The interaction of a charged particle beam with radio-frequency (RF) systems in most linear or circular accelerators is an non-linear process. The large longitudinal electric fields for acceleration and longitudinal beam manipulations can only be generated thanks to the resonant build-up of the field in a high quality oscillator, the RF cavity, driven by a sinusoidal and hence inherently non-linear excitation. The course gives an introduction to linear and non-linear longitudinal beam dynamics, deriving the equations of motion, as well as the RF potential and the Hamiltonian of the longitudinal beam dynamics. Profiting from the non-linear dynamics, longitudinal beam manipulations to control bunch length, distance with multiple RF systems are shown as examples. Additionally, the distribution of the synchrotron frequencies of the particles in a bunch can be modelled thanks to the non-linearity of the applied RF voltage. Double- or even multi-harmonic RF systems are a powerful technique to improve the longitudinal stability of high-intensity beams.
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.
I describe a new scheme for selectively isolating high density low longitudinal emittance beam particles in a storage ring from the rest of the beam without emittance dilution. I discuss the general principle of the method, called longitudinal momentum mining, beam dynamics simulations and results of beam experiments. Multi-particle beam dynamics simulations applied to the Fermilab 8 GeV Recycler (a storage ring) convincingly validate the concepts and feasibility of the method, which I have demonstrated with beam experiments in the Recycler. The method presented here is the first of its kind.
The use of Smith-Purcell radiation to measure electrons longitudinal profiles has been demonstrated at several facilities in the picosecond and sub-picosecond range. There is a strong interest for the development of non intercepting longitudinal profile diagnostics for high current proton beams. We present here results of simulations on the expected yield of longitudinal profile monitors using Smith-Purcell radiation for such proton beams.
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.
Barrier rf buckets have brought about new challenges in longitudinal beam dynamics of charged particle beams in synchrotrons and at the same time led to many new remarkable prospects in beam handling. In this paper, I describe a novel beam stacking scheme for synchrotrons using barrier buckets without any emittance dilution to the beam. First I discuss the general principle of the method, called longitudinal phase-space coating. Multi-particle beam dynamics simulations of the scheme applied to the Recycler, convincingly validates the concepts and feasibility of the method. Then I demonstrate t…
A high peak current, flat longitudinal phase space electron beam is desirable for efficient x-ray free electron laser (FEL) radiation in next generation light sources. To attain such a beam requires the extensive design of the linear accelerator (linac) including both linear and nonlinear effects. In this paper, we propose a lumped longitudinal beam dynamics model for fast optimization of the electron beam longitudinal phase space through the accelerator. This model is much faster than available tracking programs and also shows good agreement with the fully three-dimensional element-by-element…
In this Letter, I report on a novel scheme for beam stacking without any beam emittance dilution using a barrier rf system in synchrotrons. The general principle of the scheme called longitudinal phase-space coating, validation of the concept via multi-particle beam dynamics simulations applied to the Fermilab Recycler, and its experimental demonstration are presented. In addition, it has been shown and illustrated that the rf gymnastics involved in this scheme can be used in measuring the incoherent synchrotron tune spectrum of the beam in barrier buckets and in producing a clean hollow beam …
The use of coherent transition radiation autocorrelation methods to determine bunch length and profile information is examined with the compressed electron beam at the BNL ATF. A bi-gaussian fit is applied to coherent transition radiation auto-correlation data to extract the longitudinal current distribution. The effects of large transverse beam sizes are studied in theory and compared to experimental results. A suitable form of the correction factor is derived for beams with large transverse-longitudinal aspect ratios.
In this paper, we get the 1D approximate analytical solution of the plasma electrostatic wake driven by the laser, and get the oscillating frequency of the wake modified by the nonlinear high order terms, and find that the frequency depends on the oscillating amplitudes of the wake and the laser which is the general nonlinear phenomenon. Finally we analyze the longitudinal beam dynamics in this electrostatic wake, and find that the high order terms don't change the topology of the longitudinal phase space.
This paper analyzes the longitudinal space charge (LSC) impedances of a round uniform beam inside a rectangular and parallel plate chambers using the image charge method. This analysis is valid for arbitrary wavelengths and the calculation converges fast. The research shows that only a few of the image beams are needed to obtain a relative error less than 0.1%. The beam offset effect is also discussed in the analysis.
A longitudinal beam properties, data acquisition system has been commissioned to operate in the Fermilab booster ring. This system captures real time information including beam synchronous phase, bunch length, and coupled bunch instability amplitudes as the beam is accelerated from 400 MeV to 8 GeV in 33 ms. The system uses an off-the-shelf Tektronix oscilloscope running Labview software and a synchronous pulse generator. This paper describes the hardware configuration and the software configuration used to optimize the data processing rate.
