1. Accelerator Basics or 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 B 0 → 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 B x ~ -y will focus y-motion Magnet acquires dipole and quadrupole components combined function magnet

7. Let’s introduce a field gradient magnetic field component B x ~ -y will focus y-motion Magnet acquires dipole and quadrupole components Problem! Maxwell demands B y ~ -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 λ i 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 f 2 = -f 1, 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: β ψ α γ = 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: β ψ α γ = Courant-Snyder functions or Twiss-parameters properties of lattice design properties of particle (beam)

16. 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 Phase-space ellipse

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

18. particle → beam different particles have different values of ε and ψ 0 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 s 1 to s 2 in terms of α 1,2 β 1,2 γ 1,2 ψ 1,2 then one-turn map from s to s+L with α=α 1 =α 2, β=β 1 =β 2, γ=γ 1 =γ 2, Φ=ψ 1 -ψ 2 = phase advance per turn, is given by: obtain one-turn map at s by multiplying all elements can get α, β, γ at different location by: betatron tune

21. Example 1: beta-function in drift space

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

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. Closed orbit distortion due to dipole error Consider dipole field error at s 0 producing an angular kick θ integer resonances ν = integer

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

26. beta-beat and half-integer resonances quadrupole error at s 0 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 s 0 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 ν x + ν y = n sum resonance: unstable ν x - ν y = n difference resonance: stable

29. Linear coupling and resonances So far, x- and y-motion were decoupled Coupling due to skew quadrupole fields ν x + ν y = n sum resonance: unstable ν x - ν y = n difference resonance: stablemymy mxmx mymy mxmx 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 quadrupoleFODO cell …Φ = horizontal betatron phase advance per cell x

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)BQD(1-x)BQFxBQDxBQF/2

33. (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” Longitudinal motion 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: (ω rf, R/L, δ skin ) ↔ (ω rf, Q, R s ) Quality factor Q = stored field energy / ohmic loss per RF oscillation volume surface area Shunt impedence R s = (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 relative phase between adjacent cavities large frequency spacing → stable mode

37. cavity array field pattern: 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 coupling

38. Synchrotron equation of motion synchronous particle moves along design orbit with exactly the design momentum Principle of phase stability: pick ω rf → beam chooses synchronous particle which satisfies ω rf = hω 0 other particles will oscillate around synchronous particle synchronous particle, turn after turn, sees RF phase of other particles at cavity location: h = integer C = circumference v = velocity

39. Synchrotron equation of motion η = phase slippage factor α c = 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 θ s and stay away from (π – θ s ) (remember: Φ ↔ z) Beams in a synchrotron with RF acceleration are necessarily bunched! bucket area = bucket area(Φ s =0)·α(Φ s )