This study explores various neural network approaches for simulating beam dynamics, with a particular focus on non-linear space charge effects. We introduce a convolutional encoder-decoder architecture that incorporates skip connections to predict transversal electric fields. The model demonstrates robust performance, achieving a root mean squared error (RMSE) of $0.5\%$ within just a few minutes of training. Furthermore, this paper explores the feasibility of replacing traditional ellipsoidal methods with Gaussian envelope models for improved non-linear space-charge calculations. Our findings indicate that these advancements could provide a more efficient alternative to numerical space-charge methods in beam dynamics simulations.

Crab crossings are designed to increase the luminosity of accelerators by ensuring beam interactions are closer to a head on collision. One will be implemented at the Electron Ion Collider (EIC) at Brookhaven National Laboratory. It is then important to examine how the crab cavity will affect beam dynamics at the EIC. Methods such as Frequency Map Analysis (FMA) have been shown to be helpful in examining the phase space of accelerators in order to find properties such as resonances and the dynamic aperture. An alternative to such methods is an iterative method based on square matrix method that has been shown to reveal similar properties as FMA while reducing the computational power needed*,**. This method has been applied to the crab crossing scheme in order to find and explain effects of the higher order mode of crab cavities on the particle dynamics of the EIC.

For the Recycler Ring at Fermilab, space charge tune shifts of almost 0.1 will have to be dealt with under the Proton Improvement Plan (PIP-II) framework. This will lead to the excitation of third order resonances. The minimization of Resonance Driving Terms (RDTs) allows to mitigate the harmful effect of these betatron resonances. Past work has shown that previously-installed sextupoles can compensate the RDTs of individual third order resonance lines, thus reducing particle losses in these operational regimes. Nevertheless, trying to compensate multiple resonances of the same order simultaneously with these existing sextupoles is limited due to current constraints in the magnets. The following study showcases the procedure to install additional sextupoles in order to aid the compensation of multiple resonances. This includes the optimization of the new sextupoles' locations in order to cancel out multiple RDTs while minimizing the currents needed. This is followed by a verification of their effectiveness by means of the RDT response matrix.

Beam lengthening is an effective and commonly used method to improving the beamlife of storage rings. Based on the previously proposed design of a room temperature conducting bimodal RF cavity. we conducted relevant dynamic simulations. Tracking study on a simulated storage ring lattice with the beam energy of 2 GeV and the synchronous radiation energy of 357 KeV, the results show that, the bimodal RF cavity which contains an accelerating field and a third harmonic field can effectively lengthen beam length, the beam lengthening effect similar to the double RF system which consists of main RF cavity and third harmonic cavity.

In December 2021, damage to a couple of RHIC power supplies forced one of two Siberian Snakes in the Blue ring to operate as a partial Siberian Snake and with a different snake axis of rotation. The time-averaged polarization for that run actually ended up higher than in the Yellow ring, after casting the undamaged snake as a partial snake as well. In this work, we simulate polarization transmission through a series of increasingly realistic models of the Blue ring in the “dangerous region” of polarization loss. At first the bare lattice has a perfect closed-orbit and ideal magnet strengths. Then the measured magnet-to-magnet field strength variations were added to the lattice. Finally, the six Interaction Region 5mm closed orbit bumps were implemented. Each of these model lattices compared the use of a pair of partial snakes against a pair of a full snakes, and in simulations with realistic emittances, realistic polarization losses were not reproducible without inclusion of nonzero RMS lattice misalignments.

Transverse resonance island buckets (TRIBs) near a third-order resonance are observed at the Cornell Electron Storage Ring with a newly-designed lattice. Hamiltonian perturbation theory and map-based PTC method have been successfully implemented to design the TRIBs lattice and find the fixed points near the third-order resonance. For even higher order resonances which are intrinsically weaker, the effects of radiation damping and excitation may not be negligible in the lepton machine such that the Hamiltonian approach could break down. In this paper, we study TRIBs near the fourth-order resonance with tracking simulations, which predict the existence of islands. Then experimental observations confirmed the islands formation near the fourth-order resonance. The positions of the fixed points extracted from experimental results agree reasonably well with those predictions from both tracking and PTC calculations. A unique method to push all particles into a single island is also demonstrated and discussed.

