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This paper makes use of a one-dimensional kinetic model to investigate the nonlinear longitudinal dynamics of a long coasting beam propagating through a perfectly conducting circular pipe with radius r_{w}. The average axial electric field is expressed as ⟨E_{z}⟩=-(∂/∂z)⟨ϕ⟩=-e_{b}g_{0}∂λ_{b}/∂z-e_{b}g_{2}r_{w}^{2}∂^{3}λ_{b}/∂z^{3}, where g_{0} and g_{2} are constant geometric factors, λ_{b}(z,t)=∫dp_{z}F_{b}(z,p_{z},t) is the line density of beam particles, and F_{b}(z,p_{z},t) satisfies the 1D Vlasov equation. Detailed nonlinear properties of traveling-wave and traveling-pulse (soliton) solutions with time-stationary waveform are examined for a wide range of system parameters extending from moderate-amplitudes to large-amplitude modulations of the beam charge density. Two classes of solutions for the beam distribution function are considered, corresponding to: (i) the nonlinear waterbag distribution, where F_{b}=const in a bounded region of p_{z}-space; and (ii) nonlinear Bernstein-Green-Kruskal (BGK)-like solutions, allowing for both trapped and untrapped particle distributions to interact with the self-generated electric field ⟨E_{z}⟩.

The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy is parametrized using a generalized Courant-Snyder (CS) theory, which extends the original CS theory for one degree of freedom to higher dimensions. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or a U(2) element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimensions. Other components of the original CS theory, such as the transfer matrix, Twiss functions, and CS invariant (also known as the Lewis invariant) all have their counterparts, with remarkably similar expressions, in the generalized theory. The gauge group structure of the generalized theory is analyzed. By fixing the gauge freedom with a desired symmetry, the generalized CS parametrization assumes the form of the modified Iwasawa decomposition, whose importance in phase space optics and phase space quantum mechanics has been recently realized. This gauge fixing also symmetrizes the generalized envelope equation and expresses the theory using only the generalized Twiss function β. The generalized phase advance completely determines the spectral and structural stability properties of a general focusing lattice. For structural stability, the generalized CS theory enables application of the Krein-Moser theory to greatly simplify the stability analysis. The generalized CS theory provides an effective tool to study coupled dynamics and to discover more optimized lattice designs in the larger parameter space of general focusing lattices.

The transverse dynamics of an intense charged particle beam propagating through a periodic quadrupole focusing lattice is described by the nonlinear Vlasov-Maxwell system of equations, where the propagation distances play the role of time. To determine matched-beam quasiequilibrium distribution functions, one needs to determine a dynamical invariant for the beam particles moving in the combined applied and self-generated fields. In this paper, a perturbative Hamiltonian transformation method is developed which is an expansion in the particle’s vacuum phase advance ϵ[over ¯]∼σ_{v}/2π, treated as a small parameter, which is used to transform away the fast particle orbit oscillations and obtain the average Hamiltonian accurate to order ϵ[over ¯]^{3}. The average Hamiltonian is an approximate invariant of the original system, and can be used to determine self-consistent beam quasiequilibrium solutions that are matched to the focusing channel. The equation determining the average self-field potential is derived for general boundary conditions by taking into account the average contribution of the charges induced on the boundary. It is shown for a cylindrical conducting boundary that the average self-field potential acquires an octupole component, which results in the average motion of some beam particles being nonintegrable and their trajectories chaotic. This chaotic behavior of the beam particles may significantly change the nature of the Landau damping (or growth) of collective excitations supported by an intense charged particle beam.

Self-consistent Vlasov-Poisson simulations of beams with high space-charge intensity often require specification of initial phase-space distributions that reflect properties of a beam that is well adapted to the transport channel—both in terms of low-order rms (envelope) properties as well as the higher-order phase-space structure. Here, we first review broad classes of kinetic distributions commonly in use as initial Vlasov distributions in simulations of unbunched or weakly bunched beams with intense space-charge fields including the following: the Kapchinskij-Vladimirskij (KV) equilibrium, continuous-focusing equilibria with specific detailed examples, and various nonequilibrium distributions, such as the semi-Gaussian distribution and distributions formed from specified functions of linear-field Courant-Snyder invariants. Important practical details necessary to specify these distributions in terms of standard accelerator inputs are presented in a unified format. Building on this presentation, a new class of approximate initial kinetic distributions are constructed using transformations that preserve linear focusing, single-particle Courant-Snyder invariants to map initial continuous-focusing equilibrium distributions to a form more appropriate for noncontinuous focusing channels. Self-consistent particle-in-cell simulations are employed to show that the approximate initial distributions generated in this manner are better adapted to the focusing channels for beams with high space-charge intensity. This improved capability enables simulations that more precisely probe intrinsic stability properties and machine performance.

