Download Limiting instabilities in multibunch : review and cures

A. Mosnier, Review & Cures of CBI
Limiting Instabilities in Multibunch :
Review and Cures
Alban Mosnier, CEA/DAPNIA - Saclay
Since very high beam currents are distributed among many tightly spaced bunches
unstable coupling between bunches through long-range wakefields has become
the main limiting instability
Conventional Coupled-bunch mainly driven by :
• long-range parasitic modes of rf cavities
• resistive wall (transverse)
New recently discovered collective effects :
• fast ion instability (for e- rings)
• photo-electron instability (for e+ rings)
A. Mosnier, Review & Cures of CBI
Energy & position oscillations spoil :
 Luminosity in colliders (wrong time/position collisions)
 Brilliance in SLS (undulators strongly sensitive to
increase in effective beam energy spread or emittance)
2
Brightness (ph/s/mm /0.1%bw)
Ex. effect of a coupled-bunch longitudinal instability on the brightness
of a typical undulator in the SOLEIL Light Source
7 10
14
6 10
14
5 10
14
4 10
14
3 10
14
2 10
14
1 10
14
0
5250
Undulator U34 (n=7)
w/o oscillation
1. E-03
2. E-03
3. E-03
 (eV)
5300
5350
5400
5450
A. Mosnier, Review & Cures of CBI
General theory for multi-bunch instabilities exists for more than 20 years
(Sacherer '73, Pellegrini & Sands '77, …)
Rigid bunch approximation (Coherent motion of bunch as a whole)
 stability of the system = eigenvalue problem


Single-particle equation of longitudinal motion :

2
Ý
Ýk (t)  2 s 
Ýk (t)   sk  k (t) 

for M equally spaced and equally populated rigid bunches,T0 E
ˆ e j( t k  )
 k (t)  
Signals add up coherently (synchrotron sidebands) with
  2 n M
total induced voltage = sum of the currents of the M individual bunches
coherent oscillation of the k-th bunch described by


e
Vk (t)
Vk (t)  j M Ib

 p Z( p )  e j k   (t)
Impedance sampled at
p frequencies
 p  ( pM  n) 0   s
A. Mosnier, Review & Cures of CBI
For evenly filled rings  analytical expression
well-know coherent frequency shift j  
and growth rate 1   e  j 
 I0
Zeff (n  0   s )
4 E e  s
Zeff = aliasing of Z//() into the band from 0 to M0
Zeff ( )   (p M  0   ) Z/ / ( p M  0   )
p
Transverse coupled-bunch instabilities (very similar)
 0 I0 e
j  
Z eff (n  0    )
4 E e
Z eff ( )   Z ( p M  0   )
p
For unevenly filled rings  eigenvalues of a MM coupling matrix
(K. Thompson & R. Ruth '89, S. Prabhakar '00)
Prabhakar : more convenient to expand the uneven-fill modes into the set of
the M basis vectors formed by the even-fill modes
proposed modulation coupling of strong even-full modes to alleviate CBI
A. Mosnier, Review & Cures of CBI
CBI growth rate strongly dependent on fill pattern
(observed at various storage rings, ex. APS '97)
Main idea : • for each unstable mode n corresponds an highly stabilised counterpart m = M-n
• create then a coupling of unstable modes to stable modes through uneven fills
• find the best current distribution among the RF buckets
which minimises the largest instability growth rate
107
108
Fres = 850.26 MHz
Fres = 851.15 MHz
10
6
3
Q = 2.6 10
10
4
10
3
10
2
10
1
Z eff (f)
Z eff (f)
105
100
10
7
10
6
10
5
10
4
10
3
10
2
10
1
3
Q = 2.6 10
100
0
50
100
150
200
N x f (MHz)
250
300
350
0
50
100
150
200
250
300
350
N x f (MHz)
Simplest case : 1 HOM and its effective impedance with uniform filling M = h buckets = 396
• couple the unstable mode (n=165) to the stable mode (m=231) by uneven filling (same I0)
ex. only every Nth bucket is filled so that (m-n) ≈ M / N ( max. coupling ) N=6
but demands that HOM frequencies be well controlled ex. freq shift excite next mode n=164
A. Mosnier, Review & Cures of CBI
Usual Cures against Coupled-Bunch Instabilities
attempts to
• Landau damping
 destroy the coherence of the beam
• HOM frequency control  avoid the overlap of HOMs
with beam spectrum
• Heavy mode damping
 reduce the resonant buildup of fields
(grapples directly with the source)
• Active feedback
 apply a correction signal
from a sensed error signal
A. Mosnier, Review & Cures of CBI
K E K-B P E P-II CESR-I I I D ANE E S RF
ELET TRA
ALS
S OLEI L
E (GeV)
3.5
3.1
5.3
0.510
6
2
1.9
2.5
c (10-4)
1.7
12
114
180
1.8
16
16
4.8
s (10-2)
1.5
2.5
5.2
1.
0.55
0.987
0.8
0.67
s (m s)
23
29.3
20
17.8
3.6
7.94
6.6
4.33
SLCB I
3.7
22.7
41.4
3141
1
32.2
34.7
6.2
Id esi g (A)
2.6
2.14
0.5
5.2
0.2
0.2
0.4
0.5
1658
45
120
992
432
328
396
 (S C)


