduced. A short bunch is necessary to minimise the negative impact of wakefields (see Sec. 2.1.8) in the main linac accelerating structures. Main linac The main linac is where particles are accelerated to the final very high energy. The main linac consists of a repeating pattern of accelerating structures, 14.

(227) quadrupole magnets, dipole correctors and Beam Position Monitors (BPMs). The large number of structures necessary to accelerate the beam to the desired energy means that the linacs have to be very long (of the order of 10 − 20 km). During the transport through the long linac, the beam is prevented from diverging by the focusing effect of a special arrangement of quadrupole magnets, a so-called Focusing-Defocusing (FODO) lattice, see Sec. 2.1.2. The dipole correctors and the BPMs are used for a number of alignment and beam control techniques. Such techniques will be described in detail in Sec. 3. Beam Delivery System At the end of the linac, the accelerated beam is injected into the BDS, which consists of three subsystems: collimation, chromatic correction, and Final Focus System (FFS). Collimation is necessary to remove particles which would otherwise cause unacceptable background in the detectors at the IP. Particles that deviate too much from the nominal trajectory or have a large momentum offset are therefore eliminated. In the FFS, the beam has to be strongly demagnified and directed onto the target beam to meet the demanding luminosity requirements. The demagnification is achieved using very strong focusing. In a region of strong focusing, chromatic aberrations (deviations in focusing due to beam energy spread) are important, and a number of sextupole magnets have to be used for chromatic correction. In order to optimise the collision with the target beam, tuning of a number of parameters, for instance beam offsets, angles, and waists, is necessary, see Sec. 3.3.4. In addition, so-called beam-beam feedback is foreseen to keep the beams into collision on a nano-second timescale, see Sec. 3.4. Interaction Point and post-collision line The high-energy, low-emittance beams are brought into collision at the IP. In order to study the new particles produced in the collision, a particle detector has to be used. Not only should this detector be able to separate and track particles in space and time, it must also identify each particle and precisely determine its energy and momentum. Modern detectors are in general composed of a number of subdetectors specialised for different purposes. Several detector concepts are under study for future linear colliders, but a description of them is not within the scope of this thesis. The spent beam has to be cleanly removed from the IP in order to minimise background. This is done using the post-collision line, which will also be used for diagnostics needed to optimise the collision parameters. This includes the measurement of the deflection angle of the spent beam, which is important for beam-beam feedback. Determination of beamstrahlung or coherent pair signals may also be important, cf. Sec. 3.3.4 and Paper III.. 15.

(228) 1.2.4. The International Linear Collider (ILC). Until 2004, five proposals for a future linear collider were being prepared in parallel (CLIC, JLC/GLC, NLC, TESLA, and VLEPP). In 2004, many institutes involved in linear collider R&D agreed to join their efforts and work towards an International Linear Collider (ILC) based on superconducting RF accelerating cavities. During the next few years, the basic parameters and layout of ILC were defined and a detailed design report prepared, the ILC Reference Design Report [4]. The design is based on acceleration technology pioneered by the TESLA collaboration with an average accelerating gradient of 31.5 MV/m in 1.3 GHz superconducting RF cavities. In the baseline ILC design, the centre-of-mass energy is 500 GeV, and the luminosity is 2 × 1034 cm−2 s−1 . The requirement to achieve very high luminosity imply the use of high power and low-emittance beams. Layout In the ILC baseline design, the 500 GeV centre-of-mass energy is achieved using approximately 17000 cavities operated at 2◦ K in the two 11.3 km long main linacs. The total length of the ILC complex is 31 km, see Fig. 1.3. In order to upgrade the centre-of-mass energy to 1 TeV, both main linacs would have to be extended to twice their length. In addition, the BDS would have to be augmented, and the total ILC length would be more than 50 km.. Figure 1.3: Schematic layout of the ILC complex for a centre-of-mass energy of 500 GeV [12].. As indicated in Fig. 1.3, the beams are brought into collision with a crossing angle. This is required to cleanly get rid of the spent beam and to minimise parasitic bunch crossing effects. In the baseline design, this crossing angle is 14 mrad. So-called crab cavities are used to kick the beam bunches before collision such that they are rotated and collide as if no crossing angle ever existed. While the general layout of a linear collider was explained in Sec. 1.2.3, a few ILC characteristics of importance for this thesis will be pointed out below. 16.

