Lecture 5: Beam optics and imperfections

1. Lecture 5: Beam optics and imperfections Errors in magnetic lattices dipole errors quadrupole errors Nonlinear Perturbations Resonances Dynamic aperture

2. Linear betatron equations of motion (recap) In the magnetic fields of dipoles magnets and quadrupole magnets (without imperfections) the coordinates of the charged particle w.r.t. the reference orbit are given by the Hill’s equations These are linear equations (in y = x, z). They can be integrated and give

3. Transfer lines or circular accelerators are made of a series of drifts and quadrupoles for the transverse focussing and accelerating section for acceleration. • Real lattices will always unavoidably be different from the model lattices. This is due to • misalignments and rolls of the magnetic elements • gradient error • additional unwanted magnetic field multipoles (stray fields) • time varying magnetic errors (due to change in magnetic fields, ground motion, etc) Errors in lattices

4. Dipole errors (I) Dipole field errors arise due to calibration errors current vs magnetic fields, misalignment and rolls. 4/26

5. Dipole field errors and misalignment of the magnetic elements produce orbit distortions. They generate an additional term in the Hill’s equation Dipole errors (II) The general solution can be found as the sum of a particular integral and the general solution of the imhogeneous equation and reads The effect is stronger where the  functions are larger The orbit gets unstable if the betatron tune y is close to an integer

6. With many distributed dipole errors the orbit will be distorted. The beam position monitors (BPMs) will read a non zero orbit. Orbit corrections (I) To correct the orbit we use a system of horizontal and vertical corrector dipoles. We need to know what is the effect of each correctors on the orbit itself, first. Then we can use this information to correct orbit distortion.

7. V V H H The orbit response matrix collects all this information in one matrix The orbit response matrix R is the change in the orbit at the BPMs as a function of changes in the steering magnets strength. Orbit correction (II) orbit RM at diamond 168*2 rows 168*2 columns orbit reading at all BPMs dipole corrector angle kick

8. Once the matrix R is known we can correct an orbit distortion by inverting the matrix Orbit correction (III) The matrix R generally is not a square matrix so it might be non-invertible but we can use the Singular Value Decomposition (SVD) of the response matrix R to invert it and correct the closed orbit distortion

9. Quadrupole errors (I) Quadrupole magnet are due to calibration errors in current vs field gradient, misalignment and rolls. A misaligned quadrupole adds a dipole kick.

10. Consider a lattice with a quadrupole distribution K(s) and a gradient error K(s) This is still a homogeneous Hill’s equation and can be solved with the matrix method Quadrupole errors (III) Let us consider a localised quadrupole error and let us put M0 the transfer matrix without error M the transfer matrix with the error m the transfer matrix of the localised section where the error is m0 the transfer matrix of the same section without error We can compute the matrix M with the relation m0 m M0 M

11. We use the Twissparameterisation for M and M0 Quadrupole errors (IV) and the thin lens approximation for m and m0 we get to first order in ds Using again the Twissparameterisation for the transfer matrix we get the new phase advance

12. If the perturbation is distributed around the ring we have Quadrupole errors (V) this can be eventually expressed with an integral over the ring circumference Therefore a gradient error leads to a betatron tune shift The effect is stronger where the beta functions are larger (final focus - IPs). A gradient error changes the beta functions as well The optics get unstable if the betatron tune Qy is close to an half-integer

13. The shift in the betatron tunes has to be controlled to avoid creating resonance conditions between the horizontal and the vertical tune Resonances m Qx + n Qz = p |m| + |n| is called the order of the resonance We have already met two example of resonances: integer resonances which make the closed orbit unstable and half integer resonances which make the optics unstable. More examples when we will consider nonlinear elements.

14. Multipolar expansion of the magnetic field inside a magnet Nonlinear magnetic errors The on axis magnetic field can be expanded into multipolar components (dipole, quadrupole, sextupole, octupoles and higher orders)

15. Including higher order terms in the expansion of the magnetic field normal multipoles Hill’s equation with nonlinear terms skew multipoles the Hill’s equations acquire additional nonlinear terms No analytical solution available in general: the equations have to be solved by tracking or analysed perturbatively

16. Example: nonlinear errors in the LHC main dipoles Finite size coils reproduce only partially the cos- desing necessary to achieve a pure dipole fields LHC main dipole cross section Multipolar errors up to very high order have a significant impact on the nonlinear beam dynamics.

