Coherent beam-beam effects X. Buffat Content Coherent vs. - - PDF document
Intensity limitations in Particle Beams Coherent beam-beam effects X. Buffat Content Coherent vs. incoherent Self-consistent solutions Coherent modes of oscillation Decoherence Impedance driven instabilities Summary
Coherent vs. incoherent
Self-consistent solutions
Coherent modes of oscillation
Decoherence Impedance driven instabilities
Summary
The other beam is not
The electromagnetic
Δ x ' (x)=−2 r0 N γr 1 x (1−e
−x
2
2σ
2)≈4 π ξ x
Optics Beam parameters
Disturbed optics Disturbed beam
Disturbed optics Disturbed beam
Disturbed optics Disturbed beam
Weak-strong : δ x'
Δ x' (x)=−2 r0 N γr 1 x (1−e
−x
2
2σ
2)≈4 π ξ x
The average force felt by
−Δ x
2
4σ
2 )≈ 4 π ξ
Δ x'coh(Δ x)=∫
−∞ ∞
Δ x'(Δ x−X)ρ(X)dX
Strong-strong :
Similar treatment applies to the optical functions
These effects were already covered in T. Pieloni's lectures, but :
Iterative methods are used to evaluate these effects (3) Prohibits several single beam measurement techniques
The solution of the non-linear equations is not always unique Weak-strong : δ x' δ x'
Displacement of the luminous region
Different bunches experience different beam-beam
Also observed in LEP with bunch trains
Low ξ : The two beams have identical transverse sizes High ξ : Two equivalent equilibrium configurations : Electron beam is blown up Positron beam is blown up
x1 x1')t+1 =( cos(2πQ) sin(2πQ) −sin(2πQ) cos(2πQ))( x1 x1')t
t+1
t
t+1
t
(Small amplitude approximation)
−Δ x
2
4σ
2 )≈k(xB1−xB2)
t+1
t
(Small amplitude approximation)
−Δ x
2
4σ
2 )≈k(xB1−xB2)
x1= x2 at every
x1= -x2 at every
(*) ξ << 1 and for tunes away from resonances
The rigid dipole mode
xi xi')t+1 =Mlattice⋅M BB( xi xi')t
Higher order
(5)
Δ x' coh=−2r 0 N γr 1 Δ x (1−e
−Δ x
2
4σ
2 )
Δ x 'coh= 4 πξ 2 Δ x
Rigid bunch model :
Each beam centroid position and momentum x1,x'1 and x2,x'2
Non Linear beam-beam map :
Linearized kick :
xi xi')t+1 =Mlattice⋅M BB( xi xi')t
Write one turn matrix and find eigenvalues / eigenvectors
Equation of motion : Vlasov perturbation theory :
Each beam phase space distribution
F
(1), F (2)
Liouville's thorem :
∂F
(1)
∂t +[F
(1), H (F (2))]=0
∂F
(2)
∂t +[F
(2), H(F (1))]=0
(1), Ψ (1), F (2),t)
Hamiltonian (lattice + beam-beam)
First order perturbation
(i)=F0+F1 (i)
Formulate the linearized system as a linear operator → find eigenvalues / eigenfunctions
The Yokoya factor Y is usually between 1.0 and 1.3
Rigid bunch :
The Yokoya factor Y is usually between 1.0 and 1.3
Self-consistent Model :
The non-linearity of beam-beam
The single particles generate a
The non-linearity of beam-beam
The single particles generate a
The non-linearity of beam-beam
The single particles generate a
Both the σ and π
The non-linearity of beam-beam
The single particles generate a
Both the σ and π
TRISTAN PETRA LEP RHIC LHC
SPPS ? Tevatron ?
(6)
xi xi')t+1 =Mlattice⋅M BB( xi xi')t
Non-linear beam-beam map
Gaussian fit : soft-Gaussian
approximation
−xi
2
2σ
2)
Model the beam distribution with a discrete set of macro-
Track the particles, solving for each beam's fields at each
The soft-Gaussian approximation underestimate
Soft-Gaussian approximation Self-consistent field solver
Multiparticle tracking simulation, with a single
Start the simulation with a beam offset with
2
2
Similar setup but :
Two independent beams Non-linear beam-beam map based on the charge
Start the simulation with both beams offset in the same
Let the mode decohere ?
The single particle motion is the linear composition of the
The incoherent and coherent motion are decoupled
Identical setup :
Two independent beams Non-linear beam-beam map based on the charge
Start the simulation with both beams offset in opposite
Let the mode decohere ?
Due to the particles frequency spread, the beam distribution is
The bunch centroids remain out of phase
This is a consequence of the decoupling of the incoherent
Again, the single particle motion is 'regular' with
A slight emittance growth still exists due to the
Since there is no decoherence, any external source of
2(1−s0)
2
1 ϵ0 d ϵ dt =⟨ Δ
2
2 4 π
2(1− g
2)
2
ΔQ
2
4 π
2(1− g
2)
2
ΔQ
2+(
g 2)
2⟩
When ξ << g : The strong-strong mechanism slowed
When g >> ξ: Both formalisms lead to similar results
(7)
Assumptions :
First order perturbation in ξ
Absence of other sources of amplitude detuning
Symmetric optics and beam parameters
Real configurations are not that simple !