(229) Beam generation and injection systems While the electron beam is produced using a thermionic gun, the production of positrons is, as shown in Fig. 1.3, based on photo-production in an undulator parallel to the electron linac. After acceleration to 150 GeV, electrons are diverted into the 150 meter long undulator, before being returned to the linac. In the undulator, the electrons follow a slalom course through a special arrangement of magnets and produce a beam of 10 MeV photons (synchrotron radiation). These photons are directed onto a Ti-alloy target where electron-positron pairs are created. The electrons and the remaining photons are separated off and dumped, while the positrons, after a 5 km transport, are accelerated and injected into the positron damping ring. In the damping rings, the emittance of the beam has to be significantly reduced (by five orders of magnitude for the vertical emittance). This is done using a system of superconducting wiggler magnets and RF cavities. The damped bunches are then transported to the bunch compressor at the entrance of the main linac, where the length of each bunch is compressed by a factor 30 − 45 to match the main linac requirements. During the compression, the bunches are also accelerated from 5 to 15 GeV. The normalised vertical emittance of the beam injected into the main linac is 20 nm. Main linac In the main linac, the bunches are accelerated from the initial energy of 15 GeV to the final energy of 250 GeV. The acceleration is obtained by RF units each containing 26 cavities. The two 11.3 km long main linacs consist of 278 and 282 such RF units for the positron and electron linac respectively. The additional RF units in the electron linac are required to compensate for the energy lost in the undulator. At the centre of each RF unit, there is also a superconducting quadrupole, a BPM and superconducting horizontal and vertical corrector magnets. The quadrupoles of the linac form the FODO lattice required to achieve the necessary focusing of the beam. The BPMs and the corrector magnets are used for trajectory control and emittance preservation throughout the linac, see Sec. 3. In order to simplify the transport of liquid helium to the superconducting elements, the main linacs should follow the Earth curvature. Beam dynamics issues related to the use of a curved linac were studied in detail in Paper VII.. 1.2.5. The Compact Linear Collider (CLIC). A detailed study of the possibility of building a new linear collider to complement the LHC has been going on at CERN, and elsewhere, since almost 20 years. This so-called Compact Linear Collider (CLIC) is designed to produce electron-positron collisions with a centre-of-mass energy of 0.5-5 TeV and will provide a sufficiently high luminosity (1034 − 1035 cm−2 s−1 ) to allow precision studies of new physics discovered at the LHC and beyond. 17.

(230) General layout For CLIC, an accelerating gradient as high as 150 MeV/m has been considered [13]. Due to this extremely high gradient, CLIC may have a relatively compact layout with a total length of nearly 34 km in case of a 3 TeV centreof-mass energy, see Fig. 1.4. Observe that the layout was changed during. Figure 1.4: Overall 2005 layout of CLIC for a centre-of-mass energy of 3 TeV [13].. 2007 when cost optimisation and performance considerations [14] lead to a reduction of the accelerating gradient to 100 MeV/m and of the RF frequency to 12 GHz. See Table 1.2 for a list of important parameter changes. All the work reported in this thesis was carried out using the old CLIC layout, and the new parameters will not be further discussed. The conclusions of the work are expected to remain valid. The high gradient of CLIC is achieved by the use of very high RF frequencies. Superconducting RF cavities cannot be operated at a higher gradient than that of ILC. Consequently, room-temperature travelling wave structures have to be used in CLIC. In order to provide sufficient RF power for the highgradient structures, a novel two-beam acceleration scheme has to be used instead of conventional high-frequency RF sources. In this scheme, the power is generated by a high-current electron drive beam with very high frequency. With the use of special Power Extraction and Transfer Structures (PETS), the drive beam, which runs parallel to the main beam, is decelerated, and the extracted RF power is transferred through wave guides to the main beam. Both the main and the drive beams are produced centrally before being transported to the main linacs and the drive beam decelerators respectively. 18.

