MAC-E-Filter characterization for
PTOLEMY, a relic neutrino direct
detection experiment
Carl-Fabian Strid
Space Engineering, master's level 2019
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
MAC-E-Filter characterization for
PTOLEMY, a relic neutrino direct
detection experiment
Carl-Fabian Strid
Department of Computer Science, Electrical and Space Engineering
Luleå University of Technology
Luleå, Sweden
Supervisors:
Dr. Ferella Alfredo Davide and Prof. Dr. Conrad Jan
Stockholm University
Abstract
The cosmic neutrino background (CNB) can be composed of both active and hypothetical ster- ile neutrinos. At approximately one second after big bang, neutrinos decoupled from radiation and matter at a temperature of approximately one MeV. Neutrinos played an important role in the origin and evolution of our universe and have been indirectly verified by cosmological data on the BBN (Big Bang nucleosynthesis) of the Big Bang.
It was Steven Weinberg in 1962 that first theorized on the direct detection of relic neutrinos.
The signal of the relic neutrino capture on a tritium target can be observed by studying the endpoint of the electrons kinetic energy that are above the endpoint energy of the beta de- cay spectrum. The PTOLEMY project aims to archive direct detection of the relic neutrino background with a large tritium target of 100 gram, MAC-E-Filter, RF-tracking, Time of flight tracking and a cryogenic calorimetry.
In this thesis the MAC-E-Filter have been simulated in two filter configurations. In the first configuration, the electron were simulated five times in the filter. Two in the opposite side of the detector, one in the middle, and two at the detector. In the second configuration the electrons was simulated in the entrance solenoid at a fixed position of y= -0.19634954 m from the center of the filter and in random positions. Both multiple electrons and single electrons were simulated in the second configuration.
In the single electron configuration the electron had a starting position of y = -0.19634954 m from the center of the filter, and an initial kinetic energy of 18.6 KeV. The first filter config- uration successfully accomplished to simulate the electron track, as the electron was reflected back and forth between the entry and detector solenoid. The electric and magnetic field profile differed at the entry and detector solenoid. The second filter configuration successfully showed that the electron will reach the end solenoid, when the filter length was 0.5 m. When the filter length was increased to 0.7 m, then the electron was reflected in the middle of the filter. The simulation showed that the electron energy dropped below 1 eV from 18.6 KeV as the electron propagated through the filter. The magnetic and electric fields decreased exponentially in the direction of the detector solenoid. The Simulation of multiple electrons showed mixed results and would need more modifications in order to come to a final conclusion.
ii
Acknowledgements
I would like to thank my supervisors Dr. Ferella Alfredo and Professor Jan Conrad for allow- ing me to pursue my master thesis at Albanova University Center, at Stockholm University.
Without Alfredo’s expertise and guidance this thesis would not have been possible. I would also like to thank the Dark Matter and Astroparticle Physics Group for allowing me to go to Gran Sasso National Laboratory in Italy.
iii
Contents
1 – Introduction 1
2 – Cosmic Neutrino Background (CNB) 3
2.1 CNB signal . . . 4 2.2 Sterile neutrinos . . . 6
3 – Detection approach of MeV Dark Matter 9
3.1 Evidence for dark matter . . . 10
4 – The PTOLEMY project 14
4.1 An Overview . . . 14 4.2 MAC-E-Filter . . . 20 4.3 New filter design . . . 22
5 – Simulation 27
5.1 Kassiopeia package . . . 28
6 – Results 31
6.1 Default filter configuration . . . 31 6.2 New filter configuration . . . 34
7 – Discussion 36
References 37
Appendix A – Plots of the default filter configuration 41
Appendix B – Plots of the new filter configuration 54
Appendix C – EvectorB new2.xml 81
Appendix D – EvectorB new2 geom.xml 90
Appendix E – flat world single point.xml 100
Appendix F – flat world geom.xml 108
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1
Introduction
When the early hot universe expanded during the Big Bang, the radiation cooled to become the cosmic microwave background (CMB). It’s now known that CMB is a rest from the Big Bang, and can be detected from every direction due to the early universe was in thermal equilibrium [1]. Nucleosynthesis (also known as Big Bang Nucleosynthesis, BBN) of the lightest elements which has a larger mass than hydrogen began to be produced during the first ≈ 20 min after the Big Bang [2, 3]. Light neutrinos are predicted to have thermally decoupled from other form of matter in the early thermal universe at approximately ∼ 1-0.1 seconds after Big Bang [4]. The light neutrinos are predicted to have cooled down to 1.9 K in the present day and are predicted to have the number density of 56/cm3 per lepton flavor. Steven Weinberg proposed a method for direct detection of the relic electron neutrinos in 1962 [5]. The method Weinberg proposed is based on neutrino capture on a tritium target
There are many nuclei that are β-unstable and can be used for direct detection of the cosmic neutrino background (CNB). Of the nuclei that are β-unstable, tritium is considered the best option based on the more favourable combination of capture cross section and half-life [27].
To separate CNB interaction events from the natural β decay background events, a filter with high luminosity (which is the light collecting power measurement of a spectrometer) and low background is needed. These requirements can be fulfilled with the MAC-E-Filter (Magnetic Adiabatic Collimation combined with an Electrostatic Filter). The MAC-E-Filter can best be described as a spectrometer which acts as a high-pass filter. The MAC-E-Filter has been used by the KATRIN collaboration to reduce the upper bound mass of neutrinos [6]. The magnetic fields of the MAC-E-Filter are created by two superconducting solenoids which adiabatically guides electrons to the detector [7].
The aim of the PTOLEMY (Pontecorvo Tritium Observatory for Light, Early-Universe, Mas- sive Neutrino Yield) project is to directly detect neutrinos from the cosmic neutrino background (CNB). This will be achieved with a three-steps-approach:
1. In the proof-of-principle phase the technology will be validated in a small prototype that is currently (2019) being assembled;
2. The intermediate phase will be devoted to the search of low mass (∼MeV/c2) Dark Matter
1
2
particles using graphene as target [8];
3. Finally the full scale experiment with a target of 100 g of tritium and fully functional en- ergy measurement will be realized in a currently not yet identified underground location.
This thesis will be using computer simulations to run simulations on electrons as they propa- gates through the MAC-E-Filter. The files that was used in the simulations is written in XML, and the output files are analyzed with the aid by the ROOT program.
