Coherent transverse oscillations of a bunch in a ring accelerator are considered. Three different models of the bunch are used: a hollow bunch in a square potential well, a square bunch in a parabolic potential well, and a parabolic bunch in a parabolic well. The bunch space-charge field, wakefield, and chromaticity are taken into account in all the cases. The bunch eigenmodes and various instabilities are studied. The results are compared and analyzed to establish the range of applicability of the models. Role of the Landau damping due to dependence of the SC field on the longitudinal coordinate is discussed.
Space charge effects play significant role in modern-day accelerators. These effects frequently constrain attainable beam parameters in an accelerator - or - in an accelerator chain. They also could limit the luminosity of hadron colliders operating either at low energies or with a sub-TeV high brightness hadron beams. A number of schemes for compensating space charge effects in a coasting (e.g. continuous) hadron beam were proposed and some of them had been tested. But all of these methods do not address the issue of tune spread compensation of a bunched hadron beam, e.g. the tune shift depen…
Longitudinal oscillations of bunched beams in synchrotrons have been analyzed by accelerator physicists for decades, and a closed theory is well-known [1]. The first modes of oscillation are the coherent dipole mode, quadrupole mode, and sextupole mode. Of course, these modes of oscillation are included in the general theory, but for developing RF control systems, it is useful to work with simplified models. Therefore, several specific models are analyzed in the paper at hand. They are useful for the design of closed-loop control systems in order to reach an optimum performance with respect to…
We discuss simulations and experiments on creating micro-bunch beams for generating narrow band THz radiation at the Fermilab Accelerator Science and Technology (FAST) facility. The low-energy electron beamline at FAST consists of a photoinjector-based RF gun, two L-band superconducting accelerating cavities, a chicane, and a beam dump. The electron bunches are lengthened with cavity phases set off-crest for better longitudinal separation and then micro-bunched with a slit-mask installed in the chicane. We carried out the experiments with 30 MeV electron beams and detected signals of the micro…
Slow extraction of bunched beams is widely used for mitigating the formation of micro structures due to ripples of the accelerator magnets power supplies on the extracted beam which is also referred to as spill. That helps, in particular, experiments with slow detectors or with low extraction rates. Furthermore, this technique is widely used for avoiding long gaps in the spill measured in ionisation chambers which in turn would trigger interlocks. On the other hand, the bunches create spill structures on time scales defined by the rf frequency. In addition, macroscopic spill structures of dura…
In the usual parameter regime of accelerator physics, particle ensembles can be treated as classical. If we approach a regime where $ε_xε_yε_s \approx N_{particles}λ_{Compton}^3\$, however, the granular structure of quantum-mechanical phase space becomes a concern. In particular, we have to consider the Pauli exclusion principle, which will limit the minimum achievable emittance for a beam of fermions. We calculate these lowest emittances for the cases of bunched and coasting beams at zero temperature and their first-order change rate at finite temperature.
This paper presents expected THz radiation spectra emitted by micro-bunched electron beams produced using a slit-mask placed within a magnetic chicane in the FAST (Fermilab Accelerator Science and Technology) electron injector at Fermilab. Our purpose is to generate tunable narrow-band THz radiation with a simple scheme in a conventional photo-injector. Using the slit-mask in the chicane, we create a longitudinally micro-bunched beam after the chicane by transversely slicing an energy chirped electron bunch at a location with horizontal dispersion. In this paper, we discuss the theory related …
Recent experiments [1] have explored the use of a free-electron laser (FEL) as a buncher for a microwave two-beam accelerator, and the subsequent driving of a standing-wave rf output cavity. Here we present a deeper analysis of the longitudinal dynamics of the electron bunches as they are transported from the end of the FEL and through the output cavity. In particular, we examine the effect of the transport region and cavity aperture to filter the bunched portion of the beam. [1] T. Lefevre, et. al., Phys. Rev. Lett. 84 (2000), 1188.
