CERN ACCELERATOR SCHOOL ADVANCED ACCELERATOR PHYSICS The Queen's College ...

429 Pages·2010·6.83 MB·English

CERN 87-03
2\ APRIL
ORGANISATION EUROPÉENNE POUR LA RECHERCHE NUCLÉAIRE
CERN
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
C AS
CERN ACCELERATOR SCHOOL
ADVANCED ACCELERATOR PHYSICS
The Queen's College, Oxford, England
16-27 September 1985
PROCEEDINGS
Editor: S. Turner
Vol. 11
GL NEVA
198?
t'URN - Service d'Information scientifique — RD'127- Í500- Avril I1)«?
ABSTRACT
This advanced course on general accelerator physics is the second of the biennial series given
by the CERN Accelerator School and follows cn from the first basic course qiven at Gif-sur-Yvette,
Paris, in 1984 (CERN Yellow Report 85-19). Stress is placed on the mathematical tools of Hani Kon i an
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, chromât icily, motion in
longitudinal phase space, growth and control of longitudinal and transverse bean emittaiice, space-
charge effects and polarization. The seminar progranne treats some specific accelerator techniques,
devices, projects and future possibilities.
/
CONTENTS
Page NO.
Foi fiwui d . 1
Opening address 1
Local coordinates for the beam and frequently used symbols 3
j.s. 3eil
Ham i I Ionian mechanics 5
Introduction 5
Poisson brackets 8
Stationary and varying action S
Poincaré invariant 10
Lagrange invariant 11
Symplectics U
Liouvilie's theorem 14
Conserved quadratic form 15
Charatlerislk 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
Smalt 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 Hamilton i an 43
The magnetic vector rotential for injltipoies 45
Linear dynamics in action angle variables 46
Perturbation theory 52
Effect of nonlinearities far from a resonance 55
Resonances 55
The third-integer resonance 57
The trajectory of a third-integer resonance 59
The effect of m octupole 62
Phase-space topology with amplitude frequency variation 63
Amplitude growth on crossinq a resonance 65
Synchrotron resonances 66
Beam lifetime due to maq.iet imperfections 6B
The effect of two dimensions of transverse motion 69
Three dimensions af magnetic field 12
Conclusions 73
B. W. Montague
Chromatic effects and their first-order correction
Introduction
Basic ideas
Chromatic perturbation equations
T*ro dimensions
G. Guignard
Chromaticity: nonlinear aberrations
Introduction
Description of the nonlinear perturbations
Perturbation theory in the canonical variables
Dynamic aperture and analytical approach
G, Dôme
Theory of RF acceleration
Energy gain and transit time factor
Harmonic number
Finite difference equations
Differential equations for an arhitrary RF voltage
Hamiltonian vith reduced variables
Small oscillations around the stable fixed point
Wjtion m the vicinity o, the fixed points
Stationary bucket with a harmonic cavity
Formulae for a sinusoidal RF voltaqe
Adiabatk damping of phase oscillations
Back Lo finite difference equations. Stochasticity
Pfcise displacement acceleration
Linear accelerators
L.R. h'voni} and J. Carey t,e
Beam-bean effects
Introduction
The beam-beam force
Experimental and numerical data fron-, f+e" machines
Experimental data from hadron machines
Nonlinear beam-beam resonances
Beam disrupt ion
Conclusions
A. Piitinaki
Synchro-betatron resonances
Introduction
Dispersion in a cavity
Transverse fields with longitudinal variation
Beam-beam interaction with a crossing anqle
C. Guignard
Betatron coupling with radiation
Introduction
Perturbation treatment of linear betatron couplina
/implitude variation due to radiation and acceleration
Equilibrium in the case of betali Dn coupling with radiali
Application to emittancç control
C. N • La xIimorn-Dav ies
Kinetic theory and the Vlasov equation
Introduction
The Vlasov equation
The effect of binary collisions
Fluid models
C. -V. Lashmove-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 magnet i2ed plasma 245
Raman scattering 247
H. O. Hereward
Landau damping 255
Spectrum of linear oscillations 255
Longitudinal instability 259
Nonlinear oscillations 261
J.L. Laclare
Bunched beam coherent instabilities 264
Introduction 254
Longitudinal instabititiss 264
Transverse instabilities 3(16
Conclusion 325
I. Hofmcuin
Space charge dominated bean transport 327
Introduction 327
Basic properties 327
Efjiïttance and field energy 329
Application to eniitta.'ice growth 332
L, Palumbo caul V.G. Vaacaro
Halte fields, inpedarxes 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
C, Dôme
Diffusion due to RF noise 370
Statistical properties of random variables 370
FokJcer-Planck equation 372
Differential equations for a stationary bucket with amplitude and phase noise 374
Computation of the coefficients A¿, A2 ™ tne Fokker-Planck equation 377
Case of a sinusoidal RF voltage 382
Hnite difference equations 386
Diffusion equation 388
Contribution of RF noise to the finite bean lifetime in the SPS collider 392
A. Pivinnki
Jntra-beam scattering 402
Introduction 402
Calculation of rise times and damping times 403
Experimental results 411
D. Bous sard
SchotLky noise and bean transfer function diagnostics 415
Schottky signals "17
Beam detectors 426
Observation of Schottky signals fl40
Bean transfer functions 445
D. !-iohl
Stochastic cooling 453
Introduction 453
Simplified theory, Lime-domain picture 455
A more detailed presentation of betatron cooling, frequency domain oicture 47.g
Distribution function equations (Fokker-Planck) and momentum scalina 510
//. Polh
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) 53R
Experimental realization of electron cooling 543
More on the theory of electron cooh'nq 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. Jewell
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 ¡amping 534
Quantum fluctuations and Fokker-Planck equations 537
Nonlinear wigglers 591
J. Le IM ff
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 UD 621
Beam loading 626
Introduction 626
Sinqle bunch passaqe in a cavity 627
Multiple bunch passaqes 629
Limiting case í> = 0 630
The case of a trüavelling wave structure 633
Transient correction 635
RF drive generation 637
J. iíuatt
Polarization in electron arid 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
li. Ma~'-iîi 0. Ripken, A. it'yu I i ah caul t'. 5'jisnidt
Particle tracking 690
Introduction 690
Hamiltonian 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, i'ugit-ji
The radiofrequency quadrupole linear accelerator 706
Introduction 706
The accelerating structure 706
Outline of the T-K expansion 709
The vane tips shaping 71?
