A highly accurate self-consistent particle code to simulate the beam-beam collision in $e^{+}{e}^{-}$ storage rings has been developed. It adopts a method of solving the Poisson equation with an open boundary. The method consists of two steps: assigning the potential on a finite boundary using Green's function and then solving the potential inside the boundary with a fast Poisson solver. Since the solution of Poisson's equation is unique, our solution is exactly the same as the one obtained by simply using Green's function. The method allows us to select a much smaller region of mesh and therefore increase the resolution of the solver. The better resolution makes more accurate the calculation of the dynamics in the core of the beams. The luminosity simulated with this method agrees quantitatively with the measurement for the PEP-II B Factory ring in the linear and nonlinear beam current regimes, demonstrating its predictive capability in detail.

The collimation system for the Spallation Neutron Source accumulator ring is designed for a capture efficiency close to 95% of the proton beam halo, dissipating about 2 kW of beam power. The collimation system consists of a two-stage collimation system (one scraper and two absorbers) cleaning the transverse halo and a beam-in-gap kicker system cleaning the gap residual and longitudinal halo. Preliminary studies indicate that a maximum level of uncontrolled loss of 0.01% of the total beam is achievable. On the other hand, the energy lost in the primary scraper may kick protons outside the rf bucket concentrating uncontrolled losses in areas of maximum dispersion. We use Monte Carlo simulations to clarify some beam dynamic issues that may compromise the high efficiency required. The material interacting with the beam and the shape of the scraper and absorbers have been carefully chosen to maximize the collimation efficiency and to minimize radioactivation. Furthermore, a realistic distribution of losses around the machine is used to identify potential hot areas. Finally, we determine the sensitivity of the collimation efficiency to misalignments and closed orbit errors. This paper describes the latest design of the collimation system and summarizes the results of these numerical studies.

This comment takes up the suggestion of the authors that it could be informative to model the excitation of the bunch shape oscillations using multiparticle tracking. The experiment they report employs an 80% modulation of the rf amplitude at approximately twice the synchrotron frequency; the first-order perturbation analysis given in the paper is not suitable for quantitative comparison to the experimental observations. The tracking code ESME is used to model the conditions of the experiment. The potential well distortion arising from image charge and current and a broadband longitudinal impedance of $Z_{\parallel }/n=6\mathrm{}\Omega$ was considered as an additional perturbation which also varies at the modulating frequency.

Response is made to J. A. MacLachlan's preceding Comment [Phys. Rev. ST Accel. Beams 4, 017001 (2001)]. We are gratified to see that high quality simulations, as presented by MacLachlan, verify our experimental results.

Photoemission and secondary emission are known to give rise to a quasistationary electron cloud inside the beam pipe through a beam-induced multipacting process. We investigate the electron-cloud build up and related effects via computer simulation. In our model, macroparticles representing photoelectrons are emitted synchronously with the passing proton or positron bunch and are subsequently accelerated in the field of the beam. As they hit the beam pipe, new macroelectrons are generated, whose charges are determined by the energy of the incoming particles and by the secondary emission yield of the beam pipe. A quasistationary state of the electron cloud is eventually reached due to space charge. The equilibrium density is used as an input parameter for a second program that analyzes the electron-cloud driven single-bunch instability. The electron cloud simulation also allows the evaluation of the heat load on the cold Large Hadron Collider beam screen, which must stay within the available cooling capacity, and the electron charge deposited on or emitted from the electrodes of the beam-position monitors.

A new computer code for simulation of multipacting phenomenon in axisymmetrical radio frequency structures is presented. The existence of resonant electron trajectories can lead to electron multiplication under given field and surface conditions and can damage these accelerator components or prevent their use. The goal of the simulation is to detect such occurrences. Therefore, large parameter sets have to be tested; a parallel version of the code has been developed to reduce computing time while dealing with highly demanding cases in terms of accuracy and parameter sweeping. Both superconducting cavities and power coupler components have been studied. Simulation results are compared with experimental measurements.

