52ae25a81abe6baa4adb6d4b8e5156f8.ppt

- Количество слайдов: 93

Beam Delivery Andrei Seryi SLAC International Accelerator School for Linear Colliders 19 -27 May 2006, Sokendai, Hayama, Japan

Linear Collider – two main challenges • Energy – need to reach at least 500 Ge. V CM as a start • Luminosity – need to reach 10^34 level 2

The Luminosity Challenge at SLC • Must jump by a Factor of 10000 in Luminosity !!! (from what is achieved in the only so far linear collider SLC) • Many improvements, to ensure this : generation of smaller emittances, their better preservation, … • Including better focusing, dealing with beam-beam, safely removing beams after collision and better stability 3

How to get Luminosity • To increase probability of direct e+e- collisions (luminosity) and birth of new particles, beam sizes at IP must be very small • E. g. , ILC beam sizes just before collision (500 Ge. V CM): 500 * 5 * 300000 nanometers (x y z) Vertical size is smallest 4 5 50 0 000 3

BDS: from end of linac to IP, to dumps BDS 5

BDS subsystems • As we go through the lecture, the purpose of each subsystem should become clear 6

Beam Delivery System challenges • Focus the beam to size of about 500 * 5 nm at IP • Provide acceptable detector backgrounds – collimate beam halo • Monitor the luminosity spectrum and polarization – diagnostics both upstream and downstream of IP is desired • Measure incoming beam properties to allow tuning of the machine • Keep the beams in collision & maintain small beam sizes – fast intra-train and slow inter-train feedback • Protect detector and beamline components against errant beams • Extract disrupted beams and safely transport to beam dumps • Minimize cost & ensure Conventional Facilities constructability 7

How to focus the beam to a smallest spot? • If you ever played with a lens trying to burn a picture on a wood under bright sun, then you know that one needs a strong and big lens (The emittance e is constant, so, to make the IP beam size (e b)1/2 small, you need large beam divergence at the IP (e / b)1/2 i. e. short-focusing lens. ) • It is very similar for electron or positron beams • But one have to use magnets 8

Recall couple of definitions • Beta function b characterize optics • Emittance e is phase space volume of the beam • Beam size: (e b)1/2 • Divergence: (e/b)1/2 • Focusing makes the beam ellipse rotate with “betatron frequency” • Phase of ellipse is called “betatron phase” 9

What we use to handle the beam Etc… Just bend the trajectory Focus in one plane, defocus in another: x’ = x’ + G x y’ = y’– G y Second order effect: x’ = x’ + S (x 2 -y 2) y’ = y’ – S 2 xy Here x is transverse coordinate, x’ is angle 10

Optics building block: telescope final doublet (FD) Essential part of final focus is final telescope. It “demagnify” the incoming beam ellipse to a smaller size. Matrix transformation of such telescope is diagonal: f 1 A minimal number of quadrupoles, to construct a telescope with arbitrary demagnification factors, is four. If there would be no energy spread in the beam, a telescope could serve as your final focus (or two telescopes chained together). 11 f 2 IP f 2 (=L*) Use telescope optics to demagnify beam by factor m = f 1/f 2= f 1/L* Matrix formalism for beam transport:

Why nonlinear elements • As sun light contains different colors, electron beam has energy spread and get dispersed and distorted => chromatic aberrations • For light, one uses lenses made from different materials to compensate chromatic aberrations • Chromatic compensation for particle beams is done with nonlinear magnets – Problem: Nonlinear elements create geometric aberrations • The task of Final Focus system (FF) is to focus the beam to required size and compensate aberrations 12

How to focus to a smallest size and how big is chromaticity in FF? Size: (e b)1/2 Angles: (e/b)1/2 Size at IP: L* (e/b)1/2 L* IP • The final lens need to be the strongest • ( two lenses for both x and y => “Final Doublet” or FD ) • FD determines chromaticity of FF • Chromatic dilution of the beam size is Ds/s ~ s. E L*/b* Typical: s. E -- energy spread in the beam ~ 0. 002 -0. 01 L* -- distance from FD to IP b* -- beta function in IP ~ 3 - 5 m ~ 0. 4 - 0. 1 mm + (e b)1/2 s. E Beta at IP: L* (e/b)1/2 = (e b* )1/2 => b* = L*2/b Chromatic dilution: (e b)1/2 s. E / (e b* )1/2 = s. E L*/b* • For typical parameters, Ds/s ~ 15 -500 too big ! • => Chromaticity of FF need to be compensated 13

