Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 1 Lecture 3 - Magnets and Transverse Dynamics I ACCELERATOR PHYSICS MT 2014 E. J. N. Wilson

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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:

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 3 Relativistic definitions Energy of a particle at rest Total energy of a moving particle (definition of  ) Another relativistic variable is defined: Alternative axioms you may have learned You can prove:

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 5 Magnetic rigidity Fig.Brho 4.8 dd dd

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 6 Equation of motion in a cyclotron  Non relativistic  Cartesian  Cylindrical

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 7  For non-relativistic particles (m = m 0 ) and with an axial field B z = -B 0  The solution is a closed circular trajectory which has radius  and an angular frequency  Take into account special relativity by  And increase B with  to stay synchronous! Cyclotron orbit equation

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 9  From previous slide  Taylor expansion of field about orbit  Define field index (focusing gradient)  To give horizontal focusing Cyclotron focusing – field gradient

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 12  Dipoles bend the beam  Quadrupoles focus it  Sextupoles correct chromaticity Magnet types x By x ByBy

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 13 Multipole field expansion Scalar potentialobeys Laplace whose solution is Example of an octupole whose potential oscillates like sin 4  around the circle

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 14 Taylor series expansion Field in polar coordinates: To get vertical field Taylor series of multipoles Fig. cas 1.2c

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 15 Diamond dipole

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 16 Bending Magnet  Effect of a uniform bending (dipole) field  If then  Sagitta

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 17 Magnetic rigidity Fig.Brho 4.8

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 18 Diamond quadrupole

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 20 Transverse coordinates

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 21 Weak focussing in a synchrotron  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 The Cosmotron magnet

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 22 Gutter

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 23 Transverse ellipse

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 24 Alternating gradients

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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)   Physical meaning (H or V planes) Envelope Maximum excursions

Lecture 3 - E. Wilson - 22 Oct 2014 –- Slide 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