2
Cosmic Neutrino Background
(CNB)
The existence of CNB is a generic prediction of the standard Big Bang model. Experimental observations show that neutrinos decoupled approximately one second after Big Bang and are expected to have a spectrum similar to a black-body radiation [10]. When the early universe was hot, the neutrinos were in thermal equilibrium by weak interactions with other particles before they decoupled [9]. The Cosmic neutrino background (CNB) can both be composed of active neutrinos (there are three types of neutrino flavors: electron neutrinos, muon neutrinos, and tau neutrinos and their corresponding antineutrinos) and the hypothetical sterile neutrinos, which only interact by gravity (but cosmological data is in disfavor of relic sterile neutrinos) [11]. Relic neutrinos was an important factor in the evolution and formation of our universe right after the Big Bang [12]. Neutrinos ceased to interact with the other particles when their temperature dropped bellow a certain critical value. This is because the universe expanded faster than the neutrino interaction rate and the number density was diluted during the expan- sion, and the neutrinos decoupled from other matter [13]. The neutrino number density and neutrino temperature can be derived by (neglecting lepton asymmetries) setting the entropy proportional to the number of degrees of freedom gνand the temperature Tν cubed [14]:
s ∝ gT3 (2.1)
We then set the neutrino entropy equal to the entropy of the photons and solve for neutrino temperature,
gνTν3 = gγTγ3 (2.2)
Which then becomes:
Tν = Tγ(gγ
gν)1/3 = Tγ( 2
2+ 2 · 7/8 + 2 · 7/8)1/3 = Tγ( 4
11)1/3 ≈ 1.945K (2.3) Where 2 · (7/8) (fermions) in the denominator is for the electrons and positrons and 2 for the photons (bosons) in the numerator and denominator. Here is Tγthe photon temperature which is equal to 2.725 K. [18]
3
2.1. CNB signal 4
The number density for each neutrino plus anti-neutrino flavors is given by[16], nν =
Z d3p
2π3 fν(p, Tν)=
Z d3p ep/Tν+ 1
1 2π3 ≈ 3
11nγ ≈ 112cm−3 (2.4) Where nγis number density of photons and fν(p, Tν) is the form factor.
After the neutrinos decoupled from other form of matter, the temperature started to drop below the electron mass and the electron-positron pairs started to annihilate:
e++ e− ⇐⇒γ + γ
The entropy and the energy of the electron and positrons are transferred to the photons after the annihilation, but not to the decoupled neutrinos. This means that photons are heated up relative to the neutrinos [17].
2.1 CNB signal
Consider both active and sterile neutrinos 3+Ns, where 3 is the number of active neutrinos and Nsis the number of sterile neutrinos. The eigenstate of the flavor can then be written as,
νe
νµ
ντ ...
=
Ue1 Ue2 Ue3 . . . Uµ1 Uµ2 Uµ3 . . . Uτ1 Uτ2 Uτ3 . . . ... ... ... ...
ν1 ν2
ν3 ...
(2.5)
The mass eigenstate of the active (1 ≤ i ≤ 3) and sterile (4 ≤ i ≤ 3+ Ns) neutrinos are given by νi· Uαi, where νiis the mass eigenstate of active and sterile neutrinos, while Uαiis the neutrino mixing matrix. The natural beta decay process of tritium with mass number A and atomic number Z is given by:
N(A, Z) → N0(A, Z+ 1) + e−+ ¯νe (2.6) where N(A,Z) is the parent nucleus while N0(A, Z+ 1) is the daughter nucleus with the mass number A, and atomic number Z.
2.1. CNB signal 5
Figure 2.1: Electron spectrum (idealized) of the tritium beta decay
A measurement of the neutrino mass can be archived by the kinematics of β-decay. The end point energy at the end of the β-decay spectrum is sensitive to the neutrino mass. This can best be seen in 2.1, where
K(Ee)=
s dN(Ee)/dEe
F(Z0, Ee) · Ee· pEe2− m2ec4 (2.7) is plotted as a function of the electron energy Ee. Where the number of electrons in the energy interval [Ee, Ee + dEe] is given by dN(Ee), while F(Z0, Ee) is the Fermi function [20]. The spectrum of the natural tritium decay can be seen from the Figure 2.1, where the spectral line to the left of the dashed line is for non-massless neutrinos, while the dashed line is for massless neutrinos. The Q value, Qβ is the total energy released during the decay of the tritium nucleus,
Qβ= [m(3H) − m(3He) − me− ¯νe]c2 (2.8) The sharp peak to the right of the spectral line is the relic neutrino capture signal [17],
νe+ N(A, Z) → N0(A, Z+ 1) + e− (2.9) CNB can be detected by measuring the distance between the decay and capture process. The differential neutrino capture rate is give by,
dλν
dTe
=X
i
|Uei|2σνivνinνiR(Te, Te0i) (2.10) where nνi = ζi < nνi > is the number density of relic neutrinos (νi) around the Earth, and the sum index is over all neutrino mass eigenstates. While
2.2. Sterile neutrinos 6
R(Te, Te0i)= 1
√
2πσexp
− (Te− Te0i) 2σ2
(2.11) is the Gaussian energy resolution function, implemented in equation (2.10) to include the finite energy resolution [17]. The number density of active neutrinos for each species of active neutrinos are predicted to be < nνi > ≈ < nνi > ≈ 56 cm−3, its also true for sterile neutrinos if they could thermalize after Big Bang. Gravitational clustering may also enhance the number density (ζi) when the neutrino mass is > 0.1 eV [18]. The capture cross-section times neutrino velocity is given by,
σνivνi = 2π2ln 2
A · T1/2 (2.12)
where T1/2 is the half-life of the parent-nucleus and A is the nuclear form factor. It’s very difficult in a laboratory environment to directly detect relic neutrinos. The PTOLEMY is the only one currently (2019) that can ’look’ at the first second after Big Bang.
The PTOLEMY project is designed to have 100 gram of tritium as target coupled with a MAC- E-Filter, RF tracking, time-of-flight (TOF) systems and cryogenic colorimetry. The event rate of the PTOLEMY project is given by:
Nν(PT OLE MY) ' 8.0 ·X
i
|Uei|2ζiyr−1 (2.13) According to equations (2.6) and (2.9) only electron neutrinos can be captured on a tritium target. It has been established through experiments in the last thirty years that there are two neutrinos that are massive of the three neutrinos in the standard model [11]. It should be noted that we don’t know the absolute mass of the neutrinos only their squared mass differences.
What also is unclear is the mass hierarchy of the neutrino masses, whatever the ν3mass eigen- state is heavier or lighter than the other neutrino eigenstates [19].
