Simulation of transverse multi-bunch instabilities of proton beams in LHC

Transcription

1 Simulation of transverse multi-bunch instabilities of proton beams in LHC Alexander Koschik Technische Universität Graz, Austria & CERN Geneva, Switzerland TU Graz supervisor: CERN supervisors: B. Schnizer D. Brandt F. Ruggiero B. Zotter

2 Outline Introduction & Motivation Simulation Techniques & Approximations Resistive Wall Impedance Models Measurements in CERN SPS & Comparison to Simulation LHC simulated Summary A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.1

3 Introduction CERN currently builds the Large Hadron Collider (LHC) in the old 27km long LEP tunnel. Foreseen startup in 27. Proton-proton collisions with center-ofmass energy o4 TeV. 288 bunches per beam. A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.2

4 Motivation Physics demands: High Event Rate (N Event = L σ Event ) + High Energy (7 TeV) High Single Bunch Intensity Multiple Bunches High Bunch Repetition Rate L = N2 p n b f γ r 4 π ε n β Large Luminosity Hadrons Strong Bending Magnetic Fields needed Superconducting Magnets Small Structure Sizes Machine Protection needed (Collimators) Resulting Problems: Instabilities No Radiation Damping Collective Effects Multi-Bunch Effects Multi-Turn Effects Long-Range A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.3

5 Motivation Stability analysis normally done in frequency domain. Simulation additionally allows: Non-equidistant filling schemes Investigate Transition Effects Interplay between different effects (impedances) Long-Range Effects Correct and efficient implementation of corresponding impedances A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.4

6 The Simulation r x ε new new r r = x+ δx = ε+ δε r x M = β β i+ 1 i 1 β β Other Resonators i+ 1 i ε new new cos sin r x t new new r = x+ δ Start t r = M( ε) x = t+ α r x T E = ε+ δε+ ev Arc Impedance (Resistive Wall) RF Cavity RF r x ε new new r r = x+ δx = ε+ δε RF sin( ϕ + ω t) ( µ () ε) β ( ()) i+ 1βisin µ ε β ( µ () ε) cos( µ () ε) i βi+ 1 ξ η µ () ε = µ 1 + ε δx r = E x' p ε Transverse Kick s δε t= t Longitudinal Kick ε = E Classical Tracking Code Linear Transfer Matrices + Kicks Long-Range regime: τ σ τ σ τ... bunch length τ... time interval for wake calc. Impedances with correspondingly long-lasting wake fields: Wake W V Cm Wake W V Cm Resistive Wall Impedance (with inductive bypass ) Narrow-Band Impedances (HOMs of cavities, wakes of cavity-like structures) Transverse Resistive Wall Wake Function c L W m 1 Τ Π 3 2 b 3 Μ Μ r 1 Σ c Τ Time Τ s Transverse Resonator Wake Function W m 1 Τ Ω r R Im exp jω 1Τ Q' Ω 1 Ω r Q j 2 Q ', Q' Q Time Τ s W (τ) 1/ τ W (τ + τ) =? W (τ) Fast summation via FFT Convolution W (τ) exp(j ω 1 τ) W (τ + τ) = exp (j ω 1 τ) W (τ) Time evolution and summation using Phasors = Resonator Model (R, Q, ωr ) A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.5

7 The Simulation / Approximations 1 Detailed Bunch Multiple Bunches represented by 1 super-particle rigid bunch approximation W pot (τ) W(τ) W pot(τ) = W pot(τ) = 1 ξ wake function approximation dt W (t) λ(τ t) dt W (t) ξ(τ t) λ(τ t) W (τ) for τ σ t W (τ) for τ σ t Impedance modelled by 1 kick per turn a lumped impedance approximation a K. Thompson and R. D. Ruth. Transverse coupled bunch instabilities in damping rings of high-energy linear colliders. Phys. Rev., D43: , A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.6

8 The Simulation / Wake Summation Problem The kick x on bunch j at turn n, in the case of the resistive wall impedance: x j n = j 1 i= x i n (j i) τbuc. } {{ } sum over preceding bunches at current turn + n 1 k= n n mem N b 1 i= x i k (n k) τrev. + (j i) τ buc. } {{ } sum over preceding bunches over previous turns } {{ } O(Nb 2) Computation Time a.u direct summation FFT convolution Number of Bunches A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.7

