Opening

• Frank Tecker (CERN)

Recap of Introductory course I

• Hermann Schmickler

The Advanced General CAS course is thought to be the continuation of the Introductory Course, which is held every year. Some of the students might not have had the chance to assist to this course and therefore on the first day a two hours recapitulation is foreseen. The main focus of this recapitulation is the theoretical description of particle motion in electromagnetic fields. Since the longitudinal motion is treated an an extra two hours recap, the main focus of this presentation is transverse beam motion, mathematical models, different accelerator types and collective effects.

Intro to RF measurement techniques I

• Manfred Wendt

RF measurement techniques are a vital part in today's accelerator design and engineering. In this lecture we cover the measurement of radio-frequency (RF) signals and the characterization of RF components, systems and subsystems deployed in particle accelerators, based on the use of modern measurement instruments, like the RF vector network analyzer (VNA) and the spectrum analyzer (SA). In addition, the handling of more general purpose instruments, like the oscilloscope and others will be covered in this course.

Intro to Beam Instrumentation and Diagnostics I

• Michal Krupa (CERN)

Beam instrumentation and diagnostics combines the disciplines of accelerator physics with mechanical, electronic and software engineering, making it an extremely interesting field in which to work. The aim of the beam instrumentation physicist or engineer is to design, build, maintain and improve the diagnostic equipment for the observation of particle beams with the precision required to tune, operate and improve the accelerators and their associated transfer lines. This introduction is intended to give an overview of the instrumentation in use in modern accelerators and how they can be used to diagnose problems or improve machine performance.

Recap of Introductory course II

• Hermann Schmickler

The Advanced General CAS course is thought to be the continuation of the Introductory Course, which is held every year. Some of the students might not have had the chance to assist to this course and therefore on the first day a two hours recapitulation is foreseen. The main focus of this recapitulation is the theoretical description of particle motion in electromagnetic fields. Since the longitudinal motion is treated an an extra two hours recap, the main focus of this presentation is transverse beam motion, mathematical models, different accelerator types and collective effects. Accelerator technologies are only treated in a summary approach.

Intro to Optics Design

• Guido Sterbini (CERN)

This lecture provides an overview of the principles and methodologies involved in linear optics design. It aims to introduce key concepts such as the matrix formalism for linear optics, the use of symplectic matrices in particle evolution, the Courant-Snyder invariant. It also covers the concept of beam emittance and matching conditions for ensemble of particles. The material delves into using xsuite, a general-purpose tracking code, for simulating beam dynamics and optimizing lattice design for large-scale projects like the LHC. The goal is to equip readers with the foundational tools needed for both theoretical understanding and practical application in linear optics and accelerator design.

1S1M

Lattice Cells

• Guido Sterbini (CERN)

This lecture presents an analysis of lattice cell design, with a particular focus on the FODO topology and its various applications in particle accelerator optics. The design methodology explores the stability conditions, phase advances, and chromaticity of the FODO lattice under both theoretical and practical constraints. Additionally, the optics of the CERN accelerators are presented, demonstrating the principles in practical scenarios.

Accelerator issues overview

• Frank Tecker (CERN)

This lecture summarises the status and the recent advances on muon collider facility design. The challenges in producing intense proton beams, using them to make pions and muons and subsequent cooling, acceleration and collision are described.

Intro to RF measurement techniques II

• Manfred Wendt

RF measurement techniques are a vital part in today's accelerator design and engineering. In this lecture we cover the measurement of radio-frequency (RF) signals and the characterization of RF components, systems and subsystems deployed in particle accelerators, based on the use of modern measurement instruments, like the RF vector network analyzer (VNA) and the spectrum analyzer (SA). In addition, the handling of more general purpose instruments, like the oscilloscope and others will be covered in this course.

