The flip-flop effect with the linearized beam-beam force is formulated through self-consistent $\beta$ functions and equilibrium emittances which are both affected by collision. We give the results of two models of emittance dependence. The effect of finite bunch length is also discussed.

A weak-strong simulation code is used to study the single-particle stability in the presence of triplet field errors, head-on collisions, and long-range beam-beam interactions at the Large Hadron Collider. We present the dependence of the simulated transverse diffusion rate on various parameters, such as starting amplitude, working point in tune diagram, crossing angle, beta function at the interaction points (IPs), beam current, triplet nonlinearities, tune modulation, and a transverse offset at one of two IPs. For several examples, we perform a frequency map analysis à la Laskar, to obtain tune footprints and the tune variation in time. A cursory look at the effect of a Möbius lattice is also reported.

This paper will explore the possibility of the formation of localized modes in the coupled transverse oscillations of bunches in an accelerator. Such modes are characterized by a large amplitude excitation of one of the bunches, with little or no excitation of the other bunches. They are very similar to intrinsic localized modes predicted to appear in lattice excitations of anharmonic condensed matter systems. The existence of the phenomenon in accelerators requires long-range coupling between the bunches, and quartic terms in the effective potential experienced by the bunches. The discussion in this paper considers the specific cases of a long-range coupling due to the resistive wall impedance, and nonlinearities in the transverse force caused by octupoles. The general equations for the localized mode frequencies and amplitudes are derived. The equations are applied to the specific cases of 9 and 21 equally spaced, equally populated bunches in the Cornell electron storage ring. For both cases, with the currently available octupole strength, and with currents of 8 mA per bunch, localized modes with amplitudes of about 2 mm, and with frequency shifts of $50–100\mathrm{s}{}^{-1\mathrm{}}$ relative to the linear coupled-bunch mode frequencies, may be possible.

Single bunch instabilities for the $50\mathrm{GeV}\times 50\mathrm{GeV}$ muon collider are discussed. An impedance budget of the collider is estimated. A phase-slip factor of $|\eta |=1\mathrm{}\times {10}^{-6\mathrm{}}$ is desired to avoid excessive rf systems. Potential-well distortion of a smooth bunch can be compensated by rf cavities. Accumulated growth in energy due to imperfections and noises in the muon bunch can be reduced by smoothing the bunch distribution before injection. The growth due longitudinal microwave instability is small because of the compensated rf cavities, the finite lifetime of the muons, and the choice of a small $|\eta |$. Beamloadings in the compensating rf cavities are large and suitable feed-forward cancellation is required. Transverse microwave instability can be damped by chromaticities and octupoles. Beam breakup can be cured by Balakin-Novokhatsky-Smirnov damping in principle, but is nontrivial in practice. When beam breakup is small, it can possibly be damped by a betatron tune spread in the beam.

Space charge debunching is a major issue for future high-gradient, high-frequency accelerator techniques. Space charge will set limits on the maximum six-dimensional phase space density obtainable in optical or plasma based accelerators. These accelerators will have short microbunches a fraction of an optical wavelength, in which space charge debunching is unmitigated by two-dimensional effects. The element of an accelerator system most vulnerable to space charge is the drift space between the prebuncher and the acceleration sections. A self-consistent model coupling the energy and phase modulation in the drift space is developed. It is shown that both space charge effects and coherent energy spread can be offset by adjusting the prebuncher and beam optics parameters. In the accelerator sections, the large relativistically corrected inertia together with two-dimensional effects combine to make space charge debunching unimportant. The analytical results compare well with PARMELA code simulations.

We describe an instability diagnostic that exploits the information contained in the angular evolution of coupled-bunch oscillations in phase space. In addition to enabling measurement of coherent tunes and bunch tunes with accuracy of a few hertz, phase space tracking allows new kinds of comparisons between instability theory and experiment. Phase space evolution of bunches participating in a low-threshold vertical instability in the high energy ring of the Stanford Linear Accelerator Center B factory (PEP-II) is used to distinguish between the fast beam-ion instability and conventional instabilities. Tracking of longitudinal instabilities at the LBNL Advanced Light Source and PEP-II is used to measure coherent tunes and gain new insights into uneven-fill instabilities.

