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

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### 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

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

How to design a storage ring?

- Uniform magnetic field B0 → circular trajectory
- Cyclotron frequency:

Why not electric bends?

What about slight deviations?

- 6D phase-space
- stable in 5 dimensions
- beam will leak out in y-direction

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 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

Hill’s equation:

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

- 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

- quadrupole doublet separated by distance d:
- if f2 = -f1, net focusing effect in both planes:

FODO cell

stable for |f| > L/2

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 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

- 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

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

- 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

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?

- 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

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

Consider dipole field error at s0 producing an angular kick θClosed orbit distortion due to dipole error

integer resonances

ν= integer

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

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 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

- 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 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

- 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

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

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

- (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

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

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

- 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

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

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 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

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

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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