Transverse Accelerator Dynamics Ralph J. Steinhagen Special acknowledgements and credits to: B. Goddard, B. Holzer & R. Steerenberg Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Outline Part I – Linear Beam Dynamics & Hill's Equation – Periodic Focusing System in Circular Accelerators – Phase Space Ellipse – Emittance & Acceptance – Machine Imperfections • Betatron Tune & Beam Stability Part II – Non-Linear Dynamics & Injection/Extraction – Non-linear dynamics: • limits of stable motion – Separatrix • Dispersion & Chromaticity • Space charge effects – Injection & Extraction: • Fast extraction, Multiturn Injection (phase-space painting) • Basics of resonance-, KO-extraction GSI Accelerator Operator Training – Transverse Dynamics, Darmstadt, Germany, R.Steinhagen@gsi.de, 2016-01-26
Question: Does the Moon revolve around the Earth or the Sun? Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Moon's Trajectory around Sun F=ma x F gravity s ρ(s) Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26 not to scale!*
Transverse Beam Dynamics Hill's equation 1,2 : K ( s )= ( p B dipole ) ∂ B y 2 q q − z '' + K ( s ) ⋅ z = f ( s ,t ) ⏟ ⏟ p ∂ x strong focusing : k ( s ) weak focusing : 1 ρ 2 – k(s): focusing strength, defines: G.W. Hill 1838-1914 • betatron function β(s) → envelope of the oscillation • dispersion function D(s) → trajectory for off-momentum Δp/p 0 particles – f(s,t): external driving force 1 George William Hill, “On the part of the motion of the lunar perigee which is a function shorthand: x ' = dx of the mean motions of the sun and moon” , Acta Mathematica, 8:1–36, 1886 & z : = ' x ' or ' y ' 2 coordinate 'z' being place holder for either x,y ds Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Hill's Equation I/II Simplified Hill's equation 1,2 : ∂ B y K ( s )= − q z '' + K ( s ) ⋅ z = 0 ∧ (I) ⏟ p ∂ x Cournat, (Cosmotron) , Livingston, Snyder, strong focusing Blewet (LINACs) – If the restoring force K(s) would be constant in ‘s’ → Simple Harmonic Oscillator • usually K(s) varies strongly with 's' (discrete magnets, FODO arrangement, ..) How to solve? – Try the following Ansatz 1,2 : z s = i s ⋅ sin s i (II) ε i ,Φ i : initial particle state – derive (II) twice to obtain z'' and insert into (I) – Don't worry, KISS → solution typically (nearly always) done numerically using tools like: MAD-X, ORBIT, TRANSPORT, ELEGANT, MIRKO, ... 1 Richard Q. Twiss and N. H. Frank, “Orbital stability in a proton synchrotron”, Rev. Sci. Instr., 20(1):1–17, January 1949. 2 E. D. Courant and H. S. Snyder, “Theory of the Alternating-Gradient Synchrotron”, Annals of Physic, 3, 1, 1958. shorthand: x ' = dx ds Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
MAD-X Typical Output MQF MQD MQF MQD MQF MQD MB MB MB MB MB MB BPM BPM BPM BPM BPM BPM BPM Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
FoDo alternatives → FD Doublet Lattice More space between quads Stronger quad strengths Round beams Used e.g. in CTF3 linac FoDo FD Doublet Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Many Alternatives a) Weak focusing (dipole only) b) FODO line (w/o dipoles) c) FODO cell d) Low emittance cell ‐ e) CF low emittance cell ‐ f) Low emittance FODO ‐ g) Dispersion match h) Periodic dispersion match i) Double bend achromat ‐ j) Triple bend achromat ‐ k) ... Very good course on low‐emittance lattice design: A.Streun, PSI Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Hill's Equation II/II Usually define add. 'Twiss' functions 1 : – betatron phase advance μ(s), α(s) & γ(s) 1 α( s ) : =−β ' ( s ) γ( s ) : = 1 +α 2 ( s ) s Δμ( s ) : = ∫ 0 β( s ' ) ds ' β( s ) 2 – typically stored in look-up tables (e.g. LSA) and re-used for other computations More general first-order solution to Hill's equation: ⏟ ⏟ z ( s ) = z co ( s ) + D ( s ) ⋅ Δ p + z β ( s ) ⏟ p 0 betatronoscillations closedorbit dispersion orbit → sinusoidal particle motion in accelerators: z s = i s ⋅ sin s i N.B. discussed later: Dispersion function D(s) – ↔ trajectory dependence for off-momentum particles 1 Richard Q. Twiss and N. H. Frank, “Orbital stability in a proton synchrotron”, Rev. Sci. Instr., 20(1):1–17, January 1949. shorthand: x ' = dx 2 E. D. Courant and H. S. Snyder, “Theory of the