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Control of Transverse Multibunch Instabilities in the First Stage of the VLHC Valeri Lebedev Fermilab VLHC mini-workshop SLAC, March 21-23, 2001. Content 1. Machine Parameters 2. Multibunch Transverse Instability 3. Feedback System for Suppression of the Transverse Multibunch Instability 4. Emittance Growth Suppression 5. Conclusions VLHC Parameters Stage 1 Stage 2 Beam energy Ep 20 TeV 87.5 TeV Luminosity L 1034 cm-2s-1 21034 cm-2s-1 Magnetic field B0 1.96 T 10 T Injection energy Einj 0.9 TeV 10 TeV Bunch spacing b 18.9 ns Circumference C=2R 232 km Revolution frequency f0 1294 Hz Number of bunches Nb 36943 RF frequency fRF 477.9 MHz Betatron tunes ~214 Momentum compaction 2.110-5 Beta-function at IP 30 cm 50 cm Head-on beam-beam tune shift per IP 1.810-2 1.810-2 Number of particles per bunch N 2.51010 Beam current Ib 0.190 A 0.175 A RF voltage per turn, top/injection energy V0 50/50 MeV 50/50 MeV Synchrotron frequency, top/injection energy fs 2.32/10.9 Hz 1.1/3.28 Hz Rms momentum spread, top/injection energy p 1.5/14.910-4 0.5/2.410-4 Rms bunch length, top/injection energy s 6.6/14.2 cm 4.5/7.8 cm 2 2. Transverse Impedance for Vacuum Chamber with Thin Wall Impedance calculation for round vacuum chamber 1 i c k , , y 2 d 4 Z0 377 . c b a Solutions for the vector potential in different regions 2 I 0 x0 x C1 r cr I II AI III C 2 e k ( r a ) C 3 e k ( r a ) it Az r , , t AII e cos AIII C4 I0 – the beam current r x0 – amplitude of transverse beam motion dA dA dA dA Matching solutions at boundaries: ; yields four linear d r d r 0 dr r dr r 0 equations for coefficients Ci . The solution is: 3 2I x e ikd bk 1 e ikd bk 1 C1 0 2 0 2 ikd e ak 1bk 1 e ikd ak 1bk 1 1 , ca 4 I 0 x0 e ikd bk 1 C2 , ca e ak 1bk 1 e ak 1bk 1 ikd ikd 4 I 0 x0 e ikd bk 1 C3 , ca e ikd ak 1bk 1 e ikd ak 1bk 1 8I 0 x0 b2k C4 . ca e ak 1bk 1 e ak 1bk 1 ikd ikd Taking into account contribution from electric field of the beam we obtain an expression for the vacuum chamber transverse impedance (case of positive frequency): C1 2 Z0 eikd 1 kb eikd 1 kb 1 Z i 2 i 2 ikd 1 (1) I x ca 0 0 2a 2 e 1 ka 1 kb eikd 1 ka 1 kb 4 There are following asymptotes for the transverse impedance ( > 0): c1 i 2a 3 2 , d R c2 Z Z0 2 i... , ad d 4 Ra d 3 Rd 1 2 i , ad c a 2a 2 There is no simple expression for imaginary part of the impedance in the case ad d and Eq. (1) has to be used To simplify the solution the expression for the vector potential inside conductor was chosen for the flat geometry (instead of Bessel functions of imaginary argument) and therefore the result is not quite accurate if the wall is sufficiently thick. In particular the equation yields incorrect value for imaginary part of the impedance at very small frequencies, ad The following correction term is applied to fix the problem Z d Z i 0 2 2a (a d )1 k 4 d 2 a 2 Note that although this term corrects asymptotic behavior of the impedance at low frequencies it does not change the impedance at frequencies of interest of VLHC. 5 Z0 c ( 1 i) Z0 c ( 1 i) ZtrAH ZtrAH 2 3 2 3 a 2 R a 2 R 100 100 f0 f0 f0 f0 10 2 10 2 Ztr [Ohm/cm/cm] Ztr [Ohm/cm/cm] 1 1 0.1 0.1 0.01 0.01 100 1 10 1 10 1 10 1 10 1 10 1 10 100 1 10 1 10 1 10 1 10 1 10 1 10 3 4 5 6 7 8 3 4 5 6 7 8 1 10 1 10 f [Hz] f [Hz] d = 0.1 mm d = 1 mm Real (red) and imaginary (brown) parts of the transverse impedance and asymptotes for the real part of the impedance as function of frequency for round vacuum chamber: Al, 300 K, = 2.74 cm, a = 9 mm. Skin-layer thickness is 3.3 mm at f0/2=646 Hz. Peak is achieved when 2 ad and its value does not depend on vacuum chamber thickness 6 Transverse Impedance of Elliptic Vacuum Chamber Impedance of flat vacuum chamber is about half of the round vacuum chamber impedance. For an estimate we will introduce the effective radius of the vacuum chamber: a 2a b 3 3 1/ 3 b aeff 3 a b3 so that we would get correct result 3 for cases a b , a b and a b . Al For impedance estimates we will resistivity[microohm -cm] substitute aeff instead of a into Eq.