Presentation on theme: "Olivier Napoly CEA-Saclay, France Final Version 17 September 2009"— Presentation transcript:
1 Olivier Napoly CEA-Saclay, France Final Version 17 September 2009 Fourth International Accelerator School for Linear Colliders Beijing, China September 2009Lecture A3Beam Delivery System and Beam-Beam EffectsPart 1Olivier NapolyCEA-Saclay, FranceFinal Version17 September 2009
2 Collider Parameters, Physical Constants and Notations E , energy and p , momentumB , magnetic rigidity (B = p/e)L , luminosityQ , bunch chargeN , number of particles in the bunch (N = Q/e)nb , number of bunches in the trainfrep , pulse repetition ratex and y , horizontal and vertical emittancessx* and sy*, rms horizontal and vertical beam sizes at the IPsz , bunch length,, , Twiss parameters , tune , phase advance,c = m/se = Cmec2 = eVµ0 = 4 10-7 N A-2 permeability0 = 1/µ0c permittivityre = mwith
3 Beam Delivery System BDS ILC The Beam delivery system is the final part of the linear collider which transports the high energy beam from the high energy linac to the collision point (Interaction Point = IP).ILCBDSMost Important Functions:Final Focus: Focus the beams at the interaction point to achieve very small beam sizes.Collimation: Remove any large amplitude particles (beam halo).Tuning: Ensure that the small beams collide optimally at the IP.Matching: Precise beam emittance measurement and coupling correction.Diagnostics: Measure the key physics parameters such as energy and polarization.Extraction: Safely extract the beams after collision to the high-power beam dumps.
5 Contents: Four (+ Two) Outstanding Questions ? Part 1Q1: do we need a Final Focus System ?Luminosity, Emittance,Q2: do we need High Field Quadrupole Magnets ?Quadrupole Magnets, Multipoles, Superconducting Quadrupoles,Part 2Q4: do we need Flat Beams ?Beam-beam forces, Beamstrahlung, e+e- Pairs, Fast FeedbackQ3: do we need Corrections Systems ?Beam Optics, Achromat, Emittance GrowthPart 3 (if time permits)Q5 : do we need a Crossing Angle ?Beam extraction, Kink Instability, Crab-crossingQ6: do we need Collimation ?Synchrotron radiation, SR collimation, Beam collimation
6 Contents: Fundamentals in Beam Physics Part 1LuminosityEmittanceMagnetismPart 2Beam-Beam EffectBeam OpticsSynchrotron RadiationPart 3Collimation
8 This is ½ of a Linear Collider: luminosity L ~1034 cm-2s-1 Collider SystematicThis is ½ of a Linear Collider: luminosity L ~1034 cm-2s-1Collisions of beams directly from RF Linac luminosity L ~ cm-2s-1ElectronsIPXPositrons
9 Collisions of Linac Beams IPXElectronsPositronsLuminosity:Typical electron Linacs are producing:Q = 1 nC bunch charge (N = 6109) at xy =1 µm2 normalized transverse emittances, transported through a FODO lattice with L = 38 m cell length, leading to xy 382 m2 .Assuming nb = 2625 bunches at frep = 5 Hz and E = 250 GeV per beam L = cm-2s-1 luminositygenerated by x*y* = 80 µm2 round beam sizes at the IP.
10 Collisions of Linac Beams from Dedicated Sources IPXElectronsPositronsLuminosity:Adding powerful electron and positron sources withQ = 3.2 nC bunch charge (N = 21010) L = 5 1029 cm-2s-1 luminosityassuming xy =1 µm2 normalized emittances (not realistic for 3.2 nC, especially for the positron source !!)
11 Linac Beams from Dedicated Sources and Damping Rings IPXElectronsSymetric layoutfor PositronsAdding Damping Rings to reduce the normalized transverse emittances of the high charge electron and positron beams, to xy =0.4 µm2 L = 8.5 1029 cm-2s-1 luminositygenerated by x*y* = 50 µm2 flat beam sizes at the IP
12 De-magnification from Final Focus Systems IP waistβ*l*ElectronsXFinal Focus Systems are implementing the necessary ~104 demagnification factors to reduce the x*y* = 50 µm2 flat beam sizes to µm2 L ~ 1034 cm-2s-1 luminosityThe final focusing lens is implemented via a doublet (or triplet) of strong Quadrupole Magnets located at distance l* from the IP, typically 3 to 4 m.
