Download Development of a Resistive Plate Chamber detector simulation

Faculty of Sciences
Department of Physics and Astronomy
Academic year 2015–2016
Development of a Resistive Plate Chamber detector
simulation framework
Frederic Van Assche
Promoter: Dr. Michael Tytgat
Supervisor: Alexis Fagot
Thesis submitted in partial fulfilment of the requirements for the degree of
Master of Science in Physics and Astronomy
Acknowledgements
This thesis marks the end of a five year journey as a physics student, and while it ultimately
bears my name on the cover, I would not have been at this point now without the support of
many people whom I would like to mention personally.
Prof. Ryckbosch was the first to spark my interest in experimental particle physics and has
been responsible for a significant fraction of my education since.
Building and testing a detector prototype for my bachelor’s thesis under the supervision of
Alexis and Michael set me down the path of detector hardware research, and I have spent
countless hours continuing this endeavour over the past two and a half years. My heartfelt
gratitude goes out to you for giving me the perfect mix of guidance and freedom to make my
time in the RPC group such a rewarding experience.
The other thesis and PhD students at the INW have contrasted the countless hours in the lab
and behind my desk with chats, tea, lunches, and laughs. They bring colour to what would
otherwise be a very boring and grey building.
While my prior educational path has seen its fair share of hardships, my parents have never
withdrawn their love and support. The trust and opportunities you have given me over the
years can not be repaid, but will be paid forward.
A wholehearted thank you goes to my friend and uncle, Wim, who has often times offered
me a different perspective during the difficult moments. Your persistent support has helped
me through a few key moments.
To Charis, my wonderful better half: Your enthousiasm and encouragement has been a continuing source of motivation throughout my studies, even though my appreciation for it might
not always have been apparent. You and I make a great team, during the holidays just as
well as the sleepless nights before exams.
Frederic
iii
Nederlandstalige samenvatting
Het Standaardmodel van de deeltjesfysica is grotendeels vervolledigd sinds de ontdekking van
het Higgsdeeltje in 2012. Hiermee is het verhaal van de hoge-energie deeltjesfysica echter
nog niet afgelopen. Er zijn meerdere aanwijzingen, onder meer in de sterrenkunde en in de
deeltjesstroom uit kernreactoren, die doen vermoeden dat er nog fysica Voorbij het Standaardmodel (BSM) te ontdekken is. Deze volgende uitdagingen vereisen echter steeds krachtigere
deeltjesversnellers en steeds gevoeligere detectoren.
Rond 2018 zal de Large Hadron Collider (LHC) de huidige periode van verzamelen van data
stopzetten voor een volgende upgrade. Het is voorzien dat de aanpassingen gemaakt aan de
deeltjesversneller de luminositeit (een maat voor de productiviteit van een versneller) met
een factor tien zullen doen toenemen. Dit plaatst de onderzoekers verantwoordelijk voor het
onderhoud en de ontwikkeling van de grote detectorexperimenten aan de LHC voor een nieuwe
uitdaging, want het aantal deeltjes dat per seconde te verwerken is zal met een gelijkaardige
factor tien toenemen.
Resistive Plate Chambers, de detectorsoort onderzocht in deze thesis, bestaan in hun basisvorm uit twee parallelle platen gemaakt uit een hoogresistief materiaal, zoals glas of bakeliet. Over deze platen wordt hoogspanning aangelegd, wat zorgt voor een elektrisch veld
tussen de elektroden.
Wanneer een geladen deeltje door het werkingsgas tussen de elektroden passeert, ioniseert
het onderweg enkele gasmoleculen. De elektronen die vrijgekomen zijn zullen vervolgens door
het elektrisch veld versneld worden richting kathode. Indien het veld sterk genoeg is, zullen
de versnelde elektronen genoeg kinetische energie krijgen om zelf opnieuw een gasmolecule te
ioniseren. Dit zorgt voor een exponentiële toename aan bewegende elektronen die men een
Townsendlawine noemt. Het bewegen van deze elektronen kan, dankzij het resistief zijn van
de elektroden, buiten het gasvolume waargenomen worden door de signaalelektroden.
Aan dit basisprincipe van de RPC-detector dienen enkele correcties gemaakt te worden wanneer er een groot aantal deeltjes per seconde door passeert. Hierdoor daalt de efficiëntie en
levensduur van een RPC sterk. De deeltjestempo’s voorzien na de upgrade van de LHC zorgen
er voor dat de bestaande modellen niet meer voldoen. Er zijn nieuwe ontwerpen, materialen
v
vi
en werkingsgassen nodig om deze volgende stap in de deeltjesfysica mogelijk te maken.
In deze thesis wordt een simulatieframework opgesteld om zonder prototypes te bouwen
nieuwe materialen en werkingsgassen te kunnen testen op een geparametriseerd RPC-model.
Deze simulatiecode wordt vervolgens gebruikt om het gedrag van enkele bestaande modellen
te bestuderen.
Contents
1 Introduction
1.1
1.2
1.3
1
High-energy physics and collider experiments . . . . . . . . . . . . . . . . . .
1
1.1.1
The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.3
Particle detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.1.4
The Compact Muon Solenoid . . . . . . . . . . . . . . . . . . . . . . .
7
Resistive Plate Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1
Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.2
Current problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Goals of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2 RPC physics primer
2.1
13
Detector construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.1
Electrode materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.2
Working gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.1.3
Construction variants . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
The Townsend amplification mechanism . . . . . . . . . . . . . . . . . . . . .
17
2.3
Alternative working mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4
Induced signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.5
Limitations in high-rate environments . . . . . . . . . . . . . . . . . . . . . .
20
2.5.1
Space charge effects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.5.2
Aging of electrode materials . . . . . . . . . . . . . . . . . . . . . . . .
21
3 Electrostatics and field calculations
3.1
3.2
23
Classical electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1.1
Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.1.2
Computation of electric fields . . . . . . . . . . . . . . . . . . . . . . .
25
Relative permittivity measurements
. . . . . . . . . . . . . . . . . . . . . . .
29
3.2.1
Parallel plate capacitors . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2.2
Capacitance measurement . . . . . . . . . . . . . . . . . . . . . . . . .
31
ix
x
Contents
3.2.3
Measurement setup and results . . . . . . . . . . . . . . . . . . . . . .
4 Simulation setup
4.1
4.2
4.3
37
Basic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.1.1
Detector geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.1.2
Electric field calculations . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.1.3
Physics simulation and signal generation . . . . . . . . . . . . . . . . .
42
Garfield++ simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.1
Primary cluster generation . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.2
Gas performance calculation
. . . . . . . . . . . . . . . . . . . . . . .
43
4.2.3
Physics simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Hybrid simulation code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.3.1
Initialisation of the code . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.2
Preparation of a simulation . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.3
Primary tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.3.4
Avalanche generation . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3.5
Signal calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3.6
Positional signal weighting . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.3.7
Comparison to the Garfield++ code . . . . . . . . . . . . . . . . . . .
51
4.3.8
Overview of approximations . . . . . . . . . . . . . . . . . . . . . . . .
51
5 Verification of simulated data
5.1
33
53
Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.1.1
The data acquisition system . . . . . . . . . . . . . . . . . . . . . . . .
53
5.1.2
Collected data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.2
Gas parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.3
Efficiency curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.3.1
Simulation of the glass RPC and RE2/2 detector . . . . . . . . . . . .
56
5.3.2
Simulation of the multigap RPC . . . . . . . . . . . . . . . . . . . . .
57
5.3.3
Physics of the multigap prototype . . . . . . . . . . . . . . . . . . . .
61
6 Conclusions and outlook
63
6.1
Permittivity measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.2
Simulation results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.3
Minor points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.4
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.4.1
Simulation modifications . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.4.2
Measurements and prototypes . . . . . . . . . . . . . . . . . . . . . . .
64
6.4.3
Permittivity measurements . . . . . . . . . . . . . . . . . . . . . . . .
65
Contents
xi
A Parameters of tested RPC models
67
B Permittivity measurement results
69
List of Figures
71
List of Tables
75
Bibliography
77
List of Abbreviations
83
Chapter 1
Introduction
The particle physics community has been shaped by the Standard Model (SM) since the
1960s. This theory describes three of the four fundamental interactions: the electromagnetic,
weak and strong forces, and classifies all known subatomic particles. With the discovery
of the Higgs boson by the CMS [1] and ATLAS [2] collaborations in 2012 and subsequent
confirmation one year later, the Standard Model is now mostly complete.
The high-energy physics (HEP) field now faces a new set of challenges in the near future. Our
understanding of the SM and its parameters can further be improved by new, more precise
measurements, and there are many clear signs1 of physics beyond the Standard Model. These
new searches call for many key advances in our current detector and accelerator designs.
1.1
1.1.1
High-energy physics and collider experiments
The Standard Model
Figure 1.1 shows an overview of the Standard Model of particle physics [3]. The electromagnetic, weak and strong interactions are unified in this theory, which provides a classification
for all known subatomic particles. The two major particle categories are the fermions, out of
which all known matter is built up, and the bosons, which mediate the interactions between
the fermions.
The fermions are further divided into leptons, which are only affected by the electroweak
interaction (i.e. the unification of electromagnetic and weak interactions), and the quarks
which also interact through the strong force. Both leptons and quarks are split into three
generations, shown as columns in figure 1.1. Generation I contains all particles necessary
for construction of everyday matter, and nuclear decay products. The second generation is
still encountered on a more microscopic scale, and in cosmic radiation as muons. The third
1
A lack of dark matter candidates in the SM, the non-zero neutrino masses, the matter-antimatter assymetry, unification of the SM and gravity, ...
1
2
Chapter 1. Introduction
generation particles have the largest masses and consequently shortest lifetimes of the three
generations, and are not easily encountered in nature.
Bosons
I
II
III
u
c
t
g
H
1968
1974
1995
1978
2012
d
s
b
γ
1968
1968
1977
1923
e
µ
τ
Z
1897
1937
1976
1983
νe
νµ
ντ
W
1956
1962
2000
1983
Gauge bosons
Leptons
Quarks
Fermions
Figure 1.1: The Standard Model of particle physics with years of discovery. The first observation of
photon momentum by Arthur Compton can be considered the discovery of the photon
as a particle.
Mass-energy equivalence
The famous law of mass-energy equivalence, formulated by Albert Einstein, is:
E = mc2
Through this relation unstable particles can decay into lighter particles. Since energy and
momentum are both conserved, the mass difference between the heavy parent particle and
the sum of the decay particle masses must be converted into kinetic energy. One example of
an unstable elementary particle is the muon, which decays as:
µ− → W ∗− νµ → e− ν e νµ
The final state of an electron and two neutrinos is lighter than the parent muon, but all
three decay products will be moving away from eachother through the gained kinetic energy.
While this reaction only happens spontaneously when mass can be converted into energy, it
is possible to reverse this process given enough energy input. This happens at large collider
experiments, where particle beams are accelerated up to high energies and are brought to
collison. Elementary particles colliding with their corresponding antiparticles are annihilated,
Chapter 1. Introduction
3
providing a short-lived packet of energy. This intermediate state then converts back into mass,
creating multiple elementary particles.
Since energy is conserved, the elementary particles created in collider experiments can never
have masses exceeding the energies given initially to the collided beam particles. This has
the consequence that the discoveries of particles from the Standard Model have generally
followed technological advances in particle accelerators. Heavier particles, and ones that only
form through rare types of collisions generally require more powerful machines to be detected
in sufficient numbers, or to be seen at all.
Shortcomings of the Standard Model
With the discovery of the tau lepton in 1995 [4], the tau neutrino in 2000 [5] and confirmation
of the Higgs boson discovery in 2013 [6], all particles predicted by the Standard Model have
now been observed. Even though the SM might now appear to be complete, the story of
particle physics is far from over.
The Standard Model has a number of obvious and less obvious shortcomings that need to be
resolved. A big omission in figure 1.1 is the gravitational interaction, which is entirely absent.
Unifying relativity and the SM is a very active field in theoretical physics with hopes of one
day combining the four known forces into a Theory Of Everything (TOE). An example of a
candidate TOE is the string theory framework.
An intermediate step between a Theory Of Everything and the Standard Model would be a
Grand Unified Theory (GUT), which unifies the electroweak and the strong forces, which are
still separate in the SM.
The Standard Model still leaves a number of questions unanswered, including:
• Its 19 parameters and the mathematical structure itself do not currently have a verified,
more fundamental explanation.
• The cosmological standard model is at odds with the particle physics SM in different
ways [7]. The SM gives equal importance to matter and antimatter yet we dominantly
observe the former in the visible universe, and Dark Matter is widely accepted as a requirement to explain our astronomical observations, but the SM provides no acceptable
candidate particle.
• Neutrinos oscillate between their different flavours as they propagate, as observed in
experiments [8]. This is only possible if they have non-zero (though small) masses,
which are not explained by the mass mechanisms in the Standard Model.
4
Chapter 1. Introduction
Some of the shortcomings could find a possible solution in the theoretical framework called
supersymmetry (SUSY). Theories based on this framework predict each Standard Model
particle to have a supersymmetric “partner”, with fermions having boson partners and vice
versa. This would solve a number of problems in cosmology, for example by providing Dark
Matter candidate particles, and by giving a way for matter to dominate over antimatter.
The search for physics Beyond the Standard Model (BSM), e.g. by looking for SUSY particles,
as well as further refinement of our understanding of the SM and its parameters will require
continuous development of more powerful accelerators and more sensitive particle detectors
in the coming years.
1.1.2
The Large Hadron Collider
Figure 1.2: Schematic representation of the LHC layout. The storage ring houses two particle beams
running in opposite directions. Four collision points are provided, each housing one large
detector experiment.
At the time of writing, the most powerful particle accelerator in the world is the Large Hadron
Collider (LHC) at CERN, which accelerates particles up to energies of 6.5 TeV in two beams.
At four points, shown on figure 1.2, the beams are brought to collision to give particle-particle
interactions at up to 13 TeV.
For particle physics research the LHC accelerates and collides protons, the results of which are
Chapter 1. Introduction
5
analysed by the CMS [9], ATLAS [10] and LHCb detectors in three of the four collision points.
