Paul Görgen TEMF / GSI

Overview 1.Introduction to Beam-Beam 2.The code used and simulation of BTF 3.Demonstration of different beam beam simulations and comparison with a measurement 4.Conclusion 2

Motivation General direction of this work: transverse BTF of bunched beams Desire to gain information about the tune distribution from the BTF RHIC case: Beam-Beam dominant  fast simulation possible by using 1-turn map from madx for the machine 3

The beam-beam effect A bunch of charge density For example a circular gaussian distribution Will, colliding head-on with another bunch act on that bunches particles with a force: 4

Beam-Beam parameter The field is linear in the center where the tune change corresponds to a tune change of ξ Beam-beam parameter Yellow line indicates ideal quadrupole like kick 5 Beam-Beam tune change for a particle of a given single particle emittance

Coherent Modes In general colliding bunches can coherently oscillate against each other π mode: out of phase σ mode: in phase (nominal tune) Spectrum cartoon (for equal tunes in both rings): Amplitude σ π Modes illustration by W. Herr See: Tune shifts of coherent beam-beam oscillations, Yokoya and Haruyo, Part. Accel. 1990, Spectrum see W.Herr CAS on Beam-Beam Incoherent spectrum 6

Patric Simulation flow (initial, single bunch in synchrotron) Output particle coordinates and beam parameters On each lattice position kick particles for different effects Translate to next lattice position Patric originally has parallelization based on slicing bunches, objective was coherent space charge simulation. 7

Code flow modified for beam beam Bunch sliced longitudinally for coherent space charge simulation On each lattice position kick particles for different effects Translate to next lattice position At interaction points: Exchange beam beam fields with partner bunch (lookup table for partners) Each bubble is a group of processes running the original patric code for a single bunch 8

2D Interaction model for beam-beam Bunches compute their charge density projected onto the h/v-plane Bunches compute fields (electro- statically) Field exchange with collision partner Particle x’ change according to received field ExEx ρ Patric specific 128x128 grid size (not fixed Solving for fields: ~ ms Aggregation of 100k particles on grid: ~20 ms Poisson Solver currently not parallelized! 9

Round gaussian simplified model for the beam-beam interaction Bunches compute their RMS σ and center of charge μ Parameter exchange with partner Analytic kick assuming round gaussian with received μ and σ Intuitive understanding: μ  responsible for coherent modes σ  incoherent spectrum 10

BTF BTF: Response/ex citation BTF: Response/ex citation Electronics Delay -> phase shift Electronics Delay -> phase shift Electronics Delay -> phase shift Electronics Delay -> phase shift 11

BBQ Step through excitation frequencies, excite at each for 33 ms, step to next frequency Pick up beam offset for every bunch every turn (through analogue filters) Record beam response at excitation frequency in phase and amplitude, division by exciting signal gives BTF ★ M. Bai et al. RHIC Spin Flipper Status and Simulation Studies. Proceedings of PAC 2011,

BBQ BTF simulation Time is crucial: – Gauss approximation can run in 30 ms per interaction for 100k particles, scales linearly with number of processes per bunch – Poisson needs ms per interaction for 100k particles – Was worse by a factor of 10 when I arrived due to a few peculiarities in the code that came from a parallelization for space charge and output data Step through excitation Frequencies, excite at each for a number of turns, then reset particle distribution Pick up beam center of charge Record beam response in phase and amplitude, division by exciting signal gives BTF Excitation period >> bunch length  all particles in a bunch receive the kick in their transverse velocity For efficiency BTF kicks take place at an IP (no additional tracking required) Excitation period >> bunch length  all particles in a bunch receive the kick in their transverse velocity For efficiency BTF kicks take place at an IP (no additional tracking required) 13

