Transcript ppt - Cern

Review of IBS: analytic and simulation studies
A. Vivoli*
Thanks to : M. Martini, Y. Papaphilippou and F. Antoniou
* E-mail : [email protected]
CONTENTS
•
•
•
•
•
•
Motivation
Conventional Calculation of IBS
SIRE code
Zenkevich-Bolshakov algorithm
Results of simulations
Conclusions & Outlook
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Introduction: Intra-Beam Scattering in DR
IBS is the effect due to multiple Coulomb scattering between charged particles in the beam:
P1
P2
P1’
P2’
F. Antoniou, IPAC10
Evolution of the emittance:
IBS Growth Times
Radiation Damping
IBS
Quantum Excitation
Tk contain the effect of all the scattering processes
in the beam at a given time.
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Introduction: Conventional theories of IBS
Growth rates are calculated at different points of the lattice and then averaged over the ring:
s6
s1
s2
s5
s4
s3
Conventional calculation of IBS effect in Accelerator Physics derive an estimation of T k from the theories of Bjorken-Mtingwa or
Piwinski.
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Bjorken-Mtingwa theory of IBS
Useful references:
• J.D.
Bjorken, S.K. Mtingwa, Part. Acc. Vol. 13, pp. 115-143 (1983).
• M. Conte, M. Martini, Part. Acc. Vol. 17, pp. 1-10 (1985).
• M. Zisman, S. Chattopadhyay, J. Bisognano, LBL-21270, ESG-15 (1986).
• K. Kubo, K. Oide, PRST-AB 4,124401 (2001).
• K.L.F. Bane, EPAC’02, Paris (2002).
• F. Zimmermann, CERN-AB-2006-002.
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Bjorken-Mtingwa theory of IBS
P1
P1’
Let assume the bunch distribution is:
P2
P2’
From arguments of relativistic quantum mechanics it is possible to estimate the
transition rate for scattering from
to
Energy-momentum conservation
Invariant Coulomb scattering amplitude:
Derived from the Mott cross section in the non-relativistic limit and small-angle scattering
approximation.
Momentum change
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Scattering angle
Momentum in the center-of mass
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Invariants of motion in the ring
The motion of the particles in the ring can be expressed through 3 invariants (and 3 phases).
Transversal invariants:
Longitudinal invariant:
Emittance:
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Bjorken-Mtingwa theory of IBS
The rms emittance is evaluated as:
We can then estimate the emittance growth due to IBS as:
In conclusion we have:
We still need a distribution. We choose a Gaussian:
Normalization constant
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Bjorken-Mtingwa theory of IBS
With this choice almost all the integrals reduce to gaussian integrals:
In the end we have:
With:
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Bjorken-Mtingwa theory of IBS
Conte and Martini found out that the formula can be reduced to:
With:
are functions of:
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Coulomb logarithm
It is not a well defined quantity. It comes from the integration over the scattering angle. There is not
complete agreement to define it. One option is as follows.
Debye length
Maximum impact parameter
Minimum impact parameter
Transverse temperature
Particle volume density
Classical distance of closest approach
Quantum mechanical diffraction limit
from the nuclear radius
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Piwinski theory of IBS
Useful references:
• A.
Piwinski, Proc. 9th Conf. on High Energy Accelerators, SLAC, Stanford (1974).
• M. Martini, CERN PS/84-9 (AA) (1984).
• A. Piwinski, CERN-87-03-V-1, pp. 402-415 (1985).
• A. Piwinski, CERN-92-01, pp. 226-242 (1991).
• K.L.F. Bane, EPAC’02, Paris (2002).
Classical approach in the determination of
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Software for IBS and Radiation Effects
Goals:
1.
2.
Follow the evolution of the particle distribution in the DR (we are not sure it remains Gaussian).
Calculate IBS effect for any particle distribution (in case it doesn’t remain Gaussian).
s6
s1
s2
s5
s4
• The lattice is read from a MADX file containing the Twiss functions.
• Particles are tracked from point to point in the lattice by their invariats (no phase
tracking up to now).
• At each point of the lattice the scattering routine is called.
s3
• 6-dim Coordinates of particles are calculated.
• Particles of the beam are grouped in cells.
• Momentum of particles is changed because of scattering.
• Invariants of particles are recalculated.
• Radiation damping and excitation effects are evaluated at the end of every loop.
• A routine has also been implemented in order to speed up the calculation of IBS effect.
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Intra-beam Scattering
Relativistic Center of Mass Frame:
Laboratory Frame:
Tranformation Matrix:
Lorentz Tranformation:
Caracterization of the Center of Mass frame:
We conclude that:
Finally:
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Intra-beam Scattering
Let us take 2 colliding particles in the beam:
The transformation matrix to the Beam Rest Frame is:
Assuming the BRF is the CMF of the particles, we derive:
In conclusion:
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Intra-beam Scattering
Applying the rotation of the system:
In conclusion, we have:
z’
Energy-Momentum conservation imposes:
s’
x’
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Rutherford Cross Section
z’ Pf
Pi
b
q
r
s’
Rutherford formula:
Rutherford Cross Section:
Cut off of angle/impact parameter:
Distribution of scattering angles:
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Intra-beam Scattering
Average momentum change:
sIBS
Coulomb logarithm:
vCM
-vCM
2 VCM Dt’
Number of particles met in CMF:
Relativistic effects:
Number of collisions in LF:
Statistical approximation:
Total momentum change:
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Energy Conservation
Energy is not conserved!
