thesis

TRANSMISSION PROPERTIES OF THE KATRIN MAIN SPECTROMETER
WHEN OPERATED TO SEARCH FOR STERILE NEUTRINOS
A thesis presented to the faculty of
San Francisco State University
In partial fulﬁlment of
The Requirements for
The Degree
Master of Science
In
Physics
by
Alexander V. Pan
San Francisco, California
August 2015

Copyright by
Alexander V. Pan
2015

CERTIFICATION OF APPROVAL
I certify that I have read TRANSMISSION PROPERTIES OF THE
KATRIN MAIN SPECTROMETER WHEN OPERATED TO SEARCH
FOR STERILE NEUTRINOS by Alexander V. Pan and that in my
opinion this work meets the criteria for approving a thesis submitted in
partial fulﬁllment of the requirements for the degree: Master of Science
in Physics at San Francisco State University.
Dr. Susan Lea
Professor of Physics ,
Dr. Joesph Barranco
Associate Professor of Physics
Dr. Alan Poon
Group Leader at LBNL

TRANSMISSION PROPERTIES OF THE KATRIN MAIN SPECTROMETER
WHEN OPERATED TO SEARCH FOR STERILE NEUTRINOS
Alexander V. Pan
San Francisco State University
2015
The Karlsruhe Tritium Neutrino (KATRIN) Experiment is a next-generation,
large-scale tritium β-decay experiment. It is targeted at measuring the absolute
neutrino mass scale with a sensitivty of 200 meV (90 % C.L.). Its unique source and
spectroscopic properties allows KATRIN to search for sterile neutrinos in the keV
mass range. The goal of the project is to investigate non-adiabatic systematic eﬀects
associated with this novel measurement. In particular, this work will make use of
the KATRIN general simulation software Kassiopeia 3.0 to study the transmission
properties of the large KATRIN main spectrometer at low retarding potentials, as
this will be the setting used in the search for sterile neutrinos in the keV mass
range.
I certify that the Abstract is a correct representation of the content of this thesis.
Chair, Thesis Committee Date

ACKNOWLEDGMENTS
I would like to express the deepest appreciation to my thesis advisor Dr. Alan Poon
and Dr. Susanne Mertens. They continually helped and guided me in my physics
research eﬀorts. Without their guidance and persistent help this thesis would not
have been possible.
I would also like to thank my committee members, Dr. Susan Lea and Dr. Joesph
Barranco, whose works demonstrate to me the epitome of physics professors and
scientists. Their guidance, lessons, and time have provided me the requirements
needed to complete my degree and thesis.
v

TABLE OF CONTENTS
Intro and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Introduction to Neutrino Physics . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Active Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Discovery of the neutrino . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Discovery of a non-zero neutrino mass . . . . . . . . . . . . . 6
1.1.3 Direct measurement of the neutrino mass . . . . . . . . . . . . 8
1.2 Sterile Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 keV-scale Sterile Neutrinos . . . . . . . . . . . . . . . . . . . . 13
1.2.3 keV-scale Sterile Neutrinos in Tritium β-decay . . . . . . . . . 15
2 The KATRIN Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The tritium source . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 The transport and pumping sections . . . . . . . . . . . . . . 20
2.1.3 The spectrometers . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.4 Aircoil system . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.5 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 MAC-E Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Transmission Function . . . . . . . . . . . . . . . . . . . . . . 29
vi

2.3 KATRIN’s search for sterile neutrinos . . . . . . . . . . . . . . . . . . 30
3 Kassiopeia 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Purpose of Kassiopeia . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 Optimization of electromagnetic design . . . . . . . . . . . . . 33
3.1.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 Investigation of systematic effects and uncertainties of KATRIN 34
3.2 Simulation organization . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Configuring Kassiopeia . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Basic configuration initialization components . . . . . . . . . . 37
3.4 Overview of Physical Modules . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Particle creation . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.2 Particle terminators . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.3 Particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.4 Field calculation methods . . . . . . . . . . . . . . . . . . . . 48
3.4.5 Data output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Simulation of non-adiabatic effects in the main spectrometer . . . . . . . . 54
4.1 Adiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Adiabatic transport . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.2 Non-adiabatic transport . . . . . . . . . . . . . . . . . . . . . 57
vii

4.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Conﬁguration settings . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Single Particle Trajectory . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Magnetic ﬁeld dependence of transmission probability . . . . . . . . . 70
4.5 Surplus energy dependence of transmission probability . . . . . . . . 72
4.5.1 Aircoil current at 50 A . . . . . . . . . . . . . . . . . . . . . . 72
4.5.2 Aircoil current at 100 A . . . . . . . . . . . . . . . . . . . . . 76
4.5.3 Aircoil current at 150 A . . . . . . . . . . . . . . . . . . . . . 79
4.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
viii

Table Page
LIST OF TABLES
4.1 For aircoil current = 50 A, the probability results for electrons that
reached the detector, term max z, reﬂected towards the spectrome-
ter’s entrance, term min z, or became trapped within the spectrom-
eter, term trapped, are displayed as a function of energy (eV). . . . . 75
A.1 The transmission probability results for electrons that reached the
detector, term max z, for θ = 0◦
− 90◦
at aircoil current 50 A . . . . . 87
A.2 The probability results for electrons reﬂected towards the spectrom-
eter’s entrance, term min z, for θ = 0◦
− 90◦
at aircoil current 50
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A.3 The probability of an electron trapped within the spectrometer, term trapped,
θ = 0◦
− 90◦
at aircoil current 50 A . . . . . . . . . . . . . . . . . . . 88
ix

− 90◦
at aircoil current 50 A . . . . 90
− 90◦
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
θ = 60◦
− 90◦
at aircoil current 50 A . . . . . . . . . . . . . . . . . . 91
− 90◦
− 90◦
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
θ = 0◦
− 90◦
− 90◦
at aircoil current 100 A . . . 96
− 90◦
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
θ = 60◦
− 90◦
x

− 90◦
− 90◦
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
θ = 0◦
− 90◦
− 90◦
at aircoil current 150 A . . . 105
− 90◦
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
θ = 60◦
− 90◦
xi

Figure Page
LIST OF FIGURES
1.1 Continuous energy spectrum of the beta electrons from radium decay.
[25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The complete (a) and zoomed in near the endpoint (b) electron energy
spectrum of tritium. With a nonzero neutrino mass the spectrum
around the endpoint energy is diﬀerent from the spectrum with a
zero neutrino mass. The end point energy is shifted toward lower
energies. [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 The Standard Model (left) shows the current theory of subatomic
particles. The neutrino minimal standard model, νMSM, (right) is
an extension of the Standard Model of particle physics. All fermions
have both left and right-handed components. [15] . . . . . . . . . . . 14
1.4 The spectrum corresponding to the light active neutrinos is repre-
sented by the curve that covers the energies E = 0 − 18 keV. The
spectrum corresponding to the sterile neutrinos is represented by the
curve that covers the energies E = 0 − 8 keV. The sterile neutrino
has mass ms = 10 keV. The superposition of these spectra is shown
in Fig. 1.5. [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
xii

1.5 Visualization of the kink in the tritium decay spectrum that is caused
by the existence of a sterile neutrino. The dashed curve shows the
spectrum associated with the light neutrinos. The solid line shows
the superposition of the spectrum associated with the light neutrinos
and heavy neutrinos. The example uses a mass of the sterile neutrino
is ms = 10 keV and mixing angle θ = 20◦
. The endpoint region is
depicted separately. [24] . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Composition of the KATRIN experiment: Windowless Gaseous Tri-
tium Source (WGTS) (a), the transport and pumping sections (b),
the pre-spectrometer (c), the main spectrometer (d), and the detec-
tor (e). [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 This is a photo of the main spectrometer. The main spectrometer
is 23.3 m long and has a diameter of 10 m. The photo shows the
spectrometer surrounded by the aircoil system. . . . . . . . . . . . . . 24
2.3 The figure shows the MAC-E filter. In the bottom, the vectors in-
dicate the electron’s momentum transformation due to the adiabatic
invariance of the magnetic orbit momentum µ in the inhomogeneous
magnetic field. Due to the electron’s momentum in the longitudinal
direction in the center of the filter, the electrons with energies lower
than |qU0| can be filtered out by the electric field because they lack
the velocity required to overcome the electrostatic barrier. [22] . . . 26
xiii

3.1 General Structure of Kassiopeia. The information produced in a
Kassiopeia simulation is stored in abstraction layers: Run, Event,
Track, and Step. The physical modules are labeled, such as KPAGE
for particle generation, KNavi for particle navigation, and KESS and
KTrack for step computation. The physical modules produce the
information stored in their respective abstraction levels shown in the
figure. [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Illustration of the exact trajectory (a) and the adiabatic trajectory
(b). In the adiabatic trajectory the guiding center position is propa-
gated, which allows larger step size. The exact particle’s position is
reconstructed afterwards. [16] . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Coils c1 and c2, with field point F, source point S, and central (ρ <
ρcen) and (ρ > ρrem) convergence regions. [13] . . . . . . . . . . . . . 50
4.1 Figure shows trajectories of stored electrons with kinetic energies of
(a) 5 keV and (b) 25 keV. Both trajectories show some degree of
non-adiabaticity. In both cases, the z-position of the reflection points
changes. As the electron’s energy increases, the more chaotic the
trajectory becomes. [26] . . . . . . . . . . . . . . . . . . . . . . . . . 58
xiv

