Beam and spin dynamics studies in eRHIC ERL · ERL2015 Workshop, BNL-STONY BROOK UNIVERSITY, 7-12...

ERL2015 Workshop, BNL-STONY BROOK UNIVERSITY, 7-12 June 2015

Beam and spin dynamics studies in eRHIC ERL

François Ḿeot, BNL C-AD

Contents

1 Basic principles 31.1 Particle dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 31.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 6

2 eRHIC FFAG optics 9

3 Dynamical effects of SR in eRHIC FFAG-II ERL 123.1 Working conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 123.2 Orders of magnitude of the effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Longitudinal phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Transverse phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 End-to-end bunch tracking, cumulated effects (initialǫx ≈ ǫy = 0 and dp/p=0). . . . . . . . . . . 163.4 End-to-end 9-D tracking, nominal initial transverse emittances50πµm . . . . . . . . . . . . . . . . 18

4 Dispersion suppressors 20

5 Dynamical acceptance of eRHIC FFAG lattice 22

6 Injecting alignment defects 276.1 Horizontal misalignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276.2 Vertical misalignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 296.3 Investigating alignment error correction of a 11-beam set. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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1 Basic principles

1.1 Particle dynamics

• Electrons circulating in eRHIC FFAG arcs loose energy by synchrotron radia-tion (SR)

Over a trajectory arc ∆θ,with constant curvature1/ρ, relative average energyloss :

∆E

E= 1.879× 10−15 γ

3

ρ∆θ

Energy loss at top energy,starting with E=21.16 GeV :

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Zgoubi|Zpop 16/03/**

* # COORDINATES - STORAGE FILE, 16/03/14 08:56:13. *

ELoss (MeV) vs. s (m)

* Ex. : at E=21.16 GeV, loss over a 6-arc ring, 2138 m distance, is ∆E ≈ 100 MeV .

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• SR is a stochastic process, photons fluctuate in number and energy,thus inducing

- energy spread,

σEE

= 3.794× 10−14 γ5/2

ρ

√∆θ

* Ex. : σE = 5.2 MeV af-ter 11 passes in eRHIC from7.94 to 21.16 GeV.

Energy spreadσE after 11 passesin eRHIC from 7.944 to 21.16 GeV :

20835 20845 20855 208650

10

20

30

40

50

60

70



dN/dE vs. E (MeV)

- and bunch lengthening [*],

σl =(σEE

)

[

1

Lbend

∫ sf

s

(Dx(s)T51(sf ← s) +D′x(s)T52(sf ← s)− T56)2ds

]1/2

(the integral is taken over the bends).

* Ex. : bunch lengthening is σl ≈ 5µm at 21.16 GeV in a 6-arc ring.

——–[*] G. Leleux et al., Synchrotron Radiation Perturbations in Long Transport Lines, PAC 91.

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• The energy loss causes a spiraling of the beam centroid.

Over a distance[si, sf ] :

[

x(sf)

x′(sf)

]

= T (sf ← si)×[

x(si)

x′(si)+

σEE

{

< U >

< V >

}]

[

U(se)

V (se)

]

=

Dx(si)−∫ sesi

T12(s← si)ρ(s)

ds

D′x(si) +∫ sesi

T11(s← si)ρ(s)

ds

Inward spiraling of the electron beamat 21 GeV in a 6-arc ring

0 500 1000 1500 2000

0.00424

0.00426

0.00428

0.0043

0.00432

0.00434



x (m) vs. s (m)

* Ex. : ∆x = −0.12mm at 21.16 GeV, over 6 arcs, 2138 m distance.

• and horizontal emittance growth,

∆σ(sf) = T (sf ← si)×

(σEE

)2[

< U 2 > < UV >

< UV > < V 2 >

]

× T̃ (sf ← si)

Horizontal phase space after 21 passes7.944→ 21.16→ 7.944 GeV.

-.00542 -.0054 -.00538 -.00536 -.00534 -.00532

0.005

0.00502

0.00504

0.00506

0.00508

0.0051

0.00512

0.00514



x’ (rad) vs. x (m)

* Ex. : SR induced emittance isβγǫx ≈ 2 πµm , after 11 passes in eRHIC, from7.94 to 21.16 GeV.

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1.2 Spin

• Averages and second momenta build up upon stochastic SR.

• Spin precession around the vertical dipole fieldsamounts to

φ = aγθ

a = 1.1610−3, anomalous gyromagnetic factor

γ is the Lorentz relativistic factor

θ is the particle deflection angle.

* Over a turn at E=21.16 GeV, spin undergoesaγ ≈45 precessions around the vertical axis.

• Stochastic SR causes stochastic change inprecession rate, hence spin diffusion

Complete ERL ring,arcs+straights+DS.

Starting 6-D emittance zero.Spin angle density after accelerationfrom 7.9 to 21.16 GeV in 11 passes :

40 60 80 100 120 1400

20

40

60

80

100

120

140

160

Zgoubi|Zpop 07-06-2015 COUNT vs.

σφ = 14.89 degcos(σφ) = 0.966

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• Evolution of the diffusion [*]

In constant magnetic field :

∆E2

∆E∆φ

∆φ2

=

1 0 0

αs 1 0

α2s2 2αs 1

∆E2

∆E∆φ

∆φ2

s=0

+ω×

s

αs2/2

α2s3/3

ω = Cρ3λ̄creγ

5E2 ≈ 1.44× 10−27γ5

ρ3E2 (λ̄c = ~/mec electron Compton wavelength, E in GeV),

C = 110√3/144,

α = aρE0≈ 10.4406ρ (with a = 1.16× 10

−3, electron massE0 = 0.511× 10−3 GeV).

• Cumulated spread, over a six-arc ring (2138 m circumference), from7.9 to 21.2 GeV :

0

2

4

6

8

10

12

20 25 30 35 40 45 0

2

4

6

8

10

12

8 10 12 14 16 18 20

σ φ [deg.]

