Details of space charge calculations for J-PARC rings

J-PARC accelerator complex

– Phase 1 + Phase 2 = 1,890 Oku Yen (= $1.89 billions if $1 = 100 Yen).

– Phase 1 = 1,527 Oku Yen (= $1.5 billions) for 7 years.– JAERI: 860 Oku Yen (56%), KEK: 667 Oku Yen (44%).

JAERI Portion

KEK Portion

Repetition of 3GeV Synchrotron

• injection 　　 500μs• injection turns 　 ~350• particles per pulse 　 8.3e13

• acceleration 　　 20 ms

• extraction 　　　 <1μs

injection

extraction

acceleration

Repetition of 50 GeV Synchrotron

• injection 　　 0.17s• particles per pulse 　　 3.3e14

• acceleration 　 　 1.96 s

• extraction (slow) 　 0.7sinjection

extraction

acceleration

Two approaches

• A whole cycle of 3 GeV synchrotron takes 20 ms.– Full simulation with self-consistent model is possible.– Tracking parameters (# of macro particles, grid size, etc)

have to be optimized.

• Only injection period of 50 GeV synchrotron takes 0.6 s (or a bit less).– Not realistic to make self-consistent simulation.– Frozen space charge model might be justified because of well defi

ned particle distribution.

Examples of full tracking for 3GeV Syn.

Different colors shows results of differentnumber of macro particles.

Things are included.• Injection painting• Multipole errors• Misalignment• Acceleration• Aperture of all elements• Image in a circular pipe

3 months (100,000)

5 weeks (50,000)

2 weeks (20,000)

Results within 3 months (1,000,000~200,000)

Things are not included.• Scattering at foil.• RF jitter• Impedance

Other tracking parameters

Number of azimuthal mode

Number of z grids

Max. mode= 4, 8, 16

z grids= 10, 20, 30, 40, 50

Detailed study results

• Correlated and anti-correlated painting• COD and beam loss

– Coupled with strong chromaticity correction sextupole, COD introduces nonlinearity of all harmonics.

• Beam intensity dependence

Correlated and anti-correlated painting

There is particle loss even during injection period.

correlatedanti-correlated

0.5 s for injection

Phase space density right after injection and at 3 ms later

horizontal vertical

at 0.5 ms

at 3 msat 3 ms

at 0.5 ms

correlatedanti-correlated

COD and beam loss

rms COD 0 mm 0.2 mm 0.5 mm 1.0 mm

Coupled with strong chromaticity correction sextupole, COD introduces nonlinearity of all harmonics.

Phase space density for different COD

• No difference in core density.• Tails are developed with COD.

rms COD 0 mm 0.2 mm 0.5 mm 1.0 mm

rms COD 0 mm 0.2 mm 0.5 mm 1.0 mm

Hor. Ver.

Beam intensity dependence

30mA

20mA

cf. 30mA is design value which deliver 0.6 MW beam from RCS with tune spread of ~0,25.

Intensity dependence

• Core density is reduce with 30 mA. (lower order resonance is involved?)• Tails are also developed.

20mA30mA

20mA30mA

Summary of self-consistent simulation

• A whole cycle of 3 GeV Syn can be simulated even though it takes a few months.

• Horizontal and vertical coupling is the source which makes anti-correlated painting worse.

• Increase of particle loss due to larger COD is attributed to tail development. Higher order effects are involved.

• Intensity limitation may be explained with lower order resonance. That is a regime where coherence picture is applicable.

Example of beam loss during injection with frozen space charge model

Model assumes• Particle distribution is Gaussian.• Emittance is constant.• dp/p is finite and there are synchrotron oscillations.• Transverse space charge force depends on longitudinal position.

Tracking model

• “Frozen model” of space charge is adopted.– Space charge potential is fixed throughout a tracking.– No self-consistency.– No coherent oscillations.– Gaussian charge distribution in 3D is assumed.

• Lattice nonlinearities and misalignment errors are included.• Aperture of magnets and collimator are included so that we can

estimate beam loss.• Macro particles (1,000) of 3D Gaussian distribution with 2 sigma

cut are tracked for 0.12s (original design value for accumulation) or more.

Some numbers

• Emittance(2sigma) 54 pi mm-mrad (36pi, 45pi, 64pi)

• Acceptance at collimator 71 pi mm-mrad for H and V• Acceptance at magnets > 81 pi mm-mrad• Circulating current 10 A (3.3E14 ppp)• Incoherent tune shift -0.16• Bare tune (22.42, 20.80)

COD

• Chromaticity sextupoles coupled

with COD introduce beta modulation

and higher harmonics of nonlinearity.

• Survival at 0.12s after injection.

• COD shows a rms value.

Maximum is about 3 times.

• Collimator aperture is adjusted

taking a local COD into account.

• We expect COD(rms) is less than

0.5mm after correction.

• The loss is not linear as COD.

sur

viva

l at 0

.12s

(%)

80

85

90

9

5

10

0

0 0.5 1.0 1.5 2.0 COD (rms) (mm)

Different lattices

• Although rms COD is almost same,

different lattices (seeds) give

different results.• Previous example is the worst

case among three.

sur

viva

l (%

)96

97

9

8

99

10

0

0 0.05 0.1 time (s)

Beam current

• The pattern of COD is the

same for both. Magnitude is

different.

• The design current is 10A.

Blue: COD=0.5mm

Red: COD=1.0mm

sur

viva

l at 0

.12s

(%)

80

85

90

9

5

10

0

0 5.0 10 15 beam current (A)

Initial emittance

• Acceptance at collimator is fixed at

71 pi mm-mrad.