Intro to Beam Instrumentation and Diagnostics II

• Michal Krupa (CERN)

Beam instrumentation and diagnostics combines the disciplines of accelerator physics with mechanical, electronic and software engineering, making it an extremely interesting field in which to work. The aim of the beam instrumentation physicist or engineer is to design, build, maintain and improve the diagnostic equipment for the observation of particle beams with the precision required to tune, operate and improve the accelerators and their associated transfer lines. This introduction is intended to give an overview of the instrumentation in use in modern accelerators and how they can be used to diagnose problems or improve machine performance.

Insertions & Dispersion Suppressors

• Guido Sterbini (CERN)

This lecture presents a comprehensive exploration of the design principles for low-beta insertions and dispersion suppressors in particle colliders. The primary focus is on optimizing beam optics to enhance collider performance, particularly in high-luminosity environments. Both sections emphasize theoretical models, practical implementation, and the comparative advantages of alternative topological configurations.

Recap Longitudinal Beam Dynamics I

• Frank Tecker (CERN)

The course gives a summary of longitudinal beam dynamics for both linear and circular accelerators. After discussing different types of acceleration methods and synchronism conditions, it focuses on the particle motion in synchrotrons.

Recap Longitudinal Beam Dynamics II

• Frank Tecker (CERN)

The course gives a summary of longitudinal beam dynamics for both linear and circular accelerators. After discussing different types of acceleration methods and synchronism conditions, it focuses on the particle motion in synchrotrons.

RF Manipulations I

• Helga Timko (CERN)

The lectures will cover different ways of controlling the longitudinal beam parameters, such as bunch length, bunch emittance, and bunch profile through the RF system. The RF parameters used for this control include RF cavity voltage and phase, as well as RF frequency. The manipulations discussed cover, among others, beam splitting, bunch rotation, and controlled emittance blow-up. More advanced schemes, such as momentum slip stacking and longitudinal painting, will also be discussed. Finally, the lectures touch on beam-loading compensation and RF cycle design.

RF Manipulations II

• Helga Timko (CERN)

The lectures will cover different ways of controlling the longitudinal beam parameters, such as bunch length, bunch emittance, and bunch profile through the RF system. The RF parameters used for this control include RF cavity voltage and phase, as well as RF frequency. The manipulations discussed cover, among others, beam splitting, bunch rotation, and controlled emittance blow-up. More advanced schemes, such as momentum slip stacking and longitudinal painting, will also be discussed. Finally, the lectures touch on beam-loading compensation and RF cycle design.

RF Feedbacks

• Heiko Damerau (CERN)

"Feedback systems are commonly applied to stabilise charged particle beams in accelerators. In the longitudinal plane radio-frequency (RF) feedback reduces the beam induced voltage in an RF system, or it allows to damp bunch oscillations. While local feedback decreases the cavity impedance at the origin, global feedback cures the consequence when the source driving the instability is not identified or cannot be removed. The general stability criterion for bandwidth-limited RF feedback is derived and defines the maximum gain due to loop delay. Profiting from the periodicity of bunches passing through an RF system in a circular accelerator, this electrical stability limit can be overcome with 1-turn delay or narrow-band multi-harmonic feedback. For a multi-bunch beam in a synchrotron, global feedback systems can either stabilise the beam bunch-by-bunch in the time domain, or mode-by-mode in the frequency domain. Examples for RF feedback implementations are compared, and criteria to choose the appropriate architecture for a given application are established. "

Seminar: Going bananas : a journey to the tropics

• Hervé Vanderschuren (KU Leuven)

KU Leuven has a long tradition of performing research relevant for the Global South and more specifically in the tropics. Recent technological advances in genetics offer unique opportunities to improve the performance of tropical crop plants, to increase food security as well as to empower communities and scientists in the Global South. There are numerous challenges to overcome in order to bring those benefits to smallholders and consumers. It ranges from colonial views on tropical agriculture to building laboratories with advanced technologies.

