Search Results (34 total)

We study the microbunching instability in a bunch compressor by a parallel code with some improved numerical algorithms. The two-dimensional charge/current distribution is represented by a Fourier series, with coefficients determined through Monte Carlo sampling over an ensemble of tracked points. This gives a globally smooth distribution with low noise. The field equations are solved accurately in the lab frame using retarded potentials and a novel choice of integration variables that eliminates singularities. We apply the scheme with parameters for the first bunch compressor system of FERMI@Elettra, with emphasis on the amplification of a perturbation at a particular wavelength and the associated longitudinal bunch spectrum. Gain curves are in rough agreement with those of the linearized Vlasov system at intermediate wavelengths, but show some deviation at the smallest wavelengths treated and show the breakdown of a coasting beam assumption at long wavelengths. The linearized Vlasov system is discussed in some detail. A new 2D integral equation is derived which reduces to a well-known 1D integral equation in the coasting beam case.

Direct numerical methods for solving the Vlasov equation offer some advantages over macroparticle simulations, as they do not suffer from the consequences of the statistical fluctuations inherent in using a number of macroparticles smaller than the bunch population. Unfortunately, these methods are more time consuming and generally considered impractical in a full 6D phase space. However, in a lower-dimension phase space they may become attractive if the beam dynamics is sensitive to the presence of small charge-density fluctuations and a high resolution is needed. In this paper we present a 2D Vlasov solver for studying the longitudinal beam dynamics in single-pass systems of interest for x-ray FELs, where characterization of the microbunching instability stemming from self-field amplified noise is of particular relevance.

We are concerned with coherent longitudinal motion in a storage ring, especially with situations in which coherent synchrotron radiation (CSR) can influence stability of the beam. The collective force from CSR is usually described by an impedance or a wake function in such a way that the force depends only on the charge distribution at the present time. This description is exact only for a rigid bunch, since causality demands that the force depend on the prior history of the bunch. We show how to treat a deforming bunch by applying the “complete impedance” Z(n,ω), a function of wave number and frequency. We derive this impedance and study its analytic properties for a special model: radiation from circular orbits shielded by parallel plates representing the metallic vacuum chamber. We analyze the corresponding collective force, obtaining the usual formula as a first approximation, plus easily computed corrections that depend on present and prior values of the time derivative of the charge density. In related papers we have applied these results in numerical simulations of instabilities induced by CSR.

We examine the effect of the collective force due to coherent synchrotron radiation (CSR) in an electron storage ring with small bending radius. In a computation based on time-domain integration of the nonlinear Vlasov equation, we find the threshold current for a longitudinal microwave instability induced by CSR alone. The model accounts for suppression of radiation at long wavelengths due to shielding by the vacuum chamber. In a calculation just above threshold, small ripples in the charge distribution build up over a fraction of a synchrotron period, but then die out to yield a relatively smooth but altered distribution with eventual oscillations in bunch length. The instability evolves from small noise on an initial smooth bunch of rms length much greater than the shielding cutoff.

We present a model describing high power stable broadband coherent synchrotron radiation (CSR) in the terahertz frequency region in an electron storage ring. The model includes distortion of bunch shape from the synchrotron radiation (SR), which enhances higher frequency coherent emission, and limits to stable emission due to an instability excited by the SR wakefield. It gives a quantitative explanation of several features of the recent observations of CSR at the BESSY II storage ring. We also use this model to optimize the performance of a source for stable CSR emission.

We study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in 1 degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove the existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models.

Evidence of coherent synchrotron radiation has been reported recently at the electron storage rings of several light source facilities. The main features of the observations are (i) a radiation wavelength short compared to the nominal bunch length, and (ii) a coherent signal showing recurrent bursts of duration much shorter than the radiation damping time, but with spacing equal to a substantial fraction of the damping time. We present a model of beam longitudinal dynamics that reproduces these features.

