SIMULATION OF RHIC EXPERIMENTS1 Introduction Motivation Wire Compensation 2 Simulations BBSIM Model Gold Injection Results Gold Collision Results ... ( 1 n D bbsim theory 2 4 8 10

SIMULATION OF RHIC EXPERIMENTS(Beam-Beam Simulation for RHIC Wire Compensator)

H. J. Kim and T. Sen

Accelerator Physics Center, Fermilab

LARP Mini-Workshopon Beam-Beam Compensation

July 2-4, 2007

H. J. Kim and T. Sen (Fermilab) BBSIM LARP Meeeting 1 / 23

Outline

1 IntroductionMotivationWire Compensation

2 SimulationsBBSIM ModelGold Injection ResultsGold Collision Results

3 Summary / future work


Introduction Motivation

Motivation

Compensation of beam-beam effects using current carrying wires in RHIC isinteresting subject. During 2007, tesing long range beam-beam compensationusing wire compensation is scheduled.

Unnderstanding the nature of the wire effects in RHIC is neededexperimentally/theoretically to apply the wire effectively.

For this reason a set of experiments were recently conducted in RHIC. Theseresults can be compared with our simulations and we can then correctlyimplement wire compensation.

Benchmark simulations against observations.Test adequacy of model.Test accuracy of the model


Introduction Wire Compensation

Wire Compensation

Beam-beam kicks by a round beam:

∆x′

=qtqb

2Nbrbγb

cosθBTrBT

{1−exp

[−

r2BT2σ2

]}

∆y′

=qtqb

2Nbrbγb

sinθBTrBT

{1−exp

[−

r2BT2σ2

]}

Kicks due to a current carrying wire:

∆x′

=µ02π

Iw Lw(Bρ)

cosθWTrWT

∆y′

=−µ02π

Iw Lw(Bρ)

sinθWTrWT

wire

beam

test beamθBT

θWT

y

x

rWT

rBT

At large beam separation, beam-beam kicks can be canceled by wire kicks:

θBT = π +θWT , rWT = rBT , IwLw = (qbcNb)(ptc/Eb)

In reality:

Aspect ratio of beam is not constant.Wire location is limited due to lattice components.


Introduction Wire Compensation

Wire Compensators for a Beam Test in RHIC

Wire Locations:

quantity unit valueIL, single interaction Am 9.6

(IL)max Am 125Lw m 2.5rw mm 3.5

rpipe mm 60

quantity unit injection collisiongold energy Gev/n 9.795 100

bunch intensity 109 0.7 1.0emittance εx ,y (95%) mm mrad 5.8 18βy at wire location m 124 372σy at wire location mm 3.4 3.3

tunes (νx ,νy ) B (0.230,0.216) B (0.220,0.231)Y (0.220,0.230) Y (0.232,0.2228)

vertical separation σ 5A: 2.3-4.4 5A: 6-950A:3.8-7.4 50A: 6-9


Simulations BBSIM Model

BBSIM Model

6D weak-strong model, includes synchrotron oscillations etc

At injection: nonlinearities are chromaticity sextupoles & wire

At collision: above + IR quad nonlinearities

Head-on and Long-range interactions when present

Implementation by Fortran 90/95

Parallelization of I/O and computation (PETSc, HDF5, SPRNG)


Simulations Gold Injection Results

Results of Gold Injection Energy



Tune Shift at Zero Amplitude by Wire

Gold injection, I = 50A

2 4 6 8 10 12 14

y )σ(d

0

1

2

3

4

5

6

7x

)3− 01(

ν∆

bbsimtheory

2 4 6 8 10 12 14

y )σ(d

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

y)2

− 01(ν

∆

bbsimtheory

Analytic tune shift ∆ν is evaluated by

∆νx ,y =± µ0IwLw

8π2 (Bρ)βx ,y

d2y −d2

x(d2y +d2

x

)2 .

Similiar result at Iw = 5A.



Tune Footprint (separation distance)


Blue beam base tunes is (0.230, 0.216).

Tune spread becomes wide as the separation is decreased.

The closest resonances are the 9th , 13th , 14th and 17th order resonances.

