accelerator physics and ion optics summary longitudinal optics

accelerator physics and ion optics summary longitudinal optics Sytze Brandenburg sb/accphys007_5/1

coupling energy difference acceleration phase stability when accelerating on slope of sine low energy: rising slope high energy (above transition): falling slope sb/accphys007_5/

coupling energy difference acceleration phase stability when accelerating on slope of sine low energy: rising slope high energy (above transition): falling slope stable region depends on phase reference particle sb/accphys007_5/3

coupling energy difference acceleration phase stability when accelerating on slope of sine low energy: rising slope high energy (above transition): falling slope stable region depends on phase reference particle sb/accphys007_5/4

concept longitudinal optics same as transverse motion transverse: quadrupole longitudinal: buncher phase difference energy difference at time focus phase difference strongly reduced initial energy spread blurs time focus sb/accphys007_5/5

accelerator physics and ion optics distortions and resonances Sytze Brandenburg sb/accphys007_5/6

dipole error induced by dipole magnet different strength different length dipole kick ( ) quadrupole magnet BL = BL+ B L y y y positioning error x and/or y B y (x) = gx; B x (y) = -gy ( BL y ) dipole kick = g xl sb/accphys007_5/7

dipole error single error along circumference angle kick x = (B y L)/Bρ largest effect when close to F-quadrupole minimum divergence, maximum beamsize smallest effect when close to D-quadrupole maximum divergence, minimum beamsize sb/accphys007_5/8

dipole error at F-quadrupole sb/accphys007_5/9

error at D-quadrupole sb/accphys007_5/10

closed orbit distortion δ-function kick ( BL y ) q ( BL y ) x' = = Bρ requirement for closed orbit: after one turn x x = R, R matrix for full turn ' ' ' x x x p sb/accphys007_5/11

solution in space coordinates R cos π Q +α sinπq β sin πq 0 0 = γ0sinπq cosπq α0sinπq α 0, β 0 and γ 0 depend on location field error (Twiss matrix) ( 1) ( π Q ) x cosπ Q+α sinπ Q + x' β sinπ Q= 0 0 0 0 0 x γ sinπ Q + x ' cos πq α sin 1 0 0 0 0 = x' solution x' β0 x0 = tanπq x' α0 x' 0 = 1 tanπq Q integer amplitude explodes resonance sb/accphys007_5/1

solution in Floquet coordinates Floquet coordinate transformation (see also lecture 3) ( ) ( s) s x s 1 d σ η( s) ψ ( s) = β Q β σ ψ changes by π per turn 0 ( ) field perturbations along orbit given by ( ) Fs = B y Bρ ( s) equation of motion of η 3 d η Q s Q F s + η=β dψ ( ) ( ) driven harmonic oscillator sb/accphys007_5/13

δ-function perturbation equation inhomogeneous in one point homogeneous solution applies except at perturbation ( ) ( ) ( ) η s =η cosq ψ s ψ π =η cosφ K K K c phase at perturbation -Qπ decreasing φ c from zero: approaching from above Qπ increasing φ c from zero: approaching from below sb/accphys007_5/14

at perturbation dη dψ dη dψ + ( ) = η Qsinφ = ηqsin Qπ =ηqsinqπ K c 0 0 = η Qsinφ = ηqsinqπ K K c 0 relate to kick in coordinate space dx ds dψ 1 dx dη 1 dη = = β K = ds Qβ ds ds Q β dψ ( BL y ) x' dx = = = K sinq π Bρ ds βk η K = sinqπ ( BL y ) Bρ η β K K sb/accphys007_5/15

graphical representation complete turn - perturbation: φ c = πq orbit should be closed: φ c = πn η K = 0 (trivial solution, no kick) kick closes orbit φ K = π(n-q) φ = Qπ φ = Qπ dη/dψ η sb/accphys007_5/16

distorted orbit ( s) β K ( BL y ) β x( s) = β( s) η ( s) = cosq ψ s ψ π sinqπ Bρ ( ( ) K ) distortions at locations s Ki, corresponding to ψ Ki sum contributions β ( BL) Ki y x( s) = β( s) cosq ψ( s) ψki π i sinqπ Bρ ( ) sb/accphys007_5/17