We explore an algorithm for the construction of symplectic maps to describe nonlinear particle motion in circular accelerators. We emphasize maps for motion over one or a few full turns, which may provide an economical way of studying long-term stability in large machines such as the Superconducting Super Collider (SSC). The map is defined implicitly by a mixed-variable generating function, represented as a Fourier series in betatron angle variables, with coefficients given as B-spline functions of action variables and the total energy. Despite the implicit definition, iteration of the map proves to be a fast process. The method is illustrated with a realistic model of the SSC. We report extensive tests of accuracy and iteration time in various regions of phase space, and demonstrate the results by using single-turn maps to follow trajectories symplectically for 10⁷ turns on a workstation computer. The same method may be used to construct the Poincaré map of Hamiltonian systems in other fields of physics.

We describe a nonperturbative numerical technique for solving the Hamilton-Jacobi equation of a nonlinear Hamiltonian system. We find the time-periodic solutions that yield accurate approximations to invariant tori. The method is suited to the case in which the perturbation to the underlying integrable system has a periodic and not necessarily smooth dependence on the time. This case is important in accelerator theory, where the perturbation is a periodic step function in time. The Hamilton-Jacobi equation is approximated by its finite-dimensional Fourier projection with respect to angle variables, then solved by Newton’s method. To avoid Fourier analysis in time, which is not appropriate in the presence of step functions, we enforce time periodicity of solutions by a shooting algorithm. The method is tested in soluble models, and finally applied to a nonintegrable example, the transverse oscillations of a particle beam in a storage ring, in two degrees of freedom. In view of the time dependence of the Hamiltonian, this is a case with ‘‘21/2 degrees of freedom,’’ in which phenomena like Arnol’d diffusion can occur.

A general method to compute precise approximations to invariant tori of Hamiltonian systems is presented. For illustration, a strongly nonlinear example from accelerator theory is treated, in 21/2 degrees of freedom. Accuracy, computation time, and effectiveness near resonances are found to be highly favorable in comparison to previous methods.

By constructing action variables that are very nearly invariant in a region Ω of phase space, and by examining their residual variation, we set long-term bounds on any orbit starting in an open subregion of Ω. A new and generally applicable method for constructing the required high-precision invariants is applied. The technique is illustrated for transverse oscillations in a circular accelerator, a case with 21/2 degrees of freedom and strong nonlinearity.

We evaluate the longitudinal coupling impedance of a smooth toroidal vacuum chamber in the domain of frequencies below the first synchronous resonant mode. The chamber has rectangular cross section. With infinite wall conductivity, as assumed here, the nonresonant impedance is purely reactive. It consists of a space-charge term proportional to 1 / γ² and a curvature term which survives even when γ→∞. In the entire subresonant domain, the curvature term is well represented as a quadratic function of frequency: namely, Z / n=iZ₀(h / πR)²[A-3B(ν / π)²], where h is the height of the chamber and R is the trajectory radius and ν=ωh / c. The constants A and B are of order 1, being nearly equal to 1 if the width of the chamber is somewhat greater than its height. Thus, ImZ / n from curvature is typically a very small fraction of an ohm below the resonance domain, which begins when ν>(R / h)^{1 / 2}. Consequences for beam stability, if any, arise from high-frequency resonances which can produce values of several ohms for Z / n.

Integral equations for construction of a crossing-symmetric unitary Regge theory are extended to allow two coupled two-body channels. As in the case of a single channel, spectral functions are represented as Watson-Sommerfeld integrals over continued partial waves. A new type of partial wave is needed to represent one of the spectral functions in a region where one channel is open and the other is closed. This leads to certain difficulties in allowing realistic Regge poles.

In a crossing-symmetric Regge theory with several coupled two-particle channels, there are many-particle absorptive effects due to crossed two-particle processes. Also, some of the partial-wave amplitudes have overlapping left-hand and right-hand cuts in the s plane. To enforce unitarity in such a theory, one needs a coupled-channel N / D method allowing arbitrary absorption parameters and overlapping cuts. These requirements lead to a nonlinear marginally singular N / D equation, which is proposed as part of a scheme to construct a coupled-channel Regge theory with exact crossing symmetry.

The Roy equations, combined with unitarity, can be regarded as a system of integral equations for the π-π scattering amplitude in a finite energy region. Even when the partial-wave absorptive parts above this finite range are prescribed, and the two S-wave scattering-length parameters are held fixed, the singular equations have multiple solutions, some of which could be missed in a direct numerical study. We regularize the system by a modified N / D method, in which the full manifold of solutions is parametrized explicitly. If δ(s₀) is the phase shift of a particular wave at the eutoff point, then that wave carries a number of arbitrary, real parameters equal to the integer part of 2δ(s₀) / π, provided δ(s₀)≥-π / 2. We suggest that the N / D formulation is appropriate for applications of the Roy equations.