(231) Table 1.2: Important differences between the 2005 and 2007 layout [9, 13].. Parameter Centre of mass energy. 2005 layout. 2007 layout. 3000. 3000. Unit GeV 1034. cm−2 s−1. Luminosity. 6.5. 5.9. Loaded acc. gradient. 150. 100. MV/m. RF frequency. 30. 12. GHz. Particles per bunch. 2.56. 3.72. 109. Bunches per pulse. 220. 312. 1. Pulse length. 58.4. 156. ns. Pulse repetition rate. 150. 50. Hz. Site length. 33.6. 47.9. km. After being accelerated through the 14 km long main linac, the main beam enters the BDS and the beam is collimated, strongly focused and directed onto the target beam. Drive beam generation The most distinct feature of CLIC is the use of the two-beam acceleration scheme. As mentioned above, the high-frequency RF power required to feed the CLIC accelerating structures is obtained from a high-current electron drive beam with very high frequency. This drive beam is generated with a thermionic gun, then bunched and accelerated with conventional accelerating structures operating at low frequency (approximately 1 GHz). After being accelerated to 2.4 GeV, the beam is frequency multiplied and compressed in three stages: first in a delay line providing a factor 2 frequency multiplication and then in two combiner rings each with a factor 4 multiplication, see Fig. 1.4. The drive beam is finally transported to the PETS where its power is extracted and then transferred to the accelerating structures. The RF power that this drive beam offers is far higher than that of any conventional RF source at 30 GHz. Main beam generation and injection system A 10-MeV electron beam is produced using a thermionic gun. This beam is then accelerated to approximately 2.4 GeV before being injected into a damping ring. Next, the beam is transferred to a first bunch compressor, then accelerated to 9 GeV in a booster linac and transferred to a second bunch compressor at the entrance of the main linac. The production of positrons starts with a 10-MeV RF electron gun generating electrons which are accelerated to 2 GeV and directed onto a positron target. The released positrons are accelerated to 2.4 GeV, in the same linac 19.

(232) as the electron main beam, before being injected into a pre-damping ring and subsequently into the positron damping ring. The rest of the positron injection system consists of the same type of bunch compressors and transfer lines as for the electron system. It also uses the same booster linac to accelerate the positrons to 9 GeV. According to the 2005 design, a beam injected into the main linac consists of 58.4 ns long pulses, which is significantly shorter than for ILC. The pulse repetition rate is 150 Hz, and each pulse consists of 220 bunches. The normalised vertical emittance of a bunch is 5 nm at the linac entrance. Main linac In the nominal design, each of the two CLIC main linacs will accelerate beams from 9 GeV up to 1.5 TeV. Each main linac is made of more than 6800 2.23 m long modules. Each such module consists of up to four accelerating structures precisely positioned on a girder. In some modules, between one and four of the accelerating structures are replaced by quadrupoles to provide beam focusing. At the start of each girder there is a BPM. As previously mentioned, the information from the BPMs is used for alignment and beam control. In the first part of the linac, the RF acceleration phase is adjusted such that a certain energy spread is introduced in the bunch. This energy spread is maintained throughout the main part of the linac by accelerating at another RF phase. At the end of the linac, the RF phase is adjusted to reduce the energy spread such that the bunch can pass through the FFS. The energy spread is required for BNS damping, which is particularly important for high-frequency normal-conducting structures with strong wakefield effects. Beam Delivery System Out of the three subsystems of the BDS, only the FFS was of importance for this thesis. The FFS has to demagnify the beam to transverse rms spot sizes of 60 nm horizontally and 0.7 nm vertically at the IP. These very small beam spot sizes are necessary in order to meet the demanding luminosity requirements. The demagnification is achieved using very strong focusing. In a region of strong focusing, chromatic aberrations cause problems, and a number of sextupole magnets need to be used for chromatic correction. In Paper III, these sextupole magnets were used to optimise certain collision parameters in order to maximise luminosity. Interaction Point and post-collision line At the IP, the beams collide with a crossing angle of 20 mrad, and the collision products are detected and analysed using the experimental detector setup. The spent beam is directed into a post-collision line containing diagnostic tools used for optimisation of the collision. A conceptual design of the CLIC postcollision line is described in detail in [15].. 20.

