Search Results (27 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.

It has been predicted and found experimentally that the polarization direction of particles on the closed orbit of a circular accelerator can be manipulated, without a noticeable reduction of polarization, by means of a slow variation of magnetic fields. This feature has been used to avoid imperfection resonances where the spin precession frequency is close to a multiple of the circulation frequency. As a first step we show that this property is related to an adiabatic invariant of spin motion. The proof is relatively simple since it involves only two frequencies, the spin-rotation frequency and the particle’s rotation frequency on the closed orbit. The invariant spin field (ISF) describes a periodic polarization state of a beam’s phase-space distribution. This ISF leads to a very useful parametrization of coupled spin and orbit dynamics. We prove that this ISF gives rise to an adiabatic invariant of spin-orbit motion. This proof is much more complicated since the orbital frequencies are involved. Because of this adiabatic invariance, a beam’s spin field follows slow changes of the accelerator’s ISF that can occur during a slow acceleration cycle. This feature is essential when high-order spin-orbit resonances are crossed, since it allows polarization that has been reduced at the resonance condition to be recovered, to a large degree, after the resonances have been crossed.

We reply to Lee and Mane’s foregoing Comment [Phys. Rev. ST Accel. Beams 8, 089001 (2005)]. In particular, we discuss how an adherence to certain notions of spin-orbit resonance and spin tune can limit the analysis and understanding of phenomena. Since the Comment has very little to do with the main thrust of our paper we take the opportunity to point out the main features of the “proper uniform precession rate,” a concept introduced in our paper and based on the concept of quasiperiodicity. We also respond to other material in the Comment.

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 an in-depth analysis of the concept of spin precession frequency for integrable orbital motion in storage rings. Spin motion on the periodic closed orbit of a storage ring can be analyzed in terms of the Floquet theorem for equations of motion with periodic parameters, and a spin precession frequency emerges in a Floquet exponent as an additional frequency of the system. To define a spin precession frequency on nonperiodic synchrobetatron orbits we exploit the important concept of quasiperiodicity. This allows a generalization of the Floquet theorem so that a spin precession frequency can be defined in this case, too. This frequency appears in a Floquet-like exponent as an additional frequency in the system in analogy with the case of motion on the closed orbit. These circumstances lead naturally to the definition of the uniform precession rate and a definition of spin tune. A spin tune is a uniform precession rate obtained when certain conditions are fulfilled. Having defined spin tune we define spin-orbit resonance on synchrobetatron orbits and examine its consequences. We give conditions for the existence of uniform precession rates and spin tunes (e.g., where small divisors are controlled by applying a Diophantine condition) and illustrate the various aspects of our description with several examples. The formalism also suggests the use of spectral analysis to “measure” spin tune during computer simulations of spin motion on synchrobetatron orbits.

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.

Luminosity-driven channeling extraction has been observed for the first time in a 900 GeV study at the Fermilab Tevatron. This experiment, Fermilab E853, demonstrated that useful TeV level beams can be extracted from a superconducting accelerator during high luminosity collider operations without unduly affecting the background at the collider detectors. Multipass extraction was found to increase the efficiency of the process significantly. The beam extraction efficiency was about 25%. Studies of time dependent effects found that the turn-to-turn structure was governed mainly by accelerator beam dynamics. Based on the results of this experiment, it is feasible to construct a parasitic 5–10 MHz proton beam from the Tevatron collider.

We develop the method of weighted macroparticle tracking (WMPT) for simulating the time evolution of the moments of the phase space densities of two beams which are coupled via the collective (strong-strong) beam-beam interaction in the absence of diffusion and damping. As an initial test we apply this method to study the π mode and the σ mode in three different 1D limits of the beam-beam interaction. The three limits are flat beams and transverse motion in the direction of the small width, round beams, and flat beams and motion in the direction of the large width. We have written a code (BBDeMo1D) based on WMPT, which allows testing of all three limits and is suited for extension to 2 degrees of freedom.

Luminosity-driven channeling extraction has been observed for the first time using a 900 GeV circulating proton beam at the superconducting Fermilab Tevatron. The extraction efficiency was found to be about 30%. A 150 kHz beam was obtained during luminosity-driven extraction with a tolerable background rate at the collider experiments. A 900 kHz beam was obtained when the background limits were doubled. This is the highest energy at which channeling has been observed.

