Beam losses along the ESS LINAC due to nonlinear effects

107 Pages·2016·5.18 MB·English

Beam losses along the ESS LINAC due to nonlinear effects
- A Statistical review
Alexander Lauge Pedersen
A thesis presented for the degree of
Master of Science in Engineering - Engineering Physics
Centre for Mathematical Sciences
Lund University, LTH, in collaboration with the European Spallation
Source
Sweden
2016-12-21
Abstract
In this project, a comprehensive statistical review is done regarding beam
distribution evolution and losses of protons in the ESS LINAC. It aims to
clarifysomeunansweredquestionsregardingthechangeinthedistributionas
it evolves through the LINAC and how sudden changes can effect the losses
of protons downstream. The correlation between the sub phase-spaces were
removed to be able to study the nonlinear forces acting on the distribution.
Several statistical tools were used to analyse these distributions and visualize
them in a comprehensive manner. Furthermore, this report aims to give an
overview of the maximum losses that one may expect in the current lattice
configurationintheESSLINAC.AnEVT(ExtremeValueTheory)approach
was used to model these extreme losses to highlight extreme scenarios in the
LINAC. Many interesting features arise regarding the distribution evolution
which can be confirmed in theory and the maximum losses could be regarded
as within the limit set by the ESS design report. The results illustrates that
changes in the distribution can be found with the KS-test and can act com-
plementarytothetoolsusedbythephysiciststoday, withalittlerefinements.
Furthermore, the GMM may indicate that the distributions are multimodal,
thus opening the questions around the interpretation of the moments of the
distributions and their relation to the tools used for analysis. The EVT ap-
proach showed that no concern is needed regarding the exceedances of the
proposed highest limit of losses, 1 W/m. In conclusion, the report highlights
statistical tools for beam monitoring and further research in this subject may
be of scientific interest.
Keywords: ESS,Beam Physics, linac, Linear Accelerator, High in-
tensitylincas,particledistributions,statistics,Kolmogorov-Smirnov
tests, mathematics, Monte Carlo, EVT, GEV, TraceWin, Particle
Simulation
Contents
1 Acknowledgements 1
2 Introduction 2
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Project specification . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 European Spallation Source 5
3.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 The Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2.1 The LINAC configuration . . . . . . . . . . . . . . . . 6
3.3 Lattice in simulation . . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1 The FODO - lattice . . . . . . . . . . . . . . . . . . . . 9
3.3.2 The realistic lattice - periodic approach . . . . . . . . . 10
3.3.3 The realistic approach . . . . . . . . . . . . . . . . . . 11
4 Theory 12
4.1 Basic Accelerator Physics . . . . . . . . . . . . . . . . . . . . 12
4.1.1 The equation of motion - single particle . . . . . . . . . 12
4.1.2 ThecoordinatesystemandHill’sequation-Transverse
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1.3 Longitudinal dynamics . . . . . . . . . . . . . . . . . . 23
4.1.4 Multiparticle view - the beam . . . . . . . . . . . . . . 23
4.2 Fundamental statistics . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 Hypothesis testing . . . . . . . . . . . . . . . . . . . . 32
4.2.2 Parametric tests . . . . . . . . . . . . . . . . . . . . . . 33
4.2.3 Non-parametric tests . . . . . . . . . . . . . . . . . . . 34
4.2.4 Parameter estimation - Maximum likelihood . . . . . . 37
i
4.2.5 Finite Mixture Models - the Gaussian fit . . . . . . . . 37
4.2.6 The EM-algorithm . . . . . . . . . . . . . . . . . . . . 41
4.2.7 The randomness . . . . . . . . . . . . . . . . . . . . . . 42
4.2.8 Extreme Value Theory . . . . . . . . . . . . . . . . . . 42
4.2.9 Profile likelihood . . . . . . . . . . . . . . . . . . . . . 45
4.2.10 Empirical bootstrapping . . . . . . . . . . . . . . . . . 46
5 Method 50
5.1 Design of the lattice . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.1 TraceWin . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2.3 Python . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 The scientific approach . . . . . . . . . . . . . . . . . . . . . . 52