 (S C)


Nbun c h e s ~5000
de-Qing


Feedback



M-shifting
b-b Fsp r ad
e
¸


¸
SLCBI 
 s
2 E / eQs
A. Mosnier, Review & Cures of CBI
Landau Damping
successfully used for the operation at ESRF
When oscillators (either particles in a bunch or different bunches in the train)
have a finite spectrum of natural frequency
 net beam response to the driving force due to WFs 
beam stable again if frequency spread large enough.
 Dispersion Relation
Coherent frequency shift
w/o Landau & radiation damping
( ) d
 0  2  s  2
2 1
  2 j s   
0  
 I0
j  p Z( p )

4 E / e Qs
p
A. Mosnier, Review & Cures of CBI
 rf voltage modulation
easily provided by beam loading in the rf cavity with partial filling
frequency distribution ≈ rectangular spectrum

for phase modulation
 

2V
total spread

1 V

 tan   tan s
 2 V
R Q I0 Tgap
At ESRF : instability threshold increased
from ≈ 60 mA  beyond nominal intensity of 200 mA with a 1/3 filling
1
15
100 mA
SOLEIL :
100 mA
10
0,5
0
V cav (kV)
100 mA
5
 cav (deg)
2/3 filling
0
-5
-0,5
-10
bunch index
-1
bunch index
-15
0
100
200
300
400
500
0
100
200
300
400
500
A. Mosnier, Review & Cures of CBI
Stability diagram for the SOLEIL ring
assuming 352 MHz LEP Cu cavities 1st HOM at ≈ 500 MHz (R/Q=75, Q=3.104)
 radiation damping only + HOM with 16 mA
 rectangular spectrum (spread = 6.3 %) + HOM with 100 mA.
But frequency spread of only 0.3 % for 2/3 filling and 100 mA
 method impractical for the SOLEIL ring
plot in complex plane : - locus of the inverse of the integral as  is swept from - to +
- frequency shift w/o Landau and radiation dampings
(HOM frequency, not exactly known, also scanned  0 looks like resoannce curve of the HOM
A. Mosnier, Review & Cures of CBI
Bunch-to-bunch frequency splitting
can also be achieved by driving the normal RF cavities at a frequency (h±1) f0
 used at CERN to suppress longitudinal instability in PS ('71)
 tested at ESRF by driving 2 of the 4 installed cavities
at one revolution harmonic above the rf frequency
n=1 instability prevents cavities from being tuned close to h+1 rev. Harmonic
 tradeoff between modulation level & reflected power  170 mA max
A. Mosnier, Review & Cures of CBI
 Landau Cavity
non-linearities in focusing force  some spread in synchrotron frequency
Max. Freq. spread in bunchlengthening mode: slope total voltage ≈ 0 at bunch loc
Quartic bucket potential
2

1
  n 4
2
 ( )  K  e
maximum generally much lower
than natural synchrotron frequency
Ex. SOLEIL freq. Spread of 200%,
But center-freq. dramatically decreased
 net result = poor improvement
 radiation damping only + HOM with 16 mA
 spread from 3rd harm. cav. + HOM with 18 mA.
 s
A. Mosnier, Review & Cures of CBI
 Betatron spread (transverse plane)
significant spread easily obtained :
 non-linearities in the focusing system
 with non-zero chromaticity, together with energy spread
  multi-bunch instability after // instability
on most existing rings (crude threshold calculation gives the inverse)
With Gaussian distribution in energy
stability recovered for rms betatron freq. spread
  

