Electron Cooling Code Development and Simulation Studies He Zhang, He Huang, Rui Li, Jie Chen, Li-Shi Luo JLEIC Collaboration Meeting, 10/06/2016 Newport News, VA Outline Twostage electron cooling scheme for JLEIC A new electron cooling simulation program Electron cooling simulation for JLEIC Particle tracking simulation for electron cooling Summary He Zhang ---2--- JLEIC Staged Cooling Scheme Multi-phased scheme takes advantages of high electron cooling efficiency at low energy and/or small 6D emittance Low energy DC cooler (Within state-of-art) at the booster: Reduce the emittance 2GeV/u ion beam, 1.6 MeV electron beam High energy bunched beam cooler (needs R&D) at the collider ring: Maintain the emittance

Up to 100 GeV/u ion beam, 55 MeV electron beam ERL single pass cooler (Qe = 420 pC/buncn, baseline design, no circulator ring) ERL circulator cooler (Qe = 2 nC/bunch, lower emittance, higher luminosity) JLEIC ion complex layout He Zhang ---3--- A New Electron Cooling Simulation Code Goals: Fulfill the specific needs for JLEIC electron cooling scheme Different scenarios: DC cooling, bunched beam cooling More flexibility, higher efficiency. Use for electron cooling simulation and further code development Approaches: Following the models in BETACOOL whenever applicable Revise the models whenever needed Improve the efficiency by strategically arrange the computation Adaptive to the modern multi-core platform Formulas and models implemented: IBS: Martini model (no vertical dispersion lattice) Friction force: Parkhomchuk formula (magnetized cooling) Cooling rate: single particle model, Monte Carlo model

Cooling dynamics: Four-step procedure Other formulas and models can be added in future Codes and a tutorial are available online: He Zhang ---4--- Electron Cooling Rate Single particle model: sample the ion beam with particles with the same dynamic invariant (single particle emittance) Sample the ion beam on a regular grid Calculate the electron charge density around each ion Convert into beam frame Calculate the friction force on each ion (Parkhomchuk formula) Convert back to lab frame Each ion feels a kick as . Calculate the new dynamic invariant and thus the emittance for each ion (as in the previous page) and take average on all of them. Calculate the cooling rate (the change of emittance in a unit time) He Zhang ---5---

Electron Cooling Rate Monte Carlo model : sample the ion beam with particles of Gaussian distribution Sample the ions as described in the previous page Calculate the electron charge density around each ion Convert into beam frame Calculate the friction force on each ion Convert back to lab frame Each ion feels a kick as . Emittace is calculated statistically as Calculate the cooling rate (the change of emittance in a unit time) He Zhang ---6--- Cooling Dynamic Four-step procedure: 1. Create sample ions 2. Calculate the expansion rate under the IBS and/or electron cooling effect 3. Update the beam parameters and the sample ions, and 4. repeat from the second step till the end time The RMS dynamic method and the model beam method in BETACOOL fit into this four-step procedure.

He Zhang ---7--- Cooling Dynamic RMS dynamic method in the four-step procedure: 1. Create sample ions (using the single particle model or Monte Carlo model) w.r.t. the given emittance 2. Calculate the expansion rate under the IBS and/or electron cooling effect 3. Update the beam parameters using , create new sample ions w.r.t. the new emittance, and 4. repeat from the second step till the end time He Zhang ---8--- Cooling Dynamic Model beam method in the four-step procedure: 1. Create sample ions with the given initial distribution 2. Calculate the IBS rate and/or the friction force. 3. Apply the IBS kick on each particle as ,,

Gaussian random number ; apply the electron cooling kick as ; apply a random phase advance for each particle; calculate new emittance and expansion rate; and 4. repeat from the second step till the end time. He Zhang ---9--- Benchmark with BETACOOL Thoroughly benchmarked with BETACOOL (a few examples below) RMS dynamic Monte Carlo RMS dynamic Single particle Model beam Model beam He Zhang

---10--- Benchmark DC cooling and IBS for JLEIC Bunched cooling and IBS for JLEIC collider ring with KE = booster ring with KE = 800 100 GeV MeV Time cost 133 s (BETACOOL 3060 s) Time cost 30.7 s (BETACOOL 422 s) He Zhang ---11--- Parallelization Good efficiency of the serial code, so not necessary to run the code on a cluster machine. Further improve the efficiency for shared memory structure hardware: GPU, Intel Xeon Phi, or just a multi-core PC. Parallelization is based on thrust, a parallel algorithm

library that supports CUDA, TBB (Threading Building Blocks) and OpenMP. IBS calculation, 100100100 grid: NVidia GTX 660 ti GPU: 7.76 s CPU AMD Phenom II X4 840T Processor 2.9GHz: 62 s Electron cooling 200,000 samples: GPU 0.03 s CPU 0.151 s 40,000 samples: GPU 0.024 s CPU 0.023 s He Zhang ---12--- DC Cooling at the Booster Proton beam: KE = 2 GeV Emit = 2.15 mm mrad dp/p = 0.001 N = 2.8E12 IBS ECOOL IBS+ECOOL RH 1/s