The other detector experiment, ALICE, investigates proton-lead and lead-lead collisions to
examine the quark-gluon plasma, a state of matter believed to have existed for a brief period
following the Big Bang. Three additional smaller detectors (TOTEM, MoEDAL and LHCf)
perform more specialised research and share collision points with the larger experiments.
Future development: the HL-LHC
In light of the future challenges discussed in section 1.1.1, the LHC is scheduled to receive a
number of upgrades throughout its lifetime [11]. While the particle energies have recently been
brought close to the original 14 TeV design value, no further increases in energy are planned
for the forseeable future. To still be able to provide the collisions required for BSM searches
the next long shutdown will see the luminosity increased by roughly an order of magnitude to
around 1035 cm−2 s−1 , with the new name High Luminosity Large Hadron Collider (HL-LHC).
Luminosity has dimensions of events per time per area, and is a measure of particle collider
performance. It is used to calculate the rate at which processes of interest happen, and a
tenfold increase in luminosity will generally lead to a similar increase in process rates. This
will increase the chances at observing processes which are currently very rare, and will also
improve the statistics of current measurements through more data.
While this luminosity increase is a challenge for accelerator construction itself, it also demands
new developments in particle detectors. The different experiments at the LHC have to be
upgraded to cope with the higher rates of incident particles, without decreasing detector
sensitivity.
1.1.3
Particle detectors
Any direct detection of particles happens through interaction of said particles with matter.
Particles which do not interact with matter can only be detected indirectly, for example by
observing their decay products if they are unstable, or by making up the balance of energies
and momenta according to the preservation laws and noticing missing energy or momentum.
Neutrinos are excellent examples of hard to detect particles, which interact only weakly with
matter. They can travel through lightyears of lead without interacting, so detection of single
neutrinos in experiments such as CMS is generally done by looking for missing momentum.
Figure 1.3 shows a plot of the Bethe-Bloch formula for muons in copper. This formula carries
a large importance in detector design, since it quantifies empirically how charged particles
interact with matter in function of their energies:
−
dE
dx
1 1 2me c2 β 2 γ 2 Tmax
δ(βγ)
2
= Kz
ln
−β −
Z β2 2
I2
2
2A
(1.1)
6
Chapter 1. Introduction
Figure 1.3: The Bethe-Bloch curve of a muon travelling through copper. The material density has
here been factored out of the expression and divided out, giving the Y-axis units of
MeV cm2 g−1 . Figure taken from [12]
In equation (1.1) K is proportional to the electron density of the material. The dependency
on material type is rather weak, and is often factored out. The energy loss or stopping power
gives the energy transferred to the material per unit of distance. This energy transfer can
happen in a variety of ways, such as by ionisation in a gas, or by knocking particles out of a
crystal lattice.
While many particle detector concepts use the Bethe-Bloch energy loss of charged particles in
their active medium, it is not the only possible detection mechanism. Other examples include
detection of recoil of a nucleus after interaction with an incident particle, or Cherenkov light
produced by particles travelling faster than the local phase velocity of light in a detection
medium.
Two examples of alternate detection methods are the Large Underground Xenon experiment
(LUX) [13], which tries to perform direct detection of dark matter particles using their proposed recoil off the atoms in a vat of liquid xenon, and the Oscillation Project with EmulsiontRacking Apparatus (OPERA) experiment [14] which looks for neutrinos using bricks of
photographic film interleaved with lead plates.
Chapter 1. Introduction
1.1.4
7
The Compact Muon Solenoid
The Compact Muon Solenoid (CMS) is one of the two general purpose detector experiments
installed in one of the beam crossing points of the LHC. The detector is roughly cylindershaped, with the main volume called the barrel region, closed off by the endcap regions.
Figure 1.4 shows a schematic cross-section of the barrel region, laying out the different layers
of the detector.
Figure 1.4: A slice through the barrel region of the Compact Muon Solenoid, showing the different
detector layers. Example tracks for the different particle types show possible tracks
through the detector structure.
Like most general purpose particle detectors, CMS follows a layered approach at particle identification and tracking. The innermost layer is a silicon tracker, providing very high resolution
particle tracks. The tracker is followed by electromagnetic and hadron calorimeters, which are
intended to fully absorb and measure the energy of impacting particles. Around these layers
a superconducting solenoid generates a solenoidal magnetic field, in which charged particle
tracks get curved. These curvatures make it possible to determine the momentum and the sign
of the charge of a particle. This information, combined with the energy measurements in the
calorimeters and charge measurements based on ionisation density enable full identification
of particle masses and types.
8
Chapter 1. Introduction
Outside the solenoid an additional set of layers are placed, alternating muon detectors with
iron return yokes. These latter shape and confine the magnetic field outside the solenoid,
enabling measurement of muon momenta and charge signs. By having these muon chambers
on the outer layers of the detector the flux through the muon chambers will largely consist of
muons and neutrinos, with the latter type passing through without interactions.
Figure 1.5: A quadrant of the CMS detector, highlighting the muon detection system in its planned
form. The dashed box indicates the region of high background rates where new GEM
detectors and RPCs types are required. Station 4 is scheduled to be installed during
Long Shutdown 2. Figure taken from [15]
The structure of the endcaps is similar to that of the barrel region, with layers of calorimeters,
followed by return yokes and muon detectors. Figure 1.5 shows a cut through a quadrant
of the CMS detector. The blue detectors in the figure are resistive plate chamber (RPC)
detectors, with the RE4/3 type RPCs manufactured in part in Ghent, and installed during
Long Shutdown 1 from 2013 to 2015.
For the proposed Phase-II upgrade [16] of CMS to accompany the HL-LHC, a great many
parts of the detector will have to be upgraded to cope with the increased particle flux. This
is scheduled to happen during Long Shutdown 2 and Long Shutdown 3, which are expected
to start in mid-2018 and around 2022–2023 respectively.
Chapter 1. Introduction
1.2
9
Resistive Plate Chambers
The basic principle behind a resistive plate chamber (RPC) detector [17] is explained here,
with a more thorough discussion reserved for chapter 2. For now it suffices to have a basic
understanding of the structure and operation of RPC detectors, so the goals of this thesis can
be laid down in section 1.3.
Figure 1.6 shows the minimal structure of an RPC: two resistive electrode plates with a high
voltage over them. This creates an electric field inside the gap between the electrodes, which
is filled with a working gas. While multiple variants of this simple design are found in realworld detectors built out of a larger number of components, the basic operating principle
remains the same.
1.2.1
Working principle
When a charged particle passes through the gas gap of an RPC detector, it generates an
ionising track according to the Bethe-Bloch formula for the working gas, as discussed in
section 1.1.3. The ionisation clusters in this track will each contain one or more free electrons,
~
which will experience an acceleration in the electric field E.
If this electric field is above a certain threshold depending on gas type and detector geometry,
the accelerated electron can reach a high enough energy to ionise another gas molecule in a
collision. This frees another electron, which can in turn attain sufficient kinetic energy to
ionise yet another molecule, etc. This effect of an exponentially growing electron cloud is
called a Townsend avalanche, and is the main working mode for an RPC. In chapter 2 this
simple picture will be expanded into a more realistic model.
µ−
−
+
~
E
Electrodes
HV
Townsend avalanche
Figure 1.6: Simple view of the working principle of a resistive plate chamber detector. The outer
~
plates are electrodes, over which a high voltage is applied, generating an electric field E
between the plates. The inside of this detector is filled with a working gas, in which the
passing muon will leaves an ionisation track. One of the freed electrons has developed
into a Townsend avalanche.
10
1.2.2
Chapter 1. Introduction
Current problems
Current generations of RPC detectors commonly suffer from two important problems. Both
of these shortcomings will have to be overcome in light of the future HL-LHC upgrade.
The first problem puts an upper limit on the particle flux where resistive plate chamber
detectors can be competitive with other detector types. As shown in figure 1.6, the detection
principle relies on the development of a cloud of charged electrons and ions inside the gas gap.
The electrons have a fairly high drift velocity, and they quickly impact the electrode plate at
one end of the gap. Ionised gas molecules are much larger and heavier than electrons, and
as such have far lower drift velocities. This means that clouds of positive ions linger in the
gas after passage of a charged particle. Recombination of ions and electrons and forced gas
flow removing the avalanche products will eventually diffuse the ion cloud, but this takes a
fair amount of time. In the mean time, this small region of detector volume will see a slight
shielding of the electric field, causing the detection efficiency to locally decrease.
The electrons deposited into the electrode plates will have a similar effect. Local charging of
the electrodes will cause a departure from an idealised homogeneous electric field, leading to
a local drop in field strength. The electrode plates are made of a material of high resistivity
(i.e. an insulator), which means that this deposited charge will not quickly dissipate.
Both the space charge and the charge deposition effect will, in high particle flux environments,
cause the detector to stay below its optimal efficiency [18]. This is especially relevant in the
case of the High Luminosity Large Hadron Collider where particle rates are expected to
increase tenfold over their current fluxes.
The second problem is related to the first, by means of the charge deposited into the electrode
material. While insulators are able to slowly even out local charge excesses, they generally
only contain a very limited amount of free charge carriers due to their high resistivity. Over
time this supply of free carriers will actually deplete [19], exacerbating the rate limiting effects
of the deposited electrons. In the environment created by the future HL-LHC the lifespan
of RPC detectors will be reduced by a factor of ten, causing the RPCs to potentially reach
end-of-life before the end of the experiments.
1.3
Goals of this thesis
To solve the problems outlined in the previous section, multiple avenues are being explored.
New electrode materials with lower resistivity generally show greatly reduced aging effects.
Tests with low resistive doped glasses, as well as ceramics with tuneable resistivity have shown
great promise.
Chapter 1. Introduction
11
Similarly, changing the working gas could reduce the space charge effect, and the current mixture consists mostly of R134a which is a greenhouse gas. For improved detector performance
and decreased environmental impact [20], new gas mixtures are being considered.
The goal of this thesis is to develop a simulation framework with a flexible detector geometry
and interchangeable materials and working gas. These simulations can then be used to both
verify and explain the behaviour and performance of current prototypes, and to assist in
development of new detector geometries and material choices through predictions of detector
performance, reducing the need for a large number of different detector prototypes.
Chapter 2
RPC physics primer
Resistive plate chamber are a type of charged particle detectors. They can be constructed to
cover large areas, and are able to reach sub-nanosecond time resolutions. This makes them
excellent timing and triggering detectors, especially in muon systems for general purpose
detector experiments. They are less well suited as tracking detectors due to their fairly poor
spatial resolution. In high particle fluxes their efficiencies can drop, and detector lifetimes can
drop down to single digit years. They are not well adapted to mobile applications, as they
require extensive support hardware in the form of high voltage supplies and gas systems.
These advantages and disadvantages are direct results of their construction and mode of
operation, as will now be discussed.
2.1
Detector construction
As was previously discussed in section 1.2 and shown in figure 1.6, an RPC requires the
following essential components:
• Two electrodes made out of a resistive material.
• A high voltage, typically between 5 and 10 kV, applied over the electrodes. This results
in an electric field of around 50 kV cm−1 to 100 kV cm−1 , depending on RPC construction.
• Some type of working gas filling the space between the electrodes.
While these elements are sufficient to enable the basic working principle of an RPC detector,
real-world models require more components to be useful:
• A varying number of support elements are required: spacers to keep the electrode plates
at a uniform separation distance, gas inlet and flow control structures, high voltage
connections, etc.
13
14
Chapter 2. RPC physics primer
• Signal generation in RPC detectors happens inductively, so some form of signal pick-up
electrodes are required outside the active volume.
These extra components can cause deviations form the ideal behaviour to varying degrees,
for example by introducing regions of non-uniformity in the electric field.
2.1.1
Electrode materials
The most defining characteristic of resistive plate chambers when compared to other types of
detectors that use ionisation of a working gas by charged particle tracks, is the use of resistive
materials for the high voltage electrodes [21]. Most other detector types would use conductive
electrodes with the dual function of electric field generation and signal pick-up, or place the
signal electrodes inside the active volume.
Separating these functions in RPCs has a number of key advantages:
• High voltage electrodes and signal electrodes no longer have to share the same geometry.
One can for example use grids of conducive pads to get positional readouts of signals.
• Read-out electronics are fully separated from the high-voltage supply, removing the
need for large high-voltage coupling capacitors.
• The electric field inside the gap is not disturbed by any read-out structures.
Seen from a small-signal analysis (i.e. small deviations from the steady-state working point)
view, the electrodes are connected to ground. They effectively form a high-pass filter RCnetwork, defined by the capacitance of the electrode plates and the resistivity of the signal
paths through their volumes to ground. The cut-off frequency of a high-pass filter, defined as
the frequency above which the filter attenuates more than 3 dB, is given by:
fc =
1
1
=
2πτ
2πRC
(2.1)
This fc has to remain well below the lowest frequency components of the signals. Conductive
electrodes have very low resistivity and derived resistance, and would shield the short-lived
signals from the pick-up electrodes. Materials with higher resistivity, such as glass or plastics,
will be transparent to the signals of interest.
Selecting a good electrode material is an exercise in balancing two opposed requirements.
Insulators are very poor at equalizing the charges applied over them, and will lead to very
inhomogeneous electric fields if used as high voltage electrodes directly. The low resistivity
required for uniform charge distributions and quick equalisation of deposited charges is at
odds with the high resistivity required for signal transparency.
Chapter 2. RPC physics primer
15
A common solution to this problem is to coat one side of each electrode with a less resistive
layer. Figure 2.1 shows the paint layer on a multigap RPC prototype, created by mixing a
conductive (Electrodag 6017SS [22]) and resistive (Electrodag PM-404 [23]) paint.
Figure 2.1: Example RPC electrode, showing a paint layer of lower resistivity on top of higher
resitivity glass. A copper strip is attached onto the paint layer by means of conductive
glue and provides a connection point for the high voltage cabling.
Finding materials with the perfect compromise resistivity is challenging, as most materials
tend to fall on either the conductive or the insulating end of the spectrum. Some materials
provide tuneable resistivity, e.g. by doping glass to form low-resistance silicate (LRS) glass,
though resistivities of interest are often in a region where small material changes result in
large resistivity deviations.
