Physics 598ACC Accelerators: Theory and Applications
|
|
- Hilda Kelly
- 5 years ago
- Views:
Transcription
1 Physics 598ACC Accelerators: Theory and Instructors: Fred Mills, Deborah Errede Lecture 4: Betatron Oscillations 1
2 Summary A. Mathieu-Hill equation B. Transfer matrix properties C. Floquet theory solutions D. CSL invariant and emittance 2
3 Betatron Oscillations (see Theory of the Alternating-Gradient Synchrotron, E.D. Courant and H.S. Snyder) The equations of motion for betatron oscillations were found to be of the type 4.1 y + Ks ( )y = 0 where K(s) is a periodic function with period C = 2πR, or higher periodicity if the accelerator "lattice" has higher periodicity, as will usually be the case. In terms of the magnetic field, its gradient and the "magnetic rigidity Bρ = P 0c, K = B 2 0 B + e Βρ Bρ for the horizontal plane, and K = B for the vertical plane. Bρ Then 4.1 is a Mathieu-Hill equation for y, and Floquet's theorem applies to the solutions, that is that the two linearly independent solutions can be written in the form exp[±iψ(s)]w(s), where w(s) is a periodic function of s. We will use this fact later, but first let us look at some general properties of the solutions. note B ' B = x z 3
4 Let N be the number of periods per revolution, and be the length of one period. Since the y equation is linear, the solutions at two points s and s 0 are linearly related. If Y is the column vector we can write 4.2 y y Y()= s M s s 0 ( )Ys 0 ( ) L= C N ( ) M( s 2 s 1 ) where M s 2 s 1 is a 2x2 matrix. The determinant of is unity, since the * Wronskian determinant is constant because the coefficient of y' is zero in 4.1. We note that the transformations M form a group, since the identity Μ( s s)= I exists, the inverse M 1 ( s s 0 )= M( s 0 s) exists because det Μ 0, and M( s s 0 )= M( s s 1 )M ( s 1 s 0 ). Some useful examples of M are when K is constant. For K=0 (drift), 4.3 M( s s 0 )= 1 s s *see notes 4
5 Wronskians: (see, for example, Morse and Feshbach p. 524) for a differential equation of the form " ' y + f(x)y + g(x)y= 0 y y W(y,y ) = = y y y y 1 2 ' ' 1 2 ' ' y1 y2 (y 1,y 2 solutions to the eqn above) for f(x), g(x) continuous on an open interval I, two solutions y 1, y 2 are linearly independent if their Wronskian W is nonzero for any range of x in I. Also Abel s Theorem states that for the above differential eqn the Wronskian of the two solutions is f (z)dz W(y,y )(x) = W(y,y )(x )e = ce note that if f(x)=0, the Wronskian is constant. (canonical Poisson brackets = 1, and components of the brackets almost correspond to the elements of the matrix M. The Poisson bracket is the determinant of M) x 0 0 x f (z)dz 5
6 When K is positive (F or focusing), 4.4 M( s s 0 )= cosφ Ksinφ sinφ K cosφ while if K is negative (D or defocusing), 4.5 M( s s 0 )= coshψ -Ksinhψ sinhψ -K coshψ ( ) ψ= -K( s s ) 0 Here φ= K s s 0, and. 6
7 Now let us define the matrix for one period 4.6 a b M(s + L s ) = M( s) = c d The matrix for one revolution is Μ N ( s), and for k revolutions is Μ kn ( s). The motion will remain bounded if the matrix elements remain bounded as kn->. Consider the eigenvalues λ of M 4.7 Μ Y =λy Solutions exist if 4.8 The equation for λ becomes 4.9 det Μ λi = 0 λ 2 λ( a + d)+ 1 = 0 7
8 Let 4.10 cosμ = 1 2 Tr M = a + d 2 The two solutions of 4.8 become µ is real if 4.12 λ=cosμ ± isinμ 4.11 *see homework a + d 2 Now define α, β, and γ in the following way; a d = 2αsinμ 4.13 b =βsinμ c = γsinμ det Μ =1 implies that 4.14 γ = 1+α 2 β 8
9 for transfer matrix on p.10 cosμ= αsin μ= a+ d 2 a d 2 b and c are given in a+ d a d a = + = cosμ+αsinμ 2 2 d= cosμ αsinμ 9
10 The transfer matrix is now * see notes p Μ()= s a b c d cosμ + αsinμ = βsinμ γsinμ cosμ - αsinμ = I cosμ+jsinμ 4.16 J = α β, γ α det J = M Nk = ( Icosμ +Jsinμ) Nk = IcosNkμ +JsinNkμ Then if µ is real, the matrix elements of M Nk remain bounded, and the motion is stable. We note that M()= 0 I, 4.19 M 1 ()= μ M( -μ), M( μ 1 +μ 2 )= M( μ 1 )M( μ 2 ) 10
11 The definition of µ does not depend on the point s where the matrix is defined, for, if we calculate the matrix between s 1 and s 2 +L, first by going from s 1 to s 2, then by going from s 2 to s 2 +L, if M s 2 s 1 is the matrix connecting s 1 and s 2, while if we first go from s 1 to s 1 +L, and then from s 1 +L to s 2 +L, Then ( ) M( s 2 + Ls 1 )= M( s 2 )M( s 2 s 1 ) ( ) ( ) (see definition 4.6) M(s + L s) = M s + L s + L M s = M(s s) M(s) M( s 2 )= M( s 2 s 1 )M( s 1 )M 1 s 2 s 1 ( ) Thus the two matrices are related by a similarity transformation. Thus if M(s )Y = λy, M ' ' (s 2)Y =λy ( ) also, where Y = M 1 s 2 s 1 Y. If det M( s 2 ) λi = 0 then det M( s 1 ) λi = 0 also, and the two matrices have the same eigenvalues, hence the same value of µ. 1 11