2.2 Sterile neutrinos
The PTOLEMY project can also help to detect sterile neutrinos by analyzing the β decay electron spectrum. Sterile neutrinos with mass range in keV are a promising warm dark matter candidate. The mass of the sterile neutrino is unknown and can range from 1015 GeV/c2 to 1 eV [11]. The former mass is predicted by the seesaw mechanism. Heavy neutrinos are an attractive idea postulated by the so called seesaw mechanism to explain the smaller neutrino masses of the light neutrinos relative to the quarks[16]. The idea behind the seesaw mechanism is that neutrinos can be their own antiparticles (Majorana neutrinos). If the model is correct, then one could postulate that for each light neutrino νe, νµ and ντ, there would be additional heavy neutrinos (Majorana fermions). The mass of the neutrinos are, according to the theory, generated by the neutrinos interacting with each other and therefore changing their mass. The
2.2. Sterile neutrinos 7
relationship between a light neutrino and a heavy neutrino can be loosely formulated in the form [21]:
mν ≈ m2S M MN
= mS M
mS M
MN
(2.14) where mS M is the Dirac neutrino. The interaction of both neutrinos (N and ν) leads to the suppression of the mass of the active neutrinos. This could explain why the mass is much lighter than the other fermions in the standard model, suppressed by a factor of mS M/MN. One can see that from equation (2.14) that mν becomes smaller when MN becomes larger. Hence the name seesaw mechanism [21]. If one takes the mass of the Dirac neutrino equal to the mass of the tau neutrino (mS M ≈ mt), and mν ≈
q
m2ν3 − m2ν
2
as the largest mass of the neutrino one gets MN ≈ 1015Gev/c2. However sterile neutrinos should not be confused with the majorana neutrinos. Other interesting aspects that could arise from the sterile neutrino with masses in range of eV to KeV, is that it could mix with active neutrinos. That in turn could explain some issues regarding anomalies in neutrino oscillation experiments [22].
Figure 2.2: Spectrum of the tritium beta decay without (black dashed) and with (red solid) light sterile neutrino mixing[17].
Since neutrinos oscillate between flavors and mass states, one could say that the sterile neutrino carries away a bit of the energy. This causes the β decay spectrum to be altered, as shown by the red line in figure 2.2. If there were no mixing with the sterile neutrino, then one would get the dotted line (shown in figure 2.2). The tritium β-decay spectrum is given by,
2.2. Sterile neutrinos 8
dΓ
dE = N G2F
2π3~7c5 cos(θC)|M|2F(E, Z0) · p · (E+ mec2)·
X
j
Pj(E0− Vj− E) · q
(Eo− Vj− E)2− m2ic4,
(2.15)
where N is the number of tritium atoms, E is the kinetic energy, θC is the Cabbibo angle, GF
the Fermi constant, M the matrix element, F(E, Z0) the Fermi function, pj the probability to decay into excited rotational-vibrational and electronic state with the excitation energy of Vj, and the endpoint energy of the β-spectrum is given by E0. The beta-decay spectrum for the electron neutrino is given by,
dΓ dE =
3
X
i=1
|Uei|2dΓ
dE(mi) (2.16)
which is an incoherent superposition of the contributions of the three mass eigenstates of the neutrinos. Here is Ueithe Pontecorvo–Maki–Nakagawa–Sakata matrix or the neutrino mixing matrix, given by equation (2.5). If there existed a sterile neutrino in KeV range, then a fourth mass state m4may be introduced, but with lower mixing with the electron neutrino. Then one can combine the spectrum of the tritium β-decay as superposition of the sterile neutrino and the active neutrinos [23],
dΓ
dE(mνe)= sin2θdΓ
dE(mh)+ cos2 dΓ
dE(ml), (2.17)
where mh= m4is the sterile neutrino mass, and mlis the light neutrino mass, ml = P3i=1|Uei|2m2i .
3
Detection approach of MeV Dark
Matter
Although PTOLEMY is primarily designed for direct detection of relic neutrinos, its interme- diate phase is designed to detect dark matter particles in MeV/c2 mass range. There are cur- rently two proposed dark matter searches that may be included in the PTOLEMY prototype.
The PTOLEMY-G3 has a graphene target based on G-FETs with a sensitivity to a single elec- tron. The other is called PTOLEMY-CNT that uses carbon nanotubes (CNT) tightly aligned as means to detect dark matter. With a volume of 103 cm3, consisting of 100 stacked 4-inch
× 4-inch planes based of G-FET, the PTOLEMY prototype will have a sensitivity of approxi- mately σe = 10−33 cm2 for dark matter candidates of 4 MeV/c2. Figure 3.1 shows the path of propagation of the low energy scattered electrons trough the carbon nanotubes target[24].
Figure 3.1: Left figures shows the directionality from the carbon nanotubes depends on the orientation of the carbon nanotubes relative to the dark matter wind. The right figure shows a densely packed array of nanotubes oriented vertically[24]
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3.1. Evidence for dark matter 10
3.1 Evidence for dark matter
When all of the mass of the luminous objects in our galaxy (the thin and the thick disks, the interstellar medium, the galactic bulge, the stellar halo and the bar) are added together, one obtain the value 9 · 1010 M [25], where M is the Solar mass. The orbital motion of the Sun relative to the galactic center is in agreement with the value 9 · 1010 M . The mass 9 · 1010 M is not in agreement with the orbital motion of other stars and the interstellar medium that lies beyond the distance greater than from our Sun to the center of the Milky Way galaxy [26]. It appears that there is another form of matter that is not visible, but can make its presence observable through gravitational interaction or by the weak interaction. This invisible matter or
’dark matter’ seems to be spherically distributed around our galaxy, extending out to a distance of 230 kpc. By observing the gravitational influence that the dark matter halo has over ordinary matter, one obtains the following mass distribution of the dark matter halo:
ρ(r) = ρ0
(r/a)(1+ r/a)2 (3.1)
This function behaves as an 1/r when r a and 1/r3 when r a. The estimated mass of the dark matter halo in the Milky Way galaxy is approximately 5.4 · 1011M within 50 kpc from the galactic center. And 1.9 · 1012M within 230 kpc from the galactic center. It is estimated that the dark matter halo is composed of approximately 95 percent of the total mass of the galaxy [25]. What the dark matter halo is composed of is one of the key questions today in physics. It cannot be dust since dust diffuse light and gas would create absorption lines when observed. There are many candidates of what dark matter is composed of, one of such candidates are the weakly interacting massive particles or WIMPs. If WIMPs exists they would only interact gravitationally and by the weak interaction (however WIMPs would not contribute to the luminosity of the galaxy). From the theoretical framework of the formation of our galaxy and from the Big Bang model, it seems that all of the dark matter halos mass is made of non-baryonic matter. Relic Neutrinos can be a warm dark matter candidate, however they are not massive enough to explain the mass of the dark matter halo. There is a other hypothesis that dark matter is composed of massive compact halo object (MACHOs). Masses that can’t be detected such as white dwarfs, neutron stars, black holes or brown dwarfs are includes as MACHOs (MACHOs are made of ordinary baryons as opposed to WIMPs). But according from statistical analysis, there are too few MACHO objects that could explain dark matter.[28]
To measure the orbital velocity for distances greater than the distance of the galactic center to the Sun, we have to relay on a object in the galactic plane for which we can measure the distance. Observations does not support the claim that the orbital velocity will decrease with distance beyond Earths orbit.
