Introduction to Transverse. Beam Optics. Bernhard Holzer, DESY-HERA

Transcription

1 Introduction to Tranvere Beam Optic Bernhard Holzer, DESY-HERA

2 Larget torage ring: The Solar Sytem atronomical unit: average ditance earth-un 1AE 150 *10 6 km Ditance Pluto-Sun 40 AE AE

3 Luminoity Run of a typical torage ring: HERA Storage Ring: Proton accelerated and tored for 12 hour ditance of particle travelling at about v c L = km... everal time Sun - Pluto and back guide the particle on a well defined orbit ( deign orbit ) focu the particle to keep each ingle particle trajectory within the vacuum chamber of the torage ring, i.e. cloe to the deign orbit.

4 Tranvere Beam Dynamic: 0.) Introduction and Baic Idea... in the end and after all it hould be a kind of circular machine need tranvere deflecting force r r r r Lorentz force F = q*( E + v B) typical velocity in high energy machine: v 8 c 3*10 m old greek dictum of widom: if you are clever, you ue magnetic field in an accelerator wherever it i poible. But remember: magn. field act allway perpendicular to the velocity of the particle only bending force, no beam acceleration

5 The ideal circular orbit ẑ θ ρ circular coordinate ytem condition for circular orbit: Lorentz force F = e* v* B L centrifugal force F Zentr = γ m0 v ρ 2 p = B* ρ e 2 γ mv 0 = e* v* B ρ

6 I.) The Magnetic Guide Field Dipole Magnet: define the ideal orbit homogeneou field created by two flat pole hoe court. K. Wille ρ α Magnetic field of a dipole magnet: B0 = μ 0 ni d h Normalie to momentum:... remember p/ e = B*ρ field map of a torage ring dipole magnet B0 [T ] e B0 m 1 = = ρ p p[gev / c] 1 radiu of curvature, bending trength

7 Quadrupole Magnet: required: focuing force to keep trajectorie in vicinity of the ideal orbit linear increaing Lorentz force linear increaing magnetic field B = g x B = g z z x at the location of the particle trajectory: no iron, no current r r r r B = 0 B= V themagneticfieldcanbe expreed a gradient of a calar potential! V( x, z)= g xz equipotential line (i.e. the urface of the iron contour) = hyperbola

8 Calculation of the Quadrupole Field: r r r r Hd = jda = N * I A B ( r) = g * r gradient of a quadrupole field: 2μ0nI g = 2 r normalied quadrupole trength: g k = p / e Separate Function Machine: Split the magnet and optimie them according to their job: bending, focuing etc Example: heavy ion torage ring TSR

9 II.) The equation of motion: Linear approximation: * ideal particle deign orbit * any other particle coordinate x, z mall quantitie x,z << ρ magnetic guide field: only linear term in x & z of B have to be taken into account Taylor Expanion of the B field: 2 3 dbz 1 d Bz 2 1 d Bz 3 z x Bz 0 x x x 2 3 B ( ) = dx 2! dx 3! dx what about the vertical plane: r r r r E Maxwell: B= j + = 0 t Bx B = z x z B (, x, z) x Bx = z z

10 Equation of Motion: ẑ Conider local egment of a particle trajectory... and remember the old day: (Goldtein page 27) θ ρ x z radial acceleration: a r 2 2 d ρ dθ d ρ = ρ 2 Ideal orbit: ρ = cont, = 0 dt dt dt Force: dθ 2 F = mρ = mρω dt 2 general trajectory: 2 / F = mv ρ 2 2 = d ( + ) mv F m x ρ = eb 2 zv dt x + ρ

11 develop for mall x: x << ρ m 2 2 d x mv x (1 ) = eb 2 zv dt ρ ρ guide field in linear approx. B = B + x z 0 B x z 2 2 d x mv x m (1 ) = ev 2 B0 + x dt ρ ρ B z x independent variable: t dx = * dt dx d d dt x = dx d 1 x eb (1 ) 0 exg x ρ ρ = mv + mv 1 x 1 x + = + kx 2 ρ ρ ρ 1 x + x( k) = 0 2 ρ

12 Remark: * 1 x + ( k) x= 0 2 ρ there eem to be a focuing even without a quadrupole gradient weak focuing of dipole magnet Ma pectrometer: particle are eparated according to their energy and focued due to the 1/ρ effect of the dipole * Equation for the vertical motion: z + k z =0