12 Modern accelerators have lattices which are composed of successive regions of constant values of K (which might include curvature). The matrix for each region is one of the forms 4.3 to 4.5. Then we can find the one period matrix M(s) by matrix multiplication, and find µ from the trace of that matrix, and α, β, γ by using On the other hand, to find α, β, γ at every point in the lattice by this method is tedious and time consuming. We will develop better means to calculate these functions, but first we need to learn more about them. Let us attempt a solution to 4.1 in the phase-amplitude form 4.22 where w and ψ are real, and w is periodic. Then 4.23 y + Ky = [ w ± i2 ( w ψ + w ψ ) w ( ψ ) 2 + Kw]exp( iψ)= 0 The exponent is not zero, in general, so both the real and imaginary parts of the bracket must vanish. y = ws ()exp[±iψ( s)] 12
13 For the imaginary part, ( ) = w 2 ψ, or ψ = 1 w 2 where we have taken the arbitrary constant of integration to be 1 by absorbing it into the definition of w. For the real part, w + Kw 1 = 0 w (using 4.24) We are now in a position to express M in terms of w and ψ. Any solution can be written as a linear combination of the two linearly independent solutions 4.26 y = Awcosψ + Bwsinψ y = A w cosψ sinψ w + B w sinψ+cosψ w 13
14 We can evaluate A and B at s 1, let ψ=0, y=y 1, and y ' =y 1. Then A = y 1, and B = w 1 y 1 w 1 y 1. Introduce the values of A and B into 4.26 and collect coefficients of y 1 and y 1 ' to find 4.27 M( s 2 s 1 )= cosψ w 2 w 1 cosψ w 2 w 2 w 1 sinψ w 1 w 1 w 2 sinψ w 1 w 1 w 2 sinψ 1 + w 1 w 2 cosψ w 1 + sinψ w 1 w 2 w 1 w 2 w 2 Now we evaluate M in the case s 2 = s 1 +L, and require w 1 = w 2 = w 4.28 M( s 2 )= cosψ w w sinψ w 2 sinψ 1 w + ( w 2 )2 sinψ cosψ+w w sinψ 14
15 Now we compare 4.28 with 4.15, and we can find the following relations 4.29 w 2 = β w w = α α= β 2 μ=ψ( s + Ls) ψs ()= α = Kβ γ 1 γ= + 2 w s + L s ds β ' ( w ) 2 s+ L = s dψ ds ds (see 4.24) (use 4.25) The last equation follows from 4.25 and the relations between w, α, and β. Another useful differential relationship exists for γ, although α, β, and γ are related by 4.14, 4.30 γ =2Kα 15
16 We can now use 4.29 and 4.30 in 4.27 to find a general transfer matrix between any two points 1 and 2 in the lattice, 4.31 M( s 2 s 1 )= β 2 β 1 ( cosψ +α 1 sinψ ) β 1 β 2 sinψ cosψ α 1 α 2 sinψ 1+α 1α 2 β 1 β 2 β 1 β 2 β 1 β 2 ( cosψ α 2 sinψ) We will use these differential relationships together with matrix properties to solve for lattice parameters. First we wish to define an important quantity, ν, which is related to the phase advance µ, but is a property of the whole accelerator ν= Nμ 2π = s s+ C ds 2πβ ν is the total number of betatron oscillations in one revolution (in the y coordinate). see
17 We can also find a constant of the linear motion, W, called the CSL (Courant, Livingston, and Snyder) invariant. Any solution can be written, by virtue of 4.29, 4.33 y= Wβcos( ψ+δ) W β [ ] ' y = sin( ψ+δ) αcos( ψ+δ) where W and δ are constants. Solving for cos and sin, squaring and adding, 4.34 W = y2 +β ( y +αy) 2 β =γy 2 + 2αy y +β y 2 17
18 This is the equation of an ellipse of area πw in y,y ' space. The ellipse is upright when α = 0 and tilted otherwise. The maximum value of y at a given point in the lattice is βw because of If βw is the maximum amplitude of betatron oscillation of particles in a beam, then the beam "emittance", which is the area enclosing all the beam particles, is πw. Note that πwp is the projected area of the orbit in phase space, and will tend to remain constant as the particles are accelerated (adiabatic invariant). 18
19 End of Lecture 19
20 20
Transverse Dynamics II
Transverse Dynamics II JAI Accelerator Physics Course Michaelmas Term 217 Dr. Suzie Sheehy Royal Society University Research Fellow University of Oxford Acknowledgements These lectures have been produced
More informationIntroduction to Accelerator Physics 2011 Mexican Particle Accelerator School
Introduction to Accelerator Physics 20 Mexican Particle Accelerator School Lecture 3/7: Quadrupoles, Dipole Edge Focusing, Periodic Motion, Lattice Functions Todd Satogata (Jefferson Lab) satogata@jlab.org