This paper gives an introduction of longitudinal beam dynamics for circular accelerators. After briefly discussing some types of circular accelerators, it focuses on particle motion in synchrotrons. It summarizes the equations of motion, discusses phase-space matching during beam transfer, and introduces the Hamiltonian of longitudinal motion.
For a single hadron bunch in a circular accelerator at zero chromaticity, without multi-turn wakes and without electron clouds and other beams, only one transverse collective instability is possible, the mode-coupling instability, or TMCI. For sufficiently strong space charge (SC), the instability threshold of the wake-driven coherent tune shift normally increases linearly with the SC tune shift, as independently concluded by several authors using different methods. This stability condition has, however, a very strange feature: at strong SC, it is totally insensitive to the number of particles…
A new algorithm for calculating the spin tune and the n-axis for circular accelerators is presented. The method resembles the one employed in the existing program code SODOM in that one-turn numerical spin maps at equally spaced orbit angle variables are used but it is more efficient than the latter. Furthermore, it is applicable at large openning angles of the n-axis, whereas the existing SODOM only converges for small angles.
Recent progress in particle accelerator tracking has shown that the field representation is one of the major limits of the prediction accuracy especially for machines, whose aperture is fully filled by the beam and thus higher the artefacts created by higher order modes have to be thoroughly understood. The standard tool for field presentation today are cylindrical circular multipoles due to their straight forward correspondence to the Cartesian coordinates. In this paper we extend the standard approach to other coordinate systems, show how these can be measured next to their realisation in me…
The electron cloud effect (ECE) causes beam instabilities in accelerator structures with intense positively charged bunched beams. Reduction of the secondary electron yield (SEY) of the beam pipe inner wall is effective in controlling cloud formation. We summarize SEY results obtained from flat TiN, TiZrV and Al surfaces carried out in a laboratory environment. SEY was measured after thermal conditioning, as well as after low energy, less than 300 eV, particle exposure.
The resonant laser cooling of circular accelerator beams of relativistic charged particle is studied. It is shown that in the approximation of the given external electromagnetic wave amplitude (small gain free electron laser) the emittance of a beam of charged particles decreases. In the field of particle energy about 100 in the mass energy units the beam energy losses are negligible. The discovered effect can be used for cooling of charged particle beams in various accelerators. The significant cooling rates are possible to achieve by placing of the cooling device in the appropriate points of…
By considering delta function sextupole, octupole, and decapole perturbations and using difference action-angle variable equations, we find some useful analytical formulae for the estimation of the dynamic apertures of circular accelerators due to single sextupole, single octupole, single decapole (single 2$m$ pole in general). Their combined effects are derived based on the Chirikov criterion of the onset of stochastic motions. Comparisons with numerical simulations are made, and the agreement is quite satisfactory. These formulae have been applied to determine the beam-beam limited dynamic a…
The leading Lie generators, including the chromatic effects, due to hard-edge fringe field of single multipole and solenoid are derived from the vector potentials within a Hamiltonian system. These nonlinear generators are applied to the interaction region of PEP-II to analyze the linear errors due to the feed-down from the off-centered quadrupoles and solenoid. The nonlinear effects of tune shifts at large amplitude, the synchro-betatron sidebands near half integer and their impacts on the dynamic aperture are studied in the paper.