This was not expected from astronomers, to discover the orbital velocity beyond radius of the distance between the Earth and galactic center to be constant. According to classical mechan- ics, is that if all of the mass was inside the radius of the distance of Earth to the galactic center, the orbital velocity would be proportional to θ = R−1/2. In other words the orbital velocity
3.1. Evidence for dark matter 11
Figure 3.2: Rotational speed as a function of the galactocentric radius RG ( R0 ≈ 8 kpc for our Sun) for the Milky Way galaxy [29]
would decrease as you go further out in the galaxy. Keplerian motion would imply that most mass would be located inside the radius of the distance from the galactic center to the Sun. This was proven to be false, we know now that a significant amount of mass is located beyond that distance (Galactocentric radius). This was a surprise for many astronomers as they expected most of the mass of the galaxy would be in the Galactocentric radius. By observing other spiral galaxies, one get a nearly identical result as for our own galaxy. This confirms that dark matter is also present in other galaxies in form of dark matter halos. In the outermost regions of the galaxies the rotational curve is nearly flat. By studying these curves we can obtain information about the matter in the galaxies. That is because rotational velocity are a function of the total mass distribution in the galaxy. If the rotational curve is flat, that implies that the mass distri- bution in the outer region of the Galaxy are spherical distributed with density proportional to r−2. A rigid body rotation near the center of the galaxy implies that the mass must be spherical distributed and have a nearly constant density.
To see why this is the case put θ(r) = V, where V is a constant and θ(r) is the orbital velocity of the Sun around the Milky Way galaxy. Then by Newtons second law and Newtons gravitational law, the force acting on mass m due to mass Mrof the inner region of the Galaxy at the position r of the star:
mV2
r = GMrm
r2 (3.2)
Assume spherical symmetry and solve for Mr, Mr = V2r
G (3.3)
and differentiating with respect to r,
dMr
dr = V2
G (3.4)
3.1. Evidence for dark matter 12
(a) Label 1 (b) Label 2
Figure 3.3: Label 1 and Label 2 (Label 2 are for 21 Sc Galaxies) are showing the mean rota- tional velocities of galaxies versus their radius. [30][31]
The mass conservation for a spherical symmetric mass distribution for a stellar structure is described by[15]:
dMr
dr = 4πr2ρ (3.5)
By inserting equation (3.5) in equation (3.4) and solve for the mass density ρ,
ρ(r) = V2
4πGr2 (3.6)
We can see that the mass density is proportional to r−2. The observed number density of stars in luminous stellar halo is proportional to r−3.5
nhalo(r) = n0,halo(r/a)−3.5 (3.7)
We can see that the number density has a much higher drop-off (r−3.5) than the observed rotation curve. This lead astronomers to modify the equation (3.1), so that the equation was modified to approach a constant value near the galactic center. They did so because they didn’t want the equation to diverge. The density function for our Galaxy’s dark matter halo then becomes,
ρ(r) = ρ0
1+ (r/a)2 (3.8)
where a and ρ0 are parameters to fit the rotation curve. The same equation can be used for other Galaxies, but one have to have different parameter values. In 1996 Julio Navarro, Carlos Frenk and Simon White proposed an alternative form of the distribution of the dark matter halo. They ran numerical simulations over formation of dark matter halos over a wide range of object, from dwarf galaxies to galaxy clusters. The simulation reveled that the density function is proportional to,
ρNFW(r)= ρ0
(r/a)(1+ r/a)2 (3.9)
3.1. Evidence for dark matter 13
This density equation is applicable over large distances and the total mass contained in it is not bound[32].
What dark matter is composed of is one of the key question today facing cosmologists and physicists. Its know from previous studies that dark matter is making up approximately 85%
of the total matter in the universe [33]. The information gathered from Wilkinson Microwave Anisotropy Probe (WMAP) and the Big Bang nucleosynthesis (BBN) are inconsistent with larger amount of baryonic matter. Some dark matter, which is composed of baryonic matter may be hiding in galactic halos, and can be detected by gravitational microlensing. But statis- tical analyses currently shows that only 19% of the mass of our Galaxy’s dark matter halo are explained by MACHOs. Dark matter can be made into two groups, cold dark matter (CDM) and hot dark matter (HDM). Hot dark matter particles are relativistic and cold dark matter are non-relativistic. Massive neutrinos are most likely hot dark matter while WIMPs are likely cold dark matter. Another cold dark matter candidate are the low mass axion (mc2 ≈ 10−5eV).
The difference between hot and cold dark matter is important because hot dark matter rarely gravitationally group together and didn’t participate in the formation of galaxies. This is why cold dark matter is favored. The standard model of cosmology are incorporating the cold dark matter and the cosmological constant asΛCDM model[34].
4
The PTOLEMY project
4.1 An Overview
The conceptual design of the PTOLEMY project is shown in figure 4.1. The tritium target area is where the electrons originate from, before entering the RF antenna. The RF antenna mea- sures the radio frequency-signal emitted by the electron moving in a magnetic field, before the electron reaches the transverse drift filter (MAC-E-Filter). In the filter the electrons are slowed down, only the most energetic ones can make it to the calorimeter, where the measurement of the electron energy takes place. An example of an electron track through the MAC-E-Filter is also shown in the Figure 4.1.
The PTOLEMY project can be divided up into three phases, the first phase is the proof-of- principle concept, the second is an scalable prototype model, and the third phase is the full size construction design. In the first “proof-of-principle” phase, a precise electron source will be used to assess the feasibility of the full scale experiment: the two main instruments, a high precision MAC-E-Filter and a TES microcalorimeter with an energy resolution as good as 0.05 eV will be proved. In the second phase, the unique properties of graphene is planned to be exploited for direct detection of MeV dark matter [24]. The main results of this thesis will be instrumental for the proof-of-principle concept. At present time (2019) the PTOLEMY project is a technological demonstrator that will converge to a final design.