Space charge in linear machines

• Massimo Ferrario

Space charge forces are those generated directly by the charge distribution within th bunch. In this lecture we introduce, from basic principles, the main concepts of beam focusing and transport in high brightness linacs and injectors under the effects of space charge forces using the beam envelope equation as a convenient mathematical tool. Matching conditions suitable for preserving beam quality are derived from the model for significant beam dynamics regimes.

Wakefields and Impedances

• Giovanni Rumolo (CERN)

In beam dynamics, low intensity particle beams can be satisfactorily modelled using the single particle approach, in which it is sufficient to describe the motion of particles through the external electromagnetic fields, while neglecting all other types of interactions. Conversely, the evolution of high intensity particle beams must be described including not only the externally applied electromagnetic fields but also the mutual interactions between beam particles as well as the interactions of beam particles with their surrounding environment. All the perturbations to the motion induced by the additional driving terms associated to these interactions are known as collective effects. The powerful tools of the kinetic theory in plasma physics and self-consistent multi-particle simulations including the beam-induced fields are needed to model the dynamics of particle beams in this regime.

Space charge in circular machines

• Massimo Ferrario

Space charge forces are those generated directly by the charge distribution, with the inclusion of the image charges and currents due to the interaction of the beam with a perfectly conducting smooth pipe. Space charge forces are responsible of several unwanted phenomena related to beam dynamics, such as energy loss, shift of the synchronous phase and frequency, shift of the betatron frequencies, and instabilities. We will discuss in this lecture the main feature of space charge effects in high energy storage rings.

Beam Instabilities - Longitudinal

• Giovanni Rumolo (CERN)

In beam dynamics, low intensity particle beams can be satisfactorily modelled using the single particle approach, in which it is sufficient to describe the motion of particles through the external electromagnetic fields, while neglecting all other types of interactions. Conversely, the evolution of high intensity particle beams must be described including not only the externally applied electromagnetic fields but also the mutual interactions between beam particles as well as the interactions of beam particles with their surrounding environment. All the perturbations to the motion induced by the additional driving terms associated to these interactions are known as collective effects. The powerful tools of the kinetic theory in plasma physics and self-consistent multi-particle simulations including the beam-induced fields are needed to model the dynamics of particle beams in this regime. Beside the beam's own space charge, collective effects are typically triggered by the direct electromagnetic interaction of the beam with the external chamber and equipment, described through wake fields and beam-coupling impedances, and the interaction of the beam with electron or ion clouds generated in the vacuum chamber through vacuum or surface processes. They manifest themselves as macroscopic responses of the particle beams to intensity dependent excitations, resulting in observables such as coherent tune shifts, coherent instabilities or emittance growth. Collective effects are important because they define the machine and beam parameter space, outside of which instabilities or incoherent processes cause intolerable beam quality degradation. For example, the intensity threshold due to impedance represents the upper limit to the number of particles that can circulate in an accelerator or storage ring.

Free Study Time

Beam Instabilities - Transverse

• Kevin Shing Bruce Li (CERN)

In beam dynamics, low intensity particle beams can be satisfactorily modelled using the single particle approach, in which it is sufficient to describe the motion of particles through the external electromagnetic fields, while neglecting all other types of interactions. Conversely, the evolution of high intensity particle beams must be described including not only the externally applied electromagnetic fields but also the mutual interactions between beam particles as well as the interactions of beam particles with their surrounding environment. All the perturbations to the motion induced by the additional driving terms associated to these interactions are known as collective effects. The powerful tools of the kinetic theory in plasma physics and self-consistent multi-particle simulations including the beam-induced fields are needed to model the dynamics of particle beams in this regime. Beside the beam's own space charge, collective effects are typically triggered by the direct electromagnetic interaction of the beam with the external chamber and equipment, described through wake fields and beam-coupling impedances, and the interaction of the beam with electron or ion clouds generated in the vacuum chamber through vacuum or surface processes. They manifest themselves as macroscopic responses of the particle beams to intensity dependent excitations, resulting in observables such as coherent tune shifts, coherent instabilities or emittance growth. Collective effects are important because they define the machine and beam parameter space, outside of which instabilities or incoherent processes cause intolerable beam quality degradation. For example, the intensity threshold due to impedance represents the upper limit to the number of particles that can circulate in an accelerator or storage ring.

