oliver boine-frankenheim, he-lhc, oct. 14-16, 2010, malta simulation of ibs (and cooling) oliver...

Oliver Boine-Frankenheim, HE-LHC, Oct. 14-16, 2010, Malta 1

Simulation of IBS (and cooling)

Oliver Boine-Frankenheim, GSI, Darmstadt, Germany


Contents

o Intra-beam scattering rates for high-energy ion beams

o Why/when do we need kinetic simulations ?

o Interplay of IBS, cooling, collective effects

o 3D IBS simulations: Langevin equations

o ‘Local’ IBS model

o IBS code validation

‘Disclaimer:’ I will report about IBS and cooling simulation schemes developed and validated for the HESR (15 GeV pbars), SIS-300 (35 GeV/u U92+) and RHIC at BNL.


Multiple Coulomb Collisions in a PlasmaFokker-Planck equation

Di, j (rv) =4π

q2

4πÚ0m

⎛

⎝⎜⎞

⎠⎟LC f(

r′v )u2δi, j −uiuj

u3 d3 ′v∫

df (rv)

dt=−

∂∂vi

( fK j ) +12

∂2

∂vi∂vj

( fDi, j )i, j∑

j∑

rK(

rv) =8π

q2

4πÚ0m

⎛

⎝⎜⎞

⎠⎟

2

LC f(r′v )

ruu3d

3 ′v∫

τ j , j−1 ≈

D j , j

v j2

A. Sorensen, CERN Acc. School (1987)

ru =

rv−

r′v

LC =lnrbeam

b900

⎛

⎝⎜

⎞

⎠⎟ ≈10 −20

rv r

′vf (

rx,

rv)

(small angle collisions dominate)

Coulomb logarithm:

friction vector:

diffusion tensor: f (rv,t) =

nπ 2π ΔPΔ⊥

2exp −

v⊥2

Δ⊥2

⎛

⎝⎜⎞

⎠⎟exp −

vP2

2ΔP2

⎛

⎝⎜

⎞

⎠⎟

Δ⊥ ? ΔP

Gaussian velocity distribution:

Anisotroptic velocity distribution:

Diffusion rate:DP ≈n

q2

4πÚ0m

⎛

⎝⎜⎞

⎠⎟

2LcΔ⊥

τP−1 ≈ n

q2

4πÚ0m

⎛

⎝⎜⎞

⎠⎟

2Lc

Δ⊥ΔP2

Longitudinal diffusion coefficient:

Longitudinal diffusion rate:


IBS rates for high energy beams

vP

v⊥

=β̂⊥

%δ 2

γ2 %ε⊥

= 1

J.D. Bjorken, S.K. Mtingwa, Part. Accel. 13 (1983)RHIC: A.V. Fedotov et al., PAC 2005HESR: O. Boine-Frankenheim et al., NIM 2006

Ratio of longitudinal/transverse velocities in the beam frame:

%δ =Δpp

(rms momentum spread)

with

DPibs =

ri2cNLC

πRβ03γ0

3 β̂⊥

1/2%ε⊥

3/2

Longitudinal ‘plasma’ diffusion in the lab frame:

τP,loss−1

( )ibs

=DP

ibs

LCδmax2Touschek loss rate:

τ h−1

( )ibs

≈ τ P−1( )

ibs %δ2

ε⊥

Dx2 + %Dx

2

β̂x

%Dx =Dxα̂x + ′Dxβ̂x

Transverse IBS heating rates:

with

%δ 2 =τ cΛP

ibs

ε⊥3/2

∝ N 2 /5

ε⊥ = τ cΛPibs Dx

2 + %Dx2

βx

⎛

⎝⎜⎞

⎠⎟

2/5

Cooling equilibrium for a constant cooling time :

τ c = τ PIBS :

τ c = τ ⊥IBS :

Longitudinal ‘plasma’ IBS heating rate:

τP−1

( )ibs

=1%δ 2

d%δ 2

dt=DP

IBS

%δ 2=

ΛPIBS

%ε⊥3/2 %δ 2

(corresponds to the Bjorken-Mtingwa result for high energies)

ΛPIBS =

Ncri2LC

8 π Rβ03γ 0

3 β̂⊥1/2

with


Why/When do we need kinetic IBS simulation ?