The status of the research on muon colliders is discussed and plans are outlined for future theoretical and experimental studies. Besides work on the parameters of a 3–4 and 0.5 TeV center-of-mass (COM) energy collider, many studies are now concentrating on a machine near 0.1 TeV (COM) that could be a factory for the $s$-channel production of Higgs particles. We discuss the research on the various components in such muon colliders, starting from the proton accelerator needed to generate pions from a heavy-$Z$ target and proceeding through the phase rotation and decay ($\pi \to \mu {\nu }_{\mu }$) channel, muon cooling, acceleration, storage in a collider ring, and the collider detector. We also present theoretical and experimental R&D plans for the next several years that should lead to a better understanding of the design and feasibility issues for all of the components. This report is an update of the progress on the research and development since the feasibility study of muon colliders presented at the Snowmass '96 Workshop [R. B. Palmer, A. Sessler, and A. Tollestrup, Proceedings of the 1996 DPF/DPB Summer Study on High-Energy Physics (Stanford Linear Accelerator Center, Menlo Park, CA, 1997)].

The beam-beam interaction in the Tevatron collider sets limits on bunch intensity and luminosity. These limits are caused by a tune spread in each bunch which is mostly due to head-on collisions, but there is also a bunch-to-bunch tune spread due to parasitic collisions in multibunch operation. We propose to compensate these effects with the use of a countertraveling electron beam, and we present general considerations and physics limitations of this technique.

This paper considers an intense non-neutral ion beam propagating in the $z$ direction through a periodic-focusing quadrupole or solenoidal field with transverse focusing force, $-\left[{\kappa }_x\right(s)x{\hat{\mathbf{e}}}_x+{\kappa }_y(s\left)y{\hat{\mathbf{e}}}_y\right]$, on the beam ions. Here, $s={\beta }_b\mathrm{ct}$ is the axial coordinate, $({\gamma }_b-1\mathrm{})m_b{c}^2$ is the directed axial kinetic energy of the beam ions, and the (oscillatory) lattice coefficients satisfy ${\kappa }_x(s+S)={\kappa }_x\left(s\right)$ and ${\kappa }_y(s+S)={\kappa }_y\left(s\right)$, where $S=\mathrm{const}$ is the periodicity length of the focusing field. The theoretical model employs the Vlasov-Maxwell equations to describe the nonlinear evolution of the distribution function $f_b({x,y,x}^{\prime }{,y}^{\prime },s)$ and the (normalized) self-field potential $\psi (x,y,s)$ in the transverse laboratory-frame phase space $\left({x,y,x}^{\prime }{,y}^{\prime }\right)$. Here, $\hat{H}({x,y,x}^{\prime },y^{\prime },s)=\left(1/2\right)\left(x^{\prime 2}{+y}^{\prime 2}\right)+\left(1/2\right)\left[{\kappa }_x\right(s)x^2+{\kappa }_y(s\left)y^2\right]+\psi (x,y,s)$ is the (dimensionless) Hamiltonian for particle motion in the applied field plus self-field configurations, where $(x,y)$ and $\left(x^{\prime }{,y}^{\prime }\right)$ are the transverse displacement and velocity components, respectively, and $\psi (x,y,s)$ is the self-field potential. The Hamiltonian is formally assumed to be of order $\epsilon$, a small dimensionless parameter proportional to the characteristic strength of the focusing field as measured by the lattice coefficients ${\kappa }_x\left(s\right)$ and ${\kappa }_y\left(s\right)$. Using a third-order Hamiltonian averaging technique developed by P. J. Channell [Phys. Plasmas 6, 982 (1999)], a canonical transformation is employed that utilizes an expanded generating function that transforms away the rapidly oscillating terms. This leads to a Hamiltonian, $H(\tilde{X},\tilde{Y},{\tilde{X}}^{\prime },{\tilde{Y}}^{\prime },s)=\left(1/2\right)({\tilde{X}}^{\prime 2}+{\tilde{Y}}^{\prime 2})+\left(1/2\right){\kappa }_f({\tilde{X}}^2+{\tilde{Y}}^2)+\psi (\tilde{X},\tilde{Y},s)$, correct to order ${\epsilon }^3$ in the “slow” transformed variables $(\tilde{X},\tilde{Y},{\tilde{X}}^{\prime },{\tilde{Y}}^{\prime })$. Here, the transverse focusing coefficient in the transformed variables satisfies ${\kappa }_f=\mathrm{const}$, and the asymptotic expansion procedure is expected to be valid for a sufficiently small phase advance ($\sigma <\pi /3=60°$, say). Properties of axisymmetric beam equilibrium distribution functions, $F_b^0\left(H^0\right)$, with $\partial /\partial s=0=\partial /\partial \Theta$, are calculated in the transformed variables, and the results are transformed back to the laboratory frame. Corresponding properties of the periodically focused distribution function $f_b({x,y,x}^{\prime }{,y}^{\prime },s)$ are calculated correct to order ${\epsilon }^3$ in the laboratory frame, including statistical averages such as the mean-square beam dimensions, $〈x^2〉\left(s\right)$ and $〈y^2〉\left(s\right)$, the unnormalized transverse beam emittances, ${\epsilon }_x\left(s\right)$ and ${\epsilon }_y\left(s\right)$, the self-field potential, $\psi (x,y,s)$, the number density of beam particles, $n_b(x,y,s)$, and the transverse flow velocity, ${\mathbf{V}}_b(x,y,s)$. As expected, the beam cross section in the laboratory frame is a pulsating ellipse for the case of a periodic-focusing quadrupole field or a pulsating circular cross section for the case of a periodic-focusing solenoidal field.