Example of traditional Final Focus Sequence of elements in ~100 m long Final Focus Test Beam beam Focal point Dipoles. They bend trajectory, but also disperse the beam so that x depend on energy offset d Necessity to compensate chromaticity is a major driving factor of FF design 14 Sextupoles. Their kick will contain energy dependent focusing x’ => S (x+ d)2 => 2 S x d +. . y’ => – S 2(x+ d)y => -2 S y d +. . that can be used to arrange chromatic correction Terms x 2 are geometric aberrations and need to be compensated also

Final Focus Test Beam Achieved ~70 nm vertical beam size 15

Synchrotron Radiation in FF magnets v

Beam-beam (Dy, d. E , ) affect choice of IP parameters and are important for FF design also • Luminosity per bunch crossing • “Disruption” – characterize focusing strength of the field of the bunch (Dy ~ sz/fbeam) • Energy loss during beam-beam collision due to synchrotron radiation 17 • Ratio of critical photon energy to beam energy (classic or quantum regime)

Beam-beam effects HD and instability Dy~12 Nx 2 Dy~24 18

Factor driving BDS design • Chromaticity • Beam-beam effects • Synchrotron radiation – let’s consider it in more details 19

Let’s estimate SR power Energy in the field left behind (radiated !): Field left behind the volume v = c The field v

Let’s estimate typical frequency of SR photons For g>>1 the emitted photons goes into 1/g cone. During what time Dt the observer will see the photons? A B Observer v e

Let’s estimate energy spread growth due to SR We estimated the rate of energy loss : The photon energy And the characteristic frequency where (per angle q : Number of photons emitted per unit length The energy spread DE/E will grow due to statistical fluctuations ( ) of the number of emitted photons : Which gives: Compare with exact formula: 22 )

Let’s estimate emittance growth rate due to SR /E DE When a photon is emitted, the particle starts to oscillate around new equilibrium orbit h E or qui bi lib t ri fo um r E E fo qui r lib E+ ri DE um or bi t Dispersion function h shows how equilibrium orbit shifts when energy changes Emit photon Amplitude of oscillation is Compare this with betatron beam size: And write emittance growth: Resulting estimation for emittance growth: Compare with exact formula (which also takes into account the derivatives): 23

Let’s apply SR formulae to estimate Oide effect (SR in FD) Final quad IP divergence: Energy spread obtained in the quad: IP size: R L L* Radius of curvature of the trajectory: Growth of the IP beam size: Which gives This achieve minimum possible value: ( where C 1 is ~ 7 (depend on FD params. )) When beta* is: Note that beam distribution at IP will be non-Gaussian. Usually need to use tracking to estimate impact on luminosity. Note also that optimal b may be smaller than the sz (i. e cannot be used). 24

Concept and problems of traditional FF • Chromaticity is compensated by sextupoles in dedicated sections • Geometrical aberrations are canceled by using sextupoles in pairs with M= -I X-Sextupoles Y-Sextupoles Chromaticity arise at FD but pre-compensated 1000 m upstream Problems: • Chromaticity not locally compensated – Compensation of aberrations is not ideal / since M = -I for off energy particles – Large aberrations for beam tails – … 25 Traditional FF Final Doublet

FF with local chromatic correction • Chromaticity is cancelled locally by two sextupoles interleaved with FD, a bend upstream generates dispersion across FD • Geometric aberrations of the FD sextupoles are cancelled by two more sextupoles placed in phase with them and upstream of the bend 26

Local chromatic correction • The value of dispersion in FD is usually chosen so that it does not increase the beam size in FD by more than 10 -20% for typical beam energy spread 27

Chromatic correction in FD x+hd sextup. quad • Straightforward in Y plane • a bit tricky in X plane: IP KS KF Quad: chromaticity Sextupole: 28 Second order dispersion If we require KSh = KF to cancel FD chromaticity, then half of the second order dispersion remains. Solution: The -matching section produces as much X chromaticity as the FD, so the X sextupoles run twice stronger and cancel the second order dispersion as well.

