Un m´etodo de detecci´on de errores magn´eticos de alta precisi´on

Un m´etodo de detecci´on de errores magn´eticos de alta precisi´on

Un método de detección de errores magnéticos de
alta precisión en RHIC y el LHC
Javier Cardona
Mayo 2010
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p.1/33
Contenido
Introducción
Método de Salto de Acción y Fase
Resultados de las simulaciones
Experimentos
• Órbitas con errores cuadrupolares conocidos
• Determinación de errores cuadrupolares en
RHIC
• Experimentos no lineales en el SPS
• Algunos resultados en el LHC
• Planes para el LHC
•
•
•
•
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p.2/33
Los Aceleradores y su Importancia
• Fundamental para comprender la estructuras
elementales de la naturaleza.
• Producen la llamada radiación de sincrotrón
ampliamente utilizada en muy diversos campos
de la ciencia.
• Diagnóstico y tratamiento de cáncer en medicina.
• Los aceleradores pueden ser lineales o circulares
siendo los sincrotrones los más populares.
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p.3/33
Dipolo del Sincrotrón
Para una partícula “ideal” se tiene que:
p
= BR
e
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p.4/33
Lente Magnética: El Cuadrupolo
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p.5/33
Campo Magnético del Cuadrupolo
Un par de cuadrupolos enfocan en X y Y
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p.6/33
Principio del Sincrotrón
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p.7/33
Trayectorias del Haz de Partı́culas
2
x[mm]
Oscilacion de Betatron
Trayectoria Disenada
0
-2
800
850
900
950
1000
Coordenada longitudinal "s" [m]
x(s) = A ∗
p
βx (s) sin (ψx (s) − ϕ)
Todas las partículas oscilan con fases diferentes dando
lugar al perfil longitudinal característico del haz.
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p.8/33
Efecto de un Error Magnético en el Haz
α + ∆ x’
α
α = eBl/p
α
Después de un error magnético, el haz como un todo
realiza oscilaciones de betatron.
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p.9/33
Errores Magnéticos
• Errores Dipolares: ∆x′ =
e∆Bl
p
= B0
• Errores Cuadrupolares: ∆x′ = A1 y − B1 x
• Errores Sextupolares:
∆x′ = 2A2 xy + B2 (−x2 + y 2 )
• En General:
∆x′ = B0 −B1 x+A1 y+2A2 xy+B2 (−x2 +y 2 )+...
Estos errores siempre están presentes en los imanes de
un acelerador. Cómo se miden ?.
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p.10/33
Métodos de Detección de Errores
• Matriz respuesta [J. Safranek,1997 ]
• Términos resonantes [R. Bartolini, 1998]
• Análisis independiente del modelo [J. Irwin,
1999]
• Corrección de Modelo Iterativo [M. Aiba et al,
2009]
• Acción y Fase [J. Cardona, Physical Review ST
AB 12, 2009]
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p.11/33
Principle of Action and Phase Jump (1)
Assume a particle is launched in a lattice with a
gradient error
x (mm)
5
0
-5
0
1000
2000
s (m)
3000
4000
0
1000
Quad Error
2000
s (m)
3000
4000
2
1
0
-1
-2
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p.12/33
Principle (2)
There are two possibilities to describe the particle
trajectory:
p
x(s) = 2JβN (s) sin(ψN (s) − δ)
βN and φN are the new lattice functions generated by
the magnetic error.
Or,
x(s) =
x(s) =
p
p
2J0 βD (s) sin(ψD (s) − δ0 ) f or s < s1
2J1 βD (s) sin(ψD (s) − δ1 ) f or s > s1
βD and φD are the designed beta functions. Here, “constants” J and δ change instead of lattice functions.
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p.13/33
x (mm)
Principle (3)
(a)
5
0
-5
-10
0
1000
3000
4000
2
(b)
1
0
-1
-2
0
2000
s (m)
1000
2000
s (m)
3000
4000
3000
4000
3000
4000
action (nm)
370
(c)
J0
360
350
340
J1
0
1000
phase (rad)
3.2
(d)
3.1
2000
s (m)
δ0
δ1
3
2.9
0
1000
s = sθ
2000
s (m)
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p.14/33
Error Strength
Not only the location of the error can be easily
determined but also the strength of the error:
′
∆x = θz =
s
√
2J0 + 2J1 − 4 ∗ J0 J1 cos(δ1 − δ0 )
βz (sθ )
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p.15/33
Multipole Components of Magnetic Errors
The magnetic error is a contribution from different
multipole components:
θx =
B0 − B1 x(sθ ) + A1 y(sθ ) + 2A2 x(sθ )y(sθ )
+B2 [−x2 (sθ ) + y 2 (sθ )] + ...,
θy =
A0 + A1 x(sθ ) + B1 y(sθ ) + 2B2 x(sθ )y(sθ )
+A2 [x2 (sθ ) − y 2 (sθ )] + ...
where Bn and An are values related with the normal
and skew multipole components of the error ∆B.
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p.16/33
Estimating Errors (One Multipole)
θx y(sθ ) + θy x(sθ )
A1 = 2
x (sθ ) + y 2 (sθ )
θy y(sθ ) − θx x(sθ )
B1 = 2
x (sθ ) + y 2 (sθ )
If only one error multipole component is present, only
one particle trajectory is needed.
