Accelerator Basics or things you wish you knew while at IR-2 and talking to PEP-II folks

1. Accelerator Basicsor things you wish you knew while at IR-2 and talking to PEP-II folks Martin Nagel University of Colorado SASS September 10, 2008

3. Outline • Introduction • Strong focusing, lattice design • Perturbations due to field errors • Chromatic effects • Longitudinal motion

4. How to design a storage ring? • Uniform magnetic field B0 → circular trajectory • Cyclotron frequency: Why not electric bends?

5. What about slight deviations? • 6D phase-space • stable in 5 dimensions • beam will leak out in y-direction

6. Let’s introduce a field gradient • magnetic field component Bx~ -y will focus y-motion • Magnet acquires dipole and quadrupole components combined function magnet

7. Let’s introduce a field gradient • magnetic field component Bx~ -y will focus y-motion • Magnet acquires dipole and quadrupole components • Problem! Maxwell demands By ~ -x • focusing in y and defocusing in x combined function magnet

8. Equation of motion Hill’s equation:

9. Equation of motion Hill’s equation: natural dipol focusing

10. Weak focusing ring K ≠ K(s) • define uniform field index n by: • Stability condition: 0 < n < 1 natural focusing in x is shared between x- and y-coordinates

11. Strong focusing • K(s) piecewise constant • Matrix formalism: • Stability criterion: eigenvalues λi of one-turn map M(s+L|s) satisfy 1D-system: drift space, sector dipole with small bend angle quadrupole in thin-lens approximation

12. Alternating gradients • quadrupole doublet separated by distance d: • if f2 = -f1, net focusing effect in both planes:

13. FODO cell stable for |f| > L/2

14. Courant-Snyder formalism • Remember: K(s) periodic in s • Ansatz: ε = emittance, β(s) > 0 and periodic in s • Initial conditions • phase function ψ determined by β: • define: βψαγ = Courant-Snyder functions or Twiss-parameters

15. Courant-Snyder formalism • Remember: K(s) periodic in s • Ansatz: ε = emittance, β(s) > 0 and periodic in s • Initial conditions • phase function ψ determined by β: • define: βψαγ = Courant-Snyder functions or Twiss-parameters properties of lattice design properties of particle (beam)

16. Phase-space ellipse • ellipse with constant area πε • shape of ellipse evolves as particle propagates • particle rotates clockwise on evolving ellipse • after one period, ellipse returns to original shape, but particle moves on ellipse by a certain phase angle • trace out ellipse (discontinuously) at given point by recording particle coordinates turn after turn

17. Adiabatic damping – radiation damping With acceleration, phase space area is not a constant of motion • energy loss due to synchrotron radiation • SR along instantaneous direction of motion • RF accelerartion is longitudinal • ‘true’ damping Normalized emittance is invariant:

18. particle → beam • different particles have different values of ε andψ0 • assume Gaussian distribution in u and u’ • Second moments of beam distribution: beam size (s) = beam divergence (s) =

19. Beam field and space-charge effects uniform beam distribution: beam fields: • E-force is repulsive and defocusing • B-force is attractive and focusing relativistic cancellation beam-beam interaction at IP: no cancellation, but focusing or defocusing! Image current: beam position monitor:

20. How to calculate Courant-Snyder functions? • can express transfer matrix from s1 to s2 in terms of α1,2β1,2γ1,2ψ1,2 • then one-turn map from s to s+L with α=α1=α2, β=β1=β2, γ=γ1=γ2, Φ=ψ1-ψ2 = phase advance per turn, is given by: • obtain one-turn map at s by multiplying all elements • can get α, β, γat different location by: betatron tune

21. Example 1: beta-function in drift space

22. Example 2: beta-function in FODO cell discontinuity in slope by -2β/f QF/2 QD QF/2

23. Perturbations due to imperfect beamline elements • Equation of motion becomes inhomogeneous: • Multipole expansion of magnetic field errors: • Dipole errors in x(y) → orbit distortions in y(x) • Quadrupole errors → betatron tune shifts → beta-function distortions • Higher order errors → nonlinear dynamics

