Beam Delivery

1
Beam Delivery

• Andrei Seryi
• SLAC

International Accelerator School for Linear
Colliders 19-27 May 2006, Sokendai, Hayama, Japan
2
Linear Collider two main challenges

• Energy need to reach at least 500 GeV CM as a
  start
• Luminosity need to reach 1034 level

3
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

4
How to get Luminosity

• To increase probability of direct ee- collisions
  (luminosity) and birth of new particles, beam
  sizes at IP must be very small
• E.g., ILC beam sizes just before collision
  (500GeV CM) 500 5 300000 nanometers (x
  y z)

Vertical size is smallest
5
BDS from end of linac to IP, to dumps
BDS
6
BDS subsystems

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

7
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

8
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
• It is very similar for electron or positron
  beams
• But one have to use magnets

(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.)
9
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

10
What we use to handle the beam
Etc
Second order effect x x S (x2-y2) y y
S 2xy
Focus in one plane,defocus in anotherx x
G x y y G y
Just bend the trajectory
Here x is transverse coordinate, x is angle
11
Optics building block telescope
final

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
doublet

(FD)

IP

f

f

f

2
1
2
f1
f2 (L)
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).
Use telescope optics to demagnify beam by factor
m f1/f2 f1/L
Matrix formalism for beam transport
12
Why nonlinear elements

• As sun light contains different colors, electron
  beam has energy spread and get dispersed and
  distorted gt 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

13
How to focus to a smallest size and how big is
chromaticity in FF?
Size at IP L (e/b)1/2 (e b)1/2 sE
Size (e b)1/2 Angles (e/b)1/2
IP
L
Beta at IP L (e/b)1/2 (e b )1/2 gt b
L2/b

• The final lens need to be the strongest
• ( two lenses for both x and y gt Final Doublet
  or FD )
• FD determines chromaticity of FF
• Chromatic dilution of the beam size is Ds/s
  sE L/b
• For typical parameters, Ds/s 15-500 too big
  !
• gt Chromaticity of FF need to be compensated

Chromatic dilution (e b)1/2 sE / (e b )1/2
sE L/b
sE -- energy spread in the beam 0.002-0.01L
-- distance from FD to IP 3 - 5 m b --
beta function in IP 0.4 - 0.1 mm
Typical
14
Example of traditional Final Focus
Sequence of elements in 100m 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
Sextupoles. Their kick will containenergy
dependent focusing x gt S (x d)2 gt 2S
x d .. y gt S 2(x d)y gt -2S y d ..
that can be used to arrange chromatic
correction Terms x2 are geometric
aberrationsand need to be compensated also
Necessity to compensate chromaticity is a major
driving factor of FF design
15
Final Focus Test Beam
Achieved 70nm vertical beam size
16
Synchrotron Radiation in FF magnets

• Bends are needed for compensation of chromaticity
• SR causes increase of energy spread which may
  perturb compensation of chromaticity
• Bends need to be long and weak, especially at
  high energy
• SR in FD quads is also harmful (Oide effect) and
  may limit the achievable beam size

Field left behind
v c
v lt c
Field lines
Energy spread caused by SR in bends and quads is
also a major driving factor of FF design
17
Beam-beam (Dy, dE , ?) 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
• Ratio of critical photon energy to beam energy
  (classic or quantum regime)

18
Beam-beam effectsHD and instability
Dy12
Nx2 Dy24
19
Factor driving BDS design

• Chromaticity
• Beam-beam effects
• Synchrotron radiation
• lets consider it in more details

20
Lets estimate SR power
21
Lets estimate typical frequency of SR photons
For ggtgt1 the emitted photons goes into 1/g cone.
Photons emitted during travel along the 2R/g arc
will be observed.
22
Lets estimate energy spread growth due to SR
23
Lets estimate emittance growth rate due to SR
Dispersion function h shows how equilibrium orbit
shifts when energy changes
24
Lets apply SR formulae to estimate Oide effect
(SR in FD)
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).
25
Concept and problems of traditional FF
Final Doublet

• Chromaticity is compensated by sextupoles in
  dedicated sections
• Geometrical aberrations are canceled by using
  sextupoles in pairs with M -I

Y-Sextupoles
X-Sextupoles
Chromaticity arise at FD but pre-compensated
1000m upstream
Problems

