code development and polarized beam simulation...

IAS HEP Conference, HKUST

Mini-workshop on Accelerator, Jan. 18-19, 2019

Code Development and PolarizedBeam Simulation Studies

François Méot

Collider-Accelerator Department

Brookhaven National Laboratory

Contents

1 ZGOUBI 2

2 Hadron polarization at RHIC complex 7

3 Electron polarization in eRHIC 16

4 Hadron polarization in eRHIC 20

5 JLEIC EIC 23

6 Cornell-BNL CBETA FFAG ERL 25

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21 ZGOUBI

• SEEN FROM SPACE

https://sourceforge.net/projects/zgoubi/files/stats/map

⋄ Zgoubi was first written in 1972... 45+yrs !

⋄ Installed in sourceforge in 2007

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3https://sourceforge.net/projects/zgoubi/files/stats/timeline

⋄ China seems poised to catch up !

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4• A DOE FUNDED SBIR (2018-2020)

⋄ [1] Dan Abell (Radiasoft) et als., “Zgoubi Status: Improved Performance, Features,and Graphical”, IPAC 19, Melbourne.

⋄ This includes development of a graphical interface under SIREPO (Paul Moeller/Ra-diasoft)

Ref.: https://beta.SIREPO.com/#/accel

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5• ZGOUBI NUMERICAL INTEGRATOR

⋄What needs be solved:

d(mṽ)dt

= q ṽ × b̃

dS̃dt

=qm S̃× ω̃

(

ω̃ = (1 + γG)b̃ +G(1− γ)b̃//

)

G : gyromagnetic factor, γ : Lorentz relativistic factor,

c : velocity of light, q : charge, m :mass.

u (M1)

M1

R (M 1

)

u (M0

)

R (M

0)

z

x

y

Z

Y

X

M

0

Reference

⋄ ODE slver: Taylor series in the step size ∆s:

ã(M1) ≈ ã(M0) +dã

ds(M0)∆s + ... +

dnã

dsn(M0)

∆sn

n!(1)

- Solving particle motion : ã stands for position R̃ or velocity ṽ.

- Solving spin motion : ã stands for the spin S̃

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•A note in passing: Zgoubi is as good at accelerating in a cyclotron ...

⋄ 184 inch cyclotron at

Berkeley, 100MeV/u ions

(1940s)

⋄ Double-Dee acceleration to 10 MeV in 50 turns.Accuracy on time computation is critical:

-100

-80

-60

-40

-20

0

20

40

60

80

100

-100-80-60-40-20 0 20 40 60 80 100

R sin

θ [cm]

R cos θ [cm]

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.5 1 1.5 2 2.5 3 3.5 4

∆W [M

eV]

RF phase [rad]

starting point (φ =π/2)0__

synchronism at one point(f = f ) rf rev

|

∆φ0

[cosφ](W) = cosφ0 + π[

1− ωRFωrev0(1 + W

2m0c2)]

W

qV̂

• ... as it is at tracking polarization in large colliders... Let’s see that!

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72 Hadron polarization at RHIC complex

• Polarized beam simulations in RHIC complex using Zgoubi- include p, He3

- start at the booster, and include AGS, AtR, RHIC

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8• Booster

⋄ Zgoubi simulations support the eRHIC polarized 3He project

⋄ AC dipole & tune kickerat booster E3 straight:

⋄ fosc ≈ 250 kHz

⋄ This year:- Will establish crossing Gγ = 0 + Qy(Qy ≈ 4.2) with polarized protons .- Spin dynamics conditions similar to 3He

crossing 12−Qy for eRHIC.

Refs.: K. Hock et als., BNL C-AD,https://public.bnl.gov/docs/cad/Documents/Overcoming%20proton%20and%203He%20intrinsic%20resonances%20in%20the%20AGS%20Booster%20with%20an%20ac%20dipole.pdf;https://public.bnl.gov/docs/cad/Documents/Intrinsic%20resonances%20and%20AC-dipole%20simulation%20of%203He%20in%20the%20AGS-Booster.pdf.

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9• AGS (1): On-line AGS acceleration cycle model

⋄ Zgoubi is one of the two engines of the on-line model of the AGS (other is MADX).

It uses the 3-D OPERA field maps of the helical snakes

The 240 main dipoles in the AGS may also be represented by their measured field maps.

