DOCSLIB.ORG

Electromagnets in Synchrotron Design and Fabrication

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electromagnets in Synchrotron Design and Fabrication 3rd ILSF Advanced School on Synchrotron Radiation and Its Applications September 14-16, 2013 Electromagnets in Synchrotron Design and Fabrication Prepared by: Farhad Saeidi, Jafar Dehghani Mechanic Group ,Magnet section, Institute for Research in Fundamental Sciences ILSF 1 References • S.Fatehi, M.khabazi, “2nd ILSF school on synchrotron radiation and its applications” October2012. • Th.Zickler, CERN Accelerator School , Specialized Course on Magnets, Bruges, Belgium, 16‐25 June 2009. • D. Einfeld, ‘’Magnets’’, CELLS CAS, Frascati, Nov. 2008. • Jack Tanab, “Iron Dominated Electromagnets Design, Fabrication, Assembly and Measurements”, January 6, 2005. • G.E.Fisher,”Iron Dominated Magnets”, Stanford Linear Accelerator Center, 1985. • H.Ghasem, F.Saeidi, first ILSF school on synchrotron radiation and its applications. Sepember 2011. • H. Wiedemann, Particle Accelerator Physics I, Springer, 1999. ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd • K.Wille, The physics of Particle Accelerators, Oxford University, 2000. 3 ILSF-IPM, Sep. 2013 2 OutLine Light sources and magnets Magnet types Dipoles Quadrupoles Sextupoles Combined magnets Design Procedure ILSF Prototype magnets Fabrication Procedure ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 3 Light sources and magnets Storage ring Booster LTB BTS Linac Diamond Accelerator Complex ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 4 Magnet types Use in medium energy ALBA,TPS, PAL, SLS,… light sources (≈ 3 GEV), magnetic field ≤ 2 T Use in high energy light CERN sources (>3 GEV), magnetic field > 2 T Brazilian light source Rarely used because of high prices, aging & etc ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 5 Electromagnet types ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 6 Each magnet has 2 main parts : 1. Iron yoke( includes back leg , pole root, pole ,…) 2. Coils ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 7 Dipole magnet(Bending magnet) •Has 2 poles,2 coils •Bends the electrons in a single curved trajectory. •Injection of particles into the accelerator •Extraction of particles from the accelerator •Production of synchrotron radiation B (magnetic field at center) Pole profile eq.: y=h Dipole main L (length) parameters h (half gap height) GFR (good field region), By = ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF By (x) const rd 3 x ILSF-IPM, Sep. 2013 8 Standard Dipole Geometries C‐core H‐type Window frame High ampere turn ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF High ampere turn rd 3 ILSF-IPM, Sep. 2013 9 Normal Conducting Dipole Magnet B magnetic flux h (gap height) ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd µo air permeability 3 ILSF-IPM, Sep. 2013 10 Magnetic length Coming from ∞, B increases towards the magnet center (stray flux) ‘Magnetic’ length > iron length +∞ ∫ B dz − ∞ Lmagnetic = Magnetic length: B0 = + For a dipole: L magnetic Liron 2*(2h) * K 2h: gap height k: geometry specific constant (≈ 0.56) Pole width < gap height Beam direction K gets smaller in case of: Saturation ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 11 Quadrupole magnet • 4 poles, 4 coils • Focus the beam and prevent deviating • Zero field in center • Strong gradient g (T/m) • Placed in straight sections between bending magnets R 2 Pole profile eq.: xy = g (magnetic gradient field ) 2 Quadrupole main L ( length) parameters R (Aperture) GFR (good field region) By By (x) = g x ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 x ILSF-IPM, Sep. 2013 12 Standard Quadrupole Geometries ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 13 Normal Conducting Quadrupole Magnet integration path split in 3 sections field defined by gradient g along s1: along s3: along s2: & ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF g field gradient rd 3 ra Aperture µoair permeability ILSF-IPM, Sep. 2013 14 Magnetic length ‘Magnetic’ length > iron length Magnetic length For a Quadrupole: Lmagnetic = Liron + 2RK R: Aperture k: geometry specific constant (≈ 0.45) Beam direction ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 15 Sextupole magnet • Sextupole magnet correct chromatic aberration due to focusing errors on particles with different energy B‘‘ (Sextupole component) L ( length) Sextupole main R (Aperture) parameters GFR (good field region) Pole profile: 3yx2− y3=R3 2 ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF 1 By (x) = B′′x rd 2 3 x 0 0 ILSF-IPM, Sep. 2013 16 Normal Conducting Sextupole Magnet B´´ Sextupole component ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF R Aperture rd 3 µ0air permeability ILSF-IPM, Sep. 2013 17 Combined magnets Functions generated by pole shape (sum a scalar potentials): Amplitudes cannot be varied independently • Dipole + quadrupole h(0) y = gx 1+ B0 • Dipole + quadrupole + sextupole h(0) y = gx B′′x2 1+ + B0 2B0 ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF • Quadrupole + sextupole rd 3 3 y gxy + B′′(x2 y − ) = cte. 3 ILSF-IPM, Sep. 2013 18 Summary Bend the electron beam focuse the electron beam correct chromatic aberration ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 19 Magnet Design ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 20 Magnet Design Procedure Define Specifications ( by beam dynamics) Perform computer design a) obtain the needed c) profile designing b) choosing material current from the and pole d) Saturation test e) Checking field for the yoke defined parameters optimization tolerances in the good field region Electrical and Cooling Design ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 Mechanical Design ILSF-IPM, Sep. 2013 21 2D design: (Poisson Superfish ,FEMM, Opera2D …) •Use pre‐processor or modeler to build geometry •Profit from symmetries to reduce number of elements ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 22 Yoke materials Today’s standard: cold rolled, non‐oriented electro‐steel sheets • Massive iron only for dc magnets •Magnetic and mechanical properties can be adjusted by final annealing in decarbonized atmosphere – the smaller carbon content results in better magnetic properties (Increase in permeability Decrease in hysteresis loss and aging) • Magnetic properties (permeability, coercivity) should be within small tolerances . • Homogeneity and reproducibility among the magnets of a series can be enhanced by selection, sorting or shuffling. • Organic or inorganic coating for insulation and bonding. • Material is usually cheaper, but laminated yokes are commonly used esp. in Booster rings. (Eddy current) • Packing factor should be kept bellow 98%. Common used materials: • AISI1010 (max 0.1 % carbon content & max 0.3% silicon content ) •1200-100A(less than 0.003% carbon content & less than 1.3% silicon content) ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd •M400-50A( 0.02% carbon content & 2.4 % silicon content) 3 •M800-50A( 0.01% carbon content & 1.7 % silicon content) ILSF-IPM, Sep. 2013 23 •Sheet thickness: 0.3 ≤ t ≤ 1.5 mm •Specific weight: 7.60 ≤ δ≤ 7.85 g/cm3 ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF •Coercivity: Hc< 70 (±10) A/m rd 3 •Electrical resistivity @ 20°C: 0.16 (low Si) ≤ ρ≤ 0.61 μΩm (high Si) ILSF-IPM, Sep. 2013 24 Pole Optimization Rising the amount of these multipoles ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF leads to BAD field quality i.e. more than rd 3 0.1% SHIMMING is needed ILSF-IPM, Sep. 2013 25 Pole Optimization‐Shimming process ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd Pole Optimization is an iteration process & should be continued till 3 one reaches the desire field quality i.e. < 10-4 ILSF-IPM, Sep. 2013 26 ILSF Dipole, Quadrupole & Sextupole shims Dipole Sextupole Quadrupole ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 27 Field Quality Calculations B − B ∆B = 0 B0 Dipole Quadrupole Sextupole B = const central field B = g x B = 1 B′′x2 0 0 0 0 2 0 B(x) Is the field at each point calculated by simulation ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep. 2013 28 0.002 X[mm] 0 0.001 ‐40 ‐30 ‐20 ‐10 0 10 20 30 40 ‐0.005 0 ‐30 ‐20 ‐10 0 10 20 30 0 ‐0.001 0 ‐0.01 B'/B' B/B ∆ ∆ ‐0.002 ‐0.015 ‐0.003 ‐0.02 ‐0.004 ‐0.025 ‐0.005 X (mm) ILSF School on Synchrotron Radiation and Its Applications Its and Radiation Synchrotron on School ILSF rd 3 ILSF-IPM, Sep.
