Impedance, Damper, Radial Modes, Coupled Bunches, Beam- Beam and more - - PowerPoint PPT Presentation
Nested Head Tail Vlasov Solver: Impedance, Damper, Radial Modes, Coupled Bunches, Beam- Beam and more A. A. Burov ov Fermilab milab special thanks - to V.Danilov, E.Metral, N.Mounet, S.White, X.Buffat, and T.Pieloni JAI, Oxford, 10/10/13
A.
Fermilab milab
JAI, Oxford, 10/10/13
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special thanks - to V.Danilov, E.Metral, N.Mounet, S.White, X.Buffat, and T.Pieloni
Intensity ensity Effects ects
each other: – By means of direct Coulomb fields (space charge). By itself, this cannot drive an instability. – By means of image charges and currents. By itself, this cannot drive an instability. – By means of fields left behind (wake fields due to radiation). This may drive collective instability.
mutual Coulomb scattering (intra-beam scattering). This is possible due to huge number of particles inside the beam, like ~1E11 /bunch.
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Liuville ville/Collisionless /Collisionless Boltzmann/ tzmann/Jeans/Vlasov Jeans/Vlasov Equation ation
medium in the phase space:
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[ , ] ; ; [ , ] [ , ] [ , ] f H f t f f f H H H H f f H f H f t , H f
, H f
( ) (z z ) ( ) H W f d
As a result, we have a linear integro-differential equation to solve. See details in e.g. A.Chao, “Physics of Collective Beam Instabilities”
; f f f t H H f H f H f t r p r p r p p r p r r p
Nested ted Head-Tail Tail Basis is
4 z
p z
exp( cos ) ; ;
l b
il i i t Q r c
I am using equally populated rings which radii are chosen to reflect the phase space density.
r
n r
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My basis functions for transverse oscillations
Starting rting Equation, ation, single gle bunch ch
can be presented as in Ref [A. Chao, Eq. 6.183]: where is a vector of the HT mode amplitudes, is the damper gain in units of the damping rate, time is in units of the angular synchrotron frequency.
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ˆ ˆ ˆ X S X Z X D X X
ˆ ˆ ( ) ( ) ( ) ( ) ˆ ( ) ( )
l m lm lm l m r m l lm l m r
S Z il i d Z J J n d D i J J n
2
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b b s
N r R Q Q d
AB
0.4 0.2 0.2 0.4 Re Q Qs 0.01 0.02 0.03 Im Q Qs
Analysis lysis of solutions utions
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impedance table (N. Mounet).
mode, then max is taken.
l l
l
Stability diagram at +200 A of octupoles Impedances
108 109 1010 1011 f, Hz 1 104 2 104 5 104 1 105 2 105 5 105 Z, Ohm cm
Gaussian, transverse only
AB
Im /
s
dQ Q Re /
s
dQ Q
Coupled pled Equidistant idistant Bunches ches
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Main idea: For LHC, wake field of preceding bunches can be taken as flat within the bunch length. The only difference between the bunches is CB mode phase advance,
Thus, the CB kick felt by any bunch is proportional to its own offset, so the CB matrix has the same structure as the damper matrix :
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ˆ ˆ ˆ ; ˆ ˆ ˆ ( ) ( ); 2 ( ) / ; 2 ( { }) ( ) ( )exp( ) 1. ˆ ; ;
m l lm l m r x b k b
X S X Z X D X d D i J J C i W D d n Q W W ks ik M C M X
ˆ D
ˆ C Wake and impedance are determined according to A. Chao book.
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Old damper per gain
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Old narrow-band ADT gain profile (W. Hofle, D. Valuch) . At 10 MHz it drops 10 times. The new damper is bbb for 50ns beam. Below gain is measured in omega_s units, max gain=1.4 is equivalent to 50 turns of the damping time. ( ) g
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CB Mode e Damping ping Rate e
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With as the frequency response function of the previous plot, the time- domain damper’s “wake” is assuming this response to be even function of time (no causality for the damper!). From here (equidistant bunches!): where is the rate provided for low-frequency CB zero-head-tail modes at zero chromaticity. ( ) g
( ) ( )cos / , G g d
1 1
(0) 2 ( )cos( ) ; (0) 2 ( )
k k
G G k k d d G G k
d
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CB Wake e and Gain n Factors tors for the Old ADT
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2(S (SB B and CB), , flat t ADT, , Tunes es at the Plateau teau
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plateau.
