oliver boine-frankenheim, high current beam physics group simulation of space charge and impedance...

Oliver Boine-Frankenheim, High Current Beam Physics Group

Simulation of space charge and impedance effects

Funded through the EU-design study ‘DIRACsecondary beams’

Longitudinal beam dynamics simulations (LOBO code)o Motivation: ‘Loss of Landau damping’ and longitudinal beam stabilityo LOBO physics model and numerical schemeo Longitudinal bunched beam BTF: Experimental and numerical results

3D simulations with PATRIC (PArticle TRakIng Code) • Motivation: Transverse (space charge) tune shifts and ‘loss of Landau damping’• Numerical tracking scheme with space charge and impedance kicks• Application 1: Damping mechanisms in bunches with space charge• Application 2: Head-tail-type instabilities with space charge

General motivation (in the context of the SIS 18/100 studies): Effect of space charge on damping mechanisms and instability thresholds Study possible cures (double RF, octupoles, passive/active feedback,...)

O. Boine-Frankenheim, O. Chorniy, V. Kornilov


Longitudinal incoherent + coherent space charge effects

‘Loss of Landau damping’

Σ =1

V0

RF / V0

sc −1> 0

e.g. Boine-F., Shukla, PRST-AB, 2005

ωs(φ, Σ) ≈ω

s 0(φ) 1−G(φ)

Σ

2

⎛

⎝⎜

⎞

⎠⎟

ω

s 0(φ) ≈ω

s 0− ′′ω φ2

G(φ) =1

ω

s 0(φ) ≈ ′ω φ

synchrotron frequency (oscillation amplitude ):

Space charge factor:

Elliptic bunch distribution:

single rf wave: double rf wave:o Intensities ∑ >∑th require active damping.o Analytic approaches with space charge and nonlinearities are usually limited.o Use simulation code to determine ∑th

o Compare with experiments

Ω1

ωs 0

≈ 1−φ

m

2

10

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1/ 2

Ω1

ωs 0

≈5

4φ

m

Coherent (dipole) frequencies (bunch length m):

(single rf) (double rf)

φ

ω

s

min φm, Σ( ) < Ω

1φ

m( ) < ωs

max φm, Σ( )

γL∝

∂f

∂ωs ω

s=Ω

Landau damping rate:

Landau damping will be lost above some ∑th if the (coherent) dipole frequency is outside the band of (incoherent) synchrotron frequencies.


BTF measurement in the SISPhD student Oleksandr Chorniy (with help by S.Y. Lee)

V

RF(φ,t ) =V

0sin(φ + ε sinΩt )

Motivation:Measure Landau damping with space charge

Measure syn. frequency distribution f(ωs)

Measure coherent modes Ωj

Measure the effective impedance Zeff

Further activities:

Double rf and voltage modulation.

Nonlinear response with space charge.

Supporting simulation studies.

Measure bunch response: φ(t ), ψ

rf phase modulation:

γ

L(Ω

1) ∝ φ

max

−1

τ

P

cool ≈100 msXe48+, 11.4 MeV, Nb=108

Ω1

ωs0

≈ 1−φ

m

2

10

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1/ 2

Results of the first measurement (Dec. 2005):


Longitudinal beam dynamics simulationsMacro-particle scheme

&δj=

q

p0

Vrf(z

j,t ) +V

ind(z

j,t )( ) −F

c(δ

j) + ξ(t ) 2D

ibsΔt +K

&z

j=−η

0δ

j

I (z,t ) =β0c S (z −z

j)Q

jj

∑

V

ind(z,t ) =−FFT -1 Z

P(ω

n)I (ω

n)⎡

⎣⎤⎦

LOBO code:Macro-particle schemeAlternative ’noise-free’ grid-based schemeflexible RF objects and impedance librarymatched bunch loading with (nonlinear) space chargee-cooling forces, IBS diffusion, energy loss (straggling)C++ core, Python interface

The LOBO code has been used (and benchmarked) successfully in a number of studies: -microwave instabilities -rf manipulations -beam loading effects -collective beam echoes (!) -bunched beam BTF -e-cooling equilibrium

Position kick for the j-th particle:

Momentum kick:

(slip factor η, momentum spread δ)

Current profile:

Induced voltage:

(linear and higher order interpolation)

e-cooling+IBS+internal targets


LOBO example: beam loading effects

Matched ‘sausage’ bunch with space charge and broadband (Q=1) rf cavity beam loading.


