International Journal of Modern Physics A

The masses of charged atoms and molecules were first investigated by laterally dispersive sector field mass analyzers, which early on already achieved high mass resolving powers. Equally, high mass resolving powers were achieved by time-of-flight mass analyzers during the last decades. These measurements became possible when fast and precise electronic circuitries became available. Such techniques have been developed and used extensively for the mass analysis of short-lived nuclei, whose mass values reveal insight in processes that describe the formation of elements in star explosions. Precise mass determinations of short-lived ions have been performed for energetic ions in large accelerator storage rings as well as for low-energy ions in time-of-flight mass spectrographs with long flight paths. Similarly, precise mass measurements can also be performed for molecular ions that help to reveal the structure of molecules. In case of very high mass resolving powers, the mass determination of molecular ions can be so high that the measured ion mass directly reveals the molecule’s sum formula.

There are many applications in electron microscopy, electron spectroscopy, as well as accelerator physics that require the combination of minimizing a focused electron beam’s probe size while maximizing its beam current. This paper describes how this can be done through the use of annular focused electron beam column designs, where an electron beam is propagated and focused in the form of a ring beam. For probe semi-angles of $\pi /4$ radians, a column consisting of two identical electric sectors and two identical focusing lenses functioning with odd symmetry will be presented, together with a wide-angle cold field emission gun design. The column is designed to cancel energy dispersion while limiting geometric aberrations to the third order at the point of final focus. This design is predicted to have over two orders of magnitude higher beam current than the corresponding conventional on-axis focused electron beam columns for the same final probe size. For a 1.2 keV annular electron beam, simulation results predict a spot radius of 4.8 nm at a working distance of 3.4 mm, final beam angle of $45^{\circ }\pm 2$ mrad, and energy spread of $\pm 0.15$ eV.

The quadrupole linear Paul trap is one of the key instruments in building highly stable atomic clocks. However, a frequency reference based on a single trapped ion is limited in stability due to the time needed for the interrogation cycle which cannot be further shortened. A promising strategy is the utilization of multiple trapped ions. The ions of the same kind then repulse each other with the Coulomb force, which is countered by the ponderomotive force of the time depended field in the trap. A few ions form a chain along the axis of a linear Paul trap. Adding more ions (a few tens or hundreds) gives rise to Coulomb crystals. We created an efficient simulation code which calculates the motion of such collections of ions in quasi-static radiofrequency fields of real linear quadrupole traps (including the micromotion). We include a model for laser cooling of the ions. The simulation tool can be used to study the formation and the dynamics of Coulomb crystals under conditions corresponding to various experimental setups.

In the optimization of the quadrupole mass spectrometer (QP-MS), the understanding of ion motion in terms of the phase space (the combined representation of the trajectory coordinate and momentum) is useful. The phase space representation gives an “ensemble” behavior of ions inside the filter. Even though each ion trajectory does not have the RF periodicity of the applied field, the phase space evolution does. It is only when appropriate ensemble ions are considered together that a proper QP filter characterization is possible. We here report a new framework for the phase space calculation of the QP-MS. The Mathieu–Hill equation is first solved for “complex eigen-trajectory” that has pseudo RF periodicity (the Bloch function). It is then shown that the acceptance phase space can be derived from the Bloch function without a need to calculate each ion trajectory. The ensemble behavior of ions can be estimated from a single Bloch function by expressing the trajectory phase space point by the complex amplitude (coefficient) of the Bloch function. The application of the Bloch function method to the auxiliary (pre-rod) field reveals that the ion injection efficiency may significantly be improved by optimizing the number of RF periods the ions spend in the pre-rod section.