Gaussian beam-beam Run simulation with small transverse offset in space – Observe oscillation resulting from initial mismatch / numerical noise – s Position of the Pi-Mode as computed by*: Position of the Pi-Mode as computed by*: Poisson beam-beam Code-Benchmark: 1 Interaction spectrum σ mode (tune) * Tune shifts of coherent beam-beam oscillations, Yokoya and Haruyo, Part. Accel

Coherent BTF 1 interaction Position of the Pi-Mode as computed by: Position of the Pi-Mode as computed by: Btf simulation for 1x1 bunches – Positions of peaks match with kick-BTF and theory 15

Model for 2 IP’s Interaction Point Clock 8 1.Beam-beam kick 2.BTF: Record center of charge amplitude 3.BTF: Kick beam Clock 8 1.Beam-beam kick 2.BTF: Record center of charge amplitude 3.BTF: Kick beam Clock 6 Beam-beam kick only Clock 6 Beam-beam kick only Use linear transport matrices from madx between IP’s 16 Currently: Chromaticity is ignored! To take it into account either: Cellwise kicks for chrom (slow, tracking through a few 100 cells instead of 2) Chrom matrix (not simplectic)

Multiple collisions Clock 6 Clock

3x3 bunches collision scheme Like in the BBQ, the excitation should be applied to all bunches in the beam taking into account their phase shift with respect to each other. Resulting signal: 3 samples per turn t x Time signals: bunch position and BTF kick amplitude output at clock 6 18

Thought experiment: if we push the first bunch, when will the other bunches know? Excitation symbolized by ball initially owned by yellow bunch 1 being passed from bunch to bunch at each IP t x Time signals: bunch position and BTF kick amplitude output at clock 6 Simulation reproduces the results of the thought experiment: Yellow bunch 1 needs to wait till Blue bunch 3 has the fields ready => 5x slower than 1x1 beam-beam simulation Bunch number 19

Plausibility check for 3x3 model Nominal rhic lattice, expect 6 modes (middle three possibly weak Poisson solver Round Gaussian approximation h.tune v.tune 20 Exact mode structure depends on phase shift between IPs

Physics comparison: Split tunes BTF (Split tunes measurement results provided by Simon White) Qh Qv Qh Qv N1.8e11 εx20 mm mrad εy20 mm mrad E100GeV Warning: I compare the h plane simulation to the v plane measurement. In measurement the features are unique to the blue h-plane. Simon Whites rigid bunch simulation shows modes in all planes. Data from split tunes fill #16465 Emittances in “RHIC units” 21

Conclusion Code was much to slow upon arrival (was optimized for a different kind of problem) Benchmark of the beam-beam interaction and BTF shows expected results BTF simulation compares more or less well with Split tunes measurement 22

Outlook Further simulate split tunes / tune sweep measurements Wolfram: Try to find a way to calculate the tune distribution from the BTF (Simulated BTF are good for this  we know the tune distribution) – Possibly need to take into account other effects like chromaticity, amplitude detuning In parallel: Use experience with beam-beam to try and simulate BTF with space charge, (experience bephy group at GSI) 23

Thank you For your attention

Seperator slide 25

3x3 nominal lattice BTF (Gaussian approximation) 26

Beating of the central dip BTF amplitude oscillates over time in the central dip (Plot shows how a btf over 2000 turns changes when the first n turns are left out in analysis.) 27

Amplitude response of the central dip Periodicity is identical for all bunches when viewed seperately 28

Beam-Beam parameter and Yokoya Factor Beam-Beam parameter: (N= particle number, = particle charge radius, =relativistic parameter) Yokoya factor: Shift of pi-mode in units of beam-beam parameter: 29

Plots for round beam-beam s 30

31

Objectives for my stay Get the code to simulate BTF comparable to those obtained using BBQ Benchmark the implementation of Beam-Beam Compare simulated BTF with measurement – See how I can get experimental data from RHIC databases – See BTF of beam-beam and space charge (independently). Return with full implementation of BBQ and Beam-Beam in order to further analyze BTF in simulation upon my return 32