Transcript Simulation of space charge and impedance effects

Simulation of space charge and impedance effects
Funded through the EU-design study ‘DIRACsecondary beams’
O. Boine-Frankenheim, O. Chorniy, V. Kornilov
Longitudinal beam dynamics simulations (LOBO code)
o Motivation: ‘Loss of Landau damping’ and longitudinal beam stability
o LOBO physics model and numerical scheme
o 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,...)
Oliver Boine-Frankenheim, High Current Beam Physics Group
Longitudinal incoherent + coherent space charge effects
‘Loss of Landau damping’
e.g. Boine-F., Shukla, PRST-AB, 2005
Space charge factor:

1
V 0 /V 0  1
RF
sc
0
synchrotron frequency
(oscillation amplitude ̂ ):


 s (̂, )   s 0 (̂ )  1  G(̂ ) 
2

Elliptic bunch distribution: G(̂ )  1
single rf wave:
 s 0 (̂ )   s 0   ̂ 2
double rf wave:
 s 0 (̂ )   ̂
Coherent (dipole) frequencies (bunch length m):

m2 
1
 1  
 s 0 
10 
1/2
(single rf)
1
s 0
Landau damping rate:  L 
  

5
4
m (double rf)
f
 s
s  
 
 smin m ,   1 m   smax m , 
Landau damping will be lost above some ∑th
if the (coherent) dipole frequency is outside
the band of (incoherent) synchrotron
frequencies.
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
Oliver Boine-Frankenheim, High Current Beam Physics Group
BTF measurement in the SIS
PhD student Oleksandr Chorniy (with help by S.Y. Lee)
Results of the first measurement (Dec. 2005):
rf phase modulation:
VRF (,t) V 0 sin(  ̂ sint)
Xe48+, 11.4 MeV, Nb=108
 Pcool  100 ms
Measure bunch response:  (t ), 
Motivation:
Measure Landau damping with space charge
1
 L (1 )  max
Measure syn. frequency distribution f(s)
Measure coherent modes Ωj
Measure the effective impedance Zeff
Further activities:
Double rf and voltage modulation.
Nonlinear response with space charge.
Supporting simulation studies.
Oliver Boine-Frankenheim, High Current Beam Physics Group
1
s 0

2 
 1  m 

10 
1/2
Longitudinal beam dynamics simulations
Macro-particle scheme
Position kick for the j-th particle:
Momentum kick:
Current profile:
&j 
q
p0
z&j  0 j
V (z ,t) V
rf
j
ind

(z j ,t)  Fc ( j )  (t) 2Dibs t  K
I (z,t )   0c  S(z  z j )Q j
j
Induced voltage:
(slip factor , momentum spread )
e-cooling+IBS+internal targets
(linear and higher order interpolation)
Vind (z,t)  FFT -1 Z P( n )I ( n ) 


LOBO code:
Macro-particle scheme
Alternative ’noise-free’ grid-based scheme
flexible RF objects and impedance library
matched bunch loading with (nonlinear) space charge
e-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
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 scans
single rf wave, phase modulation, long bunch m=±900
Sawtooth field:
VRF (,t)  V 0 (  ̂ sint)
1   s 0
s 
s 0
+ Space charge:
V ind  Z Psc I
No difference between Parabolic or Gaussian
bunches.
-> No damping due nonlinear space charge.
1 
Single rf wave:
VRF (,t) V 0 sin(  ̂ sint)
1
s 0

m2 
 1  

10 
1/2
+ space charge:
V ind  Z Psc I
Loss of Landau damping for ∑th≈0.2.
No significant difference between Gaussian
and Elliptic distribution.
->weak influence of nonlinear space charge.
Oliver Boine-Frankenheim, High Current Beam Physics Group
Gaussian vs. Elliptic Bunch Distribution
Double rf wave, phase modulation, long bunch m=±900


1
ˆ
VRF (,t) V0 sin(   sin t)  sin( 2 )