An ion optical design of a high resolution multi-turn time-of-flight mass analyzer (MT-TOF MA) is presented. The analyzer has rotationally symmetric main electrodes with additional mirror symmetry about a mid-plane orthogonal to the axis of symmetry. Rotational symmetry allows a higher density of turns in the azimuthal (drift) direction compared to MT-TOF MAs that are linearly extended in the drift direction. Mirror symmetry about a mid-plane helps to achieve a high spatial isochronicity of the ions’ motion. The analyzer comprises a pair of polar-toroidal sectors S₁ and S₃, a pair of polar (trans-axial) lenses, and a pair of conical lenses for longitudinal and lateral focusing. A toroidal sector S₂ located at the mid-plane of the analyzer has a set of embedded drift focusing segments providing focusing and spatial isochronicity in the drift direction. The ions’ drift in the azimuthal direction can be reversed by using dedicated reversing deflectors. This gives the possibility of several operational modes with different numbers of turns and passes in the drift direction. According to numerical simulations, the mass resolving power of the analyzer ranges from $\sim$40 k (fwhm) at small (typically below ten) numbers of turns to $\sim$450 k (fwhm) at 96 turns.

In this paper, a monochromator (MC) with double cylindrical lenses (CLs) is evaluated. The MC achieved an energy resolution of 37 meV with high repeatability and stability. The double CLs realized the coincidence of the beam positions at the exit. The beam profiles were evaluated in a relative manner. Comparing two conditions with or without the MC, the beam kept equivalent sizes and showed sufficiently fine profiles. This is because the first-order aberration was a dominant factor under the experimental condition, where the second-order aberration was canceled by the middle plane symmetry of the MC optics and the beam was limited in small angles by the incident aperture. The experimental results mean that the MC with double CLs functioned properly as designed. The MC has the additional advantage of a simple structure. The results confirm that this MC has a high possibility for application to electron microscopy.

The paper deals with space-charge interactions in the ion population trapped in the Orbitrap${}^{\mathrm{\text{TM}}}$ mass analyzer where the ions perform multiple quasi-harmonic oscillations in the axial direction. The many-particle problem for interacting ions is mathematically complicated and its solution, even numerical, is obstructed by the required precision of one per million to be maintained on a large number $(10^5-10^6)$ of oscillation periods. We develop a perturbation method based on the Bogoliubov–Krylov–Mitropolsky theory and derive averaged Hamiltonian equations in perturbations, which describe the evolution of the ions’ oscillation amplitudes and phases in so-called “slow” time. This approach provides a semi-analytical comprehensive model of resonant and nonresonant space-charge effects and allows fast and accurate numerical computation. Practical mitigation strategies for most deteriorating space-charge effects like coalescence and frequency shifts are considered.

The Muon $g-2$ Experiment at Fermilab (E989) is directed toward measuring the muon magnetic anomaly, $a_{\mu }=(g-2)/2$, with total statistical and systematic errors of 0.14 ppm. This new measurement will serve as strong probe of effects of as yet undiscovered particles beyond the Standard Model (SM), and perhaps validate or disprove other theoretical models beyond the SM. Of special interest is the reduction of muon losses from the storage ring to achieve the precision needed at the Muon $g-2$ Experiment. For this purpose, we have developed a detailed and precise symplectic model of the Muon $g-2$ Storage Ring using COSY INFINITY that considers measured inhomogeneities of the magnetic field; high-order representation of the Electrostatic Quadrupole System (EQS) electrostatic field at different stages of the experiment including fringe fields; injection to the ring based on measurements; and beam collimation. Specifically, we have performed numerical analyses of the rate of muons that are lost before they have a chance to decay for several possible configurations of the EQS in order to find the best possible scenarios that minimize muon losses and understand the resonance mechanisms that contribute to betatron and possibly spin resonances. Additionally, comparisons with measurements have permitted the determination of whether observed resonances come from anticipated features of the $g-2$ storage ring or from unexpected sources of error whose effect could be detrimental to the precision of E989.

The code COSY INFINITY uses a beamline coordinate system with a Frenet–Serret frame relative to the reference particle, and calculates differential algebra-valued transfer maps by integrating the ODEs of motion in the respective vector space over a differential algebra (DA).

We described and performed computation of the DA transfer map of an electrostatic spherical deflector in a laboratory coordinate system using two conventional methods: (1) by integrating the ODEs of motion using a numerical integrator and (2) by computing analytically and in closed form the properties of the respective elliptical orbits from Kepler theory. We compared the resulting transfer maps with (3) the DA transfer map of COSY INFINITY’s built-in electrostatic spherical deflector element $ESP$ and (4) the transfer map of the electrostatic spherical deflector computed using the program GIOS, which uses analytic formulas from a paper¹ by Hermann Wollnik regarding second-order aberrations.