• Chromaticity not locally compensated
• Compensation of aberrations is not ideal since M
  -I for off energy particles
• Large aberrations for beam tails

Traditional FF
/
26
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

27
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

28
Chromatic correction in FD
quad
sextup.
x h d

• Straightforward in Y plane
• a bit tricky in X plane

IP
KS
KF
Quad
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.
Second order dispersion
chromaticity
Sextupole
29
Traditional and new FF
Traditional FF, L 2m

• A new FF with the same
• performance can be
• 300m long, i.e. 6 times shorter

New FF, L 2m
new FF
30
New Final Focus

• One third the length - many fewer components!
• Can operate with 2.5 TeV beams (for 3 ? 5 TeV
  cms)
• 4.3 meter L (twice 1999 design)

1999 Design
2000 Design
31
IP bandwidth
Bandwidth is much better for New FF
32
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
Halo beam at the FD entrance. Incoming beam is
100 times larger than nominal beam
Traditional FF
Incoming beam halo
New FF
33
Beam halo collimation

• Even if final focus does not generate beam halo
  itself, the halo may come from upstream and need
  to be collimated

• Halo must be collimated upstream in such a way
  that SR g halo e- do not touch VX and FD
• gt VX aperture needs to be somewhat larger than
  FD aperture
• Exit aperture is larger than FD or VX aperture
• Beam convergence depend on parameters, the halo
  convergence is fixed for given geometry gt
  qhalo/qbeam (collimation depth) becomes tighter
  with larger L or smaller IP beam size
• Tighter collimation gt MPS issues, collimation
  wake-fields, higher muon flux from collimators,
  etc.

34
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
• Collimation depth (esp. in x) can be only 10 or
  even less
• It is not unlikely that not only halo (1e-3
  1e-6 of the beam) but full errant bunch(s) would
  hit the collimator

35
MPS and collimation design

• The beam is very small gt single bunch can punch
  a hole gt the need for MPS (machine protection
  system)
• Damage may be due to
• electromagnetic shower damage (need several
  radiation lengths to develop)
• direct ionization loss (1.5MeV/g/cm2 for most
  materials)
• Mitigation of collimator damage
• using spoiler-absorber pairs
• thin (0.5-1 rl) spoiler followed by thick
  (20rl) absorber
• increase of beam size at spoilers
• MPS divert the beam to emergency extraction as
  soon as possible

Picture from beam damage experiment at FFTB. The
beam was 30GeV, 3-20x109 e-, 1mm bunch length,
s45-200um2. Test sample is Cu, 1.4mm thick.
Damage was observed for densities gt 7x1014e-/cm2.
Picture is for 6x1015e-/cm2
36
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.
Need the spoiler thickness increase rapidly, but
need that surface to increase gradually, to
minimize wakefields. The radiation length for Cu
is 1.4cm and for Be is 35cm. So, Be is invisible
to beam in terms of losses. Thin one micron
coating over Be provides smooth surface for
wakes.
37
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 4Tstress, 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.
38
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

39
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.
40
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

energy
betatron

• Beam Delivery System Optics, an earlier version
  with consumable spoilers

41
ILC FF Collimation

• Betatron spoilers survive up to two bunches
• E-spoiler survive several bunches
• One spoiler per FD or IP phase

E- spoiler
betatron spoilers
42
MPS in BSY
sigma (m) in tune-up extraction line
skew correction
Energy diag. chicane MPS energy collimator
MPS betatron collimators
4-wire 2D e diagnostics
kicker, septum
polarimeter chicane
betatron collimation
tune-up dump
43
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

Single octupole focus in planes and defocus on
diagonals. An octupole doublet can focus in all
directions !
44
Strong focusing by octupoles

• Two octupoles of different sign separated by
  drift provide focusing in all directions for
  parallel beam

Next nonlinear term focusing defocusing depends
on j
Focusing in all directions
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