tune shift as well.
tunes tune shifts all tunes AB
NHT vs BeamBeam3D mBeam3D tracking cking (S. White) te)
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Highest growth rates for single bunch, gain=1.4 and nominal impedance
2(S (SB B and CB), , flat t ADT, , MUM
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Growth rate and -tune shift of the most unstable mode (MUM) vs chroma and gain. Both are in units of Qs. Note that at the plateau the rate (Im[dQ_c]) is ~20-30 times smaller than the shift (Re[dQ_c]).
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Q’ Q’ d
d
2(S (SB B and CB), , flat t ADT, , MUM CM and Coupling pling
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Center of mass (CM) and head-tail coupling parameters for MUM. Note strong suppression of CM at the plateau by the damper. Note that at plateau the weak head-tail approximation is well-justified.
( ) / ; .
l l l r
A i J n x X A
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Q’ Q’
d d
2 2 2 1 2 2 2
| | | | ; | | 1; :| | max | | ; HTC 1 | |
r m
n l l l l m l l l lm
X X X l X X X
2 4 6 8 10 ImpFactor 0.00 0.02 0.04 0.06 0.08 Rate ImpFactor
Q' 0, 5, 15, 20
Intensity ensity scan, n, flat t ADT, , MUM: : where re is TMCI? I?
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Gain=1.4
Q’
Gain=0
2 4 6 8 10 ImpFactor 0.00 0.01 0.02 0.03 0.04 Rate ImpFactor
Q' 0, 5, 15, 20
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Q’
Rate/ImpFactor Rate/ImpFactor
Coherent erent Beam-Beam Beam
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Main assumption: bunch length << beta-function. For transversely dipolar modes, CBB is a cross-talk of bunch CM – thus, intra-bunch matrix structure is similar to the ADT and CB: Here 2 identical opposite IRs are assumed (IR1 and IR5 for LHC) with 2K+1 LR collisions for each, every one with its beta-function and separation . Alternating x/y collision for IR1/IR5 is assumed with as a difference between the two phase advances, while is the incoherent beam-beam tune shift per IR.
12 2 1 1 1 1 1 2 2 2 2 2 bb 2 2 * 1 2 1 2 21 1
ˆ ˆ ˆ ˆ ; ˆ ˆ ˆ ˆ ; ˆ ˆ ( / ) cos( ) ; 1 ). ˆ ˆ exp(
/
K K k k k K k K k k
b B X b B X X S X Z X D X C X X S X Z X D X C X B i D d k b b i
,
k k
bb
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Cohere erent nt BB at Platea teau: u: effect ect ~30% 0%
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3 bb
2.5 10 , / 2, 7 Q K
1 beam 1 beam
Dispersion persion Equation ation
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Let’s consider a small fraction of the beam described by an NHT amplitude vector : Due to the frequency spread eigenvalues are slightly changed, but eigenvector at the first approximation are the same (similar to QM). From here
ˆ ˆ
p p p p p
x i x S x Z X
p
x
c
1
ˆ ˆ ˆ ˆ ( )
p p p c
x I iS I iS X
1
ˆ ˆ ˆ ˆ ( )
p p c
X I iS I iS X
1 †
ˆ ˆ ˆ ˆ 1 ( )
p p c
X I iS I iS X
2
| ( ) | 1 ( )
l s x c s l s x x
X J J F l d l i J
2
| ( ) | 1
l s
X J Fd Fd
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ˆ ˆ ˆ ( )
c
Z X i I iS X
Weak k Head-Tail Tail case e
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This derivation assumes frequency spread can be treated as a perturbation. This is justified when the resonant particles are at the tails of the distribution. With the damper, weak HT approximation can be applied at many cases. If so (true for LHC), the DE is simplified: When the LD is provided by the far tails, the mode form-factor can be
2 2
| ( ) | 1 ( ) ; | ( ) | 1
l s x c s l s x x l s
X J J F l d l i J X J Fd Fd
2
| ( ) | 1 ( ) .
l s x c s s x x
X J J F l d l i J
2
| |
l
X
/ 1 ( ) .
x x c s s x
J F J l d l i
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Stability bility Diagram gram
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Stability diagram (SD) is defined as a map of real axes on the complex plane: To be stable, the coherent tune shift has to be inside the SD.
