KEKB Accelerator Review Committee, 15-16 Feb 2010, KEK, Japan

Systematic study of nonlinear dynamics
and dynamic aperture optimization
Haruyo Koiso1), Eugene Levichev2), Akio Morita1), Kazuhito Ohmi1),
Yukiyoshi Ohnishi1), Katsunobu Oide1), Pavel Piminov2)
1) KEK, Tsukuba, 2) BINP, Novosibirsk

15 February 20101

Outline

1. Relevant SuperKEKB lattice parameters
2. Nonlinearities
4. Phase space plots
5. Nonlinear detuning
6. Dynamic aperture study
7. DA optimization
8. Conclusion2SuperKEKB LER Lattice

IR with final focus and vertical chromatic section. Bymax = 2.7 km
3SuperKEKB LER relevant parameters

All plot data below are given for IP
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Nonlinearities

Strong chromatic sextupoles (FF horizontal andvertical correction sections) + arc chromatic
sextupoles. All sextupoles are arranged in pairs separated by –I
Fringe fields in quadrupoles (third order)
Fringe fields in bends (second order)
Kinematic terms (third order)5

Nonlinearities - sextupoles
For kick sextupoles all high orderaberrations are cancelled exactly

For thick sextupoles only the secondorder terms are cancelled but third andhigher appear

6

Nonlinearities – quad fringe fields
Particle transport through the quadrupole fringe (leading term)

For simulation we use more complex but symplectic expressions from SAD
Rough comparison of two effects:
FF QC1LPH
Thick sextupole pair in vertical
IR chromatic section
Quadrupole fringe field effect is stronger than the tick sextupole effect7

Nonlinearities – kinematic term

As we expect, a bulk of the kinematic effects

should come from IP with Beta_Y = 270 um.
The kinematic effects are included in the simulation.

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Phase space plot (horizontal)

Extremely small seed of the vertical motion reduces the horizontal
aperture by factor 2 through the nonlinear coupling term9

Phase space plot (vertical)

1D and 2D (with small seed of horizontal motion) vertical apertures
have the same size. Vertical DA is defined by the vertical nonlinear
terms

10

Tune amplitude dependence
Qx(Jx)Qx(Jy)

Sextupoles only

Main effect is due to thequadrupole fringe fields!

Qy(Jx)Qy(Jy)

S + fringe fields

S + fringe fields+ kinematic terms 11

Dynamic aperture - I
Illustration: dynamic aperture for thin and thick sextupoles arranged inpairs separated by -I map. Quadrupole fringe fields are switched off.
Thin = geometry aperture

Thin
Thick

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Dynamic aperture - II
Dynamic aperture decrease due to the kinematic effects.
Off-energy
On-energy---- Without kinematic terms---- With kinematic terms

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Dynamic aperture - III
W/o synch.osc.

With synch.osc.

All nonlinearities are included

60 Sigma_y, ±15 Sigma_x

80 Sigma_y, ±30 Sigma_x

60 Sigma_y, ±15 Sigma_x

14Dynamic aperture - IVConclusion before optimization
DA in rms beam size

NyNx
(1) DA due to the quadrupole fringe field and thick sextupoles have thesame order of magnitude(2) Among all the quadrupoles influence of QC1 is most crucial15

Dynamic aperture optimization - I(1) Correction of the thick sextupole effect (by weak sextupolecorrectors)(2) Correction of the quadrupole fringe fields (how?)
One can try to cope with the quadrupole edges (third orderaberration) by octupole correctors compensating relevant
detuning terms
To improve vertical aperture we have to correct verticaldetuning term
To improve horizontal aperture we have to correct coupling
detuning term
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Dynamic aperture optimization - II
Separate octupole correctors,
detuning coefficients

Vertical DA increase by separate octupoles(1) In IR, where the bulk of the perturbation sources are concentrated, By>>Bx(2) In any casecoupling term

so we can not control effectively the(3) But with the vertical one everything works well17

Dynamic aperture optimization - III
Octupole correctors incorporated in the quadrupole

Transformation through this simple model shows that we can effectively
compensate either direct or coupling terms for both planes. The
octupole corrector strength for each octupole kick is

Later we found that a single kick or the octupole corrector distributed along thequadrupole length work as well.
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Dynamic aperture optimization - IV
Example of the local correction scheme

DA reduced by the edges of strong
quadrupole
Octupole corrector open it either
horizontally (~10 times) (coupling termcorrection) or vertically (~3 times)
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Dynamic aperture optimization - VAs we have two knobs to control the horizontal (by localcorrection) and vertical (by global correction) aperture, thefollowing procedure was applied(1) Optimization of the horizontal aperture in main quadrupoles(2) Optimization of the vertical aperture by two octupoles in thevertical section of the FF chromaticity correction(3) Numerical optimization of all octupole correctors because
in real lattice every corrector not only suppresses a required
term but also affects others (though slightly)

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Dynamic aperture optimization - VI
Optimization of the quadrupole fringe fields by octupole correctors

Octupole correctors

Black – initial apertureBlue – FF quadrupoles corrected
Red – all quadrupoles corrected21

Dynamic aperture optimization - VII
Tune vs amplitude before and after optimization

Qx(Jx)Qy(Jx)Qy(Jy)Qy(Jx)Qy(Jy) zoomed22

Dynamic aperture optimization - VIII
Thick sextupole correction by two additional weak (strength = 3-7% of
the main one) sextupole correctors (not applied yet to SuperKEKB)Black – initial aperture
Red – improved by sextupolecorrectors

23Conclusion(1) In SuperKEKB LER the dynamic aperture is limited in thefirst place by the fringe field in the strong FF quadrupoles(2) In the second place by the finite length strong chromaticsextupoles(3) The quadrupole fringe fields can be corrected by octupole
correctors, individual and incorporated in the FF quadrupoles(4) The next steps are: (a) including and optimization of thesextupoles, (b) global optimization of all effects for on- and offmomentum particle, (c) optical errors (!)
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