14
4.1. An Overview 15
Figure 4.1: The present (2019) conceptual design of the PTOLEMY project, where electrons originate in the tritium target, drift to the RF antenna and enter the Transverse Drift Filter (MAC-E-Filter) before the electron hits the calorimeter[11].
As already presented in Chapter 2, it is expected that the average number density of electron relic neutrinos in the Universe is 56 per cubic centimeter. Gravitational clustering may increase the number up to a factor of 1-100 depending on the neutrino mass [26]. The target mass of tritium needed in order to archive the sensitivity to detect a significant number of relic neutrino events within a reasonable exposure time is 100 gram [27]. The required energy resolution of the calorimeter is in the order of 0.05 eV and is a fundamental new approach to measure the neutrino mass. Moreover as stated above, PTOLEMY could aid in the search for keV-scale dark matter candidates. One of such candidates is the sterile neutrino which can mix with the active neutrino. Figure 4.2 shows a possible scenario of the mixing of the three active neutrinos with one sterile neutrino.
Figure 4.2: Mass spectrum of three active neutrinos and one sterile neutrino
The PTOLEMY project will have a unique sensitivity to one of the possible weakly interacting components of the matter in the universe with a potential to discover unexpected properties in the early evolution of the universe through anomalous contributions to the relic neutrino
4.1. An Overview 16
capture energy spectrum, and also increase our knowledge in elementary particle physics as in the possibly finite neutrino lifetime.
The target geometry, the cryostat, and the calorimeter of PTOLEMY need to be simulated in order to characterize the detector response and model in detail all the various background components. Simulations are also used to evaluate the signal efficiency for end point electrons to enter the aperture of the MAC-E-Filter. The precise cut-off and the flux of the β-decay electrons from tritium decay and 14C on the calorimeter, and the MAC-E-Filter are modeled after the simulations.
Three different types of simulations are used in order to characherize the behaviour of the apparatus:
• The GEANT4 package tracks and generates background sources. Particles that scatter or are captured on the material, or decay to secondary electrons may contribute as back- ground signals.
• The COMSOL package computes the electric and magnetic field maps that are then input in the GEANT4 simulation for charged particle tracking.
• Electron trajectories are tracked in the MAC-E-Filter with the Kassiopeia package [37].
The Princeton Plasma Physics Lab (PPPL) has the expertise to handle the tritium requirements of the PTOLEMY project. Surface-deposited tritium sources were studied extensively by the Mainz neutrino mass experiment [38]. The advantages of a tritium source were found to be: a low backscattering rate, an atomic flat surface and high purity. The total inelastic cross-section was measured in the Mainz neutrino mass experiment to be 3 · 10−18cm2 for E = 18.6 KeV, with a maximum loss per scatter of approximately 12-14 eV. The column density of molecular tritium (T2) is limited by the total inelastic cross section. The column density is thus limited to 5 · 1017molecules/cm2before scattering in the source begins to dominate the energy resolution of the tritium beta decay at the endpoint energy.
When a neutrino is captured by a tritium nucleus, it will induce a decay with a kinetic energy that is above the endpoint energy of the natural β decay of tritium. The energy spectrum of the decay electron must be as much as possible free from fluctuations such that a sizable separation between the natural tritium β-decay and the neutrino capture can be obtained. A shift is produced by the binding energy of the tritium atom and the possible 3Hebound states:
this causes the electron endpoint kinetic energy to be spread. This effect can be minimized by weakly binding the tritium atoms to a nearly inert chemical substrate with binding energy comparable to the required energy resolution. One of such substrate is graphene with a single atomic-layer, with weakly bound tritium in the sp-3 configuration with the carbon lattice. The scattering probability of the outgoing electrons with other tritium or carbon atoms limits the thickness of the tritium target. A fully tritiated graphene has tritium attached on both sides of the graphene, with 3 · 1015tritium nuclei/cm2.
4.1. An Overview 17
Figure 4.3: Hydrolyzed graphene places an individual tritium atom at every carbon atom
There are numerous non-electron backgrounds that can directly interact with the calorimeter and would easily overwhelm the signal. To reduce the non-electron background, one can use a narrow energy window at the endpoint with limit in the width by the energy resolution of the calorimeter. The information of the pulse shape and regional hit multiplicity is a limit- ing factor when one wants to suppress the background with the calorimeter data. There are different sources of unwanted electron background, ranging from tritium beta-decay, target scattering, beta-decays from other elements, electron capture electrons from an unstable iso- tope and electron secondaries from cosmic rays. The cosmic ray flux on the target tritium is approximately 20 kHz with a planer surface of 100m2. There is a small chance that cosmic rays might interact to directly liberate low energy electrons into the magnetic aperture of the PTOLEMY spectrometer. For an experiment located on/near the surface the required rejection factor protection from the background are enormous. A background rejection factor of 1012is needed for the PTOLEMY project. An energy of 0.5 eV at the tritium endpoint is estimated to suppress approximately 107− 108. The reminding factor will require to move the experiment underground. Solar neutrinos as a potential source for neutrino background to the relic neu- trino capture process was found to be negligible. Neutrinos from neutron decay and nuclear reactors only produces electron anti-neutrinos and will not interact with the target. The only background source is from the natural beta-decay of tritium [7].
Trajectories of the electrons emitted from the tritium target are restricted by the geometry of the MAC-E-Filter. Electrons are restricted (In the MAC-E-Filter) from within the cyclotron radius of the adiabatically varying magnetic field lines which are threading (”linked”) with the tritium target. The technology to develop the graphene-held atomic tritium target are being developed at PPPL.
The sensitivity of the PTOLEMY project is determined by the product of the total tritium target
4.1. An Overview 18
area, the total tritium mass, the exposure time and the product acceptance times efficiency for endpoint electrons which enter the filter without scattering. The tritium target is designed as a monolayer of graphene with one atomic tritium bounded to one carbon atom with a weak sub-eV binding energy. The area of the graphene that is exposed is taken to be normal to the magnetic field lines. The magnetic field lines in the tritium target containers are drawn into the magnetic ducts through the RF-tracker, fed into that feeds the MAC-E-Filter and finally into calorimeter.
