Beam Delivery System and Beam-Beam Effects
Fourth International Accelerator School for Linear Colliders Beijing, China 7-18 September 2009
Lecture A3 Beam Delivery System and Beam-Beam Effects
Part 2
Olivier Napoly CEA-Saclay, France
Preliminary Version 17 September 2009 Collider Parameters, Physical Constants and Notations
E , energy and p , momentum c = 299 792 458 m/s Bρ , magnetic rigidity (Bρ = p/e) e = 1.602 177 33 10-19 C 2 L , luminosity mec = 510 999 eV = 4π 10-7 N A-2 permeability , bunch charge µ0 Q 2 ε0 = 1/µ0c permittivity N , number of particles in the bunch (N = Q/e) -15 re = 2.818 10 m e 2 nb , number of bunches in the train 2 with e cm = 4πε 0 re frep , pulse repetition rate
εx and εy , horizontal and vertical emittances * * σx and σy , rms horizontal and vertical beam sizes at the IP σz , bunch length α,β,γ , Twiss parameters μ , tune ψ , phase advance,
Beam Delivery System 2 Beam Delivery System
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).
ILC
BDS
Most 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.
Beam Delivery System 3 ILC Beam Delivery System Layout (RDR)
Linac Exit Beam Diagnostics Switch Yard Sacrificial β-collimator collimators
E-collimator Final Focus Tune-up & Emergency extraction Tune-up 14mr IR dump
Muon wall
grid: 100m*1m Main dump Extraction
Beam Delivery System 4 Contents: Four (+ Two) Outstanding Questions ? Part 1 Q1: do we need a Final Focus System ? Luminosity, Emittance, Q2: do we need High Field Quadrupole Magnets ? Quadrupole Magnets, Multipoles, Superconducting Quadrupoles
Part 2 Q3: do we need Flat Beams ? Beam-beam forces, Beamstrahlung, e+e- Pairs, Fast Feedback Q4: do we need Corrections Systems ? Beam Optics, Achromat, Emittance Growth
Beam Delivery System 5 Contents: Fundamentals in Beam Physics
Part 1 Luminosity Emittance Magnetism (continued) Part 2 Beam-Beam Effects Beam Optics Synchrotron Radiation
Beam Delivery System 6 Magnetic Measurements
Magnet fields are measured to checks: 1. the field integral of the main component (Bdl for a dipole, Gdl for a quadrupole) 2. the field harmonics content (~ field homogeneity, or quality)
These measurement are usually performed by a magnetic probe located at a Reference Radius Rref along the magnet bore.
Hence, magnet experts use the following expansion:
n-1 ∞+ ⎛ z ⎞ ⎜ ⎟ i )( ∑ ()−= i abzB nn ⎜ ⎟ for < Rz i n=1 ⎝ Rref ⎠
(sometimes with a +)
n- )1( B )0( n−1 so that i ab =− R for n ≥ 1 nn n − )!1( ref
Beam Delivery System 7 Quadrupole Magnets: the Engineer Standpoint
Given a Central Axis Oz which coincides with the Reference Trajectory, we introduce an “Engineering Symmetry” RS(θ ) which allows categorize the Magnets into “Engineering Multipoles”. y (θ ) Notations: R R(θ ) is the rotation with angle θ around Oz . zˆ θ • S is the symmetry with respect to the Oxy plane. O x
S θ ≡ θ )()( = oo RSSRR θ )( is the combination of the two. y y Note that : B 1) For a transverse B field (pseudo-vector) r r S −=⋅ BB zˆ zˆ B • • 2) For a longitudinal electrical current J O O r r ⊗ J x • J x S −=⋅ JJ 3) S S θ o S θ = RRR θ )2()()(
Beam Delivery System 8 Symmetric Current Distributions y Consider a distribution of longitudinal electrical current J(r,θ) . J(r,θ)
If a current distribution is invariant with respect to RS zˆ • RS θ0 ⋅ θ = rJrJ θ ),(),()( O x then: y 1) the integrated current through the plane is zero, since J(r,θ) rJ θ +R θ ⋅ θ ),()(),( θ − rJrJrJ θ +θ ),(),( rJ θ ),( = S 0 = 0 zˆ + 2 2 • O x 2) 2 is a sub-multiple of 2 , since − θ0 π
rJ θ = S θ0 oRR S θ0 ⋅ rJ θ = R θ0 ⋅ θ = rJrJ θ − θ0 )2,(),()2(),()()(),( 2π ⇒ 2either θ = θ ≡ rJrJ )(),(or constant zero offunction θ. 0 n
Beam Delivery System 9 Multipolar Current Distributions We introduce the following categories of current distributions J(r,θ) :
Monopole (n=0): )0( rJ )( , azimuthally independent distributions
Dipole ( =1), such that: )1( )1( n RS θπ=⋅ rJrJ θ ),(),()(
Quadrupole (n=2): )2( )2( RS π θ =⋅ rJrJ θ ),(),()2( Sextupole (n=3): )3( )3( RS π θ =⋅ rJrJ θ ),(),()3(
Octupole ( =4): )4( )4( n RS π θ =⋅ rJrJ θ ),(),()4(
Decapole ( =5): )5( )5( n RS π θ =⋅ rJrJ θ ),(),()5( …. pole ( ): n)( n)( 2n- n RS π θ =⋅ rJrJn θ ),(),()(
)0( )0( Note that , for the Monopole : RS θ rJrJ )()()( ∀−=⋅ θ
Beam Delivery System 10 Decomposition of Current Distributions The space J of longitudinal current distributions can be decomposed over the sum of symmetric subspaces as follows. 1. Since , one can decompose any current S π o S π = RRR π = 1)2()()( distribution into symmetric (n=1) and antisymmetric components w.r.t. RS(π):
JJ R( S π JJ R( S π ⋅−+⋅+= J 2))(2))( n= )1( R( π ⋅++= JJJ 2))( n= )1( += JJ π where J π is invariant under the rotation R π )(
2. Since R S π o R S π = R π )()2()2( , one can decompose the current distrib- utionJπ into symmetric (n=2) and antisymmetric components w.r.t. RS(π/2): JJ ππR( S π π 2))2( JJ π R( S π ⋅−+⋅+= Jπ 2))2( n= )2( JJ π R( π ⋅++= Jπ 2))2( n= )2( += JJ etc, etc,… π 2 leading to : n= ( n= ( n= ⊕⊕⊕= n= )8()4()2()1( ...))) + JJJJJJ =0)(n dipole + quadrupole + octupole + 16-pole + …. + monopole
Beam Delivery System 11 Natural Harmonics Question: Where are the Sextupoles, and Decapoles, … ? Answer: Sextupoles (n=3) and Decapoles (n=5) are Dipole (n=20) Natural Harmonics:
n= n= )3()3( 3 n= )3( n= n= )3()3( RS π )3( JJ ⇒=⋅ R S π )3( J RS π )( =⋅=⋅ JJ n= n= )5()5( 5 n= )5( n= n= )5()5( RS π )5( JJ ⇒=⋅ R S π )5( J RS π )( =⋅=⋅ JJ Dodecapoles (n=6) and 20-poles (n=10) are Quadrupole (n=21) Natural Harmonics: π )6( n= JJ n= )6()6( ⇒=⋅ 3 π )6( J n= )6( π )2( n= =⋅=⋅ JJ n= )6()6( RS R S RS π )10( n= JJ n= )10()10( ⇒=⋅ 5 π )10( J n= )10( π )2( n= =⋅=⋅ JJ n= )10()10( RS R S RS In general, 2n-poles with n =(2m+1)2p are 2.2p-pole (n=2p) Natural Harmonics: p R π )( JJn nn )()( ⇒=⋅ R m+12 π )( Jn n)( =⋅ R π )2( =⋅ JJ nn )()( S S S . In other words, using the example of Dipoles, the ensemble of Dipole (RS(π)- invariant) current distributions includes that of Sextupole (RS(π/3)-invariant) ones. As a consequence, Dipole magnets built from a Dipole invariant current distribution most likely contains Sextupole, Decapole, etc… components . Other harmonics are excluded, unless induced by non-invariant fabrication errors.