(233) 2. Beam dynamics. Both for CLIC and ILC, test facilities have been built where the feasibility of key issues can be demonstrated. In addition, the test facilities may be used to test critical components (in case of CLIC, eg. the accelerating structures and their nominal gradient and pulse length) [16]. However, during the early stages of the design studies, and for more complete investigations of the accelerator performance, theoretical arguments and, above all, simulations are of great importance. A number of tools exist which facilitate beam dynamics simulations. The simulations described in Papers IVIII were mainly carried out using the programs PLACET [17] (for particle tracking) and GUINEAPIG [18] (for beam-beam interaction simulations). In this section, a brief overview of some important beam dynamics concepts is presented. For a more complete description see [8, 19].. 2.1 2.1.1. Beam optics The Lorentz force. Charged particles in an electromagnetic field are subject to the Lorentz force dp = F = q(E + v × B). dt. (2.1). By using Eq. (2.1), it can be argued that it is much easier to produce a magnetic rather than an electric field that efficiently steers (or focuses) a highly relativistic (v ≈ c) beam. As a consequence, magnets are used for these purposes in almost all modern accelerators. In many cases, like in the main linac, it may be assumed that the magnets produce fields with transverse (perpendicular to the ideal beam trajectory) components only.. 2.1.2. Magnetic fields. By expanding the horizontal magnetic field component in the vicinity of the ideal trajectory and multiplying by e/p, the field may be considered as a sum of multipoles: e e dBx e 1 e d 2 Bx 2 y + O(y3 ) Bx (y) = Bx (0) + y+ p p p dy 0 2! p dy2 0 (2.2) 1 1 2 3 = + O(y ). + ky + my R 2! 21.

(234) These terms correspond to the dipole, quadrupole, sextupole, and higher multipole effects of the field. In the first (dipole) term, R is the radius of curvature obtained by equalling the Lorentz force and the centrifugal force. The second (quadrupole) term is linear in y and causes a focusing effect if it is properly oriented. A dipole field can be created using a dipole magnet. Ideally, such a magnet produce a perfectly uniform field as shown in Fig. 2.1. This magnet will bend the trajectory of the beam vertically and may be used as a dipole corrector for various emittance preservation techniques.. ey N. ex. ez. S. B. Figure 2.1: Ideal dipole field. Dipole magnets are often used for correction of the beam trajectory. The field shown here causes a vertical deflection of a particle moving into the plane of the paper.. The quadrupole field term of Eq. (2.2) can similarly be produced using a quadrupole. By using Maxwell’s equations, it may be shown that the only way to produce a field with the desired focusing effect is by using four hyperbolic pole faces as shown in Fig. 2.2. The focusing effect achieved along one axis is accompanied by a defocusing effect along the other. By rotating the quadrupole by 90◦ in the plane of the paper, the effect of the quadrupole will be reversed. A quadrupole that focuses the beam in the vertical plane will in this thesis be referred to as a focusing quadrupole, while one that focuses the beam in the horizontal plane (defocuses in the vertical) will be referred to as a defocusing quadrupole. Observe that a quadrupole that is transversely displaced by a distance Δy from its ideal position will give rise to an additional dipole field (e/p)Bx (y) = −kΔy. In order to transport a beam over long distances without the beam crosssection becoming too large, a lattice that focuses in both planes is required. This is achieved using a FODO lattice consisting of alternating focusing and defocusing quadrupoles. Since the quadrupole field is stronger farther away 22.

