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The problem of dynamics optimization for charged particle beams in RFQ accelerators is discussed in this article. The problems of the optimization of particle capture into the acceleration mode are analyzed. Accelerators with low injection energies are considered.

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.

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.

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 simulation and optimization of an electron injector that operates a high-intensity polarized electron beam are presented. The electron injector would provide a single-bunch electron beam at a repetition rate of 1 Hz with a bunch charge of 10 nC, kinetic energy of 400 MeV and an electron spin polarization of 85$\%$. A direct current (DC) high-voltage photogun was employed to produce the 10 nC polarized electron beam with a kinetic energy of 350 keV, beam diameter of 1.44 cm (Gaussian distribution with $\sigma$ of 0.36 cm) and full bunch length of 1.3 ns (Gaussian distribution with $\sigma$ of 0.325 ns). The beam was compressed to 6 ps of RMS bunch length using one 114.24 MHz standing-wave sub-harmonic buncher, two 571.2 MHz standing-wave sub-harmonic bunchers and one 2.856 GHz traveling-wave buncher. The beam was accelerated finally to 400 MeV via eight 2.856 GHz traveling-wave Linacs. We performed the beam simulation, and the simulated results showed that the optimized RMS bunch length was 4.65 ps, RMS relative energy spread was 0.48$\%$ and normalized RMS transverse emittance was $30.13mm\cdot mrad$ at a beam energy of 400 MeV.

A Low Energy Beam Transport (LEBT) has been designed and optimized to inject both a 50keV/u, 50mA $H^{+}$ beam and a 20keV/u, 1mA ${}^3{\mathrm{\text{He}}}^{2+}$ beam into the downstream Radio Frequency Quadrupole (RFQ) at the Lanzhou Light Ion Cancer Therapy Facility (LLICTF). The overall aim is to increase the operation efficiency and simplify the complexity of the LLICTF. This dual-beam operation mode creates phase space distortions as a result of the intense space charge, and causes transverse asymmetry due to a mixture of solenoids and a dipole magnet. To mitigate the phase space distortions of the high-intensity proton beam, two beam scrapers have been placed at locations determined through careful analysis of the space charge forces and external focusing force of the beam. Additionally, the Particle Swarm Optimization (PSO) algorithm has been utilized to obtain an approximate solution for the dipole parameters, which helps maintain the symmetry of different symmetrical beams passing through it. Optics designs that implement these solutions are verified with multi-particle simulations.

In classical electrodynamics, the well-known Lorentz force law falls short of providing a satisfactory result for the trajectory of point-like charged particles when considering that particle’s own self-force. While there have been many historical attempts, Gralla, Harte and Wald developed a new model for classical charged particles that is free from pathologies while being consistent with Maxwell’s equations and conserves stress-energy. Expanding upon this approach, we derive a relativistically correct, modified Lorentz force law in vector form, which includes radiation reaction, and spin- and magnetic moment-dependent correction terms, suitable to be included in classical electrodynamics lectures and beam dynamics simulation tools. As by-products we obtain evolution equations for mass, spin angular momentum and the radiated power. We compare the new equations to the classical ones and use the new equations to conduct numerical simulations, showing that the results are free of any nonphysical artifacts, and which might be possible to test in future experiments at particle accelerators. The new equations foster improved insight into beam dynamics.

Proton therapy is an advanced particle radiotherapy technology. Lanzhou University recently designed a new linear accelerator system for proton therapy. In the system, a 750MHz radio frequency quadrupole (RFQ) accelerator was used to complete the initial acceleration of particles. The RFQ can accelerate a proton beam intensity of 1mA from 30keV to 3MeV kinetic energy within a length of 2m. This paper aims to complete the physical design of this high-frequency RFQ, including beam dynamics design, radio frequency (RF) design, and multiphysics analysis. In the dynamics design, a novel design strategy for controlling the longitudinal emittance was adopted to improve the output beam quality of the RFQ. The simulation results showed that the RFQ outlet longitudinal emittance was controlled below 0.1 $\pi$ mm mrad. In the RF design, the pi-mold rod structure was applied to RFQ at such high frequencies for the first time. After the whole cavity simulation, it was found that the separation between the operating frequency of the RFQ cavity and its closest dipole mode reached 58.3MHz, and its quality factor reached 6352. Finally, a multiphysics analysis was performed. The analysis showed that the maximum temperature rise of the RFQ cavity was less than $2.5^{\circ }$C and the frequency drift due to heating was 18kHz. This paper presents the current status of the design of this new linear accelerator. Compared with previous RFQs, the new RFQ has a very high transmission efficiency and higher RF stability.

