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An algorithm is presented to calculate the field components of the magnetic field [ B x ( x , y , z ) , B y ( x , y , z ) , B z ( x , y , z ) ] at a point ( x , y , z ) in space, from the knowledge of the components [ B x ( x , y = 0 , z ) , B y ( x , y = 0 , z ) , B z ( x , y = 0 , z ) ] on a “reference plane”, which is normal to the y-axis at y = 0 . The algorithm, which is a general one and is not restricted to fields with mid-plane symmetry is based on the Maclaurin series expansion of the magnetic field components at any point in space in terms of the distance (y) of the point from the reference plane. The coefficients of the Maclaurin series expansion are expressed in terms of the on-plane field components and their partial derivatives with respect to spatial coordinates ( x , z ). The field components are usually generated from magnetic field measurements on a rectangular grid on the plane. This algorithm was employed in 1986 in the RAYTRACE computer code to help calculate the optical properties of magnets and of the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory (BNL). A general mathematical formulation of this algorithm based on the differential algebraic method was presented by Makino in 2011. This paper presents the step by step derivation of the algorithm and provides the necessary formulas to be introduced by the reader in any computer code which requires the field components generated by magnetic devices. In addition provides an example of the use of the algorithm and its limitations as applied to a Halbach type magnet with or without median plane symmetry.

The acceleration process of charged particle beams often requires the use of few acceleration stages to provide the beam with the desired energy. The extraction of the beam from one acceleration stage and the injection to the next, both require a special type of magnet which comes under the name septum magnet. Such a magnet generates a strong field in one region and a very low field in another region with the two regions separated by a very thin material (septum). The septum thickness of such a magnet should be as thin as possible to reduce the strength of other devices, like kickers, which are involved in the extraction or injection processes. A thin septum is also advantageous during the slow beam extraction process to reduce the beam losses at the septum. One of the methods which in theory can generate very large differences in field strength in adjacent field regions separated by a thin septum, is the use of two thin parallel current sheets of infinite dimensions. In practice we use other devices that can approximate such an abrupt change of the magnetic field within the septum thickness. In this paper we describe such a device, the DC current septum, we present results from the study of its electromagnetic properties, and we discuss a method to minimize the magnetic field in the region which requires a very small magnetic field. We also provide some results from the experimental measurements of the magnetic field generated by the “D6” current septum magnet which is installed in the experimental beam line of the NASA’s Space Radiation Laboratory (NSRL) facility (Tsoupas et al., 2007; Brown et al., 2010) at Brookhaven National Laboratory (BNL). This septum magnet is part of the beam’s extraction system from the AGS-Booster into the NSRL beam line for material and biological studies.

Energy recovery has been achieved in a multipass linear accelerator, demonstrating a technology for more compact particle accelerators operating at higher currents and reduced energy consumption. Energy delivered to the beam during the first four passes through the accelerating structure was recovered during four subsequent decelerating passes. High-energy efficiency was achieved by the use of superconducting accelerating cavities and permanent magnets. The fixed-field alternating-gradient optical system used for the return loop successfully transported electron bunches of 42, 78, 114, and 150 MeV in a common vacuum chamber. This new kind of accelerator, an eight-pass energy recovery linac, has the potential to accelerate much higher current than existing linear accelerators while maintaining small beam dimensions and consuming much less energy per electron.

The physics program at the future electron ion collider (EIC) calls for polarized neutron beams at high energies. The best neutron carriers are $^{3}\mathrm{He}$ nuclei and deuterons. Both neutron carriers are expected to be utilized by the spin physics program in the EIC. Owing to the small magnetic moment anomaly of a deuteron particle, significantly higher magnetic fields are required for spin rotation, so that full Siberian snakes are not feasible. However, the resonance strengths are weak, and the number of resonances is small. It is possible to deal with individual resonances using conventional methods such as a betatron tune jump and vertical ac dipole for intrinsic depolarizing resonances, and a weak partial snake for imperfection resonances. This study employed a realistic magnet ramp rate and parameters (both transverse and longitudinal emittances, as well as realistic lattices) to show that it is possible to accelerate polarized deuterons beyond 100 GeV/n. This paper summarizes the details of the feasibility of a polarized deuteron beam for the future EIC at Brookhaven National Laboratory.