2.1.2
Working gas
The working gas used in RPCs has a large impact on the performance of the detector. Noise
rates, sensitivity, spatial resolution and power consumption can all be influenced by the choice
of gas mixture.
The CMS and ATLAS experiments use a mixture [20] of 95.2 % R134a (CH2 FCF3 ), 4.5 %
isobutane (i−C4 H10 ) and 0.3 % SF6 . These components each serve a separate function: The
R134a, a Freon gas, is a fairly dense gas that provides most of the ionisation targets as main
component. The isobutane strongly absorbs UV photons which can get emitted when a gas
ion and electron recombine. This UV quencher is important, otherwise one ionisation track
could cause further ionisations in other parts of the detector by sending out these photons. A
third component, SF6 , is added in a very small quantity. This gas is strongly electronegative,
which absorbs some of the electrons freed during avalanche development. SF6 is a spark
quencher, and is often used in larger concentrations as an industrial gaseous insulator.
16
2.1.3
Chapter 2. RPC physics primer
Construction variants
RPC detectors come in many different configurations, but two geometries are frequently used
in this thesis and deserve special mention.
The double chamber RPC uses two complete detectors, with the pick-up electrodes sandwiched
in between. All prototypes mentioned are built using this base design, shown in figure 2.2a.
This effectively combines the efficiencies of the separate RPCs, allowing detection even if one
of the chambers does not generate a signal above detection treshold.
The multigap resistive plate chamber is a modification [24] of the basic RPC design, where
the gap between the outer electrode plates is further divided into subgaps, usually by means
of electrically floating resistive plates placed between the main electrodes. An example of
such a configuration is shown in figure 2.2b.
~
E
-HV
Signal electrodes
(a) A double chamber structure. The inner electrodes are referenced to ground, with a negative high
voltage connected to the outer. A PCB with pick-up electrodes is placed between the chambers.
-HV
Subgaps
~
E
(b) Multigap RPC configuration. The plates between the main electrodes are electrically floating and
get charged by the electric field.
Figure 2.2: Two specific variants of RPC construction.
Chapter 2. RPC physics primer
2.2
17
The Townsend amplification mechanism
When a free electron in a gas gets accelerated under the influence of an electric field, it will
scatter off the surrounding gas molecules. This repeated scattering will, over many collisions
and paths, cause electrons to reach a terminal velocity called the drift velocity vd . This
velocity is a function of electric field strength and gas density (or pressure).
Depending on the energy of the moving electrons, which is given by vd , collisions with gas
molecules can have a number of outcomes:
a) Completely elastic scattering: the gas molecule will remain mostly unaffected through its
much higher relative mass, and the electron will be scattered into a new direction.
b) Excitation of the gas molecule using part of the electron’s kinetic energy.
c) Ionisation of the gas molecule, forming a positive ion and a new free electron.
d) Capture of the electron by an electronegative gas molecule, forming a negative ion.
Possibility a) and b) only contribute to determining the drift velocity and spread of the
electrons. Possibility c) introduces new electrons, and option d) removes electrons from the
medium. These effects are energy dependent, with elastic scattering happening at all energies,
and new ionisation often requiring the highest Ekin .
At high enough electric fields, ionisation rates will overtake attachment rates, and a free
electron will create an avalanche of secondary electrons. This process is called Townsend
amplification, after Johh Sealy Townsend who discovered the mechanism during the 1890s.
Quantifying Townsend amplification
If one takes the assumption that any scattering event is independent of any previous events, it
is possible to define a Townsend coefficient α, which gives the number of ionisations per unit
distance. Similarly, the attachment coefficient η is defined as the number of attachments per
unit distance. This approximation is valid in the case of normal gas densities at atmospheric
pressure. The electrons only reach low drift velocities on the order of 100 µm ns−1 , and scattering energy losses are easily compensated by the electric fields encountered (O(10 kV cm−1 )).
Since the electrons can be considered to move at their drift velocities most of the time, combined with the fact that their energies determine the scattering cross-sections, the ionisation
and attachment rates can reasonably be approximated as constants.
With this we can approach avalanche multiplication from a statistical point of view. If a
given avalanche has n electrons at position x, then there is a nαdx probability that it will
18
Chapter 2. RPC physics primer
contain n + 1 electrons at position x + dx. Similarly, the probability of n − 1 electrons after
an attachment in the interval dx is nηdx. This gives us the following relation:
dn
= (α − η)n
dx
(2.2)
Setting n(0) = 1 and solving for n gives:
n(x) = e(α−η)x
(2.3)
This shows that the average avalanche will show exponential growth for α > η, and exponential decay for η > α. Figure 2.3 shows this exponential growth for single-electron avalanches
with α > η. Once the number of electrons grows to about 100, the avalanche grows exponentially. This figure also demonstrates how the beginning of the avalanche determines the
eventual size.
Figure 2.3: Simulated avalanches for n(x) = 0, α = 13 mm−1 and η = 3.5 mm−1 . The e(α−η)x
exponential growth is clearly visible once the avalanche contains about 100 electrons.
Figure taken from [25].
2.3
Alternative working mode
The avalanche mode of operation is not the only way an RPC can generate signals. Historically, the first RPCs were operated in streamer or spark mode [26].
While most of the discussion focuses on the electrons with relatively high drift velocities, the
amount of positive ions can be very large for strong electric fields or high values of α. The
ions will slowly drift towards the cathode, where their space charge can locally disturb the
Chapter 2. RPC physics primer
19
electric field to the point where they accelerate electrons out of the cathode material. These
electrons will in turn start avalanching, which further feeds the positive ion cloud.
Once the ion density between the anode and cathode reaches high enough levels to form a
conductive plasma filament, a spark will form across the gap. This locally discharges the
electrodes, causing a local dead time inside the detector.
These streamers and sparks are delayed with respect to the original avalanche signal, and
cause large disturbances in the charge distribution of the electrodes. This mode of operation
is therefore not preferred, as it introduces large time jitter and can bring the rate capability
of an RPC down by several orders of magnitude [27].
2.4
Induced signals
As mentioned in section 2.1.1, the most defining feature of resistive plate chamber detectors
are their resistive electrodes. Because these are chosen to allow signals to pass through to
signal pick-up electrodes outside the active volume, collection of the avalanche charges is not
the primary signal generation mechanism.
In 1938 William Shockley published [28] a method to calculate the current induced in a
conductor by a charge moving nearby. One year later Simon Ramo published [29] a more
mathematically detailed description of what would become known as the Shockley-Ramo
theorem.
This theorem is based off the principle that instantaneous electric currents in a conductor
induced by outside moving charges are caused by the instantaneous change in electrostatic
flux lines through that electrode. Based on Green’s Theorem Shockley and Ramo arrive at
the formula:
i(t) =
~ w (~x(t)) · ~v (~x(t), t)
E
q
Vw
(2.4)
~ w is a
In this expression, v(t) is the instantaneous velocity vector of a point charge q, and E
~ w is calculated for each
weighting field, with weighting potential Vw . This weighting field E
electrode by setting it to potential Vw (most logically 1 V). All other conductors are grounded.
~ w /Vw has dimension m−1 , although equation (2.4) is often cited in a
The vector quantity E
way that drops Vw , suggesting the wrong V m−1 as dimension instead. The weighting vector
field’s physical significance is how well a moving charge at position ~x with velocity ~v (~x, t)
couples into the electrode, giving a current per coulomb of charge.
~ w (~x) = Ew,z = Ew , uniform and with a zero x and y component,
In the simple case where E
and the charge velocities approximated as the (z-aligned) drift velocity vd in a medium, the
20
Chapter 2. RPC physics primer
equation simplifies to:
i(t) =
Ew
vd qe N (t)
Vw
(2.5)
with electron charge qe and N (t) electrons at time t.
The method described for calculation of the weighting fields could cause confusion at first
when applied to RPCs. It is tempting to also ground the main high voltage electrodes of the
detector, which would completely shield the weighting field out of the detector. The solution
here is to keep the main electrodes electrically floating, which is in line with the physical
situation. As mentioned in section 2.1.1, they are essentially transparent to fast signals such
as the nanosecond induced current pulses.
For a more thorough discussion on signal generation in RPCs, refer to [30] which includes
effects from detector geometries and an equivalent circuit model for induced voltage calculation.
2.5
Limitations in high-rate environments
In environments where a high flux of charged particles travels through an RPC detector,
the simple approximation where avalanches do not interact with eachother and the detector
properties are static will no longer hold.
Two main effects can be considered. One works on a microscopic timeframe, directly reducing
detector efficiency as a result of the high particle rate. The other effect works on a macroscopic
timeframe, as a reduced lifetime of the detector.
2.5.1
Space charge effects
As an avalanche grows, two charged clouds develop. A fast moving distribution of electrons
moves towards the cathode, while a slower cloud of positive ions moves towards the anode. If
the avalanches are sufficiently far apart these clouds will both have been recombined, absorbed
by the electrodes, or diffused into the gas. If instead the particle rate is high enough that one
avalanche starts to overlap with the remnants of the last, the electric field in the gap seen by
one avalanche will be shielded by the ones before it.
If one or both of the charge clouds have deposited onto an electrode surface, they will locally
recombine with the surface charge responsible for the normal electric field in the gap. This
local reduction of surface charge will cause a diminished electric field until the charge deficit
has been refilled. Given that the electrode materials are insulators, this process can be slower
than the appearance of new tracks and avalanches.
Chapter 2. RPC physics primer
21
The space charge effect will limit the size of avalanches, as their own positive ion cloud will
gradually reduce the field strength seen by the moving electrons. The effect on coinciding
avalanches will be similar. The deposited charges on the electrodes will cause a temporary
and local reduction of detector sensitivity. Depending on particle rates, the overlap between
tracks can become high enough to see these effects.
2.5.2
Aging of electrode materials
Figure 2.4: The electrode aging effect shown for different material types. On average, materials with
higher resitivity will see worse aging effects over time. Figure taken from [19]
Figure 2.4 shows a rate limiting effect that occurs on macroscopic timescales. Because insulating materials have only very limited amounts of mobile charges available, these will gradually
be depleted over time by recombination of avalanche remnants at the surface. The aging
effect is usually quantified as a total charge absorbed per unit of surface area.
This effect can be greatly reduced by choosing materials with lower resistivity. Different kinds
of doped low-resistance silicate (LRS) glasses have shown great promise for future detectors.
22
Chapter 2. RPC physics primer
Ferrite ceramics are even better, as seen on figure 2.4, showing virtually no aging effects over
a long range of absorbed charge.
The GIF++ facility at CERN [31] has been constructed to enable accelerated aging of detector
prototypes intended for the HL-LHC, delivering integrated doses corresponding to ten year
lifetimes in less than one year real time. This is done using a 14 TBq 137 Cs gamma source
and high energy muon beams.
Chapter 3
Electrostatics and field calculations
3.1
Classical electrostatics
All materials can be very broadly categorised into two classes. Those whose internal charges
do not flow freely and in which approximately no electric currents flow when an electric field
is applied over them are called insulators. The second class of materials allow free movement
of charge carriers to a much greater degree, allowing currents to flow within. Conductors and
semiconductors make up this second class.
I
The key observable distinguishing these materials is the electrical resistivity ρ, expressed in
Ω m. One might be tempted to consider electrical resistance R instead, expressed in Ω, but
this is not an intrinsic property of a material.
A
l
Figure 3.1: A piece of resistive material with cross-sectional area A and length l. Charge carriers
flow through this material as current I.
Referring to figure 3.1, both properties are related through:
R=ρ·
l
A
(3.1)
This clearly shows that the resistance depends on both the cross-sectional area A and the
traversed length l of a piece of material, and is thus an extrinsic property. This also follows
the naı̈ve physical interpretation where electrical currents are seen in similar fashion as fluid
currents. Here it is easy to see that an increased cross-section will allow for a greater flow of
23
24
Chapter 3. Electrostatics and field calculations
charge carriers, while the “friction” loss experienced by these charges is a function of how far
they have travelled inside a material.
Typical values for ρ are 1.68 × 10−8 Ω m for copper, an excellent conductor, 6.40 × 102 Ω m
for silicon, a typical semiconductor, and 1.00 × 1011 Ω m to 1.00 × 1015 Ω m for glass, a good
insulator. Resistive plate chamber detectors are usually built using electrode materials with
resistivities that classify them as insulators. Additionally, electric fields play a key role in
the functioning of these devices. It is therefore desirable to discuss the behaviour of these
materials when they are subjected to electric fields.
3.1.1
Dielectrics
-
-
~
E
-
+
-
-
-
(a) Without applied electric field.
- - P~
-
+
(b) With electric field.
Figure 3.2: This shows the simplistic model where dielectric materials are represented as atoms with
a positive nucleus, surrounded by a negatively charged electron cloud. Under an electric
field this cloud will locally counteract the electric field by deforming and generating a
dipole moment P~ .
To understand the behaviour of insulators when placed under the influence of an external
electric field, it helps to picture a simple atomic model. As shown in figure 3.2a an insulator
can be approached as a collection of atoms with a small positively charged core, surrounded
by a negatively charged electron cloud. Since these atoms are unpolarised and carry no charge
as seen from an outside macroscopic perspective, the charge distributions inside must cancel
eachother out.
This picture changes when the insulator is placed inside an external electric field, as shown
~ which
in figure 3.2b. Positive and negative carriers experience forces of opposite sign inside E,
deforms the atom. At the same time the electron cloud is bound to the nucleus with a strength
depending on material properties. An equilibrium will be reached once the deformation is
balanced out by the binding force. This spatial separation of charges generates a dipole
moment and is called induced polarisation. It is quantified by the polarisability of a material.
Insulators which exhibit non-zero polarisability are called dielectrics, and their polarisation
is defined as:
~
P~ = ǫ0 χe E
(3.2)
Chapter 3. Electrostatics and field calculations
25
with polarisation density P and electric susceptibility χe . The vacuum permittivity ǫ0 gives
the capability of the vacuum to permit electric field lines. Its role will become clearer in section 3.1.2.