In a circular accelerator, a horizontal-vertical coupling can occur as a purely geometrical effect, in the absence of any coupling device (skew-quadrupole magnets, etc.). The coupling between horizontal and vertical coordinates (and momenta) is represented by a rotation [SO(2)]. The rotation angle is called the twist angle. It comes from the geometry of the reference orbit, and is related to the possible nonintegrability of parallel transport on it. We give an analytic expression of the twist and discuss some simple dynamical consequences of the twist on betatron tune.

Estimates of random field-shape errors induced by cable mispositioning in superconducting magnets are presented and specific applications to the Large Hadron Collider (LHC) main dipoles and quadrupoles are extensively discussed. Numerical simulations obtained with Monte Carlo methods are compared to analytic estimates and are used to interpret the experimental data for the LHC dipole and quadrupole prototypes. The proposed approach can predict the effect of magnet tolerances on geometric components of random field-shape errors, and it is a useful tool to monitor the obtained tolerances during magnet production.

We report the development of different algorithms for the calculation of quality factors of eigenmodes in accelerating cavities, which have resonance frequencies above the cutoff frequency of the beam tubes. The analysis is based on a discretization of such cavity structures by the finite integration technique, and the radiation at the open boundaries is systematically taken into account by different approaches in time and frequency domain. Results indicate that even single cell cavities of the TESLA type show $Q$ values of ${10}^3$ and multicell cavities values in excess of ${10}^4$. Thus these modes may cause considerable beam instabilities. Comparison with the conventional method of analyzing closed cavities and identifying modes with little change in frequency as a function of boundary condition shows qualitative differences. Some modes from the closed cavity model do not exist in the open structure and thus would be misinterpreted as trapped modes when only a closed cavity analysis is employed.

The six-dimensional beam-beam interaction as developed in 1992 by Hirata, Moshammer, and Ruggiero has been extended to include linear coupled motion and an arbitrary crossing plane. The technique of symplectic mapping in the six-dimensional phase space, called synchrobeam mapping, is applied to investigate the beam-beam kick within a solenoid. A linear beam-beam model including coupling is discussed in detail, also in the framework of a six-dimensional symplectic dispersion formalism.

In this paper, we present a series of analytic expressions to predict the beam dynamics in a long linear accelerator. These expressions can be used to model the linac optics, calculate the magnitude of the wakefields, estimate the emittance dilution due to misaligned accelerator components, and estimate the stability and jitter limitations. The analytic expressions are based on the results of simple physics models and are useful to understand the parameter sensitivities. They are also useful when using simple codes or spreadsheets to optimize a linac system.

This paper describes a semianalytical hysteresis model for hard superconductors. The model is based on the critical state model considering the dependency of the critical current density on the varying local field in the superconducting filaments. By combining this hysteresis model with numerical field computation methods, it is possible to calculate the persistent current multipole errors in the magnet taking local saturation effects in the magnetic iron parts into consideration. As an application of the method, the use of soft magnetic iron sheets (coil protection sheets mounted between the coils and the collars) for partial compensation of the multipole errors during the ramping of the magnets is investigated.

Electron bunches of high charge (up to 10 nC) are compressed in length in the Compact Linear Collider Test Facility magnetic chicane to less than 0.4 mm rms. The short bunches radiate coherently in the chicane magnetic field, and the horizontal and longitudinal phase space density distributions are affected. This paper reports the results of beam emittance and momentum measurements. Horizontal and vertical emittances and momentum spectra were measured for different bunch compression factors and bunch charges. In particular, for 10 nC bunches, the mean beam momentum decreased by about 5% while the rms momentum spread increased from 2% to 8%. The experimental results are compared with simulations made with the code TraFiC4.

