Presentation on theme: "Introduction to Accelerator Physics"— Presentation transcript:
1 Introduction to Accelerator Physics Beam Dynamics for „Summer Students“Bernhard Holzer,CERN-LHCThe Ideal WorldIP5I.) Magnetic Fields and Particle TrajectoriesIP8IP2IP1*
2 Luminosity Run of a typical storage ring: LHC Storage Ring: Protons accelerated and stored for 12 hoursdistance of particles travelling at about v ≈ cL = km... several times Sun - Pluto and backintensity (1011)3 -2 -1 -guide the particles on a well defined orbit („design orbit“)focus the particles to keep each single particle trajectorywithin the vacuum chamber of the storage ring, i.e. close to the design orbit.
3 typical velocity in high energy machines: 1.) Introduction and Basic Ideas„ ... in the end and after all it should be a kind of circular machine“ need transverse deflecting forceLorentz forcetypical velocity in high energy machines:Example:technical limit for el. field:equivalent el. field ...
4 The ideal circular orbit old greek dictum of wisdom:if you are clever, you use magnetic fields in an accelerator whereverit is possible.The ideal circular orbitρθ●scircular coordinate systemcondition for circular orbit:Lorentz forcecentrifugal forceB ρ = "beam rigidity"
5 2.) The Magnetic Guide Field Dipole Magnets:define the ideal orbithomogeneous field createdby two flat pole shoesNormalise magnetic field to momentum:convenient units:Example LHC:
6 The Magnetic Guide Field ραdsfield map of a storage ring dipole magnet2πρ = 17.6 km≈ 66%rule of thumb:„normalised bending strength“
7 2.) Focusing Properties – Transverse Beam Optics classical mechanics:pendulumthere is a restoring force, proportionalto the elongation x:general solution: free harmonic oszillationStorage Ring: we need a Lorentz force that rises as a function of the distance to ?the design orbit
8 Quadrupole Magnets:required: focusing forces to keep trajectories in vicinity of the ideal orbitlinear increasing Lorentz forcelinear increasing magnetic fieldnormalised quadrupole field:simple rule:LHC main quadrupole magnetwhat about the vertical plane:... Maxwell
9 Focusing forces and particle trajectories: normalise magnet fields to momentum(remember: B*ρ = p / q )Dipole MagnetQuadrupole Magnet
10 * man sieht nur 3.) The Equation of Motion: only terms linear in x, y taken into account dipole fieldsquadrupole fieldsSeparate Function Machines:Split the magnets and optimisethem according to their job:bending, focusing etcExample:heavy ion storage ring TSR* man sieht nurdipole und quads linear
11 * * The Equation of Motion: ● ρ y θ ● x s Equation for the horizontal motion:ρyθ●xs*Equation for the vertical motion:yno dipoles … in general …quadrupole field changes signx
12 4.) Solution of Trajectory Equations Define … hor. plane:… vert. Plane:Differential Equation of harmonic oscillator … with spring constant KAnsatz:Hor. Focusing Quadrupole K > 0:For convenience expressed in matrix formalism:
13 hor. defocusing quadrupole: s = s1hor. defocusing quadrupole:Ansatz: Remember from schooldrift space:K = 0! with the assumptions made, the motion in the horizontal and vertical planes areindependent „ ... the particle motion in x & y is uncoupled“
14 Transformation through a system of lattice elements combine the single element solutions by multiplication of the matricesfocusing lensdipole magnetdefocusing lenscourt. K. Willein each accelerator element the particle trajectory corresponds to the movement of aharmonic oscillator „x(s)typical valuesin a strongfoc. machine:x ≈ mm, x´ ≤ mrads
15 5.) Orbit & Tune: 64.31 59.32 Relevant for beam stability: Tune: number of oscillations per turn64.3159.32Relevant for beam stability:non integer partLHC revolution frequency: kHz
16 LHC Operation: Beam Commissioning First turn steering "by sector:"One beam at the timeBeam through 1 sector (1/8 ring), correct trajectory, open collimator and move on.16
17 Question: what will happen, if the particle performs a second turn ? ... or a third one or turnsxs
18 Particle Trajectories, Beams & Bunches II.) The Ideal World:Particle Trajectories, Beams & BunchesBunch in a Storage Ring
19 Astronomer Hill:differential equation for motions with periodic focusing properties„Hill‘s equation“Example: particle motion withperiodic coefficientequation of motion:restoring force ≠ const, we expect a kind of quasi harmonick(s) = depending on the position s oscillation: amplitude & phase will dependk(s+L) = k(s), periodic function on the position s in the ring.
