SIMPSONS for beam modelling in high intensity rings Shinji Machida - - PowerPoint PPT Presentation

SIMPSONS for beam modelling in high intensity rings

Shinji Machida STFC/Rutherford Appleton Laboratory 19 September 2019

Many thanks to H. Hotchi for his contribution to the code and letting me show his results. Based on the presentation at the workshop on the frontiers of Intense Beams Physics Modelling 2016.

Contents

• Introduction: a couple of remarks
• Single particle tracking
• Space charge kick
• Study examples
• Summary

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A couple of remarks as an introduction

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Hotchi (J-PARC) at HB2016

Observations

KEK-PS ~2000

Goals of SIMPSONS

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Provide as a realistic model of high intensity rings as possible to understand the observation, especially regarding beam loss of a few % or below. Predict what the crucial problem are in each ring and the way to improve the performance even reducing beam loss much below a few %.

Current limitation of SIMPSONS

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Computational algorithm is not fully optimised, only by OpenMP. It is unrealistic to simulate a time period

• f ~1s self-consistently.

Mainly used for 10~100 ms (real time) simulation, but frozen and hybrid models are available for longer time scale.

Basic structure of SIMPSONS

single particle tracking

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space charge kick

Push 6D particle coordinates within a certain time step including kicks from the lattice elements. At the end of each time step, include space charge effects as an integrated kick. For 100k ~ 1M macro particles, both processes take almost equal time

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Single particle tracking

Accurate modelling of ring accelerators

single particle tracking

• Injection
• Phase space painting with programable bump orbits.
• Scattering and energy loss at charge exchange foil.
• Acceleration
• Lattice imperfections
• Physical aperture to define beam loss

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Anti-correlated painting for the first 307 turns.

Emittance evolution during injection

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This wipes out detailed structure of injection beams.

Random or pseudo-random (like Halton sequence) generator do not make difference.

1st turn: Inject N particles. 2nd turn: Inject another N particles. Throw away randomly selected N particles. Increase charge per particle by 2. 3rd turn: Inject N/2 particles. Throw away randomly selected N/2 particles. Increase charge per particle by 3/2. n-th turn: Inject N/(n-1) particles. Throw away randomly selected N/(n-1) particles. Increase charge per particle by n/(n-1). The same number of macro particles N are used during injection.

N N N N/2 N/2 N/2 N/2 N/2 N/3 N/3 N/3

Simulate multi turn injection without increasing # of macro particles

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Instantly becomes 2N

Accurate modelling of ring accelerators

single particle tracking

• Injection
• Acceleration
• RF gymnastics to capture and accelerate.
• 1st and 2nd (or even higher) RF harmonics voltage.
• Lattice imperfections
• Physical aperture to define beam loss

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V2/V1=0

φ2='100(deg φ2='50(deg φ2=0 V2/V1=80%

φ((Degrees) RF(poten:al(well((Arb.)

φ2=#1000&deg

Manipulation of rf bucket

Change relative phase of the fundamental and 2nd harmonic rf.

• 0.010
• 0.005

0.000 0.005 0.010 dp/p

• 40

40 phase [deg.]

Single particle trajectory

Time dependent RF voltage and phase

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Accurate modelling of ring accelerators

single particle tracking

• Injection
• Acceleration
• Lattice imperfections
• Imperfections of individual magnet, both static and

dynamic.

• Alignment errors.
• Influence of neighbouring beam line components.
• Physical aperture to define beam loss

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Simpsons!(PIC!par)cle!tracking!code!developed!by!Dr.!Shinji!Machida)( Imperfec.ons(included:(

! Time!independent!imperfec)ons! !!!B!Mul)pole!field!components!for!all!the!main!magnets:! !!!!!!!!!BM!(K1~6),!QM!(K5,!9),!and!SM!(K8)!obtained!from!field!measurements! B!Measured!field!and!alignment!errors! ! Time!dependent!imperfec)ons! B!Sta)c!leakage!fields!from!the!extrac)on!beam!line:! !!!!!!K0,1!and!SK0,1!es)mated!from!measured!COD!and!op)cal!func)ons! B!Edge!focus!of!the!injec)on!bump!magnets:! !!!!!!!!!K1!es)mated!from!measured!op)cal!func)ons! B!BMBQM!field!tracking!errors! !!!!!!!!!es)mated!from!measured!tune!varia)on!over!accelera)on! B!1BkHz!BM!ripple! !!!!!!!!!es)mated!from!measured!orbit!varia)on! B!100BkHz!ripple!induced!by!injec)on!bump!magnets! !!!!!!!!!es)mated!from!turnBbyBturn!BPM!data! ! !Foil!scaWering:! !!!Coulomb!&!nuclear!scaWering!angle!distribu)on!calculated!with!GEANT!