4.2 Figure shows a test of non-adiabaticity by an investigation of the
starting condition dependence of two electron trajectories. Shown
is the distance between two trajectories when varying the starting
position by 10−14
m. (a) Linear dependence of the adiabatic motion
of a 100 eV electron. (b) Exponential dependence of a non-adiabatic
(chaotic) motion of a 15 keV electron. [26] . . . . . . . . . . . . . . . 59
4.3 The coordinate system used within Kassiopeia. The z-axis points
from source to the detector, the y-axis points towards the top, and
the x-axis points in the direction to satisfy the right handed system.
The origin (0,0,0) of the coordinate system is in the center of the main
spectrometer. The polar coordinates θ and φ are defined as shown. [16] 64
4.4 The trajectory for a particle to reach the detector is shown with
values E0 = 500 eV and θ = 51◦
. The rings around the spectrometer
represent the aircoils surrounding it. The color variation shows the
longitudinal kinetic energy of the particle. . . . . . . . . . . . . . . . 67
4.5 The 2D model of figure 4.4 is shown. . . . . . . . . . . . . . . . . . . 68
4.6 The trajectory for a particle to be reflected is shown with values
E0 = 14, 000 eV and θ = 5◦
. The rings around the spectrometer
represent the aircoils surrounding it. The color variation scale shows
the longitudinal kinetic energy of the particle. . . . . . . . . . . . . . 68
xv

4.7 The trajectory for a particle trapped in the spectrometer is shown
with values E0 = 18, 600 eV and θ = 15◦
. The rings around the
spectrometer represent the aircoils surrounding it. The longitudinal
kinetic energy of the particle is shown in the color variation scale. . . 69
4.8 With an electron starting energy E0 = 18.6 keV, the transmission
probability as a function of the magnetic field in Gauss is shown
for all electron events that terminated at the detector (term max z).
Nominal KATRIN field B = 3 G, which has aircoil currents set to
approximately 11 A. Maximum KATRIN field B = 10 G, shown by
the vertical line in the figure, when aircoil currents are set to approx-
imately 88 A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.9 The transmission properties for electrons transmitted to the detector
are shown for aircoil currents for 50, 100, and 150 A. . . . . . . . . . 73
4.10 The aircoil currents are set to 50 A and the starting angles for par-
ticles are between θ = 0◦
− 60◦
. The transmission probability as a
function of the starting energy in eV is shown. The figure shows re-
sults for electrons that reached the detector, term max z, electrons
that reflected backwards, term min z, and electrons trapped within
the spectrometer, term trapped. . . . . . . . . . . . . . . . . . . . . 74
xvi

− 60◦
− 60◦
A.1 The aircoil currents are set to 50 A and the starting angles for par-
− 90◦
xvii

− 90◦
. The transmission probability as
a function of the starting energy in eV is shown. The ﬁgure shows
results for electrons that reached the detector, term max z, electrons
A.3 The aircoil currents are set to 50 A and the plot shows the transmis-
sion probability for the electrons to reach the detector, term max z,
for all angle θ ranges: 0◦
− 60◦
, 0◦
− 90◦
, and 60◦
− 90◦
. . . . . . . . 92
− 90◦
− 90◦
xviii

sion probability of the electrons that reach the detector, term max z,
− 60◦
, 0◦
− 90◦
, and 60◦
− 90◦
. . . . . . . . 100
− 90◦
− 90◦
sion probability of the electrons that reach the detector, term max z,
− 60◦
, 0◦
− 90◦
, and 60◦
− 90◦
. . . . . . . . 107
xix

1
Introduction and Objectives
The next generation KATRIN experiment (KArlsruhe TRItium Neutrino experi-
ment) is designed to measure the neutrino mass down to mνe = 200 meV/c2
(90%
C.L.) [4]. It also has the potential to search for sterile neutrinos. KATRIN’s dimen-
sions are massive: its total length is 70 meters and the main spectrometer weighs
over 200 tons. It is located on the Karlsruhe Institute of Technology’s (KIT) cam-
pus. The experiment requires expertise in several ﬁelds of science and engineering,
such as molecular and nuclear physics, vacuum and cyrogenic technology, and so-
phisticated programs for data analysis. Many students, scientists, and engineers are
committed to working for the success of this project.
My involvement with the KATRIN experiment is with the group at Lawrence
Berkeley National Laboratory in Berkeley, California. The group is headed by
Dr. Alan Poon.
In this thesis the transmission properties are investigated in the sterile neutrino
search mode of KATRIN. In this mode the spectrometer runs at a low or zero
retarding potential and the transmission properties are diﬀerent from its normal

2
mode for the neutrino absolute mass scale measurement. The simulation software
Kassiopeia 3.0 of the Kasper framework uses Monte Carlo simulations to deter-
mine the transmission properties of the large KATRIN main spectrometer at low
retarding potentials. In the simulation, the β-electrons are created at the entrance
of the KATRIN main spectrometer, and their motions through the electromagnetic
field are tracked through the spectrometer, which is a so-called MAC-E-Filter (Mag-
netic Adiabatic Collimation combined with a Electrostatic Filter). The transmission
probability is determined by the fraction of particles reaching the detector at the
end of the spectrometer for various electromagnetic field settings. It was found that
the electromagnetic field settings have to be drastically modified compared with the
standard settings in order to adiabatically guide the electrons to the detector before
a high sensitivity sterile neutrino search with KATRIN can be realized.
In this thesis an introduction to neutrino physics is presented in the first chapter.
Chapter 2 gives an explanation of the KATRIN experiment; its technical aspects
and associated physics are illustrated. Chapter 3 goes into detail about the syntax
and structure of the Kassiopeia 3.0 software used for data analysis and simulation.
Chapter 4 includes the simulation setup, data analysis, and results from the project.
Finally, the conclusion is given in chapter 5.

3
Chapter 1
Introduction to Neutrino Physics
During the past two decades, neutrino experiments have provided compelling evi-
dence for neutrino mass through the discovery of neutrino ﬂavor oscillations [3] [6].
The well-established standard neutrino oscillation framework is comprised of three
light active neutrino mass eigenstates (Sec. 1.1.2). However there is a broad phe-
nomenology addressing additional neutrino mass eigenstates. These new states
would be predominantly sterile (i.e. would not take part in Standard Model inter-
actions) but could have a small admixture of active neutrinos. This sterile neutrino
is interesting due to its potential as a dark matter candidate (Sec. 1.2.2).
This chapter gives an overview of the basics of neutrino physics. In Sec. 1.1, a
brief history of active-neutrino physics is illustrated, and neutrino oscillations are
explained. Finally, in Sec. 1.2 the theory and motivation for sterile neutrinos are
introduced.

4
1.1 Active Neutrinos
The energy spectrum of the β-decay electrons was first investigated by James Chad-
wick in 1914 [1] [2]. He observed a continuous spectrum, as shown in Fig. 1.1. The
observation could not be explained by the two-body nuclear model popular at the
time due to the lack of energy and angular momentum conservation. If the β-decay
was an electron emission, then the energy of the emitted electron should equal the
energy difference between the initial and final nuclear states. The continuous spec-
trum suggests that energy is lost in the decay process.
Figure 1.1: Continuous energy spectrum of the beta electrons from radium decay.
[25]
In a ”desperate move” not to abandon the fundamental conservation laws the
neutrino was postulated by Wolfgang Pauli [21] in 1930 in a letter to the ”Dear
Radioactive Ladies and Gentlemen” in Tübingen. He used an electrically neutral
spin 1/2 particle emitted during the decay to share the decay energy with the elec-

5
tron. The three-body final state solves the energy conservation issue. In 1934 Enrico
Fermi [11] developed a theory to describe the β-decay as
n → p + e−
+ ¯νe (1.1)
The neutron decays into a proton, electron, and electron anti-neutrino. The anti-
neutrino is the anti-particle of the neutrino. It has a right-handed helicity, while
neutrinos are left-handed. Both interact only via weak forces, and as a result their
interactions have small cross sections.
1.1.1 Discovery of the neutrino
A neutrino was first detected in 1956 by Clyde Cowan and Frederick Reines, more
than 20 years after Fermi’s neutrino theory. ”Herr Auge” located at the Hanford
reactor site as the famous Poltergeist project was the first detector to ”see” the
neutrino. The background in this experiment overwhelmed the signal. Cowan and
Reines definitely proved the existence of the neutrino with an improved detector
at the Savannah river reactor [5] [23]. The neutrino was detected by the classical
inverse β-decay
¯νe + p → n + e+
(1.2)
The Savannah River detector consisted of liquid scintillator tanks with a Cadmium-

6
loaded (Cd) water target. The positron from Eqn. 1.2 annihilates with an electron
and produces two gammas immediately. The neutron is thermalized on a millisec-
ond timescale and captured by the 108
Cd, which releases gammas when the ex-
cited 109
Cd-state decays to the ground state. The gammas are detected by their
Compton-scattered electrons. The scintillation light from the energy deposited by
the Compton electrons is detected by photomultiplier tubes. This delayed light
signal represents a distinct signature of a neutrino interacting in a detector.
1.1.2 Discovery of a non-zero neutrino mass
In the Standard Model of particle physics there are three different flavors of neu-
trinos: the electron, muon, and tau neutrinos. The effect of neutrinos changing
their flavors in a vacuum is called neutrino oscillation. The discovery of neutrino
oscillations proved that neutrinos are not massless, since this quantum mechanical
effect requires neutrinos to be massive. The first indication of neutrino oscillation
occurred at the Homestake experiment in South Dakota [6] [3], but was validated in
the atmospheric neutrino data by the Super-Kamiokande experiment in 1998 [14].
A neutrino flavor eigenstate of the weak interaction, with α = e, µ, τ is defined
as a superposition of mass eigenstates, with k = 1,2,3.
|να =
3
k=1
Uαk |νk (1.3)

7






νe
νµ
ντ






=






Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3












ν1
ν2
ν3






(1.4)
where U is a unitary matrix [20]. Consider the case in which an electron neutrino
νe is produced. The state at t = 0 can be written as
|ν(t = 0) = |νe = Ue1 |ν1 + Ue2 |ν2 + Ue3 |ν3 (1.5)
The mass eigenstates νk are the physical states that propagate through space
with an energy Ek and momentum pk. After some time t > 0 the state evolves to
|ν(t > 0) = |νe = Ue1e−iE1t
|ν1 + Ue2e−iE2t
|ν2 + Ue3e−iE3t
|ν3 (1.6)
|να(t) =
3
k=1
Uαke−iEkt
|νk , (1.7)
where the quantities expressed are in natural units (c = 1, = 1). Therefore one
finds a non-vanishing probability of measuring the neutrino in a different flavor than
at the origin t = 0. The neutrino interacts only by the weak force and can only be
detected in a flavor eigenstate.
The probability P of finding a flavor state νβ is given by a projection of the state

