LONGITUDINAL SPACE CHARGE SIMULATIONS WITH BLOND AT INJECTION IN THE CERN PS BOOSTER D. Quartullo, V. Forte Acknowledgments: E. Benedetto, A. Lasheen, G. Rumolo, E. Shaposhnikova, H. Timko, L. Wang, C. Zannini 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 1
Motivation (1/2) The plans of the LHC Injectors Upgrade (LIU) project for the CERN PS Booster: Increase of injection and extraction energy to 160 MeV and 2 GeV respectevely (with Linac4) Replacement of the existing RF cavities with FINEMET cavities Analysis of longitudinal beam stability with new RF system using realistic impedance model Since the space charge impedance is dominant in the longitudinal plane it’s very important to have: A reliable and fast code to simulate longitudinal dynamics A good evaluation of the longitudinal space charge impedance Studies of the new injection scheme from Linac4 to reduce the transverse space charge Longitudinal simulations needed for transverse beam dynamics 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 2
Motivation (2/2) PyOrbit has been used extensively to study longitudinal dynamics for transverse purposes (see slides by V. Forte at LIU-PSB Injection Working Group, CERN) . PyOrbit (full 6D tracking code) can take one day to simulate 10000 turns with longitudinal space charge and PTC tracking using parallelization with 48 cores. BLonD is a pure longitudinal code recently developed at CERN; it needs 20 minutes with one core for the same simulation. The benchmark between the two codes was aimed to check if the two software give the same results Expected target parameters for LHC standard beams from Linac4 have been taken as reference: 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 3
CONTENTS Motivation Part I: Benchmark between BLonD and PyOrbit (with V. Forte) Part II: SC impedance calculation with LSC code Conclusions Bibliography 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 4
Starting scenario • Longitudinal space charge simulations done with PyOrbit to study the injection of Linac4 bunches in the PS Booster. • Time frame considered: the first 10 ms of the injection. uniform waterbag like t = 0 ms t = 10 ms • Evolution in phase space of 5000 particles distributed uniformly between –π and π without space charge. • Acceleration implies bucket shrinkage and synchronous phase displacement. • Double RF. 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 5
