beam transfer lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 m ... 0 50 100 0 250 500 750 1000 ......
TRANSCRIPT
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Beam Transfer Lines• Distinctions between transfer lines and circular machines• Linking machines together• Trajectory correction• Trajectory correction• Emittance and mismatch measurement• Blow-up from steering errors, optics mismatch and thin screens• Phase-plane exchange
Brennan GoddardCERN
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Injection, extraction and transfer
CERN Complex• An accelerator has limited dynamic range.
• Chain of stages needed to h hi hreach high energy
• Periodic re-filling of storage rings, like LHCExternal experiments like• External experiments, like CNGS
Transfer lines transport thebeam between accelerators,and onto targets, dumps,instruments etc.
LHC: Large Hadron ColliderSPS: Super Proton SynchrotronAD: Antiproton DeceleratorISOLDE: Isotope Separator Online DevicePSB: Proton Synchrotron BoosterPS P t S h tPS: Proton SynchrotronLINAC: LINear AcceleratorLEIR: Low Energy RingCNGS: CERN Neutrino to Gran Sasso
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Normalised phase space• Transform real transverse coordinates x, x’ by
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡'
011' x
xxx
SSS βαβN
'XX
1 xβ
⋅=X
⎦⎣⎦⎣⎦⎣⎦⎣ SSS ββ
'1 xx
S
βα
β
+'X 'xx SSS
βαβ
+⋅=X
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Normalised phase space
Real phase space Normalised phase space
1
γαε−x’
ε='X
'X
γε 1
βε 1
γε=max'x
ε=maxX
⇒x
β
Area = πεArea = πε
⇒X
βε=maxxε=maxX
22 ''2 xxxx ⋅+⋅⋅+⋅= βαγε 22 'XX +=ε
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General transport
y
s
Beam transport: moving from s1 to s2 through n elements, each with transfer matrix Mi
xx
y ss
1
2
⎥⎤
⎢⎡
⋅⎥⎤
⎢⎡
=⎥⎤
⎢⎡
⋅=⎥⎤
⎢⎡
21'2 xSCxx
M
1
∏=n
21 MM⎥⎦
⎢⎣
⎥⎦
⎢⎣
⎥⎦
⎢⎣
⎥⎦
⎢⎣
→ ''''21'2 xSCxx
M ∏=
→i
n1
21 MM
( )
( ) ( )[ ] ( )⎥⎥⎥⎤
⎢⎢⎢⎡
ΔΔΔΔ
ΔΔ+Δ=→ β
μββμαμββ
ii1
sinsincos 2111
2
21MTwiss parameterisation
( ) ( )[ ] ( )⎥⎥⎦⎢
⎢⎣
Δ−ΔΔ+−Δ− μαμββμααμααββ sincossin1cos1
22
12121
21
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Circular Machine
Circumference = L
( ) ⎥⎤
⎢⎡ +
==QQQ πβπαπ
12sin2sin2cos
2MMOne turn ( ) ⎥⎥⎦⎢
⎢⎣
−+−== →→ QQQL παππαβ 2sin2cos2sin11 2021 MMOne turn
• The solution is periodic p• 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
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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 βvalues ⇒ Matched Ellipse (for this location)
x’
βα 2
max1' += ax βαγ 22 ''2 ⋅+⋅⋅+⋅= xxxxa
xArea = πa β
αγ21+=
βax =max
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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(via filamentation) a large ellipse with the original α, β, but larger ε
x’
α, β
x’
α, β
xx
Turn 1 Turn 2
After filamentation
εο, α∗ , β∗
After filamentation
εο, α∗ , β∗
ε > εο, α, βε > εο, α, βTurn 3 Turn n>>1
Matched ellipse determines beam shape
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Transfer line
⎥⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡→ '21'
2
2
xx
xx
MOne pass:
⎥⎦
⎤⎢⎣
⎡'1
xx
⎥⎦
⎤⎢⎣
⎡'2
2
xx
⎤⎡ β
⎦⎣ 1x ⎦⎣ 2x
( )
( ) ( )[ ] ( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
Δ−ΔΔ+−Δ−
ΔΔ+Δ=→
μαμββμααμααββ
μββμαμββ
sincossin1cos1
sinsincos
22
12121
21
2111
2
21M
⎦⎣
• No periodic condition existsp• 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
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Transfer line• On a single pass there is no regular motion
– Map single particle coordinates at entrance and exit.