In addition to the electrostatic spherical deflector, we studied an electrostatic cylindrical deflector, where the Kepler theory is not applicable. We computed the DA transfer map by the ODE integration method (1), and we compared it with the transfer maps by (3) COSY INFINITY’s built-in electrostatic cylindrical deflector element $ECL$ and (4) GIOS.

The transfer maps of electrostatic spherical and cylindrical deflectors obtained using the direct calculation methods (1) and (2) are in excellent agreement with those computed using (3) COSY INFINITY. On the other hand, we found a significant discrepancy with (4) the program GIOS.

Nonlinear effects of the various electric field and magnetic field components of storage rings to confine the particles and bend their trajectory can cause substantial amplitude-dependent tune shifts within the beam. Furthermore, tune shifts are often sensitive to variations of system parameters, e.g. total particle momentum offsets $\delta p$. Such amplitude- and parameter-dependent tune shifts influence the dynamics and stability of a beam in particle storage rings. Thus, it is critical for high precision measurements to analyze and understand these influences. On this basis, we present normal form methods for the calculation of high order amplitude and system parameter dependencies of the horizontal and vertical tunes in storage rings using the differential algebra (DA) framework within COSY INFINITY. A storage ring is simulated using COSY INFINITY to generate a DA Poincaré return map describing the transverse phase space behavior after each revolution in the storage ring. The map is expanded around the parameter-dependent closed orbit of the system before transforming the resulting map into normal form coordinates to extract the high order tune dependencies on the phase space amplitude and variation in the system parameters. As a specific example, a storage ring similar to the Storage Ring of the Muon $g$-2 Experiment at Fermilab (E989) is investigated.

To establish an accelerator facility based on superconducting linear accelerator (linac) for nuclear and applied sciences in Korea, the rare isotope science project started in December 2011 under the auspices of the Institute for Basic Science (IBS). The layout of the facility has been mostly frozen since 2013 and civil construction began in 2016. On the other hand, an alternative linac design option was recently investigated in collaboration with the linac development group of Argonne National Lab in search of a further optimized configuration. A new linac lattice was developed and evaluated against realistic machine errors using TRACK. The beam optics simulations and error analysis then proved the soundness of this alternative design. In addition, beam optics of the injector was studied for simultaneous acceleration of both stable and rare isotope beams with the use of an electron beam ion source (EBIS). Also considered are alternative options of high-power cyclotrons as ISOL driver in order to enhance the capability of rare isotope beam production and fully exploit the superconducting linac facility built at high cost.

The Proton Improvement Plan, Stage Two (PIP-II), is a program of upgrades proposed for the Fermilab injection complex, the central part of which is an 800 MeV, 2 mA CW-compatible SRF linac. A prototype of the PIP-II linac front-end called the PIP-II Injector Test (PIP2IT) is being built at Fermilab. As of now, a 15 mA DC, 30 keV H${}^{-}$ ion source, a 2 m long Low Energy Beam Transport (LEBT), a 2.1 MeV CW Radio-Frequency Quadrupole (RFQ), followed by a 10 m Medium Energy Beam Transport (MEBT) have been installed and tested. A Fast Faraday Cup (FFC) installed in the MEBT measures the length of a beamlet cut out of the bunch by a small-size entrance hole of the FFC. The information about the bunch length measured at various settings allows for the reconstruction of the longitudinal beam dynamics and optimization of injection into the first cryomodule.

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.

Due to the unique “Bragg peak” dose-distribution characteristics of proton beams, the proton therapy (PT) is recognized as one of the most precise and effective radiotherapy methods for tumors. A gantry is required to project the beam onto a tumor at various angles for multiple-field irradiation, and a superconducting beamline can significantly reduce the size and weight of the gantry. A PT system is under development at Huazhong University of Science and Technology (HUST), and this paper presents a comparison study of the beam optics and related design considerations for normal conducting and superconducting gantry beamlines.