20 6.) The Beta Function „it is convenient to see“ o ... after some beer general solution of Mr Hill can be written in the form:Ansatz:ε, Φ = integration constantsdetermined by initial conditionsβ(s) periodic function given by focusing properties of the lattice ↔ quadrupolesε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter,cannot be changed by the foc. properties.scientifiquely spoken: area covered in transverse x, x´ phase space … and it isconstant !!!Ψ(s) = „phase advance“ of the oscillation between point „0“ and „s“ in the lattice.For one complete revolution: number of oscillations per turn „Tune“o
21 7.) Beam Emittance and Phase Space Ellipse x´Liouville: in reasonable storage ringsarea in phase space is constant.A = π*ε=const●●●●x●●x(s)sε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter,cannot be changed by the foc. properties.Scientifiquely spoken: area covered in transverse x, x´ phase space … and it is constant !!!
22 ● Particle Tracking in a Storage Ring Calculate x, x´ for each linear acceleratorelement according to matrix formalismplot x, x´as a function of „s“●
23 note for each turn x, x´at a given position „s1“ and plot in the … and now the ellipse:note for each turn x, x´at a given position „s1“ and plot in thephase space diagram
25 Emittance of the Particle Ensemble: Particle Distribution: GaußParticle Distribution:particle at distance 1 σ from centre↔ 68.3 % of all beam particlessingle particle trajectories, N ≈ per bunchLHC:aperture requirements: r 0 = 12 * σ
26 III.) The „not so ideal“ World Lattice Design in Particle Accelerators 1952: Courant, Livingston, Snyder:Theory of strong focusing in particle beams
27 Recapitulation: ...the story with the matrices !!! Equation of Motion:Solution of Trajectory Equations… hor. plane:… vert. Plane:
28 8.) Lattice Design: „… how to build a storage ring“ Geometry of the ring:p = momentum of the particle,ρ = curvature radiusBρ= beam rigidityCircular Orbit: bending angle of one dipoleThe angle run out in one revolutionmust be 2π, so for a full circle… defines the integrated dipole field around the machine.
29 Example LHC:7000 GeV Proton storage ringdipole magnets N = 1232l = 15 mq = +1 e
30 FoDo-LatticeA magnet structure consisting of focusing and defocusing quadrupole lenses inalternating order with nothing in between.(Nothing = elements that can be neglected on first sight: drift, bending magnets,RF structures ... and especially experiments...)Starting point for the calculation: in the middle of a focusing quadrupolePhase advance per cell μ = 45°, calculate the twiss parameters for a periodic solution
32 β-Function in a Drift: * l l β0 At the end of a long symmetric drift space the beta function reaches its maximum value in the complete lattice.-> here we get the largest beam dimension.-> keep l as small as possible7 sima beam size iside a mini beta quadrupole
33 ... unfortunately ... in general high energy detectors that are installed in that drift spacesare a little bit bigger than a few centimeters ...... clearly there is another problem !!!Example: Luminosity optics at LHC: β* = 55 cmfor smallest βmax we have to limit the overall lengthand keep the distance “s” as small as possible.
34 The Mini-β Insertion: production rate of events is determined by the cross section Σreactand a parameter L that is givenby the design of the accelerator:… the luminosity
35 10.) Luminosity p2-Bunch p1-Bunch ± σ IP 10 11 particlesp1-BunchIP± σ10 11 particlesExample: Luminosity run at LHC
36 Mini-β Insertions: Betafunctions A mini-β insertion is always a kind of special symmetric drift space.greetings from Liouvillex´the smaller the beam sizethe larger the bam divergence●●●●x●●
37 Mini-β Insertions: some guide lines * calculate the periodic solution in the arc* introduce the drift space needed for the insertion device (detector ...)* put a quadrupole doublet (triplet ?) as close as possible* introduce additional quadrupole lenses to match the beam parametersto the values at the beginning of the arc structureparameters to be optimised & matched to the periodic solution:8 individuallypowered quadmagnets areneeded to matchthe insertion( ... at least)
38 IV) … let´s talk about acceleration Electrostatic Machines(Tandem –) van de Graaff Acceleratorcreating high voltages by mechanicaltransport of charges* Terminal Potential: U ≈ MVusing high pressure gas to suppress discharge ( SF6 )Problems: * Particle energy limited by high voltage discharges* high voltage can only be applied once per particle ...... or twice ?