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/ D
x x s x
J F J d l i
D
c s
l
AB
0.4 0.2 0.2 0.4Re Q Qs 0.01 0.02 0.03 0.04 Im Q Qs
Stability Diagram: LO 200A, Gauss
T T LO 2 2
, ; ( , ) ; ˆ ˆ / ; ; 100A 1.8 10 / (2μm); 1.3 10 / (2μm);
x y x y xx xy s yx yy xx yy xy yx
Q Q J J a a I Q a a a a a a
Q J Q A J A
For LHC, with Landau octupoles (LO):
(E.Metral, N.Mounet, B.Salvant, 2010)
Im /
s
dQ Q Re /
s
dQ Q
0.10 0.05 0.00 0.05 0.10 0.15 0.20 Re 0.005 0.010 0.015 0.020 Im
2D SD, MO 100A, H
Stability bility Diagrams grams
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( , ) 1 ; ( , ) 1; ( , ) 1.
n x y n x y n n x y x y x n x y x y n
J J F J J a F J J dJ dJ J F J J dJ dJ b
see more on SD with at E. Metral & A. Verdier, 2004
n
F
Gauss Gauss 3 F2
LHC stability diagrams for both emittances 2m and 100A of the octupole current.
Im 0.04Re Im /
s
dQ Q Re /
s
dQ Q
Couple ple Bunch ch Factor tor: : LO+, , bbb ADT, , 2Imp p
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5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
Almost no difference at Plateau. All the plots – for 1.5E11 p/b, emittances of 2m, 50ns beam. Plateau Alps
5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
single bunch
Q’ Q’
d d
Tails ls Factor tor: : LO+, , CB, bbb ADT, , 2Imp p
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5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
F2
Almost no difference at this polarity. Q’ Q’
d d
Tails ls Factor: tor: LO-, , bbb ADT, , 2Imp p
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5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
F2
5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
Gauss
About a factor of 3 difference at the plateau!
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Q’ Q’
d d
Beam Beam-Bea Beam m Factor tor: : 2Imp, p, CB, CBB =/2 /2, , LO+, +, bbb ADT
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5 5 10 15 20 Q' 100 200 300 400 Joct
gain 0.2, 0.7 and 1.4
CBB effect ~ 30% at the Plateau.
5 5 10 15 20Q' 100 200 300 400 Joct
gain 0.2, 0.7, 1.4
3 IR1
2.5 10 Q
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Q’ Q’
d d
Impeda edance nce Factor tor: : CB, CBB =/2 /2, , LO+, +, bbb ADT
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5 5 10 15 20 Q' 100 200 300 400 Joct
gain 0.2, 0.7 and 1.4
2·Imp 1·Imp
5 5 10 15 20 Q' 50 100 150 200 Joct
gain 0.2, 0.7 and 1.4
At the Plateau it scales ~ linearly
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Q’ Q’
d d
Long Long-Range Range Beam-Beam Beam Tune e Spread ead
spread is Here is the linear LR bb tune shift per IR, is beam separation in units of their rms size at that point. Round betas are assumed.
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bb 4 2
2 3 .
x y x
J J Q Q r
bb
Q 1 r
3 bb
2.5 10 Q
9.5 r
Stability bility Diagrams grams with h Long g Range ge Beam-Beam Beam
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LO=140A – computed threshold BB only, LO=0 LO=500A, no BB BB and LO=500A LO=1000A, no BB
At the end of the squeeze and LO+, BB is equivalent to +500A of LO. For the black curve, where we are now, we must be very stable, being 7 times in effective LO above the threshold. However, we are unstable! A big beast is still overlooked…
E-cloud in the IRs? Big drift of Q’? …? Any idea can be checked with NHT.
Im /
s
dQ Q
Re /
s
dQ Q
Summary: mary: power er of the model el
solving the eigenproblem at its 4D set: azimuthal radial coupled-bunch beam-beam.
distribution functions and nonlinearities, beam-beam scheme.
computed.
representative CB modes it takes only 1s on my 3 years old laptop.
tracking; it is unsolvable for a thousand bunches.
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Next t steps ps
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