High brightness electron beams play an important role in accelerator-based applications such as driving X-ray free electron laser (FEL) radiation. In this paper, we report on advances in global beam dynamics optimization of an accelerator design using start-to-end simulations and a new parallel multi-objective differential evolution optimization method. The global optimization results in significant improvement of the final electron beam brightness.

We present the High-Rigidity Spectrometer (HRS) planned at the Facility for Rare Isotope Beams (FRIB). The current status of the layout and the ion-optical design of the HRS is discussed.

The CERN SPS 200 MHz traveling-wave (TW) accelerating structures pose an intensity limitation for the planned High Luminosity (HL-) LHC upgrade. Higher-order modes (HOMs) around 630 MHz have been identified as one of the main sources of longitudinal multi-bunch instabilities. Improved mitigation of these HOMs with respect to today’s HOM-damping scheme is therefore an essential part of the LHC injectors upgrade (LIU) project.

The basic principles of HOM-couplers in cavities and today’s damping scheme are reviewed. Before illustrating the numerous requirements, an improved damping scheme for the future 33-cell structures must be fulfilled. These are, among others, the mitigation of HOMs situated in the lower part of the structure where there are no access ports for extraction, a sufficient overall damping performance and an acceptable influence on the fundamental accelerating passband (FPB). Different approaches tackling these challenges are investigated and their performance, advantages and pitfalls are evaluated by ACE3P and CST electromagnetic (EM) field solver suites.

The application of machine learning and nature-inspired optimization methods, like for example genetic algorithms (GA) and particle swarm optimization (PSO) can be found in various scientific/technical areas. In recent years, these approaches are finding application in accelerator physics to a greater extent. In this paper, nature-inspired optimization as well as the machine learning will be shortly introduced and their application to the accelerator facility at GSI/FAIR will be presented. For the heavy-ion synchrotron SIS18 at GSI, the multi-objective GA/PSO optimization resulted in a significant improvement of multi-turn injection performance and subsequent transmission for intense beams. An automated injection optimization with genetic algorithms at the CRYRING@ESR ion storage ring has been performed. The usage of machine learning for a beam diagnostic application, where reconstruction of space-charge distorted beam profiles from ionization profile monitors is performed, will also be shown. First results and the experience gained will be presented.

In electrostatic charged particle lens design, optimization of a multi-electrode lens with many free optimization parameters is still quite a challenge. A fully automated optimization routine is not yet available, mainly because the lens potential calculations are often done with very time-consuming methods that require meshing of the lens space. A new method is proposed that improves on the low speed of the potential calculation while keeping the high accuracy of the mesh-based calculation methods. This is done by first using a fast potential calculation based on the so-called Second-Order Electrode Method (SOEM), at the cost of losing some accuracy, and then using a Genetic Algorithm (GA) for the optimization. Then, by using the parameters of the approximate systems found from this optimization based on SOEM, an accurate GA optimization routine is performed based on potential calculation with the commercial finite element package COMSOL. A six-electrode electrostatic lens was optimized accurately within a few hours, using all lens dimensions and electrode voltages as free parameters and the focus position and maximum allowable electric fields between electrodes as constraints.

Statistical Coulomb interactions in conventional scanning electron microscopy mostly affect the probe size via energy spread and virtual source broadening in the emitter vicinity. However, in a multi-beam probe forming system such as a multi-beam scanning electron microscopes (MBSEM), the trajectory displacement due to interactions in the whole column can give a contribution to the final probe size. For single-beam systems, this can be expressed using approximate formulae for the total trajectory displacement in a beam segment (Jansen’s theory) or by integrating contributions of infinitesimally thin beam slices (the slice method). We build on Jansen’s theory of statistical Coulomb interactions and develop formulae for the trajectory displacement in a multi-beam system. We also develop a more precise semi-analytical result using the slice method. We compare both approaches with a Monte Carlo simulation and show a good agreement with the results of the slice method. Finally, we discuss the implications of our results for the optical design of multi-beam SEM. In a multi-beam with probe size dominated by Coulomb interactions, an increase in the number of beamlets does not necessarily provide an increase of throughput, because the probe size is limited by the total current. Furthermore, we disprove the notion of “the fewer the crossovers — the less the Coulomb interactions” by showing the quadratic dependence of trajectory displacement on segment length.