– Infinite number of equally valid possible starting ellipses for single particleInfinite number of equally valid possible starting ellipses for single particle……transported to infinite number of final ellipses…
L→0Mx’
x’
α1, β1
α∗ β∗α2, β2
⎥⎤
⎢⎡ 2x
⎥⎦
⎤⎢⎣
⎡'1
1
xx
x
α∗1, β∗1 ⎥⎦
⎢⎣
'2x
x xTransfer Line
βα∗2, β∗2Entry Exit
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Transfer Line• Initial α, β defined for transfer line by beam shape at entrance
x’ x’α∗, β∗x’ x’α∗, β∗
α, βα, β
x xx x
Gaussian beamNon-Gaussian beam(e.g. slow extracted)
Gaussian beamNon-Gaussian beam(e.g. slow extracted)
• Propagation of this beam ellipse depends on line elementsPropagation of this beam ellipse depends on line elements
• A transfer line optics is different for different input beams
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Transfer Line
350
• The optics functions in the line depend on the initial values
Design β functions in a transfer line
200
250
300
[m]
- Design βx functions in a transfer line− βx functions with different initial conditions
50
100
150
Bet
aX [
1500 2000 2500 30000
50
S [m]
• Same considerations are true for Dispersion function: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
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Transfer Line
• Another difference….unlike a circular ring, a change of an element in a line affects only the downstream Twiss values (including di i )
300
350
dispersion)- Unperturbed βx functions in a transfer line− βx functions with modification of one quadrupole strength
150
200
250
Bet
aX [m
]
10% change in this QF strength
1500 2000 2500 30000
50
100
1500 2000 2500 3000S [m]
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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)
• Other important constraints can includeOther important constraints can include– Minimum bend radius, maximum quadrupole gradient, magnet aperture,
cost, geology
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Linking Machines
Extraction
Matched Twiss at extraction propagated to matched Twiss at injection
Transferα1x, β1x , α1y, β1yαx(s), βx(s) , αy(s), βy(s)
s
α2x, β2x , α2y, β2y
Injection
⎥⎤
⎢⎡
⎥⎤
⎢⎡ −
⎥⎤
⎢⎡ 1
222
''''2 ββ
SSSCCSCCSCSC
The Twiss parameters can be propagated when the transfer matrix M is known
⎥⎤
⎢⎡
⎥⎤
⎢⎡
⎥⎤
⎢⎡
⎥⎤
⎢⎡ 2 xSCxx
M⎥⎥⎥
⎦⎢⎢⎢
⎣
⋅⎥⎥⎥
⎦⎢⎢⎢
⎣ −−+−=
⎥⎥⎥
⎦⎢⎢⎢
⎣ 1
122
2
2
'''2'''''
γα
γα
SSCCSSSCCSCC⎥
⎦⎢⎣
⋅⎥⎦
⎢⎣
=⎥⎦
⎢⎣
⋅=⎥⎦
⎢⎣
→ ''''21'2
2
xSCxxM
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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– 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, i i t iti t i l dexperience, intuition, trial and error, …
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Linking MachinesF l t f li i lif th bl b d i i th• 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
i ( b di di l ) ith t d i ispacing, (+ bending dipoles), with magnets powered in series
– Initial and final matching sections – independently powered quadrupoles, with sometimes irregular spacing.