Load more

Recommended publications
  • Measurement of Sextupole Orbit Offsets in the Aps Storage Ring ∗
    Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 MEASUREMENT OF SEXTUPOLE ORBIT OFFSETS IN THE APS STORAGE RING ∗ M. Borland, E.A. Crosbie, and N.S. Sereno ANL, Argonne, IL Abstract However, we encountered persistent disagreements be- tween our model of the ring and measurements of the beta Horizontal orbit errors at the sextupoles in the Advanced functions. Hence, a program to measure the beam posi- Photon Source (APS) storage ring can cause changes in tion in sextupoles directly was undertaken. Note that while tune and modulation of the beta functions around the ring. we sometimes speak of measuring sextupole “offsets” or To determine the significance of these effects requires “positions” and of “sextupole miscentering,” we are in fact knowing the orbit relative to the magnetic center of the sex- measuring the position of the beam relative to the sextupole tupoles. The method considered here to determine the hor- center for a particular lattice configuration and steering. izontal beam position in a given sextupole is to measure the tune shift caused by a change in the sextupole strength. The tune shift and a beta function for the same plane uniquely 2 PRINCIPLE OF THE MEASUREMENT determine the horizontal beam position in the sextupole. The measurement relies on the quadrupole field component The beta function at the sextupole was determined by prop- generated by a displaced sextupole magnet. It also makes agating the beta functions measured at nearby quadrupoles use of the existence of individual power supplies for the to the sextupole location. This method was used to measure 280 sextupoles and 400 quadrupoles in the APS.
    [Show full text]
  • Optimal Design of the Magnet Profile for a Compact Quadrupole on Storage Ring at the Canadian Synchrotron Facility
    OPTIMAL DESIGN OF THE MAGNET PROFILE FOR A COMPACT QUADRUPOLE ON STORAGE RING AT THE CANADIAN SYNCHROTRON FACILITY A Thesis Submitted to the College of Graduate and Postdoctoral Studies in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mechanical Engineering, University of Saskatchewan, Saskatoon By MD Armin Islam ©MD Armin Islam, December 2019. All rights reserved. Permission to Use In presenting this thesis in partial fulfillment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of the material in this thesis in whole or part should be addressed to: Dean College of Graduate and Postdoctoral Studies University of Saskatchewan 116 Thorvaldson Building, 110 Science Place Saskatoon, Saskatchewan, S7N 5C9, Canada Or Head of the Department of Mechanical Engineering College of Engineering University of Saskatchewan 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada i Abstract This thesis concerns design of a quadrupole magnet for the next generation Canadian Light Source storage ring.