⋄ Full polarized proton bunch acceleration cycles in the AGS are routinely simulated : thousands

of particles, 150000 turns. (on NERSC at the moment, plans for GPU platforms [V. Ranjbar])

5 10 15 20 25 30 35 40 45

-.003

-.002

-.001

0.0

0.001

0.002

0.003

0.004

x, z (m) vs. G γ

H: red V: blue

Particle excursion over full AGS cycle.

8.7

8.75

8.8

8.85

8.9

8.95

9

5 10 15 20 25 30 35 40 45

Tun

es

Energy (Gγ)

Qy ZgoubiQz Zgoubi

Qy MeasuredQz Measured

Tunes, measured and modelled.

⋄ Optimization of AGS parameter settings routinely basedon Zgoubi simulations.

[Y. Dutheil, BNL C-AD, PhD dissertation, April 2015].

0.72

0.74

0.76

0.78

0.8

0.82

5 10 15 20 25 30 35 40 45

Ave

rage

bea

m p

olar

izat

ion

(<S>

⋅n0)

Energy (Gγ)

Without tune jumpsWith tune jumps

Evolution of the polarization along AGS acceleration cycle,

103 particles, 6-D bunch.

Loss at Gγ ∼ 5 (a pair of weak vertical intrinsic resonances)and at Gγ = 36 +Qy in the present hypotheses.

Ref.: Y. Dutheil, BNL C-AD, PhD dissertation,https://public.bnl.gov/docs/cad/Documents/Spin%20dynamics%20modeling%20in%20the%20AGS%20based%20on%20a%20stepwise%20ray-tracing%20method.pdf.

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• AGS (2): Polarization profiles

⋄ Simulating the AGS acceleration cycle provides guidance through thesubtleties of the manipulation of polarization, for optimized transmission

Yann Dutheil

(PhD dissertation, 08/2015)

Polarization profile R =σPσl

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• RHIC ring (1)

⋄ Zgoubi has been used over the past 10 years to study polarization inRHIC.

⋄ A benchmarking example: crossing the strong snake resonance Gγ =393 +Qy

⋄ Theory: < Sy >= 1−

8|ǫ|2

λ2 sin2 πλNs(1− |ǫ|

2

λ2 sin2 πλNs)

λ =√

δ2 + |ǫ|2

δ = GγR −Gγ -1-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

410 415 420 425 430 435 440 445

<Sy>

G.gamma

Zgoubi tracking (Nov. 2011)

1000 prtcls2000 prtcls

Model

⋄ Routinely studies include sensitivity to defects (e.g., spin rotation anglein snakes or in rotators), studies accompanying RHIC runs, optimization

of the AtR beam line, etc.

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• RHIC (2): A full acceleration cycle to 100 GeV×100 GeV pp physics

⋄ A 1 million turn tracking⋄ Goal: comparison of polarization transport with two different optics:

- a new, Run 15 optics- earlier Run 13 optics

8 particles on 10 rms invariants are plotted.Envelopes (blue) are from theory [SYL].

30 40 50 60 70 80 90-1.

-.8

-.6

-.4

-.2

0.0

0.2

0.4

0.6Zgoubi|Zpop 20-04-2015 S_y .vs. Kinetic E (GeV)

93-

69+

75+

81+174-

30 50 70 90-1.

-.8

-.6

-.4

-.2

0.0

0.2

0.4

Zgoubi|Zpop 29-04-2015 S_y vs. kinetic E (GeV)

93- 69+

75+

150+

Normalized invariants over 500,000 turns :

2.38

2.4

2.42

2.44

2.46

2.48

2.5

2.52

2.54

2.56

2.58

0 100 200 300 400 500 600 2.4

2.42

2.44

2.46

2.48

2.5

2.52

2.54

2.56

2.58 26 28 30 32 34 36 38 40

βγεx (

µm)

βγεy (

µm)

TURN (103)

E (GeV)

βγεxβγεy

x×√

(p/p0) and y ×√

(p/p0)

over 2 million turns :

3.E+4

4.E+4

5.E+4

6.E+4

7.E+4

8.E+4

-.003

-.002

-.001

0.0

0.001

0.002

0.003

Zgoubi|Zpop 20-04-2015 norm. x (m) vs. KinEnr (MeV)

3.E+4

4.E+4

5.E+4

6.E+4

7.E+4

8.E+4

-.004

-.002

0.0

0.002

0.004

0.006

Zgoubi|Zpop 20-04-2015 norm. y (m) vs. KinEnr (MeV)

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• RHIC (3): Re-visiting RHIC snakes

Everything You Always Wanted to Know About Snakes (But Were Afraid to Ask)