13 Depth of Focus and Chromaticity l*IP waistβ*ElectronsXThe Beam Waist is realized over a finite region of longitudinal space called the Depth of Focus (like for a camera), measured by the β* parameter.The collision with the opposite beam is optimum only within this region.Like all magnets (cf. spectrometer magnets) the final quadrupoles create an energy dependant focusing effect, a 2nd order effect called Chromaticity.As a consequence, the sizes and positions of the beam waist vary with the energy of the particles : this is called Chromatic Aberrations.
14 The six important concepts introduced so far LuminosityBeam emittance ( -functions)Final doublet of quadrupolesBeam optics (FODO lattice, 1st order beam optics)Chromatic aberrations (2nd order beam optics)Flat beams
16 The Collider Luminosity The Rate NE of Physics Event E generated during a beam-beam collision is given by:whereE is the Cross-Section of the Physics Event EL is the Integrated Luminosity over the bunch crossing.NE , E , and L are Lorentz invariant quantities !
17 Luminosity for Fixed Target Collision 1d3x2e+vdtThe integrated luminosity of an elementary volume element d3x of particle density 2 moving through a fixed target of density 1 during the time dt , is given by:The particle densities are defined such that:
18 The Luminosity for Mono-kinetic Beams 12v2v1d3xe+The integrated luminosity generated by the elementary volume element d3x of particle density 1 moving through 2 with homogeneous velocities v1 and v2 , is:Introducing the Lorentz current 4-vector , the integrated luminosity is expressed in an explicitly Lorentz invariant way:
19 The Luminosity for Mono-kinetic Beams +vv+e+For relativistic beams , this simplifies to:First example: the head-on collision of Gaussian bunchesleads to with
20 The Luminosity for Mono-kinetic Beams 2nd example: head-on collision of mirror Gaussian bunches+e+ce-c average luminosity over the bunch trains and collider pulse:3rd example: idem with transverse offsets x , y :+e+ce-cx
21 The Luminosity for Mono-kinetic Beams 4th example: case n°3, with a crossing anglex+e+ce-cswith , ‘Piwinski angle’or ‘diagonal angle’5th example: case n°3, with a crab crossing angle+e+ce-cxs
22 The Luminosity of Realistic Colliding Beams b* = “depth of focus”+Θ*y±*e+z–Θ*The above “academic” results are modified by the reality of 2 hard factors of Linear CollidersReal beams are not mono-kinetic: include angular distribution (cf. left pict.)Real particle trajectories are not straight during the collision: they are strongly deviated by the electromagnetic field generated by the opposite beam, the so-called “Beam-Beam Forces” (cf. right pict. and Beam-Beam section)
23 The Luminosity for Colliding Beams 12v2v1d3xe+Introducing time and velocity dependent distributions such that:the integrated luminosity for colliding relativistic beams is given by:
24 The Hour Glass effect y z b* = “depth of focus” +Θ* ±* e+ –Θ* Beams are naturally converging to their focus point, reaching their minimum sizes x , y , and diverging from there, with angular spreads : x , y .The focus point is described by a local ‘parabola’ of finite ‘depths of focus’Introducing the angle variablesand assuming Gaussian bunches:
25 The Hour Glass Reduction Factor It is desirable to fulfil the condition in order to longitudinally contain most of the bunch particles within the depths of focus x and y , hence the 1st need for a longitudinal bunch compression system to reduce z . ILC (Ay =3/4)
26 The Pinch EffectIn e+e- colliders, electrons are focused by the strong electromagnetic forces generated by the positron beam ( f = focal length).This is the well known beam-beam effect limiting the luminosity of circular colliders.In linear collider (single pass), the beams are ‘pinched’ and the luminosity is enhanced by this Pinch Effect:HD= disruption enhancement factor (usually includes the hour-glass reduction).Nota Bene:HD > 1 in e+e- collidersHD < 1 in e-e- collidersfHowever, for long bunches, several trajectory oscillations can develop and lead to an unstable disruption regime.The Disruption Parameter is defined ashence the 2nd need for a short bunches and a longitudinal bunch compression system.