Instabilities in Linacs

• Massimo Ferrario

When a charged particle travels across the vacuum chamber of an accelerator, it induces electromagnetic fields, which are left mainly behind the generating particle. These electromagnetic fields act back on the beam and influence its motion. Such an interaction of the beam with its surroundings results in beam energy losses, alters the shape of the bunches, and shifts the betatron and synchrotron frequencies. At high beam current the fields can even lead to instabilities thus limiting the performance of the accelerator in terms of beam quality and current intensity. We discuss in this lecture the general features of the electromagnetic fields, introducing the concepts of wake fields and giving few simple examples of them in cylindrical geometry. We then show the effect of the wake fields on the dynamics of a beam in a LINAC, dealing in particular with the beam breakup instability and the way to cure it.

Electron Cloud and Instabilities

• Kevin Shing Bruce Li (CERN)

In beam dynamics, low intensity particle beams can be satisfactorily modelled using the single particle approach, in which it is sufficient to describe the motion of particles through the external electromagnetic fields, while neglecting all other types of interactions. Conversely, the evolution of high intensity particle beams must be described including not only the externally applied electromagnetic fields but also the mutual interactions between beam particles as well as the interactions of beam particles with their surrounding environment. All the perturbations to the motion induced by the additional driving terms associated to these interactions are known as collective effects. The powerful tools of the kinetic theory in plasma physics and self-consistent multi-particle simulations including the beam-induced fields are needed to model the dynamics of particle beams in this regime. Beside the beam's own space charge, collective effects are typically triggered by the direct electromagnetic interaction of the beam with the external chamber and equipment, described through wake fields and beam-coupling impedances, and the interaction of the beam with electron or ion clouds generated in the vacuum chamber through vacuum or surface processes. They manifest themselves as macroscopic responses of the particle beams to intensity dependent excitations, resulting in observables such as coherent tune shifts, coherent instabilities or emittance growth. Collective effects are important because they define the machine and beam parameter space, outside of which instabilities or incoherent processes cause intolerable beam quality degradation. For example, the intensity threshold due to impedance represents the upper limit to the number of particles that can circulate in an accelerator or storage ring.

Discussion on Instabilities

• Kevin Shing Bruce Li (CERN)
• Giovanni Rumolo (CERN)

Overview of Wakefield Acceleration

• Massimo Ferrario

Recent years have seen spectacular progress in the development of innovative acceleration methods that are not based on traditional RF accelerating structures. These novel developments are at the interface of laser, plasma and accelerator physics and may potentially lead to much more compact and cost effective accelerator facilities. While primarily focusing on the ability to accelerate charged particles with much larger gradients than traditional RF, these new techniques have yet to demonstrate comparable performances to RF in terms of both beam parameters and reproducibility. In this lecture will review the most promising developments in new acceleration methods and it will present the status of ongoing projects including the European EuPRAXIA project.

Beam loading

• Heiko Damerau (CERN)

Radio-frequency (RF) systems in particle accelerators are usually designed to transfer energy to the beam or to define its longitudinal structure. However, charged particles passing through an RF cavity induce a voltage which acts back on themselves and on subsequent particles. The additional contribution of the beam to the cavity voltage moreover changes the effective properties of the RF system and is generally referred to as beam loading. The fundamental theorem of beam loading is introduced to derive the effect of a single bunch passage through an RF cavity. The choice of the cavity parameters, notably shunt impedance divided by quality factor, plays an important role to reduce the beam induced voltage. Extending the single bunch case to the periodic passage of bunches allows to calculate the steady state cavity detuning due to beam loading for a continuous bunch pattern. Special emphasis is given to the partially filled ring, with gaps in the filling pattern, which is the most common case of transient beam loading in electron and hadron synchrotrons.