8
|να(t) onto the ﬂavor state |νβ .
P(να→β(t)) = | νβ|να(t) |2
=
3
k=1
U∗
βkUαke−iEkt
2
(1.8)
=
kj
U∗
αkUβkUαjU∗
βje−i(Ek−Ej)t
(1.9)
Using a relativistic approximation assumption t ≈ L, one expresses the proba-
bility as
P(να→β(L)) =
kj
U∗
αkUβkUαjU∗
βje−i
∆m2
jkL
2E (1.10)
3
k=1
U∗
βkUαk
2
= δβα (1.11)
with ∆m2
jk = m2
j −m2
k corresponding to the mass splittings, L denoting the distance
between the source and detector, and E corresponding to the energy of the neutrino.
It holds that for vanishing masses of neutrinos m1 = m2 = m3 the observed neutrino
ﬂavor states could not exist.
1.1.3 Direct measurement of the neutrino mass
Presently, the most sensitive direct searches for the electron neutrino mass are based
on the investigation of the electron spectrum of tritium β-decay.
3
H →3
He+
+ e−
+ ¯νe (1.12)

9
The electron energy spectrum for a neutrino mass eigenstate mν is given by
dΓ
dE
= C · F(E, Z = 2) · p(E + mec2
)(E − E0) (E − E0)2 − m2
ν (1.13)
where Γ represents an arbitrary unit count rate, E denotes the kinetic energy of the
electron, E0 is the maximum electron energy for mν = 0, or the endpoint energy,
me is the mass of the electron, mν is the mass of the neutrino, F(Z, E) is the Fermi
function that takes into account the Coulomb interaction of the emitted electron
and the daughter nucleus, and C is a constant [4]. Normally, the speed of light c
and the Planck constant are set to unity. The normalization constant C is given
by
C =
G2
F
2π3
cos2
(ΘC)|M|2
(1.14)
with the Fermi constant GF , the Cabbibo angle ΘC, and the energy independent
nuclear matrix element |M|2
, which describes the probability of a neutron turning
into a proton inside a nucleus.
Since the electron neutrino is a superposition of mass eigenstates. The spectrum
must be a superposition of spectra corresponding to each mass eigenstate m(νi),
weighted by its fraction |Uei| within the electron ﬂavor [20],
dΓ
dE
= C ·F(E, Z = 2)·p(E +mec2
)(E −E0)
i
|Uei|2
(E − E0)2 − m(νi)2 (1.15)
By measuring and analyzing the electron energy spectrum of the tritium β-

10
decay near the endpoint energy, as shown in Fig. 1.2, the mass of the neutrino can
be determined.
Figure 1.2: The complete (a) and zoomed in near the endpoint (b) electron energy
spectrum of tritium. With a nonzero neutrino mass the spectrum around the end-
point energy is diﬀerent from the spectrum with a zero neutrino mass. The end
point energy is shifted toward lower energies. [4]
Tritium has the following advantages as a β-emitter in ν-mass investigations [4]:
1. Tritium has a low endpoint energy of E0 = 18.6 keV.
2. Tritium has a relatively short half life t1/2 = 12.3 years.
3. The hydrogen isotope tritium and its daughter, the 3
He+
ion, have simple
electron conﬁgurations. The atomic corrections for the β decaying atom and
corrections due to interaction of the out-going β-electron with the tritium
source can be calculated simply.

11
4. The inelastic scattering of out-going β-electrons within the β source is small
5. The tritium β decay is a super-allowed nuclear transition. Therefore, no cor-
rections from the nuclear transition matrix elements have to be taken into
account.
Using this method of analyzing the tritium β-decay spectrum and combining the
data from the Mainz and Troitsk experiments [4], an upper limit to the neutrino
mass of
mν < 2.0eV (1.16)
was determined.
1.2 Sterile Neutrinos
Besides the known light active neutrinos, many theoretical models predict the exis-
tence of sterile neutrinos. These would not even take part in the weak interaction
but could mix with the active neutrinos, which would make detection possible. In
this thesis sterile neutrinos in the keV mass range are investigated, which could
manifest themselves in a tritium β-decay.

12
1.2.1 Theoretical Framework
The well-established neutrino oscillation framework from previous sections comprises
three light active neutrino mass eigenstates, which would be left-handed. However,
there have been observations for a fourth neutrino mass observed to be right-handed,
which is called a sterile neutrino [19]. The neutrino oscillation framework would be
modiﬁed to include a fourth neutrino mass eigenstate.
|να =
4
k=1
Uαk |νk (1.17)
As opposed to the active neutrinos, the sterile neutrino does not participate in weak
interactions. This makes the neutrinos very diﬃcult to detect.
However, it is not impossible to detect sterile neutrinos. The detection of these
neutrinos depends on the their ability to mix with active neutrinos. In a seesaw-type
model, the neutrino mass matrix is given as
¯νL ¯νR



0 mD
mD M






νL
νR


 (1.18)
where the indices L and R represent the left and right-handed neutrinos, M is the
mass of νR, and mD = yv (y is the Yukawa coupling and v = 174 GeV) [24]. The
eigenvalue for the active neutrino is mactive =
m2
D
M
. If the right-handed neutrino
exists, then it may mix with the light active neutrino via coupling to the Higgs.

13
1.2.2 keV-scale Sterile Neutrinos
There are typically three mass scales considered for sterile neutrinos. The lightest
sterile neutrinos are expected in the sub-eV scale. The heavy sterile neutrino (GeV)
is postulated in the see-saw mechanism and explains the lightness of the active
neutrinos. And ﬁnally, there are hints that there may be keV-scale sterile neutrinos
[8][9]. These are good candidates for both warm and cold dark matter.
Dark Matter Candidate
It is theorized that the universe is composed of 68.3% dark energy, 26.8% dark
matter, and 4.9% baryonic matter [19]. The nature of dark matter is continually
being sought in physics because the Standard Model does not provide a suitable
dark matter candidate. A candidate must be electrically neutral, at most weakly
interacting, and stable with respect to the age of the universe [19]. Fig. 1.3 shows
the Standard Model of elementary particles, but extended to include the possibility
of three right-handed sterile neutrinos.
At the time of structure formation in the early universe, light neutrinos had
relativistic velocities. This led to the elimination of small-scale structures, which
disagrees with current observations [7] [17]. As a result, active neutrinos, also known
as hot dark matter, were eliminated as candidates. The most favored candidate is
a cold dark matter particle called a weakly interacting massive particle (WIMP).
WIMPs are actively being sought in direct and indirect experiments, but no solid

14
Figure 1.3: The Standard Model (left) shows the current theory of subatomic par-
ticles. The neutrino minimal standard model, νMSM, (right) is an extension of the
Standard Model of particle physics. All fermions have both left and right-handed
components. [15]
evidence has been produced.
Interestingly, heavy sterile neutrinos in the keV mass range are a candidate
for both warm and cold dark matter. The warm and cold dark matter scenarios
ﬁt well with large-scale structure. The next generation of β-decay experiments can
search for keV-scale sterile neutrinos. Tritium β-decay is the most advantageous due
to its being super-allowed, which gives a precise theoretical spectral shape. Also,
the 12.3 year half-life of tritium is short, allowing high signal rates with relatively
small amounts of tritium and consequently low source densities, which minimizes
systematic eﬀects due to inelastic scattering in the source. Finally, the low endpoint
energy of 18.6 keV provides the search range for keV-scale sterile neutrinos up to
that mass [19].

15
1.2.3 keV-scale Sterile Neutrinos in Tritium β-decay
The super-allowed β-decay of tritium is shown in Eqn. 1.12. The decay rate is
given in Eqns. 1.13 and 1.15. The three light mass eigenstates are indistinguishable
relative to the keV-scale sterile neutrino of interest. The mass eigenstates can be
written as a single effective light neutrino mass
m2
light =
3
k=1
|Uek|2
m(νk)2
Let us assume a heavy mass eigenstate ms and light mass eigenstate mlight. The
superposition of the mass eigenstates no longer forms a single effective mass. As a
result, the differential spectrum dΓ/dE can be rewritten as a function of the two
mass terms ms and mlight
dΓ
dE
= cos2
θ
dΓ
dE
(mlight) + sin2
θ
dΓ
dE
(ms) (1.19)
The mixing angle θ between the light and heavy states determines the size of the
effect on the observed β-electron spectrum.
If the heavy sterile neutrinos mix with the light active neutrinos, there can be
detectable traces in β-decay experiments. The spectra for the light active neutrinos
and the heavy sterile neutrinos are shown in Fig. 1.4. The tritium spectrum would be
a superposition of the spectrum associated with the light neutrino and the spectrum

16
corresponding to the heavy neutrino. This is seen visually as a ”kink” in the β-
spectrum as shown in Fig. 1.5 and should be visible in the KATRIN experiment.
Figure 1.4: The spectrum corresponding to the light active neutrinos is represented
by the curve that covers the energies E = 0 − 18 keV. The spectrum corresponding
to the sterile neutrinos is represented by the curve that covers the energies E = 0−8
keV. The sterile neutrino has mass ms = 10 keV. The superposition of these spectra
is shown in Fig. 1.5. [20]

17
Figure 1.5: Visualization of the kink in the tritium decay spectrum that is caused by
the existence of a sterile neutrino. The dashed curve shows the spectrum associated
with the light neutrinos. The solid line shows the superposition of the spectrum
associated with the light neutrinos and heavy neutrinos. The example uses a mass
of the sterile neutrino is ms = 10 keV and mixing angle θ = 20◦
. The endpoint
region is depicted separately. [24]

18
Chapter 2
The KATRIN Experiment
KATRIN is a next generation experiment to measure the neutrino mass with a sen-
sitivity of m(νe) = 200 meV/c2
(90% C.L.) by precisely analyzing the area near
the endpoint of the tritium β-electron energy spectrum. The sensitivity will be an
improvement of one order of magnitude compared with past experiments [4]. KA-
TRIN uses the MAC-E-Filter principle, which is explained in the following sections,
in order to explore the sub-eV region of neutrino masses.
In this chapter an overview of KATRIN will be given. After an introduction to
each component of the KATRIN experiment, the MAC-E-Filter will be explained.
Finally, the transmission function will be introduced, and the motivation for KA-
TRIN’s potential in the search for sterile neutrinos will be discussed.