More informationEquations of motion in an accelerator (Lecture 7)
Equations of motion in an accelerator (Lecture 7) January 27, 2016 130/441 Lecture outline We consider several types of magnets used in accelerators and write down the vector potential of the magnetic
More informationS5: Linear Transverse Particle Equations of Motion without Space Charge, Acceleration, and Momentum Spread S5A: Hill's Equation
S5: Linear Transverse Particle Equations of Motion without Space Charge, Acceleration, and Momentum Spread S5A: Hill's Equation For a periodic lattice: Neglect: Space charge effects: Nonlinear applied
More informationS5: Linear Transverse Particle Equations of Motion without Space Charge, Acceleration, and Momentum Spread S5A: Hill's Equation
S5: Linear Transverse Particle Equations of Motion without Space Charge, Acceleration, and Momentum Spread S5A: Hill's Equation Neglect: Space charge effects: Nonlinear applied focusing and bends: Acceleration:
More informationTransverse dynamics. Transverse dynamics: degrees of freedom orthogonal to the reference trajectory
Transverse dynamics Transverse dynamics: degrees of freedom orthogonal to the reference trajectory x : the horizontal plane y : the vertical plane Erik Adli, University of Oslo, August 2016, Erik.Adli@fys.uio.no,
More informationAccelerator Physics Homework #3 P470 (Problems: 1-5)
Accelerator Physics Homework #3 P470 (Problems: -5). Particle motion in the presence of magnetic field errors is (Sect. II.2) y + K(s)y = B Bρ, where y stands for either x or z. Here B = B z for x motion,
More informationPractical Lattice Design
Practical Lattice Design S. Alex Bogacz (JLab) and Dario Pellegrini (CERN) dario.pellegrini@cern.ch USPAS January, 15-19, 2018 1/48 D. Pellegrini - Practical Lattice Design Purpose of the Course Gain a
More informationE. Wilson - CERN. Components of a synchrotron. Dipole Bending Magnet. Magnetic rigidity. Bending Magnet. Weak focusing - gutter. Transverse ellipse
Transverse Dynamics E. Wilson - CERN Components of a synchrotron Dipole Bending Magnet Magnetic rigidity Bending Magnet Weak focusing - gutter Transverse ellipse Fields and force in a quadrupole Strong
More informationIntroduction to Transverse Beam Dynamics
Introduction to Transverse Beam Dynamics B.J. Holzer CERN, Geneva, Switzerland Abstract In this chapter we give an introduction to the transverse dynamics of the particles in a synchrotron or storage ring.
More informationLattice Design in Particle Accelerators
Lattice Design in Particle Accelerators Bernhard Holzer, DESY Historical note:... Particle acceleration where lattice design is not needed 4 N ntz e i N( θ ) = * 4 ( 8πε ) r K sin 0 ( θ / ) uo P Rutherford
More informationAccelerator Physics. Elena Wildner. Transverse motion. Benasque. Acknowldements to Simon Baird, Rende Steerenberg, Mats Lindroos, for course material
Accelerator Physics Transverse motion Elena Wildner Acknowldements to Simon Baird, Rende Steerenberg, Mats Lindroos, for course material E.Wildner NUFACT08 School Accelerator co-ordinates Horizontal Longitudinal
More information06. Orbit Stability and the Phase Amplitude Formulation *
06. Orbit Stability and the Phase Amplitude Formulation * Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU) US Particle Accelerator
More informationTransverse Beam Dynamics II
Transverse Beam Dynamics II II) The State of the Art in High Energy Machines: The Theory of Synchrotrons: Linear Beam Optics The Beam as Particle Ensemble Emittance and Beta-Function Colliding Beams &
More informationPhysics 598ACC Accelerators: Theory and Applications
hysics 598ACC Accelerators: Theory and Instructors: Fred Mills, Deborah Errede Lecture 3: Equations of Motion in Accelerator Coordinates 1 Summary A. Curvilinear (Frenet-Serret) coordinate system B. The
More informationPhysics 598ACC Accelerators: Theory and Applications
Physics 598ACC Accelerators: Theory and Instructors: Fred Mills, Deborah Errede Lecture 6: Collective Effects 1 Summary A. Transverse space charge defocusing effects B. Longitudinal space charge effects
More informationMagnets and Lattices. - Accelerator building blocks - Transverse beam dynamics - coordinate system
Magnets and Lattices - Accelerator building blocks - Transverse beam dynamics - coordinate system Both electric field and magnetic field can be used to guide the particles path. r F = q( r E + r V r B
More informationAccelerator Physics Final Exam pts.