24. Consider dipole field error at s0 producing an angular kick θ Closed orbit distortion due to dipole error integer resonances ν= integer

25. Tune shift due to quadrupole field error quadrupole field error k(s) leads to kick Δu’ q = integrated field error strength tune shift • can be used to measure beta-functions (at quadrupole locations): • vary quadrupole strength by Δkl • measure tune shift

26. beta-beat and half-integer resonances quadrupole error at s0 causes distortion of β-function at s: Δβ(s) (1,2)-element of one-turn map M(s+L|s) β-beat:

27. beta-beat and half-integer resonances quadrupole error at s0 causes distortion of β-function at s: Δβ(s) (1,2)-element of one-turn map M(s+L|s) β-beat: twice the betatron frequency half-integer resonances

28. Linear coupling and resonances • So far, x- and y-motion were decoupled • Coupling due to skew quadrupole fields νx + νy = n sum resonance: unstable νx - νy = n difference resonance: stable

29. Linear coupling and resonances • So far, x- and y-motion were decoupled • Coupling due to skew quadrupole fields mx νx + νy = n sum resonance: unstable my mx νx - νy = n difference resonance: stable my nonlinear resonances ν= irrational!

30. Chromatic effects • off-momentum particle: • equation of motion: • to linear order, no vertical dispersion effect • similar to dipole kick of angle • define dispersion function by • general solution:

31. Calculation of dispersion function transfer map of betatron motion inhomogeneous driving term Sector dipole, bending angle θ = l/ρ << 1 quadrupole FODO cell x …Φ = horizontal betatron phase advance per cell

32. Dispersion suppressors at entrance and exit: after string of FODO cells, insert two more cells with same quadrupole and bending magnet length, but reduced bending magnet strength: QF/2 (1-x)B QD (1-x)B QF xB QD xB QF/2

33. Longitudinal motion • (z, z’) → (z, δ = ΔP/P) → (Φ = ω/v·z, δ) • allow for RF acceleration • synchroton motion very slow • ignore s-dependent effects along storage ring • avoid Courant-Snyder analysis and consider one revolution as a single “small time step” Synchroton motion

34. RF cavity Simple pill box cavity of length L and radius R Bessel functions: Transit time factor T < 1: Ohmic heating due to imperfect conductors:

35. Cavity design 3 figures of merit: (ωrf, R/L, δskin) ↔ (ωrf, Q, Rs) Quality factor Q = stored field energy / ohmic loss per RF oscillation volume surface area Shunt impedence Rs = (voltage gain per particle)2 / ohmic loss

36. Cavity array • cavities are often grouped into an array and driven by a single RF source • N coupled cavities → N eigenmode frequencies • each eigenmode has a specific phase pattern between adjacent cavities • drive only one eigenmode , m = coupling coefficient large frequency spacing → stable mode relative phase between adjacent cavities

37. cavity array field pattern: coupling pipe geometry such that RF below cut-off (long and narrow) side-coupled structure in π/2-mode behaves as π-mode as seen by the beam

38. Synchrotron equation of motion synchronous particle moves along design orbit with exactly the design momentum h = integer • Principle of phase stability: • pick ωrf → beam chooses synchronous particle which satisfies ωrf = hω0 • other particles will oscillate around synchronous particle synchronous particle, turn after turn, sees RF phase of other particles at cavity location: C = circumference v = velocity

39. Synchrotron equation of motion η = phase slippage factor αc = momentum compaction factor transition energy: …beam unstable at transition crossing • linearize equation of motion: • stability condition • synchrotron tune: “negative mass” effect

40. Phase space topology Hamiltonian: • SFP = stable fixed point • UFP = unstable fixed point • contours ↔ constant H(Φ, δ) • separatrix = contour passing through UFP, • separating stable and unstable regions bucket=stable region inside separatrix

41. RF bucket Particles must cluster around θs and stay away from (π – θs) (remember: Φ↔ z) Beams in a synchrotron with RF acceleration are necessarily bunched! bucket area = bucket area(Φs=0)·α(Φs)