CERN 87-03 21 April 1987 ORGANISATION EUROPÉENNE POUR LA RECHERCHE NUCLÉAIRE CERN EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CAS CERN ACCELERATOR SCHOOL ADVANCED ACCELERATOR PHYSICS The Queen's College, Oxford, England 16-27 September 1985 PROCEEDINGS Editor: S. Turner Vol.1 .GENEVA 1987 © Copyright CERN, Genève, 1987 Propriété littéraire et scientifique réservée pour Literary and scientific copyrights reserved in ail tous les pays du monde. Ce document ne peut countries of the world. This report, or any part of être reproduit ou traduit en tout ou en partie sans it, may not be reprinted or translated without l'autorisation écrite du Directeur général du written permission of the copyright holder, the CERN, titulaire du droit d'auteur. Dans les cas Director-General of CERN. However, permission appropriés, et s'il s'agit d'utiliser le document à will be freely granted for appropriate non des fins non commerciales, cette autorisation commercial use. sera volontiers accordée. If any patentable invention or registrable design Le CERN ne revendique pas la propriété des is described in the report, CERN makes no claim inventions brevetables et dessins ou modèles to property rights in it but offers it for the free use susceptibles de dépôt qui pourraient être décrits of research institutions, manufacturers and dans le présent document; ceux-ci peuvent être others. CERN, however, may oppose any attempt librement utilisés par les instituts de recherche, by a user to claim any proprietary or patent rights les industriels et autres intéressés. Cependant, le in such inventions or designs as may be des CERN se réserve le droit de s'opposer à toute cribed in the present document. revendication qu'un usager pourrait faire de la propriété scientifique ou industrielle de toute invention et tout dessin ou modèle décrits dans le présent document. - iii - ABSTRACT This advanced course on general accelerator physics is the second of the biennial series given by the CERN Accelerator School and follows on from the first basic course given at Gif-sur-Yvette, Paris, in 1984 (CERN Yellow Report 85-19). Stress is placed on the mathematical tools of Hamiltonian mechanics and the Vlasov and Fokker-Planck equations, which are widely used in accelerator theory. The main topics treated in this present work include: nonlinear resonances, chromaticity, motion in longitudinal phase space, growth and control of longitudinal and transverse beam emittance, space- charge effects and polarization. The seminar programme treats some specific accelerator techniques, devices, projects and future possibilities. ADVANCED ACCELERATOR PHYSICS COURSE, OXFORD SEPTEMBER 1985 Time Monday Tuesday Wednesday Thursday Friday Saturday Sunday Monday Tuesday Wednesday Thursday Friday 16 September 17 September 18 September 19 September 20 September 21 September 22 September 23 September 24 September 25 September 26 September 27 September R E A K F A ST (served 08.00 - 08.30) R E A K F A ST 09.00 Registration Non-Linear Non-Linear Landau Waves in Wake Fields Instabilities Instabilities Synchro- Diffusion 09.30 Resonances Resonances Damping Plasmas Betatron due to Welcome Resonances R.F. Noise and Introduction J. Mulvey C. Lashmore- K. Johnsen E. Wilson E. Wilson H. Hereward Davies V.G. Vaccaro J.-L. Laclare O.