2
Elliptic bunch distribution
+ space charge: V ind  Z Psc I
Gaussian bunch

th  0.05
1
s 
 max
s 0

5
4
m
th  0.5
1 
->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 !
Oliver Boine-Frankenheim, High Current Beam Physics Group
Bunched beam BTF simulation scans
single rf wave, voltage modulation, long bunch m=±900
Quadrupolar mode in a short bunch:
VRF (,t) V 0 (1  ̂ sint)
s 
2
s 0
 3
1
1 
VRF (,t) V 0 (1  ̂ sint) sin 
s 0
1 
Quadrupole modes and their damping in long bunches with space charge needs more study !
Oliver Boine-Frankenheim, High Current Beam Physics Group
Transverse incoherent + coherent space charge effects
‘loss of Landau damping’
e.g. K.Y. Ng, ‘Transverse Instability in the Recycler’, FNAL, 2004
Incoherent space charge tune shift:
Qyinc  
Z r0 N gt
2
A  2 B f  n, y   n, y n, x
qIR
i
Z coh
y
4 Q 00E 0
Space charge impedance:
Zy  
sc
RZ 0 1
 02 20 b 2
Stability condition
(or ‘Loss of Landau damping’):
Q yinc  Q coh

y
Incoherent tune spread:


Q inc    (n  Q)0  HWHM
Coherent tune shift:
Q coh

y
Damping mechanisms
1
3
F Q inc
Incoherent space charge tune spread:
inc
Q sc
 Q J
Coherent tune spread along the bunch:
Q coh  Q coh
(I )
y
Goal: resolving all these effects in a 3D tracking code
Damping mechanisms and resulting instability thresholds
Study beam behavior close to the thresholds.
Oliver Boine-Frankenheim, High Current Beam Physics Group
PATRIC: ‘Sliced’ tracking model
and self-consistent space charge kicks
sm: position in the lattice
y
z: position in
the bunch
M(sm|sm+1) ∆sm << betatron wave length
s
Sliced bunch
The transfer maps M are
‘sector maps’ taken from MADX.
z
slice-length: ∆z∆s
(N macro-slices for MPI parallelization)
x
2D space charge field for each slice:
r
r
(x, y, z, s m )   Q jS( x  x j ) (3D interpolation)
j
E x
x

Ez  
E y
y

(x, y, {z, s m }) (fast 2D
0
L (z, {s m })
g
4 0 0
2
z
Poisson solver)
(L line density)
x 
 j
 x 
 j
y 
 j
 y j 
 
z j 
 
  j 
s
m1
space charge kick:




xj




qE x (x j , y j , z j , {s m }) 

 M(s m |s m 1 )  x j 
s 
2 3
mv 0






M




s
Oliver Boine-Frankenheim, High Current Beam Physics Group
m
Transverse Impedance Kicks
Implementation in PATRIC
Dipole moment
times current: (t)
localized impedance
 j  (n  Q) 0
Impedance kick:
V.Danilov, J. Holmes, PAC 2001
O. Boine-F., draft available
In the bunch frame (∆s=L for localized impedance):
Coherent
frequencies
Slowly varying dipole amplitude:
Numerical implementation:
Coasting beam:
 (z,t m )   0c  Q jS(z  z j )x j
j
Oliver Boine-Frankenheim, High Current Beam Physics Group
PATRIC benchmarking
Presently ongoing !
Benchmark the 3D sliced space charge solver and the impedance module
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’.
Oliver Boine-Frankenheim, High Current Beam Physics Group
Coasting beam transverse instability
PATRIC example run
SIS 18 bunch parameters
(in the compressed bunch center):
o U73+ 1 GeV/u
o dp/p: m=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
N=106 macro-particles
T=100 turns in SIS 18 (ca. 2 hours CPU time)
Grid size Nx=Ny=Nz=128
Example run on 4 processors (dual core Opterons)
Without space charge the beam is
stabilized by the momentum spread
(in agreement with the analytic
dispersion relation).
Oliver Boine-Frankenheim, High Current Beam Physics Group
Decoherence of a kicked compressed bunch
PATRIC test example
‘Decoherence rate’ due to the
SIS 18 compressed bunch parameters:
coherent tune spread along the bunch:
U73+ 1 GeV/u
1
Q coh  Q coh
(I )   dec
  0Q coh  (0.02ms) 1
Ions in the bunch: 3x1010
y
Duration: 300 turns (0.2 ms)
dp/p: m=5x10-3
Bunch length: m=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.
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):
x
x
y
Oliver Boine-Frankenheim, High Current Beam Physics Group
Head tail instability of a compressed bunch
due to the SIS 18 kicker impedance ?
Z(fr)≈0.6 MΩ
Result of the PATRIC simulation:
SIS 18 kicker
impedance
(one of 10
modules):
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 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 Outlook
Simulation 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