The Courant-Snyder theory gives a complete description of the uncoupled transverse dynamics of charged particles in electromagnetic focusing lattices. In this paper, the Courant-Snyder theory is generalized to the case of coupled transverse dynamics with two degrees of freedom. The generalized theory has the same structure as the original Courant-Snyder theory for one degree of freedom. The four basic components of the original Courant-Snyder theory, i.e., the envelope equation, phase advance, transfer matrix, and the Courant-Snyder invariant, all have their counterparts, with remarkably similar expressions, in the generalized theory presented here. In the generalized theory, the envelope function is generalized into an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. The generalized theory gives a new parametrization of the 4D symplectic transfer matrix that has the same structure as the parametrization of the 2D symplectic transfer matrix in the original Courant-Snyder theory. All of the parameters used in the generalized Courant-Snyder theory correspond to physical quantities of importance, and this parametrization can provide a valuable framework for accelerator design and particle simulation studies. A time-dependent canonical transformation is used to develop the generalized Courant-Snyder theory. Applications of the new theory to strongly and weakly coupled dynamics are given. It is shown that the stability of coupled dynamics can be determined by the generalized phase advance developed. Two stability criteria are given, which recover the known results about sum and difference resonances in the weakly coupled limit.

A random noise-induced beam degradation that could affect intense beam transport over long propagation distances has been experimentally investigated by making use of the transverse beam dynamics equivalence between an alternating-gradient focusing system and a linear Paul trap system. For the present study, machine imperfections in the quadrupole focusing lattice are considered, which are emulated by adding small random noise on the voltage waveform of the quadrupole electrodes in the Paul trap. It is observed that externally driven noise continuously increases the rms radius, transverse emittance, and nonthermal tail of the trapped charge bunch almost linearly with the duration of the noise. The combined effects of collective modes and colored noise are also investigated and compared with numerical simulations.

Large-space-scale and long-time-scale plasma flow simulations are executed in order to study the spatial and temporal evolution of plasma parameters for two types of plasma sources used in the neutralized drift compression experiment (NDCX). The results help assess the charge neutralization conditions for ion beam compression experiments and can be employed in more sophisticated simulations, which previously neglected the dynamical evolution of the plasma. Three-dimensional simulations of a filtered cathodic-arc plasma source show the coupling efficiency of the plasma flow from the source to the drift region depends on geometrical factors. The nonuniform magnetic topology complicates the well-known general analytical considerations for evaluating guiding-center drifts, and particle-in-cell simulations provide a self-consistent evaluation of the physics in an otherwise challenging scenario. Plasma flow profiles of a ferroelectric plasma source demonstrate that the densities required for longitudinal compression experiments involving ion beams are provided over the drift length, and are in good agreement with measurements. Simulations involving azimuthally asymmetric plasma creation conditions show that symmetric profiles are nevertheless achieved at the time of peak on-axis plasma density. Also, the ferroelectric plasma expands upstream on the thermal expansion time scale, and therefore avoids the possibility of penetration into the acceleration gap and transport sections, where partial neutralization would increase the beam emittance. Future experiments on NDCX will investigate the transverse focusing of an axially compressing intense charge bunch to a sub-mm spot size with coincident focal planes using a strong final-focus solenoid. In order to fill a multi-tesla solenoid with the necessary high-density plasma for beam charge neutralization, the simulations predict that supersonically injected plasma from the low-field region will penetrate and partially fill the high-field region of the solenoid. Because of the magnetic mirroring effect, the on-axis plasma density in the solenoid depends on the injection velocity and magnetic field strength.

The Paul trap simulator experiment is a compact laboratory Paul trap that simulates a long, thin charged-particle bunch coasting through a kilometers-long magnetic alternating-gradient (AG) transport system by putting the physicist in the beam’s frame of reference. The transverse dynamics of particles in both systems are described by similar equations, including all nonlinear space-charge effects. The time-dependent quadrupolar electric fields created by the confinement electrodes of a linear Paul trap correspond to the axially dependent magnetic fields applied in the AG system. Results are presented for experiments in which the lattice period and strength are changed over the course of the experiment to transversely compress a beam with an initial depressed tune of 0.9. Instantaneous and smooth changes are considered. Emphasis is placed on determining the conditions that minimize the emittance growth and the number of halo particles produced by the beam compression process. Both the results of particle-in-cell simulations performed with the warp code and envelope equation solutions agree well with the experimental data.

Longitudinal bunch compression of intense ion beams for warm dense matter and heavy ion fusion applications occurs by imposing an axial velocity tilt onto an ion beam across the acceleration gap of a linear induction accelerator, and subsequently allowing the beam to drift through plasma in order to neutralize its space-charge and current as the pulse compresses. The detailed physics and implications of acceleration gap effects and focusing aberration on optimum longitudinal compression are quantitatively reviewed using particle-in-cell simulations, showing their dependence on many system parameters. Finite-size gap effects are shown to result in compression reduction, due to an increase in the effective longitudinal temperature imparted to the beam, and a decrease in intended fractional tilt. Sensitivity of the focal plane quality to initial longitudinal beam temperature is explored, where slower particles are shown to experience increased levels of focusing aberration compared to faster particles. A plateau effect in axial compression is shown to occur for larger initial pulse lengths, where the increases in focusing aberration over the longer drift lengths involved dominate the increases in relative compression, indicating a trade-off between current compression and pulse duration. The dependence on intended fractional tilt is also discussed and agrees well with theory. A balance between longer initial pulse lengths and larger tilts is suggested, since both increase the current compression, but have opposite effects on the final pulse length, drift length, and amount of longitudinal focusing aberration. Quantitative examples are outlined that explore the sensitive dependence of compression on the initial kinetic energy and thermal distribution of the beam particles. Simultaneous transverse and longitudinal current density compression can be achieved in the laboratory using a strong final-focus solenoid, and simulations addressing the effects of focusing aberration in both directions are presented.