To recover the energy conservation (at the 1st order):
For consistency:
Total momentum change:
(P.R. Zenkevich, O. Boine-Frenkenheim, A. E. Bolshakov, A new algorithm for the kinetic analysis of inta-beam scattering in storage
rings, NIM A, 2005)
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Lattice Recurrences
Elements of the lattice with twiss functions differing of less than 10% are considered equal.
Lattice:
First reduction:
Second reduction:
+3X
+
+2X
( +3X
CLIC DR LATTICE: 14400 elements
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+3X
+
+
)
+
420 elements
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SIRE: Benchmarking (Gaussian Distribution) CLIC DR
F. Antoniou, IPAC10
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SIRE: Benchmarking (Gaussian Distribution) on LHC
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SIRE: Distribution Study
Case studies:
• A – Damping + QE
• B – Damping + IBS + QE
• C – Damping + IBS
• D – IBS
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Parameter
A
B
C
D
INITIAL
gex, gey,szsp
74.3e-6,1.8e-6,
1.71e+5
74.3e-6, 1.8e-6,
1.70e+5
74.3e-6, 1.8e-6,
1.71e+5
229.7e-9,3.7e-9,
2.87e+3
229.7e-9, 3.76e-9,
2.88e+3
435.6e-9, 5.54e-9,
3.65e+3
458.5e-9, 3.61e-9,
1.58e+3
1.12e-6, 1.16e-8,
9.61e+3
(m,m,eV m)
FINAL
gex, gey,szsp
(m,m,eV m)
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CLIC DR A: Damping + QE
Simulation of the CLIC Damping Rings case A:
Beam parameters
ex (m)
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ey (m)
ez (eV m)
Injection
13.27e-9
321.6e-12
1.71e+5
Extraction
(SIRE)
4.104e-11
6.72e-13
2.88e+3
Extraction
(MAD-X)
4.102e-11
6.69e-13
2.87e+3
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SIRE: IBS Distribution study A: Damping + QE
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Parameter
c2999
Confidence
DP/P
964.2251
0.7876
X
976.2195
0.6988
Y
957.4559
0.8290
Parameter
Value
Eq. ex (m rad)
4.1039e-011
Eq. ey (m rad)
6.7113e-013
Eq. sd
1.0901e-3
Eq. sz (m)
9.229e-4
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CLIC DR D: IBS
Simulation of the CLIC Damping Rings case D:
Beam parameters
ex (m)
ey (m)
ez (eV m)
1/Tx (s-1)
1/Ty (s-1)
1/Tz (s-1)
Injection
4.104e-11
6.663e-13
2871
Bjorken-Mtingwa
29.6
21.0
28.9
Extraction
(SIRE)
2.001e-010
2.064e-12
9609
SIRE compressed (Gauss)
21.6
17.8
20.6
SIRE not compressed (Gauss)
18.1
18.0
19.3
SIRE compressed
17.0
14.6
17.2
SIRE not compressed
18.3
15.3
16.5
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SIRE: IBS Distribution study D: IBS
Parameter
c2999
Confidence
Dp/p
3048.7
<1e-15
X
1441.7
<1e-15
Y
1466.9
<1e-15
Parameter
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Value
Eq. ex (m rad)
2.001e-10
Eq. ey (m rad)
2.064e-12
Eq. sd
1.992e-3
Eq. sz (m)
1.687e-3
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SIRE: IBS Distribution study D: IBS
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Parameter
c237
Confidence
Sample
%
DP/P
38.81
0.39
26
X
36.73
0.48
25
Y
46.83
0.13
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Parameter
Value
ap
5.281e+7
bp
1.568
ax
3.840e+10
bx
1.280
ay
4.557e+12
by
1.196
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SIRE: SLS simulations
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1/Tx (s-1)
1/Ty (s-1)
1/Tz (s-1)
MADX (B-M)
20
37
59
SIRE (compressed)
15.6
24.5
47.2
SIRE (not compressed)
14.4
23.4
42.2
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SIRE: SLS simulations
IBS ON
IBS ON
IBS ON
IBS OFF
IBS OFF
IBS OFF
Beam parameters
ex (m)
ey (m)
ez (eV m)
Initial
1.68e-8
6.01e-12
71571
Final
6.08e-9
2.33e-12
8945
Equilibrium
5.59e-9
2.02e-12
7921
(NO IBS)
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Conclusions & Outlook
•
Simulation code SIRE has been developed to simulate IBS effect in
storage/damping rings.
– Benchmarking with conventional IBS theories gave good agreement.
– Evolution of the particle distribution shows deviations from Gaussian
behaviour due to IBS effect.
•
Comparison with data from SLS could provide the possibility of
–
–
–
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Benchmarking with real data
Tuning of code parameters (number of cells, number of interactions, etc.)
Revision of the theory or theory parameters (Coulomb log, approximation used, etc.)
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THANKS.
The End
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