A formalism for describing the coupled two-dimensional motion of high energy particle beams in a storage ring is developed and extended to circumstances where the coupling is very strong, such as for the Möbius twist accelerator.

In the context of future accelerators and, in particular, the beam vacuum of the Large Hadron Collider (LHC), a 27 km circumference proton collider to be built at CERN, VUV synchrotron radiation (SR) has been used to study both qualitatively and quantitatively candidate vacuum chamber materials. Emphasis is given to show that angle and energy resolved photoemission is an extremely powerful tool to address important issues relevant to the LHC, such as the emission of electrons that contributes to the creation of an electron cloud which may cause serious beam instabilities and unmanageable heat loads on the cryogenic system. Here we present not only the measured photoelectron yields from the proposed materials, prepared on an industrial scale, but also the energy and in some cases the angular dependence of the emitted electrons when excited with either a white light (WL) spectrum, simulating that in the arcs of the LHC, or monochromatic light in the photon energy range of interest. The effects on the materials examined of WL irradiation and /or ion sputtering, simulating the SR and ion bombardment expected in the LHC, were investigated. The studied samples exhibited significant modifications, in terms of electron emission, when exposed to the WL spectrum from the BESSY Toroidal Grating Monochromator beam line. Moreover, annealing and ion bombardment also induce substantial changes to the surface thereby indicating that such surfaces would not have a constant electron emission during machine operation. Such characteristics may be an important issue to define the surface properties of the LHC vacuum chamber material and are presented in detail for the various samples analyzed. It should be noted that all the measurements presented here were recorded at room temperature, whereas the majority of the LHC vacuum system will be maintained at temperatures below 20 K. The results cannot therefore be directly applied to these sections of the machine until measurements at cryogenic temperatures, i.e., in the presence of cryosorbed gas layers, are obtained. However, these results are directly relevant to all the warm regions of the LHC vacuum system, such as the experimental vacuum chambers and warm element vacuum chambers in the insertion regions.

The inductance of the vacuum chamber of the Proton Storage Ring at Los Alamos National Laboratory was intentionally increased by the introduction of ferrite rings to counteract the longitudinal space-charge effect of the intense beam. The magnetic permeability of the ferrite could be adjusted by introducing current into solenoids wound around the ferrite. Results show that the minimum rf voltage necessary to stabilize the beam against $e-p$ instability may be reduced over that previously measured. The injected bunch length was observed to be longer when the ferrite was heavily biased so that its effect was reduced.

In an x-ray free electron laser like the Linear Coherent Light Source being designed at SLAC, electron bunches about $70\mu \mathrm{m}$ FWHM long are sent into a beam tube only 5 mm in internal diameter and more than 100 m in length. Because of the surface roughness of the beam tube, wakefields can be generated that catch up to the bunch and interact with it, causing energy spread and emittance growth. The strength of this effect depends on the details of the roughness of the surface. We present here a study in which the roughness of the beam tube was measured and the longitudinal impedance of the tube was calculated. Our result shows that a commercially available beam tube can be made smooth enough so the resulting wakefield effects are within the tolerance determined by the requirement that the induced relative energy spread of the beam be less than $5\times {10}^{-4\mathrm{}}$.