150
200
250
300
β [m
]
BETXBETY
0
50
100
0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000S [m]Regular lattice (FODO)
SPS LHC(elements all powered in serieswith same strengths)
Final matching
section
Initial matching
section
SPS to LHC Transfer Line
Extractionpoint
Injectionpoint
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Trajectory correction
• Magnet misalignments, field and powering errors cause the trajectory to deviate from the design
• Use small independently powered dipole magnets (correctors) toUse 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 )corrector (π/2, 3π/2, …)
Corrector dipole Pickup Trajectory
QF QF
π/2
QF
QD QD
QF
• Horizontal and vertical elements are separated
• H-correctors and pick-ups located at F-quadrupoles (large βx )
V t d i k l t d t D d l (l β )• V-correctors and pick-ups located at D-quadrupoles (large βy)
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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– D and β functions can be large → bigger beam size
– Often very limited in aperture
– Injection offsets can be detrimental for performance
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Trajectory correction
Uncorrected trajectory.
y growing as a result f d i thof random errors in the
line.
The RMS at the BPMs is 3.4 mm, and ymax is , ymax12.0mm
Corrected trajectory.
The RMS at the BPMs is 0.3mm and ymax is 1mm
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Trajectory correction
• Sensitivity to BPM errors is an important issue– If the BPM phase sampling is poor, the loss of a few key BPMs can
allow a very bad trajectory, while all the monitor readings are ~zero
Correction with some monitors disabled
y j y, g
With poor BPM phase sampling the correction algorithm produces a trajectory with 185mmtrajectory with 185mm ymax
Note the change of vertical scale
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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 targetsFor stability on secondary particle production targets
• Convenient to express injection error in σ (includes x and x’ errors)
Δ [ ] √ √ 2 2 β 2
Δa
Δa [σ] = √((X2+X’2)/ε) = √((γx2 + 2αxx’+ βx’2)/ε)
X
'X
Septum
XBumpermagnets kicker Mis-steered
injected beam
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Steering (dipole) errors
• Static effects (e.g. from errors in alignment, field, calibration, …) are dealt with by trajectory correction (steering).
B t there are also d namic effects from• 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 batchvary from batch to batch, or even within a batch.
• An injection damper system is used to minimise effect on emittance
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Blow-up from steering error• Consider a collection of particles with amplitudes A
• 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• For an injection error Δay (in units of sigma = √βε) the mis-injected beam is offset in normalised phase space by d = Δax√ε
'X Mi i j t dMatched X Misinjectedbeam
Matchedparticles
X
A
d
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Blow-up from steering error• The new particle coordinates in normalised phase space are
θcosnew dXX 0 +=
θsinnew dXX '0
' +=
Mi i j t dMatched 'X
• For a general particle distribution, where A denotes amplitude in
Misinjectedbeam
Matchedparticles
X
'222 XXA +=
where A denotes amplitude in normalised phase space A
θ X
2/2A=ε
d
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Blow-up from steering errorS if l i th di t
( ) ( )'00
'222 dXdXXXA +++=+=22
θθ sincosnewnewnew
• So if we plug in the new coordinates….
( )
( ) 2'00'2002
2'00'200
dXXDXXA
dXXdXX
++++=
++++=
2
2
2
2
θθ
θθ
sincos
sincos
new ( )
( ) 2'00
0000
dXXD +++= 0 22 θθε sincos
new
0 0
2d+= 02ε
Giving for the emittance increase
2/2/ 022 dA εε +== newnew
• Giving for the emittance increase
( )2/102aΔε +=
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Blow-up from steering error
A numerical example….
Consider an offset Δa of 0.5 sigma for ginjected beam
Misinjected beam( )0 2/1 Δεε += 2anew
'X
0ε1.125=X
0.5√ε√ε
X
MatchedBeam
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Blow-up from betatron mismatch
• Optical errors occur in transfer line and ring, such that the beam can be injected with a mismatch.
• Filamentation will produce an emittance increase.