Fourier analysis of perturbations equation of motion of η 3 d η Q s Q F s + η=β dψ ( ) ( ) expand β 3/ F(s) in Fourier series (periodic system) d η + η=β = dψ 3 ( ) ikψ Q s Q F ( s) Q fke k= 1 π π 3 ik 1 1 ψ 1 ikψ( s) fk = β ( s) F( s) e dψ = β ( s) F( s) e ds π πq 0 0 solution η= Re k= 1 Q Q fk k e ikψ dψ 1 = ds Q β K sb/accphys007_5/18

η= Re k= 1 Q Q fk k e ikψ amplitude very large for Q close to k resonance 10 100 5 Q = 6.4 Q / Q /(Q -k -k ) 0 10-5 -10 10 5 10 15 0 3.6 3.8 4.0 4. 4.4 k Q sb/accphys007_5/19

closed orbit bumps correction of field perturbations displacement of closed orbit for injection and extraction ( BL y ) x( s) = β s βk sin φ φ Bρ ( ) ( ) simplest case: betatron phase advance π between two bumps K sb/accphys007_5/0

more general case: three field bumps: control x at specific location four field bumps: control x and x at specific location x' δ1 δ3 δ x sb/accphys007_5/1

general transfer matrix β ( cos φ + α1sin φ) β1β sin φ x β1 x x' = 1 x' β1 ( α α ) cos φ+ ( 1+α α ) sin φ ( cos φ α sin φ) 1 1 1 1 ββ β 1 x() related to x (1) ( ) =δ ββ sin( φ φ ) x 1 1 1 x (3) related to x(): by going backward ( ) =δ ββ sin( φ φ ) x δ1 β 1 = sinφ 3 3 3 δ β 3 3 sinφ 3 1 sb/accphys007_5/

bump δ determined by change of x at x() dx dx dφ = ds d φ ds dx 1 β =δ ββ cos φ ϕ =δ φ ϕ ds 1 ( ) cos( ) 1 1 1 1 1 β β + dx β =δ cos φ ϕ ds ( ) 3 3 3 β + dx dx δ = ds + ds after some algebra δ1 β1 δ β δ β = = sinφ sinφ sinφ 3 3 3 13 1 sb/accphys007_5/3

closed orbit correction profile monitors at each quadrupole sensitivity proportional to β horizontal at F-quadrupole vertical at D-quadrupole steering magnets at each quadrupole effect proportional to β horizontal at F-quadrupole vertical at D-quadrupole use magnets n-1, n and n+1 to correct error n linear problem N equations, N unknowns matrix inversion sb/accphys007_5/4

injection: transversal stacking displace closed orbit with injection magnets in ring inject one pulse reduce strength injection magnets inject next pulse etc. sb/accphys007_5/5

quadrupole error matrix of nominal quadrupole magnet (thin lense) 1 0 g MQ = By( x) = gx k = kl 1 Bρ matrix of perturbed quadrupole magnet M Q+ Q 1 0 ( k+ k) L 1 matrix one turn with perturbed quadrupole 1 1 0 M= MQ+ QMQM0 = M0 kl 1 sb/accphys007_5/6

cosφ 0 +α0sinφ0 β0sinφ0 M = kl( cos 0 0sin 0) 0sin 0 kl 0sin 0 cos 0 0sin φ +α φ γ φ β φ + φ α φ0 α 0, β 0, γ 0 and φ 0 : betatron functions and phase at quadrupole Tr(M) = cos φ cosφ= cosφ klβ sinφ 0 0 0 1 cosφ= sinφ φ= klβ sinφ φ klβ0 Q = = π 4 π 0 0 0 sb/accphys007_5/7

why resonance for Q = N/ coordinates at passage of quadrupole x 0, x 0 kick (kl)x 0 after one turn for Q = N coordinates x 1 = x 0 + δx, x 1 = x 0 +(δx), kick (kl)x 1 after one turn for Q = M coordinates -x 1 = -(x 0 + δx), -x 1 = -(x 0 +(δx) ), kick - (kl)x 1 mirror symmetry kicks coherent for both Q = N and Q = M sb/accphys007_5/8

second order resonance angle kick proportional to position ( ) dx s = x s kl = klη0 β s cosφ s ds ( ) ( ) ( ) in (η, φ)-coordinates dη = klηβ 0 cosφ dφ dη dη dφ dη = =β dφ ds ds ds φ = π Q dη/dφ η sb/accphys007_5/9

