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Lecture 3 - Magnets and Transverse Dynamics I

Lecture 3 - Magnets and Transverse Dynamics I. ACCELERATOR PHYSICS MT 2012 E. J. N. Wilson. Slides for study before the lecture. Please study the slides on relativity and cyclotron focusing before the lecture and ask questions to clarify any points not understood More on relativity in:

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Lecture 3 - Magnets and Transverse Dynamics I

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  1. Lecture 3 - Magnets and Transverse Dynamics I ACCELERATOR PHYSICS MT 2012 E. J. N. Wilson

  2. Slides for study before the lecture Please study the slides on relativity and cyclotron focusing before the lecture and ask questions to clarify any points not understood More on relativity in: http://cdsweb.cern.ch/record/1058076/files/p1.pdf

  3. Relativistic definitions Energy of a particle at rest Total energy of a moving particle (definition of g) Another relativistic variable is defined: Alternative axioms you may have learned You can prove:

  4. Newton & Einstein • Almost all modern accelerators accelerate particles to speeds very close to that of light. • In the classical Newton regime the velocity of the particle increases with the square root of the kinetic energy. • As v approaches c it is as if the velocity of the particle "saturates" • One can pour more and more energy into the particle, giving it a shorter De Broglie wavelength so that it probes deeper into the sub-atomic world • Velocity increases very slowly and asymptotically to that of light

  5. Magnetic rigidity dθ dθ Fig.Brho 4.8

  6. Equation of motion in a cyclotron • Non relativistic • Cartesian • Cylindrical

  7. Cyclotron orbit equation • For non-relativistic particles (m = m0) and with an axial field Bz = -B0 • The solution is a closed circular trajectory which has radius • and an angular frequency • Take into account special relativity by • And increase B with g to stay synchronous!

  8. Cyclotron focusing – small deviations • See earlier equation of motion • If all particles have the same velocity: • Change independent variable and substitute for small deviations • Substitute • To give

  9. Cyclotron focusing – field gradient • From previous slide • Taylor expansion of field about orbit • Define field index (focusing gradient) • To give horizontal focusing

  10. Cyclotron focusing – betatron oscillations • From previous slide - horizontal focusing: • Now Maxwell’s • Determines • hence • In vertical plane • Simple harmonic motion with a number of oscillations per turn: • These are “betatron” frequencies • Note vertical plane is unstable if

  11. Lecture 3 – Magnets - Contents • Magnet types • Multipole field expansion • Taylor series expansion • Dipole bending magnet • Magnetic rigidity • Diamond quadrupole • Fields and force in a quadrupole • Transverse coordinates • Weak focussing in a synchrotron • Gutter • Transverse ellipse • Alternating gradients • Equation of motion in transverse coordinates

  12. By By x x Magnet types • Dipoles bend the beam • Quadrupoles focus it • Sextupoles correct chromaticity

  13. Multipole field expansion Scalar potential obeys Laplace whose solution is Example of an octupole whose potential oscillates like sin 4around the circle

  14. Taylor series expansion Field in polar coordinates: To get vertical field Taylor series of multipoles Fig. cas 1.2c

  15. Diamond dipole

  16. Bending Magnet • Effect of a uniform bending (dipole) field • If then • Sagitta

  17. Magnetic rigidity Fig.Brho 4.8

  18. Diamond quadrupole

  19. Fields and force in a quadrupole No field on the axis Field strongest here (hence is linear) Force restores Gradient Normalised: POWER OF LENS Defocuses in vertical plane Fig. cas 10.8

  20. Transverse coordinates

  21. Weak focussing in a synchrotron The Cosmotron magnet • Vertical focussing comes from the curvature of the field lines when the field falls off with radius ( positive n-value) • Horizontal focussing from the curvature of the path • The negative field gradient defocuses horizontally and must not be so strong as to cancel the path curvature effect

  22. Gutter

  23. Transverse ellipse

  24. Alternating gradients

  25. Equation of motion in transverse co-ordinates • Hill’s equation (linear-periodic coefficients) where at quadrupoles like restoring constant in harmonic motion • Solution (e.g. Horizontal plane) • Condition • Property of machine • Property of the particle (beam) e • Physical meaning (H or V planes) Envelope Maximum excursions

  26. Summary • Magnet types • Multipole field expansion • Taylor series expansion • Dipole bending magnet • Magnetic rigidity • Diamond quadrupole • Fields and force in a quadrupole • Transverse coordinates • Weak focussing in a synchrotron • Gutter • Transverse ellipse • Alternating gradients • Equation of motion in transverse coordinates

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