Circular particle accelerators at the energy frontier are based on superconducting magnets that are extremely sensitive to beam losses as these might induce quenches, i.e.\ transitions to the normal-conducting state. Furthermore, the energy stored in the circulating beam is so large that hardware integrity is put in serious danger, and machine protection becomes essential for reaching the nominal accelerator performance. In this challenging context, the beam halo becomes a potential source of performance limitations and its dynamics needs to be understood in detail to assess whether it could b…
Electron cloud interaction with high energy positive beam are believed responsible for various undesirable effects such as vacuum degradation, collective beam instability and even beam loss in high power proton circular accelerator. An important uncertainty in predicting electron cloud instability lies in the detail processes on the generation and accumulation of the electron cloud. The simulation on the build-up of electron cloud is necessary to further studies on beam instability caused by electron cloud. China Spallation Neutron Source (CSNS) is the largest scientific project in building, w…
The efficiency of the beam extraction and collimation systems in circular accelerators with the use of the channeling effect in a bent crystal is determined. The dependences of the extraction efficiency on the geometrical parameters of the crystal (its length, thickness, and bending angle), the azimuthal location of the system components, and the offset of the septum or scraper are presented. The influence of crystal imperfections (amorphous layers, misorientation, and torsion) on the efficiency of the systems is considered, and their tolerances are proposed. It is shown that an extraction eff…
Accelerator-based X-ray free-electron lasers (XFELs) are the latest addition to the revolutionary tools of discovery for the 21st century. The two major components of an XFEL are an accelerator-produced electron beam and a magnetic undulator which tend to be kilometer-scale long and expensive. Here, we present an ultra-compact scheme to produce 10s of attosecond X-ray pulses with several GW peak power utilizing a novel aspect of the FEL instability using a highly chirped, pre-bunched and ultra-bright electron beam from a plasma-based accelerator interacting with an optical undulator. The self-…
We give an overview of our current/future analytical and numerical work on the spin polarization in high-energy electron storage rings. Our goal is to study the possibility of polarization for the CEPC and FCC-ee. Our work is based on the so-called Bloch equation for the polarization density introduced by Derbenev and Kondratenko in 1975. We also give an outline of the standard approach, the latter being based on the Derbenev-Kondratenko formulas.
We consider acceleration of protons in a cavity with such dimensions as length l = 10 m, width b = 20 m, height a = 1.01 m, located between two bending (180)degree magnets with isochronous magnetic field. The cavity is loaded with rows of drift plates (tubes), being excited by a wave of E110 type at the frequency fo = 148.5 MHz, excitation power P1 = 5.65 MW. Additionally, the cavity is fed with the power P2 = 8 MW, transmitted into a beam of protons with such parameters as the final energy 0.8 GeV, pulse beam current I = 10 mA, pulse width 25 microseconds, repetition rate 10 Hz. We discuss th…
Electron cloud effects have become one of the main performance limitations for circular particle accelerators operating with positively-charged beams. Among other machines worldwide, the CERN Super Proton Synchrotron (SPS), as well as the Large Hadron Collider (LHC) are affected by these phenomena. Intense efforts have been devoted in recent years to improve the understanding of electron cloud (EC) generation with the aim of finding efficient mitigation measures. In a different domain of accelerator physics, non-linear resonances in the transverse phase space have been proposed as novel means …
Energy-frontier accelerators provide powerful tools performing high precision measurements confirming the fundamental of the physics and broadening new research horizons. Such machines are either driven by circular or linear accelerators. The circular machines, having the center-mass (CM) energy values reaching 200 GeV (for leptons) and above, experience beam energy loss and quality dilution, for example, due to synchrotron radiation, limiting the overall CM energy achievable and requiring a constant energy top-up to compensate the loss and the beam quality dilution. Linear colliders overcome …
Many advanced techniques have been developed, tested and implemented in the last decades in almost all circular accelerators across the world to measure the linear optics. However, the greater availability and accuracy of beam diagnostics and the ever better correction of linear magnetic lattice imperfections (beta beating at 1% level and coupling at 0.1%) are reaching what seems to be the intrinsic accuracy and precision of different measurement techniques. This paper aims to highlight and quantify, when possible, the limitations of one standard method, the harmonic analysis of turn-by-turn b…
The transverse instability of a bunch in a circular accelerator is elaborated in this paper. A new tree-modes model is proposed and developed to describe the most unstable modes of the bunch. This simple and flexible model includes chromaticity and space charge, and can be used with any bunch and wake forms. The dispersion equation for the bunch eigentunes is obtained in form of a third-order algebraic equation. The known head-tail and TMCI modes appear as the limiting cases which are distinctly bounded at zero chromaticity only. It is shown that the instability parameters depend only slightly…