9 FFT Convolution Analogy between wake sum and (discrete) convolution: x n = n 1 k= x k (n k) τrev. = n 1 k= g(k) f(n k) (single bunch case) Continuous Convolution h(t) = g f Discrete Convolution h(n) = g f = N 1 k= dτ g(τ)f(t τ) g(k) f(n k) = N 1 k= g k f n k Convolution Theorem F [f g] = F[f]F[g] or f g = F 1 [F[f]F[g]] Compute wake sum via the FFT convolution in frequency domain: x n = h n = (g f) n = N 1 k= g k f n k FFT ===== IFFT G j F j = H j A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.8

10 FFT Convolution Circular Convolution Linear Convolution f(i) g(i) f(-i) f(1-i) f(2-i) f(3-i) h(i) i eg: f -3 f -2 f f f f f -2 f -3 f f f -2 f -3 f f f -2 f -3 f f f -2 f -3 h h 1 g g 1 g 2 g 3 g g 1 g 2 g 3 h 2 h 3 f f f -2 h 1=g+ g1f + g2f + g3f -2 f(i) g(i) f(-i) f(1-i) f(2-i) f(3-i) f(4-i) f(5-i) f(6-i) f(7-i) i f -7 f -6 f -5 f g g 1 g 2 g f f -7 f f -5 f -6 f -7 f f -5 f -6 f -7 f f -5 f -6 f f -5 f f f -5 f -6 f f -7 f -6 f -5 f -7 f -6 f -5 f -7 f -6 f -5 f -7 f -6 f -7 h(i) h h 1 h 2 h 3 h 4 h 5 h 6 h 7 To get the correct wake sums, linear convolution has to be realized by zero padding. A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.9

11 FFT Convolution f(i) g(i) Multi-Bunch Multi-Turn Convolution Scheme i f f f g g 1 g 2 g 3 f(-i) f(1-i) f(2-i) f(3-i) f(4-i) f(5-i) f(6-i) f(7-i) f(8-i) f(9-i) f(1-i) f(11-i) f(12-i) f(13-i) f(14-i) f(15-i) f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f h(i) h h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 1 h 11 h 12 h 13 h 14 h 15 A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.1

12 FFT Convolution Wake Summation Algorithm Search GCD (greatest common divisor) of given bucket layout. Defines new, equidistant bunch pattern. mask. Set up extended zero-padded arrays f c. Precalculate FFTs of f c for c =,..., n mem, this gives arrays F c = F [f c ] Speed Considerations Direct Summation (2n mem + 1) Nb 2 + N b FFT Convolution At turn n do the following: Write (signal, offsets) to array g i mask. FFT of g, G c= = F [g] use (n mem + 2) 4N b log 4N b + (5n mem + 5)N b Turn Memory n mem Multiply c [, n mem ] : H c = F c G c FLOPS a.u..5 Inv. FFT c [, n mem ] : h c = F 1 [H c ] Sum over n mem turns to get the kicks, use the mask: x j n = n mem c=1 hc j+n b + h c= j Number of Bunches 4 5 A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.11

13 Resistive Wall Impedance Models Classical thick & thin wall formula known to be incorrect for ω., thick Zm=1 (ω) = (sgnω + j) Z Lδ µ r ω 2π b 3 ω, thin Zm=1 (ω) = c L π b 3 σ c d ω Transverse Resistive Thick Wall Impedance Z Ω Transverse Resistive Thin Wall Impedance Z Ω Re Z m resistive wall with inductive bypass classical resistive wall Re Z m resistive wall with inductive bypass classical resistive wall Frequency Ω 2Π Hz Frequency Ω 2Π Hz A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.12

14 Resistive Wall Impedance Models Measured a Transverse Resistive Wall Impedance Real (left) and Imaginary (right) part a A. Mostacci, F. Caspers, and U. Iriso. Bench measurements of low frequency transverse impedance. CERN-AB RF. Proc. of PAC 3, Portland, Oregon, 126 May 23. A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.13

