effects of quad fringe fields and magnetic …effects of quad fringe fields and magnetic...
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Effects of Quad Fringe FieldsEffects of Quad Fringe Fieldsand Magnetic Interferenceand Magnetic Interference
Demin ZhouIHEP, Beijing
Joint DESY and University of Hamburg Accelerator Physics Seminar
Feb. 26, 2008, DESY, Hamburg
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
The BEPCIIThe BEPCIIAn upgrade project of the Beijing ElectronPositron Collider (BEPC)
A factory-like double-ring collider@τ-Charmenergy regionProviding beams to both high energy physicsexperiments and synchrotron radiation usersConstructed in the same tunnel for BEPCKeeping all previous beamlines unchanged
““ThreeThree”” rings in the same tunnelrings in the same tunnel
BSR: Bepcii Synchrotron RingBER: Bepcii Electron RingBPR: Bepcii Positron Ring
Main parameters of BEPCIIMain parameters of BEPCIIParameters Unit Colliding mode SR mode
Operation energy (E) GeV 1.0−2.1 2.5 Injection energy (Einj) GeV 1.55−1.89 1.89
Circumference (C) m 237.53 241.13 β*-function at IP (βx
*/ βy*) cm 100/1.5
Tunes (νx/νy/νs) 6.53/5.58/0.034 7.28/5.18/0.036 Hor. natural emittance (εx0) mm⋅mr 0.14 @1.89 GeV 0.12 Damping time (τx/τy/τe) 25/25/12.5 @1.89 GeV 12/12/6
RF frequency (frf) MHz 499.8 499.8 RF voltage per ring (Vrf) MV 1.5 1.5~3.0
Bunch number (Nb) 93 Bunch spacing m 2.4 Beam current mA 910 @1.89 GeV 250
Bunch length (cm) σl cm ~1.5 Impedance |Z/n|0 Ω ~ 0.2 Crossing angle mrad ±11
beam-beam parameter 0.04/0.04 Beam lifetime hrs. 3.0 15
[email protected] GeV 1033cm-2s-1 1
MilestonesMilestonesMar. 2006: Ring installation startedNov. 12, 2006: Commissioning startedNov. 18, 2006: Beam accumulated in BSR Dec. 25, 2006: Beam provided to SR usersFeb. 09, 2007: e- beam stored in BERMar. 04, 2007: e+ beam stored in BPRMar. 25, 2007: First collision observedJun. 15, 2007: Second SR runJan. 29, 2008: 500mA*500mA collisionLuminosity exceeded 1x1032cm-2s-1
Critical success factors in the commissioning Critical success factors in the commissioning (personal viewpoints)(personal viewpoints)
Motivated teamGood preparation and managementTeam work spirit
Components with very high quality assuranceMagnetsPower supplyBeam instrumentation: BPM, BLM, ...Feedback system
Control and diagnostics tools with high efficiencyBBA and COD correctionOptics correction based on response matrixInjection controlCollision tuning
Challenges (personal viewpoints)Challenges (personal viewpoints)
Complicated design schemeNon-symmetric lattice for collisionMirror symmetric lattice for SR
Very tight schedule for installation and commissioningCommissioning
Hardware fault detection with beam, such as Magnet, BPM, etc.Improvised softwaresRamping with wigglers (E=1.89GeV->2.5GeV)Sensitivity of the lattice to imperfectionsBeam instabilities and intensity limitationsLifetime
x
Difference between design and measured Difference between design and measured opticsoptics
0
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35
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Design model
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LOCO model
Design model
Beta function of Beta function of BSRBSR (half of the mirror symmetric ring)(half of the mirror symmetric ring)LOCO model: fitting model based on response matrix measurementLOCO model: fitting model based on response matrix measurementLOCO: Linear Optics from Closed OrbitsLOCO: Linear Optics from Closed Orbits
Y.Y. Wei
Difference between design and measured Difference between design and measured optics (cont)optics (cont)
-0.08
0
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AF-1
QuadrupoleQuadrupole fudge factor AF:fudge factor AF:The change of The change of quadrupolequadrupole strengths to restore the opticsstrengths to restore the optics
AFKK ∆=∆ 11 1−=∆ AFAF
Y.Y. Wei
SR opticsSR optics BSR_07jan01:Nominal tunes: (Nominal tunes: (7.27, 5.377.27, 5.37)) Measured: (Measured: (7.205, 5.2817.205, 5.281))Negative tune shift: (Negative tune shift: (--0.065, 0.065, --0.090.09))
BSR fudge factors
Fudge factors indicate an overall positive shift of quad strengtFudge factors indicate an overall positive shift of quad strengthshs
(0.28, 0.18)
(0.53, 0.58)
Resonances up to 4th order
Tuning BSR @ nominal tune (7.28, 5.18) ...Tuning BER/BPR @ nominal tune (6.53, 5.58)...