Accelerator Physics Final Exam - 170 pts. S. M. Lund and Y. Hao Graders: C. Richard and C. Y. Wong June 14, 2018 Problem 1 P052 Emittance Evolution 40 pts. a) 5 pts: Consider a coasting beam composed of
More informationLow Emittance Machines
Advanced Accelerator Physics Course RHUL, Egham, UK September 2017 Low Emittance Machines Part 1: Beam Dynamics with Synchrotron Radiation Andy Wolski The Cockcroft Institute, and the University of Liverpool,
More information07. The Courant Snyder Invariant and the Betatron Formulation *
07. The Courant Snyder Invariant and the Betatron Formulation * Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU) US Particle
More informationBeam Dynamics. D. Brandt, CERN. CAS Bruges June 2009 Beam Dynamics D. Brandt 1
Beam Dynamics D. Brandt, CERN D. Brandt 1 Some generalities D. Brandt 2 Units: the electronvolt (ev) The electronvolt (ev)) is the energy gained by an electron travelling, in vacuum, between two points
More informationAccelerator Physics Homework #7 P470 (Problems: 1-4)
Accelerator Physics Homework #7 P470 (Problems: -4) This exercise derives the linear transfer matrix for a skew quadrupole, where the magnetic field is B z = B 0 a z, B x = B 0 a x, B s = 0; with B 0 a
More information4. Statistical description of particle beams
4. Statistical description of particle beams 4.1. Beam moments 4. Statistical description of particle beams 4.1. Beam moments In charged particle beam dynamics, we are commonly not particularly interested
More informationD. Brandt, CERN. CAS Frascati 2008 Accelerators for Newcomers D. Brandt 1
Accelerators for Newcomers D. Brandt, CERN D. Brandt 1 Why this Introduction? During this school, you will learn about beam dynamics in a rigorous way but some of you are completely new to the field of
More informationThe A, B, C and D are determined by these 4 BCs to obtain
Solution:. Floquet transformation: (a) Defining a new coordinate η = y/ β and φ = (/ν) s 0 ds/β, we find ds/dφ = νβ, and dη dφ = ds dη dφ d 2 η dφ 2 = ν2 β ( β y ) ( 2 β 3/2 β y = ν β /2 y ) 2 β /2 β y,
More informationCERN Accelerator School. Intermediate Accelerator Physics Course Chios, Greece, September Low Emittance Rings
CERN Accelerator School Intermediate Accelerator Physics Course Chios, Greece, September 2011 Low Emittance Rings Part 1: Beam Dynamics with Synchrotron Radiation Andy Wolski The Cockcroft Institute, and
More informationTheory of the Alternating-Gradient Synchrotron 1, 2
Annals of Physics 281, 3648 (2) doi:1.16aphy.2.612, available online at http:www.idealibrary.com on Theory of the Alternating-Gradient Synchrotron 1, 2 E. D. Courant and H. S. Snyder Brookhaven National
More informationThree Loose Ends: Edge Focusing; Chromaticity; Beam Rigidity.