15 Resistive Wall Impedance Models Resistive Wall Impedance with inductive bypass (L.Vos) I beam inductive bypass Z ind [, LV Zm=1(ω) = Z L 2 π b bωµ 1 + Z Z 1 tanh γ 1 d 1 j 2 2Z 1 + Z 1 Z tanh γ 1 d 1 ] 1 Quasi-Static Beam Model (Burov/Lebedev) Solving Maxwell Equations, Poisson equation for electric dipole and vector potential for magnetic dipole Z ibp wall impedance Z = 2c b 2 ω vacuum chamber wall d Z Z ind Z + Z ind Z,BL m=1(ω) = j Z βl πb 2 s 1 + κ 2,1 s 1 s 1 + κ 2,1 κ 1, c 1 + κ 2,1 s 1 + κ 1, c 1 Field Matching (B.Zotter) Solution of Maxwell s equation by matching 4 components at every layer. Most rigorous but lengthy or only numeric expressions. a 1 a 2 a m d m d 2 a L, Z, m m m, Z, d 1 1, Z 1, 1 beam }multilayered wall beam axis A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.14

16 Resistive Wall Impedance Models Standard Parameter Range: b d Transverse Resistive Wall Impedance Z Ω Transverse Resistive Wall Impedance Z Ω Re Z m t w.15 m b.5 m Ρ m skin t w.15m 2Π s 1 thick wall classical thin wall classical thick wall ind. bypass thin wall ind. bypass Burov Lebedev n 2 Burov Lebedev Approx. L.Vos general n 2 Im Z m t w.15 m b.5 m Ρ m skin t w.15m 2Π s 1 thick wall classical thin wall classical thick wall ind. bypass thin wall ind. bypass Burov Lebedev n 2 Burov Lebedev Approx. L.Vos general n Frequency Ω s Frequency Ω s 1 A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.15

17 Resistive Wall Impedance Models LHC Collimators: b d skin d.25m Hz skin d.25m Hz d.25 m a.2 m skin a.2m Hz skin a.2m Hz Re Z Ω m Ρ m Classical Thick Wall Graphite Classical Thin Wall Graphite Thick Wall w. Inductive Bypass Burov Lebedev n 2 L.Vos n 2 Im Z Ω m d.25 m a.2 m Ρ m Classical Thick Wall Graphite Classical Thin Wall Graphite Thick Wall w. Inductive Bypass Burov Lebedev n 2 L.Vos n Frequency f Hz Frequency f Hz A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.16

18 Resistive Wall Impedance Models Usual regime : d, d < a d New regime : d >> a, d d beam beam a d a Þ a eff >> a Þ a eff» a when d d Induced Induced current current Standard parameter range: Wall thickness d is smaller than beam pipe radius a, and the skin depth δ for all frequencies under concern is smaller than d. New regime: Wall thickness d is larger than beam pipe radius a, and the skin depth δ is in the order of d for low frequencies. A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.17

19 Resistive Wall Wake Function, thick,ibp Zm=1 (ω) = (1 + j sgn ω) c µ L 1 ( 2 π b 2 j + sgn ω 1 + b σc ) µ 2 µ r ω Fourier Transform (not straightforward!), thick,ibp Wm=1 (t) = + cl µo µ r 1 π 3/2 b 3 σ c t }{{} classic thick wall wake function [ ] ( ) 4µr 2cLµr 4µr exp t 1 Erf t b 2 σ c µ b 4 πσ c b 2 σ c µ }{{} correction term due to inclusion of inductive bypass A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.18

20 SPS Measurement & Simulation Fixed Target Beam Measurement Parameters inj. Momentum p [GeV/c] 14 Revolution time τrev. [µs] 23.7 Tunes Q H /Q V / Gamma transition γ T 23.2 Maximum # of batches 2 # of bunches per batch 21 Bunch Intensity Np Total Intensity N p,tot Batch spacing [ns] 15 Bunch spacing [ns] 5 Full bunch length [ns] 4 Trans. emittance ǫ H,V [µm] <1 / <7.5 Long. emittance ǫ L [evs].2 -signal (position) [a.u.] Typical vertical BPM readings for growth rate measurements Latency time ~5 turns Feedback off vertical damper (feedback) switched on after 3ms Feedback on Number of Turns 1 Batch (21 bunches) Growth rate 2π/τ [turns] Coherent tune Q Vertical Plane 77 ± ±.79 Horizontal Plane ± ±.29 A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.19

