Lecture 6 - Transverse Optics III

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Lecture 6 - Transverse Optics III
ACCELERATOR PHYSICS MT 2004 E. J. N.
Wilson
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Contents of previous lecture - Transverse Optics
II

• Equation of motion in transverse co-ordinates
• Check Solution of Hill
• Twiss Matrix
• Solving for a ring
• The lattice
• Beam sections
• Physical meaning of Q and beta
• Smooth approximation

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Lecture 6 - Transverse Optics III Contents

• Effect of a drift length and a quadrupole
• Focusing in a sector magnet
• The lattice
• Calculating the Twiss parameters
• Meaning of Twiss parameters
• Stability diagram
• Liouvilles theorem

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Effect of a drift length and a quadrupole
Drift length
Quadrupole
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Focusing in a sector magnet
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The lattice (1 of SPS)

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Calculating the Twiss parameters
THEORY
COMPUTATION (multiply elements)
Real hard numbers
Solve to get Twiss parameters
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Meaning of Twiss parameters

• e is either
• Emittance of a beam anywhere in the ring
• Courant and Snyder invariant fro one particle
  anywhere in the ring

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Example of Beam Size Calculation

• Emittance at 10 GeV/c

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Stability diagram for FODO
Conditions
bounded
Just like
If motion is bounded then m must be real
HENCE
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Liouvilles theorem

• The area of a contour which encloses all the
  beam in phase space is conserved
• This area pe is the emittance
• It is the same all round the ring
• NOT TRUE
• during acceleration
• in an electron machine where synchrotron emission
  damps

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Summary

• Effect of a drift length and a quadrupole
• Focusing in a sector magnet
• The lattice
• Calculating the Twiss parameters
• Meaning of Twiss parameters
• Stability diagram
• Liouvilles theorem