Beam Delivery System 12 Quadrupoles Have Four Poles A Quadrupole magnet can be designed as follows:
1. Define a current distribution J(r,θ) over a 90° wide sector : θ ∈[θ θ00 + π 2, ] nd 2. Rotate it by the symmetry RS(π/2) over the 2 sector θ ∈[θ0 + π ,2 θ0 + π ] y y J (1)(r,θ) J (2)(r,θ) J (1)(r,θ) − − − + + + RS(π/2) x x
rd 3. Repeat the RS(π/2) operation to the 3 sector θ ∈[θ0 + π θ0 + π 23, ] 4. Repeat the R (π/2) operation to the 4th sector θ ∈[θ + π θ + 2,23 π ] S y 0 0 J (2)(r,θ) J (1)(r, ) − θ − + + x + + (4) J (3)(r,θ) − − J (r,θ)
Beam Delivery System 13 The Simplest Monopole Magnet
The simplest Monopole magnet is given by longitudinal current line transporting a total current I ^z . r By rotation symmetry : = rBB )( θ ˆ . 2π μ I From Ampère’s law : 0 . )( μθ0 rBIrBrd )( =⇒= ∫0 2π r y B
• I zˆ x
μ I The complex magnetic field = + iBBB is then given by zB )( = 0 xy 2π z Nota Bene 1: The magnetic field inside of the conductor is an ideal focusing field ! Nota Bene 2: Point-like Magnetic Monopoles do not exist on earth !
Beam Delivery System 14 A ‘Dipole’ Magnet The simplest Dipole Field configuration is given by a pair of linear conductors: y μ I μ I n= )1( zB )( = 0 − 0 π − Rz 0 π + Rz 0 )(2)(2
μ RI 00 −I +I = − • + 2 2 x π − Rz 0 )( zˆ B R0 This expression can be expanded for < Rz 0 : m μ RI μ I μ I ∞ ⎛ z 2 ⎞ n= )1( zB )( = 00 −= 0 −= 0 ⎜ ⎟ 2 2 2 2 ∑⎜ 2 ⎟ π − Rz 0 π 0 − RzR 0 )1()( π R0 m=0 ⎝ R0 ⎠ Comparing to the mathematician expressions (page 71) one gets :
μ0 I μ0 I m 0 →= Dipole : b 1 −= ; m 2 →= Decapole : b5 −= 5 π R0 π R0 μ I μ I Sextupole: 0 ; (2m+1)-pole : 0 m 1 →= b3 −= 3 m → b m+12 −= m+12 π R0 π R0 Nota Bene: This ‘Dipole’ Magnet still has zero curvature radius !
Beam Delivery System 15 A Quadrupole Magnet The simplest Quadrupole Field configuration is given by a quadruplet of conductors:
n= )2( μ0 I μ0 I μ0 I μ0 I zB )( = − iπ 2 + iπ − i π 23 π − Rz 0 π − 0eRz π − 0eRz π (2)(2)(2)(2 − 0eRz ) m μ I ∞ ⎛ z ⎞ 0 mi +− π mi +− π mi +− π 23)1(22)1(2)1( < Rz −= ∑⎜ ⎟ 1( − + ee − e ) 0 2π R0 m=0 ⎝ R0 ⎠ m y μ I ∞ ⎛ z ⎞ − 0 m+1 mm ++ 11 I −= ∑⎜ ⎟ i −−+−− i ))1()(1( − 2π R0 m=0 ⎝ R0 ⎠ m +I +I ∞ ⎛ ⎞ ⎧4 if km += 14 + • + μ0 I z x −= ∑⎜ ⎟ × ⎨ zˆ 2π R0 m=0 R0 0 if km +≠ 14 ⎝ ⎠ ⎩ R0 −I − Comparing to the mathematician expressions (page 71) one gets :
2μ0 I 2μ0 I m 1 →= Quadrupole: b 2 −= 2 ; m 5 →= Dodecapole: b6 −= 6 π R0 π R0 Beam Delivery System 16 A Better Quadrupole Magnet An improved Quadrupole Field configuration is given by an octuplet of conductors: y −−I −I − +I + + +I α R • 0 zˆ x + + +I The calculation proceeds like the preceding one +I leading to: −−I −I − m ∞ n= )2( μ I ⎛ z ⎞ ⎧ m + )1cos(4 α if km += 14 zB )( −= 0 ⎜ ⎟ × ⎨ ∑⎜ ⎟ < Rz 0 π R0 m=0 ⎝ R0 ⎠ ⎩ 0 if km +≠ 14 Comparing to the mathematician expressions (page 71) one gets :
2μ0 I m 1 →= Quadrupole: b 2 −= 2 α )cos( ; π R0 2μ I π Dodecapole: 0 cancels for α = m 5 →= b6 −= 6 α)cos(6 π R0 12
Beam Delivery System 17 A Sector Quadrupole Magnet This quadrupole is realised with 4 finite sectors of current density J . One can integrate elementary currents by using the preceding result : R α k +14 4μ 00 ∞ ⎛ z ⎞ zB )( = 0 ∫∫ drdJ θ ∑⎜ ⎟ k + )24cos( θ π k =0 r Ri 0 ⎝ ⎠ R k +14 α 4μ J ∞ 00⎛ z ⎞ = 0 ∑ ∫∫dr⎜ ⎟ θ kd + )24cos( θ π k =0 r Ri ⎝ ⎠ 0 2μ ⎛ RJ ⎞ μ J ∞ ⎛ 11 ⎞ k + )24sin( α = 0 ⎜ 0 ⎟ α )2sin(ln z + 0 ⎜ − ⎟ 0 z k + )14( ⎜ ⎟ 0 ∑⎜ 4 4kk ⎟ π ⎝ Ri ⎠ π k =1 ⎝ i RR 0 ⎠ kk + )24( < Rz 2μ ⎛ RJ ⎞ i 0 ⎜ 0 ⎟ m 1 →= Quadrupole : b2 = ln⎜ ⎟ α 0 )sin(2 ; π ⎝ Ri ⎠ μ J ⎛ 11 ⎞ π m 5 →= Dodecapole: b −= 0 ⎜ − ⎟ α )6(sin , cancels for α = 6 ⎜ 4 4 ⎟ 0 0 6 6π ⎝ i RR 0 ⎠ Beam Delivery System 18 An Ideal Quadrupole Magnet: Cos(2θ ) distribution The engineer meets the mathematician when he tries to build current lines following a “cos(2θ )” angular distribution : θ = rJrJ θ )2cos()(),( with < RrJ i = 0)( . The complex magnetic field is given by the following integral: μ ∞ 2π 1 zB )( = 0 θ rJdrdr θ )2cos()( 2π ∫∫ − erz iθ )( Ri 0 μ ∞ 2π 1 −= 0 θθ)2cos()( edrJdr −iθ 2π ∫∫ − −iθ rze )1( Ri 0 m μ ∞ 2π ∞ ⎛ z ⎞ −= 0 θθ)2cos()( −iθ eedrJdr −miθ < Rz ∫∫ ∑ ⎜ ⎟ i 2π m=0 r Ri 0 ⎝ ⎠ m μ ∞ ∞ ⎛ z ⎞ 2π μ z ∞ rJ )( −= 0 ∑ ∫∫rJdr )( ⎜ ⎟ θθ)2cos( eed −− mii θθ −= 0 ∫ dr 2π m=0 r 2 r Ri ⎝ ⎠ 0 Ri This is a Pure Quadrupole, in the mathematician sense: μ ∞ rJ )( m 1 →= Quadrupole b −= 0 dr 2 2 ∫ r Ri
Beam Delivery System 19 The Cos(2θ ) Quadrupole Magnet Assuming a uniform radial distribution of currents :