Random fluctuations in the tune, beam offsets, and beam size in the presence of the beam-beam interaction are shown to lead to significant particle diffusion and emittance growth in hadron colliders. We find that far from resonances high frequency noise causes the most diffusion while near resonances low frequency noise is responsible for the large emittance growth observed. Comparison of different fluctuations shows that offset fluctuations between the beams cause the largest diffusion for particles in the beam core.

A recent experiment at the Indiana University Cyclotron Facility with electron cooling showed that rf phase modulation near the 1:1 resonance leads to longitudinal beam splitting. Here we explain this by applying the method of averaging, a powerful tool from the study of dynamical systems, to the underlying equation of motion—a pendulum equation with small damping and periodic forcing. The beam splitting is explained by showing that the associated Poincaré map has two attracting fixed points, each with a well-defined basin of attraction. Our approach can be immediately applied to other accelerator-physics problems governed by a similar equation.

Strictly one-dimensional theories of particle loss due to longitudinal diffusion model the loss by an absorbing boundary condition at the separatrix of the underlying unperturbed motion. Particle loss always occurs at a physical aperture and the loss is always coupled to the betatron motion. A theory of particle loss which includes the effect of betatron motion is presented. Results are compared with Monte Carlo simulations.

Catastrophic planar dechanneling in strained-layer superlattices provides for a critical evaluation of fundamental-particle channeling phenomena, as well as a method of strain assessment. Using continuum-model theory, the equation of projectile motion is solved numerically. The depth and angular dependences of catastrophic dechanneling are examined as a function of the parameters: incident energy and angle, superlattice strain, crystal potential, and the minimum impact parameter for channeling. The catastrophic dechanneling increases with the strain-induced tilt angle Δψ and contains well-defined structures in the angular and depth dependences. These features are particularly sensitive to strain for the region Δψ∼0.1 to 1.0 times the critical angle for channeling. Also, there is a range of energies (∼±10%) about the resonant value for which good catastrophic dechanneling behavior is retained. Using the thermally averaged Doyle-Turner potential and data from a GaAs_xP_1-x/GaP superlattice, we obtain a minimum impact parameter for planar channeling of 1.3 times the Thomas-Fermi screening length. When scaled by thermal vibrational amplitude, this value (2.3u₁, where u₁ is the one-dimensional amplitude) is consistent with recent computer-simulation results for the steering of particles by single-crystal planes.

A resonance effect is reported for planar channeling of MeV helium ions in a strained-layer superlattice. In addition to the well-documented catastrophic dechanneling which maximizes dechanneling, a novel resonance channeling effect has been observed that minimizes the dechanneling. These two contrasting phenomena are demonstrated in a single sample by varying the projectile energy for channeling along the inclined {110} planes of a GaAs_0.8P_0.2/GaAs superlattice of equal-thickness layers. The resonance phenomena are measured and modeled theoretically as a function of ion energy and crystal strain.

We report systematic studies of energetic particle channeling along the inclined (110) crystal planes in a strained-layer superlattice (SLS) under conditions of catastrophic dechanneling. This phenomenon is distinguished by the rapid enhancement in dechanneling after the penetration of several superlattice layers due to a resonance between the channeled particle wavelength and the SLS period. Catastrophic dechanneling conditions are achieved in a (100) GaAs_0.09P_0.91/GaP superlattice with layers of equal thickness (34 nm) along the inclined (110) planes for a 1.2-MeV ⁴He beam. Due to the alternating tilts at each interface, a rapid increase in the scattering yield to a value close to that for random scattering occurs after penetration of 1–5 superlattice layers, and a strong asymmetric dependence on incident angle is observed. The theoretically predicted depth and angular dependences of the catastrophic dechanneling are calculated by numerically integrating for various initial conditions the differential equation describing planar channeled particle motion in SLS’s. A two-parameter, least-squares fit gives excellent agreement between theory and experiment for a tilt angle of Δψ=0.153° and a critical impact parameter of r_c=1.25a_T. Steps are found in the depth and angular dependence of the catastrophic dechanneling. This structure provides the greatest sensitivity to the relative strain between superlattice layers of any of the ion-channeling methods.