The transverse compression of a long charge bunch is investigated in the Paul trap simulator experiment (PTSX), which is a linear Paul trap that simulates the nonlinear transverse dynamics of an intense charged particle beam propagating through an equivalent kilometers-long magnetic alternating-gradient (AG) focusing system. Changing the voltage amplitude at fixed focusing frequency in the PTSX device corresponds to changing the field gradient of the quadrupole magnets with fixed axial periodicity in the AG transport system. In this work, we present experimental results on transverse compression of the charge bunch in which the amplitude of the applied oscillatory focusing voltage is changed instantaneously, and adiabatically. The experimental data are also compared with analytical estimates and 2D WARP particle-in-cell simulations.

Collective effects with strong coupling between the longitudinal and transverse dynamics are of fundamental importance for applications of high-intensity bunched beams. The self-consistent Vlasov-Maxwell equations are applied to high-intensity finite-length charge bunches, and a generalized δf particle simulation algorithm is developed for bunched beams with or without energy anisotropy. The nonlinear δf method exhibits minimal noise and accuracy problems in comparison with standard particle-in-cell simulations. Systematic studies are carried out under conditions corresponding to strong 3D nonlinear space-charge forces in the beam frame. For charge bunches with isotropic energy, finite bunch-length effects are clearly evident by the fact that the spectra for an infinitely long coasting beam and a nearly spherical charge bunch have strong similarities, whereas the spectra have distinctly different features when the bunch length is varied between these two limiting cases. For bunched beams with anisotropic energy, there exists no exact kinetic equilibrium because the particle dynamics do not conserve transverse energy and longitudinal energy separately. A reference state in approximate dynamic equilibrium has been constructed theoretically, and a quasi-steady state has been established in the simulations for the anisotropic case. Collective excitations relative to the reference state have been simulated using the generalized δf algorithm. In particular, the electrostatic Harris instability driven by strong energy anisotropy is investigated for a finite-length charge bunch. The observed growth rates are larger than those obtained for infinitely long coasting beams. However, the growth rate decreases for increasing bunch length to a value similar to the case of a long coasting beam. For long bunches, the instability is axially localized symmetrically relative to the beam center, and the characteristic wavelength in the longitudinal direction is comparable to the transverse dimension of the beam.

The Paul Trap Simulator Experiment (PTSX) is a linear Paul trap whose purpose is to simulate the nonlinear transverse dynamics of intense charged particle beam propagation in periodic-focusing quadrupole magnetic transport systems. Externally created cesium ions are injected and trapped in the long central electrodes of the PTSX device. In order to have well-matched one-component plasma equilibria for various beam physics experiments, it is important to optimize the ion injection. From the experimental studies reported in this paper, it is found that the injection process can be optimized by minimizing the beam mismatch between the source and the focusing lattice, and by minimizing the number of particles present in the vicinity of the injection electrodes when the injection electrodes are switched from the fully oscillating voltage waveform to their static trapping voltage.

Heavy ion drivers for warm dense matter and heavy ion fusion applications use intense charge bunches which must undergo transverse and longitudinal compression in order to meet the requisite high current densities and short pulse durations desired at the target. The neutralized drift compression experiment (NDCX) at the Lawrence Berkeley National Laboratory is used to study the longitudinal neutralized drift compression of a space-charge-dominated ion beam, which occurs due to an imposed longitudinal velocity tilt and subsequent neutralization of the beam’s space charge by background plasma. Reduced theoretical models have been used in order to describe the realistic propagation of an intense charge bunch through the NDCX device. A warm-fluid model is presented as a tractable computational tool for investigating the nonideal effects associated with the experimental acceleration gap geometry and voltage waveform of the induction module, which acts as a means to pulse shape both the velocity and line density profiles. Self-similar drift compression solutions can be realized in order to transversely focus the entire charge bunch to the same focal plane in upcoming simultaneous transverse and longitudinal focusing experiments. A kinetic formalism based on the Vlasov equation has been employed in order to show that the peaks in the experimental current profiles are a result of the fact that only the central portion of the beam contributes effectively to the main compressed pulse. Significant portions of the charge bunch reside in the nonlinearly compressing part of the ion beam because of deviations between the experimental and ideal velocity tilts. Those regions form a pedestal of current around the central peak, thereby decreasing the amount of achievable longitudinal compression and increasing the pulse durations achieved at the focal plane. A hybrid fluid-Vlasov model which retains the advantages of both the fluid and kinetic approaches has been implemented to describe the formation of pedestals in the current profiles. The comparison between the experimental measurements and the various theoretical models is excellent.