⎧ for i << RrRJ o θ = rJrJ θ )2cos()(),( rJ )(with = ⎨ ⎩0 elsewhere the complex magnetic field is given by the following expression:
μ0 J ⎛ Ro ⎞ zB )( −= z ln⎜ ⎟ 2 ⎝ Ri ⎠
This is a Pure Quadrupole, in the mathematician sense: μ ⎛ RJ ⎞ Quadrupole 0 ⎜ o ⎟ m 1 →= 2 Gb −=≡ ln⎜ ⎟ 2 ⎝ Ri ⎠
Beam Delivery System 20 Superconducting Quadrupoles
Beam Delivery System 21 The LHC Cos(θ ) Dipole Magnet LHC Superconducting Dipoles have been wound from the “cos(θ )” model, using two layers of cables and anglular wedge to fit the “cos(θ )” distribution.
Internal Diameter : Φ = 56 mm Current Density: J = 500 A.mm-2 → Dipole field : B = 8.33 T Current Lines: NI = ??? A.t
angular wedges
pole n°1
Beam Delivery System 22 LHC Arc Quadrupole Magnet LHC Superconducting Arc Quadrupoles have been wound from the “sector” model. Note that current sectors encompass the [0°, 30°] azimuth interval to eliminate the b6 “dodecapole” harmonics. Their main parameters are: Internal Diameter : Φ = 56 mm Current Density: J = 500 A.mm-2 → Quadrupole gradient : G = 223 T/m Current Lines: NI =??? A.t
pole n°1
Beam Delivery System 23 LHC Low-β Quadrupole Magnets LHC Superconducting Low-β Quadrupoles are realized from the “cos(2θ )” model. Internal Diameter : Φ = 70 mm Current Density: J = 545-700 A.mm-2 → Quadrupole gradient : G = 215 T/m Current Lines: NI = ??? A.t Fermilab Low-β Quadrupole KEK Low-β Quadrupole
pole n°2 pole n°1
pole n°3 pole n°4
Beam Delivery System 24 NbTi Superconducting Cables
NbTi ingot (136 kg) → mono-filament rod in Cu can (10 km) → multi-filaments cable in Cu can multi-filament cable → multi-wire Rutherford cable
mono-filament hex. rods
~1mm ∅ NbTi Wire (Courtesy Alstom/MSA)
“Rutherford” type cable
(Courtesy Oremet Wah Chang) (Courtesy Furukawa Electric Co.) 2 High performance mono-filament rods achieve JC of 3000 A/mm at 4.2 K and 5 T. Copper is necessary to flow the electric current in case of a magnet quench : → J = 500 A/mm2 at 2 K on average in the conductor. Electric Resistivities: Copper NbTi • at room temperature: 1.6×10-8 Ω.m • in normal state : 5-6×10-7 Ω.m -10 • at 10 K, 0 T (RRR =100): 1.6×10 Ω.m • in superconducting state : ∝ • at 10 K, 7 T (RRR = 100): 4.3×10-10 Ω.m
Beam Delivery System 25 B vs. JC Limitations of SC Conductors
Material TC (K) 10000 YBCO B⊥ Tape Plane Nb 9.2 YBCO B|| Tape Plane NbTi 9.5 SuperPower tape RRP Nb3Sn used in record Nb3Sn 18 Nb-Ti breaking NHMFL Complied from insert coil 2007 ASC'02 and 1000 ICMC'03 papers (J. Parrell OI-ST) 427 filament strand with Ag alloy outer sheath 2212 tested at NHMFL (A/mm²) YBCO Insert Tape (B|| Tape Plane) E
J Maximal JE for entire LHC Nb-Ti YBCO Insert Tape (B⊥ Tape Plane) strand production (– 100 ) CERN-T. Boutboul MgB2 19Fil 24% Fill (HyperTech) '07, and (- -) <5 T Bronze MgB2 data from Boutboul Nb Sn et al. MT-19, IEEE- 3 2212 OI-ST 28% Ceramic Filaments TASC’06) NbTi LHC Production 38%SC (4.2 K)
4543 filament High Sn Nb3Sn RRP Internal Sn (OI-ST) 18+1 MgB2/Nb/Cu/Monel Bronze-16wt.%Sn- Courtesy M. Tomsic, 2007 0.3wt%Ti (Miyazaki- Nb3Sn High Sn Bronze Cu:Non-Cu 0.3 MT18-IEEE’04) 10 0 5 10 15 20 25 30 35 40 45 Applied Field (T) Nb3Sn conductors allow for increase of field and gradient, but: 1. Cables are more expensive × 5-10 and production volume ÷ 100, 2. Fabrication of accelerator magnets still at the R&D stage.
Beam Delivery System 26 Normal Conducting Magnets
Normal Conducting are generally built with water cooled Cu conductors. In “Air magnets”, the magnetic field is generated by the current lines through the Maxwell-Ampère’s equation: r r =∧∇ μ0 JB “Air magnets” are very ineffective, as illustrated by this example: Internal Diameter : Φ =300 mm Current Density: J = 4 A/mm2 → Quadrupole gradient : G = 0.39 T/m Current Lines: NI =13000 A.t Air Conductors
r B r B
Beam Delivery System 27 Normal Conducting Magnets: the Role of the Iron Yoke
The Iron Yoke increases the magnetic field in the Iron thanks to its much higher Permeability µ r r =∧∇ μ JB with μ ≈ 6000 μ 0 , It is shaped to optimize the Magnetic Circuit through the Air internal chamber, and to reduce the Stray Fields at large radii. → Quadrupole gradient : G = 5.7 T/m Iron Yoke Conductors
r B r B
Beam Delivery System 28 Magnetic Saturation r r Introducing the auxiliary magnetic field H such that =∧∇ JH , one gets r r r == μμμHHB where is the relative permeability. r 0 µr 2 T Saturation occurs around 2 T ±10% for high quality electrical steel or permalloy (nickel-iron).