Design G.γ

Design energy [GeV]

beam Etheory

σφ (εx=0)

[*] cf. V. Ptitsyn, EIC’14

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• Ray-tracing

Particle and spin motion

u (M1)M1

R (M 1

)

u (M0)

R (M

0)

z

xy

Z

Y

X

M

0

Reference

Position and velocity of a particle, pushed from lo-cationM0 to locationM1 in Zgoubi frame.

d(m~v)

dt= q (~e + ~v ×~b) ,

d~S

dt=

q

m~S × ~ω

with ~ω = (1 + γa)~b + a(1− γ)~b//a : gyromagnetic factor,γ : Lorentzrelativistic factor, c : velocity of light,q : charge,m :mass.

• Both equations are solved using a truncated Taylor series in the step size∆s,

~a(M1) ≈ ~a(M0) +d~a

ds(M0)∆s + ... +

dn~a

dsn(M0)

∆sn

n!(1)

- Solving particle motion : ~a stands for position ~R or normalized velocity ~u = ~v/v,- Solving spin motion :~a stands for the spin~S.

• The local magnetic field and its derivatives fully determine the coefficients a(n) = dna/dsn inthis Taylor series.

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2 eRHIC FFAG optics

•Stepwise methods allow detailed insight in the FFAG cell:

• Orbits in defocus-ing quad and focusingquad :

0.0 0.2 0.4 0.6 0.8

-.008

-.007

-.006

-.005

-.004

-.003

-.002

-.001

0.0

05/11/**

BD

x (m) vs. s (m) 21.16 GeV

7.94 GeV

0.0 0.2 0.4 0.6 0.8 1.

-.002

0.0

0.002

0.004

0.006

0.008

QF

x (m) vs. s (m) 21.16 GeV

7.94 GeV

• Field along orbits :

Trajectory curvature varies continuouslyacross the magnets.

0.0 0.5 1. 1.5 2. 2.5

-.1

0.0

0.1

0.2

0.3

0.4

05/11/** BZ (T) vs. s (m)

7.9 GeV

13.2 GeV 21.2 GeV

7.9 GeV

21.2 GeV

BD QF

•While we are here : hard edged magnet models will be used throughout, no such thing as “fringefields”, here.

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• Cell : geometry, optics

A continuousaγ scan of the high energy FFAG eRHIC cell (7.944→ 21.16 GeV) :

Energy dependence of devia-tion and curvature radius inarc cell quads :

Cell tunes and chromaticities;the vertical bars materializethe 11 design energies :

-15

-10

-5

0

5

10

6 8 10 12 14 16 18 20 22-1000

-500

0

500

1000

Deviation [mrd]

Curv. radius [m]

θquadDθquadFρquadDρquadF

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

20 25 30 35 40 45-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1 8 10 12 14 16 18 20

Qx, Qy

ξ x,

ξ y

a.γ

E (GeV)

beam EQxQyξyξx

•Matching polynomials can be devised :Qx(aγ) = −0.002262268 + 8.60741/aγ − 120.24058/(aγ)2 + 1545.2231/(aγ)3

Qy(aγ) = −0.140723255 + 11.113263/aγ − 153.6170/(aγ)2 + 1457.3424/(aγ)3

• Note in passing :aγ is close to an integer at all pass⇐ linac kick multiple of mass/a=440.5MeV,for longitudinal alignment of polarization at IP6, IP8.

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• More on optical functions

Horizontal phase-space,middle of drift,matched ellipses,E : 7.9 → 21.2 GeV,step0.95 GeV.

-.004 -.002 0.0 0.002 0.004

-.001

0.0

0.001

0.002

0.003

0.004

0.005

Zgoubi|Zpop 03/02/2014 x’ (rad) vs. x (m)

No SR

21.2 GeV

7.9 GeV

SR induced ellipse mismatch,after one sextant at 21.16 GeV :

0.004333 0.004335 0.004337

-.001521

-.001521

-.001521

-.001521

-.001521

-.00152

-.00152

-.00152

-.00152

-.00152

Zgoubi|Zpop 17/03/** x’ (rad) vs. x (m)

w/o SR

w/ SR

21 GeV arc

Middle of drift.Matched vertical ellipses,E : 7.9 → 21.2 GeV,step0.14 GeV.

-.0004 -.0002 0.0 0.0002 0.0004

-.0002

-.0001

0.0

0.0001

0.0002

0.0003

Zgoubi|Zpop 03/02/2014 y’ (rad) vs. y (m)

No SR

21.2 GeV

7.9 GeV

• Tight comparisons have been performed between codes : SYNCH,MUON1, MADX[-PTC].The conclusion is : good in general, some discrepancies are observed(some may be serious)

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3 Dynamical effects of SR in eRHIC FFAG-II ERL

3.1 Working conditions

• A complete FFAG-II ring ( 7.944→ 21.16 GeV) is considered in the simulations discussed here.

• A ring is built from 6 arcs, 6 long straight sections (LSS), and 12 dispersion suppressors (DS)

6 ×[

12LSS − DS − ARC − DS − 12LSS

]

+ Linac

It includes one, zero-length, 1.322 GeV energy kick in a straight section (possibly accountingfor phase), in order to simulate the linac.

- An arc is a series of FD 138 cells. The deviation is almost2π/6 (2π/6.74).

- A DS has 17 arc-type cells, but for vanishing quadrupole axis shift from arc to LSS

- An LSS is made of 70 such cells, with quadrupole axes aligned

• The bunch to be tracked will in general be launched from the center of a long straight section,- where theoretical local orbit has the virtue of being zero, whatever the bunch energy

- with initial transverse/longitudinal emittances either zero or nominal, depending on the exer-cise

- beam centroid is in general (artificially) re-centered on the LSS axis, at each LSS.