6 Results 54
6.1 Courant-Snyder parameter calculations . . . . . . . . . . . . . 54
6.1.1 FODO lattice . . . . . . . . . . . . . . . . . . . . . . . 54
6.1.2 The “real” lattice . . . . . . . . . . . . . . . . . . . . . 57
6.2 Kolmogorov-Smirnov tests . . . . . . . . . . . . . . . . . . . . 58
6.2.1 The periodic FODO-lattice . . . . . . . . . . . . . . . . 58
6.2.2 The “real” lattice . . . . . . . . . . . . . . . . . . . . . 60
6.3 Statistical model . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3.1 The Gaussian Mixture Model . . . . . . . . . . . . . . 71
6.4 EVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4.1 General statistics of the sections in the ESS LINAC . . 72
6.4.2 The GEV model . . . . . . . . . . . . . . . . . . . . . 74
7 Discussion 78
7.1 Distribution analysis . . . . . . . . . . . . . . . . . . . . . . . 78
7.2 The Gaussian input . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 Gaussian mixture model . . . . . . . . . . . . . . . . . . . . . 80
7.4 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . 81
7.5 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8 Appendix A - TraceWin 84
8.1 Elements and error options . . . . . . . . . . . . . . . . . . . . 85
8.2 User interface for analysis . . . . . . . . . . . . . . . . . . . . 85
ii
8.3 Computation routine - PARTRAN . . . . . . . . . . . . . . . 86
9 Apendix B - matLab 87
9.1 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9.2 The statistical toolbox . . . . . . . . . . . . . . . . . . . . . . 87
10 Appendix C - Code 89
10.1 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.1.1 Twiss (Courant-Snyder) calculations . . . . . . . . . . 89
10.1.2 Distribution analysis . . . . . . . . . . . . . . . . . . . 93
10.1.3 EVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
iii
Chapter 1
Acknowledgements
I would first like to thank my thesis advisors Dragi Anevski (LTH), Moham-
mad Eshraqi (ESS) and Ryoichi Miyamoto (ESS). The office and all of the
office supplies provided by ESS gave me a privileged situation to work from.
The presence of my supervisors has been very comforting throughout the
project and I am very grateful for the huge amount of space I was given to
pursue the subjects I found interesting.
Finally, I can not enough emphasize the gratitude I hold for Emilia, my
bethrothed,andherunfailingsupportandcontinuousencouragementthrough-
out my years at LTH. The patience and understanding, when I spent days
and nights studying, has been invaluable to me and for that, I will be ever
grateful. Thank you!
The author
Alexander Lauge Pedersen
1
Chapter 2
Introduction
2.1 Motivation
TheEuropeanSpallationSource, ESS, isthefutureleaderinNeutron-science
and the most intense neutron source in the world with its 2.0 GeV protons
producing 5 MW of power. These protons can be lost along the LINAC due
to non-linear forces and may cause damage on electrical components and/or
cause radiation. The radiation caused by the loss of protons complicates
hands-on maintenance and the mitigation of lost protons becomes a priority
[22]. The beam physics group within the accelerator division simulates the
accelerator using TraceWin [25] and studies the impact of unknown errors
that might occur as a result of the sensitivity of the system to internal and
externalfactors. Thisisdonebyintroducingstatic(mostlycausedbyinternal
factors) and dynamic (mostly external factors) errors like miss-alignments in
the location of the quadrupoles (QPs) and/or degradation in the electrical
fields within the accelerating cavities. This problem is highly prioritised and
thus simulations before commissioning are important in order to understand
how, why and where losses are concentrated and how to mitigate them in
an effective manner. Many scientists and engineers are currently working
on how to identify these losses and many simulations are made in order to
understand the physics behind the deviating protons and the requirements
for keeping them at a controllable level. Much research is done in this area
and specially in-house research at ESS and other accelerator facilities [19,
2
22]. Thesesimulationsconstitutesthefoundationforthespecificationswhich
are sent to the in-kind partners and manufacturers for development of the
different elements in the LINAC.