    [Show full text]
  • Lecture Notes For: Accelerator Physics and Technologies for Linear Colliders
    Lecture Notes for: Accelerator Physics and Technologies for Linear Colliders (University of Chicago, Physics 575) ¢¡ ¢¡ February 26 and 28 , 2002 Frank Zimmermann CERN, SL Division 1 Contents 1 Beam-Delivery Overview 3 2 Collimation System 3 2.1 Halo Particles . 3 2.2 Functions of Collimation System . 4 2.3 Collimator Survival . 4 2.4 Muons . 7 2.5 Downstream Sources of Beam Halo . 8 2.6 Collimator Wake Fields . 10 2.7 Collimation Optics . 12 3 Final Focus 14 3.1 Purpose . 14 3.2 Chromatic Correction . 14 3.3 £¥¤ Transformation . 20 3.4 Performance Limitations . 22 3.5 Tolerances . 24 3.6 Final Quadrupole . 26 3.7 Oide Effect . 27 4 Collisions and Luminosity 29 4.1 Luminosity . 29 4.2 Beam-Beam Effects . 29 4.3 Crossing Angle . 31 4.4 Crab Cavity . 33 4.5 IP Orbit Feedback . 35 4.6 Spot Size Tuning . 36 5 Spent Beam 42 2 1 Beam-Delivery Overview The beam delivery system is the part of a linear collider following the main linac. It typically consists of a collimation system, a final focus, and a beam-beam collision point located inside a particle-physics detector. Also the beam exit line required for the disposal of the spent beam is often subsumed under the term ‘beam delivery’. See Fig. 1. In addition, somewhere further upstream, there might be a beam switchyard from which the beams can be sent to several different interaction points, and possibly a bending section (‘big bend’) which provides a crossing angle and changes the direction of the beam line after collimation, so as to prevent the collimation debris from reaching the detector.
    [Show full text]
  • FS&T Template
    A COMPACT STORAGE RING FOR THE PRODUCTION OF EUV RADIATION R.M. Bergmann, T. Bieri, P. Craievich, Y. Ekinci, T. Garvey, C. Gough, M. Negrazus, L. Rivkin, C. Rosenberg, L. Schulz, T. Schmidt, L. Stingelin, A. Streun, V. Vrankovic. A. Wrulich, A. Zandonella Callegher, R. Zennaro. Paul Scherrer Institut,: 5232 Villigen, Switzerland. We present the design of a compact low emittance synchrotron radiation source which will synchrotron light source for the production of EUV meet the needs of industry for mask inspection. Hereafter, the source will be referred to as COSAMI (Compact radiation for metrology applications in the semiconductor Storage ring for Actinic Mask Inspection). industry. Stable, high brightness EUV light sources are of great potential interest for this industry. The recent II. SOURCE REQUIREMENTS availability of highly reflective mirrors at 13.5 nm As the storage ring (SR) is operated for wavelength makes EUV lithography a strong candidate commercial purposes it is mandatory that it has a very for future generation semiconductor manufacture. Our high reliability. The Swiss Light Source (SLS) at PSI has design is based on a storage ring lattice employing design a typical reliability figure of ~ 99% and we believe that principles similar to those used in the new family of this is sufficient for the semiconductor industry. However, diffraction limited synchrotron radiation sources. The as reliability is of such great importance our SR design will use only well-established accelerator technologies. In 430 MeV storage ring of circumference 25.8 m would addition, as the SR will have a single user (i.e. it is not a have an emittance of ~ 6 nm-rad.
    [Show full text]
  • Magnet Design for the Storage Ring of TURKAY
    Turkish Journal of Physics Turk J Phys (2017) 41: 13 { 19 http://journals.tubitak.gov.tr/physics/ ⃝c TUB¨ ITAK_ Research Article doi:10.3906/fiz-1603-12 Magnet design for the storage ring of TURKAY Zafer NERGIZ_ ∗ Department of Physics, Faculty of Arts and Sciences, Ni˘gdeUniversity, Ni˘gde,Turkey Received: 09.03.2016 • Accepted/Published Online: 16.08.2016 • Final Version: 02.03.2017 Abstract:In synchrotron light sources the radiation is emitted from bending magnets and insertion devices (undulators, wigglers) placed on the storage ring by accelerating charged particles radially. The frequencies of produced radiation can range over the entire electromagnetic spectrum and have polarization characteristics. In synchrotron machines, the electron beam is forced to travel on a circular trajectory by the use of bending magnets. Quadrupole magnets are used to focus the beam. In this paper, we present the design studies for bending, quadrupole, and sextupole magnets for the storage ring of the Turkish synchrotron radiation source (TURKAY), which is in the design phase, as one of the subprojects of the Turkish Accelerator Center Project. Key words: TURKAY, bending magnet, quadrupole magnet, sextupole magnet, storage ring 1. Introduction At the beginning of the 1990s work started on building third generation light sources. In these sources, the main radiation sources are insertion devices (undulators and wigglers). Since the radiation generated in synchrotrons can cover the range from infrared to hard X-ray region, they have a wide range of applications with a large user community; thus a strong trend to build many synchrotrons exists around the world.