1. Ultra-cold beams, strong correlations (LC<1).

2. Interplay of IBS with: - nonlinear resonances (beam-beam, e-clouds, space charge) - impedances and wakes - internal targets - (nonlinear) cooling force and particle losses-> non-Gaussian distribution functions


Coulomb strings in coasting ion beams

Experiment: M. Steck et al., Phys. Rev. Lett. 1996Theory/Simulations: R. Hasse Phys. Rev. Lett. 1999, 2001, 2003

Equilibrium momentum spread vs. number of ions

Equilibrium between IBS and e-cooling

Δpp

∝ N 0.3

Ultra-cold beams

LC ≤1:-Molecular dynamics simulations-Tree codes


Why/When do we need kinetic IBS simulation ?

1. Ultra-cold beams, strong correlations (LC<1).

2. Interplay of IBS with: - nonlinear resonances (beam-beam, e-clouds, space charge) - impedances and wakes - internal targets - (nonlinear) cooling force and particle losses-> non-Gaussian distribution functions


Interplay of IBS, cooling and impedances1D coasting beam example

Self-bunching in a coasting beam (observed in the ESR) due to the rf cavity impedance:

O. Boine-Frankenheim, I. Hofmann, G. Rumolo, Phys. Rev. Lett. 1999

ESR circumference

time

[s]

current profile from BPM

Vlasov-F-P simulation results

‘space charge waves’

∂f∂t

−η 0v0δ∂f

∂z+qV (z, t)

p0

∂f

∂δ= −

∂

∂δFPe(δ) f( )+DP

ibs ∂2 f∂δ2

f (z,δ,t)1D Vlasov-Fokker-Planck equation for

Direct numerical solutionof the V-F-P equation on agrid in (z,δ) phase space.

Impedances (harmonic n): Vn (t) = Znsc + Zn

cav( ) In(t)


Simplified 3D Fokker-Planck approach

Di, j =

Dx,x 0 0

0 Dy,y 0

Dz,x 0 Dz,z

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Assumptions:- Gaussian distribution- Constant diffusion coefficients Di,j - No vertical dispersion

Di, j =A0LC

u2δ i, j −uiuj

u3Ki

=A0LC

2ui

2

u3

P. Zenkevich, O. Boine-Frankenheim, A. Bolshakov, NIM A (2006)

Averaging of the FP coefficients over the field and test particles:

Diffusion tensor:

Di, j =AN (δ i, j INTi,ii=1

3

∑ −INTi, j )

INTz ,z

=4πλ(a2 + λ)dλ

[(a1 + λ)(a2 + λ) −α 2 ]3/ 2(a3 + λ)1/ 20

∞

∫

INTy , y

=4πλdλ

[(a1 + λ)(a2 + λ) −α 2 ]1/ 2(a3 + λ)3/ 20

∞

∫

INTz ,x

=−8παλdλ

[(a1 + λ)(a2 + λ) −α 2 ]3/ 2(a3 + λ)1/ 20

∞

∫

INTx ,x

=4πλ(a1 + λ)dλ

[(a1 + λ)(a2 + λ) −α 2 ]3/ 2(a3 + λ)1/ 20

∞

∫

a1

2=

1εz

+γ 2(Dx

2 + %Dx2)

β̂xεx,

a2

2=β̂x

εx,

a3

2=

2β̂y

εy

α =2γ %Dx

εx, %Dx = ′D̂xβ̂x +D̂xα̂x

Calculation of the coefficients using the B-M formalism:

with

and


3D numerical solution of the Fokker-Planck equation: Langevin equations

Pi (t + Δt) =Pi (t) −K iPi (t)Δt + Δt Ci, jj=1

3

∑ ξ j

Ci,kC j ,k =Di, jk=1

3

∑

rP =

′x′y

1γ

Δpp

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟ ξi : Random numbers with Gaussian distribution

P. Zenkevich, O. Boine-Frankenheim, A. Bolshakov, NIM A (2006)General diffusion tensor: Meshkov et al., BNL Report, 2007, BETACOOL code

