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TRANSCRIPT
1
Beam Transfer Lines
Verena Kain CERN
(based on lecture by B. Goddard and M. Meddahi)
What is the purpose of "transfer lines”?
CERN Complex
LHC: Large Hadron Collider SPS: Super Proton Synchrotron AD: Antiproton Decelerator ISOLDE: Isotope Separator Online Device PSB: Proton Synchrotron Booster PS: Proton Synchrotron LINAC: LINear Accelerator LEIR: Low Energy Ring CNGS: CERN Neutrino to Gran Sasso
Transfer lines transport the beam between accelerators, and onto targets, dumps, instruments etc.
• An accelerator has limited dynamic range
• Chain of stages needed to reach high energy
• Periodic re-filling of storage rings, like LHC
• External experiments, like CNGS
2
Transport of good quality beams • Transfer lines transport beams
• The challenge: preserve “GOOD QUALITY”
Examples:
• Position stability on a target – Trajectory in transfer line needs to be under control
• Not below/above certain beam size at a window/target – Optics in transfer line AND exit of last machine need to be under control
• Preserve emittance between machines – Trajectory, optics, tilt angles etc. need to be under control
Challenges with high energy/intensity • Machine protection becomes design constraint for extraction/
injection/ transfer lines
• Sophisticated and reliable active protection with surveillance of power supply currents, beam losses, etc.
• Passive protection with absorbers – Dedicated areas in the transfer lines with optics requirements to install a
collimation system
3
Example: CNGS Target • CNGS transfer line from SPS is ~ 1 km long • At the end of the line the CNGS target needs to be hit
Ø = 5mm
CNGS target
Example: Emittance and LHC Luminosity
• Proton machines è emittance can only grow
• Emittance has to be preserved through the LHC cycle • One of most critical moments for emittance:
– Beam Transfer from Injectors
– è NO ERRORS FROM THE BEAM TRANSFER
Want to measure rare events with high statistics Collision rate = Luminosity x cross-section
σ 2 =εn ⋅β
*
γ
4
Contents of this lecture • Distinctions between transfer lines and circular machines • Linking machines together • Emittance Blow-up from steering errors • Correction of injection oscillations • Emittance Blow-up from optics mismatch • Optics measurement • Blow-up from thin screens
General transport
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡→
1
1
1
121'
2
2
'''' xx
SCSC
xx
xx
M
x x
y
y
s s
1
2
∏=
→ =n
in
121 MM
Beam transport: moving from s1 to s2 through n elements, each with transfer matrix Mi
( )
( ) ( )[ ] ( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
Δ−ΔΔ+−Δ−
ΔΔ+Δ=→
µαµββµααµααββ
µββµαµββ
sincossin1cos1
sinsincos
22
12121
21
2111
2
21MTwiss parameterisation
5
Circular Machine
Circumference = L
• The solution is periodic • Periodicity condition for one turn (closed ring) imposes α1=α2, β1=β2, D1=D2 • This condition uniquely determines α(s), β(s), µ(s), D(s) around the whole ring
( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+−+
== →→ QQQQQQ
L παππαβ
πβπαπ
2sin2cos2sin112sin2sin2cos
2021 MMOne turn
Circular Machine
• Periodicity of the structure leads to regular motion – Map single particle coordinates on each turn at any location
– Describes an ellipse in phase space, defined by one set of α and β values ⇒ Matched Ellipse (for this location)
x
x’
βα 2
max1' +
= ax
βax =max
Area = πa βα
γ
βαγ
2
22
1
''2
+=
⋅+⋅⋅+⋅= xxxxa
6
Circular Machine
• For a location with matched ellipse (α, β), an injected beam of emittance ε, characterised by a different ellipse (α*, β*) generates (via filamentation) a large ellipse with the original α, β, but larger ε
x
x’
After filamentation
α, β
εο, α∗ , β∗
ε > εο, α, β
x
x’
After filamentation
α, β
εο, α∗ , β∗
ε > εο, α, β
Matched ellipse determines beam shape
Turn 1 Turn 2
Turn 3 Turn n>>1
Transfer line
( )
( ) ( )[ ] ( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
Δ−ΔΔ+−Δ−
ΔΔ+Δ=→
µαµββµααµααββ
µββµαµββ
sincossin1cos1
sinsincos
22
12121
21
2111
2
21M
⎥⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡→ '
1
121'
2
2
xx
xx
M
⎥⎦
⎤⎢⎣
⎡'1
1
xx
• No periodic condition exists • The Twiss parameters are simply propagated from beginning to end of line • At any point in line, α(s) β(s) are functions of α1 β1
One pass:
⎥⎦
⎤⎢⎣
⎡'2
2
xx
7
Transfer line
L→0M
• On a single pass… – Map single particle coordinates at entrance and exit.