Numerical(simula.on(setup

TimeBdependent!imperfec)ons! can!be!included!easily,! because!“Simpsons”!takes!“)me”! as!an!independent!variable.

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Hotchi (J-PARC) at SC2013

Accurate modelling of ring accelerators

single particle tracking

• Injection
• Acceleration
• Lattice imperfections
• Physical aperture to define beam loss
• Each component has its own shape of aperture.

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Single particle tracking in SIMPSONS

• thers
• Time T as the independent variable.
• Easy to implement time dependent parameters such

as RF voltage and phase, orbit bump for painting, ripple in magnets, etc.

• Lattice is described by thin lenses created by TEAPOT.
• Symplectic tracking of TEAPOT to the first order of dT.
• Space charge effects are included each time step, not at

fixed position. No spurious harmonic is excited.

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Space charge kick

In cylindrical coordinate system

Solving Poisson equation (1)

cylindrical coordinate

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(r, z, ') =

∞

X

m=−∞

m (r, z) exp (im') , − ⇢ ✏0 (≡ n (r, z, ')) =

∞

X

m=−∞

nm (r, z) exp (im') . 1 r ∂ ∂r ✓ r ∂ ∂rφm (r, z) ◆ − m2 r2 φm (r, z) = nm (r, z) Fourier decompose in azimuthal direction Potential is solved separately in each mode. 1 r @ @r ✓ r @ @r ◆ + 1 r2 @2 @'2 + @2 @z2 − ✏0µ0 @2 @t2 = − ⇢ ✏0

Assumption here is that the beam is nearly cylindrical.

m=0 m=1 m=2 m=3 y y y y x x x x

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Solving Poisson equation (2)

mode expansion Typically up to m=8.

Apply, if necessary, 2D Savitzky-Golay smoothing filters for charge n(r,z) and/or field d\phi(r,z)/dr, d\phi(r,z)/dz, d\phi(r,z)/dt.

Solving Poisson equation (3)

smoothing filter

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φm,c (r, z) = 1 2π Z 2π φ (r, z, ϕ) cos (mϕ) dϕ, φm,s (r, z) = 1 2π Z 2π φ (r, z, ϕ) sin (mϕ) dϕ, nm,c (r, z) = 1 2π Z 2π n (r, z, ϕ) cos (mϕ) dϕ, nm,s (r, z) = 1 2π Z 2π n (r, z, ϕ) sin (mϕ) dϕ,

Note: Converging value depends on parameters of the filtering!

Savitzky-Golay smoothing filter

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2.8D? space charge

neither 2.5D or 3D Longitudinal space charge is local gradient of n

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Ez, m (r) ∝ ∂nm (r) ∂z

where

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Study examples with SIMPSONS

Beam survival Li pulse length (s)

539 kW 433 kW 326 kW 217 kW 104 kW

539 kW (Li pulse 500 s) 433 kW (Li pulse 400 s) 325 kW (Li pulse 300 s) 217 kW (Li pulse 200 s) 104 kW (Li pulse 100 s) Beam survival : ratio of output intensity (DCCT) to input intensity (SCT76) Painting parameter ID8 : ‐ 100 transverse painting ‐ Full longitudinal painting

Measurements vs. Calculations : Intensity dependence of beam loss

• Measurements

○ Calculations

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Hotchi (J-PARC) at SC2013

J-PARC RCS

Identifying beam loss mechanism

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• Space charge effects in rings are relatively small (< ~5%). Unlike in

linacs, it is a small perturbation in rings.

• Beam loss of a percent or less should be understood as incoherent

phenomenon, not coherent one.

• Single particle trajectory out of macro particles would be a relevant

measure and tell us what is happening.

Qx=6 dQx,y=-0.25 J-PARC, RCS (H. Hotchi) Qy=6

• None of the PIC codes preserves phase space position of

the particle starting at (x=0, xp=0, y=0, yp=0) even in an ideal lattice.

Can we believe single particle trajectory?

small amplitude particles

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physical aperture physical aperture

• Amplitude of a lost particle is

(or have to be) large.

Can we believe single particle trajectory?

large amplitude particles (1)

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• Field evaluation of large amplitude particle is easier

because strength is weaker and gradient is smaller.

Orbit of some particle looks peculiar. Faulty trajectory due to PIC tracking?

• r it is real?

Can we believe single particle trajectory?

large amplitude particles (2)

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Particle is trapped in space charge driven resonances at Qx=6, where 2Qx=12 and 4Qx=24 are structure resonances.