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3.2 Orders of magnitude of the effects

3.2.1 Longitudinal phase space

• Turn-by-turn energy loss and spread :

0

20

40

60

80

100

20 25 30 35 40 45 0

1

2

3

4

5

6 8 10 12 14 16 18 20 22

∆E [MeV]

σ E [MeV]

aγ

E [GeV]

∆E, theor.Ray-tracing

σE, theor.Ray-tracing

•What is monitored here is energydensity,

average and spreading.

Example, 5000 particles at end of pass#11 (21 GeV) :

2.1075E+4

2.1085E+4

2.1095E+4

2.1105E+4

0

20

40

60

80

100

120

140

160

Zgoubi|Zpop 08-06-2015 COUNT vs. KinEnr (MeV)

• Approximation to ∆E ∝ E2B2∆t, taking ρQF = lQF/θQF, ρBD = lBD/θBD :

∆E[MeV ] = ∆EQF +∆EBD ≈ 0.96× 10−15γ4(θQF|ρQF|

+θBD|ρBD|

) per cell (2)

• Assuming< (1/ρ)2 >≈ 1/ρ2, then :

σE ≈√

σ2E,QF + σ2E,BD ≈ 1.94× 10−14γ7/2

√

θQFρQF

2 +θBDρBD

2 per cell (3)

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• Bunch lengthening (still zero initial ǫx, ǫy and dp/p)

Longitudinal bunch distribution at end of full 7.944 → 21.16 GeV cycle :σl = 0.11 mm.

0.0 0.0002 0.0004 0.0006 0.00080

20

40

60

80

100

120

140

160

Zgoubi|Zpop 08-06-2015 COUNT vs. s (m)

• Bunch lengthening : σl =(σEE

)

[

1

Lbend

∫ Lbend

0

(Dx(s)T51(sf ← s) +D′x(s)T52(sf ← s)− T56)2ds

]1/2

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3.2.2 Transverse phase space

• Energy loss causes drift of bunch centroid, stochasticity causeshorizontal emittance increase.

x-drift along the 6 arcs in eRHIC ring,at E = 21.164 GeV

0 500 1000 1500 2000 2500 3000 35000.00348

0.0035

0.00352

0.00354

0.00356

0.00358

0.0036

0.00362

0.00364Zgoubi|Zpop 11-01-2015 x (m) vs. s (m)

Horizontal phase space after theE =21.164 GeV pass

xf = −15 µm, σxf = 4.3 µm

x′f = −1.1. µrad, σx′f = 1.8 µrad.

-0.25E-4

-0.2E-4

-0.15E-4

-0.1E-4

-0.5E-5

-0.6E-5

-0.4E-5

-0.2E-5

0.0

0.2E-5

0.4E-5

0.6E-5

Zgoubi|Zpop 11-01-2015 x_f’ (rd) vs. x_f (m)

• Beam centroid motion :

[

x(sf)

x′(sf)

]

= T (sf ← si)×[{

x(si)

x′(si)

}

+σEE

{

< U >< V >

}]

with[

U(s)V (s)

]

=

Dx(si)−∫ s

si

T12(s← si)ρ(s)

ds

D′x(si) +∫ s

si

T11(s← si)ρ(s)

ds

• Perturbation of the beam matrix : ∆σ(sf) = T (sf ← si) 4(σEE

)2[

< U 2 > < UV >< UV > < V 2 >

]

× T̃ (sf ← si) (4)

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3.3 End-to-end bunch tracking, cumulated effects (initialǫx ≈ ǫy = 0 and dp/p=0)

Qualitatively, orbits in arcs,pass 1 to 21 :

1.E+4

3.E+4

5.E+4

7.E+4

-.006

-.004

-.002

0.0

0.002

0.004

Zgoubi|Zpop 05-06-2015

* # COORDINATES - STORAGE FILE, 05-06-0015 09:01:27 *

Y (m) vs. s (m)

Mi-ma H/V: 163. 8.056E+04/ -6.795E-03 4.113E-03

(x,x’) phase space, at8, 10.6, 12 and 21 GeV (end of pass # 1, 3, 4, 11)

-0.38E-4

-0.36E-4

-0.34E-4

-0.32E-4

0.54E-4

0.56E-4

0.58E-4

0.6E-4

0.62E-4

0.64E-4

0.66E-4



Y’ (rad) vs. Y (m)

Mi-ma H/V: -3.853E-05 -3.086E-05/ 5.270E-05 6.664E-05 Eps/pi, Beta, Alpha : 5.5871E-13 2.3413E+00 4.4792E+00




Mi-ma H/V: -3.853E-05 -3.086E-05/ 5.270E-05 6.664E-05

-0.17E-4

-0.16E-4

-0.15E-4

-0.14E-4

-0.13E-4

-0.12E-4

-0.11E-4

-0.6E-5

-0.5E-5

-0.4E-5

-0.3E-5

-0.2E-5

-0.1E-5

0.0




Mi-ma H/V: -1.769E-05 -1.048E-05/ -6.968E-06 2.185E-07 Eps/pi, Beta, Alpha : 7.2218E-13 1.5617E+00 1.3484E+00




Mi-ma H/V: -1.769E-05 -1.048E-05/ -6.968E-06 2.185E-07

-0.5E-4

-0.4E-4

-0.3E-4

-0.2E-4

-0.1E-4

0.0 0.1E-4

0.4E-4

0.5E-4

0.6E-4

0.7E-4

0.8E-4

0.9E-4

0.0001

0.00011Zgoubi|Zpop 05-06-2015



Mi-ma H/V: -5.491E-05 1.237E-05/ 3.292E-05 1.101E-04 Eps/pi, Beta, Alpha : 1.7260E-11 6.7243E+00 7.6983E+00




Mi-ma H/V: -5.491E-05 1.237E-05/ 3.292E-05 1.101E-04

-0.8E-4

-0.6E-4

-0.4E-4

-0.2E-4

0.5E-5

0.1E-4

0.15E-4

0.2E-4

0.25E-4

0.3E-4

0.35E-4




Mi-ma H/V: -9.040E-05 -4.022E-06/ -9.772E-07 3.945E-05 Eps/pi, Beta, Alpha : 4.4430E-11 3.1694E+00 1.1592E+00




Mi-ma H/V: -9.040E-05 -4.022E-06/ -9.772E-07 3.945E-05

• Note : the source of the orbit error that triggers the apparentemittance increase, due to the strong chromaticity, is the

residual steering of the adiabatic DS.

