Introduction to particle accelerators

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Introduction to particle accelerators Walter
Scandale CERN - AT department Roma, marzo 2006
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Lecture II - single particle dynamics

• topics
• Guiding fields and transverse motion
• Weak versus strong focusing
• Equation of motion
• Unperturbed case
• Orbit errors
• Quadrupole errors
• Chromaticity
• Resonances and dynamic aperture
• Low-ß insertion
• Longitudinal stability

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Synchrotron guiding field

• Synchrotron ring
• particle trajectories at fixed radius r
• to keep r constant B should increase as p
  increases during acceleration,
• RF frequency synchronized to the particle
  revolution
• bending and focusing fields

Dipole the Lorenz force provides the centripetal
acceleration
bending radius
magnetic rigidity
bending angle
x
Quadrupole the Lorenz force focuses the
trajectories
?
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Weak focusing

• Weak focusing of the transverse particle motion
• to get vertical stability, the bending field
  should decrease with r, as in cyclotrons,
• to get horizontal stability the the decrease of B
  with ? should be moderate, so that, for ? gt ?0,
  the Lorenz force exceeds the centripetal force.

horizontal stability
centripetal force
Lorenz force

weak focusing
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Strong focusing
Horizontal and vertical focusing for a large
range of f1 f2 and d

• separated functions the alternate gradient is
  made with quadrupoles of opposite focusing
  strength
• combined functions the alternate gradient is
  made with dipoles with radial shape of opposite
  sign

normalized quadrupole gradient
quadrupole strength
Examples
Ring p GeV/c B0 T r m Br Tm 1/r2 m-2 weak focus Lquad m B T/m K m-2
CERS 5.2 0.18 96.4 17.3 10-4 0.5 5 0.298
Tevatron 1000 4.4 758 3335 1.710-6 1.7 76 0.0228
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Particle equation of motion
B-field expansion
Maxwell equations and quadrupolar gradients
g
gskew
Equation of motion
transport matrix approach
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Weak versus strong focusing
Equation of motion
Strong focusing

• Strong focusing
• Smaller pipe
• Smaller magnet
• Reduced cost

cosmotron
AGS
Weak focusing
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Unperturbed equation of motion
Hill equation
periodicity condition
envelop
solution Floquet theorem

• b and f depend on the lattice arrangement
• e and f0 depend on the initial conditions of the
  trajectory

cos-like orbit
Here ß is NOT the relative speed v/c
b-function (periodic)
phase advance between s0 and s
sin-like orbit
phase advance variation
envelop equation
several orbits
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Courant Snyder invariant and more
Courant-Snyder parameters
Courant-Snyder invariant
emittance

• e is the particle emittance
• ep is the area of the ellipse mapped turn by turn
  in the phase plane (z,z)
• ebeam is the (1rms) beam emittance if the area
  pebeam encloses 39 of the circulating particles
  N

Liouville theorem

• ebeam/N is a constant of the motion (Liouville
  theorem)
• The Liouville theorem holds in absence of
  acceleration, losses, scattering effects and
  radiation emission

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Adiabatic invariant
The Courant-Snyder invariant emittance e
decreases if we the accelerate the particle.
This is called adiabatic damping (a pure
cinematic effect, since there is no damping
process involved).
The slope of the trajectory is z pz/ps.
Accelerate the particle ps increases to ps?ps,
but pz doesnt change gt slope changes.
b -gt Courant Snyder parameter
normalized emittance

• Invariant of the motion
• In a stationary Poincaré section -gt e
• In an accelerating Poincaré section -gt ebg

b the relative speed g the relativistic factor
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Stability of the motion
Transfer matrix from s0 to s
One turn transfer matrix
Condition for the stability of the motion
Condition for the invariance of e
4D resonance condition --gt order
tune
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Exact approximate solution
Exact solution of the Hill equation
Exact solution in compact form
with

• This is a pseudo-harmonic oscillation modulated
  both in amplitude and in frequency
• Q is the total number of oscillation per turn
• The phase advances faster in the sections with a
  smaller b

Approximate solution (smooth approximation)
with

• Ten cell lattice
• Cell length L 1 m
• Ring length C 10 m
• Focal length f 0.45 m

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Perturbed equation of motion
Solution with dipoles, quadrupoles sextupoles and
octupoles
dispersion
Uncoupled motion (x-plane)
orbit distortion
with
dispersive orbit
with
betatron oscillation
natural chromaticity aberration
chromaticity correction by sextupoles
geometric aberration
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Dipolar and quadrupolar field errors
Dipole error
DB localized in sk over the length L (kick
approximation)
Periodicity of the closed orbit
At every turn the perturbation is compensated
At every turn the perturbation is enhanced
Avoid tune close to integer
Quadrupole error
DB localized in sk over the length L (thin lens
approximation)
first order
second order
Avoid tune close to 1/2 integer - the range of
forbidden tunes is called stop-band
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Momentum dispersion and chromaticity
First order solution
Dispersion function
Divergent for Qinteger
Gradient error induced by momentum dispersion
in a FODO lattice (thin lens approximation)
Chromaticity correction with sextupoles
Sextupole strength
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Why chromaticity should be corrected

• The beam rigidity increases with p
• K decreases with p
• the tune decreases with p
• Q is negative

• Q non zero produces a tune shift with p
• In a beam Q produces a tune spread
• Be aware of resonance crossing

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Dynamic aperture

• Nonlinear fields imply multiple traversal of
  resonances
• Emittance distortion and growth
• Tune shift and spread with the amplitude
• Coupling of the degrees of freedom
• Chaotic motion
• Particle loss -gt dynamic aperture

Phase space with only linear fields
Distortion due to sextupoles
Distortion due to octupoles
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Low ß insertion

• A low-ß insertion is used to focalize the beams
  at the collision point
• This is achieved with triplets or doublets of
  quadrupoles
• In the drift space where the experimental devices
  sit ß growth with the square of the length

since Lgtgtb
The chromaticity induced by the triplet can be
large (local correction scheme may be needed)
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Acceleration mechanism
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Longitudinal stability
a measures how closely packed orbits with
different momenta are
Momentum compaction
h measures how how much off-momentum particles
slip in time relative to on-momentum ones
Slip factor

• Phase stability principle
• ??lt ?tr
• B is late respect to A
• B will receive a larger voltage and will increase
  its speed
• B will be closer to A one turn later
• ??gt ?tr
• B is late respect to A
• B will receive a smaller voltage and will see a
  shorter circumferential path
• one turn later B will be closer to A

small oscillations
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LHC luminosity
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Lecture II - single particle dynamics

• reminder
• In a circular accelerator the guiding fields
  provide the required forces to keep the particles
  in a closed orbit along the magnet axis
• Strong focusing allows building much smaller
  magnets and is a fundamental progress respect to
  weak focusing
• The particle trajectory is a pseudo-harmonic
  function modulated both in amplitude and phase
  rather well approximated by a sinusoidal function
  oscillating at the betatron frequency
  (tunerevolution frequency) with an amplitude
  proportional to the square root of the emittance
• The imperfections of the guiding field and of the
  momentum particles produce resonances and
  eventually chaotic motion
• The low ß insertions are basic devices to focus
  the beam size at the collision point of a
  collider ring
• The phase stability principle guarantees the
  stability of the longitudinal motion