Physical considerations 717
The structure of an RFQ 720
Design and technical considerations
Recent developments 733
//. HcL
Fundamental features of superconducting cavities for high energy accelerators 736
Introduction 7?fi
Some cavity fundamentals 737
Superconducting cavities ,'H\
Cavity design 716
Anomalous losses 751
Cavities covered with superconduclinq thin films 761
Current accelerator projects and achievcnenls 76S
J* La Duff
High field electron linacs 7-V
Introduction 7!¿
Extrapol aUun of present technologies 7 7?
RF compression scheme 775
Ultimate acceleratinq qradienls in conventional structures 7M3
A survey of acceleralinq structures 7Hrj
RF power source: the laserIron w
0. Ut.it to Li, /I. l.cnicvi (JUti A. "JVI><S
Free electron lasers: a short review of the theory and experiments 79?
Introduction 79?
FEL; theory and desiqn criteria ;qv>
FEL storage ring operation SO.!
Single passage FEL operation RHU
Conclusions
U.A. u't'iiy <i't't '*'.''. /iff-;;
ISIS, the accelerator based neutron source at RAL 81'
Introduction R(7
Linac and synchrotron B17
Target station Pl9
Experimental facilities $2\
High intensity performance of the ISIS synchrotron 02?
Heavy ions; Lhe present and future*)
Linear colliders versus storage rings*)
Contribution not received
SCHOTTKY MOISE AMD BEAM TRAHSFEH FUNCTION DIflGHOSTICS
D Boussard
CEflV, Geneva. Switzerland
ABSTRACT
Following the analysis of Schottky signals for the unbunci-ed and
the bunched beam cases, a general study of electromagnetic
detectors is presented. Here the image-current approach and the
Lorentz reciprocity theorem will be used to evaluate the detector
(or pick-up> performance for several typical examples. Then,
signal-processir.g techniques, which play an important role in the
Schottky signal analysis, will be reviewed. The beam transfer
function which relates the beam response to an external excitation
also provides very useful information about the accelerator
behaviour. It requires an element to excite the beam (kicker)
which will be shot.-n to be equivalent to a detector working in
reverse. With beam-transfer-function measurements an assessment
of beam stability limits can be made, leading to the determination
of the overall ring impedance.
The noise generated In an old fashioned electron tubu is governed by the Schottky
formula which simply reflects the fact that the anode current Is composed of individual
electrons randomly emitted by the cathode. Very similarly, the beam current in a circular
particle accelerator, also exhibits a random component, called the Schottky noise, which
results from the large, but finite, number of particles in the beam. In the absence of
random quantum emissions (i.e. for hadron machines} the analysis of Schottky noLse signals
(or Schottky signals, for brevity) is a very powerful tool to study the accelerator
behaviour . Historically, Schottky signals have been observed first on unbunched beam
machines (CERN ISR),1'^ leading to the development of the very successful stochastic
cooling technique. For bunched beams, the presence of strong "macroscopic" beam signals
renders the observation of the tiny Schottky signals more difficult. However improved
signal processing techniques have recently made their observation possible.
Following the analysis of Schottky signals For the unbunched and the bunched beam
cases, a general study of electromagnetic detectors is presented. Here the image-current
approach and the Lorentz reciprocity theorem will be used to evaluate the detector (or
pick up) performance for several typical examples. Then, signal-processing techniques,
which play an important role in the Schottky signal analysts, will be reviewed.
The beam transfer function which relates the beam response lo an external excitation
also provides very useful information about the accelerator behaviour. It requires an
element to excite the beam (kicker) which will be dtiown to be equivalent to a detector
working in reverse. With beam -1 ransfor-function measurements an assessment of beam
stability limits can be made, leading to the determination of the overall ring Impedance.