-L. Laclare A. Piwinski G. Dome 10.00 C O F F EE C O F F EE 10.30 Hami Hon i an Chromaticity Chromat i city Beam-Beam Beam-Beam Betatron Space-Charge Beam Schottky Noise Beam Dynamics Effect Effect and Coupling Dominated Break-up and Beam Loading Beam with Beam Transport Transfer Disruption Radiation Function Diagnostics J. Bell B. Montague G. Guignard L. Evans L. Evans (Presented b. J. Gareyte) G. Guignard I. Hofmann J. LeDuff D. Boussard D. Boussard 11.30 Hamiltonian Hamiltonian Kinetic Hamiltonian Dynamics with Stochastic Stochastic Electron Intra-beam Seminar Dynamics Treatment of Theory and Treatment of Radiation and Cooling Cooling Cooling and Multiple High Intensity Longitudinal Vlasov Longitudinal Non-linear Scattering Neutron Beams Mot i on Equation Motion Wigglers C. Lashmore- FREE DAY FREE DAY J. Bell G. Dome Davies G. Dome J. Jowett D. Mohl D. Mohl H. Poth A. Piwinski G. Rees 12.30 L U N CH L U C H 14.00 STUDY SESSI 0 N 15.00 Study Study Fi Im on Seminar Discussion Discussion Discussion Discussion Session Session Polarization and and and RFQ V I S IT Discussion Discussion Discussion Closing M. Puglisi TO Remarks 16.00 T E A RUTHERFORD TEA K. Johnsen 16.30 TEA Seminar Seminar Seminar Seminar APPLETON Seminar Polarisation Seminar 17.00 in Electron Particle S.C. R.F. High-Field Heavy Ions; LABORATORY Free-electron and Proton Linear 17.30 Tracking Systems Linacs The Present Laser Beams Colliders Welcome and Future versus Reception Storage Rings H. Mais H. Piel J. LeDuff H. Gutbrod A. Renieri J. Buon U. Amaldi 18.00 E V E N I NG M E AL (start 19.00 prompt) E V E N I NG 20.00 Organ Recital Bingham String CocktaiIs The Queen's Quartet + College Chapel Holywell Music Banquet M.S. Gautrey Rooms CERN ACCELERATOR SCHOOL Rutherford Appleton Laboratory & Department of Nuclear Physics, Oxford will jointly organize a course on A D V A N C ED A C C E L E R A T OR P H Y S I CS 16-27 SEPTEMBER 1985 at The Queen's College, Oxford, England Lectures: Hamiltonian Dynamics Kinetic Theory and Vlasov Equation Non-linear Resonances Waves in Plasma Chromaticity Landau Damping Betatron Coupling with Radiation Dynamics with Radiation and Non-linear Wigglers Synchro-Betatron Resonances Polarization in Electron and Proton Beams Hamiltonian Treatment of Longitudinal Motion Diffusion due to R.F. Noise Seminars: Beam Loading Beam-Beam Effect RFQ Beam Disruption Particle Tracking Beam Break-up Superconducting Radiofrequency Systems Space-Charge Dominated Beam Transport High-Field Linacs Wake Fields Heavy Ions Instabilities Free-electron Laser Stochastic Cooling Linear Colliders versus Storage Rings Electron Cooling High Intensity Neutron Beams Intra-beam and Multiple Scattering Discussions and visit to Schottky Noise and Beam Rutherford Appleton Transfer Function Diagnostics Laboratory General information Students should have a basic knowledge of accelerators. Persons wishing to attend this course can obtain further information and application forms from the CERN Accelerator School, Mrs. S. von Wartburg, LEP Division, CH 1211 Geneva 23. Application forms must be returned no later than 1st June 1985. The registration fee of 970.- FS includes full board and lodging in The Queen's College, Oxford, England. All participants will receive a copy of the proceedings. The number of participants is limited to 130. Administration Head of School: K. Johnsen; Programme Committee: P. Bryant, J. Buon, K. Johnsen, W. Joho, R. Kohaupt, J. Lawson, K. Nielsen, S. Tazzari; Local Organizing Committee: P. Bryant, S. Collier, K. Johnsen, J. Maidment, J. Mulvey, S. von Wartburg. - ix - CONTENTS Page No. Foreword vi i Opening address 1 Local coordinates for the beam and frequently used symbols 3 J.S. Bell Hamiltonian mechanics 5 Introduction 5 Poisson brackets 8 Stationary and varying action 8 Poincaré invariant 10 Lagrange invariant 11 Synplectics 13 Liouville's theorem 14 Conserved quadratic form 15 Characteristic exponents 16 Canonical transformations 19 Point transformation 20 Change of independent variable 22 Scaling 23 Dynamical evolution as canonical transformation 24 Poincaré invariant 25 Lagrange invariant 25 Liouville invariant 27 One degree of freedom 29 Action and angle variables 30 Small deviations from closed orbit 31 Adiabatic invariance of J 35 Small canonical