19
2.1 Experimental Overview
The basic idea of the KATRIN experiment is to implement a molecular tritium
source of the highest stability and luminosity in combination with a variable retard-
ing potential. The filter transmits only those electrons which have more energy than
the filter retarding voltage to a detector for counting. By measuring the count rate
for different retarding voltages, the shape of the integrated energy spectrum can be
determined. Figure 2.1 shows a schematic view of the KATRIN setup.
Figure 2.1: Composition of the KATRIN experiment: Windowless Gaseous Tritium
Source (WGTS) (a), the transport and pumping sections (b), the pre-spectrometer
(c), the main spectrometer (d), and the detector (e). [20]
According to Fig. 2.1, the KATRIN experiment with a total length of 70 m can
be separated into four main parts:
• The Windowless Gaseous Tritium Source (WGTS) described in Sec. 2.1.1
• The transport and pumping section including the Differential Pumping Section
(DPS) and the Cryogenic Pumping Section (CPS), described in Sec. 2.1.2

20
• The spectrometers with the smaller pre-spectrometer and the large main spec-
trometer described in Sec. 2.1.3
• The detector counting the transmitted electrons as described in Sec. 2.1.5
2.1.1 The tritium source
The WGTS has a cylindrical geometry with a length of 10 m and radius of 45 mm
which is filled with ultra-cold, highly isotropically pure (> 95%) molecular tritium
gas at a temperature of 27 K. The cold tritium allows a column density of up to
ρd = 5 × 1017
molecules/cm2
. The tritium is injected through an injection port
located at its center and diffuses over a length of 5 m from both ends of the WGTS.
These systems contain turbomolecular pumps to reduce the tritium gas at both
ends of the WGTS by more than 99%. The collected tritium is led to a purification
section before it is reinjected into the WGTS. The constant pumping and reinjecting
of the tritium lowers the density. The transport time of tritium molecules through
the WGTS is of the order of 1 s. The decay probability of a single tritium molecule
is about 10−9
. Electrons from the β-decay process are adiabatically guided by the
WGTS magnetic field Bsource = 3.6 T to both ends of the tube.
2.1.2 The transport and pumping sections
The task of the transport section is to guide the β-decay electrons adiabatically
from the WGTS to the spectrometers. Since the spectrometer section must be

21
tritium free, the tritium flow must be reduced from an injection rate of 1.8 mbar· /s
to 10−14
mbar· /s at the end of the transport section. This is achieved through
differential and cyrogenic pumping. In order to transport the electrons emitted in
the WGTS through the transport and pumping section adiabatically, the beam tube
is surrounded by superconductive magnets that generate magnetic fields up to 5.6
T.
Differential Pumping Section (DPS)
The DPS contains four turbomolecular pumps to reduce the tritium flow. The beam
tube is designed as a ”dog-leg” chicane to avoid beaming effects of the neutral tritium
molecules. Together with the turbomolecular pumps at the end of the tritium source,
the tritium flow can be reduced by seven orders of magnitude.
Cyrogenic Pumping Section (CPS)
The inner surface of the CPS is cooled down to about 3 K using liquid helium. This
allows for the trapping of single tritium molecules on the surface of the tilted beam
tube, which is covered with argon snow to passively cryosorb tritium molecules.
Any molecule hitting the argon frost surface is cyrosorbed and fixed. The CPS
will accumulate about 1017
tritium molecules per day and is expected to reduce the
tritium flow by another seven orders of magnitude.

22
2.1.3 The spectrometers
The spectrometer section consists of two electrostatic retarding filters: the pre- and
main spectrometers, which are of the MAC-E filter type. The pre-spectrometer acts
as a pre-filter, reflecting electrons 300 eV below the endpoint. The pre-spectrometer
reduces the electron flux from the tritium souce into the main spectrometer by
about six orders of magnitude to reduce scattering on residual gas molecules in the
eXtreme High Vacuum (XHV) of the main spectrometer (10−11
mbar or less), thus
minimizing background activity. All electrons transmitted through this stage are
guided to the main spectrometer for precise energy analysis.
The large dimensions of the main spectrometer allow it to operate as a precise
high energy filter. The highest electrostatic potential is located in the central plane,
perpendicular to the beam axis. This location is commonly referred to as the ”ana-
lyzing plane.” The β-electrons from the WGTS are guided along the magnetic field
lines into the spectrometer. Due to their isotropic emission, the electrons orbit the
field lines. The cyclotron motion is fully transformed into motion parallel to the
magnetic field lines in order to achieve high energy resolution. This is achieved by
dropping the magnetic field by four orders of magnitude. The magnetic flux Φ is
conserved, and as a result, the cross section of the flux tube in the center plane is
4 orders of magnitude larger than the entrance. This explains the large size of the
spectrometer (length L = 23.8 m, diameter d= 9.8 m, cross sectional area A = 650
m2
, and volume V = 1,400 m3
).

23
2.1.4 Aircoil system
To compensate Earth’s magnetic field, the spectrometer is surrounded by a huge
aircoil system. The magnetic field in the analyzing plane is dominated by the two
pre-spectrometers (ps1,ps2) and the two detector solenoids. Due to the distance of
the coils to the analyzing plane (more than 12 m), their magnetic field contribution
is only about B = 0.179 mT [16]. The low field values lead to problems:
• At the center of the main spectrometer, the flux tube has a radius more than
11 m, which does not fit in the spectrometer anymore.
• The Earth’s magnetic field is not negligible. Its horizontal field Bhor =
20.6 × 10−6
T and vertical component Bver = 43.6 × 10−6
T have strong in-
fluences on the orientation and strength of the magnetic field in the analyzing
plane.
The aircoil system helps solve these issues. The system consists of two units:
the Earth magnetic field compensation system (EMCS) and the low-field coil system
(LFCS). The EMCS compensates for the vertical and horizontal, non-axially sym-
metric components of Earth’s magnetic field. The EMCS consists of 10 horizontal
current loops and 16 vertical ones. The LFCS produces an axially symmetric mag-
netic guiding field with 14 large coils, which are shown in Fig. 2.2. They surround
the spectrometer and are individually powered in order to optimize and precisely
adjust the magnetic field inside the spectrometer.

24
Figure 2.2: This is a photo of the main spectrometer. The main spectrometer is 23.3
m long and has a diameter of 10 m. The photo shows the spectrometer surrounded
by the aircoil system.

25
2.1.5 Detector
All β-electrons passing the retarding potential of the main spectrometer are re-
accelerated to their initial energy and magnetically guided by the 2-solenoid trans-
port system (DTS) to the focal plane detector (FPD). The FPD is located inside a
separate superconducting solenoid with a large warm bore.
The detector is a semi-conductor based silicon PIN diode. Its main goal is to
detect transmitted electrons with a detection efficiency of > 90%. The electrons
passing the analyzing plane at different radii will experience different retarding po-
tentials, and to account for this the detector is subdivided into 148 pixels to achieve
good spatial resolution. This is accomplished with 12 concentric rings subdivided
azimuthally into 12 pixels each and the center ”bullseye” is divided into 4 segments.
This allows for precise mapping of the inhomogeneities of the retarding potential.
Each pixel measures an independent tritium β-spectrum, which has to be corrected
for the actual retarding potential.
The detector is situated in a superconducting magnet of about 3 - 6 T. The
magnet is adjacent to the pinch magnet that provides the maximum magnetic field.
All electrons that started in the source with an angle greater than the maximum
angle θmax = 51◦
will be reflected by the pinch magnet. This is required because the
electrons with large angles perform larger cyclotron motions, which increases path
length and scattering probability.

26
2.2 MAC-E Filter
KATRIN’s measurement principle relies on the MAC-E filter (Magnetic Adiabatic
Collimation combined with an Electrostatic filter) to enable a high precision mea-
surement of the electron energy near the tritium endpoint energy. The filter principle
is shown in Fig. 2.3. Two superconducting solenoids produce magnetic fields that
Figure 2.3: The figure shows the MAC-E filter. In the bottom, the vectors indicate
the electron’s momentum transformation due to the adiabatic invariance of the
magnetic orbit momentum µ in the inhomogeneous magnetic field. Due to the
electron’s momentum in the longitudinal direction in the center of the filter, the
electrons with energies lower than |qU0| can be filtered out by the electric field
because they lack the velocity required to overcome the electrostatic barrier. [22]

27
initially guide the electrons from the tritium source. The electrons emitted isotrop-
ically in the tritium source orbit around the magnetic field lines. As we approach
the center of the spectrometer the magnetic field drops to a minimum magnetic
field Bmin. The magnetic field changes slowly along the longitudinal axis, and the
transformation can be approximated as an adiabatic process in which the magnetic
moment µ is kept constant
µ =
E⊥
|B|
(2.1)
As the magnetic field changes, the perpendicular kinetic energy component E⊥
changes proportionally as well. This is how most of the perpendicular energy is
transformed into longitudinal motion. The total kinetic energy can be written as
Ekin = E⊥ + E (2.2)
Only the longitudinal part E is analyzed by the electrostatic filter. With the
electrons traveling parallel to the magnetic field lines, those with less energy than
the retarding potential E = |qU0| are filtered and reflected backwards. The electrons
with greater energy are accelerated and travel toward the detector because as the
electron moves from low to high magnetic fields its longitudinal energy is transformed
into the original transverse energy.
An electron passing through the analyzing plane depends not only on its starting
energy but also on its starting angle. The maximum magnetic field is applied at the