2
m( 0 )
   Q 0  E
Ex. SOLEIL with LEP Cu cavities
1st deflecting HOM
fr=614 MHZ
R/Q=360 /m
Q=6.104
E
current threshold ≈ 6 mA
 240 mA with  = 0.1
E /E <10-3
A. Mosnier, Review & Cures of CBI
HOM Frequency Control
CB modes spaced one revolution frequency apart
 some latitude to escape HOMs from beam spectrum lines
small rings & HOMs not damped
developed and routinely used at ELETTRA :
HOM tuning by precise cavity temperature control
Procedure :
 find temperature settings which give largest stability windows for all cavities
 refine by direct measurement of CBM spectrum on the machine
Frequency of cavity mode k
 k
 k
 k (T,  f )   k (T0 ) 
(T  T0 ) 
( f   f 0 )
T
 f
Temperature
Fundamental tuning = F(beam current)
A. Mosnier, Review & Cures of CBI
But difficulty to find temperature intervals
stable for both longitudinal and transverse planes
 movable plungers designed at ELETTRA
for allowing additional degree of freedom
W/o plunger
after plunger adjustment
long.
trans.
10
5 MV rf voltage and 400 kW rf power
No stability intervals for 25%
( over 100 different seeds )
cavity # 6
-1
Growth rate (s )
6 ELETTRA-type cavities in SOLEIL
4
10
3
T (°C)
40
45
50
55
60
65
70
A. Mosnier, Review & Cures of CBI
Heavy Mode Damping
cavity modes damped as much as possible to lower the resonant buildup of fields
2 technologies SC & NC developed to meet
high power & low impedance challenges
SC advantages :
fewer cells  lower overall impedance for given voltage
due to the high CW gradient capability
higher achievable deQing
large beam holes allowed, while keeping very high Rs
HOMs propagate out & easily damped
 Mode Damping used alone for SC cavities
used with feedback system for NC cavities
SC drawbacks :
larger complexity (cryogenics)
precautions against risk of cavity & coupler pollution
A. Mosnier, Review & Cures of CBI
 Normalconducting cavities
Dampers mounted directly on cavity walls at proper locations (max. coupling)
HOM power carried out & dissipated on external rf loads
Waveguide couplers : cut-off frequency ≥ fundamental mode frequency
 natural FM rejection & higher deQing than coaxial couplers
3 ridged waveguides generally placed symetrically around the cell
 additional power dissipation, due to field penetration into the waveguide
Ex. DANE cavity
includes 2 additional WGs
A. Mosnier, Review & Cures of CBI
 Superconducting cavities
Dampers cannot be directly mounted on the cavity walls
(risk of multipactor, magnetic quench and surface contamination)
But; beam tubes made large enough for efficient coupling to the cavity modes
2 approaches :
 Dampers = beam pipes themselves (CESR, KEK-B)
rf lossy material (ferrite) to the inner surface of both pipes, outside the cyostat
 More classical HOM dampers mounted on beam pipes
in the vicinity of the cavity (LHC, SOLEIL)

needs large openings to ensure the propagation of all modes
with high HOM powers  outgassing rate of ferrite (surface contamination)

more challenges on HOM couplers (power & de-Qing)
optimized in combination with string of cavities
A. Mosnier, Review & Cures of CBI
cryostat of KEK-B SC cavity
Wide beam pipe & closer iris ( modes)
coaxial high power input coupler
ferrite HOM loads
cryostat of CESR SC cavity
fluted beam pipe ( modes)
WG high power input coupler
ferrite HOM loads
A. Mosnier, Review & Cures of CBI
Ex. Cavity-pair arrangement for SOLEIL
accelerating mode
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longitudinal HOM
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Features : weak coupling for the accelerating mode & strong coupling for HOMs
A. Mosnier, Review & Cures of CBI
Coupler optimization with RF codes
A. Mosnier, Review & Cures of CBI
Results of calculation
(2 couplers / cavity)
Highest impedance
(at optimal coupler location)
versus inner tube length and
for different tube radii
Conclusion :
diameter of 400 mm and
cavity spacing ≈ 3l/2
seem optimal
Fundamental mode :
R/Q = 45  / cavity
Epeak/Eacc = 2
Hpeak / Eacc = 4.2 mT/(MV/m)
A. Mosnier, Review & Cures of CBI
schematic drawing of the SOLEIL cryostat
developed within the framework of a collaboration with CERN
Tuning system
(180 kHz/mm
resolution ≈ 50 nm)
Cryo transfer lines
phase separator
Power coupler
(200 kW)
Conduction break
4°K  300°K
Vacuum tank
352 MHz
Nb/Cu cavity
HOM couplers
He tank
A. Mosnier, Review & Cures of CBI
Assembly & Power tests at CERN
Eacc > 7 MV/m Qo > 109
main coupler Pinc = 160 kW w/o beam
static losses = 20 W @ 4°K
A. Mosnier, Review & Cures of CBI
Feedback Systems
Developed for more than 20 years
 first in frequency domain, on a mode-by-mode basis
(Ex. CERN PS booster)
 more recently in time domain, on a bunch-by-bunch basis
thanks to the advent of commercially available fast DSPs
complementary to passive mode damping
can damp definitely all coupled bunch modes
impedances arising from strong HOMs first sufficiently reduced
correction kick voltage needed :
Ex.
V  I 0 