transformation 36 Canonical perturbation theory 37 E.J.N. Wilson Nonlinear resonances 41 Introduction 41 The general form of the Hamiltonian 43 The magnetic vector potential for multipoles 45 Linear dynamics in action angle variables 46 Perturbation theory 52 Effect of nonlinearities far from a resonance 55 Resonances 56 The third-integer resonance 57 The trajectory of a third-integer resonance 59 The effect of an octupole 62 Phase-space topology with amplitude frequency variation 63 Amplitude growth on crossing a resonance 65 Synchrotron resonances 66 Beam lifetime due to magnet imperfections 68 The effect of two dimensions of transverse motion 69 Three dimensions of magnetic field 72 Conclusions 73 - X - B. W. Montague Chromatic effects and their first-order correction 75 Introduction 75 Basic ideas 76 Chromatic perturbation equations 80 Two dimensions 84 G. Guignavd Chromaticity: nonlinear aberrations 91 Introduction 91 Description of the nonlinear perturbations 92 Perturbation theory in the canonical variables 96 Dynamic aperture and analytical approach 102 G. Dome Theory of RF acceleration 110 Energy gain and transit time factor 110 Harmonic number 112 Finite difference equations 112 Differential equations for an arbitrary RF voltage 117 Hamiltonian with reduced variables 119 Small oscillations around the stable fixed point 123 Motion in the vicinity of the fixed points 124 Stationary bucket with a harmonic cavity 127 Formulae for a sinusoidal RF voltage 128 Adiabatic damping of phase oscillations 131 Back to finite difference equations. Stochasticity 136 Phase displacement acceleration 139 Linear accelerators 143 L. R. Evans and J. Gareyte Beam-beam effects 159 Introduction 159 The beam-beam force 159 Experimental and numerical data from e+e" machines 165 Experimental data from hadron machines 171 Nonlinear beam-beam resonances 175 Beam disruption 184 Conclusions 185 A. Piwinski Synchro-betatron resonances 187 Introduction 187 Dispersion in a cavity 188 Transverse fields with longitudinal variation 192 Beam-bean interaction with a crossing anqle 194 G. Guignard Betatron coupling with radiation 203 Introduction 203 Perturbation treatment of linear betatron coupling 203 Amplitude variation due to radiation and acceleration 209 Equilibrium in the case of betatron coupling with radiation 213 Application to emittance control 217 C. H. Lashnore-Davies Kinetic theory and the Vlasov equation 219 Introduction 219 The Vlasov equation 220 The effect of binary collisions 224 Fluid models 229 - xi - C. N. Lashmore-Davies Haves in plasmas 235 Introduction 235 Waves in a field-free plasma 236 Waves in a magnetized plasma 240 Low-frequency waves in a magnetized plasma 245 Raman scattering 247 H.G. Hereward Landau damping 255 Spectrum of linear oscillations 255 Longitudinal instability 259 Nonlinear oscillations 261 J.L. Laclare Bunched beam coherent instabilities 264 Introduction 264 Longitudinal instabilities 264 Transverse instabilities 306 Conclusion 325 J. Hofrnann Space charge dominated beam transport 327 Introduction 327 Basic properties 327 Emittance and field energy 329 Application to emittance growth 332 L. Palumbo and V.G. Vocear o Wake fields, impedances and Green's function 341 Longitudinal wake potential and impedance 341 Longitudinal impedance and wake potential for simple structures 344 General analysis 357 Circular accelerators 361 Transverse-wake potential and impedance 363 Remarks 368 Conclusions 369 G. Dôme Diffusion due to RF noise 370 Statistical properties of random variables 370 Fokker-Planck equation 372 Differential equations for a stationary bucket with amplitude and phase noise 374 Computation of the coefficients k A in the Fokker-Planck equation 377 it 2 Case of a sinusoidal RF voltage 382 Finite difference equations 386 