Dynamic Aperture (DA) studies, based on single-particle tracking simulations including important non-linear fields such as beam-beam effects, have played a crucial role in guiding the operation of the Large Hadron Collider (LHC) in proton-proton collisions. The correspondence between DA computed through simulations and the actual beam lifetime during operation has been established for proton beams through dedicated experiments at the LHC. However, such an approach has not yet been applied to the Pb ion operation of the accelerator, as the simulation tools have not been rigorously benchmarked against experimental data yet. The present paper presents the simulation studies and experimental tests performed to establish the correlation between DA and beam lifetime for ions. The main focus lies on exploring the beam-beam limit when the crossing angle is significantly reduced in all LHC experiments as compared to the nominal configuration. This approach opens the possibility to operate with reduced crossing angles or reduced $\beta^*$ within the beam-beam limit, potentially leading to an enhanced performance of the accelerator with ions.

A general theory of symplectic tracking under the influence of space charge force is not yet available, even if some specific solution was proposed [1, 2]. In this paper we will first review how the pull-back of the Lie transform can be used to self-transport the beam distribution and its associated electromagnetic potential under the effect of the space-charge. We will then classify the functions suitable for an iterative algorithm with the Lie transform. Those functions will be used to describe the electromagnetic potential of the space charge.

This study explores various neural network approaches for simulating beam dynamics, with a particular focus on non-linear space charge effects. We introduce a convolutional encoder-decoder architecture that incorporates skip connections to predict transversal electric fields. The model demonstrates robust performance, achieving a root mean squared error (RMSE) of $0.5\%$ within just a few minutes of training. Furthermore, this paper explores the feasibility of replacing traditional ellipsoidal methods with Gaussian envelope models for improved non-linear space-charge calculations. Our findings indicate that these advancements could provide a more efficient alternative to numerical space-charge methods in beam dynamics simulations.

Recently exact Hamiltonian integrators have been added for drift, multipoles and dipoles in Accelerator Toolbox. This paper reports the tracking simulations benchmarks performed to compare with the results provided by MADX-PTC for four lattices: FODO, DBA, H7BA and FCC-HFD@Z. Tracking times are also reported for completeness. The agreement in 4D is complete while small discrepancies persist for 6D tracking. Fringe fields models were not included in the comparison and are known to be different for the two codes.

Crab crossings are designed to increase the luminosity of accelerators by ensuring beam interactions are closer to a head on collision. One will be implemented at the Electron Ion Collider (EIC) at Brookhaven National Laboratory. It is then important to examine how the crab cavity will affect beam dynamics at the EIC. Methods such as Frequency Map Analysis (FMA) have been shown to be helpful in examining the phase space of accelerators in order to find properties such as resonances and the dynamic aperture. An alternative to such methods is an iterative method based on square matrix method that has been shown to reveal similar properties as FMA while reducing the computational power needed*,**. This method has been applied to the crab crossing scheme in order to find and explain effects of the higher order mode of crab cavities on the particle dynamics of the EIC.

In this work, we investigate the transverse dynamics of a single particle in a model integrable accelerator lattice, based on a McMillan axially symmetric electron lens. Although the McMillan e-lens has been considered as a device potentially capable of mitigating collective space charge forces, some of its fundamental properties have not been described yet. The main goal of our work is to close this gap and understand the limitations and potential of this device. We classify possible regimes with stable trajectories and provide set of canonical action-angle variables, along with an evaluation of the dynamical aperture, Poincar\'e rotation numbers as functions of amplitudes, and spread in nonlinear tunes.

In this study, we reassess the dynamics within a simple accelerator lattice featuring a single degree of freedom and incorporating a sextupole magnet. In the initial segment, we revisit the H\'enon quadratic map, a representation of a general transformation with quadratic nonlinearity. In the subsequent section, we unveil that a conventional sextupole is essentially a composite structure, comprising an integrable McMillan sextupole and octupole, along with non-integrable corrections of higher orders. This fresh perspective sheds light on the fundamental nature of the sextupole magnet, providing a more nuanced understanding of its far-from-trivial chaotic dynamics. Importantly, it enables the description of driving terms of the second and third orders and introduces associated nonlinear Courant-Snyder invariant.

