Studies and measurements of linear coupling and nonlinearities in hadron circular accelerators [Elektronische Ressource] / von Andrea Franchi

Studies and Measurements
of Linear Coupling and Nonlinearities
in Hadron Circular Accelerators
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
¨der Johann Wolfgang Goethe-Universitat
in Frankfurt am Main
von
Andrea Franchi
aus Teramo (Italien)
Frankfurt am Main 20062
vom Fachbereich Physik der Johann Wolfgang Goethe-Universitat¨ als
Dissertation angenommen.
Dekan: Prof. Dr. Wolf Assmus
Erster Gutachter: Prof. Dr. Ingo Hofmann
Zweiter Gutachter: Prof. Dr. Ulrich Ratzinger
Datum der Disputation:La brevita,` gran pregio!
La Boheme,` Atto 1Contents
1 Introduction 7
2 Motivations 13
2.1 The heavy ion synchrotron SIS-18 at GSI . . . . . . . . . . . . . . . . 13
2.2 Motivations in view of the upgrade and FAIR . . . . . . . . . 16
2.2.1 Measurement of nonlinearities . . . . . . . . . . . . . . . . . . 16
2.2.2 Linear coupling studies . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Analysis of RHIC and SPS data . . . . . . . . . . . . . . . . . 17
3 Resonance driving terms (RDT): theoretical basis 19
3.1 Physics assumptions and hypothesis . . . . . . . . . . . . . . . . . . . 20
3.2 One-turn map, Hamiltonian coe cien ts and nonlinearities . . . . . . 20
3.3 Normal form, resonance driving terms and BPM spectrum . . . . . . 23
3.4 Resonance classi cation and nomenclature . . . . . . . . . . . . . . . 25
4 Magnet strength measurement from BPM data 31
4.1 From BPM spectrum to magnet strength . . . . . . . . . . . . . . . . 31
4.2 Magnet strengths from RDT variation along the ring . . . . . . . . . 32
4.3 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.1 BPM calibration factor . . . . . . . . . . . . . . . . . . . . . . 34
4.3.2 Do we really need to reconstruct the momentum? . . . . . . . 35
4.3.3 Upon the dependence on the model . . . . . . . . . . . . . . . 37
4.4 Analysis of existing SPS data . . . . . . . . . . . . . . . . . . . . . . 37
5 Betatron coupling and emittance transfer: static case 43
5.1 Short review of previous theory . . . . . . . . . . . . . . . . . . . . . 44
5.2 Betatron motion close to the (1,-1) resonance . . . . . . . . . . . . . 45
5.3 Single particle emittances . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 RMS emittances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.5 From f to formulae in the literature . . . . . . . . . . . . . . . . . 491001
5.6 Computing and measuring f . . . . . . . . . . . . . . . . . . . . . 511001
5.7 Generalized coordinates and decoupled motion . . . . . . . . . . . . . 53
5.8 Emittance variation along the ring . . . . . . . . . . . . . . . . . . . 54
34 CONTENTS
6 Betatron coupling and emittance transfer: dynamic case 57
6.1 Short review of previous theory . . . . . . . . . . . . . . . . . . . . . 57
6.2 Single particle emittances . . . . . . . . . . . . . . . . . . . . . . . . 58
6.3 RMS emittances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7 Optics and emittance measurements in the SIS-18 63
7.1 SIS-18 turn-by-turn BPM acquisition system . . . . . . . . . . . . . . 63
7.2 Tune measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.3 Nonlinear chromaticity measurement . . . . . . . . . . . . . . . . . . 68
7.4 The SIS-18 residual gas pro le monitor (RGM) . . . . . . . . . . . . 70
7.4.1 From RGM data to RMS emittance . . . . . . . . . . . . . . . 71
7.4.2 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . 73
7.5 Measurement of emittance sharing . . . . . . . . . . . . . . . . . . . . 74
7.6t of exchange . . . . . . . . . . . . . . . . . . . 75
7.7 Betatron coupling resonance compensation in the SIS-18 . . . . . . . 82
8 Measuring and correcting betatron coupling 85
8.1 Betatron coupling correction . . . . . . . . . . . . . . . . . . . . . . . 85
8.2 From f tojCj ( Q ) . . . . . . . . . . . . . . . . . . . . . . . 871001 min
8.3 From f to phase of C ( ) . . . . . . . . . . . . . . . . . . . . . . 891001
8.4 Measurement and correction of C in RHIC during 2005 . . . . . . . . 91
9 Space charge and emittance transfer 97
9.1 Multi-particle PIC simulations . . . . . . . . . . . . . . . . . . . . . . 99
9.2 Case with unsplit tunes . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.2.1 Suppressing the space charge driven emittance exchange . . . 103
9.3 Case with split tunes . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9.3.1 Emittance dependent detuning . . . . . . . . . . . . . . . . . 108