4 head-on collisions ~120 long-range
2808 bunches
Rigid bunch model : Find the eigenvalues of the
The σ/π modes become a forest of modes with
The coherent modes
t+1
t
Most symmetry breaking between the beams
In realistic configurations, several parameters
Intensities, emittances, β*, tunes (phase advances
(8)
Treating consistently the motion of the two beams
Simple configuration : Two discrete coherent modes of
Complex configurations : Multiple coherent modes
What happens in the presence of beam coupling
84
t+1
t
s Slice 1 Slice 2
(9)
85
t+1
t
s Slice 1 Slice 2
(9)
86
t+1
t
s Slice 1 Slice 2
After half a synchrotron period, particles one and
(9)
87
t+1
t
s Slice 1 Slice 2
After half a synchrotron period, particles one and
(9)
88
t+1
t
s Slice 1 Slice 2
The effect over one turn is described with a fraction of a
After half a synchrotron period, particles one and
(9)
2Qs
89
xB1 s1 xB1 s1' xB1 s2 xB1 s2' xB2 s1 xB2 s1' xB2 s2 xB2 s2')
t+1
=
1 k 1 − k 2 − k 2 1 k 1 − k 2 − k 2 1 − k 2 − k 2 k 1 1 − k 2 − k 2 k 1) ⋅M lattice ,SB
xB1 s1 xB1 s1' xB1 s2 xB1s 2' xB2 s1 xB2 s1' xB2 s2 xB2 s2')
t
90
Synchrotron motion is slow with respect to betatron motion
s
Δ xB1s2'=W dip(ss2−ss1)x B1s1 ss1 ss2
t+1
t
91
Synchrotron motion is slow with respect to betatron motion
t+1
t
s
Δ xB1s2'=W dip(ss2−ss1)x B1s1 ss1 ss2 +W quad(ss2−ss1)xB1 s1
92
xB1 s1 xB1 s1' xB1 s2 xB1 s2' xB2 s1 xB2 s1' xB2 s2 xB2 s2')
t+1
=
1 1 1 W dip W quad 1 1 1 1 W dip W quad 1) ⋅M BB⋅Mlattice ,SB
xB1 s1 xB1 s1' xB1 s2 xB1 s2' xB2 s1 xB2 s1' xB2 s2 xB2 s2')
t
The stability of the system is given by the normal mode
93
The wake and the beam-beam force feeds back a
94
Identical formalism, with N slices
Derive matrices for different
beam interactions, transverse feedback, …)
The flip matrix becomes a circulant matrix
Analyse the stability of the one
(10)
95
The coupling of coherent beam-beam modes
Fully self-consistent
Rigid bunch model :
96
At non-zero chromaticity, each beam is
The coherent beam-beam forces changes the
97
Mode coupling
Syncro-betatron beam-
98
Single beam stability requires
Usually through amplitude
Lattice non-linearities are
In specific cases, other
99
Single beam stability requires
Usually through amplitude
Lattice non-linearities are
In specific cases, other
100
When colliding beams of similar strength (strong-strong)
Orbit effect Dynamic β effect Coherent beam-beam modes
Several model exists to describe coherent beam-beam
tracking simulations)
Fully self-consistent treatment, allowing for non-Gaussian
The decoherence mechanism is very different in the strong-
101
Complicate the estimation (on paper and experimentally) of
Coherent beam-beam modes may be driven unstable by :
Resonances
The beam coupling impedance
External excitations / noise
Coherent beam-beam modes may break stabilisation
They were observed in several colliders, stabilised through :
Landau damping (asymmetric configurations, lattice non-linearities, chromaticity, ...)
Transverse feedback
(1) E. Keil, Beam-beam dynamics, CERN Accelerator School, Rhodes, Greece, 1993 (2) Dynamic β effect
PAC99, New York, USA (3) Self-consistent methods
CERN SL/95-75 (1995)
workshop 2001, Fermilab (4) Flip-Flop effect M.H.R. Donald, et al, An Investigation of Flip-Flop beam-beam effect in SPEAR, IEEE Trans. Nuc. Sci. NS-26, 3580 (1979) J.F Tennyson, Flip-flop modes in Symmetric and Asymmetric colliding beam storage rings, LBL-28013 (1989) D.B. Shwartz, Recent beam-beam effect at VEPP-2000 and VEPP-4M,
Workshop on beam-beam effects in hadron colliders, Geneva, Switzerland, 2013
(5)Vlasov perturbation theory
(1990)
perturbation theory, Nucl. Instrum. Methods Phys. Res. A 480, 253 (2002) (6)Observations of beam-beam modes
26, 3 (1979)
accumulator ring, Part. Acc. 27, 83 (1990)
the Particle Accelerator Conference 2003, Portland, USA
effects in hadron colliders, Geneva, Switzerland, 2013 (7)Emittance growth V.A. Lebedev, Emittance growth due to noise and methods for its suppression with the feedback system in large hadron colliders, AIP Conf. Proc. 326, 396 (1995)
(8) T. Pieloni, A study of beam-beam effects in hadron colliders with a large number of bunches, EPFL PhD thesis, 2008 (9) A.W, Chao, Physics of collective beam instabilities, John Wiley and Sons Inc, New York, 1993 (10) Circulant matrix model V.V. Danilov, et al, Feedback system for the elimination of transverse mode coupling instability, Nucl. Instum. Methods Phys. Res. A 391, 77 (1997) E.A. Perevedentsev, et al, Simulation of the head-tail instability of colliding bunches, Phys. Rev. ST Accel. Beams 4, 024403 (2001)
(11) W. Herr, et al, Landau damping of coherent beam-beam modes by overlap with synchrotron sidebands, LHC Project Note 304