Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 1 Accelerator Physics Topic III Perturbations and Nonlinear Dynamics Joseph Bisognano Synchrotron Radiation Center University of Wisconsin

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 2 Chromaticity From form, it’s clear tune will depend on momentum

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 3 Sextupoles A sextupole field can remove much of this Tune change

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 4 Natural Chromaticity

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 5 Dispersion to the Rescue We can move to orbit at energy offset by canonical transform

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 6 Chromaticity Correction Judicious choice of SD vs K’s can cancel chromaticity Price: NONLINEARITY

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 7 Linear Coupling

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 8 Linear Coupling with Skew Quads See Wiedemann II

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 9 Linear Coupling/cont. periodic

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 10 Linear Coupling/cont.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 11 Difference Resonance l=-1

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 12 Difference Resonance/cont. Implies measurement scheme for  tunes quad

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 13 Sum Resonance

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 14 Action Angle Variables Ruth/Wiedemann

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 15 Action Angle Variables/cont.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 16 Action Angle Variables/cont.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 17 Canonical Perturbation Theory Following R. Ruth

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 18 Canonical Perturbation Theory/cont.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 19 Canonical Perturbation Theory/cont.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 20 Canonical Perturbation Theory/cont.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 21 Octopole

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 22 Isolated Resonance

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 23 Fixed Points

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 24 Island Structure From Ruth

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 25 Resonance Widths Expanding around unstable fixed point at a resonance action J r yields an equation for the separatrix, and, on expanding, a “bucket height” or width

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 26 R Ruth Avoiding Low Order Resonances

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 27 Dynamic Aperture

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 28 Eigenvalues

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 29 Surface of Section For an nD time independent Hamiltonian, energy is conserved, and motion is on shell, a (2n-1)D set Condition q n =constant gives (2n-2) surface, a surface of section Let’s take a look at Henon map, with the Hamiltonian having a cubic nonlinearity, sort of sextupole like

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 30 Position Plot of Henon Map

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 31 E=1/12

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 32 E=1/8

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 33 E Almost 1/6

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 34 Dynamic Aperture Determines usable aperture of accelerator, which must be consistent with emittance, injection gymnastics Determines whether intrabeam scattered particles survive and be damped in electron machines Definition: Region in phase space where particles have stable motion, will be stored indefinitely More practically, will particles remain in the machine for the planned storage time; e.g., turns in proton accelerators, or synchrotron damping times (10 4 turns in electron storage rings For higher dimensional systems Arnold diffusion adds further complications, but we will take a practical approach

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 35 Tools Tracking (approximate computer mapping) is primary game But tracking for “storage time” is still beyond computational limits, so some “numerically derived” criteria to extrapolate are essential Since systems are “chaotic,” they are very sensitive to initial conditions and numerical error, so one has to be careful Scandale, et al.

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 36 Tracking Tools Work-horse programs such as MAD, SIXTRACK use transfer maps for linear part of mapping, but “thin lens” approximation for nonlinearities. This maintains symplecticity of transforms Extensions of transfer maps of finite length (or turn) for nonlinearities using differential algebra techniques with Taylor expansions, etc. used for “analysis.” “Symplectification” is issue that limits initially perceived advantages of maps over element by element approach

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 37 Indicators of Chaos

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 38 Survival Plots Plot maximum number of turns that survive as function of starting amplitude Plots are interpolated with fitting on functional form

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 39 A Survival Plot Scandale, Todesco

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 40 Implications of Dynamic Aperture Studies Sources of nonlinearities: chromatic sextupoles, multipoles in dipoles, multipoles in lattice quads, multipoles in low-  quads, long-range beam-beam kicks For hadron colliders, multipoles of dipoles can dominate at injection; at collision, low-  quads can dominate Target aperture roughly 12  at 10 5, which implies a 6  with safety margins Yields limits on multipole content, suggests multipole correction schemes, optimized optics, beam separation

Topic Three: Perturbations & Nonlinear Dynamics UW Spring 2008 Accelerator Physics J. J. Bisognano 41 Homework for Topic III From S.Y. Lee –2.5.1 –2.5.3 –2.5.8 –2.6.1 –2.6.2 –2.7.3