(235) ey S. N. ex. ez a S. N. Figure 2.2: Ideal quadrupole field. For positive charges moving into the plane of the paper the quadrupole has a focusing effect in the vertical plane and a defocusing in the horizontal plane.. from the magnetic centre, the overall effect of such a FODO lattice is the desired focusing in both planes. Similarly, sextupole magnets can be constructed using six pole faces. The sextupoles are used for chromatic correction. Higher multipoles are less common but may be used to correct higher order geometric aberrations.. 2.1.3. Equation of motion. A lattice consisting of only dipoles and quadrupoles may be referred to as linear since the force they apply to the beam is either constant or grows linearly with the distance from the reference trajectory (the trajectory of the nominal particle). By introducing a reference frame moving with the particle, centred at its projection on the reference trajectory, Eq. (2.1) may be reformulated as 1 1 Δp y (s) + + k(s) y(s) = (2.3) R2 (s) R(s) p for the vertical particle position. Here, R(s) is the bending radius of the reference trajectory, k(s) is the focusing strength, and s is the distance along the reference trajectory. This is the equation of motion for a particle moving through a lattice of dipole and quadrupole magnets. While Eq. (2.3) only describes the vertical motion of the particle, the only difference for the horizontal motion would be a sign change of the focusing strength k(s). Note that k(s) is positive in a focusing quadrupole. 23.

(236) 2.1.4. Particle tracking. In a linear lattice, it can be shown that each dipole and quadrupole magnet may be described by a transfer matrix acting on a six-component vector containing the coordinates of a particle. Similarly, a matrix may be used to describe the drift of a particle through vacuum and to describe the effect of an accelerating structure. The coordinates of a particle are horizontal position and angle, vertical position and angle, deviation in momentum from the nominal particle, and longitudinal position (or phase) relative to the nominal particle. This way of tracking particles is used in a large number of codes. A further simplification of the tracking may be performed under the assumption that the tracked particles are highly relativistic (v ≈ c) and move nearly parallel to the ideal trajectory. The longitudinal position relative to the nominal particle may then be considered constant during the tracking, and the beam, consisting of a large number of particles, may be cut longitudinally into slices. Each slice consists of macroparticles that have the same longitudinal position. A macroparticle represents a large number of particles with nearly the same energy and phase space position. The macroparticle is described by the six coordinates mentioned above, by a number of second moments, and by a weight proportional to the number of particles it represents. The second moments are the covariances of transverse phase space coordinates for all particles represented by the macroparticle. In the CLIC main linac, it has been observed that as few as 21 slices and 7 macroparticles per slice gives a sufficiently accurate description of the beam, see eg. Paper IV.. 2.1.5. The beta function and the phase space ellipse. In order to describe the motion of a complete beam rather than that of single particles, the so-called beta function (amplitude function) β (s) may be introduced. This can be done by writing the homogeneous solution to Eq. (2.3) as √ y(s) = ε β (s) cos (Ψ(s) + φ ), (2.4) where ε is the vertical emittance and Ψ is the phase. By inserting this expression into Eq. (2.3), for Δp/p = 0, it is found that Ψ(s) =. s dσ 0. β (σ ). .. (2.5). By differentiating Eq. (2.4), it may be shown that y(s) and y (s) are correlated according to γ(s)y2 (s) + 2α(s)y(s)y (s) + β (s)y2 (s) = ε, 24. (2.6).

(237) where β (s) , 2 1 + α 2 (s) γ(s) = . β (s) α(s) = −. (2.7). The expression in Eq. (2.6) describes an ellipse with an area A = πε (see Fig. 2.3), but with a shape and an orientation that vary with the distance along the trajectory. Using Liouville’s theorem [19], it is possible to show that. Figure 2.3: The phase space ellipse describes the movement of a particle in phase space (yy -plane). The ellipse changes shape and orientation along the trajectory of the beam but has a constant area. The maximum amplitude of the beam position is seen to be proportional to β .. a particle travelling in a magnetic field will move along an ellipse with an invariant area. The advantage of introducing the beta function and the phase space ellipse is not obvious when a single particle is studied. However, by noting that all particles of the beam move along ellipses with the same shape and orientation but with different ε and φ , the value of this formalism becomes apparent. The whole beam may now be represented by one single ellipse for the proper value of ε . 25.