13 III.) Solution of Trajectory Equation Define hor. plane: vert. Plane: K =1 ρ2 k K =k y + K * y = 0 K = cont within a magnet Differential Equation of harmonic ocillator with pring contant K Anatz: x ( ) = a1 co(ω t ) + a 2 in(ω t ) general olution: linear combination of two independent olution

14 Hor. Focuing Quadrupole K > 0: 1 x( ) = x0 co( K ) + x0 in( K ) K x ( ) = x K in( K ) + x co( K ) 0 0 x(0) = x 0 x (0) = x 0 tarting condition For convenience expreed in matrix formalim: x x = M foc x x 0 M foc 1 co( K ) in( K = K in( ) co( ) K K K 0

15 hor. defocuing quadrupole: K < 0 M defoc 1 coh Kl inh Kl = K inh coh K K l K l drift pace: K = 0 M drift 1 l = 0 1! with the aumption made, the motion in the horizontal and vertical plane are independent... the particle motion in x & z i uncoupled

16 Thin Len Approximation: matrix of a quadrupole len M 1 co kl in kl = k in co k k l k l in many practical cae we have the ituation: f 1 = >> l q kl q... focal length of the len i much bigger than the length of the magnet lime: l 0 while keeping kl = cont M x 1 0 = 1 1 f M z 1 0 = 1 1 f... uefull for fat (and in large machine till quite accurate) back on the envelope calculation... and for the guided tudie!

17 Tranformation through a ytem of lattice element combine the ingle element olution by multiplication of the matrice M M M M M M x x total = QF * D * QD * Bend * D*... = M (, x )* C = S x * x C S x 1 1 focuing len dipole magnet defocuing len court. K. Wille C and S = in- and co- like trajectorie of the lattice tructure, in other word the two independent olution of the homogeneou equation of motion x() 0 typical value in a trong foc. machine: x mm, x mrad

18 IV.) Orbit & Tune: Tune: number of ocillation per turn Relevant for beam tability: non integer part HERA revolution frequency: 47.3 khz 0.292*47.3 khz = khz

19 Quetion: what will happen, if the particle perform a econd turn? x... or a third one or turn 0

20 19th century: Ludwig van Beethoven: Mondchein Sonate Sonate Nr. 14 in ci-moll (op. 27/II, 1801)

21 Atronomer Hill: differential equation for motion with periodic focuing propertie Hill equation Example: particle motion with periodic coefficient equation of motion: x () k()() x = 0 retoring force cont, we expect a kind of quai harmonic k() = depending on the poition ocillation: amplitude & phae will depend k(+l) = k(), periodic function on the poition in the ring.

22 V.) The Beta Function General olution of Hill equation: (i) x () = ε β()co( ψ() + φ) ε, Φ = integration contant determined by initial condition β() periodic function given by focuing propertie of the lattice quadrupole β( + L) = β( ) Inerting (i) into the equation of motion ψ () = 0 d β() Ψ() = phae advance of the ocillation between point 0 and in the lattice. For one complete revolution: number of ocillation per turn Tune Q y 1 d 2 π β( ) = o

23 VI.) Beam Emittance and Phae Space Ellipe general olution of Hill equation (1) x ( ) = ε * β( ) *co( ψ( ) + φ) ε (2) x ( ) = * ( )*co( ( ) + ) + in( ( ) + ) β() { α ψ φ ψ φ } from (1) we get x () co( ψ( ) + φ) = ε * β( ) Inert into (2) and olve for ε 1 α() = β () α( ) γ () = β() 2 ε = γ()* x() + 2 α()() xx () + β() x () 2 2 * ε i a contant of the motion it i independent of * parametric repreentation of an ellipe in the x x pace * hape and orientation of ellipe are given by α, β, γ

24 Beam Emittance and Phae Space Ellipe ε = γ()* x() + 2 α()() xx () + β() x () 2 2 x α ε γ Liouville: in reaonable torage ring area in phae pace i contant. εγ α ε β A = π*ε=cont εβ x x() ε beam emittance = woozilycity of the particle enemble, intrinic beam parameter, cannot be changed by the foc. propertie. Scientifiquely poken: area covered in tranvere x, x phae pace and it i contant!!!

