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**Beam Delivery System and Beam-Beam Effects**

Fourth International Accelerator School for Linear Colliders Beijing, China September 2009 Lecture A3 Beam Delivery System and Beam-Beam Effects Part 2 Olivier Napoly CEA-Saclay, France Preliminary Version 17 September 2009

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**Collider Parameters, Physical Constants and Notations**

E , energy and p , momentum B , magnetic rigidity (B = p/e) L , luminosity Q , bunch charge N , number of particles in the bunch (N = Q/e) nb , number of bunches in the train frep , pulse repetition rate x and y , horizontal and vertical emittances sx* and sy*, rms horizontal and vertical beam sizes at the IP sz , bunch length ,, , Twiss parameters , tune , phase advance, c = m/s e = C mec2 = eV µ0 = 4 10-7 N A-2 permeability 0 = 1/µ0c permittivity re = m with

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**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). 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.

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**ILC Beam Delivery System Layout (RDR)**

Linac Exit Beam Switch Yard Diagnostics Sacrificial collimators b-collimator E-collimator Final Focus Tune-up & Emergency extraction 14mr IR Tune-up dump Muon wall grid: 100m*1m Main dump Extraction

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**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

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**Contents: Fundamentals in Beam Physics**

Part 1 Luminosity Emittance Magnetism (continued) Part 2 Beam-Beam Effects Beam Optics Synchrotron Radiation

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**Magnetic Measurements**

Magnet fields are measured to checks: the field integral of the main component (Bdl for a dipole, Gdl for a quadrupole) 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: (sometimes with a +) so that

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**Quadrupole Magnets: the Engineer Standpoint**

Given a Central Axis Oz which coincides with the Reference Trajectory, we introduce an “Engineering Symmetry” RS( ) which allows categorize the Magnets into “Engineering Multipoles”. Notations: R( ) is the rotation with angle around Oz . S is the symmetry with respect to the Oxy plane. is the combination of the two. Note that : 1) For a transverse B field (pseudo-vector) 2) For a longitudinal electrical current J 3) x y R( ) O x y B x y B O O J J S

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**Symmetric Current Distributions**

x y Consider a distribution of longitudinal electrical current J(r,) . If a current distribution is invariant with respect to RS then: 1) the integrated current through the plane is zero, since 2) 20 is a sub-multiple of 2, since J(r,) O x y J(r,) + O − zero

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**Multipolar Current Distributions**

We introduce the following categories of current distributions J(r,) : Monopole (n=0): , azimuthally independent distributions Dipole (n=1), such that: Quadrupole (n=2): Sextupole (n=3): Octupole (n=4): Decapole (n=5): …. 2n-pole (n): Note that , for the Monopole :

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**Decomposition of Current Distributions**

The space J of longitudinal current distributions can be decomposed over the sum of symmetric subspaces as follows. Since , one can decompose any current distribution into symmetric (n=1) and antisymmetric components w.r.t. RS(): Since , one can decompose the current distrib- ution into symmetric (n=2) and antisymmetric components w.r.t. RS(/2): etc, etc,… leading to : dipole + quadrupole + octupole + 16-pole + …. + monopole where is invariant under the rotation

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Natural Harmonics Question: Where are the Sextupoles, and Decapoles, … ? Answer: Sextupoles (n=3) and Decapoles (n=5) are Dipole (n=20) Natural Harmonics: Dodecapoles (n=6) and 20-poles (n=10) are Quadrupole (n=21) Natural Harmonics: In general, 2n-poles with n =(2m+1)2p are 2.2p-pole (n=2p) Natural Harmonics: . In other words, using the example of Dipoles, the ensemble of Dipole (RS()-invariant) current distributions includes that of Sextupole (RS(/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.