Linear colliders are an attractive platform to explore high-precision physics of newly discovered particles. The recent significant progress in advanced accelerator technologies has motivated their applications to colliders which has been discussed in the {\sc alegro} workshop. In this paper we discuss one promising scheme, collinear wakefield acceleration. We especially discuss available drive and witness beam sources based on L and S-band radiofrequency technology, and also summarize available and forthcoming longitudinal shaping techniques to improve the overall acceleration efficiency via …
This paper studies the space charge impedances of a rectangular beam inside a rectangular chamber, and the limiting case, e.g., a rectangular beam between parallel plates, respectively. The charged beam has uniform density in vertical direction and arbitrary distribution in horizontal direction. The method of separation of variables is used to calculate the space charge potentials, fields, and impedances which are valid in the whole perturbation wavelength spectrum. Comparisons between the theoretical calculations and the numerical simulations are also provided and they match quite well. It is…
We show that if in the particle beam there are linear correlations between energy of particles and their transverse positions and momenta (linear beam dispersions), then the transverse projected emittances always can be reduced by letting the beam to pass through magnetostatic system with specially chosen nonzero lattice dispersions. The maximum possible reduction of the transverse projected emittances occurs when all beam dispersions are zeroed, and the values of the lattice dispersions required for that are completely defined by the values of the beam dispersions and the beam rms energy spre…
We present measurement and analyses of longitudinal timing jitter of Beijing Electron Positron Collider (BEPCII) linear accelerator electron gun. We simulated longitudinal jitter effect of the gun using PARMELA about beam performance including beam profile, average energy, energy spread, longitudinal phase of reference particle and XY emittance. The maximum percentage difference of the beam parameters are calculated to be; 100%, 13.27%, 42.24%, 7.79% and 65.01%, 86.81%, respectively due to which the bunching efficiency is reduced to 54%. The simulation results are in agreement with test and ar…
In DAΦNE, the Frascati e+/e- collider, the crab waist collision scheme has been successfully implemented in 2008 and 2009. During the collision operations for Siddharta experiment, an unusual synchrotron damping effect has been observed. Indeed, with the longitudinal feedback switched off, the positron beam becomes unstable with beam currents in the order of 200-300 mA. The longitudinal instability is damped by bringing the positron beam in collision with a high current electron beam (~2A). Besides, we have observed a shift of \approx 600Hz in the residual synchrotron sidebands. Precise measur…
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.
The low-level RF system (LLRF) generates the drive sent to the high-power equipment. In synchrotrons, it uses signals from beam pick-ups (radial and longitudinal) to minimize the beam losses and provide a beam with reproducible parameters (intensity, bunch length, average momentum and momentum spread) for either the next accelerator or the physicists. This presentation is the first of three: it considers synchrotrons in the lowintensity regime where the voltage in the RF cavity is not influenced by the beam. As the author is in charge of the LHC LLRF and currently commissioning it, much materi…
This paper presents the approximate analytical solutions to the longitudinal space charge (LSC) impedances of a round beam with uniform transverse distribution and sinusoidal line density modulations under two boundary conditions: (a) between parallel plates (b) inside a rectangular chamber, respectively. When the ratio of beam diameter to chamber height is small, the image charge fields of the round beam can be approximated by those of a line charge, the approximate analytical LSC impedances can be obtained by image method. The derived theoretical LSC impedances are valid at any perturbation …
The development of compact accelerator facilities providing high-brightness beams is one of the most challenging tasks in field of next-generation compact and cost affordable particle accelerators, to be used in many fields for industrial, medical and research applications. The ability to shape the beam longitudinal phase-space, in particular, plays a key role to achieve high-peak brightness. Here we present a new approach that allows to tune the longitudinal phase-space of a high-brightness beam by means of a plasma wakefields. The electron beam passing through the plasma drives large wakefie…
To achieve high focal spot intensities in heavy ion fusion, the ion beam must be compressed longitudinally by factors of ten to one hundred before it is focused onto the target. The longitudinal compression is achieved by imposing an initial velocity profile tilt on the drifting beam. In this paper, the problem of longitudinal drift compression of intense charged particle beams is solved analytically for the two important cases corresponding to a cold beam, and a pressure-dominated beam, using a one-dimensional warm-fluid model describing the longitudinal beam dynamics.