(238) 2.1.6. Emittance. A practical definition of the vertical emittance εy of a beam is obtained by setting σy2 = (y − y) ¯ 2 = βy εy , (y − y¯ )2 = γy εy , (y − y)(y ¯ − y¯ ) = −αy εy , (2.8). and εy =. (y − y) ¯ 2 (y − y¯ )2 − (y − y)(y ¯ − y¯ )2 ,. (2.9). where y¯ and y¯ are the average vertical position and angle, and σy is the vertical rms spot size. Observe that the emittance measure discussed so far is not truly invariant. During acceleration, the transverse beam size and the divergence scale inversely to the square root of the particle momentum. A proper invariant is instead the normalised emittance εN = βr γr ε ≈ γr ε,. (2.10). where the subscript r refers to relativistic beta and gamma in order to distinguish from the parameters of the phase space ellipse. Observe that all emittance values stated in this thesis and the included papers refer to normalised emittance since this is the relevant parameter for accelerated beams. For a macroparticle beam, the vertical normalised emittance is εN = γr [(∑ wk (yk − y) ¯ 2 + σ¯ yy )(∑ wk (yk − y¯ )2 + σ¯ y y ) k. k. ¯ k − y¯ ) + σ¯ yy )2 ]1/2 , − (∑ wk (yk − y)(y. (2.11). k. where wk is the weight of macroparticle k and σ¯ yy , σ¯ y y , and σ¯ yy are the weighted average second moments of all macroparticles. The emittance is a very important measure of the quality of the beam. While it is invariant under ideal conditions, it may increase due to imperfections generating dispersion and wakefield effects. Efficient means of preserving the emittance along a machine are crucial. Various techniques are presented in Chapter 3.. 2.1.7. Dispersion. Since the beam that is accelerated through the main linac will inevitably have a certain momentum spread, it is important to make sure that this will not cause a large spread in the trajectories of the individual particles. The equation of motion, Eq. (2.3), is inhomogeneous in regions where 1/R = 0. By determining the particular solution to this inhomogeneous differential equation and normalising with respect to the momentum deviation Δp/p, the so-called dispersion function is obtained. This function, in other words, describes the 26.

(239) normalised deviation of an off-momentum particle trajectory from the reference trajectory: Δy(s) . (2.12) D(s) = Δp/p Similarly its derivative, with respect to s, describes the deviation in y . Apparently, particles follow different trajectories depending on their momentum, and as a consequence, the emittance increases in a dispersive region. Since the main linac is straight and 1/R = 0, the dispersion is ideally equal to zero along the whole linac. As mentioned in Sec. 2.1.2, misaligned quadrupoles will, however, give rise to dipole fields and, consequently, to a non-zero dispersion. In the ILC main linac, dispersion caused by such misalignments is the main source of emittance growth. It is also important for CLIC, where, however, wakefield effects are more important.. 2.1.8. Wakefield effects. A wakefield is an excitation of modes induced by a bunch when passing through an accelerating structure [10,20]. For a bunch which is displaced with respect to the central axis of the structure, the excited field will deflect the bunch itself. Since the bunch is moving at the speed of light, the wakefields will only deflect particles later in the bunch. The forces acting on these trailing particles can be described by longitudinal and transverse wake potentials, WL and WT . These potentials scale with frequency as WL ∼ f 2 and WT ∼ f 3 and are consequently much stronger for CLIC than for ILC. The transverse wakefields cause a deflection of particles which varies along the bunch and leads to a certain distortion of the beam. This distortion causes severe emittance growth in the CLIC main linac. An efficient way of counteracting this effect is by displacing structures by a carefully tuned distance in order to induce wakefield kicks that cancel the undesired wakefield effects.. 2.2. Beam-beam interaction. Together with the centre-of-mass energy, the most important parameter of a collider is its luminosity. Due to the complexity of a collision involving a very large number of particles, simulations are always necessary to be able to predict the outcome of the collision. The luminosity, may however to some extent be estimated using reasonably simple expressions.. 2.2.1. Luminosity. When two beam bunches collide, the rate at which a certain physical process occurs is given by σ L , where σ is the interaction cross-section of the physical process and L is the luminosity. The luminosity may be thought of as a 27.