27 Beam-Beam Simulations ILC parametersDy ~ 12Luminosity enhancementHD ~ 1.4Not much of an instability.Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).
28 Luminosity as a Function of Vertical Beam Offset Small disruption is beneficial to the luminosity because the bunches are attracting and focusing each other smoothly.Large disruption is detrimental at small offsets because a kink instability develops.
29 ILC Beam Delivery System Parameters Notice:‘Razor blade’ transverse aspect of the beams at collision:aspect ratio (cf. next page)The 14 mrad crossing angle is larger that Piwinski angleThe reduction of luminosity is too large: , hence crab-crossing is mandatory (cf. dedicated section on crossing angle).The vertical disruption is large Dy =18.5 ~ one oscillation in the vertical plane.
30 Flat Beams : Luminosity and Horizontality 5th example: error on ‘horizontality’ of flat beamsye+xeThe horizontality of flat beams is inherited from their respective Damping Rings.In case of an error, the overlap of their distributions is not perfect and induces a reduction of the luminosity:sfor large aspect ratiosTolerance:
32 Laplace Equation and Hamiltonian Mechanics The motion of individual trajectories in an external electromagnetic fieldderives from the Laplace Equationwith , and particle momentum.It obeys the laws of Hamiltonian Mechanics:(Hamilton-Jacobi equations)withand is the conserved energy,where is the electromagnetic potential 4-vector.Hamilton equations derives from a Least Action Principle based on the Action:
33 Hamiltonian Mechanics and Beam Distribution Hamiltonian Mechanics is valid as long asThere is no Collective Effects (space charge, wake fields)Synchrotron Radiation is ignoredThere is no beam collimationThese conditions are realized ‘most of the time’ along the Beam Delivery System.Hamiltonian Mechanics allows a powerful description of the beam dynamics using Symplectic Transformations of the beam particle 6-dimensional Phase SpaceCoordinatesIntroducing the 6D Skew Matrix , with the 3D unit matrix,Hamilton-Jacobi equations read:Solving this system of equations amounts to deriving the time evolution of particletrajectories, and therefore the expressions of the transformation which maps thecoordinates intoThis map is noted , for convenience.
34 Some Symplectic Algebra in Even Dimensions D=2,4,6,… The skew matrix is anti-symmetric and such thatThe Symplectic Group Sp(D) is composed of the symplectic matrices S such that :In other words (and by analogy with orthogonal matrices), they leaves the anti-symmetric quadratic form invariant N.B:The matrix itself is symplectic:If , thenIf , then ; in other wordsThe group Sp(D) is generated by the matrices such that T is a symmetric matrixIn 2 dimensions: Sp(2) Sl (2).Indeed, for any regular 2D matrix:
35 Symplectic Maps 1st property: Map Symplecticity The transformation which maps coordinates at t0 into coordinates at t , is symplectic in the sense that its Jacobian matrix is a symplectic matrix.Proof:Let us note the Jacobian matrix of the mapSince is a symplectic matrix, we need to show thatsince is symmetric
36 Liouville Theorem P0 P P Q0 Q Q 2nd property: Phase Space Volumes are invariant of Motion3rd property: Particle Densities are invariant of MotionGiven a particle density distribution such that , then following the motion of the particles from t0 to t :P0PPYESNOQ0QQ
37 Global Phase Space Volumes (‘areas’) are invariant of Motion: Liouville TheoremGlobal Phase Space Volumes (‘areas’) are invariant of Motion:Gaussian-like ellipsoidlinear motionnon-linear motionP0PPt0 tQ0QQRectangular collimatedP0PPt0 tQ0QQ
38 The Particle Density and its Moments The beam is modelled by a particle density distribution which generates a set of characteristic real numbers, its Moments.The first three moments0th order moment: beam charge (or population):1st order moment: centre of mass (or 6-vector beam centroid):2nd order moment: beam matrix (6x6 matrix):are widely used to characterize the beam transport.