-80

-70

-60

-50

-40

-30

-20

-10

20 25 30 35 40 45

0

1

2

3

4

5

6

7

8 10 12 14 16 18 20

∆E : actual-theor.

(MeV)

σ E (MeV)

Gγ

ENERGY AND SPREAD Initial zero 6-D emittance

E (GeV)

∆EσE

0

1

2

3

4

5

6

7

20 25 30 35 40 45

-1

-0.5

0

0.5

1

8 10 12 14 16 18 20

ε x/

π, norm (

µm)

ε y/

π, norm (

µm)

Gγ

EMITTANCE EVOLUTION Initial zero 6-D emittance

E (GeV)

εx/π, norm.εy/π, norm.

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• Evolution of the polarization

Case initial ǫx, ǫy and dp/p0 all zero

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

20 25 30 35 40 45 0

2

4

6

8

10

12

14

16 8 10 12 14 16 18 20

Polar. (<cos

∆ φ>)

σ φ (deg.)

G

γ α [360] (deg.)

G.γ

Initial εx≈ εy≈50π, σdp/p=0

E (GeV)

σφGγα

40 60 80 100 120 1400

20

40

60

80

100

120

140

160

Zgoubi|Zpop 08-06-2015 SX counts vs. ANGLE (deg)

Density of horizontal spin componentSx at end of pass 11

σφ = 14.9 deg.

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3.4 End-to-end 9-D tracking, nominal initial transverse emittances50πµm

Case initial dp/p=0

-90

-80

-70

-60

-50

-40

-30

-20

-10

20 25 30 35 40 45

0

1

2

3

4

5

6

7

8 10 12 14 16 18 20


(MeV)

σ E (MeV)

Gγ

ENERGY AND SPREAD Initial Gaussian εx≈ εy≈50π rms, σdp/p=0

E (GeV)

∆EσE

48

50

52

54

56

58

60

62

64

66

20 25 30 35 40 45

48.5 49 49.5 50 50.5 51 51.5 52 52.5 53 53.5

8 10 12 14 16 18 20

ε x/

π, norm (

µm)

ε y/

π, norm (

µm)

Gγ

EMITTANCE EVOLUTION Initial Gaussian εx≈ εy≈50π rms, σdp/p=0

E (GeV)


Case initial dp/p0 ∈ [−3,+3]× 10−4

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

20 25 30 35 40 45

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8 10 12 14 16 18 20


(MeV)

σ E (MeV)

Gγ

ENERGY AND SPREAD Initial Gaussian εx≈ εy≈50π rms, σdp/p∈ ± 3 × 10

-4

E (GeV)

∆EσE

45

50

55

60

65

70

75

80

20 25 30 35 40 45

49

49.5

50

50.5

51

51.5

52

52.5

53

8 10 12 14 16 18 20

ε x/

π, norm (

µm)

ε y/

π, norm (

µm)

Gγ

EMITTANCE EVOLUTION Initial Gaussian εx≈ εy≈50π rms, σdp/p∈ ± 3 × 10

-4

E (GeV)


ER

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June2015

Polarization

Case initial ǫx ≈ ǫy ≈ 50 πmm.mrad anddp/p0 = 0

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

20 25 30 35 40 45 0

2

4

6

8

10

12

14

16 8 10 12 14 16 18 20

Polar. (<cos

∆ φ>)

σ φ (deg.)

G

γ α [360] (deg.)

G.γ

Initial εx≈ εy≈50π, σdp/p=0

E (GeV)

σφGγα

-.1 -.05 0.0 0.05 0.10

50

100

150

200

250

Zgoubi|Zpop 01-06-2015 COUNT vs. SZ Zgoubi|Zpop 01-06-2015 COUNT vs. SZ

Vertical component of spinat end of pass 11 (end ofpass 11 is 2 IPs beyond

IP6 :

Avrg. Sigma6.304E-05 2.331E-02

Asin(σSZ ) ≈ 1.3 deg.

Case initial ǫx ≈ ǫy ≈ 50 πmm.mrad anddp/p0 ∈ ±3× 10−4

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

20 25 30 35 40 45 0

5

10

15

20

25

30

35 8 10 12 14 16 18 20

Polar. (<cos

∆ φ>)

σ φ (deg.)

G

γ α (deg.)

G.γ

SPIN DIFFUSION Initial Gaussian εx≈ εy≈50π rms, σdp/p∈ ± 3 × 10

-4

E (GeV)

σφGγα

-.1 -.05 0.0 0.05 0.10

50

100

150

200

250

Zgoubi|Zpop 02-06-2015 COUNT vs. SZ

Vertical component ofspin at end of pass 11(end of pass 11 is 2 IPs

beyond IP6 :

Avrg. Sigma-4.207E-04 2.495E-02

Asin(σSZ ) ≈ 1.43 deg.

ER

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IVE

RS

ITY,7-12

June2015

4 Dispersion suppressors

• The dispersion suppressors are based on a “missing magnet” style scheme, where the relative displacement of thetwo cell quadrupoles (the origin of the dipole effect in the FFAG cell) is brought to zero over a series of cells.

• From orbit viewpoint, a quadrupole displacement is equivalent to a kick θk at entrance and exit.

• Due to the varying displacement of the quads (from their misalignement in the arc to aligned configuration in thestraight) the orbit builds along the DS (with origin at upstream arc, endat downstream straight, or reverse) following

yorb(s)√

β(s)=

yorb(0)√

β(0)cos(φ) +

α(0)yorb(0) + β(0)y′orb(0)

√

β(0)sin(φ) +

∑

k

√

β(sk)θk sin(φ− φk)α(s)yorb(s) + β(s)y

′orb(s)

√

β(s)= −yorb(0)√

β(0)sin(φ) +

α(0)yorb(0) + β(0)y′orb(0)

√

β(0)cos(φ) +

∑

k

√

β(sk)θk cos(φ− φk)

• Case of the 11.9GeV pass. The orbit is shown from end of first LSS to upstream region of first arc.