21 SPS Measurement & Simulation SPS Simulation Model Injection and BPM Typical simulation result of resistive wall instability Resistive Wall Impedance Accelerating cavity -signal (position) [a.u.] Growth Rate 1/τ = 78 turns Coherent Tune Q = Number of Turns 1 Batch (21 bunches) Growth rate 2π/τ [turns] Coherent tune Q Vertical Plane 78 ± ±.1 Horizontal Plane 15 ± ±.1 SPS elliptic vacuum chamber geometry Yokoya factors included! Wake is not only dependent on exciting charge s offset, but also on witness particle offset: W pot,x W pot,y (x, y, x, ȳ, s) x W pot (x, y, x, ȳ, s) y W pot pot (x, s) + x W ( x, s) pot (x, s) + ȳ W (ȳ, s) A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.2

22 SPS Measurement & Simulation Vertical Amplitude Growth, SPS FT beam, n b = 21 (1 batch) Number of turns Amplitude [m] Bunch number m Vertical Amplitude Growth, SPS FT beam, n b = 42 (2 batches) Number of turns Amplitude [m] Bunch number m A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.21

23 SPS Measurement & Simulation Coupled Bunch Modes, SPS FT beam, n b = 21 (1 batch) Amplitude [a.u.] Number of turns Mode number n Amplitude [a.u.] Mode number n Coupled Bunch Modes, SPS FT beam, n b = 42 (2 batches) Number of turns Amplitude [a.u.] Amplitude [a.u.] Number of turns Mode number n Mode number n Number of turns A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.22

24 45 Simulation of LHC LHC Simulation Model 335 HOM Injection and BPM 22.5 Octupoles Impedances HOMs (undamped modes of the 2MHz cavities) Resistive Wall (Machine Resistance = mostly beam screen, collimators) Resistive Wall Impedance Accelerating cavity Parameter Injection Collision Momentum p [GeV/c] 45 7 Circumference C [m] Rev. frequency f [Hz] Dipole field B [T] Hor./Ver. Tune 64.28/59.31 Harmonic number h 3564 RF Frequency f RF [MHz] 4.8 RF Voltage V RF [MV] Particles per bunch Np [1 11 ] 1.15 Number of bunches n b 288 Bunch spacing τsp [ns] 25. Total # of particles N tot [1 14 ] 3.23 Total DC beam current I [A].582 Luminosity L [cm 2 s 1 ] mm Non-Linearities Octupoles cold bore 22 mm Beam screen stainless steel + 55 µm copper coating LHC vacuum chamber geometry δx = k 3 l 3! δy = k 3 l 3! (x 3 3 x y 2) (3 x 2 y y 3) A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.23

25 Simulation of LHC Results Amplitude growth of individual bunches vs. Turns Machine Resistance only, LHC Injection Energy, nominal Intensity Machine Resistance + Collimators, LHC Injection Energy, nominal Intensity A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.24

26 Simulation of LHC Results Coupled-bunch mode spectra vs. Turns at injection energy and nominal intensity Machine Resistance + Collimators, LHC Injection Energy, nominal Intensity A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.25

27 Simulation of LHC Results Amplitude growth and coupled-bunch mode spectra vs. Turns HOMs omhz cavities, LHC Injection Energy, nominal Intensity A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.26

28 Simulation of LHC Results Coupled-bunch mode spectra vs. Turns Machine Resistance + Collimators + HOMs and Octupoles LHC Top Energy, ultimate Intensity Non-linearities through octupoles tune spread Landau damping A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.27

29 Summary Simulation code MultiTRISIM developed Efficient implementation of wake summation via FFT Convolution Resistive wall impedance models in a new parameter regime δ skin,d > b Resistive wall wake function with inductive bypass computed and used in simulation Code benchmarked with measurements in CERN SPS Simulation of LHC. Present octupole design should provide enough Landau damping to stabilize beam at top energy. A. Koschik, Simulation of transverse multi-bunch instabilities of proton beams in LHC p.28