Many applications of high brightness, highly relativistic electron beams carry strict requirements on longitudinal phase space quality. To meet these requirements, accelerator systems typically utilize dispersive elements to manipulate the correlated energy spread acquired during acceleration or via collective effects. The many free variables in these systems make finding an optimal configuration difficult. We present here an analytical method for tracking a desired final longitudinal phase space backwards through an accelerator section composed of an acceleration or drift section followed by …
Present-day and next-generation accelerators, particularly for applications in driving wakefield-based schemes, require longitudinal beam shaping and attendant longitudinal characterization for experimental optimization. Here we present a diagnostic method which reconstructs the longitudinal beam profile at the location of a wakefield-generating source. The methods presented derive the longitudinal profile of a charged particle beam solely from measurement of the time-resolved centroid energy change due to wakefield effects. The reconstruction technique is based on a deconvolution algorithm th…
The longitudinal Schottky spectra of a radio-frequency (RF) bunched and electron cooled 22Ne10+ ion beam at 70 MeV/u have been studied by a newly installed resonant Schottky pick-up at the experimental cooler storage ring (CSRe), at IMP. For an RF-bunched ion beam, a longitudinal momentum spread of has been reached with less than 107 stored ions. The reduction of momentum spread compared with coasting ion beam was observed from Schottky noise signal of the bunched ion beam. In order to prepare the future laser cooling experiment at the CSRe, the RF-bunching power was modulated at 25th, 50th an…
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…
Increasing proton beam power on neutrino production targets is one of the major goals of the Fermilab long term accelerator programs. In this effort, the Fermilab 8 GeV Booster synchrotron plays a critical role for at least the next two decades. Therefore, understanding the Booster in great detail is important as we continue to improve its performance. For example, it is important to know accurately the available RF power in the Booster by carrying out beam-based measurements in order to specify the needed upgrades to the Booster RF system. Since the Booster magnetic field is changing continuo…
Motivated by the desire to understand the longitudinal effects of beam-beam forces, we study the longitudinal dynamics of protons and anti-protons at injection and top energy in the Tevatron. Multi-turn data of the longitudinal profiles are captured to reveal information about frequencies of oscillation, and changes in the bunch distributions. Tomographic reconstruction is used to create phase space maps which are subsequently used to find the momentum distributions. Changes in these distributions for both proton and anti-proton beams are also followed through the operational cycle. We report …
In DAFNE, the Frascati $e^+e^-$ collider operating since 1998, an innovative collision scheme, the crab waist, has been successfully implemented during the years 2008-09. During operations for the Siddharta experiment an unusual synchrotron oscillation damping effect induced by beam-beam collisions has been observed. Indeed, when the longitudinal feedback is off, the positron beam becomes unstable with currents above 200-300 mA due to coupled bunch instability. The longitudinal instability is damped by colliding the positron beam with a high current electron beam (of the order of 2 A). A shift…
This paper presents an analysis of the emittance preservation during a heavy ion bunching process. This analysis aims at finding the correlation between the rms emittance of a coasting beam and that of the resulting bunches. The emittance of a Gaussian coasting beam bunched adiabatically in synchrotrons is investigated using theoretical considerations and particle tracking simulations. New results are presented on the relation between the emittance before and after the bunching process. It is found that a constant factor can be determined to describe this relation. This factor can be used to a…
In carrying out calculations of the effect of longitudinal space charge on longitudinal motion, the transverse beam size appears in a form factor which is usually written as $g=1+2\ln (b/a)$. In fact, this expression applies to particles with vanishing betatron amplitude in a beam with uniform transverse distribution. It is argued that an average over the transverse distribution should be used instead of the value on axis. It is shown that for the realistic `binomial' family of distributions the 1 in the above expression for $g$ should be replaced by a value near 0.5 if $a$ is interpreted as t…
In this paper we present several techniques for characterizing longitudinal impedances based on transient measurements of the growth rates and tune shifts of unstable coupled-bunch modes. These techniques are applicable to measurement of both fundamental and higher-order mode impedances and allow characterization of shunt impedances and quality factors of the HOMs. Methods presented here are complementary to lab bench measurements of RF cavities, in that the beam based measurements directly sense the physical impedance in the installed configuration. In contrast to a single-bunch integrated im…
The J-PARC Main Ring (MR) has achieved the delivery of the 30 GeV proton beam with the beam power of 500 kW to the neutrino experiment in May 2018. The longitudinal coupled-bunch instabilities have been observed in the MR for the beam power beyond 470 kW. Since more significant coupled-bunch oscillation was observed in higher beam power, the mitigation of the coupled-bunch instabilities is necessary for the stable beam delivery with the beam power beyond 500 kW. A new feedback system was developed to suppress the coupled-bunch instabilities. The feedback system consists of a wall current monit…