9.3.2 Static case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.3.3 Dynamic case . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.3.4 How to avoid overcompensation of betatron coupling . . . . . 116
10 Conclusions 119
Acknowledgments 121
Zusammenfassung 123
Bibliography 131
A From magnet strength to Hamiltonian coe cien ts 135CONTENTS 5
B Hamiltonian coe cien ts from RDT variation 137
B.1 The shadow e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.2 RDT close to a resonance . . . . . . . . . . . . . . . . . . . . . . . . 140
B.3 beta functions at the multipoles . . . . . . . . . . . . . . . . . . . . . 141
C Lie series and RDT close to the (1,-1) resonance 143
D Betatron coupling: equivalence of RDT and matrix approaches 147
D.1 Resonance driving term formalism . . . . . . . . . . . . . . . . . . . . 147
D.2 Matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
D.3 Relating the C matrix and the RDT’s . . . . . . . . . . . . . . . . . 149
D.4 C matrix and skew quadrupole strengths . . . . . . . . . . . . . . . . 150
2D.5 Measurement ofjCj= in RHIC during 2005 . . . . . . . . . . . . . . 151
E Software for the analysis of turn-by-turn BPM data 153
Curriculum Vitae of the Author 1616 CONTENTSChapter 1
Introduction
The future hadron circular accelerators SIS-100/SIS-300 of the FAIR complex of
GSI are superconducting machines and shall accelerate high-intensity beams. Su-
perconducting magnets are known to drive nonlinear elds usually up to one order
of magnitude higher than room-temperature magnets. This is due to a limited accu-
racy in cabling the coils and to persistent currents after each energy ramp. Magnet
nonlinearities are of main concern because particles having large oscillations or mov-
ing close to the pipe are subject to chaotic motion resulting in unstable trajectories
and eventually in beam loss. In high energy proton colliders the beam size is usually
small compared to the dimensions of the pipe. The region of stability (dynamic
aperture) is also large enough to contain the entire beam. This is not the case for
the heavy ion synchrotron SIS-100 where the beam occupies transversely a large
fraction of the pipe and the dynamic aperture is close to both the beam contour and
the wall. A continuous monitoring of the \nonlinearity budget" is mandatory not
only to reach the expected beam quality, but also to avoid radiation and quenching
damages driven by losses of high energy particles at the dipole walls.
The commissioning of new large accelerators, as well as of existing machines
after the periodic maintenance, might become a tedious task in presence of uncor-
rected magnet polarities or problems in the power supply connections. While for
dipoles and quadrupoles established beam-based methods for detecting wrong mag-
net strengths already exist (closed orbit and linear optics), state-of-art techniques
for skew quadrupoles and sextupoles are either time consuming or limited to the
measurement of global quantities (amplitude dependent detuning, minimization of
tune split and nonlinear chromaticity). A beam-based method to infer on-line both
the strength and the polarity of corrector magnets in few machine cycles is therefore
desirable especially in machines with a large number of correctors.
Furthermore, high-gradient superconducting quadrupoles induce linear coupling
between the transverse planes because of both skew quadrupole eld errors and
limited accuracy installing the magnets in the beam line (tilting angle). Betatron
coupling is of concern because it makes the beam rotate in the x y plane. In
high-intensity heavy ion synchrotrons any rotation would lead to beam scraping, as
the beam lls almost entirely the elliptical pipe at injection energy. On the other
78 CHAPTER 1. INTRODUCTION
hand, it is under consideration to operate the existing SIS-18 as booster for the
SIS-100 with equal transverse emittances at at top. During multi-turn injection
a partial exchange of the beam emittance from the horizontal plane to the vertical
is also foreseen to protect the injection septum in high-intensity operations. Both
manipulations can be obtained with controlled betatron coupling to be arti cially
driven by external skew quadrupoles.
Both the heavy ions synchrotrons SIS-100 and SIS-300 shall operate in a regime of
beam current and energy where the repulsive space-charge forceF is not negligible.s
The latter one scales with the beam parameters according to
ZI
F / ;s 3A(