Linear Dynamics, Lecture 5 Three Loose Ends: Edge Focusing; Chromaticity; Beam Rigidity. Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. November, 2012 What we Learned
More informationNonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 4: Canonical Perturbation Theory Nonlinear Single-Particle Dynamics in High Energy Accelerators There are six lectures in this course
More informationBeam Dynamics with Space- Charge
Beam Dynamics with Space- Charge Chris Prior, ASTeC Intense Beams Group, RAL and Trinity College, Oxford 1 1. Linear Transverse Review of particle equations of motion in 2D without space-charge - Courant-Snyder
More informationLow Emittance Machines
CERN Accelerator School Advanced Accelerator Physics Course Trondheim, Norway, August 2013 Low Emittance Machines Part 1: Beam Dynamics with Synchrotron Radiation Andy Wolski The Cockcroft Institute, and
More informationTWISS FUNCTIONS. Lecture 1 January P.J. Bryant. JUAS18_01- P.J. Bryant - Lecture 1 Twiss functions
TWISS FUNCTIONS Lecture January 08 P.J. Bryant JUAS8_0- P.J. Bryant - Lecture Slide Introduction These lectures assume knowledge of : The nd order differential equations of motion in hard-edge field models
More informationBernhard Holzer, CERN-LHC
Bernhard Holzer, CERN-LHC * Bernhard Holzer, CERN CAS Prague 2014 Lattice Design... in 10 seconds... the Matrices Transformation of the coordinate vector (x,x ) in a lattice x(s) x = M 0 x'(s) 1 2 x' 0
More informationaccelerator physics and ion optics summary longitudinal optics
accelerator physics and ion optics summary longitudinal optics Sytze Brandenburg sb/accphys007_5/1 coupling energy difference acceleration phase stability when accelerating on slope of sine low energy:
More informationWed Jan 25 Lecture Notes: Coordinate Transformations and Nonlinear Dynamics
Wed Jan 25 Lecture Notes: Coordinate Transformations and Nonlinear Dynamics T. Satogata: January 2017 USPAS Accelerator Physics Most of these notes kindasortasomewhat follow the treatment in the class
More informationMagnetic Multipoles, Magnet Design
Magnetic Multipoles, Magnet Design Alex Bogacz, Geoff Krafft and Timofey Zolkin Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-21, 2013 1 Maxwell s Equations for Magnets - Outline Solutions
More informationMedical Linac. Block diagram. Electron source. Bending magnet. Accelerating structure. Klystron or magnetron. Pulse modulator.
Block diagram Medical Linac Electron source Bending magnet Accelerating structure Pulse modulator Klystron or magnetron Treatment head 1 Medical Linac 2 Treatment Head 3 Important Accessories Wedges Dynamic
More informationELECTRON DYNAMICS WITH SYNCHROTRON RADIATION
ELECTRON DYNAMICS WITH SYNCHROTRON RADIATION Lenny Rivkin Ecole Polythechnique Federale de Lausanne (EPFL) and Paul Scherrer Institute (PSI), Switzerland CERN Accelerator School: Introduction to Accelerator
More informationParticle Accelerators: Transverse Beam Dynamics
Particle Accelerators: Transverse Beam Dynamics Volker Ziemann Department of Physics and Astronomy Uppsala University Research Training course in Detector Technology Stockholm, Sept. 8, 2008 080908 V.
More informationThu June 16 Lecture Notes: Lattice Exercises I
Thu June 6 ecture Notes: attice Exercises I T. Satogata: June USPAS Accelerator Physics Most o these notes ollow the treatment in the class text, Conte and MacKay, Chapter 6 on attice Exercises. The portions
More informationPBL SCENARIO ON ACCELERATORS: SUMMARY
PBL SCENARIO ON ACCELERATORS: SUMMARY Elias Métral Elias.Metral@cern.ch Tel.: 72560 or 164809 CERN accelerators and CERN Control Centre Machine luminosity Transverse beam dynamics + space charge Longitudinal
More informationThe FFAG Return Loop for the CBETA Energy Recovery Linac
The FFAG Return Loop for the CBETA Energy Recovery Linac BNL-XXXXX J. Scott Berg 1 Brookhaven National Laboratory; P. O. Box 5; Upton, NY, 11973-5; USA Abstract The CBETA energy recovery linac uses a single
More informationAccelerator Physics. Tip World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI BANGALORE. Second Edition. S. Y.
Accelerator Physics Second Edition S. Y. Lee Department of Physics, Indiana University Tip World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI BANGALORE Contents Preface Preface
More informationMatrix formalism of synchrobetatron coupling. Abstract
SLAC-PUB-12136 Matrix formalism of synchrobetatron coupling Xiaobiao Huang Stanford Linear Accelerator Center, Menlo Park, CA 9425 (Dated: January 3, 27) Abstract In this paper we present a complete linear
More informationParametrization of the Driven Betatron Oscillation
Parametrization of the Driven Betatron Oscillation R. Miyamoto and S. E. Kopp Department of Physics University of Texas at Austin Austin, Texas 7872 USA A. Jansson and M. J. Syphers Fermi National Accelerator
More informationIntroduction to particle accelerators