This article discusses the main building blocks of a superconducting (SC) linac, the choice of SC resonators, their frequencies, accelerating gradients and apertures, focusing structures, practical aspects of cryomodule design, and concepts to minimize the heat load into the cryogenic system. It starts with an overview of design concepts for all types of hadron linacs differentiated by duty cycle (pulsed or continuous wave) or by the type of ion species (protons, H^-, and ions) being accelerated. Design concepts are detailed for SC linacs in application to both light ion (proton, deuteron) and heavy ion linacs. The physics design of SC linacs, including transverse and longitudinal lattice designs, matching between different accelerating–focusing lattices, and transition from NC to SC sections, is detailed. Design of high-intensity SC linacs for light ions, methods for the reduction of beam losses, preventing beam halo formation, and the effect of HOMs and errors on beam quality are discussed. Examples are taken from existing designs of continuous wave (CW) heavy ion linacs and high-intensity pulsed or CW proton linacs. Finally, we review ongoing R&D work toward high-power SC linacs for various applications.

The aim of our contribution is to present the analogy between the Hamiltonian Analytical Optics and the Beam Dynamics. We follow the Hamilton’s memoir by means of a revision of the text, where it is discussed an application of Lagrange δ-calculus. After an philological introduction we consider the correspondence between the trajectory of particles in a circular accelerator and rays in an optical system or a wave guide. An experiment, with a laser on an optical system with mirrors and lenses simulating a FODO cell, showed that the ray bundles move on an ellipse in agreement with the linear theory. We consider here a cylindrical optical fiber with a radially varying refraction index such that n²(r) is quadratic to reproduce the betatronic beam dynamics between two subsequent nonlinear equally spaced thin elements. The nonlinear effect can be introduced with short nonlinear insertions where n² depends on the transverse coordinates x; y and is polynomial of degree three or higher. The Poincaré map for the rays is approximated by the Hénon map, which describes the beam dynamics, provided that d the rays do not reach the boundary where the refections would introduce a different dynamic behavior. Indeed the billiard like dynamics of rays reflecting at boundaries has been analyzed in the literature for laser beams propagating in a waveguide or an optical cavity and the chaotic behaviour was found to be relevant for applications, but the correspondence with beam dynamics is lost.

Celestial and beam dynamics are similar and share the same methods for the stability analysis. The perturbation theory is the main analytical tool and its version for the symplectic maps, which describe the transverse motion in circular accelerators, are the Birkhoff normal forms. They allow to evaluate the dynamic aperture when the frequency is close to an unstable resonance and to estimate the stability time when it is far from low order resonances. In celestial mechanics the numerical analysis of stability is usually based on the Lyapunov error (for instance, the Fast Lyapunov Indicator). In the case of symplectic maps the forward and reversibility errors induced by random perturbations have been proposed. The use of non-resonant normal forms allows to establish the power law asymptotic behavior of errors with the corresponding pre-factors. The error plots in a phase plane, obtained by fixing the missing coordinates, allow to establish the stability properties of each orbit. The reversibility error due to round-off jointly with the Lyapunov error provide an adequate stability pattern for the Hénon map and their use is suggested for more realistic models such as the one turn map in a ring.

Caustics are a common occurrence in optics, describing the bright lines seen in a well-lit coffee cup or the dancing networks of light seen at the bottom of a swimming pool on a sunny day. Recently, caustic formation in electron trajectories was identified as the mechanism driving strong current modulations in accelerated charged-particle beams. Under certain conditions, neighboring electron trajectories coalesce to form caustics, resulting in current spikes. Caustic lines and surfaces are regions of maximum electron density, and are often witnessed in accelerator physics as folds in the transverse or longitudinal phase space distribution. Knowledge of caustics can allow us to shape the longitudinal beam shape to our advantage.

This article discusses the main building blocks of a superconducting (SC) linac, the choice of SC resonators, their frequencies, accelerating gradients and apertures, focusing structures, practical aspects of cryomodule design, and concepts to minimize the heat load into the cryogenic system. It starts with an overview of design concepts for all types of hadron linacs differentiated by duty cycle (pulsed or continuous wave) or by the type of ion species (protons, H⁻, and ions) being accelerated. Design concepts are detailed for SC linacs in application to both light ion (proton, deuteron) and heavy ion linacs. The physics design of SC linacs, including transverse and longitudinal lattice designs, matching between different accelerating-focusing lattices, and transition from NC to SC sections, is detailed. Design of high-intensity SC linacs for light ions, methods for the reduction of beam losses, preventing beam halo formation, and the effect of HOMs and errors on beam quality are discussed. Examples are taken from existing designs of continuous wave (CW) heavy ion linacs and high-intensity pulsed or CW proton linacs. Finally, we review ongoing R&D work toward high-power SC linacs for various applications.