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**Quadrupoles Have Four Poles**

A Quadrupole magnet can be designed as follows: Define a current distribution J(r,) over a 90° wide sector : Rotate it by the symmetry RS(/2) over the 2nd sector Repeat the RS(/2) operation to the 3rd sector Repeat the RS(/2) operation to the 4th sector x y x y J (1)(r,) J (2)(r,) J (1)(r,) − − − + + + RS(/2) x y J (2)(r,) J (1)(r,) − − + + + + − J (4)(r,) J (3)(r,) −

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**The Simplest Monopole Magnet**

The simplest Monopole magnet is given by longitudinal current line transporting a total current I z . By rotation symmetry : From Ampère’s law : The complex magnetic field is then given by ^ x y B I 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 !

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**A ‘Dipole’ Magnet x y I +I B R0 +**

The simplest Dipole Field configuration is given by a pair of linear conductors: x y I +I + B R0 This expression can be expanded for : Comparing to the mathematician expressions (page 71) one gets : Dipole : ; Sextupole: ; Decapole : (2m+1)-pole : Nota Bene: This ‘Dipole’ Magnet still has zero curvature radius !

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**A Quadrupole Magnet y I +I +I x R0 I + +**

The simplest Quadrupole Field configuration is given by a quadruplet of conductors: x y I +I +I + + R0 I Comparing to the mathematician expressions (page 71) one gets : Quadrupole: Dodecapole:

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**A Better Quadrupole Magnet**

An improved Quadrupole Field configuration is given by an octuplet of conductors: x y I I +I + + +I R0 +I + + +I The calculation proceeds like the preceding one leading to: I I Comparing to the mathematician expressions (page 71) one gets : Quadrupole: Dodecapole: cancels for

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**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 : Quadrupole : Dodecapole: , cancels for

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**An Ideal Quadrupole Magnet: Cos(2 ) distribution**

The engineer meets the mathematician when he tries to build current lines following a “cos(2 )” angular distribution : with The complex magnetic field is given by the following integral: This is a Pure Quadrupole, in the mathematician sense: Quadrupole

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**The Cos(2 ) Quadrupole Magnet**

Assuming a uniform radial distribution of currents : the complex magnetic field is given by the following expression: This is a Pure Quadrupole, in the mathematician sense: Quadrupole

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**Superconducting Quadrupoles**

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**The LHC Cos( ) Dipole Magnet**

LHC Superconducting Dipoles have been wound from the “cos( )” model, using two layers of cables and anglular wedge to fit the “cos( )” distribution. Internal Diameter : F = 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

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**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 : F = 56 mm Current Density: J = 500 A.mm-2 → Quadrupole gradient : G = 223 T/m Current Lines: NI =??? A.t pole n°1

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**LHC Low- Quadrupole Magnets**

LHC Superconducting Low- Quadrupoles are realized from the “cos(2 )” model. Internal Diameter : F = 70 mm Current Density: J = A.mm-2 → Quadrupole gradient : G = 215 T/m Current Lines: NI = ??? A.t Fermilab Low- Quadrupole KEK Low- Quadrupole pole n°1 pole n°2 pole n°4 pole n°3

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**NbTi Superconducting Cables**

NbTi ingot (136 kg) mono-filament rod in Cu can (10 km) multi-filaments cable in Cu can multi-wire Rutherford cable multi-filament cable mono-filament hex. rods ~1mm NbTi Wire (Courtesy Alstom/MSA) “Rutherford” type cable (Courtesy Oremet Wah Chang) (Courtesy Furukawa Electric Co.) High performance mono-filament rods achieve JC of 3000 A/mm2 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 at room temperature: 1.610-8 .m at 10 K, 0 T (RRR =100): 1.610-10 .m at 10 K, 7 T (RRR = 100): 4.310-10 .m NbTi in normal state : 5-610-7 .m in superconducting state :

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**B vs. JC Limitations of SC Conductors**