One can see by the geometry that the graphene substrate spans the total sectional area of the tritium target container. This is what defines the total area per storage container and its area times the number of containers feeding the MAC-E-Filter is the total tritium target area. The endpoint acceptance rate starts with the graphene substrate with a solid angle of 2π which are oriented normally to the magnetic field. Electrons with large opening angles that originated in the tritium target will not make it though the MAC-E-Filter (where they are subjected to the magnetic mirror effect), if the cut-off is not sharp or low enough. The RF-tracker will not be able to detect the electron signature if its transverse kinetic energy is too low. Even if those electrons can be recorded by the calorimeter, the non-electron background will be substantial.
The electrons can lose efficiency, if the endpoint electrons scatter with gas in the vacuum, or if the electrons are inefficiently detected by the calorimeter mostly due to geometric efficiency or something else that degrades the energy resolution of the endpoint electrons.
Figure 4.4: The tritium target area uses a E × B field configuration to drift endpoint electrons into shared magnetic flux lines that feed the MAC-E-Filter[7]
A large tritium target uses a series of monoatomic graphene area elements, which uses elec- trodes to drift endpoint electrons out of the magnetic field lines into the MAC-E-Filter. The gyromotion of an electron is accomplished by an E × B field configuration which leads to a non-relativistic drift velocity:
VDri f t = E × B
B2 (4.1)
4.1. An Overview 19
The drifting stops when the electron drifts out of the electric field region and enters a magnetic flux lines that feed the MAC-E-Filter. The electron will undergo cyclotron motion at the given frequency:
fe = qB
2πγmec2 (4.2)
Where γ is the Lorentz factor γ = 1/ p1 − β2, q is the electric charge, B the magnetic field strength, and methe electron mass. The frequency arise when the electron goes into a uniform magnetic field. The electron radiates power when the electron undergoes cyclotron motion,
Ptot = 1 4π0
8πq2fc2 3c
β2⊥
1 − β2 (4.3)
Which depends on the cyclotron frequency fc and the transverse velocity β⊥. The energy of an endpoint electron is 18.6 KeV, which corresponds approximately to the total velocity of β= 0.26c. An endpoint electron will radiate approximately Ptot= 3 · 10−14W of coherent RF power when the electron is moving transversely to the magnetic field with β⊥≈β.
What is central to the design of the PTOLEMY is the high precision cryogenic microcalorime- try. The electron energy resolution estimated at 100 eV is 0.15 eV. The energy resolution is estimated to be in the order of the highest neutrino mass eigenstate and provides an opportunity to measure the energy spectrum of the tritium endpoint over range of ∼ 100 eV.
The response time (of an individual calorimeter cell) is proportional to the heat capacity of the calorimeter cell and is inversely proportional to the thermal conductance G of the transition- sensor to the thermal reservoir. The energy range is set bellow the endpoint, this is done by the cut-off precision of the MAC-E-Filter. This would be needed if one wants to maintain high efficiency at the endpoint, which determines the total rate of β-decay electrons collected on the calorimeter.
The RF tracker and the calorimeter are recording time domain data. Longitudinal and trans- verse velocities of the electrons are constrained by the Time-of-flight (TOF) information and thus the production angle at the target. Information of the velocity is largely independent of the amplitude measured from the calorimeter. Background sources that are not electrons are suppressed by requiring consistency in the measurements within the experimental resolutions [7].
4.2. MAC-E-Filter 20
4.2 MAC-E-Filter
During this thesis, I have simulated two filter configurations. The first filter design is the default configuration and the second configuration is based on the modifications that I have made on the first filter. The first filter design is described below and the second design is described in the next section.
In order to make measurements at the end of the β spectrum, one would need a spectrometer with high luminosity coupled with a low background reduction. The spectrometer that satisfies these requirements is called: Magnetic Adiabatic Collimation combined with an Electrostatic Filter, abbreviated MAC-E-Filter. A MAC-E-Filter is used by the KATRIN collaboration to re- duce the upper bound on the neutrino mass, and was first proposed by Beamson et al. [36][6].
Two superconducting solenoids are situated opposite side of each other, which generates a magnetic field that adiabatically guides electrons into the detector. The detector is situated at the opposite side of the entry solenoid. As the electron propagates through the filter, the magnetic field will start to decrease when the electron reaches the analyzing plane (middle of the filter), where the magnetic field has its minimum value Bmin. The magnetic field will start to expand, when the electron continue to propagate towards the detector solenoid, from the analyzing plane. The perpendicular cyclotron momentum is transformed into longitudi- nal cyclotron momentum, when the electron moves to a region of lower magnetic field. An electrostatic retarding potential is imposed at the analyzing plane. Only electrons which have a minimal energy to the parallel field lines can pass though, Ek > qV. Other electrons are reflected back. One can say that the MAC-E-Filter appears to act as a integrating high-pass filter. The flux enclosed by the cyclotron orbit is constant due to adiabatic conditions. This is accomplished by the magnetic fields along the circular orbit which are slowly changing. The cyclotron radius changes to preserve the constant-flux condition when the magnetic field varies along the trajectory. The motion is an invariant and is adiabatic
φ = Z →−
B · d→−
S = Constant (4.4)
Equation (4.4) can also be expressed as a conserved quantity µ = E⊥
B (4.5)
To derive the transmission functions we can take advantage of the properties of the idealized MAC-E-Filter: only the parallel velocity can overcome the potential barrier, electrons total energy is conserved (energy contributions from the magnetic field is negligible). The endpoint electron initial non-relativistic energy is
E0= 1
2mv20− eV0 = Ek0+ E⊥0 − eV0 = 1
2mv20(cos2θ0+ sin2θ0) − eV0 (4.6)
4.2. MAC-E-Filter 21
Where V0 is the source potential, θ is the initial angle between electrons momentum and the guiding magnetic field lines, e is the electric charge and m is the electron mass. To express this at any point in the filter one uses the conservation of energy and the adiabatic condition
E0 = Ek+ E⊥− eV = Ek+ B B0
E0⊥− eV (4.7)
To see if an electron can pass through at the analyzing plane and hit the detector at the other end. By solving for Ek, one can determine if an electron can reach the end of the filter
Ek = 1
2mv20− B
B0E0⊥+ e(V − V0)= E0(1 − B
B0 sin2θ0)+ e(V − V0) (4.8) When Ek= 0 the electron will fail to reach the detector and reverse direction along the field line.
