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The effect of space charge on bunches stored for long term in a synchrotron can be severe for beam survival. This may be the case in projects as SIS100 at GSI or LIU at CERN. In the past decade systematic simulation studies and experiments performed at CERN and GSI have highlighted the space charge induced periodic crossing of one dimensional resonances as an underlying mechanism of long term beam loss or emittance growth. However only in 2012, for the first time, the effect of space charge on a normal third order coupled resonance was investigated at the CERN-PS. The experimental results have highlighted an unprecedented asymmetric beam response where in the vertical plane the beam exhibits a thick halo, whereas the horizontal profile has only core growth. The quest for explaining these results requires a journey through the 4 dimensional dynamics of the coupled resonance investigating the fixed-lines, and requires a detailed code-experiment benchmarking also including beam profile benchmarking. This study shows that the experimental results of the 2012 PS measurements can be explained by the dynamics the of the fixed-lines with space charge also including the effect of the chromaticity. THE ACCELERATOR CASE Space charge effects on beam circulating in a synchrotron becomes stronger for an increased beam intensity, and the problems arising with the increased intensity define an intensity limit of performance. The space charge limit is a figure of merit used to characterize the maximum intensity tolerated by a machine. It is usually defined with respect to the beam loss, although other criteria can be used to define the limit performance: for example, emittance growth, halo formation. For projects the space charge limit plays a significant role, because it defines the ultimate accelerator performance in terms of stored particles. Usually the space charge limit is defined by a constraint on the maximum allowed incoherent space charge tune-shift. This is dictated by the working point the machine for avoiding the overlapping of the space charge tune-spread with machine resonances. For a well placed machine working point, the maximum allowed incoherent tune-shift is usually ΔQsc ∼ −0.25 to avoid second, third, and fourth order resonances. Clearly this argument mainly depends on which machine resonances are significant for beam loss. If the 3rd and 4th order resonances are very weak one can use more optimistically relax the constraint to ΔQsc ∼ −0.5. The assessment of the real effect of the space charge on machine resonances requires a detailed study of the dynamics, and it is strongly related to the storage time. The investigation of the effect of space charge for long term storage of a bunched beam has been motivated from the planned operations with U28+ ions in the SIS100 synchrotron of the FAIR project [1]. In this scenario SIS18 will operate with harmonics 2, and 4 cycles will be injected into SIS100 for a total of 8 bunches transferred in 10 buckets. An amount of 6×1011 ions will be accumulated during 1 second at the injection energy of 200 MeV/u. This means that the first 2 bunches from the SIS18 cycle injected into SIS100 will be stored for 1 second, namely for 1.5 × 105 turns. During this time the beam with transverse rms emittances of x,rms = 8.75 mm-mrad, y,rms = 3 mm-mrad, will be bunched with bunching factor Bf = 0.33, generating an incoherent space charge tune-shift ofΔQx −0.2,ΔQy −0.33. In the SIS100 RF buckets U28+ ions follow a longitudinal dynamics with a synchrotron tune of Qs ∼ 4.27 × 10−3, and over 1 second storage, a particle will perform 643 or less synchrotron oscillations. This accelerator regime is very unique with no established operational experience in the accelerator community. The synchrotron motion pushes periodically bunched particles from regions of small transverse space charge, to the region of large space charge tune-shift (at z = 0). Therefore, according to the initial condition, a particle may cross a machine resonance each quarter of a synchrotron oscillation. A fast single crossing of a weak resonance typically produces a small emittance growth: in SIS100 for U28+ ions a quarter of synchrotron oscillation happens in ∼ 58 turns. However, over one second storage the number of resonance crossings becomes as high as ∼ 2700. The integrated effect arising from this large number of repeated resonance crossing may create a large impact generating a beam diffusion to large transverse amplitudes, hence may lead to a steady beam loss throughout the storage time. Note that for this effect to be significant requires many resonance crossings, hence it disappears, or becomes insignificant for short term storage. The assessment of the beam dynamics in this regime is challenging. The simulations are difficult: standard PIC codes are subjected to artificial noise, which affects the long term result of the simulations [2–5]. In order to mitigate this artifact one should use a very large number of macroparticles [5], which makes unfeasible systematic studies to understand the properties of the dynamics. Especially it is difficult to develop tools which may be used to predict the effects, or to develop a theory that may allow to assess how the different beam and machine features influence the emittance growth. Proceedings of IPAC2017, Copenhagen, Denmark WEXB1 05 Beam Dynamics and Electromagnetic Fields D07 High Intensity Circular Machines Space Charge, Halos ISBN 978-3-95450-182-3 2503 Co py rig ht © 20 17 CC -B Y3. 