– Infinite number of equally valid possible starting ellipses for single particle ……transported to infinite number of final ellipses…
x
x’
x
x’
α1, β1
α*1, β*1
Transfer Line
α*2, β*2
α2, β2 ⎥⎦
⎤⎢⎣
⎡'2
2
xx
⎥⎦
⎤⎢⎣
⎡'1
1
xx
Entry Exit
Transfer Line • Initial α, β defined for transfer line by beam shape at entrance
• Propagation of this beam ellipse depends on line elements
• A transfer line optics is different for different input beams
x
x’
α, β
x
x’α∗, β∗
Gaussian beamNon-Gaussian beam(e.g. slow extracted)
x
x’
α, β
x
x’α∗, β∗
Gaussian beamNon-Gaussian beam(e.g. slow extracted)
8
1500 2000 2500 30000
50
100
150
200
250
300
350
S [m]
Beta
X [m
]
Horizontal optics
Transfer Line
• The optics functions in the line depend on the initial values
• Same considerations are true for Dispersion function: – Dispersion in ring defined by periodic solution → ring elements
– Dispersion in line defined by initial D and D’ and line elements
- Design βx functions in a transfer line - βx functions with different initial conditions
1500 2000 2500 30000
50
100
150
200
250
300
350
S [m]
Beta
X [m
]
Horizontal optics
Transfer Line
• Another difference….unlike a circular ring, a change of an element in a line affects only the downstream Twiss values (including dispersion)
10% change in this QF strength
- Unperturbed βx functions in a transfer line - βx functions with modification of one quadrupole strength
9
Linking Machines
• Beams have to be transported from extraction of one machine to injection of next machine – Trajectories must be matched, ideally in all 6 geometric degrees of freedom
(x,y,z,θ,φ,ψ)
– Otherwise emittance blow-up
• Other important constraints can include – Minimum bend radius, maximum quadrupole gradient, magnet aperture,
cost, geology
Linking Machines
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−+−
−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1
1
1
22
22
2
2
2
'''2'''''
2
γ
α
β
γ
α
β
SSCCSSSCCSCCSCSC
Extraction
Transfer
Injection
α1x, β1x , α1y, β1y αx(s), βx(s) , αy(s), βy(s)
s
The Twiss parameters can be propagated when the transfer matrix M is known
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡→ '
1
1'1
121'
2
2
'' xx
SCSC
xx
xx
M
α2x, β2x , α2y, β2y
10
Linking Machines • Linking the optics is a complicated process
– Parameters at start of line have to be propagated to matched parameters at the end of the line
– Need to “match” 8 variables (αx βx Dx D’x and αy βy Dy D’y)
– Maximum β and D values are imposed by magnet apertures
– Other constraints can exist • phase conditions for collimators,
• insertions for special equipment like stripping foils
– Need to use a number of independently powered (“matching”) quadrupoles
– Matching with computer codes and relying on mixture of theory, experience, intuition, trial and error, …
Linking Machines • For long transfer lines we can simplify the problem by designing the
line in separate sections – Regular central section – e.g. FODO or doublet, with quads at regular
spacing, (+ bending dipoles), with magnets powered in series
– Initial and final matching sections – independently powered quadrupoles, with sometimes irregular spacing.