• Space charge force is fixed according

to the initial emittance.

• We expect 54 pi mm-mrad emittance

shaped at the 3-50BT collimator.

• Collimator acceptance should be

optimized to have the maximum

survival.

sur

viva

l at 0

.12s

(%)

80

85

90

9

5

10

0

30 40 50 60 70 80 initial emittance (pi mm-mrad)

Location of loss

COD Magnet acceptance

Collimator acceptance

Initial emittance

Loss at collimator

Loss at magnet

Total number

0mm > 81 pi

mm-mrad

71 pi

mm-mrad

54 pi

mm-mrad

3 (100%) 0 (0%) 3/1000

0.2 > 81 pi 71 pi 54 pi 4 (100%) 0 (0%) 4/1000

0.5 > 81 pi 71 pi 54 pi 11 (92%) 1 (0%) 12/1000

1 > 81 pi 71 pi 36 pi 33 (97%) 1 (3%) 34/1000

1 > 81 pi 71 pi 45 pi 33 (92%) 3 (8%) 36/1000

1 > 81 pi 71 pi 54 pi 21 (88%) 3 (12%) 24/1000

1 > 81 pi 71 pi 64 pi 59 (88%) 8 (12%) 67/1000

Beam loss at collimator (h=18)total 0.72 MW

COD 0 mm 0.2 mm 0.5 mm

Loss 670 W 800 W 1370W

COD (rms) = Red: 0mm Yellow: 0.2mm Green: 0.5mm

1000

980

960

940

920

900

survival

0.60.50.40.30.20.10.0

time [s]

All the particles hit collimator first.

Beam loss at collimator (h=18)total 0.58 MW

COD 0 mm 0.2 mm 0.5 mm

Loss 350 W N/A 690 W

COD (rms) = Red: 0mm Yellow: 0.2mm Green: 0.5mm

1000

980

960

940

920

900

survival

0.60.50.40.30.20.10.0

time [s]

All the particles hit collimator first.

Frozen model with acceleration

phis

bunch length dp/p

300

280

260

240

220

200

2.52.01.51.00.50.0-0.5

horizontal vertical

300

280

260

240

220

200

2.52.01.51.00.50.0-0.5

horizontal vertical

1000

990

980

970

960

950

940

1.51.00.50.0-0.5

1000

990

980

970

960

950

940

1.51.00.50.0-0.5

99% emittance and beam loss

Acceleration starts right after injection.

Acceleration starts at 0.16 s after injection.

300

280

260

240

220

200

2.52.01.51.00.50.0-0.5

horizontal vertical

1000

990

980

970

960

950

940

1.51.00.50.0-0.5

Acceleration starts at 0.6 s after injection and h=18.

Single particle behavior

• Tracking without aperture limit

to see single particle behavior.

• Slow growth of amplitude.• Not obvious correlation with

synchrotron oscillations. Trapping?

Timing ofhitting collimator

0 0.01 0.02 0.03 0.04 0.05 time (s)

hori

zont

al p

osit

ion

(m)

-0.1

-0.0

5

0

0.5

0

0

.1

Single particle behavior

• Track a single particle which is lost in 0.6 s.• Look at betatron oscillation amplitude and transverse tune as a f

unction of turn until a particle is lost.

• For example, there are– 38 lost particles (out of 1000) when rms COD is 0mm.– 40 lost particles (out of 1000) when rms COD is 0.2 mm.– 52 lost particles (out of 1000) when rms COD is 0.5 mm.

#1 #2 #3

#5 #6#4

H

V

H

V

turn number (~ 10,000 turns)

ampl

itud

e

rms COD is 0.5 mm

H

V

H

V

#7 #8 #9

#11 #12#10

turn number (~ 10,000 turns)

ampl

itud

e

rms COD is 0.5 mm

H

V

• Horizontal amplitude always increases and gets to the aperture limit.• Vertical amplitude always decreases. Coupling between H and V is manifest.

#16

#15#14#13

H

V

turn number (~ 10,000 turns)

ampl

itud

e

rms COD is 0.5 mm

In tune space

Blue points are intermediatetune of lost particles.Red points are tune just beforeparticles are lost.

Tune before particle loss aresame with and without COD.

2x-y=24

x-2y=-19

bare tune

x-22 x-22

y-2

0

y-2

0

Coupling between H and V is manifest, but

• Tune space plot does not show resonance driving term.– 2x-y=24 is skew and cannot be excited even with finite dp/

p and dispersion in a lattice.• If there is any way to reduce a driving term.

– Since the source is not identified, it is difficult.

Summary of frozen space charge simulation

• Particle loss occurs because horizontal amplitude increases and hits the collimator aperture. The source of the increase is a coupling between H and V.

• With finite COD, particle loss occurs with less turns. However, transverse tune when a particle loss occurs does not depend on COD magnitude.

• Loss is very slow process: the order of 104 turns. Time scale of horizontal and vertical coupling is also same order.

Basic loop of calculation

Advance particle coordinatesdo ip=1,np (200,000)

Calculate space chargepotential based onparticle positions.

do imode=1,nmode (16)

Apply space charge kicksto all particles

do ip=1,np (200,000)

Simpsons uses Fourierexpansion in azimuthaldirection.

Make parallel processingof Fourier modes.

Distribution of workload (4 CPUs) with MPI

imode=3,4,5,6

imode=0,1,2

imode=7~11

imode=12~16

Add up all E-fields4 CPU worksin the same way