• In normalised phase space, consider Mismatchedbeam'Xp p ,
the matched beam as a circle, and the mismatched beam as an ellipse.
beamX
X
Matchedbeam
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Blow-up from betatron mismatch
[ ])sin()cos('),sin( 2222222 ooo xx φφαφφβεφφβε +−+=+=
General betatron motion – coordinates of particles on mismtached ellipse
⎤⎡⎤⎡⎤⎡ 2011 xX
applying the normalising transformation for the matched beam (subscript 1)
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡
2
2
111 '011
xx
βαβ22'X
X
an ellipse is obtained in normalised phase space
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
2
121
1
2
1
22
2
121
1
2
2
12 2ββαα
ββ
ββ
ββαα
ββ
ββ
2222
22 'XX'XXA
an ellipse is obtained in normalised phase space
2121212 ⎟⎞
⎜⎛
⎟⎞
⎜⎛− ββββ
βββ
characterised by γnew, βnew and αnew, where
2
121
1
2
2
1
1
2
2
121
1
2⎟⎟⎠
⎜⎜⎝
−+==⎟⎟⎠
⎜⎜⎝
−=ββ
ααββ
ββ
γββ
βββ
ααββ
α newnewnew , ,
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Blow-up from betatron mismatch
Mismatched( ) ( )112
112
−−+=−++= HHAbHHAa ,
From the general ellipse properties
'Xbeam
b
22
where
X
ab
( )
⎟⎞
⎜⎛
+⎟⎟⎞
⎜⎜⎛
+
+=
2
2
1211
21
ββααββ
βγ newnewH
AX
⎟⎟
⎠⎜⎜
⎝+⎟⎟
⎠⎜⎜⎝
−+=1
2
2
121
1
2
2
1
2 ββαα
ββ A
λAa
MatchedBeam generally
giving
( ) ( )112
11112
1 −−+=−++= HHHHλ
λ ,
λλ
⋅==
AA
ba
)cos(1),sin( 1new1new φφλ
φφλ +=+⋅= A'XAX
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Blow-up from betatron mismatch
)(cos1)(sin 2202
220
2211newnew φφ
λφφλ +++⋅=+= AA'XXA 22
new
We can evaluate the square of the distance of a particle from the origin as
The new emittance is the average over all phases
⎟⎟⎠
⎞⎜⎜⎝
⎛ +++== 22
22 )(cos1)(sin21
21 φφ
λφφλε 11new
20
20
2 AAAnew
⎟⎟⎠
⎞⎜⎜⎝
⎛ +++=
⎟⎠
⎜⎝
22
22
2
)(cos1)(sin21
22
φφλ
φφλ
λ
1120A
0.5 0.5
⎟⎟⎠
⎞⎜⎜⎝
⎛ +=2
20
121
λλε
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+==⎟
⎠⎞
⎜⎝⎛ += 2
2
121
21002
20 2
1121
ββ
ββαα
ββ
ββεε
λλεε Hnew
If we’re feeling diligent, we can substitute back for λ to give
⎟⎠
⎜⎝ ⎠⎝⎠⎝ 121222 ββββλ
where subscript 1 refers to matched ellipse, 2 to mismatched ellipse.
![Page 32: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/32.jpg)
Blow-up from betatron mismatchA numerical example….consider b = 3a for the mismatched ellipse
3/ abλ
Mismatched
3/ == abλ
'X
( )22 11 λλεε +
beamThen
X
( )0
0
67.1
12
ε
λλεε
=
+=new
oεa b=3a
X
MatchedBeam
![Page 33: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/33.jpg)
Emittance and mismatch measurement• A profile monitor is need to measure the beam size
– E.g. beam screen (luminescent) provides 2D density profile of the beam
• Profile fit gives transverse beam sizes σ• Profile fit gives transverse beam sizes σ.