Bunched beam BTF simulation scanssingle rf wave, phase modulation, long bunch m=±900

Ω

1=ω

s 0

V

RF(φ,t ) =V

0(φ + ε sinΩt )

ωs=

ωs 0

1 + Σ

No difference between Parabolic or Gaussian bunches. -> No damping due nonlinear space charge.

V

ind=−Z

P

scI

Sawtooth field: + Space charge:

Ω1

ωs0

≈ 1−φ

m

2

10

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1/ 2

V

RF(φ,t ) =V

0sin(φ + ε sinΩt )

V

ind=−Z

P

scI

Loss of Landau damping for ∑th≈0.2.No significant difference between Gaussian and Elliptic distribution. ->weak influence of nonlinear space charge.

Single rf wave: + space charge:


Gaussian bunch

Σ

th≈0.5

Gaussian vs. Elliptic Bunch DistributionDouble rf wave, phase modulation, long bunch m=±900

€

VRF (φ,t ) =V0 sin(φ + ˆ ε sinΩt ) −1

2sin(2φ)

⎛

⎝ ⎜

⎞

⎠ ⎟

Σ

th≈0.05

->Nonlinear space charge strongly increases Landau damping in a double rf wave->Analytic calculations for the double rf wave with space charge are difficult ?!->This effect can be very beneficial for cooler storage rings. Experiments needed !

Elliptic bunch distribution

Ω1

ωs 0

≈5

4φ

m

ωs=

ωmax

1 + Σ

V

ind=−Z

P

scI+ space charge:


Bunched beam BTF simulation scanssingle rf wave, voltage modulation, long bunch m=±900

Ω2

ωs 0

= 3 +1

1+ ΣQuadrupolar mode in a short bunch:

V

RF(φ,t ) =V

0(1 + ε sinΩt ) sinφ

V

RF(φ,t ) =V

0(1 + ε sinΩt )φ

Quadrupole modes and their damping in long bunches with space charge needs more study !

ωs=

ωs 0

1 + Σ


Transverse incoherent + coherent space charge effects

‘loss of Landau damping’

ΔQyinc = −

Z

A

r0N

βγ 2

gt

B f

2

ε n,y + ε n,yε n,x

ΔQy

coh =−i

4πqI R

Q0β

0E

0

Zy

coh

Zy

sc =−RZ

0

β0

2γ0

2

1

b2

ΔQy

inc −ΔQy

coh ≤1

3FδQ inc

e.g. K.Y. Ng, ‘Transverse Instability in the Recycler’, FNAL, 2004

δQ inc = ξ + (n−Q )η

0( )δHWHM

δQ coh ≈ ΔQ

y

coh(I )

Damping mechanisms

Incoherent tune spread:

δQ

sc

inc ≈ ′Q J

Incoherent space charge tune spread:

Coherent tune spread along the bunch:

Goal: resolving all these effects in a 3D tracking codeDamping mechanisms and resulting instability thresholdsStudy beam behavior close to the thresholds.

Incoherent space charge tune shift:

Coherent tune shift:

Space charge impedance:

Stability condition (or ‘Loss of Landau damping’):


PATRIC: ‘Sliced’ tracking model and self-consistent space charge kicks

∆sm << betatron wave length

The transfer maps M are

‘sector maps’ taken from MADX.

x

z

y

s

M(sm|sm+1)

Sliced bunch

xj

′xj

yj

′yj

zj

δj

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

sm+1

=M(sm|s

m+1)

xj

′xj+

qEx(x

j,y

j,z

j, s

m)

mv0

2γ3Δs

M

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

sm

space charge kick:

2D space charge field for each slice:

∂Ex

∂x+∂E

y

∂y=ρ(x,y, z,s

m)

ε0

Ez≈−

g

4πε0γ

0

2

∂ρL(z , s

m)

∂z

ρ(x,y,z,sm) = Q

jS (

rx −

rx

j)

j

∑ (3D interpolation)

(fast 2D Poisson solver)

(ρL line density)

sm: position in the lattice

z: position in the bunch

slice-length: ∆z∆s (N macro-slices for MPI parallelization)


Transverse Impedance KicksImplementation in PATRIC

Dipole moment times current: ψ(t)

ω

j=(n±Q )ω

0

V.Danilov, J. Holmes, PAC 2001O. Boine-F., draft available

Coherent frequencies

localized impedance

Impedance kick:

Coasting beam:

In the bunch frame (∆s=L for localized impedance):

Slowly varying dipole amplitude:

Numerical implementation:

ψ(z,tm) =β

0c Q

jS (z −z

j)x

jj

∑


PATRIC benchmarkingPresently ongoing !