Outline of the algorithm:1. Calculation of Ci,j at every lattice elements2. Three random numbers ξ for each macro-particle3. Apply Langevin kick.4. Transport particles through the lattice element

Langevin equation:

Relation between Langevin and diffusion coefficients:

C33 = Dz,z

C 11= Dx,x −C132

C13

=1

C33

C22 = Dy,y

C31 =0

Choice of Cij gives correct growth of the emittances (B-M theory)

Di, j =

Dx,x 0 0

0 Dy,y 0

Dz,x 0 Dz,z

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⇒

(e.g. H. Risken, The Fokker Planck equation, 1984)


Local IBS modela 1D illustration

Plasma: Manheimer, Lampe, Joyce, J. of Comput. Phys. (1997)RHIC: Meshkov et al., BNL Report, 2007; Fedotov, Proc. of ICFA-HB2010

DP(z,vP) ≈A0LC f(z, ′vP)u⊥

2 − vP − ′vP( )2

u⊥3∫ d ′vP

Motivation:-accurately treat tail/halo particles in non-Gaussian distributions r

u =rv−

r′v

Outline/problems of the algorithm:-Diffusion coefficient has to be calculated every time step for every macro-particle- Nloc large enough to reduce numerical noise

Longitudinal diffusion function:

with

z

Δpp

Nloc

Δztest particle (z,v)

Nloc field particles

DPΔz(vP) ≈A0LC

1Nloc

u⊥2 − vP − ′vP, ′m( )

2

u⊥3

′m =0

Nloc

∑

Local integration over field particles (index m’):

vP,m(z,v,t+ Δt) =vP,m(z,v,t) −ξm ΔtDP(z,vP)

Langevin equation for macro-particles (index m):


IBS and internal targets

Momentum kick:e-cooling force

IBS diffusion(Langevin force)

target

Integrated probability:

Random number:

‘Thin target’ requirement:

Δδ j = −FP

e(δ i ) +Q j (t) DPibsΔt −

ΔÚj (t)

β02E0

Target energy kick:

Landau tails

Simulation of the interaction of an electron cooled bunch in a barrier bucket with an internal target.

Beam equilibrium and beam loss with internal targets in the HESR O. Boine-Frankenheim, R. Hasse, F. Hinterberger, A. Lehrach, P. Zenkevich Nucl. Inst. and Methods A 560 (2006) 245.


Validating IBS simulation modules

Global IBS: IBS emittance growth rates for Gaussian beams

Simulation of beam echoes and code validation:Sorge, Boine-Frankenheim, W. Fischer, Proc. ICAP 2006Al-khateeb, Boine-Frankenheim, Hasse, PRST-AB 2003

Aecho,rel =⟨x⟩echo,max

⟨x⟩0

=q

α 3

ττdec

α =2

3

τ

τ diff

τ

τ dec

⎛

⎝⎜⎞

⎠⎟

2

+1 D⊥ =ε⊥

τdiff

Echo amplitude depends very sensitive on diffusion !

Problem in beam dynamics simulations with IBS modules: Artificial numerical diffusion vs. IBS diffusion in (detailed) IBS simulations

dipolar kick

quadrupolar kick

dipolar echo

tracking with constant diffusion

τ dec

τ

τ echo = 2τ

Aecho ∝1

D⊥3

Test case: (transverse) beam echo amplitudes

∂∂v

KPcool (v)f0(v)( )+

∂2

∂v2 DPibs(v)f0(v)( )=0 ⇒ f0(v)

IBS for non-Gaussian distributions: Integration of the stationary F-P equation


Conclusions

IBS simulation schemes based on the Fokker-Planck equation have been presented.

a. Direct solution of the Vlasov-Fokker-Planck equation in 1D

b. Langevin equations in 1D-3D: - Constant diffusion coefficients from B-M theory - Local coefficents for non-Gaussian distributions

Code validation for IBS effect is an important issue (numerical noise vs. IBS) !

For most applications the Langevin equations with constant diffusion coefficients might the method of choice.

oliver boine-frankenheim, he-lhc, oct. 14-16, 2010, malta simulation of ibs (and cooling) oliver...

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