Tomography is now a very broad topic with a wealth of algorithms for the reconstruction of both qualitative and quantitative images. In an extension in the domain of particle accelerators, one of the simplest algorithms has been modified to take into account the nonlinearity of large-amplitude synchrotron motion. This permits the accurate reconstruction of longitudinal phase space density from one-dimensional bunch profile data. The method is a hybrid one which incorporates particle tracking. Hitherto, a very simple tracking algorithm has been employed because only a brief span of measured profile data is required to build a snapshot of phase space. This is one of the strengths of the method, as tracking for relatively few turns relaxes the precision to which input machine parameters need to be known. The recent addition of longitudinal space charge considerations as an optional refinement of the code is described. Simplicity suggested an approach based on the derivative of bunch shape with the properties of the vacuum chamber parametrized by a single value of distributed reactive impedance and by a geometrical coupling coefficient. This is sufficient to model the dominant collective effects in machines of low to moderate energy. In contrast to simulation codes, binning is not an issue since the profiles to be differentiated are measured ones. The program is written in Fortran 90 with high-performance Fortran extensions for parallel processing. A major effort has been made to identify and remove execution bottlenecks, for example, by reducing floating-point calculations and recoding slow intrinsic functions. A pointerlike mechanism which avoids the problems associated with pointers and parallel processing has been implemented. This is required to handle the large, sparse matrices that the algorithm employs. Results obtained with and without the inclusion of space charge are presented and compared for proton beams in the CERN protron synchrotron booster. Comparisons of execution times on different platforms are presented and the chosen solution for our application program, which uses a dual processor PC for the number crunching, is described.

Residual vertical dispersion can be a significant performance limitation for the LEP collider because the associated vertical emittance increase reduces the luminosity of the machine. To make the search for orbits yielding small vertical emittances fast and deterministic, a simultaneous correction of the closed orbit and the residual dispersion was implemented at LEP. The principle of the correction and the resulting performance gains are discussed.

Air-core printed-circuit (PC) quadrupoles and dipoles have been developed for the University of Maryland electron ring, currently under construction. The quadrupoles and dipoles are characterized by very small magnetic fields (about 15 G at the aperture edge) and small aspect ratios (length/diameter $<1$). We review the theory behind the design of the PC lenses and bending elements, and present general expressions for estimating the values of integrated field and integrated field gradient as functions of design parameters. The new quadrupole magnet represents an improvement over an earlier version which was based on an empirical approach. Further, we summarize the results of multipole content of the magnet fields as measured with a rotating coil apparatus of special construction. The results are compared with calculations with an iron-free magnetics code and are related to different types of errors in the manufacture and assembly of the PC magnets.

A cw high power linear accelerator can only work with very low particle losses and structure activation. At low energy, the radio frequency quadrupole (RFQ) is an accelerator element that is very sensitive to losses. To design this structure, a good understanding of the beam dynamics is required. Generally, the reference code PARMTEQM is enough to design the accelerator. TOUTATIS has been written with the goals of cross-checking results and obtaining more reliable dynamics. This paper relates the different numerical methods used in the code. It is time based, using multigrids methods and adaptive mesh for a fine description of the forces without being time consuming. The field is calculated through a Poisson solver and the vanes are fully described, allowing it to properly simulate the coupling gaps and the RFQs extremities. Theoretical and experimental tests are also described and show a good agreement between simulations and reference cases.

Recently there has been renewed interest in the influence of fringe fields on particle dynamics, due to studies that revealed their importance in some cases, as, for example, the proposed Neutrino Factory and muon colliders. In this paper, we present a systematic study of generic fringe field effects. Using as an example a lattice of the proposed Neutrino Factory, we show that fringe fields influence the dynamics of particles at all orders, starting with the linear motion. It is found that the widely used sharp cutoff approximation leads to divergences regardless of the specific fall-off shape of the fields. The results suggest that a careful consideration of fringe field effects in the design stage of small machines for large emittances is always recommended.

The soft-Gaussian approximation is often employed in computer simulations of strong-strong beam-beam interactions in storage rings. Its defect on the coherent oscillation frequency is pointed out and a possible remedy using Hermitian polynominals is discussed.