    [Show full text]
  • Iron Dominated Electromagnets Design, Fabrication, Assembly and Measurements
    SLAC-R-754 June 2005 Iron Dominated Electromagnets Design, Fabrication, Assembly and Measurements Jack Tanabe January 6, 2005 Stanford Linear Accelerator Center, Stanford Synchrotron Radiation Laboratory, Stanford, CA 94025 Work supported in part by Department of Energy contract DE-AC02-76SF00515 2 Dedicated with Love to Sumi, for her support and devo- tion. For our beautiful grand-daughter, Sarah. 3 Abstract Medium energy electron synchrotrons (see page 15) used for the produc- tion of high energy photons from synchrotron radiation is an accelerator growth industry. Many of these accelerators have been built or are under construction to satisfy the needs of synchrotron light users throughout the world. Because of the long beam lifetimes required for these synchrotrons, these medium energy accelerators require the highest quality magnets of various types. Other accel- erators, for instance low and medium energy boosters for high energy physics machines and electron/positron colliders, require the same types of magnets. Because of these needs, magnet design lectures, originally organized by Dr. Klaus Halbach and later continued by Dr. Ross Schlueter and Jack Tanabe, were organized and presented periodically at biennual classes organized under the auspices of the US Particle Accelerator School (USPAS). These classes were divided among areas of magnet design from fundamental theoretical consider- ations, the design approaches and algorithms for permanent magnet wigglers and undulators and the design and engineering of conventional accelerator mag- nets. The conventional magnet lectures were later expanded for the internal training of magnet designers at LLNL at the request of Lou Bertolini. Because of the broad nature of magnet design, Dr.
    [Show full text]
  • 1 Lattices for Synchrotron Radiation Sources*+ Max
    SLAC–PUB–6459 May 1994 (N) LATTICES FOR SYNCHROTRON RADIATION SOURCES*+ MAX CORNACCHIA [email protected] Stanford Linear Accelerator Center Stanford Synchrotron Radiation Laboratory Stanford, CA 94309, USA 1. Introduction This chapter introduces the reader to the motion of electrons1 in a storage ring, and to the connection between electron beam dynamics and the properties of synchrotron radiation. The system of magnetic lenses that guides and focuses an electron beam is called the lattice. The choice of a lattice for a synchrotron radiation source is, arguably, the single most important decision in the history of a project. The lattice determines the emittance of the electron beam, the brightness of the photon beam, the beam lifetime, the quality of the experimental conditions, the number of insertion devices that can be accommodated in the straight sections, and the size and cost of the accelerator. The best choice of lattice is not a straightforward affair, involving complex performance and cost trade-offs, and a certain amount of intuition and subjectivity. The various types of lattices, and future directions in lattice design, will be covered, however briefly, in this review. Synchrotron radiation is emitted from the bending sections of the electron trajectory and in the straight sections, where insertion devices might be installed (see Chapter 12). From the source points the radiation is channeled into a beam line for experimental use. The lattice and the electron beam energy define the trajectory and, together with the natural divergence of the radiation, the size and divergence of the source. After a broad overview in Section 2, the magnetic forces acting on the electrons and the associated differential equations of motion are discussed in Section 3.