Extracting the coefficients of Fourier–Bessel series, known as pseudo-multipoles or generalized gradients, from magnetic measurements of accelerator magnets involves technical and mathematical challenges. First, a novel design of a short, rotating-coil magnetometer is required that does not intercept any axial field component of the magnet. Moreover, displacing short magnetometers, step-by-step along the magnet axis, yields a convolution of the local multipole field errors and the sensitivity (test function) of the induction coil. The deconvolution must then contend with the limited signal-to-noise ratio of the measured quantities, which are integrated voltages corresponding to spatial flux distributions. Finally, the compensation schemes, as implemented on long coils and based on scaling laws derived for the integrated field harmonics, cannot be applied to short magnetometers intercepting only a local field distribution. All this requires careful design of experiment to derive the optimal length of the induction coil, the step-size of the scan, and the highest order of pseudo-multipoles in the field reconstruction. This paper presents the theory of the measurement method, the data acquisition and deconvolution, and the design and production of a saddle-shaped, rotating-coil magnetometer.

A novel concept for a superconducting, fixed-field bending section is presented for use in a proton therapy gantry. The large momentum acceptance of this design allows for treatment over the full proton energy range of 70–220 MeV with fixed field in the superconducting magnets, eliminating the technical risks associated with fast-field ramping to match beam energy changes during treatment. A combined study of beam dynamics and magnet design is shown for such a system in which a simple magnet geometry with straight Nb–Ti racetrack coils is used to produce the desired fields. Particle tracking through this design is compared with clinical requirements for beam spot shape and size at isocenter over the full range of proton energy.

In proton therapy, the last part of the beam transport system is installed on a rotatable gantry, so that the beam can be aimed at the tumor from different angles. Since such a gantry system consists of many dipole and quadrupole magnets, it is typically a 100–200 tons device of more than 10 m in diameter. The use of superconducting (SC) magnets for proton therapy allows gantries to be significantly lighter and potentially smaller, which is attractive for this medical application. In addition to that, SC combined function magnets enable beam optics with a very large momentum acceptance. The latter can be advantageous for patient treatment, since the irradiation time can then be significantly reduced by avoiding magnet current changes. To design such an achromatic system, we performed precise high-order calculations. To reach the required accuracy and to check consistency of the obtained results, we have used different simulation tools in our iterative design approach. Here, we will describe our beam optics calculations in the code COSY Infinity and particle tracking using OPAL (open source software from PSI) in 3D field maps obtained from detailed magnet calculations performed in Opera. Our method has shown to be advantageous in a complicated beam optics study and it reduces the risk of systematic errors in a design.

The first conference in what would become the International Computational Accelerator Physics (ICAP) series was held in 1988. At that time the most powerful computer in the world was a Cray YMP with 8 processors and a peak performance of 2 gigaflops. Today the fastest computer in the world has more than 2 million cores and a theoretical peak performance of nearly 200 petaflops. Compared to 1988, performance has increased by a factor of 100 million, accompanied by huge advances in memory, networking, big data management and analytics. By the time of the next ICAP in 2021 we will be at the dawn of the Exascale era. In this talk I will describe the advances in Computational Accelerator Physics that brought us to this point and describe trends in regard to large-scale accelerator simulation in the future.

The Physics Beyond Colliders study investigates the feasibility of future projects and experiments using CERN facilities. In the scope of this study, a future operation of the NA62 experiment using the existing K12 beam line operated in beam dump mode is discussed. Such a setup allows for a search for dark sector particles, e.g. heavy neutral leptons, dark photons and axions. Production of these hypothetical particles requires the 400GeV/c proton beam extracted from the Super Proton Synchrotron (SPS) to be dumped on a massive dump collimator located in the first part of the K12 beam line. The decay of these dark sector particles, for example, into muons could be detected by the NA62 experimental setup. However, muons created in particle interactions with material and from in-flight decays of secondary hadrons pose a critical background for this kind of experiment. Magnetic sweeping serves as an efficient tool for reducing this background. This paper discusses the simulation studies conducted with the software framework G4beamline to reduce the expected single-muon rate for this experiment. The required beam line modifications, in particular the removal of the beryllium target and repositioning of the dump, have been implemented into this model. A comparison of simulated muon distributions and spectra available experimental data, already taken in test runs, has been made in order to validate the beam line simulation. Finally, additional beam line modifications to further suppress the muon background are elaborated upon. These studies show a potential further reduction of a factor four of the total single-muon rate relative to the nominal setup over the full momentum range, and up to a factor twenty considering only muons with momenta $p>15GeV/c$.