Ref.: F. Méot, Re-visiting RHIC snakes: OPERA fields, ñ0 dance, C-A/AP/590, https://public.bnl.gov/docs/cad/Documents/Re-visiting%20RHIC%20snakes%20OPERA%20fields%20n0%20dance.pdf

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• Theory: particle coordinates across the helical dipole satisfy:

x = x0 +B0

k2Bρ[cos(ks + α)− cos(α)] +

[

x′0 +B0kBρ sin(α)

]

s

y = y0 +B0

k2Bρ[sin(ks + α)− sin(α)] +

[

y′0 −B0kBρ cos(α)

]

s

α=0−→

{

x = x0 +B0

k2Bρ[cos(ks)− 1] + x′0s

y = y0 +B0

k2Bρsin(ks) +

[

y′0 −B0kBρ

]

s

⋄ x0, , x′0, y0, y

′0 the initial coordinates (taken wrt snake axis),

⋄ α the angle of the field to the vertical at entrance,

⋄ k = pitch = 2π/L = 2.61800, length L = 2.4 m.

µφS

S

• From a field map model (i.e., field map of a 10.4 meter, 4-module RHIC snake) :

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0 1 2 3 4 5 6-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03 0 1 2 3 4 5 6

x, y [m]

z [m]

Half-RHIC snake, trajectories

x, th.track

y, th.track

x and y vs. distance

at injection energy, half the snake

(5.2 meter).

-2

-1

0

1

2

-2 -1 0 1 2

y (cm)

x (cm)

RHIC snakes: projected helix

300250

theor. " 100

22.9theor. "

Non-circular, (x,y) projection ,

theory and numerical,

23, 100, 250, 300 GeV.

0 2 4 6 8 10 12

-.8

-.6

-.4

-.2

0.0

0.2

0.4

0.6

0.8

1.Zgoubi|Zpop 13-12-2016

Sx

Sy

Sz

Sx, S_y, S_z vs. s (m)

Spin components along orbits .

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• RHIC (4): Spin flipper

⋄ Sweep Qosc through Qs =12

+:

a matter of a 104 ∼ 5 × 105 turn trackingaround RHIC

⋄ AC-dipole + DC-rotator assembly excites asingle, isolated resonance, at Qs = Qosc.

ACD bump 1 ACD bump 2

DC bump

1 2 3 41 2 3 4 5

Spin flip efficiency vs. crossing speed :Very good accord obtained between

measurements and numerical simulations

-100

-98

-96

-94

-92

-90

-88

-86

50000 100000 150000 200000 250000

0.5 1 1.5 2 2.5 3 3.5

Pf/Pi[%]

τX [turn]

τX [sec]

APEX (∆D’=3.45)fit

Y, ∆D’=3.45fit

∆D’=3fit

∆D’=0fit

Image at 1−Qs is not excited:

Frequency mis-match of ACD1-3 (or

ACD3-5) oscillator excites Qs = 1−1

2

+:

0 10000 20000 30000 40000 50000

turn

0.492

0.494

0.496

0.498

0.500

0.502

0.504

0.506

0.508

ν osc,ν s

−1.0

−0.5

0.0

0.5

1.0

Sy

Multiple crossing:

δνs =1 +Gγ

π∆D′

∆p

p

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3 Electron polarization in eRHIC

Studies in eRHIC include SR induced spin diffusion :

SPIN DYNAMICS

+STOCHASTIC ENERGY LOSS BY SR

=⇒ SPIN DIFFUSION, FOR FREE !

Ref.: eRHIC pCDR, BNL C-AD (2018), to be published.

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• eRHIC (1): Monte Carlo SR benchmarked in the late 1990s

⋄ Thousands of particles, tracked over tens of thousands of turns

0 10 20 30 40 500

5

10

15

20

25

30

35

BETA-LNS v5.10 /01/10/99/ 17-Jun-10 11:11:40 1st 0.000E+00OPTICAL FUNCTIONS (m) VERSUS S (m)

Qx = 36.200Qy = 11.200 betX

betY

Dx

Chasman-Green ESRF cell.

Benchmarking virtue : provides

analytical formulas for damping

times, emittance damping, etc.

⋄ Coupled lattice. 2000 particles,20,000 turns. 6 GeV ←−

Good agreement withǫyǫx =

|κ|2

|κ|2 +∆2/2and with ǫx + ǫy = ǫ0

1e-05

0.0001

0.001

0 1000 2000 3000 4000 5000

εl (eV.s)

turn

18 GeV

12 GeV

9 GeV

6 GeV

Longitudinal damping, 6, 9, 12 and 18 GeV.