28
pinch magnet close to the detector but not at the source. The electrons starting
with an angle larger than θmax are reflected by the maximum magnetic field. To
calculate the maximum acceptance angle,
Es
⊥
|B|s
=
Ep
⊥
|B|p
(2.3)
where the index indicates position (s = source and p = pinch magnet). An electron
is reflected by the maximum magnetic field, if all its kinetic energy is transverse at
the pinch magnet Ep
= Ep
⊥ or earlier. With a starting pitch angle θ,
E⊥ = E sin2
(θ) (2.4)
one finds
E sin2
(θmax)
|B|s
=
Ep
⊥
|B|p
→ sin(θmax) =
|B|s
|B|p
=
Bs
Bmax
(2.5)
Using values |B|s
= 3.6 T and |B|p
= 6 T, one obtains θmax = 51◦
.
Whether an electron can or cannot pass the analyzing plane depends only on
its starting energy and starting angle θ. A β-electron created with a kinetic energy
larger than the retarding potential, but greater than θmax might be reflected. An
electron may have some transverse energy remaining in the analyzing plane. The

29
finite energy resolution of a MAC-E filter due to a small remaining transverse mo-
mentum at the analyzing plane depends on the minimum and maximum magnitude
of the magnetic field, and the filtering energy is given as
∆E =
Bmin
Bmax
· E0 = .93 eV (2.6)
with the design KATRIN values Bmin = 3 × 10−4
T, Bmax = 6 T and E0 = 18.6
keV.
2.2.1 Transmission Function
The transmission function T of a MAC-E filter is analytically given for electrons
with a maximum accepted starting angle stated in equation 2.5. It describes the
probability of transmitting electrons through the MAC-E filter. From equation 2.1,
the relative sharpness ∆E/E of this filter is given by the ratio of the minimum
magnetic field in the analyzing plane Ba and the maximum magnetic field Bmax
between the β-electron source and the spectrometer.
∆E
E
=
Ba
Bmax
(2.7)

30
Following 2.1, 2.3, and 2.7, the normalized transmission function of the MAC-E
filter is
T(E, qU) =



0 , E − eU < 0
1− 1−E−eU
E
· Bs
Ba
1− 1−∆E
E
· Bs
Ba
, 0 ≤ E − eU ≤ ∆E
1 , E − eU > ∆E
(2.8)
with a retarding potential U, isotropic electron source energy E, the magnetic field
in the analyzing plane Ba, and electron charge e.
The transmission function depends only on the two field ratios Ba/Bmax and
Bs/Ba. The total width ∆E of the transmission function from T = 0 to T = 1 is
given by Eqn. 2.7. The shape of T in this interval is determined by Bs/Ba, as the
ratio defines the maximum accepted electron starting angle θmax. The transmission
function does not account for interactions at the source.
2.3 KATRIN’s search for sterile neutrinos
Tritium β-decays have an endpoint energy of 18.6 keV. After KATRIN has achieved
its primary goal, it can extend its physics reach to search for keV-scale sterile neu-
trinos. In this case, the entire tritium β-decay spectrum is of interest and therefore
the main spectrometer would operate at very small retarding energies to allow the
electrons of the interesting part of the spectrum to reach the detector. In this thesis
it has been shown that to guarantee an adiabatic transport of electrons with high

31
surplus energy through the spectrometer, the magnetic ﬁeld at the center of the
spectrometer has to be increased by a factor of 3-4 times that of KATRIN’s normal
measuring mode. This is achieved by making use of the large air coil system around
the main spectrometer [18] [19].

32
Chapter 3
Kassiopeia 3.0
The simulation package Kassiopeia [20] [22] was developed by members of the
KATRIN collaboration. The most recent version 3.0 was used in the simulations
performed in this thesis. The software is written in the C++ programming language.
It is able to track trajectories of multiple charged particles in electromagnetic fields
using Monte Carlo simulations. It is very customizable by the user by offering plug-
in modules for physical effects like energy losses due to synchrotron radiation or
scattering events. Configurations and settings of the simulations are stored in XML
language files, which are handed to Kassiopeia.
3.1 Purpose of Kassiopeia
The Monte Carlo simulations are performed with Kassiopeia for the following
purposes:

33
3.1.1 Optimization of electromagnetic design
Kassiopeia is a tool for the optimization of electromagnetic design of KATRIN.
The electric potential as well as the electric and magnetic fields in the KATRIN
setup can be calculated using Kassiopeia using various calculation methods. In
my project the entire β-spectrum is of interest. The use of zero retarding potential
meant that the electric field calculation was unnecessary. In order to calculate the
magnetic field a Legendre polynomial expansion is used (more detail in Sec. 3.4.4).
3.1.2 Monte Carlo simulations
Aside from precise, fast field calculation methods, Kassiopeia provides algorithms
to compute particle trajectories in electromagnetic fields down to the level of ma-
chine precision. The tool also allows the user to perform Monte Carlo simulations
of specific measurements. The user can make full use of Kassiopeia to better un-
derstand the results of test experiments during the design and commissioning phase
of KATRIN. The test measurements of interest in this project are the transmission
properties of the spectrometer.

34
3.1.3 Investigation of systematic effects and uncertainties of KA-
TRIN
Kassiopeia is part of a bigger software package called Kasper. Kasper provides
a detailed tritium source model that allows simulations of the actual neutrino mass
measurements. The source model includes the final state distribution of tritium,
scattering in the source, and more. Kassiopeia also includes classes for fitting the
integrated tritium β-spectrum, which can be used to determine systematic effects
and neutrino mass sensitivity.
With the functionality of Kasper it is possible to investigate systematic effects,
like shifts due to magnetic fields, electric potentials, etc. KATRIN’s statistical
uncertainty can be studied using Kassiopeia.
3.2 Simulation organization
An overview of the basic structure and organization of Kassiopeia is given. The
description is divided into two parts: the information produced during simulation
and how that information is produced.
3.2.1 Physical States
Kassiopeia is divided into four levels of abstraction representing physical states
of the simulated experiment. Each of these abstraction levels offers bindings to the

35
configuration layer to allow user-defined manipulations of settings, such as which
information is written to the output file with the simulation results. Fig. 3.1 shows
the workflow of the abstraction levels during a simulation.
Figure 3.1: General Structure of Kassiopeia. The information produced in a
Kassiopeia simulation is stored in abstraction layers: Run, Event, Track, and Step.
The physical modules are labeled, such as KPAGE for particle generation, KNavi
for particle navigation, and KESS and KTrack for step computation. The physical
modules produce the information stored in their respective abstraction levels shown
in the figure. [20]

36
Runs
A run is the highest level of abstraction in the Kassiopeia software. It represents
an executed Kassiopeia program of a given experimental setup. Each run includes
a user-defined number of events. Multiple program instances can be submitted and
run in parallel, and the output files can be merged.
Events
Kassiopeia events represent particles. The total number of events created in the
simulation run is defined by the user in the configuration file. The run creates
these events in a loop, following the single trajectories and particle states using
the abstraction levels below, until all created subparticles and the particle itself are
terminated.
Tracks
A track is the representation of a physical particle including physical properties for
the initial and final states of the particle. Multiple tracks can thereby be assigned
to one event since a tracked particle can create sub-particles (e.g. through inelastic
scattering) belonging to the same event.

37
Steps
Steps are the lowest abstraction level in Kassiopeia. They represent the current
state of a particle. By handing each step’s information on the initial or final state
to the output, the particle’s physical properties as well as its trajectory between
creation and termination can be examined, which is especially useful for debugging.
3.3 Configuring Kassiopeia
Kassiopeia is set up by writing a configuration file in a markup language heavily
based on the EXtensible Markup Language (XML). This section explains and shows
examples of the configuration components using the project’s configuration file.
3.3.1 Basic configuration initialization components
The simulation tool requires some basic components in the configuration file. The
following contains the essential parts needed for a simulation, including sample
syntax.
• Geometry Information: In the configuration file all magnets and electrodes
used must be defined and initialized in the geometry tag to allow Kassiopeia
to calculate the electric and magnetic fields during simulation. The KATRIN
geometry, including all magnets and electrodes, is predefined in the KATRIN
Specific Code (KSC) section of Kassiopeia.