0
50
100
150
200
250
300
0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000S [m]
β [m
]
BETXBETY
SPS LHC Regular lattice (FODO)
(elements all powered in series with same strengths)
Final matching section
SPS to LHC Transfer Line (3 km)
Extraction point
Injection point
Initial matching section
11
• Magnet misalignments, field and powering errors cause the trajectory to deviate from the design
• Use small independently powered dipole magnets (correctors) to steer the beam
• Measure the response using monitors (pick-ups) downstream of the corrector (π/2, 3π/2, …)
• Horizontal and vertical elements are separated • H-correctors and pick-ups located at F-quadrupoles (large βx )
• V-correctors and pick-ups located at D-quadrupoles (large βy)
Trajectory correction
π/2
Corrector dipole Pickup Trajectory
QF
QD QD
QF
Trajectory correction
• Global correction can be used which attempts to minimise the RMS offsets at the BPMs, using all or some of the available corrector magnets.
• Steering in matching sections, extraction and injection region requires particular care – D and β functions can be large → bigger beam size
– Often very limited in aperture
– Injection offsets can be detrimental for performance
12
Some pictures – LHC Transfer Line TI 8
TI 8
Some pictures – LHC Transfer Line TI 8
13
Some pictures – LHC Transfer Line TI 8
Some pictures – LHC Transfer Line TI 8
14
Some pictures – LHC Transfer Line TI 8
Steering (dipole) errors
• Precise delivery of the beam is important. – To avoid injection oscillations and emittance growth in rings
– For stability on secondary particle production targets
– Injection oscillations = if beam is not injected on the closed orbit, beam oscillates around closed orbit and eventually filaments (if not damped)
Septum
kicker Mis-steered injected beam
15
Reminder – Filamentation – Emittance Growth
Reminder - Normalised phase space • Transform real transverse coordinates x, x’ by
'1
1
xx
x
SSS
S
βαβ
β
+⋅=
⋅=
'X
X
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡
'011
' xx
xx
SSS βαβN
'XX
16
Reminder - Normalised phase space
a 1γ
−a αγ
x
x’
−a αβ
a 1β
x 'max = a γ
xmax = a β
Area = πε
Xmax = a
X'max = a
Area = πε ⇒
a = γ ⋅ x2 + 2α ⋅ x ⋅ x '+β ⋅ x '2 a = X 2 + X' 2
Real phase space Normalised phase space
X
'X
Blow-up from steering error • Consider a collection of particles • The beam can be injected with a error in angle and position.
• For an injection error Δa (in units of sigma = √βε) the mis-injected beam is offset in normalised phase space by L = Δa√ε
X
'X Misinjected beam
Matched particles
L
17
Blow-up from steering error • The new particle coordinates in normalised phase space are
θ
θ
sin
cos
new
new
LXX
LXX
'0
'
0
+=
+=
2/2
'222
A
XXA
=
+=
ε
• For a general particle distribution, where A denotes amplitude of a paricle in normalised phase space
Misinjected beam
Matched particles
L θ X
'X
Blow-up from steering error
Anew2 = Xnew
2 + Xnew'2 = X0 + Lcosθ( )
2+ X0
' + Lsinθ( )2
= X02 + X0
'2 + 2L X0cosθ + X0' sinθ( )+ L2
Anew2 = X0
2 + X0'2 + 2L X0cosθ + X0
' sinθ( ) + L2
= 2ε0 + 2L cosθX0 + sinθX0'( )+ L2
= 2ε0 + L2
• So if we plug in the new coordinates….
