Vertical dynamic aperture with radiation from quadrupoles - PowerPoint PPT Presentation

Vertical dynamic aperture with radiation from quadrupoles FCCee_z_202_nosol_13.seq at 45.6 GeV A. Bogomyagkov, E. Levichev, S. Glukhov, S. Sinyatkin Budker Institute of Nuclear Physics Novosibirsk July, 2017 A. Bogomyagkov (BINP) FCC-ee DA 1 / 15

6d (SR from BEND, QUAD) and 4d tracking: XY 6d(SR) 4d σ σ σ σ σ σ =6.3e-006 m, =3.1e-008 m, =3.8e-004 =6.3e-006 m, =3.1e-008 m, =3.8e-004 x y e x y e 60 150 40 100 20 50 0 y y σ σ 0 Y/ Y/ − 20 − 50 − 40 − 100 − 60 − 150 − 80 − − − − − − − 150 100 50 0 50 100 150 40 30 20 10 0 10 20 30 40 σ σ X/ X/ x x R x = 109 σ x R y = 142 σ y R x = 35 σ x R y = 40 σ y A. Bogomyagkov (BINP) FCC-ee DA 2 / 15

6d (SR from BEND, QUAD) and 6d tracking: XY 6d(SR) 6d σ σ σ σ σ σ =6.3e-006 m, =3.1e-008 m, =3.8e-004 =6.3e-006 m, =3.1e-008 m, =3.8e-004 x y e x y e 60 150 40 100 20 50 0 y y σ σ 0 Y/ Y/ − 20 − 50 − 40 − 100 − 60 − 150 − 80 − − − − − − 100 50 0 50 100 40 30 20 10 0 10 20 30 40 σ σ X/ X/ x x R x = 109 σ x R y = 142 σ y R x = 35 σ x R y = 40 σ y A. Bogomyagkov (BINP) FCC-ee DA 3 / 15

Introduction Problem Why vertical dynamic aperture drops from R y = 142 σ y to R y = 40 ÷ 50 σ y ? PTC ptc_create_layout, time=true, model=2, exact=true, method=6, nst=10; ptc_setswitch, fringe=true, debuglevel=1,radiation=true; Yin = y 0 + j ∗ dy ; Tin = 0 . 225 ∗ (( Yin / y 0 ) 2 ) ∗ σ t ; ptc_start, x=0, px=0, y=Yin, py=0,pt=0, T=Tin; ptc_track,icase=6,closed_orbit,dump,maxaper={1,1,1,1,1,1}, turns=5200,ffile=1,element_by_element; A. Bogomyagkov (BINP) FCC-ee DA 4 / 15

6d (SR from BEND, QUAD) PX:X PY:Y PT:T 5 150 0 0 100 − 1 − 5 50 − 2 px py 0 σ t σ σ − 10 PT/ PX/ PY/ − − 3 50 − 15 − 100 − 4 − 20 − 150 − 5 − 200 − 2 − 1 0 1 2 − 150 − 100 − 50 0 50 100 150 1 2 3 4 5 6 7 σ σ σ X/ Y/ T/ x y t σ σ σ =6.3e-006 m, =3.1e-008 m, =3.8e-004 PT:turn T:turn x y e 58 0 0.7 57 − 0.6 56 1 55 0.5 − 2 σ t y 54 σ t σ PT/ 0.4 T/ Y/ − 3 53 0.3 52 0.2 − 4 51 0.1 − 5 50 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 − 1 − 0.5 0 0.5 1 σ turn turn X/ x A. Bogomyagkov (BINP) FCC-ee DA 5 / 15