Laser cooling of relativistic heavy ion beams of Li-like C$^{3+}$ and O$^{4+}$ is being in preparation at the experimental Cooler Storage Ring (CSRe). Recently, a preparatory experiment to test important prerequisites for laser cooling of relativistic $^{12}$C$^{3+}$ ion beams using a pulsed laser system has been performed at the CSRe. Unfortunately, the interaction between the ions and the pulsed laser cannot be detected. In order to study the laser cooling process and find the optimized parameters for future laser cooling experiment, a multi-particle tracking method was developed to simulate…
Differential energy structure of a micro multi-charged-particle system and the beam internal potential energy is derived with consequent property and ecessary inference. Then by combining the energy differential structure with differential transfer relations of physical flux density between different time domains, we present a general explicit formula for energy exchange in the beam element's transfer or accelerating process between source and observer, further give an analysis on relativistic beam's mass-energy relation and beam's energy up limits for resonant frequency (rf) and DC accelerati…
We develop a universal framework which allows quickly solve a wide class of problems for longitudinal space charge effects in beams and plasmas in cylindrical geometry. We introduce the longitudinal dielectric permittivity for the beam of charged particles, which describes its collective space charge response. The analyis yields an effective plasma frequency, which depends on the transverse geometry of the system. This dielectric permittivity mirrors the dielectric permittivity of plasma and matches the one dimensional (1D) expression once the transverse size of the beam is large. Several part…
Electron accelerators with higher and higher longitudinal field gradients are desirable, as they allow for the production of high energy beams by means of compact and cheap setups. The new laser-plasma acceleration technique appears to constitute the more promising breakthrough in this direction, delivering unprecedent field gradients up to TV/m. In this article we give a quantitative description of the impact of longitudinal wake fields on the electron beam. Our paper is based on the solution of Maxwell's equations for the longitudinal field. Our conclusions are valid when the acceleration di…
The successful operation of X-ray free-electron lasers (FELs), like the Linac Coherent Light Source or the Free-Electron Laser in Hamburg (FLASH), makes unprecedented research on matter at atomic length and ultrafast time scales possible. However, in order to take advantage of these unique light sources and to meet the strict requirements of many experiments in photon science, FEL photon pulse durations need to be known and tunable. This can be achieved by controlling the FEL driving electron beams, and high-resolution longitudinal electron beam diagnostics can be utilized to provide constrain…
Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameter-less integral equation. For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency. In other words, notwithstanding the RF bucket nonlinearity and potential well distortion, the Landau damping is lost. Hence, even a tiny c…
An experimental program to demonstrate a novel phase space manipulation in which the horizontal and longitudinal emittances of a particle beam are exchanged has been completed at the Fermilab A0 Photoinjector. A new beamline, consisting of a TM110 deflecting mode cavity flanked by two horizontally dispersive doglegs has been installed. We report on the first direct observation of transverse and longitudinal emittance exchange.
This chapter gives an overview of the longitudinal dynamics of the particles in an accelerator and, closely related to that, the issue of synchronization between the particles and the accelerating field. Beginning with the trivial case of electrostatic accelerators, the synchronization condition is explained for a number of driven accelerators like Alvarez linacs, cyclotrons and finally synchrotrons and storage rings, where it plays a crucial role. In the case of the latter, the principle of phase focusing is motivated qualitatively as well as on a mathematically more correct level and the pro…
We investigate influence on the storage ring beam dynamics of the coherent Synchrotron Radiation (SR) self fields produced by an electron bunch. We show that the maximum energy gain in the RF cavity must far exceed the energy loss of electrons due to the coherent SR.
This paper gives a very brief summary of longitudinal beam dynamics for both linear and circular accelerators. After discussing synchronism conditions in linacs, 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.
A longitudinal coupled bunch mode diagnostic was built to identify and characterize coupled bunch modes in the Max IV 3 GeV ring. A longitudinal coupled bunch Mode 0 dipole instability was identified and a longitudinal Mode 0 dipole damper was built which cured the instability. As a separate project, a radio frequency current transformer was built as an emergency backup to the DC current transformer. All the devices where built using the open-source Blinky-Lite Internet of Things platform that enabled quick and inexpensive deployment.
In the Brazilian synchrotron light source, we observed that modulating the phase of the accelerating fields at approximately twice the synchrotron frequency suppressed remarkably well a longitudinal coupled-bunch mode of the beam driven by a higher order mode (HOM) in one of the radiofrequency (RF) cavities. In this work, we present the results of a set of systematic measurements, in single and multibunch mode, aimed at characterizing the effects of RF phase modulation on the beam. We compare those experiments with the results of tracking simulations and of a theoretical model in which Landau …
Longitudinal charge profile measurements or bunch shape measurements is a challenge for temporally short non-relativistic bunches. The field profile has a larger longitudinal extent compared to the charge profile for such beams. This affects the ability of field sensing devices such as phase pick-ups or wall current monitors to measure charge distribution. Here we evaluate the feasibility for usage of coherent transition and diffraction radiation from non-relativistic beams for bunch shape measurements.