Beam Dynamics with Synchrotron Radiation

• Ian Martin

Synchrotron radiation affects the motion of particles in storage rings in various ways. In the absence of radiation, particle motion is symplectic, and the beam emittances are conserved. In this lecture, it is shown that the inclusion of radiation effects in a classical approximation leads to emittance damping: expressions for the longitudinal and transverse damping times are derived. Then, it is shown that quantum radiation effects lead to excitation of the beam emittances. General expressions for the equilibrium longitudinal and horizontal (natural) emittances in a synchrotron storage ring are derived.

Insertion devices - Radiation

• Jim Clarke

This lecture reviews the options for building insertion device magnets. It highlights why permanent magnets are the most popular choice and explains how more complex magnetic field shapes, such as helical, can be generated with simple rectangular permanent magnet blocks. The use of iron is discussed to enhance the performance of insertion devices, as are the engineering challenges of building and adjusting these devices. Finally more exotic examples are discussed, including superconducting magnets and cryogenic permanent magnets.

Low emittance lattices

• Ian Martin

The impact of lattice design on the natural emittance in an electron storage ring is discussed. The general expression for the equilibrium horizontal emittance in a storage ring is applied to the specific cases of storage rings constructed from FODO cells, double bend achromats, multi-bend achromats and theoretical minimum emittance lattices. Additional methods to reduce the emittance are discussed, including the use of transverse gradient bends, longitudinally varying bends and reverse bends. Finally, practical storage ring designs and existing light sources facilities are reviewed.

Insertion devices - Technology

• Jim Clarke

This lecture reviews the options for building insertion device magnets. It highlights why permanent magnets are the most popular choice and explains how more complex magnetic field shapes, such as helical, can be generated with simple rectangular permanent magnet blocks. The use of iron is discussed to enhance the performance of insertion devices, as are the engineering challenges of building and adjusting these devices. Finally more exotic examples are discussed, including superconducting magnets and cryogenic permanent magnets.

Optics calculations

• Guido Sterbini (CERN)

This lecture provides hands-on guidance for performing optics calculations related to a simple synchrotron. It walks users through the steps needed to execute these calculations using the xsuite, a software for accelerator simulations. This resource is aimed at students and researchers interested in accelerator physics, particularly in understanding basic concept of resonances, chromatic correction, optimal tune and dynamic aperture.

High Brightness Beam Diagnostics

• Lorraine Bobb

FEL I

• Wolfgang Hillert

The subject of this advanced course is a simplified treatment of the light amplification process in a free electron laser (FEL). The first lecture deals with the so-called low gain FEL, where a “short” undulator is surrounded by an optical resonator. Starting with the calculation of the energy exchange between the electron beam and the radiation field, the ponderomotive phase is defined which remains constant when operating on resonance. The pendulum equations, linking this phase with the relative energy deviation from the resonance energy are derived and interpreted under the assumption that the radiation field will not change significantly during one pass through the undulator. Electron injection above resonance energy leads to an amplification of the radiation field. For low intensities, this is quantitatively described by the Madey theorem, which links the FEL gain curve with the spectrum of the spontaneous emission. For strong radiation, the gain curve will deviate significantly from the predictions of the Madey theorem and the gain will be reduced. Saturation and therewith stationary operation is achieved when the resonator losses are compensated by the FEL amplification. The second lecture deals with an introduction to the dynamics of a high-gain FEL. Now dropping the assumption of a constant radiation field and assuming a slowly varying field amplitude and phase,  these variations are derived from a simplified one-dimensional treatment. The additional field equation supplements the pendulum equations extending them to a system of coupled differential equations. Further insight is gained by the definition of normalized parameters simplifying this system of coupled equations to a single cubic differential equation. The cubic equation is investigated in detail by solving it for two different cases: amplification starting from an existing radiation field and amplification starting from an initial density modulation. Both situations will finally lead to an exponential growth of the light intensity for a sufficiently long undulator and beam injection on resonance. Important parameters like the gain length and the Pierce parameter are introduced and explained in detail. Finally, the SASE process is presented and the properties of the generated radiation is discussed. State of-the-art methods for decreasing the spectral width of the generated radiation and increasing its longitudinal coherence are sketched briefly.