If an electron should be able to reach the detector, it would need to overcome two obstacles: the potential barrier in the analyzing plane and the magnetic mirror effect. The Magnetic mirror effect occurs when an electron with a specific momentum at an angle to a magnetic field line moves to a higher magnetic field region. By using Equation (4.8) and assuming Ek = 0, then we can rearrange for θ0so that we get
θ0 = sin−1(B0
B
E0+ V − V0
E0 ) (4.9)
Consider the case when V= V0 (see Equation 4.9). There is no limit of the pitch angle if the electron moves to a region of lower magnetic field. In the case of when the electron moves to a region of higher magnetic field B > B0, then there is a maximum initial angle larger than which the electron will reverse its direction along the magnetic field lines. The initial angle is determined by the values in Equation 4.9. It’s recommended to have the magnetic fields in the detector solenoid that is smaller or equal to the magnetic field in the source solenoid, this is needed if one wants to maximize the count rate of electrons in the detector.
If one want to maximize the count rate of electrons that will reach the detector, then one needs first to maximize the range of the electron emission angles. To do this one can make sure that the field at the detector is smaller or equal to the field in the source. One can also accelerate electrons to compensate for this mirror effect when the electric potential is in consideration.
The transmission functions shown bellow, describes what fraction of the 2π solid angle that will the reach the detector for a given initial energy level. For the transmission functions, one will need to consider both the electrostatic cutoff and the magnetic mirror effect. The maximum available solid angles is reduced by the magnetic mirror effect and the slope and position decreases from 1 to 0 by the electrostatic cutoff. The transmission functions is given by
T(E0)Mirror = Ω(E0) 2π = 1 −
r 1 − B0
Bd
E0− V0+ Vd
E0 (4.10)
4.3. New filter design 22
T(E0)Filter = Ω(E0) 2π = 1 −
r 1 − B0
Bm
E0− V0− Vm
E0 (4.11)
where the subscripts is the initial values (X0), midpoint values is (Xm) and the detector values is (Xd). The PTOLEMY project does not need a sharp cutoff due to the use of a detector with precision calorimetry. The MAC-E-Filter’s main role is to remove enough below the endpoint of the spectrum, so that the detector may not be spammed with large signal rates [7].
4.3 New filter design
The new filter design has been redesigned so that there is no need to collimate the electrons transverse kinetic energy into longitudinal velocity. Instead of climbing a voltage potential as in the default configuration, the magnetic drift gradient does all of the work on the electrons, in order to reduce the transverse kinetic energy. If one starts with an initial estimate of the orbital magnetic moment (µ), the E × B drift is then configured to balance the grad-B drift of the electron in such a way that the electron is guided into a voltage potential along the analyzing plane of the filter.
The electron will undergo cyclotron motion when it propagates perpendicular to the magnetic field lines. We can use guiding center system (GCS) variables to explain the central axis of the trajectory of an electron in gyromotion with respect to the magnetic field, by setting all cyclotron phase forces acting on the electron to zero. The GCS directory (GCS is a non-inertial frame, e.g the transverse plane is oriented orthogonal to the direction of magnetic field) does not have to be in the direction of the magnetic field line and will in principle deviate from the magnetic field direction in presence of four fundamental drift terms,
VD= V⊥= (qE + F − µ∇B − mdV dt ) × B
qB2 (4.12)
where V = V⊥ + Vk is the total phase-average velocity, while q is the electric charge and m is the rest mass of the electron, and VD is the drift velocity. There are four terms in equation 4.12, going from left to right. They are described by i) the E × B drift; ii) external force; iii) gradient-B drift and; iv) inertial force drift.
GCS is only valid when the spatial motion of electric and magnetic varies slowly relative to the cyclotron radius and in time. The cyclotron radius ρc and the cyclotron period τc is given by:
ρc
B
∇B ,
E
∇E
; (4.13)
τc
B dB/dt
,
E dE/dt
; (4.14)
4.3. New filter design 23
where the time variation ”seen” by the electron come from the variation in time at a fixed point in space, and the variation is due to displacement while the field is fixed in time: d/dt= ∂/∂t + V · ∇. The motion of the electron is described by the adiabatic invariant.
The orbital magnetic moment µ in GCS frame is given by,
µ = mv∗2⊥
2B (4.15)
where v∗2⊥ is the electrons instantaneous velocity (in GCS frame) perpendicular to the magnetic field line. It’s related to the inertial frame instant velocity v = v⊥+ vk by v∗⊥ = v⊥− VDand v∗k = vk− Vk≈ 0. And the pitch angle of the electron is given by,
α = arccosvk
v (4.16)
where α is the angle between v and B. When a non uniform magnetic field is present. Then the Hamiltonian is given by the term U= −µ · B, which give rise to a total net force:
f = −∇U = −µ∇B (4.17)
The mirror force is given by the fk which is responsible for magnetic bottle effect for trapping electrons in non-uniform magnetic fields and the magnetic adiabatic collimation. The gradient B-drift is given by f⊥. Drifts which are caused by the electric fields, are always perpendicular to E and will not do any work. The electrons follow surfaces of constant voltage under E × B drift. The gradient-B drift can do work on the electron and reduce the kinetic energy of the gyromotion when under E × B drift, for an increase in voltage potential. This is described by Equation (4.12), by inserting some terms:
dT⊥
dt = −qE · VD= −qE · (qE − µ∇B) × B qB2 = µ
B2E ·(∇B × B) (4.18) where the internal kinetic energy in GCS frame of gyromotion is T⊥. Properties q, m and µ is the electric charge, the electron mass and the orbital magnetic moment.
The measurements of tritium endpoint energy is a combination of two measurements, the first is the electrical potential energy difference between the tritium target and the calorimeter, and the second is the calorimetric measurement of the electron total kinetic energy at the end of the filter. The target resolution is 0.05 eV, which is a combination of the two measurements. This is equal to the total neutrino mass difference, taken from oscillation experiments.
The expression that satisfies the adiabatic invariance is given by:
B
∇B
∼ρc = q
r2mµ
B (4.19)
4.3. New filter design 24
where 1. The magnetic field is chosen so that its component along the x dimension, Bx, declines exponentially with a characteristic length scale λ,
Bx(z)= B0e−z/λ (4.20)
The current density j−(z) in the y-direction starts at z= 0 m and goes to z = L (where L is the thickness in the z-direction), has an exponential fall off in the z-direction with the same length scale λ. The current density is j−[A/m2] and is given by:
J++ Z L
0
j−(z)dz = 0 (4.21)
The current density integrated with respect of z is opposite in sign and equal in absolute mag- nitude to J+. Bx is zero for z > L and z < −z0. B0 is constant in the interval −z0 < z < 0 and its exponential is decreasing for 0 ≤ z ≤ L, according to equation (4.20).