Two special cases should be mentioned. A perfect vacuum has χe = 0, as expected from a
total absence of charge distributions. A vacuum will always have P~ = 0, and is an example
of a medium that is an insulator, yet not a dielectric1 .
The second interesting case is that of a perfect dielectric, a material with zero conductivity (or
infinite resistivity). It exactly counteracts an external electric field using only displacement of
charge distributions, and returns this stored energy perfectly. This corresponds to a perfect
capacitor, and also to a perfect insulator through which no currents flow.
Since real materials will never be perfect dielectrics, a sufficiently high electric field will cause
the electrons to eventually break away from their nuclei. At this point an insulator will
become conductive, usually with a sharp drop in resistivity. This effect is called breakdown,
and happens at a material-dependent field strength.
Conductors generally do not exhibit any polarisation, as their charge carriers are uncoupled
from the nuclei (usually arranged in a lattice structure).
3.1.2
Computation of electric fields
Classical electromagnetism is one of the earliest examples of a unifying theory, following
the publication of James Clerk Maxwell’s A Treatise on Electricity and Magnetism in 1873.
Maxwell demonstrated that electricity and magnetism were both regulated by one force –
electromagnetism – and described its behaviour with a set of four equations, most famously
known as Maxwell’s equations.
The equations, represented in differential form for a vacuum, are [32]:
~ ·E
~ = ρ
∇
ǫ0
~ ·B
~ =0
∇
(3.3)
(3.4)
~
~ ×E
~ = − ∂B
∇
∂t
~ ×B
~ = µ0
∇
~
∂E
J~ + ǫ0
∂t
(3.5)
!
(3.6)
~ and B
~ are the electric and magnetic fields, J~ electric current and ρ the spatial charge
Here E
1
This only applies in a classical picture where a true vacuum has no charge carriers, and a dielectric is
defined as a material that has a nonzero induced dipole moment under influence of an external electric field.
26
Chapter 3. Electrostatics and field calculations
distribution. As mentioned in section 3.1.1 and now evident from equation (3.3), ǫ0 represents
the proportionality constant between space charges and electric fields in a vacuum.
~ = µ0 J,
~ and equation (3.5)
In absence of time-varying fields, equation (3.6) reduces to ∇ × B
~ = 0.
becomes ∇ × E
~ now irrotational one can define an electrostatic potential such that:
With E
~ = −∇φ
~
E
(3.7)
Combining equation (3.7) and equation (3.3) then leads to a form of Poisson’s equation:
∇2 φ = −
ρ
ǫ0
(3.8)
or in the absence of any charge distributions:
∇2 φ = 0
(3.9)
Macroscopic representation
The form of Maxwell’s equations presented until now is called the microscopic representation.
Referring to section 3.1.1, one can include macroscopic effects such as polarisation of dielectrics
in what is known as the macroscopic representation of Maxwell’s equations.
To this end, the electric displacement is defined as:
~ + P~
~ = ǫ0 E
D
(3.10)
Substituting P~ from equation (3.2) one gets:
~ + ǫ 0 χe E
~
~ = ǫ0 E
D
(3.11)
~ = ǫ0 (1 + χe )E
~
D
(3.12)
~ = ǫ0 ǫr E
~
D
(3.13)
Here we have introduced ǫr = 1 + χe , the relative electric permittivity of a material. Given
equation (3.13), (3.7) and (3.3), one gets the macroscopic equivalent of equation (3.8):
∇2 φ = −
ρ
ǫ0 ǫr
(3.14)
Solving Maxwell’s equations
The “solving” of Maxwell’s equations for an electrostatics problem means finding a potential
φ that satisfies a given spatial charge distribution ρ(x, y, z) according to equation (3.14).
Additionally, boundary conditions can be imposed:
• Dirichlet boundary conditions specify a value or function for the potential φ(x, y, z) on
a certain boundary. An electrode set to a fixed voltage is an example of a Dirichlet
condition.
Chapter 3. Electrostatics and field calculations
27
~ ·
• Neumann boundary conditions give a flux condition through a boundary, as −ǫ0 ǫr ∇φ
~ through a boundary.
~n = g. These conditions fix the electric field E
A physical interpretation of Dirichlet conditions is that of a perfect conductor. An electrode
could for example be modeled as a rectangular boundary, which is placed at a fixed potential
φ. This boundary then acts as a perfect frictionless conductor, sourcing or sinking charges as
ρ to ensure the potential φ satisfies the condition.
The finite element method
Solving the linear partial differential equation (PDE) from equation (3.14) can be done analytically for simple geometries. Calculating a potential map for a more complicated real-world
model of a particle detector poses a greater challenge, necessitating the use of numerical methods. The finite element method (FEM) is one such method, where the solution of a general
problem happens in the following steps:
1. The problem domain is divided into a collection of subdomains, called elements. For 1D
domains these elements are simple line segments. Higher dimensional problems require
elements with both size and shape, such as triangles for 2D problems and tetrahedra
for 3D problems.
2. The partial differential equations are transformed into integral form and in each of the
problem domain elements, a local approximation of the PDEs is determined. These
are the elemental equations. For steady-state problems these take the form of algebraic
equations. If the underlying PDEs are linear, the elemental equations will be too.
3. The full set of elementary equations is combined into a global system. This usually
means performing a coordinate transformation from the element-local system to the
global coordinate system.
4. The set of global equations, the original problem’s initial values and any boundary
conditions are used as input for a varying set of numerical solver methods.
The strength of the finite element method is how it trades a computationally complex problem
into a (large) set of more trivial calculations. Through the discretisation of the problem
domain, this technique lends itself really well to large scale parallelisation.
The specific application of FEM to electrostatics then solves equation (3.14) iteratively, starting from the problem geometry and initial values. The potential is defined in all mesh points,
and the elemental equations are applied to them. The new potential values are plugged into
the main PDE and a global error value is calculated. The procedure is then repeated with
these new potential values until the error is below a set threshold. This way the initial values
and boundary conditions are “propagated” through the system until a steady-state solution
is converged upon.
28
Chapter 3. Electrostatics and field calculations
Mesh generation
Figure 3.3: This figure shows how a triangular 3D mesh is built. Points (also called nodes or vertices)
are first defined. These are then connected into edges (or lines). Closed loops of edges
form faces, which in turn combine into elements (volumes).
Step one in the FEM process – dividing the problem domain into elements – is often done
separately using a mesh generator or mesher. This is an application that uses specialised
meshing algorithms to turn the geometrical model of a problem into a mesh of elements.
Figure 3.3 shows how a ususal tetrahedral mesh is constructed through successive combination
of points into edges, edges into faces and faces into volume elements.
There are a variety of ways to construct a mesh. One can choose different shapes for the
faces and volume elements, with resp. triangles and tetrahedra being a common choice.
The way the geometry is divided into mesh points can also differ. Similar to how spatial
coordinates are often divided into a grid with regular spacings, one can have a structured
mesh, as demonstrated in figure 3.4a. Alternatively, placement of mesh points dividing a
geometry can be randomised as seen in figure 3.4b, termed an unstructured mesh.
(a) Structured mesh.
(b) Unstructured mesh.
Figure 3.4: Demonstration of 2D triangular mesh types.
In the rare case where a problem geometry is fully aligned with one single regular grid, a
structured mesh can be superior. It is clear from figure 3.4 that an unstructured mesh would
Chapter 3. Electrostatics and field calculations
29
give a clear advantage2 representing more general geometries, even one as simple as a circle.
A core strength of meshes, especially when unstructured, is how the element sizes can be
adjusted to give more precision in regions of interest. This focuses computation time into
actual useful regions (e.g. a turbulent boundary layer in fluid dynamics) while keeping the
mesh coarse in areas where the solution is not expected to have smaller features. It also allows
mesh points to be placed exactly on boundaries relevant to the model geometry, regardless of
how complex this geometry is. This in turn provides clear and unambiguous ways to define
boundary conditions.
3.2
Relative permittivity measurements
As mentioned in section 3.1.2, determining the electric field inside a detector requires solving
the following PDE:
ρ
(3.15)
∇2 φ = −
ǫ0 ǫr
This requires knowledge of the detector geometry and of the relevant ǫr for all materials
involved. The geometry is either known by design, or easily measured from the prototypes
under study. The relative permittivities of the materials used for prototype construction is
usually less well defined, necessitating the development of a measuring technique for ǫr of
arbitrary materials.
The RPC models available for study use three different electrode materials: a bakelite, 1.1 mm
thick float glass and 0.7 mm float glass. None of these materials have a manufacturer specified
static relative permittivity, and literature is only able to specify a broad range of values for
each class of material.
Other construction elements such as mylar sheets [33] and the FR4 PCB [34] with pick-up
strips have more precisely specified material parameters. These materials are used outside
the active area of the detector, where variations in electric field would not strongly influence
detector performance. The permittivity values obtained from the manufacturers or literature
are therefore acceptable, and precise measurements will not add much value to the simulations.
The relative permittivity of materials is frequency dependent, but for electrostatic calculations only the zero-frequency static value is relevant. Most measuring techniques concern
measurement of ǫr over frequency intervals, and as such do not apply to the requirements for
this work. A simple method was constructed based on the physics of parallel plate capacitors,
and was used to measure ǫr for the bakelite and two different glass types.
2
This is not strictly correct, as there is a third class called a conformal mesh. Here the grid is not fixed
but “conforms” to the problem geometry. A simple example would be the meshing of a cylinder according to
a regular grid of cylindrical coordinates.
30
3.2.1
Chapter 3. Electrostatics and field calculations
Parallel plate capacitors
+ + + + + + + +
~
E
d
- - - - - - - l
Figure 3.5: A simple parallel plate capacitor geometry. Between the electrodes (yellow), separated by
a distance d, a slab of dielectric material (blue) is placed with static relative permittivity
ǫr . The electrodes are square, with an area A = l2 .
Figure 3.5 shows the theoretical model used for the measurements: a simple parallel plate
capacitor with two electrodes. These electrodes have a surface area A = l2 , and are separated
by a distance d. Between these electrodes we place a dielectric material with unknown ǫr .
By applying a voltage U over the capacitor plates a charge Q will be stored in them that is
~
equal and of opposite sign. These charge distributions will in turn lead to an electric field E
inside the dielectric.
The electric field between two parallel plates with equal but opposite charges Q is given by:
E=
σ
ǫ
(3.16)
The charge density σ is defined as the amount of charge per surface area, or σ = Q/A.
Plugging this into equation (3.16) and keeping in mind that the electric field will exist in a
dielectric with ǫr > 1, we get:
Q
E=
(3.17)
ǫ0 ǫr A
Expressing voltage as work done on a test charge moving a distance d gives:
V =
Fd
= Ed
q
(3.18)
With the above equations, and the definition of capacitance as charge stored per volt of
potential difference, we obtain the formula for capacitance of a parallel plate capacitor with
a dielectric:
Q
Q
ǫr ǫ0 A
C=
=
=
(3.19)
V
Ed
d
Given a way to measure a capacitance C of a test setup with a material of interest inserted
into a parallel plate capacitor of known dimensions, one can calculate ǫr .
Chapter 3. Electrostatics and field calculations
31
B
C
R
I
Figure 3.6: A simple RC circuit.
3.2.2
Capacitance measurement
To measure capacitances in a reproducable and precise way, one can place the capacitor to be
measured in an RC circuit, as seen in figure 3.6. When the capacitor C is charged to an initial
voltage U0 = Q0 /C, a current I will flow as indicated once button B is closed. According
to Kirchhoff’s current law, the current into the resistor must equal the current out of the
capacitor. Kirchhoff’s voltage law states that the voltage over capacitor and resistor must be
equal, given their parallel configuration. This leads to the expression for the time-dependent
behaviour of an RC-circuit:
U
dU
+
=0
(3.20)
C
dt
R
Solving for U (t) gives an exponential decay:
t
U (t) = U0 e− RC
(3.21)
Therefore, if one has freedom to choose the value of R, this circuit can be used to determine
C. Additionally, the RC time constant is defined as τ = RC, or the time required for the
voltage to fall (or rise) to a fraction 1/e of its initial (resp. final) voltage.
Figure 3.7 shows a more complete picture, where the capacitor is charged and discharged
using a square wave voltage source Us . Measuring the time necessary for the capacitor to
reach 1/e of the amplitude of the square wave then gives the time constant τ , from which the
capacitance can be calculated.
R
Us
A
Umeas
C
B
Figure 3.7: Circuit for capacitance measurements using a function generator Us and an oscilloscope
measuring across points A and B.
32
Chapter 3. Electrostatics and field calculations
Measurement technique
There is a complication with the measurement sketched in figure 3.7, which is a result of the
behaviour of real-world circuits. One can generally assume all components to have some degree
of parasitic elements: capacitors have small series resistances, resistors also show a small
amount of capacitance connected in parallel, wires and connections have some resistance and
inductance in series, but also capacitance in parallel, etc. Parasitic inductances are typically
too small to influence these measurements, especially at low frequencies. In a similar vein,
series resistances can have their effect minimised by choosing a large enough value of R, such
that any series resistance is orders of magnitude smaller.
The only type of parasitic that can not be neglected is the extra capacitance connected in
parallel with the test capacitance, as is modeled on figure 3.8. These parasitics are generally
on the order of a few picofarad, while the actual target capacitances are of order:
C = ǫr ǫ0
A
d
∼ 5 · 10 pF
∼ 25 pF
(5 × 10−2 m)2
5 × 10−3 m
This parasitic component will thus be of similar magnitude as the target capacitance, and
will impact the measurements. A way is needed to measure this parasitic value separately
with minimal disturbance to the test setup.
Since capacitances connected in parallel act as one capacitor with the sum of their capacitance,
we have:
Cmeas = Cpar + Ctest
(3.22)
RCmeas = RCpar + Ctest
(3.23)
1
(3.24)
d
This is a simple linear function f (x) = a · x + b, with f (x) = τmeas , a = ǫr · (ǫ0 AR), b = τpar
and x = 1/d. By measuring τmeas at different values of d, the resulting curve should show a
τmeas = τpar + ǫr · (ǫ0 AR) ·
R
Us
A
Cpar
Ctest
Umeas
B
Figure 3.8: Circuit for capacitance measurements including parasitic component Cpar , parallel to
the target capacitance.