In parts I and II of this series, a system of partial-wave equations for construction of a crossing-symmetric unitary Regge theory of meson-meson scattering was described. Here we show that the sum of the partial waves of a solution has a representation in which crossing symmetry is apparent, all integrals converge without subtractions, double-spectral functions have the correct support, and the contributions of Regge poles in all three channels are displayed simultaneously. We obtain the Regge asymptotic limit for s→∞ at arbitrary fixed t by a method which avoids a difficulty in the usual heuristic argument. We also discuss the consequences at high energy of a new method of avoiding ghost poles at l=0 on even-signature trajectories.

Equations for the construction of a crossing-symmetric unitary Regge theory of meson-meson scattering are described. In the case of strong coupling, Regge trajectories are to be generated dynamically as zeros of the D function in a nonlinear N / D system. This paper is concerned mainly with writing the inputs to the N / D system in such a way that a convergent theory with exact crossing symmetry is defined. The scheme demands elimination of ghosts, i.e., bound-state poles at energies below threshold where trajectories pass through zero. A method for ghost elimination is proposed which entails an s-wave subtraction constant, and allows the physical s wave to be different from the l-analytic amplitude evaluated at l = 0. A dynamical model is suggested in which the subtraction constant alone generates the meson-meson interaction. An alternative ghost-elimination scheme proposed by Gell-Mann, in which only l-analytic amplitudes are involved, can be discussed in a formalism including channels with spin.

A program for construction of a crossing-symmetric unitary Regge theory of meson-meson scattering is proposed. The construction proceeds through solution of a nonlinear functional equation, ψ = G(ψ), for certain partial-wave scattering functions ψ. The functional equation is analogous to a conventional dynamical equation, in that the scattering amplitude is generated from input functions which describe the primary forces between mesons and possible inelastic effects. A solution of the equation provides a scattering amplitude having Mandelstam analyticity, exact crossing symmetry, exact unitarity below the production threshold, and meromorphy of partial waves in the right-half l plane, with the consequent Regge asymptotics. Inelastic unitarity [0 ≤ η(l,s) ≤ 1] is not guaranteed, but may perhaps be achieved through constraints on inputs. In any case, the partial waves are bounded throughout the physical region; such a bound was not ensured in earlier schemes based on the Mandelstam iteration. In this first paper of a series, the equations are formulated for the case of weak coupling, in which no Regge poles enter the right-half l plane. Inelastic effects are described by crossed two-particle processes and assigned input functions. Later papers will treat the case of strong coupling, in which Regge trajectories are generated dynamically, and the extension of the formalism to include many coupled channels.

The Mandelstam construction of crossing-symmetric, unitarity amplitudes for pion-pion scattering is reformulated in terms of partial waves. The amplitude is obtained by solving a set of nonlinear equations for the physical partial-wave amplitudes. A conformal mapping of the cosθ plane, introduced by Ciulli, Cutkosky, and Deo, is used to implement crossing symmetry. It is shown that the equations have a solution, which may be constructed by means of a convergent iteration.

Solutions of the Ball-Zachariasen equation for high-energy scattering are calculated unmerically. For small values of the parameter c, which measures the strength of particle production, the solutions coincide with those obtained in the existence proof of paper I. When cσ_el (σ_el=elastic cross section ) is much smaller than the physical value, the solutions are obtained successfully by either plain iteration or Newton-Kantorovich iteration, but they disagree qualitatively with experiment. A continuation to larger c is attempted by using the result of a successful iteration as the starting point for an iteration with slightly larger c. The continuation proceeds only a short distance before coming to a halt, because of a singularity of the Fréchet derivative of the nonlinear integral operator. The singularity is circumvented by an excursion into the complex c plane. On return to the real axis the solutions are complex, however, which is not allowed physically. A proposed approximate solution of Ball and Zachariasen, quite different from those of paper I, is also investigated. Plain or Newton-Kantorovich iteration starting with this function fails to converge. The failure is explained again in terms of a nearby singularity of the Fréchet derivative. The singularity makes it difficult to decide whether the Ball-Zachariasen function is close to a true solution.