In a first beam dynamics validation experiment for a new Pulse Line Ion Acceleration (PLIA) concept, the predicted energy amplification and beam bunching were experimentally observed. Beam energy modulation of -80 to +150 keV was measured using a PLIA input voltage waveform of -21 to +12 kV. Ion pulses accelerated by 150 keV, and bunching by a factor of 4 were simultaneously achieved. The measured longitudinal phase space and current waveform of the accelerated beam are in good agreement with 3D particle-in-cell simulations.

The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant, a fundamental concept in accelerator physics, is fundamentally a result of the corresponding symmetry admitted by the harmonic oscillator equation with linear time-dependent frequency. It is demonstrated that the Lie algebra of the symmetry group for the oscillator equation with time-dependent frequency is eight dimensional, and is composed of four independent subalgebras. A detailed analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. As an application to accelerator physics, the symmetries of the envelope equation enable a fast numerical algorithm for finding matched solutions without using the conventional iterative Newton’s method, where the envelope equation needs to be numerically integrated once for every iteration, and the Jacobi matrix needs to be calculated for the envelope perturbation.

The transverse compression and dynamics of an intense beam propagating through an alternating-gradient quadrupole lattice, plays an important role in many accelerator physics applications. Typically, the compression can be achieved by means of increasing the focusing strength of the lattice along the beam propagation direction. However, beam propagation through the lattice transition region inevitably leads to a certain level of beam mismatch and halo formation. In this work we present a detailed analysis of these phenomena using the envelope equations in the smooth-focusing approximation, which describe the average effects of an alternating-gradient lattice, and full particle-in-cell numerical simulations using the WARP code, taking into account the effects of the alternating-gradient quadrupole field. Simulations are presented for both space-charge–dominated beams, and beams with a moderate space-charge strength.

The classical electrostatic Harris instability is generalized to the case of a one-component intense charged particle beam with anisotropic temperature including the important effects of finite transverse geometry and beam space charge. For a long, coasting beam, the eigenmode code bEASt have been used to determine detailed 3D stability properties over a wide range of temperature anisotropy and beam intensity. A simple theoretical model is developed which describes the essential features of the linear stage of the instability. Both the simulations and the analytical theory clearly show that moderately intense beams are linearly unstable to short-wavelength perturbations provided the ratio of the longitudinal temperature to the transverse temperature is smaller than some threshold value. The delta-f particle-in-cell code BEST has been used to study the detailed nonlinear evolution and saturation of the instability.

A kinetic model based on the Vlasov equation is used to describe the axial drift compression and transverse focusing of an intense ion charge bunch propagating along the axis of a solenoidal focusing field B^{sol}(x). The space charge and current of the ion charge bunch are assumed to be completely neutralized by the electrons provided by a dense background plasma. In the absence of self-field forces, the Vlasov equation is solved exactly for general initial distribution function f_{b}(x,v,0), using the method of characteristics. It is shown that the Vlasov equation possesses a class of exact, dynamically evolving solutions f_{b}(W_{⊥},W_{z}), where W_{⊥} and W_{z} are transverse and longitudinal constants of the motion. Detailed dynamical properties of the charge bunch are calculated during axial compression and transverse focusing for several choices of distribution function f_{b}(W_{⊥},W_{z}).

The authors disagree with Dr. D. V. Pestrikov’s assertion that “the results obtained in the commented paper are not true,” based on how Volterra’s integral equations are treated in the paper. The authors would agree, for clarity, that the equal sign (=) in Eqs. (34), (36), and (38) in the paper be replaced by some special symbol or the approximate sign (≈) to indicate the omission of initial conditions. These and similar changes together with the revision to an error found by the authors have been published in a recent erratum.

This paper presents a survey of the present theoretical understanding of collective processes and beam-plasma interactions affecting intense heavy ion beam propagation in heavy ion fusion systems. In the acceleration and beam transport regions, the topics covered include discussion of the conditions for quiescent beam propagation over long distances; the electrostatic Harris-type instability and the transverse electromagnetic Weibel-type instability in strongly anisotropic, one-component non-neutral ion beams; and the dipole-mode, electron-ion two-stream instability driven by an (unwanted) component of background electrons. In the plasma plug and target chamber regions, collective processes associated with the interaction of the intense ion beam with a charge-neutralizing background plasma are described, including the electrostatic electron-ion two-stream instability, the electromagnetic Weibel instability, and the resistive hose instability. Operating regimes are identified where the possible deleterious effects of collective processes on beam quality are minimized.

In the currently envisioned configurations for heavy ion fusion, it is necessary to longitudinally compress the beam bunches by a large factor after the acceleration phase. Because the space-charge force increases as the beam is compressed, the beam size in the transverse direction will increase in a periodic quadrupole lattice. If an active control of the beam size is desired, a larger focusing force is needed to confine the beam in the transverse direction, and a nonperiodic quadrupole lattice along the beam path is necessary. In this paper, we describe the design of such a focusing lattice using the transverse envelope equations. A drift compression and final focus lattice should focus the entire beam pulse onto the same focal spot on the target. This is difficult with a fixed lattice, because different slices of the beam may have different perveance and emittance. Four time-dependent magnets are introduced in the upstream of drift compression to focus the entire pulse onto the same focal spot. Drift compression and final focusing schemes are developed for a typical heavy ion fusion driver and for the integrated beam experiment being designed by the Heavy Ion Fusion Virtual National Laboratory.