38
The following excerpt is an example of the KATRIN geometry. The geom-
etry is put into a world space formed by a cylindrical shape by nesting a
global assembly tree into the cylinder. In this thesis, the pre-defined geome-
try configurations ”axial main spec assembly” and ”magnet sds assembly” for
the KATRIN setup are used:
<geometry>
<disk_surface name="disk_surface" z="{0.}" r="{0.1}"/>
<cylinder_space name="world_space" z1="-50" z2="50" r="20"/>
<space name="world" node="world_space">
<space name="magnet_sds" tree="magnet_sds_assembly"/>
<space name="axial_main_spec" tree="axial_main_spec_assembly"/>
</space>
</geometry>
• Simulation Information: The single Kassiopeia simulation settings are
defined inside a Kassiopeia tag and arranged together in a command group
which is attached to a geometry space.
– Field Solvers: Field solvers are used to calculate the magnetic or elec-
tric fields according to the assembly specified in the geometry section.
In order to calculate the KATRIN magnetic field, an electromagnet field

39
solver must be configured to calculate the fields of the geometry shapes
marked with a magnet tag. This is an example of how to configure the
field solvers:
<ksfield_electromagnet
name="field_magnet_sds"
file="MagnetSDSMagnets.kbd"
system="world/magnet_sds"
spaces="world/magnet_sds/@magnet_tag"
>
<zonal_harmonic_field_solver/>
</ksfield_elecromagnet>
The nested zonal harmonic field solver, as described in Sec. 3.4.4, can be
configured for precision and calculation time.
– Generators: In order to create a Kassiopeia particle generator, value
generators for the particle’s direction, position, energy, and starting time,
as described in Sec. 3.4.1, are composed of a single composite generator:
<ksgen_composite_generator name = "particle_generator"
energy = "energy_generator" position = "position_generator"
direction = "direction_generator" time = "time_generator"/>

40
– Terminators: Terminators can be chosen from the complete terminator
list described in Sec. 3.4.2 and are created using an individual terminator
tag and required parameter setting. Below is an example which termi-
nates the particle after 106
steps:
<ksterm_max_steps name="term_max_steps" steps="{10e6}"/>
– Trajectory Settings: Kassiopeia has two options between the ex-
act or the adiabatic tracking method, as described in Sec. 3.4.3, for the
trajectory. The most common settings use an eighth order Runge-Kutta
integrator to solve the ordinary diﬀerential equations written in Sec. 3.4.3.
Below is an example of the exact method setting that calculates the cy-
clotron trajectory motion using a step size of 1/64:
<kstraj_trajectory_exact name="trajectory_exact">
<integrator_rk8 name="integrator_rk8"/>
<term_propagation name="term_propagation"/>
<control_cyclotron name="control_cyclotron" fraction="{1./64.}"/>
</kstraj_trajectory_exact>
– Navigation Settings: A space navigator must be deﬁned for determin-
ing whether a space was entered or left during the calculated step using

41
tolerance parameters to control its precision:
<ksnav_space name="nav_space" tolerance="1.e-3"/>
<ksnav_surface name="nav_surface"/>
– Output Information: In the configuration file, the particle values
needed in the output ROOT file as described in Sec. 3.4.5 can be called
and grouped. The output file’s path and filename can be specified as
well. Below is an example:
<ks_component_member name="output_step_final_particle"
field="final_particle" parent="step"/>
<ks_component_group name="output_step_world">
<component_member name="position" field="position"
parent="output_step_final_particle"/>
</ks_component_group>
<kswrite_root name="write_root" path="[PATH]" base="[FILE]"/>
– Command Groups: After defining the single components of the con-
figuration, the settings can be arranged to command groups which can
be attached to a space defined in the geometry. Using multiple command
groups attached to separate spaces, different physical properties can be

42
simulated while writing only the values needed in the individual space to
the output ﬁle:
<ksgeo_space name="space_world" spaces="world">
<command parent="root_trajectory" field="set_trajectory"
child="trajectory_exact"/>
</ksgeo_space>
– Simulation Settings: In the simulation settings, the simulation is set
up deﬁning the number of simulated runs, the number of particle events,
the seed value to initialize the Kassiopeia unit that produces random
values, the space that contains trajectory and output information, and
navigation:
<ks_simulation
run="[RUN]"
seed="[SEED]"
events="[NEVENTS]"
spaces="space_world"
generator="entrance_uniform"
space_navigator="nav_space"
surface_navigator="nav_surface"

43
writer="write_root"
/>
3.4 Overview of Physical Modules
In this section, the modules that are responsible for the creation, tracking, and
detection of particles within Kassiopeia will be explained. Also, the magnetic
field solving method will be presented.
3.4.1 Particle creation
The module responsible for particle generation is the KAssiopeia PArticle GEnerator
(KPAGE). The generic parameters: position, energy, time, and direction determine
the starting conditions of a particle. The user can adjust KPAGE to create par-
ticles at a fixed position, homogenously distributed across a surface, in a volume,
or adjusted according to the tritium source dynamics. Particles can also be cre-
ated at fixed time with a constant rate or exponential decay time distribution. A
particle’s energy may be monoenergetic, equally or Gaussian distributed within a
defined interval, or adjusted to a decay energy spectrum like radon, krypton, or tri-
tium. Finally, the starting direction can be chosen to be fixed, isotropic, or emitted
isotropically from a surface or angularly distributed like an e-gun.

44
Particle generators
The particle generators require values for energy, position, direction, and time. The
following list of particle generators were used in the course of this thesis:
• Energy
– Composite: The composite energy generator can generate particles with
a fixed value of initial kinetic energy or a range of initial values. For
simulating the transmission probability as a function of initial energy,
ranged values between a lower and upper limit were used. For simulating
transmission probability as a function of magnetic field, a fixed value for
the initial energy was used.
• Direction
– Composite: The composite direction generator takes two parameters
for the polar and azimuthal angles. The angles can be fixed or ranged
between a lower and upper limit.
• Position
– Cylindrical: Particles are created in a cylindrical volume described by
a radius, polar angle, and z values. The values can be fixed or ranged
between a lower and upper limit.
• Time

45
– Composite: Since the time information was not relevant for this thesis,
the only parameter used is a fixed time starting at 0 seconds.
3.4.2 Particle terminators
Terminators define rules and conditions that are checked at the beginning of each
step, and they determine whether the calculation of a track should be continued or
be canceled. The following lists and describes the terminators in Kassiopeia used
in this thesis.
• Min z and max z: Termination occurs if the particle’s z-position is less than
or greater than the specified value.
• Max steps: Terminates after calculating a specified number of steps. This
saves computation time in certain cases.
• Trapped: Terminates if the particle performs a specified number of axial
turns. A particle can be trapped indefinitely and computation time can be
saved before maximum number of steps are reached.
3.4.3 Particle tracking
KTrack is a program package that computes particle trajectories and provides two
main computation methods. The first method is an exact trajectory calculation
using the Lorentz force. The second method uses the adiabaticity of the motion,

46
Figure 3.2: Illustration of the exact trajectory (a) and the adiabatic trajectory (b).
In the adiabatic trajectory the guiding center position is propagated, which allows
larger step size. The exact particle’s position is reconstructed afterwards. [16]
i.e. the conservation of orbital magnetic moment, to compute the position of the
guiding center of the particle. The guiding center is the center of the particle’s orbit
about the magnetic ﬁeld. The exact and adiabatic trajectory methods are shown in
Fig. 3.2. The adiabatic method has the potential to be much faster than the exact
method due to its larger step sizes.
Exact method
For exact tracking, the equation of motion
FL = q(E + v × B)
˙p = FL
(3.1)

47
is solved, where FL is the Lorentz force , q is the particle charge, E is the electro-
static field, B is the magnetostatic field, and v is the particle velocity [26]. The
most common algorithm used by Kassiopeia is an eighth order Runge-Kutta in-
tegrator (RK8). Because the RK8 can only solve first order differential equations,
the equation of motion 3.1 must be written as
˙x = v (3.2)
with p being
p =
m0v
1 − v2
c2
(3.3)
[26].
Adiabatic method
In the adiabatic method the ordinary differential equations for calculating the guid-
ing center position can be written as
˙xGC = ˆB · vL
˙vL = −
µ
γ
( |B|) + qE · ˆB
(3.4)
with the orbital magnetic moment µ and the particle velocity component parallel
to the magnetic field line vL. The advantage of using the adiabatic method over
the exact method in simulations is the large step sizes that reduce the computation

48
significantly while sustaining comparable accuracy [26]. In this thesis, I do not use
the adiabatic method since the focus of this thesis is to investigate non-adiabatic
effects.
3.4.4 Field calculation methods
Kassiopeia provides a comprehensive number of electric and magnetic field calcula-
tion methods. In this thesis, the electric field calculation was omitted to allow faster
computation times and smaller file sizes. The zonal harmonic expansion method was
used to compute the magnetic field throughout the KATRIN spectrometer.
Magnetic field calculation
The main sources of magnetic fields in KATRIN are normal conducting and super
conducting coils, which are axially symmetric. For the axially symmetric magnetic
field calculations, the simulations use a Legendre polynomial expansion method
known as zonal harmonic expansion [13]. This method can be 100-1000 times faster
than the more widely known elliptic integral method. The zonal harmonic method
has several advantages. First, the field and source equations are separated, which
means the source constant computations use only source points and parameters,
but not field point parameters. During the field computations the source constants
contain the information of the magnetic sources. Second, the method has speed
and accuracy, which makes it appropriate for charged particle trajectories. When

49
the magnetic field is computed, these properties do not allow interpolation. Third,
it is more general for practical applications, and the series formulas are easy to
differentiate and integrate [13].
Zonal Harmonic Expansion
The zonal harmonic expansion is the most appropriate method for most cases as it
is much faster compared with other methods offered by Kassiopeia.
Let’s define an arbitrary reference point on the symmetry axis with axial coordi-
nate z0 called the source point S. An arbitrary point where we want to calculate the
magnetic field will be the field point F. The field point can be defined by cylindrical
coordinates z and r, by the distance ρ between the source point S and the field point
F and the angle θ between the symmetry axis z and the direction vector connecting
the source and field points.
ρ = (z − z0)2 + r2
u = cos θ = (z − z0)/ρ
s = sin θ =
√
1 − u2 = r/ρ
(3.5)
Fig. 3.3 shows an example of an axisymmetric magnetic system with 2 coils. As-
suming the magnetic system is constrained inside a spherical shell with the source
point S as the center, there is no current and magnetization inside the sphere with

50
Figure 3.3: Coils c1 and c2, with field point F, source point S, and central (ρ < ρcen)
and (ρ > ρrem) convergence regions. [13]
center S and central convergence radius ρcen and outside the sphere with center S
and remote convergence radius ρrem. ρcen is the smallest distance between the source
point S and the coil. ρrem is the maximum distance between the source point S and
the coil. [13].
Inside a source-free region, the magnetic vector potential satisfies the Laplace
equation and can be written as an expansion in spherical harmonics. In the case of
axial symmetry, the spherical harmonics are restricted to zonal harmonics: ρn
Pn(u)
in the central region and p−(n+1)
Pn(u) in the remote region, with Pn(u) being the
Legendre polynomical of order n.
Inside the central convergence region ρ < ρcen, the magnetic field components