Material TC (K) Nb 9.2 NbTi 9.5 Nb3Sn 18 10 100 1000 10000 5 15 20 25 30 35 40 45 Applied Field (T) JE (A/mm²) YBCO Insert Tape (B|| Tape Plane) YBCO Insert Tape (B Tape Plane) MgB2 19Fil 24% Fill (HyperTech) 2212 OI-ST 28% Ceramic Filaments NbTi LHC Production 38%SC (4.2 K) Nb3Sn RRP Internal Sn (OI-ST) Nb3Sn High Sn Bronze Cu:Non-Cu 0.3 YBCO B|| Tape Plane YBCO B Tape Plane 2212 RRP Nb3Sn Bronze Nb3Sn MgB2 Nb-Ti SuperPower tape used in record breaking NHMFL insert coil 2007 18+1 MgB2/Nb/Cu/Monel Courtesy M. Tomsic, 2007 427 filament strand with Ag alloy outer sheath tested at NHMFL Maximal JE for entire LHC NbTi strand production (–) CERN-T. Boutboul '07, and (- -) <5 T data from Boutboul et al. MT-19, IEEE-TASC’06) Complied from ASC'02 and ICMC'03 papers (J. Parrell OI-ST) 4543 filament High Sn Bronze-16wt.%Sn-0.3wt%Ti (Miyazaki-MT18-IEEE’04) Nb3Sn conductors allow for increase of field and gradient, but: Cables are more expensive 5-10 and production volume 100, Fabrication of accelerator magnets still at the R&D stage.

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**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: “Air magnets” are very ineffective, as illustrated by this example: Internal Diameter : F =300 mm Current Density: J = 4 A/mm → Quadrupole gradient : G = 0.39 T/m Current Lines: NI =13000 A.t Air Conductors

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**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 with , 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

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**µr H Magnetic Saturation**

Introducing the auxiliary magnetic field H such that , one gets where µr is the relative permeability. First magnetization curve, then hysteresis takes over. Saturation occurs around 2 T 10% for high quality electrical steel or permalloy (nickel-iron). µr 2 T The relative permeability can reach max. values from to H In the saturation regime, B vs. J ‘linearity’ is lost and increasing the electric current is not effective any more. As a consequence: Normal Conducting Magnets are limited to ~2 T fields In Superconducting Magnets, the Iron Yoke is mostly used to contain the transverse stray fields.

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**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 G=144 T/m, Ri =10 mm Head-on, ‘LHC –Arc’ G=250 T/m, Ri =28 mm 2 mrad Xing, ‘LHC low-’ G=250 T/m, Ri =35 mm

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**ILC : 14 mrad Final Focus System**

NbTi cable : serpentine winding technology l* = 3.5 m Quadrupole QD0 Combined sextupole -octupole

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**ILC : Head-on Final Focus System**

IP st nd 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.

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**ILC : 2 mrad Final Focus System**

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**Reasons for the Strong Quadrupoles**

Strong Quadrupoles allow for a short focal distance f ~ l*, needed for 3 reasons : 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. Spent beam : in the Head-on and 2 mrad cases, the main constraint is to clear the outgoing beam and beamstrahlung after the collision: 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*

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**Question n°3: Do we need a Flat Beams ?**

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Beam Beam Effects

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**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

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**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 (qv ): with an internal focusing field for same sign current: i.e. the perfect magnetic focusing lens. x y J B Nota Bene: principle of a Lithium lens used for proton and muon focusing.

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**Focusing Effect of a Round Line Charge Distribution**

A simple circular charge distribution with uniform density of radius R generates: an external field which repels same-sign charge q: with an internal defocusing field for same-sign charge q: x y E

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**Focusing Effect of a Round Positron Beam**

A circular Positron beam (travelling towards us) of radius R is generating both charge and current J distribution with the relation: Effect on a Positron travelling with the beam, at r < R : the electric and magnetic forces cancel, yielding a repelling force vanishing for Effect on an Electron travelling against the beam: the electric and magnetic forces add, yielding a non-vanishing Beam-Beam attracting force for x y (E , B) ( , J)

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**Beam-Beam Effect : Round Case**

Incoherent Beam-Beam Force (assuming ) This beam-beam force can be easily extended to the case of a round Positron beam with a non-uniform circular charge distribution The force acting on an individual Electron is given by: It is called the Incoherent beam-beam force. x y ( (r) , J (r)) Coherent Beam-Beam Force The Coherent beam-beam force is the force acting on the electron bunch, i.e. the average of the forces over the electron distribution : (E , B) However, two ingredients are still missing ! realistic positron and electron bunches are not infinite lines : i.e. colliding positron and electron bunches are not round:

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**The Electromagnetic Field of a Relativistic Particle**

This will be worked out during the Home Work session. Er γ = 0 Lorentz Boost → B 1/c2 v E v = 0 Electrostatic field in the rest frame v < c γ = 1/γ F = q(E + v B) (1 vv/c2) / r No Intra-beam Forces for γ = Strong Beam-Beam Forces No Wake: the Front and Rear parts of the bunch do not contribute to the beam-beam forces ! v = c v = c “Shock Wave” in the lab frame

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**Beam-Beam Effect : Round Gaussian Case**

Incoherent Beam-Beam Force This incoherent beam-beam force generated by a round Gaussian Positron beam , with on an individual Electron is given by: Incoherent Beam-Beam Kick The incoherent beam-beam kick is given by integrating the effect of the force over the bunch collision: assuming that the position r stays constant over the collision ! Hence the following expression of the weak incoherent beam-beam kick :

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**Beam-Beam Effect : Round Gaussian Case**

Coherent Beam-Beam Kick Without demonstration, the expression of the weak coherent beam-beam kick is: The Disruption Parameter In the linear regime, , the incoherent beam-beam kick can be linearized as a thin focusing lens: Coulombic Interaction: In the far distance regime, , both the individual electron (incoherent kick) and the electron bunch (coherent kick) reach the point-like Coulombic interaction.

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**Incoherent Beam-Beam Effect : Gaussian Flat Case**

Incoherent Beam-Beam Kick For flat Gaussian beams the incoherent beam-beam kick is given by: where is the complex error function. Using the following expansions: and the round beam case can be retrieved and the linearized form, projected on each x and y plane, is given by: , with the disruption parameters : Coherent Beam-Beam Kick The coherent beam-beam kick is not amenable to easy analytic treatment.

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**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 whose motion is generating surface current JS= c . x y E B B E The electric and magnetic fields are given by the well known expressions: The force acting on an individual electron is given by for 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 , the vertical range of the beam- beam forces is set by the smallest ‘infinite’ size, i.e. by

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**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 y* 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: 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 , 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. the Kink Instability: in the high disruption regime, , individual trajectories are kinked, i.e. oscillate through the opposite bunch leading to unstable regime.

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**The Pinch Effect Dy ~12 HD ~ 1.4 ILC parameters:**

Luminosity enhancement HD ~ 1.4 Not much of an instability. Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).

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**Beam-Beam Instability Effects**

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).

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**Beam-Beam Simulations**

Dy ~ 12 Nx2 Dy ~ 24 Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).

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**γ Beamstrahlung e- e+ Ne+**

Electrons radiate against the coherent field of the positron bunch synchrotron radiation – called ‘Beamstrahlung’. Ne+ e- γ e+ The average magnetic field of a Gaussian bunch is given by: Introducing Schwinger’s critical field: B = 0.32 LEP, 60 SLC , 440 ILC ILC 500 0.05 ng 1.3 dBS 2 % ‘Beamstrahlung’ is traditionnally characterized by the parameter: with the electron Compton wavelength.

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**The Beamstrahlung Parameter**

Beamstrahlung causes a spread in the e e center of mass energy. This effect is characterized by the Beamstrahlung parameter dBS : Since the luminosity is given by : one would like sxsy small to maximise L , AND (sx+sy) large to reduce dBS . This is achieved by using Flat Beams : sx >>sy in such a way that does not depend on sy . Thus, one can make sy as small as possible to achieve high luminosity, while dBS is fixed by sx .

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**Beamstrahlung Numbers**

The average value for Gaussian beams : Photons per electron : Average energy loss : ILC 500 0.05 ng 1.3 dBS [%] 2 dBS and ng (and hence ) directly relate to luminosity spectrum and backgrounds.

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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.

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Beam-Beam Deflection Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte).

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**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: A high sensitivity to small offsets: Long range forces: the range of vertical beam-beam is set by x* ~ 1 µm

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**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

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**e e Pair Production Spectrum of pairs**

Beamstrahlung photons and beam particles interact, and create e+ e pairs. Three processes are important for incoherent pair production ( <0.6 ): Breit-Wheeler process (gg e+e-) Bethe-Heitler process (e±g 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. Luminosity Monitor Calorimeter 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.