In heavy-ion inertial-confinement fusion systems, intense beams of ions must be transported from the exit of the final-focus magnet system through the fusion chamber to hit spots on the target with radii of about 2 mm. For the heavy-ion-fusion power-plant scenarios presently favored in the U.S., a substantial fraction of the ion-beam space charge must be neutralized during this final transport. The most effective neutralization technique found in numerical simulations is to pass each beam through a low-density plasma after the final focusing. To provide quantitative comparisons of these theoretical predictions with experiment, the Virtual National Laboratory for Heavy Ion Fusion has completed the construction and has begun experimentation with the neutralized-transport experiment. The experiment consists of three main sections, each with its own physics issues. The injector is designed to generate a very high-brightness, space-charge-dominated potassium beam, while still allowing variable perveance by a beam aperturing technique. The magnetic-focusing section, consisting of four pulsed quadrupoles, permits the study of magnet tuning, as well as the effects of phase-space dilution due to higher-order nonlinear fields. In the final section, the converging ion beam exiting the magnetic section is transported through a drift region with plasma sources for beam neutralization, and the final spot size is measured under various conditions of neutralization. In this paper, we discuss the design and characterization of the three sections in detail and present initial results from the experiment.

This paper employs a one-dimensional kinetic model to investigate the nonlinear longitudinal dynamics of a long coasting beam propagating through a perfectly conducting circular pipe with radius r_{w}. The average axial electric field is expressed as ⟨E_{z}^{s}⟩=-e_{b}g_{0}∂λ_{b}/∂z-e_{b}g_{2}r_{w}^{2}∂^{3}λ_{b}/∂z^{3}, where g_{0} and g_{2} are constant geometric factors, and λ_{b}(z,t)=∫dp_{z}F_{b}(z,p_{z},t) is the line density. Assuming a waterbag distribution for the longitudinal distribution function F_{b}(z,p_{z},t), it is shown that weakly nonlinear disturbances moving near the sound speed evolve according to the Korteweg–deVries equation.

This paper describes a self-consistent kinetic model for the longitudinal dynamics of a long, coasting beam propagating in straight (linear) geometry in the z direction in the smooth-focusing approximation. Starting with the three-dimensional Vlasov-Maxwell equations, and integrating over the phase-space (x_{⊥},p_{⊥}) transverse to beam propagation, a closed system of equations is obtained for the nonlinear evolution of the longitudinal distribution function F_{b}(z,p_{z},t) and average axial electric field ⟨E_{z}^{s}⟩(z,t). The primary assumptions in the present analysis are that the dependence on axial momentum p_{z} of the distribution function f_{b}(x,p,t) is factorable, and that the transverse beam dynamics remains relatively quiescent (absence of transverse instability or beam mismatch). The analysis is carried out correct to order k_{z}^{2}r_{w}^{2} assuming slow axial spatial variations with k_{z}^{2}r_{w}^{2}≪1, where k_{z}∼∂/∂z is the inverse length scale of axial variation in the line density λ_{b}(z,t)=∫dp_{z}F_{b}(z,p_{z},t), and r_{w} is the radius of the conducting wall (assumed perfectly conducting). A closed expression for the average longitudinal electric field ⟨E_{z}^{s}⟩(z,t) in terms of geometric factors, the line density λ_{b}, and its derivatives ∂λ_{b}/∂z,… is obtained for the class of bell-shaped density profiles n_{b}(r,z,t)=(λ_{b}/πr_{b}^{2})f(r/r_{b}), where the shape function f(r/r_{b}) has the form specified by f(r/r_{b})=(n+1)(1-r^{2}/r_{b}^{2})^{n} for 0≤r

The linearized Vlasov-Maxwell equations are used to investigate detailed properties of the wall-impedance-driven instability for a long charge bunch (bunch length ℓ_{b}≫bunch radius r_{b}) propagating through a cylindrical pipe with radius r_{w} and wall impedance Z[over ˜](ω). The stability analysis is carried out for perturbations about a cylindrical Kapchinskij-Vladimirskij beam equilibrium with a flattop density profile in the smooth-focusing approximation. The perturbations are assumed to be of the form δψ(x,t)=δψ^{ℓ}(r)exp⁡(iℓθ+ik_{z}z-iωt), where (r,θ) are the radial and azimuthal coordinates in the transverse direction, and z is the coordinate in the longitudinal direction. Here, ℓ=1,2,… is the azimuthal mode number of the perturbation in the transverse direction, k_{z} is the wave number in the longitudinal direction, and ω is the oscillation frequency. As an example, detailed stability properties are determined for dipole-mode perturbations (ℓ=1) assuming negligibly small axial momentum spread of the beam particles. The stability analysis is valid for a general value of the normalized beam intensity s_{b}=ω[over ^]_{pb}^{2}/2γ_{b}^{2}ω_{β⊥}^{2} in the interval 0