Traditional and new FF Traditional FF, L* =2 m A new FF with the same performance can be ~300 m long, i. e. 6 times shorter New FF, L* =2 m new FF 29

New Final Focus • One third the length - many fewer components! • Can operate with 2. 5 Te. V beams (for 3 5 Te. V cms) • 4. 3 meter L* (twice 1999 design) 1999 Design 30 2000 Design

IP bandwidth Bandwidth is much better for New FF 31

Aberrations & halo generation • Traditional FF generate beam tails due to aberrations and it does not preserve betatron phase of halo particles • New FF has much less aberrations and it does not mix phases particles Beam at FD Incoming beam halo Traditional FF New FF 32 Halo beam at the FD entrance. Incoming beam is ~ 100 times larger than nominal beam

Beam halo & collimation • Even if final focus does not generate beam halo itself, the halo may come from upstream and need to be collimated Vertex Detector AFD Halo qhalo= AFD / L* Beam qbeam= e / s* Final Doublet (FD) 33 L* • Halo must be collimated upstream in such a way that SR g & halo e+- do not touch VX and FD • => VX aperture needs to be somewhat larger than FD aperture • Exit aperture is larger than FD IP or VX aperture • Beam convergence depend on parameters, the halo convergence is fixed for given geometry => qhalo/qbeam (collimation depth) becomes tighter with larger L* or smaller IP beam size

More details on collimation • Collimators has to be placed far from IP, to minimize background • Ratio of beam/halo size at FD and collimator (placed in “FD phase”) remains collimator • Collimation depth (esp. in x) can be only ~10 or even less • It is not unlikely that not only halo (1 e-3 – 1 e-6 of the beam) but full errant bunch(s) would hit the collimator 34

MPS and collimation design • The beam is very small => single bunch can punch a hole => the need for MPS (machine protection system) • Damage may be due to – electromagnetic shower damage (need several radiation lengths to develop) – direct ionization loss (~1. 5 Me. V/g/cm 2 for most materials) • Mitigation of collimator damage – using spoiler-absorber pairs • thin (0. 5 -1 rl) spoiler followed by thick (~20 rl) absorber – increase of beam size at spoilers – MPS divert the beam to emergency extraction as soon as possible 35 Picture from beam damage experiment at FFTB. The beam was 30 Ge. V, 3 -20 x 109 e-, 1 mm bunch length, s~45 -200 um 2. Test sample is Cu, 1. 4 mm thick. Damage was observed for densities > 7 x 1014 e-/cm 2. Picture is for 6 x 1015 e-/cm 2

Spoiler-Absorber & spoiler design Thin spoiler increases beam divergence and size at the thick absorber already sufficiently large. Absorber is away from the beam and contributes much less to wakefields. 36 Need the spoiler thickness increase rapidly, but need that surface to increase gradually, to minimize wakefields. The radiation length for Cu is 1. 4 cm and for Be is 35 cm. So, Be is invisible to beam in terms of losses. Thin one micron coating over Be provides smooth surface for wakes.

Spoiler damage Temperature rise for thin spoilers (ignoring shower buildup and increase of specific heat with temperature): The stress limit based on tensile strength, modulus of elasticity and coefficient of thermal expansion. Sudden T rise create local stresses. When DT exceed stress limit, micro-fractures can develop. If DT exceeds 4 Tstress, the shock wave may cause material to delaminate. Thus, allowed DT is either the melting point or four time stress limit at which the material will fail catastrophically. 37

Survivable and consumable spoilers • A critical parameter is number of bunches #N that MPS will let through to the spoiler before sending the rest of the train to emergency extraction • If it is practical to increase the beam size at spoilers so that spoilers survive #N bunches, then they are survivable • Otherwise, spoilers must be consumable or renewable 38

Renewable spoilers This design was essential for NLC, where short inter-bunch spacing made it impractical to use survivable spoilers. This concept is now being applied to LHC collimator system. 39

BDS with renewable spoilers • Location of spoiler and absorbers is shown • Collimators were placed both at FD betatron phase and at IP phase • Two spoilers per FD and IP phase • Energy collimator is placed in the region with large dispersion • Secondary clean-up collimators located in FF part • Tail folding octupoles (see below) are include 40 energy betatron • Beam Delivery System Optics, an earlier version with consumable spoilers