We examine the transverse impedance of a periodic array of cavities in a beam pipe at high frequency. The calculation is an extension of a previous one for the longitudinal impedance of a periodic array of azimuthally symmetric pillboxes, for which only TM modes were needed. In the present case, we must include TE modes as well. In addition, we extend the applicability of the previous calculation by including an extra term in the coupling kernel so that the results are valid for all values of the ratio of the cavity length to the period of the structure (all values of the ratio of iris thickness to structure period). In spite of the presence of TE modes, we find that the high frequency limit of the transverse impedance is simply $\left(2/{\mathrm{ka}}^2\right)$ times the corresponding limit of the longitudinal impedance, just as it is for the resistive wall impedances, a relation which occurs frequently for azimuthally symmetric structures. Finally, we present numerical results as well as approximate expressions for the impedance per period, valid for all ratios of cavity length to structure period.

We recently created a snake depolarizing resonance using an rf solenoid magnet in a ring containing a nearly 100% Siberian snake. We found that the primary snake rf resonance also had two weaker synchrotron sidebands, which are second-order snake resonances; they were probably caused by the energy-dependent strength of the solenoid snake due to the Lorentz contraction of its longitudinal $\int Ḃ\mathrm{dl}$. This was the first observation of an rf synchrotron-sideband depolarizing resonance in the presence of a nearly full Siberian snake.

We discuss a new layout of a cavity chain (called superstructure) allowing, we hope, a significant cost reduction due to a simplification of the rf system of the TESLA linear collider. The proposed scheme increases the fill factor and thus makes an effective gradient of the accelerator higher. In this paper computations and preliminary measurements on existing copper models of the TESLA Test Facility accelerating structures are presented. A new copper model of the scheme has been ordered, made of four 7-cell standing wave cavities, which according to the results of computations and measurements seems to be the most promising version. Experiments with a beam will be necessary to prove that the proposed layout can be used for the acceleration.

Bent solenoids can transmit charged particle beams while providing momentum dispersion. While less familiar than quadrupole and dipole systems, bent solenoids can produce superficially simple transport lines and large acceptance spectrometers for use at low energies. Design issues such as drift compensation and coupling sections between straight and bent solenoids are identified, and aberrations such as shears produced by perpendicular error fields are discussed. Examples are considered which provide the basis for the design of emittance exchange elements for the cooling system of a muon collider.

Simulations of the longitudinal instability in the $50\mathrm{GeV}\times 50\mathrm{GeV}$ muon collider ring have been performed. Operation of the ring close to the slippage factor ${\eta }_1\simeq {10}^{-6\mathrm{}}$, such that synchrotron motion is frozen, minimizes the need for rf to maintain the bunch length. However, there is still an energy spread due to the bunch wake. For design parameters of the ring, this induced energy is too large and must be controlled. This paper demonstrates that the bunch wake may be compensated for by two rf cavities with low rf voltages. These studies were made at the nominal design point, and sensitivities to errors were explored. It is seen that the small energy spread of the beam ($\delta E/E=3\mathrm{}\times {10}^{-5\mathrm{}}$) in the $50\mathrm{GeV}\times 50\mathrm{GeV}$ muon collider ring can be maintained during the 1000 turn lifetime of the muons. Controlled beam dynamics requires proper choice of rf parameters (rf voltage, rf frequency, and phase offset) for two cavities; these parameters depend on the ring design through the impedance, beam pipe radius, and momentum compaction. The simulation also shows that the computation of wake field using bins of variable width (each with a constant number of macroparticles in each bin) gives an accurate wake and also yields reduced computing time compared to an evaluation of the wake as the direct sum over the wakes of all preceding macroparticles.

Beam halo formation issues are important for the design of high current linear ion accelerators. Various mechanisms can potentially cause beam halo. Some recent studies suggested that Coulomb collisions in the beam bunch can contribute significantly to beam bunch growth and halo development in linear accelerators. Despite the general belief that collisions are not important, it is clear that a rigorous treatment of this question is needed. In an effort to explore this issue in detail we have undertaken an analysis of the effects of Coulomb scattering between ions in a self-consistent spherical bunch.