Can we believe single particle trajectory?

large amplitude particles (3)

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Qy=0.25 Qx-Qy=0 Qx-2Qy=0 position tune turn number turn number turn number red: hori, blue: vert

Tune analysis of single particle trajectory

2D plot of turn-by-turn betatron actions The horizontal and vertical actions gradually grow up along the line of 2Jx−Jy=const., while oscillating in a direction parallel to the line of Jx+Jy=const. Characteristic emittance blow-up that implies the combined effect

• f the two resonances, x+2y=19 and x−y /2x−2y=0;

This analysis confirmed :

• Most of the beam halo particles

are generated through such a single-particle behavior caused by the two resonances.

• The contribution of the x+2y=19 resonance

is more critical for the observed extra beam loss, because the resonance causes more severe beam halo formation on the vertical plane.

• QDTs act to mitigate the x+2y=19 resonance

through the recovery of the super-periodic condition, which results in the beam loss reduction achieved in this beam test.

Typical sample of incoherent motion

• f one macro-particle that forms beam halo

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Hotchi (J-PARC) at HB2016 and

• Phys. Rev. AB 19, 010401 (2016)

Detailed analysis of single particle trajectory in PIC is useful to see what is going on.

Some examples show

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In an accelerator like JPARC RCS, “space charge effects” mean incoherent tune spread. Interplay between resonances and depressed incoherent tune is the source of beam loss.

Finding by Hotchi-san

update in 2019

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incoherent (Teng, Laslett)

1960 1970 1980 1990 2000 2010 2020

coherent (Schecher, Gluckstern) incoherent (Holmes, Ankenbrant) coherent (Hofmann, Machida, Bartmann) incoherent (Franchetti) incoherent (Hotchi) coherent (Okamoto)

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For long time scale (~1 s) simulation

• If the particle distribution does not change, no need to update to

space charge potential.

• Frozen model assumes that.
• It is much faster computationally.
• Frozen space charge model is not an approximation of PIC.
• CERN PS and SPS are good examples to apply.
• Emittance can be updated with some interval if necessary.
• In practice, beam shape changes a lot in longitudinal direction

especially at injection unless bunch to basket transfer is employed.

Efforts to make faster calculation

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• Use reasonable number of macro particles to simulate transient

behaviour of longitudinal bunch evolution.

• Make slices in longitudinal and calculate local line density.
• Apply frozen space charge model in each slice.
• Keep the transverse distribution function same with updating

emittance.

Hybrid space charge model

self-consistent in longitudinal and frozen in transverse

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AGS Booster example

suggesting new operating point

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Joanne&Beebe)Wang&&&&&&7/28/2016&&&&&&&&&15& RHIC%Retreat%2016%%

Tune shifts and Possible Improvement

Conclusion: 1. Moving the tunes upward faster can reduce beam loss; 2. Reverse final tunes to Qx>Qy can largely reduce horizontal emittance. However, it has a small increase in vertical emittance and beam loss.

Time [msec]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

• norm. RMS ϵH [πmm-mr]

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

∆TQ=0, Qx< Qy ∆TQ=-0.3ms, Qx< Qy ∆TQ=-0.8ms, Qx< Qy ∆TQ=-1.0ms, Qx< Qy ∆TQ=-1.5ms, Qx< Qy ∆TQ=-2.0ms, Qx< Qy

Time [msec]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

• norm. RMS ϵV [πmm-mr]

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

∆TQ=0, Qx< Qy ∆TQ=-0.3ms, Qx< Qy ∆TQ=-0.8ms, Qx< Qy ∆TQ=-1.0ms, Qx< Qy ∆TQ=-1.5ms, Qx< Qy ∆TQ=-2.0ms, Qx< Qy

Beam loss rate [%]

3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Beam loss rate [%]

3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Time [msec]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

• norm. RMS ϵH [πmm-mr]

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

∆TQ=0, Qx< Qy ∆TQ=-0.8ms, Qx> Qy ∆TQ=-1.0ms, Qx> Qy ∆TQ=-1.5ms, Qx> Qy ∆TQ=-2.0ms, Qx> Qy

Time [msec]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

• norm. RMS ϵV [πmm-mr]

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

∆TQ=0, Qx< Qy ∆TQ=-0.8ms, Qx> Qy ∆TQ=-1.0ms, Qx> Qy ∆TQ=-1.5ms, Qx> Qy ∆TQ=-2.0ms, Qx> Qy

Beam loss rate [%]

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2

Beam loss rate [%]

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2

• SIMPSONS is optimised for modelling of high intensity beams in rings

with realistic operational conditions.

• Calculation and measurement agree with the order of a percent or

less (that is the Hotchi-san’s achievement).

• Despite a belief that single particle trajectory in PIC is not trustful, the

study shows it is essential to understand the beam loss mechanism of % or below.

• The beam loss estimate is less affected by PIC noise because lost

particles have large amplitude.

• Frozen model is not an approximation of PIC. For some applications,

it is useful than PIC.

• For long term tracking, hybrid model is implemented and applied to

AGS Booster as a test case.

Summary

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Thank you for your attention