The Cornell-BNL Electron Test Accelerator (CBETA), a 150 MeV energy recovery linac (ERL) now in construction at Cornell, employs a fixed-field alternating gradient optics return loop: a single beam line comprised of FFAG cells, which accepts four recirculated energies. CBETA FFAG cell uses Halbach permanent magnet technology, its design studies have covered an extended period of time supported by extensive particle dynamics simulations using computed 3-D field map models. This approach is discussed, and illustrated here, based on the final stage in these beam dynamics studies, namely the validation of a ultimate, optimized design of the Halbach cell.

The CBETA (Cornell Brookhaven Experimental Test Accelerator) project described in reference C. Mayes (2015) is a prototype electron accelerator for the proposed eRHIC (electron Relativistic Heavy Ion Collider) project, described in reference E. C. Aschenauer (2014). An Energy Recovery Linac (ERL) in combination with a recirculating beam line which is based on the Fixed Field Alternating Gradient (FFAG) principle constitute the electron accelerator. The FFAG beam line of the electron accelerator is comprised of cells consisting of one focusing and one defocusing quadrupoles which are designed as Halbach type permanent magnet quadrupoles which are discussed in reference K. Halbach (1980). Window frame electromagnets surrounding the Halbach magnets will be used as normal and skew dipoles correctors and quadrupole correctors. A Halbach magnet in combination with low field electromagnets to act as dipole and/or quadrupole correctors can also be used in the proposed eRHIC project as well as other hadrons facilities like RHIC. We will discuss some of the advantages the Halbach type permanent magnets have over electromagnets and provide experimental and theoretical evidence of these advantages.

In polarized proton collision experiments, spin flip is needed to reduce the systematic errors. At high energy colliders with Siberian snakes, a single magnet spin flipper does not work . A more sophisticated spin flipper, constructed of nine-dipole magnets, was used to flip the spin in the BNL Relativistic Heavy Ion Collider. A 97% spin-flip efficiency was measured at both 24 and 255 GeV. The spin flip experiment results are presented in this paper.

The proposed eRHIC project is an electron hadron collider to be built in the existing tunnel of the RHIC collider at Brookhaven National Laboratory (BNL). Polarized 3He+2 ions are one of the hadron species to collide with the polarized electrons. To overcome the spin resonances during the pre-acceleration of polarized 3He+2 ions in the AGS-Booster, an AC Dipole [2] is being built to create an artificial vertical intrinsic spin resonance which will eliminate the naturally occurring vertical intrinsic spin resonances during the acceleration. We will present an overview of the intrinsic spin resonances in the AGS-Booster, discuss the physics and the requirements of overcoming the spin resonances using an AC dipole, and present the results from the 2D and 3D electromagnetic study of the AC Dipole using the OPERA computer code [3].The proposed eRHIC project is an electron hadron collider to be built in the existing tunnel of the RHIC collider at Brookhaven National Laboratory (BNL). Polarized 3He+2 ions are one of the hadron species to collide with the polarized electrons. To overcome the spin resonances during the pre-acceleration of polarized 3He+2 ions in the AGS-Booster, an AC Dipole [2] is being built to create an artificial vertical intrinsic spin resonance which will eliminate the naturally occurring vertical intrinsic spin resonances during the acceleration. We will present an overview of the intrinsic spin resonances in the AGS-Booster, discuss the physics and the requirements of overcoming the spin resonances using an AC dipole, and present the results from the 2D and 3D electromagnetic study of the AC Dipole using the OPERA computer code [3].

We have developed an intrinsic resonance free circular electron accelerator. This lattice could be placed in the existing RHIC tunnel and accelerate electrons from 400 MeV to 18 GeV avoiding all major polarization loss usual in such machines.

In polarized proton collision experiments, it is highly advantageous to flip the spin of each bunch of protons during the stores to reduce the systematic errors. Experiments done at energies less than 2 GeV have demonstrated a spin-flip efficiency over 99%. At high energy colliders with Siberian snakes, a single magnet spin flipper does not work because of the large spin tune spread and the generation of multiple, overlapping resonances. A more sophisticated spin flipper, constructed of nine-dipole magnets, was used to flip the spin in the BNL Relativistic Heavy Ion Collider. A special optics choice was also used to make the spin tune spread very small. A 97% spin-flip efficiency was measured at both 24 and 255 GeV. These results show that efficient spin flipping can be achieved at high energies using a nine-magnet spin flipper.

This review summarizes projects and studies on circular accelerators proposed for driving subcritical reactors. The early isochronous cyclotron cascades, proposed about 20 years ago, and the evolution of these layouts up to the most recent solutions or designs based on cyclotrons and fixed field alternating gradient accelerators, are reported. In addition, the newest ideas and their prospects for development are discussed.