(1) 2 Cu 0.6 Vacuum Chamber Conductivity Vacuum chamber cooling Skin-depth[cm] 1 Ag 0.4 significantly reduces transverse impedances 0 0.2 0 100 200 300 T[K] 0 0 7 Detuning Wake For round vacuum chamber the beam excites s dipole field behind him. That corresponds to x x 0 normal wake field W(s). For non-round vacuum chamber there are quadrupole field and higher multipoles. *A.Burov, V.Danilov “Suppression of transverse instabilities by asymmetries in the chamber geometry,” Contribution corresponding to quadrupole PRL, 82, 1999, p.2286. field is described by detuning wake D(s): Fx ds e 2 x0Wx ( s ) xD( s) L Fy ds e 2 y 0W y ( s ) yD( s ) L Detuning wake changes betatron tunes for tail particles and can cause non-linear resonances. For round vacuum chamber D(s) = 0 . For two parallel plates R c W x s W x s D x s 3 a Rs 8 1 f0 f0 2 Attenuation 0.1 0.01 1 10 1 10 3 4 10 100 f [Hz] Attenuation of the beam magnetic field by the elliptic aluminum vacuum chamber at 300 K, a = 9 mm, b = 14 mm; red line - d=1 mm, blue line - d=2 mm There is only about 30% attenuation for thickness of 1 mm at the first unstable betatron sideband That implies that the beam sees everything around vacuum chamber and can acquire additional contribution to the impedance from the surroundings if appropriate measures are not taken. 2 mm vacuum chamber attenuates the beam field by more than 2 times That is better but still requires careful attitude to what can be located in near vicinity of vacuum chamber 9 Multibunch Transverse Instability due to Finite Wall Resistivity Estimate of tune shift and instability increment due to vacuum chamber impedance i i 2 W ( s) dZ ( )e is / v , Z ( ) dsW ( s)e is / v v0 i q n t n Let the beam position for mode n of continuous beam be as x(t , ) x0 e o Then, the force acting on the reference particle is eI b s W s x 0 e i q n s / v ds ieI b Z q n 0 x 0 v F0 (t ) 0 0 eI b R 2 n i Z q n 0 2 Pv The exact result for the betatron tune shift of n-th mode for the bunched beam eI b R 2 n i Z nk , 2 Pv k (2) nk 0 [q] n Nk [q] – the fractional part of the betatron tune N – the number of bunches, n N / 2, ( N 1) / 2 Z – the transverse impedance per unit length averaged over the ring 10 The Dimensionless Increment n eI b R 2 2 Im Re Z nk (3) f0 Pv k The definition of the transverse impedance yields that impedance at a negative frequency is related to the impedance at positive frequency through its complex conjugated value Z ( ) Z ( ) * or ReZ ( ) ReZ ( ) ImZ ( ) ImZ ( ) At low frequencies impedance is completely determined by the wall resistivity. For > 0 real part of the impedance is positive coherent motion is damped n 0 [q] n , n 0...( N 1) / 2 For < 0 real part of the impedance is negative coherent motion is unstable n 0 n [q] , n 1...N / 2 If the impedance grows with frequency decrease preferable fractional tune is in the range [0 – 0.5] 11 Asymptotics for the dimensionless increment c 2a 3 2 , d R n n eI b R 2 c2 Z 0 sign n 2 , ad d f0 Pv 4 R n a d 3 d R 2n , ad c a 12 Einj Einj rw1 Re c i aeff ( .9 1.4) .1 R rw2 Re c i aeff ( .9 1.4) 0.2 R i mp i mp 10 1 f0 f0 f0 f0 Dimensionless increment 1 2 0.1 2 Tune shift 0.1 0.01 1 10 3 0.01 1 10 3 1 10 4 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 3 4 5 6 7 8 3 4 5 6 7 8 1 10 100 1 10 100 f [Hz] f [Hz] Dependence of the dimensionless decrement (left) and the betatron tune shift (right) on the mode frequency for elliptic aluminum vacuum chamber at 300 K, a = 9 mm, b = 14 mm; red line - d=1 mm, blue line - d=2 mm; brown dashed line - a = 12 mm, b = 20 mm, d=2 mm. 9 mm vacuum chamber yields instability growth time about one revolution Increase of wall thickness