In plasmas with strongly anisotropic distribution functions (T_{∥b}/T_{⊥b}≪1) a Harris-like collective instability may develop if there is sufficient coupling between the transverse and longitudinal degrees of freedom. Such anisotropies develop naturally in accelerators and may lead to a deterioration of beam quality. This paper extends previous numerical studies [E. A. Startsev, R. C. Davidson, and H. Qin, Phys. Plasmas 9, 3138 (2002)] of the stability properties of intense non-neutral charged particle beams with large temperature anisotropy (T_{⊥b}≫T_{∥b}) to allow for nonaxisymmetric perturbations with ∂/∂θ≠0. The most unstable modes are identified, and their eigenfrequencies, radial mode structure, and nonlinear dynamics are determined. The simulation results clearly show that moderately intense beams with s_{b}=ω[over ^]_{pb}^{2}/2γ_{b}^{2}ω_{β⊥}^{2}≳0.5 are linearly unstable to short-wavelength perturbations with k_{z}^{2}r_{b}^{2}≳1, provided the ratio of longitudinal and transverse temperatures is smaller than some threshold value. Here, ω[over ^]_{pb}^{2}=4πn[over ^]_{b}e_{b}^{2}/γ_{b}m_{b} is the relativistic plasma frequency squared, and ω_{β⊥} is the betatron frequency associated with the applied smooth-focusing field. A theoretical model is developed based on the Vlasov-Maxwell equations which describes the essential features of the linear stages of instability. Both the simulations and the analytical theory predict that the dipole mode (azimuthal mode number m=1) is the most unstable mode. In the nonlinear stage, tails develop in the longitudinal momentum distribution function, and the kinetic instability saturates due to resonant wave-particle interactions.

This paper makes use of the Vlasov-Maxwell equations to investigate collective excitations in an intense sheet beam, infinite in the y and z directions, propagating in the z direction with directed kinetic energy (γ_{b}-1)m_{b}c^{2}. The beam is confined in the x direction by the smooth-focusing force F[over →]_{foc}=-γ_{b}m_{b}ω_{β⊥}^{2}xe[over →]_{x}, and perfectly conducting walls are located at x=±x_{w}. A self-consistent water bag equilibrium f_{b}^{0} satisfying the steady-state (∂/∂t=0) Vlasov-Maxwell equations is shown to be exactly solvable for the beam density n_{b}^{0}(x) and electrostatic potential φ^{0}(x). A closed Schrödinger-like eigenvalue equation is derived, assuming small-amplitude perturbations (δf_{b},δφ) about the self-consistent water bag equilibrium, and the eigenfrequency spectrum is shown to be purely real. The WKB approximation is employed to determine the eigenfrequency spectrum as a function of the normalized beam intensity s_{b}=ω[over ^]_{pb}^{2}/γ_{b}^{2}ω_{β⊥}^{2}, where ω[over ^]_{pb}^{2}=4πn[over ^]_{b}e_{b}^{2}/γ_{b}m_{b} and n[over ^]_{b}=n_{b}(x=0) is the on-axis number density of beam particles.

The dispersion relation for the resistive hose instability in a charged particle beam with a flattop density profile is derived from the linearized Vlasov-Maxwell equations. Stability properties of the resistive hose instability where the perturbations are initiated at the beam entrance are investigated. In particular, the complex eigenfrequency Ω in the dispersion relation is expressed as a function of the real oscillation frequency ω of the excitation at the beam entrance. As expected, the growth rate ImΩ=Ω_{i} decreases rapidly as the conducting wall approaches the beam (r_{w}/r_{b}→1). The growth rate also decreases substantially as the frequency ratio ω/ν_{c} increases, where ν_{c} is the electron collision frequency. Stability properties for perturbations propagating through the beam pulse from its head to tail are also investigated. In this case, the growth rate Imω is calculated in terms of the real oscillation frequency Ω of each beam segment. It is shown that the resonance frequency Ω=Ω_{r} corresponding to the infinite growth rate detunes considerably from the betatron frequency ω_{β} of the beam particles. It is also found that the bandwidth corresponding to instability is narrow when the plasma electron collision time (1/ν_{c}) is long compared with the magnetic decay time (τ_{d}).

An intense charged particle beam with directed kinetic energy (γ_{b}-1)m_{b}c^{2} propagates in the z direction through an applied focusing field with transverse focusing force modeled by F_{foc}=-γ_{b}m_{b}ω_{β⊥}^{2}x_{⊥} in the smooth-focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F_{b}(r,p_{⊥})=Aexp⁡(-H_{⊥}/T[over ^]_{⊥b})⊕(H_{⊥}-E_{b}), where A, T[over ^]_{⊥b}, and E_{b} are positive constants, and H_{⊥} is the Hamiltonian for transverse particle motion. The equilibrium profiles for beam number density, n_{b}(r)=∫d^{2}pF_{b}(r,p_{⊥}), and transverse temperature, T_{⊥b}(r)=[n_{b}(r)]^{-1}∫d^{2}p(p_{⊥}^{2}/2γ_{b}m_{b})F_{b}(r,p_{⊥}), are calculated self-consistently including space-charge effects. Several properties of the equilibrium profiles are noteworthy. For example, the beam has a sharp outer edge radius r_{b} with n_{b}(r≥r_{b})=0, where r_{b} depends on the value of E_{b}/T[over ^]_{⊥b}. In addition, unlike the choice of a semi-Gaussian distribution, F_{b}^{SG}=Aexp⁡(-p_{⊥}^{2}/2γ_{b}m_{b}T[over ^]_{⊥b})⊕(r-r_{b}), the truncated thermal equilibrium distribution F_{b}(r,p) depends on (r,p) only through the single-particle constant of the motion H_{⊥} and is therefore a true steady-state solution (∂/∂t=0) of the nonlinear Vlasov-Maxwell equations.