39 The Gaussian Particle Density The Gaussian density distribution, in even D dimensionsis fully characterized by its 3 first momentsIt simplifies to the ‘usual’ Gaussian distribution functions, assuming for instancein D=2 dimensionsthenIt is one of the few typical beam distribution used in the Beam Tracking simulation codes which probe the beam transport properties.Other distributions widely used: Uniform (‘Water-bag’), Ellipsoid surface (K-V), …
40 Some Algebra about the Beam Matrix 1st property: is a symmetric definite-positive matrix2nd property: is strictly positive (except for the point-like distribution) and can be inverted.3rd property: Normal form of :
41 Some Algebra about the Beam Matrix 3rd property: Normal form ofProof:Introducing the Square Root Matrix such that , and the anti-symmetric matrix , one can find an orthogonal matrix O such that(kind of Jordan normal form for anti-symmetric matrices)Then, introducingone can show that1)2)6D generalizedTwiss parametersPhase AdvanceEmittance
42 Linearized MotionFor small deviations about the accelerator reference trajectory , the transformation map can be linearized into:witha symplectic matrixAt the first order, the beam centre and beam matrix transform as:
43 Intrinsic EmittancesThe 3 quantities are called the Intrinsic Emittances : they are invariant of the motion, in the linearized motion approximation.To calculate these intrinsic emittances, one uses the 2 properties:D=6 dimensions: ???D=4 dimensions:D=2 dimensions:where A is the area of the 2D ellipse
44 Switching Basis We (will) switch to a new coordinates basis in which is the skew matrix,and the normal form of the beam matrix is
45 Projected Emittances Case of Uncoupled Beam Matrix In some ideal locations along perfect beam lines, the beam matrix is uncoupled in the 3 physical dimensions (x,y z):Then, the 3 emittances are given byCase of 4D Transverse Coupled MatrixIn reality, either by concept or by the effect of misalignment errors, the beam matrix is x-y coupled.Accelerator instrumentation, usually built in horizontal and vertical frames, allows to measure the projected emittanceswhich are not invariant of motion, and such that
46 Beam Line Coordinate System Particle trajectories are parameterized with respect to a Reference Trajectoryassociated to a Reference MomentumOne introduces the curvilinear coordinate system defined by the Moving Frameof (planar) Reference Trajectory, such that :arc lengthtangent vectorTRANSPORTK. Brown et al = curvature radius
47 Particle Motion Coordinates Then, the particle motion is parameterized by throughwhere are the timeand position of the particle whencrossing the plane normal to thereference trajectory atNote 1: for particlesahead of the reference particle.Note 2: for a planar referencetrajectory:withTRANSPORTK. Brown et al = curvature radius
48 The Resulting Hamiltonian To study the evolution of Particle Motion, Beam Physicists like to trade the Time variable t with the Arc Length variable s defined by :After some Canonical manipulations, this leads to a new Hamiltonianin terms of the conjugate variables:Hamiltonian mechanics and Symplecticity provide powerful tools to study beam dynamics over very long times, like the beam orbits in circular colliders.They are of lesser importance for the design of linacs and transfer beam lines.
49 TRANSPORT Coordinate System For transfer lines, like BDS and linear accelerators, we turn to more familiar, but not symplectic conjugate variables, the TRANSPORT coordinates:a time variable !transverse ‘angles’relative momentum deviationwithThese coordinates are to the first order proportional to symplectic conjugate variables:
50 Geometric and Normalized Emittances Geometric emittances of uncoupled motionIn TRANSPORT coordinates, the ‘geometric’ emittances calculated from the beam matrixwithis not invariant under acceleration.Normalized emittancesNormalized emittance, invariant under linearized motion, are given byFor high energy beam lines and
51 The Beta Functions of the Beam (D=2) The 2D beam matrix is parameterized as follows:withBy definitionThe 2D beam ellipse can then be parametrized byCourant-Snyder InvariantThe remarkable points can be recovered from the above equation, and also from the search for extrema:e.g.
52 Normalized Variables (D=2) The 2D beam matrix takes the normal form:withThis suggests introducing the Normalized Variablessuch thatIn these variables, the particle motion is described by simple rotations around a circle in (u,v) ‘phase space’ (cf. Beam Optics section).But most of the information is retained in the - functions.
53 The General Form of the Beam Matrix (D=6) In Final Focus Systems, several simplifications are in force :the longitudinal motion is frozen: the arc length differences between trajectories are negligible and the bunch length is constant.the transverse (x,y) motions are not coupled.The resulting 6D beam matrix is parametrized as follows:with the dispersion coefficients measure the correlations between transverse position-angle and energy, e.g..