45
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.
QD6
QD0
QF1
Tail folding by means of two octupole doublets in
the ILC final focus Input beam has (x,x,y,y)
(14mm,1.2mrad,0.63mm,5.2mrad) in IP units (flat
distribution, half width) and ?2 energy spread,
that corresponds approximately to
Ns(65,65,230,230) sigmas with respect to the
nominal beam
46
Tail folding or Origami Zoo
QF5B
QF1
Oct.
IP
QD6
QD2
QD0
QF1
QD6
QD0
QF5B
IP
QD2
47
Halo collimation
Assuming 0.001 halo, beam losses along the
beamline behave nicely, and SR photon losses
occur only on dedicated masks Smallest gaps are
-0.6mm with tail folding Octupoles and -0.2mm
without them.
Assumed halo sizes. Halo population is 0.001 of
the main beam.
48
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
49
9 18 m Toroid Spoiler Walls
Long magnetized steel walls are needed to spray
the muons out of the tunnel
2.2m
50
BDS design methods examples
Example of a 2nd IR BDS optics for ILC design
history location of design knobs
51
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 errors
  (errors of strength of the elements, 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

Laser wire will be a tool for tuning and
diagnostic of FF
52
Stability tolerance to FD motion
IP

• Displacement of FD by dY 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?

53
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

7sec hum
Cultural noise geology
Power Spectral Density of absolute position data
from different labs 1989 - 2001
54
Detector complicates reaching FD stability
Cartoon from Ralph Assmann (CERN)
55
Beam-Beam orbit feedback
use strong beam-beam kick to keep beams colliding
56
Beam-beam deflection
Sub nm offsets at IP cause large well detectable
offsets (micron scale) of the beam a few meters
downstream
57
ILC intratrain simulation
ILC intratrain feedback (IP position and angle
optimization), simulated with realistic errors in
the linac and banana bunches, show Lumi 2e34
(2/3 of design). Studies continue.
Luminosity for 100 seeds / run
Angle scan
Position scan
Luminosity through bunch train showing effects of
position/angle scans (small). Noisy for first
100 bunches (HOMs).
Injection Error (RMS/sy) 0.2, 0.5, 1.0
Glen White
58
Crab crossing
x
factor 10 reduction in L!
use transverse (crab) RF cavity to tilt the
bunch at IP
x
RF kick
59
Crab cavity requirements
Crab Cavity
IP
0.12m/cell
15m
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
Slide from G. Burt P. Goudket
60
Crab cavity requirements
Phase jitter need to be sufficiently
small Static (during the train) phase error can
be corrected by intra-train feedback
Phase error (degrees) Phase error (degrees)
Crossing angle 1.3GHz 3.9GHz
2mrad 0.222 0.665
10mrad 0.044 0.133
20mrad 0.022 0.066
Slide from G. Burt P. Goudket
61
Crab cavity
Right earlier prototype of 3.9GHz 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
Omega3P, to optimize design of the LOM, HOM and
input couplers.FNAL T. Khabibouline et al., SLAC
K.Ko et al.
62
Anti-Solenoids in FD
When solenoid overlaps QD0, coupling between y
x and y E causes sy(Solenoid) / sy(0) 30
190 depending on solenoid field shape (greenno
solenoid, redsolenoid)
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
without compensation sy/ sy(0)32
with compensation by antisolenoidsy/ sy(0)lt1.01
63
Preliminary Design of Anti-solenoid for SiD
Four 24cm individual powered 6mm coils, 1.22m
total length, rmin19cm
15T Force
64
Detector Integrated Dipole

• With a crossing angle, when beams cross solenoid
  field, vertical orbit arise
• For ee- the orbit is anti-symmetrical and beams
  still collide head-on
• If the vertical angle is undesirable (to
  preserve spin orientation or the e-e-
  luminosity), 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)

65
Use of DID or anti-DID
DID field shape and scheme
DID case
anti-DID case
66
14(20)mrad IR
67
2mrad IR
Shared Large Aperture Magnets
Disrupted beam Sync radiations
Q,S,QEXF1
SF1
QF1
SD0
QD0
60 m
Beamstrahlung
Incoming beam
pocket coil quad
Rutherford cable SC quad and sextupole
68
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
20mrad (K.Busser) and an earlier version of 2mrad
IR
69
Collider hall

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

70
Tentative tunnel layout
71
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 25rem/hr and
  integrated less than 0.1 rem for 36MW of maximum
  credible incident (MCI)

72
IR rad. safety
18MW loss on Cu target 9r.l \at s-8m. No
Pacman, no detector. Concrete wall at 10m. Dose
rate in mrem/hr.