Chapter 3. Electrostatics and field calculations
33
straight line in function of 1/d, and the values a and b can both be determined using a fit.
Since A and R should both be known quantities, ǫr can be gained from a.
3.2.3
Measurement setup and results
(a) General view of the measurement setup.
(b) Detail of the electrodes and material samples and clamping setup.
Figure 3.9: View of the setup constructed to measure the static permittivities of key materials.
The constructed setup is shown in figure 3.9. The voltage source is a TTi TG5011 [35], the
oscilloscope a Tektronix TDS 3034C [36]. The electrode separation distance d is measured
with a vernier caliper with a precision of 0.05 mm, and tightly controlled by clamping the stack
of electrodes and material sample plates between two acrylic panels, as shown in figure 3.9b.
At the start of each measurement, the value of R was taken using a Fluke 175 [37] digital
multimeter (DMM) with a precision of 0.9 % plus one count. The resistor was found to be
stable between tests, well within the error margin of the DMM. For this reason R has been
considered a constant of the test setup in further discussion.
Parameter
Function generator
Resistor
Electrodes
function
frequency
amplitude
DC bias
resistance R
area A
Value
square wave
500 Hz
1V
0V
(268.8 ± 2.5) kΩ
(22.12 ± 0.02) cm2
Table 3.1: All chosen parameters and values that are constant across tests for the capacitance measurement setup.
34
Chapter 3. Electrostatics and field calculations
The parameters and specifications of the setup are shown in table 3.1. With every new set of
measurements the peak-to-peak amplitude seen over the capacitor was measured to determine
the 1/e voltage level. The time constant was then measured automatically using this value.
Figure 3.10 shows an example capacitance measurement.
TDS 3034C 6 Jun 2016 11:52:50
Figure 3.10: Oscilloscope view of a capacitance measurement. Channel 2 (blue) shows the input
square wave. Channel 1 (yellow) is measuring the 1/e rise-time, seen on the right of
the screen.
Using this method τ -versus-d curves were taken for three materials: 1.1 mm float glass and
0.7 mm float glass used in glass RPC prototypes, and bakelite used for the RE4/3 RPCs
assembled in Ghent and installed into the CMS endcaps during long shutdown 1.
With the values from table 3.1 the constant ǫ0 AR can be calculated. The value used for the
remainder of this chapter is:
ǫ0 AR = (5.26 ± 0.07) µs mm
(3.25)
Float glass measurements
A total of eight plates were used for the 0.7 mm variant, and seven for the 1.1 mm. The results
of these tests are shown respectively in figure 3.11a and figure 3.11b. It is slightly surprising
for these values to differ this much between both glass types, as they were sourced from the
same supplier. It is of course possible that these are normal variations between production
runs, which means that in future designs care should be taken not to expect very specific
permittivities.
Chapter 3. Electrostatics and field calculations
35
Bakelite measurements
Figure 3.12 shows the test results and fitted function for the bakelite material used for the
RE4/3 electrodes. These values show more deviation from the fit curve compared to the glass
species in figure 3.11. The glass was easier to clean, and has a more uniform thickness. The
bakelite samples had to be sourced from a rejected chamber and needed to have their carbon
paint layer removed, which could explain the larger variation observed due to minor surface
damage.
Relative permittivities
Table 3.2 displays the fit results for all three material types. Based on these values, and
equation (3.25), the relative permittivities can be calculated. These results are tabulated in
table 3.3.
Given the very similar values for b in table 3.2 we can conclude that the parasitic capacitance
remained nearly the same between tests. This indicates good control over the measuring
process, with minimal disturbances to the test setup other than switching out the material
samples.
Material
Parameter
Value
0.7 mm glass
a
b
(28.4 ± 0.3) µs mm
(11.5 ± 0.2) µs
1.1 mm glass
a
b
(31.3 ± 0.4) µs mm
(12.2 ± 0.2) µs
Bakelite
a
b
(37 ± 1) µs mm
(11.7 ± 0.4) µs
Table 3.2: Fit results for the tested material species.
Material
0.7 mm glass
1.1 mm glass
Bakelite
ǫr
5.40 ± 0.09
6.0 ± 0.1
7.0 ± 0.2
Table 3.3: Static relative permittivities for all three materials measured, based on the obtained fit
results.
36
Chapter 3. Electrostatics and field calculations
0.7 mm float glass
55
Fit
Data
50
1.1 mm float glass
45
Fit
Data
40
45
35
τ [µs]
τ [µs]
40
35
30
30
25
25
20
20
15
0
1
2
3
d [mm]
4
5
15
6
(a) Result for 0.7 mm float glass.
1
2
3
4
5
d [mm]
6
7
8
(b) Result for 0.7 mm float glass.
Figure 3.11: Measurement points and fits for static relative permittivity measurements of the two
float glass species.
RE4/3 bakelite
30
Fit
Data
28
26
τ [µs]
24
22
20
18
16
14
2
4
6
8
d [mm]
10
12
14
Figure 3.12: Measurement points and fits for static relative permittivity measurements of the RE4/3
bakelite material.
Chapter 4
Simulation setup
4.1
Basic structure
A simulation of an RPC detector is usually built out of four distinct elements:
1. Description of the geometry and physical properties of the detector.
2. Calculation of the electric field present inside the RPC.
3. Simulation of the physics relevant to the detection process.
4. Generation of the induced signals, read out by virtual electronics.
Each of these steps relies on the one before it to provide an accurate representation of the
modelled system. For example, the physics simulation requires knowledge of the local field
strength inside the gas, which in turn is a function of the geometry of the system under study.
One can sometimes greatly simplify parts of this process, e.g. by modelling a simple parallel
plate capacitor with infinite plane electrodes. For this study a more realistic model was
desired, the elements of which will be described in more detail next. All software choices were
motivated in large part by the desire to have the source code freely available while avoiding
large licensing costs. This guarantees transparency and easy reproduction of results.
4.1.1
Detector geometry
To better understand the choices made here, please refer to section 3.1.2 where the finite
element method (FEM) method is outlined. All three RPC models used for this work have
their relevant parameters listed in appendix A.
The first step in the simulation chain is the generation of a mesh based on the detector
geometry. The Gmsh 1 [38] utility was chosen for this task, as it provides a way to describe
1
An open-source and freely available 3D mesh generator. See http://gmsh.info/.
37
38
Chapter 4. Simulation setup
the complete geometry in a scripting language. This is essential when one wants to have a
flexible setup which can be adapted to the many different types and configurations of RPC
detectors encountered today.
(a) Top view.
(b) Detail view of the glass parts.
Figure 4.1: Cut-out view of the mesh for a double-chamber multigap structure with four internal
electrically floating glass plates as sub-electrodes inside each chamber. A PCB with 16
read-out strips is sandwiched between the chambers. The conductive paint layer can be
seen as the differently coloured rectangle on the top electrode. The potential boundary
conditions are applied to these paint surfaces, in analogy to the real prototypes.
The mesh, as generated for the double-chamber multigap glass RPC produced in Ghent
in 2014 is shown in figure 4.1. The mesh nodes (vertices, points) are connected by edges
(lines), which are in turn combined into faces (triangles). Both surfaces and volumes are
composed of a large number of faces. Overlapping elements are strictly forbidden, e.g. two
adjacent volumes will share at least once surface where all of their volume lines and faces
connect cleanly. If a mesh contains overlapping elements it is considered self-intersecting,
and will indefinitely cause undefined behaviour, either when the mesh is generated or during
calculations using FEM packages.
Mesh model parameters
The mesh model created for use during this research has been parameterized as follows:
• Thicknesses of all major components can all be changed: main electrode plates, optional
sub-electrode plates, insulating mylar sheets, and the signal readout PCB.
• The number of sub-electrodes can be either zero for a traditional RPC design, or nonzero
for a multigap configuration.
Chapter 4. Simulation setup
39
• Gap size can be tuned to any positive and non-zero number.
• Surface dimensions of the main electrodes, sub-electrodes and paint layer can be set
independently.
• Pick-up strip count, width, length and separation distance can be configured.
While the model code is set up in a way that allows fairly easy addition of new parameters,
a few important restrictions currently apply. Based on common features of the available
prototypes in Ghent the model assumes the following characteristics:
• All possible detector configurations have two chambers, one on each side of the signal
pick-up board.
• The main electrodes, sub-electrodes and paint surfaces are square, their surfaces defined
only by one dimension.
• The pick-up strips are oriented along a fixed axis (Y-axis) and are placed on top of the
PCB. They are centered in the XY-plane of the detector.
• All main-electrodes share common dimensions and material. The sub-electrodes share
a different set of dimensions and material parameters.
4.1.2
Electric field calculations
In real-world geometries calculating the internal electric field analytically quickly becomes
impractical or outright impossible. FEM software provides a way to perform these calculations
in an approximate and iterative way, such that even very complex models can be solved within
reasonable computation time.
The Elmer FEM project 2 [39] provides the necessary equation solvers and file format interfaces
to turn the detector mesh into a map of electric potentials that can be processed by the physics
simulation package. In the same spirit as employed for mesh generation, all configuration files
are human-readable and glued together using simple shell scripts.
Figure 4.2 shows the potential and Ez maps for the same detector geometry shown in section 4.1.1. A −1 kV potential has been applied over the outer electrodes, with the inner
electrodes set to the ground reference potential. From these maps two conclusions can be
drawn.
First of all, the potential map in figure 4.2a shows that the potential drop inside the 0.7 mm
glass layers is relatively minor compared to the drop inside the 0.2 mm gas gaps. On figure 4.3a
2
An open-source and freely available multiphysics finite elements suite, initially developed by the Finnish
IT Center for Science (CSC). See https://www.csc.fi/web/elmer
40
Chapter 4. Simulation setup
this is shown in detail for a line along the Z-axis in the centre of the detector. This is to be
expected from the relative permittivity of 5.4 to 6 for the glass parts. This highlights the
importance of correct material choices for the sub-electrode materials, as a poor choice will
lead to a large “loss” of electric field in regions that do not contribute to detection efficiency.
Single-gap configurations do not suffer as much from this limitation, as the main electrodes
are the only dielectric components.
The second conclusion is strongly related to the first. As a strong dielectric electrode material
can focus most of the potential drop into the gas regions, the field map in figure 4.2b shows
that Ez is by far the highest in the 0.2 mm gas regions. Even the edge effects to the sides
of the sub-electrodes will contribute very little to the detection mechanisms. Similar to the
potential profile, figure 4.3b shows a line profile of Ez through the multigap geometry.
All artifacts seen in the maps of figure 4.2 are a result of the interpolation necessary to
represent a 2D slice of an unstructured triangular grid: very few points will fall exactly
inside the plane of the 2D cut. Taking this into consideration, the field strength inside the
sub-gaps is actually extremely homogeneous, as expected from this ideal geometry. This has
implications towards the actual simulations, where the gaps can reasonably be considered to
have a uniform E throughout.
6000
0
4000
−200
2000
Ez [V]
U [V]
Potential profile of a multigap RPC
200
−400
0
−600
−2000
−800
−4000
−1000
−8 −6 −4 −2 0
2
z [mm]
4
(a) Potentials along the Z-axis.
6
8
Field profile of a multigap RPC
−6000
−8 −6 −4 −2 0
2
z [mm]
4
6
8
(b) Field strength along the Z-axis.
Figure 4.3: These figures show a 1D cut of the maps of figure 4.2, along the Z-axis in the centre of
the multigap RPC.
Chapter 4. Simulation setup
41
Potentials inside a multigap RPC
0
6
4
−250
0
−500
U [V]
z [mm]
2
−2
−750
−4
−6
−100
−50
0
x [mm]
50
100
−1000
(a) Potentials in the XZ-plane.
Electric field inside a multigap RPC
5300
6
4
2650
0
0
Ez [V/cm]
z [mm]
2
−2
−2650
−4
−6
−100
−50
0
x [mm]
50
100
−5300
(b) Z-component of the electric field in the XZ-plane.
Figure 4.2: Electric potential and derived electric field for the multigap RPC shown in figure 4.1.
These maps have been calculated with −1 kV applied on the outermost electrodes.
42
Chapter 4. Simulation setup
4.1.3
Physics simulation and signal generation
All remaining simulation steps have been functionally grouped together, taking care of gas
behaviour calculations, physics simulation and signal generation. Initial simulations have been
performed exclusively using Garfield++ 3 [40]. During the course of this thesis the need arose
for a second simulation codeset taking advantage of some facilities provided by Garfield++,
while performing all actual simulation work in custom code. Both approaches warrant a more
thorough discussion and will be handled in section 4.2 and section 4.3.
For signal calculations using the Shockley-Ramo theorem discussed in section 2.4, weighting
field maps corresponding to the different pick-up electrodes are required. These are generated
from the same detector mesh when the main field map is calculated. An example weighting
map is shown in figure 4.4. This weighting map shows how the sensitivity of the strip is
distributed spatially inside the gas gaps.
Electric field inside a multigap RPC
8.7
6
4
0.0
0
Ez [V/cm]
z [mm]
2
−2
−4
−6
−100
−50
0
x [mm]
50
100
−10.1
Figure 4.4: Z-component of the weighting field in the XZ-plane. The scale of this figure is the same
as for figure 4.2b, and shows the map for strip number 5 of 16 (counting starts at 1).
3
A toolkit for detailed simulation of particle detectors with gas or semiconductor sensitive media. See
http://garfieldpp.web.cern.ch/garfieldpp/
Chapter 4. Simulation setup
4.2
43
Garfield++ simulation
The first simulation code was developed using Garfield++, a simulation framework for detailed simulation of gas and semiconductor particle detectors. This framework is equipped
to take care of all steps of the simulation process. Garfield++ also provides classes for the
construction of analytical geometries and electric fields, but for this work the Elmer field
map import functionality was used (Garfield::ComponentElmer ) to enable processing of more
arbitrary geometries.
The functionality of Garfield::ComponentElmer was extended to provide a way to scale the
electric field map. Generating a new mapping for every potential step would be computationally expensive, so this rescaling of the potential map is very beneficial.