ILC FF & Collimation • Betatron spoilers survive up to two bunches • E-spoiler survive several bunches • One spoiler per FD or IP phase 41 E- spoiler betatron spoilers

MPS in BSY sigma (m) in tune-up extraction line skew correction Energy diag. chicane & MPS energy collimator MPS betatron collimators 4 -wire 2 D e diagnostics kicker, septum polarimeter chicane betatron collimation 42 tune-up dump

Nonlinear handling of beam tails in ILC BDS • Can we ameliorate the incoming beam tails to relax the required collimation depth? • One wants to focus beam tails but not to change the core of the beam – use nonlinear elements • Several nonlinear elements needs to be combined to provide focusing in all directions – (analogy with strong focusing by FODO) • Octupole Doublets (OD) can be used for nonlinear tail folding in ILC FF 43 Single octupole focus in planes and defocus on diagonals. An octupole doublet can focus in all directions !

Strong focusing by octupoles • Two octupoles of different sign separated by drift provide focusing in all directions for parallel beam: Focusing in all directions Next nonlinear term focusing – defocusing depends on j Effect of octupole doublet (Oc, Drift, -Oc) on parallel beam, DQ(x, y). • For this to work, the beam should have small angles, i. e. it should be parallel or diverging 44

Tail folding in ILC FF • Two octupole doublets give tail folding by ~ 4 times in terms of beam size in FD • This can lead to relaxing collimation requirements by ~ a factor of 4 Oct. QD 6 QD 0 QF 1 45 Tail folding by means of two octupole doublets in the ILC final focus Input beam has (x, x’, y, y’) = (14 mm, 1. 2 mrad, 0. 63 mm, 5. 2 mrad) in IP units (flat distribution, half width) and 2% energy spread, that corresponds approximately to Ns=(65, 230, 230) sigmas with respect to the nominal beam

Tail folding QF 5 B or Origami Zoo QD 6 QF 1 Oct. IP QD 6 QD 2 QD 0 QF 1 QD 0 QF 5 B IP QD 2 46

Halo collimation Assumed halo sizes. Halo population is 0. 001 of the main beam. 47 Assuming 0. 001 halo, beam losses along the beamline behave nicely, and SR photon losses occur only on dedicated masks Smallest gaps are +-0. 6 mm with tail folding Octupoles and +-0. 2 mm without them.

Dealing with muons in BDS Assuming 0. 001 of the beam is collimated, two tunnel-filling spoilers are needed to keep the number of muon/pulse train hitting detector below 10 Good performance achieved for both Octupoles OFF and ON 48

9 & 18 m Toroid Spoiler Walls 2. 2 m 49 Long magnetized steel walls are needed to spray the muons out of the tunnel

BDS design methods & examples Example of a 2 nd IR BDS optics for ILC; design history; location of design knobs 50

In a practical situation … Laser wire at ATF • While designing the FF, one has a total control • When the system is built, one has just limited number of observable parameters (measured orbit position, beam size measured in several locations) • The system, however, may initially have Laser wire will be a tool for errors (errors of strength of the elements, tuning and diagnostic of FF transverse misalignments) and initial aberrations may be large • Tuning of FF is done by optimization of “knobs” (strength, position of group of elements) chosen to affect some particular aberrations • Experience in SLC FF and FFTB, and simulations with new FF give confidence that this is possible 51

Stability – tolerance to FD motion IP • Displacement of FD by d. Y cause displacement of the beam at IP by the same amount • Therefore, stability of FD need to be maintained with a fraction of nanometer accuracy • How would we detect such small offsets of FD or beams? • Using Beam- beam deflection ! • How misalignments and ground motion influence beam offset? 52

Ground motion & cultural noises • Periodic signals can be characterized by amplitude (e. g. mm) and frequency • Random signals described by PSD • The way to make sense of PSD amplitude is to *by frequency range and take 7 sec hum Cultural noise & geology Power Spectral Density of absolute position 53 data from different labs 1989 - 2001

Detector complicates reaching FD stability 54 Cartoon from Ralph Assmann (CERN)