In this paper, we review three methods to construct symplectic and self-consistent multiparticle algorithms to simulate space–charge effects in particle accelerators. The first method is based on a discrete multiparticle Hamiltonian with an interaction term that depends explicitly on the coordinates of the macroparticles. The second method derives from Low’s Lagrangian for a collisionless plasma. The third method is based on a corresponding collisionless Hamiltonian. The last two methods have been mostly developed by the plasma physics community, but are equally applicable to accelerator physics problems.

We report on the concept of an innovative laser-driven plasma accelerator for polarized proton (or deuteron) beams with a kinetic energy up to several GeV. In order to model the motion of the particle spins in the plasmas, these have been implemented as an additional degree of freedom into the Particle-in-Cell simulation code VLPL. For the experimental realization, a polarized HCl gas-jet target is under construction, where the degree of proton polarization is determined with a Lamb-shift polarimeter. The final experiments, aiming at the first observation of a polarized particle beam from laser-generated plasmas, will be carried out at the 10 PW laser system SULF at SIOM/Shanghai.

The SPACE code is a parallel, relativistic, three-dimensional, electromagnetic, particle-in-cell code developed for the simulations of beam dynamics and interactions. The SPACE code can perform simulations with various beam distributions, different types of boundary conditions, and flexible beam line, as well as provide sufficient processing routines for data analyses and visualization. Self-consistent, highly resolved and realistic simulations were performed using the SPACE code to study the coherent electron cooling (CeC) experiment with two types of amplifiers, the free electron laser (FEL) and plasma-cascade micro-bunching amplifiers.

We report a detailed simulation of a bunched electron-beam accelerated in a TE${}_{113}$ cylindrical cavity immersed in a static inhomogeneous magnetic field using a relativistic full electromagnetic particle-in-cell (PIC). This type of acceleration concept is known as Spatial AutoResonance Acceleration (SARA) in which the magnetic field profile is such that it keeps the electron-beam in the acceleration regime along their trajectories. In this work, the numerical experiments are carried out including a bunched electron-beam with the concentrations in the range $10^8$–$10^9$cm${}^{-3}$ in a TE${}_{113}$ cylindrical microwave field, at a frequency of 2.45 GHz and an amplitude of 15 kV/cm. The electron energy reaches values up to 250 keV without significant unfocusing effect that can be used as a basis to produce hard X-ray. Additionally, a comparison between the data obtained from the full electromagnetic PIC simulations and the results derived from the relativistic Newton–Lorentz equation in a single particle approximation is carried out.

Dielectric Laser Acceleration (DLA) achieves the highest gradients among structure-based electron accelerators. The use of dielectrics increases the breakdown field limit, and thus the achievable gradient, by a factor of at least 10 in comparison to metals. Experimental demonstrations of DLA in 2013 led to the Accelerator on a Chip International Program (ACHIP), funded by the Gordon and Betty Moore Foundation. In ACHIP, our main goal is to build an accelerator on a silicon chip, which can accelerate electrons from below 100 keV to above 1 MeV with a gradient of at least 100 MeV/m. For stable acceleration on the chip, magnet-only focusing techniques are insufficient to compensate the strong acceleration defocusing. Thus, spatial harmonic and Alternating Phase Focusing (APF) laser-based focusing techniques have been developed. We have also developed the simplified symplectic tracking code DLAtrack6D, which makes use of the periodicity and applies only one kick per DLA cell, which is calculated by the Fourier coefficient of the synchronous spatial harmonic. Due to coupling, the Fourier coefficients of neighboring cells are not entirely independent and a field flatness optimization (similarly as in multi-cell cavities) needs to be performed. The simulation of the entire accelerator on a chip by a Particle In Cell (PIC) code is possible, but impractical for optimization purposes. Finally, we have also outlined the treatment of wake field effects in attosecond bunches in the grating within DLAtrack6D, where the wake function is computed by an external solver.