HZB is currently designing the lattice for BESSY III, the successor of the 1.7 GeV electron storage ring running in Berlin since 1998. HZB follows a deterministic lattice design strategy, where the natural substructures of a non-hybrid MBA lattice are optimized separately. The substructures consist of only a few parameters, that can be derived from the strategic goals of the project. In the next step, the focusing and de-focusing sextupole families are split up, to optimize the longitudinal and the transverse apertures. The paper compares two approaches to select the optimal sextupole strengths. The first one is multi-objective Bayesian optimization, where the dynamic aperture volume from tracking simulations is used as an objective to be maximized. The second approach does not involve tracking and minimizes the geometric and chromatic resonance driving terms. The comparison of the two results includes their quality in terms of the size of the achievable 3D dynamic aperture and the computational effort involved.

The Advanced Light Source Upgrade (ALS-U) is a 2 GeV high-brightness, nine-bend achromat storage ring, designed to reduce the natural emittance relative to the existing ALS by a factor of 20 for improved x-ray coherent flux and brightness. The upgrade includes the installation of an accumulator ring of the same energy as, and slightly smaller circumference than, the storage ring. The bending dipoles provide special challenges for accurate symplectic modeling, such as the combination of large sagitta and magnet narrow vertical aperture (in the accumulator ring) and overlapping fringe fields (in the main ring). We describe a procedure for the calculation of symplectic maps for the ALS-U dipoles using robust surface-fitting methods based on 3D finite-element field data, including a discussion of vector potential gauge choice and model-dependent effects on the lattice chromaticity.

For the Recycler Ring at Fermilab, space charge tune shifts of almost 0.1 will have to be dealt with under the Proton Improvement Plan (PIP-II) framework. This will lead to the excitation of third order resonances. The minimization of Resonance Driving Terms (RDTs) allows to mitigate the harmful effect of these betatron resonances. Past work has shown that previously-installed sextupoles can compensate the RDTs of individual third order resonance lines, thus reducing particle losses in these operational regimes. Nevertheless, trying to compensate multiple resonances of the same order simultaneously with these existing sextupoles is limited due to current constraints in the magnets. The following study showcases the procedure to install additional sextupoles in order to aid the compensation of multiple resonances. This includes the optimization of the new sextupoles' locations in order to cancel out multiple RDTs while minimizing the currents needed. This is followed by a verification of their effectiveness by means of the RDT response matrix.

The CERN Proton Synchrotron Booster (PSB) operation involves the crossing of multiple resonance lines in the tune diagram. Loss maps from dynamic tune scans are a helpful way to visualize and quantify the strength of such resonances. Sextupole and octupole correctors can be used in order to partially or fully compensate multiple resonance lines, i.e., third and fourth order lines. The following work explores the application of advanced optimization algorithms such as Bayesian Optimization and Bound Optimization By Quadratic Approximation (BOBYQA) in order to compensate these resonance lines with available correctors.

The physical realization of integrable Hamiltonian dynamics provides promising avenues for investigations of new particle accelerators, best demonstrated by the Integrable Optics Test Accelerator (IOTA) at Fermilab. The core concept of IOTA centers around the results of the Danilov-Nagaitsev paper, where taking the paraxial approximation of the Hamiltonian for a charged particle can lead to a completely integrable system for a charged particle in the transverse plane. However, certain generalizations of that paper fail to provide similar results. We provide insights into some reasons for failure, as well as discuss a set-up for establishing a 6D integrable Hamiltonian system, in order to include the possibility of acceleration.

Beam lengthening is an effective and commonly used method to improving the beamlife of storage rings. Based on the previously proposed design of a room temperature conducting bimodal RF cavity, we conducted relevant dynamic simulations. Tracking study on a simulated storage ring lattice with the beam energy of 2 GeV and the synchronous radiation energy of 357 KeV, the results show that, the bimodal RF cavity which contains an accelerating field and a third harmonic field can effectively lengthen beam length, the beam lengthening effect similar to the double RF system which consists of main RF cavity and third harmonic cavity.

In a generalized longitudinal strong focusing (GLSF) approach employed in steady-state microbunching (SSMB) storage rings, the objective is to achieve complete cancellation of modulations by ensuring that a particle's longitudinal position remains unchanged after passing through both modulators. This requires effective control over the deviation in longitudinal position, which arises from lattice nonlinearities. This paper derives analytical formulations for the mean and standard deviation of the particle position deviation, expressed in terms of the beam and lattice-dependent parameters. The aim is to provide valuable insights into the system's behavior and enable optimization of the GLSF unit's performance.