Longitudinal Beam Diagnostics

• Lorraine Bobb

FEL II

• Wolfgang Hillert

The subject of this advanced course is a simplified treatment of the light amplification process in a free electron laser (FEL). The first lecture deals with the so-called low gain FEL, where a “short” undulator is surrounded by an optical resonator. Starting with the calculation of the energy exchange between the electron beam and the radiation field, the ponderomotive phase is defined which remains constant when operating on resonance. The pendulum equations, linking this phase with the relative energy deviation from the resonance energy are derived and interpreted under the assumption that the radiation field will not change significantly during one pass through the undulator. Electron injection above resonance energy leads to an amplification of the radiation field. For low intensities, this is quantitatively described by the Madey theorem, which links the FEL gain curve with the spectrum of the spontaneous emission. For strong radiation, the gain curve will deviate significantly from the predictions of the Madey theorem and the gain will be reduced. Saturation and therewith stationary operation is achieved when the resonator losses are compensated by the FEL amplification. The second lecture deals with an introduction to the dynamics of a high-gain FEL. Now dropping the assumption of a constant radiation field and assuming a slowly varying field amplitude and phase,  these variations are derived from a simplified one-dimensional treatment. The additional field equation supplements the pendulum equations extending them to a system of coupled differential equations. Further insight is gained by the definition of normalized parameters simplifying this system of coupled equations to a single cubic differential equation. The cubic equation is investigated in detail by solving it for two different cases: amplification starting from an existing radiation field and amplification starting from an initial density modulation. Both situations will finally lead to an exponential growth of the light intensity for a sufficiently long undulator and beam injection on resonance. Important parameters like the gain length and the Pierce parameter are introduced and explained in detail. Finally, the SASE process is presented and the properties of the generated radiation is discussed. State of-the-art methods for decreasing the spectral width of the generated radiation and increasing its longitudinal coherence are sketched briefly.

Landau Damping I

• Xavier Buffat (CERN)

Charged particle beams naturally exhibit self-enhanced oscillations, or coherent instabilities, driven by the interaction between the particles in the beam through collective forces such as electromagnetic wake fields, space-charge, beam-beam, ions or electron clouds. Highlighting the fundamental difference between the collective motion and the one of individual particles, we discuss how a spread in velocities or oscillation frequencies in an ensemble of particles leads to a damping of collective oscillations through the mechanism of Landau damping. We then illustrate the Liouville theorem from which we derive the strength of Landau damping in a simplified configuration using the Van Kampen approach. Based on this model we define the concept of stability diagrams.

Muon Colliders I

• Chris Rogers

A muon collider would enable the big jump ahead in energy reach that is needed for a fruitful exploration of fundamental interactions. The challenges of producing muon collisions at high luminosity and 10 TeV centre of mass energy is being investigated by the recently-formed International Muon Collider Collaboration. This Review summarises the status and the recent advances on muon collider facility design. The challenges in producing intense proton beams, using them to make pions and muons and subsequent cooling, acceleration and collision are described.

Landau Damping II

• Xavier Buffat (CERN)

Charged particle beams naturally exhibit self-enhanced oscillations, or coherent instabilities, driven by the interaction between the particles in the beam through collective forces such as electromagnetic wake fields, space-charge, beam-beam, ions or electron clouds. Highlighting the fundamental difference between the collective motion and the one of individual particles, we discuss how a spread in velocities or oscillation frequencies in an ensemble of particles leads to a damping of collective oscillations through the mechanism of Landau damping. We then illustrate the Liouville theorem from which we derive the strength of Landau damping in a simplified configuration using the Van Kampen approach. Based on this model we define the concept of stability diagrams.