• For 36MW MCI, the concrete wall at 10m from
  beamline should be 3.1m

10m
73
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
18MW on Cu target 9r.l at s-8m Pacman 1.2m iron
and 2.5m concrete
18MW lost at s-8m. Packman has Fe 1.2m,
Concrete 2.5m
dose at pacman external wall dose at r7m
0.65rem/hr (r4.7m)
0.23rem/hr
74
Beam dump for 18MW beam

• Water vortex
• Window, 1mm thin, 30cm diameter hemisphere
• Raster beam with dipole coils to avoid water
  boiling
• Deal with H, O, catalytic recombination
• etc.

20mr extraction optics
undisrupted or disrupted beam size does not
destroy beam dump window without rastering.
Rastering to avoid boiling of water
75
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 gt need aperture r0.2m and
  have beam losses gt need warm magnets which may
  consume many MW gt may cause to look to new
  hybrid solutions, such as high T SC magnets

76
Magnet current (Ampturn) per coil and total power
Bend
I(A)B(Gs)h(cm)10/(4p)
P(W)2I(A)j(A/m2)r(Wm)l(m)
Quad
I(A)1/2B(Gs)h(cm)10/(4p)
P(W)4I(A)j(A/m2)r(Wm)l(m)
I(A)1/3B(Gs)h(cm)10/(4p)
Sextupole
P(W)6I(A)j(A/m2)r(Wm)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 18kGs, quads up to
10kGs. Length of turn l is approximately twice
the magnet length. For copper r210-8 Wm. For
water cooled magnets the conductor area chosen so
that current density j is in the range 4 to 10
A/mm2
77
ATF and ATF2
78
ATF2
Optics Design of ATF2
Beam
(A) Small beam sizeObtain sy 35nmMaintain for
long time (B) Stabilization of beam center
Down to lt 2nm by nano-BPM Bunch-to-bunch
feedback of ILC-like train
New diagnostics
existing extraction
New final focus
New Beamline
Earlier version of layout and optics are shown
79
Advanced beam instrumentation at ATF2

• BSM to confirm 35nm 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
Cavity BPMs with 2nm resolution, for use at the
IP (KEK)
Cavity BPMs, for use with Q magnets with 100nm
resolution (PAL, SLAC, KEK)
80
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!

81
Homework

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

82
HW1

• For given FD
• Estimate beam size growth due to Oide effect
  for nominal ILC parameters andother cases such
  as low P at 1TeV
• Note1 you may need to derive formula for Oide
  effect if x size in FD is larger than y size
• Note2 you need to rescale b for corresponding
  parameter set

83
HW2

• 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

84
HW3

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

85
HW4

• 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

86
HW5

• 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

87
HW6

• Estimate SR emittance growth at 1TeV in big
  bend that turns the beam to one of IRs
• Estimate SR emittance growth at 1TeV in the
  polarimeter chicane

big bend
to IR
polarimeter chicane
to IR
to dump
88
HW7

• 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 7m

89
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

90
Two more definitions of chromaticity1st
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.
is driven by the first order transfer matrix R
such that
The first one is based on TRANSPORT notations,
where the change of the coordinate vector
The second, third, and so on terms are included
in a similar manner
In FF design, we usually call chromaticity the
second order elements T126 and T346. All other
high order terms are just aberrations, purely
chromatic (as T166, which is second order
dispersion), or chromo-geometric (as U32446).
91
Two more definitions of chromaticity2nd W
functions
Lets assume that betatron motion without energy
offset is described by twiss functions a1 and b1
and with energy offset d by functions a2 and b2
Show that if in a final defocusing lens a0, then
it gives DWL/(2b)
Show that if T346 is zeroed at the IP, the Wy is
also zero. Use approximation DR34T346d , use
DR notes, page 12, to obtain R34(bb0)1/2
sin(DF), and the twiss equation for da/dF.
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Several useful formulaeTRANSPORT ? 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
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Length required for the bends in FF
We know now 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. Lets estimate the required
length of the bend, taking this effect into
account.
Parameters length of bend LB, assume total
length of the telescope is 2LB, the el-star L ,
IP dispersion is h
Example 650GeV/beam, L3.5m, h0.005, Wx2E3,
and requesting Ds/slt1 gt LB gt 110m
Energy scaling. Usually h 1/g1/2 then the
required LB scales as g7/10
Estimate LB for telescope you created in Exercise
2-4.