Difference between design and measured Difference between design and measured optics (cont)optics (cont)
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0 . 9 9 5
1 . 0 0 0
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1 . 0 2 0
1 . 0 2 5
1 . 0 3 0
1 . 0 3 5
Fudg
e fa
ctor
Q u a d
B P R 2 0 0 7 - 0 5 - 0 1
BPR AF: 1.01~1.02BPR AF: 1.01~1.02
Y.Y. Wei
Difference between design and measured Difference between design and measured optics (cont)optics (cont)
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Q03
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0 . 9 6
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1 .1 4 B E R 2 0 0 7 - 0 4 - 1 7
Fudg
e fa
ctor
Q u a d
R1OQ16R1OQ16Cable connecting to one poleCable connecting to one poleshortedshorted
BER AF: 1.01~1.02BER AF: 1.01~1.02
Y.Y. Wei
Why negative tune shifts and large fudge Why negative tune shifts and large fudge factors of 1.01~1.02?factors of 1.01~1.02?
Fudge factors: to compensate gradient errors fromMagnet alignment (random)Magnetic measurement (X)Faulty powering (X)Fringe fieldsMagnetic interference
Fringe fields and magnetic interferenceare reasonable candidates
Fringe fields neglected in the design stage but important for small ringsShort distances between quads and sextsdue to limited spaces
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
QuadrupoleQuadrupole modelingmodeling
Hard edge approximation: simple but unphysical
Trapezoidal fringe modelApproximation of fringe fieldsFringe extension: SAD definition 0L
1f
SAD: http://acc-physics.kek.jp/sad
Linear magnet imperfectionsLinear magnet imperfections
0 200 400 600 800 1000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Fiel
d
s (mm)
Measurement Trapezoid fringe
0 200 400 600 800 1000-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Fiel
d
s (mm)
Measurement Trapezoid fringe
0 200 400 600 800 1000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Fiel
d
s (mm)
Hard-edge
== ++
)()()( 0 sksKsK +=
Real magnet Ideal magnet Gradient errors
Linear magnet imperfections (cont)Linear magnet imperfections (cont)
The theory
0)]()([ 02
2
=+± usksKds
ud
BBeta function measurement and manipulationeta function measurement and manipulation
Gradient errorGradient error
20
0
02
000 )()(
)(1))((2)()( sss
ssssss −+
+−−=βααββ
PParticle directionarticle direction
EntranceEntrance ExitExit
Transformation of Courant-Snyder parameters
Linear magnet imperfections (cont)Linear magnet imperfections (cont)
Some conclusionsQuad fringe fields do lead to tune shift and beta-beatingTune shift is always negativeTune shift is proportional to quad focusing strengthTune shift is proportional to alpha function, the slope of beta functionTune shift is proportional to the square of fringe extension
Tune shift computation using SADTune shift computation using SADSAD can treat fringe fields of quads and bendsChoose SR mode: BSR_07jan01Totally 7 types of quads in SR ring
Special quads in IRSpecial quads in IRAdjacent to Sexts
Radius
Tune shift computation using SAD (cont)Tune shift computation using SAD (cont)
Tune shift for each quad (half SR ring)x10Ex10E--33 x10Ex10E--33
HHorizontal tuneorizontal tune Vertical tuneVertical tune
Estimation:
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
Simulation using OPERASimulation using OPERA--3D/TOSCA (Y. Chen)3D/TOSCA (Y. Chen)
Quad (105Q) and Sext (130S) assembly36 sexts in a ring divided into 4 groupsDistance of yokes: from 6cm to 25cm
DDistance of yokesistance of yokes
Simulation using OPERASimulation using OPERA--3D/TOSCA (Y. Chen) 3D/TOSCA (Y. Chen) (cont)(cont)
Quad field integral decay~0.6% @17.3cm of yoke distance (BEPCII case)Simulation agreed well with point-to-point measurementDistance of yokes should be larger than 25cm,if decay<0.1% required. Quad aperture radius: 5.25cm
Decay vs. yoke distanceSimulation and measurement
field gradient difference (simulation)@17.3cm of yoke distance
Measurement
Simulation using OPERASimulation using OPERA--3D/TOSCA (Y. Chen) 3D/TOSCA (Y. Chen) (cont)(cont)