Muon Colliders II

• Chris Rogers

A muon collider would enable the big jump ahead in energy reach that is needed for a fruitful exploration of fundamental interactions. The challenges of producing muon collisions at high luminosity and 10 TeV centre of mass energy is being investigated by the recently-formed International Muon Collider Collaboration. This Review summarises the status and the recent advances on muon collider facility design. The challenges in producing intense proton beams, using them to make pions and muons and subsequent cooling, acceleration and collision are described.

ERL

• Andreas Jankowiak (Helmholtz-Zentrum Berlin)

Energy-Recovery-Linacs (ERLs) are an emerging field in accelerator R&D. They are needed for the sustainable operation of future high-power electron machines, e.g. the LHeC with an anticipated electron beam-power of 1 GW. The energy used for the beam's acceleration is recycled and re-used in an ERL. Higher beam currents are feasible while maintaining the RF power consumption at a comparable level. At the same time, the beam will be dumped at lower beam energies, typically below the neutron separation threshold. Less radiation is produced, and only a moderate dump cooling is needed. Since the postulation of the principle in 1965, many ERLs have been operated, and many more are foreseen and planned for the future. Especially the R&D for superconducting multi-turn ERLs is urgently needed to path the way to future high-power ERLs. An overview of the field and the latest developments, will be presented.

Free Study Time

Non Linear Dynamics - Methods and Tools I

• Yannis Papaphilippou (CERN)

The goal of this lecture is to present the contemporary methods and numerical tools of non-linear dynamical systems in order to analyse the motion in particle accelerators. After a short introduction to non-linear effects and their impact to beam performance, the lecture will briefly review elements of classical mechanics, essential for the study of non-linear dynamics, including the Lagrangian and Hamiltonian formalism, canonical transformation and simplicity. Starting from the relativistic Hamiltonian for E/M fields, elements of canonical perturbation theory will be presented, showing its limitation, for particle accelerators. In this respect, concepts of linear and non-linear beam transport will be introduced, represented by matrices or more generally maps. The Lie formalism will be employed in order to elaborate and analyse these maps through non-linear normal form construction. An introduction to Truncated Power Series through differential Algebra will be also given, as it is essential for constructing 1-term maps. Finally, elements of symplectic integration will be reviewed.

Beam-Beam effects

• Xavier Buffat (CERN)

Particle colliders are one of the main drivers for discoveries in particle physics. Their performance is usually limited by the strong electromagnetic interactions of the two beams, so-called beam-beam interactions. Due to its non-linear and dynamical nature, this force introduces severe limitations by generating collective instabilities or distorting the single particle trajectories leading to a reduction of the beam lifetime and/or a growth of the emittances. In this lecture, we illustrate the mechanisms through which these degradations of the beam quality occur as well as basic models to describe them. We thus obtain an understanding of the fundamental properties of modern colliders.

Non Linear Dynamics - Methods and Tools II

• Yannis Papaphilippou (CERN)

The goal of this lecture is to present the contemporary methods and numerical tools of non-linear dynamical systems in order to analyse the motion in particle accelerators. After a short introduction to non-linear effects and their impact to beam performance, the lecture will briefly review elements of classical mechanics, essential for the study of non-linear dynamics, including the Lagrangian and Hamiltonian formalism, canonical transformation and simplicity. Starting from the relativistic Hamiltonian for E/M fields, elements of canonical perturbation theory will be presented, showing its limitation, for particle accelerators. In this respect, concepts of linear and non-linear beam transport will be introduced, represented by matrices or more generally maps. The Lie formalism will be employed in order to elaborate and analyse these maps through non-linear normal form construction. An introduction to Truncated Power Series through differential Algebra will be also given, as it is essential for constructing 1-term maps. Finally, elements of symplectic integration will be reviewed.