2.2 Project specification
One objective of the project is to adopt statistical methods to analyse the
particle distributions and losses along the LINAC. The aim is to analyse sta-
tistical properties of the particle distributions that can be connected to losses
of protons in the accelerator. The delimitations are formulated to keep the
project within a reasonable time frame. The study of the evolution of the
particles are done on a simple FODO-lattice first, to provide an understand-
ing of the dynamics of the beam propagation. Next, a more realistic lattice
are considered from the MEBT-HB to illustrate the part of the accelerator
where the majority of the acceleration is done. For the EVT approach, the
project is delimited to study the losses in the whole LINAC and the losses
in the MEBT. The distortion of the particle distribution affects the number
of particles located in critical areas of the distribution subjected to the risk
of getting lost. Starting from a simple lattice, where the outcome is known,
statistical tests are used to determine whether a distribution at certain loca-
tions are what would be expected to a certain degree of certainty. Different
lattices and different input distributions are used to create a full picture of
the evolution of the particle distributions. Furthermore, a Gaussian Mixture
Model (GMM) approach is adopted to study if it is possible to parametrise
the distribution evolution and to identify critical parts of the accelerator
and couple them to the behavior of the parameters (and compare to the
Courant-Snyder parameters) but also to see if one can find multiple modes
is the distribution. Another objective is to model the extreme losses to be
able to say something about the ´´worst case scenario” in terms of losses and
compare the results with the limits set by ESS. With the help from the Beam
Physics Group, a data set containing information of 20000 different simula-
tions where errors are introduced to create a more lifelike machine are used
to study the amount of losses along the LINAC. These errors are randomly
introduced under certain restrictions which produces stochastic variability
in the outcome. This allows for a parametric Generalised Extreme Value
(GEV) approach where information of the extreme values of the losses are
3
given. An empirical approach is also done and fundamental statistical tools
are used to extract information of the losses in the LINAC.
2.3 Thesis outline
Chapter 3 gives an introduction to the ESS project and puts weight on the
LINAC (LINear ACcelerator) configuration with the purpose of introducing
the reader to the structure and layout of the accelerator. Each part of the
LINAC is explained to provide an overall view of the LINAC structure. Fur-
thermore, the FODO-lattice (FOcus and DefOcus) is introduced along what
the author means with the “realistic lattice”.Chapter 4 presents the relevant
theory and because the sheer size of the project spans over a huge area,
much information and many methods are presented to enfold the spread of
the problem. First, the relevant physics are presented to introduce the reader
to the concepts often used in the accelerator community. Next, the statis-
tics relevant for the project is presented to knit the theory section together.
In chapter 5, the method which the author assumes is presented with the
purpose of spreading light on the approach which has been taken during the
project. Explanation of how the experiments are conducted and analysed for
replicating studies. The author describes the lattice design which is used in
the project and which demarcations that are done. The software utilities are
introduced and explained and lastly, a schematic over the scientific approach
is presented. Chapter 6 contains the results from the simulations and calcu-
lations and shall serve as an apparent and visual end product of the project
to ease the comprehension and connection to the latter discussion in chapter
7. In this chapter, the results are discussed with the support from theory and
the reader is guided through the arguments and conclusions of the author.
The appendices presents a brief introduction to TraceWin and MATLAB to
ensure the reader the best possible comprehension of the project and the ap-
proach it takes. Also some code, subjectively regarded as the most important
code by the author, will be presented. The whole set of scripts produced in
this project, apart from the TraceWin-code, will be available if asked for.
4
Chapter 3
European Spallation Source
3.1 Overview
The European Spallation Source (ESS) is to be the leading research facility
for frontier neutron science and is currently under construction in Lund,
Sweden. The planned construction completion year is 2018 when the actual
commissioning of the beam will start. By 2019 the facility shall be able to
produce a 570 MeV proton beam on target. The site will hold a 600 meter
longlinearacceleratorproducingabunched62.5mAprotonbeamof2.0GeV
releasing 5 MW onto a Tungsten target. In this process, called spallation (see
Figure 3.1), neutrons will be extracted for research
purposes[22]. ThemachinewillworkcomplementarytoX-raysandtherefore
to MAX IV. Both will form a strong and steadfast stronghold for a vast
variety of scientific disciplines.
3.2 The Accelerator
The accelerator, seen in Figure 3.2, consists of different sections with the
instrumentation to accelerate the protons in a controlled manner. The accel-
erator is divided into a “warm” and a “cold” section. The “cold” sections are
immersed in liquid helium and hold a temperature of 2 K. These supercon-
5