• In a ring, β is ‘known’ so ε can be calculated from a single screen
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Emittance and mismatch measurement
• Emittance and optics measurement in a line needs 3 profile measurements in a dispersion-free region
• Measurements of σ0 σ1 σ2 plus the two transfer matrices M01 andMeasurements of σ0,σ1,σ2, plus the two transfer matrices M01 and M12 allows determination of ε, α and β
Ms0 s2s1
M 32→M21→M
2
22
1
21
0
20
βσ
βσ
βσε ===
σ0 σ1 σ2
![Page 35: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/35.jpg)
Emittance and mismatch measurement⎤⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−+−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
0
0
0
2111
21
11111111
2111
21
1
1
1
'''2'''''
2
γαβ
γαβ
SSCCSSCSSCCC
SSCC
We have
( )
( ) ( )[ ] ( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−+−−
+=⎥
⎦
⎤⎢⎣
⎡
μΔαμΔββμΔααμΔααββ
μΔββμΔαμΔββ
sincossin1cos1
sinsincos
22
12121
21
2111
2
'1
'1
11
SCSC
where
( ) ( )20
0
22
0220222
20
0
21
01102
11 1212 αβ
αββαβ
αββ ++−=++−= SSCCSSCC ,so that
⎥⎦⎢⎣ βββ 221
Using 0
2
0
220
2
0
11
20
0 βσσ
ββσσ
βε
σβ ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛== , ,
W00 21 βα =we find
( ) ( ) ( ) ( )222222 ////// SCSCSS +σσσσ( ) ( ) ( ) ( )( ) ( )2211
11222
1012202
////////
SCSCSCSCSS
−+−−= σσσσ Wwhere
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Emittance and mismatch measurement
Some (more) algebra with above equations gives
( ) ( ) ( ) 4/////1 222
222
22
2020
2WW −+−= SCSCSσσβ
2 β
And finally we are in a position to evaluate ε and α0
W1 β020 βσε =
Comparing measured α β0 with expected values gives numerical
W00 2βα =
Comparing measured αo, β0 with expected values gives numerical measurement of mismatch
![Page 37: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/37.jpg)
Blow-up from thin scattererS tt i l t ti i d i th b• 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 dueThe 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:⎠⎝ radradc p ][β
βc = v/c, p = momentum, Zinc = particle charge /e, L = target length, Lrad = radiation length
![Page 38: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/38.jpg)
Blow-up from thin scatterer
Ellipse afterscattering
Each particles gets a random angle change θs but there is no effect on the positions at the scatterer
'X
θβ+=
=
'0
'
0
XX
XXnew
sθβ+0XXnew
After filamentation the particles have different amplitudes and the beam has
X
Matchedellipse
a larger emittance
2/2Anew=ε
![Page 39: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/39.jpg)
Blow-up from thin scatterer+ '222 XXA
Ellipse afterfilamentation
( )2sθβ++=
+=
'020
222
XX
XXA newnewnew'X
uncorrelated( )22
22
2
2 ss
θβθβ
βθθβ
+++
+++=
''22
'0'200
XXXA
XXX
20
22
22
2
ss
ss
θβθβε
θβθβ
++=
+++=
'0
02002
X
XXXA new
0
X
Matchedellipse
202 sθβε +=
20 2 snew θβεε +=
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.)
![Page 40: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/40.jpg)
Blow-up from charge stripping foilF LHC h i Pb53+ i t i d t Pb82+ t 4 25G V/ i• 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 liney
• Emittance increase expected is about 8%
120TT10 optics
80
100
Stripping foil
]
beta Xbeta Y
40
60
Bet
a [m
]
0 50 100 150 200 250 3000
20
S [m]
![Page 41: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/41.jpg)
Emittance exchange insertion• Acceptances of circular accelerators tend to be larger in horizontal
plane (bending dipole gap height small as possible)
• Several multiturn extraction process produce beams which have p pemittances which are larger in the vertical plane → larger losses
• We can overcome this by exchanging the H and V phase planes (emittance exchange)(emittance exchange)
Low energy machine After multi-turn Aft itt High energy machine
y
extraction After emittanceexchange
x
In the following, remember that the matrix is our friend…
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Emittance exchangePhase-plane exchange requires a transformation of the form:
⎟⎟⎞
⎜⎜⎛
⎟⎟⎞
⎜⎜⎛
⎟⎟⎞
⎜⎜⎛
0
0
2423
1413
1
1
'0000
' xx
mmmm
xx
⎟⎟⎟⎟
⎠⎜⎜⎜⎜
⎝⎟⎟⎟⎟
⎠⎜⎜⎜⎜
⎝
=
⎟⎟⎟⎟
⎠⎜⎜⎜⎜
⎝ 0
0
0
4241
3231
2423
1
1
1
'0000
00
' yyx
mmmm
mm
yyx
A skew quadrupole is a normal quadrupole rotated by an angle θ.