Beam-Beam orbit feedback use strong beam-beam kick to keep beams colliding 55

Beam-beam deflection 56 Sub nm offsets at IP cause large well detectable offsets (micron scale) of the beam a few meters downstream

ILC intratrain simulation ILC intratrain feedback (IP position and angle optimization), simulated with realistic errors in the linac and “banana” bunches, show Lumi ~2 e 34 (2/3 of design). Studies continue. Luminosity for ~100 seeds / run 0. 2 Position scan Angle scan 0. 5 1. 0 Luminosity through bunch train showing effects of position/angle scans (small). Noisy for first ~100 bunches (HOM’s). 57 [Glen White] Injection Error (RMS/sy): 0. 2, 0. 5, 1. 0

Crab crossing x factor 10 reduction in L! use transverse (crab) RF cavity to ‘tilt’ the bunch at IP x RF kick 58

Crab cavity requirements Crab Cavity IP ~0. 12 m/cell ~15 m Use a particular horizontal dipole mode which gives a phase-dependant transverse momentum kick to the beam Actually, need one or two multi-cell cavity 59 Slide from G. Burt & P. Goudket

Crab cavity requirements Phase jitter need to be sufficiently small electron bunch Δx Static (during the train) phase error can be corrected by intra -train feedback positron bunch Interaction point Phase error (degrees) Crossing angle Slide from G. Burt & P. Goudket 60 1. 3 GHz 3. 9 GHz 2 mrad 10 mrad 20 mrad 0. 222 0. 044 0. 665 0. 133 0. 022 0. 066

Crab cavity Right: earlier prototype of 3. 9 GHz deflecting (crab) cavity designed and build by Fermilab. This cavity did not have all the needed high and low order mode couplers. Left: Cavity modeled in Omega 3 P, to optimize design of the LOM, HOM and input couplers. FNAL T. Khabibouline et al. , SLAC K. Ko et al. 61

Anti-Solenoids in FD When solenoid overlaps QD 0, coupling between y & x’ and y & E causes sy(Solenoid) / sy(0) ~ 30 – 190 depending on solenoid field shape (green=no solenoid, red=solenoid) without compensation sy/ sy(0)=32 62 Even though traditional use of skew quads could reduce the effect, the LOCAL COMPENSATION of the fringe field (with a little skew tuning) is the best way to ensure excellent correction over wide range of beam energies with compensation by antisolenoid sy/ sy(0)<1. 01

Preliminary Design of Anti-solenoid for Si. D 70 mm cryostat 1. 7 m long Four 24 cm individual powered 6 mm coils, 1. 22 m total length, rmin=19 cm 15 T Force 316 mm 456 mm 63

Detector Integrated Dipole • With a crossing angle, when beams cross solenoid field, vertical orbit arise • For e+e- the orbit is anti-symmetrical and beams still collide head-on • If the vertical angle is undesirable (to preserve spin orientation or the e-eluminosity), it can be compensated locally with DID • Alternatively, negative polarity of DID may be useful to reduce angular spread of beam-beam pairs (anti-DID) 64

Use of DID or anti-DID field shape and scheme Orbit in 5 T Si. D IP angle zeroed w. DID case anti-DID case 65

14(20)mrad IR 66

2 mrad IR Shared Large Aperture Magnets SF 1 QD 0 SD 0 Disrupted beam & Sync radiations Q, S, QEXF 1 QF 1 Beamstrahlung Incoming beam pocket coil quad Rutherford cable SC quad and sextupole 67 60 m

IR design • Design of IR for both small and large crossing angles and to handle either DID or anti-DID • Optimization of IR, masking, instrumentations, background evaluation • Design of detector solenoid compensation Shown the forward region considered by LDC for 20 mrad (K. Busser) and an earlier version of 2 mrad IR 68

Collider hall • Collider hall sizes and detector assembly procedure for GLD (earlier version) 69

Tentative tunnel layout 70

Collider hall shielding design • Shielding is designed to give adequate protection both in normal operation, when beam losses are small, and for “maximum credible beam” when full beam is lost in undesired location (but switched off quickly, so only one or several trains can be lost) • Limits are different for normal and incident cases, e. g. what is discussed as guidance for IR shielding design: – Normal operation: dose less than 0. 05 mrem/hr (integrated less than 0. 1 rem in a year with 2000 hr/year) – For radiation workers, typically ten times more is allowed – Accidents: dose less than 25 rem/hr and integrated less than 0. 1 rem for 36 MW of maximum credible incident (MCI) 71