100 120 140 160 180 200

-.004

-.003

-.002

-.001

0.0



orbit x (m) vs. s (m)

• However, the orbit build-up from LLS to arc ends up, at the arc, with (x,x’) coordinates which do notfullycoincide with the periodic orbit of the arc FFAG cell.

ER

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June2015

• The orbit build-up depends on the phase advanceφ =∫ s

0dsβ(s)

. Thus it depends on the cell tune, and on energy.

Orbit along 5 ring-turns,7.9, 9.3, 10.6, 11.9GeV and 13.2 GeV.

In each case starting coordinates are (x,x’)=(0,0) at thecenter of an LSS.

-0.007

-0.0065

-0.006

-0.0055

-0.005

-0.0045

-0.004

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

x (m)

Distance (m)

ORBITS

54321

The orbit amplification is tune-dependent.For instance, case of pass #4:

Varied energy±1, 2, 3% in the vicinity ofE = 11.9GeV:

-0.0058

-0.0056

-0.0054

-0.0052

-0.005

-0.0048

-0.0046

-0.0044

-0.0042

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

x (m)

Distance (m)

ORBITS

-3-2-10123

ER

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ON

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UN

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ITY,7-12

June2015

5 Dynamical acceptance of eRHIC FFAG lattice

Simulation conditions

• The available aperture at entrance to a pass around the ring is evaluated, independently foreach one of the 11 energies in the range7.944 : 21.164 : 1.322.

• Two types of lattices are simulated, for comparison :

- either a single cell,≈ 1000 passes, this would be the dynamical acceptance at entrance into a6-arc ring, 138 cells per arc

- eRHIC ring acceptance window, observed at the center of a LSS.

RING = 6 ×[

1

2LSS − DS − ARC − DS − 1

2LSS

]

• The effects of some field defects are summarily investigated.

• There is no SR in these tracking, just geometrical effects are addressed (apart from all 11energies being investigated).

SR increases beam emittance and is expected to decrease the dynamical acceptance accord-ingly.

ER

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ITY,7-12

June2015

• 1000-cell DA, hard-edge (left) and soft-edge (right)

0

100

200

300

400

500

600

700

800

900

-1500 -1000 -500 0 500 1000 1500

y (mm)

x (mm)

DAs on FFAG2 energies, hard-edge model

E1E2E3E4E5E6E7E8E9

E10E11

-1.5 -1. -.5 0.0 0.5 1. 1.5

-1.

-.5

0.0

0.5

1.



Y’ (rad) vs. Y (m) Zgoubi|Zpop 17-02-2015



21.164

7.944

-.6 -.4 -.2 0.0 0.2 0.4 0.6-.6

-.4

-.2

0.0

0.2

0.4

0.6



Z’ (rad) vs. Z (m)

Maximum H (left) and V (right) invariants, 11 energies.

Field along BD field along QF

0.0 0.2 0.4 0.6 0.8

10

20

30

40

50

60

70

80


* # TRAJECTORIES - STORAGE FILE, 17-02-0015 14:45:26 *

BR (T) vs. X (m)

0.0 0.2 0.4 0.6 0.8 1.

20

40

60

80

100



BR (T) vs. X (m)

Smooth fields experienced by the 11 particles across thetwo cell quadrupoles

0

50

100

150

200

250

-400 -300 -200 -100 0 100 200 300 400

y (mm)

x (mm)

DAs on FFAG2 energies, soft-edge model

E1E2E3E4E5E6E7E8E9

E10E11

-.2 -.1 0.0 0.1 0.2-.3

-.2

-.1

0.0

0.1

0.2



Z’ (rad) vs. Z (m)

0.0 0.5 1. 1.5 2. 2.5

-.1

0.0

0.1

0.2

0.3

0.4



BZ (T) vs. s (m) Zgoubi|Zpop 28-01-2015


BZ (T) vs. s (m)

Maximum vertical invariants, 11 energies.Fringe field model (after H. Enge) :

V (X,Y, Z) =

(

G− G′′

12(Y 2 + Z2) +

G′′′′

384(Y 2 + Z2)2 − G

′′′′′

23040(Y 2 + Z2)3

)

Y

G(s) =G0

1 + expP (s)with G0 =

B0

R0

P (s) = C0 + C1

( s

λ

)

+ C2

( s

λ

)2

+ C3

( s

λ

)3

+ C4

( s

λ

)4

+ C5

( s

λ

)5

ER

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ON

YB

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UN

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RS

ITY,7-12

June2015

• 1000-cell DA, including dodecapole defect, random (top) and systematic (bottom)

0

2

4

6

8

10

12

14

16

-20 -15 -10 -5 0 5 10 15 20y (mm)

x (mm)

DAs on FFAG2 energies, 3G dodecapole

E1E2E3E4E5E6E7E8E9

E10E11

With random dodecapole defect,±3 Gauss at r=1 cm, uniform.

0

10

20

30

40

50

60

70

80

90

-250 -200 -150 -100 -50 0 50 100 150 200 250

y (mm)

x (mm)

DAs on FFAG2 energies, 3G dodecapole

E1E2E3E4E5E6E7E8E9

E10E11

With systematic dodecapole defect+3 Gauss at r=1 cm.

ER

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ON

YB

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UN

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RS

ITY,7-12

June2015

• eRHIC ring , available aperture into the ring as seen at middle of LSS

6 ×[

1


2LSS

]

0

100

200

300

400

500

600

-800 -600 -400 -200 0 200 400 600 800

y (mm)

x (mm)


E1E2E3E4E5E6

Available dynamic admittance of the ring,for 6 energies [7.944:21.164:2.644],

observed at the center of an LSS, defect-free lattice.