The transfer matrix S obtained by a rotation of the normal transfer matrix Mq:1S = R-1MqR
where R is the rotation matrix ⎟⎟⎟⎞
⎜⎜⎜⎛
θθθθ
sin0cos00sin0cos
⎟⎟⎟
⎠⎜⎜⎜
⎝ −−
θθθθ
cos0sin00cos0sin
( i lf f h t R d b h ki th t i t f d t(you can convince yourself of what R does by checking that x0 is transformed to x1 = x0cosθ +y0sinθ, y0 into -x0sinθ +y0cosθ, etc.)
![Page 43: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/43.jpg)
Emittance exchange⎞⎛
For a thin-lens approximation⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
=
10001000010001
δ
δqM
(where δ = kl = 1/f is the quadrupole strength)
So that⎟⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎜⎛
−⎟⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎜⎛
⎟⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎜⎛
−−
== −
0cos0sinsin0cos0
0sin0cos
01000010001
0cos0sinsin0cos00sin0cos
1
θθθθ
θθδ
θθθθ
θθ
RMRS q
⎟⎟⎞
⎜⎜⎛
⎟⎟⎠
⎜⎜⎝ −⎟⎟⎠
⎜⎜⎝ −⎟⎟⎠
⎜⎜⎝
02i120001
cos0sin0100cos0sin0
θδθδ
θθδθθ
⎟⎟⎟⎟
⎠⎜⎜⎜⎜
⎝ −
=
12cos02sin010002sin12cos
θδθδ
θδθδ
N l d45º k d
For the case of θ = 45º, ⎟⎟⎟⎞
⎜⎜⎜⎛
=0100001
δS
Normal quad45º skew quad
this reduces to ⎟⎟⎟
⎠⎜⎜⎜
⎝
=
1000100
δ
S
![Page 44: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/44.jpg)
Emittance exchangeThe transformation required can be achieved with 3 such skew quads in a lattice,
of strengths δ1, δ2, δ3, with transfer matrices S1, S2, S3
A B
Skew quad Skew quad Skew quad
δ1 δ2 δ3
The transfer matrix without the skew quads is C = B A .
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
y
x
CC
C
0000
0000
[ ]
( ) ( ) [ ]⎟⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎜⎛
−+−−
+
=xxxxxxx
xxxxxxx
x
x
φΔαφΔβφΔααφΔαα
φΔββφΔαφΔββ
sincossin1cos
sinsincos
12112
2111
2
C
and similar for C[ ]⎟⎟⎠
⎜⎜⎝
− xxxx
x
xx
φΔαφΔβββsincos 2
21
21and similar for Cy
![Page 45: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/45.jpg)
Emittance exchangeWith the skew quads the overall matrix is M = S3B S2A S1
⎟⎟⎟⎞
⎜⎜⎜⎛
⎤⎡⎤⎡
++ 23412233121121221341211 δδδδδ ababccabc
( ) ( )
⎟⎟⎟⎟⎟⎟
⎜⎜⎜⎜⎜⎜
++
+⎥⎦
⎤⎢⎣
⎡++
++⎥⎦
⎤⎢⎣
⎡++
+
=
34211234332341221134134
33423422
211234333
2332212232123422
211341343
21342221
δδδδδ
δδδδδ
δδδδδδδ
δδ
cabcbaabc
cababc
abcabcabc
abc
M
( ) ( )⎟⎟⎟⎟⎟
⎠⎜⎜⎜⎜⎜
⎝+⎥
⎦
⎤⎢⎣
⎡++
++⎥
⎦
⎤⎢⎣
⎡++
+
++
3234124421124443
23312112321244312
213422144
213412113
34211234332341221134134
δδδδδδδ
δδδδδδδδ
δδδδδ
abcabcabc
abcabcabc
cabcbaabc
⎟⎟⎟⎞
⎜⎜⎜⎛
000000
2423
1413
mmmmmm
Equating the terms with our target matrix form⎟⎟⎟
⎠⎜⎜⎜
⎝ 0000
4241
3231
mmmm
a list of conditions result which must be met for phase-plane exchangea list of conditions result which must be met for phase plane exchange.