( )2/1
2/2/
0
0
2
22
a
LA
Δ+=
+==
ε
εε newnew
• Giving for the emittance increase
0 0
18
Blow-up from steering error
A numerical example…. Consider an offset Δa of 0.5 sigma for injected beam For nominal LHC beam: εnorm = 3.5 µm allowed growth through LHC cycle ~ 10 %
Misinjected beam
Matched Beam
( )
0
0 2/1
ε
Δεε
1.125=
+= 2anew
0.5√ε √ε
X
'X
Example: LHC injection of beam 1
Display of injection oscillations in LHC control room
Transfer line TI 2 ~3 km
Injection point in LHC IR2
LHC arc 23 ~3 km
closed orbit subtracted
-6 to
+6
mm
0.5 mm injection oscillation amplitude
0.5 mm injection oscillation is GOOD. Don’t touch. è Transverse Damper
19
Example: LHC injection of beam 1 • Oscillation down the line has developed in horizontal plane • Injection oscillation amplitude > 1.5 mm
• Good working range of LHC transverse damper +/- 2 mm
• Aperture margin for injection oscillation is 2 mm
• è correct injection oscillations before continue LHC filling • Transfer line steering
Injection oscillation correction • x, x’ and y, y’ at injection point need to be corrected. • Minimum diagnostics: 2 pickups per plane, 90° phase advance apart
• Pickups need to be triggered to measure on the first turn
• Correctors in the transfer lines are used to minimize offset at these pickups.
• Best strategy: – Acquire many BPMs in circular machine (e.g. one octant/sextant of
machine)
– Combine acquisition of transfer line and of BPMs in circular machine • Transfer line: difference trajectory to reference
• Circular machine: remove closed orbit from first turn trajectory è pure injection oscillation
– Correct combined trajectory with correctors in transfer line with typical correction algorithms. Use correctors of the line only.
20
Steering (dipole) errors • Static effects (e.g. from errors in alignment, field, calibration, …) are
dealt with by trajectory correction (steering). • But there are also dynamic effects, from:
– Power supply ripples
– Temperature variations
– Non-trapezoidal kicker waveforms
• These dynamic effects produce a variable injection offset which can vary from batch to batch, or even within a batch.
• An injection damper system is used to minimize effect on emittance
SOURCES AND SOLUTIONS FORLHC TRANSFER LINE STABILITY ISSUES
L.Drosdal, V. Kain, W. Bartmann, C. Bracco, B. Goddard,G. Le Godec, M. Meddahi, J.Uythoven, CERN, Geneva, Switzerland
AbstractThe LHC is filled through two 3 km long transfer lines
from the last pre-injector, the SPS. During the LHC pro-ton run 2011 large drifts, shot-by-shot and even bunch-by-bunch trajectory variations were observed with the conse-quence of high losses at injection and frequent lengthy tra-jectory correction campaigns. The causes of these instabil-ities have been studied and will be presented in this paper.Based on the studies solutions have been proposed. The ef-fect of the solutions will be shown and the remaining issueswill be summarized.
INTRODUCTIONBeam is injected from the SPS into the LHC through
two transfer lines: TI 2 for beam 1 and TI 8 for beam 2.The trajectory in the transfer line must be well controlledin order to limit losses at the transfer line collimators andto minimize injection oscillations for the available aperturein the LHC (<1.5 mm) [1]. The main source of lossesare trajectory variations; during the 2011 run shot-by-shotvariations, bunch-by-bunch variations and long time driftswere observed [2].Frequent trajectory correction (steering) of the transfer
lines was necessary in 2011 impacting LHC efficiency.Steering the lines was complicated due to the large shot-by-shot and bunch-by-bunch variations and had to be repeatedseveral times per week taking 0.5 - 2 h per correction cam-paign [3]. The typical correction strength is about 10 µrad.