This paper presents an analytical investigation of the transverse electron-proton (e-p) two-stream instability in a proton bunch propagating through a stationary electron background. The equations of motion, including damping effects, are derived for the centroids of the proton beam and the electron cloud by considering Lorentzian and Gaussian frequency spreads for the particles. For a Lorentzian frequency distribution, we derive the asymptotic solution of the coupled linear centroid equations in the time domain and study the e-p instability in proton bunches with nonuniform line densities. Examples are given for both uniform and parabolic proton line densities.

Two-stream instabilities in intense charged particle beams, described self-consistently by the nonlinear Vlasov-Maxwell equations, are studied using a 3D multispecies perturbative particle simulation method. The recently developed Beam Equilibrium, Stability and Transport code is used to simulate the linear and nonlinear properties of the electron-proton (e-p) two-stream instability observed in the Proton Storage Ring (PSR) experiment for a long, coasting beam. Simulations in a parameter regime characteristic of the PSR experiment show that the e-p instability has a dipole-mode structure, and that the growth rate is an increasing function of beam intensity, but a decreasing function of the longitudinal momentum spread. It is also shown that the instability threshold decreases with increasing fractional charge neutralization and increases with increasing axial momentum spread of the beam particles. In the nonlinear phase, the simulations show that the proton density perturbation first saturates at a relatively low level and subsequently grows to a higher level. Finally, the nonlinear space-charge-induced transverse tune spread, which introduces a major growth-rate reduction effect on the e-p instability, is studied for self-consistent equilibrium populations of protons and electrons.

The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function f_{b}\(x,x^{′},s\) propagating through a periodic focusing lattice κ_{x}\(s+S\)=κ_{x}\(s\), where S=const is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density f_{b}\(x,x^{′},s\)=A=const inside of the simply connected boundary curves, x_{+}^{′}\(x,s\) and x_{-}^{′}\(x,s\), in the two-dimensional phase space \(x,x^{′}\). Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, x_{+}^{′}\(x,s\) and x_{-}^{′}\(x,s\), and the self-field potential ψ\(x,s\)=e_{b}φ\(x,s\)/γ_{b}m_{b}β_{b}^{2}c^{2}. The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure P_{b}\(x,s\)=const[n_{b}\(x,s\)]^{3}. Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. Specific examples of periodically focused beam equilibria are presented, ranging from a finite-emittance beam in which the boundary curves in phase space \(x,x^{′}\) correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, n_{b}\(x,s\), exhibits large-amplitude periodic oscillations. For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.

High beam current can be achieved by longitudinally compressing bunched beams. The objective of drift compression is to compress a long beam bunch by imposing an initial longitudinal velocity distribution over the length of the beam in the beam frame. The longitudinal dynamics of drift compression and pulse shaping for high intensity particle beams are studied using a one-dimensional warm-fluid model. Two self-similar drift compression solutions admitted by the one-dimensional warm-fluid equations are derived. The pulse shaping problem is also solved such that an arbitrary input pulse shape can be shaped into the pulse shapes required by the self-similar drift compression solutions.

A Hamiltonian approach to the solution of the Vlasov-Poisson equations has been developed. Based on a nonlinear canonical transformation, the rapidly oscillating terms in the original Hamiltonian are transformed away, yielding a new Hamiltonian that contains slowly varying terms only. The formalism has been applied to the dynamics of an intense beam propagating through a periodic focusing lattice, and to the coherent beam-beam interaction. A stationary solution to the transformed Vlasov equation has been obtained.

This paper provides a systematic derivation of a guiding-center kinetic model that describes intense beam propagation through a periodic focusing lattice with axial periodicity length S, valid for sufficiently small phase advance (say, σ0 and κ_{y sf}>0, and the model is readily accessible to direct analytical investigation. Similar smooth-focusing Vlasov-Maxwell descriptions are widely used in the accelerator physics literature, often without a systematic justification, and the present analysis is intended to place these models on a rigorous, yet physically intuitive, foundation.

Collective processes in intense charged particle beams described self-consistently by the Vlasov-Maxwell equations are studied using a 3D multispecies nonlinear perturbative particle simulation method. The newly developed beam equilibrium, stability, and transport (BEST) code is used to simulate the nonlinear stability properties of intense beam propagation, surface eigenmodes in a high-intensity beam, and the electron-proton (e-p) two-stream instability observed in the Proton Storage Ring (PSR) experiment. Detailed simulations in a parameter regime characteristic of the PSR experiment show that the dipole-mode two-stream instability is stabilized by a modest spread (about 0.1%) in axial momentum of the beam particles.