Similiar result at Iw = 5A.



Dynamic Aperture (separation distance)


Dynamic aperture become smaller as the separation is decreased.

Dynamic aperture is highly dependent upon particle’s position angle.



Dynamic Aperture (tune scan)

At all wire separations, the largest dynamic apertures are distributed alongthe line νx −νy ' 0.02.

The zone along νx −νy ' 0.03 has the smallest dynamic apertures.

This scan indicates that the nominal tune (28.230,29.216) is close to optimal.

A sharper drop in dynamic aperture is observed near the 5th resonance.



Diffusion Coefficients

Diffusion equation is

∂

∂ tρ

(~J , t

)=

1

2∇~J·(D(~J)

∇~J

)ρ

(~J , t

).

Assumption:

Diffusion eqution is separable, i.e, D(~J)

= Dx (Jx )Dy (Jy ).

The diffusion coefficients can be calculated numerically from

Dx (Jx ) = limN→∞

⟨(Jx (N )−Jx (0))2

⟩N

,

where Jx (0) is initial action, Jx (N ) action after N turns, and 〈〉 averageover simulation particles.



Diffusion Coefficients (I=50A)

Diffusion vs wire separation (at constant action) show that the diffusionincreases exponentially as the separation is decreased.

Diffusion vs initial action (at constant separation distance) show that thediffusion increases exponentially as the action is increased.

Diffusion due to 50A wire is one-order higher than diffusion due to 5A.



Lifetime Estimate

D (J ) is used to estimate the escape time to an absorbing boundary JA:

τescape =∫ JA

0

J dJ

D (J ),

where D (J ) is modelled by D (J ) = C exp(− J

J0

).

We interprete the escape time as a measure of beam lifetime.



Loss rate

The loss rate at dy = 3.8σ is over-estimated compared with the measuredone within few ten’s percentage of error.

The lost rate at dy = 5.5σ is under-estimated. Slight loss of particles isobserved at dy = 7.3σ in the experiment, but not in the simulation.


Simulations Gold Collision Results

Results of Gold Collision Energy



Tune Shift by Wire

Gold collision, I = 50A



Tune Footprint (separation distance)


Blue beam base tunes is (28.220, 29.231).

Tune spread becomes wide as the separation is decreased.

At 6 to 9 σ separations, the beam does not span resonances.



Dynamic Aperture (separation distance)


Dynamic aperture become smaller as the separation is decreased.

Dynamic aperture is highly dependent upon particle’s position angle.



Dynamic Aperture (tune scan)

Dynamic apertures are scanned by ∆ν = 0.01.

At all wire separations, the largest/smallest dynamic apertures are distributeddiagonally.

The zone along νx = 0.25 has the small dynamic apertures.

A sharper drop in dynamic aperture is observed near the 4th and 5th

resonances.



Diffusion Coefficients (I=50A)

Diffusion vs wire separation (at constant action) show that the diffusionincreases exponentially as the separation is decreased.

Diffusion vs initial action (at constant separation distance) show that thediffusion increases exponentially as the action is increased.



Lifetime Estimate

D (J ) is used to estimate the escape time to an absorbing boundary JA, i.e.

τescape =∫ JA0

J dJD(J ) .

We interprete the escape time as a measure of beam lifetime.

The lifetime varies exponentially with the separation distances.


Summary / future work

Summary / future work

Developed (and continue developing) new tools to address the problem.

Results with wire compensators are presented.

The results show that the dynamic aperture is highly dependent upon theangle between the wire and beam particles and mostly linearly dependent uponthe separation.From the tune scan of dynamic aperture, the optimal working points aresought and verified.Dependence of beam life time on the separation is found to be exponential.

Future works:

more exact estimate of lifetime from the solution of diffusion equation usingthe calculated diffusion coefficient.simulations with wire and long-range interaction.


Documents

SIMULATION OF RHIC EXPERIMENTS1 Introduction Motivation Wire Compensation 2 Simulations BBSIM Model Gold Injection Results Gold Collision Results ... ( 1 n D bbsim theory 2 4 8 10