First magnetization curve, then hysteresis takes over. µ The relative permeability can reach r max. values from 5000 to 10000. H In the saturation regime, B vs. J ‘linearity’ is lost and increasing the electric current is not effective any more. As a consequence: 1. Normal Conducting Magnets are limited to ~2 T fields 2. In Superconducting Magnets, the Iron Yoke is mostly used to contain the transverse stray fields.
Beam Delivery System 29 ILC-RDR Final Focus Systems To avoid parasitic collisions, the ILC FFS is based on a large 14 mrad crossing angle, with two alternate designs for zero (head-on) or 2 mrad crossing-angles. 14 mrad : the final doublet specialized to the incoming beam , another doublet is used for the outgoing beam → compact quadrupoles with small Ro outer radius. head-on : the final doublet is shared by the incoming and outgoing beams → large aperture quadrupoles with large Ri inner radius. 2 mrad : only the final D-quadrupole is shared the incoming and outgoing beams.
14 mrad Xing Head-on, ‘LHC –Arc’ 2 mrad Xing, ‘LHC low-β’ G=144 T/m, Ri =10 mm G=250 T/m, Ri =28 mm G=250 T/m, Ri =35 mm
Beam Delivery System 30 ILC : 14 mrad Final Focus System
NbTi cable : = 3.5 m l* serpentine winding technology
Quadrupole QD0 Combined sextupole -octupole
Beam Delivery System 31 ILC : Head-on Final Focus System
IP 1st 2nd etc... parasitic collisions = n×51 m (= n·c·340 ns/2) SPS ZX separator
l* = 4 m
Electrostatic separators (26 kV/cm, Ti, 7 × 4 m) needed to separate e+ and e- beams. A dipole (~30 mT) is used to keep the incoming trajectory straight.
Beam Delivery System 32 ILC : 2 mrad Final Focus System
Beam Delivery System 33 Reasons for the Strong Quadrupoles
Strong Quadrupoles allow for a short focal distance f ~ l*, needed for 3 reasons : 1. Chromaticity: it is kept under control by keeping l* < 10 m. This is a soft constraint, especially since the chromaticity is optically corrected by sextupoles. 2. Spent beam : in the Head-on and 2 mrad cases, the main constraint is to clear the outgoing beam and beamstrahlung after the collision: * <<Θ⇒ Rl ix
3. Synchrotron radiation: in all cases, the need to clear the synchrotron radiation generated by the final doublet from the inner detector region (with a vertex detector at low radius, and masks) is ensured by tight collimation of the transverse beam tails. The collimation requirements are growing with l*
Beam Delivery System 34 Question n°3:
Do we need a Flat Beams ?
Beam Delivery System 35 Beam Beam Effects
Beam Delivery System 36 Beam-Beam Interactions (in short) The beam-beam interaction in a linear collider differs from the circular collider: • High charge densities at the interaction point lead to very intense fields and to very strong interaction within single collision
• Strong mutual focusing of beams (pinch) gives rise to luminosity enhancement by a factor HD. • As electrons/positrons pass through intense field of opposite beam, they radiate hard photons ‘beamstrahlung’, and lose energy in a random fashion. • Interaction of ‘beamstrahlung photons’ with intense field causes e+e− pair production, which causes background and provides a luminosity signal ☺. ** • The beam-beam interaction is reduced with flat beams R σσ yx >>= 1 .
Beam Delivery System 37 Focusing Effect of a Round Current Line
A simple circular current line with uniform density J of radius R generates: • an external field which attracts same sign current (qvzˆ):
r μ I 2 rB )( = 0 θˆ with = π > RrJRI )( 2πr y r μ I −=⇒ qvF )( 0 rˆ J 2π r B • an internal focusing field for same sign current: r μ rJ • rB )( = 0 θˆ < Rr )( zˆ x 2 r μ J −=⇒ qvF )( 0 rr 2 i.e. the perfect magnetic focusing lens. Nota Bene: principle of a Lithium lens used for proton and muon focusing.
Beam Delivery System 38 Focusing Effect of a Round Line Charge Distribution
A simple circular charge distribution with uniform density ρ of radius R generates: • an external field which repels same-sign charge q:
r Q' 2 rE )( = rˆ with ' = ρπ > RrRQ )( 2 επ0r r Q' +=⇒ qF rˆ 2πε0 r y
ρ • an internal defocusing field for same-sign charge q: E r ρ )( = r < RrrrE )( 2ε 0 • zˆ x r ρ +=⇒ rqF r 2ε 0
Beam Delivery System 39 Focusing Effect of a Round Positron Beam
A circular Positron beam (travelling towards us) of radius R is generating both r charge ρ and current J distribution with the relation: = ρ + zvJ ˆ 2 • Effect on a Positron travelling with the beam, at r < R : με00 c = )1( the electric and magnetic forces cancel, r μ J ρ e 2 e 0 rr 2 r r y evF rer =+−= 2 + )( ρrvc =− 2 ρr 0 222 εε0c 2 0γε+ (E , B) (ρ , J) yielding a repelling force vanishing for γ + → ∞ . • Effect on an Electron travelling against the beam: the electric and magnetic forces add, • r μ0 J ρ rr zˆ x −= evF − − rer 22 ε 0
e 2 r e r −= 2 −+ )( ρrvvc 2 +−≈+ −+ )1( ρββr 2ε 0c ε 0c
yielding a non-vanishing Beam-Beam attracting force for γ ± → ∞ .
Beam Delivery System 40 Beam-Beam Effect : Round Case
• Incoherent Beam-Beam Force (assuming γ ± → ∞ β = β −+ = 1, ) This beam-beam force can be easily extended to the case of a round Positron beam with a non-uniform circular charge distribution ρ + r )( . The force acting on an individual Electron is given by: r e 2 r y rF −= ρ rr r ρ r)(with)()( = ρ uduu )( + + 2 ∫ + ε 0 r 0 It is called the Incoherent beam-beam force. (ρ (r) , J (r)) • Coherent Beam-Beam Force (E , B) The Coherent beam-beam force is the force acting on the electron bunch, i.e. the average of the forces over • zˆ x the electron distribution ρ − r )( : r r rF = ρπ ρ drrrrFdrrr '')'()'('')'(2)( C ∫ − ∫ − • However, two ingredients are still missing !