Elmer is used to produce one potential map at a fixed −1 kV applied potential, which is then
rescaled according to simulation requirements. This is possible because the equations used to
calculate the potential map and electric field are fully linear, so a −2 kV applied potential in
the FEM step will give the same result as multiplying the −1 kV map by two.
4.2.1
Primary cluster generation
To generate the primary ionisation clusters for a given charged particle track, Garfield++
uses the Heed++ 4 code.
The Heed++ (and predecessor Heed) code has been extensively tested. Figure 4.5 shows
example curves for a few different gas mixtures. The solid lines are measurements for isobutane
and methane. These curves correspond well to the Bethe-Bloch function shown in figure 1.3.
4.2.2
Gas performance calculation
The drift velocity, Townsend coefficient and attachment coefficients are important parameters
of the simulation, and depend on the gas type and electric field. Garfield++ uses the Magboltz 5 code to calculate these values using a Monte Carlo method. Electrons are transported
through a chosen gas mixture at microscopic level using the different scattering cross-sections.
Figure 4.6 shows the calculated coefficients used for the simulations of RPCs with the CMS
gas mixture. This same mixture was used in Ghent for measurements with prototype RPCs.
4
Code to calculate the energy loss of charged particles in gases. See http://ismirnov.web.cern.ch/
ismirnov/heed
5
Fortran code that solves the Boltzmann transport equations for electrons in gas mixtures, under electrical
and magnetic fields. See http://magboltz.web.cern.ch/magboltz/
44
Chapter 4. Simulation setup
Figure 4.5: Average clusters per mm as calculated by Heed++ for different gases at standard temperature and pressure. The solid lines show methane and isobutane measurements from [41].
Figure taken from [25].
Magboltz results for CMS mixture
100
Townsend
Attachment
Effective Townsend
α and η [1/mm]
80
60
40
20
0
0
20
40
60
E [kV/cm]
80
100
120
Figure 4.6: Magboltz calculation results for the CMS mixture of 95.2 % R134a, 4.5 % isobutane and
0.3 % SF6 . The effective Townsend coefficient α − η is also shown.
Chapter 4. Simulation setup
4.2.3
45
Physics simulation
Garfield++ offers three options for simulation of electron transport and avalanche generation.
The first two phenomenological methods solve the equations of motion using the statistical
properties vd , α and η. Here one can choose either the Runge-Kutta-Fehlberg (RKF) integrator (Garfield::DriftLineRKF ) or the Monte Carlo integrator (Garfield::AvalancheMC ). The
third method uses the microscopic properties (i.e. the collison cross-sections) for electron
transport.
The Monte Carlo and RKF integrators are more appropriate for calculations where all structures are much larger than the average electron mean free path, and precise excitation or
ionisation information is not required.
Computation times
While developing the Garfield++ based code, it became apparent that the computation times
of both the Monte Carlo and the microscopic transport classes would not allow simulation of
tests performed using the RPC prototypes. These tests use cosmics, which give a very broad
energy spectrum to work with. To faithfully simulate an energy spectrum, sufficient cosmic
rays have to be drawn from the energy distribution.
Figure 4.7a shows the computation times for worst-case avalanches (starting from the very
top) in a 200 µm gap at various electric field strengths. Both the Monte Carlo and microscopic algorithms take tens of seconds to simulate a single avalanche at higher field strengths,
which makes calculation O(1000) cosmic ray tracks with O(10 − 100) primary electrons each
prohibitively time consuming.
Figure 4.7b highlights another problem with direct application of the Garfield++ algorithms.
The Monte Carlo code gives almost a full order of magnitude bigger avalanches at 55 kV cm−1
field strength. It would seem that the microscopic calculations better represent the attachment
effects.
4.3
Hybrid simulation code
To circumvent this computation time problem, a new hybrid simulation code was developed.
Most of the Garfield++ functionalities are still used in the hybrid codebase, but the actual
physics simulation and signal generation has been replaced with custom code.
The problem lies in the fact that calculation of the induced signals requires tracking the
individual electrons to a certain extent, which prevents treating avalanches as a simple number
of electrons in function of distance in the gap. To overcome this problem signal generation
and avalanche calculations were decoupled into two separate steps.
46
Chapter 4. Simulation setup
Time per avalanche for a 200um gap
102
Microscopic
Monte Carlo
Hybrid
Time per avalanche [s]
101
100
10−1
15
20
25
30
35
40
E [kV/cm]
45
50
55
60
55
60
(a) Computation times for the different algorithms.
Avalanche sizes for a 200um gap
104
103
Microscopic
Monte Carlo
Hybrid
ne
102
101
100
10−1
15
20
25
30
35
40
E [kV/cm]
45
50
(b) Avalanche sizes for the different algorithms.
Figure 4.7: Comparison of the different simulation algorithms, showing the average avalanche size
and computation time for between 30 and 10 iterations per field strength, adaptively
chosen according to avalanche sizes.
Chapter 4. Simulation setup
4.3.1
47
Initialisation of the code
When the simulation is first initialised, the field maps and gas performance file are read using
the standard Garfield++ facilities. Then each of the detector’s gaps is subdivided into steps
along the z axis, from here on referred to as zsteps. The default step size is 10 µm, which is
decreased as necessary to fit an exactly integer number of steps into each gap.
The initialisation finishes by calculation a set of signal weights in the exact center of the
detector. Figure 4.8 shows this process in a graphical way. Each of the gaps has been divided
in steps of size zstep, and starting at each of these points the weighting field value is summed
over a straight line in the direction of movement of the electrons. These signal weight maps
are saved for later use, with one value per zstep per gap.
zstep
Figure 4.8: Schematic representation of the calculation of signal weigths in a gap. At every zstep,
the weighting field is summed as indicated by the arrows.
4.3.2
Preparation of a simulation
Before a simulation is run, the potential over the detector is set. The code first multiplies
the electric field maps by an appropriate number to get the desired applied potential. Then
the electric field inside each gap is determined, and this value is saved into memory for quick
access. Here the assumption is made that each individual gap has a uniform and constant
electric field, by sampling the field strength in a grid of 10 by 10 points in the XY-plane,
centered vertically in the gap. The average value of these 100 sampling points is taken as
“the” field strength of a gap.
Using the field strength determined for each gap, the Townsend coefficient α and attachment
coefficient η is determined for each gap. These values are also committed to memory for quick
retrieval.
The signal size maps
The final step of the preparation of a simulation run is a bit more involved. To be able to
calculate the induced signals, while growing the avalanche statistically without tracking single
electrons, a map of signal sizes is needed.
48
Chapter 4. Simulation setup
This map gives the total induced charge generated by an electron “dropped” from each possible height in a gap. The procedure here is very similar to what is shown in figure 4.8 for
the signal weight calculation. In the center of the XY-plane and for every zstep in the gap,
100 electrons are released and transported down until they reach the end of the gap. These
calculations use the microscopic transport code provided by Garfield++ to track the charge
and its induced signal.
The avalanche mode of the Garfield++ code is here disabled, so no extra electrons are being
added due to the Townsend coefficient. The attachment coefficient is still accounted for, and
not all of the 100 launched electrons might reach the end of the gap. This gives a reliable
measure of how big the contribution of each electron is that starts at a certain height inside
the gap. The average signal and standard deviation are then calculated, and saved in a map.
This is a computationally intensive step, even though the transport code does not generate
extra electrons. For this reason, the signal size maps are only calculated for applied potentials
in steps of 250. When a new potential is set, the closest multiple of 250 is calculated and the
code tries to open an earlier saved mapping file. If this mapping file does not exist, a new
mapping is generated and saved to disk. This saves a large amount of computation time for
repeated simulations, and can even be performed on a separate machine.
Figure 4.9 shows the result of two such signal size maps. Both the average signal sizes and
the standard deviations show a linear relationship with the height in the gap, which allows
simple linear interpolation when specific z-values are requested.
0.020
Signal size maps for a 200um gap
Induced charge [aC]
3 kV applied
10 kV applied
0.015
0.010
0.005
0.000
200
150
100
Distance from gap start [um]
50
Figure 4.9: Results of two signal size map calculations. These values show the average signal size
and standard deviation for an electron that starts at each height in the gap. The 3 kV
map is dominated by the attachment coefficient, so all electrons disappear before they
reach the end of the gap. The spread of the signal sizes decreases as the electron nears
the end of the gap, because there is less time to drift in the XY-plane.
Chapter 4. Simulation setup
4.3.3
49
Primary tracks
Once the signal weight maps and signal size maps are prepared, simulations can be run at
the previously set potential.
The hybrid code offers both simulations using cosmics as primary tracks, and fixed-energy
virtual beam particles. The cosmics are generated using the CRY library6 , which provides
cosmic muons down to about 1 MeV, and generates electrons and protons as well. All these
particle types are used in simulations with cosmics to get a more realistic picture. Kaons,
pions and neutrons are not included, since their effects are negligible. Figure 4.10 shows a
spectrum obtained using CRY for 1 000 000 cosmic rays at sea level.
The beam type of primary particles gives single-energy muons with an optional gaussian beam
spread around the center of the detector in the XY-plane.
Cosmic ray spectra at sea level
10−1
Muons
Electrons
Protons
10−2
Flux [m−2 s−1 sr−1 MeV−1 ]
10−3
10−4
10−5
10−6
10−7
10−8
10−9
100
101
102
103
Ekin [MeV]
104
105
106
Figure 4.10: Cosmic ray spectrum at sea level generated using the CRY library.
6
Generation software for correlated cosmic ray showers. See http://nuclear.llnl.gov/simulation/main.
html
50
4.3.4
Chapter 4. Simulation setup
Avalanche generation
For each generated beam or cosmic track, the primary ionisation clusters are calculated using
the Heed interface provided by Garfield++. Each of the primary electrons is then developed
into an avalanche using the custom physics code.
A new Z-axis step is calculated, based on α and η. Considering the probability of ionisation
αdz and probability of attachment ηdz, the step dz is chosen such that both probabilities
are below a treshold, by default set to 1 %. The calculation of the avalanche is then stepped
through the gap using the steps dz while keeping track of the total number of electrons ne .
At each step the probabilities αdz and ηdz are evaluated as binomial distributions. If ne 6
1000, both distributions are evaluated once for every ne . This results in two numbers, ngain
(successful draws using αdx) and nloss (successful draws using ηdx). These give the new
ionisations, resp. attachments in the last step dz.
If ne > 1000, the binomials are approximated as gaussian distributions according to the
central limit theorem, and just two random numbers are drawn: a gaussian random number
ngain with:
µ = ne αdx
(4.1)
σ = ne αdx(1 − αdx)
(4.2)
µ = ne ηdx
(4.3)
σ = ne ηdx(1 − ηdx)
(4.4)
ne = ne + ngain − nloss
(4.5)
and a gaussian nloss with:
In both cases, ne is updated as:
If ne is zero or negative, the avalanche is considered lost, and the electron transport stops.
4.3.5
Signal calculation
With every step dz, a value ngain is obtained that gives the electrons that have entered the
avalanche at that specific height in the gap. This is the point where the signal size maps that
have been obtained earlier enter the calculation.
Since we have a mean signal size and spread, we can assume a gaussian distribution and draw
a random gaussian number Cind with:
µ = ngain µsignal
√
σ = ngain σsignal
(4.6)
(4.7)
Chapter 4. Simulation setup
51
Here µsignal and σsignal are the average signal size and spread, interpolated for the specific
z-coordinate in the gap.
The induced charge Cind is then added to the total induced charge for the avalanche. Once
the avalanche reaches the end of the gap, or all electrons are lost (ne = 0), the tracking stops
and the current avalanche electron count and induced charge are returned.
4.3.6
Positional signal weighting
Because we have calculated the signal size maps in the exact center of the detector, we need
to compensate for the fact that the avalanches created by cosmics and beam tracks can
appear anywhere in the XY-plane. The signal weight map that was calculated at simulation
initialisation will now be useful.
Referring to the procedure described in section 4.3.1, we can now follow a similar procedure.
We now take the closest multiple of zstep to the starting point of the avalanche and repeat
the calculation shown in figure 4.8, but this time for the X and Y coordinate of the avalanche
start, and one specific Z coordinate.
This gives us a new line sum of the weighting field along the avalanche. We then apply this
factor to the total induced charge of the avalanche as:
Cweighted = C ·
Σ(x0 , y0 , z0 )
Σ(0, 0, z0 )
(4.8)
Here C is the induced charge obtained from the avalanche development, Σ(0, 0, z0 ) is the interpolated line sum of the weighting field obtained at simulation initialisation, and Σ(x0 , y0 , z0 )
is the line sum along the avalanche.
This final value Cweighted is then taken as the actual induced signal.
4.3.7
Comparison to the Garfield++ code
Figure 4.7 shows the computation times and avalanche sizes at different field strengths for
the three simulation methods. The new hybrid code is fastest by a large margin, and remains
compatible with the Garfield++ microscopic transport class for avalanche sizes. This simulation method now allows processing large amounts of particle tracks through various RPC
models.
4.3.8
Overview of approximations
The hybrid code makes a number of approximations and assumptions on top of what the
original Garfield++ code did. While these are justified for the detector geometries used in
52
Chapter 4. Simulation setup
this work, some might no longer apply for RPCs that strongly deviate from the regular large
area parallel plane model.
The added approximations are:
• The gaps are considered to have a uniform electric field throughout, with Ex = Ey = 0.
• Signal distributions from single electrons can be used to generate random signals for
entire avalanches.
• The signal distributions for single electrons are gaussian shaped.
• To calculate the induced charge of an avalanche with starting point (x0 , y0 , z0 ), the
avalanche is approximated as a straight line along the Z-axis out of this point, and
compensated for the signals being generated out of (0, 0, z0 ).
• Townsend multiplication can be modeled as two binomial distributions over small steps
of dz, with ionisation probability αdz and attachment probability ηdz.
• Gas parameters (α and η) remain constant for each individual gap.
Chapter 5
Verification of simulated data
5.1
5.1.1
Experimental data
The data acquisition system
(a) Overview of the setup.
(b) The test stand.
Figure 5.1: Overview of the dedicated DAQ for efficiency measurements of small RPC prototypes.
