Bernhard Holzer, DESY-HERA-PETRA III / CERN-LHC

Transcription

1 Introduction to Tranvere Beam Dynamic Bernhard Holzer, DESY-HERA-PETRA III / CERN-LHC The Ideal World I. Magnetic Field and Particle Trajectorie *

2 Larget torage ring: The Solar Sytem atronomical unit: average ditance earth-un AE 5 * 6 km Ditance Pluto-Sun 4 AE AE

3 Luminoity Run of a typical torage ring: HERA Storage Ring: Proton accelerated and tored for hour ditance of particle travelling at about v c L - km... everal time Sun - Pluto and back guide the particle on a well defined d 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:. Introduction and Baic Idea... in the end and after all it hould be a kind of circular machine need tranvere deflecting force Lorentz force F r r r q E + v B typical velocity in high energy machine: v c 8 3* 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 y θ circular coordinate ytem condition for circular orbit: Lorentz force F L e v B centrifugal force γ m F v centr p e B γ m v e v B

6 . The Magnetic Guide Field Dipole Magnet: define the ideal orbit homogeneou field created by two flat pole hoe μ n B h I Normalie magnetic field to momentum: convenient unit: p e B e B p B V m [ T ] p GeV c Eample LHC: B 8. 3T GeV p 7 c 8.3 V 8.3 3* e m 9 7* ev 7* c m 8 9 m m

7 The Magnetic Guide Field α d field map of a torage ring dipole magnet.53 km π 7.6 km 66% B... 8 T rule of thumb:.3 p B[ T] [ GeV / c] normalied bending trength

8 . Quadrupole Magnet: required: focuing force to keep trajectorie in vicinity of the ideal orbit linear increaing Lorentz force linear increaing magnetic field By g B g y normalied quadrupole field: μ ni gradient of a quadrupole magnet: g r k g p / e imple rule: k.3 g T / m p GeV / c LHC main quadrupole magnet g 5... T / m what about the vertical plane:... Mawell r r r r E B y B B j + t y

9 3. The equation of motion: Linear approimation: * ideal particle deign orbit * any other particle coordinate, y mall quantitie i,y << magnetic guide field: only linear term in & y of B have to be taken into account Taylor Epanion of the B field: B db d B eg! d 3! d y y normalie to momentum y B y p/e B d B B g * eg eg p / e B p / e! p / e 3! p / e +...

10 The Equation of Motion: B p / e 3 + k + m + n! 3! +... only term linear in, y taken into account dipole field quadrupole field Separate Function Machine: Split the magnet and optimie them according to their job: bending, focuing etc Eample: heavy ion torage ring TSR * man ieht nur dipole und quad linear

11 Equation of Motion: ŷ Conider local egment of a particle trajectory... and remember the old day: Goldtein page 7 θ y radial acceleration: a r d dθ d Ideal orbit: cont, dt dt dt Force: dθ F m mω dt general trajectory: + / F mv F m d dt + mv + e B y v

12 F m d dt mv + e By v ŷ + d d + a cont dt dt y remember: mm, m develop for mall + f Taylor Epanion f + f + f +!!. m d dt mv eb y v

13 guide field in linear appro. B y B d mv B B y y B + m ev B + dt : m d dt v e v B m + e v m g independent variable: t d dt d d d dt d dt d dt d d d dt d d d d d dt d dt d dt v + d d dv d v v v e v B v + m e v m g : v

14 e B + mv e g mv m v p B + p / e + g p / e + + k + k normalize to momentum of particle B p / e g k p / e * Equation for the vertical motion: no dipole in general y k k quadrupole field change ign y + k y

15 Remark: + k there eem to be a focuing even without t * a quadrupole gradient weak focuing of dipole magnet k even without quadrupole there i a retriving force i.e. focuing in the bending plane of the dipole magnet in large machine it i weak.! Ma pectrometer: particle are eparated according to their energy and focued due to the / effect of the dipole

16 * Hard Edge Model: + k + k thi equation i not correct!!! bending and focuing field are function of the independent variable! Inide a magnet we aume contant focuing propertie! cont k cont B l eff l mag B d

17 4. Solution of Trajectory Equation Define hor. plane: vert. Plane: K K k k + K Differential Equation of harmonic ocillator with pring contant K Anatz: a co ω + a in ω general olution: linear combination of two independent olution a ω in ω + a ω co ω a ω K ω co ω aω in ω ω general olution: a co K + a in K

18 determine a, a by boundary condition: a, a, K Hor. Focuing Quadrupole K > : co K + in K K K in K + co K For convenience epreed in matri formalim: M foc * M foc co K in K K in co K K K

19 hor. defocuing quadrupole: K Remember from chool: f coh, f inh Anatz: a coh ω + a inh ω M defoc coh Kl inh Kl K inh coh K K l K l drift pace: K M drift l! with the aumption made, the motion in the horizontal and vertical plane are independent... the particle motion in & y i uncoupled

20 Thin Len Approimation: matri of a quadrupole len M co kl in kl k in co k k l k l in many practical cae we have the ituation: f >> l q kl q... focal length of the len i much bigger than the length of the magnet lime: l q while keeping k l q cont M f M z f... ueful for fat and in large machine till quite accurate back on the envelope calculation... and for the guided tudie!

21 Tranformation through a ytem of lattice element combine the ingle element olution by multiplication of the matrice M M M M M M total QF * D * QD * Bend * D*...,* M focuing len dipole magnet defocuing len court. K. Wille typical value in a trong foc. machine: mm, mrad

22 5. Orbit & Tune: Tune: number of ocillation per turn Relevant for beam tability: non integer part HERA revolution frequency: 47.3 khz.9*47.3 khz 3.8 khz

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

24 9th century: Ludwig van Beethoven: Mondchein Sonate Sonate Nr. 4 in ci-moll op. 7/II, 8

25 Atronomer Hill: differential equation for motion with periodic focuing propertie Hill equation Eample: particle motion with periodic coefficient equation of motion: k retoring force cont, we epect 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.

26 6. The Beta Function General olution of Hill equation: i ε βco 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 d β Ψ phae advance of the ocillation between point and in the lattice. For one complete revolution: number of ocillation per turn Tune Q y π d β

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

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

29 Particle Tracking in a Storage Ring Calculate, for each linear accelerator element according to matri formalim plot, a a function of

30 and now the ellipe: note for each turn, at a given poition and plot in the phae pace diagram

31 Réumé: p q beam rigidity: B p bending trength of a dipole: focuing trength of a quadrupole:.998 B T m p GeV / c.998 g k m p GeV / c focal length of a quadrupole: f k l q equation of motion: + K Δ p p matri of a foc. quadrupole: M M co Kl in Kl K in co K Kl Kl, M f

32 6. Bibliography:. Edmund Wilon: Introd. to Particle Accelerator Oford 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, 5. Herni Bruck: Accelerateur Circulaire de Particule, pree Univeritaire de France, Pari 966 englih / francai 6. M.S. Livington, J.P. Blewett: Particle Accelerator, Mc Graw-Hill, New York,96 7. Frank Hinterberger: Phyik der Teilchenbechleuniger, Springer Verlag Mathew Sand: The Phyic of e+ e- Storage Ring, SLAC report, D. Edward, M. Sypher : An Introduction to the Phyic of Particle Accelerator, SSC Lab 99