Bias of B2 vs. Sext current Field integral decay vs. magnet aperture
Large aperture=>long fringe extension=>large field integral decay
Tune shift computation using SAD Tune shift computation using SAD ---- summarysummary
BSR_07jan01Turn on the bend fringe fields: (0.0, -0.0226)Turn on the quad fringe fields: (-0.0360, -0.0402)Turn on both the bend and quad fringe fields: (-0.0360, -0.0632)Turn on magnetic interference between quads and sexts:(-0.028, -0.037)Turn on fringe fields and magnetic interference: (-0.064, -0.102)Measured tune shift with beam: (-0.065, -0.09)
Basically, estimated tune shift agreed well withbeam based measurement ( measured tune shiftand fudge factors)
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
Lie Algebra techniqueLie Algebra technique
Hamiltonian systemSolve the problem analyticallyPerturbation treatment if necessaryPreserve the semplecticity of the solution
),( rrfr ′=′′ rrr],[ ii XHX =′
)(:),(:)( 0 itdtXHf XeXt
∫=′′−
GGenerating function:enerating function: ∫ ′′=t
tdtXHtF0
),()(
Step 1: sStep 1: s--dependent Hamiltonian in the field of dependent Hamiltonian in the field of a normal quada normal quad
Frenet-Serret coordinate systemOn-momentum particleExpand H(s) in polynomials
220
2)(),,( cmAcPcetpqH +−+=rr
φφ : scalar potential: scalar potential
Ar
: vector potential: vector potential
Ref.[1] J. Irwin and C.X. WangRef.[1] J. Irwin and C.X. Wang
Step 2: Perturbation treatmentStep 2: Perturbation treatmentSolutions for s-dependent Hamiltonian system arehard to be found, even for linear systemOffer clear physical picture of perturbationsEvaluate the significancy of fringe field effect
Ref.[1] J. Irwin and C.X. WangRef.[1] J. Irwin and C.X. Wang
Step 3: Linear map (from quad center to far Step 3: Linear map (from quad center to far right side)right side)
Map of ideal quadMap of fringeMap of drift
1s 0s 2s
Ref.[1] J. Irwin and C.X. WangRef.[1] J. Irwin and C.X. Wang
Step 4: Generating functionsStep 4: Generating functions
s-dependent dynamical variables: Taylor expansionAssumption: fringe region is short2rd BCH formula is enough
Step 4: Generating functions (cont)Step 4: Generating functions (cont)
Represented by fringe field integrals (FFI)
000 ≡+ +− II
Fringe field integralsFringe field integrals
Anti-symmetric assumption not necessary
+iI
−iI
1s 0s 2s
Fringe field integrals (cont)Fringe field integrals (cont)
Case of BEPCII SR ringR
3OQ
1B
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K0(1
/m2 )
Quad
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I 1
Quad
210111 24
1 fKIII =+= +−
Proportional to square of fringe extension
Step 5: Correction matrix of fringe fieldStep 5: Correction matrix of fringe fieldLinear fringe effects
Scale changeDriftQuadrupole
SAD linear fringe:SAD linear fringe:
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=
− 1
1
00
101
)SAD( 2I
I
x eeJ
MR
First derivation from linear fringe mapFirst derivation from linear fringe map
Corrected focal length
Recalculation of tune shift
21011 24
1 fKIJ =≈
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
Second derivation from linear fringe map Second derivation from linear fringe map ––equiv. H. E. modelequiv. H. E. model
Two parameters for symmetric longitudinal field distribution
Equivalent strengthEquivalent length
==
Equivalent strength and length in the focusing Equivalent strength and length in the focusing planeplane
AA, , BB, , CC, and , and DD: parameters on fringe profile: parameters on fringe profile
0K∝
20K∝
Equivalent strength and length in the Equivalent strength and length in the defocusing planedefocusing plane
Easily derived from results of focusing plane using analogy
Properties of equiv. H.E. modelProperties of equiv. H.E. model
Equiv. length is longerEquiv. strength is lowerDeviation of Equiv. product is small
Ref.[2] Courtesy J.G. WangRef.[2] Courtesy J.G. Wang
Ref. [3] Courtesy S. Bernal, et al.Ref. [3] Courtesy S. Bernal, et al.