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Emittance exchange
21341211
34
12
000
δδabccc
+===
( )32341244
21123433
32123422
21341211
000
δδδδδδ
abcabcabc
+=+=+=
( )( )23312112321124443
21134134321342221
00
δδδδδδδδδδ
abcabcabcabc
+++=+++=
The simplest conditions are c12 = c34 = 0.The simplest conditions are c12 c34 0.
Looking back at the matrix C, this means that Δφx and Δφy need to be integer multiples of π (i.e. the phase advance from first to last skew quad should be 180º, 360º, …))
W l h f th t th f th k d1234
33
3412
1121 ab
cab
c−=−=δδ
We also have for the strength of the skew quads3412
44
1234
2232 ab
cab
c−=−=δδ
![Page 47: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/47.jpg)
Emittance exchangeSeveral solutions exist which give M the target form.
One of the simplest is obtained by setting all the skew quadrupole strengths the same, and putting the skew quads at symmetric locations in a 90º FODO l ttilattice
A B (=A)
δ δ δFrom symmetry A = B, and the values of α and β at all skew quads are identical.
( )⎟⎟⎞
⎜⎜⎛ + xxxxx φΔβφΔαφΔ sinsincos
with the same form for yTherefore ( ) ( )⎟⎟⎟
⎠⎜⎜⎜
⎝−
−−
==xxx
x
xxxx
φΔαφΔβφΔα
sincossin1 2BA
The matrix C is similar, but with phase advances of 2Δφ
![Page 48: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/48.jpg)
Emittance exchange
⎞⎛ βα 00
Since we have chose a 90º FODO phase advance, Δφx = Δφy = π/2, and 2Δφx = 2Δφy = π which means we can now write down A,B and C:
( )
⎟⎟⎟⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎜⎜⎜⎛
−−
−
==
xx
x
xx
αβα
βα
2
001
00
BA ⎟⎟⎟⎞
⎜⎜⎜⎛
−−
=00100001
Ci.e. 180º across the insertion in both planes
( )⎟⎟⎟⎟⎟⎟
⎠⎜⎜⎜⎜⎜⎜
⎝−
−− y
y
y
yy
αβ
α
βα
2100
00BA
⎟⎟⎟
⎠⎜⎜⎜
⎝ −−
10000100 insertion in both planes
⎠⎝
sδδδδ 1321 ====we can then write down the skew lens strength as
yxs
ββ321we can then write down the skew lens strength as
12For the 90º FODO with half-cell length L, 2
1,2LL sDF ==−= δδδ
![Page 49: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/49.jpg)
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
– Trajectory correction is rather simple compared to circular machineTrajectory correction is rather simple compared to circular machine
– Emittance blow-up is an important consideration, and arises from several sources
Phase plane rotation is sometimes required skew quads– Phase-plane rotation is sometimes required - skew quads
![Page 50: Beam Transfer Lines · · 2017-06-241 2 0 1 sin2 cos2 sin2 M ... 0 50 100 0 250 500 750 1000 ... (0 '0) 2 '2 0 0 2 ' 2 0 0 '2 0 0 A X X DX X d](https://reader033.vdocument.in/reader033/viewer/2022051803/5b03f2d67f8b9a41528bc0b6/html5/thumbnails/50.jpg)
Keywords for related topics• Transfer lines
– Achromat bends
– Algorithms for optics matchingAlgorithms for optics matching
– The effect of alignment and gradient errors on the trajectory and optics
– Trajectory correction algorithms
– SVD trajectory analysis
– Kick-response optics measurement techniques in transfer lines
– Optics measurements including dispersion and δp/p with >3 screensOptics measurements including dispersion and δp/p with >3 screens
– Different phase-plane exchange insertion solutions