Effects of Quad Fringe Fields and Magnetic …Effects of Quad Fringe Fields and Magnetic...

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

luminosity@1.89 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

Difference between design and measured Difference between design and measured opticsoptics

R3OQ1A

R3OQ04

R3OQ06

R3OQ08

R3OQ10

R3OQ12

R3OQ14

R3OQ16

R2OQ17

R2OQ15

R2OQ13

R2OQ11

R2OQ09

R2OQ07

R2OQ05

R2OQ03

BetaX (m)

Measured

LOCO model

Design model

R3OQ1A

R3OQ04

R3OQ06

R3OQ08

R3OQ10

R3OQ12

R3OQ14

R3OQ16

R2OQ17

R2OQ15

R2OQ13

R2OQ11

R2OQ09

R2OQ07

R2OQ05

R2OQ03

BetaY (m)

Measured

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)

R3OQ1A

R3OQ02

R3OQ04

R3OQ06

R3OQ08

R3OQ10

R3OQ12

R3OQ14

R3OQ16

R2OQ17

R2OQ15

R2OQ13

R2OQ11

R2OQ09

R2OQ07

R2OQ05

R2OQ03

R1OQ02

R1OQ04

R1OQ06

R1OQ08

R1OQ10

R1OQ12

R1OQ14

R1OQ16

R4OQ17

R4OQ15

R4OQ13

R4OQ11

R4OQ09

R4OQ07

R4OQ05

R4OQ03

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)...

0 . 9 9 5

1 . 0 0 0

1 . 0 0 5

1 . 0 1 0

1 . 0 1 5

1 . 0 2 0

1 . 0 2 5

1 . 0 3 0

1 . 0 3 5

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

0 . 9 6

0 .9 8

1 .0 0

1 .0 2

1 .0 4

1 .0 6

1 .0 8

1 .1 0

1 .1 2

1 .1 4 B E R 2 0 0 7 - 0 4 - 1 7

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

OutlineOutline

QuadrupoleQuadrupole modelingmodeling

Hard edge approximation: simple but unphysical

Trapezoidal fringe modelApproximation of fringe fieldsFringe extension: SAD definition 0L

SAD: http://acc-physics.kek.jp/sad

Linear magnet imperfectionsLinear magnet imperfections

0 200 400 600 800 1000

s (mm)

Measurement Trapezoid fringe

0 200 400 600 800 1000-0.20

s (mm)

Measurement Trapezoid fringe

0 200 400 600 800 1000

s (mm)

Hard-edge

)()()( 0 sksKsK +=

Real magnet Ideal magnet Gradient errors

Linear magnet imperfections (cont)Linear magnet imperfections (cont)

The theory

0)]()([ 02

=+± usksKds

BBeta function measurement and manipulationeta function measurement and manipulation

Gradient errorGradient error

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:

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)

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

2)(),,( cmAcPcetpqH +−+=rr

φφ : scalar potential: scalar potential

: 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

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

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

1s 0s 2s

Fringe field integrals (cont)Fringe field integrals (cont)

Case of BEPCII SR ringR

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

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:

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

)SAD( 2I

First derivation from linear fringe mapFirst derivation from linear fringe map

Corrected focal length

Recalculation of tune shift

21011 24

1 fKIJ =≈

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

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.

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

Effective strength as variable

mD 105.0=

mL 34.00 =

LL /∆

KK /∆

KLKL /∆

2121 /TT∆

1111 /TT∆

Effective length as variableshort magnet=>full fringe

20 10 −= mK

mD 105.0=

KK /∆

LL /∆KLKL /∆

2121 /TT∆

1111 /TT∆

Fringe extension as variableshort fringe: excellentlong fringe: not so good

20 2 −= mK

mL 6.00 =

KK /∆

LL /∆

KLKL /∆

2121 /TT∆

1111 /TT∆

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

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!!

backupbackup

Optics correctionOptics correction

Comparison of SAD and LOCO correction

0 . 9 4

0 . 9 6

0 . 9 8

1 . 0 0

1 . 0 2

1 . 0 4

1 . 0 6

1 . 0 8

1 . 1 0

Q u a d

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

Effects of Quad Fringe Fields and Magnetic …Effects of Quad Fringe Fields and Magnetic...

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