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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 l.jpg

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 l.jpg
Outline

  • Introduction

  • Strong focusing, lattice design

  • Perturbations due to field errors

  • Chromatic effects

  • Longitudinal motion


How to design a storage ring l.jpg
How to design a storage ring?

  • Uniform magnetic field B0 → circular trajectory

  • Cyclotron frequency:

Why not electric bends?


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What about slight deviations?

  • 6D phase-space

  • stable in 5 dimensions

  • beam will leak out in y-direction


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Let’s introduce a field gradient

  • magnetic field component Bx~ -y will focus y-motion

  • Magnet acquires dipole and quadrupole components

combined function magnet


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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


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Equation of motion

Hill’s equation:


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Equation of motion

Hill’s equation:

natural dipol focusing


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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


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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


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Alternating gradients

  • quadrupole doublet separated by distance d:

  • if f2 = -f1, net focusing effect in both planes:


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FODO cell

stable for |f| > L/2


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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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Example 2: beta-function in FODO cell

discontinuity in slope by -2β/f

QF/2

QD

QF/2


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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


Closed orbit distortion due to dipole error l.jpg

Consider dipole field error at s0 producing an angular kick θ

Closed orbit distortion due to dipole error

integer resonances

ν= integer


Tune shift due to quadrupole field error l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg

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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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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