The well-known continuum model theory for planar channeled energetic particles in perfect crystals is extended to layered crystalline structures and applied to superlattices. In a strained-layer structure, the planar channels with normals which are not perpendicular to the growth direction change their direction at each interface, and this dramatically influences the channeling behavior. The governing equation of motion for a planar channeled ion in a strained-layer superlattice with equal layer thicknesses is a one degree of freedom nonlinear oscillator which is periodically forced with a sequence of δ functions. These δ functions, which are of equal spacing and amplitude with alternating sign, represent the tilts at each of the interfaces. Thus upon matching an effective channeled particle wavelength, corresponding to a natural period of the nonlinear oscillator, to the period of the strained-layer superlattice, corresponding to the periodic forcing, strong resonance effects are expected. The condition of one effective wavelength per period corresponds to a rapid dechanneling at a well-defined depth (catastrophic dechanneling), whereas two wavelengths per period corresponds to no enhanced dechanneling after the first one or two layers (resonance channeling). A phase plane analysis is used to characterize the channeled particle motion. Detailed calculations using the Moliere continuum potential are compared with our previously described modified harmonic model, and new results are presented for the phase plane evolution, as well as the dechanneling as a function of depth, incident angle, energy, and layer thickness. General scaling laws are developed and nearly universal curves are obtained for the dechanneling versus depth under catastrophic dechanneling.

The Krylov-Bogoliubov method of averaging is applied to the time-dependent quantum anharmonic oscillator. A regular perturbation expansion contains secular terms. The averaging approximation does not, and as a result has a validity over larger time intervals. A new variant of the usual averaging transformation is used and rigorous error bounds are derived. Rigorous averaging methods have been applied extensively to ordinary differential equations but our work appears to be the first generalization to partial differential equations.

A strong asymmetry has been observed in the angular dependence of the catastrophic dechanneling depth in strained-layer superlattices. This new effect occurs under resonance conditions, where the wavelength of the planar-channeled particles equals the period of the superlattice, and is demonstrated for {110} planar channeling in GaP/GaAs_xP_1-x. A theoretical, phase-plane analysis shows that the angular tilt required for shifting of the catastrophic depth by one layer is a direct measure of the strain in the first few layers.

Derivations of the axial- and planar-channeling continuum models from the perfect-string and perfect-plane models by the Krylov-Bogoliubov method of averaging are discussed. For the first time, error bounds are obtained relating the perfect crystal trajectories to the continuum-model trajectories. Details are presented for the axial case, because this case is easier to treat and the generalization to planes is straightforward.

A resonance effect is observed between the wavelength of a planar-channeled ion beam and the layer period of a strained-layer superlattice. Catastrophic dechanneling is observed when the wavelength and layer period are matched. A phase-rotation analysis is developed to calculate the amount of dechanneling and the amount of strain in the superlattice.

The elements of continuum-model axial channeling are discussed with an emphasis on the depth-dependent phase-space density. This density, which is the joint spatial-momentum density, does not contain effects of thermal vibration or electron multiple scattering, but does contain the ordered nature of the continuum strings. Representations for the depth-dependent momentum density are derived, and two approximations for its determination are developed. The angular dependence of the momentum density is studied using these approximations, and the results are compared with recent data from axial-channeling transmission experiments in thin crystals. The agreement between theory and experiment is quite good.

The elements of continuum-model planar channeling are discussed using a general planar continuum potential with an emphasis on the depth-dependent phase-space density. This joint spatial-momentum density contains all the information concerning continuum-model planar channeled particles and allows a unified treatment of the depth-dependent and statistical equilibrium, spatial, and momentum densities for an arbitrary initial density. The Gaussian-beam-divergence case is discussed in some detail. A simple, two-parameter planar-continuum potential, the tangent-squared potential, is then introduced. We show that this potential is physically reasonable, and, for many calculations, easier to use than previously used planar continuum potentials such as the Lindhard, Moliere, and hyperbolic cosine. It simplifies many calculations because the channeled-particle wavelength function and the solutions of the associated equation of motion can be written in terms of elementary functions (specifically the square root, inverse sine, and trigonometric functions) and because the phase-space density has a simple analytic form.

The phase-space density for planar channeled particles has been derived for the continuum model under statistical equilibrium. This is used to obtain the particle spatial probability density as a function of incident angle. The spatial density is shown to depend on only two parameters, a normalized incident angle and a normalized planar spacing. This normalization is used to obtain, by numerical calculation, a set of universal curves for the spatial density and also for the channeled-particle wavelength as a function of amplitude. Using these universal curves, the statistical-equilibrium spatial density and the channeled-particle wavelength can be easily obtained for any case for which the continuum model can be applied. Also, a new one-parameter analytic approximation to the spatial density is developed. This parabolic approximation is shown to give excellent agreement with the exact calculations.