0 500 1000 1500 2000 2500 3000 3500 4000-2.

-1.5

-1.

-.5

0.0

0.5

1.

1.5

2.Zgoubi|Zpop 18-02-2015


y (m) vs. s (m) Zgoubi|Zpop 18-02-2015


y (m) vs. s (m) Zgoubi|Zpop 18-02-2015


y (m) vs. s (m)

500 1000 1500 2000 2500 3000 3500 4000-2.

-1.5

-1.

-.5

0.0

0.5

1.

1.5

2.Zgoubi|Zpop 18-02-2015


y (m) vs. s (m)

Vertical excursion along the 6 LSS (top)and 6 arcs (bottom),

6 particles (6 energies) launched at(x=0,y=max.).

ER

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June2015

• eRHIC ring , available aperture into the ring as seen at middle of LSS, withdodecapoledefect

6 ×[

1


2LSS

]

Defect value retained here, following from prior study ofemittance growth over single ring turn : ± 3 Gauss at

1 cm (±3G/5kG = 6× 10−4 relative).

0.001

0.01

0.1

1

10

100

0 5 10 15 20

46

48

50

52

54

56

58

60

0 5 10 15 20

(ε x,dfct -

ε x,nd)/

ε x,nd

ε x,nd, no defect

(

πµm, norm.)

Pass number

DODECAPOLE DEFECT, EVOLUTION OF HORIZONTAL EMITTANCE (1000 particles)

no defect0.01G @1cm0.1G @1cm1G @1cm2G @1cm3G @1cm10G @1cm30G @1cm

0.01

0.1

1

10

100

1000

0 5 10 15 20

46

46.5

47

47.5

48

48.5

49

49.5

50

0 5 10 15 20

(ε y,dfct -

ε y,nd)/

ε y,nd

ε y,nd, no defect

(

πµm, norm.)

Pass number

DODECAPOLE DEFECT, EVOLUTION OF VERTICAL EMITTANCE (1000 particles)

no defect0.01G @1cm0.1G @1cm

1G @1cm2G @1cm3G @1cm

10G @1cm30G @1cm

Hyp. in these two graphs : bunch centroid is re-centered(average position and angle are zero-ed) at entrance to

each one of the 6 LSS.

0

1

2

3

4

5

6

7

8

9

10

-20 -15 -10 -5 0 5 10 15 20

y (mm)

x (mm)


E1E2E3E4E5E6E7E8E9

E10E11

In the presence of dodecapole defect,± 3 G at 1cm, uniform

available dynamic admittance of the ring, for 11 energies[7.944:21.164:1.322],

Observation point is at the center of an LSS, where allorbits have a common reference axis.

ER

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ITY,7-12

June2015

6 Injecting alignment defects

6.1 Horizontal misalignment

• An either 0, or 1, or 10 Gauss dipole defect is considered - random, uniform, applied to all quads around the ring• Qualitatively, below : 4 particles launched with initial ǫx = 0 and dp/p = 0 (subject to increase due to SR)and initial ǫy = 50πµm norm.

Horizontal orbit excursion along the arcs (regionsspanning large∆x values), and along the straights

(regions with∆x values in the vicinity of x=0)

db = zero

1.E+4

3.E+4

5.E+4

7.E+4

-.006

-.004

-.002

0.0

0.002

0.004Zgoubi|Zpop 02-06-2015


Y (m) vs. s (m)

db = 1 G

1.E+4

3.E+4

5.E+4

7.E+4

-.008

-.006

-.004

-.002

0.0

0.002

0.004



Y (m) vs. s (m)

db = 10 G

1.E+4

3.E+4

5.E+4

7.E+4

-.008

-.006

-.004

-.002

0.0

0.002

0.004



Y (m) vs. s (m)

Vertical excursion around the ring, over the 21 passes :

1.E+4

3.E+4

5.E+4

7.E+4

-0.5E-4

0.0

0.5E-4

0.0001



y (m) vs. s (m)

1.E+4

3.E+4

5.E+4

7.E+4

-0.5E-4

0.0

0.5E-4

0.0001



y (m) vs. s (m)

1.E+4

3.E+4

5.E+4

7.E+4

-.0001

-0.5E-4

0.0

0.5E-4

0.0001



y (m) vs. s (m)

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June2015

• Beam evolution from 7.9 to 21.1 GeV and back. Startingǫx ≈ ǫy ≈ 50πµm, dp/p ∈ ±2× 10−4

• db =1 Gauss : Equivalent displacement is∆x = ∆B/G = 2µm - given G=50 T/m.

-90

-80

-70

-60

-50

-40

-30

-20

-10

20 25 30 35 40 45

2

3

4

5

6

7

8

8 10 12 14 16 18 20


(MeV)

σ E (MeV)

Gγ

ENERGY AND SPREAD Initial Gaussian εx≈ εy≈50π rms, σdp/p=2 x 10

-4

E (GeV)

∆EσE

40

60

80

100

120

140

160

180

200

20 25 30 35 40 45

48

49

50

51

52

53

54

8 10 12 14 16 18 20

ε x/

π, norm (

µm)

ε y/

π, norm (

µm)

Gγ

EMITTANCE EVOLUTION Initial Gaussian εx≈ εy≈50π rms, σdp/p=2 x 10

-4

E (GeV)


0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

20 25 30 35 40 45 0

5

10

15

20

25 8 10 12 14 16 18 20

Polar. (<cos

∆ φ>)

σ φ (deg.)

G

γ α [360] (deg.)