6d (SR from BEND, QUAD) damping σ σ σ Y:turn {part==1, Yin=50 } Y:turn {part==5, Yin=52 } Y:turn {part==11, Yin=55 } y y y 60 60 60 40 40 40 20 20 20 y y y σ σ σ Y/ Y/ Y/ 0 0 0 − − 20 20 − 20 − − 40 − 40 40 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 turn turn turn σ σ σ Y:turn {part==16, Yin=57.5 } Y:turn {part==17, Yin=58 } Y:turn {part==18, Yin=58.5 } y y y 60 300 400 200 40 300 100 200 20 0 y y 100 y σ σ σ Y/ Y/ Y/ 0 − 100 0 − − 20 200 − 100 − 300 − 40 − 200 − 400 0 1000 2000 3000 4000 5000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 0 200 400 600 800 1000 1200 1400 1600 turn turn turn A. Bogomyagkov (BINP) FCC-ee DA 6 / 15

History References John M. Jowett (SLAC), Introductory Statistical Mechanics for Electron Storage Rings, AIP Conf.Proc. 153 (1987) 864-970 J.M. Jowett (CERN), Nonlinear Dissipative Phenomena In Electron Storage Rings, Lect.Notes Phys. 247 (1986) 343-366 In *Santa Margherita Di Pula 1985, Proceedings, Nonlinear Dynamics Aspects Of Particle Accelerators*, 343-366 J. Jowett (CERN), Dynamic aperture for LEP: Physics and calculations, Conf.Proc. C9401174 (1994) 47-71, In *Chamonix 1994, LEP performance* 47-71 Comments Some tracking plots are similar. There was no mentioning of damping turning into raising. A. Bogomyagkov (BINP) FCC-ee DA 7 / 15

Parameters (radiation ON, no tappering) Energy: E = 45 . 6 Gev. Tunes: ν s = 0 . 0413, ν y = 0 . 2217, ν s = 0 . 1366 Damping times [turns]: τ s = 1300, τ y = 2600, τ x = 2600 Energy loss: U 0 = 35 . 96 MeV/turn U q ( FF , 50 σ y ) = 4 × 0 . 5 MeV, U q ( FF , 50 σ x ) = 4 × 3 MeV envelope, U d ( B , arc ) = 12 . 4 keV, U q ( QF , 50 σ x ) = 2 . 8 keV, U q ( QF , 50 σ y ) = 2 . 5 eV, U q ( QD , 50 σ x ) = 1 keV, U q ( QD , 50 σ y ) = 10 eV A. Bogomyagkov (BINP) FCC-ee DA 8 / 15

Equations of motion: longitudinal Exact 2 − p 2 σ ′ = − K 0 x − p 2 y x 2 E 4 � − eV 0 � φ s + 2 πσ δ ( s − s 0 ) − C γ � � p ′ p 0 c K 2 0 t = sin 0 ( 1 + 2 p t ) p 0 c λ 2 π E 4 − C γ 1 ( x 2 + y 2 ) p 0 c K 2 0 2 π Average, x β = 0 σ ′ = − α p t − J y � γ � 2 E 4 � − eV 0 � φ s + 2 πσ p 0 c Π( 1 + 2 p t ) − C γ U 0 � � p ′ p 0 c Π K 2 0 1 L q y 2 t = sin − q p 0 c Π λ 2 π A. Bogomyagkov (BINP) FCC-ee DA 9 / 15

Synchronous phase No synchrotron oscillations � 2 n � σ ′ = 0 , p ′ t = 0 , J y ∝ exp τ y � 1 − J y � γ � � + U q ( σ y ) J y U 0 sin [ φ s ] = ( − eV 0 ) α ( − eV 0 ) ε y T:turn {part==1 || part==17} 0.7 0.7 0.6 0.6 0.5 2 π σ s 0.5 λ RF 0.4 y in = 50 σ y, τ y =- 2.6 10 3 0.4 0.3 t σ ϕ s T/ y in = 58 σ y, τ y = 5 10 3 0.3 0.2 0.1 0.2 0.0 0.1 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Turn turn A. Bogomyagkov (BINP) FCC-ee DA 10 / 15