The interest of high-power, fixed-field ring methods in accelerator-driven sub-critical reactor proton driver applications is discussed.

A proton-therapy hospital installation, based on multiple beam extraction systems from a fixed-field synchrotron, is presented and commented. Potential interest as hospital operation efficiency, as well as estimates of the impact of continuous, multiple-port extraction, on the cost of a session, are discussed.

In this paper, we investigate the stability of the particle trajectories in fixed field alternating gradient accelerators (FFAs) in the presence of field errors. The emphasis is on the scaling radial sector FFA type: A collaboration work is ongoing in view of better understanding the properties of the 150 MeV scaling FFA at Kyoto University Institute for Integrated Radiation and Nuclear Science in Japan and progress toward high-intensity operation. Analysis of certain types of field imperfections revealed some interesting features that required the development of an analytical model based on the scalloping angle of the orbits. This helped explain some of the experimental results as well as generalize the concept of a scaling FFA to a nonscaling one for which the tune variations obey a well-defined law. Based on this, a compensation scheme of tune variations in imperfect scaling FFAs is presented. This is the cornerstone of a novel concept of a fixed tune FFA in which the scaling is not achieved at every azimuthal position of the ring but rather in an average sense. In this paper, we investigate the stability of the particle trajectories in Fixed Field Alternating Gradient (FFAG) accelerators in the presence of field errors: The emphasis is on the scaling FFAG type: a collaboration work is on-going in view of better understanding the properties of the 150 MeV scaling FFAG at KURRI in Japan, and progress towards high intensity operation. Analysis of certain types of field imperfections revealed some interesting features about this machine that explain some of the experimental results and generalize the concept of a scaling FFAG to a non-scaling one for which the tune variations obey a well defined law. A compensation scheme of tune variations in imperfect scaling FFAGs is presented. This is the cornerstone of a novel concept of a non-linear non-scaling fixed tune FFAG that we present and discuss in details in the last part of this paper.

The ray-tracing code Zgoubi has been used in a number of projects, in the frame of high energy and nuclear physics R&D, since it was last documented in review articles. Its library of optical elements and its accelerator simulation tools have been further developed in these contexts. Its use covers design studies regarding large colliders such as LHC and RHIC, synchrotron radiation, short lived beams, acceleration of polarized lepton and hadron beams. This report gives an overview of the present state of the code and of its evolution, illustrated with examples aimed at highlighting its capabilities.

The 400 MeV LINAC1 of the proposed eRHIC collider1 is the first acceleration stage of the 18 GeV electron accelerator. The second acceleration stage of the electron accelerator is the Rapid Cycling Synchrotron (RCS), which can increase the energy of the electron bunches to 18 GeV and subsequently extract transfer and inject the 18 GeV electron bunches into the 18 GeV electron Storage Ring (SR) to collide with the hadron bunches of the existing (Relativistic Heavy Ion Collider (RHIC)) accelerator. This paper describes the beam optics of the transfer line between the 400 MeV LINAC and the RCS accelerator (LtRCS). The function of the LtRCS line is twofold, first to transfer the electron beam from the exit of the 400 MeV LINAC to the injection point of the RCS1 and second to rotate the stable spin direction of the polarized electrons from the longitudinal direction to the vertical. A detailed description of the constraints to be satisfied by the LtRCS transfer line will be given. A section will also be devoted to discuss the spin rotator2.

The Cornell-BNL Electron Test Accelerator (CBETA), a four-pass, 150 MeV energy recovery linac (ERL), is now in construction at Cornell. Commissioning commenced on March 2019. A particularity of CBETA is that a single channel loop recirculates the four energies (42, 78, 114 and 150 MeV). The return loop arcs are based on fixed-field alternating gradient (FFAG) optics. The loop is comprised of 107 quadrupole-doublet cells, built using Halbach permanent magnet technology. Spreader and combiner sections (4 independent beam lines each) connect the 36 MeV linac to the FFAG arcs. We introduce here a start-to-end simulation of the 4-pass ERL, based entirely, and exclusively, on the use of magnetic field maps to model the optical components.

Depolarization response for a system of two orthogonal snakes at irrational tunes is studied in depth using lattice independent spin integration. In particular we consider the effect of overlapping spin resonances in this system, to understand the impact of phase, tune, relative location and threshold strengths of the spin resonances. These results are benchmarked and compared to two dimensional direct tracking results for the RHIC e-lens lattice and the standard lattice. Finally we consider the effect of longitudinal motion via chromatic scans using direct six dimensional lattice tracking.