1. realistic positron and electron bunches are not infinite lines : i.e. σ z < ∞
2. colliding positron and electron bunches are not round: σ ≠ σ yx
Beam Delivery System 41 The Electromagnetic Field of a Relativistic Particle
This will be worked out Er during the Home Work γ = 0 session.
v = 0
Electrostatic field in the rest frame v < c 1/γ γ = ∞
2 F = q(E + v × B) ∝ (1 − vv/c ) / r v = c ⇒ No Intra-beam Forces for γ = ∞ v = c ⇒ Strong Beam-Beam Forces ± ⇒ No Wake: the Front and Rear parts of the bunch do not contribute to the beam-beam forces ! “Shock Wave” in the lab frame
Beam Delivery System 42 Beam-Beam Effect : Round Gaussian Case
• Incoherent Beam-Beam Force γ ± → ∞)( This incoherent beam-beam force generated by a round Gaussian Positron beam Q' ρ r)( = − r σ 22 )2exp( , with = ρπ = )()(2)(' dzzdNerrdrzQ + 2 r ∫ + + 2πσ r r eQ 1' ⎛ r 2 ⎞ on an individual Electron is given by: rF )( ⎜ exp(1 −−−= )⎟rˆ ⎜ 2 ⎟ πε 0 r ⎝ 2σ r ⎠ • Incoherent Beam-Beam Kick The incoherent beam-beam kick Δ r ' is given by integrating the effect of the force over the bunch collision: Δp 1 1 eN 2 1 ⎛ r 2 ⎞ r'=Δ r = )( dtrF = )( dzrF −= + ⎜ exp(1 −− )⎟ r ∫∫ r ⎜ 2 ⎟ pp 2 pc 2πε0 rpc ⎝ 2σ r ⎠ assuming that the position r stays constant over the collision ! Hence the following expression of the 2 rN ⎛ r 2 ⎞ r' + e ⎜ exp(1 −−−=Δ )⎟ weak incoherent beam-beam kick : ⎜ 2 ⎟ γ r ⎝ 2σ r ⎠
Beam Delivery System 43 Beam-Beam Effect : Round Gaussian Case
• Coherent Beam-Beam Kick Without demonstration, the expression of the weak coherent beam-beam kick is: 2 rN ⎛ r2 ⎞ r' + e ⎜ exp(1 −−−=Δ )⎟ C ⎜ 2 ⎟ γ r ⎝ 4σr ⎠
• The Disruption Parameter Dr In the linear regime, r <<σr , the incoherent beam-beam kick can be linearized as a thin focusing lens:
r 1 +rN e Dr r' −=Δ with 2 ≡= ff r σσγz
• Coulombic Interaction:
In the far distance regime, r >>σr , both the individual electron (incoherent kick) and the electron bunch (coherent kick) reach the point-like Coulombic interaction.
Beam Delivery System 44 Incoherent Beam-Beam Effect : Gaussian Flat Case
• Incoherent Beam-Beam Kick
For flat Gaussian beams σ ≠ σ yx the incoherent beam-beam kick is given by: ⎛ ⎛ σ σ ⎞⎞ ⎜ ⎜ y +iyx x ⎟⎟ ⎛ ⎞ 2 2 ⎛Δx'⎞ rN 2π ⎛Im⎞⎜ +iyx ⎛ yx ⎞ ⎜ σ x σ y ⎟⎟ ⎜ ⎟ = + e ⎜ ⎟ w⎜ ⎟ exp⎜ −−− ⎟w ⎜ ⎟ 22 ⎜ ⎟⎜ ⎜ 22 ⎟ ⎜ 2 2 ⎟ ⎜ 22 ⎟⎟ Δy' γ −σσyx Re 2 −σσ 22 σσyx 2 −σσ ⎝ ⎠ ⎝ ⎠⎜ ⎝ yx ⎠ ⎝ ⎠ ⎜ yx ⎟⎟ ⎜ ⎜ ⎟⎟ ⎝ ⎝ ⎠⎠ −z2 where −= izerfcezw )()( is the complex error function. Using the following expansions: 2iz 1 zw 1)( ++= 2 zz →0for)(O and zw )( .+= z−2 for))(O1( z ∞→ π z π the round beam case can be retrieved and the linearized form, projected on each x and y plane, is given by: σ 2 rN σ Δx Δy , with the disruption z + ze x −=Δ y',' −=Δ D ,yx =≡ parameters : f x f y f ,yx , +σσγσyxyx )( • Coherent Beam-Beam Kick The coherent beam-beam kick is not amenable to easy analytic treatment.
Beam Delivery System 45 Focusing Effect of a Sheet (ultra-flat) Beam
The ultra-flat limit of a flat positron bunch is given by an infinite surface density of charge σ whose motion is generating surface current JS=σ c . E y B
+++++++++++++++++++++++++++++++++++++• +++++++++++++ zˆ x E B
The electric and magnetic fields are given by the well known expressions: r σ r = ˆ and −= μ ˆ yxJByE > .0for ε 0 S 0 r The force acting on an individual electron is given by −= 2 εσ 0 yeF ˆ for y > 0 . The important result is that the e.m. fields are constant over space, and the resulting forces have an infinite vertical range !
In the realistic case where σ < < σ < < σ zxy , the vertical range of the beam- beam forces is set by the smallest ‘infinite’ size, i.e. by σ x .
Beam Delivery System 46 Strong Beam-Beam Forces
The preceding expressions assume that the beam-beam effect is weak and that the bunch particle trajectory are not perturbed during the collision. In Linear Colliders, this assumption is correct for the horizontal motion.
On the contrary, the vertical beam-beam forces are strong because σy* is very small, (a few nanometres) and the vertical charge densities are very large. Hence the particle trajectories are bent towards the other beam. There are 3 main consequences: 1. the Pinch Effect: the bunch are not rigid but they are pinched during the collision.
This induces an enhancement of the luminosity by a factor HD~1.4 –2 , 2. the Breamstrahlung effect : the individual electrons and positrons are emitting Synchrotron Radiation hard Photons while their trajectories are curved. This generates energy loss (~2-3 %), energy spread and secondary background.
3. the Kink Instability: in the high disruption regime, D y > 20 , individual trajectories are kinked, i.e. oscillate through the opposite bunch leading to unstable regime. 2 nD yd Δy' y nσ z y 2 ≈ osc. ≈⇒−= 2 σπλyz nfn osc. =≈⇒ ds n z σσfn yz osc. 2πλ
Beam Delivery System 47 The Pinch Effect
ILC parameters:
Dy ~12
Luminosity enhancement
HD ~ 1.4
Not much of an instability.
Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).
Beam Delivery System 48 Beam-Beam Instability Effects
N × 2 Dy ~ 24
Beam-beam instability is clearly pronounced.
Luminosity enhancement is compromised by higher sensitivity to initial offsets.
Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).
Beam Delivery System 49 Beam-Beam Simulations
Dy ~ 12
Nx2
Dy ~ 24
Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).