IR & rad. safety 18 MW loss on Cu target 9 r. l at s=-8 m. No Pacman, no detector. Concrete wall at 10 m. Dose rate in mrem/hr. • For 36 MW MCI, the concrete wall at 10 m from beamline should be ~3. 1 m Wall 25 rem/hr 10 m 72

Self-shielding detector Detector itself is well shielded except for incoming beamlines A proper “pacman” can shield the incoming beamlines and remove the need for shielding wall 18 MW on Cu target 9 r. l at s=-8 m Pacman 1. 2 m iron and 2. 5 m concrete 18 MW lost at s=-8 m. Packman has Fe: 1. 2 m, Concrete: 2. 5 m dose at pacman external wall 0. 65 rem/hr (r=4. 7 m) 73 dose at r=7 m 0. 23 rem/hr

Beam dump for 18 MW beam • Water vortex • Window, 1 mm thin, ~30 cm diameter hemisphere • Raster beam with dipole coils to avoid water boiling • Deal with H, O, catalytic recombination • etc. 20 mr extraction optics undisrupted or disrupted beam size does not destroy beam dump window without rastering. Rastering to avoid boiling of water 74

Get real with magnets • Things to care: – needed aperture, L – strength, field quality, stability – losses of beam or SR in the area • E. g. , extraction line => need aperture r~0. 2 m and have beam losses => need warm magnets which may consume many MW => may cause to look to new hybrid solutions, such as high T SC magnets 75

Magnet current (Amp*turn) per coil and total power Bend I(A)=B(Gs)*h(cm)*10/(4 p) P(W)=2*I(A)*j(A/m 2)*r(W*m)*l(m) I(A)=1/2*B(Gs)*h(cm)*10/(4 p) Quad P(W)=4*I(A)*j(A/m 2)*r(W*m)*l(m) I(A)=1/3*B(Gs)*h(cm)*10/(4 p) Sextupole P(W)=6*I(A)*j(A/m 2)*r(W*m)*l(m) For dipole h is half gap. For quad and sextupole h is aperture radius, and B is pole tip field. Typical bends may have B up to 18 k. Gs, quads up to 10 k. Gs. Length of turn l is approximately twice the magnet length. For copper r~2*10 -8 W*m. For water cooled magnets the conductor area chosen so that current density j is in the range 4 to 10 A/mm 2 76

ATF and ATF 2 77

ATF 2 Optics Design of ATF 2 (A) Small beam size Obtain sy ~ 35 nm Maintain for long time (B) Stabilization of beam center Down to < 2 nm by nano-BPM New Bunch-to-bunch feedback of final focus ILC-like train New Beamline Earlier version of layout and optics are shown 78 Beam New diagnostics existing extraction

Advanced beam instrumentation at ATF 2 • • • BSM to confirm 35 nm beam size nano-BPM at IP to see the nm stability Laser-wire to tune the beam Cavity BPMs to measure the orbit Movers, active stabilization, alignment system Intratrain feedback, Kickers to produce ILC-like train IP Beam-size monitor (BSM) (Tokyo U. /KEK, SLAC, UK) Laser-wire beam-size Monitor (UK group) Laser wire at ATF 79 Cavity BPMs with 2 nm resolution, for use at the IP (KEK) Cavity BPMs, for use with Q magnets with 100 nm resolution (PAL, SLAC, KEK)

Many thanks to • Many colleagues whose slides or results were used in this lecture, namely Tom Markiewicz, Nikolai Mokhov, Brett Parker, Nick Walker, Jack Tanabe, Timergali Khabibouline, Kwok Ko, Cherrill Spencer, Lew Keller, Sayed Rokni, Alberto Fasso, Joe Frisch, Yuri Nosochkov, Mark Woodley, Takashi Maruyama, Karsten Busser, Graeme Burt, Rob Appleby, Deepa Angal Kalinin, Glen White, Phil Burrows, Tochiaki Tauchi, Junji Urakawa, and many other colleagues. Thanks! 80