54 Sources of Emittance Growth Linear Collider beams are experiencing many sources of emittance growth (or emittance degradation), in particular through the Beam Delivery Systems :Non Hamiltonian Mechanics :Synchrotron Radiation (SR) : an interesting case because it works both sides:at low energy, big injected emittances (transverse and longitudinal) are damped (i.e. reduced) down to the equilibrium emittances of Damping Rings;at high energy, the longitudinal emittance growth dominates due to the strong energy dependence of SR, and couples to the transverse motion through magnet chromaticity.Collective Effects, e.g. beam-beam interactionsNon-linear motiona source of the Linear Emittance growthbut does not violate Liouville TheoremCouplinga source of 4D projected emittance growthbut not of 4D intrinsic emittance growthTransverse Wakefieldsbut not of 4D intrinsic emittance growth.
55 Kinematics at the Interaction Point We assume a perfect beam at the IP, without any coupling, and we investigate the kinematics of the beam in the vertical plane at the 5.7 nm focus.before the IP:converging beamat the IPfocused beamafter the IP:diverging beam (no collision)yy'yy'yy'+14µrad5.7 nm 14 µradxxxs+ss = 0Linear motionIn a Drift space:
56 Beta Functions at the Interaction Point Introducing the transfer matrix of the Drift Spacebefore the IPat the IPafter the IPyy'yy'yy'+14µradconvergingdiverging5.7 nm 14 µrads = 0s+sxwe can calculate the Beam Matrix at and after the IP, using
57 Question n°2: Do we need High Field Quadrupole Magnets ?
58 The Last Focusing Lens s < -l* s = l* s > l* The 14 µrad angular spread is the highest ever downstream of the Damping Rings.The ‘high’ beam convergence is created by a focusing lens, located at a distance l* from the IP.before the lens:diverging beamfocusing lens withfocal length fafter the lens:converging beamyy'yy'l* 14µrad l* 14µradxxxs < -l*s = l*s > l*Linear motionin a focusingThin Lens:
59 The Final Lens: A Magnetic Quadrupole The angular kick y’ created by the lens is proportional to the trajectory offset y .If it is generated by a magnet of length L:IPFinal Lensl*~ fy’= y/fyThe LHC lattice quadrupoles are designed for 223 T/m.They are Superconducting Magnets (NbTi technology) with 56 mm bore diameter.
60 Long Quadrupole Magnet Quadrupole magnets create gradient of bending fieldFrom Maxwell’s equation in Vacuumthis gradient has the same value and same sign in both planes.As a consequence, for G > 0 , the Laplace forceis focusing in the horizontal plane and defocusing in the vertical plane:focusingdefocusing to focus in both planes, a quadrupole doublet, or triplet, is required.
61 Quadrupole Doublet: Thin Lens Approximation A quadrupole doublet is potentially focusing the beam in the 2 planes.lDf1f2IPAs a first approach,Short Quadrupoles can be treated like thin lenses.beam directionThe overall linear motion is given the following composition of elementary transfer matrices:This transfer matrix is focusing in both planes if the ‘R21’ terms are negative in both planes, i.e. for both signs of f1 and f2 .
62 Quadrupole Doublet: Long Quadrupoles A long quadrupoles has the following transfer matrices (cf. Beam Optics section) :in its focusing planewithin its defocusing planeIt can be treated as a Thin Lens in the limit with constant.The ILC Final Doublet focuses the beam with the RMS envelopesshown here.IPNota Bene: final doublet quadrupoles are VERY far from being thin lenses.2 m
63 The Final Focus System f1 The Final Focus System operates like an Optical Telescope:it uses 2 confocal lenses: a weak lens with a long f1 focal distance, followed by a strong lens with a short f2 focal distance;it realizes point-to-point and parallel-to-parallel imaging of the beam with a demagnification of the images by the factor M = f1/ f2 >> 1f1IP Imagey’yy’0
64 FFS Chromatic effect: the Momentum Bandwidth The Final Focus System generates chromatic aberrations of the IP beam image.Introducing the momentum dependence of thefocal distances one can show that:,the aberration for the parallel-to-parallel images (~ cosine-like trajectories) are of order ;the aberration for the point-to-point images (~ sine-like trajectories) are given by:On average, one gets:.The parameter is called the Chromaticity.It sets the Momentum Bandwitdh (acceptance) of the FFS:which demonstrates the need for Chromaticity Correction.