Beam Delivery System 50 Beamstrahlung
Electrons radiate against the coherent field of the positron bunch synchrotron radiation – called ‘Beamstrahlung’. e− The average magnetic field of a γ Gaussian bunch is given by: e+ e− + γ F 5 mcrN Ne − B == + e e 2ec e + )(6 σσσzyx
Introducing Schwinger’s critical field: γ 22 〈B〉 = 0.32 T @ LEP, 60 T @ SLC , 440 T @ ILC cm 9 BS . ⋅== T1044 eh ILC 500 ‘Beamstrahlung’ is traditionnally characterized by the Υ parameter: Υ 0.05 B 5 γ Nr D ee γ =≡ϒ nγ 1.3 B S + )(6 σσσzyx δBS 2 % with = h the electron Compton wavelength. D e mc Beam Delivery System 51 The Beamstrahlung Parameter
Beamstrahlung causes a spread in the e+ e− center of mass energy.
This effect is characterized by the Beamstrahlung parameter δBS : ΔE er 3 ⎛ 2E ⎞ N 2 δ = beam ≈ 86.0 e ⎜ 0 ⎟ BS 2 ⎜ ⎟ 2 E0 2 0cm ⎝ ⎠ +σσσyxz )(
Since the luminosity is given by : 2 b fNn rep L = H D 4 σπσyx one would like σxσy small to maximise L , AND (σx+σy) large to reduce δBS .
This is achieved by using Flat Beams : σx >>σy ⎛ 2 ⎞ NE 2 in such a way that δ ∝ ⎜ 0 ⎟ does not depend on σ . BS ⎜ ⎟ 2 y ⎝ σ z ⎠ 4σ x
Thus, one can make σy as small as possible to achieve high luminosity, while δBS is fixed by σx .
Beam Delivery System 52 Beamstrahlung Numbers
5 γ Nr The average value for Gaussian beams : =ϒ D ee + )(6 σσσzyx
max 4.2 ϒ≈ϒ
⎡ασ ⎤ ϒ n ≈ 54.2 z Photons per electron : γ ⎢ ⎥ 32 ⎣ D eγ ⎦ 1 ϒ+
2 ΔE ⎡ σα⎤ ϒ δ −= ≈ 24.1 ze Average energy loss : BS ⎢ ⎥ 32 2 E ⎣ D eγ ⎦ []ϒ+ )5.1(1
D = r α eee
ILC 500 δBS and nγ (and hence ϒ) directly relate to luminosity spectrum and backgrounds. Υ 0.05 nγ 1.3
δBS [%] 2
Beam Delivery System 53 Beam-Beam Deflection
Colliding with offset, e+ and e- beams deflect each other. Tiny offsets (~nm) at IP cause large angular kick (~100 µrad) and therefore large offsets of the beam a few meter downstream.
Beam Delivery System 54 Beam-Beam Deflection
Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).
Beam Delivery System 55 Coherent Beam-Beam Deflection Curves
Horizontal beam-beam curve Vertical beam-beam curve
The outstanding question of IP collisions is: How can one bring two 5 nm-wide spots to collide at the same point when the mechanical motions and vibrations are of the order of micrometers, or more ? The beam-beam deflection brings two fondamental ingredients of the answer: 1. A high sensitivity to small offsets: y ⇒=Δ Δy ≈ µrad20'nm1* Δ⇒ y 1@ ≈ µm20m
2. Long range forces: the range of vertical beam-beam is set by σx* ~ 1 µm
Beam Delivery System 56 Collision Orbit Feedback Two additional ingredients permit the fast feedback system to correct the IP offset: • the long pulse: the ILC pulse with 2820 bunches separated by 340 ns (~100m) is suitable to bunch- to-bunch correction system (analogy with a machine- gun shooting a moving target ☺, and the eyes motion), • beam position monitors with 1 µm resolution can be used to measure and zero the B-B deflections. As a result, no active mechanical stabilization of final doublet is required up to ~ 100 nm jitter tolerance ( f < 3 GHz).
14 mrad Collisions Head-on Collisions
Beam Delivery System 57 e+ e− Pair Production
Beamstrahlung photons and beam Spectrum of pairs particles interact, and create e+ e− pairs. Three processes are important for incoherent pair production ( ϒ<0.6 ): Breit-Wheeler process (γγ e+e-) Bethe-Heitler process (e±γ e±e+e-) Landau-Lifshitz process (e+e- e+e-e+e-) Nota Bene: the number of Real Photons in the process equals by the number of ‘r’ in the Physicists names. Calorimeter Monitor Luminosity
Pairs are low energy particles affected by the beam (focused or defocused), hence a source of background. But they are curled by the field of the detector solenoid: B-H and L-L pairs provide a powerful signal proportional to the luminosity because they are produced by overlapping the e+ and e− beam distributions.
Beam Delivery System 58 ILC Beam Parameters at Collision
Parameter Unit ILC nominal
GeV 250 E0 N 1010 2 * nm 639 σx * nm 5.7 σy μm 300 σz 2820 nb Hz 5 frep Δt ns 340 L 1034cm-2s-1 2.0 %2 δBS 1.3 nγ
Beam Delivery System 59 Question n°4:
Do we need Correction Systems?
Beam Delivery System 60 Beam Linear Optics
Beam Delivery System 61 ILC BDS Optical Functions
Beam Delivery System 62 “Horizontal” Planar Symmetry
Most accelerators are designed with simplifying assumptions : 1. a planar “horizontal” reference trajectory 2. an electromagnetic circuit symmetric with respect to the “horizontal” plane: ⎛ x⎞ ⎛ x ⎞ Sy :⎜ ⎟ → ⎜ ⎟ ⎝ y⎠ ⎝− y⎠
Under these assumptions, the 6D Transfer Map M ss 21 ),( satisfies: ⋅ X )( = ⋅ X )( → 21 yss MSSM →ssy 21
The Linearized Transfer Map (a matrix R) commute with the matrix Sy ⎛ 001 ⎞ ⎜ 2 ⎟ with =⋅ yy ⋅ RSSR S y ⎜ −= 2 010 ⎟ ⎜ ⎟ ⎝ 100 2 ⎠
Beam Delivery System 63 General Form of Transfer Matrix
The planar symmetric 6D transfer matrix simplifies to the following 2x2 blocks: ⎛ 0 DR ⎞ ⎜ x x ⎟ R = ⎜ Ry 00 ⎟ ⎜ ⎟ ⎝ z 0 RE z ⎠ in such a way that the vertical motion is decoupled from the two other planes. We make two more assumptions: 1. The BDS is a static system with no time dependence: hence trajectories do not depend on the z coordinate; 2. The energy is constant : no acceleration, energy loss due to synchrotron radiation is neglected at the first order is constant. ⇒δ = δ pp 0 Under these assumptions, the 6D transfer matrix further simplifies to ⎛ rr ⎞ ⎛0 D ⎞ ⎛ rr ⎞ ⎜ 5655 ⎜ x ⎟ 5251 Rz = ⎜ ⎟ , Dx = ⎜ ⎟ , Ez = ⎜ ⎟ ⎝ 10 ⎠ ⎝ Dx '0 ⎠ ⎝ 00 ⎠
Beam Delivery System 64 Symplecticity
The Transfer Matrix R is symplectic provided that : ⎛ Τ + Τ EJERJR 0 Τ + Τ RJEDJR ⎞ ⎜ 2 2 zzxx 2 2 zzxx ⎟ Τ Τ JRR = ⎜ 0 2 RJR yy 0 ⎟ = J ⎜ Τ Τ Τ Τ ⎟ ⎝ 2 + 2 EJRRJD zzxx 0 2 + 2 DJDRJR zzzz ⎠ Τ Note that, for a 2x2 matrix, 2 = R )det( JRJR 2 , so that ⎛ RJ )det( 0 Τ + Τ RJEDJR ⎞ ⎜ 2 x 2 2 zzxx ⎟ Τ JRR = ⎜ 0 2 RJ y )det( 0 ⎟ ⎜ Τ Τ ⎟ ⎝ 2 + 2 EJRRJD zzxx 0 2 RJ z )det( ⎠ The symplecticity condition is fulfilled if :
x = y = rRR 55 =1)det()det(
and = 1251 x ,' 52 = − 22 DrrDrr x
Notice that r 56 is ‘free’.