Controlling non-linear effects in the transverse dynamics of charged particle beams in circular accelerators opens new possibilities for controlling some of the beam properties. Beam splitting by crossing a stable 1D non-linear resonance is part of the routine operation of the CERN Proton Synchrotron. The beam undergoes trapping and transport inside stable islands created in the horizontal plane to allow multi-turn extraction towards the Super Proton Synchrotron, where the beam is used for fixed-target experiments. This process acts only on the horizontal beam emittance, inducing a reduction o…
New crystal technique - array of bent strips and a fan-type reflector, based on thin straight plates - have been used for research of extraction and collimation a circulating beam in the U-70 accelerator at the energy 50 GeV and 1.3 GeV. It is shown, that new devices can effectively steer a beam in a wide energy range. For protons with energy 50 GeV efficiency of extraction and collimation about 90 % has been achieved which is record for this method. Reduction of particle losses in 2-3 times was observed also in accelerator at application of different crystals in comparison with the usual one-…
This paper is a summary report of the ICFA Beam Dynamics Workshop 'Accelerators for a Higgs Factory: Linear vs. Circular' (HF2012). It discusses four types of accelerators as possible candidates for a Higgs factory: linear e+e- colliders, circular e+e- colliders, muon collider and photon colliders. The comparison includes: physics reach, performance (energy and luminosity), upgrade potential, technology maturity and readiness, and technical challenges requiring further R&D.
Crossing the transition energy is always a delicate process, representing a potential source of strong perturbations of the dynamics of charged particle beams in a hadron circular accelerator. Since the first generation of multi-GeV rings, intense studies have been devoted to understanding the possible harmful mechanisms involved in transition crossing and to devise mitigation measures. Nowadays, several circular particle accelerators are successfully operating across transition and this process is well mastered. In a completely different context, stable resonances of the traverse phase space …
We revise a previous version of this paper proposing a method to transfer emittances from the transverse planes to the longitudinal. It turned out the system showed non-Hamiltonian behavior because the longitudinal component of the transverse electric field at ends of the transverse resonator were not included in the model. Including those fields the 6D emittance is conserved. We review the mechanism behind the emittance conservation. It is still possible to produce an emittance transfer with the RF quadrupole field setting the tunes on synchro-betatron resonance for a short period of time the…
This paper introduces the protection of circular particle accelerators from accidental beam losses. Already the energy stored in the beams for accelerators such as the TEVATRON at Fermilab and Super Proton Synchrotron (SPS) at CERN could cause serious damage in case of uncontrolled beam loss. With the CERN Large Hadron Collider (LHC), the energy stored in particle beams has reached a value two orders of magnitude above previous accelerators and poses new threats with respect to hazards from the energy stored in the particle beams. A single accident damaging vital parts of the accelerator could…
What prevents us from building super-high intensity accelerators? The answer is case-specific, but it often points to one of the following phenomena: machine resonances, various tune shifts (and spreads), and instabilities. These three phenomena are interdependent in all present machines. In this paper we propose a path toward alleviating these phenomena by making accelerators nonlinear. This idea is not new: Orlov (1963) and McMillan (1967) have proposed initial ideas on nonlinear focusing systems for accelerators. However, practical implementations of such ideas proved elusive, until recentl…
The acceleration of ions in the interaction of circular polarized laser pulses with overdense plasmas is investigated. For circular polarization laser pulses, the quasi-equilibrium for electrons is established due to the light pressure and the electrostatic field built up at the interacting front of the laser pulse. The ions located within the skin-depth of the laser pulse can be synchronously accelerated and bunched in the charge couple processes by the electrostatic field, and thereby monoenergetic and high intensity proton beam can be generated. The dynamics equations for accelerated ions a…
Plasma accelerators can generate ultra high brightness electron beams which open the door to light sources with smaller physical footprint and properties un-achievable with conventional accelerator technology. In this paper we show that electron beams from Plasma WakeField Accelerators (PWFAs) can generate few-cycle coherent tunable soft X-ray pulses with TW peak power and a duration of tens of attoseconds, an order of magnitude more powerful, shorter and with better stability than state-of-the-art X-ray Free Electron Lasers (XFELs). Such a light source would significantly enhance the ability …
