Academic Training Lecture 4 Nonlinear Dynamics Resonance condition

Academic Training Lecture 4 : Non-linear Dynamics Resonance condition Multipole field expansion Multipole field shapes Sextupole magnet Dipole magnet errors Third integer resonance Phase space trajectory for the one-third integer resonance Definition of Hamiltonian Casting the. Hamiltonian in the correct form Other interesting forms Third integer phase space Injection studies at FNAL Field around a moving cylinder of charge Beam Force Example of tracking Crossing the stochastic limit Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 1

Resonance condition Simple resonant condition This equation just defines a set of lines in the Q diagram for each order of resonance and for each value of the integer p. Figure shows these lines for the SPS. Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 2

Multipole field expansion (polar) Scalar potential obeys Laplace whose solution is Example of an octupole whose potential oscillates like sin 4 around the circle Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 3

Taylor series expansion Field in polar coordinates: To get vertical field Taylor series of multipoles Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 4 Fig. cas 1. 2 c

Multipole field shapes Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 5

Sextupole magnet Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 6

Normal and skew sextupoles Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 7 Fig. cas 1. 24 c and Lines

Dipole magnet errors The d. N and d. S poles superimposed on the magnet poles give the effect of cutting the poles to a finite width. The remanent magnetomotive force : is weaker at the pole edges, and the field tends to sag into a parabolic or sextupole configuration Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 8 Fig. cas 1. 12 c

Third integer resonance Substituting Incidentally Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 9

Phase space trajectory for the onethird integer resonance Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 10

Definition of Hamiltonian Any old coordinates Lagrangian Equation Define is just kinetic - potential energy of motion canonical momenta Prescription Equations for Hamiltonian of motion are: Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 11

Casting the. Hamiltonian in the correct form Hamiltonian of a relativistic particle in a general electromagnetic field. In our transverse ccordinates: Turn inside out to change from t to s Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 12

Other interesting forms In x plane Small divergences: But: Substitute a Taylor series: Multipoles each have a term: Quadrupoles: Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 13

Third integer phase space Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 14

INJECTION STUDIES AT FNAL Remanent sextupole in the FNAL main ring caused serious beam loss due to non-linear resonances. This was exacerbated by magnet ripple. A three dimensional hill and dale model spanning the Q ( or ) diagram The vertical co-ordinate of this three dimensional model is the fraction of the beam which survives to be accelerated Fig. (always < 70%) FNAL-MODELL. PCT AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 15

Field around a moving cylinder of charge Take unit length of the bunch Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 16

Beam Force Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 17

Example of tracking Below the beam-beam limit and without tune modulation we see the many archipelagos of islands due to different multipole components in the beam-beam potential. Increasing the beam-beam interaction will enlarge the islands so that they overlap in stochastic regions where particle may diffuse out in 4 dimensional phase space. Increasing amplitude tune slope will merge Fig. them too TED 0. PCT AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 18

Crossing the stochastic limit Increasing Q shift causes islands to merge in a stochastic sea of chaos Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 19

Summary : Non-linear Dynamics Resonance condition Multipole field expansion Multipole field shapes Sextupole magnet Dipole magnet errors Third integer resonance Phase space trajectory for the one-third integer resonance Definition of Hamiltonian Casting the. Hamiltonian in the correct form Other interesting forms Third integer phase space Injection studies at FNAL Field around a moving cylinder of charge Beam Force Example of tracking Crossing the stochastic limit Fig. AC 01 -4. PPT- E. Wilson - 11/28/2020 - Slide 20