BUNCH-BY-BUNCH-VARIATIONSThe bunch-by-bunch analysis of the automatic LHC In-
jection Quality Check (IQC [4]) indicated large bunch-by-bunch differences of the injection oscillation amplitudes inthe horizontal plane for beam 2 (TI 8) see Fig. 1. An insuffi-cient flatness of the waveform of the SPS extraction kicker,MKE4 was suspected. A waveform scan indeed revealed alarge ripple of 3.8% (specification: 1%), see Fig. 2.Due to machine protection reasons trajectory correction
is done with 12 bunches only. In 2011 the part of thewaveform which was sampled with the first 12 buncheswas unfortunately not representative for the full batch (144bunches) as indicated also in Fig. 2. The first 12 buncheswere following a very different trajectory from the rest ofthe bunches due to the large ripple at the beginning of thewaveform. For the 2012 run the MKE delay was changedfrom 54 µs to 53.2 µs to only sample the region afterthe second overshoot. This should make steering with 12
bunches more straight forward. The waveform could how-ever not be flattened in the short shutdown between the2011 and 2012 run.
Figure 1: IQC plot of injection oscillation amplitudes as afunction of bunch in the horizontal plane for a full 50 nsbatch of 144 bunches, beam 2. Due to the large ripple ofthe SPS extraction kicker waveform, the bunch-by-bunchvariations are large.
Figure 2: Scan of the SPS extraction kicker waveform forTI 8: The difference between minimum and maximumvoltage along the waveform is 3.8%. In 2012 the kicker de-lay with respect to extracted beam was changed. The areaof the waveform a full batch sampled in 2011 is indicatedin red, for 2012 it is indicated in green.
SHOT-BY-SHOT VARIATIONSLarge trajectory variations were observed from one shot
to the next. The analysis of the 2011 proton data recordedby the IQC show that the shot-by-shot variations are partic-ularly large in the horizontal plane, around 0.6 mm for TI2 and 0.4 mm for TI 8. The variations are around 0.1 mmin the vertical plane for both lines [3].To understand the phenomenon dedicated stability stud-
ies were carried out extracting beam onto the beam stoppers
Inje
ctio
n os
c.
bunch number
LHC injected batch Beam 2
• Optical errors occur in transfer line and ring, such that the beam can be injected with a mismatch.
Blow-up from betatron mismatch
• Filamentation will produce an emittance increase.
• In normalised phase space, consider the matched beam as a circle, and the mismatched beam as an ellipse.
Mismatched beam
Matched beam
X
'X
21
Blow-up from betatron mismatch
λλ⋅=
=AA
ba
Mismatched beam
Matched Beam
a b
generally
Define parameter λ to transform original coordinates:
( ) ( )1121111
21
−−+=−++= HHHHλ
λ ,
A X
'X)cos(1),sin( 1new1new φφλ
φφλ +=+⋅= A'XAX
( ) ( )112
112
−−+=−++= HHAbHHAa ,
Frequently also use H as mismatch parameter:
Blow-up from betatron mismatch
)(cos1)(sin 2202
220
2211newnew φφ
λφφλ +++⋅=+= AA'XXA 22
new
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+==⎟
⎠
⎞⎜⎝
⎛ +=1
2
2
2
121
1
2
2
1002
20 2
1121
ββ
ββ
ααββ
ββ
εελ
λεε Hnew
We can evaluate the square of the distance of a particle from the origin as
The new emittance is the average over all phases
If we’re feeling diligent, we can substitute back for λ to give
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+++=
⎟⎟⎠
⎞⎜⎜⎝
⎛+++==
22
0
22
22
22
22
121
)(cos1)(sin21
)(cos1)(sin21
21
λλε
φφλ
φφλ
φφλ
φφλε
11
11new
20
20
20
2
A
AAAnew
where subscript 1 refers to matched ellipse, 2 to mismatched ellipse.
0.5 0.5
22
OPTICS AND EMITTANCE MEASUREMENT IN TRANSFER LINES
Dispersion measurement • Introduce ~ few permille momentum offset at extraction into transfer
line
• Measure position at different monitors for different momentum offset – Linear fit of position versus dp/p at each BPM/screens.