In this paper, we describe our work on spin polarization in high-energy electron storage rings which we base on the Full Bloch equation (FBE) for the polarization density and which aims towards the $e^{-}-e^{+}$ option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The FBE takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov–Ternov effect with its Baier–Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev–Kondratenko formulas. For our numerical and analytical studies of the FBE, we develop an approximation to the latter to obtain an effective FBE. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations (SDEs) underlying the FBE and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective FBE. This discretizes the phase space using spectral methods and discretizes time via the additive Runge–Kutta (ARK) method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

The Muon Campus at Fermilab is a system through which muons are delivered to the storage ring of the Muon $g-2$ Experiment (E989). It consists of a set of 1 km beamlines that transport and prepare a highly polarized muon beam out of secondaries produced downstream a target station. Realistic simulations of this beam delivery system (BDS) using COSY INFINITY, and presented here, contribute to the understanding and characterization of the muon beam production in relation to the statistical and systematic uncertainties of the E989 measurement, intended to be smaller than 0.14 parts per million to achieve the goals of the experiment. The impact of nonlinearities from fringe fields and high-order contributions on the BDS performance are presented, as well as detailed studies of the interactions between secondaries and the beamline elements apertures, particle decay channels, spin dynamics and beamline misalignments.

SixTrack is a single-particle tracking code for high-energy circular accelerators routinely used at CERN for the Large Hadron Collider (LHC), its luminosity upgrade (HL-LHC), the Future Circular Collider (FCC) and the Super Proton Synchrotron (SPS) simulations. The code is based on a 6D symplectic tracking engine, which is optimized for long-term tracking simulations and delivers fully reproducible results on several platforms. It also includes multiple scattering engines for beam–matter interaction studies, as well as facilities to run the integrated simulations with external particle matter interaction codes. These features differentiate SixTrack from general-purpose, optics-design software. The code recently underwent a major restructuring to merge the advanced features into a single branch, such as multiple ion species, interface with external codes and high-performance input/output. This restructuring also removed a large number of compilation flags, instead enabling/disabling the functionality with runtime options. In the process, the code was moved from Fortran 77 to Fortran 2018 standard, also allowing and achieving a better modularization. Physics models (beam–beam effects, Radio-Frequency (RF) multipoles, current carrying wires, solenoid and electron lenses) and methods (symplecticity check) have also been reviewed and refined to offer more accurate results. The SixDesk runtime environment allows the user to manage the large batches of simulations required for accurate predictions of the dynamic aperture. SixDesk supports several cluster environments available at CERN as well as submitting jobs to the LHC@Home volunteering computing project, which enables volunteers contributing with their hardware to CERN simulation. SixTrackLib is a new library aimed at providing a portable and flexible tracking engine for single- and multi-particle problems using the models and formalism of SixTrack. The library is able to run in CPUs as well as graphical processing units (GPUs). This contribution presents the status of the code, summarizes the main existing features and provides details about the main development lines SixTrack, SixDesk and SixTrackLib.

The Super Proton Synchrotron (SPS) at CERN is the injector of the Large Hadron Collider (LHC). Multi-bunch instabilities limit the intensity of the beam that can be accelerated to 450 GeV in the SPS and transferred to the LHC. Without mitigation measures, the threshold of bunch intensity of the longitudinal instability is three times below the nominal bunch intensity of the LHC beam. The High Luminosity LHC project (HL-LHC), which requires a doubling of the nominal bunch intensity, relies on improvement of beam stability in the SPS. A fourth harmonic RF system allows, presently, to stabilize the beam up to nominal LHC intensity. It increases the synchrotron frequency spread inside the bunch, providing more efficient Landau damping of beam instability. However, nonlinearities of the synchrotron frequency distribution inside the bunch pose a limitation on bunch length. This paper explores possible intensity increase in the SPS by studying the effect on beam stability of the voltage ratio between two RF systems. The results are substantiated by beam measurements and particle-tracking simulations. An optimized voltage program of the second RF system during the cycle has been tested in operation and beam stability and the quality have been successfully improved.