Injection optimisation with PyOrbit • From single Linac4 bunches it is possible to create arbitrary injection structures The goal was to optimize: bunch length to minimize the number of injected turns; energy spread to minimize filamentation and peaks in line density. 𝚬𝒒 𝚬𝒒 𝚬𝒒 𝒒 = 𝟏. 𝟒𝟗 ∙ 𝟐𝟏 −𝟒 𝒒 = 𝟐. 𝟒𝟕 ∙ 𝟐𝟏 −𝟒 𝒒 = 𝟑 ∙ 𝟐𝟏 −𝟒 Good compromise, initial Peak line density too high Short bunch length (many distribution used for the (bad for transverse space turns needed for given N). last benchmark. charge) in the first 200 turns. 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 6
Injection to CERN PS Booster at 160 MeV (with Linac4) Movie of bunch profile and distribution in phase space evolutions from 0 to 10 ms. Space charge is included, N = 295 ∙ 10 10 , # macro particles = 5 ∙ 10 5 . 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 7
Single particle tracking • Acceleration, one RF and no space charge In BLonD the phase space trajectories for small amplitudes are not symmetric with respect to ∆𝐹 = 0 , in PyOrbit with PTC-tracking they are. synchronous phase starting point of a particle at ∆𝐹 = 0 1. Kick in energy BLonD equations of motion (impulsive system) 𝑜+1 (𝑞 𝑇 𝑜+1 − 𝑞 𝑇 𝑜 ) Δ𝐹 (𝑜+1) = Δ𝐹 (𝑜) + 𝑊 sin 𝜄 (𝑜) − 𝛾 𝑇 (𝑜+1) 𝜄 (𝑜+1) = 𝛾 𝑇 2. Drift with 𝜀 ≐ Δ𝑞 Δ𝐹 (𝑜) 𝜄 (𝑜) + 2 𝜌 𝜃 (𝑜+1) 𝜀 (𝑜+1) 𝑞 𝑇 = 2 𝐹 𝑇 𝛾 𝑇 𝛾 𝑇 8 24/03/2015 EuCARD2/XBeams Workshop on Space Charge
Single particle tracking (𝑜+1) 𝛾 𝑇 (𝑜) = 1 𝛾 𝑇 But the derivation of the theta equation is clear: (𝑜+1) (𝑜+1) 𝑢 𝑜+1 = Ω 𝑆𝐺 𝑢 𝑜+1 = 𝑢 𝑜 + … (𝑜) 𝑢 (𝑜) + ⋯ Ω 𝑆𝐺 (𝑜) Ω 𝑆𝐺 Ω 𝑆𝐺 (𝑜+1) (𝑜+1) (𝑜+1) Ω 𝑆𝐺 = 𝛾 𝑇 𝜄 (𝑜+1) = 𝛾 𝑇 (𝑜) 𝜄 (𝑜) + ⋯ (𝑜) 𝑢 (𝑜) 𝜄 (𝑜) = Ω 𝑆𝐺 (𝑜) (𝑜) Ω 𝑆𝐺 𝛾 𝑇 𝛾 𝑇 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 9
Single particle tracking Why in BLonD the beta ratio implies an asymmetry in phase space at least when we are near the synchronous phase? Let’s consider the equations of motion for : Δ𝐹 (1) = 𝑊 𝑡𝑗𝑜 𝜄 (0) − 𝑊 𝑡𝑗𝑜 𝜄 𝑇 (0) Δ𝐹 (1) → 0 𝜄 (0) → 𝜄 𝑇 (0) (𝑜+1) the ellipse is 𝜄 (1) → 𝛾 𝑇 (𝑜) 𝜄 (0) > 𝜄 (0) shifted up 𝛾 𝑇 We can even estimate a priori the energy offset: 𝛾 𝑇 𝛾 𝑇 𝜄 𝑇 formula: ∆𝐹 𝑝𝑔𝑔𝑡𝑓𝑢 = 53.99 eV ∆𝐹 𝑝𝑔𝑔𝑡𝑓𝑢 = − 𝐹 𝑇 simulated: ∆𝐹 𝑝𝑔𝑔𝑡𝑓𝑢 = 57.59 eV 𝜃 𝜕 0 We will investigate further this discrepancy between the codes. Possible risk: divergenge could increase for long simulations. Red: BLonD, Blu: PyOrbit 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 10