Figure 4.5: (left) Tightly bound series of planes of current density over distance L. (right) Radius of curvature (Rc) introduced by a finite gap∆x (vacuum separation) between the planes of the current density.
The vacuum separation is shown in figure 4.5 in the x-direction∆x between the planes of the current density. Here ∇ · B is equal to zero in the vacuum. Applying it and evaluating the transverse component of the magnetic field gradient one obtains:
∇⊥B= (B · ∇)(B
B) − (∇ × B) × (B
B)= −B Rc
bn (4.22)
Wherebn is the unit vector normal to the curvature of the magnetic field lines, and Rc is the curvature radius. The components of the magnetic field that is satisfying the vacuum condition
∇ × B= 0 is given below by approximation of the central gaps’ region between the planes [35],
4.3. New filter design 25
Bx = B0cos
x λ
e−z/λ (4.23)
By = 0 (4.24)
Bz= −B0sin
x λ
e−z/λ (4.25)
These equations are implemented in the simulation code that the author is using. To set the voltage filter one can choose a voltage configuration that keeps the electron at a fixed position in the beginning of the filter (at z = 0 m), as the electron propagates slightly downward in the filter. In this configuration a voltage difference is introduced between the plates in the y-direction which is exponentially falling in the z-direction with the same length scale λ. To accomplish this one can make segments in the electrodes in the z-direction which in turn causes a constant E × B-drift. One can adjusts the voltage difference across the plates in the z-position, so that the electron maintain a constant height of y0. Evaluated at the fixed height of y= y0and as a function of position-z, the voltage is then given by:
V(y, z)|y=y0 = V0−µB0(1 − e−z/λ) (4.26) Where V0 is the initial voltage at z= 0 m. Example for the voltage configuration at V0 = 0 is shown in Figure 4.6. The voltage at the top plates is held at a constant value of: Vt = −µB0, and at the bottom plates at: Vb= µB0(2e−z/λ− 1) for z ≥ 0.
Figure 4.6: Example of a voltage configuration (y= 0) for a given starting voltage and trans- verse kinetic energy [35]
The components of the corresponding electric field is given by:
Ex = 0 (4.27)
Ey = E0cos
y λ
e−x/λ (4.28)
4.3. New filter design 26
Ez= −E0sin
y λ
e−z/λ (4.29)
Where E0 is the magnitude of the electric fields when z = 0 m. For the static B field one can see that the ∇ × E = 0 is satisfied by the E field, and in the vacuum region (between the filter plates) when ∇ · E = 0. To determine E0 we set the magnitude of the E × B drift equal and opposite to the grad-B drift (y= y0 ):
Ez
B ≈ −E0 B0 sin
y0 λ
(4.30)
= − µ qB
∂Bx
∂z (4.31)
= µ
qλ (4.32)
Solving for E0:
E0 = − µB0
qλ sin(y0/λ) (4.33)
The value of E0enforce the durability of the cyclotron-average height of the electron trajectory everywhere. This is due to the z-dependency of Ezmatches ∂Bx/∂z. This causes the magnitude of the E × B drift velocity to be constant along the trajectory of the electron path.
5
Simulation
For the simulations discussed in this thesis, I have used the supercomputer at PDC, the Center for High Performance Computing at KTH. It should be noted that both KTH and SU share the PDC supercomputer cluster. The computer at Grand Sasso National Laboratory (LNGS) was also used in the early stages (first filter configuration) of the simulations before shifting to PDC (for the new filter configuration). The benefit with using the PDC is one can run multiple simulations at the same time. The simulations were run remotely connecting from my laptop at AlbaNova University Center in Stockholm. The configuration files presented in Appendix C, D, E and F were originally written by Alfredo Cocco a researcher at the LNGS and by Wonyong Chung a post-doc at Princeton, and were modified in order to carry out the studies presented in this thesis.
The static electric and magnetic field simulations were carried out with the Kassiopeia package [37]. The output of Kassiopeia is a series of many variables disposed in ROOT ntuples that are analyzed with the ROOT analysis framework [39]. ROOT can, together with the Visualization Toolkit (VTK)[40], create a 3D view of the electron track through the filter. VTK can however only create the electron track together with a 3D view. The main input parameters of the Monte Carlo code are stored in a XML file, that can be edited by the user. When the necessary editing is done in the code, then one can use the Kassiopeia package to run the Monte Carlo code. The time to run the simulation varies between some minutes to couple of days depending on the level of computation. A ROOT file and a VTK file are created when the simulation comes to an end. Note that in the new filter configuration the E and B fields are defined analytically for all space (hard coded) instead of solving them for geometric configuration of charged object as in the old configuration.
The most common parameters in the simulation files are the events and the “number of steps”
nsteps. If, after having made all the important modifications in the file, wants to run the simula- tion with different number of steps or with the number of times one wants to run the simulation, one can then just modify the above parameters to preference. Terminators and fence are also important parameters. Terminators can be set if one wants to kill the particle in its track if no further interest in the simulation exist. Fence is a parameters that ”catches” the particle if it goes outside the configuration.
27
5.1. Kassiopeia package 28
5.1 Kassiopeia package
Kassiopeia is a software package based on C++ made for particle tracking in complex ge- ometries and electromagnetic fields. The design goal of Kassiopeia is to simulate with great precision and efficiency the physical evolution of multiple particles states. The particles prop- erties can thus be modified by built-in algorithms. During the initialization both the electron mass and the charge are fixed, while its position and momentum will evolve during the simula- tion. The data structure of Kassiopeia is organized into four structures: Track, Step, Event and Run.
Figure 5.1: A schematic representation with a total of six tracks and of a run with three events[37][41]
Step: Step is the lowest level of organization in the simulation. Step represent the evolution of a particles from initial state to the final state with a small amount of time. By solving the equations of motion and taking into consideration the interactions with other forms of matter and fields, one gets the propagation of the particles.
Track: The track is a sequencial collection of steps. It is the evolution of the particle from its origin to its termination. The track can be terminated by a section of the code called termina- tors.
Event: The event is a collection of causally related tracks. Each event corresponds to a partic- ular set of particle track set by the simulation parameters.
Run: Run is the highest order of organization in the Kassiopeia package. It is a representation of one execution of the simulation for a fixed setup.
The strength of Kassiopeia lies in its flexibility and modularity. The user can choose between modules to be used in the simulation for example: generators, field generators and interactions.
But the user can also change the composition of the simulation algorithm depending on the geometric state of the particle. It is the configuration file in Kassiopeia that is defining the run of the simulation, including the geometry. It also includes all different physical processes as well as the detail in the output and the recorded quantities.