The present analysis considers an intense non-neutral ion beam with characteristic axial velocity V_{b}=β_{b}c and directed kinetic energy (γ_{b}-1)m_{b}c^{2} propagating in the z direction through an applied focusing field which produces a transverse focusing force F_{foc}=-γ_{b}m_{b}ω_{βb}^{2}(xe[over ^]_{x}+ye[over ^]_{y}) on a beam ion (smooth focusing approximation). For a thermal equilibrium distribution function F_{b}(H_{⊥}^{0})=const×exp(-H_{⊥}^{0}/T_{b}), it is shown that the normalized radial density profile πr_{b}^{2}n_{b}^{0}( r)/N_{b}, plotted as a function of r/r_{b}, depends on a single dimensionless parameter, δ_{b}=N_{b}Z_{b}^{2}e^{2}/2γ_{b}^{2}T_{b}, which is a measure of the normalized beam intensity. Here, N_{b}≡∫dxdy n_{b}^{0}(r) is the number of beam ions per unit axial length and r_{b}^{2}≡N_{b}^{-1}∫dxdyr^{2}n_{b}^{0}(r) is the mean-square beam radius. “Universal” profiles for the beam density are presented for a wide range of system parameters. The present results are readily extended to the case of a cylindrical non-neutral plasma column confined by a uniform axial magnetic field B_{0}e[over ^]_{z}.

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, -[κ_{x}(s)xe[over ^]_{x}+κ_{y}(s)ye[over ^]_{y}], on the beam ions. Here, s=β_{b}ct is the axial coordinate, (γ_{b}-1)m_{b}c^{2} is the directed axial kinetic energy of the beam ions, and the (oscillatory) lattice coefficients satisfy κ_{x}(s+S)=κ_{x}( s) and κ_{y}(s+S)=κ_{y}( s), where S=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^{′},y^{′},s) and the (normalized) self-field potential ψ(x,y,s) in the transverse laboratory-frame phase space (x,y,x^{′},y^{′}). Here, H[over ^](x,y,x^{′},y^{′},s)=(1/2)(x^{′2}+y^{′2})+( 1/2)[κ_{x}( s)x^{2}+κ_{y}(s)y^{2}]+ψ(x,y,s) is the (dimensionless) Hamiltonian for particle motion in the applied field plus self-field configurations, where (x,y) and (x^{′},y^{′}) are the transverse displacement and velocity components, respectively, and ψ(x,y,s) is the self-field potential. The Hamiltonian is formally assumed to be of order ε, a small dimensionless parameter proportional to the characteristic strength of the focusing field as measured by the lattice coefficients κ_{x}(s) and κ_{y}(s). 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(X[over ̃],Y[over ̃],X[over ̃]^{′},Y[over ̃]^{′},s)=(1/2)(X[over ̃]^{′2}+Y[over ̃]^{′2})+(1/ 2)κ_{f}(X[over ̃]^{2}+Y[over ̃]^{2})+ψ(X[over ̃],Y[over ̃],s), correct to order ε^{3} in the “slow” transformed variables (X[over ̃],Y[over ̃],X[over ̃]^{′},Y[over ̃]^{′}). Here, the transverse focusing coefficient in the transformed variables satisfies κ_{f}=const, and the asymptotic expansion procedure is expected to be valid for a sufficiently small phase advance (σ

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 γ_{b}m_{b}β_{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 F_{foc}^{b}=-γ_{b}m_{b}ω_{βb}^{0^{2}}x_{⊥} in the smooth-focusing approximation. Here, ω_{βb}^{0}=const is the effective betatron frequency associated with the applied focusing field, x_{⊥} is the transverse displacement from the beam axis, (γ_{b}-1)m_{b}c^{2} is the ion kinetic energy, and V_{b}=β_{b}c is the average axial velocity, where γ_{b}=(1-β_{b}^{2})^{-1/2}. 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 φ 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}≪1, and sufficiently high frequency that |ω/k_{z}|≫v_{Tez} and |ω/k_{z}-V_{b}|≫v_{Tbz}, where v_{Tez} and v_{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 ω[over ^]_{pb}^{2}=4πn[over ^]_{b}Z_{b}^{2}e^{2}/γ_{b}m_{b}, and the fractional charge neutralization defined by f=n[over ^]_{e}/Z_{b}n[over ^]_{b}, where n[over ^]_{b} and n[over ^]_{e} are the characteristic ion and electron densities. The equilibrium and stability analysis is carried out for arbitrary normalized beam intensity ω[over ^]_{pb}^{2}/ω_{βb}^{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 ω[over ^]_{pb}^{2}/ω_{βb}^{0^{2}}≲ 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 ω[over ^]_{pb}^{2}/ω_{βb}^{0^{2}}≲ 2γ_{b}^{2}. The stability analysis shows that the instability growth rate Imω increases with increasing normalized beam intensity ω[over ^]_{pb}^{2}/ω_{βb}^{0^{2}} and increasing fractional charge neutralization f. In addition, the instability is strongest (largest growth rate) for perturbations with azimuthal mode number ℓ=1, 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 ω[over ^]_{pb}^{2}/ω_{βb}^{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.