– è Dispersion at the BPMs/screens
Data: 25.10.2009
x(s) = xβ (s)+D(s) ⋅dpp
Position at a certain BPM versus momentum offset
23
Optics measurement with screens • A profile monitor is needed to measure the beam size
– e.g. beam screen (luminescent) provides 2D density profile of the beam
• Profile fit gives transverse beam sizes σ. • In a ring, β is ‘known’ so ε can be calculated from a single screen
Optics Measurement with 3 Screens
• Assume 3 screens in a dispersion free region • Measurements of σ1,σ2,σ3, plus the two transfer matrices M12 and
M13 allows determination of ε, α and β
σ1 σ2 σ3
s1 s3 s2
ε =σ12
β1=σ 22
β2=σ 32
β3
M1→2
M1→3
24
Optics Measurement with 3 Screens • Remember:
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡→
1
1
1
121'
2
2
'''' xx
SCSC
xx
xx
M
β2α2
γ2
!
"
####
$
%
&&&&
=
C22 −2C2S2 S2
2
−C2C2 ' C1S1 '+ S1C1 ' −S2S2 '
C2 '2 −2C2 'S2 ' S2 '
2
!
"
####
$
%
&&&&
⋅
β1α1γ1
!
"
####
$
%
&&&&
β2 =C22 ⋅β1 − 2C2S2 ⋅α1 + S2
2 ⋅γ1
β3 =C32 ⋅β1 − 2C3S3 ⋅α1 + S3
2 ⋅γ1 ✕ ε
σ 22 =C2
2 ⋅β1ε − 2C2S2 ⋅α1ε + S22 ⋅γ1ε
σ 32 =C3
2 ⋅β1ε − 2C3S3 ⋅α1ε + S32 ⋅γ1ε
Square of beam sizes as function of optical functions at first screen
Optics Measurement with 3 Screens
• Build matrix
• We want to know Π
σ 22 =C2
2 ⋅β1ε − 2C2S2 ⋅α1ε + S22 ⋅γ1ε
σ 32 =C3
2 ⋅β1ε − 2C3S3 ⋅α1ε + S32 ⋅γ1ε
σ12 =1⋅β1ε − 0 ⋅α1ε + 0 ⋅γ1ε
Σ =N ⋅Π
Σ =
σ12
σ 22
σ 32
"
#
$$$$
%
&
''''meas
Π =
β1ε
α1ε
γ1ε
"
#
$$$$
%
&
''''
N =1 0 0C22 −2C2S2 S2
2
C23 −2C3S3 S3
2
"
#
$$$$
%
&
''''
Π =N−1 ⋅ Σ
25
Optics Measurement with 3 Screens
• Measure beam sizes and want to calculate β1, α1, ε
• with β1γ1-α12 = 1 get 3 equations for β1, α1 and ε
β1 = A / AC −B2
α1 = B / AC −B2
ε = AC −B2with
A =Π1
B =Π2
C =Π3
…and if there is dispersion at the screens • Beam size at screen i
• Di …dispersion, δ…momentum spread
• è Measure first momentum spread in circular machine before extraction and dispersion at every screen
• If you have more than 3 screens, can try to measure δ or D with screens
Remember: • Trajectory transforms with Mi transport matrix for δ ≠ 0 – ξi is the contribution to the dispersion between the first and the ith screen
xi!xiδ
"
#
$$$$
%
&
''''=
Ci Si ξi!Ci !Si !ξi0 0 1
"
#
$$$$
%
&
''''⋅
x1!x1δ
"
#
$$$$
%
&
''''
Di =CiD1 + Si !D1 +ξi
σ i2 = βi ⋅ε +Di
2 ⋅δ 2
26
6 screens with dispersion • Your measurements are • You know how βi and Di transform depending on the optical
functions of screen 1.
• Build again system Σ = NΠ
• Result is again Π = N-1Σ
• è Can measure β, α, ε, D, D’ and δ with 6 screens without any other measurements.