    [Show full text]
  • Compensation Techniques in Accelerator Physics Hisham Kamal Sayed Old Dominion University
    Old Dominion University ODU Digital Commons Physics Theses & Dissertations Physics Spring 2011 Compensation Techniques in Accelerator Physics Hisham Kamal Sayed Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/physics_etds Part of the Electromagnetics and Photonics Commons, and the Physics Commons Recommended Citation Sayed, Hisham K.. "Compensation Techniques in Accelerator Physics" (2011). Doctor of Philosophy (PhD), dissertation, Physics, Old Dominion University, DOI: 10.25777/e7ad-a731 https://digitalcommons.odu.edu/physics_etds/82 This Dissertation is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. COMPENSATION TECHNIQUES IN ACCELERATOR PHYSICS by Hisham Kamal Sayed B.S. May 2001, Cairo University M.S. December 2007, Old Dominion University A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY PHYSICS OLD DOMINION UNIVERSITY May 2011 ABSTRACT COMPENSATION TECHNIQUES IN ACCELERATOR PHYSICS Hisham Kamal Sayed Old Dominion University, 2011 Director: Dr. Geoffrey Krafft Accelerator physics is one of the most diverse multidisciplinary fields of physics, wherein the dynamics of particle beams is studied. It takes more than the under­ standing of basic electromagnetic interactions to be able to predict the beam dynam­ ics, and to be able to develop new techniques to produce, maintain and deliver high quality beams for different applications. In this work, some basic theory regarding particle beam dynamics in accelerators will be presented.
    [Show full text]
  • Nonlinear Beam Dynamics Lecture 1 Contents
    Nonlinear Beam Dynamics Lecture 1 Contents Recap on linear betatron motion Hill’s Equations linear maps for simple elements Courant-Snyder Invariant Sources of nonlinear fields fringe fields, construction errors, multipole magnets Impact on particle motion sensitivity to phase advances resonances stopbands Examples of nonlinear effects in electron storage rings injection efficiency vs. betatron tune frequency map analysis Summary Aims In these two lectures, the topic of nonlinear beam dynamics is introduced for both circular and linear machines. The intention is not to provide a rigorous analysis of all aspects, but rather to introduce the various concepts and tools used for the studying the behaviour. The first lecture concentrates on storage rings, from the description of the nonlinear magnetic elements through to the impact on the particle motion, and concludes with a few examples from Diamond which illustrate this behaviour. In the second lecture, the analysis is continued using the Hamiltonian formalism to study the perturbations from sextupoles, before moving on to look at nonlinearities in the longitudinal plane. The lecture concludes by looking at nonlinearities in linear accelerators, both from the accelerating RF wave and from a bunch compressor. We start, however, with a recap of the equations for linear betatron motion. Linear Betatron Equations of Motion If in a circular accelerator, the magnetic fields consist of purely dipolar and quadrupolar fields, then the motion of the particles will be linear. For such systems, the particle motion can be described by Hill’s equations: 푑2푥 1 1 푑퐵 + − 푦 푥 = 0 푑푠2 휌2 푠 퐵휌 푑푥 푑2푦 1 푑퐵 + 푦 푦 = 0 푑푠2 퐵휌 푑푥 These equations can be integrated, and have as the solutions: 푧 푠 = 퐴 훽(푠) cos(휇푧 푠 + 휇0) 푑푧 퐴 푧′ 푠 = = − 훼 cos 휇 푠 + 휇 + sin 휇 푠 + 휇 푑푠 훽 푠 푥 푧 0 푧 0 where 푧 can be either 푥 or 푦, and the phase advance can be computed using 푠 1 휇 푠 = න 푑푠′ 푧 훽 푠′ 푠0 푧 Linear Betatron Equations of Motion The particle motion between two points can be described in terms of a map.