G.γ

SPIN DIFFUSION Initial Gaussian εx≈ εy≈50π rms, dp/p in ± 2×10

-4

E (GeV)

σφGγα

ER

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YB

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RS

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June2015

6.2 Vertical misalignment

• Beam evolution from 7.9 to 21.1 GeV and back. Startingǫx ≈ ǫy ≈ 50πµm and dp/p ∈ ±2×10−4uniform

• From top to bottom : dipole defecta0 ∈ [−0.1,+0.1], [−1,+1] G :

-80

-70

-60

-50

-40

-30

-20

-10

20 25 30 35 40 45

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

8 10 12 14 16 18 20


(MeV)

σ E (MeV)

Gγ

ENERGY AND SPREAD, END-TO-END

E (GeV)

∆EσE

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

20 25 30 35 40 45

2

3

4

5

6

7

8

9

8 10 12 14 16 18 20


(MeV)

σ E (MeV)

Gγ

ENERGY AND SPREAD, END-TO-END

E (GeV)

∆EσE

45

50

55

60

65

70

75

20 25 30 35 40 45

48

49

50

51

52

53

54

55

8 10 12 14 16 18 20

β γ

ε x (

π µm)

β γ

ε y (

π µm)

Gγ

EMITTANCE EVOLUTION, END-TO-END

E (GeV)

β γ εxβ γ εy

45

50

55

60

65

70

75

20 25 30 35 40 45

40

50

60

70

80

90

100

110

120

130

8 10 12 14 16 18 20

β γ

ε x (

π µm)

β γ

ε y (

π µm)

Gγ

EMITTANCE EVOLUTION, END-TO-END

E (GeV)

β γ εxβ γ εy

ER

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ITY,7-12

June2015

6.3 Investigating alignment error correction of a 11-beam set

• Sets of H and/or V dipole errors are generated, in cell quadrupoles, in the 6 arcs

• Error sets are taken from uniform ∆b0 and/or ∆a0 density,over±20 Gauss (±40µm equivalent misalignment)

-40

-30

-20

-10

0

10

20

30

40

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

b0 [G]

BD2, QF2 number along ring

b0 errors along ring arcs, in BD and QF magnets

BD

QF

• No errors introduced in LSS neither DS - for simplicity

• These errors are corrected using brute force FIT procedure, that tweaks b0 and/or a0, in all QFand BD until constraints are fulfilled, namely :

- position and angle of 11 different energies

- every 23 cells in the arcs (an arc contains 138 cells, so, there are 6 such sections per arc).

• That allows 23 variables for 22 constraints (x and x’ for each oneof the 11 energies, in one go).

• Each 23-cell section, in all 6 arcs, is corrected in that manner. This is repeated 36 times to coverthe ring.

• SR is on. Each energy is represented bu a 50 particle bunch (hence, 11×50 = 550 particlestracked in one go).

ER

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June2015

• Typical convergence of the FIT, toward the 11 theoretical orbits, for one of 36 arc sections :STATUS OF VARIABLES (Iteration # 0 / 120 max.) ! THE VARIABLES ARE THE QF2’s b_0LMNT VAR PARAM MINIMUM INITIAL FINAL MAXIMUM STEP NAME LBL1 LBL2

12 1 4 -1.000E-02 4.518E-05 4.5175080247E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT18 2 4 -1.000E-02 -5.017E-05 -5.0172080247E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT24 3 4 -1.000E-02 1.340E-04 1.3397781481E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT30 4 4 -1.000E-02 -1.333E-04 -1.3327810247E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT36 5 4 -1.000E-02 4.745E-05 4.7446188889E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT42 6 4 -1.000E-02 -1.474E-05 -1.4741066667E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT48 7 4 -1.000E-02 -2.099E-04 -2.0985586296E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT54 8 4 -1.000E-02 7.811E-05 7.8105985185E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT60 9 4 -1.000E-02 -2.761E-04 -2.7612202222E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT66 10 4 -1.000E-02 6.330E-05 6.3297133333E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT72 11 4 -1.000E-02 -1.786E-04 -1.7864781852E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT78 12 4 -1.000E-02 1.816E-05 1.8164953086E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT84 13 4 -1.000E-02 1.527E-05 1.5267744444E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT90 14 4 -1.000E-02 -5.354E-05 -5.3537733333E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT96 15 4 -1.000E-02 2.148E-04 2.1484726914E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT102 16 4 -1.000E-02 -7.791E-05 -7.7908722222E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT108 17 4 -1.000E-02 2.431E-04 2.4311473333E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT114 18 4 -1.000E-02 7.990E-05 7.9898833333E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT120 19 4 -1.000E-02 8.440E-05 8.4401917284E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT126 20 4 -1.000E-02 1.636E-04 1.6362697037E-04 1.000E-02 8.230E-07 MULTIPOL QF2 MULT132 21 4 -1.000E-02 4.122E-05 4.1218400000E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT138 22 4 -1.000E-02 5.180E-05 5.1796065432E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT144 23 4 -1.000E-02 4.227E-05 4.2273506173E-05 1.000E-02 8.230E-07 MULTIPOL QF2 MULT

STATUS OF CONSTRAINTS (Target penalty = 1.0000E-08)TYPE I J LMNT# DESIRED WEIGHT REACHED KI2 NAME LBL1 LBL2 * Parameter(s)