Ball and Zachariasen have proposed a model of high-energy diffraction scattering, which entails nonshrinking diffraction peaks. The model has some resemblance to the multiperipheral model, in that it emphasizes s-channel unitarity with a simple factored form of production amplitudes. It leads to an integral equation for the elastic amplitude which is relatively simple in appearance. The equation is analyzed here with the help of methods which may also be useful in the study of more realistic high-energy models. The Hankel transform of the original equation is studied as a nonlinear equation in a certain Banach space. Existence of an infinite class of solutions is proved by means of the contraction mapping principle. These solutions are constructed by iteration for small values of a parameter c, which measures the strength of particle production. The range of allowed values for the product of c and the elastic cross section does not include the physical value. One can try, however, to continue the solutions to the physical value, since they are analytic in c. In paper II, the solutions are calculated numerically, and the continuation to large c is attempted. The continuation stops short of its goal, because of a singularity of the Fréchet derivative of the nonlinear operator. This derivative becomes a linear integral operator of the "third kind," which has no inverse in a space of continuous functions. No physically acceptable solutions of the Ball-Zachariasen equation have been found. A proposed approximate solution appears to have some difficulties, as is explained in paper II.

In a Regge theory of spinless elastic scattering, subtraction constants in physical partial waves are not always free parameters. In a partial wave where there is no Castillejo-Dalitz-Dyson (CDD) pole, the subtraction constant A_l(s₀), s₀<4m², is uniquely determined when the left-cut term and the elasticity η_l are specified. If there are CDD poles, all at finite points, then the subtraction constant is again fixed uniquely, but its value depends on the CDD parameters. If there is a CDD pole at infinity, the subtraction constant is unconstrained. These results are proved assuming that high-energy behavior is determined by a moving Pomeranchuk pole, with or without associated branch points. The analysis, although model independent, has implications for a dynamical model based on Reggeon exchange—namely, a model in which the input to the inelastic N / D equation (left- and right-cut parts) is constructed from crossed-channel Regge terms. In such a model the N / D equation is a regular Fredholm equation without high-energy truncation. In general, the phase shift obtained as output from the N / D equation has the high-energy behavior required by Regge theory if, and only if, the subtraction-constant constraint is satisfied. It is argued that new calculations are needed to test the Reggeon-exchange model. Earlier calculations in the Chew-Jones scheme are not conclusive for the formulation given here.

The relativistic equations expressing analyticity, crossing, and unitarity are analyzed without approximations. An infinite family of solutions is constructed, corresponding to a Castillejo-Dalitz-Dyson (CDD) ambiguity in the s wave. This ambiguity is in addition to the one resulting from arbitrary inelastic functions. The amplitudes constructed have nonvanishing single spectral functions, and this implies that a Kronecker δ is present in the angular-momentum plane. This relation between CDD poles and the Kronecker δ is proved only within a certain limited range of the coupling strength. A computational program for reaching the interesting domain of large couplings is outlined. In the latter domain it is not expected that all CDD poles entail Kronecker δ's in the l plane.

The CERN phase-shift analysis is used to evaluate the left-cut term of the P₁₁ partial wave of pion-nucleon scattering. The inelastic N / D equation is solved with this term and the elasticity η as input. The output and input phase shifts agree below 2 BeV if and only if at least two Castillejo-Dalitz-Dyson poles are included in the canonical D function. This result rules out single-channel bootstrap models and furnishes a constraint on many-channel models.

Fixed-point theorems are used to prove the existence of a class of solutions to the one-meson Low equation of the static-baryon model. The main result is that there exist solutions involving an arbitrary choice of narrow resonances. This is true for any crossing matrix with a finite number of channels, and for any cutoff function of a large class. For sufficiently small coupling constants, the solutions can be constructed by a convergent iteration procedure. The stable particles and the arbitrarily chosen resonances are associated with Castillejo-Dalitz-Dyson poles of an appropriate denominator function. The methods used do not suffice to show that solutions of the bootstrap type exist. Our earlier work is improved in that resonances are allowed and a bigger range of coupling constants and a weaker cutoff are permitted. The analysis is based on a crossing-symmetric N / D formulation of the Low equation.