Diffusion equation 388 Contribution of RF noise to the finite beam lifetime in the SPS collider 392 A. Piwinski Intra-beam scattering 402 Introduction 402 Calculation of rise times and damping times 403 Experimental results 411 - xii - D. Bousscœd Schottky noise and beam transfer function diagnostics 416 Schottky signals 417 Beam detectors 425 Observation of Schottky signals 440 Beam transfer functions 445 D. Möhl Stochastic cooling 453 Introduction 453 Simplified theory, time-domain picture 455 A more detailed presentation of betatron cooling, frequency domain picture 478 Distribution function equations (Fokker-Planck) and momentum scalinq 510 H. Poth Electron cooling 534 Introduction 534 What electron cooling is 534 Why electron cooling? 535 Ion-beam and storage-ring properties 535 How electron cooling works in principle 537 Introduction to electron cooling theory (for pedestrians) 538 Experimental realization of electron cooling 543 More on the theory of electron cooling 552 Recombination 555 Electron cooling experiments 557 Simulation of electron cooling in storage rings 561 Electron cooling diagnostics 562 Applications of electron cooling 564 Electron cooling projects 566 J. Jowett Electron dynamics with radiation and nonlinear wigglers 570 Introduction 570 The dynamics of electrons in a storage ring 571 Normal modes and optical functions 579 Radiation damping 584 Quantum fluctuations and Fokker-Planck equations 587 Nonlinear wigglers 591 J. Le Duff Beam break up 610 Experimental evidence 610 Transverse deflection of charge particles in radio-frequency fields 612 Deflecting modes in circular iris loaded waveguides 614 Regenerative beam break up 617 Cumulative beam break up 621 £>„ Boussard Beam loading 626 Introduction 626 Single bunch passage in a cavity 627 Multiple bunch passages 629 Limiting case 6 = 0 630 Ü The case of a travelling wave structure 633 Transient correction 635 RF drive generation 637 - xiii - J. Buon Polarization in electron and proton beams 647 Introduction 647 Generalities on polarization and spin motion 649 Acceleration of polarized protons in synchrotrons 661 Polarization of electrons in storage rings 671 H. Mais, G. Ripken, A. Wruliah and F. Schmidt Particle tracking 690 Introduction 690 Hamiltoni an description of the proton motion 690 Dynamic aperture 693 Particle tracking 694 Qualitative theory of dynamical systems 696 Studies of chaotic behaviour in HERA caused by transverse magnetic multipole fields 699 Summary 704 M. Puglisi The radiofrequency quadrupole linear accelerator 706 Introduction 706 The accelerating structure 706 Outline of the T-K expansion 709 The vane tips shaping 712 Physical considerations 717 The structure of an RFQ 720 Design and technical considerations 728 Recent developments 733 H. Piel Fundamental features of superconducting cavities for high energy accelerators 736 Introduction 736 Some cavity fundamentals 737 Superconducting cavities 741 Cavity design 746 Anomalous losses 751 Cavities covered with superconducting thin films 761 Current accelerator projects and achievements 765 J. Le Duff High field electron linacs 772 Introduction 772 Extrapolation of present technologies 772 RF compression scheme 775 Ultimate accelerating gradients in conventional structures 783 A survey of accelerating structures 785 RF power source: the lasertron 787 G. Dattoli, A. Renieri and A. Torre Free electron lasers: a short review of the theory and experiments 792 Introduction 792 FEL: theory and design criteria 795 FEL storage ring operation 803 Single passage FEL operation 808 Conclusions 814 D.A. Gray and G.H. Rees ISIS, the accelerator based neutron source at RAL 817 Introduction 817 Linac and synchrotron 817 Target station 819 Experimental facilities 821 High intensity performance of the ISIS synchrotron 822