Several indicators have pointed to the presence of an Electron Cloud (EC) in some of the CERN accelerators, when operating with closely spaced bunched beams. In particular, spurious signals on the pick ups used for beam detection, pressure rise and beam instabilities were observed at the Proton Synchrotron (PS) during the last stage of preparation of the beams for the Large Hadron Collider (LHC), as well as at the Super Proton Synchrotron (SPS). Since the LHC has started operation in 2009, typical electron cloud phenomena have appeared also in this machine, when running with trains of closely …
A diffusion model of the time evolution of loss rates caused by a step in collimator position is presented. It builds upon the model of Seidel (1994) and its assumptions: (1) constant diffusion rate within the range of the step and (2) linear halo tails. These hypotheses allow one to obtain analytical expressions for the solutions of the diffusion equation and for the corresponding loss rates vs. time. The present model addresses some of the limitiations of the previous model and expands it in the following ways: (a) losses before, during, and after the step are predicted; (b) different steady…
Here we calculate the effects of astrophysical gravitational waves (GWs) on the travel times of proton bunch test masses in circular particle accelerators. We show that a high-precision proton bunch time-tagging detector could turn a circular particle accelerator facility into a GW observatory sensitive to millihertz (mHz) GWs. We comment on sources of noise and the technological feasibility of ultrafast single photon detectors by conducting a case study of the Large Hadron Collider (LHC) at CERN.
Beam lifetime in storage rings and colliders is affected by, among other effects, lattice nonlinearities. Their control are of great benefit to the dynamic aperture of an accelerator, whose enlargement leads in general to more efficient injection and longer lifetime. This article describes a procedure to evaluate and correct unwanted nonlinearities by using turn-by-turn beam position monitor data, which is an evolution of previous works on the resonance driving terms (RDTs). Effective sextupole magnetic errors and tilts at the ESRF electron storage ring are evaluated and corrected (when possib…
Using the pre-bunched electron beam of 5-MeV linear accelerator the coherent Smith-Purcell radiation (CSPR) from a flat periodic target (made of conductive layers separated by dielectric gaps) in millimeter wavelength region has been investigated. The angular distribution of this radiation was measured with a narrow-band detector and experimental data agree with our theoretical calculations for similar kind of targets. Such properties of Smith-Purcell radiation as strong dependence of radiation wavelength on the observation angle overlapping with coherent radiation effect may be used for a non…
Recently, the first direct measurement of a full 6D accelerator beam distribution was reported [1]. That work observed a correlation between energy and transverse coordinates, for which the energy distribution becomes hollowed and double-peaked near the transverse core. In this article, a similar structure is shown to emerge in expansion of an initially uncorrelated, high density bunched beam as the result of velocity perturbation from nonlinear space charge forces. This hollowing is obscured when the 6D phase space is projected onto one- and two-dimensional axes. This phenomenon has not been …
XFELs provide X-ray pulses with unprecedented peak brightness and ultrashort duration. They are usually driven by planar undulators, meaning that the output radiation is linearly polarized. For many experimental applications, however, polarization control is critical: besides the ability to produce linearly polarized radiation, one often needs the possibility of generating circularly polarized radiation with a high, stable degree of polarization. This may be achieved by using a first part of the XFEL undulator to produce bunching and then, by propagating the the bunched beam through an "afterb…
Determining a model for the time scaling of the dynamic aperture of a circular accelerator is a topic of strong interest and intense research efforts in accelerator physics. The motivation arises in the possibility of finding a method to reliably extrapolate the results of numerical simulations well beyond what is currently possible in terms of CPU time. In earlier work, a proposal for a model based on Nekhoroshev theorem and Kolmogorov--Arnold--Moser theory was made. This model has been studied in detail and proved successful in describing the evolution of the dynamic aperture in numerical si…
Ionization injection is attractive as a controllable injection scheme for generating high quality electron beams using plasma-based wakefield acceleration. Due to the phase dependent tunneling ionization rate and the trapping dynamics within a nonlinear wake, the discrete injection of electrons within the wake is nonlinearly mapped to discrete final phase space structure of the beam at the location where the electrons are trapped. This phenomenon is theoretically analyzed and examined by three-dimensional particle-in-cell simulations which show that three dimensional effects limit the wave num…
A brief historical review is presented of progressing understanding of transverse coherent instabilities of charged particles beams in circular machines when both Coulomb and wake fields are important. The paper relates to a talk given at ICFA Workshop on Mitigation of Coherent Beam Instabilities in Particle Accelerators, 23-27 September 2019 in Zermatt, Switzerland.
