T.V. Gorlov, J.A. Holmes
ORNL, Oak Ridge, Tennessee, USA
A self-consistent beam maintains linear space charge forces under any linear transport, even with the inclusion of space charge in the dynamics. Simulation indicates that it is possible to approximate certain self-consistent distributions in a ring with the use of phase space painting. We focus on the so-called Danilov distribution, which is a uniform density, rotating, elliptical distribution in the transverse plane and a coasting beam in the longitudinal plane. Painting the beam requires measurement and control of the orbit at the injection point, and measuring the beam requires re- construction of the four-dimensional (4D) transverse phase space. We discuss efforts to meet these requirements in the Spallation Neutron Source (SNS) ring.
N. Mounet, E. Dadiani, E. Métral, C. Zannini
CERN, Geneva, Switzerland
A. Rahemtulla
EPFL, Lausanne, Switzerland
Indirect space charge contributes significantly to the impedance of non ultrarelativistic machines such as the LEIR, PSB and PS, at CERN. While general expressions exist in frequency domain for the beam coupling impedance, the time domain wake function is typically obtained numerically, thanks to an inverse Fourier transform. An analytical expression for the indirect space charge wake function, including the time dependence as a function of particle velocity, is nevertheless highly desirable to improve the accuracy of time domain beam dynamics simulations of coherent instabilities. In this work, a general formula for the indirect space charge wake function is derived from the residue theorem. Moreover, simple approximated expressions reproducing the time and velocity dependence are also provided, which can even be corrected to recover an exact formula, thanks to a numerical factor computed once for all. The expressions obtained are successfully benchmarked with a purely numerical approach based on the Fourier transform.
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A.J. Laut, A. Lasheen
CERN, Geneva 23, Switzerland
Particles in an intense bunch experience longitudinal self-fields due to space~charge. This effect, conveniently described by geometric factors dependent on a particle¿s transverse position, beam size, and beam pipe aperture, is usually incorporated into longitudinal particle tracking on a per-turn basis. The influence of transverse betatron motion on longitudinal space~charge forces is, however, usually neglected in pure longitudinal tracking codes. A dedicated tracking code was developed to characterize the CERN PS such that an effective geometric factor of a given particle could be derived from its transverse emittance, betatron phase~advance, and momentum~spread. The effective geometry factor is then estimated per particle by interpolation without the need for full transverse tracking and incorporated into the longitudinal tracker BLonD. The paper evaluates this effect under conditions representative of the PS, where space~charge is dominant at low energy and progressively becomes negligible along the acceleration ramp. The synchrotron frequency distribution is modified and the filamentation rate is moreover increased, which could suggest a stabilizing space~charge phenomenon.
I. Karpov, T. Argyropoulos, E.N. Shaposhnikova
CERN, Meyrin, Switzerland
S. Nese
University of Bergen, Bergen, Norway
Landau damping is a very important stabilization mechanism of beams in circular hadron accelerators. In the longitudinal plane, Landau damping is lost when the coherent mode is outside of the incoherent synchrotron frequency spread. In this paper, the threshold for loss of Landau damping (LLD) for constant inductive impedance ImZ/k is derived using the Lebedev matrix equation (1968). The results are confirmed by direct numerical solutions of the Lebedev equation and using the Oide-Yokoya method (1990). For more realistic impedance models of the ring, new definitions of an effective impedance and the corresponding cutoff frequency are introduced which allow using the same analytic expression for the LLD threshold. We also demonstrate that this threshold is significantly overestimated by the Sacherer formalism based on the previous definition of an effective impedance using the eigenfunctions of the coherent modes.
In the context of the LHC Injectors Upgrade (LIU) project, a series of studies have been performed in order to better understand the beam brightness limitations imposed by resonances and space charge effects. Space charge simulations using the analytic (frozen) space charge solver as implemented in the MAD-X code conducted for the CERN Proton Synchrotron (PS) show that a particle approaching the integer tune of Qx = 6 demonstrates a resonant behavior. The analysis of the single particle transverse motion reveals the excitation of a second order resonance. The interplay of the space charge effect and the optics perturbation in the regime of the integer tune on this excitation was further investigated. The simulations were complemented with the analysis of the resonance driving terms coming from the space charge potential derived in a classical perturbative approach.
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F. Schmidt, A. Latina, H. Renshall
CERN, Meyrin, Switzerland
Y.I. Alexahin
Fermilab, Batavia, Illinois, USA
In 2018 as part of a collaboration between CERN and FNAL, the space charge (SC) implementation has been upgraded in a test version of MAD-X. The goal has been to implement the 3D symplectic SC kick together with a number of new features and benchmark it with earlier MADX-SC versions. Emphasis has given to the use of the Sigma Matrix approach that allows to extend MAD-X optics calculations. In the meantime, significant effort has been made to fully debug and optimize the code and in particular to achieve a speed-up of the simulations by a factor of 2. The code has been ported to the latest MAD-X version, the elaborated set-up procedures have been automated and a user manual has been written.
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R. Ainsworth, A.V. Burov, N. Eddy, A. Semenov
Fermilab, Batavia, Illinois, USA
Recent advances in the theoretical understanding of beam stability in the presence of strong space charge, has suggested a new class of instabilities known as convective instabilities. A novel approach to excite and study these instabilities will be to install a ‘waker’ system, a dedicated wake-building feedback system. The System was installed in the Fermilab Recycler and commissioned during 2021. The first results are presented.
O. Boine-Frankenheim
TEMF, TU Darmstadt, Darmstadt, Germany
Understanding the 3D collective long-term response of beams exposed to resonances is of theoretical interest and essential for advancing high intensity synchrotrons. This study of a hitherto unexplored beam dynamical regime is based on 2D and 3D self-consistent particle-in-cell simulations and on careful analysis using tune spectra and phase space. It shows that in Gaussian-like beams Landau damping suppresses all coherent parametric resonances, which are of higher than second order (the "envelope instability"). Our 3D results are obtained in an exemplary stopband, which includes the second order coherent parametric resonance and a fourth order structural resonance. They show that slow synchrotron oscillation plays a significant role. Moreover, for the early time evolution of emittance growth the interplay of incoherent and coherent resonance response matters, and differentiation between halo and different core regions is essential. In the long-term behavior we identify a progressive, self-consistent drift of particles toward and across the resonance, which results in effective compression of the initial tune spectrum. However, no visible imprint of the coherent features is left over, which only control the picture during the first one or two synchrotron periods. An intensity limit criterion and an asymptotic formula for long-term rms emittance growth are suggested. Comparison with the commonly used non-self-consistent "frozen space charge" model shows that in 3D this approximation yields a fast and useful orientation, but it is a conservative estimate of the tolerable intensity. HB’21 talk on "Effect of Space Charge on Bunch Stability and Space Charge Compensation Schemes" based on this APS PR-AB published contribution.
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