Introduction to particle accelerators Walter Scandale CERN - AT department Lecce, 17 June 2006 Introductory remarks Particle accelerators are black boxes producing either flux of particles impinging on
More informationLecture 2: Modeling Accelerators Calculation of lattice functions and parameters. X. Huang USPAS, January 2015 Hampton, Virginia
Lecture 2: Modeling Accelerators Calculation of lattice functions and parameters X. Huang USPAS, January 2015 Hampton, Virginia 1 Outline Closed orbit Transfer matrix, tunes, Optics functions Chromatic
More informationaccelerator physics and ion optics summary longitudinal optics
accelerator physics and ion optics summary longitudinal optics Sytze Brandenburg sb/accphys003_5/1 feedback energy difference acceleration phase stability when accelerating on slope of sine low energy:
More informationS9: Momentum Spread Effects and Bending S9A: Formulation
S9: Momentum Spread Effects and Bending S9A: Formulation Except for brief digressions in S1 and S4, we have concentrated on particle dynamics where all particles have the design longitudinal momentum at
More informationEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH AN INTRODUCTION TO TRANSVERSE BEAM DYNAMICS IN ACCELERATORS. M. Martini. Abstract
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN/PS 96{ (PA) March 996 AN INTRODUCTION TO TRANSVERSE BEAM DYNAMICS IN ACCELERATORS M. Martini Abstract This text represents an attempt to give a comprehensive
More informationMagnetic Multipoles, Magnet Design
Magnetic Multipoles, Magnet Design S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage Jefferson Lab and Old Dominion University Lecture 5 - Magnetic Multipoles USPAS, Fort Collins, CO, June 13-24, 2016
More informationBernhard Holzer, CERN-LHC
Bernhard Holzer, CERN-LHC * 1 ... in the end and after all it should be a kind of circular machine need transverse deflecting force Lorentz force typical velocity in high energy machines: old greek dictum
More informationSynchrotron radiation
Synchrotron radiation When a particle with velocity v is deflected it emits radiation : the synchrotron radiation. Relativistic particles emits in a characteristic cone 1/g The emitted power is strongly
More informationVERTICAL BEAM EMITTANCE CORRECTION WITH INDEPENDENT COMPONENT ANALYSIS MEASUREMENT METHOD
VERTICAL BEAM EMITTANCE CORRECTION WITH INDEPENDENT COMPONENT ANALYSIS MEASUREMENT METHOD Fei Wang Submitted to the faculty of the University Graduate School in partial fulfillment of the requirement for
More informationBasic Mathematics and Units
Basic Mathematics and Units Rende Steerenberg BE/OP Contents Vectors & Matrices Differential Equations Some Units we use 3 Vectors & Matrices Differential Equations Some Units we use 4 Scalars & Vectors
More informationAPPM 2360 Section Exam 3 Wednesday November 19, 7:00pm 8:30pm, 2014
APPM 2360 Section Exam 3 Wednesday November 9, 7:00pm 8:30pm, 204 ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) lecture section, (4) your instructor s name, and (5)
More informationPhase Space Study of the Synchrotron Oscillation and Radiation Damping of the Longitudinal and Transverse Oscillations
ScienceAsia 28 (2002 : 393-400 Phase Space Study of the Synchrotron Oscillation and Radiation Damping of the Longitudinal and Transverse Oscillations Balabhadrapatruni Harita*, Masumi Sugawara, Takehiko
More informationPhysics 663. Particle Physics Phenomenology. April 9, Physics 663, lecture 2 1
Physics 663 Particle Physics Phenomenology April 9, 2002 Physics 663, lecture 2 1 History Two Principles Electrostatic Cockcroft-Walton Accelerators Van de Graaff and tandem Van de Graaff Transformers
More informationIntroduction to Accelerators
Introduction to Accelerators D. Brandt, CERN CAS Platja d Aro 2006 Introduction to Accelerators D. Brandt 1 Why an Introduction? The time where each accelerator sector was working alone in its corner is
More informationHill s equations and. transport matrices
Hill s equations and transport matrices Y. Papaphilippou, N. Catalan Lasheras USPAS, Cornell University, Ithaca, NY 20 th June 1 st July 2005 1 Outline Hill s equations Derivation Harmonic oscillator Transport
More information88 CHAPTER 3. SYMMETRIES
88 CHAPTER 3 SYMMETRIES 31 Linear Algebra Start with a field F (this will be the field of scalars) Definition: A vector space over F is a set V with a vector addition and scalar multiplication ( scalars
More informationNon-Scaling Fixed Field Gradient Accelerator (FFAG) Design for the Proton and Carbon Therapy *
Non-Scaling Fixed Field Gradient Accelerator (FFAG) Design for the Proton and Carbon Therapy * D. Trbojevic 1), E. Keil 2), and A. Sessler 3) 1) Brookhaven National Laboratory, Upton, New York, USA 2)
More informationCalculation of matched beams under space-charge conditions