3 -1 2 147 -5.7866111E-01 2.0000E-01 -5.7928530E-01 9.9814E-02 MARKER #ECellSec * * 2 : 1.0E+00/ 5.0E+01/ Theor. x_orbit for 7.9GeV3 -1 3 147 5.2102765E+00 1.0000E+00 5.2105759E+00 9.1857E-04 MARKER #ECellSec * * 2 : 1.0E+00/ 5.0E+01/ Theor. x’_orbit for 7.9GeV3 -1 2 147 -5.6123649E-01 2.0000E-01 -5.6144094E-01 1.0709E-02 MARKER #ECellSec * * 2 : 5.1E+01/ 1.0E+02/ Theor. x_orbit for 9.2GeV3 -1 3 147 4.3105917E+00 1.0000E+00 4.3116745E+00 1.2014E-02 MARKER #ECellSec * * 2 : 5.1E+01/ 1.0E+02/ Theor. x’_orbit for 9.2GeV3 -1 2 147 -5.2088326E-01 2.0000E-01 -5.1981394E-01 2.9294E-01 MARKER #ECellSec * * 2 : 1.0E+02/ 1.5E+02/ ETC.3 -1 3 147 3.4780794E+00 1.0000E+00 3.4804963E+00 5.9862E-02 MARKER #ECellSec * * 2 : 1.0E+02/ 1.5E+02/3 -1 2 147 -4.6023083E-01 2.0000E-01 -4.6066645E-01 4.8617E-02 MARKER #ECellSec * * 2 : 1.5E+02/ 2.0E+02/3 -1 3 147 2.7061058E+00 1.0000E+00 2.7037711E+00 5.5856E-02 MARKER #ECellSec * * 2 : 1.5E+02/ 2.0E+02/3 -1 2 147 -3.8142236E-01 2.0000E-01 -3.8151658E-01 2.2743E-03 MARKER #ECellSec * * 2 : 2.0E+02/ 2.5E+02/3 -1 3 147 1.9880861E+00 1.0000E+00 1.9901914E+00 4.5421E-02 MARKER #ECellSec * * 2 : 2.0E+02/ 2.5E+02/3 -1 2 147 -2.8634189E-01 2.0000E-01 -2.8663176E-01 2.1526E-02 MARKER #ECellSec * * 2 : 2.5E+02/ 3.0E+02/3 -1 3 147 1.3189515E+00 1.0000E+00 1.3184897E+00 2.1852E-03 MARKER #ECellSec * * 2 : 2.5E+02/ 3.0E+02/3 -1 2 147 -1.7668768E-01 2.0000E-01 -1.7635203E-01 2.8863E-02 MARKER #ECellSec * * 2 : 3.0E+02/ 3.5E+02/3 -1 3 147 6.9385348E-01 1.0000E+00 6.9173531E-01 4.5978E-02 MARKER #ECellSec * * 2 : 3.0E+02/ 3.5E+02/3 -1 2 147 -5.3767999E-02 2.0000E-01 -5.3818008E-02 6.4071E-04 MARKER #ECellSec * * 2 : 3.5E+02/ 4.0E+02/3 -1 3 147 1.0858879E-01 1.0000E+00 1.1067439E-01 4.4575E-02 MARKER #ECellSec * * 2 : 3.5E+02/ 4.0E+02/3 -1 2 147 8.1093355E-02 2.0000E-01 8.0888057E-02 1.0798E-02 MARKER #ECellSec * * 2 : 4.0E+02/ 4.5E+02/3 -1 3 147 -4.4050766E-01 1.0000E+00 -4.3872210E-01 3.2672E-02 MARKER #ECellSec * * 2 : 4.0E+02/ 4.5E+02/3 -1 2 147 2.2697555E-01 2.0000E-01 2.2728052E-01 2.3828E-02 MARKER #ECellSec * * 2 : 4.5E+02/ 5.0E+02/3 -1 3 147 -9.5714321E-01 1.0000E+00 -9.5775222E-01 3.8007E-03 MARKER #ECellSec * * 2 : 4.5E+02/ 5.0E+02/3 -1 2 147 3.8260641E-01 2.0000E-01 3.8335722E-01 1.4442E-01 MARKER #ECellSec * * 2 : 5.0E+02/ 5.5E+02/3 -1 3 147 -1.4428212E+00 1.0000E+00 -1.4417261E+00 1.2288E-02 MARKER #ECellSec * * 2 : 5.0E+02/ 5.5E+02/

Fit reached penalty value 9.7583E-05

0

0.001

0.002

0.003

0.004

0.005

0.006

0 5 10 15 20 25 30 35

penalty

section number

Matching 6 sections per arc, 6 arcs, to 11 theoretical orbits in 1 go. Penalties.

penalties

ER

L2015W

orkshop,BN

L-ST

ON

YB

RO

OK

UN

IVE

RS

ITY,7-12

June2015

• Sample results

Horizontal misalignment (b0 ∈ ±20 G, uniform)

Initial dp/p = 0, initial ǫx, ǫy zero. Initial dp/p ∈ ±2 10−4, initial ǫx, ǫy zero.BEFORE CORRECTION

-.006 -.004 -.002 0.0 0.002

-.004

-.002

0.0

0.002

0.004

0.006

0.008




-.006 -.004 -.002 0.0 0.002 0.004

-.006

-.004

-.002

0.0

0.002

0.004

0.006

0.008




AFTER CORRECTION (Note the scales : big effect)

-.0002 0.0 0.0002 0.0004 0.0006 0.0008

-.0004

-.0002

0.0

0.0002

0.0004

0.0006




-.0002 0.0 0.0002 0.0004 0.0006

-.0004

-.0002

0.0

0.0002

0.0004

0.0006




ER

L2015W

orkshop,BN

L-ST

ON

YB

RO

OK

UN

IVE

RS

ITY,7-12

June2015

Initial dp/p = 0, initial ǫx, ǫy 50pi. Initial dp/p ∈ ±2 10−4, initial ǫx, ǫy 50pi.

BEFORE CORRECTION

-.006 -.004 -.002 0.0 0.002

-.004

-.002

0.0

0.002

0.004

0.006

0.008




-.006 -.004 -.002 0.0 0.002 0.004

-.006

-.004

-.002

0.0

0.002

0.004

0.006

0.008




AFTER CORRECTION (Note the scales : big effect)

-.0004 0.0 0.0004 0.0008

-.0004

-.0002

0.0

0.0002

0.0004

0.0006

0.0008




-.0004 0.0 0.0004 0.0008

-.0006

-.0004

-.0002

0.0

0.0002

0.0004

0.0006

0.0008





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Beam and spin dynamics studies in eRHIC ERL · ERL2015 Workshop, BNL-STONY BROOK UNIVERSITY, 7-12...

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Transcript of Beam and spin dynamics studies in eRHIC ERL · ERL2015 Workshop, BNL-STONY BROOK UNIVERSITY, 7-12...