The present analysis makes use of the Vlasov-Maxwell equations to develop a fully kinetic description of the electrostatic, electron-ion two-stream instability driven by the directed axial motion of a high-intensity ion beam propagating in the $z$ direction with average axial momentum ${\gamma }_b{m}_b{\beta }_{b}c$ through a stationary population of background electrons. The ion beam has characteristic radius $r_b$ and is treated as continuous in the $z$ direction, and the applied transverse focusing force on the beam ions is modeled by ${\mathbf{F}}_{\mathrm{foc}}^b=-{\gamma }_b{m}_b{\omega }_{\beta b}^{0^2}{\mathbf{x}}_{\perp }$ in the smooth-focusing approximation. Here, ${\omega }_{\beta b}^0=\mathrm{const}$ is the effective betatron frequency associated with the applied focusing field, ${\mathbf{x}}_{\perp }$ is the transverse displacement from the beam axis, $({\gamma }_b-1\mathrm{})m_b{c}^2$ is the ion kinetic energy, and $V_b={\beta }_{b}c$ is the average axial velocity, where ${\gamma }_b=(1-{\beta }_b^2{)}^{-1\mathrm{}/2}$. Furthermore, the ion motion in the beam frame is assumed to be nonrelativistic, and the electron motion in the laboratory frame is assumed to be nonrelativistic. The ion charge and number density are denoted by ${+Z}_{b}e$ and $n_b$, and the electron charge and number density by $-e$ and $n_e$. For $Z_b{n}_b>n_e$, the electrons are electrostatically confined in the transverse direction by the space-charge potential $\phi$ produced by the excess ion charge. The equilibrium and stability analysis retains the effects of finite radial geometry transverse to the beam propagation direction, including the presence of a perfectly conducting cylindrical wall located at radius ${r=r}_w$. In addition, the analysis assumes perturbations with long axial wavelength, $k_z^2{r}_b^2\ll 1$, and sufficiently high frequency that $|\omega /k_z|\gg v_{\mathrm{Tez}}$ and $|\omega /k_z{-V}_b|\gg v_{\mathrm{Tbz}}$, where $v_{\mathrm{Tez}}$ and $v_{\mathrm{Tbz}}$ are the characteristic axial thermal speeds of the background electrons and beam ions. In this regime, Landau damping (in axial velocity space $v_z$) by resonant ions and electrons is negligibly small. We introduce the ion plasma frequency squared defined by ${\hat{\omega }}_{\mathrm{pb}}^2=4\mathrm{}\pi {\hat{n}}_b{Z}_b^2{e}^2/{\gamma }_b{m}_b$, and the fractional charge neutralization defined by $f={\hat{n}}_e/Z_b{\hat{n}}_b$, where ${\hat{n}}_b$ and ${\hat{n}}_e$ are the characteristic ion and electron densities. The equilibrium and stability analysis is carried out for arbitrary normalized beam intensity ${\hat{\omega }}_{\mathrm{pb}}^2/{\omega }_{\beta b}^{0^2}$, and arbitrary fractional charge neutralization $f$, consistent with radial confinement of the beam particles. For the moderately high beam intensities envisioned in the proton linacs and storage rings for the Accelerator for Production of Tritium and the Spallation Neutron Source, the normalized beam intensity is typically ${\hat{\omega }}_{\mathrm{pb}}^2/{\omega }_{\beta b}^{0^2}\lesssim 0.1$. For heavy ion fusion applications, however, the transverse beam emittance is very small, and the space-charge-dominated beam intensity is much larger, with ${\hat{\omega }}_{\mathrm{pb}}^2/{\omega }_{\beta b}^{0^2}\lesssim 2{\gamma }_b^2$. The stability analysis shows that the instability growth rate $\mathrm{Im}\omega$ increases with increasing normalized beam intensity ${\hat{\omega }}_{\mathrm{pb}}^2/{\omega }_{\beta b}^{0^2}$ and increasing fractional charge neutralization $f$. In addition, the instability is strongest (largest growth rate) for perturbations with azimuthal mode number $\ell =1\mathrm{}$, corresponding to a simple (dipole) transverse displacement of the beam ions and the background electrons. For the case of overlapping step-function density profiles for the beam ions and background electrons, corresponding to monoenergetic ions and electrons, a key result is that there is no threshold in beam intensity ${\hat{\omega }}_{\mathrm{pb}}^2/{\omega }_{\beta b}^{0^2}$ or fractional charge neutralization $f$ for the onset of instability. Finally, for the case of continuously varying density profiles with parabolic profile shape, a semiquantitative estimate is made of the effects of the corresponding spread in (depressed) betatron frequency on stability behavior, including an estimate of the instability threshold for the case of weak density nonuniformity.