Excellent match with expected ǫl,eq =αEsΩs

Cqγ2

Jl ρ.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10 0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

ε y/ε

x

(εx+

ε y)/

ε x,0

|κ| / ∆

ratiofit

invariantfit

Ref.: F. Méot, Simulation of radiation damping in rings, using stepwise ray-tracing methods, 2015 JINST 10

T06006.

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• eRHIC (2.1): Polarization lifetime in eRHIC storage ring

• Typically:

As produced using SITROS at DESY

[J. Kewisch, DESY Rep. 83-032, 1983]

(i) track particles and spins, including Monte

Carlo SR,

(ii) produce polarization landscape,

i.e., Peq versus ring rigidity setting (aγ units,here):

• Spins tracked over 240 damping

times in eRHIC, 18 GeV

0.99

0.992

0.994

0.996

0.998

1

40.5 40.55 40.6 40.65 40.7 40.75 40.8 40.85

Sl

aγ

Polarization, eRHIC e-storage ring, 18 GeV From polSpectrumFromFai.out

240τSR 23τSR

• Polarization decay with time:

The diffusion time constant τDis obtained from linear regression

P/P0 = exp(−t/τD) ≈ 1− t/τD.

Then, Peq stems from

Peq = PST × τeq/τST ,

given that τeq = (1/τST + 1/τD)−1.

0.9992

0.9993

0.9994

0.9995

0.9996

0.9997

0.9998

0.9999

0 20 40 60 80 100 120 140

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 0×100 1×108 2×108 3×108 4×108 5×108 6×108 7×108 8×108

<Sl> zoom

<Sl> eq.

τ [×103 turn]

Polarization vs. time at store, 18 GeV

τ [turn]

x1-y1: 40.66, 63 min.x1-y1: 40.615, 48 min.x1-y1: 40.705, 46 min.

x2y2: e-t/τEq

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• eRHIC (2.2): An ergodic approach to electron spin diffusion

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 50 100 150 200 250 300 350 400 450

S y

turn [x103]

Sy vs. turn#, from zgoubi.fai. gnuplotfaiSy.vs.Turn.eps

0.9984

0.9986

0.9988

0.999

0.9992

0.9994

0.9996

0.9998

1

0 50 100 150 200 250 300 350 400 450

<Sl>

τ [103 turn]

Polarization vs. time at store, 18 GeV From TBTAverage.out. 2048 folders.

P0 = 0.9999

τD = 76.5708

trackedfit

Stochastic spin motion observed at IP, single particle. A linear regression on

P/P0 = exp(−tτD) ≈ 1− tτD provides the diffusion time constant τD.

10-4

10-2

100

102

104

39.8 40 40.2 40.4 40.6 40.8 41 41.2

τ D [min.]

aγref

τD, eRHIC e-storage ring, 18 GeV RunXXX/zgoubi.fai | TBTLinRegTauFromRun.FaiFiles

39.8 < aγref < 41.2

40k 80k 450k

An energy scan of the diffusion time

constant provides the diffusion landscape.

0.001

0.01

0.1

1

10

100

1000

39.8 40 40.2 40.4 40.6 40.8 41 41.2

τ D [min.]

aγref

τD, eRHIC e-storage ring, 18 GeV RunXXX/zgoubi.fai | TBTLinRegTauFromRun.FaiFiles

39.8 < aγref < 41.2

40 bins, 0.5xbeam aσE80 bins, full aσE

120 bins, 1.5xbeam aσE

Sliding average of the diffusion time

constant over the energy scan.

Ref.: F. Méot, An ergodic approach to polarization in eRHIC electron storage ring,https://public.bnl.gov/docs/cad/Documents/An%20ergodic%20approach%20to%20polarization%20in%20eRHIC%20electron%20storage%20ring.pdf

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4 Hadron polarization in eRHIC

• Studies in eRHIC include im-proving proton polarization: 70%

up to 275 GeV, using 6 snakes.

Today, with 2 snakes: close to 60%

at best, top energy is 255 GeV.

• And 3He polarization to 70%,also requiring 6 snakes.

Ref.: eRHIC pCDR, BNL C-AD (2018), to be published.

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• 3He in AGS⋄ AGS with 3He : G= −4.18. AGS partial snakes are 2.3

times stronger than for protons (14% and 25%).

⋄This is ok for overcoming 1.5 times stronger resonances.

⋄ Issue here : snakes induce

~n0 tilt resulting in harmful

81 resonance series Gγ =

int. ± Qx, stronger with

3He.