Σ =
σ 21
σ 22
σ 23
σ 42
σ 52
σ 62
"
#
$$$$$$$$$
%
&
'''''''''
,Π =
β1ε +D12δ 2
α1ε −D1 *D1δ2
γ1ε + *D12δ 2
D1δ2
*D1δ2
δ 2
"
#
$$$$$$$$$
%
&
'''''''''
,N =
1 0 0 0 0 0C22 −2C2S2 S22 2C2ξ2 2S2ξ2 ξ2
C23 −2C3S3 S23 2C3ξ3 2S3ξ3 ξ3
C24 −2C4S4 S24 2C4ξ4 2S4ξ4 ξ4
C25 −2C5S5 S25 2C5ξ5 2S5ξ5 ξ5
C26 −2C6S6 S26 2C6ξ6 2S6ξ6 ξ6
"
#
$$$$$$$$$
%
&
'''''''''
σ i2 = βi ⋅ε +Di
2 ⋅δ 2
More than 6 screens… • Fit procedure…
• Function to be minimized: Δi…measurement error
• Equation (*) can be solved analytically see – G. Arduini et al., “New methods to derive the optical and beam
parameters in transport channels”, Nucl. Instrum. Methods Phys. Res., 2001.
χ 2 (Π) = Σi − (NΠ)iΔi
%
&'
(
)*
i=1
Nmon
∑2
∂χ 2
∂Πi
= 0 (*)
27
In Practice….
Difficulty in practice is fitting the beam size accurately!!!
Matching screen • 1 screen in the circular
machine • Measure turn-by-turn profile
after injection
• Algorithm same as for several screens in transfer line
• Only allowed with low intensity beam
• Issue: radiation hard fast cameras
Profiles at matching monitor after injection with steering error.
28
Blow-up from thin scatterer • Scattering elements are sometimes required in the beam
– Thin beam screens (Al2O3,Ti) used to generate profiles.
– Metal windows also used to separate vacuum of transfer lines from vacuum in circular machines.
– Foils are used to strip electrons to change charge state
• The emittance of the beam increases when it passes through, due to multiple Coulomb scattering.
θs
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅+=
radradinc
cs L
LLLZ
cMeVpmrad 10
2 log11.01]/[
1.14][β
θrms angle increase:
βc = v/c, p = momentum, Zinc = particle charge /e, L = target length, Lrad = radiation length
Blow-up from thin scatterer
sθβ+=
=
'0
'
0
XX
XX
new
new
Ellipse after scattering
Matched ellipse
Each particles gets a random angle change θs but there is no effect on the positions at the scatterer
After filamentation the particles have different amplitudes and the beam has a larger emittance
2/2Anew=ε
X
'X
29
Ellipse after filamentation
Matched ellipse
20 2 snew θ
βεε +=
uncorrelated
Blow-up from thin scatterer
( )
( )
20
20
22
22
2
2
22
2
2
s
ss
ss
ss
s
θβε
θβθβε
θβθβ
βθθβ
θβ
+=
++=
+++=
+++=
++=
+=
'0
'0
'200
2
'0
'200
'0
20
'222
X
XXXA
XXX
XX
XXA
new
newnewnew
0
Need to keep β small to minimise blow-up (small β means large spread in angles in beam distribution, so additional angle has small effect on distn.)