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p.17/33
θx (µrad), θy (µrad), y( sθ) (mm)
Estimating Errors (Several Multipoles)
Several error multipole components -> several orbits
with different amplitudes
0
θy
θx
θy
y(sθ)
y(s θ)
-10
θx
-20
-40
-30
-20
x(sθ) (mm)
–
-10
0
p.18/33
Estimating Errors (Several Multipoles)
θx
θy
= C1x x(sθ ) + C2x x2 (sθ ) + cte
2
= C1y x(sθ ) + C2y x (sθ ) + cte
y(sθ ) = mx(sθ ) + b
A1 =
B1 =
A2 =
B2 =
C1x m + C1y
1 + m2
C1x − C1y m
−
1 + m2
−C2y − 2C2x m + C2y m2
−
1 + 2m2 + m4
C2x − 2C2y m − C2x m2
−
1 + 2m2 + m4
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p.19/33
Results of Simulations
Simulations of particle trajectories with a gradient
error, a skew quadrupole error, and a sextupole error
were done. The difference between set errors and and
errors estimated from action and phase analysis were:
• 0.03% or less for gradient errors.
• 0.01% or less for skew quadrupole errors.
• 3% or less for sextupole errors.
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p.20/33
Linear Experiments with Beam in RHIC
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p.21/33
Experimental Conditions Required
• Large betatron oscillations usually excited by
dipole correctors.
• The trajectory should have a maximum at the
place where the error is being estimated.
• Systematic errors and dipole errors are eliminated
using difference orbits.
• Difference orbits are now made with two turns of
the same multiturn orbit.
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p.22/33
y (mm)
Action and Phase Analysis on a Difference Orbit
6
3
0
-3
-6
(a)
2000
action (nm)
2
(b)
1
0
-1
-2
2000
80
ARC
IR
ARC
3000
2500
s (m)
(c)
J1
60
40
20
J0
2000
3
3000
2500
s (m)
3,2
phase (rad)
3000
2500
s (m)
(d)
2,8
δ1
δ0
2,6
2000
3000
2500
s (m)
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p.23/33
Experiments with Known Gradient Errors (1)
• Large beam orbits were excited changing a
quadrupole corrector bi8-qs3 at IR8.
• Action and phase analysis of first turn orbits for
each setting of bi8-qs3 were done.
• Difference orbits were built using two different
methods.
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p.24/33
Experiments with Known Gradient Errors (1)
-3
-1
measured skew error (10 m )
20
expected values
previous analysis
current analysis
10
0
-10
-20
-20
-10
0
10
-3 -1
set skew error (10 m )
–
20
p.25/33
Skew Quad Error Measurements (1)
• Skew quad errors (A1 ) were measured for all
RHIC IRs using the action and phase analysis.
• Roll angles of the quads were also measured
during the 2002 RHIC shutdown period.
•
q
P3
φi
i βi
β
)
(−2
x y
i=1
fi
eq
p sc sc
A1 =
βx βy
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p.26/33
Skew Quad Error Measurements (2)
2
Measured A1
A1 from measured rolls
A1 [1/m]
1
0
Blue - 6
Blue - 8
Blue-1
Blue -2
Blue - 5
Yell - 6
Yell - 8
Yell-1
Yell-2
Yell - 5
Triplets
-1
-2
Local skew quadrupole correctors were set according
to these values.
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p.27/33
Nonlinear Experiments with Beam in the SPS (1)
Sextupoles were intentionally turn on at specific
locations
4
turn 89
turn 96
turn 102
phase (rad)
3
2
sext. 32
sext. 83
1
sext. 72
0
0
sext. 12
sext. 63
2000
4000
s (m) –
6000
p.28/33
Nonlinear Experiments with Beam in the SPS (2)
30
θx (µrad)
25
measured strengths
nonlinear fit
20
15
10
5
0
5
x(sθ) (mm)
10
K2L = (0.438 ± 0.032)m−2 from the fit of the experimental points which is in good agreement with the set
value.
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Correction of Quadrupole Errors in the LHC (1)
After B2 Correction
Before Correction
Horizontal Phase [rad]
3
2
1
0
5000
10000
20000
15000
s [m]
–
25000
p.30/33
Correction of Quadrupole Errors in the LHC (2)
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p.31/33
Plans for the LHC
• Action and phase analysis of experimental orbits
to localize main errors.
• Estimation of magnetic errors from the
experimental orbits .
• What correctors should be used and how they
should be set.
• Software automatization.
• Comparisons with other methods.
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p.32/33
References
[1] J. Cardona, S. Peggs, “Linear and Nonlinear Magnetic Error Measurements using Action
and Phase Jump Analysis”, Physical Review ST AB 12, 014002 (2009).
[2] J. Cardona, “Local Magnetic Error Estimation using Action and Phase Jump Analysis of
Orbit Data”, PAC’07, Albuquerque, New Mexico (2007).
[3] J. Cardona, R. T. Garcia, “Non Linear Error Analysis from Orbit Measurements in SPS
and RHIC”, PAC’05, Knoxville, Tennesse (2005).
[4] J. Cardona, S. Peggs, F. Pilat, V. Ptitsyn, “ Measuring Local Gradient and Skew
Quadrupole Errors in RHIC IRs” ,EPAC’04, Lucerne, Switzerland (2004).
[5] J. Cardona, “Linear and Non Linear Studies at RHIC Interaction Regions and Optical
Design of the Rapid Medical Synchrotron”, Ph.D. thesis, Stony Brook University, (2003).
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