The proposed electron-hadron collider (eRHIC) to be built at Brookhaven National Laboratory (BNL) will allow the collisions of 20 GeV polarized electrons with 250 GeV polarized protons, or $100\text{ }\mathrm{GeV}/\mathrm{n}$ polarized ${^{3}\mathrm{He}}^{+2}$ ions, or other unpolarized ion species. The large value of the anomalous magnetic moment of the $^{3}\mathrm{He}$ nucleus ${G}_{\mathrm{He}}=(g\ensuremath{-}2)/2=\ensuremath{-}4.184$ (where $g$ is the $g$-factor of the $^{3}\mathrm{He}$ nuclear spin) combined with the peculiar layout of the transfer line which transports the beam bunches from the Alternate Gradient Synchrotron (AGS) to the Relativistic Heavy Ion Collider (RHIC) makes the transfer and injection of polarized $^{3}\mathrm{He}$ ions from AGS to RHIC (AtR) a special case as we explain in the paper. Specifically in this paper we calculate the stable spin direction of a polarized $^{3}\mathrm{He}$ beam at the exit of the AtR line which is also the injection point of RHIC, and we discuss a simple modifications of the AtR beam-transfer-line, to perfectly match the stable spin direction of the injected polarized $^{3}\mathrm{He}$ beam to that of the circulating beam, at the injection point of RHIC.

In a fixed-field alternating-gradient (FFAG) accelerator, eliminating pulsed magnet operation permits rapid acceleration to synchrotron energies, but with a much higher beam-pulse repetition rate. Conceived in the 1950s, FFAGs are enjoying renewed interest, fuelled by the need to rapidly accelerate unstable muons for future high-energy physics colliders. Until now a 'scaling' principle has been applied to avoid beam blow-up and loss. Removing this restriction produces a new breed of FFAG, a non-scaling variant, allowing powerful advances in machine characteristics. We report on the first non-scaling FFAG, in which orbits are compacted to within 10 mm in radius over an electron momentum range of 12-18 MeV/c. In this strictly linear-gradient FFAG, unstable beam regions are crossed, but acceleration via a novel serpentine channel is so rapid that no significant beam disruption is observed. This result has significant implications for future particle accelerators, particularly muon and high-intensity proton accelerators. © 2012 Macmillan Publishers Limited. All rights reserved.

This paper addresses an attempt to start investigating the use of the superconducting ring cyclotron (SRC) developed for the DAEδALUS experiment, for ADS-reactor/transmuter applications.

A new experiment is described to detect a permanent electric dipole moment of the proton with a sensitivity of $10^{-29}e\cdot$cm by using polarized "magic" momentum $0.7$~GeV/c protons in an all-electric storage ring. Systematic errors relevant to the experiment are discussed and techniques to address them are presented. The measurement is sensitive to new physics beyond the Standard Model at the scale of 3000~TeV., Comment: 8 pages, 3 figures

Due to the combination of fixed magnetic field operation with strong focusing, non-scaling FFAGs have a significant potential for future particle accelerator applications. However, this technology has a number of unique features, which must be fully studied before this potential can be realised. To do this, a proof-of-principle non-scaling FFAG, called EMMA – Electron Model for Many Applications – has been constructed at the STFC Daresbury Laboratory in the UK. It has been designed by an international collaboration of accelerator scientists and engineers. It will demonstrate the principle of non-scaling FFAGs and be used to study the features of this type of accelerator in detail.

International audience; The FFAG method is nowadays seen as a potential candidate for the acceleration of protons and light ions for hadrontherapy. This has motivated the design of a principle protontherapy installation, in the frame of the RACCAM project. This article presents the design study, a medical spiral scaling FFAG assembly, capable of producing variable energy proton beams, with potentially high repetition and dose delivery rates.

The first results of commissioning for the French Synchrotron Radiation Facility SOLEIL at 2.75 GeV are presented. Perspectives for the fourth generation light source based on the ARC-EN-CIEL project are described.

This paper gives a brief overview of various beam and spin dynamics investigations undertaken in the framework of the design studies regarding the FFAG lattice based electron energy recovery re-circulator ring of the eRHIC electron-ion collider project., Proceedings of the 56th ICFA Advanced Beam Dynamics Workshop on Energy Recovery Linacs, ERL2015, Stony Brook, NY, USA