17. Nonlinear magnetic fields are introduced in the lattice (chromatic sextupoles) Normal sextupole Sextupole magnets Normal sextupole Skew sextupole

18. Example: nonlinear elements in small emittance machines Small emittance  Strong quadrupoles  Large (natural) chromaticity  Strong sextupoles (sextupoles guarantee the focussing of off-energy particles) strong sextupoles have a significant impact on the electron dynamics  additional sextupoles are required to correct nonlinear aberrations

19. x’ Turn 1 Turn 2 x Turn 3 Turn 2 Turn 1 Turn 3 The orbit in phase space for a system of linear Hill’s equation are ellipses (or circles) The frequency of revolution of the particles is the same on all ellipses Phenomenology of nonlinear motion (I) x’ The orbit in the phase space for a system of nonlinear Hill’s equations are no longer simple ellipses (or circles); The frequency of oscillations depends on the amplitude x

20. Turn 2 Turn 1 Turn 5 Turn 3 Turn 4 m = 5; n = 0; p =1 When the betatron tunes satisfy a resonance relation the motion of the charged particle repeats itself periodically Nonlinear resonances If there are errors and perturbations which are sampled periodically their effect can build up and destroy the stability of motion The resonant condition defines a set of lines in the tune diagram The working point has to be chosen away from the resonance lines, especially the lowest order one (example CERN-SPS working point) 5-th order resonance phase space plot (machine with no errors)

21. Phase space plots of close to a 5th order resonance Stable and unstable fixed points appears which are connected by separatrices Islands enclose the stable fixed points On a resonance the particle jumps from one island to the next and the tune is locked at the resonance value region of chaotic motion appear The region of stable motion, called dynamic aperture, is limited by the appearance of unstable fixed points and trajectories with fast escape to infinity Phenomenology of nonlinear motion (II) Qx = 1/5

22. Phenomenology of nonlinear motion (III) • The orbits in phase space of a non linear system can be broadly divided in • Regular orbit  stable or unstable • Chaotic orbit  no guarantee for stability but diffusion rate may be very small The particle motion on a regular and stable orbit is quasi–periodic The betatron tunes are the main frquencies corresponding to the peak of the spectrum in the two planes of motion The frequencies are given by linear combination of the betatron tunes. Only a finite number of lines appears effectively in the decomposition.

23. Most accelerator codes have tracking capabilities: MAD, MADX, Tracy-II, elegant, AT, BETA, transport, … Typically one defines a set of initial coordinates for a particle to be tracked for a given number of turns. The tracking program “pushes” the particle through the magnetic elements. Each magnetic element transforms the initial coordinates according to a given integration rule which depends on the program used, e.g. transport (in MAD) Tracking (I) Linear map Nonlinear map up to third order as a truncated Taylor series

24. A Hamiltonian system is symplectic, i.e. the map which defines the evolution is symplectic (volumes of phase space are preserved by the symplectic map) M is symplectic transformation Tracking (II) non symplectic Symplectic integrator If the integrator is not symplectic one may found artificial damping or excitation effect The well-known Runge-Kutta integrators are not symplectic. Likewise the truncated Taylor map is not symplectic. They are good for transfer line but they should not be used for circular machine in long term tracking analysis Elements described by thin lens kicks and drifts are always symplectic: long elements are usually sliced in many sections.

25. Strongly excited resonances can destroy the Dynamic Aperture. The Frequency Map Analysis is a technique introduced in Accelerator Physics form Celestial Mechanics (Laskar).It allows the identification of dangerous non linear resonances during design and operation. Dynamic Aperture and Frequency Map To each point in the (x, y) aperture there corresponds a point in the (Qx, Qy) plane The colour code gives a measure of the stability of the particle (blu = stable; red = unstable) The indicator for the stability is given by the variation of the betatron tune during the evolution: i.e. tracking N turns we compute the tune from the first N/2 and the second N/2 Engineering aperture.

26. E. Wilson, CAS Lectures 95-06 and 85-19 E. Wilson, Introduction to Particle Accelerators G. Guignard, CERN 76-06 and CERN 78-11 J. Bengtsson, Nonlinear Transverse Dynamics in Storage Rings, CERN 88-05 J. Laskar et al., The measure of chaos by numerical analysis of the fundamental frequncies, Physica D65, 253, (1992). Bibliography