0 an d by th er es pe ct iv ea ut ho rs SPACE CHARGE AND ONE DIMENSIONAL RESONANCES In the past decade, the interplay of space charge and machine nonlinearities has been studied for 1D resonances, i.e. type nxQx = N . Experimental and numerical studies on beam survival and emittance growth in this regime have been carried out in the CERN-PS in the year 2003 for the resonance 4Qx = 25 (see Ref. [6, 7]), and later in the SIS18 at GSI for the resonance 3Qx = 13 (see Ref. [8]). The mechanism leading to emittance growth and beam loss is explained in terms of instantaneous stable islands in the two-dimensional phase space and their crossing the particle orbits because of the combined effect of space charge and synchrotron motion [9]. The one dimensional resonances allow to discuss the dynamics in one plane. The key features understood from past studies are summarized in the following qualitative points: a) The space charge detuning determines the position of the fixed-points generated by the 1D resonance, which is approximately given by the condition n[Qx0 + ΔQx(X)] = N, (1) where here X is the amplitude of the resonant particle, and ΔQx(X) is the incoherent amplitude dependent space charge detuning. For X = 0 the quantity ΔQx(0) corresponds to the usual incoherent tune-shift. b) The strength of the resonance determines the tune of particles around the fixed-points, and consequently how large is the island. The island size has folded-in also the detuning created by space charge: stronger gradients in the amplitude dependent detuning leads to smaller islands. c) The synchrotron tune determines how fast is the resonance crossing. A figure of merit on how fast the resonance crossing is, is given by the parameter T obtained as the ratio between the speed of migration of the fixed-points to the maximum speed of rotation of the particle in the island. If this ratio is small (T 1) it means that the motion is adiabatic, and the particles remain locked to the island (trapping); If instead T > 1 then a single resonance crossing produces a “kick” to the particle invariant (scattering). The standard accelerators settings bring the dynamics mainly to scattering processes, which are highly dependent from the strength of the resonance, space charge tune-shift, distance from the resonance, and synchrotron tune. THE DIFFICULTY OF THE TWO DIMENSIONAL RESONANCES For a two dimensional resonance as nxQx+nyQy = N , the dynamics are coupled. That means that a two dimensional analysis of the Poincaré section of the coordinates x, x ′ does not provides the proper understanding. The major difficulty is in understanding how the space charge modify the tori embedded in the four dimensional phase space. Usually the tune-spread is not the same in horizontal and vertical planes, hence it is particularly difficult to conclude on the process of halo formation, and on its extension in the x − y plane. The meaning of trapping/scattering is also not straightforward to define. MEASUREMENTS ON THE 3RD ORDER COUPLED RESONANCE Recently the LIU project at CERN [10] has engaged in studies to explore space charge effects [11], and in investigating the long term storage for high intensity bunches as well. In the framework of the CERN-GSI collaboration, studies on the effect of space charge on the resonance Qx + 2Qy = 19, have been carried out in the CERN-PS in 2012. We present here a short summary of the full experiment analysis. Amore comprehensive presentation of the experimental results and discussion is part of a future publication in Ref. [12]. In the experiment, in a resonance-free region of the tune diagram the third order resonanceQx+2Qy = 19was excited with two nearby sextupoles of strength K2 0.015 m−2. The experimental campaign used a beam with 55 × 1010 protons per bunch, which produced an incoherent space charge tuneshift of ΔQx −0.05,ΔQy −0.07. The beam was stored at the energy of 2 GeV for 1.1 seconds corresponding to ∼ 0.5×106 turns. During this time, the synchrotron tuneQs ∼ 8.57 × 10−4 induces ∼ 1700 possible resonance crossings, and their effect on the beam was assessed by measuring the beam profiles at the beginning and at the end of the storage time. The measurements were repeated for several machine tunes so to explore the dependence of the effects on the distance from the resonance. The comparison of initial and final beam profiles revealed an unexpected beam response to the distance from the resonance. When

Studies and observations of beam dynamics near a sum resonance