⋄ A possibility : stronger cold snake/wider spin tune gap

to put both Qx,Qy in the gap

⋄Proof-of-principle simulation :

a 2000-3He bunch in 15π µm emittance :

⋄With 3He, snakes are |G3He|/Gp=2.3 timesstronger than for protons :

⋄ At identical normalized emittance,strengths of intrinsic resonances satisfy

|ǫ3He||ǫp|

=

√

|G3He|Gp

≈ 1.5

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

5 10 15 20 25 30 35 40 45

|ε| (10

π µm)

Gγ

AGS, RESONANCE STRENGTHS, INTRINSIC

3Heproton

⋄Qx and Qy in spin gap⇒ Gγ = int.±Qx,y never happens :

-1

-0.5

0

0.5

1

32 34 36 38 40 42 44 0.5

0.6

0.7

0.8

0.9

1

n 0,y

frac([1-] Q s), Q x, Q

y

G g

Qx Qy n0,y Qs

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• RHIC with 6 Snakes

⋄ Polarization goal at store, 42∼275 GeV : 70%

⋄ RHIC Blue and Yellow rings today reach 60% polar-

ization in 255 GeV proton runs, with 2 snakes per ring.

⋄ Stronger 3He resonance strength (×1.5) require more

snakes.

⋄ Foreseen scheme :

- 6 snakes ensure Nsnakes > 5|ǫint|max ≈ 4,

- 2π/6 distance ensures energy-independent Qs,

- snake axes at φk = ±45o yield Qs =

1π

∑

6

k=1(−)kφk = 3/2

- build 4 additional snakes from existing, like-helicity, ro-

tator modules

A glimpse on snake resonance crossing simulations,

411-Qy and 393+Qy

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5 JLEIC EIC

Ref.: F. Lin, JLEIC Ion and Electron Polarization, EIC collaboration meeting, JLab, Oct. 2018

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JLEIC EIC

Ref.: V. Morozov, Spin Transparency Study and Test at RHIC, ibidem

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6 Cornell-BNL CBETA FFAG ERL

Exclusive use of OPERA maps to model the 4-pass ERL in Zgoubi⋄ 6 MeV injection, 36 MeV linac

⋄ 4 energies reciculated: 42, 78, 114 and 150 MeV

CBETA commissioning starts in March 2019

⋄ Zgoubi model in prepara-

tion. Will be used for off-

line studies during CBETA

commissioning.

⋄ On the other hand, Cor-

nell Wilson Lab has a polar-

ized electron source in de-

velopment.

⋄ There is plans to install

as CBETA source. An op-

portunity tp perform spin

transport studies.

Ref.: F. Méot, Procs ICAP 18 Conference, Key West, USA.

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THANK YOU FOR YOUR ATTENTION

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BACKUP SLIDES

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Cyclotron data file

USING A FIELD MAP

Uniform field sector’OBJET’64.62444403717985 ! Rigidity2 ! Generate a1 1 ! single particle.4.087013 0. 0. 0. 0. 0.3162126 ’o’1

’TOSCA’0 21. 1. 1. 1.HEADER_8315 151 1 22.1 1.geneSectorMap_180deg.out0 0 0 021.20. 0. 0. 0.

’CAVITE’30. 0.100e3 1.57079632679

’REBELOTE’60 1.1 99

’SYSTEM’6cp gnuplot_zgoubi.plt.cmd temp.cmdsed -i ’s@pause 2@pause 0@g’ temp.cmdgnuplot < temp.cmdokular gnuplot_zgoubi.plt_XYLab.eps &rm -f temp.cmd

’END’

USING ZGOUBI’s ANALYTICAL FIELD MODEL

’OBJET’64.6244440371798521 14.087013 0. 0. 0. 0. 0.3162126 ’o’1

’DIPOLE’2120. 50.60. 9.1475315 -1. 0. 0.10. 0.4 .1455 2.2670 -.6395 1.1558 0. 0. 0.+30. -30. 1.E6 -1.E6 1.E6 1.E610. 0.4 .1455 2.2670 -.6395 1.1558 0. 0. 0.-30. +30. 1.E6 -1.E6 1.E6 1.E60. 0.0 0. 0. 0. 0. 0. 0. 0.0. 0. 1.E6 -1.E6 1.E6 1.E6 0.2 20..22 43.879742 0. 43.879742 0.

’CAVITE’30. 0.100e3 1.57079632679

’REBELOTE’60 1.1 99