HL-LHC Challenges

• Markus Zerlauth (CERN)

The contribution recalls the HL-LHC design parameter and upgrade goals. The main remaining challenges and potential performance limitations that became visible with the re-start of the third operational run of the LHC after the second long shutdown (LS2) are recalled. This includes the beam performance in the injector complex following the LHC Injector upgrade, the e-cloud build-up and related heat load in the LHC arcs, the effect of dust particles in the vacuum chambers and radiation effects to electronics on machine availability as well as the status of magnet training for the LHC dipoles. The goals and planning of the IT String as one of the important intermediate milestones of the HL-LCH upgrade are presented along with the status of the global project planning and the current performance ramp-up scenario following the deployment of the project in LS3.

Non Linear Dynamics - Phenomenology I

• Yannis Papaphilippou (CERN)

This lecture will review concepts for representing non-linear particle motion starting from analysing dynamics on phase space through fixed point identification. After introducing 1-turn (Poincaré) maps, the motion close to a resonance will be reviewed and its role on the onset of chaotic motion, leading to particle diffusion. Finally, methods for analysing and detecting chaotic motion will be presented, through direct particle tracking and the concept of dynamic aperture estimation. Other methods will be briefly introduced, including Lyapunov exponent and Frequency map analysis. Several examples of the application of these methods in design studies and improvements in the performance of operating hadron and lepton synchrotrons, storage rings, and colliders will finally be given.

Collimation

• Stefano Redaelli (CERN)

Modern accelerators cannot operate high intensity hadron beams without adequate beam collimation systems that are designed to dispose safely and efficiently of unavoidable losses in each phase of the beam operation. This is in particular the case for hadron colliders that rely on super-conducting magnets to push beam energy and current, like the Large Hadron Collider (LHC). In this case, the tiny energy amounts sufficient to spoil the super-conducting state is several orders of magnitude below the total stored beam energy, calling for very efficient beam collimation. Uncontrolled beam losses also risk to damage permanently accelerator equipment, which must be avoided to ensure a high-availability operation. In this contribution, the design of a multi-stage collimation system is reviewed, discussing typical target specifications, design criteria and performance. Modern tools used to study and characterize collimation systems are briefly introduced. The LHC collimation system is presented as a case study, covering also its technical implementation and collimation operational aspects.

Non Linear Dynamics - Phenomenology II

• Yannis Papaphilippou (CERN)

This lecture will review concepts for representing non-linear particle motion starting from analysing dynamics on phase space through fixed point identification. After introducing 1-turn (Poincaré) maps, the motion close to a resonance will be reviewed and its role on the onset of chaotic motion, leading to particle diffusion. Finally, methods for analysing and detecting chaotic motion will be presented, through direct particle tracking and the concept of dynamic aperture estimation. Other methods will be briefly introduced, including Lyapunov exponent and Frequency map analysis. Several examples of the application of these methods in design studies and improvements in the performance of operating hadron and lepton synchrotrons, storage rings, and colliders will finally be given.

Discussion on Non Linear Dynamics

• Yannis Papaphilippou (CERN)

Collimation + technical implementation

• Stefano Redaelli (CERN)

Modern accelerators cannot operate high intensity hadron beams without adequate beam collimation systems that are designed to dispose safely and efficiently of unavoidable losses in each phase of the beam operation. This is in particular the case for hadron colliders that rely on super-conducting magnets to push beam energy and current, like the Large Hadron Collider (LHC). In this case, the tiny energy amounts sufficient to spoil the super-conducting state is several orders of magnitude below the total stored beam energy, calling for very efficient beam collimation. Uncontrolled beam losses also risk to damage permanently accelerator equipment, which must be avoided to ensure a high-availability operation. In this contribution, the design of a multi-stage collimation system is reviewed, discussing typical target specifications, design criteria and performance. Modern tools used to study and characterize collimation systems are briefly introduced. The LHC collimation system is presented as a case study, covering also its technical implementation and collimation operational aspects.

RF Show

Closing

• Frank Tecker (CERN)