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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