Equations of motion: longitudinal Average, x β = 0 σ ′ = − α p t − J y � γ � 2 y 2 � � p 0 c Π( 1 + 2 p t ) − U q ( σ y ) − eV 0 φ s + 2 πσ U 0 � � q p ′ t = sin − p 0 c Π λ p 0 c Π σ 2 q , y β q , y cos( ψ q + k y s ) = Af q + A ∗ f ∗ � y q = 2 | A | q Solution p t = Be − α t s cos ( k s s ) − J y � γ � − U q ( σ y ) J y β ′ q , y k 2 s σ 2 2 α p 0 c Π q , y e i 2 k y s − F ∗ − F y y e − i 2 k y s 4 k 2 4 k 2 y y J y f q f ′ F y = − U q ( σ y ) k y = 2 π { ν y } q , σ 2 p 0 c Π Π y A. Bogomyagkov (BINP) FCC-ee DA 11 / 15

Equations of motion: vertical Exact y ′ = p y ( 1 − p t ) E 4 E 4 C γ 0 − p y y 2 C γ p ′ p 0 c K 2 0 p 0 c K 2 0 y = K 1 y − p y 1 2 π 2 π Map of quadrupole radiation E 4 E 4 C γ C γ 1 L q = U q ( σ y ) ∆ p y = − p y , 0 y 2 p 0 c K 2 0 p 0 c K 2 0 ≈ 0 . 7 m − 2 1 L q , 0 E 0 σ 2 2 π 2 π y Map of quadrupole fringe K 1 ∆ p y = − p y , 0 y 2 K 1 / 4 ≈ 0 . 15 m − 2 4 , 0 A. Bogomyagkov (BINP) FCC-ee DA 12 / 15

Vertical dynamic aperture limit Solving: parameter variation and averaging y ( s ) = A ( s ) f q ( s ) + A ( s ) ∗ f q ( s ) ∗ , p y ( s ) = A ( s ) f ′ q ( s ) + A ( s ) ∗ f ′ q ( s ) ∗ Solution � − 1 + ( β ′ y / 2 ) 2 � J 2 y = − 2 α y J y + U q ( σ y ) 4 p 0 c cos( 2 ψ q ) { ν y } y J ′ Π σ 2 q , y k 2 ν y β y y DA limit J ′ y = 0 4 σ 2 q , y k 2 K 2 U 0 y 0 J y = � ∝ � − 1 +( β ′ y / 2 ) 2 K 2 U q ( σ y ) cos( 2 ψ q ) { ν y } 1 L q ν y β y J y ≈ 50 σ y A. Bogomyagkov (BINP) FCC-ee DA 13 / 15

Parametric resonance and Van der Pol oscillator E 4 E 4 C γ 0 − ( p y y 2 ) ′ C γ p ′′ y = K 1 p y ( 1 − p t ) − p ′ p 0 c K 2 0 p 0 c K 2 0 Exact: y 1 2 π 2 π y ′′ + k 2 1 − F 1 y 2 cos( 2 k y s ) y + 2 α y ′ = 0 � � Illustration: y 1500 6000 y [ 0 ]= 1.5 10 3 y [ 0 ]= 1.73 10 3 1000 4000 500 2000 0 0 y y - 500 - 2000 - 1000 - 4000 - 6000 - 1500 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Turn Turn y ′′ + k 2 y y + 2 α y ′ � 1 − F 1 y 2 � Van der Pol oscillator: = 0 A. Bogomyagkov (BINP) FCC-ee DA 14 / 15

Conclusion for vertical plane at 45.6 GeV Observed and studied a new effect limiting dynamic aperture. 1 Radiation from FF quadrupoles modulates p t at double betatron 2 frequency. Map of radiation from FF quadrupole is similar to quadrupole 3 fringe and of the same value. Parametric resonance in vertical motion changes damping. It is 4 observed in tracking and obtained by equations. Vertical dynamic aperture is limited by this effect. 5 Chromaticity of vertical beta function in the quadrupole will 6 change estimations, because particle amplitude will depend on its energy. Minimization of β and β ′ chromaticity in the quadrupole is beneficial. A. Bogomyagkov (BINP) FCC-ee DA 15 / 15