Superconducting radio frequency cavities meet the demanding performance requirements of modern accelerators and high-brilliance light sources. Their design requires a precise knowledge of their electromagnetic resonances. A numerical solution of Maxwell’s equations is required to compute the resonant eigenmodes, their frequencies and losses due to the complex cavity shape. The consideration of resonances damped by external losses leads to a nonlinear eigenvalue problem. Previous work showed that, using State-Space Concatenation to construct a reduced order model and Newton iteration to solve the arising eigenvalue problem, solutions can be obtained on workstation computers even for large-scale problems without extensive simplification of the structure itself. In this paper, we augment the solution workflow by Beyn’s contour integral algorithm to increase the number of found eigenmodes. Numerical experiments are presented for one academic and two real-life superconducting cavities and partially compared to measurements.

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.

The Cornell-Brookhaven CBETA machine is a four-pass Energy Recovery Linac (ERL) with a Non-scaling Fixed-Field Alternating gradient (NS-FFA) arc. For online modeling of single particle dynamics in CBETA, a customized version of the Tao program, which is based upon the Bmad toolkit, has been developed. This online program, called CBETA-V, is interfaced to CBETA’s EPICS control system. This work describes the online modeling system and initial experience during machine running, as well as subsequent developments since its first implementation.

We discuss the longitudinal beam dynamics in storage rings in the presence of a higher-harmonic cavity (HHC) system for bunch lengthening. First, we review the general conditions for HHC operations, both in active or passive modes, assuming the stability of the system. For uniform filling patterns, a distinction is made between operations with a normal-conducting HHC, where optimal conditions for bunch lengthening can be satisfied, and operations with a super-conducting HHC, where optimal conditions can be met only approximately. Second, we discuss the option to operate the NSLS-II storage ring with a passive, super-conducting third harmonic cavity (3HC) system. The stability and performance of the system in the presence of a gap in the uniform filling, which corresponds to the present mode of operation of the NSLS-II storage ring, is investigated with self-consistent Vlasov–Fokker–Planck simulations performed with the code SPACE. [G. Bassi, A. Blednykh and V. Smaluk, Phys. Rev. Accel. Beams 19, 024401 (2016).]

We developed a highly accurate and fully Maxwellian conformal mapping method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We used this conformal mapping method to calculate the multipole terms of the high voltage quadrupoles in the storage ring of the Muon $g-2$ Experiment (FNAL-E-0989). Next, we demonstrate that an effect where the observed tunes correspond to a voltage that is about $4\%$ higher compared to the voltage to which the Muon $g-2$ quadrupoles are set is explained by the conceptual and quantitative differences between the beam optics quadrupole voltage and the quadrupole voltage at the plates. Completing the methodological framework for field computations, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculated the quadrupole strength falloff and its effective field boundary (EFB) for the Muon $g-2$ quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to $2\%$.

In the ultra-fast electron microscopy community, electron bunches with much smaller longitudinal widths than transverse widths are emitted from the cathode surface. The community has believed that these bunches evolve to a uniform ellipsoid, but recent simulations by our group and others suggest that if the bunch has an initially transverse Gaussian profile, a ring-like density “shock” emerges at the median of the bunch during evolution. To explain these results, we generalized Reed’s 1D fluid model of charged bunch expansion to cylindrical and spherical geometries demonstrating such a shock emerges analytically under these symmetric geometries. Mathematically, the shock in these models occurs when particles more toward the middle “catch-up” to outer particles, and eventually the trajectory of the more central particle crosses-over the outer particle’s trajectory. This cross-over marks the transition from the laminar to nonlaminar regime. However, this theory has been developed for cold-bunches, i.e. bunches of electrons with zero initial momentum. Here, we briefly review this new theory and extend it to the cylindrically- and spherically-symmetric cases that have nonzero initial momentum. This formulation elucidates how charge-dominated bunches may be manipulated to maintain laminar conditions even through focussing of the bunch.