Calculation of matched beams under space-charge conditions Jürgen Struckmeier j.struckmeier@gsi.de www.gsi.de/~struck Vortrag im Rahmen des Winterseminars Aktuelle Probleme der Beschleuniger- und Plasmaphysik
More informationS2E: Solenoidal Focusing
S2E: Solenoidal Focusing Writing out explicitly the terms of this expansion: The field of an ideal magnetic solenoid is invariant under transverse rotations about it's axis of symmetry (z) can be expanded
More informationS2E: Solenoidal Focusing
S2E: Solenoidal Focusing The field of an ideal magnetic solenoid is invariant under transverse rotations about it's axis of symmetry (z) can be expanded in terms of the on axis field as as: solenoid.png
More informationAccelerator School Transverse Beam Dynamics-2. V. S. Pandit
Accelerator School 8 Transverse Beam Dnamics- V. S. Pandit Equation of Motion Reference orbit is a single laner curve. Diole is used for bending and quadruole for focusing We use coordinates (r, θ, ) Diole
More informationEtienne Forest. From Tracking Code. to Analysis. Generalised Courant-Snyder Theory for Any Accelerator Model. 4 } Springer
Etienne Forest From Tracking Code to Analysis Generalised Courant-Snyder Theory for Any Accelerator Model 4 } Springer Contents 1 Introduction 1 1.1 Dichotomous Approach Derived from Complexity 1 1.2 The
More information12. Acceleration and Normalized Emittance *
12. Acceleration and Normalized Emittance * Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU) US Particle Accelerator School Accelerator
More informationNonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 1: Introduction Examples of nonlinear dynamics in accelerator systems Nonlinear Single-Particle Dynamics in High Energy Accelerators
More informationLecture 1 - Overview of Accelerators I ACCELERATOR PHYSICS MT E. J. N. Wilson
Lecture 1 - Overview of Accelerators I ACCELERATOR PHYSICS MT 2011 E. J. N. Wilson Lecture 1 - E. Wilson 13-Oct 2011 - Slide 1 Links Author s e-mail: ted.wilson@cern.ch Engines of Discovery : http://www.worldscibooks.com/physics/6272.html
More informationLow Emittance Machines
Advanced Accelerator Physics Course Trondheim, Norway, August 2013 Low Emittance Machines Part 3: Vertical Emittance Generation, Calculation, and Tuning Andy Wolski The Cockcroft Institute, and the University
More informationELECTRON DYNAMICS with SYNCHROTRON RADIATION
ELECTRON DYNAMICS with SYNCHROTRON RADIATION Lenny Rivkin École Polythechnique Fédérale de Lausanne (EPFL) and Paul Scherrer Institute (PSI), Switzerland CERN Accelerator School: Introduction to Accelerator
More informationSuppression of Radiation Excitation in Focusing Environment * Abstract
SLAC PUB 7369 December 996 Suppression of Radiation Excitation in Focusing Environment * Zhirong Huang and Ronald D. Ruth Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 Abstract
More informationUE SPM-PHY-S Polarization Optics
UE SPM-PHY-S07-101 Polarization Optics N. Fressengeas Laboratoire Matériaux Optiques, Photonique et Systèmes Unité de Recherche commune à l Université Paul Verlaine Metz et à Supélec Document à télécharger
More informationAccelerator Physics. Accelerator Physics Overview: Outline. 1. Overview
* Overview: Outline 1 Overview 2 Quadrupole and Dipole Fields and the Lorentz Force Equation 3 Dipole Bending and Particle Rigidity 4 Quadrupole Focusing and Transfer Matrices 5 Combined Focusing and Bending
More informationAccelerator Physics. Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU)
* Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU) Exotic Beam Summer School Michigan State University National Super Conducting
More informationXiaobiao Huang Accelerator Physics August 28, The Dipole Passmethod for Accelerator Toolbox
STANFORD SYNCHROTRON RADIATION LABORATORY Accelerator Physics Note CODE SERIAL PAGE 021 8 AUTHOR GROUP DATE/REVISION Xiaobiao Huang Accelerator Physics August 28, 2009 TITLE The Dipole Passmethod for Accelerator
More informationTools of Particle Physics I Accelerators
Tools of Particle Physics I Accelerators W.S. Graves July, 2011 MIT W.S. Graves July, 2011 1.Introduction to Accelerator Physics 2.Three Big Machines Large Hadron Collider (LHC) International Linear Collider
More informationLecture 37: Principal Axes, Translations, and Eulerian Angles
Lecture 37: Principal Axes, Translations, and Eulerian Angles When Can We Find Principal Axes? We can always write down the cubic equation that one must solve to determine the principal moments But if
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Dynamic Systems Spring 2011
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.24: Dynamic Systems Spring 20 Homework 9 Solutions Exercise 2. We can use additive perturbation model with
More informationLecture 38. Almost Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 38. Almost Linear Systems April 20, 2012 Konstantin Zuev (USC) Math 245, Lecture 38 April 20, 2012 1 / 11 Agenda Stability Properties of Linear