p
p
e Z( p )
 rf


1st HOM of 2 LEP Cu cavities in SOLEIL ring
Full coupling  84 kV / turn (assuming mode amplitude 1.5°)
required power
> 5 MW !!!
2
P  V 2 Rs
A. Mosnier, Review & Cures of CBI
Model
2
Ý
Ýk (t)  2 s 
Ýk (t)   sk

 k (t)
Driving term = correction kick


T0 E e
Vk (t)  G  rf  k (t  t)
FB loop gain (V/rad)
Complex frequency shift
Vk (t)
Delay time
    s

 G  rf
  G  rf
  i  s 
sin(t)
cos(t)
4 Qs E e

 4 Qs E e
/2
Max. damping : phase shift   t
for G > 0
3 / 2 for G < 0
A. Mosnier, Review & Cures of CBI
 mode-by-mode feedback
for only a few troublesome coupled-bunch modes
 bunch-by-bunch feedback
for a large number of bunches
bunches treated as individual oscillators
minimum bandwidth = half the bunch frequency
PEP-II, ALS, DANE, etc… :
common longitudinal feedback system design
based on fast ADC/DAC converters & DSP chips for digital filtering
 digitizing of the baseband error signal
 N-taps FIR : max. gain at fs + zero dc response
 Downsampling (low fs)
 Efficient diagnostics tool :
measurements of growth & damping rates
by means of time domain transient techniques
A. Mosnier, Review & Cures of CBI
Resistive Wall Instability
About the required BW of a transverse feedback
Resistive wall impedance
  1/ 2
 only modes with spectrum lines close to the origin, will be excited
 feedback system
generally sufficient
with limited bandwidth (few revolution harmonics)
averaged measurements over several bunches
for high current rings, with large number of bunches
 many coupled-bunch modes are unstable at zero chromaticity
 > 0 : m=0 mode stable
But  not too large :
 transverse dynamic acceptance spoiling
 emergence of higher order head-tail modes
A. Mosnier, Review & Cures of CBI
growth rates of head-tail modes (+ higher order radial modes)
easily evaluated by solving the Sacherer’s integral
Ex. SOLEIL RING
growth time of most unstable modes vs. chromaticity
number of unstable modes for the first 3 head-tail modes
140
10
SOLEIL ring
Nb of unstable modes
120
m=2
100
SOLEIL ring
growth time (ms)
80
m=1
m=0
1
60
m=1
40
m=0
20
m=2
0,1
0
0
0,1
0,2
0,3
0,4
0,5
0
0,1
0,2
0,3
0,4
0,5
Conclusion : transverse feedback of, typically, a few tens of MHz bandwidth
with a proper chromaticity setting
(not too large to avoid head-tail modes, but large enough
to reduce the number of unstable rigid bunch modes m=0 )
A. Mosnier, Review & Cures of CBI
fast ion instability (for e- rings)
Analog as single-pass BBU in Linacs, except
coupling between bunches due to ions intead of wakefields
n M t
y

e
Linear theory : displacement
n
1 c   i L2

  gas ionization rate per unit length

2 
But with ion frequency spread around ring : exp. growth and
NÝi   i pgas
  i L
Not very severe for usual gas pressure
easily cured by fast feedback or Landau damping (induced by octupoles / choma)
photo-electron instability (for e+ rings)
CBI instability caused by photo-electrons created by SR at pipe wall (Ohmi)
Coupling between bunches due to primary e- (interaction with several bunches
before hitting the opposite wall) or due to electron cloud buildup in steady-sate
Cures
e- cloud dominated : TiN coating (secondary e- yield  reduction ex.PEP-II)
primary photo-e- : magnetic field to maintain e- far from beam (KEK-B)