To collect efficiency measurements of the glass based RPC prototypes built in Ghent, a
dedicated data acquisition system (DAQ) was built. This setup uses cosmics as primary
particles, and places the target detector between two scintillators which are used in coincidence
as triggers. The efficiency is then simply defined as the number of good detections inside the
test device, divided by the number of coincidence triggers generated by the scintillator pair.
The setup provides a high voltage source for the RPC that can provide negative potentials
of 0 kV to 15 kV, and a gas mixer with R134a, isobutane and SF6 lines. Dedicated control
software has been written that allows data to be taken over ranges of high voltages, interleaved
by optional noise measurements. These noise measurements use a built-in pulser of the VME1
1
A data bus frequently used for communication between scientific electronics modules.
53
54
Chapter 5. Verification of simulated data
controller, which is set to generate triggers at 2.5 kHz. This rate is sufficient to be able to
neglect any cosmics without any veto mechanism, since these will trigger at a rate of around
0.3 Hz at best.
Figure 5.1a shows an overview of the DAQ setup, with the multigap RPC prototype connected.
The test stand is shown in figure 5.1b, where the bottom and top trigger scintillators can be
seen, with the RPC prototype in the middle.
5.1.2
Collected data
Efficiency scan of different RPCs
100
Multigap RPC
Glass RPC
RE2/2
Efficiency [
80
60
40
20
0
0
2
4
6
8
Applied potential [kV]
10
12
14
Figure 5.2: Efficiency scans of three different RPCs. The multigap RPC shows a profoundly different
curve shape compared to the other two types. The RE2/2 type RPC is the same model
as has been installed in station 2 of the CMS endcaps.
Efficiency scans have been collected for three RPC types: the multigap prototype and glass
single-gap prototypes made in Ghent, and an RE2/2 detector as used in the CMS endcaps
(see figure 1.5). The multigap prototype has been tested at high voltage resolution specifically
for this simulation, while the other two tests have been performed earlier while new read-out
electronics were tested [42].
Chapter 5. Verification of simulated data
5.2
55
Gas parameters
The simulation code can output a number of gas performance parameters which can easily
be compared to measurements and calculations in literature. The Townsend and attachment
coefficients were already shown in figure 4.6. Figure 5.3 shows these same parameters for a
similar gas mixture. The values calculated for this thesis are in good agreement with the
results by Riegler et al. [25], especially given the 50 % extra isobutane in the CMS mixture
used for this work. The effective Townsend coefficient α − η shows a very similar pick-up
around 40 kV cm−1 , while the Townsend coefficient α rises more sharply.
The drift velocities are also calculated by Magboltz, as shown in figure 5.4a. The calculations
done by Riegler et al. [25] are again in good agreement, as shown in figure 5.4b, where the
square data point representing a measurement for an R134a/isobutane/SF6 mix of 96.9/3/0.1
is closer to the calculated curve in figure 5.4a.
Figure 5.3: Example Townsend and attachment coefficients for a similar mix as used in CMS. Figure
taken from [25]
56
Chapter 5. Verification of simulated data
300
Drift velocities for CMS mixture
250
vd [um/ns]
200
150
100
50
0
0
20
40
60
80
E [kV/cm]
100
120
(a) Drift velocities calculated for this work for
the CMS mixture (95.2 % R134a, 4.5 %
isobutane and 0.3 % SF6 ).
(b) Drift velocities calculated using Magboltz
by Riegler et al. [25] The upper curve
most closely resembles the CMS mixture.
Figure 5.4: Comparison of calculated drift velocities.
5.3
Efficiency curves
The only simulated observables that can be compared to data taken using the DAQ setup
in Ghent, are efficiency scans as shown in figure 5.2. The multigap prototype is the most
interesting detector to study, since the shape of the efficiency curve is very different from the
usual sharply rising sigmoid seen for the glass RPC and RE2/2 detectors.
All efficiency curves have been calculated from generated cosmic rays. The electronics were
simulated by applying a 120 fC detection threshold on the total induced charge per cosmic
track. This corresponds to the 215 mV threshold set on the electronics used for the measurements [43].
5.3.1
Simulation of the glass RPC and RE2/2 detector
The glass RPC prototype and RE2/2 detector are simple dual-chamber single gap detectors.
Their geometry and electric field was modeled and used as input for the simulation. The
simulated efficiency curves show a good agreement with the experimental data shown in
figure 5.2. A comparison of simulation and data is given in figure 5.5.
The efficiencies of both detectors are underestimated by about 1 kV. Riegler et al. [25] describe
a similar offset. The authors propose that their underestimation of the efficiency is caused
Chapter 5. Verification of simulated data
57
by only considering single avalanches, ignoring multiple simultaneous avalanches with a total
charge higher than the threshold.
The offset in the simulations using the hybrid code, which simulates complete particle tracks
instead of single avalanches, is only half of the 2 kV shown in Riegler et al. [25]. The remaining
error could be related to the properties of the non-ideal detector construction and materials:
1. The gap size is not guaranteed to be perfectly uniform. For example, the glass RPC
has been modeled using a 1.2 mm gap, while the internal ball spacers have a diameter
of only 1 mm. Using this smaller gap size would give a 10 % increase in electric field
strength.
2. When placed under high voltage, the electrodes will tend to be drawn towards eachother
due to their opposite charges. This might cause the gap to be slightly narrower still,
especially between the spacer structures.
3. The electric field is calculated under the assumption that the electrode materials are
perfect insulators. For real-world materials, this will of course be only an approximation.
The FEM calculations are entirely static and do not include dynamic effects, where the
electrodes could pass a certain amount of charge closer to the gas gap over time.
5.3.2
Simulation of the multigap RPC
The multigap prototype has not only given unexpected experimental results, it has also proven
to be difficult to simulate. Figure 5.6 shows the average induced charge in the multigap
detector as simulated by the hybrid code. As evident from the figure, these signals are not
much bigger than 1 aC, or more than five orders of magnitude below the 120 pC threshold set
by the electronics.
Figure 5.2 shows a non-zero efficiency, even as low as 4 kV applied potential. This enormous
discrepancy means that simulated efficiency curves (obviously zero efficiency across the board)
can not be compared to the measured values. A number of different possible causes have been
investigated.
Possibility of trapped air inside the multigap protoype
A first hypothesis was the possibility of air being trapped inside the subgaps of the detector.
These subgaps are only 200 µm wide, while a 3.8 mm gap surrounds them. No specific structures are in place to force the gas flow through the subgaps, which might cause the detector
to need flushing times far longer than usual to ensure only the correct gas mixture is in place.
58
Chapter 5. Verification of simulated data
Cosmics simulations of the glass RPC and RE2/2
100
glass RPC, simulated
glass RPC, measured
RE2/2, simulated
RE2/2, measured
Efficiency [%]
80
60
40
20
0
0
2
4
6
8
Applied potential [kV]
10
12
Figure 5.5: Comparison of simulated and measured efficiency scans for the glass RPC prototype and
RE2/2 production detector.
Simulated signal sizes for the multigap RPC
Total induced charge [fC]
0.0014
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
0
2
4
6
8
Applied potential [kV]
10
Figure 5.6: Simulated average induced charges for the multigap prototype.
12
Chapter 5. Verification of simulated data
59
This possibility was disproven using a low-flow air pump connected to the gas inlet. After
flushing for 48 hours, the detector was connected to high voltage and electronics, and a careful
efficiency measurement was taken between 4 kV and 6 kV. Noise rates as well as detection
rates fell down to essentially zero. If the gaps had been filled with air during the normal
efficiency measurements, performance should have remained the same even after flushing and
while testing with 100% air inside.
Alternative working mode
As mentioned in section 2.3, resistive plate chamber detectors do not only have an avalanche
working mode. They can also be operated in streamer mode, which leads to far bigger
avalanches and induced charges.
Operation of the detector was investigated using a Tektronix TDS 3034C oscilloscope. The
signals were probed directly from the read-out strips, as shown in figure 5.7. Both a candidate
avalanche and streamer signal are shown, which have a clearly different profile [44]. The vast
majority of signals had a clear avalanche pulse shape, indicating that the multigap is most
likely working in avalanche mode.
If one assumes a 50 Ω termination load as configured on the oscilloscope, then the avalanche
signals contain on average about 1 pC of charge, which is far above what is simulated in
figure 5.6.
TDS 3034C 6 Jun 2016
14:41:28
(a) Candidate avalanche signal.
TDS 3034C
6 Jun 2016
14:57:04
(b) Candidate streamer signal.
Figure 5.7: Multigap prototype signals taken using an oscilloscope. One candidate event has been
selected for each operational mode. Channel 2 (blue) shows the coincidence trigger from
the two scintillators.
60
Chapter 5. Verification of simulated data
Excessive noise
The efficiency could be artificially increased by means of excessive noise levels inside the gaps.
This hypothesis is easy to disprove using the raw timing data received from the electronics.
When a trigger is received through coincidence of both scintillators, the electronics saves an
event containing all hits in all channels for a 1000 ns window centered on the trigger time. The
hits seen in the multigap prototype show a very sharp peak at a slight delay from the trigger
point, defined by the internal electronics and cabling of the setup. Figure 5.8 shows a raw
DAQ output plot where the sharp peak can clearly be observed for 7 kV applied potential.
Noise measurements specifically show a uniform timing spectrum, which disproves the idea
that the efficiency is influenced by noise by more than a few percent.
# of events
Arrival time profile
TimeProfile
Entries
1675
Mean
434.1
108.2
Std Dev
70
60
50
40
30
20
10
0
0
100
200
300
400
500
600
700
800
900
1000
Time [ns]
Figure 5.8: Raw DAQ analysis output showing the hit timing spectrum at 5 kV for the multigap
prototype. Due to delays in the electronics and cable lengths, the trigger signal reaches
the digitizer about 60 ns later than the signals from the RPC. The time window is
centered exactly on the arrival of the trigger.
Chapter 5. Verification of simulated data
5.3.3
61
Physics of the multigap prototype
If one directly applies the physics behind the simulation to the parameters of the multigap
RPC, it is again evident that the simple Townsend amplification model alone might not be
sufficient to explain the gain of this detector.
Electric field strength
The electric field can be verified by hand using the fact that the potential over two parallel
plates is the integral of the electric field along the length of the gap. For a uniform gap this
gives:
E0
d
(5.1)
∆U = E · d =
ǫr
Here E0 is the field in free space, d the gap siz:e, and E is the field strength in a dielectric
with permittivity ǫr .
Adding multiple different dielectrics expands the equation to:
X E0 X
∆U =
di ,
di = d
ǫi
i
(5.2)
i
For the multigap, with 2 glass plates of 1.1 mm and ǫ = 6, 4 plates of 0.7 mm and ǫ = 5.4,
and 5 gas gaps of 0.2 mm and ǫ = 1, the field E0 inside the gas with ∆U = 1 kV would be:
E0 =
1 kV
4·0.7 mm
5.4
+
2·1.1 mm
6
+ 5 · 0.2 mm
= 5.3 kV cm−1
(5.3)
This value is in agreement with figure 4.2b for the multigap prototype. A miscalculation of
the electric field is thus not likely the cause of this charge deficit.
Townsend coefficient
If one assumes perfect exponential growth of avalanches, according to:
n(z) = e(α−η)z
(5.4)
then in the absolute best case scenario of an avalanche growing for the full 200 µm distance
and producing 108 electrons, the required effective Townsend coefficient α − η would be:
α−η =
8 ln(10)
= 92 mm−1
0.2 mm
(5.5)
As seen on figure 4.6, this value is not even reached for 120 kV, which would already correspond to an applied potential of 23 kV, which is more than can be reached using the testing
equipment in Ghent.
The combined investigation would thus suggest that exponential Townsend amplification alone
can not account for the gain seen in the multigap prototype.
62
Chapter 5. Verification of simulated data
Possible explanation
Space charge effects tend to decrease avalanche sizes in single gap RPCs, where the electrons
of earlier avalanches charge the (grounded) cathode plate. This plate becomes more negatively
charged, reducing the effective electric field seen by future avalanches.
In a multigap structure, this effect might instead work to increase the gain. Charge deposited
on one of the sub-electrodes will not generate a positive charge on the opposite side of this
electrode, since it is an insulator. Instead, the electric field in the next gap would be slightly
increased by this deposit of negative charge on the anode side of the second sub-gap. This
might introduce a cascade effect, where each successive gap sees a higher field strength than
the one above it.
The problem with this line of reasoning is the fact that the avalanches in the first gap will
still only be a few tens of electrons in size. To get an increase in field strength of a similar
order of magnitude as the field already present inside the gap, one would need about 106 to
107 electrons concentrated in a spot a few micrometer across.
With multiple avalanches per gap, and a cascade effect over four gaps, the fifth gap could still
see an appreciable increase in field strength. Adapting the simulation code to also include
space charges and induced charges was unfortunately beyond the scope and time allotted to
this thesis.
Chapter 6
Conclusions and outlook
6.1
Permittivity measurements
The measurement setup outlined in section 3.2.2 works well to determine the static relative
permittivity of material samples, even when the capacitances created using the setup are of
a similar magnitude as the parasitic capacitance contribution of the cabling and electronics.
Using the measured permittivities in table 3.3 electric field maps were calculated for the
different resistive plate chamber models. Figure 5.5 shows a good resemblance between simulation and reality for two of the models. Given the other uncertainties involved, the measured
permittivity values are precise enough at this point.
6.2
Simulation results
A new simulation code was developed, based on some of the facilities provided by the
Garfield++ toolkit. The gas performance was evaluated in figure 5.4 and by comparing figures 4.6 and 5.3. Results for the CMS gas mixture were found to be close to other literature
and measurements.
The hybrid code enabled simulations using cosmic ray spectra generated using the CRY library
in reasonable computation times. These simulations were compared to measurements taken
using a data acquisition system, outlined in section 5.1.1, built for the purpose of testing
small RPC prototypes.
Performance of two of the three models was well reproduced in figure 5.5. The multigap
model can not work based on Townsend multiplication alone, as shown in section 5.3.2. This
problem was evaluated from a first principles point of view in section 5.3.3, and a possible
mechanism was provided to explain the behaviour of the detector.