Beam Delivery System 65 Example : Transport Matrix Through a Quadrupole
We consider a straight line reference trajectory r ref )( = sssR ˆ with = 0tvs ⎛ y ⎞ r ⎜ ⎟ passing through a quadrupole of length lQ with a field : = GB ⎜ x ⎟ which is null along the reference trajectory (no curvature, h = 0 ). ⎜ 0 ⎟ d r ⎝ ⎠ The equations of motion γ )( rr ×= Bvqvm can be simplified since: dt 1. the energy is constant (magnetic fields do accelerate, but do not ‘work’); 2. the reference frame ( ˆ , ˆ , syx ˆ ) is constant; ⎛ ' vx 0 ⎞ d d ds d d rr ⎜ ⎟ 3. == v0 and v x == ⎜ ' vy 0 ⎟ ; dt ds dt ds dt ⎜ v ⎟ − pp 2 vc 2 − 1 ⎝ z ⎠ 4. 0 0 ; δ = = 22 1 vv 0 +=⇒− δ ))(O1( p 0 vc − 1 2 222 22 5. z 0 0 +=+−= δ yxOvyxvvv )',',()''(
Beam Delivery System 66 Example : Transport Matrix Through a Quadrupole
At the first order in the variables zyyxx δ ),,',,',( the equations of motion lead to:
• Transverse motion: ⎛ x" ⎞ G ⎛ − x ⎞ ⎛ − x ⎞ ⎜ ⎟ = ⎜ ⎟ = K 1 ⎜ ⎟ ⎝ y"⎠ 0 qp ⎝ y ⎠ ⎝ y ⎠
• Longitudinal motion: The energy is constant : δ = 0' The path length difference is given by: d sdt )( r sxd )( = vz 1))()((' −=− vstst 0 ds ref 0 r sxd )( ds v 0 yx 22 +=++−= δ yx 22 )',',(O0''11 v
Beam Delivery System 67 Example : Transport Matrix Through a Quadrupole
This equation can be easily solved to obtain the following transfer matrix: ⎛ R 00 ⎞ ⎜ x ⎟ R = ⎜ 0 R y 0 ⎟ ⎜ ⎟ ⎝ 100 2 ⎠
⎛ lK )sin()cos( KlK ⎞ R = ⎜ Q1 Q1 1 ⎟ x ⎜ ⎟ with ⎝− 1 lKK Q1 lK Q1 )cos()sin( ⎠ ⎛ cosh( lK )sinh() KlK ⎞ R = ⎜ Q1 Q1 1 ⎟ y ⎜ ⎟ ⎝ 1 lKK Q1 cosh()sinh( lK Q1 ) ⎠ assuming G > 0.
For G < 0 , the matrices x and RR y are interchanged.
Beam Delivery System 68 General Form of Beam Matrix with Planar Symmetry
The general form of the Beam Matrix (cf. Part 1, page 51-53) which fulfils a) planar symmetry, b) no time correlations, and c) invariance of δ, simplifies to ⎛ − αεβε 000 ση2 ⎞ ⎜ xx xx x δ ⎟ ⎜− γεαε '000 ση2 ⎟ ⎛ Σ 0 H ⎞ xxxx x δ ⎜ x x ⎟ ⎜ 00 − αεβε 00 ⎟ ⎜ yy yy ⎟ Σ = ⎜ Σ y 00 ⎟ = ⎜ ⎟ ⎜ Τ ⎟ 00 − γεαεyyyy 00 ⎝ H x 0 Σ z ⎠ ⎜ 2 ⎟ ⎜ 0000 σ z 0 ⎟ ⎜ 2 2 2 ⎟ ⎝ x δ x σησηδ 000' σ δ ⎠
because the correlation coefficients yy δδ == 0' by symmetry Sy. Nota Bene: these assumptions are useful to design the Beam Delivery System at the 0th order (cf. page 62). They are not adapted to investigate the effects of misalignments and errors in the beam line, hence the implementation of correction systems e.g. y-orbit correction.
Beam Delivery System 69 2D Beam-Matrix Transport Assuming, for convenience, that the average beam offsets are zero, X = 0 , the beam matrix is transported from s1 to s2 as follows: Σ XX Τ Σ Τ RXX Σ ⋅⋅=⋅=→⋅= RΤ 111 2 22 →ss 21 1 →ss 21
• 2D Vertical Transport: Τ Σ y 2 Ry 21 Σ y 1 Ry →⋅⋅→= ssssss 21 )()()()(
Clearly, the emittance ε y = Σ y s ))(det( is conserved The vertical beam Beta-functions , also called Twiss Parameters, transform as follows: ⎛ β ⎞ ⎛ R2 − 2 RRR 2 ⎞⎛ β ⎞ ⎜ 2 ⎟ ⎜ 11 1211 12 ⎟⎜ 1 ⎟ ⎜α 2 ⎟ = ⎜− 2111 + 21 2112 − RRRRRR 2212 ⎟⎜α1 ⎟ ⎜ ⎟ ⎜ 2 2 ⎟⎜ ⎟ ⎝ γ 2 ⎠ ⎝ R21 − 2 2221 RRR 22 ⎠⎝ γ 1 ⎠
Beam Delivery System 70 Normalized Variables
Conversely, given the initial and final beam matrices , ΣΣ 21 , the transfer matrix can be determined up to a phase. Using the Normalized Variables (cf. Part1, page 52) ⎛u⎞ ⎛ y ⎞ ⎛ 1 β 0 ⎞⎛ y ⎞ ⎜ ⎟ S −1 ⋅= ⎜ ⎟ = ⎜ y ⎟⎜ ⎟ ⎜ v ⎟ ⎜ y'⎟ ⎜ ββα⎟⎜ y'⎟ ⎝ ⎠ ⎝ ⎠ ⎝ yy y ⎠⎝ ⎠ ⎛ε 0⎞ Τ Τ such that Σ = ⎜ ⎟ = ε SSSS , one gets: ⎝0 ε ⎠ Τ Σ 2 R ss Σ121 R →⋅⋅→= ss 21 )()( Τ ΤΤ 22 RSS ss 1121 RSS →⋅⋅→=⇒ ss 21 )()( −1 −1 Τ 2 RS ss 2121 RSS ss S121 =⋅→⋅⋅→⋅⇒ 1))()()(( −1 is a 2D orthogonal matrix. 2 RS ss 121 =⋅→⋅⇒ OS ψ 12 )()( ⎛u⎞ ⎛u⎞ The rotation matrix O ψ 12 )( is the ⎜ ⎟ s2 O ψ 12 ⋅= ⎜ ⎟ s1)()()( transfer matrix of the normalized variables: ⎝ v ⎠ ⎝ v ⎠
Beam Delivery System 71 The Phase Advance
As a result: −1 R ss 221 ψ )()( ⋅⋅=→ SOS 112
