Accelerator Physics I / PHY-MV-BE-E09 Table of Contents: 1. Introduction __________________________________________________________________ 4 1.1. Literature ________________________________________________________________________ 4 1.2. Bending radius and beam rigidity _____________________________________________________ 6 2. Magnets _____________________________________________________________________ 9 2.1. General remarks on the calculation of magnetic fields ____________________________________ 9 2.2. Particle beam guidance ____________________________________________________________ 10 2.3. Particle beam focusing ____________________________________________________________ 12 2.4. Correction of chromatic errors ______________________________________________________ 17 2.5. Multipole expansion ______________________________________________________________ 19 2.6. Superconducting Magnets __________________________________________________________ 25 2.7. Effective field length ______________________________________________________________ 29 3. Linear Beam Optics ___________________________________________________________ 31 Transverse Linear Beam Dynamics W. Hillert page 1 Accelerator Physics I / PHY-MV-BE-E09 3.1. A quick and simple first approach using geometric optics ________________________________ 31 3.2. Some considerations concerning the equations of motion ________________________________ 36 3.3. Equations of motion in a moving reference system _____________________________________ 38 3.4. Matrix formalism _________________________________________________________________ 44 3.4.1. Drift space ____________________________________________________________________________45 3.4.2. Dipole magnets ________________________________________________________________________45 3.4.3. Quadrupole magnets ___________________________________________________________________48 3.4.4. Particle orbits in a system of magnets ______________________________________________________50 3.5. Particle beams and phase space _____________________________________________________ 52 3.3.1. Beam emittance _______________________________________________________________________52 3.5.2. Twiss parameters ______________________________________________________________________54 3.5.3. Beta functions _________________________________________________________________________57 3.5.4. Transformation in phase space ___________________________________________________________61 4. Circular Accelerators __________________________________________________________ 69 4.1. Weak focusing ___________________________________________________________________ 69 4.2. Strong focusing __________________________________________________________________ 72 4.2.1. Stability criterion ______________________________________________________________________74 4.3. Periodic focusing systems __________________________________________________________ 77 4.3.1. General FODO lattice ___________________________________________________________________77 4.3.2. Periodic beta functions _________________________________________________________________81 4.4. Transverse beam dynamics _________________________________________________________ 84 4.4.1. Closed orbit __________________________________________________________________________84 4.4.2. Betatron tune _________________________________________________________________________85 4.4.2. Filamentation _________________________________________________________________________86 Transverse Linear Beam Dynamics W. Hillert page 2 Accelerator Physics I / PHY-MV-BE-E09 4.4.3. Normalized coordinates _________________________________________________________________88 4.4.4. Closed orbit distortions _________________________________________________________________89 4.4.5. Gradient errors ________________________________________________________________________97 4.4.6. Optical resonances ___________________________________________________________________ 103 4.5. Beam dynamics with acceleration __________________________________________________ 107 5. Dynamics with Off Momentum Particles _________________________________________ 108 5.1 Dispersion and dispersion functions _________________________________________________ 108 5.2 Dispersion in circular accelerators ___________________________________________________ 111 5.3. Chromaticity ____________________________________________________________________ 114 5.4 Path length and momentum compaction _____________________________________________ 119 Transverse Linear Beam Dynamics W. Hillert page 3 Accelerator Physics I / PHY-MV-BE-E09 1. Introduction 1.1. Literature S.Y. Lee: Accelerator Physics, 3rd edition, World Scientific, New Yersey 