39 * The „Tandem principle“: Apply the accelerating voltage twice ... ... by working with negative ions (e.g. H-) andstripping the electrons in the centre of the structureExample for such a „steam engine“: 12 MV-Tandem van de GraaffAccelerator at MPI Heidelberg
40 12.) Linear Accelerator + -̶ + -̶ + -̶ + 1928, Wideroe Energy Gain per „Gap“:drift tube structure at a proton linac(GSI Unilac)500 MHz cavities in an electron storage ring* RF Acceleration: multiple application ofthe same acceleration voltage;brillant idea to gain higher energies
41 13.) The Acceleration z Where is the acceleration? Install an RF accelerating structure in the ring:czB. SalvantN. Biancacci
42 14.) The Acceleration for Δp/p≠0 “Phase Focusing” below transitionideal particle particle with Δp/p > 0 fasterparticle with Δp/p < 0 slowerFocussing effect in thelongitudinal directionkeeping the particlesclose together... forming a “bunch”oscillation frequency:≈ some Hz
43 ... so sorry, here we need help from Albert: v/c... some when the particlesdo not get faster anymore.... but heavier !kinetic energy of a proton
44 15.) The Acceleration for Δp/p≠0 “Phase Focusing” above transition ideal particle particle with Δp/p > 0 heavierparticle with Δp/p < 0 lighterFocussing effect in the longitudinal directionkeeping the particles close together ... forming a “bunch”... and how do we accelerate now ???with the dipole magnets !
46 ~ 200 turns LHC Commissioning: RF RF off RF on, phase optimisation a proton bunch: focused longitudinal bythe RF field~ 200 turnsRF offBunch length ~ 1.5 ns ~ 45 cmRF on,phase optimisationRF on, phase adjusted,beam captured
47 Problem: panta rhei !!! Bunch length of Electrons ≈ 1cm U0 t (Heraklit: v. Chr.)just a stupid (and nearly wrong) example)Bunch length of Electrons ≈ 1cmU0ttypical momentum spread of an electron bunch:
48 17.) Dispersion and Chromaticity: Magnet Errors for Δp/p ≠ 0 Influence of external fields on the beam: prop. to magn. field & prop. zu 1/pdipole magnetfocusing lensparticle having ...to high energyto low energyideal energy
49 Dispersion Matrix formalism: Example: homogeneous dipole field xβ Closed orbit for Δp/p > 0xβρ.Matrix formalism:
50 ! or expressed as 3x3 matrix Example Amplitude of Orbit oscillation contribution due to Dispersion ≈ beam size Dispersion must vanish at the collision point!Calculate D, D´: ... takes a couple of sunny Sunday evenings !
51 V.) Are there Any Problems ??? sure there are Some Golden Rules to Avoid Trouble
52 I.) Golden Rule number one: do not focus the beam ! Problem: ResonancesAssume: Tune = integerInteger tunes lead to a resonant increaseof the closed orbit amplitude in presence ofthe smallest dipole field error.Qualitatively spoken:
53 m*Qx+n*Qy+l*Qs = integer Tune and Resonancesm*Qx+n*Qy+l*Qs = integerTune diagram up to 3rd orderQy =1.5… and up to 7th orderQy =1.3Homework for the operateurs:find a nice place for the tunewhere against all probabilitythe beam will surviveQy =1.0Qx =1.0Qx =1.3Qx =1.5
54 II.) Golden Rule number two: Never accelerate charged particles ! FdefTransport line with quadrupolesTransport line with quadrupoles and space chargeKSCIn a transport line, the focusing by quadrupoles is counteracted by space charge, making focusing weaker
55 Golden Rule number two: Never accelerate charged particles ! Tune Shift due to Space Charge EffectProblem at low energiesv/c... at low speed the particlesrepel each otherLinac 2 Ekin=60 MeVLinac 4 Ekin=150 MeVEkin of a proton
56 III.) Golden Rule number three: Never Collide the Beams !25 nsthe colliding bunches influence each other change the focusing properties of the ring !!most simple case:linear beam beam tune shiftQxand again the resonances !!!Courtesy W. HerrQx
57 IV.) Golden Rule Number four: Never use Magnets “effective magnetic length”
58 Clearly there is another problem ... ... if it were easy everybody could do itAgain: the phase space ellipsefor each turn write down – at a givenposition „s“ in the ring – thesingle partilce amplitude xand the angle x´... and plot it.●A beam of 4 particles– each having a slightly different emittance:
59 Installation of a weak ( !!! ) sextupole magnet The good news: sextupole fields in acceleratorscannot be treated analytically anymore. no equatiuons; instead: Computer simulation„ particle tracking “●
60 Catastrophy ! ● Effect of a strong ( !!! ) Sextupole … „dynamic aperture“
61 Golden Rule XXL: COURAGE and with a lot of effort from Bachelor / Master / Diploma / PhDand Summer-Students the machine is running !!!thank’x for your help and have a lot of fun