Linear Imperfections Oliver Bruning / CERN AP ABP

Transcription

1 Linear Imperfection CAS Fracati November 8 Oliver Bruning / CERN AP ABP

2 Linear Imperfection equation of motion in an accelerator Hill equation ine and coine like olution cloed orbit ource for cloed orbit perturbation dipole perturbation cloed orbit repone diperion orbit integer reonance BPM & dipole corrector quadrupole perturbation one turn map & tune error beta beat half integer reonance orbit correction local orbit bump

3 Variable Definition Variable in moving coordinate ytem: x y ρ x = d d x d dt = d dt v d d x = p p x Hill Equation: d x d + K() x = ; K() = K( + L); K() = ρ drift / dipole.3 B[T/m] p[gev] quadrupole Perturbation: d x + K() x = G(); G() = d F() Lorentz v p

4 Sinelike and Coinelike Solution ytem of firt order linear differential equation: y = x x y + K y = K = cont Y () = in ( K ) K co ( K ) Y () = co ( K ) K in ( K ) initial condition: Y () = Y Y = and Y () = Y Y = general olution: y () = a Y () + b Y () tranport map: y () = M( ) y ( ) with: = co ( K [ ]) K in ( K [ ]) in ( K [ ]) K co ( K [ ])

5 Sinelike and Coinelike Solution Floquet theorem: Y () = β( ) in ( φ( ) + φ) [co ( φ( ) + φ ) + α( ) in( φ( ) + φ )] / β( ) Y () = β( ) co ( φ( ) + φ ) [in ( φ( ) + φ )+ α( ) co( φ( ) + φ )]/ β( ) β() = β ( + L); φ () = d; α () = β β () inelike and coinelike olution: C () = a Y () + b Y () S() = c Y () + d Y () with: C ( ) = ( ) C( ) C ( ) = and S( ) = S S ( ) = one can generate a tranport matrix in analogy to the cae with contant K()!

6 Sinelike and Coinelike Solution inelike and coinelike olution: S () = β( ) β( )β( ) [co ( φ( ) + φ ) + α( ) in( φ( ) + φ )]/ in ( φ( ) + φ ) β( ) C () = β( ) (+αα ) [co ( φ( ) + φ ) + α( ) in( φ( ) + φ )]/ [in ( )+(α α) co( φ( ) + φ )]/ φ( ) + φ β( ) β β tranport map from to : y () = M(, ) y ( ) with: M = C () C () S () S () tranport map for = + L: M = I co( π Q) + J in( π Q) I = J = α γ β α ; γ = [ + α ] / β

7 Cloed Orbit particle ocillate around an ideal orbit: B F B D B F D B F B D B additional dipole field perturb the orbit: error in dipole field energy error α = l ρ q B l = p + Δ p Δ p p q B l p offet in quadrupole field B = g y x B = g x y x = x + x B = g y dipole component x B = g x + g x y

8 Quadrupole Magnet B = g y x y B = g x y S N x R F x = q v B y F = q v B y x N S d x d + K() x = G(); G() = F() Lorentz v p normalized field: dipole: k () =.3 quadrupole: k () =.3 B [T] p [GeV] g [T/m] p [GeV] quadrupole mialignment: Δ k () =.3 g[t/m] p[gev] x

9 Dipole Error and Orbit Stability Q: number of β Q = N ocillation per turn Kick the perturbation add up amplitude growth and particle lo watch out for integer tune! Q = N +.5 Kick the perturbation cancel after each turn

10 Quadrupole Error and Orbit Stability Quadrupole Error: orbit kick proportional to beam offet in quadrupole Q = N +.5. Turn: x > F kick amplitude increae. Turn: x < F amplitude increae watch out for half integer tune!

11 Source for Orbit Error Quadrupole offet: alignment +/-. mm ground motion low drift Error in dipole trength power upplie calibration Energy error of particle civiliation moon eaon civil engineering injection energy (RF off) RF frequency momentum ditribtion

12 Example Quadrupole Alignment inlep

13 Problem Generated by Orbit Errror injection error: aperture filamentation beam loe beam ize cloed orbit error: x y coupling aperture energy error field imperfection diperion beam ize at IP beam eparation Aim: Δ Δ x, y < 4 mm rm <.5 mm beam monitor and orbit corrector

14 Synchrotron: the orbit determine the particle energy! aume: L > deign orbit V b) a) t energy increae Equilibrium: f RF = h f rev f = rev π q m γ B E depend on orbit and magnetic field!

15 tidal motion of the earth: tide Moon Earth orbit and beam energy modulation: f = 4 h; h mod Δ E MeV.% aim: Δ E <.3% require correction!

16 energy modulation due to tidal motion of earth Δ E [MeV] 5 November th, : 3: 7: : 5: 9: 3: 3: Δ E [MeV] 5 Augut 9 th, 993 October th, : 3: 5: 7: 9: : 3: 8: : : 4: : 4: 6: 8: Daytime Δ E MeV

17 energy modulation due to lake level change change in the water level of lake Geneva change the poition of the LEP tunnel and thu the quadrupole poition orbit and energy perturbation Δ E MeV

18 energy modulation due current perturbation in the main dipole magnet TGV line between Geneva and Bellegarde

19 correlation of NMR dipole field meaurement with the voltage on the TGC train track LEP Polarization Team Geneve TGV RAIL Meyrin Zimeya LEP beam pipe LEP NMR ΔE 5 MeV for LEP operation at 45 GeV

20 ground motion due to human activity quadrupole motion in HERA p (DESY Hamburg) RMS peak to peak

