Ingo Hofmann

Performance of solenoids versus quadrupoles in focusing and energy selection of laser accelerated protons

Using laser accelerated protons or ions for various applicationschar22{}for example in particle therapy or short-pulse radiographic diagnosticschar22{}requires an effective method of focusing and energy selection. We derive an analytical scaling for the performance of a solenoid compared with a doublet/triplet as function of the energy, which is confirmed by TRACEWIN simulations. Generally speaking, the two approaches are equivalent in focusing capability, if parameters are such that the solenoid length approximately equals its diameter. The scaling also shows that this is usually not the case above a few MeV; consequently, a solenoid needs to be pulsed or superconducting, whereas the quadrupoles can remain conventional. It is also important that the transmission of the triplet is found only 25% lower than that of the equivalent solenoid. Both systems are equally suitable for energy selection based on their chromatic effect as is shown using an initial distribution following the RPA simulation model by Yan et al. [Phys. Rev. Lett. 103, 135001 (2009].

Space-Charge Structural Instabilities and Resonances in High-Intensity Beams.

The existence of a structural resonance stop band caused by space charge in high-current beams, where the resonance frequency is associated with 90° phase advance per focusing period, is well known and alternatively referred to in the literature as envelope instability or as fourth-order resonance. We show, however, that this stop band is actually a coincidence of a structural fourth-order resonance and the much stronger envelope instability as competing mechanisms--depending on the time scale and initial matching. A similar complexity of behavior--dependent on the distribution function--is also found between a third-order instability and a sixth-order resonance in a 60° stop band. We claim that these findings are a generic property of high-intensity beams in periodic focusing; they also allow a reinterpretation of the 90° linear accelerator stop band previously observed experimentally at the UNILAC accelerator.

Revisiting the Longitudinal 90° Limit in High Intensity Linear Accelerators

Parametric envelope and sum envelope resonances are analyzed to revisit the validity of an assumed stop band and design limit of high intensity linear accelerators at a longitudinal phase advance of 90° per focusing lattice period. While the 90° limit is unquestioned in the transverse plane, we show here that it can be dropped as longitudinal limit for lattices with two or more rf gaps per focusing period. A new limit arises, however, from a novel transverse-longitudinal parametric sum envelope instability. The resulting sum instability rule allows the phase advance to exceed 90° longitudinally provided that transversely it remains correspondingly under 90°. We suggest that the additional design freedom opens the possibility for larger accelerating gradients and stronger longitudinal focusing with potential length and cost saving in the design of advanced superconducting linear accelerator concepts-as long as technological cavity limits are not reached.

Parametric sum envelope instability of periodically focused intense beams

The envelope instability is a second order parametric resonance with the periodic focusing and known to appear in space charge dominated beams near 90° phase advance per focusing period. We show in 2d approximation that space charge may also induce parametric “sum envelope instabilities” leading to simultaneous growth of envelopes or skew angles as well as emittances. This can happen by two-plane envelope coupling or by exciting a skew (“odd”) mode in an otherwise fully uncoupled linear lattice. At resonance, the two individual phase advances are split more or less symmetrically away from 90°, and exponential growth occurs. Results from perturbation theory are compared with full envelope models, particle-in-cell simulations, and smooth approximation stopband calculations, all showing very good agreement for realistic space charge parameters.

Screening of the resistive-wall impedance by a cylindrical electron plasma

The effect of an electron cloud on the longitudinal coupling impedance is studied by idealizing it as a cold and uniformly distributed non-neutral plasma of electrons. The beam pipe is assumed to be of circular cross section with a thick resistive wall and the beam charge is idealized as a uniform disk. The electron contribution to the charge and current densities is obtained from the collective electron response to the beam passage through the pipe. In the presence of the electron background, a general closed formula for the longitudinal coupling impedance is obtained. The screening of the coupling impedance with the density of the electron plasma is studied and discussed for typical parameters in an accelerator beam pipe for the under-dense and the over-dense plasma regions.