Alternating-Gradient Synchrotron”, Annals of Physic, 3, 1, 1958. ds Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Free Betatron Oscillations Free Betatron Oscillations: z β ( s ) = √ ϵ i β ( s ) ⋅ sin (μ ( s ) +ϕ i ) here: Q = 3.31 q = .31 '1' '2' '3' '4' Betatron Phase Advance: Δμ(s) Tune defined as betatron phase advance over one turn: 1 Q : = [ μ ( C ) −μ ( 0 ) ] common: Q = Q int q frac 2 π integer tune fractional tune Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Free Betatron Oscillations Example: LHC Betatron Oscillations z β ( s ) = √ ε i β ( s ) ⋅ sin (μ ( s ) +ϕ i ) vertical orbit [mm] ~ √β(s) position in ring [m] Beam size* : σ≈ √ ε i β( s ) Emittance (beam quality) ε i Beta-function β(s) specific for accelerator lattice/ constant around ring → discussed later magnet arrangement *ignores momentum & dispersion dependence Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Phase Space & (Single-Particle) Emittance Additional result from our Ansatz: Courant-Synder invariant of motion 2 + 2 α( s ) 2 ϵ=β( s ) ⋅ x ' ⋅ xx ' +γ( s ) ⋅ x Interpretation – particle motion describe ellipse in phase space: μ = 3 π /2 x’ μ = 0 = 2 π ε . γ ε / β − α ε / γ − α ε / β ε π x = A ε / γ ε . β Liouville’s Theorem: 'conservative system' (no ‘friction’, i.e. no energy loss/gain) , the phase-space area is invariant/preserved (↔ energy conservation) – N.B. if energy changes 'normalised emittance' ε* is preserved: ε*=ε·β rel γ rel Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Phase-Space & Twiss-Parameter inside a FoDo Lattice y x x' x y' y Courtesy A. Wolski Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Emittance & Acceptance II/II E.g. circular beam pipe of radius r ap By analogy with emittance σ≈ √ εβ( s ) – N.B. beam size: 2 A : = r ap Acceptance β( s ) – N.B. sometimes given in units of ' σ ' (beam widths) σ< √ A β( s ) Acceptance chosen such that: A >> ε or – ie. “ beam pipe needs to be larger than beam – duh ” LHC: LHC: cathode Beam 3 σ envel. ~ 1.8 mm @ 7 TeV 50.0 mm Beam 6 σ envel. ~ 12 mm @ 450 TeV ~ 140.0 mm Beam screen ~ third of aperture if filled @ injection (rest kept for orbit/optics/injection error uncertainties) SIS18: 36 mm 300 kV design acceptance 325 umrad @ injection anode design emittance <299 umrad @ injection (wires) (± ~14 mm margin for injection, optics and orbit errors) SIS18 electrostatic injection septum Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Normalised Phase Space often more conveniently one defines a 'normalised phase space' , ) = √ β ( β ) ( , ) = N ⋅ ( ⋅ ( , ) 1 x x x 1 0 α x x x real phase space normalised phase space x’ − − √ ε α x' √ γ − √ ε 1 − √ ε 1 √ ϵ − √ ε √ γ √ γ √ β − √ ε α √ β N x x Area = πε Area = πε − √ ε √ β √ ϵ 2 + 2 α( s )⋅ xx ' +γ( s ) 2 ϵ=β( s ) ⋅ x ' ⋅ x 2 +¯ ,2 ϵ=¯ x x Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Recap: Transfer of Optics Parameter = M ( ( x ' ) s x ' ) 0 x x Conservation of emittance 2 + 2 α 0 x 0 x ' 0 + γ 0 x 0 2 ϵ=β 0 x ' 0 (1) 2 + 2 α 1 x 1 x ' 1 +γ 1 x 1 2 ϵ=β 1 x ' 1 Express x 0 , x' 0 as a function of x,x' ( x ' ) 0 − 1 ( x ' ) → x 0 = m 22 x − m 12 x ' x x = M (2) x 0 ' =− m 21 x + m 11 x ' Inserting (2) into (1), sorting via x,x', the rest is math ... 2 ) γ ) = ( 2 2 − 2 m 11 m 12 m 11 m 12 ( ⋅ ( γ ) 0 β β − m 11 m 21 m 12 m 21 + m 22 m 11 − m 12 m 22 α α 2 m 12 − 2 m 22 m 21 m 22 Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Difference: Transfer-Line (LINAC) vs. Ring I/II Circular Machine (ring): ● β, α & γ are not free parameter but need to fulfill special periodic boundary condition: ● β(s+C) = β(s) ● α(s+C) = α(s) ● γ(s+C) = γ(s) One-pass Machine (LINAC/transfer-lines): ● Causality! ● Need to provide initial parameter for β, α & γ: ● β(s=0) = β 0 ● α(s=0) = α 0 ● γ(s=0) = γ 0 ● otherwise, trans. particle transfer the same Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
Difference: Transfer-Line (LINAC) vs. Ring II/II Circular Machine (ring): ● as before One-pass Machine (LINAC/transfer-lines): ● 10% different initial conditions → β-beating → causes problems with matching and emittance preservation at injection (later more) Transverse Dynamics / OP-Training, R.Steinhagen@gsi.de, 2016-01-26
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