More informationTransverse beam dynamics studies from turn-by-turn beam position monitor data in the ALBA storage ring
ACDIV-217-13 October 217 Transverse beam dynamics studies from turn-by-turn beam position monitor data in the ALBA storage ring Michele Carla - ALBA Synchrotron Abstract The purpose of the thesis was testing
More informationMechanics Physics 151
Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential
More informationBernhard Holzer, CERN-LHC
Bernhard Holzer, CERN-LHC * Bernhard Holzer, CERN CAS Prague 2014 x Liouville: in reasonable storage rings area in phase space is constant. A = π*ε=const x ε beam emittance = woozilycity of the particle
More informationPoisson Brackets and Lie Operators
Poisson Brackets and Lie Operators T. Satogata January 22, 2008 1 Symplecticity and Poisson Brackets 1.1 Symplecticity Consider an n-dimensional 2n-dimensional phase space) linear system. Let the canonical
More informationIntroduction to Transverse Beam Optics. II.) Twiss Parameters & Lattice Design
Introduction to Transverse Beam Optics Bernhard Holzer, CERN II.) Twiss Parameters & Lattice esign ( Z X Y) Bunch in a storage ring Introduction to Transverse Beam Optics Bernhard Holzer, CERN... don't
More informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More informationMATH 189, MATHEMATICAL METHODS IN CLASSICAL AND QUANTUM MECHANICS. HOMEWORK 2. DUE WEDNESDAY OCTOBER 8TH,
MATH 189, MATHEMATICAL METHODS IN CLASSICAL AND QUANTUM MECHANICS. HOMEWORK 2. DUE WEDNESDAY OCTOBER 8TH, 2014 HTTP://MATH.BERKELEY.EDU/~LIBLAND/MATH-189/HOMEWORK.HTML 1. Lagrange Multipliers (Easy) Exercise
More informationSpin Feedback System at COSY
Spin Feedback System at COSY 21.7.2016 Nils Hempelmann Outline Electric Dipole Moments Spin Manipulation Feedback System Validation Using Vertical Spin Build-Up Wien Filter Method 21.7.2016 Nils Hempelmann
More informationSmall Synchrotrons. Michael Benedikt. CERN, AB-Department. CAS, Zeegse, 30/05/05 Small Synchrotrons M. Benedikt 1
Small Synchrotrons Michael Benedikt CERN, AB-Department CAS, Zeegse, 30/05/05 Small Synchrotrons M. Benedikt 1 Contents Introduction Synchrotron linac - cyclotron Main elements of the synchrotron Accelerator
More informationarxiv: v1 [physics.acc-ph] 22 Aug 2014
THE SYNCHROTRON MOTION SIMULATOR FOR ADIABATIC CAPTURE STUDY IN THE TLS BOOSTER Cheng-Chin Chiang Taipei Taiwan arxiv:.7v [physics.acc-ph] Aug Abstract The synchrotron motion simulator is invented to simulate
More information04.sup Equations of Motion and Applied Fields *
04.sup Equations of Motion and Applied Fields * Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU) S2: Transverse Particle Equations
More informationSynchrotron Motion. RF cavities. Charged particles gain and lose energy in electric field via
217 NSRRC FEL Longitudinal Motion (SYL) 1 Synchrotron Motion RF cavities Charged particles gain and lose energy in electric field via Δ. For DC accelerators such as the Cockcroft-Walton and Van-der- Graaff
More informationPhysics 610. Adv Particle Physics. April 7, 2014
Physics 610 Adv Particle Physics April 7, 2014 Accelerators History Two Principles Electrostatic Cockcroft-Walton Van de Graaff and tandem Van de Graaff Transformers Cyclotron Betatron Linear Induction
More informationSpace Charge in Linear Machines
Space Charge in Linear Machines Massimo.Ferrario@LNF.INFN.IT Egham September 6 th 017 Relativistic equation of motion dp dt = F p = γm o v γm o dv dt + m ov dγ dt = F β = v c dγ dt = d dt " a v % m o γ
More informationODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class
ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is
More informationStorage Ring Optics Measurement, Model, and Correction. Yiton T. Yan SLAC, 2575 Sand Hill Road, Menlo Park, CA 94025, USA.
SLAC-PUB-12438 April 2007 Storage Ring Optics Measurement, Model, and Correction Yiton T. Yan SLAC, 2575 Sand Hill Road, Menlo Park, CA 94025, USA 1 Introduction To improve the optics of a storage ring,
More informationLinear coupling parameterization in the betatron phase rotation frame
C-A/AP/#165 September 004 Linear coupling parameterization in the betatron phase rotation frame Yun Luo Collider-Accelerator Department Brookhaven National Laboratory Upton, NY 11973 CAD/AP/165 September
More information06.lec Acceleration and Normalized Emittance *
06.lec Acceleration and Normalized Emittance * Prof. Steven M. Lund Physics and Astronomy Department Facility for Rare Isotope Beams (FRIB) Michigan State University (MSU) PHY 905 Lectures Accelerator
More informationPBL (Problem-Based Learning) scenario for Accelerator Physics Mats Lindroos and E. Métral (CERN, Switzerland) Lund University, Sweden, March 19-23,
PBL (Problem-Based Learning) scenario for Accelerator Physics Mats Lindroos and E. Métral (CERN, Switzerland) Lund University, Sweden, March 19-23, 2007 As each working day, since the beginning of the
More information