65 Synchrotron Radiation: Oide Effect The energy spread generated by SR in the final quadrupole, given bywithinduces an irreducible growth of the IP beam size:Final quadLQl*IPwhere C1 is ~ 7 (depends on FD parameters).The minimum beam size (Oide limit) :is realized for
66 Magnetism (or “Why four poles in a quadrupole magnet ?”)
67 Quadrupole Magnets: the Optician Standpoint For the beam optician a Quadrupole is a device which creates a Dipole Gradient G, that is:1) does not bend the reference trajectory,2) soft kicks particles with small offsets from the reference trajectory3) hard kicks particles with large offsets:HenceQuadrupole = Dipole Gradient (along a straight line)In the same spirit, from the beam optics standpoint a Sextupole is a device which creates a Quadrupole Gradient G’ in the transverse plane:Sextupole = Quadrupole GradientHexapôle = Gradient Quadripolaire (in French)etc…
68 Quadrupole Magnets: the Mathematician Standpoint xyB(x,y)Two Dimensional Static Magnetismin an Electrical Current Free Region(representing the accelerator beam pipe)We specialize on Time Independent Transverse Magnetic Fields,which is realized by infinitely long magnets with a uniform field distribution along the longitudinal axis, and no end-fields.Such a field derives from a longitudinal vector potentialMaxwell-Ampère’s equation in the vacuum imposes that the scalar field is an harmonic function obeying Laplace’s equation:
69 Some Symmetry : Normal and Skew Magnetic Fields Symmetry with respect to the ‘horizontal’ plane :xyBxyBTransverse magnetic fields can be decomposed into the sum of symmetric (normal) and antisymetric (skew) components:such thatThe implementation of this symmetry simplifies the design of accelerator since it eliminates (x,y) transverse coupling by construction (cf. Beam Optics section).
70 Solving the Vector Potential A We introduce the complex variable such that:The solution of Laplace’s equation isand, for a real potential :with a an analytic complex function (essentially, an infinite polynomial).By the symmetry
71 Harmonic Expansion of the Vector Potential A Introducing the notationwith real coefficientsthe vector potential can be expanded as followsIn polar coordinates ( r, ), the vector potential can be expanded in the multi-polar harmonic expansion:Nota BeneA Skew Field derives from a Rotation of the Normal Field:
72 Harmonic Expansion of the Magnetic Field B Introduce the complex magnetic field such that,the complex field can be expanded as followsIn polar coordinates ( r, ), and the magnetic field can be expanded in the multi-polar harmonic expansion:
73 The Ideal Dipole Magnetic Field : n = 1 xyxyNormal ComponentBN is a uniform “vertical” field which bends trajectory in the “horizontal” planeBN(x,y)xySkew ComponentBS is a uniform “horizontal” field which bends trajectory in the “vertical” planeBS(x,y)Unlike this mathematical 2D model, most of real accelerator dipole magnets are not straight, but are bent around the reference trajectory with a curvature radius given byDipole magnet transfer maps are the most complex elementary map to modelize !!
74 The Ideal Quadrupole Magnetic Field : n = 2 Normal ComponentSkew ComponentBS is derived BN from by a rotation ofMagnetic field lines of a quadrupole magnet
75 The Ideal Sextupole Magnetic Field : n = 3 Normal ComponentSkew ComponentBS is derived BN from by a rotation ofMagnetic Field lines of a sextupole magnet
76 Misaligned Sextupole Magnet We consider only the Normal Sextupole Magnet component.As expected, the sextupole field can be expressed as a superposition of gradients of quadrupole :but the ‘bad’ surprise is that it includes both normal and skew components.As a consequence:the effect of a sextupole on a horizontally misaligned particle is that of a normal quadrupole ,the effect of a sextupole on a vertically misaligned particle is that of a skew quadrupole .Conversely:an horizontally misaligned sextupole acts as a normal quadrupole,a vertically misaligned sextupole acts as a skew quadrupole.