(240) measure of how efficiently the collider produces collisions. A statistical estimate of the luminosity may be obtained under the assumption that the charge distributions of the colliding bunches are Gaussian in all directions, that the bunches are rigid, and that there is no crossing angle: L =. N 2 nb fr N 2 nb fr = . 4πσx σy 4π βx εx βy εy. (2.13). Here, N is the number of particles per bunch, nb is the number of bunches per pulse, and fr is the repetition frequency, i.e. the number of pulses per time unit. The two bunches are assumed to have the same horizontal rms spot size σx , and also the same vertical rms spot size σy . In the second step of Eq. (2.13), the definition of the emittance was used, cf Eq. (2.8). It is clearly seen that the luminosity is larger for small spot sizes and that this requires well preserved emittance and strongly focused bunches. The minimum spot size is, however, limited by the hour glass effect and by beamstrahlung. By focusing the beam too strongly such that the beta function is very small at the IP, the luminosity may due to the hour glass effect decrease. The effect is explained by the fact that the beta function varies like β (s) = β ∗ +. s2 β∗. (2.14). close to its minimum, the beam waist. This gives each bunch the appearance of a horizontal hour glass at the IP. For a very low β ∗ , the beta function is seen to increase rapidly on both sides of the waist. As a consequence, β ∗ should not be smaller than the bunch length σz [21]. Due to the very strong fields of the colliding bunches, a mutual focusing will occur at the IP. As a consequence, the bunch spot size is reduced, and the luminosity is enhanced. However, the so-called beamstrahlung emitted as the particles are deflected in the field of the oncoming bunch will cause an undesired spread of the centre-of-mass energy and will also create detector background. It may be shown that beamstrahlung effects can be reduced by increasing σx + σy [21]. In order to maximise luminosity (Eq. (2.13)) and keep beamstrahlung at a minimum, flat beams (σx σy ) are in general used. The much larger horizontal spot size is the reason why in general only emittance preservation in the vertical plane has to be considered.. 2.2.2. Collision parameters. Obviously, colliding beams with very small spot sizes does not guarantee that luminosity is high. For the tiny beams used, even a small relative offset may mean that they do not collide. The offset of the beams must be minimised, both in the horizontal and in the vertical plane. In addition, the angle with which beams collide and the longitudinal position of the beam waists should be optimised. 28.

(241) Maximisation of luminosity by optimisation of collision parameters is ideally carried out using the luminosity itself as a tuning signal. Luminosity can, however, not be measured directly fast enough. For CLIC, potentially useful signals had been identified in [22], namely beamstrahlung, coherent pairs and incoherent pairs. In Paper III, the use of beamstrahlung was further investigated. While the beamstrahlung is not proportional to the luminosity, it was found that by using the signals from each beam in the proper way, the collision parameters could be tuned and luminosity maximised. In case of the vertical offset and angle of the beams, the tuning was performed by minimising the total beamstrahlung signal. For the corresponding horizontal parameters, on the other hand, the total beamstrahlung signal had to be maximised. The optimisation of the longitudinal waist position was found to be efficiently carried out by minimising the difference between the two beamstrahlung signals. Another collision parameter that may require optimisation is dispersion.. 29.