from 1 mm to 2 mm decreases the growth time by about 1.5 times the coherent tune shift by about 2 times to ~ 0.1 To decrease decrement to about 2 revolutions one needs to increase vacuum chamber size to about 12 by 20 mm That also decreases the coherent tune shift to ~ 0.03 (30 deg per turn) 13 Coherent and incoherent tune shifts The interaction of the beam with round vacuum chamber causes instabilities and changes frequencies of coherent beam motion but does not change incoherent particle frequencies and cannot cause non-linear resonance In the case of elliptic vacuum chamber detuning wake is not zero. That means that the interaction of the beam with vacuum chamber creates additional focusing fields which changes incoherent particle tunes In the case of flat vacuum chamber and uniformly filled ring the frequency of all particles is eI b R 2 shifted by the same amount: ic 0 mc 3 3 a 2 That can be compensated by betatron frequency adjustments proportional to the beam current If the ring is partially filled then there is tune variations proportional to “AC” components in the beam current For flat vacuum chamber the wake and the detuning wake are equal and therefore tune variations can be computed using results obtained for coherent betatron tune shift eI b R 2 ic Re n Im Z n , n n 0 n 2 Pv k To compensate tune variations along the beam the modulation of lattice betatron tune on revolution frequency is required 14 Interaction of the beam DC current with iron of the dipoles causes additional incoherent tune shift eI b R 2 icB mc 3 2 a d 2 ad – is the dipole half-gap That tune shift is close to the incoherent tune shift due to vacuum chamber ic 0 If the ring is partially filled then there is tune variations proportional to “AC” components in the beam current. Their amplitude will depend on the thickness of vacuum chamber and dipole laminations. 15 4. Feedback System for Suppression of the Transverse Multibunch Instability Standard system 0 0 T 80) 90 Vn V0 exp i n 0 (90+ n1 N 90 cos( 0 ) sin( 0 ) 1 k ( ) 0 Vn N 0 1 k ( ) Vn sin( 0 ) cos( 0 ) T iT k ( ) g j exp j j 0 N N N – number of bunches T – revolution period g(t) – response function Perturbation theory solution exp i T 1 k ( ) exp i n n ~ n 0 nT 0 2n ~ Damping decrement: n ln n T 16 Integrating type narrow band system t 10 g (t ) g 0 exp Numerical solution 1 g0 Perturbation theory k ( ) Decr./f0 T 1 1 exp i 0.1 Resistive w all instability N g0=0.1, /T=9.48510-5, 0=20.46, 0.01 0.1 1 10 100 f/f0 1 10 3 1 10 4 1 10 5 N=105 1 1 Perturbation theory 0.015 Eigen-number module Eigen-number module 0.5 0.01 0.005 Numerical solution 0 0 1 10 5 10 5 10 1 10 5 4 4 5 200 100 0 100 200 0 f/f0 f/f0 17 There is no principal limits to reach damping faster than the revolution time Narrow band system can be stable in the entire frequency range Technical problems Signal transfer from BPMs to kickers Practical system for the case of Forward transfer aluminium vacuum chamber at 78 K Expensive tunnel for signal transfer 1 One turn delay Large betatron phase shift after one 0.1 Feedback system turn Large delay digital system Decr./f0 0.01 Practical system needs subtraction of Resistive w all instability orbit offset 1 10 3 Slow feedback to nullify BPMs Or notch filter at revolution frequency 1 10 0.1 1 4 10 100 1 10 3 1 10 4 1 10 5 f/f0 which will reduce maximum achievable gain g0=0.005, /T=510-4, 0=20.46, N=105 Limits of the system gain Final accuracy of electronics Emittance growth due to BPM noise Kicker voltage 5 m, 10 kV kicker suppresses 2.5 mm injection oscillations at 1 turn 18 System with delayed response Single system with delayed response T Vn V0 exp i n N cos( 0 ) sin( 0 ) 0 0 Vn N Vn k ( ) Vn N sin( 0 ) cos( 0 ) 0 1 T iT k ( ) g j exp j j d j 0 N N N – number of bunches T – revolution period g(t) – response function jd – number of skipped bunches