    [Show full text]
  • High Field Gradient Multi-Pole Magnets With
    Proceedings of the 8th Annual Meeting of Particle Accelerator Society of Japan (August 1-3, 2011, Tsukuba, Japan) HIGH FIELD GRADIENT MULTI-POLE MAGNETS wW ITH PERMENDUR ALLOY FOR THE SPring-8 UPGRADE Y. Okayasu∗, C. Mitsuda, K. Fukami, K. Soutome, T. Watanabe and T. Nakamura Japan Synchrotron Radiation Research Institute 1-1-1, Koto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan Abstract ring in SPring-8 II. A site view of the SPring-8 accelerator complex is shown One of the major goals for the SPring-8 upgrade is to in Fig. 1. At present, SPring-8 consists of 1 GeV linac as reduce the beam emittance in the storage ring down the an injector, 1 - 8 GeV booster synchrotron, 8 GeV storage diffraction limit in the X-ray region using the same tun- ring and SACLA (XFEL). Normal operational beam en- nel for the ring. In order to achieve the above goal while ergy of SACLA is designed as 8 GeV. In SPring-8 II, the keeping the circumference the same, a magnet lattice with operational beam energy of the storage ring is designed to six bending magnets in a cell is designed. Although the be lowered from 8 to 6 GeV taking into account a required final design has not been fixed yet, the designed lattice re- energy range of radiation spectrum. In addition, SACLA is quires extremely strong magnetic fields, especially for the also considered to be an injector candidate for the SPring-8 sextupole magnets, such as ≥10000 T/m2. It is difficult to II storage ring.
    [Show full text]
  • Lecture 2 Aspects of Transverse Beam Dynamics
    Lecture 2 Aspects of Transverse Beam Dynamics Chandra Bhat 1 Accelerator and Beamline Magnets Dipole Magnet: Dipole magnet is a device used to bend the path of charged particles during beam transport. The radius of curvature of a charged particle in a constant magnetic field perpendicular to its path is, 1 1 eB 0.2998 B[T ] 0.04 I [amp].n Iron Yoke = [m-1] = 0 = ;B = total R ρ p p[GeV / c] 0 h[cm] 1 n=number of turns h=pole gap Quadrupole Magnet: A device used to focus charged particle beam during beam transport. Particle trajectory Let us see what is the relationship between focal length, f, and the in a magnetic field A quadrupole strength. Fig. A shows bending of a charged particle in a magnetic field perpendicular to the plane of the paper and “B” l α ρ shows optical analogue of focusing. Then the deflection angle, l r eB eB α = − = − = φ l = − φ l l 2 α = − ρ f p βE ρ But the total bending field Bφ is given by, B dB Optics B = φ r = gr φ dr Then, egrl r 1 eg eg α = − = − or = kl; k = = 3 βE f f βE p f= focal length Quad strength 2 0.2998 g[Tesla / m] 2µ nI k[m−2 ] = ; g = 0 2 4 Field free region βE[GeV / c] R The quadrupole magnets provide material free aperture and focusing. A conventional quadrupole magnet used in synchrotrons has four iron poles with hyperbolic contours. By = −gx Bx = −gy Interesting features: The horizontal force component depends only on the horizontal position of the particle trajectory.
    [Show full text]
  • APS-U Definitions of Signs and Conventions Related to Magnets Revision 0
    APSU-2.03-TN-001 APS-U Definitions of Signs and Conventions Related to Magnets Revision 0 Advanced Photon Source About Argonne National Laboratory Argonne is a U.S. Department of Energy laboratory managed by UChicago Argonne, LLC under contract DE-AC02-06CH11357. The Laboratory’s main facility is outside Chicago, at 9700 South Cass Avenue, Argonne, Illinois 60439. For information about Argonne and its pioneering science and technology programs, see www.anl.gov. DOCUMENT AVAILABILITY Online Access: U.S. Department of Energy (DOE) reports produced after 1991 and a growing number of pre-1991 documents are available free via DOE's SciTech Connect (http://www.osti.gov/scitech/) Reports not in digital format may be purchased by the public from the National Technical Information Service (NTIS): U.S. Department of Commerce National Technical Information Service 5301 Shawnee Rd Alexandra, VA 22312 www.ntis.gov Phone: (800) 553-NTIS (6847) or (703) 605-6000 Fax: (703) 605-6900 Email: [email protected] Reports not in digital format are available to DOE and DOE contractors from the Office of Scientific and Technical Information (OSTI): U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 37831-0062 www.osti.gov Phone: (865) 576-8401 Fax: (865) 576-5728 Email: [email protected] Disclaimer This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor UChicago Argonne, LLC, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.
    [Show full text]