Space charge in BLonD and PyOrbit • Different ways to calculate the space charge kick in BLonD and PyOrbit: ∞ ∞ |𝑎 𝑇𝐷 | 𝐽𝑤 𝑇𝐷 𝑢 = −𝑓 ℱ −1 𝑎 𝑇𝐷 𝜏 = − 𝑓 𝑎 𝑇𝐷 𝜏 𝑓 𝑗 𝜕 𝑢 𝑒𝜕 = 𝑓 𝑜 𝜏 𝑓 𝑗 𝜕 𝑢 𝑒𝜕 2 𝜌 2 𝜌 𝑗 𝑜 −∞ −∞ BLonD PyOrbit Specific routine for constant |𝑎 𝑇𝐷 | imaginary 𝑜 ∞ ℱ −1 𝑎 𝑇𝐷 𝜏 is calculated after the |𝑎 𝑇𝐷 | = 𝑓 1 𝑗 ω 𝜏 𝑓 𝑗 𝜕 𝑢 𝑒𝜕 2 𝜌 𝑜 𝜕 𝑠𝑓𝑤 beam spectrum computation as if it −∞ were a general impedance. ∞ 𝑒 |𝑎 𝑇𝐷 | 𝑓 𝑒𝑢 𝑓 𝑗 𝜕 𝑢 𝜏 𝑒𝜕 = 2 𝜌 𝜕 𝑠𝑓𝑤 𝑜 −∞ ∞ |𝑎 𝑇𝐷 | 𝑓 𝑒 𝑓 𝑗 𝜕 𝑢 𝜏 𝑒𝜕 = 𝑒𝑢 2 𝜌 𝜕 𝑠𝑓𝑤 𝑜 −∞ |𝑎 𝑇𝐷 | |𝑎 𝑇𝐷 | 𝑓 𝑒 𝑓 𝑒 𝑒𝑢 ℱ −1 𝜏 = = 𝑒𝑢 𝜇(𝑢) 𝜕 𝑠𝑓𝑤 𝑜 𝜕 𝑠𝑓𝑤 𝑜 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 11
Benchmark N1: realistic case no space SC • 6 different longitudinal distributions tracked with acceleration, double RF and no space charge, # macro particles ≈ 5 ∙ 10 5 good qualitative agreement. PyOrbit BLonD 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 12
Benchmark N2: parabolic bunch • Parabolic bunch at 160 MeV in the PSB, single RF, no acceleration. • The bunch was not matched with space charge effects. • Simulation parameters: ℎ = 1, 𝑔 𝑠𝑓𝑤 = 991968.3 Hz, 𝑊 = 8 kV, 𝜃 = -0.67, 𝛾 𝑇 = 0.52, 𝐹 𝑇 = 1098.2 MeV 𝑓 = proton charge, 𝑂 = number of particles, 𝐷 = 157 m, 𝑚 𝑗𝑜𝑗𝑢 = 148.16 ns (~23m) 𝑎 𝑇𝐷 = 795.8 W same value for the two codes, 𝑜 # macro particles = 5 ∙ 10 5 • Syncrotron frequency of the syncronous particle: No significant shifts up to 𝑂 = 10 10 2 ℎ 𝑔 𝑊 |𝜃| 𝑠𝑓𝑤 No space charge. 𝑔 𝑡0 = 2 𝐹 𝑇 2 𝜌 𝛾 𝑇 𝑔 𝑡 < 𝑔 𝑡 [𝐼𝑨] 𝑡0 𝑔 With space charge 3 1 − 3 𝑓 𝑂𝑔 𝐷 𝑎 𝑠𝑓𝑤 (for a matched 𝑔 𝑡 = 𝑔 𝑡0 𝜌 2 ℎ 𝑊 𝑚 𝑜 parabolic bunch 𝑇𝐷 𝑂 below transition). 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 13
Benchmark N2: synchronous frequency diagram 𝑂 = 2.95 ∙ 10 10 𝑂 = 2.95 ∙ 10 9 𝑂 = 0 𝑔 𝑔 𝑔 𝑡 𝑡 𝑡 𝑚 𝑗𝑜𝑗𝑢 𝑚 𝑗𝑜𝑗𝑢 𝑚 𝑗𝑜𝑗𝑢 Max θ [rad] Max θ [rad] Max θ [rad] • Good quantitative correspondence between the codes and the analytical formula up to 𝑂 = 2.95 ∙ 10 10 , where space charge effect is negligible in comparison to the RF voltage and there is no blow-up. 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 14
Benchmark N2: synchronous frequency diagram 𝑂 = 2.95 ∙ 10 11 𝑂 = 2.95 ∙ 10 12 𝑔 𝑡 𝑚 𝑗𝑜𝑗𝑢 𝑚 𝑗𝑜𝑗𝑢 Max θ [rad] Max θ [rad] For 𝑂 > 2.95 ∙ 10 10 blow up occurs and the formula is not anymore applicable. • • However the codes give practically the same values for the bunch lengthening and the synchronous particle frequency shift. 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 15
Benchmark N2: phase space • Parabolic bunch evolution in phase space, PyOrbit (up), BLonD (down). 𝑂 = 2.95 ∙ 10 9 𝑂 = 2.95 ∙ 10 10 𝑂 = 0 Blow-up does not occur up to 𝑂 = 2.95 ∙ 10 10 . • 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 16
Benchmark N2: phase space • Bunch blow-up. 𝑂 = 2.95 ∙ 10 11 𝑂 = 2.95 ∙ 10 12 24/03/2015 EuCARD2/XBeams Workshop on Space Charge 17
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