Coasting beam:

Coherent tune shifts with pure imaginary impedance (analytic, passed)

o Decoherence of a kicked beam with/without space charge and imaginary impedance (analytic, talk by V. Kornilov)

o Instability threshold and growth rate for the transverse microwave instability with/without space charge driven by a broadband oscillator (analytic, ongoing)

Bunched beam:

o Decoherence of a kicked bunch with space charge and imaginary impedance (analytic ?)

o Headtail-type instabilities with space charge driven by a broadband oscillator (compare with CERN codes and experimental data).

Headtail-type instabilities might be of relevance for the compressed bunches foreseen in SIS 18/100: Use PATRIC/HEADTAIL to check ‘impedance budget’.

Benchmark the 3D sliced space charge solver and the impedance module


Coasting beam transverse instabilityPATRIC example runSIS 18 bunch parameters

(in the compressed bunch center):

o U73+ 1 GeV/u

o dp/p: δm=5x10-3

o ‘DC current’: Im≈25 A

o SC tune shift: ∆Qy=-0.35

o SC impedance: Z=-i 2 MΩ

(∆Qcoh=-0.01)o Resonator: Q=10,

fr=20 MHz, Re(Z)=10 MΩ

Without space charge the beam is stabilized by the momentum spread (in agreement with the analytic dispersion relation).

N=106 macro-particlesT=100 turns in SIS 18 (ca. 2 hours CPU time)Grid size Nx=Ny=Nz=128Example run on 4 processors (dual core Opterons)


Decoherence of a kicked compressed bunch PATRIC test example

SIS 18 compressed bunch parameters:U73+ 1 GeV/uIons in the bunch: 3x1010

Duration: 300 turns (0.2 ms)

dp/p: δm=5x10-3

Bunch length: τm=30 ns

Peak current: Im≈25 A

SC tune shift: ∆Qy=-0.35

SC impedance: Z=-i 2 MΩ (∆Qcoh=-0.01)

Horizontal offset: 5 mm

Remark: For similar bunch conditions G. Rumolo in CERN-AB-2005-088-RF found a fast emittance increase due to the combined effect of space charge and a broad band resonator.

δQ coh ≈ ΔQ

y

coh(I ) ⇒ τdec

−1 =ω0δQ coh ≈(0.02ms)−1

‘Decoherence rate’ due to the coherent tune spread along the bunch:


Decoherence of a kicked bunch with space charge

With space charge (∆Qy=-0.5):Without space charge (only image currents):

ε

x

ε

y

x


Head tail instability of a compressed bunch

due to the SIS 18 kicker impedance ?

In the simulation test runs one peak of the kicker impedance is approximated through a resonator centered at 10 MHz: fr=10 MHz, Q=10, Z(fr)=5 MΩ

SIS 18 kicker impedance (one of 10 modules):

Result of the PATRIC simulation:Z(fr)≈0.6 MΩ

SIS 100 kicker impedances (per meter) will be larger !

Fast head tail due to broadband impedance ?


Conclusions and OutlookSimulation of collective effects

Longitudinal studies with the LOBO code:Benchmarked, versatile code including most of the effects relevant for the FAIR rings.Detailed studies of the ‘Loss of Landau damping’ thresholds for different rf wave forms.RF phase modulation experiments with space charge and e-cooling started

Transverse and 3D simulation studies with PATRIC:3D (‘sliced’) space charge and impedance kicks have been added recently. Estimations of coasting beam instability thresholds (resistive wall and kickers): next talk.Simulation studies for bunched beams (headtail-type modes) have just been started.

To do:

o PATRIC benchmarking with dispersion relations, HEADTAIL and CERN data.o Soon we have to come up with conclusions related to bunch stability and feedback (EU study).o Implement feedback schemes in LOBO and PATRIC.o .........

oliver boine-frankenheim, high current beam physics group simulation of space charge and impedance...

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