second order resonance dη 1 η 0 = sin kl 0 cos sin kl 0 sin d φ= η β φ φ= η β φ φ 1 dη klβ φ= cosφ= klβcos φ= ( 1+ cosφ) η0 dφ φ klβ Q= = 1+ cosφ π 4π ( ) φ = π Q dη/dφ resonance for Q = N/ η sb/accphys007_5/30

width second order resonance dη 1 η 0 = sinφ= klη0βcos φ sinφ= klη0β sin φ d φ per turn φ= πq resonance for φ = 4πQ= πn Q = N/ φ klβ Q= = ( 1+ cosφ) π 4π klβ Q oscillates in band with width Q = 4π if Q - N/ < Q locking in resonant condition at some moment in Q-oscillation Q = N/ sb/accphys007_5/31

sextupole error second order effect angle kick proportional to x ( ) dx s db L db L x s 0 s cos s ds = dx Bρ = dx Bρ η β φ y y ( ) ( ) ( ) analysis analogous to quadrupole error in (η, φ)-coordinates dby L 0 cos 3 dη d = β η φ φ dx Bρ dη dφ dη dφ dη = =β ds ds ds sb/accphys007_5/3

third order resonance 3 dη dby L η 0 = sin 0cos sin d φ= β η φ φ φ dx Bρ 3 1 dη db y L 3 φ= cosφ= β η 0 cos φ η0 dφ dx Bρ dby 3 L = β η0 φ+ φ dx 8Bρ 3 dby ( cos3 3cos ) φ β L Q = = η0 cos3φ+ 3cosφ π 16π dx Bρ ( ) Q close to N/3 resonance second term averages out in three turns sb/accphys007_5/33

width third order resonance 3 dby β L Q= η0 cos3φ+ 3cosφ 16π dx Bρ ( ) β dby L Q moves in band Q Q < η 16π dx Bρ 3 0 0 amplitude changes given by dby L a =β a sin3 dx 8Bρ φ Q = N/3 amplitude grows quadratically no locking because Q amplitude dependent amplitude growth faster than change in Q beam losses sb/accphys007_5/34

third order resonant extraction Q= dby β L a 16π dx Bρ Q = N/3 - Q: particles will not hit resonance for a < 16πBρ Q dby βl dx stable area reduce Q with quadrupoles particles pushed into unstable area can be extracted from ring sb/accphys007_5/35

third order resonant extraction stable area particles on closed orbit Q periods per turn unstable area jump from one trajectory to another Q = p/3 step π/3 Q = p/3 step 4π/3 Q = p step π sb/accphys007_5/36

Fourier analysis Fourier expansion of perturbation around ring P ( ψ ) = k p coskψ response on single perturbation n = quadrupole; n = 3 sextupole 1 Q= βpcosnqψ π overall response π 1 Q= βp( ψ) cosnqψdψ π 0 π 1 = βpk cosnqψcoskψdψ π 0 large for nq = k driven by single harmonic of distribution sb/accphys007_5/37

resonance zoo simple resonance mq x = p; nq y = p coupling resonances drive e.g. rotated quad mq x ± nq y = p + sign: beamloss - sign exchange of amplitude sb/accphys007_5/38

chromaticity dependence tune on momentum (second order) Q = Q p/p 0 caused by momentum dependence focussing 1 db k = y k p = B ρ dx k p 0 treatment analogous to quadrupole error π π 1 1 p Q = β( s) k( s) ds= β( s) k( s) ds 4π 4π p 0 0 0 typical value Q 1.3 Q; Q 5 p/p 0 = x 10-3 ; Q 0.15 too large to escape resonances momentum dependent additional focussing needed sb/accphys007_5/39

dispersion function: radius depends on momentum magnet with radial dependent gradient = sextupole locations with large dispersionfunction h By x,y = x y Bx x,y = hxy sextupole ( ) ( ) ( ) sb/accphys007_5/40

focussing strength hd k = B ρ p p 0 0 0 0 ( ) ( ) π π 1 1 p hsds Q = β( s) k( s) ds= β( s) ds 4π 4π p Bρ separate correction for Q x and Q y two sets of sextupoles width third order resonance amplitude dependent constraints on useful acceptance sb/accphys007_5/41

next lecture damping and cooling reading material Wilson: chapter 8 Electrons Wilson: chapter 1 Cooling CERN Accelerator School 199, CERN report 94-01 Chapter 18 Synchrotron radiation Chapter 4 Cooling techniques presentations and excercises available in PDF-format on http:\\www.kvi.nl\~brandenburg sb/accphys007_5/4