We discuss the goals, the designs, the state of technical readiness, and the critical R&D needs of the accelerators that are currently under discussion as Higgs and electroweak factories. We also address the respective staging options enabling future energy-frontier colliders. The accelerators covered are based on many different techniques and approaches. They include several circular colliders, various linear colliders, colliders based on energy recovery linacs (ERLs), ERL-ring combinations, as well as gamma-gamma colliders. The linear colliders proposed consist of options for the Internation…
The equations of classical spin-orbit motion can be extended to a Hamiltonian system in 9-dimensional phase space by introducing a coupled spin-orbit Poisson bracket and a Hamiltonian function. After this extension and by establishing connections between initial and extended systems it becomes possible to apply the methods of the theory of Hamiltonian systems to the study of polarized particle beam dynamics in circular accelerators and storage rings. Some of those methods have been implemented in the computer code FORGET-ME-NOT.
Fermilab is constructing the Integrable Optics Test Accelerator (IOTA) as the centerpiece of the Accelerator R&D Program towards high-intensity circular machines. One of the factors limiting the beam intensity in present circular accelerators is collective instabilities, which can be suppressed by a spread of betatron frequencies (tunes) through the Landau damping mechanism or by an external damper, if the instability is slow enough. The spread is usually created by octupole magnets, which introduce the tune dependence on the amplitude and, in some cases, by a chromatic spread (tune dependence…
CERN has recently launched the Future Circular Collider (FCC) study to deal with all aspects of an ambitious post-LHC possible programme. The emphasis of the study is on a 100 TeV proton collider to be housed in a 80-100 km new ring in the Geneva region. An electron machine will also be considered as a possible intermediate first step (FCC-ee). The study benefits from earlier work done in the context of TLEP and has already published a parameter table, to serve as the basis for the work to be done. The study aims to publish a conceptual design report at around 2018. The recent discovery of a l…
The Integrable Optics Test Accelerator (IOTA) at Fermilab is a small machine dedicated to a broad frontier accelerator physics program. An important aspect of this program is to investigate the potential benefits of the resonance free tune spread achievable with integrable optics to store and accelerate high intensity proton beams for which space charge is significant. In this context, a good understanding of proton beam emittance growth and particle loss mechanisms is essential. Assuming nominal design parameters, simulations show that for a bunched beam, the bulk of emittance growth takes pl…
High peak power, tunable, narrowband terahertz emitters are becoming sought after given their portability, efficiency, and ability to be deployed in the field for industrial, medical, and military applications. The use of accelerator systems producing THz frequencies via Cherenkov radiation, generated by passing an electron beam through a slow-wave wakefield structure, is a promising method to meet future THz requirements. To date, efforts have been dedicated to analysis and design of sources utilizing laser seeded bunched electron beam drivers with relativistic energies beyond 5 MeV. Presente…