51
Bz and Br and the azimuthal magnetic vector potential A = Aφ can be expressed
by the following zonal harmonic central expansion [13]:
Bz =
∞
n=0
Bcen
n
ρ
ρcen
n
Pn(u) (3.6)
Br = −s
∞
n=1
Bcen
n
n + 1
ρ
ρcen
n
Pn(u) (3.7)
A = sρcen
∞
1
Bcen
n−1
n(n + 1)
ρ
ρcen
n
Pn(u) (3.8)
Using the appendix in [12], it has been checked that these formulas satisfy the fun-
damental static magnetic field equations, where A is the magnetic scalar potential:
· B = 0 × B = 0 B = × A (3.9)
The central magnetic source constants Bcen
n represent the central region’s mag-
netic field sources, such as coils and magnetic materials. The constants are propor-
tional to the higher derivatives of the on-axis magnetic field at the source point.
The convergence ratio is expressed as ρ
ρcen
[13].
In the remote region ρ > ρrem, the magnetic field can be written with the
following remote zonal expansion formulas:
Bz =
∞
n=2
Brem
n
ρrem
ρ
n+1
Pn(u) (3.10)

52
Br = −s
∞
n=2
Brem
n
n
ρrem
ρ
n+1
Pn(u) (3.11)
The coefficients Brem
n are remote source constants that represent the remote region’s
magnetic field sources. The remote zonal harmonic expansions correspond to the
multipole expansion of a magnetic field for an axisymmetric system. The first term
(n = 2) corresponds to the magnetic dipole, the second (n = 3) quadrupole, etc., and
the remote source constants are proportional to the multipole magnetic moments.
The convergence ratio is expressed as ρrem
ρ
[13].
Many source points are used in order to get double precision accuracy for the
field components. The series expansions stop if the sum of the absolute values of
the last 4 terms in the series of the field components are 1015
times smaller than
the sum of the corresponding series. Also, more source points allow the algorithm
to find the source point with the smallest convergence ratio, which will be used as
the actual field point. As a result, the computation time is reduced.
3.4.5 Data output
Kassiopeia writes the initial and final states of a particle’s tracks and steps to an
output file for analysis using CERN’s data analysis framework ROOT. The output
file is a ROOT TFile, which includes ROOT TTrees representing each output group
defined in the Kassiopeia configuration file among other TTrees needed by the
internal Kassiopeia file reader. Each particle property marked for output in the

53
configuration file is represented as a branch in the specified TTree while vector type
values are converted into three double type values: one for event, one for track, and
one for step data allowing the output file to be investigated using ROOT’s TBrowser
class. Index values are added to the output file to allow allocation of corresponding
Kassiopeia abstraction levels.
Merging output files
The simulations calculated in this thesis were executed on a multi-core grid com-
puting system. However, the single-thread design of Kassiopeia runs allows the
individual simulations to be split into many runs, each including a fraction of the
particles. Using the program called ”ROOTFileMerge,” multiple output files are
represented as multiple runs in a single output file while adapting stored step, track,
event, and run IDs to ensure readability by the Kassiopeia ROOT file readers.

54
Chapter 4
Simulation of non-adiabatic effects in the
main spectrometer
The measurement of the tritium β-decay spectrum with a source similar to that
of the KATRIN experiment will reach a statistical sensitivity to keV-scale sterile
neutrinos with mixing amplitudes down to sin2
θ = 10−8
[19]. To assess the final
sensitivity of the experiment, detailed studies of experimental systematic effects are
necessary. I explore and analyze non-adiabatic systematic effects that could affect
the transmission of β-electrons to the detector.
The previous chapters give the necessary tools and backgrounds in order to
accomplish this task. In this chapter the motivation and setup for the task are de-
scribed. Finally, the Monte Carlo simulations of non-adiabatic effects are presented
and a solution based on an increased magnetic field will be demonstrated.

55
4.1 Adiabaticity
As mentioned in Sec. 2.3, the retarding potential of the main spectrometer has to
be set from a high to low or zero potential when searching for sterile neutrinos. As
a result, the electrons will have high surplus energies with respect to the retarding
potential of the main spectrometer.
Electrons with large surplus energies will experience non-adiabatic transport
when propagating through the low magnetic field region in the center of the main
spectrometer. As will be explained in this section, non-adiabatic transformation
leads to the fact that electrons no longer are transmitted through the spectrometer.
Instead, they are are reflected at the pinch magnet and either leave the spectrom-
eter through the entrance (and are lost) or they are magnetically trapped in the
spectrometer (i.e. they are lost too). These electrons can be trapped for long pe-
riods of the order of minutes. The trapping condition will be broken by further
non-adiabatic effects or by rare elastic scattering processes of the electrons with the
residual gas in the spectrometer volume, which is held at a pressure of 10−11
mbar.
Eventually, these electrons leave the spectrometer through either the source or the
detector side. Due to the asymmetric magnetic field caused by Earth’s magnetic
field and the detector and source side superconducting coils, approximately 60% of
the electrons will leave the spectrometer toward the source side due to the magnetic
mirror effect. In the search for sterile neutrinos, both lost and trapped electrons due
to non-adiabatic motion have to be completely eliminated. The main objective of

56
this investigation is to determine which fraction of the electrons are lost or trapped
as a function of their surplus energy, and secondly to find out what magnetic field
setting is needed to assure 100% transmission. In order to adiabatically transport
these electrons, the use of the large air coil system surrounding the KATRIN main
spectrometer is necessary.
4.1.1 Adiabatic transport
Due to the Lorentz force
F = q(E + v × B)
the electrons that enter the main spectrometer follow the guiding magnetic field
lines in helical trajectories (cyclotron motion). As the electrons move towards the
analyzing plane, the magnetic field decreases. This causes the polar angle θ of the
electrons to decrease and causes the transverse energy to decrease. After passing
the analyzing plane, the magnetic field increases toward the pinch solenoid magnet.
Hence, the longitudinal energy is transformed back into transverse energy and the
pitch angle increases respectively. If the initial pitch angle of the electron is bigger
than
θmax = sin−1 Bs
Bp
(4.1)
the polar angle gets bigger than 90◦
at the pinch magnet and the electron gets
reflected. All electrons with θ < θmax will be transmitted. In equation 4.1, Bs is

57
the source magnetic field and Bp is the pinch magnetic field. However, this is only
valid if the change of magnetic field and electric field
∆B
B
<< 1 and
∆E
E
<< 1 (4.2)
are small within one cyclotron length
lcyc = 2π
γ · me
e · B
· v , (4.3)
where me is the electron mass, γ is the relativistic factor, e is the electron charge,
and v is the electron’s parallel velocity. If these conditions are fulfilled, then the
motion is considered adiabatic, and the adiabatic invariant is conserved
γµ =
γ + 1
2γ
·
E⊥
B
. (4.4)
For tritium β-decay electrons γ < 1.04, the adiabatic invariant can be approximated
as the orbital magnetic moment
µ ≈
E⊥
B
. (4.5)
4.1.2 Non-adiabatic transport
When operating the main spectrometer at low or zero potentials, the electrons travel
at higher velocities since their velocity is not reduced by the retarding potential. As

58
Figure 4.1: Figure shows trajectories of stored electrons with kinetic energies of (a)
5 keV and (b) 25 keV. Both trajectories show some degree of non-adiabaticity. In
both cases, the z-position of the reﬂection points changes. As the electron’s energy
increases, the more chaotic the trajectory becomes. [26]
shown in equation 4.3, as the velocity v increases the cyclotron length lcyc increases
as well. Consequently, the electrons see a larger change of the magnetic ﬁeld ∆B
within one orbit. The adiabatic condition (equation 4.2) is no longer valid, and the
electron now experiences non-adiabatic conditions:
µ = constant
This means that the transverse energy E⊥ no longer decreases proportionally to
B [26]. Analogously, the polar angle θ changes chaotically because the electron
motion becomes chaotic. Fig. 4.1 shows that electron trajectories display some
degree of chaotic behavior when the electron’s kinetic energy is increased. In an
adiabatic behavior, the distance between two calculated trajectories should remain

59
Figure 4.2: Figure shows a test of non-adiabaticity by an investigation of the starting
condition dependence of two electron trajectories. Shown is the distance between two
trajectories when varying the starting position by 10−14
m. (a) Linear dependence
of the adiabatic motion of a 100 eV electron. (b) Exponential dependence of a
non-adiabatic (chaotic) motion of a 15 keV electron. [26]
constant. Fig. 4.2 shows a simulation of two different energy electrons with iden-
tical starting parameters except for a change in starting position by 10−14
m. The
simulation showed that the high-energy electron displayed an exponential increase
of the distance observed, which is typical for a chaotic system [26].
This implies that there is a chance that an electron that started with θmax < 51◦
(an electron that would be transmitted in an adiabatic scenario) can have a polar
angle of 90◦
on the way to the exit of the main spectrometer and be magnetically
reflected as a result. As the electron arrives at the entrance of the spectrometer
it can either exit or again be reflected. Therefore, the non-adiabatic effect can
be reduced by significantly increasing the magnetic field in the center of the main

60
spectrometer to account for the higher velocities.
4.2 Simulation setup
This section describes the setup used to run the simulations. The first step was
to determine the number of events required for the Monte Carlo simulations. The
number of transmitted electrons follows a Poisson distribution. The relative error
σrel is given by
σrel =
√
N
N
(4.6)
To achieve the desired error of the order of 1% for transmitted particles, N = 10, 000
particles were started per bin. For 10 bins, a total of 100,000 particle events is
required. These 100,000 particle events were separated into 100 runs with 1000
particle events per run, which were combined in the end. This step spreads 1000
jobs over many machines compared to just one job on one machine.
4.2.1 Configuration settings
Before running the simulations, the XML configuration file must be modified de-
pending on the simulation.