2012, ISBN 978-981-4374-94-1 Bryant / Johnson: The Principles of Circular Accelerators and Storage Rings, Cambridge University Press, Cambridge 2005, ISBN 978-0-521-61969-1 Edwards / Syphers: An Introduction to the Physics of High Energy Accelerators, John Wiley & Sons, New York 1992, ISBN 978-0-471-55163-8 K. Wille: The physics of particle accelerators, Oxford Univ. Press 2005, Oxford, ISBN 0-19-850550-7 H- Wiedemann: Particle Accelerator Physics, 4th edition, Springer 2015, Berlin, ISBN 978-3-319-18316-9 Chao / Tigner: Handbook of Accelerator Physics and Engineering, 2nd edition, World Scientific, Singapore 2013, ISBN 987-4417-17-4 Transverse Linear Beam Dynamics W. Hillert page 4 Accelerator Physics I / PHY-MV-BE-E09 F. Hinterberger: Physik der Teilchenbeschleuniger und Ionenoptik, 2. Ausgabe, Springer 2008, Berlin, ISBN 978-3-540-75281-3 K. Wille: Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen, 2. überarb. und erw. Ausgabe, Teubner 1996, Stuttgart, ISBN 978-3-519-13087-1 Rossbach / Schmüser: Basic Course on Accelerator Optics, CAS 5th general accelerator physics course CERN 94-01 Transverse Linear Beam Dynamics W. Hillert page 5 Accelerator Physics I / PHY-MV-BE-E09 1.2. Bending radius and beam rigidity Particle guidance and focusing based on beam deflection by Lorentz force F qEvB Ultra-relativistic particles move with speed very close to speed of light! Impact of magnetic fields is enhanced by enormous factor: vc B 1Tesla E 3 108 V /m Only magnetic fields are used for beam deflection! Bending radius from balance of forces mm r 0 : v2 Bv: m qvB pmvqB Leads to the definition of the magnetic rigidity B! In circular accelerators, the magnetic rigidity defines the momentum of the beam: p GeV Bp 1Tm 0.3 q c Transverse Linear Beam Dynamics W. Hillert page 6 Accelerator Physics I / PHY-MV-BE-E09 Example LHC: bending radius: = 2.8 km magnetic field: B = 8.3 Tesla Magnetic rigidity: B = 23.2⸱103 Tm → momentum: p[GeV/c] = 0.3⸱ B → kin. energy: E ≈ pc = 7 TeV Transverse Linear Beam Dynamics W. Hillert page 7 Accelerator Physics I / PHY-MV-BE-E09 Magnets Beam Guidance Beam Focusing Correction of Chromatic Errors Multipole expansion Transverse Linear Beam Dynamics W. Hillert page 8 Accelerator Physics I / PHY-MV-BE-E09 2. Magnets 2.1. General remarks on the calculation of magnetic fields Maxwell’s Equations: Hj (coils) H 0 (gap) → H !!! 0 nc Magnets: = const. defines the pole’s contour! Magn. Induction from BH 0 r Taylor Expansion of the Magnetic Field: BB2 B (x,)y By(0, )x yy (0, y )x2 (, 0 y ) y y xx2 Dipoles Quadrupoles Sextupoles Transverse Linear Beam Dynamics W. Hillert page 9 Accelerator Physics I / PHY-MV-BE-E09 2.2. Particle beam guidance Deflection of particles → homogenous field: BBe0 ˆy const. Corresponding magnetic potential: (,x y ) B0 y defining the pole’s profile to be flat and parallel: Dipole Magnets! parallel poles nI Hds H ds H , HH H H 0 E rE00 E gap yoke nI 1 qqnI B , Curvature: B 0 , m1 00h pph0 Transverse Linear Beam Dynamics W. Hillert page 10 Accelerator Physics I / PHY-MV-BE-E09 Dipole Magnets: Iron dominated: Superconducting: field determined by field determined by geometry of poles geometry of coils → 2 flat poles → j() ~ cos Transverse Linear Beam Dynamics W. Hillert page 11 Accelerator Physics I / PHY-MV-BE-E09 2.3. Particle beam focusing Restoring force, linearly increasing with increasing distance from the axis: B B B g x, B g y with g y x const . yx xy Corresponding potential: (,xy ) g x y, solves B 0 defining the pole’s profile to four hyperbolic poles: Quadrupole Magnets! y iron yoke coils hyperbolic poles a2 yx()0 at a distance ag2 from the axis. gx 2 x 0 Transverse Linear Beam Dynamics W. Hillert page 12 Accelerator Physics I / PHY-MV-BE-E09 The “restoring” force acting on the particles is F qBqvv g xeˆˆx yey A quadrupole magnet is therefore focusing only in one plane and defocusing in the other; depending on the sign of g. The g-parameter may be related to the current of the coils by evaluating the closed 12 01 loop integral n I H ds H ds H ds H ds H ds, 000E 01 20 y g One obtains with Hds rdr : 0 coil 2 nI g 0 , normalized: a2 Quadrupole Strength qqnI2 0 2 kg 2 , k m ppa Transverse Linear Beam Dynamics W. Hillert page 13 Accelerator Physics I / PHY-MV-BE-E09 The focal length of a thin quadrupole magnet of length L can be derived from the de- flection angle of the particles beam and its relation to the quadrupole strength k, x tan f Lqq tan LB gxLxkL Rppy 1 → Gives a better understanding of the quadrupole strength: kL f Here we have assumed the length L to be short compared to the focal length f such that R does not change significantly within the quadrupole magnetic field. Transverse Linear Beam Dynamics W. Hillert page 14 Accelerator