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

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(No Transcript)
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Outline

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

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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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Lets 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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Lets 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
Hills equation
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Equation of motion
Hills equation
natural dipol focusing
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Weak focusing ring K ? K(s)

• define uniform field index n by
• Stability condition 0 lt n lt 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 ?i of one-turn
  map M(sLs) 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 gt L/2
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Courant-Snyder formalism

• Remember K(s)
  periodic in s
• Ansatz
• e emittance, ß(s) gt 0 and periodic in s
• Initial conditions
• phase function ? determined by ß
• define
• ß ? a ? Courant-Snyder functions or
  Twiss-parameters

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Courant-Snyder formalism

• Remember K(s)
  periodic in s
• Ansatz
• e emittance, ß(s) gt 0 and periodic in s
• Initial conditions
• phase function ? determined by ß
• define
• ß ? a ? Courant-Snyder functions or
  Twiss-parameters

properties of lattice design
properties of particle (beam)
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Phase-space ellipse

• ellipse with constant area pe
• 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

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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
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particle ? beam

• different particles have different values of e
  and ?0
• assume Gaussian distribution in u and u
• Second moments of beam distribution

beam size (s) beam divergence (s)
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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
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How to calculate Courant-Snyder functions?

• can express transfer matrix from s1 to s2 in
  terms of a1,2 ß1,2 ?1,2 ?1,2
• then one-turn map from s to sL with aa1a2,
  ßß1ß2, ??1?2, F?1-?2 phase advance per
  turn, is given by
• obtain one-turn map at s by multiplying all
  elements
• can get a, ß, ? at different location by

betatron tune
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Example 1 beta-function in drift space
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Example 2 beta-function in FODO cell
discontinuity in slope by -2ß/f
QD
QF/2
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

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Closed orbit distortion due to dipole error

• Consider dipole field error at s0 producing an
  angular kick ?

integer resonances
? integer
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Tune shift due to quadrupole field error
quadrupole field error k(s) leads to kick ?u
q integrated field error strength
tune shift

• can be used to measure beta-functions (at
  quadrupole locations)
• vary quadrupole strength by ?kl
• measure tune shift

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beta-beat and half-integer resonances
quadrupole error at s0 causes distortion of
ß-function at s ?ß(s)
(1,2)-element of one-turn map M(sLs)
ß-beat
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beta-beat and half-integer resonances
quadrupole error at s0 causes distortion of
ß-function at s ?ß(s)
(1,2)-element of one-turn map M(sLs)
ß-beat
twice the betatron frequency
half-integer resonances
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Linear coupling and resonances

• So far, x- and y-motion were decoupled
• Coupling due to skew quadrupole fields

?x ?y n sum resonance
unstable
?x - ?y n difference
resonance stable
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Linear coupling and resonances

• So far, x- and y-motion were decoupled
• Coupling due to skew quadrupole fields

?x ?y n sum resonance
unstable
my
mx
?x - ?y n difference
resonance stable
my
mx
nonlinear resonances
? irrational!
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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

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Calculation of dispersion function
transfer map of betatron motion
inhomogeneous driving term
Sector dipole, bending angle ? l/? ltlt 1
quadrupole
FODO cell
x
F horizontal betatron phase advance per cell
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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 Q
D (1-x)B QF xB QD xB QF/2
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Longitudinal motion

• (z, z) ? (z, d ?P/P) ? (F ?/vz, d)
• 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
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RF cavity
Simple pill box cavity of length L and radius R
Bessel functions
Transit time factor T lt 1
Ohmic heating due to imperfect conductors
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Cavity design
3 figures of merit (?rf, R/L, dskin) ? (?rf, 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
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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
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cavity array field pattern
coupling
pipe geometry such that RF below cut-off (long
and narrow)
side-coupled structure in p/2-mode behaves as
p-mode as seen by the beam
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Synchrotron equation of motion
synchronous particle moves along design orbit
with exactly the design momentum
h integer

• Principle of phase stability
• pick ?rf ? beam chooses synchronous particle
  which satisfies ?rf h?0
• 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
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Synchrotron equation of motion
? phase slippage factor ac momentum
compaction factor
transition energy
beam unstable at transition crossing

• linearize equation of motion
• stability condition
• synchrotron tune

negative mass effect
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Phase space topology
Hamiltonian

• SFP stable fixed point
• UFP unstable fixed point
• contours ? constant H(F, d)
• separatrix contour passing through UFP,
• separating stable and unstable regions

bucket stable region inside separatrix
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RF bucket
Particles must cluster around ?s and stay away
from (p ?s)
(remember F ? z)
Beams in a synchrotron with RF acceleration are
necessarily bunched!
bucket area bucket area(Fs0)a(Fs)