The parameter ψ 12 such that : ⎛ ψ ψ )sin()cos( ⎞ ⎜ 12 12 ⎟ O ψ 12 )( = ⎜ ⎟ ⎝− 12 ψψ12 )cos()sin( ⎠ is called the phase advance. It allows to parameterize the transfer matrix as follows: ⎛ β ⎞ ⎜ 2 + ψαψ))sin()(cos( ψββ)sin( ⎟ ⎜ β 112 12 21 12 ⎟ R ss )( =→ 1 21 ⎜ − ++ ψααψαα)sin()1()cos()( β ⎟ ⎜− 12 12 21 12 1 − ψαψ))sin()(cos( ⎟ ⎜ β 212 12 ⎟ ⎝ ββ21 2 ⎠ One can check that when R i → ss 212 = ∂ R is → 211 iss = 2,1),()( 1 s2 1 ψ ss )( ψ )( =→⇒=→∂ dsss s2 21 β s )( 21 ∫ β s)( 2 s1 Beam Delivery System 72 2D Chromatic Filamentation Ideal beam transport of a polychromatic bunch assumes that: 1) the centroids of all energy slices are aligned (no banana shape) 2) the emittance and Twiss parameters of all energy slices are identical (no butterfly shape) 3) the beam is matched to the ideal beam-line (requires a precise definition for single pass beam lines). Chromatic Filamentation is an emittance growth effect which occurs as soon as one of these conditions is broken, allowing an accumulation of phase errors along very long beam lines like the Main Linac or the BDS . v Σ s1 δ ),( We consider the 3rd source of filamentation, pictured as shown in the normalized phase space of the matched beam at the position s1 u where the mismatch shows up.
Σ s1M δ ),(
Beam Delivery System 73 2D Chromatic Filamentation
The outgoing beam matrix at the position s2 can be calculated as follows: Σ = Σ = dsds δρδδδρδ⋅ XX Τ ss )()()();()()( 2 ∫∫2 2 2 δ = δρδR → δ ⋅ Τ RXX Τ → ssssssd δ );()()();()( ∫ 21 1 1 21 Τ = d s SOS −1 ;())(();()( δδψδδρδΣ S −1 sss O ΤΤ S Τ s δδψδ);())(();()() ∫ 2M 12 M 1 M1 1 12 2M Τ = d δρδ s δψSOS −1 Σ −1 sss OS Τ δψS Τ + os δ )()())(()()()())(()()( ∫ 2M 12 M M11 1 12 2M −1 −1Τ Τ Τ = SM 2 ()ds 12 δψδρδSO M Σ M11 sss 1 OS 12 δψSM 2 + os δ )()())(()()()())(()()( ∫ v Τ ≈ S ()ds ψψ⋅[]SO −1 Σ −1 sss ⋅ Τ ψ SOS Τ s )()()()()()()( M 2 ∫ M M11 1 M 2
⎛ 1 −1 −1Τ ⎞ Τ u = SM s2 )( ⎜ []SM Σ M11 21 ⎟S1S M ssss 2 )()()()(trace ⎝ 2 ⎠ 1 −1 1 = []Σ ()Τ SSSS Τ sssss )()()()()(trace = ()−1 ΣΣΣ sss )()()(trace 2 1M1M1 M 2 M 2 2 M1 2M1
1 -1 After filamentation, the emittance is given by εε tr( Σ⋅Σ= ) M 2 M
Beam Delivery System 74 The Perturbative Approach to Higher Order Optics
In a single pass, based on TRANSPORT notations, the change of the co- ordinate vector is given by
⎛ x ⎞ ⎜ ⎟ ⎜ x' ⎟ ⎜ y ⎟ X = ⎜ ⎟ ⎜ y' ⎟ ⎜ ⎟ ⎜ z ⎟ ⎜ ⎟ ⎝ δ ⎠
out in in in in in in i jji += jkji k + jnkji k n + ...xxxUxxTxRx {j,k,n=1,6}
All terms with one subscript equal to 6 are referred to as chromatic terms, since the effect depends on the momentum deviation of the particle.
All terms without subscript equal to 6 are referred as geometric terms, since the effect depends only on the central momentum.
Beam Delivery System 75 Chromatic Aberrations at the IP
The Final Doublet develops the highest magnetic field, and hence the biggest chromaticity GG G K1 δ )( == = 0 + Bqpqp +δρδ)1()1(
+δ δ = 0 s
−δ
Energy spread ⇒ chromatic aberrations
Beam Delivery System 76 Local Chromatic Correction
Final Quadrupole: G K1 δ )( = ΔK1(+δ) Bρ +δ )1( )( −=Δ⇒ KK δδ +δ Δx(+δ) 1 1
δ = 0 s Dispersion
Δ δ )( = Dx xδ −δ−δ Δx (-δ) Sextupole: ΔK (-δ) 1 Δ 1(δ = 2ΔxKK δ )() Sextupole Magnet in the doublet ⇒ Local Chromatic Correction
for = 12 DKK x
Beam Delivery System 77 Local Chromaticity Correction Scheme
A single sextupole cures the chromaticity but introduces dramatic geometric aberrations, unless they are paired with π-phase advance. IP dipole Dx
sextupoles
⎛⎞m 000 ⎜⎟FD mm21 1/ 0 0 R = ⎜⎟ ⎜⎟00m 0 l* ⎜⎟ ⎝⎠00mm43 1/
Local chromatic corrections introduced in the NLC FFS, and now adopted by all LC designs.
Beam Delivery System 78 End of Part 2
Beam Delivery System 79 Conclusions
The Beam Delivery System of a Superconducting Linear Collider is the scene of many accelerator physics outstanding issues:
• Non linear optics and high order correction systems • High field magnets and superconducting technology • Collimation and wakefields • Synchrotron radiation and emittance growth • Fast feedback system • Luminosity optimisation and monitoring • Strong beam-beam interaction • Novel beam-physics concepts • ...
These topics can only be exhausted with longer lectures and more thorough lecturer.
Thank you for your attention.
Beam Delivery System 80 Acknowledgements
I would like to thank:
• Weiren Chou for extending this invitation,
• Deepa Angal-Kalinin for the permission to re-use her lecture from 2008,
• My colleagues Antoine Chancé, Olivier Delferrière, Jacques Payet, Pierre Védrine for providing expertise, material, corrections and proof-reading,
• My young colleague Reine Versteegen for all the above, plus considerable help in assembling the material presented.
Beam Delivery System 81