3.86E-4 -9.05E-3 -9.10E-3 RV 1/s 3.86E-4 -9.00E-3 -8.71E-3 RL 1/s 2.27E-4 -15.3E-3 -15.3E-3 Electron beam:

I=2A Ttr = 0.1 eV, Ts = 0.1 eV DC cooler: L = 10 m B=1T He Zhang ---13--- Strong Cooling at the Collider Ring (Qe = 2 nC) Proton beam: KE = 30/60/100 GeV RH 1/s Emit = 0.45/0.35/0.35 mrad RV 1/s dp/p = 5/4/4 RL 1/s = 2.5/2.75/2.75 cm N = 6.56E9 KEp=30 GeV Electron beam: Qe = 2 nC Ttr = 0.5 eV, Ts = 0.1 eV IBS ECOOL

4.74E-3 -8.05E-3 -1.59E-5 -8.05E-3 2.10E-3 -1.31E-2 Bunched cooler: L = 302 m B=1T He Zhang ---14--- Strong Cooling at the Collider Ring (Qe = 2 nC) IBS ECOOL IBS ECOOL RH 1/s 6.21E-3

-3.78E-3 4.43E-3 -2.65E-3 RV 1/s -4.08E-5 -3.75E-3 -1.53E-5 -2.60E-3 RL 1/s 2.00E-3 -8.65E-3 0.97E-3

-7.29E-3 KEp=60 GeV KEp=100 GeV He Zhang ---15--- Weak Cooling at the Collider Ring (Qe = 420 pC) Lower electron beam current, within the state-of-art technique. Electron bunch smaller than the proton bunch. Higher proton beam emittance, lower IBS. Larger proton beam momentum spread. Proton beam: Electron beam: Bunched cooler: KE = 100 GeV Q = 420 pC L = 60 m Emit_x = 1.2 Ttr = 0.5 eV, Ts = 0.1 eV B=1T mrad

= 0.035 cm Emit_y = 0.6 = 0.84 cm mrad Proton beam: Parkhomchuk formula = 2.5 cm Horizontal expansion rate decreases as dp/p increases, with = N = 6.56E9 dp/p (1E-4) dp/p (1E-4) IBS (1E-4 1/ IBS s) (1E-4 1/ s) Cool (1E-4 Cool 1/s) (1E-4 1/s) Total (1E-4 5.03 5.03 1.011.01 4.02 5 5

2.5 cm 0 0 1.401.40 - 8 8 0.01 0.01 4.14- 0.85- 1.254.14 0.85 1.25 - He2.54 Zhang 2.26 2.26 3.39 3.39 0.63 0.63 2.35

2.35 12 12 0.01 0.01 0.20 0.20 2.452.45 - 0.71- 0.980.71 0.98 1.64 ---16--- 1.341.34 - Weak Cooling at the Collider Ring (Qe = 420 pC) time to double 1.5h 2.8h dp/p=8E-4

dp/p=1.2E-3 Lower proton beam current Np = 4.4E9 6.6h or 20% transverse coupling 4.0h dp/p=1.2E-3 dp/p=1.2E-3 He Zhang ---17--- Issues with Bunched Beam Cooling Cooling with correlated electron beam The momentum of the electron is correlated with its longitudinal position, modeled as Friction force is calculated in the electron beam frame. Ions got a various velocity shift according to their positions. Assume . Non-correlated

Correlated eV 1 0.00511 1.59 1.59 1.66 1.60 1.60 1.60 2 0.02044 1.54 1.54 1.50 1.53

1.53 1.30 3 0.04599 1.48 1.48 1.34 1.43 1.43 1.02 4 0.08176 1.41 1.41 1.19 1.33

1.33 0.79 5 0.12775 1.34 1.34 1.06 1.23 1.23 0.61 Issues with Bunched Beam Cooling Cooling rate vs. electron bunch size Electron number per bunch is limited by the capacity of the cathode and the instability during transport Two competing parameters affect the cooling rate: how many ions covered by the

electron bunch & local electron density around an ion. Calculate cooling rate for Gaussian ion bunch with cylindrical electron bunch of various sizes. Best cooling rate at . Issues with Bunched Beam Cooling Cooling rate vs Larmor emittance Magnetic electron beam Drift emittance , relates to the transverse bunch size Larmor emittance , relates to the transverse temperature Important parameter for cooler and electron source design Turn by Turn Simulation Initial distribution 40,000 turn 10,000 turn 80,000 turn 20,000 turn Particle Tracking Simulation (in progress)

Simulate the motion of ions and electrons inside the cooler based on the first principles. Focus on the interactions between individual particles. Single pass, not the whole cooling process. Friction force calculation. Fast multipole method for Coulomb interaction using Cartesian tensor. Fast multipole method for other non-oscillating interaction. MPI based parallel integrators: (RK4 and Hermite) Order 2 4 6 8 Linear scale with particle number 10 Relative error

Summary A new electron cooling and IBS effect simulation program has been developed. Code available on online Benchmarked with BETACOOL. Significant improvement in efficiency, which brings the possibility for more sophisticated physical model or algorithm Actively implemented in JLEIC electron cooling simulations Lowe energy DC cooling is within the state of art. Bunched beam cooling at the collision energy is challenging, but possible.

A particle tracking program is in development He Zhang ---23--- He Zhang --24--