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
Numerical test of the equiv. H.E. modelNumerical test of the equiv. H.E. model
Purpose of numerical testAccuracy of linear fringe mapValidity of equiv. H.E. model
ComparisonNumerical calculation using “slicing” methodNon-truncated equiv. H.E. modelTruncated equiv. H.E. model
slicing
Numerical test of the equiv. H.E. model (cont)Numerical test of the equiv. H.E. model (cont)
Cases with variables asEffective strengthEffective lengthFringe extensionFull fringe
Focusing functionsEnge function (Default Enge coefficients used in COSY INFINITYGaussian function
Numerical test of the equiv. H.E. model (cont)Numerical test of the equiv. H.E. model (cont)
Effective strength as variable
mD 105.0=
mL 34.00 =
LL /∆
KK /∆
KLKL /∆
2121 /TT∆
1111 /TT∆
Numerical test of the equiv. H.E. model (cont)Numerical test of the equiv. H.E. model (cont)
Effective length as variableshort magnet=>full fringe
20 10 −= mK
mD 105.0=
KK /∆
LL /∆KLKL /∆
2121 /TT∆
1111 /TT∆
Numerical test of the equiv. H.E. model (cont)Numerical test of the equiv. H.E. model (cont)
Fringe extension as variableshort fringe: excellentlong fringe: not so good
20 2 −= mK
mL 6.00 =
KK /∆
LL /∆
KLKL /∆
2121 /TT∆
1111 /TT∆
Numerical test of the equiv. H.E. model (cont)Numerical test of the equiv. H.E. model (cont)
Case of full fringefringe map estimation is no quite good for “long” full fringe magnet
20 2 −= mK2121 /TT∆ 1111 /TT∆
Proposal of a simple H.E. modelProposal of a simple H.E. model
First order correctionApplicable for both focusing and defocusing planesEasy implemented in codes not include fringe fields,such as MAD and ATMore effective for cases of small quad field integraland short fringe extension
Proposal of a simple H.E. model (cont)Proposal of a simple H.E. model (cont)
Test of the simple model using SADBSR_07jan01, nominal: (7.28, 5.18)R3OQ02:
R2OQ06:
Turn on fringe:Turn on fringe:
SSimple model:imple model:
Turn on fringe:Turn on fringe:
SSimple model:imple model:
405.00 =K
46.10 =K
IntroductionEstimation of tune shift
Quad fringe fieldsMagnetic interference
Linear fringe mapEquivalent hard edge modelNumerical testSummary
OutlineOutline
SummarySummaryTune shift in BEPCII rings was well explained by effects of fringe fields and magnet interference.A simple method was found to calculate the tune shift due toquad fringe fields.Perturbation treatment based on Lie technique is a good approach for quad fringe field effects. It is easy to be extendedto calculate nonlinear fringe maps, even magnetic interference included.The work will also help to estimate the significancy of fringe fields and magnetic interference in small rings such as CSNS, Proton Therapy Accelerator, etc.The simple H.E. model may be applied in MAD and LOCO for linear optics design and compensation.
AcknowledgementsAcknowledgements
Thanks to Prof. J.Y. Tang, Dr. Y. Chen, Dr. Y.Y. WeiThanks to the BEPCII commissioning team
[1] J. Irwin and C.X. Wang, Explicit soft fringe maps of a quadrupole, PAC95.
[2] J.G. Wang, Particle optics of quadrupole doublet magnets in Spallation Neutron Source accumulator ring, Phys. Rev. ST Accel. Beams 9, 122401 (2006).
[3] S. Bernal, et al., RMS envelope matching of electron beams from “zero” current to extreme space charge in a fixed lattice of short magnets, Phys. Rev. ST Accel. Beams 9, 064202 (2006).
References:References:
Thank you for your attention!!Thank you for your attention!!
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Optics correctionOptics correction
Comparison of SAD and LOCO correction
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Fudg
e fa
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S A D _ m a t c h w i t h f r i n g e ( B + Q ) L O C O
SAD: with fringe fields of quads and bends, without magnetic interferenceLOCO: based on beam measurement, include all imperfections
The figure shows similar correction scheme in SAD and LOCO