63
64
Chapter 6. Conclusions and outlook
6.3
Minor points
A few remarks have been gathered here for completeness’ sake:
• The electric field inside an RPC, even a multigap model, can be calculated by hand
quite accurately and easily using equation (5.2).
• Larger gains can be reached by focussing the electric field into the gas gap, by choosing
electrode materials with a higher ǫr . This is especially important in multigap structures,
where a large part of a detector’s thickness can consist of electrode plates.
6.4
6.4.1
Outlook
Simulation modifications
The mechanism described at the end of section 5.3.3 could be added into the hybrid simulation
code. Space charge effects for single gap configurations could be modeled at the same time.
This work should ideally be accompanied by a new prototype with variable geometry to more
easily verify simulation results.
More complete implementation of the electronics transfer function could be useful for accurate
simulations, especially if electronic noise is added to the model. Adding detector noise could
happen through this transfer function as well.
6.4.2
Measurements and prototypes
Verification of simulation results would be greatly simplified by building a detector prototype
with configurable geometry. This model would ideally have a smoothly variable gas gap, and
a way to insert sub-electrodes to create a multigap configuration.
Switchable electrode materials would be a bonus, though the main advantage would be in
the variable geometry. The same 200 µm gap size of the multigap could be investigated in a
single gap configuration using this prototype, followed by adding extra subgaps in steps.
The single-gap configuration would quickly show if the discrepancy is a result of the multigap
configuration, or of the very narrow gap. Adding extra subgaps one by one could help verifying
the hypothesis outlined at the end of section 5.3.3.
Beam tests have already been performed using the single gap models, and performance was
found to be largely the same as with cosmics. No such tests have been performed using the
multigap prototype. The results of measurements using mono-energetic particles, for example
minimally ionising and maximally ionising species in separate tests, could help clarify the
multigap’s behaviour.
Chapter 6. Conclusions and outlook
6.4.3
65
Permittivity measurements
The permittivity measurement setup could be adapted to allow gas measurements. This
would allow obtaining a more precise permittivity value of the gas mixture, which has until
now been assumed to be similar to that of air and most other gasses. This permittivity
ǫair = 1.000 59 has been simplified to ǫair = 1 given the other experimental uncertainties.
This same adaptation would also allow measuring the permittivity of air directly, which is
well known and could give a further improved estimate of the parasitic components involved.
Appendix A
Parameters of tested RPC models
All simulation detectors were modelled as having two chambers with a 1.55 mm PCB between
them. The signal pick-up strips are located on the top side. The PCB and strips are insulated
on both sides using 0.2 mm mylar sheets between them and the chambers.
Multigap prototype
1.1 mm glass
5× 0.7 mm glass
0.2 mm
1.1 mm glass
Figure A.1: Geometry of one chamber of the glass multigap prototype.
Glass RPC prototype
1.1 mm glass
1.2 mm
1.1 mm glass
Figure A.2: Geometry of one chamber of the glass RPC single-gap prototype.
67
68
Appendix A. Parameters of tested RPC models
Bakelite RE2/2 detector
2 mm bakelite
2 mm
2 mm bakelite
Figure A.3: Geometry of one chamber of the RE2/2 production RPC.
Appendix B
Permittivity measurement results
Bakelite
d [mm]
τ [µs]
2.05 ± 0.05
4.10 ± 0.05
6.10 ± 0.05
8.05 ± 0.05
10.10 ± 0.05
12.10 ± 0.05
29.3 ± 0.1
21.3 ± 0.1
18.2 ± 0.1
16.1 ± 0.1
15.1 ± 0.1
14.3 ± 0.1
Table B.1: Time constant measurements for bakelite samples of 2 mm.
1.1mm float glass
d [mm]
τ [µs]
1.10 ± 0.05
2.15 ± 0.05
3.25 ± 0.05
4.35 ± 0.05
5.40 ± 0.05
6.45 ± 0.05
7.55 ± 0.05
40.8 ± 0.1
26.3 ± 0.1
22.0 ± 0.1
19.2 ± 0.1
18.2 ± 0.1
17.1 ± 0.1
16.3 ± 0.1
Table B.2: Time constant measurements for main electrode float glass samples of 1.1 mm.
69
70
Appendix B. Permittivity measurement results
0.7 mm float glass
d [mm]
τ [µs]
0.70 ± 0.05
1.35 ± 0.05
2.05 ± 0.05
2.80 ± 0.05
3.45 ± 0.05
4.10 ± 0.05
4.80 ± 0.05
5.35 ± 0.05
51.8 ± 0.1
32.9 ± 0.1
25.8 ± 0.1
21.7 ± 0.1
19.6 ± 0.1
18.2 ± 0.1
17.1 ± 0.1
16.8 ± 0.1
Table B.3: Time constant measurements for sub-electrode float glass samples of 0.7 mm.
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
The Standard Model of particle physics with years of discovery. The first
observation of photon momentum by Arthur Compton can be considered the
discovery of the photon as a particle. . . . . . . . . . . . . . . . . . . . . . . .
2
Schematic representation of the LHC layout. The storage ring houses two particle beams running in opposite directions. Four collision points are provided,
each housing one large detector experiment. . . . . . . . . . . . . . . . . . . .
4
The Bethe-Bloch curve of a muon travelling through copper. The material
density has here been factored out of the expression and divided out, giving
the Y-axis units of MeV cm2 g−1 . Figure taken from [12] . . . . . . . . . . . .
6
A slice through the barrel region of the Compact Muon Solenoid, showing the
different detector layers. Example tracks for the different particle types show
possible tracks through the detector structure. . . . . . . . . . . . . . . . . .
7
A quadrant of the CMS detector, highlighting the muon detection system in its
planned form. The dashed box indicates the region of high background rates
where new GEM detectors and RPCs types are required. Station 4 is scheduled
to be installed during Long Shutdown 2. Figure taken from [15] . . . . . . . .
8
Simple view of the working principle of a resistive plate chamber detector. The
outer plates are electrodes, over which a high voltage is applied, generating an
~ between the plates. The inside of this detector is filled with a
electric field E
working gas, in which the passing muon will leaves an ionisation track. One of
the freed electrons has developed into a Townsend avalanche. . . . . . . . . .
9
2.1
Example RPC electrode, showing a paint layer of lower resistivity on top of
higher resitivity glass. A copper strip is attached onto the paint layer by means
of conductive glue and provides a connection point for the high voltage cabling. 15
2.2
Two specific variants of RPC construction. . . . . . . . . . . . . . . . . . . .
2.3
13 mm−1
Simulated avalanches for n(x) = 0, α =
and η =
The
(α−η)x
e
exponential growth is clearly visible once the avalanche contains about
100 electrons. Figure taken from [25]. . . . . . . . . . . . . . . . . . . . . . .
71
16
3.5 mm−1 .
18
72
List of Figures
2.4
3.1
3.2
The electrode aging effect shown for different material types. On average,
materials with higher resitivity will see worse aging effects over time. Figure
taken from [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
A piece of resistive material with cross-sectional area A and length l. Charge
carriers flow through this material as current I. . . . . . . . . . . . . . . . . .
23
This shows the simplistic model where dielectric materials are represented as
atoms with a positive nucleus, surrounded by a negatively charged electron
cloud. Under an electric field this cloud will locally counteract the electric field
by deforming and generating a dipole moment P~ . . . . . . . . . . . . . . . . .
24
3.3
This figure shows how a triangular 3D mesh is built. Points (also called nodes
or vertices) are first defined. These are then connected into edges (or lines).
Closed loops of edges form faces, which in turn combine into elements (volumes). 28
3.4
Demonstration of 2D triangular mesh types. . . . . . . . . . . . . . . . . . . .
28
3.5
A simple parallel plate capacitor geometry. Between the electrodes (yellow),
separated by a distance d, a slab of dielectric material (blue) is placed with
static relative permittivity ǫr . The electrodes are square, with an area A = l2 .
30
3.6
A simple RC circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.7
Circuit for capacitance measurements using a function generator Us and an
oscilloscope measuring across points A and B. . . . . . . . . . . . . . . . . . .
31
Circuit for capacitance measurements including parasitic component Cpar , parallel to the target capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . .
32
View of the setup constructed to measure the static permittivities of key materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.10 Oscilloscope view of a capacitance measurement. Channel 2 (blue) shows the
input square wave. Channel 1 (yellow) is measuring the 1/e rise-time, seen on
the right of the screen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.11 Measurement points and fits for static relative permittivity measurements of
the two float glass species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.12 Measurement points and fits for static relative permittivity measurements of
the RE4/3 bakelite material. . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.8
3.9
4.1
4.3
Cut-out view of the mesh for a double-chamber multigap structure with four
internal electrically floating glass plates as sub-electrodes inside each chamber.
A PCB with 16 read-out strips is sandwiched between the chambers. The
conductive paint layer can be seen as the differently coloured rectangle on the
top electrode. The potential boundary conditions are applied to these paint
surfaces, in analogy to the real prototypes. . . . . . . . . . . . . . . . . . . . .
38
These figures show a 1D cut of the maps of figure 4.2, along the Z-axis in the
centre of the multigap RPC. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
List of Figures
4.2
4.4
4.5
4.6
4.7
4.8
73
Electric potential and derived electric field for the multigap RPC shown in figure 4.1. These maps have been calculated with −1 kV applied on the outermost
electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Z-component of the weighting field in the XZ-plane. The scale of this figure
is the same as for figure 4.2b, and shows the map for strip number 5 of 16
(counting starts at 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Average clusters per mm as calculated by Heed++ for different gases at standard temperature and pressure. The solid lines show methane and isobutane
measurements from [41]. Figure taken from [25]. . . . . . . . . . . . . . . . .
44
Magboltz calculation results for the CMS mixture of 95.2 % R134a, 4.5 % isobutane and 0.3 % SF6 . The effective Townsend coefficient α − η is also shown. .
44
Comparison of the different simulation algorithms, showing the average avalanche
size and computation time for between 30 and 10 iterations per field strength,
adaptively chosen according to avalanche sizes. . . . . . . . . . . . . . . . . .
46
Schematic representation of the calculation of signal weigths in a gap. At every
zstep, the weighting field is summed as indicated by the arrows. . . . . . . . .
47
Results of two signal size map calculations. These values show the average
signal size and standard deviation for an electron that starts at each height
in the gap. The 3 kV map is dominated by the attachment coefficient, so all
electrons disappear before they reach the end of the gap. The spread of the
signal sizes decreases as the electron nears the end of the gap, because there is
less time to drift in the XY-plane. . . . . . . . . . . . . . . . . . . . . . . . .
48
4.10 Cosmic ray spectrum at sea level generated using the CRY library. . . . . . .
49
4.9
5.1
Overview of the dedicated DAQ for efficiency measurements of small RPC
prototypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Efficiency scans of three different RPCs. The multigap RPC shows a profoundly
different curve shape compared to the other two types. The RE2/2 type RPC
is the same model as has been installed in station 2 of the CMS endcaps. . .
54
Example Townsend and attachment coefficients for a similar mix as used in
CMS. Figure taken from [25] . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.4
Comparison of calculated drift velocities. . . . . . . . . . . . . . . . . . . . . .
56
5.5
Comparison of simulated and measured efficiency scans for the glass RPC prototype and RE2/2 production detector. . . . . . . . . . . . . . . . . . . . . .
58
5.6
Simulated average induced charges for the multigap prototype. . . . . . . . .
58
5.7
Multigap prototype signals taken using an oscilloscope. One candidate event
has been selected for each operational mode. Channel 2 (blue) shows the
coincidence trigger from the two scintillators. . . . . . . . . . . . . . . . . . .
59
5.2
5.3
74
List of Figures
5.8
Raw DAQ analysis output showing the hit timing spectrum at 5 kV for the
multigap prototype. Due to delays in the electronics and cable lengths, the
trigger signal reaches the digitizer about 60 ns later than the signals from the
RPC. The time window is centered exactly on the arrival of the trigger. . . .
60
A.1 Geometry of one chamber of the glass multigap prototype. . . . . . . . . . . .
A.2 Geometry of one chamber of the glass RPC single-gap prototype. . . . . . . .
A.3 Geometry of one chamber of the RE2/2 production RPC. . . . . . . . . . . .
67
67
68
List of Tables
3.1
All chosen parameters and values that are constant across tests for the capacitance measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fit results for the tested material species. . . . . . . . . . . . . . . . . . . . .
Static relative permittivities for all three materials measured, based on the
obtained fit results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
B.1 Time constant measurements for bakelite samples of 2 mm. . . . . . . . . . .
B.2 Time constant measurements for main electrode float glass samples of 1.1 mm.
B.3 Time constant measurements for sub-electrode float glass samples of 0.7 mm.
69
69
70
3.2
3.3
75
33
35
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List of Abbreviations
ATLAS A Toroidal LHC ApparatuS. 1, 5, 15
BSM Beyond the Standard Model. 4, 5
CERN European Organization for Nuclear Research. 4
CMS Compact Muon Solenoid. 1, 5, 7, 8, 15, 34, 54–56, 71, 73
DAQ data acquisition system. 53, 54, 56, 60, 63, 73, 74
DMM digital multimeter. 33
FEM finite element method. 27, 28, 37–39, 43, 57
GUT Grand Unified Theory. 3
HEP high-energy physics. 1
HL-LHC High Luminosity Large Hadron Collider. 5, 8, 10, 22
LHC Large Hadron Collider. v, 4, 5, 7, 71
LRS low-resistance silicate. 15, 21
LUX Large Underground Xenon experiment. 6
OPERA Oscillation Project with Emulsion-tRacking Apparatus. 6
PCB printed circuit board. 16, 29, 38, 39, 67, 72
PDE partial differential equation. 27, 29
RKF Runge-Kutta-Fehlberg. 45
83
84
List of Abbreviations
RPC resistive plate chamber. v, vi, 8–10, 13–16, 18–20, 24, 29, 34, 37, 38, 40, 41, 43, 45,
51–54, 56–64, 68, 71–74
SM Standard Model. 1, 3, 4
SUSY supersymmetry. 4
TOE Theory Of Everything. 3