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(243) 3. Emittance and luminosity preservation. As mentioned initially, the luminosity required in future linear leptonantilepton colliders must be extremely high (6 × 1034 cm−2 s−1 for CLIC, 2 × 1034 cm−2 s−1 for ILC). According to Eq. (2.13), the luminosity is √ inversely proportional to εx εy . In order to achieve the luminosity goal in CLIC, the emittance, in particular in the vertical plane, has to be preserved throughout the machine. For CLIC, a vertical emittance of approximately εy = 10 nm is required at the IP. The horizontal emittance can be significantly higher and will be achieved rather easily. Similarly, the nominal vertical emittance at the IP of ILC is εy = 40 nm. Both for CLIC and ILC, it will be very challenging to achieve the emittance target in the vertical plane. The main linac is the part of a collider contributing most to emittance growth. Due to the numerous elements of the linac, even small element displacements may cause significant emittance growth. The nominal vertical emittance at the entrance to the CLIC main linac is 5 nm, and the emittance budget requires that the average emittance growth along the linac for a large number of machines is less than 100%, i.e. Δεy < 5 nm. For ILC, the initial emittance is 10 nm, and the emittance budget is often formulated as: in at least 90% of all machines the emittance growth should be less than 10 nm (Δεy (90%) < 10 nm).. 3.1. Imperfections. The future linear colliders are very sensitive to imperfections. In this thesis, the focus will be on imperfections affecting the main linac. These imperfections can in principle be divided into two types: static and dynamic imperfections. The static imperfections are time-independent (or very slowly changing) imperfections, while the dynamic imperfections are time-dependent. Static effects include, eg., displaced magnets, BPMs or complete girders, incorrect accelerating structure phase or gradient, quadrupole strength errors, and tilted elements. As mentioned in Sec. 2.1.7, displaced quadrupoles give rise to dispersion, which is the main source of emittance growth in the ILC main linac. Similarly, displaced quadrupoles cause emittance growth in the CLIC main linac where, however, displaced accelerating structures cause wakefield effects which are even more severe. In particular after the 31.

(244) installation of elements in the main linac, large misalignments may be expected. In order to keep static element misalignments at a reasonable level, an active prealignment system is planned for CLIC. The achievable precision of such element alignment is 10 μ m (vertically) over a distance of 200 m [23]. The prealignment is, however, not nearly sufficient to keep emittance growth below Δεy = 5 nm in the CLIC main linac. In order to reduce emittance growth further, beam-based alignment (Sec. 3.2) and emittance tuning bumps (Sec. 3.3) have to be used. In case of ILC, a survey system is being studied which will prealign elements with an accuracy of 200 μ m over 600 m [24]. In addition to this prealignment, beam-based alignment and possibly also tuning bumps are necessary to meet the ILC main linac emittance requirements. The other type of imperfections, the dynamic (time-dependent) imperfections, are variations of, for instance, magnet strength, accelerating phase or gradient, the position of elements, and the offset of the injected beam. The two latter were studied in detail in Paper V. The most important dynamic effect is, however, ground motion. As shown for CLIC in Paper V, and also in [25], movements in the ground cause increasing element misalignments and emittance that increases with time and quickly becomes unacceptable. Observe that emittance growth refers to the amount with which the emittance of a pulse increases along the linac. The term emittance growth rate will be used to denote how the emittance at the end of the linac changes with time. Studies at different sites have shown that ground motion is very site-dependent [26]. While some sites are relatively quiet, others are more noisy and may not be appropriate for future linear colliders with extreme stability requirements. It has been shown that the diffusive part of the ground motion can be accurately described by the simple ATL model [27–29]. According to this model, the rms relative displacement Δy of two points separated by a distance L grows with time T: Δy2 = A × T × L,. (3.1). where A is a site-specific parameter with units μ m2 /s/m and values typically in the 10−7 − 10−5 range. A more accurate model of the ground motion is obtained if its temporal and spatial characteristics are described by a twodimensional power spectrum, see eg. [30]. While ground motion may be efficiently corrected using feedback, a source of noise which cannot be corrected is so-called jitter. Here, jitter refers to noise which is uncorrelated from one pulse to the next. Examples of jitter effects studied in Paper V include injected beam jitter, and jitter in the vertical position of quadrupoles and accelerating structures. Because of the short pulse length in CLIC, feedbacks can only act on a pulse-to-pulse basis, not on an intrapulse time-scale. Consequently, jitter effects may not be counteracted using feedback. On the contrary, feedbacks will actually indirectly induce some additional emittance growth due to jitter effects disturbing its operation. Another 32.

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