21 Cloed Orbit Repone inhomogeneou equation: d x d + K() x = G(); G() = Δk () y + K y = G; G = G y() = a S() + b C() + ψ( ) we need to find only one olution! variation of the contant: ψ( ) = c() S() + d() C()

22 Cloed Orbit Repone variation of the contant in matrix form: ψ( ) = φ( ) u(); with φ( )= C() C () S() S () ubtitute into differential equation: φ( ) u() = u () = G() φ( t ) G(t) dt y() = a S() + b C() + φ( ) φ( t ) G(t) dt

23 Cloed Orbit Repone periodic boundary condition: y() = a S() + b C() + φ( ) φ( t ) G(t) dt with y() = x() x () ; x() = x( + L); x () = x ( + L) periodic boundary condition determine coefficient a and b β( ) x() = β( t ) G(t) co[ φ( t ) φ( ) π Q] dt in( π Q) +circ

24 Cloed Orbit Repone Example: particle momentum error normalized dipole trength: k () =.3 B[T] p[gev] k () = ρ( ) t ρ( t ) Δ p p G(t) = ρ( t ) Δ p p x() = β( ) in( π Q) β( t ) G(t) co[ φ( t ) φ( ) π Q] dt with β( ) D() = in( π Q) x() = D() β( t ) ρ( t ) Δ p p co[ φ( t ) φ( ) π Q] dt Diperion Orbit

25 Orbit Correction the orbit error in a torage ring with conventional magnet i dominated by the contribution from the quadrupole alignment error orbit perturbation i proportional to the local β -function at the location of the dipole error alignment error at QF caue mainly horizontal orbit error alignment error at QD caue mainly vertical orbit error

26 Orbit Correction aim at a local correction of the dipole error due to the quadrupole alignment error place orbit corrector and BPM next to the main quadrupole horizontal BPM and corrector next to QF vertical BPM and corrector next to QD BPM BPM QF MB QD HX HV orbit in the oppoite plane? relative alignment of BPM and quadrupole?

27 Horizontal Orbit: Vertical Orbit: LEP Orbit beam offet in quadrupole: energy error Lake Geneva moon beam offet in quadrupole beam eparation orbit deflection depend on particle energy vertical diperion [D()] σ = ε β + δ y y y D mall vertical beam ize relie on good orbit 994: 3 vertical orbit correction in phyic

28 Quadrupole Gradient Error one turn map: can be generated by matrix multiplication: x z = M z n+ n z = x and can be expreed in term of the C and S olution M = I co( π Q) + J in( π Q) I = J = α γ β α ; γ = [ + α ] / β remember: co( π Q) = trace M the coefficient of: π M I co( Q) in( π Q) provide the optic function at

29 Quadrupole Gradient Error tranfer matrix for ingle quadrupole: m = k l matrix for ingle quadrupole with error: m = [k + Δ k ] l one turn matrix with quadrupole error: trace M M = m m M π π co( Q) = co( Q ) β Δ k l in( π Q )

30 Quadrupole Gradient Error ditributed perturbation: co( πq) = co( Q ) π in( π Q ) β Δk d Δ Q = 4 π β Δk d chromaticity: momentum error k = e g p Δ k = k Δ p p Δ Q = 4 π β k d Δ p p = ξ Δ p p

31 β Beat quadrupole error: z = M z n+ n M = m m m m with M = I co( π Q) + J in( π Q) I = J = α γ β α ; γ = [ + α ] / β calculate: m in( π Q) +circ β( ) Δβ( ) = β( t ) Δk(t) co[[ φ( t ) φ( )] π Q] dt in( π Q) β beat ocillate with twice the betatron frequency

32 Local Orbit Bump I deflection angle: θ = i dipole G (t) dt i =.3 B i[t] l p[gev] trajectory repone: [no periodic boundary condition] x() = β β( ) i θ i in[ φ( ) φ i ] x () = β / i β( ) θ i co[ φ( ) φ i ]

33 Local Orbit Bump II cloed orbit bump: compenate the trajectory perturbation with additional corrector kick further down tream cloure of the perturbation within one turn local orbit excurion poibility to correct orbit error locally cloure with one additional corrector magnet π - bump cloure with two additional corrector magnet three corrector bump

34 Local Orbit Bump III x π - bump: (quai local correction of error) QF QD QF QD QF θ θ θ 3 θ = β β θ limit / problem: cloure depend on lattice phae advance o require 9 lattice enitive to BPM error enitive to lattice error require horizontal BPM at QF and QD require large number of corrector

35 x QF QD Local Orbit Bump IV 3 corrector bump: (quai local correction of error) QF QD QF θ θ θ 3 θ = β β Δφ 3- Δφ 3- in( ) in( ) θ θ = 3 Δφ 3- β in( ) - co( Δφ 3- ) θ tan( Δφ 3- ) β 3 work for any lattice phae advance require only horizontal BPM at QF limit / problem: enitive to BPM error large number of corrector can not control x

36 Summary Linear Imperfection avoid machine tune near integer reonance: they amplifie the repone to dipole field error a cloed orbit perturbation propagate with the betatron phae around the torage ring dicontinuitie in the derivative of the cloed orbit repone at the location of the perturbation avoid torage ring tune near half integer reonance: they amplifie the repone to quadrupole field error betafunction perturbation propagate with twice the betatron phae advance around the torage ring integral expreion are mainly ued for etimate numerical program mainly rely on map cloed orbit = fixed point of turn map diperion = eigenvector of extended turn map tune i given by the trace of the turn map twi function are given by the matrix element