Two topics in particle accelerator beams - Desy

392 Pages·2012·2.58 MB·English

Two topics in particle
accelerator beams
by
Klaus Heinemann
Diplom Physiker, University of Hamburg, 1986
Committee Chair
James A. Ellison
DISSERTATION
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
Mathematics
The University of New Mexico
Albuquerque, New Mexico
May, 2010
(cid:2)c 2010, Klaus Heinemann
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Dedication
To my family
iv
Acknowledgments
I would like to thank my advisor, Professor James Ellison, for his advisement
and continued support of my work. I would also like to thank Dr. Gabriele Bassi,
Professor Georg Hoffstaetter, Dr. Mathias Vogt, Dr. Robert Warnock, and Dr.
Jack Zhang for the many discussions we had over the years. I am also grateful
to Dr. Desmond Barber and Professors Charles Boyer, Thomas Hagstrom, and
Stephen Lau for their time and efforts on my dissertation committee. Additional
thanks are extended to Professors Alex Buium, Todd Kapitula, and Cristina Pereyra
for their support as Graduate Chairs. Moreover I am indebted to Dr. Desmond
Barber for his support over the years. Finally many thanks to everybody else from
the Mathematics and Statistics Department at UNM, who helped me directly or
indirectly. In particular I have to thank Dr. Oksana Guba for her advice concerning
several aspects of my dissertation.
This work has been supported by the Deutsches Elektronen Synchrotron (DESY)
and the U.S. Department of Energy contract DE-FG02-99ER41104.
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Two topics in particle
accelerator beams
by
Klaus Heinemann
ABSTRACT OF DISSERTATION
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
Mathematics
The University of New Mexico
Albuquerque, New Mexico
May, 2010
Two topics in particle
accelerator beams
by
Klaus Heinemann
Diplom Physiker, University of Hamburg, 1986
Ph.D., Mathematics, University of New Mexico, 2010
Abstract
This thesis hastwo parts. In the first part I present results frommy studies ofthe
Vlasov-Maxwell system which was developed, together with a code, in collaboration
with Bassi, Ellison and Warnock. The emphasis is on the link between the theory
and the self-consistent numerical computations performed by the code. The Vlasov-
Maxwell system models electron beams, typically insynchrotron light sources. Inthe
secondpartIpresentresultsfrommystudiesofthedynamicsofspinpolarizedbeams.
Here the emphasis is on improvements of the theoretical basis of beam simulations
by using topological methods.
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Contents
List of Figures xvi
1 General Introduction 1
2 Introduction to the Vlasov-Maxwell system 3
3 The Vlasov-Maxwell system 8
3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Beam frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Exact formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 The FERMI@Elettra first bunch compressor system . . . . . . . . . . 27
3.4 The Monte Carlo particle method . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.2 Causality issue . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 40
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Contents
4 Summary of Vlasov-Maxwell system and outlook 54
5 Introduction to spin-orbit tori 56
5.1 Physical context and mathematical approach . . . . . . . . . . . . . . 56
5.2 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Scope and limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 The spin-orbit tori 69
6.1 Introducing the spin-orbit tori (ω,A) . . . . . . . . . . . . . . . . . . 69
6.2 Introducing the Z-action L associated with every spin-orbit torus
ω,A
(ω,A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Introducing the polarization fields of every spin-orbit torus (ω,A) and
the associated Z-action L(PF) . . . . . . . . . . . . . . . . . . . . . . 75
ω,A
6.4 Homotopy Theory relevant for spin-orbit tori . . . . . . . . . . . . . . 79
7 Transforming spin-orbit tori 83
7.1 Introducing the transformations of spin-orbit tori and the right group
action R on SOT(d,ω) . . . . . . . . . . . . . . . . . . . . . . . . 84
d,ω
7.2 Introducing weakly trivial spin-orbit tori . . . . . . . . . . . . . . . . 90
7.3 Introducing weak coboundaries . . . . . . . . . . . . . . . . . . . . . 92
7.4 Introducing spin tune and spin-orbit resonance of first kind . . . . . . 95
7.5 Yokoya’s uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . 97
7.6 Putting weak coboundaries into perspective . . . . . . . . . . . . . . 98
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Contents
7.7 Transformations between weakly trivial spin-orbit tori . . . . . . . . . 100
8 Quasiperiodic functions as tools for studying spin-orbit tori 106
8.1 Relations between polarization fields and spin trajectories . . . . . . . 107
8.2 Simple precession frames . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.3 Uniform precession frames . . . . . . . . . . . . . . . . . . . . . . . . 112
8.4 Introducing spin tune and spin-orbit resonance of second kind . . . . 116
8.5 The SPRINT theorem and a corresponding spin tune algorithm . . . 119
8.6 The impact of Homotopy Theory on spin tunes of first kind . . . . . . 125
8.7 Further properties of invariant spin fields . . . . . . . . . . . . . . . . 128
9 Reconsidering the Z-actions L and L(PF) 129
ω,A ω,A
9.1 Carving out the topological Z-spaces (Rd+3,L ) as skew products of
ω,A
the topological Z-spaces (Rd,L ) . . . . . . . . . . . . . . . . . . . . 129
ω
9.2 Carving out the topological Z-spaces (Rd+3,L ) as extensions of the
ω,A
topological Z-spaces (Td ×R3,L(T)) . . . . . . . . . . . . . . . . . . . 130
ω,A
9.3 A principal SO(3)-bundle underlying SOT(d) . . . . . . . . . . . . . 133
9.3.1 The principal SO(3)-bundle λ . . . . . . . . . . . . . . . 134
SOT(d)
9.3.2 The bundle λ [R3,L(3D)] associated with λ . . . . . 139
SOT(d) SOT(d)
9.3.3 How λ carries the Z-action L . . . . . . . . . . . . . 140
SOT(d) ω,A
9.3.4 How λ carries the Z-action L(PF) . . . . . . . . . . . . . 143
SOT(d) ω,A
9.3.5 Reducing the structure group SO(3) . . . . . . . . . . . . . . 146
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Contents
9.3.6 Closing remarks on λ . . . . . . . . . . . . . . . . . . . 150
SOT(d)
10 Summary of spin-orbit tori and outlook 152
Appendices 154
A 155
A.1 Details on the self field . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.2 Derivation of the 4D Vlasov equation . . . . . . . . . . . . . . . . . . 164
A.3 Kernel density estimation . . . . . . . . . . . . . . . . . . . . . . . . 167
A.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.3.2 Algorithmic aspects of the kernel density estimator . . . . . . 170
A.3.3 Estimators of MISE and of related quantities . . . . . . . . . . 176
A.3.4 Asymptotic approximation of MISE . . . . . . . . . . . . . . 180
A.3.5 Least squares cross validation - general properties . . . . . . . 184
A.3.6 Least squares cross validation - Silverman’s algorithm . . . . . 188
A.3.7 Practical considerations . . . . . . . . . . . . . . . . . . . . . 199
A.4 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
B Group actions 203
C Topological concepts and facts 208
C.1 Bundles, fiber structures and Hurewicz fibrations . . . . . . . . . . . 209
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