Homework • There are 7 tasks • Some of them sequential, some independent • Very rough estimations would be ok 81

HW 1 • For given FD: • Estimate beam size growth due to Oide effect for nominal ILC parameters and other cases such as “low P” at 1 Te. V • Note 1: you may need to derive formula for Oide effect if x size in FD is larger than y size • Note 2: you need to rescale b for corresponding parameter set 82

HW 2 • For the FD shown, and your favorite vertex detector radius, find the required collimation depth in x and y – take both nominal and other cases such as “low P” 83

HW 3 • For FD shown above, estimate effect on the beam size due to – second order dispersion – geometrical aberrations • if they would not be compensated upstream 84

HW 4 • For the FD shown above, and for the collimation depth that you determined, • choose the material for thin spoiler • find the minimal beam size so that spoiler survive N (choose between 1 and 100) bunches – (ignore thermal diffusion between bunches) • find the gap opening for the spoiler 85

HW 5 • For the FD considered above, find the min length of the bend that creates dispersion, to limit beam size growth caused by SR – see appendix for a similar example 86

HW 6 • Estimate SR emittance growth at 1 Te. V in “big bend” that turns the beam to one of IRs • Estimate SR emittance growth at 1 Te. V in the polarimeter chicane 87 big bend to IR polarimeter chicane to IR to dump

HW 7 • Estimate allowable steady state beam loss in IR, from the radiation safety point of view, for – IR hall with shielding wall as shown above – for self-shielding detector assuming it is fenced out at 7 m 88

Appendix: • Couple of definitions of chromaticity, suitable for single pass beamlines • Formulae connecting Twiss functions and transfer matrices • Example of calculation of the min length of the bend in FF system 89

Two more definitions of chromaticity 1 st : TRANSPORT You are familiar now with chromaticity defined as a change of the betatron tunes versus energy. This definition is mostly useful for rings. In single path beamlines, it is more convenient to use other definitions. Lets consider other two possibilities. The first one is based on TRANSPORT notations, where the change of the coordinate vector is driven by the first order transfer matrix R such that The second, third, and so on terms are included in a similar manner: 90 In FF design, we usually call ‘chromaticity’ the second order elements T 126 and T 346. All other high order terms are just ‘aberrations’, purely chromatic (as T 166, which is second order dispersion), or chromo-geometric (as U 32446).

Two more definitions of chromaticity 2 nd : W functions Lets assume that betatron motion without energy offset is described by twiss functions a 1 and b 1 and with energy offset d by functions a 2 and b 2 Let’s define chromatic function W (for each plane) as And where: Using familiar formulae And introducing Can you show this? where and where we obtain the equation for W evolution: knowing that the betatron phase is can see that if DK=0, then W rotates with double betatron frequency and stays constant in amplitude. In quadrupoles or sextupoles, only imaginary part changes. Show that if in a final defocusing lens a=0, then it gives DW=L*/(2 b*) 91 Show that if T 346 is zeroed at the IP, the Wy is also zero. Use approximation DR 34=T 346*d , use DR notes, page 12, to obtain R 34=(bb 0)1/2 sin(DF), and the twiss equation for da/d. F.

Several useful formulae TRANSPORT Twiss 1) If you know the Twiss functions at point 1 and 2, the transfer matrix between them is given by 2) If you know the transfer matrix between two points, the Twiss functions transform in this way: And similar for the other plane 92

Length required for the bends in FF bend We know that there should be nonzero horizontal chromaticity Wx upstream of FD (and created upstream of the bend). SR in the bend will create energy spread, and this chromaticity will be ‘spoiled’. Let’s estimate the required length of the bend, taking this effect into account. Parameters: length of bend LB, assume total length of the telescope is 2*LB, the el-star L* , IP dispersion’ is h’ Can you show this ? Dispersion at the FD, created by bend, is approximately Recall that we typically have hmax~ 4 L* h’ , therefore, the bending radius is The SR generated energy spread is then And the beam size growth Example: 650 Ge. V/beam, L*=3. 5 m, h’=0. 005, Wx=2 E 3, and requesting Ds/s<1% Energy scaling. Usually h’ ~ 1/g 1/2 then the required LB scales as g 7/10 93 => LB > 110 m Estimate LB for telescope you created in Exercise 2 -4.