Javier Cardona. Mayo 2010

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1 Un método de detección de errores magnéticos de alta precisión en RHIC y el LHC Javier Cardona Mayo 2010 p.1/33

2 Introducción Contenido 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 p.2/33

3 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. p.3/33

4 Dipolo del Sincrotrón Para una partícula ideal se tiene que: p e = BR p.4/33

5 Lente Magnética: El Cuadrupolo p.5/33

6 Campo Magnético del Cuadrupolo Un par de cuadrupolos enfocan en X y Y p.6/33

7 Principio del Sincrotrón p.7/33

8 Trayectorias del Haz de Partículas 2 Oscilacion de Betatron x[mm] 0 Trayectoria Disenada Coordenada longitudinal "s" [m] x(s) = A β x (s) sin (ψ x (s) ϕ) Todas las partículas oscilan con fases diferentes dando lugar al perfil longitudinal característico del haz. p.8/33

9 Efecto de un Error Magnético en el Haz α α + x α = ebl/p α Después de un error magnético, el haz como un todo realiza oscilaciones de betatron. p.9/33

10 Errores Magnéticos Errores Dipolares: x = e Bl p = B 0 Errores Cuadrupolares: x = A 1 y B 1 x Errores Sextupolares: x = 2A 2 xy + B 2 ( x 2 + y 2 ) En General: x = B 0 B 1 x+a 1 y+2a 2 xy+b 2 ( x 2 +y 2 )+... Estos errores siempre están presentes en los imanes de un acelerador. Cómo se miden?. p.10/33

11 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] p.11/33

12 Principle of Action and Phase Jump (1) Assume a particle is launched in a lattice with a gradient error 5 x (mm) s (m) Quad Error s (m) p.12/33

13 Principle (2) There are two possibilities to describe the particle trajectory: x(s) = 2Jβ N (s) sin(ψ N (s) δ) β N and φ N are the new lattice functions generated by the magnetic error. Or, x(s) = 2J 0 β D (s) sin(ψ D (s) δ 0 ) for s < s 1 x(s) = 2J 1 β D (s) sin(ψ D (s) δ 1 ) for s > s 1 β D and φ D are the designed beta functions. Here, constants J and δ change instead of lattice functions. p.13/33

14 Principle (3) x (mm) action (nm) phase (rad) s (m) 2 (b) s (m) 370 (c) s (m) 3.2 (d) 3.1 δ 0 3 (a) J s = s θ s (m) J 1 δ 1 p.14/33

15 Error Strength Not only the location of the error can be easily determined but also the strength of the error: x = θ z = 2J 0 + 2J 1 4 J 0 J 1 cos(δ 1 δ 0 ) β z (s θ ) p.15/33

16 Multipole Components of Magnetic Errors The magnetic error is a contribution from different multipole components: θ x = B 0 B 1 x(s θ ) + A 1 y(s θ ) + 2A 2 x(s θ )y(s θ ) +B 2 [ x 2 (s θ ) + y 2 (s θ )] +..., θ y = A 0 + A 1 x(s θ ) + B 1 y(s θ ) + 2B 2 x(s θ )y(s θ ) +A 2 [x 2 (s θ ) y 2 (s θ )] +... where B n and A n are values related with the normal and skew multipole components of the error B. p.16/33

17 Estimating Errors (One Multipole) A 1 = θ xy(s θ ) + θ y x(s θ ) x 2 (s θ ) + y 2 (s θ ) B 1 = θ yy(s θ ) θ x x(s θ ) x 2 (s θ ) + y 2 (s θ ) If only one error multipole component is present, only one particle trajectory is needed. p.17/33

18 Estimating Errors (Several Multipoles) Several error multipole components -> several orbits with different amplitudes θ x (µrad), θ y (µrad), y( s θ ) (mm) θ x θ y y(s θ ) θ x θ y y(s θ ) x(s θ ) (mm) p.18/33

19 Estimating Errors (Several Multipoles) θ x = C 1x x(s θ ) + C 2x x 2 (s θ ) + cte θ y = C 1y x(s θ ) + C 2y x 2 (s θ ) + cte y(s θ ) = mx(s θ ) + b A 1 B 1 = C 1xm + C 1y 1 + m 2 = C 1x C 1y m 1 + m 2 A 2 = C 2y 2C 2x m + C 2y m m 2 + m 4 B 2 = C 2x 2C 2y m C 2x m m 2 + m 4 p.19/33

20 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. p.20/33

21 Linear Experiments with Beam in RHIC p.21/33

22 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. p.22/33

23 Action and Phase Analysis on a Difference Orbit y (mm) action (nm) phase (rad) ,2 3 2,8 2,6 (a) s (m) (b) ARC IR ARC s (m) (c) (d) J s (m) δ s (m) J 1 δ 1 p.23/33

24 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. p.24/33

25 Experiments with Known Gradient Errors (1) measured skew error (10-3 m -1 ) expected values previous analysis current analysis set skew error (10-3 m -1 ) p.25/33

26 Skew Quad Error Measurements (1) Skew quad errors (A 1 ) 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. A eq 1 = 3 i=1 ( 2φ i f i ) β sc x βsc y βxβ i y i p.26/33

27 Skew Quad Error Measurements (2) 2 A1 [1/m] Measured A1 A1 from measured rolls 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. p.27/33

28 Nonlinear Experiments with Beam in the SPS (1) Sextupoles were intentionally turn on at specific locations 4 3 turn 89 turn 96 turn 102 phase (rad) 2 sext. 32 sext sext. 72 sext sext p.28/33 s (m)

29 Nonlinear Experiments with Beam in the SPS (2) measured strengths nonlinear fit θ x (µrad) x(s θ ) (mm) K2L = (0.438 ± 0.032)m 2 from the fit of the experimental points which is in good agreement with the set value. p.29/33

30 Correction of Quadrupole Errors in the LHC (1) After B2 Correction Before Correction 3 Horizontal Phase [rad] s [m] p.30/33

31 Correction of Quadrupole Errors in the LHC (2) p.31/33

32 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. p.32/33

33 References [1] J. Cardona, S. Peggs, Linear and Nonlinear Magnetic Error Measurements using Action and Phase Jump Analysis, Physical Review ST AB 12, (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). p.33/33

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