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

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accelerator basics or things you wish you knew while at ir 2 and talking to pep ii folks

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

outline
Outline
  • Introduction
  • Strong focusing, lattice design
  • Perturbations due to field errors
  • Chromatic effects
  • Longitudinal motion
how to design a storage ring
How to design a storage ring?
  • Uniform magnetic field B0 → circular trajectory
  • Cyclotron frequency:

Why not electric bends?

what about slight deviations
What about slight deviations?
  • 6D phase-space
  • stable in 5 dimensions
  • beam will leak out in y-direction
let s introduce a field gradient
Let’s introduce a field gradient
  • magnetic field component Bx~ -y will focus y-motion
  • Magnet acquires dipole and quadrupole components

combined function magnet

let s introduce a field gradient7
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

equation of motion
Equation of motion

Hill’s equation:

equation of motion9
Equation of motion

Hill’s equation:

natural dipol focusing

weak focusing ring k k s
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

strong focusing
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

alternating gradients
Alternating gradients
  • quadrupole doublet separated by distance d:
  • if f2 = -f1, net focusing effect in both planes:
fodo cell
FODO cell

stable for |f| > L/2

courant snyder formalism
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

courant snyder formalism15
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)

phase space ellipse
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
adiabatic damping radiation damping
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:

particle beam
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) =

beam field and space charge effects
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:

how to calculate courant snyder functions
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

example 2 beta function in fodo cell
Example 2: beta-function in FODO cell

discontinuity in slope by -2β/f

QF/2

QD

QF/2

perturbations due to imperfect beamline elements
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
tune shift due to quadrupole field error
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
beta beat and half integer resonances
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:

beta beat and half integer resonances27
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

linear coupling and resonances
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

linear coupling and resonances29
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!

chromatic effects
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:
calculation of dispersion function
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

dispersion suppressors
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

longitudinal motion
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

rf cavity
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:

cavity design
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

cavity array
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

slide37
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

synchrotron equation of motion
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

synchrotron equation of motion39
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

phase space topology
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

rf bucket
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)

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