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**ILC Beam Parameters at Collision**

Unit ILC nominal E0 GeV 250 N 1010 2 sx* nm 639 sy* 5.7 sz mm 300 nb 2820 frep Hz 5 Dt ns 340 L 1034cm-2s-1 2.0 dBS % ng 1.3

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**Question n°4: Do we need Correction Systems?**

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Beam Linear Optics

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**ILC BDS Optical Functions**

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**“Horizontal” Planar Symmetry**

Most accelerators are designed with simplifying assumptions : a planar “horizontal” reference trajectory an electromagnetic circuit symmetric with respect to the “horizontal” plane: Under these assumptions, the 6D Transfer Map satisfies: The Linearized Transfer Map (a matrix R) commute with the matrix Sy with

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**General Form of Transfer Matrix**

The planar symmetric 6D transfer matrix simplifies to the following 2x2 blocks: in such a way that the vertical motion is decoupled from the two other planes. We make two more assumptions: The BDS is a static system with no time dependence: hence trajectories do not depend on the z coordinate; The energy is constant : no acceleration, energy loss due to synchrotron radiation is neglected at the first order is constant. Under these assumptions, the 6D transfer matrix further simplifies to

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**Symplecticity The Transfer Matrix R is symplectic provided that :**

Note that, for a 2x2 matrix, , so that The symplecticity condition is fulfilled if : and Notice that is ‘free’.

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**Example : Transport Matrix Through a Quadrupole**

We consider a straight line reference trajectory with passing through a quadrupole of length lQ with a field : which is null along the reference trajectory (no curvature, ). The equations of motion can be simplified since: the energy is constant (magnetic fields do accelerate, but do not ‘work’); the reference frame is constant; and ; ;

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**Example : Transport Matrix Through a Quadrupole**

At the first order in the variables the equations of motion lead to: Transverse motion: Longitudinal motion: The energy is constant : The path length difference is given by:

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**Example : Transport Matrix Through a Quadrupole**

This equation can be easily solved to obtain the following transfer matrix: with assuming G > 0. For G < 0 , the matrices and are interchanged.

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**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 , simplifies to because the correlation coefficients 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.

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**2D Beam-Matrix Transport**

Assuming, for convenience, that the average beam offsets are zero, , the beam matrix is transported from s1 to s2 as follows: 2D Vertical Transport: Clearly, the emittance is conserved The vertical beam Beta-functions , also called Twiss Parameters, transform as follows:

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Normalized Variables Conversely, given the initial and final beam matrices , the transfer matrix can be determined up to a phase. Using the Normalized Variables (cf. Part1, page 52) such that , one gets: is a 2D orthogonal matrix. The rotation matrix is the transfer matrix of the normalized variables:

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**The Phase Advance As a result: The parameter such that :**

is called the phase advance. It allows to parameterize the transfer matrix as follows: One can check that when

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**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 We consider the 3rd source of filamentation, pictured as shown in the normalized phase space of the matched beam at the position s1 where the mismatch shows up. u

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**2D Chromatic Filamentation**

The outgoing beam matrix at the position s2 can be calculated as follows: After filamentation, the emittance is given by u v

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**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 {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.

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**Chromatic Aberrations at the IP**

The Final Doublet develops the highest magnetic field, and hence the biggest chromaticity +d -d d = 0 s Energy spread chromatic aberrations

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**Local Chromatic Correction**

Final Quadrupole: DK1(+d) DK1(-d) -d Dx (-d) +d Dx(+d) -d +d Dx(+d) Dx (-d) d = 0 s Dispersion Sextupole: Sextupole Magnet in the doublet Local Chromatic Correction for

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**Local Chromaticity Correction Scheme**

A single sextupole cures the chromaticity but introduces dramatic geometric aberrations, unless they are paired with -phase advance. Local chromatic corrections introduced in the NLC FFS, and now adopted by all LC designs.

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End of Part 2

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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.

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**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.

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