61
Terminators
As mentioned in 3.4.2, I am concerned with terminators max steps, max z, min z,
and trapped. The max steps terminator ends the particle’s simulation if the number
of steps exceeds the deﬁned value. The max z terminator ends the simulation if
the particle reaches the detector. The min z terminator ends the simulation if the
particle is reﬂected and exits through the source side of the spectrometer. Finally,
the trapped terminator ends the simulation when the particle completes a number
of complete turns between the source and detector sides. The termination settings
used for all the simulations are
<ksterm_min_z name="term_min_z" z="-12.5"/>
<ksterm_max_z name="term_max_z" z="12.1835"/>
<ksterm_max_steps name="term_max_steps" steps="{10e6}"/>
<ksterm_trapped name="term_trapped" max_turns="10"/>
where the units for the z position are in meters (m).
Spectrometer settings
The spectrometer settings are divided into hull, ground, and dipole power sup-
ply potentials, aircoil currents, and solenoid currents. Below are the solenoid cur-
rent settings used throughout the thesis. The values for the pre-spectrometer
(ps 1 current and ps 2 current), pinch (pinch magnet current), and detector (de-
tector magnet current) currents are common settings used in KATRIN simulations

62
<external_define name="ps_1_current" value="157.0"/>
<external_define name="ps_2_current" value="157.0"/>
<external_define name="pinch_magnet_current" value="72.625"/>
<external_define name="detector_magnet_current" value="54.5942"/>
where the units for current are amperes (A).
The ground, hull, and dipole electric potential settings are set to zero voltage
(V). We are interested in the entire spectrum as mentioned in Sec. 1.2.3. As a result,
the spectrometer must have no energy ﬁlter in order to detect possible keV-scale
sterile neutrinos.
<external_define name="ground_potential" value="0.0"/>
<external_define name="hull_potential" value="0.0"/>
<external_define name="dipole_potential" value="0.0"/>
The simulation spectrometer settings are the aircoil current settings. The spec-
trometer is surrounded by 14 aircoils. The aircoils can be individually set or set
together to determine default magnetic ﬁelds. The aircoil settings are set up in the
form:
<global_define name="ac_1_current" value = "[ACCURRENT]">

63
<global_define name="ac_14_current" value = "-[ACCURRENT]">
where [ACCURRENT] is the value of the aircoil current in amperes (A). The mag-
netic field inside the spectrometer is asymmetric with respect to the middle plane
because of the asymmetry of the Earth’s magnetic field and detector and source
side solenoids. Since the stray field at the detector side is larger than at the source
side, we must compensate for this asymmetry. The purpose of coil 14 is to act as a
counter coil with a current direction opposite the other coils, and as a result it must
have negative value.
In this thesis, I study the transmission probability as a function of magnetic
field and particle energy. The transmission probability as a function of the magnetic
field has different aircoil current settings, where [ACCURRENT] = 0 - 160 A. The
aircoil settings determine the magnetic fields throughout the main spectrometer.

64
The transmission probability as a function of starting particle energy is investigated
with constant aircoil settings in Sec. 4.5.
Particle settings
The particle settings are initialized with starting energy, position, and direction.
The coordinate system is shown in Fig. 4.3. The particle energy, position, and
direction settings are shown for the transmission probability as a function of the
magnetic field. The energy creation generator uses a fixed value for the energy in
electron volts (eV). The position and direction creation generators use range values
with upper and lower limits for the radius in meters (m) and angles in degrees.
Figure 4.3: The coordinate system used within Kassiopeia. The z-axis points from
source to the detector, the y-axis points towards the top, and the x-axis points in
the direction to satisfy the right handed system. The origin (0,0,0) of the coordinate
system is in the center of the main spectrometer. The polar coordinates θ and φ are
defined as shown. [16]

65

<energy_composite>
<energy_fix value="18600"/>
</energy_composite>

<position_cylindrical_composite>
<r_cylindrical radius_min="0.0" radius_max="3.0e-2"/>
<phi_uniform value_min="0." value_max="360."/>
<z_fix value="-12.10375"/>
</position_cylindrical_composite>

<direction_spherical_composite>
<theta_spherical angle_min="0." angle_max="60."/>
<phi_uniform value_min="0." value_max="360."/>
</direction_spherical_composite>
For the transmission probability as a function of particle starting energy, I use
a set of ranged values for energy with an upper and lower limit.

<energy_composite>
<energy_uniform value_min="0" value_max="18600"/>

66
</energy_composite>
Other simulation settings
The remaining settings are shown in the code in Appendix B. The settings include
the generators for the trajectory, writers, navigators, and output. Finally, the sim-
ulation combines all the settings.
4.3 Single Particle Trajectory
In this section simulations for a single particle were conducted in order to under-
stand a particle’s trajectory inside the main spectrometer. The trajectories will
show particles that reached the detector, reflected back to the entrance of the spec-
trometer, or are trapped within the spectrometer. I use the exact method to track
the particle’s position in the spectrometer. The settings for the different trajectories
differ only by starting energy E0 and angle θ. Otherwise, the simulations share the
settings
• Aircoil settings: ACCURRENT = 50 A
• Cylindrical position settings: φ = 0, r = 3 × 10−2
, and z = −12.10375.
• Spherical position settings: φ = 0

67
Figs. 4.4 and 4.5 show a 3D and 2D model of a trajectory of an electron with
low surplus energy that reaches the detector. The starting energy and angle are
E0 = 500 eV and θ = 51◦
, respectively. Fig. 4.6 shows the 3D model of a
Figure 4.4: The trajectory for a particle to reach the detector is shown with values
E0 = 500 eV and θ = 51◦
. The rings around the spectrometer represent the aircoils
surrounding it. The color variation shows the longitudinal kinetic energy of the
particle.
trajectory of an electron with medium surplus energy that is reﬂected back towards
the spectrometer’s entrance. The starting energy and angle are E0 = 14, 000 eV
and θ = 5◦
, respectively. Fig. 4.7 shows the 3D model of a trajectory of an electron
with a very large surplus energy that is trapped within the main spectrometer. The
starting energy and angle are E0 = 18, 600 eV and θ = 15◦
, respectively.

68
Figure 4.5: The 2D model of ﬁgure 4.4 is shown.
Figure 4.6: The trajectory for a particle to be reﬂected is shown with values E0 =
14, 000 eV and θ = 5◦
. The rings around the spectrometer represent the aircoils
surrounding it. The color variation scale shows the longitudinal kinetic energy of
the particle.

69
Figure 4.7: The trajectory for a particle trapped in the spectrometer is shown with
values E0 = 18, 600 eV and θ = 15◦
. The rings around the spectrometer represent
the aircoils surrounding it. The longitudinal kinetic energy of the particle is shown
in the color variation scale.

70
4.4 Magnetic field dependence of transmission probability
The following section investigates the non-adiabatic impact on the transmission
probability for an electron to reach the detector. As mentioned in the simulation
setup, the number of events used in the MC simulations is N = 100, 000 particle
events, which are divided into 10 bins. By counting the number of transmitted
particles per bin, the error calculation can be expressed as
σ =
(k + 1)(k + 2)
(n + 2)(n + 3)
−
(k + 1)2
(n + 2)2
(4.7)
with k being the number of transmitted particles per bin and n being the total
number of the particles per bin [22].
Using an electron starting energy of E0 = 18.6 keV, I calculate the transmission
probability as a function of magnetic field and aircoil currents shown in Fig. 4.8.
This result shows that at a nominal magnetic field B = 3 G the transmission proba-
bility is practically zero and at a maximum magnetic field B = 10 G the transmission
probability is approximately 80%.
In order to reach 100% electron transmission probability KATRIN would need
to adjust the aircoil currents to values much higher than what is currently allowed
for the experiment, which is limited to between 100 and 125 A per coil [10]. This
corresponds to a maximum magnetic field in the analyzing plane of 1 mT (10 G) [16].

71
Figure 4.8: With an electron starting energy E0 = 18.6 keV, the transmission
probability as a function of the magnetic field in Gauss is shown for all electron
events that terminated at the detector (term max z). Nominal KATRIN field B =
3 G, which has aircoil currents set to approximately 11 A. Maximum KATRIN field
B = 10 G, shown by the vertical line in the figure, when aircoil currents are set to
approximately 88 A.

72
4.5 Surplus energy dependence of transmission probability
To investigate the non-adiabatic effect in more detail, the transmission probability
has been simulated as a function of electron starting energy with errors computed
from eq. 4.7. The three values for the aircoil currents [ACCURRENT] = 50, 100,
and 150 A correspond to magnetic fields in the analyzing plane BA = 6.43, 11.21, and
15.98 G, respectively. The MC simulations only considered particles with θ < 60◦
because larger angles are not transmitted even in an adiabatic case, as explained in
Sec. 2.2.
Fig. 4.9 shows the transmission probability for electrons that reached the detec-
tor for the three different aircoil currents used in the investigation. For the three
aircoil currents, the non-transmitted electrons (i.e. electrons that are reflected or
trapped) are discussed.
In appendix A, a further discussion on angles θ > 60◦
is presented, and the
precise results and errors are shown in tables as well.
4.5.1 Aircoil current at 50 A
The following MC simulation results show the transmission probability as a function
of the starting energy of an electron. In the simulations, the 14 aircoil currents are
set to a magnitude of 50 A. The results show transmission probabilities for a particle
that is terminated from conditions: max z, min z, and trapped. The termination
of a particle at max z represents a particle that has reached the detector, min z

73
Figure 4.9: The transmission properties for electrons transmitted to the detector
are shown for aircoil currents for 50, 100, and 150 A.

74
Figure 4.10: The aircoil currents are set to 50 A and the starting angles for particles
are between θ = 0◦
−60◦
. The transmission probability as a function of the starting
energy in eV is shown. The figure shows results for electrons that reached the
detector, term max z, electrons that reflected backwards, term min z, and electrons
trapped within the spectrometer, term trapped.
represents a particle reflected back toward the entrance of the spectrometer, and
trapped represents a particle being reflected back and forth in the spectrometer for
long time periods.

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