Long Term Simulations of Space Charge and Beam Loss Observed in the CERN Proton Synchrotron

Long term storage of high intensity beams with small loss is required in the FAIR project at GSI as well as for JPARC. In this paper we discuss that an important contribution to the loss in bunched beams can be explained it terms of particles trapped into lattice and space charge driven islands. Dedicated experiments at the CERN‐Proton Synchrotron to confirm the theoretical model have shown the existence of an emittance growth dominated regime for working points sufficiently far from the lattice resonance, and of a beam loss dominated regime for tunes very close to the resonance. While the emittance growth dominated regime has been investigated in previous studies, we focus here on the beam loss dominated regime and compare simulation results with measurements made in the CERN‐Proton Synchrotron ring.

Fluid description of the longitudinal instability in high current coasting beams

The longitudinal instability of an intense coasting beam in the ESR (Experimentier–Speicherring) [G. Rumolo et al., Nucl. Instrum. Methods A415, 411 (1998)] has been investigated far away from the stability boundary for several values of the machine impedance. Experiments have clearly shown the growth of the slow wave, generation of higher harmonics, asymmetric wave steepening, and saturation. A one-dimensional fluid model has been successfully employed to explain the growth of the slow wave as well as the nonlinear steepening and the harmonic generation. Taking into account the effects of the initial momentum spread of the beam through a modified space charge impedance, the model predicts with high accuracy the rise time and the frequency shift of the unstable wave. Subsequently, nonlinear convective effects are shown to give rise to wave steepening and harmonic generation. Predictions of the fluid model are compared to experimental data and with those obtained under a full kinetic model as well as from ...

Comparison between theory and simulations for longitudinal instabilities of coasting beams

Abstract By using the linear kinetic theory and the particle-in-cell code SCOP-RZ/PATRIC the longitudinal instability observed in the ESR has been investigated for different values of the longitudinal impedance of the machine. The numerical investigations show that initially the instability grows exponentially as predicted by the linear theory. When the perturbation reaches large values, significant high-order harmonics are produced: steepening of the density profile as well as saturation of the instability growth appear first, and turbulence later on. Analytical and numerical studies with a small RF voltage applied at the detuned cavity gap have also been carried out in order to meet the conditions in which actual measurements were made at the ESR.

Grid dependent noise and entropy growth in anisotropic 3d particle-in-cell simulation of high intensity beams

The numerical noise inherent to particle-in-cell simulation of 3D high intensity bunched beams is studied with the TRACEWIN code and compared with an analytical model by Struckmeier. The latter assumes that entropy growth can be related to Markov type stochastic processes due to temperature anisotropy and the artificial collisions caused by using macro-particles and calculating the space charge effect. The resulting noise can lead to growth of the six-dimensional rms emittance and a suitably defined rms entropy. We confirm the dependence of this growth on the bunch temperature anisotropy as predicted by Struckmeier. However,we also find an apparently modified mechanism of noise generation by the non-Liouvillean effect of the Poisson solver grid, which exists in periodic focusing systems even in the complete absence of temperature anisotropy. Our findings are applicable in particular to high current linac simulation, where they can provide guidance to estimate noise effects and help finding an effective balance between the number of simulation particles and the grid resolution.

Halo coupling and cleaning by a space charge resonance in high intensity beams

We show that the difference resonance driven by the space charge pseudo-octupole of high-intensity beams not only couples the beam core emittances; it can also lead to emittance exchange in the beam halo, which is of relevance for beam loss in high intensity accelerators. With reference to linear accelerators the main resonance kz/kxy =1 (corresponding to the Montague resonance 2Qx-2Qy=0 in circular accelerators) may lead to such a coupling and transfer of halo between planes. Coupling of transverse halo into the longitudinal plane - or vice versa - can occur even if the core (rms) emittances are exactly or nearly equal. This halo argument justifies additional caution in linac design including consideration of avoiding an equipartitioned design. At the same time, however, this mechanism may also qualify as active dynamical halo cleaning scheme by coupling a halo from the longitudinal plane into the transverse plane, where local scraping is accessible. We present semi-analytical emittance coupling rates and show that previously developed linac stability charts for the core can be extended - using the longitudinal to transverse halo emittance ratio - to indicate additional regions where halo coupling could be of importance.