25 Enemble of many (...all) poible particle trajectorie x () = ε * β()*co( ψ() + φ) max. amplitude of all particle trajectorie x () = ε * β() Beam Dimenion: determined by two parameter HERA beam ize σ = ε * β Example: tranvere beam profile meaured uing a wirecanner

26 VII.) Tranfer Matrix M ye we had the topic already general olution of Hill equation { } x () = ε β()co ψ() + φ ε x () = ()co{ () + } + in { () + } () α ψ φ ψ φ β remember the trigonometrical gymnatic: in(a+b)= etc ( ) x () = ε β coψ coφ inψ inφ ε x ( ) = [ α co ψ co φ α in ψ in φ + in ψ co φ + co ψ in φ ] β tarting at point (0) = 0, where we put Ψ(0) = 0 0 co φ = x, εβ0 1 α0x0 in φ = ( x0 β0 + ) ε β 0 inerting above

27 β x () = co + 0in x0+ 0 in x0 β 0 { ψ α ψ} { ββ ψ} 1 β0 x ( ) = {( α α) co ψ (1 + α α)inψ} x + { coψ αinψ} x ββ β which can be expreed... for convenience... in matrix form x x = M x x 0 M β ( coψ + α0inψ) ββ0 inψ β0 = ( α0 α)co ψ (1 + α0α)inψ β0 ( coψ in ) α ψ ββ β 0 * we can calculate the ingle particle trajectorie between two location in the ring, if we know the αβγat thee poition. * and nothing but the α βγat thee poition. *!

28 Stability Criterion: Quetion: what will happen, if we do not make too many mitake and your particle perform one complete turn? Matrix for 1 turn: co ψturn + α in ψturn β in ψturn M= γ inψturn co ψturn α inψturn 1 0 α β = coψ + in ψ 0 1 γ α Matrix for N turn: N ( ) M = 1co ψ + J inψ = 1co Nψ+ J innψ N 1 J The motion for N turn remain bounded, if the element of M N remain bounded ψ = real co ψ < 1 Trace( M) < 2

29 VIII.) Tranformation of α, β, γ conider two poition in the torage ring:, x x 0 = M x x 0 ince ε = cont: ε = βx + 2αxx + γx 2 2 ẑ ε = β + α + γ 2 2 0x0 2 0x0x0 0x0 expre x 0, x 0 a a function of x, x.... remember W = CS -SC = 1 θ ρ 0 x x 0 = M 1 C S M = C S x * x M 1 S = C S C x0 = Sx Sx x = Cx + Cx 0 inerting into ε ε = βx + 2αxx + γx 2 2 ε = β + α + γ ort via x, x and compare the coefficient to get ( Cx C x) 2 0( S x Sx )( Cx C x) 0( S x Sx )

30 in matrix notation: β () = C β 2SCα + S γ α() = CC β + ( SC + SC ) α SS γ γ () = C β 2SC α + S γ β C 2SC S β0 α = CC SC + CS SS α γ C SC S γ 0! 1.) thi expreion i important 2.) given the twi parameter α, β, γ at any point in the lattice we can tranform them and calculate their value at any other point in the ring. 3.) the tranfer matrix i given by the focuing propertie of the lattice element, the element of M are jut thoe that we ued to calculate ingle particle trajectorie. 4.) go back to point 1.)

31 IX.) Réumé Réumé: beam rigidity: B ρ = p q bending trength of a dipole: focuing trength of a quadrupole: 1 m = ρ B ( T ) pgev ( / c) g k m = p( GeV / c) k m = μ ni pgev ( / c) ar focal length of a quadrupole: f 1 = k l q equation of motion: x + Kx= 1 Δp ρ p matrix of a foc. quadrupole: x2 = M x1 M 1 co Kl in Kl = K in co K Kl Kl, M 1 0 = 1 1 f

32 1.) Aterix der Gallier, Uderzo & Gozini, Verlag Egmont EHAPA / Ed. Albert-René 2.) Edmund Wilon: Introd. to Particle Accelerator Oxford Pre, ) Klau Wille: Phyic of Particle Accelerator and Synchrotron Radiation Facilictie, Teubner, Stuttgart ) Peter Schmüer: Baic Coure on Accelerator Optic, CERN Acc. School: 5 th general acc. phy. coure CERN ) Bernhard Holzer: Lattice Deign, CERN Acc. School: Interm.Acc.phy coure, 6.) Herni Bruck: Accelerateur Circulaire de Particule, pree Univeritaire de France, Pari 1966 (englih / francai) 7.) M.S. Livington, J.P. Blewett: Particle Accelerator, Mc Graw-Hill, New York, ) Frank Hinterberger: Phyik der Teilchenbechleuniger, Springer Verlag ) Mathew Sand: The Phyic of e+ e- Storage Ring, SLAC report 121, ) D. Edward, M. Sypher : An Introduction to the Phyic of Particle Accelerator, SSC Lab 1990