X
'X
Blow-up from charge stripping foil • For LHC heavy ions, Pb53+ is stripped to Pb82+ at 4.25GeV/u using a
0.8mm thick Al foil, in the PS to SPS line • Δε is minimised with low-β insertion (βxy ~5 m) in the transfer line
• Emittance increase expected is about 8%
0 50 100 150 200 250 3000
20
40
60
80
100
120
Stripping foil
S [m]
Beta
[m]
TT10 optics
beta Xbeta Y
30
Kick-response measurement • The observable during kick-response measurement are the
elements of the response matrix R
– ui is the position at the ith monitor
– δj is the kick of the jth corrector
• Cannot read off optics parameters directly • A fit varies certain parameters of a machine model to reproduce the
measured data è LOCO principle
• The fit minimizes the quadratic norm of a difference vector V
Reference: K. Fuchsberger, CERN-THESIS-2011-075
Rij =uiδ j
Vk =Rijmeas − Rij
model
σ i
k = i ⋅ (Nc −1)+ j
Rijmodel =
βiβ j sin(µi −µ j )
0
for µi>µj
otherwise
σi… BPM rms noise Nc… number of correctors
Example: LHC transfer line TI 8 • Phase error in the vertical plane
• Traced back to error in QD strength in transfer line arc:
8. Optimization of the Injection Iines
strength2. These plots show a better agreement of the measurement- and model-datathan those comparing the same measured data with a model that uses the samestrength as in the machine (Figs. 8.4). As the change is in the defocussing quadrupolestrength, the effect on the horizontal plane is minimal (and therfore not shown here).
(a) MDMV.400097 (b) MDSV.400293
Figure 8.3.: Vertical responses for 2 correctors with reduced kqid.80500 in machine,compared with model using original 2007 optics (”ti8-2007-09-13”).
(a) MDMV.400097 (b) MDSV.400293
Figure 8.4.: Vertical responses for 2 correctors with reduced kqid.80500 in machine,compared with model using the same settings as in the machine (”ti8-2008-05-24-2125”).
As illustrated in Figs. 8.5, the mean rms difference for each monitor betweenmeasurement- and model- responses, is only 5.9m/rad when using the original model
2Data taken: 2008-05-24 21:33 to 22:18
74
8. Optimization of the Injection Iines
strength2. These plots show a better agreement of the measurement- and model-datathan those comparing the same measured data with a model that uses the samestrength as in the machine (Figs. 8.4). As the change is in the defocussing quadrupolestrength, the effect on the horizontal plane is minimal (and therfore not shown here).
(a) MDMV.400097 (b) MDSV.400293
Figure 8.3.: Vertical responses for 2 correctors with reduced kqid.80500 in machine,compared with model using original 2007 optics (”ti8-2007-09-13”).
(a) MDMV.400097 (b) MDSV.400293
Figure 8.4.: Vertical responses for 2 correctors with reduced kqid.80500 in machine,compared with model using the same settings as in the machine (”ti8-2008-05-24-2125”).
As illustrated in Figs. 8.5, the mean rms difference for each monitor betweenmeasurement- and model- responses, is only 5.9m/rad when using the original model
2Data taken: 2008-05-24 21:33 to 22:18
74
31
Summary • Transfer lines present interesting challenges and differences from
circular machines – No periodic condition mean optics is defined by transfer line element
strengths and by initial beam ellipse
– Matching at the extremes is subject to many constraints
– Emittance blow-up is an important consideration, and arises from several sources
– The optics of transfer line has to be well understood
– Several ways of assessing optics parameters in the transfer line have been shown
EXTRA SLIDES
32
Blow-up from betatron mismatch
x2 = a2β2 sin(ϕ +ϕo ), x '2 = a2 β2 cos(ϕ +ϕo )−α2 sin(ϕ +ϕo )[ ]
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡
2
2
111 '011
xx
βαβ2
2
'XX
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
2
121
1
2
1
22
2
121
1
2
2
12 2ββ
ααββ
ββ
ββ
ααββ
ββ
2222
22 'XX'XXA
General betatron motion
applying the normalising transformation for the matched beam
an ellipse is obtained in normalised phase space
2
2
121
1
2
2
1
1
2
2
121
1
2⎟⎟⎠
⎞⎜⎜⎝
⎛−+==⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−=
ββ
ααββ
ββ
γββ
βββ
ααββ
α newnewnew , ,
characterised by γnew, βnew and αnew, where