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
Oliver Boine-Frankenheim, High Current Beam Physics Group
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.
Oliver Boine-Frankenheim, High Current Beam Physics Group
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):
Oliver Boine-Frankenheim, High Current Beam Physics Group
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
Oliver Boine-Frankenheim, High Current Beam Physics Group
LOBO example: beam loading effects
Matched ‘sausage’ bunch with space charge and broadband (Q=1) rf cavity beam loading.
Oliver Boine-Frankenheim, High Current Beam Physics Group
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:
Oliver Boine-Frankenheim, High Current Beam Physics Group
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:
Oliver Boine-Frankenheim, High Current Beam Physics Group
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
Oliver Boine-Frankenheim, High Current Beam Physics Group
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’):
Oliver Boine-Frankenheim, High Current Beam Physics Group
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)
Oliver Boine-Frankenheim, High Current Beam Physics Group
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
∑
Oliver Boine-Frankenheim, High Current Beam Physics Group
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
Oliver Boine-Frankenheim, High Current Beam Physics Group
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)
Oliver Boine-Frankenheim, High Current Beam Physics Group
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:
Oliver Boine-Frankenheim, High Current Beam Physics Group
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
Oliver Boine-Frankenheim, High Current Beam Physics Group
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 ?
Oliver Boine-Frankenheim, High Current Beam Physics Group
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 .........
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