Perturbation theory solution 1 k ( n ) 2 1 k ( n ) 2 2 1 n exp iT cos 0 1 k ( ) cos 2 0 1 k ( n ) n 1 k ( n ) n T 0 2n 19 t g0 Integrating type system g (t ) g 0 exp , k ( ) T 1 1 exp i N 0.5 1 1 Re(k()) || 0 0.9 Im(k( )) 0.5 0.8 1 10 1 10 100 50 0 50 100 3 4 0.1 1 10 100 f/f0 f/f0 g0=210-4, /T=.025, 0=20.27, N=105, jd =50 – 50 skipped bunches (105/(4500)) Optimization strategy Suppress resistive wall instability at low frequencies Achieve minimum increments at higher frequencies 20 Double system with delayed response 0 cos( 0 ) sin( 0 ) 90 M , M M 1M 2 , sin( 0 ) cos( 0 ) l 1,2 k2() T i T k l ( ) g l j exp j j d M2 N N M1 j 0 1 2 1 0 kl , 0 1 k l ( ) 1 MV n k 1 M 2 k 2 M 2 k 1 Vn N k1() N – number of bunches T – revolution period 0 90 g(t) – response function jd – number of skipped bunches n 1 k1 ( n ) k 2 ( n ) k1 ( n )k 2 ( n ) 2 k ( ) k 2 ( n ) k ( )k ( ) 2 n cos 0 1 1 n cos1 cos 2 1 n 2 n 1 0 2 2 21 Comparison of single (0, g0) and double systems (1=2=0/2, g1=g2=g0) 1 1 =0.2*2 =0.4*2 Single system Single system 0.5 0.5 1 1 e e Double s ystem 1 1 Double s ystem 2 2 e e 0 0 0 2 4 6 8 0 2 4 6 8 gain gain Dependence of absolute values of eigen-numbers on system gain, g, for different betatron tunes. 22 1 1 Single system Double system Resistive w all 0.1 0.1 Single system decr/f0 decr/f0 Double system Resistive w all 0.01 0.01 1 10 1 10 3 3 1 10 1 10 1 10 1 10 3 4 3 4 0.1 1 10 100 0.1 1 10 100 f/f0 f/f0 Damping decrement Residual instability increment N=10 , jd =50; Single system: g0=210-4, /T=2.510-2, 0=20.27, 5 Double system: g0=810-8, /T=210-2, 0=20.46. Conclusions for a system with delayed response for VLHC System allows to reduce a requirement for the damping decrement of the bunch- bunch dumping system by an order of magnitude Installation of multiple systems does not allow to increase the total damping due to anti-damping at high frequency. Decreasing the number of skipped bunches one can improve performance but it increases noise effects due to smaller beta-functions 23 5. Emittance Growth Suppression Emittance growth due to noises N /2 d , n 0 n , 3.3 f0 67 2 S noise n 2 2 N 2n / f 0 2 2 dt n 1 2 Spectral density of transverse kicks, S d , for the feedback system 2 noise n 2 is determined by spectral density of BPM noise x , system gain and beta-function at BPM, 1 : 2 x 2 2 S noise n k n . 1 1 – beta-function at BPM 2 – beta-function at kicker Emittance growth due to noise and its suppression with the feedback system in large hadron colliders, V. Lebedev, et.al. V.V.Parkhomchuk, V.D.Shiltsev, G.V.Stupakov; SSCL-Preprint-188, Dallas, March 1993 and Particle Accelerators, 1994, v.44, pp.147-164 24 Theoretical limit for the standard narrow band system 2 10 4 Effective BPM noise of the feedback d dt d system is determined by the thermal noise of amplifiers is 1 10 4 kT x 2a 2 2 I b Z BPM 2 2 0 Factor of 2 appeared because the 0 5000 1 10 4 f/f0 1.5 10 4 2 10 4 2.5 10 4 system consists of two independent Relative contribution of different frequencies systems shifted by 90 deg in betatron into emittance growth phase = 50 , T=300 K, Ib=0.19 A, ZBPM=12 x 2 = 0.02 m (f=25 MHz) 2 x =10-20 cm2 s, 0 1.3105 days - at the injection d / dt Required BPM accuracy Emittance growth time > 100 hour bunch position measured with x 2 = 3 m (180 times of theoretical limit) 25 Conclusions Multi-bunch instabilities can be suppressed To suppress the resistive wall instability Increase vacuum chamber thickness to 2 mm and vertical size to 212 mm will be extremely profitable Two feedback systems required Low frequency system with high gain, narrow band and delayed response to damp low frequencies Unstable at high frequencies High frequency system to restore stability at high frequencies wide band, moderate gain and one turn delay Unstable at low frequencies 26

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